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Full text of "The western practical arithmetic : wherein the rules are illustrated, and their principles explained : containing a great variety of exercises, particularly adapted to the currency of the United States : with an appendix containing the canceling system, abbreviations in multiplication, mensuration, and the roots : designed for the use of schools and private students"

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IRLF 





S CIN ' /. ] i. 





IN MEMORIAM 
FLOR1AN CAJOR1 




THE WESTERN^, . 

PRACTICAL ARITHMETIC, 



WHEBE1N THE 

RULES ARE ILLUSTRATE, AND THEIR PRINCIPLES 
EXPLAINED: 

CONTAINING 

A GREAT VARIETY OF EXERCISES, 

PART^CULAJILY-ADAPTED TO THE 

CURRENCY OF THE UNITED STATES. 

WITH 

AN APPENDIX: ""S 

CONTAINING THE CANCELING SYSTEM, ABBREVIATIONS IN MULTIPLl-f 
CATION, MENSURATION, AND THE ROOTS. -) 

DEIQNEO FOB THE USE OF 

SCHOOLS AND PRIVATE STUDENTa 

COMPILED 



BY JOHN L. TALBOTT. 



PUBLISHED BY E. MORGAN & CO., 

Ill MAIN STREET. 
1 853. 



District of Ohio, to wit : ? 
District Clerk's Office. 5 

BE IT BEMEMBEBED, that on the seventeenth day of January, An> 
no Domini eighteen hundred and forty-five, E. MOBGAN & Co., of 
the said District, have deposited in this office the title of a book, the 
title of which is in the words following, to wit : 

u The Western Practical Arithmetic, wherein the rules are illus- 
trated, and their principles explained, containing a great variety of 
exercises, particularly adapted to the currency of the United States : 
with an Appendix, containing the Canceling System, Abbreviations 
in Multiplication, Mensuration and the roots ; designed for the use 
of schools and private students; compiled by JOHN L.TALuoTT;"the 
right whereof they claim as proprietors, in conformity with an act of 
Congress, entitled "An act to amend the several acts respecting copy- 
rights." WM. MINER, Clerk of District. 



PREFACE. V&^T'f-' 

I.v presenting this work to the public, the author makes no pre- 
tensions to having discovered any new spring by which to put the 
youthful mind into action, nor any new method of communicating 
a knowledge of Arithmetic. He has founded his work on the belief 
that labor and labor only, can insure success in any pursuit ; and 
-that labor should always be bestowed upon those objects which pro- 
duce the greatest useful result. 

In the selection and arrangement of matter, therefore, those rules 
that are of the most general use, have been presented first, and their 
exercises made extensive, that the pupil many early become familiar 
with their principles, and expert in their application. 

The explanations accompanying the rules, are designed to facili- 
tate the progress of private students, and to diminish the labor of 
teachers, especially in large schools, where they are unable to give 
to each pupil the necessary explanations. 

The MESSUUATIOW of Carpenters', Masons', Plasterers' and 
Pavers' work, &c., will be found an acceptable part of Arithmetic, 
to every man of business, and a practical knowledge of it will con- 
tribute much to the security and satisfaction of both workmen and 
employers, in estimating amounts of work. This has been intro- 
duced in consequence of numerous applications to the author to 
measure various kinds of work, and for instruction in particular 
rules of Mensuration. 

The system of Book Keeping, is thought to be sufficient for all 
the purposes of farmers, mechanics and retailers, in that necessary 
branch of a business education. a* - 

How far the author has succeeded in his attempts to compile a 
useful work, particularly adapted to the circumstances of the Western 
People, remains for them to judge, and for experience to determine. 



NOTICE. 

THE favorable reception of this treatise and the increasing demand 
for it, have induced the publishers to revise, enlarge, and otherwise 
improve the work. Such alterations and amendments have been 
made as the experience of the author and of other intelligent and 
successful teachers has suggested; it is therefore presumed, that the 
work will be found more useful, and consequently more acceptable 
than heretofore. 

Numerous testimonials to the merits of the work, have been re- 
ceived; but its general adoption without any efforts to force its 
introduction, and its intrinsic worth, are our main reliance; we have 
therefore given it a thorough revision, and now submit the result of 
our labors to a discerning public. 

Febr ii ajy T J841. P MORGAN & Co. 



_ -J.elk 

v5j 



CONTENTS. 

PACK. 

Numeration, ~ 5 

Simple Addition, 12 

Simple Subtraction, 15 

Simple Multiplication, 18 

Simple Division, 23 

Addition of Federal Money, - 30 

Subtraction of Federal Money, 32 

Multiplication of Federal Money, 34 

Division of Federal Money, 36 

Reduction, 38 

Compound Addition, 60 

Compound Subtraction, 66 

Compound Multiplication, 72 

Compound Division, 80 

Simple Proportion, 88 

Compound Proportion, 100 

Practice, 103 

Tare and Tret, 109 

Interest, 112 

Compound Interest, 118 

Insurance, Commission and Brokage, 121 

Discount, 122 

Equation of Payments, 124 

Barter, 126 

Loss and Gain, 128 

Fellowship, 132 

ar Fractions, 135 

Reduction of Vulgar Fractions, 136 

Addition of Vulgar Fractions, 144 

Subtraction of Vulgar Fractions, 146 

Multiplication of Vulgar Fractions,. 147 

Division of Vulgar Fractions, 148 

Decimal Fractions, . 149. 

Addition of Decimals, 150 

Subtraction of Decimals, ib. 

Multiplication of Decimals, 151 

Division of Decimals, - ib. 

Reduction of Decimals, 153 

Proportion in Decimals, 155 

Compound Proportion in Decimals, 156 

Mensuration, > ib. 

Involution, . . 167 

Evolution, 169 

Square Root, ib. 

Cube Root, 174 

Roots of All Powers, 179 

Arithmetical Progression 180 

Geometrical Progression 184 

Appendix 194 

Exchange *.* 189 

Promiscuous Exercise . . .... ,190 



ARITHMETIC. 



ARITHMETIC is that part of MATHEMATICS which 
treats of numbers. It is both a science and an art; 
the science explains the nature of numbers, and the 
principles upon which the rules are founded^ while the 
art relates merely to the application of the various 
rules. 

All the operations of arithmetic are conducted by 
means of FIVE fundamental rates, viz., Numeration, 
(which includes Notation,} Addition, Subtraction, 
Multiplication, and Division. 

NUMERATION AND NOTATION. 

Numeration is the art of representing figures or num- 
bers by words ; Notation is the art of representing num- 
bers by characters called figures. 

' All numbers are represented by the following charac- 
ters, which are called figures or digits. 

0, 1, 2, 3, 4, 5, 6, 7, 8, 0. 
nought, one, two, three, four, five, six, seven, eight, nine. 

The one is often called a unit, it signifies a whole 
tb"ig of a kind ; two signifies two units or ones ; three 
s ^nifies three units or ones, <fcc. 

The value which the figures have when standing 
alone is called their simple value ; but in order to denote 
numbers higher than 9, it is necessary to give them ano- 
ther value called a local value, which depends entirely 
on the order or place in which they stand. Thus, when 
we wish to write the number ten in figures, we do it 
by combining the characters already known, placing a 
1 on the left hand of the 0, thus, 10, which is read ten. 
This 10 expresses ten of the units denoted by 1, but 
as it is only a single ten it is called a unit, and the 
1 being written in the second order or second place 
from the right hand to express it, it is called a unit of 
the second order, the first place being called the place 



6 NUMERATION AND NOTATION. 

of units, and the second, the place of tens ; ten units 
of the first order making one unit of the second order. 

When units simply are named, units of the first order 
are always meant, when units of any other order are 
intended, the name of the order is always added. 

Two tens or twenty, are written 20. 

Three tens or thirty, " " 30. 

Four tens or forty, " " 40. 

Five ten* or fifty, " " 50. 

Six tens or sixty, " " 60. 

Seven tens or seventy, " " 70. 

Eight tens or eighty, " " 80. 

Nine tens or ninety, " " 90. 

Ten tens or one hundred, " " 100. 

The numbers between 10 and 20, between 20 and 
30, between 30 and 40, <fec. may easily be expressed 
by considering the tens and units of which they are 
composed. Thus, eleven being composed of one ten 
and one unit, is expressed thus, 11, twenty-thcee being 
composed of two tens, and three units, is expressed 
thus, 23. &c. 

Sixteen being 1 ten and 6 units, is written thus, 16. 

Thirty-nine being 3 tens and 9 units, is written 39. 

Sixty-four being 6 tens and 4 units, is written 64. 

Ninety-five being 9 tens and 5 units, is written 95. 

Ten tens or one hundred forms a unit of the third 
order ; it is expressed by placing a 1 in the third pin e, 
and filling the first and second places with cyphei . 
thus, 100. Two hundred is expressed thus, 200 
Three hundred thus, 300, <fcc. 

With the orders of units, tens, and hundreds, all the 
numbers between one and one thousand may be readily 
expressed. For example, in the number four hundred 
and twenty-seven, there are 4 hundreds, 2 tens, and 7 
units, that is, 4 units of the third order, 2 units of the 
second order, and 7 units of the first order. 

= I 

Hence the number is written thus, 427 

In the number three hundred and five, there are 3 
hundreds, no tens, and 5 units, or 3 units of the third, 



NUMERATION AND NOTATION. 7 

none of the second, and five of the first order, hence the 
number is written thus, * I 

305 

Ten units of the order of hundreds, that is ten hun- 
dreds form a unit of the fourth order, called thousands, 
written thus, 1000. 

In the same manner ten units of the fourth order form 
a unit of the fifth order, called tens of thousands. 

The following may be regarded as the principles of 
Notation and Numeration. 

1st. Ten units of the first or lowest order, make one 
unit of the second order; ten units of the second order, 
make one unit of the third order, and universally ten 
units of any order make a unit of the next higher 
order. 

2d. Jill numbers are expressed by the nine digits, 
and the cypher, and this is effected by giving to the 
same figure different values according to the place it 
occupies. Thu^, 4 in the first place is 4 units, in the 
second place 4 tens or forty, and so on. This tenfold 
increase of value by changing the place of the same 
figure is usually expressed by saying that figures in- 
crease from right to left in a tenfold proportion. The 
names of the orders are to be learned from the 

NUMERATION TABLE. 



3 2 o 

S | 

**" 3 **"" 

JS aa rS "^ 

| | | | ^ (| 

987654321 

The orders are likewise divided into periods of ei 
places each, according to the following table. 



NUMERATION AND NOTATION. 



of Billions. 



ofMilli 



llioni. 



of Unit*. 



f^ 

I 
ii 



II 



g 2 

' 



. . 

s o 'S S .3 

S3 X! P o 5 



"5 

W2 "5 "^ 

!<= - 



The periods succeeding those in the table, are Tril- 
lions, Quadrillions, Quintillions, Sextillions, Sepiil- 
lions, Octillions, and Nonnillions, and analogical names 
might be formed for the succeeding higher periods. 

From the preceding remarks the pupil will readily 
understand the reason of the following rule for numer- 
ating or expressing figures by words. 

RULE. Commence at the right hand, and separate 
the given number into periods, then beginning at the left 
hand, read the figures of each period as if they stood 
alone, and then add the name of the period. 

Thus, the number 8304000508245, when divided 
into periods, becomes 8,304000,508245, and is read, 
Eight billion, three hundred and four thousand mil- 
lion, Jive hundred and eight thousand two hundred 
and forty-Jive. The name unit of the right hand pe- 
riod is commonly omitted in reading. 

EXERCISES IN NUMERATION. 

Ex. 1. 35 10. 3700054 



2. 204 

3. 513 

4. 2000 

5. 3054 

6. 7428 
7 10345 
8. 40024 

X 9. 61304 



11. 6130425 

12. 2701030 

13. 3705423 

14. 6803217 

15. 2003005 

16. 70032004 

17. 62003005 

18. 91010010 



19. 
20. 
21. 
22. 
23. 
24. 
25. 
26. 
27. 



20031025 
68723145 
901023406 
820302008 
310275603 
600000501 
3000400230024 
80000102051003 
50000021375604 



28. 4000012000040250014 
29. 1000982000375000482000354000271000032561804 



NUMERATION AND NOTATION. 9 

From the preceding tables and remarks, the pupil will 
likewise readily understand the reason of the following 
rule for notation, or expressing numbers by figures. 

RULE. Make a sufficient number of cyphers or dots, 
and divide them into periods, then underneath these 
dots write each figure in its proper order and fill the 
vacant orders with cyphers. 

NOTE. The object of the dots or cyphers, being to 
guide the learner at first, after a little practice he may 
dispense with them. 

Ex. 1. Write down in figures the number twenty 
millions three hundred and four thousand and forty. 
Here millions being the highest period named, we 
write cyphers to correspond with that, and the period 
of units, and then underneath these place the significant 
figures in their proper order, and afterwards fill the 
vacant orders with cyphers. 

000000, 000000 
20304040 

The pupil must recollect that cyphers being of no 
use except to fill vacant orders, are never to be placed 
to the left of whole numbers. 



EXERCISES IN NOTATION. 

Express the following numbers in figures. 

EXAMPLES. 

2. Seventy-five. 

3. Ninety. 

4. One hundred and five. 

5. Three hundred and twenty. 

6. Nine hundred and four. 

7. Eight hundred and ninety. 

8. Two thousand three hundred and five. 

9. Six thousand and forty. 

10. Seven thousand and four. 

11. Eight thousand and ninety-five. 

12. Ten thousand five hundred and fifty-six 

13. Forty thousand and forty. 

14. Ninety-five thousand two hundred and sixty-seven. 

15. Eighty thousand one hundred and nine. 



j] 10 NUMERATION AND NOTATION. 

16. One hundred and thirty-six thousand two hundred 

and seventy five. 

)i 17. Three hundred and seven thousand and sixty-four. 
Ji 18. Five hundred thousand and five. 
|| 19. One million, two hundred and forty-seven thousand, 

four hundred and twenty-three. 
|i 20. Ten millions, forty thousand and twenty. 

21. Sixty millions, seventeen thousand and two. 

22. One hundred and four millions two hundred and 

four thousand and sixty- five. 

23. Five hundred and three millions, one hundred and 

two thousand and nine. 

24. Ninety one thousand and two millions, and four. 

25. Sixty billions, three millions and forty-one thousand. 

26. One billion, one hundred million, one thousand and 

one. 

The Roman method of representing nnmbers, is by 
means of certain capital letters of the Roman alphabet. 
Thus: 



I 

II 

III 

IV 

V 

VI 

VII 

VIII 

IX 

X 

XI 

XII 

XIII 

XIV 

XV 

XVI 

XVII 



one 

two 

three 

four 

five 

six 

seven 

eight 

nine 

ten 

eleven 

twelve 

thirteen 

fourteen 

fifteen 

sixteen 

seventeen 



xvni 

XIX 

XX 

XXX 

XL 

L 

LX 

LXX 

LXXX 

xc 
c 

cc 

ccc 

cccc 

D 

M 

MDCCCXXXVIII 



eighteen 

nineteen 

twenty 

thirty 

forty 

fifty 

sixty 

seventy 

eighty 

ninety 

one hundred 

two hundred 

three hundred 

four hundred 

five hundred 

one thousand 

1838 



NOTE 1 . As often ag any letter is repeated, so often is its vilue re- 
peated. 

NOTE 2. A less character before a greater one, diminishes its value 
NOTE 3. A les* character after a greater one, increases its value. 



EXPLANATION OF CHARACTERS. 11 

QUESTIONS. 

What is Arithmetic ? When is it a science ? When 
is it an art ? What are the fundamental rules of arith- 
metic ? What is numeration ? What is notation ? What 
does a unit signify ? What does two signify ? Three, 
<fec. ? What is meant by the simple value of a unit? 
What does the local value of a figure depead on ? How 
do you write the number ten in figures ? Why is the 
one in this case called a unit of the second order? How 
many units of the first order does it take to make a unit 
of the second order ? How many units of the second 
order does it require to form a unit of the third order? 
&c. Repeat the principles of notation and numeration. 
l| Repeat the names of each of the first nine orders as ex- 
[i pressed in the numeration table. Repeat the name of 
each of the periods. Repeat the Rule for numeration. 
Repeat the^ Rule for notation. 



EXPLANATION OF CHARACTERS. 

Signs. Significations. 

= equal; as 20s. = 1. 

-f- more ; as 6 -f- 2 = 8. ^ 

less ; as 8 2 = 6. 

X into, with, or multiplied by ; as 6 X 2 = 12. 
-J- by (i. e. divided by ;) as 6 -f- 2 = 3 ; or, 2)6(3. 
: : : : proportionality; as 2 : 4 : : 6 : 12. 
J or, J Square Root ; as J 64 = 8. 
J Cube Root; as J 64 = 4. 
iy Fourth Root ; as J 16 = 2, &c. 
A vinculum ; denoting the several quantities 

over which it is drawn, to be considered jointly 

as a simple quantity. 



12 SIMPLE ADDITION. 

SIMPLE ADDITION. 

SIMPLE ADDITION is the art of collecting several nuin- 
jers, of the same name, into one sum. 

RULE. 

Place the numbers with units under units, tens under 
tens, &,c. Begin the addition at the units, or right hand 
column, and add together all the figures in that column ; 
then, if the amount be less than ten, set down the whole 
sum: but if greater than ten, see how many tens there 
are, and set down the number above the even tens, and 
carry one for each ten to the next column, and proceed 
with it as in the first. 

Proof. Begin the addition at the top of each column, 
and proceed as before, and if the result be the same, it 
is presumed to be right, 

EXAMPLES. 

(1) (2) (3) (4) 

432 231 214 4 

213 413 121 2 

121 121 312 5 

213 132 321 3 



979 sum 897 sum 968 sum 1 4 sum 



(5) Here 4, 2, 1, Sand 6 make 15. In fifteen there 

27636 * s one ^ en aiu ' ^ ve lln ' ls * Set down *h e f' ve units 
7 Q ft Q 2 un ^ er tne umts c l uinn * an d carry one for the ten 
' * to the next or tens column. 

38941 

67832 Then 1,4, 3, 4, 9 and 3 make 24; in 24 there are 
59244 two tens, and four over: set down the foui under 
the column of tens, and carry two to the next or 
hundreds column &c., to the last, where thr whole 



273545 amount may be set down. 



SIMPLE ADDITION. 13 


(6) 


(7) 


(8) 


47386 


99786 


72752 


29492 


86937 


37823 


18583 


27849 


78794 


89294 


49878 


23567 


28887 


72937 


98372 


74392 


48732 


12345 


288034 


386119 


323653 


(9) 


(10) 


(11) 


47823 


72683 


84736 


73714 


95892 


78928 


27834 


82783 


27849 


23925 


94973 


63782 


67883 


76892 


28637 


62734 


43987 


73862 


(12) 


(13) 


(14) 


73684 


9376 


7379 


7 5 


723 


7463 


473 


8 7 


729 


6893 


9 


489 


7 


4 8 


7 2 


483 


937 


6 8 


9 6 


9 2 


432 




APPLICATION. 




1. Add 224 dollars, 365 dollars, 


427 dollars, and 784 


dollars, together. 










224 






365 






427 






784 




Answer^ 


$1800 Dollars 





14 SIMPLE ADDITION. 

2. Add 3742 bushels, 493 bushels, 927 bushels, 643 
bushels, and 953 bushels, together. 

Answer, 6758 bushels. 

3. Add 7346 acres, 9387 acres, 8756 acres, 8394 
acres, 32724 acres. Ans. 66607 acres. 

4. Henry received at one time 15 apples, at another 
115, at another 19. How many did he receive? 

Ans. 149. 

5. A person raised in one year 724 bushels of corn, 
in another 3498 bushels, in another 9872. How much 
in all? Ans. 14094 bushels. 

6. A rnan on a journey, travelled the first day 37 
miles, the second 33 miles, the third 40 miles, the fourth 
35 miles. How far did he travel ill the four days? 

Ans. 145 miles. 

7. A has a flock of sheep containing 34. B has a 
flock of 47, and C of fifty-four. How many sheep are 
there in the three flocks? Ans. 135. 

8. The distance from Philadelphia to Bristol is 20 
miles; from Bristol to Trenton, 10 miles; from Trenton 
to Princeton, 12 miles; from Princeton to Brunswick, 
18 miles ; from Brunswick to New York, 30 miles. How 
many miles from Philadelphia to New York? Ans. 90. 

9. A person bought of one merchant, 10 barrels of 
flour, of another 20 barrels, of another 95 barrels. 
How many barrels did he buy ? Ans. 125 barrels. 

10. A wine-merchant has in one cask 75 gallons, in 
another 65, in a third 57, in a fourth 83 ; in a fifth 74, 
and in a sixth 67 gallons. How many gallons has he 
in all f Ans. 421 gallons. 

Questions. 

How many primary rules of Arithmetic are there f 

What are they called ? 

What is addition? 

How do you place numbers to be added? 

Where do you begin the addition * 

Why do you carry one for ten, in preference to any 
other number? 

Ans. Because it takes ten ones to make one ten, ten 
tens to make one hundred, &c. (See table, page 9.) 



SDIPLE SUBTRACTION. 15 | 

SIMPLE SUBTRACTION. 

SIMPLE SUBTRACTION is taking a less number from a 
greater, of the same name, to show the difference be- 
tween them. 

The greater number is called the minuend. 

The less number is called the subtrahend. 

The difference, or what is left, is called the remainder 



RULE. 

Place the less number under the greater, with units 
under units, tens under tens, &LC. 

Then draw a line under them; begin at the right 
hand or units place, and subtract each figure of the sub- 
trahend from the figure of the minuend that is above 
it, and set the remainder below. When the figure in 
the subtrahend is greater than the one above it, borrow 
one (which is one ten) from the next figure, and add it 
to the figure of the minuend; then subtract from the 
sum. 

Proof. Add the remainder and the subtrahend to- 
gether, and if the sum equal the minuend, the work is 
! presumed to be right. 

EXAMPLES. 

(1) (2) 

79252743 Minuend 9738476 

34120312 Subtrahend 2614253 



451324!' 1 Remainder 7124223 



(3) Here we cannot take seven from two ; then we 

726398 -j must borrow one from the 8: that one is one fen; 
R A 9 Pi r Q >y tnen len an< ^ two are * vve l ve > now ta ^ e seven from 
' twelve, and five remain. 

One is borrowed from the 8, leaving only 7 ; 

838345 then take 3 from 7, and 4 remain : or, suppose 8 

to remain untliminished ; and to cancel the one which is borrowed from 

the 8, add one to the 3 below, making four; then four from eight and 

four remais-j as before, &c. 



16 SIMPLE SUBTRACTION. 

(4) (5) 

9273847 82703682 
2641386 27341237 



6632461 55362445 



(6) (7) 

7837286 273683070 
3273195 4321725 



4564091 269361345 



(8) W 

68427362 593784283 
34613524 54321432 



(10) (11) 

792836842 92037842 
24653128 41372761 



APPLICATION. 

1. From 78 take 32 and what will remain? 

Answer, 46. 

2. From 478 take 324. What will remain? 

Ans. 154. 

3. Charles had 723 apples, and sold 421. How ma- 
ny has he left? Ans. 302 

4. James had 9768 dollars, and gave for a house and 
lot 3453 dollars. How many has he left? Ans. 6315, 

5. A farmer had 3849 acres of land; he gave to his 
sons 2135 acres. How many acres has he left for 
himself? Ans. 1714. 

6. There are two piles of bricks, one contains 7SUH, 
and the other 4389. How many more are there in the 
oae than in the other? Ans. 3507. 



SIMPLE SUBTRACTION. 17 

7. Bought 100 bags of coffee, weighing 14510 Ibs., 
and sold thereof 63 bags weighing 6871 pounds; how 
many bags, and how many pounds remain unsold? 

Ans. 37 bags, and 7639 Ibs. 

8. A man bought a chaise for 175 dollars, and to pay 
for it gave a wagon worth 37 dollars, and the rest in 
money. How much money did he pay ? 

Ans. 138 dollars. 

9. A man deposited in bank 8752 dollars, and drew 
out at one time 4234 dollars, at another 1700 dollars, 
at another 962 dollars, and at another 49 dollars. How 
much had he remaining in bank? Ans. 1807 dollars. 

10. A merchant bought 4875 bushels of wheat, and 
sold 2976 bushels. How many bushels remain in his 
possession? Ans. 1899. 

1 1. A grocer bought 25 hogsheads of sugar, containing 
250 hundred weight, and sold 9 hogsheads, containing 
75 hundred weight. How many hogsheads and how 
many hundred weight had he left ? 

Ans. 16 hogsheads, and 175 hundred weight. 

12. A traveller who was 1300 miles from home, trav- 
elled homeward 235 miles in one week; in the next 275 
miles; in the next 325 miles; and in the next 290 miles. 
How far had he still to go, before he would reach home ? 

Ans. 175 miles. 

Questions. 

What is subtraction? 

What is the greater number called ? 

What is the less number called? 

What is the difference called? 

How do you place numbers for subtraction? 

Where do you begin the subtraction? 

When the lower figure is greater than the upper one, 
how do you proceed? 

Why is the one you borrow, one ten. 

Ans. Because ten ones make one ten; and if I borrow 
one ten it will make ten ones again, &c. 

How do you prove subtraction' 

M^MMMMHWBMMMMMHMMHMMMMMBIMMHB^^ 

Vi* 



18 SIMPLE 


MULTIPLICATION. 


SIMPLE MULTIPLICATION. 


I SIMPLE MULTIPLICATION is a short method of perform- 


ing particular cases of addition. 


The number to be multiplied, is the multiplicand. 


The number to 


be 


multiplied 


by, 


is the multiplier. 


The number produced is the product. 






The multiplicand and multiplier are sometimes called 


factors. 
















MULTIPLICATION 


TABLE. 


Twice 


3 times 


4 times 


5 times 


6 times 


7 times 


1 make 2 


1 make 3 


1 make 4 


1 make 5 


1 make 6 


1 make 7 


2 4 


2 6 


2 


8 


2 


10 


2 


12 


2 14 


3 6 


3 9 


3 


12 


3 


15 


3 


18 


3 21 


4 6 


4 12 


4 


16 


4 


20 


4 


24 


4 28 


5 10 


5 15 


5 


20 


5 


25 


5 


30 


5 35 


6 12 


6 18 


6 


24 


6 


30 


e 


36 


6 42 


7 14 


7 21 


7 


28 


7 


35 


7 


42 


7 49 


8 16 


8 24 


8 


32 


8 


40 


8 


48 


8 56 


9 18 


9 27 


9 


36 


q 


45 


9 


54 


9 63 


10 20 


10 30 


10 


40 


10 


50 


10 


60 


10 70 


11 22 


LI 33 


11 


44 


11 


55 


11 


66 


11 77 


12 24 


12 36 


12 


48 


J2 


60 


12 


72 


12 84 


8 times 9 times 


10 times 


11 times 


12 times 


1 make 


\ 1 make 


9 


1 make 10 


1 make 11 


1 make 12 


2 1( 


5 2 


18 


2 


20 


2 


22 


2 24 


3 & 


[ 3 


27 


3 


30 


3 


33 


3 36 


4 3S 


} 4 


36 


4 


40 


4 


44 


4 48 


5 4( 


5 


45 


5 


50 


5 


55 


5 60 


6 ' 46 


} 6 


54 


G 


60 


6 


66 


6 72 


7 5( 


5 7 


63 


7 


70 


7 


77 


7 84 i 


8 W 


[ 8 


72 


8 


80 


8 


88 


[ 


3 96 


9 7$ 


I 9 


81 


9 


90 


9 


99 


9 108 


10 8( 


1 10 


90 


10 


100 


10 


110 


10 120 


11 8* 


j 11 


99 


11 


110 


11 


121 


11 132 


12 9( 


i 12 


108 


12 


120 


12 


132 


12 144 


CASE 1. 


When the multiplier does 


not 


exceed 


12. 


RULE. Place the multiplier under 


the units 


figure of 


he multiplicand; and 


multiply each figure of the multi- 


I'licand in succession, 


and 


set down the amount, and 


:arry, as in addition. 


Proof. Multiply the multiplier by 


the multiplicand. 



SOIFLE MULTIPLICATION. 19 
EXAMPLES. 

4231 Multiplicand 34253 7342 

2 Multiplier 3 4 



8462 Product 102759 29368 



36563 8375 4378 9286 
567 8 



182815 50250 30646 74288 



4375 7862 3724 7482 
9 10 11 12 



39375 78620 40964 897^4 



EXERCISES 

1 Multiply 4218 by 2 Product 8436 

2 7321 by 3 2196 I 

3 87692 by 4 3507t>e 

4 900078 by 9 8100702 

5 826870 by 10 8268700 

6 278976 by 11 3068736 

7 569769 by 12 6837228 

CASE 2. 
When the multiplier exceeds 12. 

RULE. Place the multiplier as before, with units un- 
der units, &c. Then multiply all the figures of the mul- 
tiplicand by the units figure of the multiplier, setting 
down the product as before. 

Proceed with the tens figure in the same manner, ob- 
serving to set the product of the first figure in the tens 
place and with the hundreds figure placing the first 
product in hundreds place, &.C., and add the several pro 
ducts together. 



20 



SIMPLE MULTIPLICATION. 
EXAMPLES. 



43752 Here we multiply by the 6 or units figure 

436 as before: then by the 3 or tens figure, pla- 

cing the first product in the second or tens 

262512 place, immediately under the three, in the 

131256 multiplier. In like manner we use the 4, 

175008 placing the first product in the third or Atm- 

dreds place, immediately under the 4; alter 

19075872 which we add the several products together, 

and the work is done. 



73684 
427 

515788 
147368 
294736 

31463068 



1 Mu tiply 

2 

3 

4 

5 

6 

7 

8 

9 
10 
11 
12 
13 
14 
15 
16 > 
17 
18 



EXEBCISES. 

4736 by 

5762 by 

6483 by 

7368 by 

4327 ,y 

7382 by 

4728 by 

7584 by 

5678 by 

7G83 by 

4962 by 

7384 by 

4376 by 

7923 by 

6842 by 

7648 by 

8473 by 

9372 by 



37462 
563 



112386 
224772 
187310 

21091106 



34 Product 161024 



43 
54 
45 
56 
67 
76 
87 
78 
89 
98 
87 
97 
78 
89 
523 
456 
567 



247766 
350082 
331560 
242312 
494594 
359328 
659808 
442884 
683787 
486276 
642408 
424472 
617994 
608938 
3999904 !| 
3863688 1 
5313924 | 



SIMPLE MULTIPLICATION. 21 

NOTE 1. When either or both of the factors have 

noughts on the right hand, they may be omitted in the 

operation, and annexed to the product. Thus : 

47 I 000 734 I 00 

42 00 42 000 



94 1468 

188 2936 



-Product 197400000 Product 3082800000 

NOTE 2. When the multiplier is the exact product 
of any two factors in the multiplication table, the opera- 
tion may be performed by separating th multiplier into 
its components, and multiplying first by the one, then 
its product by the other. Thus : 

754 by 36 754 754 754 

9 3 6 36 

2272 4524 4524 

12 6 2262 



27144 27144 27144 27144 

9 and 4, or 3 and 12, or 6 and 6, multiplied together, 
produce 36, and by using either pair, according to the 
above note, the true result is obtained. 

EXERCISES. 

1 Multiply 756 by 42 Product 31752 

2 645 by 24 15480 

3 876 by 48 42048 

4 963 by 56 53028 

5 827 by 72 59544 

6 946 by 81 76626 

7 875 by 84 73500 

8 948 by 96 91008 

9 795 by 108 85860 
Tiie pupil may work these by all the several pairs of 

components that he can find in the multiplier. 



22 



SIMPLE MULTIPLICATION. 



NOTE 3. When the multiplier is not the exact pro 
duct of any two numbers in the table, use two factors 
whose product is short of the multiplier, then multiply 
the sum by the number required to supply the deficiency 
and add its product to that obtained by the two .factors. 



583X3 
4 



11680 
1749 

13409 

1 Multiply 

2 

3 

4 

5 



EXAMPLES. 

583X2 

7 



12243 
1166 



583X5 
3 

1749 
6 

10494 
2915 




13409 



13409 



13409 



EXERCISES. 



846 by 26 

784 by 29 

975 by 34 

859 by 43 

794 by 59 



Product 21996 
22736 
33150 
36937 
46846 



PROMISCUOUS EXERCISES. 



1. Charles has 24 marbles, and John has 13 times as 
many; how many has John? - Ans. 312. 

2. A gentleman owns 17 houses, for each of which he 
receives 250 dollars rent; how much does he receive 
for them all? Ans. 4250. 

3. A laborer hired himself to a farmer for 1 1 years, 
at 150 dollars a year; how much did he receive? 

Ans. 1650 dollars. 

4. A person wishes to purchase 26 shares of Bank 
stock at 75 dollars a share ; what must he pay ? 

Ans. 1950 dollars. 

5. A mason having built a house, found that 98470 
bricks were in it; suppose he desires to build 19 such 
houses, how many bricks must he obtain for the pur- 



pose 



Ans. 1870930. 



SIMPLE DIVISION. 23 

SIMPLE DIVISION. 

SIMPLE DIVISION is a short method of performing sev- 
eral subtractions. 

The number to be divided is called the dividend. 

The number by which it is to be divided is called the 
divisor. 

The number of times that the divisor is contained in 
the dividend, is called the quotient. 

So many figures of the dividend as are taken to be 
divided at one time, is called a dividual. 

If any thing remain when the operation is completed, 
it is called the remainder. 



CASE i. SHORT DIVISION. 
When the divisor does not exceed 12. 

RULE. Place the divisor on the left hand side of the 
number to be divided. 

Consider how often the divisor is contained in the first 
figure or figures of the dividend, and set down the result 
below ; observing how many remain, if any. If there 
be no remainder, consider how often the divisor is con- 
tained in the next figure : but if there be a remainder, 
call it so many tens, and add the next figure to it, and 
divide the sum, placing the result beneath, as before. 

Proof. Multiply the quotient by the divisor; add the 
remainder, if any, and the product will equal the divi- 
dend. 

EXAMPLES. 

Dividend. 
Divisor 2)182 2)648 3)963 4)484 

Quotient 241 324 



24 SIMPLE DIVISION. 

3)741851 Here 3 are contained In 

7 two times* and one re- 

o/iTOQO I Q T*^ n \*A^ mains ; place the two under 

247283-H Remainder theseven , and suppose the 

one that remains to be one 

ten, and add the next fig- 

Proof 741851 ure ( 4 ) to *> which make3 

fourteen. 

Now 3 are contained in fourteen 4 times, and 2 remain. Set the 
4 down under the 4 in the dividend, and suppose the two that remain 
to be two fen$, and add the next figure (1) to it, which make twenty- 
one. Now 3 into 21 go 7 times, and no remainder. Place the 7 un- 
der the 1 in the dividend, and proceed in the same manner with the 
other figures. 



4)65270167 5)6572686 

163175414-3 Rem. 1314537+1 Rem 

4 5 



Proof C5270167 Proof 6572G86 

6)8739627 7)4873692 

8)9273684 0)8379286 

10)946873 11)893726 12)98796 



SIMPLE DIVISION. 25 

EXERCISES. 

Divide 7893762 by 6 Ans. 1315627 

9387984 by 7 , 13411404 

6928437 by 8 8660545 

9276874 by 9 10307637 

8672934 by 10 8673934 

6873842 by 11 6248948 

7369287 by 12 6141073 

CASE 2. LONG DIVISION. 
When the divisor exceeds 12. 

RULE. Place the divisor to the left hand of the divi- 
dend, as in case 1. 

Consider how often the divisor is contained in the 
least number of figures into which it can be divided; 
and set down the result at the right hand of the dividend. 

Multiply the divisor by the quotient figure thus 
found, and set the product under the dividual or figures 
supposed to be divided. 

Subtract the product from the dividual, and set down 
what remains. Bring down the next figure of the divi- 
dend, and proceed as before, till all the figures are 
brought down and divided. 

EXAMPLES. 
Divisor. Dividend. Quotient. 

27)984376(36458 Twenty-seven into 98 go 3 

81 times: multiply the divisor 

(27) by 3 and set the product 
under the dividual (98) and 
subtract. To the remainder 
(17) bring down the next fig- 
ure (4) of the dividend. Now 
27 into 174 go 6 times. Place 
the 6 in the quotient and mul- 
tiply (27) the divisor, by 6, 
and set the product under 174 

and subtract as before, &,c. 

226 
216 
10 Remainder. 




26 SIMPLE DIVISION. 

Divisor. Dividend. Quotient. 
42) 98754 (2351 
84 42 Divisor 


147 4702 
126 9404 
12 Remainder 
215 
210 98754 Proof 


54 
42 

12 Remainder 

32)789627(24675 65)1827538(28115 
64 130 

149 527 
128 520 




216 
192 

242 
224 

187 


75 
65 

103 
65 

388 




160 






325 






27 Rem. 






63 Rem. 




EXEltCIgES. 




Divide 8769 


by 


13 


Quo. 674 


Rem. 7 




476 


by 


15 


31 


11 




958 


by 


18 


53 


4 




1475 


by 


28 


52 


19 




4277 


by 


31 


137 


30 




25757 


by 


37 


696 


5 


1 


63125 


by 


123 


513 


26 


1 


253622 


by 


422 


601 





SIMPLE DIVISION. 



27 



NOTE 1. Cyphers on the rigkt^hand of the divisor 
may be omitted in the operation, observing to separate 
as many figures from the right of the cttvidend, which 
must be annexed to the remainder. 



EXA3TPLES. 



54 



00)1463 
108 

383 
378 

Rem. 540 



40(27 



32 | 0)7617 
64 

121 
96 

257 
256 



3(238 



Rem. 13 

EXERCISES. Jf| 

Divide 40220 by 1900 Ans. 21 Keai. 320 

137000 by 1GOO 85 1000 

99607765 by 27000 3689 4765 

2304108 by 5800 397 1508 

NOTE 2. When the divisor is the exact product of 
any two numbers in the multiplication table, the opera- 
tion may be performed by dividing first by one of the 
component parts, and then the quotient by the other. 

To get the true remainder, multiply the last remain- 
der by the first divisor, and add the first remainder. 



EXAMPLES. 



7 98754 



42 



[o | 14107 5 first remainder 

2351 1 last remainder 
7 first divisor 



add 



|i 



5 first remainder 
12 true remainder 



28 SIMPLE DIVISION. 

984376 
27 * 

' 3281251 



36458~-3 X 3 + 1 =10 Rem 

EXERCISES. 

9756 by 35 Quotient 278 Rcm. 26 

8491 by 81 104 67 

44767 by 18 2487 1 

92017 by 56 1643 9 

38751 by 48 807 15 

7S4071 by 72 10195 31 

APPLICATION. 

1. SevWn boys have 161 apples, which they divide 
equally among them. How many does each have? 

Answer, 23. 

2. What is the quotient, if 8736 be divided by 8, and 
that quotient by 4? Ans. 273. 

3. If 350 dollars be equally divided among 7 men, 
what will be the share of each ? Ans. 50. 

4. How many times are 27 contained in 952? 

Ans. 35 times and 7 over. 

5. Suppose 2072 trees planted in 14 rows. How ma- 
ny trees will there be in each row? Ans. 148. 

6. Several boys who went to gather nuts, collected 
4741, of which each boy received 431. How many 
boys were there? Ans. 11. 

7. If the expense of erecting a bridge, which is 15036 
dollars, be equally defrayed by 179 persons, what must 
each pay? Ans. 84 dollars. 

8. Suppose a man receive in one year 2920 dollars; 
how much a day is his income at that rate; and sup- 
pose that his expenses for the year amount to 1769 dol- 
lars. How much will he save in a year? 

Ans. His income will be 8 dollars a day; he will save 
1151 dollars in a year. 



SIMPLE DIVISION. 29 

Questions. 

What is division? 

What do you call the number that is to be divided? 

What do you call the number you divide by ? 

What do you call the number obtained by division? 

What do you call that which is left when the work is 
done ? 

When the divisor does not exceed 12, how do you per- 
form the operation? 

When the divisor exceed 12, how do you proceed? 

How do you prove division? 

How may the operation be performed when there are 
cyphers at the right hand of the divisor? 

How may it be performed when the divisor is the exact 
product of two numbers in the multiplication table? 

How do you obtain the true remainder in the last case? 



PROMISCUOUS EXERCISES iN THE PRECEDING RULES. 

1. If the contents of five bags of dollars, containing 
$295, $410, $371, $355, and $520, be divided 
equally among 25 persons, how much is the share of 
each? Ans. $78.04 

2. A man possessed of an esnue of $30,000, disposed 
of it in the following manner: to his brother he gave 
$1500, and the balance to his 5 sons, to be equally di- 
vided among them. What was each one's share ? 

Ans. $5700. 

3. What number is it, which being added to 9709 
will make 110901? Ans. 101192. 

4. Add up twice 3?7, three times 794, four times 
31196, five times 15S80, six times 95280, and once 
33304. Ans. 812,344. 

5. Three merchants have a stock of 14876 dollars, 
of which A owns 4963 dollars, B 5188, and C the re- 
mainder. How much does C own? Ans. 4725 dolls. 



30 FEDERAL MONEY. 

FEDERAL MONEY, 

OR MONEY OF THE UNITED STATES. 

TABLE. 

10 mills make 1 cent 

10 cents 1 dime 

10 dimes 1 dollar 

10 dollars 1 eagle 

These denominations bear the same relation to each 
other as those of units, tens, hundreds, &c. Federal 
money is therefore added, subtracted, multiplied, and 
divided by the same rules as Simple Addition, Subtrac- 
tion, Multiplication, and Division. 

ADDITION OP FEDERAL MONEY. 

Rule. 

Place the numbers one under another, with mills on 
the right, cents, dimes, &,c., in succession ; observing to 
keep mills under mills, cents under cents, &c. Then 
proceed as in simple addition. 

When halves or- fourths of a cent occur, find their 
amount in fourths, and consider how many cents these 
fourths will make, and carry them to the column of cents. 

EXAMPLES. 

Eagles. Dolls. Dimes. Cents Mills. Dolls. Ds. Cts. 

789 
978 
684 
637 
482 

E. 

27 3 6 453570 

NOTE. In common business transactions, eagles, 
dimes, and mills are not used: dollars, cents, and frac- 
tions of a cent, are the only denominations kept in 
j accounts. 



3 


7 


8 


9 


5 


7 


4 


9 


8 


7 


2 


3 


8 


7 


9 


8 


8 


9 


8 


6 


4 


7 


8 


9 


8 



FEDERAL MONEY. 



EXAMPLES. 



Ds. cts. 
34 62 



56 
27 
23 
27 



31 

82 
68 
42 



169 , 85 

Ds. cts. 
468 , 31 
723 , 62 

845 , 92 
736 , 25 

846 , 31 
428 , 62 



EXERCISES. 

(2) 
Ds. cts. 

927 , 24 
768 , 32 
427 , 56 
792 , 34 

587 , 62 
842 , 27 



Ds. cts. 
427 , 68 
342 , 31 
427 , 26 
793 , 84 
273 , 42 

2264 , 51 

(3) 

Ds. cts, 
273 45 



846 
283 
846 
674 
273 



37 
75 
91 
75 
25 



Ds. cts. 

437 , 62i 
386 , 814 
243 , 18| 

427 , 37* 

428 , 12i 



One halfis two-fourths; and one half more make 
four fourths, and three fourths more make seven 
fourths, and one fourth more make eight fourths, and 
one half (or two fourths) more make ten fourtiis. 
Four fourths make one cent, then ten fourtiis make 
two cents, and leave two fourths, or one half cent. 
Set down the i cent, and carry the two cents to the 
next column. 



1923 , 12* 



Ds. cts. 
274 , 814 
362 , 87* 
421 , 184 
625 , 314 
241 , 561 



Ds. cts. 

27 , 68| 

36 , 81* 

28 , 62i 

37 , 934 
24 , 62i 



Ds. cts. 
56 , 064 
32 , 12i 
36 , 25 

42 , 62i 
54 , 814 



32 FEDERAL MONEY. 

APPLICATION. 

1. Add 48 dollars 20 cents; 14 dollars 58 cents; 100 
dollars 25 cents; and 84 dollars 36 cents. 

Ans. 247 dollars 39 cents. 

2. Add $7,62*, $34,314, $72,064, $41,314, $25, 
68|, and $87,43| together, and tell the amount. 

Ans. $268,43|. 

3. Bought a hat for $4,25 cents ; a pair of shoes for 
$2,25; a pair of stockings for $1,25, and a pair of gloves 
for 75 cents. What is the cost of the whole ? 

Ans. $8 50 cents. 

4. If I buy coffee for $1,18|, tea for $2,50, cloves 
for 87 ft, mace for 93|, cinnamon for $l,87i, raisins for 
$2,68|, nutmegs for 37i, candles for 87, and wine for 
$1,93|, what must I pay for them? Ans. $13,25. 

Questions. 

What relation do mills, cents, dimes, tec., bear to each 
other? 

How are the addition, subtraction, multiplication, and 
division of Federal money performed? 

How do you place the numbers to be added? 

How do you proceed when halves, fourths, &,c., occur? 

SUBTRACTION OF FEDERAL MONEY. 

RULE. Place the less under the greater, with dollars 
under dollars, and cents under cents; then, if there are 
no fractions, proceed as in simple subtraction. 

If there is a fraction in the upper sum and none in 
the lower, set it down as a part of the remainder, and 
proceed as before. 

If there is a fraction in each sum, and the lower be 
less than the upper, subtract the lower from the upper, 
and set down the difference. 

If the lower fraction be greater than the upper one, 
borrow one cent, and call it four fourths, and add them 
to the upper fraction, and subtract the lower one from 
the sum. 

Proof. As in simple subtraction. 



FEDERAL MONEY. 33 

EXAMPLES. 

Ds cts. Ds. cte. Ds. cts. 
32 ,62 43 , 68| 75 , 68* 
21 ,31 21 , 25 24 , 12i 

$11 , 31 $22 , 43| 51 , 564 

NOTE. TJiree fourths cannot be taken from two 

Ds. cts. fourths : then borrow one cent from the two cents, 

271 62* which has four fourths in it: add the/our fourths to 

, or> ' QQ 3 tne t wo fourths, this makes six fourths ; subtract 

lo/5 , Jof three fourths from six fourths, and three fourths (|) 

remain. Set down the and add one to the next 

138 , 68| 3, as in simple subtraction. 

EXERCISES. 

Ds. cts. Ds. cts. Ds. cts. 
65 , 49 520 , 314 436 , 31* 
35 , 12i 210 , 12* 243 , 18| 



Ds. cts. Ds. cts. Ds. cts. 
273 , 62* 237 , 564 732 , 314 
124 . 37* 142 , 874 261 , 68| 



APPLICATION. 

1. Subtract $432,68| from 1000,93|. 

Ans. $568,25. 

2. Subtraction shows the difference between two 
numbers; what is the difference between $37,62 * and 
$93,87*. Ans. $56,25. 

3. Bought goods to the amount of $545,95, and paid 
at the time of purchase $350; How much remains un- 
paid? Ans. $195,95. 

4. A merchant bought a quantity of cotil?e, for which 
he paid $560. He afterwards sold it foi $610,87* 
How much did he gain by the transaction ? 

Ans. $50,87*. 

n > 



34 FEDERAL MONEY. 

Questions. 

How do you place the numbers in subtraction of 
Federal Money ? 

How do you perform the operation? 

If a fraction occur in the upper line or minuend, what 
do you do with it? 

If a fraction occur in each, how do you proceed ? 

Suppose the lower fraction is greater than the upper 
one, how do you proceed ? 

How do you prove subtraction of Federal Money ? 

MULTIPLICATION OF FEDERAL MONEY. 

RULE. Set the multiplier under the multiplicand, 
and if there be no fractions, proceed as in simple multi- 
plication; observing to separate the cents from the dollars 
in the product. 

If there is a fraction in the sum, multiply it, and see 
how many cents are in the product; set down the frac- 
tion that is over, and proceed as before. 

Or if the multiplier exceeds 12, multiply the sum, 
omitting the fractions; then multiply the fraction, and 
add the number of cents contained in the product, to the 
product of the rest of the sum. 

EXAMPLES. 

Ds. cts. 
10 , 56* 
2 

$50 , 00 $21 , 12* $118 , 12* 

Ds. cts. Ds. cts. 

10,87* 125 times 4,18* 24 times * are 

125 one half make 24 72 fourths: four 

125 halves: 2 fourths are con- 

5435 into 125 go 62 1672tained 18 times in 

2174 times, leaving 836 72 fourths, rnak- 

1087 one; that is, 18 ing 18 cents. 

62* one half, mak- 

-ing 62* cents. $100,50 

$1359,37* 





FEDERAL MONEY. 
EXERCISES. 



1 Multiply 

2 

3 

4 

5 

G 



$145,18| 

7,874 

28,684 

42,314 

137,62* 

79,004 



by 
by 
by 
by 
by 
by 207 



7 

47 
68 
58 
67 



35 



Ans. $1016,314 

370,124 

1950,75 

2454,124 

9220,874 

16354,034 



APPLICATION. 



1. What will 8 pounds of cheese come to, at 18 cents 
a pound? Ans. $1 44 cts. 

2. What is the value of 12 yards of linen, at 35 cents 
;a yard? Ans. $4 20 cts. 

3. What cost 29 yards of cloth at $2 25 cts. a yard? 

Ans. $65 25 cts. 

4. What wMl 213 barrels of flour cost, at $5 25 cents 
a barrel? Ans. $1118 25 cts. 

5. Bought 321 barrels of cider at $1 25 cts. a barrel. 
What did it amount to? Ans. $401 25 cts. 

6. What will 580 bushels of salt cost at $1 124 cts. a 
bushel.. Ans. $652 50 cts. 

7. What is the value of 2 pieces of cloth, one contain- 
ing 38 yards, and the other 26 yards, at $3 874 cts. a 
yard? ' Ans. $248. 

8. What will be the cost of 132 pieces of linen at 
$17 374 cts. each? Ans. $2293 50 cts. 

9. What will 8 cords of wood amount to, at 4 dollars 
{ 50 cents a cord? Ans. 36 dollars. 

10. Sold 213 barrels of flour for 6 dollars 25 cents per 
barrel. What is the amount? Ans. 1331 dols. 25 cts. 

11. Bought )08 pounds of coffee at 21 cents a pound. 
What is the amount? Ans. 64 dols. 68 cts. 

12. Bought 217 gallons of brandy at $1 18| cts. per 
gallon; and sold it for $1 374 cts. per gallon. What was 
the amount paid for the whole; the sum it sold for; and 
the gain? 

Ans. Prime cost, $257 68|: sold for $298 374; gain, 
$40,681 



36 



FEDERAL MONEY 



DIVISION. 

RULE. Divide as in simple division. When a re- 
mainder occurs, multiply it by 4; and add the number 
of fourths that are in the fraction of the sum (if any) to 
its product: divide this product by the divisor, and its 
quotient will be fourths, which annex to the quotient. 

Proof. As in simple division. 



Ds. cts. 

2)45,22 



22,61 
Ds. cts. 

25)629,68|(25,1S| 
50 



EXAMPLES. 

Ds. cts. 
3)63.181 

21,064 



Ds. cts. 
2)25,374 

12,684 

32)78800(24,624 
64 



129 148 

125 128 

46 200 

25 192 

218 80 

200 64 

18 Here 18 cents remain; multiply 16 

4 18 cents by four, brings them to 4 
fourths of a cent; add the |, this 
makes 75 fourths: divide 75 fourths 

by 25, and are obtained, whicii 3*2/4(2 or 4 

25)7 ''(<* place in the quotient. 64 



Divid 



D. cts. 

56,15 

96,00 
156,00 

58,14 
417,96 
494,45 
627,38 



by 
by 
by 
by 
by 
by 
by 



EXERCISES. 

10 
5 

4 

38 

129 

341 

508 



Quotient 5,61 4 

- 19,20 

- 39,00 

- 1,53 
-- 3,24 



1,234 



FEDERAL MONEY. 37 

APPLICATION. 

1. If 7 pounds of butter cost $1,89 cts., what is the 
value of 1 pound? Ans. 27 cts. 

2. If 8 Ibs. of coffee cost $2,04 cts., what is the price 
of one pound? Ans. 25i cts. 

3. Bought 29 yds. of fine linen for $65,25 cts., what 
was the price per yard? Ans. $2,25. 

4. Paid $58,75 cts. for 235 yds. of muslin, what 
was it per yard? Ans. 25 cts. 

5. A piece of cloth containing 72 yds. cost $450, 
what was it per yard? Ans. $6,25. 

Questions. 

How do you perform division of Federal Money? 
How do you proceed when a remainder occurs? 



PROMISCUOUS EXERCISES IN THE PRECEDING RULES. 

1. Bought 18 barrels of potatoes, each containing 
3 bushels, at 25 cts. a bushel, what did they cost? 

Ans. $13,50. 

2. A farmer sold 30 bushels of rye at 87 cts. a 
bushel, 30 bushels of corn at 53 cts. a bushel; 8 bushel 
of beans at $1,25 cts. a bushel; 2 yoke of oxen at 
$62 a yoke; 10 calves at $4 a piece; 15 barrels of 
cider at $2,37i a barrel, what was the amount of the 
whole? Ans. $251,62*. 

3. What will be the price of four bales of goods, each 
bale containing 60 pieces, and each piece 49 yards, at 
374 cents a yard? Ans. $4410. 

4. Add $324,43* cts. $208,09* cts. and $507,90* cts. 
together, and divide the sum by 2, and what will be 
the result? Ans. $520,21|. 

5. Divide 400 dollars, equally, among 20 persons. 
What will be the portion of each person? Ans. $20. 

6. Divide 1728 dollars, equally among 12 persons. 
What does each one of them share? Ans. $144. 

7. If 240 bushels cost 420 dollars; what is the cost 
of one bushel at the same rate? Ans. $1.75. 



38 REDUCTION. ' 

REDUCTION. 

REDUCTION is the changing o f a sum, or quantity, 
from one denomination to another, without altering the 
value. 

CASE 1. 

To reduce a sum, or quantity, to a lower denomination 
than its own. 

RTTLE. Multiply the sum, or quantity, by that num- 
ber of the lower denomination which makes one of its 
own. 

If there are one or more denominations between the 
denomination of the given sum, and that to which it is 
to be changed, first change it to the next lower than its 
own; then to the next lower, and so on to the deno- 
mination required. 



DRY MEASURE. 

TABLE. 

2 pints (pts.) make 1 quart, qt. 

8quarts - 1 peck, pc. 

4 pecks - 1 bushel, bu. 

NOTE. This measure is used for measuring grain, 
salt, fruit, &,c. 

EXAMPLES. 

NOTE. 1. To reduce bushels to pecks, multiply by 
4, because each bushel has 4 pecks in it. 

1. Reduce 23 bushels to pecks. 

bit. 

23 

4 

Amt. 92 pecks. 

2. Reduce 35 bushels to pecks. Amt. 140 peeks. 
NOTE. 2. To reduce pecks to quarts, multiply by 8, 

because each peck has 8 quarts in it. 



REDUCTION. 39 

3. Reduce 27 pecks to quarts. 



8 

Amt. 216 quarts. 

4. Reduce 43 pecks to quarts. Amt. 344 quarts. 
NOTE. 3. To reduce quarts to pints multiply by 2, 

because each quart has 2 pints in it. 

5. Reduce 43 quarts to pints. 

qt. 

43 

2 

Amt. 86 pints. 

6. Reduce 32 quarts to pints. Amt. 64 pints. 
Reduce 34 bushels to pints. . 

fctt. 
34 
4 Multiply the bushels by 4 to bring 

them to pecks. 
136 

8 Multiply the pecks by 8 to bring 

- them to quarts. 
1088 

2 And multiply the quarts by 2 to bring 

- them to pints. 
Amt. 2176 pints. 



7. Reduce 56 pecks to pints. Amt. 896 pints- 

8. Reduce 47 bushels to quarts. Amt. 1504 qt 

9. Reduce 85 bushels to pints. Amt. 5440 pt 

10. Reduce 63 pecks to quarts. Amt. 504 qt. 

11. Reduce 132 bushels to quarts. Amt. 4224 qt. 

12. Reduce 234 bushels to pints. Amt. 14976 pt. 
NOTE. 4. When several denominations occur, reduce 

the highest denomination to the next lower one, and this 
again to the next lower, and so on ; observing to add 
the amount of each denomination, the number there is 
of that denomination in the given sum. 



40 REDUCTION. 

EXAMPLES. 

1. Reduce 23 bushels, 3 pecks, 5 quarts, 1 pint, to 
pints. 

bu. pe. qt. pt. 
23-3-5-1 

4 Multiply the bushels by 4 

to bring them to pecks, and 

92 add the 3 pecks to the amount, 

3 which makes 95 pecks. 



Multiply the pecks by 8 to 
bring them to quarts, and add 
the 5 quarts, which makes 
765 quarts. 



Multiply the quarts by 2 
to bring them to pints, and 
add the 1 pint which makes 
1531 pints. 



1531 amt. 



Or thus: 
bu. pe. qt. 
23 3 - 5 - 

4 Multiply by 4 as above; 

add the 3, and set down the 

95 amount, &c. 

8 

765 
2 

1531 Amt. as before. 

EXERCISES. 

1. Reduce 13 bushels, 2 pecks, 7 quarts, 1 pint to 
pints. Amt. 879 pints. 



REDUCTION. 41 

2. Reduce 24 bushels, 3 pecks, 1 quart to quarts. 

Amt. 793 qt. 

3. Reduce 7 bushels, 3 pecks to quarts. Amt. 248 qt. 

4. Reduce 3 pecks, 2 quarts to pints. Amt. 52 pt. 

5. Reduce 7 quarts, 1 pint, to pints. Amt. 15 pt. 

6. Reduce 32 bushels, pecks, 1 quart to pints. 

Amt. 2050. 
7* Reduce 5 bushels, 1 peck, quarts, 1 pint to pints. 

Amt. 337 pt. 
8. Reduce 43 bushels, 1 peck to pints. 

Amt. 2768 pt. 

Question*. 

What is reduction? 
For what is case first used ? 

How do you reduce a sum to a lower denomination 
than its own? 

How do you reduce bushels to pecks? 

Why do you multiply by 4 ? 

How do you reduce pecks to quarts? 

Why do you multiply by 8? 

How do you reduce quarts to pints? 

How do you reduce bushels to pints? 



AVOIRDUPOIS WEIGHT. 

TABLE. 

16 drams (dr.) make 1 ounce, oz. 

16 ounces 1 pound, Ib. 

28 pounds 1 quarter of a cwt. qr. 

4 quarters, (or 112 lb.)* 1 hundred weight, cwt 

20 hundred weight 1 ton, T. 

NOTE. By this weight are weighed, tea, sugar, cof- 
fee, flour and other things subject to waste, and all the 
metals, except silver and gold. 



* The gross hundred weight of 112 pounds is nearly out cf use: 
the decimal hundred weight of 100 pounds is taking its place. 



42 



REDUCTION. 



EXAMPLES. 

1. Reduce 23 tons to hundred weight. 

tons. 
23 
20 

Amt. 460 cwt. 

2. Reduce 34 hundred weight to quarters. 

cwt. 

34 

4 

Amt. 136 quarters. 

3. Reduce 42 quarters to pounds, qrs. 

42 

28 



336 

84 



Amt. 1176 pounds. 

4. Reduce 73 pounds to ounces. 

Ibs. 

73 

16 

438 
73 

Amt. 1168 ounces. 

5. Reduce 54 ounces to drams. 

oz. 
54 

16 t 

324 
54 

Amt. 864 drams. 



REDUCTION. 43 

6. Reduce 35 tons to drams. 

tons. 
35 
20 

700 cwt. 
4 



2800 qr. 

28 

22400 
5600 

78400 Ib. 
16 

470400 
78400 

1254400 oz. 



7526400 
1254400 

Amount. 20070400 drams. 



EXERCISES. 

7 Reduce 24 pounds to drams. Amt. 6144 dr. 

8 Reduce 36 hundred weight to pounds. 

Amt. 4032 Ib. 

9. Reduce 73 quarters" to ounces. Amt. 32704 oz. 
10. Reduce 2 tons to pounds. Amt. 4480 Ib. 

11 Reduce 4 tons to drams. Amt. 2293760 dr. 



44 REDUCTION. 

12. Reduce 3 tons, 13 cwt., 2 qu., 14 Ibs., to pounds. 

T. cwt. qr. Ib. 
3-13-2-14 
20 

60 Or thus: 

13 T. cwt. qr. Ib. 

3-13-2-14 

73 20 

4 

73 

292 4 

2 

294 

294 28 

28 

2366 

2352 588 

588 

8246 pounds 

8232 

14 

8246 pounds. 

13. Reduce 2 tons. 15 cwt. 2 qr. to quarters. 

Amt. 222 qr. 

14. Reduce 3 tons. 25 Ib. to pounds, ^nt. 6745 Ib. 

15. Reduce 5 cwt. 3 qr. 14 Ib. to ounces. 

Amt. 10528 oz. 

16. Reduce 2 cwt. 2 qr 14 ounces to drams. 

Amt. 71,904 dr. 

TROY WEIGHT. 

TABLE. 

24 grains (gr.) make 1 pennyweight, dwt. 

20 pennyweights - 1 ounce, oz. 

12 ounces - 1 pound, Ib. 

NOTE. By this weight, jewels, gold, silver, and 
I liquors, are weighed. 



REDUCTION. 45 

EXASEPLES. 

1. Reduce 32 pounds to ounces. Ib. 

' 32 
12 

Amt. 384 ounces. 

2. Reduce 23 ounces to pennyweights. oz. 

23 
20 

Amt. 460 dwt. 

3. Reduce 43 pennyweights to grains, dwt. 

43 
24 



172 

86 



Amt 1032 grains. 

4. Reduce 53 pounds to grams. Ibs. 

53 
12 




50880 
25440 






Amt. 305280 grains. 

EXERCISES. 

1. Reduce 24 ounces to grains. Amt. 11520 gr. 

2. Reduce 32 pound* to pennyweights. Amt. 7680 dwt. 

3. Reduce 132 pounds to ounces. Amt. 1584 oz. 

4. Reduce 234 ounce's to grains. Amt. 112320 gr. 



46 REDUCTION. 

5. Reduce 463 pounds to grains. Amt. 2666880 gr. 

6. Reduce 47 pounds, 10 ounces, 15 pennyweights to 
pennyweights. Amt. 11495 dv/t. 

7. Reduce 5 pounds, 6 ounces, 4 pennyweights, 20 
grains to grains. Amt. 31796 gr. 



APOTHECARIES WEIGHT. 

TABLE. 

20 grains (gr.) make 1 scruple, sc. 9 

3 scruples - 1 dram-, dr. 3 

8 drams - 1 ounce, oz. 3 

12 ounces - 1 pound, Ib. 

NOTE. By this weight apothecaries mix their medi- 
cines, but they buy and sell by Avoirdupois Weight. 

EXERCISES. 

1. Reduce 32 pounds to ounces. Amt. 384 oz. 

2. Reduce 43 ounces to drams. Amt. 344 dr. 

3. Reduce 27 drams to scruples. Amt. 81 sc. 

4. Reduce 37 scruples to grains. Amt. 740 gr. 

5. Reduce 28 pounds to drams. Amt. 2688 dr. 

6. Reduce 36 ounces to scruples. Amt. 864 sc. 

7. Reduce 27 drams to grains. Amt. 1620 gr. 

8. Reduce 23 pounds to grains. Arrt. 132480 gr. 

9. Reduce 3 pounds, 5 ounces, 2 scruples to scru- 
ples. Amt. 986. sc. 

10. Reduce 7 ounces, 5 drams, 14 grains to grains. 

Amt. 3674 gr. 

11. Reduce 27 pounds, 7 ounces, 2 drams, 1 scruple, 
2 grains, to grains. Amt. 159022 gr. 

CLOTH MEASURE. 

TABLE. 
4 nails (na.) make 1 quarter of a yard, qr. 

4 quarters - 1 yard, yd, 

3 quarters - 1 Ell Flemish, E. Fl 

5 quarters - 1 Ell English, E. E. 

6 quarters - 1 Ell French, E. Fr. 
NOTE. By this measure cloth, tapes, linen, muslin, 

&c., are measured. 



REDUCTION. 47 

EXERCISES. 

1. Reduce 24 yards to quarters. Amt. 96 qr. 

2. Reduce 32 quarters to nails. Amt. 128 na. 

3. Reduce 27 yards to nails. Amt. 432 na. 

4. Reduce 46 Flemish ells to quarters. 

Amt. 138 qr. 

5. Reduce 27 English ells to quarters. 

Amt. 135 qr. 

6. Reduce 34 French ells to quarters. 

Amt. 204 qr. 

7. Reduce 45 Flemish ells to nails. Amt. 540 na. 

8. Reduce 36 English ells to nails. Amt. 720 na. 

9. Reduce 54 French ells to nails. Amt. 1296 na. 

10. Reduce 13 yards, 3 quarters to quarters. 

Amt. 55 qr. 

11. Reduce 3 quarters, 2 nails, to nails. Amt. 14 na. 

12. Reduce 24 yards, 2 nails to nails. Amt. 386 na. 

13. Reduce 13 E. ells, 2 qrs., 3 nails to nails. 

Amt. 271 na. 



LONG MEASURE. 

TABLE. 

12 inches (in.) make 1 foot, ft. 

3 feet 1 yard, yd. 

5* yards - 1 Rod, Pole, or Perch, p. 

40 poles 1 Furlong. 

8 Furlong 1 Mile. 

3 Miles 1 League. 

60 Geographic, orJ , Hpjrrpp 
694 Statute Miles j 

NOTE. This measure is used for length and di 
tances. 

A Hand is a measure of four inches, and is used in 
measuring the height of horses. 

A Fathom is 6 feet, and is chiefly used in measuring 
the depth of water. 



REDUCTION. 



EXERCISES. 



1. Reduce 23 leagues to miles. Amt. 69 m. 

2. Reduce 43 miles to furlongs. Amt. 344 f. 

3. Reduce 27 furlongs to poles. Amt. 1080 p. 

4. Reduce 56 poles to yards. Amt. 308 yd. 

5. Reduce 132 yards to feet. Amt. 396 ft. 

6. Reduce 76 feet to inches. Amt. 912 in 

7. Reduce 24 miles to poles. Amt. 7680 p. 

8. Reduce 32 furlongs to yards. Amt. 7040 yd. 

9. Reduce 86 poles to inches. Amt. 1 7028 in. 

10. Reduce 26 leagues to yards. Amt. 137280 yd. 

11. Reduce 52 miles to feet. Amt. 274560* ft. 

12. Reduce 5 leagues to inches. Amt. 950400 in. 

13. Reduce 24 degrees to statute miles. Amt. 1668 m. 

14. Reduce 12 miles, 3 furlongs, 25 poles to poles. 

Amt. 3985 po. 

15. Reduce 14 leagues, 2 furlongs to poles. 

Amt. 13520 po. 

16. Reduce 3 leagues, 2 miles, 6 furlongs, 18 poles to 
yards. , Amt. 20779 yds. 



LAND, OR SQUARE MEASURE. 



TABLE. 



144 square inches make 

9 square feet 
304 square yards 
40 square perches - 

4 roods 



square foot, 
square yard, 
square perch, 
rood, 
acre, 



ft. 
yd. 

P- 
r. 



NOTE. This measure is used to ascertain the quan- 
tity of lands, and of other things having length and 
breadth to be estimated. 

EXERCISES. 

1. Reduce 27 acres to roods. . Amt. 108 r. || 

2. Reduce 53 roods to perches. ' Amt. 2120 p. |] 



REDUCTION. 49 

3. Reduce 28 perches to square yards. 

Amt. 847 sq. yds. 

4. Reduce 36 square yards to square feet. 324 ft. 
5 Reduce 27 square feet to square inches. 

Amt. 3888 in. 

6. Reduce 34 acres to perches. Amt. 5440 p. 

7. Reduce 42 roods to square yards. 

Amt. 50820 sq. yds. 

8. Reduce 24 square perches to square feet. 

6534 ft. 

9. Reduce 32 roods to square feet. Amt. 348480 ft. 

10. Reduce 23 acres to square inches. 

Amt. 144270720 sq. in. 

11. Reduce 11 acres, 2 roods, 19 perches to perches. 

Amt. 1859 p. 

12. Reduce 17 acres, 3 roods to perches. 

Amt. 2840 p. 

13. Reduce 12 acres, 12 roods, 12 perches to square 
yards. Amt. 60863 sq. yd. 

CUBIC, OR SOLID MEASURE. 

TABLE. 

1728 cubick inches make 1 cubic foot 

27 feet 1 cubic yard 

40 feet of round timber, or ) , m 
50 feet of hewn timber, | l Ton or load 
128 solid feet 1 Cord of wood 

NOTE. This measure is employed in measuring 
solids, having length, breadth, and thickness to ba esti- 
mated. 

EXERCISES. 

1 Re Ace 29 cords of wood to cubick feet. 

Amt. 3712 c. i 

2 Reduce 32 cubic yds. to feet. Amt. 864 c. f. 

3 Reduce 23 cubic feet to inches. Amt. 39744 c. in. 

4 Reduce 32 cubic yds. to inches. Amt. 1492992 c. in. 

5 Reduce 2 cords of wood to inches. Amt. 442368 c. in 

6 Reduce 3 cords, 10 feet to feet. Amt. 394 f> 

7 .Reduce 1 cord. 3 feet, 136 inches fo inches. 

Amt. 226504 ta- 



50 REDUCTION. 

LIQUID MEASURE. 

TABLE. 

4 gills make 1 pint pt. 

2 pints (pts) 1 quart qt. 

4 quarts 1 gallon gal. 

42 gallons 1 tierce te. 

63 gallons 1 hogshead hhd. 

2 hogsheads 1 pipe or butt pi. 

2 pipes 1 tun. T 

NOTE. This measure is employed in measuring 
cider, oil, beer, &c. 

EXERCISES. 

1 Reduce 23 tuns to pipes. Amt. 46 pi. 

2 Reduce 43 pipes to hogsheads. Amt. 86 hhd. 

3 Reduce 34 hogsheads to gallons. Amt. 2142 gal. 

4 Reduce 27 tierces to gallons. Amt. 1134 gal. 

5 Reduce 53 gallons to quarts. Amt. 212 qt. 

6 Reduce 724 quarts to pints. Amt. 1448 pt. 

7 Reduce 37 pints to gills. Amt. 148 g. 

8 Reduce 12 pipes to gallons. Amt. 1512 gal. 

9 Reduce 4 hogsheads to quarts Amt. 1008 qt. 

10 Reduce 32 gallons to gills. Amt. 1024 g. 

11 Reduce 2 tuns to gills. Amt. 16128 gills 

12 Reduce 32 gals 3 qts. to pints. Amt. 262 pt. 

13 Reduce 2 hogsheads, 27 gals. 3 qts to quarts. 

Amt. 615 qt. 

14 Reduce 3 tons, 1 hogshead, 15 gals. 1 qt to pints. 

Amt. 6674 pt. 

MOTION, OR CIRCLE MEASURE. 

TABLE. * 

60 seconds ("sec) make 1 minuta ^in. 

60 minutes 1 degree deg. 

30 degrees 1 sine , sin. 

12 sines (or 360 degrees) 1 revolution 

NOTE. This measure is employed by astronomers, 
navigators, &c. 



REDUCTION. 51 

EXERCISES. 

1 Reduce 5 sines to degrees. Ami. 150 

2 Reduce 8 degrees to minutes. Amt. 480 1 

3 Reduce 6 minutes to seconds. Amt. 360 sec. 

4 Reduce 12 sines to seconds. Amt. 1296000 sec. 

5 Reduce 3 sines 15 degrees to minutes. 

Amt. 6300 min. 

TIME. 

TABLE. 

60 seconds (sec) make 1 minute min. 
60 minutes 1 hour H. 

24 hours 1 day 

7 days 1 week 

12 months (or 365 days) 1 year. 

NOTE. The true year, according to the latest and 
most accurate observations, consists of 365 d. 5 h. 48 m. 
and 58 sec : this amounts to nearly 365$ days. The com- 
mon year is reckoned 305 days, and every fourth or 
leap year one day more on account of the fraction omit- 
ted each year, which being put together, every fourth 
year is added to it, making leap year 366 days. 
The year is divided into 12 months as follows. 

The fourth, eleventh, ninth and sixth, 
Have thirty days to each affixed, 
And every other thirty-one, 
Except the second month alone, 
Which has but twenty-eight in fine, 
Till leap year gives it twenty-nine. 

OR THUS: 

Thirty days hath September, 
April, June, and November, 
February hath twenty-eight alone, 
And each of the rest has thirty one. 
When the year can be divided by four, without a re- 
mainder, it is bissextile, or leap year. 



52 REDUCTION. 

EXERCISES. 

1 Reduce 42 years to months. Amt. 504 m. 

2 Reduce 23 days to hours. Amt. 552 h. 

3 Reduce 36 hours to minutes. Amt. 2160 min. 

4 Reduce 25 minutes to seconds. Amt. 1500 sec. 

5 Reduce 14 days to minutes. Amt. 20160 min 

6 Reduce 52 hours to seconds. Amt. 187200 sec. 

7 Reduce 13 weeks to hours. Amt. 2184 h. 

8 Reduce 12 weeks to minutes. Amt. 120960 min. 

9 Reduce 3 years to minutes, allowing 365 days to 
each year. Amt. 1576800 min. 

10 Reduce 15 years and 6 months to months. 

Amt. 186 m. 

11 Reduce 4 weeks, 3 days, 22 hours, to hours. 

Amt. 766 h. 

12 Reduce 7 years, 24 days, 43 minutes, to seconds. 

Amt. 222828180 sec. 

STERLING MONEY. 

TABLE. 

4 farthings (qr) make 1 penny d. 
12 pence 1 shilling s. 

20 shillings 1 pound 

Farthings are usually written as fractions of a penny, 
thus- i one farthing 

4 two farthings or a half penny. 
I three farthings. 

EXERCISES. 

1 Reduce 14 pounds to shillings. Amt. 280 s* 

2 Reduce 23 shillings to pence. Amt. 276 d. 

3 Reduce 34 pence to farthings. Amt. 136 qr. 

4 Reduce 4 pounds to pence. Amt. 960 d. 

5 Reduce 13 shillings to farthings. Amt. 624 qr. 

6 Reduce 16 pounds to farthings. Amt. 15360 qr. 

7 Reduce 13 pounds 14 shillings, to pence. 

Amt. 3288 d. 

8 Reduce 3 pounds 15 shillings 6 pence to farthings. 

Amt. 3624 qr. 



REDUCTION. 53 

FEDERAL MONEY. 

TABLE. 

10 mills make 1 cent 
10 cents 1 dime 

10 dimes 1 dollar 

10 dollars 1 eagle 

EXERCISES. 

1 Reduce 5 eagles to cents. Amt. 5000 ct. 

2 Reduce 3 dollars to mills. Amt. 3000 m. 

3 Reduce 15 dimes to cents. Amt. 150 ct. 

4 Reduce 3 eagles, 5 dollars to cts. Amt.3500 ct. 

5 Reduce 7 dollars, 3 dimes, 6 cents, to mills. 

Amt. 7360 m. 

As eagles, dimes and mills are not used in accounts, 
they will generally be omitted in the subsequent exer- 
cises of this work. 

4 fourths, or 3 thirds, or 2 halves, make 1 cent. 
100 cents - 1 dollar. 

6 Reduce 125 cents to halves of a cent. 

Amt. 250 halves. 

7 Reduce 32 cents to fourths of a cent. 

Amt. 128 fourths. 

8 Reduce 23 dollars to cents. Amt. 2300 ct. 

9 Reduce 25 dollars 15 cents to cents. Amt. 2515 ct. 

10 Reduce 15 dollars 374 cents to halves of a cent. 

Amt. 3075 halves. 

11 Reduce 21 dollars 15 cents to thirds of a cent. 

Amt. 6345 thirds. 

12 Reduce 5 dols. 37i cents to fourths of a cent. 

Amt. 2150 fourths. 

13 Reduce 15 dollars 33* cts. o thirds of- a cent. 

Amt. 4600 thirds 

NOTE. To reduce dollars to cenrs annex two cyphers : 
thus 53 dollars are 5300 cents. 

To reduce dollars and cents to cents, place them to- 



54 lllIDUCTiOX. 

gather without any separating point, and the amount 
will be cents. Thus 35 dollars 24 cents are 3524 cents. 

Questions. 

For what purpose is Dry measure used ? 
For what is Avoirdupois weight used? 

For what is Troy weight employed? 

For what is Apothecaries weight employed? 

For what is Cloth measure employed? 

For what is Long measure used? 

For what is Land or Square measure used? 

For what is Cubick -measure employed? 

For what is Liquid measure employed? 

For wha.t is Sterling currency used? 

For what is Federal currency used? 

CASE 2. 

To reduce a sum or quantity to a HIGHER denomina- 
tion than its own 

RULE. Divide the sum or quantity by that number of 
its own denomination which makes one of the denomina- 
tion to which it is to be changed. 

When there are one or more denominations between the 
denomination of the given sum and that to which it is 
to be changed; first change it to the next higher than its 
own, and then to4he next higher, antl so on. 

Remainders are always of the same denominations as 
the sums divided. 

DRY MEASURE. 

EXAMPLES. 

1 Reduce 25 pints to quarts. pts. 
NOTE. Divide by 2, because every 2 pints 2)25 
make one quart. In 25 are 12 two's and 

1 over, that is 12 quarts and 1 pint. ql.12 Ipt 

2 Reduce 43 quarts to pecks. qt. 
Divide by 8, because every 8 qts. make 8)43 

1 peck. In 43 are 5 eights and 3 over, 

that is 5 pecks and 3 quarts. pe. 5 3qt 



^ REDUCTION. 55 

3 Reduce 2G pecks to bushels. bu. 
Divide by four because every 4 pecks 4)26 

make 1 bushel. In 26 are 6 fours and 

2 over; that is 6 bushels and 2 pecks. bu. 6 2 pecks 

4 Reduce 359 pints to bushels. pi. 
Divide pints by 2, brings them 2)359 

to quarts; divide quarts by 8, brings 

them to pecks, and divide pecks by 8)179 1 pt. 

4 brings them to bushels. 

4) 22 3qt 



5 b. 2 p. 3 qt. 1 pt. 

5 Reduce 81 quarts to bushels. A. 2 bu. 2 pe. 1 qt. 

6 Reduce 134 pints to pecks. 8 pe. 3 qt. 

7 Reduce 194 pints to bushels, 3 bu, pe. 1 qt. 

Questions, 

What is reduction? 

For what is case second used ? 

How do you reduce a sum to a higher denomination 
than its own? 

When there are one or more denominations between 
the denomination of the given sum and the one to 
which you wish to reduce it, how do you proceed ? 

Of what denomination is the remainder always! 

How do you bring pints to quarts ? 

How do you bring quarts to pecks? 

How do you bring pecks to bushels? 

How do you bring pints to bushels? 



AVOIRDUPOIS WEIGHT. 

1 Reduce 65 cwt. to tons. Result 3 tons 5 cwt. 

2 Reduce 27 quarters to cwt. Res. 6 cwt. 3 qr. 

3 Reduce 109 pounds to qr. Res. 3 qr. 25 Ib. 

4 Reduce 123 ounces to pounds. Res. 7 Ib. 11 oz. 

5 Reduce 234 drams to ounces. Res. 14 oz. 10 dr. 

6 Reduce 4274 drams to pounds. Res. 16 Ib. 11 oz. 2 dr. | 

7 Reduce 175 quarters to tons. Res. 2 tons 3 cwt. 3 qr. j 

8 Reduce 6745 pounds to tons. Res. 3 tens 25 Ib. i 



56 REDUCTION. 

TROY WEIGHT. 

1 Reduce 378 ounces to pounds. Result, 31 Ibs. 6 oz. 

2 Reduce 235 pennyweights to ounces. 

Res. 11 oz. 15 dwt. 

3 Reduce 748 grains to pennyweights. 

Res. 31 dwt. 4 grains. 

4 Reduce 678 pennyweights to pounds. 

Res. 2 Ibs. 9 oz 18 dwt. 

5 Reduce 732 grains to ounces. -* 

Res. 1 oz. 10 dwt. 12 grains. 

6 Reduce 14752 grains to pounds. 

Res. 2 Ibs. 6 oz. 14 dwt. 16 gr. 

APOTHECARIES WEIGHT. 

1 Reduce 432 ounces to pounds. Res. 36 Ibs. 

2 Reduce 782 drams to ounces. Res. 97 oz. 6 dr. 

3 Reduce 91 scruples to drams. Res. 30 dr. 1 scr. 

4 Reduce 192 grains to scruples. Res. 9 sc. 12 gr. 

5 Reduce 258 scruples to ounces. 

Res. 10 oz. 5 dr. 1 scr. 

6 Reduce 12660 grains to pounds. 

Res.. 2 lb.2 oz. 3 drs. 

CLOTH MEASURE. 

1 Reduce 60 quarters to yards. Res. 15 yds. 

2 Reduce 60 quarters to English ell&. 

Res. 12 E. ells. 

3 Reduce 60 quarters to French ells. 

Res. 10 Fr. ells. 

4 Reduce 60 quarters to Flemish ells. 

Res. 20 Fl. ells. 

5 Reduce 52 nails to quarters. Res. 13 qr. 

6 Reduce 123 nails to yards. Res. 7 yds. 2 qr. 3 na. 

7 Reduce 543 nails to English ells. 

Res. 27 ells. qr. 3 nails. 

LONG MEASURE. 

1 Reduce 36 miles to leagues. Res. 12 1. 

2 Reduce 75 furlongs to miles. Res. 9 m. 3f. 



KEDUCTION. 57 

3 Reduce 295 poles to furlongs. Res. 7f. 15 p. 

4 Reduce 286 yards to poles. Res. 52 p. 

5 Reduce 365 feet to yards. Res. 121 yds. 2 ft. 

6 Reduce 759 inches to feet. Res. 63 ft. 3 in. 

7 Reduce 253 inciies to yards. Res. 7 yds. ft 1 inch. 

8 Reduce 2792 poles to leagues. Res. 2 1. 2 m. 5 f. 32 p. 

SQUARE MEASURE. 

1 Reduce 287 roods to acres. Result 71 a. 3 r. 

2 Reduce 245 perches to roods. Res. 6 r. 5 p. 

3 Reduce 756 square feet to yards. Res. 84 yds. 

4 Reduce 4731 square yards to perches. 

Yds. Res. 156 p. 12 yds. 



304 

4 



121 




18924 
121 

682 
605 

774 
726 



156 p. 



Bring the 30^ yards and the 4731 yards 
both to fourths, and divide. The remainder 
48, is fourths of a yard; divided by four, 
brings it to yards, the true icmaiader. 



Rem. 



12 yards. 

5 Reduce 3575 square inches to feet. 

Res. 24 feet 119 inches. 

6 Reduce 1728 square perches to acres. 

Res. 10 a. 3 r. 8 p. 

CUBIC MEASURE. 

1 Reduce 789 cubic feet to cords. Result, c. 21 ft. 

2 Reduce 343 cubic feet to yards. Res. 12 yds. 19 ft. 

3 Reduce 9386 cubic inches to feet. Res. 5 "fi. 746 in. 

4 Reduce 70353i> cubic inches to cords. 

Res. 3 c. 23 ft. 243 in. 



c 2 



58 REDUCTION. 

LIQUID MEASURE. 

1 Reduce 25 pipes tq.tuns. Res. 12 T. 1 P. 

2 Reduce 34 hogsheads to pipes. Res. 17 P. 

3 Reduce 1575 gallons to hogsheads. Res. 25 hhds. 

4 Reduce 163 quarts to gallons. Res. 40 gal. 3 at. 

5 Reduce 6048 pints to tuns. Res. 3 tuns. 

MOTION. 

1 Reduce 1440 seconds to minutes. Result, 24 min 

2 Reduce 720 minutes to degrees. Res. 12 deg. 

3 Reduce 342 degrees to sines. Res. 11 sines 12 deg. 

4 Reduce 443907 seconds to sines. 

Res. 4 sines 3 deg. 18 min. 27 sec. 

TIME. 

1 Reduce 1800 seconds to minutes. Result, 30 m. 

2 Reduce 720 minutes to hours. Res. 12h. 

3 Reduce 744 months to years.- Res. 62 yrs. 

4 Reduce 4649 minutes to days. Res. 3d. 5h. 29 min- 

5 Reduce 48888 minutes to weeks. 

Res. 4w. 5d. 22hrs. 48 min. 

STERLING MONEY. 

1 Reduce 78 shillings to pounds Res.3. 18.9. 

2 Reduce 93 pence to shillings. R^p. 7*. 9d. 

3 Reduce 39 farthings to pence. Res. JW. 3qr 

4 Reduce 656 pence to pounds. Res. 2 14s. Sd. 

5 Reduce 781 farthings to shillings. Res. 16s. 3d. Iqr. 

6 Reduce 6529 farthings to pounds. 

Res. 6. 16s. OcZ. 1 qr. 

FEDERAL MONEY. 

1 Reduce 250 halves to cents. Res. 125 cts. 

2 Reduce 128 fourths to cents. Res. 32 cts. 
Reduce 2343 cents to dollars. Res. 23 dol. 43 cts. 

4 Reduce 15371 cents to dollars. Res. 15 dol. 374 cts. 

5 Reduce 6150 half cents to dollars. Res. 30 dol. 75 cts. 
NOTE. To reduce cents to dollars, separate two figures 

01 1 the right hand for cents; those on the left will be dollars. 



REDUCTION. 



59 



PROMISCUOUS EXERCISES. 

1 How many bushels in 738 quarts? 

Ans. 23 bushels 2 quarts. 

2 In 7 bushels, how many pints? Ans. 448 pints. 

3 How many cwt. in 5356 ounces? . 

Ans. 2 cwt. 3qr. 2Glb. 12 oz. 

4 How many drams in 3 qr. 23 Ib. 14 oz. ? 

Ans. 27610 drams. 

5 How many grains in 9 oz. 14 dwt. 3 gr.? 

Ans. 4659 gr. 

6 How many pounds in 7432 dwt. ?Ans. 

301b. lloz. 12 dwt. 

7 How many scruples in 15 Ib. 1 oz. 6 drams? 

Ans. 4362 scr. 

8 How many ounces in 218 scruples? Ans. 9 oz. 

9 How many furlongs in 2346 yards? Ans. 

lOf. 26p. 3yd. 

10 How many poles in 3 leagues? Ans. 2880 poles. 

1 1 How many yards in 84 nails? Ans. 5 yd. 1 qr. 

12 How many perches in 4719 square yards 

Ans. 156 perches. 

13 How many square yards in one acre? 

Ans. 4840 sq. yds. 

14 How many hogsheads in 9728 gills? 

Ans. 4 hhds. 52 gal. 

15 How many pints in 2 pipes? Ans. 2010 pints. 

16 How many minutes in 3 days 6 hours? Ans. 4680m. 

17 How many hours in 2 weeks and 4 days? 

Ans. 432 hours. 

18 How many shillings in 27 four pences? Ans. 9 s. 

19 How many cords of wood in 9334 cubic feet? 

Ans. 73 cords 20 feet. ( 

.20 How many cubic feet in 9 cords? Ans. 1152 feet.j 
21 How many inches round the globe, which is 360 de- 
grees of 69i miles each? Ans. 1,565,267,200 inches, 
Enumerate the answer. 



6t) CO3IPOUXD ADDITION. 



COMPOUND ADDITION. 

COMPOUND ADDITION is the art of collecting several 
numbers of different denominations into one sum. 



RULE. 
i 

Place the numbers so that those of the same denom- 
ination may stand directly under each other, observing 
to set the lowest denomination on the right, the next 
lowest next, &,c. 

Then add up the several columns beginning with the 
lowest denomination : divide the sum by as many of the 
number of that denomination as it takes to make one of 
the next; and so on. 

Proof. As in Simple Addition. 



DRY MEASURE. 

EXAMPLES. 

bu. pe. qt pt. The first column on the right makes 

3271 ^ ve P U) ts. F ' ve pi nts make two quarts. 

and leaves one pint. Set down the one 

pint under the column of pints and carry 

the two quarts to the column of quarts. 

6321 The column of quarts with the two quarts 
2261 added makes twenty four quarts. Twen- 
ty-four quarts make three pecks and leave 

T~ ~ _ no quarts. Set down undei the column 
^* <i U 1 O f quarts and carry the three pecks to the 



column of pecks. 



The column of pecks with the three pecks added makes fourteen 
pecks. Fourteen pecks make three bushels and leave two pecks. 
Set down the two pecks under the column of pecks and carry the 
three to the column of bushels. 

The column of bushels with the three bushels added makes 
twenty-four bushels. Here set down the whole amount. 



COMPOUND ADDITION. 61 

bu. pe. qt. pt. bu. pe. qt. bu. pe. qt. pt. 

23 371 437 371 

34 261 526 261 

42 351 615 050 

51 141 .704 141 

23 231 833 331 

14 1 2 1 412 021 

11 3 4 1 324 211 



202 3 2 1 40 3 7 3270 

APPLICATION. 

1 Add 2 bu. 3 pe.;7 bu. 3qt.;4 bu. 1 pe. 1 pt.; 6 bu. 
4 qt. 1 pt.; and 3 pe. 1 qt. together. 

Amount 21 bu. pe. 1 qt. pt. 

2 Add 3 bu. 2 pe.Sqt. 1 pt.; 7 bu. 7 qt. 1 pt.; 3 pe. 1 pt.; 
4 bu. 5 qt.; 4 bu. 3 pe.; 8 bu. 3 pe. 7 qt. 1 pt. together. 

Amt.29 bu. 2 pe. qt. pt. 

3 Add 7 bu. 1 pt.; 3 pe. 7 qt. 1 pt.; 6 qt. 1 pt.; 9 bu. 3 
pe. 6 qt. 1 pt.; 3 bu.3 qt.; 4 bu. 1 pe. Amt. 25 bu. 2 pe. 

4 In a wagon load of grain contained in seven sacks, 
viz: in the first 4 bu. 3 pe. 1 qt.; in the second 5 bu. 7 
qt. 1 pt. the third 3 bu. 1 pe. 1 pt. fourth, 3 bu. 2 pe. 
6 qt. fifth, 5 bu. sixth, 4 bu. 1 pe. 1 pt.: and in the 
seventh 6 bu. 1 pe. 1 pt. How many bushels? 

Questions. 

What is Compound Addition? 

How do you place the numbers to be added? 

Do you place the greater or smaller denominations 
in the right hand column? 

Where do you begin the addition? 

When the first column is added, how do you proceed 
with the sum? 

When you divide the sum by as many of that denom- 
ination as make one of the next; which do you set down, 
the remainder or the quotient? 

What do you do with the quotient? 



62 COMPOUND ADDITION. 

In what particular does compound addition differ from 
simple addition? 

Do you carry one for every ten in compound addition? 

Since you do not carry one for every, ten, how many 
do you always carry? A. One fo* as many of any de- 
nomination as make one of the next. 

Here the pupil will have something with which to compare simple 
addii-ion, in which he carries one for every ten. This comparison will 
improve and correct his understanding of the elementary rules. 

AVOIRDUPOIS WEIGHT. 

T. cut. qr. Ib. T. cwt. qr. Ib. oz. dr. 

(1) 15 3 2 15 (2) 7 11 2 16 4 13 

4839 15 7 3 8 16 7 

82 19 1 10 13S 19 1 12 8 13 

163 8 3 17 42 8 3 19 12 4 

34 15 2 24 357 6 2 8 3 3 



300 16 1 19 

APOTHECARIES' WEIGHT. 

339 & 3 3 9 sr- 

(1) 6 3 L 2 (2) 84 7 t> 12 

19 9 5 1 132 5 2 

182 7 3 2 16 2 2 2 8 

57 6 1 1427 6 7 19 

40 5 14 6 1 9 



306 7 3 2 

TROY WEIGHT. 

Ib. oz. dwt. Ib. oz.dwt. gr. 

(1) 47 10 12 (2) 185 2 19 20 

38 8 6 56 9 15 6 

16 11 4 1472 11 2 17 

7 2 16 385 8 5 

13 9 11 10 8 7 12 



124 6 9 



C02IPOUND ADDITION. 




63 


CLOTH MEASURE. 






yds. qr. na. E.E. qr. na. 


EFL qr. 


na. 


(1) 75 3 2 (2) 72 3 2 (3) 19 2 


3 


163 13 536 2 1 


728 1 


2 


245 2 847 1 3 


142 


1 


738 3 1 1453 2 


,816 





1785 23 41 2 


32 1 


2 


3009 1 1 






LONG MEASURE. 






L. M.fur. P. yd. ft. iji. 
(1) 5 2 4 17 (2) 3 2 11 


16 1 3 10 1 


1 9 




72 5 24 2 


8 




526 3 12 3 


1 10 




834 2 6 34 2 


4 




38 3 12 6 


2 7 




1493 2 2 29 






SQUARE MEASURE 






A. R. P. A. 


R. P. 




(1) 39 2 37 (2) 487 


2 17 




G2 1 17 25 


3 28 




68 38 67 


32 




129 3 12 45 


1 16 




532 1 18 26 


29 




832 2 2 






o< 







04 



COMPOUND ADDITION. 



CUBIC MEASURE. 



yd. ft. in. 

(I) 75 22 1412 

9 26 195 

3 19 1091 

28 15 1110 

49 24 218 

18 17 1225 



cords, feet. in. 

(2) 37 119 1015 

9 110 159 

48 127 1071 



8 

21 

9 



111 956 

9 27 
28 1091 



186 18 



67 



135 122 863 



3 In four piles of wood; the first containing 32 feet 149 
inches; the second 121 feet 1436 inches; the third 97 
feet 498 inches; the fourth 115 feet 1356 inches; how 
much did the whole amount to? 

Ans. 2 cords, 110 ft. 1711 in. 

4 In six boat-loads of wood: the first containing 22 
cords 114 feet, 987 in.; the second 18 cords, 121 feet, 
1436 in.; the third 21 cords, 109 feet, 1629 in.; the 
fourth 15 cords, 82 >et, 1321 in.; the fifth 16 cords, 98 
feet, 1111 in.; the sixth 24 cords, 89 feet, 987 in. How 
much did they contain? 

Ans. 120 c. 105 ft. 559 in. 



LIQUID MEASURE. 

T. hhd. gal lihd. gal. qt. pt. 

(1) 18 2 54 (2) 385 42 3 1 

62 1 39 27 36 2 

327 4 132 17 

46 1 19 729 25 

285 3 28 173 47 2 1 



740 1 18 



COMPOUND ADD/TION. 65 

MOTJOIV 

i ' STNf o i // 

(1) 17 55 48 ^) 1 25 49 51 

1 37 51 2 4 21 36 
28 19 45 4 19 47 18 
19 19 37 1 25 25 39 



67 13 1 10 15 24 24 



3. Add 5sm 10 46' 38'; 11 37' 18"; Isin. 1712' 
18"; Isin. 52"; Isin. 15 12' 23"; and 11 57' 29" to- 
gether. Ans. II sin. 6 46' 58". 

4. Add 45'; Isin. 9 18"; 14 21' 34"; 2sin. 8 13' 
54"; 4sin. 7 12' 19"; and 47' 32" together. 

Ans. Sain. 10 20' 37". 

TIME, 

we. d. h. H. min. sec. 

(2) 3 5 20 (3) 20 52 40 

2 3 17 k 122 12 35 

3 6 22 68 9 17 
4 16 135 37 12 
3 19 24 35 28 




231 7 11 3 22 371 7 12 

STERLING MONEY. 

8. d. s. d. .?. d. 

(1) 2 3 4 (2) 7 9 44 (3) 4 6 4 

712 13 7 6| 47 19 7 

973 452 159 53 

5 2 24 10 18 10$ 78 6 11 



23 13 114 



4 Add 703 Is. 4d.-, 39 4*. 9<Z.; 162 17s. 2d ; 
459 15*.; 473 12*. Sd together. 

Ans. 1898 16*. lid. 



66 COMPOUND SUBTRACTION. 

5 Add the following sums : viz. 69 18s. 7d.; 175 
2s. Qd. ; 1582 19s 4cZ.; 175 13s. 9d.; 143 13s. 8d.-> 
and 212 Os. 7d Ans. 2359 8s. 5d. 

COMPOUND SUBTRACTION. 

COMPOUND SUBTRACTION is 'the art of finding the dif- 
ference between two numbers consisting of several de- 
nominations. 

RULE. 

Place the numbers, as in compound, addition with the 
less under the greater: then begin at the right hand 
denomination and subtract the lower number from the 
upper, and set down the remainder. 

If the upper number of any denomination be less than 
the lower one, add to the upper one as many as it takes 
of that to make one of the next; subtract the lower num- 
ber from the amount and set down the remainder as 
before. 

Proof. As in simple subtraction. 

EXAMPLES. 

bu. pe. qt. pt bu. pe. qt. pt. 

7241 42 361* 

3120 31 231 



4121 11 1 3 



bush. pe. qt. 

925 We cannot take 7 quarts from 5 quarts ; then 
237 borsow 1 from the 2 pecks. One peck has 8 
__________^_ quarts in it: 8 quarts added to the 5 quarts, 

f* n (* make 13 quarts. Take 7 qts. from 13 qts. and 

~ .6 qts. remain. Sot down the 6 qt. 

Because I borrowed 1 from the 2 quarts, I 

/nust add one to the 3 below it, which makes the lower figure 4. Now 
[ pecks from 2 pecks we cannot take : then borrow one bushel from the 
I ; that bushel has 4 pecks in it; 4 p. and 2 p. make 6 p. Now 4 p. 
"rom 6 p. and 2 pecks remain, which set down. 

Because I borrowed 1 from the 9, I must add 1 to the figure below 
t. 1 to 2 make 3. Take 3 from 9 and 6 remain. Set down the 6, 
aid the work is done. 



COMPOUND SUBTRACTION. 67 

bu. pe. qt. pt. bu. pe. qt. pt. 
8271 8130 
4361 4251 







bu. pe. qt. bu. pe. qt. pt. 

95 3 2 28 2 2 

22 1 14 3 5 1 



APPLICATION. 

1. From a granary containing 94 bushels, 2 pecks, 7 
quarts, have been taken 43 bush. 3 pe. 5 qr. How much 
remains? Ans. 50 bush. 3 pe. 2 qt. 

2. From a wagon load of corn containing 63 bushels, 
3 pecks, 4 qts., have been sold 27 bush. 3 pe. 7 qt. 1 pt. 
How much remains unsold? Ans. 35 bu. 3 p. 4qt, 1 pt. 

Questions. 

What is compound subtraction? 

In what particular does it differ from simple subtrac- 
tion? 

How do you place the numbers in compound subtrac 
tion? 

Where do you begin the operation? 

When the upper number of any denomination is less 
than the lower one, how do you proceed? 

Do you borrow one from the next? 

Do you call the number you borrow, one ten, as in 
simple subtraction? 

What do you call it? 

Ans. I call it one peck, or one yard, or one mile, as 
the case may be ? 

What do you do with it then? 

Ans. I reduce it to quarts, or to feet, or to furlongs, 
&c. according to the nature of the case; then add these 



68 COMPOUND SUBTRACTION. 

to the upper figure on the right, subtract the lower figure 
from the sum, and set down the remainder. 

When you borrow one from the upper figure, why do 
you add one to the figure below it? 

NOTE. Upon a clear conception of the principles involved in these 
questions, depends the pupil's correct knowledge of the science of 
Arithmetic. 

AVOIRDUPOIS WEIGHT. 
tons cwt. qr. tons civt. qr. Ib. cwt. qr. Ib. oz. dr. 
From 45 11 3 52 12 3 15 17 
Take 15 10 2 24 10 26 6 3 21 15 9 

Rem. 30 1 1 28 2 2 17 10 6 0. 7 

1. Subtract 76 tons, 18 hundred weight, 3 quarters, 
from 195 tons, 2 hundred weight, 2 quarters. 

Ans. 118 tons, 3 cwt. 3 qr. 

2. Subtract 14 pounds, 6 ounces, 3 drams from 20 
pounds, 2 ounces. Ans. 5 Ibs. 11 oz. 13 dr. 

APOTHECARIES WEIGHT, 

ft g 3 ft 3 3 & gr. 

1090 16 48 9 6 1 4 

106 2 7 1 10 2 8 



983 10 7 

3. From 59ft 1 23 take 53ffi 73 63. Ans. 5fc 5g 5J. 

4. Subtract 14fc 1>3 13 from G9. Ans. 54ft x>3 73. 

TROY WEIGHT. 

Ib. oz. dwt. gr. Ib. oz. dwt. gr. Ib. oz. dwt. gr. 

10 6 18 8 3 2 106 15 

4 2 20 2 1 18 6 10 6 2 20 



6 6 15 4 



4. Subtract I4lb. Goz. \\dwt. from 92lb. I2dwt. 6 gr. 

Ans. lib. Qoz. Idwt. 6 3 *r. 

5. From 16Z&. take I2lb. lloz. iQdwt. \\gr. 

Ans. 3Z&. Ooz. Qdwt. I3gr. 



COMPOUND SUBTRACTION. 60 

CLOTH MEASURE. 

yds. qrs. na. E.E. qrs. na. E. FL qrs. na 

From 71 3 1 42 2 51 2 2 

Take 14 2 3 19 2 3 42 2 1 



Rem. 57 2 22 2 3 901 

4. Subtract 95 yards, 3 quarters, 2 nails, from 156 
yards, 2 quarters, 3 nails. Ans. 60 yds. 3 qr. 1 nail. 

5. Subtract 14 English ells, 1 quarter, 2 nails, from 
52 English ells, 3 quarters, 2 nails. Ans. 38 yds. 2 qr. 

LONG MEASURE. 

L. M.fur. L. M. fur. P. yds. ft. in. 

From 24 I 7 58 1 19 6 2 10 

Take 18 2 4 10 7 20 327 



Rem. 523 46 00 39 303 

4. Subtract 45 miles, 5 furlongs, 20 poles, from 320 
miles, 3 furlongs, 36 poles. Ans. 274 M. 6 F. 16 P. 

5. Subtract 15 yards, 2 feet, 6 inches, from 36 yards, 
1 foot, 11 inches. Ans. 20 yds. 2 ft. 5 inches. 

LAND, OR SQUARE MEASURE. 
A. R. P. A. R. P. Yds. ft. in. 

From 96 3 36 195 22 25 2 72 

Take 25 2 39 36 3 1 14 7 10 



Rem. Tl 37 158 3 1 10 4 62 

4. Subtract 36 acres, 2 roods, from 900 acres, 3 
roods, 16 perches. 864 A. 1 R. 16 P. 

5. Subtract 72 acres, from 360 acres, 2 roods, 29 
perches. 288 A. 2 R. 29 P. 

CUBICK MEASURE. 

yds. ft. in. cords, ft. in. 

79 11 917 349 97 1250 

17 25 1095 192 127 1349 



61 12 1550 156 97 1629 



70 COMPOUND SUBTRACTION. 

1. From a pile of wood containing 432 cords, 27 feet, 
and 1432 inches, have been hauled 156 cords, 92 feet, 
946 inches: how much remains? 

Ans. 275 cords, 63 feet, 486 in. 

2. From a bank of earth containing 2984 yards, 18 
feet, have been taken 143G yards, 21 feet, what re- 
mains? Ans. 1547 yds. 24 feet. 

LIQUID MEASURE. 

T. hhd.gal.qt.pt. T. hhd.gal.qt.pt. 

2 3 50 1 100 1 19 2 1 

1 2 16 3 1 99 1 28 3 1 



1 1 33 1 1 

3. If I purchase 2hhd. of wine, and to oblige a friend 
send him 29^aZ., what quantity have I left? 

Ans. \hlid. 34gal. 

4. Bought 1 pipe of wine, 4hhd. of brandy, 2 barrels 
of beer; I have since sold 93 gallons of wine, 29 of 
brandy, 1 barrel of beer: how much of each have I 
remaining? 

Ans. 33gal. of wine, 223gal. of brandy, and 1 
barrel of beer. 

MOTION. 

O f n gi n> O , ,/ 

79 21 31 6 10 12 48 
41 41 52 38 39 2 



37 39 39 3 1 33 19 

A circle being 12 sines, how far has the hand of ? 
watch to pass, after having gone through 4 sines, 23' 
.5' 29" ? 

Sin. ' " 

12 

4 23 15 29 



COMPOUND SUBTRACTION. 



71 



2 A person resioing in latitude 27 C 32' 45" north, 
wishes to visit a place 52 24' 18' north. How many 
degrees, minutes, and seconds northward must he 
travel? Ans. 24 51' 33'. 



TIME 

Y. M. w. d. ho. min. sec. 
69 3 1 3 40 20 
16 2 6 2 57 36 



H.min.sec. Y. M. 

16 29 33 18 11 

7 36 44 9 10 



53 



2 42 44 



4. From 900 Y. take 111F. 6m. 

Ans. 788 Y. 6m. 

5. If I take IF. 1M. from 6Y. what space of time 
will still remain? 

Ans. 4F. 11 M. 

NOTE. To ascertain the amount of time passed be- 
tween two events, set down the year, month, and day of 
the latter event, and place those of the former below it, 
and subtract. 

6. A bond was given 24th July, (7th month) 1809, and 
paid off 13th August, 1821. 

yrs. mo. ds. 
1821 8 13 
1809 7 24 



12 20 

7. The declaration of independence of the United 
States passed Congress, 4th July, (7th month) 1776; 
and the declaration of the late war with Great Britain, 
18th June, (6th mo.) 1812. How many years, &,c. be- 
tween them? Ans. 35yr. llmo. 14d. 

STERLING MONEY. 

. ,. d. s. d. s. d. 
146 19 lOi 47 6 71 419 7 6 



7 19 9| 
139 0| 



28 



lOi 



227 8 94 



72 COMPOUND MULTIPLICATION. 

4. Subtract 200 9s. from 1000 Us. 

Ans. 800 2s. 

5. I have a purse of money containing 1000 2s. 
4id.: it I take out 60 7*. 8|d. what sum will be left? 

Ans. 939 14*. 7|d. 



COMPOUND MULTIPLICATION. 

COMPOUND MULTIPLICATION is the art of multiplying 
numbers composed of several denominations. 

CASE 1. 
When the multiplier does not exceed 12. 

RULE. 

Place the number to be multiplied as directed in 
compound addition,- and set the multiplier undes-the 
lowest denomination. 

Multiply as in simple multiplication, and divide the 
product of each denomination by as many as it takes of 
that to make one of the next greater; set down the re- 
mainder (if any) and carry the quotient to the product 
of the next denomination. 

Proof. Double the multiplicand and multiply by half 
the multiplier. 

EXAMPLES. 

Bu. pe. qt. pt. 7 times 1 pint make 7 pints ; 2 pt. 
7 2 5 1 make 1 qt. ; then 7 pt. make 3 qt. and 

~ leave 1 pt. Set down the 1 pt. and 
carry the 3 qt. to the product of the 
next figure. 



53 261 7 times 5 qt. make 35 qt. to which 
add the 3 qt. which make 38 qt. ; 8 qt. make one peck ; then 38 qt. 
make 4 pecks and leave 6 qts. Set down the six quarts and carry the 
4 pecks. 

7 times 2 pecks make 14 pecks ; add the 4 pecks, makes 18 pe. ; 
4 pecks make one bushel ; then 18 pe. make 4 bushels and leave 2 
pecks. Set down the 2 pecks and carry the 4 bushels. 

7 times 7 bushels make 49 bushels ; add the 4 bushels, makes 53 
bushels, which set down, and the work is done. 



COMPOUND MULTIPLICATION. 73 

Bu. pe. qt. pt. Bu. pe. qt. pt. 

9361 23 2 5 1 

5 8 



49 3 1 189 1 4 

1. In one vessel are contained 29 bushels 2 pecks and 
5 quarts : how many in 9 such vessels ? 

Ans. 266 bu. 3pe.5qt. 

2. If one tub will contain 8 bu. 3 pe. 5 qt. how much 
will 11 such tubs contain? Ans 97 bu. 3 pe. 7 qt. 

CASE 2. 

When the multiplier exceeds 12, and is the exact product 
of two factors in the multiplication table. 

RULE. 

Multiply the given sum by one of the factors, and the 
product by the other factor. 
Proof. Change the factors. 

EXAMPLES. 

1. Multiply 3 bushels, 2 pecks, 7 qt. by 24. Product. 
bu. pe. qt. bu. pe. qt. 

327 327 

6 4 



22 1 2 14 3 4 

4 6 



89 1 Proof 89 1 

OR THUS: 

bu. pe. qt. bu. pe. qt. 

327 327 

3 8 



11 5 o 3 

8 3 



89 1 89 1 

2. Multiply 7 bushels, 3 pecks, 5 quarts, by 36. 

Product, 284 bu. 2 pe. 4 qt. 

3. Multiply 19 bushels, 2 pecks, 3 quarts, bv 42 

D 



74 COMPOUND MULTIPLICATION. 

CASE 3. 

When the multiplier exceeds 12, and is NOT the product 
of any two factors in the multiplication table. 

RULE. 

Multiply by the two factors whose product is the least 
short of the given multiplier; then multiply the given 
sum by the number which supplies the deficiency; and 
add its product to the sum produced by the two factors. 

EXAMPLES. 

1. Multiply 21 bushels, 1 peck, 7 quarts, by 23. Prod. 

bu. pe. qt. bu. pe. qt. 

21 1 7X3 21 1 '7x2 

5 3 



OR THUS: 



107 1 3 64 1 5 

4 7 



429 1 4 product of 20 450 3 3 product of 21 
64 1 5 product of 3 42 3 6 product of 2 

493 3 1 product of 23 493 3 1 product of 23 

2. Multiply 19 bushels, 3 pecks, 7 quarts, by 34. 

Product, 678 bu. 3 pe. 6 qt. 

3. Multiply 7 bushels, 3 pecks, 4 quarts, by 59. 

4. Multiply 9 bushels, 3 pecks, 2 quarts, by 47. 

5. Multiply 15 bushels, 1 peck, 7 quarts, by 78. 

6. Multiply 12 bushels, 2 pecks, by 92. 

7. Multiply 1? bushels, 3 quarts, by 98. 

8. How many bushels in 104 sacks, each containing 
7 bushels, 2 pecks, 3 quarts? 

9. How many bushels of wheat on 125 acres, con- 
taining 21 bushels, 3 pecks each? 



L. 



COMPOUND MUI/TIPLICATIOX. 75 

CASE 4. 

Wlien the multiplier is greater than the product iff any 
two numbers in the multiplication table. 

RULE. 

Multiply the given number by 10, as many times 
less one as there are figures in the multiplier. 

Multiply that product by the left hand figure of the 
multiplier. 

Multiply the given sum by the units figure of the 
multiplier; the product of the first 10 by the tens figure 
of the multiplier; the hundreds product by the hundreds 
figure of the multiplier, and so on, till you have multi- 
plied by all the figures of the multiplier except the left 
hand one. 

Add all the products together, and you have the pro- 
duct required. 

EXAMPLES. 

1. Multiply 3 bushels, 3 pecks, 1 quart, by 45G. 
bu. pc. qt. Product 1724 bu. 1 pe. 

3 3 1X6 

10 Because there are 3 figures, multiply 

. 2 times by 10. Multiply that product by 

ty- o v E the kft hand figure (4) of the multiplier. 

^ X Multiply the given number by the units 
figure (6) and set the product beneath. 
Multiply the 10's product by the tens 



378 4 figure (5) of the multipliar. 

Add the several products. 



1512 2 

22 2 6 

189 2 

1724 1 Product 



76 



COMPOUND MULTIPLICATION. 



2. Multiply 53 bushels, 2 pecks, 7 quarts, by 2345. 
bu. pe. qt. 
53 2 7X5 
10 



537 



6+4 
10 



5371 



4+3 
10 



53718 



107437 2 

161 < 5 2 

2148 3 

268 2 



product of the 2000 

4 " " 300 

" 40 

3 " "5 



125970 



1 



7 Product of the 2345 



NOTE. Let the pupil try experiments, by multiplying simple num- 
bers in this way. 

3. Multiply 72 bushels, 1 peck, 2 quarts, by 4723. 

Product, 341531 bu. 3 pe. 6 qt. 

4. Multiply 13 bushels, 2 pecks, 4 quarts, by 5124. 

Questions. 

What is compound multiplication? 

In what does it differ from simple multiplication? 

When the multiplier does not exceed 12, how do you 
proceed ? 

How many do you always carry ? 

How do you prove compound multiplication? 

How do you proceed when the multiplier exceeds 12, 
and is the exact product of two numbers in the multi- 
plication table? 

When the multiplier exceeds 12, and is not the exact 
product of any two numbers in the table, how do you 
proceed? 



COMPOUND MULTIPLICATION. 77 

How do you proceed when the multiplier is greater 
than the product of any two numbers in the table ? 

AVOIRDUPOIS WEIGHT. 

tons. cwt. qrs. cwt. qr. Ib. oz. dr. 

23 12 3 7 3 14 9 6 

4 6 



94 11 47 1 3 8 



tons act. qr. cwt. qr. Ib. oz. dr. 

7 15 3 7 3 24 12 14 

8 9 



5. Multiply 7 tons, 16 cwt. 3 qr. by 24. 

Product, 188 T. 2 cwt. 

6. Multiply 3 cwt. 2 qr.21 Ib. 14 oz. by 30. 

Product, 110 cwt. 3 qr. 12 Ib. 4 oz. 

7. Multiply 3 tons, 7 cwt. 2 qr. by 34. 

Product, 114 tons 15 cwt. 

APOTHECARIES WEIGHT. 

tt g 3 9 fc g 3 9 # fc 3 3 Bgr. 
4821 53 10 2 12 17 5 6 1 4 
5 9 12 



23 5 3 2 

TROY WEIGHT. 

Ib. oz. dwt. Ib. oz. dwt. gr. Ib. oz. diet. gr. 
67 5 16 43 8 10 113 6 6 
246 



134 11 12 

4. Multiply 41 Ib. 6 oz. 18 dwt. 2 gr. by 7. 

Ans. 291 Ib. oz. 6 dwt. 14 gr. 

5. Multiply 91 Ib. 4 oz. 14 dwt. 16 gr. Ly 8. 

Ans. 731 Ib. 1 oz. 17 dwt. 8 gi 

7 



78 COMPOUND MULTIPLICATION. 

CLOTH MEASURE. 

yd. qr. na. E. E. qr. na. E.Fl. qr. na. E.Fr. qr. na. 

20 2 3 37 4 2 18 3 14 1 3 

6 8 12 9 



124 2 

5. If 19 yd. 1 qr. 2 na. be multiplied by 5, what num- 
ber of yards will there be ? Ans. 96 yds. 3 qr. 2 na. 

6. Multiply 56 Ells Eng. 3 qr. by 9. 

Ans. 509 Ells E. 2 qr. 

LONG MEASURE. 

deg.m.fur.p. 1. m.fur.p. m. fur. p. yd. ft. in, 

8 1 3 36 4 2 2 29 18 3 20 1 2 10 

12 7 5 



96 17 6 32 

4. Multiply 6 deg. 40 m. 7 fur. by 10. 

Ans. 65 deg. 61 m. 2 fur. 

5. Multiply 44 m. 6 fur. 20 p. by 7. 

Ans. 313 m. 5 fur. 20 D 

LAND, OR SQUARE MEASURE. 
a. r. p. a. r. p. a. r. p. 

49 2 17 19 3 20 10 33 

2 6 9 



99 34 

4. How many acres will 10 men reap in one day, 
allowing them 1 acre 3 roods 11 perches each? 

Ans. 18 A. OR. 30 P. 

5. Multiply 63 acres 3 roods 18 perches, by 11. 

Ans. 702 A. 1 R. 38 P. 

6. How many acres in 15 lots, containing 17 acres, 2 
roods, and 20 perches each? Ans. 264 A. 1 R. 20P. 



COMPOUND MULTIPLICATION. 79 



CUBIC MEASURE. 

cords, ft. in. yd. fir in. 

7 28 1327 19 23 1421 

6 8 



43 44 1050 159 1 1000 



cords, ft. in. yd. ft. in. 

21 56 1432 27 13 1291 

7 9 



5 In a pile of wood are 14 cords 92 feet; how much in 
24 such piles? Ans. 353 c. 32 ft. 

6 In a cellar, are contained 42 yards 25 feet ; what 
are the contents of 23 such cellars? Ans. 987 yd. 8 ft. 

LIQUID MEASURE. 

hhd. gal. qt. T.hhd.gal. qt. pt. pi. hhd.gal. qt. pt. 

8 43 2 1 2 16 3 1 4 1 19 3 1 

4 '10 5 



34 48 

4 Multiply 3 T. 2 hhd. 50 gal. 2 qt. by 8. 

Ans. 29 T. 2 hhd. 26 gal. qt. 

5 Multiply 4 hhd. 41 gal. 1 pt. by 10. 

Ans. 46 hhd. 33 gal. 1 qt. pt. 

MOTION. 

sin. ' sin. ' " 

3 27 48 1 24 48 25 

7 9 



27 14 36 16 13 15 45 

3 If a planet move through 2sin. 15 23' of its orbit 
in one day ; how far will it advance in 8 days. 

Ans. 205m. 3 4' 



80 COMPOUND DIVISION 




TIME. 

weeks rf. h. d. h. min. sec. 

o 5 23 3 14 25 36 

8 9 



8 30 5 16 32 9 50 24 

4 If a man can perform a piece of work in 2 yr. 3 mo., 
how long would it take him to perform 5 such? 

Ans. il yr. 3 mo, 

5 If a laborer dig a drain in 2 weeks, 3 days, how long 
a time would he require to dig 9 such drains? 

Ans. 21 weeks 6 days. 

STERLING MONEY. 

s. d. s. d. s. d. 

246 13 3| 14 6 04 111 11 10* 

11 9 10 



2713 6 5i 

*. d. . s d. 

4 Multiply 37 6 9i by 5 Prod. 186 13 Hi 

5 56 8 7| by 9 507 17 9| 

COMPOUND DIVISION. 

COMPOUND DIVISION is the art of dividing a sum 
which consists of several denominations. 

CASE 1. 
When the divisor does not exceed 12. 

RULE. 

Divide the several denominations of the given sum, 
one after another, beginning with the highest, and set 
their respective quotients underneath. 

When a remainder occurs, reduce it to the neXt lower 
denomination, and add it to the number of ihe next de- 
nomination, and divide the sum as before. 



COMPOUND DIVISION. 81 



EXAMPLES, 

bu. pe. qt. pt. 

7) 25 2 6 1 ^ ere ^ * nto ^ ku. ^ times and 

' 4 remain. Set down the 3. 

Reduce the 4 bushels to pecks, 

3251 which makes 16 pecks: add 16 
pecks to 2 pecks, which make 18 "peeks. 

Now 7 into 18 jpe. 2 times, and leave 4. Set down the 2. 
Reduce the 4 pecks to quarts, which makes 32 qts. Add 32 qt. to 
6 qt. makes 38 qt. 

7 into 38 qt. 5 times, and 3 remain. Set down the 5. 
Reduce the 3 qt. to pt. makes 6 pt., add 6 pt. to 1 pt. makes 7 
pints. 

7 into 7, 1 time. Set down the 1, and the work is completed. 

bu. pe. qt. bu. pe. qt. 

2) 8 2 6 3) 9 3 6 



4 1 3 

4 Divide 34 bu. 3 pe. 6 qt. between 9 persons. 

Ans. 3 bu. 3 pe. 4 qt. 

5 92 bu. 3 pe. belong equally to 7 persons; what is the 
share of each? Ans. 13 bu. 1 pe. 

CASE 2. 

When the divisor exceeds 12, and is the exact product 
of two numbers in the multiplication table. 

RULE. 

Divide the sum by one of the factors, and the quotient 
by the other. 

Multiply the last remainder by the first divisor, and 
add the first remainder for the true remainder, as in 
simple division, note 2. 

EXAMPLES. 

1 Divide 89 bu. 3 pe. 7 qt. by 28. 

Quotient 3 bu. 6 qt. 1 pt. 18 .era. 



2 



82 COMPOUND DIVISION. 



'4 



bu. 

89 






22 1 73 Rem, 



3 6 5 Rem. 

4 First divisor 

20 

3 First rem. 

True rem. 23 Quarts 
2 

28)46 pints 

Pt. 1 and a rem, of 18 pints undivided. 



3. Divide 78 bushels, 3 pecks, 4 quarts, among 32 
persons; what will be the share of each? 

Ans. 2 bu. 1 pe. 6 qt. 1 pt. and a remainder of 24 
pints undivided. 



CASE 3. 

When the divisor is more than 12, and is NOT the exact 
product of any two numbers in the multiplication table 

RULE. 

Divide the highest denomination of the given sum, as 
in case 2, simple division; and reduce the remainder, 
if any, to the next lower denomination; add the number 
of that denomination to the result, and divide as before. 



EXAMPLES. 
1. Divide 77 bushels, 1 peck, 7 quarts, by 23. 

Quotient, 3 bu. 1 pe. 3 qt. 1 pt.!3rem. 



COMPOUND DIVISION. 83 

EXAMPLES. 

bu. pe. qt.bu.pe.qt. pt. 
23)79 1 7(3 1 6 1+3 pint remaining. 
69 

10 

4 

23)41(1 peck 
23 



18 
8 

23)151(6 quarts 
138 

13 

2 

23)20(1 pint 
23 

Rem. 3 pints 

2. A boat load of corn, containing 4927 bushels, 3 
pecks, is owned equally by 29 persons : wnat is the 
share of each? 

Ans. 169 bu. 3 pe. 5 qt. 1 pt., and a rem. of 1 pint. 

Questions. 

What is compound division? 

When the divisor does not exceed 12, how do you 
proceed ? 

When a remainder occurs, what do you do with it? 

Where the divisor exceeds 12, but is the product of two 
numbers in the table, how do you perform the operation ? 

How do you find the true remainder in the latter case ? 

When the divisor is more than 12, and is not the pro- 
duel of any two numbers in the table, how do you per- 
form the operation? 



84 COMPOUND DIVISION. 

AVOIRDUPOIS WEIGHT. 
tons cwt. qr. Ib. Ib. oz. dr. 

6)37 17 3 27 7)40 12 14 

6619-1 rem. 6 10 15-5 R. 



tons cwt. qr. cwt. qr. Ib. oz. dr. 

8)92 3 3 9)75 3 23 14 12 



5. A quantity of iron weighing 473 tons, 19 cwt., 
3 quarters, is owned equally by 22 persons; what is the 
share of each? 
Ans. 21 T. 10 cwt. 3 qr. 16 Ib. 8 oz. 11 dr. Rem. 14 dr. 

APOTHECARIES WEIGHT. 

fc 3 3 9 & 3 3 9 gr. 

4)23 7 5 1 5)41 6 7 2 14 



5 10 7 1 8361 2-4rem. 

fc 5 3 9 fc 3 3 9 ST. 

6)46 912 7)93 7 5 2 14 



5. Divide 127fe 3g 63 into 17 equal parcels: how 
much in each parcel? Ans. 7fe 5g 63 2^ 16gr. 8 rem. 

TROY WEIGHT. 

Ib oz. dwt. Ib. oz. dwt. gr. 

8)34 10 15 7)45 11 16 22 



Ib. oz. dwt. Ib. oz. dwt. gr. 

9)78 9 16 8)82 7 14 21 



COMPOUND DIVISION. 85 

CLOTH MEASUEE. 
yd. qr. na. Ells E. qr. na. 
5)27 3 1 6)37 3 



o 



1 1+4 rem. 



yd. qr. na. EllsE. qr. na. 
7)45 3 2 8)37 3 1 



LONG MEASURE. 

1. m. f. m. f. p. yd. 

6)37 2 2 7)46 7 17 3 



607 6 5 25 2 

yd. ft. in. m. f. p. yd. ft. 

6)53 2 9 7)87 6 23 4 2 



, 

5. A traveller has a journey of 946 miles, 6 furlongs, 
to perform in 26 days; how far must, he travel each 
day? Ans. 36 m. 3 f. 12 p. 8 rem. 

LAND, OR SQUARE MEASURE. ; 

A. R. P. A. R. P. yds. 

7)37 3 27 9)423 3 28 2 

5 1 26+5 Rem. 47 16 13+5 Rem. 

3. A farm containing 746 acres, 3 roods, 29 poles, is 
to be divided equally between 9 heirs; what is the share 
of each? Ans. 82 A. 3 R. 38 P. and 7 rem. 

CUBIC MEASURE. 

cords ft. yd. ft. in. 

8)97 48 9)148 16 493 

12 22 16 13 1398+7R. 



86 COMPOUND DIVISION. 

3. A boat load of wood, containing 92 cords 87 feet, 
is to be divided between 3 persons; what is the share 
of each? Ans. 30 c. 114 ft. 1 rem, 

4. A quantity of earth, containing 6987 yards, 25 
feet, is to be removed by 29 carters; how much must 
each remove? Ans. 240 yd. 20 \ 

LIQUID MEASURE. 
tuns.Jihd. gal. hhd. gal. qt. pt. 

5)37 3 45 6)57 36 3 1 



7 2 . 21+3 Rem. 9 37 2 l-(-l Rem 



tuns hhd. gal. hhd. gal. qt. pt. 

7)84 2 32 8)93 43 3 1 



5. A quantity of liquor owned equally by 27 person.* 
the whole quantity being 431 hhd. 47 gals; what i 
the share of each? Ans. 15 hhd. 62 gal. 1 qt.; 17 rem. 

MOTION. 

Sin. ' " Sin. 

8)9 16 45 36 9)11 23 48 54 

1 5 50 42 



TIME. 

yr. mo. we. da. ho. min. sec 

11)848 10 12)24 6 20 32 24 



77 2 2 13 42 42 



yr. mo. da. ho. min. s*,c. 

4)375 8 7)37 16 28 32 



COMPOUND DIVISION. 87 

STERLING MONEY. 

*. d. *. d. 

6)82 14 6 8)143 7 10 



13 15 9 17 18 5| 

s. d. s. d. 

7)78 10 11 9)98 17 1 



s. d. s. d. 
19)36 16 3(1 18 9 

6 Divide 113 13s. 4d. by 31. What is the quotient? 

Ans. 3 135. 4d 

7 Divide 189 14s. by 95. Quotient, 1 19s 



PROMISCUOTIS EXERCISES 

1 In 35 dollars how many cents? Ans. 3500. 

2 How many miles are there in 98 furlongs ? 

Ans. 12M. 2fur. 

3 How many weeks are there in 365 days? 

Ans. 52we. 1 da. 

4 In 84 half cents how many cents? Ans. 42cts. 

5 In 8 tons 15 cwt., how many hundred weight? 

Ans. 175 cwt. 

6 How many perches are there in 63 roods ? 

Ans. 2520 square per. 

7 How many pounds in 157s.? Ans. 7 17s. 

8 In 175 pecks how many bushels ? Ans. 43bu. 3pe. 

9 In 7642 cents how many dollars ? Ans. $76 42cts 

10 In 103 pints how many quarts? Ans. 51qt. Ipt. 

11 How many minutes are there in 720 seconds? 

Ans. 12min. 

12 In 7 hogsheads, 33 gallons, how many gallons? 

Ans. 474 gal. 



88 SINGLE RULE OF THREE, 

PROPORTION ; 

OR, 

THE SINGLE RULE OF THREE. 

PROPORTION is an Equality of RATIOS ;* 

That is, four numbers are proportional, when the first has the 
same ratio to the second as the THIRD has to the FOURTH, Thus> 
as 12 : 4 : : 24 ; 8; or as 4 : 12 : : 8 : 24, 

The ratio of 12 to 4 is 3 and the ratio of 24 to 8 is 3. Or, the 
ratio of 4 to 12 is , and the ratio of 8 to 24 is . Then 

Four numbers are proportional, when the first is as many times the 
second or the same part of the second, as the third is of the fourth. 
Or, when the ratio of the first to the second equals the ratio of the 
third to the fourth. 

Tha two quantities compared are called the TERMS 
of the ratio: the first is called the ANTECEDENT, and 
the second the CONSEQUENT. In any series of four 
proportionals, the first and fourth terms are called the 
EXTREMES, and the second and third the MEANS. The 
product of the Means, equals the product of the Ex- 
tremes. Thus in either series above, 12X8=96, and 
24X4=96. 

Now suppose we have the three first terms of a series in propor- 
tion, and we wish to find the fourth. Divide the product of the 
second and third terms by the first, and the quotient will be the 
fourth term. In this manner let the fourth term be found in each 
of the following series. 

2 : 4 : : 8 is to what? Am. 16. 

3 : 11 : : 9 is to what? tins. 33. 

4 : 6 : : 6 is to what ? Jlns. 9. 

2 : 9 : : 8 is to what? Jlns. 30. 

5 : 7 : : 15 is to" what? Am. 21. 



RULE. 

number which is of the name or kind in 
which the answer is required, in th<? third place: 



Set that 



* RATIO is the relation of one thing to another of the same kind 
in regard to magnitude or quantity. 



SINGLE RULE OF THREE. 89 

And, if the answer must be greater than the third 
term, set the greater of the remaining two terms in the 
second, and the less in the first place ; but, if the an- 
swer must be less than the third term, set the less in the 
second, and the greater in the first place. 

When the first and second terms are not of the same 
denomination, reduce one or both of them titlt they are; 
and, if the third consist of several denominations, reduce 
it to the lowest, then 

Multiply the second and third terms together and 
divide the product by the FIRST, and the quotient will 
be the fourth term or answer. 

NOTE. The answer will be of the same denomination as the 
third term , and, in many instances, must be reduced to a greater 
denomination. 

EXAMPLES. 

1. If four pounds of sugar cost 50 cents, what will 
24 pounds cost at the same rate? Ans. $3,00. 

1st Term. 2d Term. 3d Term. 

t Lj _^_ lu _j t^^^-^j y_,- ,_j In this question the answer is 

jj j, required to be money: therefore 

** C "' money (the 50 cts.) must be in 

As 4 t he third place. Because 24 

50 pounds will cost more than 4 

_ pounds the greater (24 Ibs.) 

4)1200 must occupy the second place: 

' _ and the remaining term (4 Ibs.) 

~ 



2. If 24 pounds, cost 300 cents (or 3 dollars ;) how- 
many pounds may be purchased for 50 cents at the 
same rate? Ans. 4 Ibs. 

cts. els. Ibs. In this question the answer is re- 

As 300 : 50 : : 24 quired to be in pounds,- therefore 
50 pound 8 (th 6 24) must be in the third 
place, 

Because 50 cts. will purchase less 

300)1200 than 300 cts. the lens (50 cts.) 

- must occupy the second place : and 

4 the remaining term (300 cts.) the 

first. 






90 SINGLE RULE OF THREE. 

3. Bought a load of corn containing 27 bushels, 3 
pecks, at 50 cts. per bushel, what did it cost? 

Ans. $13,87^. 
bit. bu. pe. cts. 

1 : 27 3 : : 50 

4 4 

Because pecks occur in the second 

term, the first and second are reduced 
4 111 to pecks. 

50 

4)5550 

$13,87^ 

4. What are 42 gallons worth, if 3 gallons 2 quarts 
cost $1,20? Ans. $14,40. 

gals. qts. gals. D. cts. 

3 2 : 42 : : 1,20 

44 As quarts occur in the first term, 

the first and second are reduced to 

14 168 q uarts - 
120 

14)20160 

$14,40 

5. If 8 bushels 2 pecks cost $4,25, how many bush- 
els can I purchase with $38,25 ? Ans. 76 bu. 2 pe. 

D. cts. D. cts. bu. pe. 

4,25 : 38,25 : : 8 2 

34 4 As two denominations occur 
in the third term, it is reduced 

*A r^ t0 the leSS ; hellCe the reSult ' S 

d4pe. pecks> whicil must be re< luced 
11475 to bushels. 

425)130050(306 pe. 
1275 

pe. 
2550 4)306 

2550 

76 bu. 2 pe. 



SINGLE RULE OF THREE. 



91 



G. What will 5 Ib. 6 oz. 5 d\vt. of silver-ware cost 
at $1,50 per ounce? Ans. $99,37^. 



oz. 
1 
20 

20 



U). oz. dwt. 
: 565 
12 

66 
20 

1325 
150 

66250 
1325 



D. ds. 
1,50 



As dwts. arc in the second 
term, the first and second 
must be reduced to dwts. 



. 



20)198750 

f99,37| 

7. AVhen 3 yards and 8 feet of plastering cost $1,40, 
what will be the cost of 16 yards? Ans. $5,76. 

ydv.ff. yds. D.cts. 

38 : 16 : : 1,40 
9 9 

35 144 

140 

5760 
144 

35)20160(5,76 cts. 

8. How many yards of cloth can be purchased for 95 
collars, if 4 yd. 3 qr. cost $9,50 ? 

Ans. 47 yd. 2 qr. ; or 47| yd. 

$ ct. $ ct. yd. qr. qr. 

As? 9,50 : 95,00 : : 4 3 4)190(47 yd. 2 qr. 



92 SINGLE RULE OF THREE. 

NOTE. The operation may, in many instances, be contracted by 

dividing the second or third term by the first ; or the first by either 

|! of the others, or by any number that will divide the first and either 

of the others without a remainder; and, using the quotients instead 

of the original numbers. 

9. If 24 yards cost $96, what will 8 yards cost? 

Aus. $32. 

yds. yds. D. yds. yds. D. 

24a : *8a : : 96c or 24a : 8 : : 96a 

3c Ans. 32. 4 

32 
10 If 36 bushels cost $72; what will 12 bu. cost? 

Ans. $24. 

bu. bu. D. bu. bu. D. 

3Ga : I2a : : 72c 12)36 : 12 : : 72a 

3c 24 3a 1 24 

APPLICATION. 

1 When 4 bushels of apples cost $2,25, what must 
be paid for 20 bushels ? Ans. $11,25. 

bit. bu. D. cts. 
4 : 20 : : 2,25 

2 How many yards of cloth can I buy for $60, when 
5 yards cost $12? Ans. 25 yds. 

D. D. yds. 
12 : 60 : : 5 

3 If 6 horses eat 21 bushels of oats in a given time ; 
how much will 20 horses eat in the same time ? 

Ans. 70 bu. 

,4 If 20 horses eat 70 bushels of oats in a certain time ; 
how much will 4 horses eat in the same time? 

Ans. 14 bu. 

5 If a family ofHen persons use 7 bushels 3 pecks of 
wheat in a month; how much will serve them when 
there are 30 in the family? Ans. 23 bu. 1 pe. 

6 If 14 Ibs. of sugar cost 75 cents, how many pounds 
can be bought for three dollars? Ans. 56 Ibs. 

7 If 4 hats cost 12 dollars, what will 27 feats cost at 
the same rate? Ans. $81. 



SINGLE RULE OF THREE. 93 

8 If 20 yards of cloth cost $85, what will 324 yards 
cost at the same rate ? 

Ans. $1377. 

9 If 2 gallons of molasses cost 70 cents, what will 2 
hogsheads cost ? Ans. $44,10. 

10 If 1 yard of cloth cost $3,25 cts., what will be the 
cost of 6 pieces, each containing 12 yds. 2 qrs. ? 

Ans. $243,75 cts. 

11 If 3 paces or common steps of a person be equal to 
2 yards, how many yards will 160 paces make? 

Ans. 106 yds. 2 ft. 

12 If a person can count 300 in 2 minutes, how many 
can he count in a day ? Ans. 216000. 

13 What quantity of wine at 60 cts. per gallon can be 
bought for $37,80 cts. Ans. 63 gal. 

14 If 8 persons drink a barrel of cider in 10 days, how 
many persons would it require to drink a barrel in 4 days? 

Ans. 20. 

15 If 8 yards of cloth cost $12, what will 32 yards 
[cost? Ans. $48. 

16 If 3 bushels of corn cost $1,20, what will 13 bush- 
els cost? Ans. $5,20. 

17 If 9 dollars will buy 6 yards of cloth, how many 
yards will 30 dollars b^y ? Ans. 20. 

18 If a man drink 3 gills of spirits in a day, how much 
will he drink in a year ? Ans. 34 gal. 1 pt. 3 gi. 

19 If 12 horses eat 30 bushels of oats in a week, how 
many bushels will serve 44 horses the same time ? 

-Ans. 110. 

20 If a perpendicular staff 6 feet long, cast a shadow 5 
feet 4 inches, how high is that tree whose shadow is 
104 feet long at the same time? Ans. 117 feet. 

EXERCISES. 

1 If 12 acres, 2 roods, produce 525 bushels of corn, 
how many bushels will 62 acres, 2 roods produce ? 

Ans. 2625 bu. 

2 If 7 men plough 6 acres, 3 roods in a certain time, 
how many acres will 96 men plough in the same time ? 

Ans. 92 A. 2 R. 11 Per. 12 yd.-f 



94 SINGLE RULE OF THREE. 

3 Suppose 3 men lay 9 squares* of flooring in 2 day s ; 
low many men must be employed to lay 45 squares in 
he same time? Ans. 15 men. 

4 If 7 pavers lay 210 yards of pavement in one day; 
low many pavers would be required to lay 120 yards in 
he same time? Ans. 4 pavers. 

5 If 2 hands saw 360 square feet of oak timber in 2 
days; how many feet will 8 hands saw in the same time? 

Ans. 1440 feet. 

6 An engineer having raised a certain work one hun- 
dred yards in 24 days, with 5 men; how many men must 
be employed to perform a like quantity in 15 days? 

Ans. 8 men. 

7 If 3 paces or common steps be equal to 2 yards ; how 
many vards will 160 such paces make? 

Ans. 108 yd. 2 ft. 

8 If a carriage wheel in turning twice round, advance 
33 feet 10 inches; how far would it go in turning round 
63360 times? Ans. 203 miles. 

9 Sound flies at the rate of 1142 feet in 1 second of 
time; how far off may the report of a gun be heard in 1 
minute and 3 seconds? 

Ans. 13 miles, 5 fur., poles, 2 yd. 

10 If a carter haul 100 bushelst>f coal at every 3 loads; 
how many days will it require for him to load a boat with 
3600 bushels,* suppose he haul 9 loads a day? 

Ans. 12 days. 

bu, lu. da. da. 
As 300 : 3600 : : 1 : 12. 

11 If 8 men can reap a field of wheat in 4 days; how 
many days will it require for 16 men to do it? 

Ans. 2 days. 

12 -Sold 10 yards of linen at 5 dollars 50 cents; what 
was it a yard? Ans. 55 cents. 



*A SQUARE is 10 feet long and 10 feet wide, or 100 square feet. 
This measure is employed in estimating the quantity of flooring, roof- 
ing, weather-boarding, &c. 



SINGLE RULE OF THREE. 



95 



13 If 7 pounds of cheese cost 87 i cents; what must I 
pay for 122 pounds? Ans. 15 dol. 25 ct. 

14 If 1 ounce of silver cost 72 cents; what will 3 pounds 
5 ounces come to? Ans. 29 dol . 52 ct. 



Why do you multiply the second and third terms to- 
gether and divide by the first? 

What will 24 pounds 
of bacon cost at 50 ct. 
for every 4 pounds? 
Ib. Ib. ct. ct. 



As 4 : 24 ::50 : 300 



cts. 
4)50 

12} the price of lib. 
24 

50 
25 



If 4 pounds cost 50 cents, divide the 
50 cents by 4, gives the price of 
1 pound: thus 4 into 50 12*. If 
1 pound cost 12 cents, 2 pounds 
will cost twice that; three pounds 
three times, and 24 pounds will cost 
24 times 12! ct. ; that is 300 cts. or 3 
dol. 



If the second and third terms be 
multiplied together, and their product 
divided by the first, the result will be 
the Bame as it is when the third is di- 
vided by the first, and the quotient 
multiplied by the second. 



300 ct. the price of 24 Ib. 

15 If 15 yards of broad cloth cost 80 dollars; what will 
75 cost. Ans. 400 dollars. 

16 A man bought li yards linen for $2 50 cts.; what 
is the worth of 1 qr. 2 na. at the same rate. 

Ans. 62* ct. 

17 If 321 bushels of salt cost $240 75 cents; what 
was it per bushel? Ans. 75 cents. 

18 If the moon move 13 deg. 10 min. 35 sec. in one 
day; in what time does it perform one revolution? 

Ans. 27 da. 7 hrs. 43 min. 

deg. min. sec. deg. da. 
As 13 10 35:360 ::1 



96 SINGLE RULE OF THREE. 

19 If a staff 4 feet long, cast a shade 7 feet on level 
ground ; how high is a steeple whose shade is at the 
same time 198 feet. Ans. H3\ 

20 If a man's annual income be 1333 dollars, and he 
spend 2 dollars 14 cents a day ; what will he save at the 
end of one year? Ans. $551 90 ct. 

21 Suppose A. owes B. 791 dols. 60 ct., and can pay 
only 374 cts. on the dollar; how much must B. receive? 

Ans. $296 85 ct. 

22 Bought 3 casks of raisins, each containing 3 cwt 
1 qr. 14 lb.; how much did they cost at $6 21 ct. pei 
cwt.? Ans. $62,874 

PROPORTION DIRECT AND INVERSE. 

Hitherto, proportion has been treated in general terms ; 
it now remains to consider the two kinds, DIRECT and 
INVERSE. 

DIRECT PROPORTION is that in which more requires 
more, or LESS requires LESS. Thus: 

yd. yd. dol. If 2 yd. cost 4 dol., 124 yd. being 

As 2 * I'M: * 4 more than 2 yd., will cost more than 

4 dol. 

yd. yd. dol. And, if 124 yd. cost 248 dol.; 2yd. 

As 124 2 248 being leis wil1 cost less " 

That is, more yards require more 
money, and less yards cost less money 

INVERSE PROPORTION is that in which more requires 
less; and less requires more. Thus: 

If 12 men built a wall It is supposed that 12 men perform 
in 4 davs; how many ft . P iec J of * ork I" * da r s: w a , like 

"'i o j n P 18ce work is to be done in o days; 

men can do it in 8 days i this will require a kss uumne i of men : 
that is. more days require less men. 

STATED. Here it is supposed that 12 men 

da. dct. m. m. performed a piece of work in 8 days: 

As 4 : 8 inversely : : 12 . 6 nke . iece I s to be done in ? d *y. s J 

this will require more men. 1 hat is, 
more days require less men, and less 
da. da. m. m. days require more men. 

As 8: 4 directly:: 12.6 



SINGLE RULE OP THREE. 97 

All the past exercises in proportion are Direct the 
"ollowing will be 

INVERSE PROPORTION. 

Questions in Inverse Proportion, may be stated and 
solved by the same rule that is given for Direct Propor- 
tion. 



EXERCISES. 

1 If 6 mowers mow a meadow in 12 days; in what 
time will 24 mowers do it? Ans, 3 da. 

2 If a man perform a journey in 6 days, when the 
days are 8 hours long; in what time will he do it when 
they are 12 hours long? Ans. 4 da. 

3 If, when wheat is 83 cents a bushel, the cent loaf 
weighs 9 ounces ; what ought it to weigh when wheat is 
$ 1 244 cts. a bushel? Ans. 6 oz. 

4 If 100 dollars principal in 12 months gain 6 dollars 
interest; what principal will gain the same interest in 
8 months? Ans. $150. 

5. If 12 inches long and 12 inches wide, make 1 
square foot; how long must a board be that is 9 inches 
wide, to make 12 square feet? Ans. 16 ft. 

6 A. lent B. 500 dollars for 6 months ; how long must 
B. lend A. 220 dollars to be equivalent? 

Ans. 13 months, 19 days.-f 

7 There is a cistern having a pipe that will empty it 
in 12 hours; how many pipes of the same capacity will 
empty it in 15 minutes ? Ans. 48 pipes. 

8 A certain building was raised in 8 months by 120 
workmen, but the same being demolished, it is required 
to be rebuilt in 2 months; how many workmen must be 
employed? Ans. 480 men> 

9 If for 48 dollars 225 cwt. be carried 512 miles; how 
many hundred weight may be carried 64 miles for the 
same money? Ans. 1800 cwt. 

*A month is estimated at 30 days, unless a particular month C . 
referred to. 

_^_^____ 



98 SINGLE RULE OF THREE. 

10 If 48 men can build a wall in 24 days; how many 
men can do it in 192 days ? Ans. 6 men, 

11 How many yards of carpeting 2 ft. 6 in breadth, 
will cover a floor that is 27 feet long and 20 feet wide I 

Ans. 72. 

12 What quantity of shalloon that is 3 quarters wide, 
will line 7i yards of cloth that is U yd. wide? 

Ans. 15 yd. 

13 How many yards of matting that is 3 quarters wide, 
will cover a floor that is 18 feet w r ide and 60 feet long? 

Ans. 160. 

14 In what time will $600 gain the same interest that 
$80 will gain in 15 years? Ans. 2 years. 

Questions* 

What is proportion? 
When is the proportion direct? 
When is it inverse? 

Why is the proportion inverse in. the last question? 
A. because it is more money requiring less time. 

Why is the proportion inverse in the llth question? 
A. because the shalloon is narrower than the cloth; 
that is less width requiring more length. 

Why is the 10th question inverse? 

PROMISCUOUS EXERCISES. 

1 A certain steeple standing upon level ground, casts a 
shadow to the distance of 633 feet 4 inches, when a 
staff 3 feet long, perpendicularly erected, casts a shadow 
of 6 feet 4 inches; what is the height of the steeple? 

Ans. 300 ft. 

2 A ship's company of 15 persons is supposed to have 
bread enough for a voyage, allowing each person 8 oun- 
ces a day, when they take up a crew of 5 persons, with 
whom they are willing to share; what wi" be the daily 
allowance of each person now? Ans. 6 oz. 

3 Bought 215 yards of broad cloth at 6 dollars a yard; 



SINGLE RULE OF THREE. 99 

what was the prime cost, and how must I sell it per yard 
to gain $135 on the whole. 

Aiis. prime cost $1290,00; to be sold for $G,62f per 
yard. 

4 If 100 men can complete a piece of work in 12 days; 
how many can do it in 3 days? Ans. 400 men. 

5 If a board be 4| inches wide; how long a piece will 
it take to make 1 square foot? Ans. 32 in. 

G A pole, whose height is 25 feet, at noon casts a 
shadow to the distance of 33 feet 10 inches ; what is the 
breadth of a river which runs due East at the bottom of 
a tower 250 feet high, whose shadow extends just to the 
opposite edge of the water? Ans. 338 ft. 4 in. 

7 A plain of a certain extent having supplied a body of 
3000 horses with forage for 18 days ; how long would it 
have supplied 2000 horses ? Ans. 27 da. 

8 A piece of ground 1 rod wide and 160 rods long, 
makes 1 acre*; how wide a piece must I have across the 
end of a farm 32 rods wide to make an acre ? 

Ans. 5 rods. 

9 I have a floor 24 feet long, and 15 feet wide, which I 
wish to cover with carpeting that is 3 quarters of a yard 
wide; how many yards must I buy. 

Ans. 53 yards, 1 foot. 

10 How much land at $2,50 an acre must be given in 
exchange for3GO acres worth $3,75 an acre? 

Ans. 540 acres. 

11 What is the weight of a pea to a steel-yard, which 
is 39 inches from the centre of motion, will balance a 
weight of 208 Ibs., suspended at the draught end 3 quar- 
ters of an inch? Ans. 4 Ib. 

12 If $28 will pay for the carriage of 6 cwt. 150 
miles ; how far should 24 cwt. be carried for the same 
money? Ans. 37 i miles. 



100 DOUBLE RULE OF THREE. 

COMPOUND PROPORTION; 

OR 

THE DOUBLE RULE OF THREE. 

DIRECT AND INVERSE. 

COMPOUND PROPORTION is two or more series of pro- 
portionals combined. 

Five, seven, nine, or other odd number of terms, 
is always given to find a sixth, eighth, or tenth, &c*, or 
answer. 

Rule for the Statement. 

Place the numbers that is of the denomination in which 

the answer is required to be, in the third place. Then: 

Consider separately each pair of similar terms and 

place them agreeably to the rule for SIMPLE PROPORTION. 

An 
OR, 

Work by two separate statements in simple propor- 
tion.* 

Rule for the Solution. 

Reduce the several pairs of terms to similar denom- 
inations as in single proportion, and the last to the lowest 
denomination given : Then 

Multiply the two initials, or left hand terms togethei 
for a DIVISOR, and the other three for a DIVIDEND. 

Divide the latter by thejformer, and the quotient wil< 
be the answer, in that denomination to which the third 
term was reduced. 

EXAMPLES. 

1. If 6 men in 8 days build 40 rods of wall, how rnucl 
will 18 men build in 20 days? Aiis. 300 rods 



* It would be well for the pupil to work each sum both ways. 




DOUBLE RULE OP THREE. 101 

The answer is required to be 
rods gi ven in rods : then rods must be 
Afi the third term. If 6 men build 
40 rods, 18 men will build more,; 
then more (18 men) must occupy 
the second, and less (6 men) the 

48 360 first P lace - 

If 8 days produce 40 rods, 20 
days will produce more; then 
more (20 days) must occupy the 
48)14400(300 *eomd, and few (8 days) the>j* 

144 place. 

N. B. The first pair, or two 

upper terms must be alike. Also 

the lower pair must be alike. 

That is, both must be men or both days, both hours or both bushels, &c. 

2. If 6 men in eight days eat lOlb. of bread, how 
much will 12 men eat in 24 days? Ans 60. 

men 6 : 12) in IK 

days 8 : 24 : Contracted. 

^ 6 : 12 2j 

288 8 : 24 3 ' 

10 
6 

48)2880(60 Ans. 10 

288 

60 Ans. 



3. Suppose 4 men in 12 days mow 48 acres, how 
many acres can 8 men mow in 16 days? Ans. 128A. 

4." If 10 bushels of oats be sufficient for 18 horses 20 
days, how many bushels will serve 60 horses 36 days, 
at that rate? Ans. 60bu. 

5. Suppose the wages of six persons for 21 weeks be 
288 dollars, what must 14 persons receive for 46 weeks? 

Ans. $1472. 

6. If the carriage of 8cwt. 128 miles cost f 12.80, 
what must be paid for the carriage of 4cwt. 32 miles? 

Ans. $1.60. 

7. If 371b. of beef be sufficient for 12 persons 4 days, 
how many Ib. will suffice 38 men 16 days? 

Ans. 4681b. lOi oz. 



102 DOUBLE RULE OF THREE. 

8. If a man can travel 305 miles m 30 days, when 
the days are 14 hours long, in how many days can he 
travel 1056* miles, when the days are 12^ hours long? 

Ans. 116days.-f2540. 

9. If the carriage of 24cwt. for 45 miles be 18 dol- 
lars, how much will it cost to convey 76cwt. 121 miles? 

Ans/$153 26 cts.+720. 

10. A person having engaged to remove 000cwt. in 
9 days; removed 4500cwt. in 6 days, with 18 horses: 
how many horses will be required to remove the balance 
in the remaining 3 days? Ans. 28 horses. 

11. If 3 men reap 12 acres 3 roods in 4 days 3 hours, 
how many acres can 9 men reap in 17 days? 

Ans. 153 acres. 




Analysis. 

If 3 m. reap 12 a. 3 r. 
1 m. rea 4 a. 1 r. and 
9 m. reap 38 a. 1 r. 

If 4 d. 3h. reap 38 a. 
1 r. Id. reap 9 a. ami 
17 d. reap 153 a. Ans. 



1836 
9180 

153)93636(612 roods, or 153 acrea 
918 



183 
153 

306 
306 






*The day is here estimated at twelve hours. 



PRACTICE. 103 

12. If 40 men build 32 rods of wall in 8 days, work- 
ing 10 hours each day,- in how many days will 60 men 
build 48 rods, working 12 hours a day ? 

Ans. 6 days, 8 hours. 

Men men Multiply all the initial 

60 : 40 terms (or 60, 32, and 12) 

rods rods days together for a divisor: and 

32 : 48 : : 8 the other four for a dim- 

hours hours dend. 

12 : 10 

,. 13. If 36 men dig a cellar 60 feet long, 24 feet wide, 
and 8 feet deep, in 16 days, working 16 hours per day, 
ho\v many men can dig a cellar 80 feet long, 40 feet 
wide, and 12 feet deep, in 20 days, working 12 hours per 
day? Ans. 128. 

Questions. 

What is ccfnpound proportion? 

How do you state questions in compound proportion? 
Which terms do you multiply together for a divisor? 
Which for a dividend? 
What other method is there? 



PRACTICE. 

PRACTICE is a short and expeditious method of per- 
forming various calculations in business. 

CASE 1. 

When the given price is LESS than one dollar. 
RULE. Set down the given number as one dollar, 
and take such aliquot part* or parts of that number, as 
the price is of one dollar, for the answer. 



*An aliquot part of a number is any number that will divide A 
without a remainder; thus 4 is an aliquot part of 20; and 8 of 40: 
and 25 cents is an aliquot part of 100 cents, (or $1.) because 25 c 
are contained in 100 cts-, an even number of times, without a remain- 
der. 



104 






PRACTICE. 








TABLE OF ALIQUOT 


PARTS. 




CTS. 


CWT. 


ROODS. 




50 


1 -v 

1 




10 r 




roods 




33i 










2 i 





""3 








o 


11 


> 


25 


4 




J 




4 


P 


20 


r 
f 


-*j 


2 iV 


p 




P 


121 
10 


o 
? 


p 

1 D- 


qr. 





perches 
20 i 


o 

3 




V 

iff 


p" 


2 i 




10 ,\] 




5* 


1 

TO 




1 


o 


20 |" 


o 


4. 


1 

2J 




Ibs. 




10 ^ 


p 


2 


I 

5T- 




16 i 


o 


8 i 










14 t- 


i 


5 i 


< o 








8 


r* 


4 


^ 








7 l 
7 T6" 




2 it 










EXAMPLES 


. 






1. What will 826 bushels of wheat come to at 25 


cts. 


a bushel? 


Ans. $206 50 cts. 


cts. 


$ 












25 3 


r |8 


26 82(5 bushels, at one dollar a bushel, will 
cost 826 dollars : at 25 cents, or ^ of a 


dh 1 kc rcn dollar, it will cost one fourth as much. 
$ J <6UO,OU 


2. What will 934 gallons of molasses cost, at 50 cts. 
a gallon? Ans. $467. 


cts. 




$ 










50 \ 


r 934 




$ 467 


3. What will 1832 bushels of salt cost, at 75 cents a 


bushel? 


Ans. $1374. 


cts. 


<fc At 50 cents the cost will be 


50 \ 

_ _ 


sP 
r 1832 


i as much as at one dollar. 
At 25 cents the cost will be 


25 ] 


r - 


i as much as at 50 cts. 






916 


cost at 50 c. 












458 


cost at 25 c. 








$1374 


cost at 75 c. 









50 
25 



PRACTICE. 105 

<J As before ; the cost at 50 cts. will 

1H^2 ke i as much as at 1 dollar. 

At 25 cts. it will be i as much 



916 ct. at 50. 
458 ct. at 25. 



as at 1 dollar. 



$1374 ct. at 75. 

4. What will 680 pounds of sugar cost, at 1 cents a 
pound? $68. 

5. What will 742 pounds of pork cost, at 61- cts. a 
pound? Ans. $46 37* cts. 

6. What must I pay for 371 pounds of bacon, at 12|- 
cts. a pound? Ans. $46 37i cts. 

7. How much will 8750 bushels of rye cost, at 62| 
cts. a bushel? Ans. $5468 75 cts. 

8. How much must be paid for 4360 square feet of 
marble, at 87 i cents a foot? Ans. $3815. 

9. What will 468 square yards of plastering cost, at 
18| cents a yard? Ans. $87 75 cts. 

10. How much will be the cost of laying 856 perches 
of stone, at 93| cents a perch? Ans. $802 50 cts. 

11. What will the digging of a cellar, containing 180 
cubic yards, cost, at 20 cents a yard ? Ans. $36. 

12. What will be the cost of hauling 248 cords of 
wood, at 3H cents a cord? Ans. $77 50 cts. 

13. What must be paid for 432 perches of stone, at 
37i cts. a perch ? Ans. $162. 

14. How much must be given for 724 days labor, at 
56* cents a day? Ans. $407 25. 

15. What will 742 bushels cost at 10 cts. Ans. $71 20 

16. 732 15 10980 

17. 732 20 14640 

18. 475 25 11875 

19. 684 30 20520 

20. 756 35 26460 

21. 927 40 37080 

22. 824 50 41200 

23. 682 55 375 10 

24. 341 60 20460 

25. 784 70 54880 

E 2 '~ 



106 


PRACTICE. 


28. What will 352 busnels cost at 64 cts.? Ans.$22 00. 


29. 


436 124 54 50 


30. 


724 18| 135 75 


31. 


956 314 298 75 


32. 


742 37* 278 25 


33. 


274 43| 119 87* 


34. 


732 56* 411 75 


35. 


845 624 528 124 


36. 


684 68| 470 25 


37. 


274 814 222 624 


38. 


386 93| 361 874 




CASE 2. 


When the given price is MORE than one dollar. 


RULE. Multiply the given sum by the number of 


dollars, 


and take the aliquot part or parts for the cents. 


! as in Case 1. 




EXAMPLES. 


1. What will 342 cords of wood cost, at 3 dollars 75 


cents a 


cord? Ans. $1282 50. 


cts. 


$ 


50 


r 342 342 cords at $1, will cost $342; at $3 




3 it will cost 3 times $342; at 50 cts. it 




will cost i as much as it will at $1. ; and at 




25 cents, 4 as much as it will at 50 cts. ; 




1026 which added together, will be the cost at 


25 J 


r 171 $375. 




85 50 




1282 50 


2. What will 250 acr. cost at $4 624 AJIS. $1 156 25 


3. 


435 5 874 2555 624 


4. 


273 6 124 1672 124 


5. 


942 7 374 6947 25 


6. 


846 368| 3119624 


7. 


957 5 75 5502 75 


8. 


236 6 93| 1637 25 


9. 


754 3 564 2686 124 


10. 


932 27 25 25397 00 



PRACTICE. 
CASE 3. 



107 



When the given quantity consists of several denom- 
inations. 

RULE. Multiply the given price by the number of 
hundred weight, acres, yards, or pounds, &/c. and take the 
aliquot parts for the quarters, roods, feet, or ounces, &c. 

EXAMPLES. 

1. What will 240 acres, 2 roods, 10 perches, cost at 
$ 15 25 cents an acre ? Ans. $3668 57* cts. 



2 r. 



10 p. 



1525 
240 

61000 
3050 
762i 
951-J-2 rem. 

3668574 









2. What will 29 yards, 4 feet, of stone pavement cost, 



at $2 25 cents a yard? 
3 square feet 



Ans. $66 25 cts. 



29 

2025 
450 
,75 
25 

6625 






108 



PRACTICE. 



3. What will 32 pounds 8 ounces of silver cost, 
at $15,62i a pound? Ans. $510 41 i. 



6 oz. Troy 



2oz. 



$1562i 
32 

16 

3124 
4686 
78H 
26CH+2Rem. 

510 4H 



4. What will 27 cwt. 3 qrs. cost, at $23 50 cts. a 
cwt.? Ans. $652 12i. 

5. What will be the cost of 47 Ib. 10 oz. (Troy) at 
$1 25 cts. Ans. $59 79. 

6. What will 64 yds. 3 qrs. cost, at $2 25 a yard? 

Ans. $145 68| . 

7 Sold 83 yards 2 qrs. of cloth at $10 50 a yard; 
what does it amount to? Ans. $876 75. 

8. What will the laying of 28 squares, 75 feet of floor- 
ing cost, at $2 25 cts. a square? Ans. $64 68|. 

9. What is the cost of 27 cords, 96 feet of fire wood, at 
$3 75 a cord. Ans. $104 064 

10. What is the value of 428 gals. 3 qts. at $1 40 cts. 
a gallon ? Ans. $600 25 cts. 

11. What is the value of 765 gals. 3 qt. 1 pt. at $2 18| 
cents a gallon? Ans. $1675 34| cts. 

12. What is the value of 5 hhds. 3H gals, at $47 12 
cts. a hogshead? Ans. $259 16 cts. 

13. What is the value of 17 hhds. 15 gals. 3 qts. at 
$64 75 cts. per hogshead? Ans. $1116 93 cts. 7m. 

14. What is the value of 120 bu. 2 pecks, at 35 cents 
a bushel? Ans. $42 17 cts. 5 m. 

15. What is the value of 780 bu. 2 pecks, 2 qts. at $1 
17 cts. a bushel? Ans. $913 25 cts.+ 

16. What is the value of 1354 bu. 1 peck, 5 qts. 1 pt. 
at 25 cts. a bushel? Ans. $338 60 cts. 5m.+ 

17. What is the value of 35 acres 2 roods 18 perches, 
at 51 dollars 35 cts. an acre? Ans. $1935 53 cts. 9m. 



TARE AND TEET. 109 

Questions 

What is practice? 

What is the rule for the solution of questions in prac- 
tice? 

What is an aliquot part? 

Are 50 cts. an aliquot part of 100 cents? 

What part of $1 is fifty cents? 

What part of $1 is 33i cents? 

What part of $1 is 25 cents? 

What part of $1 is 12i cents? 

What part of $1 is 10 cents? 

What part of $1 is 20 cents? 

What part of $1 is 5 cents? 

What part of $1 is 4 cents? 

What part of $1 is 6* cents? 



TARE AND TRET. 

TARE AND TRET are allowances made on the weight 
of some particular commodities. 

Gross weight is the weight of the goods, together with 
the vessel that contains them. 

Tare is an allowance for the weight of the vessel. 

Tret* is an allowance of 4 Ib. for every 104, for 
waste, &,c. 

Neat weight is the weight of the goods, after all allow- 
ances are made. 

BULB. 

Subtract the tare from the gross, and the remainder 
is the neat weight. 

EXAMPLES. 

1 Bought a chest of tea, weighing gross 63 Ib., tare 8 
Ib. what are the neat weight and value, at 85 cents 
per Ib? 

* As tret is never regularly allowed in this country ; no account -of it 
is taken in this work. 

To 



110 TARE AND TRET. 

lb. 85 ct. 

63 gross or, weight of the chest and tea 55 lb. 

8 tare or, weight of the chest 
425 

55 neat or, weight of the tea itself 425 

$40,75 value, 

2 Bought 5 bags of coffee, weighing each 97 lb. gross, 
tare of the whole 7 lb. what are the neat weight arid 
value, at 25 cents per lb.? Ans. 478 lb. neat $119,50. 

3 The gross weight of a hogshead of sulphur is 1344 
lb.; the tare 138 lb. what are the neat weight and its 
value, at $4,75 per 100 lb.? 

Ans. 1206 lb. neat $57,28i 

4 Bought 3 barrels of sugar, weighing as follows, viz 
236 lb. gross, 23 lb. tare 217 lb. gross, 22 lb. tare 
^25 lb. gross, 23 lb. tare what are the neat weight and 
value, at $8 per 100 lb.? Ans. 610 lb. neat $48,80. 

5 Sold 3 hogsheads of sugar, weighing each 12 cwt. 2 
qrs. 14 lb. gross; tare 2 cwt. 1 qr. 27 lb. what are the 
neat weight and value, at $11,50 per cwt.? 

Ans. 35 cwt. 1 qr. 15 lb. neat $406 91 i cts. 

6 What is the neat weight of 15 tierces of rice, weigh- 
ing 48 cwt. 3 qrs. 12 lb. gross; tare 6 cwt. 12 lb., and 
what is the value, at $5,25 per cwt. ? 

Ans. 42 cwt. 3 qrs. neat $224,43$. 

7 What is the neat weight of 28 hogsheads of tobacco, 
weighing 201 cwt. 3 qrs. 12 ib. gross; tare 3140 lb.; 
and what does it come to at $5 per cwt. ? 

Ans. 173 cwt. 3 qrs. 8 lb $869 10| cts. 

8 Bought 17 bags of grain, weighing 3561 lb. gross; 
tare 2 lb. per bag what is the neat? Ans. 3527 lb. 

9 What is the neat weight of 16 bags of pepper, each 
weighing 65 lb. gross; tare 1ft lb. per bag and what is 
the amount at 30 cents per lb.? 

Ans. 1016 lb. neat $304,80. 



TARE AND TRET. Ill 

10 In 14 hogsheads of sugar, weighing 89 cwt. 3 qrs.i 
17 Ib. gross; tare 100 Ib. per hhd. how much ne&t 
weight, and what is its value, at $9 per cwt.? 

Ans. 77 cwt. 1 qr. 17 Ib. neat $696,611. 

11 What are the neat weight and value of 16 hhds. of 
tobacco, each weighing 5 cwt. 1 qr. 19 Ib. gross; tare 
101 Ib. per hhd., at 2 6s. lOd. per cwt.? 

Ans. 72 cwt. 1 qr. 4 Ib. neat 169. 5s. 4id. 

12 Bought 6 hhds. of sugar, each 1126 Ib." gross; tare 
117 Ib. per hhd. what are the neat weight and value at 
$8,75 per cwt.? Ans. 6054 Ibs. $529,72*. 

13 What are the neat weight and cost of a hogshead 
of sugar weighing gross 986 Ib. ; tare 12 per cent, (or 
12 Ib. for every WO Ib.) at $8 per neat hundred pounds? 

Ib. Ib. Ib. Ib. Ib. Ib. Ib. Ib. 



AslOO:986:: 12:118, 

Ib. 

986 gross. 
118, tare. 

868, neat weight. 



Or as 100: 88:: 986: 868, 



Ib. 

883 



8 dol. 



$69,44 the value. 



14 What are the neat weight and value of 4 hhds. of 
sugar weighing gross 45001b. tare 12 Ib. per cent, at $8, 
75 percent.? Ans. 3960 Ibs. neat $346,50. 

15 Bought 10 hhds. of sugar, each 920 Ib. gross; tare 
10 Ib. percent. what arelhe neat weight and value 
at $-9,25 per cwt.? Ans. 8280 Ib. neat- $765,90. 

16 Sold 3 casks of alum, each 675 Ib. gross; tare 13 
Ib. percent. what are the neat weight and value at $4, 
25 per cent. Ans. 1762 Ib. neat $74,87.4375. 

Or, 1762 Ib. neat $74,88. 5nearly. 

17 What is the neat weight of48001b.gross: tare 12 Ib. 
per cent.? Ans. 4224 Ib. 

18 What are the neat weight and value of 4 hhds. of 
sugar, each 12 cwt. 1 qr. 14 Ib. gross; tare 12 Ib. per 
cwt. at $12,20 per cwt.? 

Ans. 44 cwt. 22 Il> neat $539 19i cts. 



112 INTEREST. 

19 Bought 17 hhds. of sugar, weighing 201 cwt. 2 qrs. 
13 Ib. gross; tare 10 Ib. per cwt. what are the neat 
weight and value at $14 per cwt.? 

Ans. 183 cwt. 2 qrs. 13 Ib. neat- $2570 62 i cts. 

INTEREST. 

INTEREST is an allowance made for the use of money. 

Principal is the sum for which interest is to be com- 
puted. 

Rate per cent, per annum is the interest of 100 dol- 
lars for one year. 

Amount is the principal and interest added together. 

CASE 1. 
When the time is one year and the rate per cent, is any 

number of dollars. 

RULE. Multiply the principal by the rate per cent., 
and divide by 100 j the quotient will be the interest for 
one year. 

EXAMPLES. 

1. What is the interest of 500 dollars for 1 year, at 6 
per cent, per annum? 

$500 
6 

100-f-$30100 Ans. 

2. What is the interest of 225 dollars for 1 year, at Y 
dollars per cent, per annum? Ans. $15 75. 

3. What is the interest of 384 dollars 50 cents, for 1 
year, at 5 dollars per cent, per annum? Ans. $19 22i. 

4. What is the amount of $275 for 1 year, at 6 per 
cent, per annum? Ans- $291 50. 

$275 
6 

16,50 interest 
275,00 principal 

$291,50 amount 



i:\-TEKEST. 113 

5. What is the interest of 1654 dollars 81 cents for 1 
year, at 5 dollars per cent, per annum? Ans. $82 74-)-. 

6. What is the interest of 1500 dollars, for 1 year, at 
i dollar per cent, per annum? Ans. $7 50. 

7. What is the amount of $736, at 6 per cent, per 
annum, for 1 year. 780 dols. 16. 

8. What is the interest of 524 dollars, for 1 year, at 
5J dollars per cent, per annum? Ans. $27 51. 

9. What would be the interest of 842 dollars, for 1 
year, at 54 dollars per cent, per annum? Ans. $46 31. 

CASE 2. 
When the interest is required for several years. 

RULE. Find the interest for one year, and multiply 
the interest for one year by the number of years. 
EXAMPLES. 

1. What is the interest of 500 dollars, for 4 years, at 
6 dollars per cent, per annum? 

$500 
6 

< 3000 

4 

$12000 Ans, 

2. What will be the interest of 540 dollars, for 2 years, 
at 5 dollars per cent, per annum?* Ans. $54 00. 

3. What would be the interest of 482 dollars, for 7 
years, at 6 dollars per cent, per annum? Ans. $202 44. 

4 What is the amount of $736 81i with 7 years, nine 
months interest due on it, at 6 per cent, per annum? 

Ans. $1079 43|. 

Note. If the interest is required for years and months, 
multiply the interest for 1 year by the number of years, 
and take the aliquot parts of the interest for 1 year, for 
the months. 



1 



10* 



114 



Omo. 



3 mo. 



INTEREST. 

$736 814 
6 

4420,874 interest 1 year 

7 

30946,12i interest 7 years 
2210,431 interest 6 months 
1105,21!-)- 3 



34261,78 interest for 7 yr. 9 mo. 
73681,25 principal 

-$107943 03 amount 



5. What is the amount of $362 25 for 4 years 6 mo. 
at 6 per cent, per annum? Ans. $460 ()5|, 

CASE 3v 
When the interest is required for any number of months, 

weeks or days, less or more than one year. 
RULE. Find the interest of the given sum for one 
year Then, by proportion, 
As 1 year 

Is to the given time, 

So is the interest of the given sum (for 1 year) 
To the interest for the time required. 
Or take the aliquot parts of the interest for one year, 
for the given time, as in note, Case 2. 

EXAMPLES. 

1. What is the interest of $560 for 2 years and 6 mo. 
at 5 per ct. per annum? Ans $70 

560 
5 



6 mo. 



2800 interest for 1 year 
2 years 

5600 
1400 

$70 00 interest for 2 years 6 months. 



INTEREST. 115 

2. What is the interest of 325 dollars, for 4 years and 

2 months, at 4 dollars per cent, per annum? 

Ans. $54 16 cts. 6m. 

3. What is the interest of 840 dollars for 5 years and 

3 months, at 4 dollars per cent, per annum? 

Ans. $176 40. 

4. WTiat is the interest of 840 dollars, for 5 years and 

4 months, at 7 dollars per cent, per annum? 

Ans. $313 60. 

5. What is the interest of 5GO dollars, for 4 months, at 
6 dollars per cent, per annum? 

560 
6 

m. m. $ cts. $ cts. 
100)33 60 As 12 : 4 : : 33 60 : 11 20 Ans. 

6. What is the interest of 1200 dollars, for 15 weeks, 
at 5 dollars per cent, per annum? Ans. $17 30. 

7. What will be the interest of 240 dollars, for 61 
days, at 4| dollars per cent, per annum? Ans. $1 90.-J- 

8. What is the interest 'of $1000, for 14 months, at 7 
per cent, per annum? Ans. $81 60S. 

9. What is the interest of 450 dollars, for 6 months 
and 20 davs, at 5i dollars per cent, per annum? 

Ans. $1375. 

10. What is the interest of 375 dollars 25 cents, for 3 
years 2 months 3 weeks and 5 days, at 6 dollars per ct. 
per annum? Ans. $72 92.-J- 

11. What is the amount of $736 for 28 weeks, at 10 
per cent, per annum? Ans. $775 63. 

CASE 4. 

To fad the interest of any sum for any number of days^ 
as computed at banks. 

RULE. Multiply the dollars by the number of days, 
and divide by 6; the quotient will be the answer in mills. 

The interest of any number of dollars for 60 days, at 
6 per cent, will be exactly the number of cents; and if 
any other rate per cent, is required, take aliquot parts, 
and add or subtract according as the rate per cent, is 
more or less than 6. 



116 INTEREST. 

EXAMPLES. 

1. What is the interest of 563 dollars, for 60 days, at 
6 per cent, per annum and likewise at 7 per ct. per an.? 

Ans. $5,63 at 6 per cent. 
60 $6,56.8 at 7 per cent. 



6)33780 

in. at) 

6per[ 5630 mills, 
cent.) 



$5630 
938 



Interest at 7 per cent. 6568 mills. 

2. What is the interest of 854 dollars, for 30 days, at 
6 per cent, per annum? Ans. $4 27. 

3. What is the interest of 1100 dollars, for 48 days, at 
6 per cent, per annum? Ans. $8 80. 

4. What is the interest of 3459 dollars, for 75 days, at 
6 per cent, per annum? Ans. $43 23 cts. 7 m.-|- 

5. What is the interest of 1500 dollars, for 60 days, at 
5 per cent, per annum? Ans. $12 50. 

CASE 5. 
TJie amount, time, and rate per cent, given, to Jlnd the 

principal. 

RULE. Find the amount of 100 dollars for the time 
required, at the given rate per cent. 

Then, by proportion, as the amount of 100 dollars for 
the time required, (at the given rate per cent.) is to the 
amount given, so is 100 dollars to the principal required. 

EXAMPLES. 

1. What principal, at interest for 8 years, at 5 per ct. 
per annum, will amount to 840 dollars? Ans. $600. 
5 dollars 
8 years 

40 Int. of $100 for Syr. 
100 $ $ $ $ 

140 : 840 ::100 : 600 
140 Amt. of $100 for 8 yr. 



INTEREST. 117 

2. What principal, at interest, for 6 years, at 4 per 
cent, per annum, will amount to $,1240. Ans. $1000 

3. What principal, at interest for 5 years, at 6 per ct. 
per annum, will amount to 2470 dollars? Ans. $1900. 

CASE 6. 

The principal, amount, and time given, to find the rate 
per cent. 

RULE. Find the interest for the whole time given, by 
subtracting the principal from the amount. 

Then, as the principal is to 100 dollars, so is the in- 
terest of the principal for the given time, to the interest 
of 100 dollars for the same time. 

Divide the interest last found by the time, and the 
quotient will be the rate per cent, per annum, 

Or by compound proportion. 

EXAMPLES. 

1. At what rate per cent, per annum, will COO dollars 
amount to 744 dollars, in 4 years? Ans. 6 per cent. 



$744 amount As 600 : 100 : : 144 : 24. 

600 principal yr. $ 

4) 24 (6 rate per cent. 
144 interest 
Or by compound proportion : 

$ $ 
As 600 : 100 $ $ 

yr. yr. : : 144 : 6 rate per cent. 
4 : 1 

2. At what rate per cent, per annum, will $1200 
amount to $1476, in 5 years and 9 months? 

Ans. 4 per cent. 

3. If 834 dollars, at interest 2 years and 6 months, 
amount to 927 dollars 82* cents, what was the rate per 
cent, per annum? Ans. 44 per cent. 



118 COMPOUND INTEREST. 

CASE 7. 
To Jlnd the time, when the principal, amount, and rate 

per cent, are given. 

RULE. Divide the whole interest by the interest of 
the principal for one year, and the quotient will be the 
time required, or by proportion. 

EPAMPLES. 

1. In what time will 400 dollars amount to 520 dol- 
lars, at 5 per cent, per annum? Ans.6 years. 

* 9 

400 520 

5 400 

<& A Y Y 

~ as jm i. JL 



20JOO 20)120(6 20 : 120 : : 1 : 6 Ans. 

2. In what time will 1600 amount to 2048, at 4 
per cent, per annum? Ans. 7 years. 

3. Suppose 1000 dollars, at 4t per cent, per annum, 
amount to 1281 dollars 25 cents, how long was it at in- 
terest? Ans.6Y. 3rao. 



COMPOUND INTEREST. 

Compound interest is that in which the interest for 
one year is added to the principal, and that amount is 
the principal for the second year; and so on for any 
number of years. 

RULE. Find the a mount of the given sum for the first 
year by simple interest, which will be the principal for the 
second year; then find the amount of the principal for 
the second year for the principal for the third year; and 
so on for \ any number of years. 

Subtract the first principal from the amount, and the 
remainder will be the compound interest required. 

. EXAMPLES. 

1. What is the compound interest of 150 dollars for 5 
years, at 4 per cent, per annum? 

Ans. $32,49 




COMPOUND INTEREST. 119 

$150 $150 t 

4 6 inst 1st year 

6|00 int. 1 yr. 156 amount 1st year 
6,24 int. 2d year 

$156 102,24 amount 2d year 

4 6,48.9 ink 3d year 

6|24 168,72,9 amount 3d year 

6,74.9 ink 4th year. 

175,47.8 amount 4th year 
7,01.9 int. 5th year 

6|48.96 182,49.7 amount 5th year 

150,00.0 principal 

32,49.7 compound int. for 5 years. 

2. What is the compound interest of 760 dollars, for 
3 years, at 6 dollars per cent, per annum? 

Ans. $145 17 cts. 2 m.-f- 

3. What is the compound inierest of $242 50 els., 
for 4 years, at 6 per cent, per annum? 

An.s. $63 65 cents. 

4. What is the amount of 1300 dollars, for 3 years, 
at 5 dollars per cent, per annum, compound interest? 

Ans. $1504 91 cts. 2 m.-f 

5. How much is tha amount of 3127 dollars, for 4 
years, at 4i dollars per cent, per annum, compound in- 
terest? Ans. $3729 OOcts. 5m. 

Questions. 
What is interest? 
What is the principal? 
What is the rate per cent. 
What is the amount? 

How do you proceed when the interest for several 
years is required? 



120 COMPOUND INTEREST. 

What is to be noted f the interest is required for 
years and months? 

When the interest is required for any number of 
weeks or days, less or more than one year, how do you 
perform the operation? 

How do you proceed to find the interest, at 6 per cent, 
for any number of daj's, as computed at banks ? 

What is to be observed when the interest is at any 
other rate than 6 per cent.? 

How do you proceed, when the principal, amount, and 
time are given, to find the rate per cent.? 

How do you find the time, when the principal, amount, 
and rate per cent, are given ? 

What is compound interest ? 

How is compound interest computed? 



PROMISCUOUS EXERCISES. 

1. What is the interest of 620 dollars 25 cents for 5 
years, at 5i per cent, per annum? 

Ans. $170 56 8 m -f- 

2 What is the interest of $420, for 1 year, at 7 
per cent, per annum? Ans. $29 40. 

3 What is the interest of 1450 dollars, for 60 days, 
at 6 per cent, per annum? Ans. $14 50 cts. 

4 What is the compound interest of $626 25, for 3 
years, at 5i per cent, per annum? 

Ans. $103 91.+ 

5 What is the interest of $1659 for 3 weeks, at 4 per 
cent, per annum? Ans. $3 82i.-f- 

6 In what time will 500 dollars amount to 1000 dol- 
lars ut 8 per cent, per annum, simple interest? 

Ans. 12 years, 6 months. 

7 What principal, at interest for 6 years and 6 months, 
at 2 per cent, per annum, will amount to 250 dollars? 

Ans. $221 23 cts. 9 m. 

8 At what rate per cent, per annum, will $300 amount 
to $450, in 5 years? Ans. 10 per cent. 



INSURANCE, COMMISSION, AND BROKAGE. 121 

INSURANCE, COMMISSION, AND 
BROKAGE. 

INSURANCE, Commission and Brokage,are allowances 
made to insurers, factors, and brokers, at such rate per 
cent, as may be agreed on between the parties. 

RULE. 

Proceed in the same manner as though you were re- 
quired to find the interest of the given sum for one year. 

EXAMPLES. 

1 What is the commission on 625 dollars, at 4 dollars 
per cent? 

$625 
4 



Ans. $25,00 

2 What is the commission on $1320, at 5 per cent.? 

Ans. $66. 

3 What is the commission on 3450 dollars, at 44 dol- 
lars percent.? Ans. $155,25. 

4 The sales of certain goods amount to 1680 dollars: 
what sum is to be received for them, allowing 2| dollars 
per cent, for commission? Ans. $1633,80. 

5 What is the insurance of $760, at 6i per cent. ? 

Ans. $49,40. 

6 What is the insurance of 5630 dollars, at 7f dollars 
per cent.? Ans. $436 32 cts. 5 m 

7 A merchant sent a ship and cargo to sea, valued at 
17654 dollars: what would be the amount of insurance, 
at 18| dollars per cent.? Ans. $3310 12i cts 

8 What is the brokage on 2150 dollars at 2 per cent.? 

Ans. $43 

9 When a broker sells goods to the amount of 984 
dollars 50 cents, what is his commission, at 14 dollar per 
cent.? Ans. $12 30i cts.-f- 

10 If a broker buys goods for me, amounting to 1050 



h 



122 DISCOUNT. 

dollars 75 cents, what sum must I pay him, allowing him 
14 per cent.? Ans. $24 76 cts. 1 m.+ 

Questions. 

What are Insurance, Commission, and Brokage? 

How do you proceed to find the Insurance, Commis- 
sion, or Brokage? 

In what does this rule differ from interest? It takes 
no account of time. 



DISCOUNT. 

DISCOUNT is an abatement of so much money from any 
sum to be received before it is due, as the remainder 
would gain, put to interest for the given time and rate 
per cent. 

RULE. 

Find the interest of 100 dollars for the given time at 
the given rate per cent. 

Add the interest so found to 100 dollars, then by pro- 
portion, 

As the amount of 100 dollars for the given time, 

Is to the given sum, 

So is 100 dollars, 

To the present worth. 

If the discount be required, subtract the present worth 
from the given sum, and the remainder will be the dis- 
count. 

NOTE. When discount is made without regard to 
time, it is fouuti precisely like the interest for one year. 

EXAMPLES. 

1 What is the present worth of 420 dollars, due in 2 
years, discount at 6 per cent, per annum? 

Ans. $375. 



DISCOUNT. 123 

$ $ $ $ r 

6 112 :420:: UK) .375 

2 

12 
100 

112 

2 What is the present worth of 850 dollars, due in 3 
months, at 6 per cent, per annum? 

Ans. $837 43| cts.-f 

3 What is the discount of 645 dollars, for 9 months, 
at 6 per cent, per annum? Ans. $27 77i cts. 

4 What is the present worth of 775 dollars 50 cents, 
due in 4 years, at 5 per cent, per annum? 

Ans. $646,25. 

5 What is the present worth of 580 dollars, due in 8 
months, at 6 per cent, per annum? Ans. $557,69.-|- 

6 What is the present worth of 954 dollars, due in 3 
years, at 4i per cent, per annum? 

Ans. $840 52 cts. 8 m.+ 

7 What is the discount of 205 dollars, due in 15 
months, at 7 per cent per annum? 

Ans. $16 49 cts. 5 m.+ 

8 Bought goods amounting to 775 dollars, at 9 months' 
credit: how much ready money must be paid, allowing 
a discount of 5 per cent, per annum? 

Ans. $746 98 cts. 7 m. 

9 I owe A. to the value of 1005 dollars, to pay as fol- 
io^: viz. 475 dollars in 10 months, and the remainder 
in To months; what is the present worth, allowing dis- 
count at 6 per cent, per annum? 

Ans. $945 40 cts. 4 m. 

10 What is the difference between the interest of 
2260 dollars, at 6 per cent, per annum, for 5 years, and 
the discount of the same sum for the same time and rate 
percent.? Ans. $156 46 cts. 2m.-4- 



124 EQUATION OF PAYMENTS. 

11 What is the discount of 520 dollars, at 5 per 
cent.? 

$520 
5 



$26,00 Ans. % 

12 How much is the discount of $782, at 4 per cent.? 

Ans. $31, 28 

13 What is the discount of 476 dollars, at 3 per cent.? 

Ans. $14,28. 

14 Bought goods on credit, amounting to 1385 dol- 
lars : how much ready money must he paid for them, if 
a discount of 6 per cent, be allowed? Ans. $1301,90. 

15 I hold A.'s note for 650 jdollars ; but I agree to al- 
low him a discount of 4 per cent, for present payment: 
what sum mustl receive? Ans. $620,75. 

Questions. 

What is discount? 

What is first to be done ? 

After having found the interest of 100 dollars, at tht 
given time and rate per cent., what is next to be done ? 

After having added the interest so found to 100 dol- 
lars or pounds, by what rule do you work to find the dis- 
count? 

When discount is made without regard to time, how is 
it found? 



EQUATION OF PAYMENTS. 

EQUATION is a method of reducing several stated 
times, at which money is payable, to one mean, or equa- 
ted time, when the whole sum shall be paid. 
RULE. 

Multiply each payment by its time, and divide the sum 
of all the products by the whole debt, the quotient will 
be the equated time. 



EQUATION OF PAYMENTS. 125 

Proof. The interest of the sum payable at the equa- 
ted time, at any given rate, will equal the interest of the 
several payments for their respective times. 

EXAMPLES. 

1 C. owes D. 100 dollars, of which the sum of 50 
dollars is to be paid at 2 months, and 50 at 4 months ; 
but they agree to reduce them to one payment; when 
must the whole be paid? Ans. 3 months. 

50X2=100 

50x4=200 



100)300(3 months 

2 A merchant hath owing to him 300 dollars, to be 
paid as follows : 50 dollars at 2 months, 100 dollars at 5 
months, and the rest at 8 months ; and it is agreed to 
make one payment of the whole; when must that time 
be? Ans. 6 months. 

3 F. owes H. 2400 dollars of which 480 dollars are 
to be paid present, 900 dollars at 5 months, and the rest 
at 10 months; but they agree to make one payment of 
the whole, and wish to know the time? Ans. 6 months. 

4 K. is indebted to L. 480 dollars which is to be dis- 
charged at 4 several payments, that is i at 2 months, i 
at 4 months, i at 6 months, and i at 8 months; but they 
agreeing to make one payment of the whole, the equa- 
ted time is therefore demanded? Ans. 5 months. 

5 P. owes Q. 420 dollars, which will be due 6 months 
hence, but P. is willing to pay him 60 dollars now, pro- 
vided he can have the rest forborn a longer time : it is 
agreed on; the time of forbearance therefore is required? 

Ans. 7 months. 

6 A merchant bought goods to the amount of 2000 
dollars and agreed to pay 400 dollars at the time of pur- 
chase, 800 dollars at 5 months, and the rest at 10 months; 
but it is agreed to imike one payment of the whole; what 
is the mean or equated time? Ans 6 months. 



11* 



126 BARTER. 

BARTER. 

BARTER is the exchanging of one kind of goods for 
another, duly proportioning their values, &c. 

RULE. 

The questions that come under this head, may be 
done by the compound rules, the Rule of Three, or 
Practice, as may be most convenient. 

EXAMPLES. 

1 A country storekeeper bought 150 bushels of salt, 
at 56 cents per bushel ; and is to pay for it in corn, at 
33* cents per bushel ; how much corn will pay for the 
salt? ct. ct. lu. bu. 

As 33* : 56 : : 150 : 252 

OR 
Ct. 

56 334)8400 

150 3 3 



1|00)252K>0 

252 bushels of corn. 
Cost of the salt. 8400 cts. 

2 How much wheat, at 1 dollar 25 cents per bushel, 
will pay for 35 sheep, at 2 dollars 25 cents a piece? 

Ans.63bush. 

3 How much sugar, at 9 cents per Ib. will pay for i 
dozen pair of shoes, at 1 dollar 75 cents per pair? 

Ans. 233* Ibs. 

4 Row much tea, at 80 cents per Ib. will pay for 560 
Ibs. of pork, at 5 cents per Ib.? Ans. 35 Ibs. 

5 Bought 4 hats for 3 dollars 50 cents; 4 dollars; 4 
dollars 50 cents ; and 5 dollars how much corn at 32 
cents will pay for them? Ans. 53 bush. 4 qts. 

6 A. has 420 bushels of corn, which he barters with 
B. for cats, and is to receive 4 bushels of oats for 3 of 
corn how many bushels of oats must A receive? 

Ans. 560 bush. 



BARTER. 127 

7 A boy bartered 735 pears for marbles, giving 5 
pears for 2 marbles how many marbles ought he to 
have received? Ans. 294 marbles. 

8 A boy exchanges marbles for pears, and gives 2 
marbles for 5 pears how many pears should he receive 
for 294 marbles? Ans. 735 pears. 

9 A farmer bartered 3 barrels of flour, at 5 dollars 25 
cents per barrel, for sugar and coffee, to receive an equal 
quantity of each ho\v much of each must he receive, 
admitting the sugar to be valued at 9 cents per Ib. and 
the coffee at 14 cents? Ans. 68i Ib. nearly. 

10 A bartered 42 hat?, at 1 dollar 25 cents per hat, 
with B. for 50 pair of shoes, at 1 dollar 12i cents per 
pair who must receive money, and how much? 

Ans. B. $3,75. 

11 Sold 75 barrels of herrings, at 2 dollars 75 cents 
per barrel, for which I am to receive 75 bushels of wheat, 
at 1 dollar 8 cents per bushel, and the residue in money 
how much money must I receive? 

Ans. 125 dolls. 25 cts. 

12 Sold 35 yards of domestic, at 20 cents* per yard, 
and am to receive the amount in apples, at 25 cents per 
bushel how many bushels must I have? 

Ans. 28 bush. 

13 Gave 35 yards of domestic for 28 bushels of ap- 
ples, at 25 cents per bushel what was the domestic 
rated at per yard? Ans. 20 cts. 

14 What is rice per Ib. when 340 Ib. are given for 4 
yards of cloth, at 4 dollars 25 cents per yard? 

Ans. 5 cts. 

15 Gave in barter 65 Ibs. of tea for 156 gallons of 
rum, at 33i per gallon what was the tea rated at? 

Ans. 80 cts. per Ib. 

16 Q. has coffee worth 16 cents per pound, but in 
barter raised it to 18 cts.; B. has broad cloth worth 4 
dollars 64 cents per yard what must B. raise his cloth 
to, so as to make a fair barter with Q? Ans. $5,23. 



128 LOSS AND GAIN. 

17 B. had 45 hats, at 4 dollars per hat, for which A. 
gives him 81 dollars 25 cents in cash, and the rest in 
pork, at 5 cents per Ib ; how much pork will be required ? 

Ans. 1975 Ib. 

18 Two merchants barter; A. receives 20 cwt. of 
cheese, at 2 dollars 87 cents per cwt.; B. 8 pieces of 
linen, at 9 dollars 78 cents per piece; which of them 
must receive money, and how much? Ans. A. $20,84. 

1>9 If 24 yards of cloth be given for 5 cwt. 1 qr. of 
tobacco, at 5 dollars 7 cents per hundred; what is the 
cloth rated at per yard? Ans. $1. 109. 

20 A. barters 40 yards of cloth, at 98 cents per yard, 
with B. for 284 Ibs. of tea, at 1 dollar 53 cents per Ib. ; 
which must pay balance, and how much ? 

Ans. A. $4,405. 

21 A has 74 cwt. of sugar, at 8 cents per Ib., for 
which B. gave him 124 cwt. of cheese , what was the 
cheese rated at per Ib.'? Ans. $. 048. 

22 What quantity of sugar, at 8 cts. per Ib. must be 
given in barter for 20 cwt. of tobacco, at 8 dollars per 
cwt.? Ans. 17 cwt. 3 qrs. 12 Ib. 

23 P. has coffee, which he barters with Q. at 11 cts. 
per Ib. more than it cost him, against tea, which stands 
Q. in 1 dollar 33 cents the Ib., but he puts it at 1 dollar 
66 cents ; query, the prime cost of the coffee ? 

Ans. $. 443+ 



LOSS AND GAIN. 

By Loss AND GAIN, merchants and dealers compute 
their gains or losses. 

RULE. 

Work by the Compound Rules, by Proportion, or 
in Practice, as may be most convenient. 



LOSS AND GAIN. '129 

EXAMPLES. 

1 Bought 1234 Ibs. of coffee, at 12j cts. per lb., and 
sold the whole for 160 dollars ; did I lose or gain by it, 
and how much? Ans. gained $5,75. 

2 Bought 120 dozen knives, at 2 dollars 50 cents per 
dozen, and sold them at 18| cents a piece; did I gain 
or lose, and how much? Ans. lost $30. 

3 Bought 1234 yards of muslin, for 17i cents, and 
sold it at 20 cents per yard; what was the gain? 

Ans. $30,85. 

4 Bought 10 chests of tea, each 63 Ibs. neat, for 600 
dollars, and retailed it at 87i cents per lb.; did I gain 
or lose, and how much ? Ans. lost 48 dol. 75 ct. 

5 Gave 285 dollars 25 cents for 4564 Ibs. of bacon, 
and sold it for 365 dollars 12 cents ; what was the gain 
per lb? Ans. . 1| cts. 

6 Bought 1234 yards of muslin, for 246 dollars 80 
cents, and sold it for 215 dollars 95 cents; what did I 
lose per yard? Ans. $. 2i cts. 

7 Gave 25 cts. per bushel for corn, and sold it at 28 
cents; what is the gain per cent.? 

Ans. 12 dolls, per 100 dolls. 

8 Sold corn at 25 cts. per bushel, and 4 cts. loss; 
what was the loss per cent.? Ans. $13,79. 

9 Bought 13 cwt. 25 Ibs. of sugar, for 106 dollars, 
and sold it at 9i cts. per lb. ; what did I gain per cent.? 

Ans. 32 dolls. 73 cts. 

10 Bought 128 gallons of wine for 150 dollars, and 
retailed it at 20 cts. per pint; what was the gain per 
cent.? Ans. 34 dolls. 40 cts. 

11 Sold a quantity of goods, for 748 dollars 66 cents, 
and gained 10 per cent; what did I give for them.? 

Ans. 680 dols. 60 cts. 



F 2 



130 LOSS AND GAIN. 

dols. dols. dols. 
100 110 : 100: : 748,66 

10 100 



110 110)74866,00($680,60 

12 Sold goods to the amount of $'1234, and gained at 
the rate of 20 per cent.; what was the prime cost? 

Ans. $1028,33* 

13 Soldji quantity of goods, for $475, and at a loss 
of 12 per cent.; what did 1 give for them? 

dols. dols. dols. 
100 88 : 100 : : 475 

12 100 

88 88)47500(539,77+Ans. 

14 Sold hats to the amount of $136, at 20 per cent, 
loss; \vhat was the first cost? Ans. $170. 

15 Laid out $755 in salt; how much must I sell it 
for, so as to gain 12 per cent.? 

12 
100 



As 100 : : 112 : 755 : : 845,60 Ans. 

16 Bought 32 yards of mole skin for 128 dollars; 
what must I sell it for per yard, so as to gain 20 per 
cent.? Ans. 4 dols. 80 cts.-j- 

17 Bought 17 yards of silk for 21 dollars; how much 
per yard /mist I Detail it for, and gain 25 per cent. ? 

Ans. 1 dol. 54 cts.-f- 

18 Bought 64 ya.rds of muslin for 1C dollars 50 cents, 
)>ut proving a bad bargain, I am willing to lose 8 per 
cent; what must I sell it at per yard? Ans.19cts.4m.-f- 

19 When hats are bought at 48 cents, and sold at 5 i 
cents; what is the gain per cent.? Ans. 12 i 



LOSS AND GAIN. 131 

20 If, when cloth is sold for 84 cents per yard, there 
is gained 10 per cent.; what will be the gain percent, 
when it is sold for 1 dollar 2 cents per yard? 

Ans. 33 dols. 68 cts.+ 

21 Bought a chest of tea, weighing 490 Ibs. for $122 
50 ct. and sold it for $137 20 cents; what was the profit 
on each lb.? Ans. 3 cts. 

22 Bought 12 pieces of white cloth, for 16 dollars 50 
cents per piece; paid 2 dollars 87 cents a piece for 
dying; for how much must I sell them each, to gain 20 
per cent. ? Ans. 23 dols. 244. 

23 If 28 pieces of stuff be purchased at 9 dollars 00 
cents per piece, and 10 of them sold at 14 dollars 40 
cents, and 8 at 12 dollars per piece; at what rate must 
the rest be disposed of, to gam 10 per cent. by the whole? 

Ans. 5 dols. 568. 

24 Sold a yard of cloth for 1 dollar 55 cents, by 
which was gained at the rate of 15 per cent.; but if it 
had been sold for 1 dollar 72 cents ; what would have been 
the gain per cent.? Ans. 27 dols. 69-j- 

25 If, when cloth is sold at $. 935 a yard, the gain 
is 10 dollars per cent. ; what is the gain or loss per cent., 
when it is sold at 80 cents per yard ? 

*Ans. 5 dollars 88+loss. 

26 A draper bought 100 yards of broad cloth, for 
which he gave $56 I desire to know how he must sell 
it per yard, to gain $19 in the whole? 

Ans. 75 ct. per yard. 

27 A draper bought 100 yards of broad cloth for $56; 
I demand how he must sell it per yard, to gain $15 in 
laying out $100? Ans. 64 ct. 4 in. 

28 Bought knives at 11 cents, and sold them at 12 
cents; what will I gain by laying out 100 dollars in 
knives? . Ans. 9 dols. 09+ 

29 Bought knives at 11 cents, and sold them at 12 
cents; what did I gain by selling to the amount of 100 j 
dollars? . Ans. 8 dols. 333+ 



jj 132 FELLOWSHIP. 

11 - 

30 If by selling 1 Ib. of peppejr for 10i cents, there 
are 2 cents- lost; how much is the loss per cent.? 

Ans. 16 dols, 

31 A merchant receives from Lisbon, 180 casks of 
raisins, which stands him in here 2 dollars 13 cents each, 
and by selling them at 3 dollars 68 cents per cwt., he 
gains 25 per cent.; required the weight of each cask, 
one with another? Ans. 81 Ib. 



* FELLOWSHIP. 

FELLOWSHIP is a method by which merchants and 
others adjust the division of property, loss, or gain, &,c., 
in proportion to their several claims. 

CASE 1. SIMPLE FELLOWSHIP. 

When the claims are in proportion to the amount of 
stock, labor, &c., without regard to time. 

RULE. (By Proportion.) 
As the whole amount of stock or labor, 
Is to each man's portion, 
So is the whole property, loss, or gain, 
To each man's share of it. 

Proof. The sum of all the shares must equal the 
whole gain, &,c. 

EXAMPLES. 

1 Two men bought a stock of goods for 480 dollars, 
of which A. paid 320, and B. 160. They gained 128 
dollars by the transaction ; what was the share of each ? 

Ans. A. received 85 dols. 33J cts. and B. 42 dollars 
661 cts. 

$ $ $ $ ct. Proof 

A's. stock $320 As 480 : 320 : : 128 : 85,33* $85,331 
H'p. stock 160 42,601 

.^ . & (ft (JN (ft s%4- _ 

Whole st'k 480 As480:160::128 : 42,661 128,00 



FELLOWSHIP. 133 

2 Three workmen having undertaken to do a piece 
of work for 275 dollars, agreed to divide their profits in 
proportion to the amount of labor each one performed. 
M. labored 50 days, N. 65 days, and O. 85 days : what 
was the share of each? 

Ans. M. received 68 dols. 75 cts. ; N. 89 dols. 37 i cts. ; 
aridO. 116 dols. 87 i cts. 

3 A merchant being deceased, worth 1800 dollars, is 
found to owe the following sums: to A. 1200 dollars, to 
B. 500 dollars, to C. 700 dollars : how much is each to 
have in proportion to the debt? 

Ans. A. 900 dols., B. 375 dols, and C. 525 dols. 

4 Three drovers pay among them 60 dollars for pas- 
ture, into which they put 200 cattle, of which A. had 50, 
B. 80, and C. 70 : I would know how much each had to 
pay? Ans. A. 15 do!?., B. 24 dols., C. 21 dols. 

5 A man failing, owes the following sums: to A. 120 
dollars, to B. 250 dollars 75 cents, to C. 800 dollars, to 
D.208 dollars 25 cents j and his whole effects were found 
to amount to but 650 dollars : what will each one receive 
in proportion to his demand ? 

Ans. A. $ 88.73.+ C. $221.84.+ 
B. $185.42.+ D. $153.99+ 

6 A bankrupt is indebted to A. 500 dollars 37 i cents 
to B. 228 dollars to C. 1291 dollars 23 cents to D. 
709 dollars 40 cents ; and his estate is worth 2046 dol- 
lars 75 cents: how much does he pay per cent., and 
what does each creditor receive? 

Ans. He pays 75 per cent., and A. receives 375 
dollars 27! cts.; B. 171 dols.,- C. G68 dols. 42| cts.; 
and D. 532 dols. 5 cts. 

7 If a man is indebted to A. 250 dollars 50 cents, to 
B. 500 dollars, to C. 349 dollars 50 cents, but when he 
comes to make a settlement, it is found he is worth but 
960 dollars, how much will each one receive, if it be in 
proportion to their respective claims? 

(A. $218 61 cts. 8 m.+ 

Ans. {B. $436 36 cts. 3 m.+ 

(C. $305 01 ct. 8 m.+ 

12 



134 FELLOWSHIP. 

CASE 2. COMPOUND FELLOWSHIP. 

When the respective stocks are considered with rela- 
tion to time. 

RULE. (By Proportion.) 

Multiply each man's stock by its time; add the several 
products together; then: 

As the sum of the products 

Is to each particular product, 

So is the whole gain or loss 
- To each man's share of the gain or loss. 

EXAMPLES. 

1 Three merchants traded together; A put in 120 
dollars for 9 months, B. 100 dollars for 16 months, and 
C. 100 dollars for 14 months, and they gained 100 dol- 
lars; what is each man's share? 

$ mo. 

A's. stock 120 X 9 = 1080 
B's. stock 100 X 16 = 1600 
C's. stock 100 X 14 = 1400 



Sum 4080 

Sum. Prod. $ $ 

As 4080 : 1080:: 100 : 26,47+ A's. share. 

As 4080 : 1600 ::100 : 39,214 f B's. share. 

As 4080 : 1400 ::100 : 34,31+ C's, share. 

2 Three men traded together; L. put in 88 dollars for 
3 months, M. 120 dollars for 4 months, and N. 300 dol- 
lars for 6 months ; they gained 184 dollars: what will 
each man receive of the gain? 

L. $ 19 09 cts. 4 m 
Ans. M. $ 34 71 cts. 6 m. 
N. $130 18 cts. 8 m. 



VULGAR FRACTIONS. 135 

3 Two merchants entered into partnership for 16 
months : A. put in at first $600, and at the end of 9 
nonths put in $100 more; B. put in at first $750, and at 
lie end of 6 months took out $250, at the close of the 
time their gain was $386, what was the share of each? 

Ans. A's. share was $200,794; B's. share was 
$185,20. 

4 A., B., and C., made a stock for 12 months; A. put 
in at first $873,60, and 4 months after he put in $96,00 
more ; B. put in at first $979,20, and at the end of 7 
months he took out $206,40; C. put in at first $355,20, 
and 3 months after he put in $206,40, and 5 months after 
that he put in $240,00 more. At the end of 12 mouths, 
their gain is found to be $3446,40 ; what is each man's 
share of the gain? 

(A's. share is $1.334,821 
Ans. ?B's. - - $1271,61i-f 
(CTs. - - $839,96 

Questions. 

What is Fellowship? 

By what rule are its operations performed? 

When is Fellowship simple? 

When is it compound? 

In what respect is Fellowship compound? 

Ans. The proportion is compound : that is, the divi- 
sion of property, gain, &c., is founded on the compound 
proportion of the stock and time. 



VULGAR FRACTIONS. 

A VULGAR FRACTION is a part, or parts of a unit ex* 
pressed by two numbers placed one above the other with 
a line between them. As ^, *,&c. 

The number below the line is the denominator, the 
number above the line is the numerator. 

The denominator denotes the number of parts into 
which the unit is divided. 



L36 VULGAR FRACTIONS. 

The numerator shows how many of those parts are to 

taken. 

Fractions are either proper, improper, or compound. 

A proper fraction is one whose numerator is less than 
its denominator, as f- or y. 

An improper fraction is one whose numerator is 
greater than its denominator, as | or |. 

A compound fraction is a fraction of a fraction, as \ of 
3-, or | of J. 

A mixed number is a whole number and a fraction. 

REDUCTION OF VULGAR FRACTIONS. 

CASE 1. 
To reduce a fraction to its lowest terms. 

RULE. 

Divide the terms by any number that will divide both 
without a remainder, and divide the quotient in the same 
manner, and so on till no number greater than one will 
divide them : the fraction is then at its lowest terms. 

EXAMPLES. 

1. Reduce T \ 4 T to its lowest terms. 
== result. 



2. Reduce - to its lowest terms. Res. \ 

3. Reduce ~~ to its lowest terms. Res. 

4. Reduce JJ to its lowest terms. Res. |* 
NOTE. When a divisor cannot readily be found, divide 

the denominator by the numerator, and that divisor by 
the remainder, and so on, till nothing remain: the last 
divisor is the common measure of the two numbers j with 
which proceed as before. 

5. Reduce -f/ 7 to its lowest terms. Res. f- 



VULGAR FRACTIONS. 137 

5 Reduce ~~j to its lowest terms. Rss.*|. 

85 

85)136(1 Here 17 being the last divisor, 

85 is the common measure of 85 

- and 136. 

51)85(1 
51 

34)51(1 j 85 ) (5 

34 17 - = J- 

I 136) (8 
17)34(2 
34 

6. Reduce j to its lowest terms. Res. . 

7. Reduce \\\ to its lowest terms. Res. f . 

8. Reduce |f \\ to its lowest terms. Res. i-^. 

~ 

CASE 2. 

To reduce a mixed number to an improper fraction. 
RULE. 

Multiply the whole number by the denominator, and 
add the numerator to the product for the numerator of 
the improper fraction, and place the denominator under 
it. 

EXAMPLES. 

1. Reduce 12 J to an improper fraction. 

12 Res. '. 



112 Nine 12's are 108; add 

- 4 makes 112 ninths. 

9 

2. Reduce 17 ^to an improper fraction. Res. ! f 2 

3. Reduce 45 | to an improper fraction. Res. I | 7 

4. Reduce 24 ~ to an imp roper fraction. Res. 4 y T 



12* 



138 . VULGAR FRACTIONS. 

CASE 3. 
To reduce an improper fraction to its proper value. 

RULE. 

Divide the numerator by the denominator, and the 
quotient will be the whole number; the remainder, if 
any, will be the numerator of tho fraction. 

EXAMPLES. 

1 Reduce Y to its proper value. Res. 3 J. 




5 

2 Reduce l J 2 to its proper terms. Res. 12 J. 

3 Reduce l ~ 2 to its proper terms. Res. 17 |. 

4 Reduce 4 T y to its proper terms. Res. 24 ~. 

CASE 4. 

To reduce several fractions to other fractions having a 
common denominator, and retaining their value. 

RULE. 

Multiply each numerator into all the denominators 
but its own, for the respective numerators; and all the 
denominators together, for a common^ denominator. 

EXAMPLES. 
1 Reduce f- J and f to a common denominator. 

Res. 4| f|, and ?}. 
2X4X6=48) 
3X3X6=54V Numerators 
5X3X4=60) 

3X4x6=72, common denominator. 
Then we have If for f ; || for J, and 4| for f . 
Reduce each new fraction to its lowest terms, and the 
result will prove the work to be right. 
2. Reduce J, f , and ^, to a common denominator 

7> p<3 216 240 J 1 6 8 

1168 2JJJ ana 



3. Reduce J, |, f , and T \, to a common denominator. 

T> Ae 216 288 360 nn ,1 252 
, ^ 1X6S - 43 2> 432? 43 2> attd 4 3; 2' 

4. Keduce |, ^, |, and , to a common denominator. 

[_>, lv , R3 480 432 Mn J 50 

\ ixes. - f .,0, y-oo, -7-30-5 ^nci y^-^. 



VULGAK TRACTIONS. 



139 



NOTE. It is often convenient to use the least possible 
common denominator; to find which, divide the denomi- 
nators by any number that will divide two or more of 
them without a remainder, setting down those that would 
have remainders; then multiply all the divisors and all 
the quotients together. 



5 6 



8 



1 



1 51713 

4X3X2X5X7X3=2520 common denom. 
Which may be divided separately by 2, 3, 4, 5, 6, 7, 8, 
and 9, without a remainder. 



1 1 



EXAMPLES. 

5 Find the least common denominator for J, J, 
T 5 j, and ~, and compute their equivalent fractions. 

Res --- -i 2 - --- --*. 
240 com. denom. 
00 X 3=180 
30X 7=210 
20X11=220 
15 X 5= 75 
12 X 9=108 

6 Reduce J, |, |, T \, and ~, to their least common 
denominator. Res. T 9 ~* V^U r^l- i4^, and 144. 



2(7 



7 Reduce f , T \, T 9 , |xjto their least common denom- 

natnr Poo 1 5 16 8" 35 on J 110 

nator. Kes. 240, 24^, ^ T o and ?T . 

8 Reduce the above fractions to a common denomina- 
tor, by the general rule, Case 4. 



19200 21504 
3-OT20J - 



14080 
3*120' 



5X10X16X24=19200 

7X 8x16x24=21504 

9X 8X10X24=17280 

11 X 8X10X16=14080 



140 VULGAR FRACTIONS. 

CASE 5. 
To reduce a compound fraction to a simple one. 

RULE. 

Multiply the numerators together for a new numera- 
tor; and the denominators together for a new denomina- 
tor. 

EXAMPLES. 
1 Reduce of of to a single fraction. 

Res. T . 
3X5X9 135 9 



4X6X10 240 16 

2 Reduce of ~ of ~ to a single fraction. Res. T y T . 

3 Reduce j of of J to a single fraction. Res. -/^. 

4 Reduce ~J of f- of i- to a single fraction. Res. T S T . 

CASE 6. 

To reduce a fraction of one denomination to the fraction 
of another denomination, but greater, retaining the 
same value. 

RULE. 

Multiply the denominator of the fraction by the num- 
ber of that denomination which it takes to make one of 
the next, and so on to the denomination required, and 
place the numerator of the given fraction over it. 

EXAMPLES. 

1 Reduce f of a quart to the fraction of a bushel.* 

qt. 

2 2 1 

= Result, 1 T of a bushel. 
3X8X4=9(5 48 

2 Reduce J of an ounce, Troy, to the fraction of a 
pound. Res. T 3 , or ~ of a pound. 

3 Reduce j of a nail to the fraction of a yard. 

Res. \, or - 2 ~ of a yard. 

3 Reduce | of a perch to the fraction of an acre. 

Res. 



* That is, what part of a bushel are two-thirds of a quart ? 



VULGAR FRACTIONS. 141 

4 Reduce ~j of a pint to the fraction of a hogshead 

Res. T ^ T of a hhd. 

CASE 7. 

To reduce ike fraction of one denomination to the frac- 
tion of another, but less, retaining the same value. 

RULE. 

Multiply the given numerator by the parts of that be- 
tween it and that to which it is to be reduced, and place 
the product over the given denominator for the fraction 
required. 

EXAMPLES. 

1 Reduce j 7 of a bushel to the fraction of a quart. 

Res. of a quart. 

2 Reduce -^ of a yard to the fraction of a nail. 

Res. J of a nail. 

3 Reduce y/j ^ f an acre to tne fraction of a perch. 

Res. I of a perch. 

4 Reduce 7 of a hogshead to the fraction of a pint. 

Res. ~ of a pint. 

5 Reduce yJg-^ of a day to the fraction of a minute. 

Res. TT of a minute. 

CASE 8. 

To reduce a fraction to its proper value or quantity, in 
whole numbers. 

RULE. 

Multiply the numerator by the parts of the integer, 
and divide by the denominator. 

EXAMPLES. 

1 Reduce J of a yard to its proper quantity. 

Res. 3 qr. 2 na. 

7 eighths of a yd. 4 eighths of a qr. 

4 4 

8)28 8)16 

3 i quarters 2 nails 



142 VULGAR FRACTIONS. 

2 Reduce of a pound, avoirdupois, to its proper 
quantity. Res. 8 oz. 14 dr. 

3 Reduce J of a pound, Troy, to its proper quantity. 

Res. 9oz. 

4 Reduce - of a mile to its proper quantity. 

Res. 4 fur. 125 yd. 2 fLlfinch, 

5 Reduce -f-g of an acre to its proper quantity. 

Res. 1 rood, 30 pf ich. 

6 Reduce J of a dollar to its proper quantity. 

Res. GO cents. 

7 Reduce ~ of a pound to its proper value. 

Res. 6s. 8d. 

8 Reduce j~ of a year (365 days) to its proper quan- 
tity. Res. 225 days. 

9 Reduce J of a tun to its proper quantity. 

Res. 3 hhd. 7 gal. 

10 Reduce J of a ton to its proper quantity. 

Res. 15 cwt. 2 qr. 6 Ib. 3 oz. 8| dr 

CASE 9. 

To reduce a given quantity to a fraction of any greate-i 
denomination of the same kind, 

RULE. 

Reduce the given quantity to the lowest denomination 
mentioned for a numerator; and th^ integer to the same 
denomination, for a denominator. 

EXAMPLES 

1 Reduce 3 qr. 2 na. to the fraction of a yard. 

Res. J of a yard, 
qr. na. 
3 2 
4 

yd. 141 (7 

1X4X4=16) (8 

2 Reduce 2 roods 20 perches to the fraction of an 
acre. Res. | of an acre. 



VULGAR FRACTIONS. 



143 



3 Reduce 6 furlongs 16 poles to the fraction of a mile. 

Res. j of a mile. 

4 Reduce 9 ounces, Troy, to the fraction of a pound. 

Res. f of a pound. 

5 Reduce 7 hours 12 minutes to the fraction of a day. 

Res. -^g- of a day. 

ADDITION OF VULGAR FRACTIONS. 
RULE. Reduce the given fractions, if necessary, to 
single ones, or to a common denominator; add all the 
numerators together, and place the sum over the com- 
mon denominator. 

EXAMPLES. 

f- 4 W 



- 

7 ft 
3 A 

ry 9 

" To 



3 I 

7 t- 
8 



19 or 2 f 26 T 8 o= 



8 



28 






NOTE 1. When the fractions are of different denom- 
inators, reduce them to a common denominator, and 
proceed as above. (See Note, page 139.) 

4 Add J, , T i, T 5 and- A- together. Result 3^\. 
240 



60X 3=180 
30X 7=210 
20X11=22C 
15X 5= 75 
12X 9=108 



IT 



73 

2T7 



^40" - 2T7 

5 Add f , f , J, -& and Vi together. 

6 Add |, T \, T V, and II together, 

7 Add |, T 3 , and T 4 together. 

8 Add f and f together. 

9 Add -j^, ii and | together 



Res. 4ii 
Res. 2/-V 

i||= r Vo 
Res. U 8 T 

Res. 2 f VV 



144 



VULGAR FRACTIONS. 



NOTS 2. When mixed numbers occur, place them as 
in examples 2 and 3; proceed with the fractions as di- 
rected in Note 1 ; and if they amount to one or more 
integers, carry them to the integers, and proceed as in 
simple addition. 

10 Add 5|, 61, and 41 together. Res. l7, 

24 

8x2=16 
3x7=21 
12X1=12 



49) 

V =2 

24 \ 



Ti- 



11 Add 21 and 3J together. 

12 Add 74 and 5 .together. 



13 Add 171 and | together. 

14 Add 4, 6, 9, and { 



15 Add 5, 7|, | and 



T 



together. 
together. 



Result 61. 
Res. 12|J 
Res. 181. 
Res. 2011. 
Res. 13vY 



' ' 2" 

NOTE 3. When compound fractions are given, re- 
duce them to single fractions, and proceed as before. 
16 Add T V of 11, T C T of T \, and T 7 j of f together. 

Res. j" 
9240 common denominator 



77X99=7623 
60X^8=-2880 
77X35=2695 



131981 

9240) 

17 Add | of 1 and * of 1 together. 

18 Add 1|, J of 1, and 9^ together. 



Res 



Res. 11 



. - 






19 Add If -, 6J, of 1, and 71 together. 

Res. 16 T 7 2\. 

NOTE 4. When the given fractions are of several 
denominations, reduce them to their proper values or 
quantities, and add as in the following example. 



13 



G 



VULGAR FRACTIONS. 145 

20 Add of a pound, to ~ of a shilling. 

Result 15s. 10 r Vl. 
15 
s. d. 



} of a 15 6$ 
ft of a *. 3| 



5x2=10 
3X3= 9 



15 10 T 4 T 19 

V =1 
15 

21 Add 2 of a pound, to J of a shilling. 

Result 18s. 3d. 

22 Add J of a penny, to J- of a pound. 

Res. 2s. 3d. Iqr. \, 

23 Add i Ib. troy, to ^ of an ounce, 

Res. 6oz. lldwt. 16grs. 

24 Add J of a mile, to f$ of a furlong. Res. Gfu. 28p. 

25 Add j of a yard, to of a foot. Res. 2ft. 2 in. 

26 Add | of a day, to |. of an hour. Res. 8h. SOmin. 

27 Add ~ of a week, \ of a day, and \ of an hour 
together. Res. 2 days, 14 hours, 30min. 

SUBTRACTION OF VULGAR FRACTIONS. 

KT7LE. 

Prepare the given fractions as in Addition; then sub- 
tract the less from the greater, and place the difference 
over the common denominator. 

EXAMPLES. 

1 Take | from J. Rem. ^. 

2 Take T ^ from f^. Rem. {. 

3 Take T \ from T V Rem. |. 

4 Take f from f . Rem. s y 

35 



5x3=15 
7X^=14 






t 

3T 



146 VULGAR FRACTIONS. 

I tf tf ft 

V . Tf 3 V 

T T2" 2 IT* 3" 5 

12 com. cknom 60 com. denom. 




24 "t TOO 

From J of a pound take ~ of a shilling. 

15 com denom. 
s. d. 

| of a pound =15 6| [ 5x2=10 
!\ of a shilling = 3| | 3x3=9 



s. 15 3 T r j Ans. jj 
Fron* | of a take | of a shilling. 

Res. 14s. 3d. 
From I of a Ib. troy, take 1 of an ounce. 

Res. 8oz. IGdwt. 16grs. 

From 1 of a yard take I of an inch. Res. 5in. 4. 

From f of a j take S of | of a shilling. 

Res. 10s. 7d. Iqr. * 



MULTIPLICATION OF VULGAR FRACTIONS. 

RULE. 

Prepare the given fractions, if necessary; then mill 
tiply the numerators together for a new numerator, and 
the denominators together for a new denominator. 

EXAMPLES. 

1 Multiply f by T V Res. T 1 T 

12 1 

TT7 TT 



VULGAR FRACTIONS. 147 

2 Multiply T V by J. Res. ,Y 

3 Multiply f by V- Res - 2 Jf> or 2 T- 

4 Multiply 12J by 7f . Res. 96|. 

12| = V and 7f=Y; then V X V =' U 9=96 l- 

5 Multiply 71 by 8|. Res. 61|. 

6 Multiply 41 by 1. Res. T 8 T . 

7 Multiply |- "by 13 T V Res. 12i. 

8 Multiply I of | by T \ of ft. Res. y T . 

9 Multiply 4| by f of J. Res. 2f . 

10 Multiply \ of 7 by |. Res. If. 

11 Multiply 21 by 1*, and multiply the product by 
of | of f . Res. | 



DIVISION OF VULGAR FRACTIONS. 

RULE. 

Prepare the given fractions, if necessary, then invert 
the divisor, and proceed as in Multiplication. 

EXAMPLES. 

1 Divide J by 1. Res. ||. 

8X4=32 

7X9=63 

2 Divide 4 by f . Res. I 

3 Divide II by |. Res. Iff 

4 Divide H by 4 T 8 . Res. ^ 

5 Divide 3JJ- by 9. Res. 1 

6 Divide J by 4. Res. y \ 

7 Divide 4 by |. Res. 4| 

8 Divide 1 of f by f of J. Res. f 

9 Divide | of 19 by f of |. Res. 7| 

10 Divide 4| by | of 4. Res. 2-^ 

11 Divide | of 1 'by f of 7|. Res. T | T 

12 Divide 5205^ by J of 91. Res. 71| 



148 VULGAR FRACTIONS. 

PROPORTION IN VULGAR FRACTIONS. 

RULE. 

State the question, (as in page 89) reduce each term 
to its simplest form, invert the first term or terms, and 
proceed as in Multiplication of Vulgar Fractions 

EXERCISES. 

1 If J yd. cost $f ; what will | yd. cost? Ans. 50c 
yd. yd. D. 

3 . 3 . . 5 r PI-i/in 4 v 3 y 5 - " 'I 1 - ^A // 
i J j* - I1( :I1 3- A j A -g- Ta'o" *n? *Jv Ci9i 

2 If 1 bu. cost $1 ; what will f bu. cost? Ans. $2,80. 
bu. bu. D. 

1 : i :: 1. Then <X JX J-=y 1 ?=$2f J=$2,80. 

3 If A owned | of a toll-bridge, and sold f of his 
share for $681 ; what is the whole value? Ans. $1520. 

I of | : V : : 6 f 4 ' that is, -/ : 1 : : "f 4 . Then 



4 If I barter 5| cwt. of sugar at 6| cts. per lb., for 
indigo at $4 T 5 per lb. ; how much indigo must I re- 
ceive? Ans. 10 lb. 5 oz. 2j dr.-f 

D. cts. cwt. cts. cts. lb. 

4f- : 6| : : 5-; thatis 6 ^ : y : : 5 V 6 - Then 
16 4 9' i| T X V X 5 V 3 6 = YAW = 10 ^. 
5 oz. 2 dr. J^J. 

f* V 

5 If the cent roll weighs 6| oz., when wheat is 68| 

cents per bu. ; what is the cost of wheat per bu. when 
it weighs 4 oz? Ans. $1,03}. 

oz. oz. cts. 

41 : 61 : : 68|; then ^X V X a j s = 4 }|| = 103i. 

64 4 

2_5 2_5 2 ^T 5 

6 How many men will reap 417| acres in 121 dayb, 
if 5 men reap 521 i n 61 days? Ans. 20. 

a. a. men. 



1O1 . l 5 V 2 V 2 8 8 y 2 5 V 5 

1^1 . bj 2T X 25X J X TXy 

NOTE. Tn multiplying, omit the numbers that 420 *"i both the 
upper and lower series. 



DECIMAL FRACTIONS. 149 

DECIMAL FRACTIONS. 

A decimal fraction is a fraction whose denominator is 
1, w^th as many cyphers annexed as there are figures in 
I the numerator, and is usually expressed by writing the 
numerator only with a point prefixed to it: thus T \, T 7 ^, 
T 6 o 2 <ro> are decimal fractions, and are expressed by .5, 
.75, .625. 

A mixed number, consisting of a whole number and 
a decimal, as 25 T \, is written thus, 25.5. 

As in numeration of whole numbers the values of the 
figures increase in a tenfold proportion, from the right 
hand to the left; so in decimals, their values decrease in 
the same proportion, from the left hand to the right, 
which is exemplified in the following 

TABLE. 




Whole numbers. Decimals. 

NOTE. Cyphers annexed to Decimals, neither in- 
crease nor decrease their value; thus, .5, .50, .500, be- 
m g T 5 o> T S OO> T\Yo> are f tne same value: but cyphers 
prefixed to decimals, decrease them in a tenfold propor- 
I tion; thus .5, .05, .005; being T \, T f^, T \^, are of dif- 
ferent values. 



150 DECIMAL FRACTIONS. 

ADDITION OF DECIMALS. 

RULE. 

Place the given numbers according to their values, 
viz. units under units, tenths under tenths, &c., and add 
as in addition of whole numbers; observing to sot the 
point in the sum exactly under those of the given num- 
bers. 

EXAMPLES. 

.12 2.16 .14 .1 .15 

.134 3.45 .24 4.12 .75 

.21 40.02 .122 15.4 .92 

743 35.4 .36 76.36 63.25 

345 36.1 .141 120.16 25. 

.002 125.32 .567 425.04 4. 

1.554 242.45 

6 Add .5, .75, .125, 496, and .750 together. 

7 Add .15, 126.5, 650.17, 940.113, and 722.2560 
together. 

8 Add 420., 372.45, .270, 965.02, and 1.1756 to- 
gether. 



SUBTRACTION OF DECIMALS. 

RULE. 

Place the numbers as in addition, with the less under 
the greater, and subtract as in whole numbers; setting 
the point in the remainder under those in the given 
numbers. 

EXAMPLES. 

.4562 56.12 .4314 5672.1 32.456 

.316 1.242 .312 321.12 1.33 

.1402 54.878 

6 From 100.17 take 1.146. 

7 From 146.265 take 45.3278. 

8 From 4560. take .720. 



DECIMAL FRACTIONS. 151 

MULTIPLICATION OF DECIMALS. 

RULE. 

Multiply as in whole numbers, and from the right 
hand of the product, separate as many figures for deci- 
mals, as there are decimal figures in both the factors. 

EXAMPLES. 

1 Multiply .612 by 4,12 2 Multiply 1.007 by .041. 
.612 1.007 

4.12 .041 




2.52144 

3 Multiply 37.9 by 46.5 

4 36.5 by 7.27 

5 29.831 by .952 

6 3.92 by 196. 

7 .285 by .003 

8 4.001 by .004 

9 .00071 by .121 



1007 
4028 

.041287 

Product 1762.35 

265.355 

28.399112 

768.32 

_ .000855 

.016004 

.00008591 



DIVISION OF DECIMALS. 



RULE. 

Divide as in whole numbers, and from the right hand 
of the quotient, separate as many figures for decimals 
as the decimal figures of the dividend exceed those of 
the divisor. If there are not so many figures as the 
rule requires, supply the defect by prefixing cyphers. 



152 DECIMAL FRACTION*. 

EXAMPLES. 

1 Divide .863972 by .92 2 Divide 4.13 by 572.4, 
.92).863972(.9391 572.4)4.130000(.00721+ 
828 . 40068 

359 12320 

276 11448 

837 8720 

828 5724 

92 2996 

92 



3 Divide 19.25 by 38.5 Quotient .5 

4 234.70525 by 64,25 3.653 

5 1.0012 by .075 13.34+ 

6 .1606 by .44 .365 

7 .1606 by 4.4 .0365 

8 .1606 by 44. .00365 

9 9. by .9 10. 

10 .9 by 9. .1 

11 186.9 by 7.476 25. 

NOTE 1. When a whole number is to be divided by a 
greater whole number, cyphers must be affixed to the 
dividend, as decimal figures. 

12 Divide 3 by 4 Quotient .75 

13 275 by 3842 .071577+ 

14 210 by* 240 .875 

NOTE 2. When any whole number is divided by ano- 
ther, if there be a remainder, cyphers may be affixed to 
the dividend, and the quotient continued. 

15 Divide 382 by 25 Quotient 15.28 

16 13689 by 75 182.52 

17 315 by 124 2.5403+ 



DECIMAL FRACTIONS. 153 

REDUCTION OF DECIMALS. 

CASE 1. 

To reduce a vulgar fraction to a decimal. 
RULE. 

Place cyphers to the right of the numerator, until you 
can divide it by the denominator, and continue to divide 
until there is no remainder left; or if it be a number 
which will never come out without a remainder, until it 
is carried out to a convenient number of decimal places. 

EXAMPLES. 

1 Reduce j to a decimal. 

5)40 



.8 Ans. 

2 Reduce | to a decimal. Ans. .875. 

3 Reduce JJ to a decimal. Ans. .70833.-(- 

4 Reduce T 3 T y* to a decimal. Ans. .1762.-f- 

5 Reduce ^f to a decimal. Ans. .4566.-J- 

CASE 2. 

To reduce any given sum or quantity to the decimal of 
any higher given denomination. 

RULE. 

Reduce the given sum or quantity to the lowest de- 
nomination mentioned in it. 

Reduce onet>f that denomination of which you wish 
to make it a decimal, to the same denomination with the 
given sum. 

Divide the given quantity so reduced by one of the 
denomination of which you wish to make it a decimal ; 
the quotient will be the decimal required. 



154 DECIMAL FRACTIONS. 

EXAMPLES. 

1 Reduce 3s. 6d. to the decimal of a pound. 
3a. 6d.= 42 240)42.000(.175 decimals. Ans, 
1. =240 240 



1800 
1680 

1200 
1200 

2 Reduce 2R. 4P. to the decimal of an acre. 

Answer, .525. 

3 Reduce 2 qr. 2 nails to the decimal of a yard. 

Ans. .625. 

4 Reduce 5 minutes to the decimal of an hour. 

Ans. .08333. 

5 Reduce 10 graias to the decimal of an ounce, 
apothecaries' weight. Ans. .02083.-)- 

6 Reduce 2 quarts 1 pint to the decimal of a hogs- 
head. Ans. .00992.+ 

CASE 3. 
To reduce a decimal fraction to its proper value. 

RULE. 

Multiply the given fraction continually by the denom- 
ination next lower than that of which it is a decimal, for 
the proper value. 

EXAMPLES. 

1 What is the value of .375 of a dollar? Ans. 37cts. 

.375 
100 

37.500 
10 

5.000 

2 What is the value of .1361 of a .? Ans. 2s. 8Jd v 



DECIMAL TRACTIONS. 155 

3 What is the value of .235 of a day? 

Ans. 5 hours, 38 min. 24 sec. 
I 4 What is the value of .42 of a gallon? 

Ans. 1 quart, 1.36 pt. 

5 What is the value of .253 of a shilling? Ans. 3.036d. 
G What is the value of .436 of a yard? 

Ans. 1 qr. 2.976 na. 
7 What is the value of .9 of an acre? 

Ans. 3R. 24P. 



PROPORTION IN DECIMALS. 

RULE. 

State the question as the rule of three, in whole num- 
bers, only ohserve, when you multiply and divide, to 
place the decimal points according to the rules of multi- 
plication and division of decimals. 

EXA3IPLES. 

1 If 4.21b of coffee cost 8s. 2.3J., what cost 639.2511).? 
Ib. Ib. s. d. s. d. 

4.2 : 639.25 : : 8 2.3 : 62 6 9.49 Ans. 

2 When 1.4 yard cost 13s. what, will 15 yards come 
to at the same price? Ans. 6 19s. 3d. 1.71 qr. 

3 If I sell 1 qr. of cloth for 2 dollars 34.5 cents, what 
is it per yard? Ans. $9 38 cts. 

4 A merchant sold 10.5 cwt. of sugar, for 108.30 dol- 
lars, for which he paid 84 dollars 39.12 cents; what did 
he gain per cwt. by the sale? Ans. $2 27 cts. 7m.-f- 

5 How many pieces of cloth, at .20.8 dollars per 
piece, are equal in value to 240 pieces, at 12.6 dollars 
per piece? Ans. 145.38-J- pieces. 

6 If, when the price of wheat is 74.6 cents per bush- 
el, the penny roll weighs 5.2 oz., what should it be per 
bushel when the penny roll weighs 3.5 oz.? 

Ans. $1 10 cts. 8m.+ 

QueMion. 

How do you perform operations in the rule of three 
in decimals? 



156 MENSURATION. 

COMPOUND PROPORTION, IN DECIMALS. 

Questions in this rule are wrought as in whole num- 
bers, placing the points agreeably to former directions. 

EXAMPLES. 

1 If 3 men receive 8.9j for 19.5 days labor, how 
much must 20 men have for 100.25 days? 

Ans. 305. Os. 8.2d. 



i 

If 



19.5: oO.25 days ' - Os ' 



2 If 2 persons receive 1.625s. for 1 day's labor, how 
much should 4 persons have for 10.5 days? 

Ans. 4. 17s. lid. 

3 If the interest of 76.5for 9.5 months, be 15.24. 
what sum will gain 6 in 12.75 months ? 

Ans. 22 8s. 9|d. 

4 How many men will reap 417.6 acres in 12 days, 
if 5 men reap 52.2 acres in 6 days? Ans. 20 men. 

5 If a cellar 22.5 feet long, 17.3 feet wide, and 10.25 
feet deep, be dug in 2.5 days, by 6 men, working 12.3 
hours a day, how many days of 8.2 hours, should 9 men 
take to dig another, measuring 45 feet long, 34.6 wide, 
and 12.3 deep? Ans. 12 (lays. 



MENSURATION. 

MENSURATION is employed in measuring masons' and 
carpenters' work, plastering, painting and paving; also, 
for measuring timber in all its forms, and for estimating 
quantity in length, superfices, and solids, whenever 
yards, feet, inches, &.C., are employed. 

The denominations are, foot, inch, second, third, and 
fourth, 

12 Fourths ,' one 1 Third'" 
12 Thirds one 1 Second" 
12 Seconds one 1 Inch. /. 
12 Inches one 1 Foot. Ft. 



MENSURATION. 157 

ADDITION. 

RULE. 

Proceed as in Compound Addition. 

EXAMPLES. 

Ft. I. " Ft. I. " Ft. I. " '" " 

25 G 3 7'2 4 G 17 9 2 3 11 

14 2 9 54 3 2 18 11 10 8 9 

35 11 10 14 8 22 11 5 4 9 

45 10 11 2G 32 14 10 11 10 8 

600 19 4 12 4 10 

490 14 00 10 2840 



132 4 9 

4 Four floors in a certain building contain each 1084 
feet, 9in. 8"j how many feet are there in ail? 

Ans. 4339ft. 2 in. 8". 

5 There are six mahogany boards, the first measures 
27 ft. 3in., the second 25 ft. llin., the third, 23ft. lOin., 
the fourth 20ft. 9in., the fifth 20ft. Gin., and the sixth 18 
feet 5 in.; how many feet do they contain? 

Ans. 13tift. Sin. 

SUBTRACTION. 

RULE. 

Proceed as in Compound Subtraction. 

EXAMPLES. 

Ft. I. " Ft. I. " Ft. I. " '" "" 
75 9 9 84 6 4 100 10 8 10 11 
14 6 11 72 9 8 97 2 4 6 8 



61 2 10 



4 If 19ft. lOin. be cut from a board which contains 
41ft. 7in. how much will be left? Ans. 21ft. 9in. 

5 Bought a raft of boards containing 59621ft. 8in., of 
which are since sold 3 parcels, each 14905ft. 5in.; how 
many feet remain? Ans. 14905ft. 5in 



14 



158 MENSURATION. 

MULTIPLICATION. 

CASE 1. 
When the feet of the multiplier do not exceed 12. 

RULE. 

Set the feet of the multiplier under the lowest denom- 
ination of the multiplicand, as in the following example ; 
then multiply as in Compound Multiplication, by each 
denomination of the multiplier separately, observing to 
place the right hand figure, or number, of each product, 
under that denomination of the multiplier by which it is 
produced. 

EXAMPLES. 
1 Multiply 10 feet G inches by 4 feet 6 inches. 

Product 47 feet, 3 in 
Ft. I. " 

10 6 A table 10 feet 6 inches long, 

4 6 and 1 foot wide, will make 10 feet 

6 inches, or 10& feet, square meas- 

530 are. 

42 And 4 feet 6 inches, or 4i feet 

wide, will make 4d times 10i, or 

47 3 47i feet, or 47 feet 3 inches. 



OR THUS: 
ft. in. 
10 6 
44 ft. 

5 3 

42 

47 3 



NOTE 1. If there are no feet in the multiplier, sup- 
ply their place with a cypher; and in like manner sup- 
ply the place of any other denomination between the 
highest and lowest. 



MENSURATION. 159 


10ft. Gin. 


or 101 feet long. 


1 


i i 






1 1 I 1 




1 


1 1 






1 1 1 1 


4fimoa 1 OA TYinlfA 4^fi 


! | 


1 1 






1 1 1 1 


tlllJtxb J.V/S illrtlVt/ TK^li* 

and itime 104 make 5i ft. 


1 ' 


1 1 






1 1 1 1 


Added, make 474. 


CD | 


1 1 




1 1 1 1 




1 


i i 


i i i i I 




Ft. 


7. " Ft. I. 


Ft. I. " " 


\ 2 Multiply 9 


7 by 3 6 


Res. 33 6 6 


3 


3 


11 bv 9 5 


36 10 7 


4 


8 


6 9 by 7 3 


8 62 6 7 9 


5 


28 


10 6 by 3 2 


4 92 2 10 6 














CASE 2. 


When the feet of the multiplier exceed 12. 


RULE. 


Multiply by the feet of the multiplier as in Compound 
Multiplication, and take parts for the inches, &,c. 


EXAMPLES. 


1 Multiply 


112ft. 3in. 5" 


by 42ft. 4in. 6" 










Ft. I. " 


' 










112 3 5 




6X7=42 










G73 8 6 






J 






7 






4 




I 


4715 11 6 '" "" 










37 5 1 


i 8 




6 






4 8 ] 


L 8 6 


4758 9 4 G 


Ft. 


I. 


" Ft. 


7. " Ft. I. " "/ 


2 Multiply 76 7 by 19 


10 Res. 1518 10 10 


3 


127 6 by 184 


8 23545 


4 


71 


2 6 by 81 


1 8 5777 922 



160 MENSURATION 

APPLICATION. 

1 A certain board is 28ft. lOin. 6" long, and 3ft. 2 in 
4" wide; how many square feet does it contain? 

Ans. 92ft. Sin. 10" 6'-. 

2 If a board be 23ft. 3in. long, and 3ft. Gin. wide, 
how many square feet does it contain? 

Ans. 81ft. 4in. 6 

3 A certain partition is 82ft. Gin. by 13ft. 3in.; how 
many square feet does it contain? Ans. 1093ft. lin. G". 

4 If a floor be 79ft. Sin. by 38ft. llin., how many 
square feet are therein? Ans. 3100ft. 4in. 4". 

NOTE. Divide the square feet by 9, and the quotient 
will be square yards. 

5 If a ceiling be 59ft. 9in. long, and 24ft. Gin. broad, 
how many square yards does it contain? 

Ans. lG2yd. 5ft.-4- 
Ft. I. 

6 in. * 59 9 

3 

179 3 

8 

1434 
29 10 6 

9)14G3 10 6 
162yd. 5ft. 

6 How many yards are contained in a pavement 50 
feet 9 inches long, and 18 feet 4 inches wide? 

Ans. 115yd. 5ft 5in. 

7 How many yards in a ceiling 92ft. 4in. long, 22ft. 
8in. wide? Ans. 232yd. 4ft. 10in.+ 

8 How many squares in a floor 37ft. Gin. long, and 
21ft. 9in. wide? Ans. 8 squares, 15 feet.-f- 

A square is 10 feet long and 10 feet wide, or 100 
square feet. It is used in estimating flooring, roofing, 
weather-boarding, &-c. 

9 How many squares of weather-boarding on the 
side of a house 43 feet 6 in. long, and 18ft. Sin. high? 

Ans. 8 squares, 12ft. 



MENSURATION. 



161 



10 How many squares in a roof 36ft. 4in. long, 15ft. 
9in. wide? Ans. 5 sq. 72ft.-f 

NOTE 2. To measure a triangle. Multiply the base 
by one half the perpendicular height, and the product 
will be its superficial content. 

11 Let C, H, and G, represent a triangle, whose 
base is 40 feet, and perpendicular height 28 feet; how 
many feet does it contain ? Ans. 560 feet. 




40 feet 



40 feet 



feet. 
40 
14 half the perpendicular 

160 
40 

560 

12 How manv square feet in a triangle 80 feet long 
and 36 feet high? Ans. 1440ft. 

13 In a triangular pavement 46 feet long, and 24 feet 
at the place of its greatest width, how many yards ; and 
how many bricks, allowing 41 to every square yard? 

Ans. 61yd. 3ft., and 2514 bricks 

14 In the gable ends of a house, which is 63 feet long 
and 22 feet high, from the "square of the building" to 
the top, how many squares? Ans. 6sq. 93in 

NOTE 3, To find the circumference of a circle, when 
the diameter is given: Say, 

As 7 are to 22, so is the diameter to the circumfer- 
ence ; or the contrary, 

As 22 are to 7, so is the circumference to the diam- 
eter. 



U* 




162 MENSURATION. 

The diameter of a circle is 14 
feet; what is the circumference? 

Ans. 44. 

As 7 : 22 : : 14 : 44 
The circumference of a circle 
is 44ft.; what is the diameter? 

Ans. 14. 



NOTE 4. To find the superficial contents of a circle. 
Multiply half the circumference by half the diameter 

15 How many square feet in a circle whose diameter 
is 14 feet, and circumference 44? Ans 154 ft. 

half circumference, 22 

half diameter, x 7 

154 feet 

16 How many square feet in a circle whose circum- 
ference is 16 feet? Ans. 20 sq. ft. 

halfcir. 8 

As 22 : 7:: 16 : 5 halfdiam.2i 

20 

17 How many square feet in a circle whose diameter 
is 21 feet? Ans. 346* ft. 

NOTE 5. To find the superficial contents of a globe. 
Multiply the circumference by the diameter. 

18 What are the superficial contents of a globe whose 
diameter is 70 feet, and circumference 220 feet? 

Ans. 15400 sq. ft. 

19 How many square feet of cloth would be re- 
quired to cover a globe, whose diameter is 28 feet, and 
circumference 88? Ans. 2464 ft. 

20 How many yards of canvass would be required to 
make a balloon of a globular form, 20yardsin diameter? 

Ans. 1257 sq. yds. 

NOTE 6. To find the solid contents of a cube,* or of 
a square stick of timber, or a .pile of wood, fyc. 
Multiply the length by the breadth, and that product by 
the thickness. 



* A cube is a solid body, contained by 6 equal sides, ail of which are 
exact squares. 



3UENSUKATION 



163 



21 What are the solid contents of a cube whose di- 
ameter is 4 feet? Ans. 64 feet. 

4 feet 
4 

16 

4 

64 

22 What is the solid contents of a stick of timber 2 
feet thick, 3 feet wide, and 36 feet long? 

Ans. 216 solid feet. 
36 feet 
3 

108 
2 

216 

23 How many solid feet in a block of marble 3 feet 
thick, 7 feet wide, and 13 feet long? Ans. 273 sol. ft. 

24 In a cube whose diameter is 7 feet, how many solid 
feet? Ans. 343 feet. 

25 How many solid feet in a pile of wood 28 feet 
long, 8 feet wide, and 10 feet high; and how many 
cords does it contain*? Ans. 2240 feet; 17 cords 64 ft. 

or, 17 i cords. 

26 In a cellar 36 feet long, 27 feet wide, and 4i feet 
deep, how 'many solid yards? Ans. 162 yards. 

27 How many perches* of stone in a wall 42 feet 
long, 84 feet high, and 2 feet thick? Ans. 28,8 per. 

feet 
24,75)714.00(28,8 

28 In a 12 inch brick wall, 52 feet long and 36 feet 
high, how many bricks, allowing 21 to every square 
foot of wall? Ans. 



* A perch is 16* feet long, 1 ft. wide, and 1 foot hig! 
solid feet. 



164 MENSURATION. 

29 In an 8 inch brick wall, 82 feet long and 16 feet 
high; how many bricks, allowing 14 bricks for every 
square foot of wall? Ans. 18368. 

30 In a 16 inch brick wall, 148 feet long and 42 feet 
high, how many bricks, allowing 28 bricks to the square 
foot? Ans. 174048. 

31 How many bricks in 3 walls, the first 68 feet long. 
18 feet 6 inches high, 16 inches thick; the second 72 ft. 
6in. long, 19ft. 4in. high, 12in. thick; the third 43ft. 4in. 
long, 12ft. Sin. high, 8in. thick? Ans. 72343.+ 

NOTE 7. To find the solid contents of a cylinder.* 
Find the contents of one end by Note 4, and multiply 
that product by the length. 

32 What are the solid contents of a cylinder whose 
diameter is 14 feet, arid length 16 feet? Ans. 2464ft. 

half circumference 22 
half diameter 7 



2464 feet 

33 What are the contents of u circular well, 7 feet in 
diameter, and 62 feet deep? Ans. 2387 ft. 

34 What are the solid contents of a tub whose diam- 
eter is 6 feet and height 7 feet? Ans. 198 ft. 

35 How many feet in a circular well, 10 feet diame- 
ter and 20 feet deep ? Ans. 157y feet. 

NOTE 8. To find the solid contents of the f rust rum 
of a cone, To the sum of the squares of the two diame- 
ters, in inches, add their product ; multiply this sum by 
one-third of the depth, and this last product by the deci- 
mal .754. Eu. El. 

The result will be the contents in cubic inches, which 

***' s desired. 



* A c /Under is a long round body, whose diameter is every 
where t k . e same,. 



MENSURATION. 165 

The liquid gallon of Ohio contains 231 cubic inches. 
The dry gallon contains 268i cubic inches. 
The bushel, of grain, contains 2150| cubic inches. 
The bushel, of coal, lime, &c. 2688 cubic inches. 

EXAMPLES. 

1 In a circular vessel, whose greater diameter is 80 
inches, the less 71, and the depth 34, what is the contents 
in liquid gallons ; and also in bushels of grain ? 

A C 659.72-1- gallons. 
3 * I 70.86 bushels. 

80X80=6400= square of 80. 
71X71 = 5041= square of 71. 

11441 
80X71=5680= product of 80 and 71. 

17121 

in depth = lU=j of the depth. 



194038X. 7854=152397.4452 inches. 
Divide by 231 for gallons, and 2150| for bushels. 

2 The greater diameter of a tub is 38 inches, the less 
20.2, and the depth 21 ; what is the content in gallons ? 

Ans. 62. 34+ gallons. 

3 The top diameter of a tube is 22 inches, the bot- 
tom 40, and the height 60 ; what is its contents in gal- 
lons, also in bushels of grain ? . C 20 1.55-1- gals. 

*'l 21.64+bu. 

4 How many barrels, of 32 gallons each, in a cistern, 
whose greater diameter is 8 feet 6 inches, the less 8 feet, 
and depth 7 feet 9 inches ? Ans. 96 barrels 28-j-gals. 

5 How many bushels of grain in i bin that is 8 feet 
long, 4 feet wide, and 6 feet high ? Ans. 123-{-bu. 

6 How many bushels of coal in a" boat 60 feet long, 
16 feet wide, and 4% feet deep ? Ans 2777+bushels. 



1 66 MENSURATION. 

NOTE 9. To find the solid contents of a round stick 
of timber of a taper from one end to the other. Find I 
the circumference a little nearer the larger than the 
smaller end; from this, by Note 3, find the diameter: 
multiply half the diameter by half the circumference, 
and the product by the length.* 

EXAMPLES. 

1 What are the solid contents of a round stick of tim- 
ber 10 feet long, and 2.61 feet circumference? 

Ans, 5.4 feet.+ 
As 22 : 7 : : 2.61 : .83 diameter 

1,305 half circumference 
.415 half diameter 

6525 
1305 
5220 

.541575 

10 length 



5.415750 

2 How many solid feet in a log 40 feet long, which girts 
66 inches? Ans. 96.25 ft. 

As 22 : 7 : : 66 : 21 in diameter 

33 xlOiX40= 13860. 144)13860(96.25 

NOTE 10. To find the solid contents of a globe. 
Multiply the cube of the diameter by .5236. 

EXAMPLES. 

1 What are the solid contents of a globe whose diame- 
ter is 14 inches? Ans. 1436.75in.+ 

14 X 14 X 14=2744. 2744 X .5236= 1436.7584. 

2 What are the contents of a balloon of a globular 
form, 42 feet in diameter? Ans. 38792.4 ft.-{- 

3 How many solid miles are contained in the earth, or 
globe, which we inhabit? 

* This method, thoitgli not quite accurate, is sufficiently near the 
truth for the purpose of measuring timber. 



INVOLUTION. 167 

Suppose the diameter to be 7954 mhes: then, 7954 X 
7954x7954=503218686664 the cube of the earth's 
axis, or diameter; then, 

50321 8686664 X .5236=2634853(34337 

cubick miles. Ans. 

NOTE. The solidity of a globe may be found by the 
circumference, thus Multiply the cube of the circum- 
ference by .016887 the product will be the contents. 



INVOLUTION, OR THE RAISING OF 
POWERS. 

The product arising from any number multiplied by 
itself, any number of times, is called its power, as fol- 
lows t 

2x2= 4 the square, or 2d power of 2. 
2x2x2= 8 3d power or cube of 2. 
2X2X2X2=16 4th power of 2. 
The number which denotes a power is called its index. 

NOTE. When any power of a vulgar fraction is re- 
quired, first raise the numerator to the required "power, 
and then the denominator to the required power, and 
place the numerator over the denominator as before : 

thus, the 4th power of | 



Questions. 

What is the product, arising from the multiplication of 
any figure by itself a given number of times, called? 

What is the number which denotes a power, called ? 

How do you proceed to find any required power of a 
vulgar fraction? 



168 


INVOLUTION. 


i 


Table of 


the first nine Powers. \ 







8s 







] 


| 


6 


II 


1 


1 


e, 


i 
i 


1 


1 


1 


1 1 


1 


1 


1 


1 


1 


i 


2 4 


8 


16 


32 


64 


128 


256 


512 


3 9 

416 
525 
63fi 

749 
864 
981 


27 
64 
125 
216 
343 
512 
729 


81 
256 
625 
1296 
2401 
4096 
6561 


243 
1024 
3125 
77-76 
16807 
32768 
59049 


729 
4096 
15625 
46656 
117649 
262144 
531441 


2187 
16384 
78125 
279936 
823543 
2097152 
4782969 


6561 
65536 
390625 
1679616 

5764801 
16777216 
43046721 


19683 - 
262144 
1953125 
10077696 
40353607 
134217728 
387420489 


EXAMPLES. 

1 What is the square of 32? 
32 


64 
96 


1024 Ans 
2 What is the cube of 14? 
14 
14 


Ans. 2744. 












56 














196 
14 














784 
196 




2744 
3 What is the sixth power of 2.8? Ans 481.890304 
4 What is the third power of .263? Ans. .018191447. 



TUB SQUARE ROOT. 169 

EVOLUTION, OR THE EXTRACTING OF 
ROOTS. 

The root of a number, or power, is such a number, as 
being multiplied into itself a certain number of times, 
will produce that power, Thus 2 is the square root of 
4, because 2x2=4; and 4 is the cube root of 64, be- 
cause 4x4X4=64, and soon. 



THE SQUARE ROOT. 

The square of a number is the product arising from 
that number multiplied into itself. 

Extraction of the square root is the finding of such a 
number, as being multiplied by itself, will produce the 
number proposed. Or, it is finding the length of one 
side of a square. 

RULE. 

1 Separate the given number into periods of two fig- 
ures, each, beginning at the units place. 

2 Find the greatest square contained in the left hand 
period, and set its root on the right of the given number: 
subtract said square from the left hand period, and to the 
remainder bring down the next period for a dividual. 

3 Double the root for a divisor, and try how often this 
divisor (with the figure used in the trial thereto annexed) 
is contained in the dividual: set the number of times in 
the root; then, multiply and subtract as in division, and 
bring down the next period to the remainder for a new 
dividual. 

4 Double the ascertained root for a new divisor, and 
proceed as before, till all the periods are brought down. 

NOTE. If, when all the periods are brought down, there be a re- 
mainder, annex cyphers to the given number, for decimals, and pro- 
ceed till the root is obtained with a sufficient degree of exactness. 

Observe that the decimal periods are to be pointed off from the de- 
cimal point toward the right hand : and that there must be as many 
whole number figures in the root, as there are periods of whole num- 
bers, and as many decimal figures as there are periods of decimals. 



15 



170 THE SQUARE ROOT. 

PROOF. 

Square the root, adding in the remainder, (if any,) 
and the result will equal the given number. 
EXAMPLES. 

1 What is the square root of 5499025? 

5,49,90,25(2345 Ans. 

4 2345 

2345 

43)149 

129 11725 

9380 

464)2090 7035 

1856 4690 

4685)23425 5499025 Proof. 

23425 

2 What is the square root of 106929? Ans. 327. 

3 What is the square root of 451584? Ans. 672. 

4 What is the square root of 36372961? Ans. 6031. 

5 What is the square root of 7596796? 

Ans. 2756.2S+ 

6 What is the square root of 3271.4007? 

Ans. 57.19+ 

7 What is the square root of 4.372594? 

Ans. 2.091+ 

8 What is the square root of 10.4976? Ans. 3.24 

9 What is the square root of .00032754? 

Ans. .01809+ 

10 What is the square root of 10? Ans. 3.16224 

To extract the Square Root of a Vulgar Fraction. 

RULE. 

Reduce the fraction to its lowest terms, then extract 
the square root of the numerator for a new numerator, 
and the square root of the denominator for a new deno- 
minator. 

NOTE. If the fraction be a surd, that k, one whose 
root can never be exactly found, reduce it tD a decimal, 
and extract the root therefrom. 



THE SQUARE ROOT. 171 

EXAMPLES. 

1 What is the square root of |54f J ^ ns - 1- 

2 What is the square root of f 14-j-? Ans. J. 

3 What is the square root of flf ? Ans. .93309-[- 

To extract the Square Root of a Mixed Number. 

RULE. 

Reduce the mixed number to an improper fraction, 
and procee'd as in the foregoing examples : or, 

Reduce the fractional part to a decimal, annex it to 
the whole number, and extract the square root there- 
from. 

EXAMPLES. 

1 What is the square root of 37J|? Ans. 61. 

2 What is the square root of 27^? Ans. 51. 

3 What is the square root of Soli? Ans. 9.27-|- 

APPLICATION. 

1 The square of a certain number is 105G25: what 
is that number? Ans. 325. 

2 A certain square pavement contains 20736 square 
stones, all of the same size; what number is contained 
in one of its sides? Ans. 144. 

3 If 484 trees be planted at an equal distance from 
each ether, so as to form a square orchard, how many 
will be in a row each way? Ans. 22. 

4 A certain number of men gave 30s Id. for a chari- 
table purpose; each man gave as many pence as there 
were men: how many men were there) Ans. 19. 

5 The wall of a certain fortress is 17 feet high, 
which is surrounded by a ditch 20 feet in breadth; how 
long must a ladder be to reach from the outside of the 
ditch to the top of the wall? Ans. 26.24-J-feet. 

NOTE. The square of the 
longest side of a right angled 
triangle is equal to the sum of 
the squares of the othef two 
sides ; and consequently, the 
difference of the square of the 




longest, and either of the other, Ditch, 

is the square of the remaining one. 



172 THE SQUARE ROOT. 

6 A certain castle which is 45 yards high, is surroun- 
ded by a ditch 60 yards broad ; what length must a ladder 
be to reach from the outside of the Hitch to the top of the 
castle ? Ans. 75 yards. 

7 A line 27 yards long, will exactly reach from the 
top of a fort to the opposite bank of a river, which is 
known to be 23 yards broad ; what is the height of the 
fort? Ans. 14.1.42-|-yards. 

8 Suppose a ladder 40 feet long be so planted as to 
reach a window 33 feet from the ground, on one side of 
the street, and without moving it at the foot, will reach a 
window on the other side 21 feet high; what is the breadth 
of the street? Ans. 56.64+feet. 

9 Two ships depart from the same port; one of them 
sails due west 50 leagues, the other due south 84 leagues; 
how far are thev asunder?? 

Ans. 97.75+ Or, 97|+leagues. 



Questions. 

What is a square? A square- is a surface whose 
length and breadth are equal, and whose angles' (or cor- 
ners) ore right angles, (or square.) 

What is its square root? The square root is the 
length of the side of a square. 

If the square be sixteen, what is the root? 

Why is the root four? 

If the root be three, what is the SQUARE? 

What is the square root of twenty-five? 

What is the square of five ? 

What is the square root of thirty-six ? 

What is the square of six? 

How do you point off a number whose square root is 
to be extracted? 

What is the next step? What do you subtract from 
the period? What do you annex* to the remainder? 



THE SQUARE ROOT. 



173 



Illustration of the Rule for extracting the Square Root. 

The reason for pointing off the given number into 
periods of two figures each, is, that the product of any 
whole number contains just as many figures as are in 
both the multiplier and the multiplicand, or but one less ; 
consequently, the square contains just double as many 
figures as the root, or one less. 

A E B 

Suppose the figure ABCD 
contains 1849 square feet, 
and that the number consists 
of two periods; then there 
must be two figures in the 



D 



120 



11 



1600 



9 



120 



root. 

The largest root whose 
square can be taken out of 
the left hand period, is 4, (or 
as it will stand in ten's place 
in the root, it is 40,) and the 
square of this is 16 (of 1600.) 
This taken from the whole 
C square ABCD, or 1849, 
leaves 249. 



18,49(43 
16 

83)249 
249 



GIRD 
AEIIG 
HFCI 
EBFH 

ABCD 



Now double GH or HI, 
which is 40, for a divisor, 
omitting the cypher to leave 
place for the next quotient 
figure, to complete the divi- 
sor. 

80 into 249 are contained 
3 times ; this 3 is the width 
of the oblong ALHG, or 
HFCI. But the square is 
imperfect without EBFH; 
then annex the three to the 
divisor. Now multiply this 
perfect divisor by the last 
figure of the root, to get the 
= 1849 Quantity m tne two oblong 
figures, and the small square 
which comprises the great 
square ABCD. 



15* 



174 THE CUBE ROOT. 

How do you find the divisor? 
Why do you place the new quotient figure in the units 
place of the divisor? 

How do you prove the square root? 



THE CUBE ROOT. 

The cube of a number is the product of that number 
multiplied into its square ? 

Extraction of the cube root is finding such a number 
as, being multiplied into its square, will produce the 
number whose cube root is extracted. 



RULE. 

Separate the given number into periods of three fig 
ures each, beginning at the units place. Find the great- 
est cube in the left hand period, and set its root in the 
quotient; subtract said cube from the period, and to the 
remainder bring down the next period for a dividual. 

Square the root, and multiply the square by three 
hundred for a divisor. 

See how often the divisor is contained in the dividual, 
and place the result in the quotient. 

Multiply the divisor by the last found quotient figure ; 
square the last found figure multiply the square by the 
preceding figure or figures of the quotient, and this pro- 
duct by thirty; and cube the last figure. Add these 
three products together, and subtract their amount from 
the dividual. 

To the remainder add the next period, and proceed as 
before, until the periods are all brought down. 

When a remainder occurs, annex periods of cyphers 
to obtain decimals, which may be carried to any conve- 
nient number. 

NOTE 1. The cube root of a vulgar fraction is found 
by reducing it to its lowest terms, and extracting the root 
rf the numerator for a numerator, and of the denomica- 



THE CUBE ROOT. 175 

tor for a denominator. If it be a surd,* extract the root 
of its equivalent decimal. 

EXAMPLES. 

1 What is the cube root of 99252847? 

99,252,847(463 Ans. 463. 

4X4X4=64 



4X4X300=4880 

Div. 4800X6= 
6X6X4X30= 



35252 463 
463 

28800 

4320 1389 



0X6X6= 216 2778 

1852 

Subtrahend 33336 

214369 

46X46X300=634800! 1916847 463 



Div. 634800x3=1904400 643107 

3X3X46X30= 12420 ' 1286214 

3X3X3= 27 857476 



Subtrahend 1916847 Proof 99252847 

2 What is the cube root of 84604519? -Ans. 439. 

3 What is the cube root of 259694072? Ans. 638. 

4 What is the cube root of 32461759? Ans. 319. 

5 What is the cube root of 5735339? Ans. 179. 

6 What is the cube root of 48228544? Ans. 364. 

7 What is the cube root of 673373097125? Ans. 87C5. 

8 What is the cube root of 7532641? Ans. 196.02-f- 

9 What is the cube root of 5382674. Ans. 175.2-j- 

10 What is the cube root of 15926.972504? 

Ans. 25.16+ 

When decimals occur, point the periods both ways, beginning at the 
decimal point, and if the last period of the decimal be not complete, 
add one or more cyphers. 

A mixed number may be reduced to an improper 
fraction, or a decimal, and the root thereof extracted. 



* A surd is a quantity whose root cannot exactly be formed, 
quantity whose root can be found, is called a rational quantity. 



176 THE CUBE ROOT. 

1 What is the cube root of ^VV ? Ans. 



. 

2 What is the cube root of |J? Ans 5-. 

3 What is the cube root of ? Ans . 



4 What is the cube root of 12UJ Ans. 2J. 

5 What is the cube root of Sl^s ? Ans. 31. 

SURDS. 

6 What is the cube root of 7^? Ans. 1.93-(- 

7 What is the cube root of 91? Ans. 2.092+ 

APrLICATION. 

1 The cube of a certain number is 103823; what is 
that number? Ans. 47 

2 The cube of a certain number is 1728 j what num- 
ber is it? Ans. 12. 

4 There is a cistern or vat of a cubical form, which 
contains 1331 cubical feet: what are the length, breadth 
and depth of it? Ans. each 11 feet. 

4 A certain stone of a cubical form contains 474552 
solid inches j what is the superficial content of one of its 
sides? * Ans. 6084 inches. 



Questions. 

What is a cube? A cube is a solid body contained by 
six equal square sides. 

What is the cube root? It is the length of one side 
of a cube. 

What is the square of the cube root? It is the su- 
perficial contents of one side of a cube. 

How do you point off a number whose cube root is tc 
be extracted? 

What is the first figure of the root? It is the root of 
the greatest cube in the first period. 

When you subtract the cube from the first period, 
what do you do? 

How do you find the divisor? 

What is the first step towards finding the subtrahend ? 
What is the second? What is the third? 

When a remainder occurs, how do you proceed? 

How do you prove the cube root? 



THE CUBE ROOT. 



177 



Illustration of the Rule for extracting the Cube Root. 

The reason for pointing off the number into periods 
of three figures eacli, is similar to the one given in the 
Square Root; for the number of figures in any cube will 
never exceed three times the figures in the root, and 
will never be more than two figures less. 

OPERATION. 

15,625 I 25 

8 



2X2X2= 
2X2X300=1200 



5X5X2X30= 
5X5X5= 



7625 

6000 

1500 

125 

7625 



Fig. 1. 



In Ois number there are 
two pel ->ods : of course there 
will be two figures in the 
root. 

"The greatest cube in the 
left hand period (15) is 8, 
the root of which is 2;" 
therefore, % is the first figure 
of the root, and as we shall 
have another figure in the 
root, the 2 stands for 2 tens, 
or 20. But the cube root is 
the length of one of the sides 
of the cube, whose length, 
breadth and thickness are 
equal : then the cube whose 
root is 20, contahw 20X20 
X 20=8000. 

"Subtract the cube thus 
found (8) from said period, 
and to the remainder bring 
down the next period," or, 
subtract the 8000 from the 
whole given number (15625) 
and 7625 will remain. Thus 
8000 feet are disposed of in 
the cube, Fit 1. 20ft .ong, 
80 ft wide, and 20 ft. nigh. 

The cube is to be enlarged 
by the addition of 7625 feet 
which remain. In doing 

this, the figure must be enlarged on three sides, to make it longer^ 

and wider, and higher, to maintain the complete cubic form. 

The next step is, to find a divisor; and this must oe the number of 

square feet contained in the three sides to which the addition must be 

made. 

Hence we ^multiply the square of the quotient Jigure by 300."* 

That is, 2 X 2 X 300=1200 : or 20 X -0 X 3= 1 200 feet, which is the 

superficial content of the three sides, A, B, and C. 



H 2 



178 



THE CUBE ROOT. 



Fig. 2. 




Fig. 4. 




Proof. 

20X20X20= 
20X20X3X5= 
5X5X20X3= 
5X5X5= 



8000 

6000 

1500 

125 



25X25X25= 15625 



This "divisor (1200) is 
contained in, ike dividual" 
(7625) 5 times : then 5 is the 
second quotient figure ; that 
is, the addition to each of the 
three sides is 5 feet thick ; if 
1200 feet cover the three 
sides one foot thick, 5 feet 
thick will require 5 times as 
many; that is 1200X5= 
6000. 

But when the additions 
are made to the three squares 
there will be a deficienc)- 
along the whole length ol 
the sides of the squares be- 
tween the additions, which 
must be supplied before the 
cube will be complete. These 
deficiencies will be three, as 
may be seen at NNN in 
Fig. 2, therefore it is that 
we "multiply the square of 
the last figure by the prece- 
ding figure, and by 30," 
(that is, 5X5X^X30,) or 
5 X 5 X 20 X 3=1500vvhich 
is the quantity required to 
supply the three deficiencies. 

Figure 3, represents the 
solid with these deficiencies 
supplied, and discovers an 
other deficiency, where they 
approach each other at ooo. 

Lastly, "cube the last fig- 
ure;' 1 '' this is done to fill the 
deficiency left at the comer, 
in filling up the other defi- 
ciencies. This corner is 
limited by the three portions 
applied to fill the former va . 
cancies, which were 5 feet in 
breadth ; consequently the 
cube of 5 will be the solid 
contents of the corner. Fig. 
4 represents this deficiency 
(eee) supplied, and the cube 
complete. 



ROOTS OP ALL POWERS. 179 

The illustration is much better made by means of 8 blocks 'of the 
ollowing description: One cube of about 3 inches diameter; three 
lieces each 3 inches square, inch thick; three pieces each & inch 
square, 3 inches long ; and one cube % inch. A set of these should 
>elong to the apparatus of every Professional Teacher. 

A GENERAL RULE FOR EXTRACTING THE 
ROOTS OF ALL POWERS. 

1 Point the given number into periods, agreeably to 
the required root. 

2 Find the first figure of the root by the table of pow- 
ers, or by trial ; subtract its power from the left hand 
seriod, and to the remainder bring down the first figure 
in the next period for a dividend. 

3 Involve the root to the next inferior power to that 
which is given, and multiply it by the nuhiber denoting 
the given power, for a divisor; by which find a second 
figure of the root. 

4 Involve the whole ascertained root to the given 
power, and subtract it from the first and second periods 
Bring down the first figure of the next period to the re- 
mainder, for a new dividend; to which, find a new divi- 
sor, as before; and so proceed. 

Note. The roots of the 4th, 6th, 8th, 9th, and 12th 
powers, may be obtained more readily thus : 

For the 4th root take the square root of the square 
root. 

For the 6th, take the square root of the cube root. 

For the 8th, take the square root of the 4th root. 

For the 9th, take the cube root of the cube root. 

For the 12th, take the cube root of the 4lh root. 

EXAMPLES. 

1 What is the 5th root of 916132832? 
9161,32832(62 Ans. 
7776 6X6X6X6X6=7776 

6X6X6X6X5=6480 div 

6480)13853 



916132832 62x62x62x62x62=916132832 
916132832 



180 ARITHMETICAL PROGRESSION. 

2 What is the fourth root of 140283207936? Ans. 612. 

3 What is the sixth root of 782757789696 ? Ans. 96. 

4 What is the seventh root of 194754273881 ? Ans. 41. 

5 What is the ninth root of 1352605460594688 ? 

Ans. 48. 



ARITHMETICAL PROGRESSION. 

A SERIES of numbers, increasing or decreasing by a 
common difference, is called an Arithmetical Progres- 
sion. 

Thus 3, 5, 7, 9, 11, 13, 15, &c., is an ascending se- 
ries, whose common difference is 2. 

And 16, 13, 10, 7, 4, 1, is a descending series, whose 
common difference is 3. 

The three most important properties of an arithmeti- 
cal series are the following : 

I. The sum of the two extremes is equal to twice the 
mean, or to the sum of any two terms equidistant from 
the mean. 

In the above series 3 -{-15= twice 9, which is the 
mean or middle term; and 5-f-13 which are equidistant 
from the mean. 

II. The difference of the extremes, is equal to the 
common difference multiplied by the number of terms, 
less one. 

In the above, the number of terms 7 1=6; then 
the common difference 2X6 = 12, which is equal to 
15 3. Or the number of terms in the other series 
61=5 ; then 5X3 = 15, which is equal to 161. 

III. The sum of all the terms is equal to the product 
of the mean, or of half the sum of the extremes, multi- 
plied by the number of terms. 

As above, the mean is 9 ; which, multiplied by 7, the 
number of terms, gives 63= 15-j-13-|- 11 4-9-f 7+5-1-3. 

Or, 15+3 = 18; half of which is 9. Then, 9X7 
=63, as before. 



ARITHMETICAL PROGRESSION. 181 

And 16-fl = 17; half of which is 8. This multi- 
tiplied by 6, the number of terms, =51 = 16-J-13-{-10 

+7+4+1. 

NOTE 1. To find the last term, multiply the common 
difference by the number of terms, less one, and add the 
product to the first term in an increasing series ; or, sub- 
tract the product from the first term in a decreasing 
series. 

EXAMPLES. 

1 If the first term is 3, the common difference 2, and 
the number of terms 7, what is the last term ? 

Ans. 15. 
7 1=6 ; the 2X6+3=15, the last term. 

2 The first term being 16, the common difference 3, 
and the number of terms 6, what is the last term ? 

Ans. 1. 

61=5; then 3X5 = 15. Then 16 15=1, the last 
term. 

3 What is the last term in a series, whose first term 
is 5, the common difference 4, and the number of terms 
25? Ans. 10K 

4 Suppose, in the above, the first term is 3 ; what is 
the last term ? Ans. 99. 

5 A man bought 50 yards of calico at 6 cents for the 
first yard, 9 for the second, 12 for the third, &c.; what 
did he pay for the last? Ans. SI. 53. 

NOTE 2. To find the mean term, take half the sum 
of the extremes. 

EXAMPLES. 

1 The first term is 3, and the last 15 ; what is the arith- 
metical mean ? Ans. 9. 

3+15=18 ; the 18-^-2=9, the mean term. 

2 The weight of 5 packages of goods is, severally, 
180, 150, 120, 90, 60 pounds ; what is the mean or 
average weight ? Ans. 1201bs. 

NOTE 3. To find the sum of all the terras, multiply 
the mean term by the number of terms ; or, the mean 



182 ARITHMETICAL PROGRESSION. 

by the sum of the two extremes, and take half the pro- 
duct. 

EXAMPLES. 

1 The mean term is 11, and the number of terms 9; 
what is the sum of the series ? Ans. 99. 

2 The first term is 5, the last 32, and the number of 
terms 10 ; what is the sum of the series ? Ans. 185. 

3 How many strokes does the hammer of a common 
clock strike in 12 hours ? Ans. 78. 

4 What debt can be discharged in one year, by week- 
ly payments in arithmetical progression, the first being 
$12, and the last, or fifty-second, payment $1236 ? 

Ans. 32448. 

NOTE 4. To find the common difference, divide the 
difference of the extremes by the number of terms, less 
one. 

EXAMPLES. 

1 The ages of 8 boys form an arithmetical series 
the youngest is 4 years old and the oldest is 18 pwhat 
is the common difference ? Ans. 2. 

2 A debt can be discharged in one year, by weekly 
payments in arithmetical progression the first is $12, 
and the last $1236 ; what is the common difference ? 

Ans. $24. 

NOTE 5. To find the number of terms, divide the 
difference of the extremes by the common difference, and 
add 1 to the quotient. 

EXAMPLES. 

1 In a series, whose extremes are 4 and 1000, and the 
common difference 12, what is the number of terms ? 

Ans. 84. 

2 If a man, on a journey, travels 18 miles the fir^t 
day, increasing the distance 2 miles each day, and on the 
last day goes 48 miles, how many days did he travel ? 

/ Ans. 16. 



ARITHMETICAL PROGRESSION. 183 

PROMISCUOUS EXERCISES. 

1 24 persons bestowed charity to a beggar the first 
gave him 12 cents, the second 18, &c., in an aiithmeti- 
cal series ; what sum did he receive ? Ans. $19.44. 

2 Suppose 100 apples were placed in a right line, 2 
yards apart, and a basket 2 yards from the first ; how 
far would a boy travel to gather them up singly, and re- 
turn with each separately to the basket ? 

Ans. 20200 yards. 

3 In a drove of 400 hogs, 5 of the largest weighs, on 
an average, 280 pounds a-piece, and 5 of the smallest 
180 ; what is the mean Or average weight of them all, 
and what is the whole weight ? 

. C 2301bs. mean weight. 
S * I 92000 Ibs. whole wt. 

4 How many acres in a piece of land 80 rods wide 
at one end and 60 at the other, and 1 20 rods long ? 

Ans. 52. 

It may be observed, that the natural numbers 1, 2, 3, 
4, 5, 6, 7, &c., is an arithmetical series, whose first term 
is 1, and common difference 1 ; and that the last term 
is equal to the number of terms. 

From this series, we may form another by adding to 
each figure the sum of all the preceding, and we shall 
have 1, 3, 6, 10, 15,21,28, &c. 

These are called triangular numbers, because they 
may be represented by points, forming equilateral trian- 
gles, thus : 



Hence we perceive, that the sum of the natural numbers, 
to any degree, expresses the triangular number of the 
same degree. 

In the same manner, the square numbers 1, 4, 9, 16, 



184 GEOMETRICAL PROGRESSION. 

25, 36, &c., may be expressed by points, arranged in 
squares, thus : 



Natural numbers -1, 2, 3, 4, 5, 6, 7, 8, 9, &c. 
Triangular numbers 1, 3, 6, 10, 15, 21, 28, 36, &c. 
Square numbers 1, 4, 9, 16, 25, 36, 49, 64, 81, <fcc. 
Cube numbers 1, 8, 27, 64, 125, 216, 343, 512, 
729, &c. 



GEOMETRICAL PROGRESSION. 

A SERIES of numbers, increasing or decreasing by a 
common ratio, is called a Geometrical Progression. 

Thus 2, 4, 8, 16, 32, 64, 128, is an increasing series 
whose common ratio is 2 ; 

And 729, 243, 81, 27, 9, 3, is a decreasing series, 
whose common ratio is . 

The most important properties of a geometrical series 
are the following : 

I. The product of the extremes is equal to the square 
of the mean ; or, to the product of any two terms equi 
distant from the mean. 

In the above, 128X2 = 16X 16, or 32X8, &c. Also 
729X3=243X9, or 81X27, between which the mean 
falls. 

Hence, the mean term, in a geometrical series, is th 
square root of the product of the extremes, or of an) 
two terms equidistant from the mean. 

II. The last term of an increasing series, is the pro 
duct of the first term, multiplied by the ratio involved t( 
the power which is one less than the number of terms 
in a decreasing series, it is the quotient of the first term 
divided by the power. 

In an increasing series, whose first term is 2, ratio 3 



GEOMETRICAL PROGRESSION. 185 

and the number of terras 5, we have, by the natural 
method, 2, 6, 18, 54, 162, which gives 162 for the last 
term. 

Or, by the artificial method, the ratio involved in the 
4th power, which is one less than the number of terms, 
3X3X3X3=81; these, multiplied by the first term, 
2X81, gives 162, the last term, as before. 

Again, let the first term be 4 ; then we have 4, 12, 36, 
108, 324, or 3X3X3X3 = 81. Then multiply the first 
term, 4, by 81=324, the last term. 

In a decreasing series, with the first term 243, ratio , 
and the number of terms 5, we have 243, 81, 27, 9, 3 ; 
then 3X3X3X3=81, and 243-f-81 gives 3, which is 
the last term, as before. 

NOTE 1. To find the last term, involve the ratio to 
the power which is one less than the number of terms, 
and multiply the first term by the power. 

EXAMPLES. 

1 What is the eighth term of an increasing geometri- 
cal series, whose first term is 4, and ratio 2 ? 

Ans. 512. 

2X2X2X2X2X2X2 = 128; then 4X128 = 512, 
which is the eighth term. Or, 4, 8, 16, 32, 64, 128, 256, 
512, the eighth term, as before. 

2 Required the last term of an increasing series, whose 
first term is 15, ratio 2, and number of terms 10. 

Ans. 7680. 

3 A boy purchased 18 oranges, at 1 cent for the first, 
4 for the second, 16 for the third, &c.; what was the 
price of the last? Ans. $171798691.84. 

NOTE 2. To find the sum of all the terms, multiply 
I the last term by the ratio, and from the product subtract 
the first term ; then divide the remainder by the ratio, less 
one. 

EXAMPLES. 

1 What is the sum of a series, whose first term is 2, 
ratio 3, and number of terms 5 ? Ans. 242. 




186 GEOMETRICAL PROGRESSION. 

Find (he last term by Note 1 . 

162 last term. 
2 3 ratio. 



486 

2 first term. 



31=2)484 
242 

242 sum of the sums. 

2 What is the sum of the series, whose first term ii 
4, ratio 5, and number of terms 7 ? Ans. 15624. 

Find the last term by Note 1. 

12500 last term. 
4 5 ratio. 

20 

100 62500 

500 4 first term. 

2500 

12500 51=4)62496 

15624 15624 sum of the series. 

3 What is the sum of a series, whose first term is 3 
ratio 4, and number of terms 7 ? Ans. 16383. 

Last term, 

Ratio, 



First term, . 

41=3)49149 




Sum of the series 16383 

Write down the series, multiply it by the ratio, an< 
subtract the first series from the second, thus : 

3 12 192 768 3072 12288 

12 192 768 3072 12288 49152 



GEOMETRICAL PROGRESSION. 187 

Hero the terms all cancel, but the first of the upper and 
last of the lower series. Then we have 

49152 
3 



Divide by ratio, less 1 : 4 1=3)49149 

The sum of the series, 16383 

Now, as we multiplied the given series by the ratio, 
which is 4, and subtracted once the series from the pro- 
duct, the remainder is three times the given series. We, 
therefore, divide by 3, which is the ratio, less 1 : the quo- 
tient is the sum of the series. 

4 At 2 cents for the first *ounce, 6 for the second, 18 
for the third, &c., what would a pound of gold cost ? 

Ans. $5314.40. 

5 Sold 10 yard3 of velvet, at 4 mills for the first yard, 
20 for the second, 100 for the third, &c.; what did the 
piece cost ? Ans. $9765.62,4. 

6 What is the cost of a coat with 14 buttons, at 5 mills 
for the first, 15 for the second, 45 for the third, &c.? 

Ans. $11957.42. 

7 What is the cost of 16 yards of cloth, at 3 cents for 
the first yard, 12 for the second, 48 for the third, &c.? 

Ans. $42949672.95.- 

NOTE 3. The sum of a geometrical series is found 
by the extremes and the ratio, independent of the num- 
ber of terms ; hence, whether the number of terms be 
many or few, there is no variation in the rule. We may, 
therefore, require the sum of the seres, 6, 3, 1, 1, -*-, &c., 
to infinity, provided we can determine the value of the 
other extreme. Now, we see the terms decrease as the 
series advances ; and the hundredth term, for example, 
would be exceedingly small, the thousandth too small to 
be estimated, the millionth still less, and the infinite term 
would be nothing : not, as some tell us, " extremely 
small," or, " too little to be considered," &c., but abso- 
lutely nothing. 



188 GEOMETRICAL PROGRESSION. 

Now let us consider the series as inverted ; then 6 will 
be the last term, and 3 the ratio. By the rule, multiply 
the last term by the ratio, subtract the first term, and 
divide by the ratio, less 1. Here we have the sum c 

6X30 
the series, =9, the answer. 



EXAMPLES. 

1 What is the sum of the infinite series, 1-j-i-f-j 
-f-yV' &c.? Invert the series : is the last term, and 2 

X 100 
the ratio; hence, - =1, the answer. 

2 What is the sum of the infinite series y\, T f^, 

_3_xiO 
y^, &c.? -- - - = |, the answer. 

9 

3 What is the value of \, ~, T V j, &c., to infinity ? 

Ans. |. 

4 What is the value of |, 5, i, f, &c., to infinity? 

Ans. f. 

5 What is the value of 1, |, T 9 g, &c., to infinity ? 

Ans. 4. 
Here the ratio is |. 

6 What is the value of f , 5 4 T , T j, &c., to infinity ' 

Ans. |. 

7 What is the value of .777, &c., to infinity ? This 
may be expressed by T \, T J T , T \o &c - 



8 What is the sum of .6666, &c., to infinity ? 



Ans. . 



9 What is the value of .232323, &c., infinitely ex- 
tended ? Ans. ff 

This may be expressed by T 2 /o T f lo & c - 



EXCHANGE. . 189 \ 

EXCHANGE. 

THE object of exchange is to find how much of the 
money of one country is equivalent to a given sum of the 
money of another. 

By the par of exchange between two countries, is 
meant the intrinsic value of the one, compared with the 
other ; it is estimated by the weight and fineness of the 
coins. 

The course of exchange, at any time, is the sum of 
the money of one country which, at that time, is given 
for a certain sum of money of another country. The 
course of exchange varies according to the circumstances 
of trade. All the calculations in exchange can be per- 
formed by the Rule of Three. 

EXAMPLES. 

1 A sovereign, of England, is worth $4.84,6. A mer- 
chant of New York is indebted to a merchant of London 
$7520 ; how many sovereigns will it require to discharge 
the debt ? 

2 A six-ducat piece of Naples, is wprth $5.25. A 
merchant of Naples is indebted to a merchant in Boston 
$6940 ; how many such pieces shall he remit ? 

3 The current rupee of Calcutta is 44.4 cents. A 
merchant of Philadelphia has a claim against a mer- 
chant of Calcutta of $437 ; how many rupees shall he 
draw ? 

4 The piaster is 20 cents ; how many piasters in 

$1128.22? 

5 Reduce 7218 rupees to Federal money, at 46 cents 
per rupee. 

6 The dollar of Bencoolen is reckoned at $1.10, Fed- 
eral money ; how much Federal money in $2740 of 
Bencoolen ? 

7 The Prussian rix-dollar is worth 66| cents. Re- 
duce $7348.32 into Prussian money. 



190 PROMISCUOUS EXERCISES. 

Ex. 1. If l sterling is worth $4,44 cts. 4 ms. ; what 
is 65^6 sterling worth ? Ans. $288,86. 

2. What is the value of $500 in English money, at 
$4,44,4, per sterling? Ans. 112 10s. 2<L 

3. What is the value of 125^ 7s. at $4,44,4, per 
sterling ? Ans. $557,05,5. 

4. What is the value of $1000, in English money, at 
$4,44,4, per sterling Ans. 225^ 5*d. 



PROMISCUOUS EXERCISES. 
1. A merchant had 1000 dollars in bank ; he drew out 
at one time $237.50, at another time, $116.09, and at 
another, $241.061 : after which he deposited at one time 
1500 dollars, and at another time $750.50 ; how much 
had he in bank after making the last deposite ? 

Ans. $2655.84. 

2 Sold 8 bales of linen, 4 of which contained 9 pieces 
each, and in each piece was 35 yards ; the other 4 bales 
contained 12 pieces each, and in each piece was 27 
yards ; how many pieces and how many yards were in 
all ? Ans. 84 pieces, 2556 yards. 

3 If a man leave 6509 dollars to his wife and two sons, 
thus to his wife f , to his elder son | of the remainder, 
and to his other son the rest ; what is the share of each ? 

("Wife's share $2440.87 1. 
Ans. 1 Elder son's $2440.87^. 
[Other son's $1627.25. 

4 What is the commission on $2176.50, at 2 per 
cent? Ans. $54.411. 

5 If a tower is 384 feet high from the foundation, a 
sixth part of which is under the earth, and an eighth 
part under water, how much is visible above the water ? 

Ans. 272 feet. 

6 How many bricks 9 inches long and 4 inches wide, 
will pave a yard that is 20 feet square ? Ans. 1600. 

7 What is the value of a slab of marble, the length of 
which is 5 feet 7 inches, and the breadth 1 foot 10 
inches, at 1 dollar per foot? Ans. $10.232. 

8 A certain stone measures 4 feet 6 inches in length, 



PROMISCUOUS EXERCISES. 191 

2 feet 9 inches in breadth, and 3 feet 4 inches in depth; 
how many solid feet does it contain? Ans. 41 ft. 3 in. 

9 A line 35 yards long will exactly reach from the 
top of a fort, standing on the brink of a river, to the op- 
posite bank, known to be 27 yards from the foot of the 
wall; what is the height of the wall? 

Ans. 22 yards 3\ inches.-f- 

10 The account of a certain school is as follows, viz. 
~ of the boys learn geometry, f learn grammar, ~ 
learn arithmetic, o learn to write, and 9 learn to read: 
what number is there of each? 

A (5 who learn geometry, 30 grammar, 24 
I arithmetic, 12 writing, and 9 reading. 

11 A merchant, in bartering with a farmer for wood 
at $5 per cord, rated his molasses at $25 per hhd., which 
was worth no more than $'20 ; what price ought the far- 
mer to have asked for his wood to be equal to the mer- 
chant's bartering price? Ans. $6,25. 

12 A and B dissolve partnership, and equally divide 
their gain : A's share, which was $332 50 cts., lay for 
21 months; B's for 9 months only: the adventure of B. 
is required. Ans. $775 834 cts. 

13 If a water-hogshead holds 110 gals, and the pipe 
which fills the hogshead discharges 15 gal. in 3 minutes, 
and the tap will discharge 20 gal. in 5 minutes, and these 
were both left running one hour, how many gallons 
would the hogshead then contain; and if the tap was 
then stopped, in what time would the hogshead be rilled? 

Ans. 60 gal., and filled in 10 min. 

14 A has B's note for $500 75 cts., with 9 months in- 
terest, at 6 per cent., v^e on it, for which B gave him 
5064 feet of boards, a> <\ ,*,ts per foot, with 140 pounds 
of tallow, at 13 cts. pei n^Mid, and is to pay the rest in 
flax seed, at 92$ cts. per b.Vnljhow many bushels of flax 
seed must A receive, to ba,A\ce the note? 

Ans. 409Jf . bushels. 

15 A, B, an3 C,in company, had put in $5762: A's 
money was in 5 months, B's 7, and C's 9 months: they 
gained $780, which was so divided, that 1 of A's was J 
of B'SJ and j of B's was 1 of C's : but B, having Deceived 



192 PROMISCUOUS EXERCISES. 

$2087, absconded: what did each gain, and put in; and 
what did A and C gain or lose hv B's misconduct? 
f A's stock $2494,887 gain 260 
A J B's do $2227,577 clo 325 

" ] C's do $1039,536 do 195 
I^A and C would gain $465,577 

16 When 100 boxes of prunes cost 2 dollars 10 cents 
each, and by selling them at 3 dollars 50 cents per cwt. 
the gain is 25 percent., the weight of each box, one with 
another, is required. Ans. 84 Ib. 

17 There are two columns, in the ruins of Persepo- 
iis, left standing upright; one is 64 feet above the plain, 
the other 50. Between these, in a right line stands an 
ancient statue, the head whereof is 97 feet from the sum- 
mit of the higher, and 86 feet from the top of the lower 
column, and the distance between the lower column and 
the centre of the statue's base, is 76 feet; the distance 
between the top of the columns is required. Ans. 157-J-ft 

18 If I see the flash of a cannon, fired from a fort on 
the other side of a river, and hear the report 47 seconds 
afterwards, what distance was the fort from where I 
stood? Ans. 53674 feet. 

NOTE. Sound, if not interrupted, will move at the rate of about 
1142 feet in a second of time. 

19 What is the difference between the interest of 
$1000 at 6 per cent, for 8 years, and the discount of the 
same sum at the same rate, and for the same time ? 

Ans. The interest exceeds the discount by 
$155 67 cts. 5 m. 

20 If a tower be built in the following manner, 7 \\ 
of its height of stone, 27 feet of brick, and i of its height 
of wood, what was the height of the tower? 

Ans. 113 feet 4 inches. 

21 A captain, 2 lieutenants, and 30 seamen, take a 
prize worth $7002, which they divide into 100 shares, 
of which the captain takes 12, the two lieutenants each 
5, and the remainder is to be divided equally among the 
sailors; how much will each man receive? Ans. Cap- 
tain's share, $840.24, each lieutenant's, $350,10, and 
each seaman's, $182,05,2. 



PREFACE TO APPENDIX. 



THE author of the foregoing work, has long contemplated its ex- 
Jgnsion in an Ajjoejidix, which is now offered to the public in hopes | 
the usefulness of the whole may be extended, and the science of; 
arithmetic advanced. The view of numbers, and the abridged 
modes of operation herein presented, it is believed, will be found ac- 
ceptable alike to the business-man and to the scholar. 

The Appendix recognizes the scientific character of numbers, 
and gives bold and enlarged views of_arithmetical operations. The 
method of cancelling is not new^ but_for a long period it has 
scarcely been known! ITls, however, coming into general use; 
and it is carried much farther in this part of the work than in any 
we have hitherto known. 

In Europe, this system has been very generally adopted in the 
higher schools, and in this country it is fast becoming known and 
as far as it is known, it supercedes the usual modes of operation. 

To the method of stating problems in Proportion by comparing 
cause and effect, we invite special attention. On critical examina- 
tion it will be found more easy and more rational than any other 
method. 

Other methods of statement sometimes require the number of 
men to be multiplied into feet of wall days into acres of grass, &c., 
all of which, though correct as abstract proportion in numbers, is 
unnatural and void of strict philosophical expression not so witt 
this method. 

The peculiarities which the student will here find in the Extrac- 
tion of Roots, and in Mensuration, are not all new indeed there 
an be nothing new in principle but as far as the author's know- 
ledge extends, he is not aware that these abbreviations have ever 
been collected in any arithmetical work. The impression seems to 
have been, that the people could not comprehend arithmetical breV' 
ity, nor appreciate mathematical beauty"; but the author thinking 
otherwise, presents this brief, yet comprehensive, Appendix to the 
public, in the full assurance that whoever will pay due attention to 
the subject will be highly gratified and abundantly rewarded. 



I 



193 



APPENDIX. 



THE pupil having passed over the common routine of 
Arithmetic, and supposed to be able to perform all its ne^ 
cessary operations in the usual way, we now present nim 
with some new modes of practical operations, by which 
the labors will be much abridged and the science pre- 
sented with more of its roses and less of its thorns. 

We commence with a systematic study of numbers. 
The following are called prime numbers ; because no 
one can be divided by any number less than itself with- 
out producing a fraction. 123. 5. 7. ..11. 13... 

17 . 19 ... 23 .. . . 29 . 31 37 ... 41 . 43 . 

. . 47 53 . . . 57 . 59 . 61 . . . . 67 . . . 71 . 

73 79 ... 83 ... 87 . 89 97 ... 101 

The points represent the composite numbers; and 
here it can be observed that there are 29 prime numbers, 
and of course 71 composite numbers in the first hun- 
dred the prime numbers becoming fewer as the num- 
bers rise higher. Observe the following series : 

5 10 15 20 25 30 35 40 45 50, 

and so on. Every body knows that our Geometric scale 
of numbers i* 1 10 100 1000, &c. Now we wish the 
student to observe the numbers, 

5 20 25 50 75 125 500, 

as being not only in the preceding series, but aliquot 
parts of some number in our Geometric scale. For ex- 
ample, 25 is I of 100 ; 125 is 1- of 1000, &c. 

We now charge the student to make his eye familiar 
with all the preceding series the prime numbers as be- 
ing unmanageable and inconvenient, and the others the 
very reverse ; but the full importance of such a study 
can only appear in the sequel. 

194 



APPENDIX. 195 

By a little attention to the relation of numbers, we 
may often contract operations in multiplication. A dead 
unifomity of operation in all cases indicates a mechani- 
cal and not a scientific knowledge of numbers. As a 
uniform principle, it is much easier to multiply by the 
small numbers 2, 3, 4, 5, than by 7, 8, 9. 

EXAMPLES. 

^ Multiply 4532 

by 39 

Commence with the 3 tens. Mul- 13596 
tiply this 13596 by 3, because 3X3=9, 40788 

and place the product in the place of 

units. 176748 

Multiply 576 Multiply this last number, 

by. ... 186 3456, (which is 6 times 576) 

by 3, and place the product 

(6X3=18.) 3456 in the place of tens, and we 

10368 have 180 times 576. Ob- 

serve the same principle in 

107136 the following examples. 

Multiply .... 576 Multiply 40788 
by 618 by .... 497 

Commence with 6. 3456 (7X7=49.) 285516 

(6X3 = 18.) 10368 1998612 

355968 20271636 

Observe, that in this last example 497 is 3 less than 
500 ; 500 is \ of 1000, thefore 

2)40788*000 

20394 000 
Subtract (3X40788=122364.) 122 364 



20271 636 



196 APPENDIX. 

Multiply ........... 785460 

by ............. 14412 

First multiply by 12, then that 9425520 

product by "2. 113106240 

11320049520 



Multiply 86416 by 135. Observe, 135 is. 125+10 - 
125 is | of 1000, therefore 

8)8 64 1 6'0 



10802 000 
864 160 

Product 11666160 



tg are from Ray's Arithmetic, page 34 : 

Multiply 1646 by 365. As the first factor is even, 
and the last ends in 5, we mentally half the one and 
double the other, which will not affect the product, but 
very much contract the operation, we then have 

823 
730 



24690 
5761 

600790 

We can do the same in all such cases. 

Multiply 999 by 777. 

777000 
Subtract 777 



Product 776223 



APPENDIX. 197 



Multiply. 61524 
by. ... 7209 



5537 1 6 Multiply this product of 9 by 
4429728 8, because 9 times 8 are 72, and 

place the product in the place 

Product 443646516 of 100, because it is 7200. 

Multiply 1243 

by 636 

7458 First by 600. 
44748 Multiply 7458 by 6. 

Product 790548 

Multiply 624 by 85. The product will be the same 
as the half of 624 by the double of 85 that is, 312 by 
170, or 3120 by 17=53040. We do not say that these 
changes give any advantage in this particular example; 
we only make them to call out thought and attention 
from the pupil. 

Multiply 

by- 




This may be done by commencing with the 2 ; then 
that product by 2 and 3. Or we may commence with 
the 6 units, and then that product by 4 ; because 4 times 
6 are 24. 

Multiply 4386, or any other number, by 49. We 
may do this by taking the number 50 times, or ^ of 100 
times, and subtracting once the number. 

Multiply 87742 by 65. This may be done by taking 
the number 1 (100,) +10+! C( 10 ) times - That is 

4387100 
877420 
438710 



Product ..5703230 
77* 



198 APPENDIX. 

Multiply 92636 by 150. It will be much more easy 
to multiply the half of it by 300, which will give the 
same result. 

Multiply 679 by 279. Multiply first by 9, and that 
product by 3, put in the place of 10. 

Multiply 87603 by 9865. By the common formal 
rule this would be a tedious operation ; but let us observe 
that 9865 is 10000 135, 135 is 125-f-10 ; 125 is } of 
1000. Therefore, 

876030000 Subtract I, 8)87603000 
11826405 



: 10950375 

864203595 Product. 10 times, 776030 



11826405 

Multiply 818327 by 9874. But 9874 is 10000 less 
126. Therefore 

8183270000 Less 1 of 818327000 
Subtract 103108202 



102290875 

8080161798 Product. 818327 



103109202 

Multiply 188 by 135, and that product by 15. In- 
stead of doing this literally and mechanically according to 
rule, we may half 188 twice, and double each of the fac- 
tors that end in 5, and we shall have 47 . 270 . . 30 ; or, 

47 ' 
8100 



4700 
376 

380700 

How far will a ship sail in 365 days, at the rate 



APPENDIX. 199 

of 8 miles per hour? Here 365X24X8730X96; 
or, 

73000 less 730X4=2920 
2920 

Product 70080 

EXERCISES FOR PRACTICE. 

299 X 299 X299=what number? Ans. 26730899 
999X999X999= what number? Ans. 997002999 
4962 X 98 = what number ? Ans. 487276 

Multiply 8340745 by 64324. Observe, that 32 is 8 
times 4, and 64 is 2 times 32. 

Multiply 8340745 by 64432. In this last example, 
commence at the 4 in the place of hundreds. 

Multiply 24 X 25 X 12 X 5, together. Here it would be 
no index of even a decent knowledge of numbers, to obey 
literally and in the order the numbers are given ; yet how 
many even at the present day would do so ! 

Take the factor 4, out of the 24, and multiply it into 
the 25 ; also, the factor 2, out of 12, and put it with 5, 
which can be done without effort. We then have 
6X 100 X 10 X 6=36000, the answer. 



SECTION II. 

WE shall say nothing of division in whole numbers, 
as nothing new or interesting can be offered on that 
topic ; but we cannot forbear making a few comments 
on division in decimals. 

To divide, to cut into parts, will not at all times give 
a clear understanding of the operation, and confusion 
frequently arises from taking this view of the subject; 
we better consider it as one number measuring another. 
For example : how often will .5 of a foot measure 12 
feet ? In other words, divide 12 by .5 ; or, divide 12 by 



200 APPENDIX. 

t. Here, if the student should imagine that 12 must be 
cut into parts, he would make a great error. He must 
divide 120 tenths into parts ; in this case, into 5 parts 
because the 5 is .5 : or he may consider that \ of a foot 
may be laid down in 12 feet; that is, measure 12 feet 24 
times. Or he may reduce the 12 feet to half feet, and 
then divide by 1. In all cases, the divisor and dividend 
must be of the same denomination before the division 
can be effected. But in decimals, these reductions are 
made so easily, that a thoughtless operator rarely per- 
ceives them ; hence the difficulty in ascertaining the 
value of the quotient. 

We now give a few examples, for the purpose of 
teaching the pupil how to use his judgment; he will 
then have learned a rule more valuable than all others. 

EXAMPLES. 

Divide 15.34 by 2.7. Here we consider the whole 
number, 15, is to be divided by less than 3 ; the quo- 
tient must, therefore, be a little over 5. One figure then, 
in the quotient, will be whole numbers, the rest decimals. 

Divide 15.34 by .27. Here we perceive that 15 is to 
be divided, or rather measured, by less then % of 1 ; 
therefore the quotient must be more than 3 times 15. Or 
we may multiply both dividend and divisor by 100, 
which will not effect the quotient, and then we shall 
have 1534, to ba divided by 27. Now no one can mis- 
take how much of the quotient will be whole numbers : 
the rest, of course, decimals. 

Divide 45.30 by .015. Conceive both numbers to be 
multiplied by 1000 ; then the requirement will-be to di- 
vide 45300 by 15, a common example in whole num- 
bers. 

By attention to this operation, the student will have 
no difficulty in any case where the divisor is less than 
the dividend. 

Here is one of the most difficult cases : 

Divide .003753 by 625.5. In all such examples as 
this, we insist on the formality of placing a cipher in 



APPENDIX. 201 

the dividend, to represent the place of whole numbers, 
thus : 

625.5)0.0037530( 

We now consider whether the whole number in the 
divisor will be contained in the whole number in the div- 
idend, and we find it will not ; wr, therefore, write a 
cipher in the quotient to represent the place of whole 
numbers, and make the decimal on its right, thus, 0. 

We now consider, that 625 will not go in the 10's, 
nor in the 100's, nor in the 3, nor in the 37, nor in the 
375, but it will go in the 3753. 

We must make a trial at every step, that is, every 
time we take in view another place ; and we must take 
but one at a time. In this case, then, we shall have 
0.000006, the quotient. 

Divide 3 by 30. 30 will not go in 3 ; we, therefore, 
write for place of whole numbers, and then say 30 
in 30 tenths, 1 tenth times ; or, 0.1. 

Divide .55 by 11. 

11)0.55(0.05 

11 in 0, no times ; 11 in 5 tenths, times ; 11 in 55 
hundredths, 5 hundredths times. 

It will be observed, that we make the decimal point in 
the quotient as soon as we ascertain it ; not wait, and 
then find where it should be by counting, &c. a rule 
that we regard as unworthy of being followed by all 
those who can use their reason. 



EXAMPLES. 

1 Divide 9 by 450. Ans. 0.02. 

V Divide 2.39015 by .007. Anss. 341.45. 

3 Divide 100 by .25. Ans. 400.00. 

4 If 350 pounds of beef cost $12.25, what is the cost 
af one pound ? Ans. .035. 

5 At $5.75 per yard, how much cloth can be pur- 
chased with $19.50625 ? Ans. 3.375 yards. 

i 2 



202 APPENDIX. 

6 At .07 per cent, per annum, how much capital must 
be invested to yield $602 ? Ans. $8600. 

7 A benevolent individual gave away $600 per annum 
to charitable objects, which was .12 of his income. 
What was his income ? 



SECTION III. 
Multiplication and Division Combined. 

WHEN it becomes necessary to multiply two or more 
numbers together, and divide by a third, or by a product 
of a third and fourth, it must be literally done, if the 
numbers are prime. 

For example : Multiply 19 by 13, and divide that pro- 
duct by 7. 

This must be done at full length, because the numbers 
are prime ; and in all such cases there will result a frac- 
tion. 

But when two or more of the numbers are composite 
numbers, the work can always be contracted. 

Example : Multiply 375 by 7, and divide that pro- 
duct by 21. To obtain the answer, it is sufficient to di- 
vide 376 by 3, which gives 125. 

The 7 divides the 21, and the factor 3 remains for a 
divisor. 

Here it becomes necessary to lay down apian of oper- 
ation. 

Draw a perpendicular line, and place all numbers that 
are to be multiplied together under each other, on the 
right hand side, and all numbers that are divisors under 
each other, on the left hand side. 

EXAMPLES. 

1 Multiply 140 by 36, and divide that product by 
84. We place the numbers thus : 

84 14 
84 36 



APPENDIX. 203 

We may cast out equal factors from each side of the 
line without affecting the result. In this case, 12 will 
divide 84 and 36. Then the numbers will stand thus : 

140 
3 

But 7 divides 140, and gives 20, which, multiplied by 
3, gives 60 for the result. 

2 Multiply 4783 by 39, and divide that product 
by 13: 

4783 



A* 
** 



3. 



Three times 4783 must be the result. 

3. Multiply 80 by 9, that product by 21, and divide 
the whole by' the product of 60X6X 14. 



9 

to 3 

In the above, divide 60 and 80 by 20, and 14 and 21 
by 7, and those numbers will stand cancelled as above, 
with 3 and 4, 2 and 3 at their sides. 

Now the product 3X6X2, on the divisor side, is 
equal to 4 times 9 on the other, and the remaining 3 is 
the result. 

Hoping the reader now understands our forms, and 
comprehends the true philosophical principle, we will 
give no more abstract examples ; but we will give many 
practical examples, such as might occur in business, and 
we prefer taking them from books, that it may not be 
said we made them expressly for this occasion. 

Again, it may be observed, that this method of opera- 
tion may serve for only a few problems. We answer, it 
will serve for 71 out of 100, according to the theory of 
numbers, as we have seen there are 71 composite num- 
bers in the first hundred,, and more as they rise higher. 
But the prime numbers, 2, 3, and 5, are so small and 
manageable ami are factors in FO many other numbers, 



204 APPENDIX. 

that they may be considered in as favorable a light as the 
composite numbers and for this reason we may say, 
75 out of every hundred problems can be abridged. 

But, in actual business, the problems are almost all 
reduceable by short operations ; as the prices of articles, 
or amount called for, always corresponds with some 
aliquot part of our scale of computation. 

This method may not work a great many problems as 
they are found in some books, but it will work 90 out 
of every 100 that ought to be found in books. 
*In a book, we might find a problem like *his: - 

What is the cost of 21b. 7oz. 13dwt. of tea at 7s. 5d. 
per pound ? But the person who should go to a store 
and call for 31b. 7oz. and 13dwt. of tea, would be a fit 
subject for a mad-house. The above problem requires 
downright drudgery, which every one ought to be able 
to perform, but such drudgery never occurs in business. 

The following examples are extracted from books in 
common use, and we mark them in order that any one 
may find the original. For instance, T. 42, means Tal- 
bott's Arithmetic, page 42 ; R. 93, means Ray's Arith- 
metic, page 93 ; E. 123, means Emerson's Arithmetic, 
page 123, &c. 

EXAMPLES. 

4 How many bushels of apples can be bought for $3, 
at 15 cts. a peck? (R. 92.) Ans. 5. 

$ Explanation 3 in 15, 5 times ; 4 

5 times 5 are 20, and 20 in 100, 5 times. 



5 A 'farmer has 91 bushels of wheat, and he wishes to 
put it into bags, each of which holds 3 bushels 2 pecks ; 
how many bags will it take ? (R. 92.) Ans. 26. 



3bu. 2 pe. = 14pe. 



13 



6 What is the value of a piece of gold weighing lib. 
3dwt., at 12|cts. per grain ? (R. 92.) Ans. $729. 

.1243 
? \&L 3 



APPENDIX. 



205 



7 At Sets, a pound what will 6cwt. Iqr. cost? (R. 
83.) Ans. $21. 



8 At $2.25 a qr. what will 1 ton Icwt. cost? (R. 
93.) Ans. $189. 

21 cwt. 

4 

9 

9 At 5cts. per oz., what will 7 Ibs. 8oz. cost? (R. 
93.) Ans. $6. 

10 In this example, we must reduce 7lbs. 8oz. to 
ounces: 7X16+8 is the same as 14X8+8, or 15X8. 



100 



15 

8 ^=100 
5 



30 

2 

10 



1 1 A grocer bought a lot of cheese, each weighing 91bs. 
15oz., the weight of the whole amounted to 39cwt. 3qr.; 
how many cheeses were there ? (R. 93.) Ans. 448. 

qr. 



28 
16 

12 How many casks, each holding 841bs., can be fill- 
ed out of a hogshead of sugar weighing 15cwt. 3qr? (R. 
93.) Ans. 17. 

13 A bell of Moscow weighs 288000lbs.; how many 
tons? (R. 93 part of Ex. 23.) 

Ans. 128 tons. 11 cwt. Iqr. 20lbs. 

14 "VVe prefer dividing (mentally) the pounds into 
their obvious factors. 

7)900 



. , 1000 1281 

15 How many times will a wheel, which is 9ft. 2in. 
18 



206 



APPENDIX. 



in circumference, turn round in going 65 miles ? (R. 94, 
Ex. 32.) Ans. 37440. 

65 



X10 



320 
12 



16 What will 2 square yards 2 square feet of ground 
come to at Sets, a square inch ? (R. 94.) Ans. $144 



2 square yards 2 square feet=20 
square feet. 



12 
12 



17 What will one square yard of gilding cost at 12.5 
cents a square inch? (R. 94.) Ans. $162. 



ft 
8 



9 
144 



18 What will 5 yards 2qrs. of cloth cost at 12|cts. a 
nail? (R. 96.) Ans. $11. 

8 I 22 qrs. 

19 How many coat patterns, each containing 3 yards 
2qrs., can be cut out of a piece of cloth containing 70 
Ells Flemish? (R. 95.) Ans. 15. 

H 7 3 

20 What will one hhd. of wine cost at 6lcts. a gill? 
(R. 95.) Ans. $126. 

Observe, that 6| = 2 T S ; and, as 4 is a divisor to 25, it 
must be put on the opposite side of the line. 



100 
4 



63 
4 
2 
4 

25 



or, 



16 



63 
4 

2 
4 



21 If a person write 10 minutes each day, how 



APPENDIX. 



207 



much time will that amount to in 4 years ? (R. 96.) 

Ans. 10 days 85 hours. 



60 
24 



365 
4 
10 



22 How many yards of carpeting, 2 feet 6 inches in 
breadth, will cover a floor 27 feet long and 20 feet wide ? 
(T. 98.) Ans. 72. 



2=f. 2 is to divide the 5 ; it must, 5 
therefore, go over the line. 3 



20 

27 

2 



23 What quantity of shalloon, 3 quarters wide, will 
line 71 yards of cloth that is 1 yards wide? (T. 98.) 

Ans. 15. 



or, 



15 



N. B. Mixed numbers are reduced to improper frac- 
tions, and the denominators thrown over the line. 

24 How much land, at $2.50 per acre, must be given 
in exchange for 360 acres, worth $3.75. (T. 99.) 

Ans. $540. 



360 
31 



or, 



90 
3 



25. What will a bnshel of clover-seed come to at 12; 
cts. a pint? (Wilson, 41.) Ans. $8. 



12 cts.=| of a dollar. The 8 on 
one side cancels 8 on the other, and 
leaves 4X2 for the answer. 



4 pecks. 



26 Suppose a hogshead of molasses, which cost $23, 
be retailed at 125Cts. a quart; what is the profit on it? 
(W. 41.) Ans. $8.50. 



208 



APPENDIX. 



63 gallons. 



Sale 
Cost 



31.50 
23 



Profit 8.50 



27 What will 5 barrels of flour cost at 3|cts. per 
pound? Ans. $34.30. 

28 How many times is of a pint contained in i of a 
gallon ? (W. 65.) 



Ans. 6|. 



or, 



f 



20 



We have already remarked, that denominators of 
fractions must go over the line from the term to which 
they belong. 

29 How many times can a vessel, holding T 9 ^ of a 
quart, be filled from ~ of a barrel containing 31 gal- 
lons ? (W. 66.) Ans. 46f . 



81* 

5 == 



10 



30 If one acre and 20 rods of ground produce 45 
bushels of wheat ; at that rate, how much will nine acres 
produce ? (W. 90.) Ans. 360. 



la. 20r. = 180. 



45X8=360. 



45 



N. B. We shall plan no more problems in this sec- 
tion ; but the following require no real labor, save cor- 
rect reasoning. When the numbers are properly arrang- 
ed, a few clips with the pencil, and perhaps a trifling 
multiplication, will suffice. 

31 At l|cts. a gill, how many gallons of cider can be 
bought for $12 ? (R. 95.) . Ans. 25. 

32 How many casks, each containing 12 gallons, can 
be filled out of a ton of wine ? (R. 95.) Ans. 21. 



APPENDIX, 209 

33 A man retailed 9 barrels of ale, and received for 
it $129.60 ; at what price did he sell it a pint ? (R. 96.) 

Ans, Sets. 

34 How much butter, at 9cts, per pound, will pay for 
12 yards of cloth, at $2.19 per yard ? (W. 79.) 

Ans. 292 

35 At 45| dollars per acre, what will 32 rods of land 
come to? (W. 79.) Ans, $9.10. 

36 How long must a laborer work, at 62;|cts. a day, 
to earn $25 ? (W. 78.) Ans. 40 days. 

37 A merchant sold 275 pounds of iron, at 5|cts, a 
pound, and took his pay in oats, at 50cts. a bushel; how 
many bushels did he receive ? (Adams, 53.) 

Ans. 38. 

38 How many yards of ctoth, at $4.66 a yard, must 
be given for 18 barrels of flour, at $9.32 a barrel? (A. 
53.) Ans. 36. 

39 How long will it require one of the heavenly bo- 
dies to move through a quadrant, at the rate of 43' 12" 
per minute ? (R. 97.) Ans. 2~ hours. 

40 If a comet move through an arc of 7 12- per 
day, how long would it be in passing an arc of 180 ? 
(R. 97.) Ans. 25 days. 

41. What is the cost of 8hhds. of wine, at 5cts. per 
pint? Ans, $201.60. 

42 There are 30| square yards in one perch of land ; 
how many perches are there in 363 square yards ? 

Ans. 12. 

43 What will 18 1 yards cost, at 75cts. per yard ? 

Ans. $14.06|. 

44 If 16 persons receive $516 for 43 days' work, how 
much does each man earn per day ? Ans. 75cts. 

45 How many times will a wheel, which is 12 feet in 
circumference, turn round in going a mile ? Ans. 440. 

46 An auctioneer sold 45 bags of cotton, each contain- 



210 APPENDIX. 

ng 400 pounds, at 1 mill a pound ; what did the whole 
come to? (R. 61.) Ans. $18. 



47 A mechanic receives $90 for 40 days T work he 
ivorked 12 hours each day; how much was it per hour? 
(R. 73.) Ans, ISfcts. 

48. A laborer worked 26 days for a farmer, at 87|cts. 
per day, and took his pay in wheat, at 65cts. per bushel ; 
how many bushels did he receive ? Ans. 35. 



SECTION IV. 

IT is an axiom in philosophy, that equal causes pro- 
duce equal effects ; and that effects are always propor- 
tionate to their causes. 

Now, causes and effects that admit of computation, 
that is, involve the idea of quantity, may be represented 
by numbers, which will have the same relation to each 
other as the things they represent. 

Keeping these premises in view, then, we have a uni- 
versal rule, applicable to all cases which can arise under 
Proportion, simple or compound, direct or inverse 
namely : 

RULE. As any given cause is to its effect, so is any 
required cause (of the same kind] to its effect ; or, so is 
another given cause, of the same kind, to its requirea 
effect. 

The only difficulty the pupil can experience in thi 
system of proportion, is readily to determine what is 
cause, and what is effect. But this difficulty is soon over 
eome, when we consider, that all action, of whatsoevei 
nature, must be cause and whatever is accomplished by 
3 that action, or follows such action, must be effect. 

y 

EXAMPLES. 

> If 10 horses, in 50 days, consume 128 bushels oi 



APPENDIX. 211 

oats, how many bushels will 5 horses consume in 90 
days? (W. 113.) Ans. 72. 

Here it is evident, that the consumption of oats spoken 
of, in both the supposition and demand, are the true ef- 
fects ; and the action of the horses, multiplied by the 
days, must express the amount of cause. We shall 
therefore state it thus : 

Cause. Effect. Cause. Effect. 

16 : 128 : : 5 : [ ] 
50 90 

We write the factors, one under another, as above ; 
their multiplication is understood, but rarely or never 
actually accomplished. The second effect is an un- 
known term, or answer, required a bracket, or blank, 
or point, is left to represent it. When found, the four 
terms above would be a perfect Geometrical proportion, 
and the product of the extremes equal to the product of 
the means. In this example, the product of the means 
is perfect ; which product, divided by the factors in the 
extremes, will give what is wanting in the extremes, 
nainly the answer. 

Therefore, agreeably to one mode of performing mul- 
tiplication and division, we draw a line thus, and cancel 
down : 

ifi 128 
16 - 

If $480, in 30 months, produce $84 interest, what cap- 
ital, in 15 months, will produce $21 ? 

Now capital will not produce interest without time ; 
and, whatever be the rate per cent., the same capital in a 
double time, will produce a double interest. Therefore, 

Cause. Effect. Cause. Effect. 

480 : 84 : : [ ] : 21 
30 15 



212 APPENDIX. 

Here one element of the second cause is wanting ; that 
is, the answer to the question. 

In this case, the extremes are complete; we will, 
therefore, divide the product of^the extremes by the fac- 
tors in the mean, and the quotient will give the definite 
factor, or answer, namely $240. 

3 If 7 men, in 12 days, dig a ditch 60 feet long, 8 
feet wide, and 6 feet deep, in how many days can 21 
men dig a ditch 80 feet long, 3 feet wide, and 8 feet 
deep? 

. It is almost too plain for comment, that 7 men, multi- 
plied by 12 days, must be the first cause, and the con- 
tents of the ditch they dig, the effect. Therefore, 

Cause. Effect. Cause. Effect. 

7 : 60 : : 21 : 80 

12 8 [ ] 3 

6 8 

Here, as in the preceding example, one of the elements 
of the second cause is wanting ; or, rather say a factor 
in the means of a perfect proportion, and can be found 
as above. Ans. 2 days. 

5 If 6 men build a wall in 12 days, how long will it 
require 20 men to build it ? Ans. 3| days. 

Questions of this kind are usually classed under the 
single rule of three inverse ; they do in fact, however, 
belong to compound proportion : but, as one of the terms 
is the same in the supposition as in the demand, it may 
be omitted. The term, in the present example, is, one 
wall. If we make the number different in the two 
branches of the question, or connect any conditions with 
it, such as lengths, breadths, &c., it at once falls under 
compound proportion of necessity, and may be stated 
thus : 

Cause. Effect. Cause. Effect. 

6:1 : : 20 1 
12 [] 

5 If 4 men, in 2^ days, mow 6| acres of gra^c by 
- 



APPENDIX. 213 

working 8| hours a day, how many acres will 15 men 
mow in 3 1 days, by working 9 hours a day? 

Ans. 40 y acres. 

Cause. Effect. Cause. Effect. 

2' ' 6| : : l l '' C ] 
81 9 

Let the pupil observe, that when a correct statement 
is made, there will be the same number of elements, or 
factors, under the same letters, as in the above example 
under each Cause. We have men, days, and hours, to 
be multiplied together. When there are fractions in any 
of the terms, their denominators are to be placed over 
the line from where the term belongs ; mixed numbers 
being previously reduced to improper fractions. 

6 If 12 oz. of wool make 1 yards of cloth, | of a 
yard wide, how many yards, 1| wide, will 16 pounds of 
wool make ? Ans 22 1 yards. 

With this example, some might hesitate as to arrang- 
ing it under cause and effect, as the actors, those wh^> 
made the cloth, whether many or few, have nothing to 
do with the question. But we take the phrase of the 
example and say, The wool makes the cloth. 

Cause. Effect. Cause. Effect. 

12 : U : : 16 
I 16 

7 If the transportation of 12icwt. 206 miles cost 
$25.75, how far, at the same rate, may 3 tons and 3qrs. 
be carried for $243 ? Ans. 402f miles. 

In this example, it is indifferent which we take for the 
cause and which for the effect. We may say, that the 
money, $25.75, is the cause of having the weight car- 
ried ; or, we may say, that carrying the weight is the 
cause of purchasing the money. There are many ques- 
tions where it is indifferent which we take for cause, and 



214 APPENDIX. 

which for effect. The above example may be stated 



thus : 

Cause. 
25| : 

Or thus : 
Cause. 
12| 
206 

Or thus : 
Cause. 

206 T 
Or thus : 
Effect. 
243 



Effect. 
12icwt. 
206 miles. 

Effect. 



Cause. 
60| 

C3 

Effect. 



Cause. 
243 



Effect. 
60|cwt, 



Cause. Effect 
60| : 243 

CD 

Effect. Effect. 

OK3 . OJ.Q 

-O.T . x-o 



Cause. 
60! 



Cause. 

121 
206 



These changes show, conclusively, that this method 
of statement is strictly scientific and philosophical ; and, 
in all these different arrangements of the terms, the same 
terms are multiplied together. 

The most that can be said for the common modes of 
statements, in the Double Rule of Three, is, that the 
products, when the terms are multiplied out, are pro- 
portional. But the first and second terms, taken as a 
whole, express no particular idea or thing; whereas, in 
this mode of statement, the thing the philosophical 
idea is the only sure guide. Nor is this all ; it is very 
extensive and easy in its application ; it will cover every 
case that can arise under interest to find time, rate per 
cent. &c., and thus do away, or suspend, live or six 
special rules, which encumber every arithmetic. We 
give a few examp'es to apply this rule. 

8 What is the interest of $240 for 3 years at 6 per 
cent.? 

Cause. Effect. Cause. Effect. 

100 : 6 : 240 :] 



APPENDIX. 215 

To obtain the answer from this statement, we perceive 
that we must multiply the means together i. e. 240X6, 
the rate, and that by the 3 years, the time and divide 
by 100 ; and this is the common rule. 

9 The interest of a certain sum of money, at 6 per 
cent., for 15 months, was $60 ; what was the sum? 

Ans. $800. 

Cause. Effect.^ Cause. Effect. 

100 : 6 " : : [ ] : 60 
12 15 

10 If $800, in 15 months, should gain $60, what 
would be the rate per cent.? Ans. 6. 

Cause. Effect. Cause. Effect. 

800 : 60 : : 100 :[] 
15 12 

11 Eight hundred dollars was put out at interest, at 6 
per cent., and the interest received was $60; how long 
was it out? Ans. 15 months. 

Cause. Effect. Cause. Effect. 

100 : 6 : : 800 : 60 
12 [] 

12 If 12 men, working 9 hours a day for 15| days, 
were able to execute f of a job, how many men may be 
withdrawn and the residue be finished in 15 days more, 
f the laborers are employed only 7 hours a day ? (W. 
109.) Ans. 4 men. 

13 The amount of a note, on interest for 2 years and 
6 months, at 6 per cent., is $690; required the principal. 
(R. 172.) Ans. 600. 

14 What is the interest of $1248 for 16 days? (R 
162.) Ans. $3.28. 

Cause. Effect. Cause. Effect. 

100 : 6 : : 1248 : [ ] 



216 APPENDIX, 

This can be cancelled down and made very brief. 

15 What is the interest of $1200, for 15 days, at 6 
per cent.? (R. 162.) Ans. $3.00. 

16 How many men will reap 417.6 acres in 12 days, 
if 5 men reap 52.2 acres in 6 days ? (T. 156.) 

Ans. 20. 

17 If a cellar 22.5 feet long, 17.3 feet wide, and 10.25 
feet deep, be dug in 2.5 days, by 6 men working 12.3 
hours a day, how many days, of 8.2 hours, should 9 
men take to dig another 45 feet long, 44.6 wide, and 
12.3 deep ? (T. 156.) Ans. 12. 

18 What is the interest of 160 dollars for 36 days, at 7 
per cent.? Stated by cause and effect. Ans. $1.12. 

Cause. Effect. Cause. Effect. 

100 : 7 : : 160 : [ ] 
12 1.2 

19 The interest of a certain sum, for 36 days, is 
$1.12 the rate per cent, is 7 ; what is the sum ? 

Ans. $160. 

20 The interest of $160 for 36 days, is $1.12 ; wHt 
was the rate per cent.? Ans. 7. 

21 The interest of $160, at 7 per cent., was $1.1$ ; 
what was the time ? Ans. 36 days. 

22 In what time will any sum double itself at 6 per 
cent.? At any per cent.? 

Ans. At 6 per cent., in 16J years; at any per cent., 
divide 100 by the per cent 

23 If 2k yards of cloth, 1J wide, cost $3.37, how 
much will 36 yards cost, 1% yards wide? 

Ans. $52.79. 

24 If 4 men spend | of of f of i-J of ^30, in T 7 T 
of T \ of 4 of y of 9 days, how many dollars, at 6 
shillings each, will 21 men spend in ? of yf of -f of T 4 j 
of 45 days? (Burnham, 142.) Ans. $630. 

25 I lend a friend $200 for 6 months ; how long ought 



APPENDIX. 217 

he to lend me $1000, to requite the favor allowing 30 
days to a month ? Ans. 36 days. 

26 If 1000 men, besieged in a town, with provisions 
for 5 weeks, allowing each man 16 ounces per day, be 
reinforced with 500 men more and supposing that they 
cannot be relieved until the end of eight weeks how 
many ounces a day must each man have, that the provis- 
ions may last them that time ? (W. 183.) 

Ans. 6J ounces 

The advocates of this system are of opinion, that 
there are far too many rules in our common arithmetics ; 
and to reduce them, and thereby simplify the science, 
they recommend that all the problems, generally arrang- 
ed under Profit and Loss, Equation of Payments, &c., 
should be solved by proportion, and arranged under 
that head. In this light, they are more simple and in- 
telligible than they can be made by any special rules. 

EXERCISES FOR PRACTICE. 

1 If I buy cotton-cloth at 2s. per yard, and sell it at 
2s. 8d,, what do I gain per cent? (W. 134.) 

Ans. 33. 

Statement. If 24 pence gain 8 pence, what will 100 
pence gain? Or, 24 : 8 : : 100 : Ans. 
Or, 3 : 1 : : 100 : Ans. 

2 A merchant bought broadcloth, at $5.50 per yard, 
and sold it for $6.60 ; what did he gain per cent? (W. 
135.) Ans. 20. 

3 If I buy Irish linen at 2s. 3d. per yard, how must 
I sell it to gain 25 per cent? Ans. 2s. 9d. 3h. 

Statement. If 100 pence return 125 pence, how 
much must 27 pence return ? 

Or, 100 : 125 : : 27 : Ans. 
Or, 4 : 5 : : 27 : Ans. 

4 If, by selling cloth at $6.50 per yard, I lose 20 per 
cent., what was the prime cost of it ? (W. 137.) 

Ans. $8.12. 



218 APPENDIX. 

That is, if 80 I now receive originally cost me 100, 
what did 6.50 originally cost? 

5 By selling calico at 37^ cents a yard, 50 per cent 
was gained; what was the first cost? (R. 185.) 

Ans. 25cts. 

150 : 100 : : 37| : Ans. 
Or, 3 : 2 : : 37 : Ans. 

6 Sold wine at $1.36 per gallon and lost 15 per cent.; 
what per cent, would have been gained had the wine 
been sold for $1.856 per gallon ? (R. 187.) Ans. 16. 

7 Bought 126 gallons of wine for 150 dollars, and re- 
tailed it at 20cts. per pint ; what was the gain per cent.? 
(T. 129.) Ans. 34f. 

8 If $126.50 are paid for llcwt. Iqr. 25lbs. of su- 
gar, how must it be sold a pound to make 30 per cent, 
profit? (W. 135.) Ans. 12|cts. 

9 If I buy 124cwts. of sugar for $140, at how much 
must I sell it per pound to make 25 per cent.? 

Ans. 12jcts. 

10 If a firkin of butter, containing 561bs. cost $7, at 
how much must it be sold per pound to yield 30 per 
cent, profit? Ans. 16|cts. 

11 What is the whole loss, and what is the loss per 
cent., in laying out $70 for hats, at $1.75 each, and sell- 
ing them for 25cts. a-piece less than cost? (Burnham, 
173.) Ans. Whole loss $10; loss per 100, 14f. 

12 A merchant purchases 180 casks of raisins, at 16 
shillings per cask, and sells the same at 28 shillings per 
owl., and gains 25 per cent.; what is the weight of each 
cask? (B. 174.) Ans. 80lb. 

We multiply 180 by 16, and to 180 

add for 25 per cent., we multiply 16 

by 5 and divide by 4. Then di- 4 5 

vide by 28, and it gives cwt.; multiply 28 1J2 
by 112, and we have pounds; then di- 180 
vide by 180, and we have pounds in each cask That 
is, arrange the numbers as above, and cancel down. 



APPENDIX. 219 

For other examples, the student is referred to the body 
of the work. 



SECTION V. 

Square and Cube Roots. 

To work the square and cube roots with ease and fa- 
cility, the pupil must be familiar with the following pro- 
perties of numbers. 

I. A square number, multiplied by a square number, 
the product will be a square number. 

II. A square number, divided by a square number, the 
quotient is a square. 

III. A cube number, multiplied by a cube, the product 
is a cube. 

IV. A cube number, divided by a cube, the quotient 
will be a cube. 

If the square root of a number is a composite number, 
the square itself may be divided into integer square fac- 
tors ; but if the root is a prime number, the square can- 
not be separated into square factors without fractions. 

N. B. Substitute the word cube, for square, in the 
preceding sentence, and tiie same remarks apply to cube 

numbers. 

x 

No person can extract roots with any tolerable degree 
of skill, without being able to recognize the squares and 
cubes of the nine digets as soon as seen. 



Numbers, 1 


2 


3 


4 


5 


6 


N 


8 


9 


i] 


Squares, 1 


A 


9 


| 


35 


36 


H 


64 


31 


,100 


Cubes, j 1 


8 


27 


64 


125 


216 


J343| 


512 


729 


luool 



We here wish to remind the reader, that the pupil is 
supposed to understand the extraction of the roots in 



220 APPENDIX. 

the common way, and we request them not to forget 
that this is merely an appendix. 



EXERCISES FOR PRACTICE. 

1 What is the square root of 625 ? (R. 220.) 

Ans 25. 

If the root is an integer number, we may knpw, by 
the inspection of the above table, that it must be 25, as 
the greatest square in 6 is 2, and 5 is the only figure 
whose square is 5 in its unit place. 

Again, take 625 

Multiply by 4 4 being a square. 

2500 

The square root of this product is obviously 50 ; but 
this must be divided by 2, the square root of 4, which 
gives 25, the root. 

2 What is the square root of 6561 ? (R. 220.) 

Ans. 81. 

As the unit figure, in this example, is 1, and in the 
line of squares in the above table, we find 1 only at 1 
and 81, we will, therefore, divide 6561 by 81, and we 
find the quotient 81 ; 81 IS, therefore, the square root. 

3 What is the square root of 106729? (T. 170.) 

Ans. 327. 

As the unit figure, in this example, is 9, if the number 
is a square, it must divide by either 9, or 49. After di- 
viding by 9 we have 11881 for the other factor, a prime 
number, therefore its root is a prime number= 109. 1 09, 
multiplied by 3, the root of 9, gives 327 for the answer. 

4 What is the root of 451584 ? (T. 179.) 

Ans. 672. 

As the unit figure is 4, and in the line of squares we 
find 4 only at 4 and 64, the above number, if a square, 
must divide by 4, or 64, or by both. 



APPENDIX. 221 

We will divide it by 4, and we have the factors 4 and 
112896. This last factor closes in 6; therefore, by 
looking at the table, we see it must divide by 16, or 36, 
<fec. &c. 

We divide by 36, and we have the factors 36 and 
3136 ; divide this last by 16, and we have 16 and 196 ; 
divide this last fraction by 4, and we have 4 and 49. 

Take now our divisors, and last factor, 49, and we 
have for the original number the product of 4X36X 16 
X4X49; the roots of which are 2X 6X4X2X7, the 
products of which are 672, the answer. 

5 Extract the square root of 2025. (E. 163.) 

Ans. 45. 

Divide by 25, and we have its square factors, 25 and 
81. Roots of these factors are 5X9=45, the answer; 

Again, multiply by the square number 4, when a num- 
ber ends in 25, and we have 8100, root 90, half of which, 
because we multiplied by 4, the square of 2, is 45, the 
answer. 

6 What is the square root of 390625 ? (R. 220.) 

Ans. 625. 
390625 
Multiply by 4, ... 4 

1562500 
Multiply by 4 again, 4 

6250000 

As the number, independent of the ciphers, still ends 
in 25, we multiply again by 4, and we have 25000000. 
The root of this is, obviously, 5000. Divide by 2 three 
times, or by 8, and we have 625, the answer. 

So far, some may think this more curious than useful. 
However this may be, there are problems where much 
labor may be saved by attending to the foregoing princi- 
ples. The following are some of them : 

Find a mean proportional between 4 and 256. 

Ans. 32. 



222 APPENDIX. 

Find a mean proportional between 4 and 196. 

Ans. 28. 
Find a mean proportional between 25 and 81. 

Ans. 45. 

As the above are square numbers, multiply their 
square roots together for the answers. 

EXAMPLES. 

1 If 484 trees be planted at an equal distance from 
each other, so as to form a square, how many will be in 
a row each way. (T. 171.) Ans. 22. 

Factors 4 and 121 2X11 roots=22. 

2 A section of land, in the Western states, is a square, 
consisting of 640 acres ; what is the length in rods of 
one of its sides? (W. 147.) Ans. 320. 

Nine out of ten of our teachers would actually reduce 
the acres to square rods, by multiplying by 160, and ex- 
tract the square root of the product but this would 
show too little attention to numbers. Remove one of 
the ciphers from one number to the other, and we have 
64 to be multiplied by 1600, both square numbers, whose 
roots are 8 and 40 product 320, the answer. 

3 What must be the side of a square field, that shall 
contain an area equal to another field of rectangular 
shape, the two adjacent sides of which are 18 by 72 
rods. (W. 147.) Ans. 36 rods. 

18 by 72 is the same as the half of 18 by the double 
of 72, or 9 by 144, square numbers, roots 3X12=36, 
the answer. 

4 A has two fields, one containing 10 acres and the 
other 12| ; what will be the length of the side of a field 
containing as many acres as both of them ? (R. 220.) 

Ans. 60 rods. 

22 ,5 XI 60 is the same as 225X16; roots 15X4=60, 
the answer. 

5 What is the mean proportional between 24 and 96 ? 

Ans. 48. 



APPENDIX. 223 

6 What is the mean proportional between 18 and 32? 

Ans. 24. 

Problems on the Right-angled Triangle. 

1 The top of a castle is 45 yards high, and is sur- 
rounded with a ditch 60 yards wide ; required the length 
of a ladder that will reach from the outside of the ditch 
to the top of the castle. Ans. 75 yards. 

This is almost invariably done by squaring 45 and 60, 
adding them together, and extracting the square root ; but 
so much labor is never necessary when the numbers 
have a common divisor, or when the side sought is ex- 
pressed by a composite number. 

Take 45 and 60 ; both may be divided by 15, and 
they will be reduced to 3 and 4. Square these, 9+16 
=25. The square root of 25 is 5, which, multiplied by 
15, gives 75, the answer. 

2 Two brothers left their father's house, and went, 
one 64 miles clue west, the other 48 miles due north, and 
purchased farms ; how far are they from each other? 
(E. 171.) Ans. 80 miles. 

Divide by. - 16)64 48 

4 3 16+9=25,5X16=80. 

3 The hypothenuse of a right-angled triangle is 520 
feet, the base 312 feet; what is the perpendicular? 

Ans. 416. 
Divide by .... 52)520 312 

2)10 6 

5 3 25 9=16, root 4. 
Multiply by 104 

Answer . . . .^ . . 416 

4 Required the height of a May-pole, whose top be- 
ing broken off, struck the ground at the distance of 15 
feet from the foot, and measured 39 feet. 

Ans. 75 feet 



224 APPENDIX. 

5 A hawk, perched on the top of a perpendicular 
tree, 77 feet high, was brought down by a sportsman, 
standing off 14 rods, on a level with its base ; what dis- 

ance, in yards, did he shoot? (W. 149.) 

Ans. 81.154-yards 

If this problem is worked with skill, it will be requi- 
site to extract the root of 10 only. 

6 If the diagonal of a rectangular field is 40 rods, and 
one of the sides 32, what is the other ? (W. 150.) 

Ans. 24. 
Cubes and Cube Root. 

Cubes, whose roots are composite numbers, may be 
divided by cube factors. Cube numbers, whose unit 
figure is 5, may be multiplied by the cube number 8, 
and that period reduced to ciphers. 

1 What is the cube root of 91125 ? Ans. 45. 
Multiply by 8 

729000 

Now 729 being the cube of 9, the root of 729000 is 
90 ; divide this by 2, the cube root of 8, and we have 
45, the answer. 

2 The contents of a cubical cellar are 1953.125 cubic 
feet ; what is the length of one of its sides ? (R. 225.) 

Ans. 12.5 feet. 
1953.125 
Multiply by 8, - - .. 8 



15625.000 

Multiply by 8 again, 8 

125.000 

The cube root of this is 50 ; divide by 4, because we 
multiplied by 8 twice, and we have 12.5 the answer. 

3 The number 195112 is a cube; what is its root? 

Ans. 58. 



APPENDIX. 225 

The cube numbers are 

8, 27, 64, 125, 216, 343, 512, 729. 

Comparing these numbers with 195112, and we observe, 
that the root, in the place of tens, cannot be more than 5, 
and the root, in the place of units, must be some num- 
ber which, when cubed, give 2 for its unit figure and 8 
js the only figure possible ; the root of the whole is, 
therefore, 58. 

4 The number 912673 is a cube ; what is its root ? 

Ans. 97. 

Observe, the root of the superior period must be 9, 
and the root of the unit period must be some number 
which will give 3 for its unit figure when cubed, and 7 
is the only figure that will answer. 

In this manner, we can speedily and easily obtain the 
cube roots of all cube numbers containing not more than 
two periods, or determine whether they are cubes or 
surds. 

The following numbers are cubes ; required their 
roots. 

1 What is the cube root of 59319 ? Ans. 39. 

2 What is the cube root of 79507 ? Ans. 43. 

3 What is the cube root of 117649? Ans. 49. 

4 What is the cube root of 110592 ? Ans. 48. 
5. What is the cube root of 357911 ? Ans. 71. 
6 What is the cube root of 389017 ? Ans. 73. 
8 What is the cube root of 571787 ? Ans. 83. 

When a cube has more than two periods, it can gener- 
erally be reduced to two by dividing by some one or more 
^f the cube numbers, unless the root is a prime number. 

The number 4741632, is a cube; required its root. 
He re we observe, that the unit figure is 2 ; the unit fig- 
i re of the root must, therefore, be the root of 512, as 
that is the only cube of the 9 digits whose unit figure is 
2. The cube root of 512 is 8 ; therefore, 8 is the unit 
figure in the root, and the root is an even number, and 



226 APPENDIX. 

can be divided by 2 and, of course, the cube itself can 
be divided by 8, the cube of 2. 

8)4741632 

592704 

Now, as the first number was a cube, and being di- 
vided by a cube, the number 592704 must be a cube, and, 
by inspection, as previously explained, its root must be 
84, which, multiplied by 2, gives 168, the root required. 

The number 13312053, is a cube ; what is its root ? 

Ans. 237. 

As there are three periods, there must be three figures, 
units, tens, and hundreds, in the root ; the hundreds must 
be 2, the units must be 7. Let us then divide the 2d 
figure, or the tens, in the usual way, and we have 237 
for the root. 

Again, divide 13312053 by 27, and we have 493039 
for another factor. The root of this last number must 
be 79, which, multiplied by 3, the cube root of 27, gives 
237, as before. 

The number 18609625 is a cube ; what is its root? 

As this cube ends with 5, we will multiply it by 8 : 

18609625 
8 



148877000 

As the first is a cube, this product must be a cube ; and, 
as far as labor is concerned, it is the same as reduced to 
two periods, and the root, we perceive at once, must be 
530, which, divided by 2, gives 265 for the root re- 
quired. 

N. B. It a number, whose unit figure is 5, be mult 
plied by 8, -md does not result in three ciphers on the 
right, the number is not a cube. 

To find the Approximate Cube Root of Surds. 
The usual way of direct extraction, is too tedious to 



APPENDIX. 227 

be much practiced, if any shorter method can possibly 
be obtained. By the invention of logarithms, a very 
short method has been found; but, before that event, 
several eminent mathematicians bestowed much time and 
labor to obtain short practical rules and some of their 
rules are too ingenious and useful to be lost, notwith- 
standing the invention of logarithms has nearly super- 
ceded their absolute value in practice. 

There is no exact and constant relation between pow- 
ers and their roots ; for this reason, all rules (save by 
logarithms, and the direct and tedious one,) must be more 
or less approximate; but, nevertheless, with common 
judgment and care, we can arrive at results as near as by 
the direct methdd. 

In order to obtain a rule, let us take two cube numbers, 
as near in value to each other as practicable, and compare 
them with their roots. 

216000 and 226981 are cubes ; their roots are 60 and 
61. 

Now 216000 is not to 226981 as 60 to 61. But let 
us double the first and add it to the second, and double 
the second and add it to the first, and we shall have 
658981 and 669962, which are to each other very nearly 
as 60 to 61. 

Or, by the principles of proportion, the first is to the 
difference between the first and second, as is the third to 
the difference between the third and fourth. That is, 
658981 : 10981 : : 60 : 1 very nearly. Now one is 
the difference between the two roots, and if the last root, 
or 61, was unknown, this proportion would give it very 
nearly. 

EXAMPLES. 
1. Required the cube root of 66. 

The cube root of 64 is 4. Now it is manifest, 
that the cube root of 66 is a little more than 4, and 
by taking a similar proportion to the preceding, we 
have 



228 . APPENDIX. 

64X2 = 128 2X66=132 
66 64 

194 : 196 : : 4 : to root of 06. 
Or, 194 : 2 : : 4 : to a correction. 

194)8.0000(0.04124 
176 



240 
194 



460 

388 

720 

Therefore, the cube root of 66 is 4.04124 

2 Required the cube root of 123. 

Suppose it 5 ; cube it, and we have 125. 

Now we perceive, that the cube of 5 being greater 
than 123, the correction for 5 must be subtracted 

2X125=250 246 
Add 123 125 



As 373 : 371 : : 5 : root of 123. 

Or, 373 : 2 : : 5 : correction for 5. 

373)10.0000(0.02681 
746 

2 540 From 5.00000 

2238 take 0.02681 



3020 Ans. 4.97319 

2984 

360 



APPENDIX. 229 

From what precedes, we may draw the following 

RULE. Take the nearest rational cube to the given 
number, and call it the assumed cube; or, assume a 
root to the given number and cube it. Double the as- 
sumed cube and add the number to it ; also, double, the 
number and add the assumed cube to it. Take the 
difference of these sums, then say, J?s double of the as- 
sumed cube, added to the number, is to this difference^ so 
is the assumed root to a correction. 

This correction, added to or subtracted from the assum- 
ed root, as the case may require, will give the cube root 
very nearly. 

By repeating the operation with the root last found as 
an assumed root, we may obtain results to any degree of 
exactness ; one operation, however, is generally suf- 
ficient. 

3 What is the cube root of 28"? Ans. 3.03658-|-. 

4 What is die cube root of 26? Ans. 2.96249-}-. 

5 What is the cube root of 214 ? 

Ans. 5.98142-h 

6 What is the cube root of 346 ? 

Ans. 7.02034-f-. 

The above being very near integral cubes that is, 28 
and 26 are both near the cube number 27, 214 is near 
216, &c., all numbers, very near cube, numbers are easy 
of solution. 

We now give other examples, more distant from inte- 
gral cubes, to show that the labor must be more lengthy 
and tedious, though the operation is the same. 

EXAMPLES. 

1 What is the cube root of 3214? Ans. 14.75758. 

Suppose tho root is 15 its cube is 3375, which, being 
greater than 3214, shows that 15 is too great; the cor- 
rection will, therefore, be subtractive. 



230 APPENDIX. 

By the rule, 9964 : 161 : : 15 : 0.2423, the cor- 
rection. 

Assumed root, 15.0000 

Less 2423 

Root nearly ...-.. 14.7577 

Now assume 14.7 for the root, and go over the opera- 
tion again, and you will have the true root to 8 or 10 
places of decimals. 

2 What is the cube root of 14760213677 1 

Ans. 2453. 

Suppose the root 2400, &c. Take the correction to 
the nearest unit, and you will find it 53. 

3 What is the cube root of 980922617856? 

Ans. 9936. 
Suppose the root to be 10000. 

4 What is the cube root of 9 ? Ans. 2.08008. 

5 What is the cube root of 9^? Ans. 2.092-f 

6 What is the cube root of 41 ? Ans. 3.44S2-J-. 

When it is requisite to multiply several numbers to- 
gether and extract the cube root, try to change them into 
cube factors, and extract the root before the multiplica- 
tion. 

EXAMPLES. 

1 What is the side of a cubical mound equal to one 
288 feet long, 216 feet broad, and 48 feet high? (R. 
225.) 

The common way of doing this, is to multiply these 
numbers together and extract the root, a lengthy opera- 
tion. But, observe, that 216 is a cube number, and 288 
=2X12X12, and 48=4X12; therefore, the whole 
product is 216X8X12X12X12. Now the cube root 
of 216 is 6, of 8 is 2, and of 12 3 is 12, and the product 
of 6X2 X 12 = 144, the answer. 

2 Required tlie cube root of the product of 448 
X392, in a brief manner. Ans. 56. 

3 If you have a pile of wood 32 feet long, 4 feet 



APPENDIX. 231 

wide, and 4 feet high, what v/ould be the side of a cubic 
pile containing the same quantity ? Ans. 8 feet. 

Proposition The solid contents of cubes or spheres 
are to each other as the cubes of their like dimensions. 
(See Geometry.) 

EXAMPLES. 

1 Mercury is about 2000 miles in diameter, and the 
earth about 8000; what is their relative magnitudes ? 

Ans. As 1 to 64. 

2 Mars is about 4000 miles in diameter, the earth 
8000 ; what is their relative magnitudes ? 

Ans. As 1 to 8. 

In the preceding examples we, of course, do not cube 
the numbers given, but their smallest integral relations. 

3 The diameter of the earth, to that of the sun, is 
nearly as 1 to Ills ; what is their relative magnitudes, 
or bulks? Ans. As 1 to 1384472 nearly. 

4 If a ball, 6 inches in diameter, weigh 32lbs., what 
will be the weight of a ball, of the same metal, whose 
diameter is 3 inches ? (R. 225) Ans. 41bs. 

6:3:: 2:1 
8 : 1 : : 32 : 4 

5 Suppose an iron ball, of 4 inches in diameter, to 
weigh 9 pounds ; required the weight of a spherical 
shell of 9 inches in diameter, and 1 inch thick. 

Ans. 541bs. 4oz. 

6 If a cable, 12 inches about, require an anchor of 
18cwt., of what weight must an anchor be for a 15 inch 
cable? (Pike, 211.) Ans 35cwt. 15lbs. 



SECTION VI. 
Mensuration, Gauging, fyc. 

N. B. EVERY problem that follows, can be done with 
much less labor than they are usually done. 



232 APPENDIX. 

EXERCISES FOR PRACTICE 

1 What is the difference of area between a square of 
40 rods on a side, and an equilateral rhombus of 40 rods 
to a side, but of a perpendicular altitude of only 34 rods ? 
(W. 175.) Ans. 240 rods. 

2 There is a barn, 50 feel "by 36, and 20 feet high to 
the eaves ; how many boards will it take to cover the 
body, if the boards were all 15 inches wide and 10 feet 
long ? Ans. 275-j- boards. 

3 On a base of 120 rods in length, a surveyor wished 
to lay off a rectangular lot of land, to contain 60 acres ; 
what distance in rods must he run out from his base line ? 
(W. 175.) Ans. 80 rods. 

4 How many square yards in a triangle, whose base 
is 48 feet, and perpendicular height 254 feet? (W. 175.) 

Ans. 67s yards. 

5 A man bought a farm 198 rods long, and 150 rods 
wide, and agreed to give $32 per acre ; what did the 
farm come ta? Ans. $5940. 

N. B. Make no attempt to compute the number of 
acres definitely. 

6. If the forward wheels of a coach are 4 feet, and 
the hind ones 5 feet in diameter, how many more times 
will the former revolve than the latter in going a mile ? 
(W. 176.) Ans 84. 

N. B. In this problem use 7 to 22. 

7 How many square feet in a board, 2 feet wide at the 
larger, 1 foot 8 inches at the smaller end, and 14 feet 
long ? (W. 176.) Ans. 25| feet. 

8 The plate supporting the rafters of a house, being 
40 feet long, 14 inches wide, and 8 inches thick, how 
many solid feet does it contain? (W. 177.) 



Ans. 31| feet. 



12 
12 
12 



40 
12 
14 
8 



inches 



APPENDIX. 233 

Cancel down, and this is the form for all solids. 

9 If a pile of wood be 60 feet long, 12 feet high, 
and 6 feet wide, how many cords does it contain ? 

Ans 33? cords. 

10 The bin of a granary is 10 feet long, 5 feet wide, 
and 4 feet high ; allowing the cubical contents of a dry 
gallon to be 268| cubic inches, how many bushels of 
grain will it contain ? Ans. 160*. 

1 1 If you wanted a bin to contain twice as much as 
mentioned in the last problem, with a length of 12 feet, 
and a breadth of 6 feet, of what height must it be ? (W. 
177.) Ans. 5| feet. 

12 A canal contractor engaged to excavate 2 miles of 
canal across a plane, at Sots per cubic yard the canal to 
be 54 feet wide at top, 40 at bottom, and 4$ feet deep ; 
what did it amount to ? Ans. $6617.60. 

13 There is a circular cistern, of uniform diameter, 
whose depth is 8 feet, and diameter 5 feet; what is its 
capacity, allowing 231 inches to the gallon, and how 
much would its capacity be increased by adding 6 inches 
to its diameter ? . C 1175.04 gallons its capacity, 

S> I 246| gallons increase. 

14 What would be the produce of a kernel of wheat 
in 11 years, at 20 fold, the produce of each year 
being sowed the next-? allowing 5000 kernels to a 
quart? (W. 166.) Ans. 64000000 bushels. 

N. B. If we blindly perform all the labor indicated by 
set rules, the above would be a tedious operation ; but it 
is extremely brief in the hands of a skillful operator. 

15 The length of a room being 20 feet, its breadth 14 
feet 6 inches, and its height 10 feet 4 inches ; how much 
will the coloring come to at 27cts. per square yard, de 
ducting a fire-place of 4 feet by 4 feet 4 inches, and 
tw. windows, each 6 feet by 3 feet 2 inches? (R. 232.) 

Ans. $19.73. 

1 6 What will the paving of a foot-path come to, at 1 8 
cents per square yard, the length being 35 feet 4 inches, 
and the breath 8 feet 3 inches ? (R. 233.) Ans. $5.83. 



234 APPENDIX. 

17 What will it cost to roof a building 40 feet long, 
the rafters on each side being 18 feet 6 inches long, at 
$3.50 per 100 square feet? (R. 233.) Ans. $51.80. 

18 There is a block of marble, in the form of a paral- 
lelepiped, whose length is 3 feet 2 inches, breadth 2 feel 
8 inches, and depth 2 feet 6 Inches ; what will it cost at 
Slcts. per cubic foot? (R. 234.) Ans. $17.10. 

19 What will it cost to build a wall 320 feet long, 6 
feet high, and 15 inches thick, allowing 20 bricks to the 
solid foot, at $5 T 8 ^ per thousand bricks ? (R. 234.) 

Ans. $282. 

20 How many bricks 8 inches long, 4 inches wide, 
and 2^ inches thick, will it take to build a wall 120 feet 
long, 8 feet high, and 1 foot 6 inches thick ? (R. 234.) 

Ans. 34560. 

21 What will it cost to build a brick wall 240 feet 
long, 6 feet high, and 3 feet thick, at $3.25 per 1000 
bricks each brick being 9 inches long, 4 inches wide, 
and 2 inches thick ? (R. 234.) Ans. $336.96. 

22 A ship's hold is 75 feet long, 18 wide, and 7| 
deep; how many bales of goods 3 feet long, 2| deep, 
and 2| wide, may be stowed therein, leaving a gangway, 
the whole length, of 3? feet wide ? (Pike, 470.) 

Ans. 385,4-f-. 

N. B. Do this by one operation after taking out the 
gangway mentally, by subtraction. 

23. A stick of timber is 16 inches broad and 8 inches 
thick ; how many feet in length must be taken to make 
20 solid feet? Ans. 22&. 

24 There is a square pyramid, each side of whose 
base is 30 inches, and whose perpendicular height is 120 
inches, to be divided by sections, parallel to its base, into 
three equal parts ; required the perpendicular height of 
each part. (P. 371.) 

Ans. The height of the lower section is 15.2 inches ; 
the height of the middle section is 21.6 inches ; the height 
of the top section is 83.2 inches. 

N. B. In solving this problem, remember that solids, 



APPENDIX. 235 

of the same shape, are to each other as the cubes of their 
like sides. 

25 A man wishes to make a cistern of 8 feet in diameter, 
to contain 60 barrels, at 32 gallons each, and 231 cubic 
inches to a gallon ; what shall be the depth of the cis- 
tern ? 

60X32X231 gives the cubic inches the cistern is to 
contain. 

This divided by the circular end, expressed in in- 
ches, will give the depth in inches. 

8X12X22 
.JNow, = the circumference in inches. 

But the circumference of any circle, multiplied by | 
of its diameter, giver its area. 

8X12X22X24 
Then, .... = the area. 

Hence, $ 



7X7X 5=245 ; which, divided by 4, gives 6U inches 
for the answer. 



SECTION VII. 

Miscellaneous Examples. 

1 One-half, one-third, and one-fourth of a certain num 
ber, added together, make 130 ; what is the number? 

Ans. 120. 

To solve this by arithmetic, we must consider the 
number as the whole of a thing, or a unit. Then 5 -f 
-r-? = T 6 2+T 4 2"i"T 3 2 or VI- Then, by proportion, if \ 
make 130, what will 1, or 4-5, make ? That is, 



130 



T* 

-' "V 



236 APPENDIX. 

By multiplying the first and last terms by 12, the pro- 
portion becomes 

13 : 130 : : 12 
Or, .1 : 10 : : 12 

2 One-fourth of a certain number exceeds one-sixth 
of the same number by 20 ; what is the number ? 

Here, again, the number must be considered as a sin- 
gle thing ; then $ of it is not, strictly, one-fourth of a 
unit, but | of that number, or that thing. In algebra, 
this thing, or number, would be represented by some let- 
ter, as x or y and the question is strictly an algebraical 
one. But all such questions in algebra, can be solved 
by fractions and proportion in arithmetic ; and, indeed, 
all questions, that involve simple equations only, can be 
resolved by arithmetic but by algebra they are much 
easier. 

In days that we re, we always found a rule in arithme- 
tic called Position, which included such problems as the 
preceding, and some few others^ but could not take in any 
questions involving powers, or roots as. powers and 
roots are not in arithmetical proportion to each other. 

For example, 16 and 64 are square numbers, and their 
square roots are 4 and 8, or as 1 to 2 ; but the numbers 
themselves, 16 and 64, are to each other as 1 to 4, a dif- 
ferent relation from the roots. 

Example : A man, having a purse of money, being 
asked how much was in it, answered, The square root 
of it, added to the half of it, made 220 dollars ; how 
much was in the purse.? 

It is evident, that this question must be excluded from 
any proportional operation ; for, unless we first suppose 
the right number, the result of the supposition will not 
be to the given result as the supposed number to the true 
number and when this proportion fails, supposition, 
that is, "Position fails ;" and if we suppose the true 
number, it is then an operation of chance, and is, in fact, 
no problem at all. 

True it is, we can give a rule, or rather give orders 
which, followed, will reduce the problem, but it will not 



APPENDIX. 237 

be arithmetic; and, to put into an arithmetical work 
what is not arithmetic, we hold to be deceptive and im- 
proper. 

We have remarked, that the solution of these prob- 
lems are much easier by algebra than by arithmetic; 
but we would remind the pupil, that the solution of a 
problem is a small object compared with a principle in- 
volved in a solution, or with a knowledge of the science 
of numbers. As one step to secure this latter object, we 
give a few more such problems. 

3 A post is J in the earth, | in the water, and 13 feet 
above the water ; what is the length of the post ? 

Ans. 35 feet. 

Add ^ and f ; not ^ and ~ of the number 1, but i and 
f of the whole post. These parts, added together, make 
|| ; the remaining ij must be 13 feet. Then, by pro- 
portion, 



35 

35 



Or, ..... 13 : 13 : : 35 : 35 the answer. 

4 A and B have the same income. A contracts an an- 
nual debt amounting to ~ of it, B lives upon | of it ; at 
the end of ten years, B lends to A money enough to pay 
off his debts, and has 160 dollars to spare; what is the 
income of each ? Ans. $280. 

If B lives on |, he saves ^ ; out of this he pays A's 
debts, 4. Hence, from jr subtract 1 7 , and there remains 
? 2 j. This, in 10 years, is worth 160 dollars; therefore, 
in 1 year it is worth 16 dollars. Now, by proportion, 

/ T : 16 : : || : the answer ; 
Or, 2 : 16 : : 35 : the answer; 
Or, . . 1 : 8 : : 35 : 280, the answer. 

5 Of the trees in an orchard, are apple trees, -i 
pear trees, and the remainder peach trees which are 20 
more than | of the whole ; what is the whole number in 
the orchard ? Ans. 800. 

The apple, the pear, and the peach trees, make the 
whole; therefore, add + '-=+-=- 



238 APPENDIX. 

This wants /$, or ^L, of being the whole ; therefore, we 
must conclude, that the 20, not taken into the account, 
is V of the whole. Hence, 20X40=800 ; or, by pro- 
portion, 

T V : 20 : : }{. : answer. 
That is, 1 : 20 : : 40 : 800, the answer. 

6 A, B, and C, would divide $200 between them, so 
that B may have $6 more than A, and C $8 more than 
B ; how many dollars for each ? (R. 229.) 

Ans. A 60, B 66, C 74. 

Observe, that A has the least sum, B 86 more than A, 
and C $14 more than A ; hence, $20 is to be reserved, 
and the remaining 180 to be divided equally among 
them; which, of course, gives $60 to A, and $60-}-6 to 
B, and 60+6+8 to C. 

7 A gentleman bought a chaise, horse, and harness for 
$378 the horse came to twice the price of the harness, 
and the chaise to twice the price of both the horse and 
harness ; what did he give for each ? (R. 229.) 

This problem is generally given under position, but it 
is really one in simple division. Take the idea of shares. 
Divide the money into shares : it will take 1 share to 
purchase the harness, 2 shares to purchase the horse, 6 
shares to purchase the chaise ; therefore, divide the whole 
into 9 shares, and we shall have $42 for one share, i. e. 
$42 for the harness, $84 for the horse, $252 for the chaise. 

In conclusion, we would caution the young arithme- 
tician against imbibing the idea, that he understands arith- 
metic, from works on arithmetic alone. We must rise 
above a plane, to have a fair view of the objects on its 
surface, and we must rise above arithmetic before we can 
understand all its scientific relations. 

Nor must we conclude that we are perfect in num- 
bers, because we may excel our comrades in disentang- 
ling knotty questions. Science is a different thing from 
acuteness at solving intricacies ; and all men. of true 
science, more or less despise all things intended to puz- 
zle, for there are enough objects of useful inquiry and 
investigation, on which to expend all our powers of mind. 



APPENDIX. 239 

Algebra is but a continuation of arithmetic; and as 
soon as we acquire a good practical knowledge of arith- 
metic, so as to have a clear comprehension of fractions 
and general proportion, we may, yea, we should, com- 
mence algebra. But when we advance in algebra, we 
should be careful not to look down with the spirit of con- 
tempt on arithmetic we may study the one in order to 
understand the other. The following properties of num- 
bers, the student must take as facts, unless he is an alge- 
braist. They cannot be demonstrated without the aid 
of that science.* 

1 If from any number, the sum of its digits be sub- 
tracted, the remainder is divisible by 9. 

Take, for example, 34 ; subtract 7, and we have 27, 
which is divisible by 9. Again, take 438 ; subtract the 
sum of 4+3+8=15, and we have 423=9X47, and so 
with any other number. 

2 If the sum of the digits of any number be divisi- 
ble by 9, the number itself is divisible by 9. 

Thus the numbers 72, 81, 99, 171, 387, 51489, &c., 
the sum of whose digits is divisible by 9, are themselves 
divisible by 9. 

All numbers divisible by 9, are, of course, divisible 
by 3. 

3 If the sum of the digits of any number be divisi- 
ble by 3, then the number itself is divisible by 3. 

Thus 18, 27, 54, 75, 111, 123, 258, &c. &c., are all 
divisible by 3. 

4 If from any number the sum of the digits stand- 
ing in the ODD places be subtracted, and the sum of the 
digits standing in the EVEN places be added, then the 
result is divisible by 11. 

Take any number, say 785432 ; then subtract the sum 
of 2+4+8 = 14, and add 3+5+7=15, or, in this ex- 
ample, add 1, and we have 785433 = 11X71403. 



* The demonstrations of these properties may be found in Bridge's 
Algebra, section XLII. 



240 APPENDIX. 

5 If the sum of the digits standing in the EVEN pla- 
ces, be equal to the sum of the digits standing in the 
ODD places, then the number is divisible by 11. 

Thus the numbers 121, 363, 12133, 48422, &c., are 
all divisible by 11. 

Our numbers increase in a ten-fold proportion, from 
the right to the left, which is called, the root of the 
scale ; but, if the scale was 7, in lieu of ten, then, what 
is now true for 9 and 11, would be then true for 6 and 8. 

6 We may change numbers from, the scale of ten to 
any other scale, by dividing the number by the number 
denoting the scale, continually saving the remainders 
and forming a new number by them. 

Example Change 63 into an equivalent number, 
wherein the value of the digit shall increase in a five-fold 
proportion ; in other words, where the root of the scale 
shall be 5. 

5)63 

5)12(3= first remainder. 
5)2(2= second remainder. 
0(2= third remainder. 

Now 223 is the value of 63, in a scale where the num- 
bers increase in a five-fold proportion. Change 3714 to 
its equivalent value on a scale of 4. Ans. 322002. 

Numbers may also be considered as arising from the 
continued multiplication of certain factors. 

Ji perfect number, is one which is equal to the nun of 
all its divisors. They are not numerous, and may be 
expressed thus : 

2 (2 2 1)=2X3=6. 

2 2 (2 3 1) =4X7=28. 
2 4 (2 5 l) = 16X 31=496. 
2 6 (2 7 1) = 64X 127=8128. 
2 10 (2 U *) = 1024X 2047=2096128. 



A PRACTICAL SYSTEM 

OF 

BOOK-KEEPING, 

FOR 

MECHANICS AND RETAILERS. 



BOOK-KEEPING is the method of recording a system- 
atic account of business transactions. 

It is of two kinds Single and Double Entry. The 
former, only, will be noticed in this work. On account 
of the simplicity of Single Entry, it is, perhaps, the best 
wliich can be recommended to farmers, mechanics, and 
retailers. It consists of two principal books the Day 
Book, or Wast* Book, and the Lcger, and one auxiliary 
book, the Cas/trBook. 



TIIE DAY BOOK. 

This book is ruled with a column on the left hand 
for the date, and three columns on the right, the first, 
for the folio or page of the Leger, to which the account 
is transferred; and tiie last two for dollars and cents. 

This book exhibits a minute history of business trans- 
actions in the order of time in which they occ ur, with 
every circumstance, necessary to render the tra asaction 
plain and intelligible. 

17 " * 



CINCINNATI, 1833. 



Jan. 4 



6 



8 



11 



a 14 



14 



15 



David Judkins, Dr. 

To 10 Ibs. coffee at 17cts.$l 70 
" 25 Ibs. sugar, at LO cts. 2 50 



Timothy W. Coolidge, Dr. 
To 1 bl. sugar, weighing 

135 Ibs. neat, at 8* cts. $11 47i 
1 bag coffee, 98 Ibs. at 
15 cents, 14 70 



Geo. H. Eaton, Dr. 

To 1 bl. flour, $3 874 

1 Ib. Y. H. Tea, I i2i 

" 1 keg lard, neat weight 
60 Ibs., at 6i cts. 375 



David Judkins, 
By cash on account, 



Cr. 



James Wilson, Dr. 

To 3 weeks boarding, at 2 dollars 
per week, 



Hiram Ames, Dr. 

To 12 yards broadcloth at 6 dol- 
lars per yard, $72 00 
" 30 yds. muslin at 14 cts. 4 20 

Cr. 

By an order on J. Jones, for Gro- 
ceries, $51 00 
" Cash, 20 00 



Timothy W. Coolidge, Cr. 
By a bill of carpenter work, 



James Wilson, Dr. 

To 1 bl. vinegar, $3 25 

" 1 keg lOd. nails, weight 
"111 Ibs. at 8 cts. 888 



76 



71 



25 



12 



CINCINNATI, 1833. 



Jan. 19 
21 
25 



George Ha rail ton, 
By 1 set of Fancy chairs, 



George H. Eaten, 
By cash, 10 balance account, 



R >bert Young, Dr. 

To cash on account, $3 00 

" 10 Ibs. N. O. sugar, at 
10 cents, 1 00 

12 Ibs. coffee at 163 cts. 2 00 



29 T( 



" 31 
Feb. 



13 



Cr. 



Cr. 



Alb.Y.H. Tea, 



62* 



Jackson Moore, Dr. 

21 Ibs. Ham, at 10 cts. $2 10 
I box soap, 30 Ibs. at 5 



cents, 



150 



David Judkins, 



Dr. 



30 To 12 bis. apples at 1 5 cents, 



By 47 bush, corn, at cents, 



Cr. 



James Wilson, 
By cash on account, 



Cr 



Thomas Hilton, Dr 

3 To cash, $5 00 

1 Ib. Y. H. Tea, 1 06 

16 Ibs. Rice, at 64 cts. 1 00 

Cr. 
By 13 days labor, at 87* cts. 



George Hamilton, Dr 

10 To 1 canister Imperial Tea, 14 
Ibs. at $1 87 i per Ib. 



Hiram Ames, 



Cr. 



By order on E. Disney for goods, 



15 



11 



76 



iO 



CINCINNATI, 1833. 



.Feb. 15 



21 



27 



March 6 



James Wilson, Dr. 

To 10 gais. molasses, at 40 

cents, $4 00 

" 4 Ibs. Old Hyson Tea, 
at 93 cents, 372 



Robert Young, Dr. 

To 1 box sperm candles, 
"25 Ibs., at 30 cts. perlb. $7 50 
" 2 bu. dried fruit, at $1 
" 25 per bushel, 2 50 



Robert Young, \ Cr. 

By 12 cords of wood at $2 25 



Ames &, Smith, Dr 

To 18 Ibs. sole leather, at 
" 25 cents, $4 50 

" 1 side upper leather, 2 75 
"3 calf skins, $125, 375 



Hiram Ames, Dr 

To 500 feet whits pine boards, ai 
" $12 50 per M. $6 25 

" 25 bu. potatoes, at 50 cts. 12 50 
1 ton of Hay, 10 00 



David Judkins, Dr 

To 200 Ibs. flour, at $175 






15 



Thomas Hilton, 



Dr 



To cash paid his order to William 
Cuolirlge, 



George Hamilton, Dr 

To 1 copy Whelpley's Com- 

pend, $1 2i 

" 1 ream letter-paper, 4 51 

1 doz. Spelling books, 1 0( 



27 



11 



18 



cts. 
00 

72 
00 

00 

75 
50 

75 



CINCINNATI, 1833. 



Mar. 20 



23 



Ames and Smith, , Cr 
25 By 1 hhd. sugar, weight 1317 Ib 
neat, at 74 cts. 



28 



30 



31 



Theodore P. Letton, Dr. 
To sharpening his plough,*$ 1 ,00 
" Shoeing his horse, 1,6 

" Repairing chain, 25 



.Timothy W. Coolidge, Dr. 
To 2 qrs. tuition of himself at 
evening school, at $3 per qr. 



Thomas Hilton, Cr 

By the hire of his horse 10 days 
at 62 i cts. per day, 



T. P. Letton, Dr. 

To 1 ream wrapping paper, 

$1,62* 

" 1 beaver hat, 5,00 

" 1 set silver tea-spoons, 6,00 



Henry C. Sanxay, Cr. 

By my order on him in favor of 
Jno. Torrence for stationary, 



Ames and Smith, Dr. 

To 2000ft. clear pine boards, at 

$20 per M. $40,00 

500 common do. at $8, 4,00 
5000 shingles, at $2,25 1 1,25 
" Cash to balance account, 32,52 



Jackson Moore, 
By painting my house, 



Cr. 



98 



cts. 



874 



00 



25 



774 



624 



874 



77 



00 



6 CINCINNATI, 1833. 


March 31 

31 

31 
31 
31 


Thomas Hilton, Dr. 
To 4 bu. wheat, at $1 25, $5 00 
1 bl. mess pork, 9 00 
" 2 bu. salt, at 50 cts. 1 00 
" 8 Ibs. brown sugar, 11 cts. 88 


2 

4 
3 

2 

f- 


$ 
15 

14 
I) 
14 
11 


cts 

88 

00 
00 
00 
50 


George Hamilton, Dr. 
To 12 cedar posts, at 25c $300 
1 plough, 937* 
" 1 scythe, 1 62d 


Jackson Moore, Dr. 
To repairing his wagon and 
plough, 


Thomas Hilton, Cr. 
By 1 pair shoes, $1 50 
" 1 mahogany table, 12 50 


George Hamilton, Cr. 
By an order on J. Hulse, $5 00 
cash, 650 


END OF THE DAY BOOK. 



r 

THE LEGER. 

THIS book is used to collect, the scattered accounts of the Day Book, and 
toarrunpe all tliat relates to each individual, into one separate statement. 
The i-usiness of transferring the accounts from the Day Book to the Leger, 
is called posting. 
The Leger is ruled with a double line in the middle of the page, to sepa- 
rate the debits from the credits. Each side has two columns for dollar* and 
cents, one for the page of the Day Book, from which the particular item ia 
brought, and a column for the date. 
When an account is posted, the page of the Leger on which this account 
is kept, is written in the column for that purpose in the Day Book, and also 
the page of the Day Book from which the account was posted, is written in 
the 2d column of the Leger. 
Tn posting, begin with the first account in the Day Book, which you will 
perceive is the name of David Judkins. Enter his name in he first page of 
the Leger, in a large, fair hand, with Dr. on the left and Cr. on the right. 
As there are several articles charged to D. judkins on the 4th of January, 
instead of specifying each article in the Leger, we merely say, For Sundries, 
and enter the amount in the proper columns see Leger, pane 1. 
The Leger has an index or alphabet, in which the narn3 of persons are 
arranged under their initial letters, with the page in the Leger where the 
account may be found. 


ALPHABET TO THE LEGER. 


A 

Ames, Hiram - 2 
Auies & Smith - 4 


I J 

Judkins, David 1' 


R 


B 

Balance 5 


K 


s 

Sanxay, H. C. - 4 


C 

Coolidge, T. W. - 1 


L 
Letton, T. P. - 4 


T 


D 


M 
Moore, Jackson 3 


U 


E 
Eaton, George H- 1 


N 


V 


F 





W 
Wilson, Jas. - 2 


G 


P 


X 


Y 

Young, Robt. - 3 


H 

Hilton, Thomas - 2 
Hamilton, Geo. - 3 


a 


Z 



I 1 Dr. David Judkins, Cr. 


183& 
Jan. 4 
30 

Mar.7 

Apl.l 

NOTE.- 

tno sums 
and for w 
that the 
person in 


Fo sundries 2 
Apples 3 
Flour 4 

To balance 
of account 
bro't down, 

-The Dr. on the left 
entered on that side 
'liich they owe you. 
sums entered on that 
ider whose account t 


4 
9 

3 

16 

= 

4 

Land 
ofth 
TheC 
side c 

hey st 


20 
)0 
30 

70 
30 

sideo 
epage 
/r. on 
>f tlie 
and, a 


Jan. 6 
" 30 

r the page 
are those 
he right 1 
page are 
id for wl 


By cash 2 t 
Corn 3 I 
" Bal ' ^ 

11 

signifies debtor, 
for articles sold t 
)and side signifies c 
"or articles receive 
irh you owe him. 


J|00 
3 40 
1 30 

3 70 

and that 
o others, 
redit, or 
1 of the 


Timothy W. Coolidge. 


Jan. ^ 

Mar. 20 

Apl. 1 

NOTE.- 
idge has 
It appea 
which dt 
and tlier 


To sundries "2 
" Tuition 5 

To balance 

-The Dr. aide of this 
received of Die, and 
a that the total aino 
duct the $25 00 (whi 
3 will remain a balan 


R 

32 

7 

aero 
the C 
unt o 
- ti ata 
ceof 


17* 
00 

17* 
174 

unt ah 
r. side 
f my? 
ndsto 
S717i 


Jan 14 

Apl. 4 

nws the a 
shows w 
iccountaf 
his credit 
due me. 


By work 2f< 
" Balance 

\ 

\ 

mount of articles ! 
lat F have receive.^ 
'ainst liim if $32 1 
on the ri;jht of the 


,500 
7 17* 

g.17* 

Hr. Cool- 
of 1 hn. 
7J, from 
account) 


George H. Eaton. 


Jan. 6 

NOT*.- 
ttellcHi 
1* fully c 


To sundries ,$ 

-Thte account prese 
e J. H Eaton noihii 
loaed. 


1 

nts e 
g< an 


75 

iua) si 

1 tlrat 


Jan21 

ma on ho 
he owes 


By cash |3 

th sides; hence H 1 
me nothing. Tlu 


875" 

s evident 
; sccount 



Dr. 


Hiram Ames. Cr. 2 


Jan 1 i r 
Mar.O 


Fo sund.' 2 
" Corn 4 


713 

28 


2U 

75 


Jan. 11 
Feb. 13 
Apl. 1 


By sund. 
" Order 
" Bal 


2 
3 


7) 
10 
23 


00 
00 

95 






104 


95 








104 


95 


Apl. 1 


To Bal 


,23 


95 












Thomas Hilton. 


iFeb.3 
Mar 8 
" 21 


To sund. S 
Cash. 4 
do. G 


7 
18 
15 


Ot> 

75 
88 


Feb. 3 
Mar23 

31 
Apl. 1 


By labor, 
" hire of 
horse, 
Sund. 
Bal 


4 

5 
6 


11 : 

OS 
14 ( 
10( 


m 

>5 
)0 
)6i 






41 


69 








41 e 


>9 


Apl. 1 


To Bal 


10 


06* 












James Wilson. 


Jan. 11 

11 
Feb. 21 


To boar- 
ding, 
> " Sunds. 
L " do. 


2 6 

2 12 

4 7 


00 
13 
72 


Jan. 31 
April 1 


: By cash 
" Bal 


3 


15 
10 


00 

85 






25 


S5 








25 


85 


April ] 


I To bal- 
ance bro't 
down, 


10 


85 








s=s 


NOTE. When an account is settled only, and sot fully paid, as in the above, 
arid several preceding accounts, the balance, whether it he in your favor or 
against you, is brought down and placed distinctly by itself, and serves for the 
beginning of a new account, as you perceive has been done in the above exam- 
ple, the balance being f 1085. 



3 Dr. George Hamilton. Cr. 


Feb. 10 
Mar.15 


To Tea, 
" Sund. 


3 20 


25 
75 


Jan. T 
Mar 3 


J By chairs ft 
1 Sunds. 6 


11 


00 
50 


" 31 


" do. 


14 


00 


Apl. 


L Balance, 




10 


50 






47 


00 








4? 


00 


April 1 


To Bal 


10 


50 












Robert Young. 


Jan. 27 
Feb. 15 


To sund- 
ries, [ 

" do. * 


* 6< 
I 1.0 ( 


52* 
)0 


Feb.27 


By wood, 


4 


tf 


'00 


Apl. 1 


Bal. 


101 


m 


















27 ( 


)0 








2" 


^00 










Apl.l 


By J?aZ. 




It 


1374 


NOTE. I 


n the above ace 


cnmt th 


ediffi 


jrence bel 


ween the Dr. 


an 


d C 


r. side is 


$10 37^, by which I perceive that the balance against me, in favor of Robert 
Voting, is S 10 37*. 


Jackson Moore, 


Jan. 29 To Sun- 




Mar.31 J 


Jy paint- 












dries, c 


3 e 





ing, 


a 


21 


00 




Mar.31 


Sunds. t 


6C 














Apl. 1 


Bal. 


11 4 



















210 









21 


00 










~ Apl. 1 1 


}y bal- 
















a 


ince bro't 
















-, 


own, 




11 


40 





Dr. 


Ames 4* Smith. Cr. 4 


Mar. 6 
30 


To Sundries 
" do. 


4 
5 


1 i 

87 


00 

77 




By sugar, 1 


fa 

= 


77 
77 


98 


77 


NV*TK. 

amount or 


This account, like 
i the Dr. side Mng . 


the 
ust 


on 
equ 


3 on 
al to 


the first p 
that on the 


age, is fully c 
Cr. 


osed, 


the 


T. P. Lctton. 


Mar. 20 
" 20 


To Sundries 
do. 


5 

5 


'2 


87^ 
ti'2i 


Ap. 1 


By Bal. 


15 


50 










15 


50 






15 


50 


April 1 


To balance 




15 


50 










NOTE.- 

ence is, th 
tail, hi the 


In the above accoirr 
at T. P. Letton ow< 
Day Book, page 5. 


itt 

>a n 


iere 
ie $ 


is ni 
155(1 


i sum on tl 
for sundrj 


e Cr side, and 

articles ezpret 


the! 
eedii 


ifer- 
ide- 


Henry C. Sanxay. 


A P i i: 


Ho balance, 




1 s 

1C 


;p| 


Mar.28 


By my 
order, 5 


12 1 

10 f, 


371 










r = 


Apl. 1 


By bal 


121 


^7i 


NOTE. Tn f his account it will he perceived that as there in no amount char- 
ged to B.C. Sanxay, on the Dr. side, I owe hint $12 t!7|. 



5 Dr. 


Balance. Cr. 


April 1 


ToD. Judkins, 11 
" T. W. Coolidge, 1 1 
H.Ames, 2 
" T. Hilton, 2 
" J. Wilson, 2 
" G. Hamilton, 3 
" T. P. Letton, 4 


4 

2: 
lO 

10 
15 


W 
7* 

)5 ! 

So 

}.'i 
>o 


April 1 


By R. Youn, 
" J. Moore, 
" II. Sanxay, 


! 


101 

i ) 

12 


7J 
40 
-7| 








82 


54 








34 


G5 


NOTK. This account exhibits the exact state of your hooks. It is made 
from the preceding accounts in the Leser. The Dr. side is an exhibit of the 
amounts due to you by others, arid the Cr. side the amounts due by you to oth- 
ers. It is noi strictly necessary that this account 'should he introduced in tne 
Le<jer in single entry : it will he found convenient, however, to balance the 
book at stated intervals, and transfer the balances to the new accounts 1 elow, 
1 as in the preceding Leger, and when that is done, a balance account like the 
" above will be found convenient, as presenting, at one view, the exact state 
of your Leger. 
FORM OF A BILL FROM THE PRECEDING. 



January 4 

30 
March 8 

January 8 
30 


Mr. David Judkins, 
To Edward Thomson, Dr. 
To lOlbs. coffee at 17 cts. - - - - 1 7C 
25 Its. sugar at 10 cts. .... 2 5( 


4 


3 

1(5 

12 


20 
00 
50 

TO 
40 


12 bbls. apples at 75 
200 IDS. flour at $1 


cts. 


. . . . 


00 
40 


Cr. 

3 


47 bushels corn at 20 els. .... 9 




Errors excepted. Balance due 
Rec'd payment in full, EDWARD THOMSON. 


4 


30 




THE CASH BOOK. 




The Cash Book is used to record the daily receipts and payments of mo- 
ney. It ia ruled nearly the same as the Leger ; the Dr. side exhibits the 
amount of money received, and the Cr. side, the amount paid out. Subtract 
the sum of the Or. from that of the Dr. and the balance will always be equal 
to the amount of cash on hand. 




FORM OF A CASH BOOK. 




Dr. 


Cash. 


Cr. 


1833 
Jan. 1 
" 1 
1 
" 1 


To cash on ha:.d, 
Cash rec'd oC J. Young. 
" '' H. Sanxay. 
" " I). Judkins 


7; 

H 

"25 
1C 


81 
40 
GO 
00 


J833 
Jan. 1 

" 1 
" 1 


By rent of house paid 
T. P. Letton, 
Paid note to R. Hand, 
Family expenses, 
By cash on hand, 


18 
5(J 
4 
52 


UO 

00 

37 
84 








Li>5 


21 








12o21 


Jan. 2 
" 2 
" 2 


To cash on hand, 
of T. Coolidge, 
Cash found on Main St. 


59 

2: 
29 


81 
16 
00 


Jan. 2 

" 2 


By cash paid Ames & 
Smith, 
By cask an hand, 


20 
85 


00 
00 








'Or, 


00 








105 00 ! 


Jan. 3 


To cash on hand, 


65 


00 











Talbott, J.;, 




THE UNIVERSITY OF CALIFORNIA LIBRARY 



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