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IN MEMORIAM FLOR1AN CAJORI TUB WESTERN CALCULATOR, NEW AND COMPENDIOUS SYSTEM PRACTICAL ARITHMETIC; CONTAINING THE ELEMENT \RY PRINCIPLES AND RULES OP CALCULATION IN WHOLE, MIXED, AND DECIMAL NUMBERS, ARRANGED, DEFINED, AND ILLUSTRATED, IN A PLAIN AND NATURAL ORDER 5 ADAI'TEP TO THE USE OF SCHOOLS, THROUGHOUT THE WESTERN COUNTRY AND PRESENT COMMERCE OF THE UNITED STATES. % IN EIGHT PARTS. BY J. STOCKTON/ A.M. PITTSBURGH : PRINTED AND PUBLISHED BY JOHNSTON & STOCKTON MARKET STREET. STEREOTYPED BY J. HOWE- 1880. Entered according to IK af the Congress, in the year 1832, bv JOCNSTON & STOCKTWR, jfr Clerk's office of the District Court of the Western District W ' . PREFACE. AMONG the many systems of Arithmetic now used in our American schools, though each has its individual merit, yet all contain many things which are either entirely useless, or *of but little value to most beginners. It is to be regretted also, that in most of these systems, even if? those parts which are valuable and important, the authors appear not to have been sufficiently aware of giving a plain and natural arrange, ment and system to the whole. The;e is not that visible connexion * / between the parts, which enables the attentive pupil to discover, as he ] / progresses, that he is learning a system, and not a number of separate * ^ and unconnected rules. In many things, also, more attention has been given to gratify the Jr^quirifjff of tliR jirn/ffipMt^thnn to furnish plain, but necessary instruc- tion to the beginner. The age, capacity, and progress of the scholar are also overlooked ; and a mode ^f instruction _too_ learned^ and too elaborate, is pursued. It is forgotten how difficult even the most sim- ple parts are to a young mind ; nor are the instances few, in which even the variety~of ways laid down, in which the same question may be solved, leaves the learner perplexed, and swells the size of the work. To remedy, in some measure, these defects, and to furnish our nu- merous schools, in the western country, with a plain and practical trea- tise of Arithmetic, compiled and printed among ourselves, thereby saving a heavy annual expense in the purchase of such books, east of the rnnnnt?iins T and {ikewi^ tliP ./..arringp tbageof^liave been the mo- lives which induced the compiler to undertake this work. In it the following objects have been steadily kept in view : 1st. Plainness and simplicity of style, so that nothing should be in- troduced above the common" capacities of scholars, at the early age in which they are generally put to the study of Arithmetic. 2d. A natural and lucid arrangement of the whole, as a systelii, in which the connexion and dependence of all the parts may be easily discovered and understood. To accomplish this object, the work ia divided into eight parts, following each other, in what appears to tnc compiler the natural and simple divisions of the science. Each of these parts is again divided into sections, following the same connected arrangement. In each of these sections, the rules are expressed in a M305992 IV PREFACE. short and plain manner, and each rulo is illustrated, with a few easy and familiar examples, gradually proceeding from that which is sim- pie, to such as are more abstruse and difficult. 3d. Clearness and precision in the definitions, directions, and exam- ples. Carefully explaining every technical term when first used, and thereby guarding against ambiguity and uncertainty. 4th. Brevity in each part, so that every thing useless, or unimport- ant, may be excluded ; in order that the work may find its way into schools at the cheapest rate ; that parents, when examining the school- books of their children, may not find, whilst one part is worn out, the other is untouched, and half the price of the book entirely lost. How far these objects have been obtained, must be left to the deci- ' sion of time. Should the work be found to aid the progress of scholars, in acquiring a practical knowledge of this useful science to save ex- penses in a book so many of which are required ; and be found a use- ful assistant to merchants, mechanics, and farmers, as well as in some degree to lessen the labor of teachers in this branch of the sciences ; the compiler will have obtained his object. NOTICE TO THE FOURTH EDITION. THE favorable reception, and wide circulation, of the former edi- tions of the Western Calculator, stimulate the author to make it still more deserving of public patronage. He has, therefore, at the suggestion of several respectable teachers, given sundry additional questions to some of the rules ; and also some other alterations, which several years' experience in teaching has pointed out. The greatest care has been taken to prevent errors from appearing. in this edition. Pittsburgh, February 1, 1823. APHORISMS FOR THE SCHOLAR'S CAREFUL CONSIDERATION AND ATTENTION. KNOWLEDGE is the chief distinction between wise men and fools ; between the philosopher and the savage. The common and necessary transactions of business can- not be conducted with profit or honesty, without the know- ledge of Arithmetic. He who is ignorant of this science must often be the dupe of knaves, and pay dear for his ignorance. Banish from your mind, idleness and sloth, frivolity and trifling ; they are the great enemies of improvement. Make study your inclination and delight ; set your hearts upon knowledge. Accustom your mind to investigation and reflection ; de- termine to understand every thing as you go along. Commit every rule accurately to memory, and never resP satisfied until you can apply it. As much as possible do every thing yourself; one thing found out by your own study, will be of more real use than twenty told you by your teacher. Be not discouraged by seeming difficulties ; patience and application will make them plain. Endeavor to be always the best scholar in your class, and to have the fewest mistakes, or blots, in' your book. " The wise shall inherit honor, but shame shall be the promotion of fools." A2 EXPLANATION OF THE SEVERAL CHARACTERS EMPLOYED FOR THE SAKE OF BREVITY, IN THIS TREATISE. Two parallel lines, signifying equality: as, 100 cents= 1 dollar ; that is, 100 cents are equal to 1 dollar. -f- Signifying more, or addition: as, 6 + 4=10; that is, 6 arid 4 added make 10. This character is called Plus. A single line, signifying less, or subtraction : as, 6 4= 2 ; that is, 6 le^s 4 is equal to two. This character is called Minus. x Signifying Multiplication: as, 2X 4=8 ; that is, 2 mul- tiplied by 4 is equal to 8. -I- Signifying Division: as, 6-f-3=2 ; that is, 6 divided by 3 is equal to 2. :: : Signifying Proportion : as, 2 : 4 :: 6 : 12 ; that is, as 2 is to 4, so is 6 to 12 ; or, that there is the same propor- tion between 6 and 12, as there is between 2 and 4. v/ or V Signifying the square root of the number before which it is placed : as, \/64=8 ; that is, the square root of 64 is 8. <X Signifying the cube root : as, y/64=4 ; that is, the cube root of 64 is 4. A Vinculum, or chain : denoting the several quanti- ties over which it is placed, are to be considered as one simple quantitv THE WESTERN CALCULATOR* PART I ARITHMETIC IN WHOLE NUMBERS. ARITHMETIC is the art, or science, of computing by num- bers, and is generally divided into five j^incipal parts, or primary rules : viz. Numeration, Additioi* Subtraction, Multiplication, and Division. SECTION 1. OF NUMERATION. NUMERATION (or, as it is often called, Notation) is the art of expressing any given or supposed number, by the ten following characters : 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. The first of these is called a cipher, the rest are termed digits or figures. These nine digits are divided into three periods, three in each period. The first period includes units, tens, and hun- dreds. The second period includes thousands, tens of thou- sands, hundreds of thousands. The third period includes millions, tens of millions, hundreds of millions. Note. The cipher is also called nought and zero. They are all Arabic characters. NUMERATION. The relative value of each period, and the different fig- ures ID each period, may be learned from the following TABLE. 3d period. 2d period. 1st period. 9 9 9 8 8 7, 9 8 7, 6 9 8 7, 6 5 9 8 7, 6 5 4, 9 8 7, 6 5 4, 3 9 8 7, 6 5 4, 3 2 9 8 7, 6 5 4, 321, After the foregoing table, with the preceding definitions and explanations, are well explained by the teacher, and accurately committed to memory by the pupil, let him next proceed To write Numbers, Observing carefully the following RULE. Write down first, the given sum, in such figures as ex- press its value, and then supply the deficiencies therein with ciphers Application. Write down in figures the following numbers. 1. Sixteen. 2. Forty -nine. ADDITION. 9 3. Three hundred and eighty-five. 4. Two thousand six hundred and ten. 5. Sixty-four thousand, five hundred and thirty-six. 6. Two hundred and fifty-three thousand, eight hundred and forty-two. 7. Five millions, six hundred thousand and six. 8. Ninety millions, three hundred and five. 9. Eight hundred and twenty-nine millions, six thousand and two. Write dcnvn in words at length the following sums : 5, 17, 35, 458, 6829, 72348, 384721, 2683200, 50678024. Numbers are also expressed by letters, and are called numeral letters, or Roman numbers. Thus, 12.3456 7 8 9 10 11 20 30 1, II, III, IV, V, VI, VII, VIII, IX, X, XI, XX, XXX, 40 41 50 60 70 80 90 100 200 500 1000 XL, XLI, L, LX, LXX, LXXX, XC, C, CC, D, IVL When a letter of less value stands before one of a greater, it diminishes, but when placed after, it increases, the value of the greater. SECTION 2. OF ADDITION. ADDITION is of two kinds, viz. Simple and compound. Simple addition teacheth to collect two or more numbers of the same denomination into one sum : as, 6 dollars and 4 dollars, make 10 dollars. RULE 1. Write the different numbers in the given sum in such a manner, that the units may stand under units, tho tens under tens, the hundreds under hundreds, &c. till the whole is set down. 2. Begin with the column of units, and add it into OIK sum, carrying to the next column, one for every ten, and set down the remainder directly underneath ; proceed in the same manner fnnn tens to hundreds, &c. till all is finished. 10 ADDITION. Prove the work by reckoning downwards as well as up- wards, and if the amounts be equal the work is right. EXAMPLES. Dollars 2465 4832 6143 Yards 2. 468256 348928 764182 13440 added upwards. 13440 added downwards. Feet 3. 647502434 861948260 959276398 4. 258335091 5. 237680923 6. 919283746 138097653 423315687 213536978 573217809 098172635 321325687 532458976 523516533 978562313 532175633 653213563 321897553 249753290 327865309 213587921 7. 5643218624 8. 4 9. 6856789436 135940536 45 40590428 42006302 456 36491 9580469 4567 2849653210 550214 45678 540 32651 456789 34906 4168 4567890 3458000 324 45678901 300 68 456789012 9 4 4567890123 35 10. 3683678048934 11. 595536210486 2864948946496 376891345613 8498649476828 765248567446 3646280568245 684720476828 6421424H78427 852134567812 3678156496862 236889634567 76545S4964859 335678902345 SUBTRACTION. 11 Application. 1. Add 125-f 23 + 16 -f 2060 + 8009574 + 6. ^Iws. 8011804. 2. Add one hundred and twenty-nine, six hundred and fifty-four, eight thousand and seventy, ten thousand, and four millions. Ans. 4018853. 3. If I have received 125 dollars from A, 286 from B, 29 from C, 672 from D ; how much have I received from all four? Ans. 1112. 4. Bought 60 barrels of flour from one man for 480 dol- lars, 75 barrels from another for 675 dollars, 220 from an- other for 2200 dollars, and 126 from another for 1386 dol- lars ; how many barrels of flour had I, and how much did they cost me ? Ans. 481 barrels, and cost 4741 dolls. 5. A farmer raised in one year 297 bushels of wheat, 125 of rye, 754 of corn, 127 of barley, and 245 of oats ; how many bushels did he raise in all ? Ans. 1548. 6. Add one thousand two hundred and nine, four hun- dred and seventy-six, eight thousand and seventeen, three millions, one hundred and nineteen thousand, two hundred and twenty-one together. Ans. 3128923. 7. James was born in the year 1811 ; in what year will be Le 21 years old? Ans. 1832. 8. A father bequeathed to his 5 sons the following sums, viz : to George he gave 3560 dollars, to William 3240, to Samuel 2850, to Henry 2555, and to Thomas 2226 ; how much did he bequeath in all ? Ans. 14431 dolls. SECTION 3. OF SUBTRACTION. SUBTRACTION is either simple or compound. Simple subtraction is the taking a less number from a greater, and thereby finding the difference. RULE. Place the less number under the greater, with units under units, tens undei tens, &c. ; begin with the units, and take the under figure r rom the upper, and then proceed with t ic 12 SUBTRACTION. tens, &c. in the same manner. But if the under figure is the greatest, then suppose ten added to the upper figure, and take the lower from that number, carrying 1 to the next place. Or, take the lower figure from 10 and add the upper one to the remainder. PROOF. Add the remainder to the less number, and that will equal the greater. From 446875296 Take 234521173 Rem. 212354123 Proof 446875296 From 76542189768 Take 32478127130 EXAMPLES. From 86250732493, Take 37014921872 Rem. 49235610621 Proof 86250732493 From 5417630912 Take 27096470 From 90621247680 Take 34567892000 From 100000000 Take 09999999 Application. 1. What was the age of a man in the year 1818, who was born in 1777 ? Ans. 41 years old. 2. A merchant owes 5648 dollars, and pays thereof 3460 ; how much is yet to pay? Ans. 2188 dolls. 3. D having on hand 1260 barrels of flour, sells to A 320, and to B ^435 ; how many barrels are yet unsold ? Ans. 505 barrels. 4. From six thousand take six hundred, and tell what remains. Ans. 5400. 5. Suppose a boy had 145 cents given him at one time, 15 at another, and 40 at another ; and he gave 35 cents for ;i penknife, Mf) Inr a slate, (M lor paper, and 30 for ap l les ; !io\v many rents has he left? Ans. 106 cents. MULTIPLICATION. 13 SECTION 4. OF MULTIPLICATION. MULTIPLICATION is either simple or compound. Simple multiplication is a compendious way of adding numbers of the same denomination into one sum. The number to be multiplied, is called the multiplicand. The number multiplied by, is called the multiplier. The amount produced, is called the product. The multiplier and multiplicand are often called factors. MULTIPLICATION TABLE. fl| 2] 3| 4| 5| 6| 7| 8| 9| 10J 11| 12] 2| 4| 6| 8|10J12|14|16| 18| 20| 22 24 3| 6| 9|12|15|18|21|24| 27| 30| 33| 36 4! 8|12!l6|20|24|28j32| 36| 40| 44| 48 5|10|15|20|25|30|35|40| 45| 50| 55 60 6|12|18|2430|36|42|48| 54| 60| 66| 72 7|14|21|2835|42|49|56| 63| 70| 77| 84 8|16|24|32|40|48|56|64j 72| 80| 88| 96 9|18|27|36|45|54|63|72| 81| 90| 99|108 10|20|30|40|50|60|70|80| 90|100|110]120 11|22|33|44|55|66|77|88| 99|110|121|132 12|*4|36|48|60|72|84|96|108|120|132144 This table must be committed to memory, with great care and accuracy, till it can be used without difficulty or hesita- tion by the scholar. Case 1. When the multiplier does not exceed 12. RULE. Place the multiplier under the multiplicand ; units under units, and tens under tejis, and then multiply as the table directs, taking care to carry 1 for every 10. EXAMPLES. 46274963 24639576 3675432568 24 S 0-25*9920 14 MULTIPLICATION. 246H5761 4708-44753 964703024 a 5 6 74057343 74020005 2901946808 246354276 8 9 11 Case 2. When the multiplier exceeds 12. RULE. Multiply by each figure in the multiplier separately, be- ginning with units, taking care to set the first figure in each product directly under its own multiplier. Then add as if. addition. EXAMPLES. 2345601 68523047653 234 2367 9382404 7036803 4691202 .,48870634 PROOF. Method 1. Change the multiplier and multiplicand ; and then, if right, the product from this multiplication will he equal to the first. Method 2. Cast the nines out of each factor separately . set dovvn the remainders and multiply them together ; casi rhe nines out of this product, and note the remainder ; then ;ast the nines out of the product, and if right, th* 3 two last remaindors will he equal. MULTIPLICATION. 16 EXAMPLES. Method 1. Method 2. 246 425 425 425 246 246 1230 2550 2550 492 1700 1700 984 850 850 104550 104550 104550 6 Note. This last method is not absolutely certain ; yet the probability is so great, that in general it may be relied on. 3. Multiply 5221 by 145 Ans. 757045 4. 23430 230 5388900 5. 3800920 80750 306924290000 S. 89536925 735 65809639875 7. 78965987 5893 465346561391 8. What will 75 bushels of wheat come to at 1,15 cents per bushel ? Ans. 86 dolls. 25 cents. 9. Bought 3950 Ibs. of coffee, at 29 cts. per Ib. what must 1 pay? Ans. 1145 dolls. 50 cents. 10. There are 12 pence in one shilling. How many are there in 40 ? Ans. 480 pence. Case 3. When the multiplier is the exact product of any two fac- tors in the multiplication table. RULE. Multiply the given sum by one of these ; and that produci multiplied by the other, will give the number required. EXAMPLE. 1. Multiply 4236 by 16. 41 4X4=16. Product 67776 J 2. Multiply 871075 by 21 Ans. 18292575 3. 2453642 36 88331112 4. 43102 64 2758528 5. 23645 144 3401880 tt 12071 99 48.00 3600 by 400 44000 550000 663000 60000 Ans. 4800 1440000 24200000000 39780000000 16 MULTIPLICATION. Case 4. When there are ciphers at the right of one or both the factors. RULE. Omit them in the operation, but annex them to the product EXAMPLE. 1. Multiply 240 by 20. 24.0 2.0 2. 3. 4. Note. When the multiplier is 10, the product will be found by add- ing one cipher to the multiplicand ; if 100, add two ciphers ; if 1000 add three ; &c. EXAMPLE. 1. Multiply 200 by 10 Ans. 2000 2. 462 100 46200 3. 879 1000 879000 Application. 1. A gentleman owes 25 laborers 15 dollars each ; how much does the whole come to ? Ans. 375 dolls. 2. A saddler owes his journeyman for 43 days' work, ai 125 cents per day ; how much does he owe him in all ? Ans. 53 dolls. 75 cts. 3. A merchant buys 440 yards of muslin at 32 cents per yard ; how much does the whole cost? Ans. 140 dolls. 80 c. 4. A farmer sells 60 bushels of wheat at 125 cents per bushel ; 40 bushels of rye at 85 cents ; 34 of corn at 50 cents ; how much is he to receive for each, and how much does the whole amount to 7 Ans. 75,00 cents for the wheat, 34,00 cents for the rye 17,00 cents for the corn ; and the. whole amounts to 126,00 <-<-nts, or 126 dollars. .1. A dollar is equal to 10 dimes, and a ciime is equal to 10 cents ; how many dimes and cents are there in 100 dol- Ans. 1000 dimes, and 10,000 cents. 6. llo\v_ many panes of glass are then: in a house that has 32 windows, 20 of which have 2 4 lights <--ach, and the r<st hav*' J - : ca<-h .' Ann. 096 panes. 17 7. What sum is equal to 7525 multiplied by 125? Ans. 940625. 8. A has 250 dollars, B has three times as many, and C has four times as many as B ; how man) 7 dollars have B and C each, and how many have they altogether ? Ans. B has 750 dolls. C. 3000 dolls, altogether $4000. SECTION 5. OF DIVISION. Division is either simple or compound. Simple division is finding how often one number is con- tained in another of the same name, or denomination. * The number given to be divided, is called the dividend. The number given to divide by, is called the divisor. The result, or answer, is called the quotient. Case 1. When the divisor does not excised 12. RULE. Find how often the divisor is contained in the first figure or figures in the dividend, under which set the result, if any remain, conceive it as so many tens added to the next figure, and then proceed in the samp manner. Division is proved by multiplying the quotient by^the di- visor, and adding the remainder, if any : the amount will equal the dividend. EXAMPLES. Divisor 2)46578238 3)672245139 4)4756394344 Quotient 23289119 2 Proof 46578238 224081713 3 672245139 5)97036142 8)37846210 12)64381259 6)3824966 7)46825486 9)8297463813 18 DIVISION. Case 2. When the divisor exceeds 12. RULE. Begin with as many of the first figures in the dividend as will contain the divisor. Try how often the divisor is con- tained therein, and set the result in the quotient. Subtract the product of the divisor multiplied by the quotient figure from the dividual above, to this remainder annex the rvxr figure in the dividend for a new dividual, arid so proceed tii : all the figures in the dividend are brought down. Note. A dividual is when one or more figures of the dividend, (in the operation of long- division) are divided separately from the rest. EXAMPLES. Divis. 42)9870 (235 Quot. 84 42 41)94979 (2316 82 41 147 126 210 210 470 . 940 9870 proof. 123 67 41 2316 9264 94956 23 rem. 269 94979 pr. 246 23 Rem, 3. Divide 29687624 4. 47989536925 Quotient. by 64 Ans. 463869 735 65291886 and 8 Rem. 715 5. 4917968967 2359 2084768 1255 6. 5374608 671 8009 569 7. 19842712000 175296 113195 81280 M. 5704392 108 52818 43 Case 3. When the divisor is the exact amount of any two factors in the table. RULE. Divide the given sum by any one of these, and the quo- tient by the other. DIVISION. EXAMPLE. Divide 9870 by 42. 6 ) 9870 7 ) 1645 235 Ans. Case 4. Vhen one or more ciphers stand on the right of the divisor. RULE. Omit them in the operation, cutting off from the right of the dividend as many figures, taking care to annex them to the remainder. EXAMPLE. 1 . Divide 2564 by 200. 2.00)25.64 12 1 Rem. 64 Quot. 12 164 Rem. 2. Divide 87654 by 600 Ans. 146 54 Rem. 3. 28347 ' 80 354 27 4. 137000 1600 85 1000 Note. When the divisor is 10 the quotient will be had by cutting off one figure from the right of the dividend, when the divisor is 100 cut off two figures, when it is 1000 cut off three figures, &c. When the figures cut off from the right of the dividend are digits, they are to be considered as so much of a remainder. EXAMPLE. I. Divide 5640 by 10. 1.0)564.0 Ans. &64. L\ Divide 25654 byj 100 Ans. 256 54 Rem. '3. 876029 1000 876 29 4. 800000 10000 80 Application. 1. Several boys went to gather nuts, and collected 4275 : when they had divided them, each had 855 ; how many boys were in company? Ans. 5. 20 DIVISION. 2. If 2072 apple trees were planted in 28 rows, how many would there be in each row ? Ans. 74. 3. If 45000 dollars were divided among 75 persons, how many would each one receive? Ans. 600. 4. Into how many parts must I divide the number 8164. so that each part may be 27, leaving the remainder 10? ' Ans. 302. 5. There is a certain number, to the double of which if you add 12, then 5 times that sum will equal 150 ; what is that number. Ans. 9. 6. A father dying, left 13440 dollars to be divided among his 6 sons in the following manner, viz. to the eldest one- fourth part, to the second one-fifth, to the third one-sixth, to the fourth one-seventh, to the fifth one-eighth, and to the youngest the remainder ; what was each son's share ? Ans. 1st 3360, 2d 2688, 3d 2240, 4th 1920, 5th 1680, 6th 1552 dolls. 7. What number^ if multiplied by 72084, will make 5190048? Ans. 72. 8. A, B, and C, engage to do a piece of work for 228 dolls* which together they accomplish in 40 days : now it was previously agreed that A should have 1 cents per day more than B, and" B 10 cents more than C ; what was each man's share? Ans. A 80, B 76, C 72 dolls. 9. A man on counting his money, found he had an equal number of half eagles, (5 dollar pieces) half dollars, and quarter dollars, and that the whole amounted to 1437 dol- lars 50 cents ; how many pieces of each kind had he ? Ans. 250 of each kind. 10. The crew of an armed ship, consisting of the cap- tain, mate, and 40 men, took a prize worth 4550 dollars now it was agreed that the captain should have 6 shares, the mate 4, and each seaman 1 share ; what did each one receive ? Ans. The capt. 546 dolls, the mate 364, and each seaman 91 dolls. As but few examples are given under each of the foregoing rules, it is recommended that every teacher add as many similar ones, as may be found necessary to make the pupil well acquainted with their appli cation, and both expert and accurate in working such questions as properly belong to these rules. Every experienced teacher is well aware that until this knowledge is obtained by the scholar, every at- tempt at any thing farther is only a waste of time and money. When this knowledge is once acquired, the future progress of the scholar will FEDERAL MONEY. 21 oe pleasant and rapid. The teacher will then be justly rewarded lor his labor and trouble in this part, by the approbation of parents, and the gratitude of his scholars, who will have acquired the necessary qualifi- cations (accuracy and expertness) for the great variety of studies and avocations in future life, which require the aid of arithmetic and math- ernatics. PART II. ARITHMETIC IN MIXED OR COMPOUND NUMBERS. SECTION I. FEDERAL MONEY Is so called from its being the general currency establish- ed by the Federal, or United States' government, and is justly considered superior to every other kind of currency now in use for its simplicity and plainness. Standard weight as establieh- ed by law. Its denominations are, dwt. gr. 10 Mills make 1 Cent 10 Cents 1 Dime 1 16 f, > gn 10 Dimes or 100 Cents 1 Dollar 17 ' l| < 10 Dollars 1 Eagle 11 4| 5 n } , i Eagle 5 14J- $ b From this table it will readily be seen, that addition, sub- traction, multiplication, and division of federal money may be performed as if they were whole numbers. It will also be seen that to reduce any number of mills to cents, it is only necessary to point, or cut off the last figure, as 100 mills = 10,0 cents; an^l cents in the same way to dimes, as 100 eents=:10,0 dimes, and dimes to dollars, as 100 dimes=:10,0 dollars, and dollars to eagles, as 100 dollars .10,0 eagles; and also that eagles may be brought to dolls, and dolls, to dimes, &c. by adding a cipher to each one, as 10 E.= 100 dolls. = 1 OOOd. = 1 OOOOc. 1 OOOOUm. In all calculations in federal money, according to com- mon custom, it is usual to omit the names of eagles, dimes, 22 FEDERAL MONEY. and mills, and only to reckon by dollars and cents; the eagles being considered as so many 10 dollars, the dimes as so many 10 cents, and the mills as fractional parts of the cent. See the following . TABLE. i s Si Dolls. 1, 2 5 1,25 cents 3 4, 1 2 i 34,12 and a fourth cents 4 5 6, 2 5 i 456,25 and a half cents 8 2 6 4, 7 5 | 8264,75 and three-fourth cents Note. 1. In addition, subtraction, multiplication, and division of federal money, if the sums are dollars only, the amount, remainder, product, or quotient, will be dollars; but when the sums consist of dollars and cents, or cents only, the two first right hand figures arc cents, and all the rest are dollars. 2. When fractions of cents are used according to the above table, every four of them make one cent : in adding, or subtracting these, we carry one for every four ; and in multiplying, the upper figure, called the numerator, is to be multiplied by- the multiplier, and divided by the lower figure, called the denominator. EXAMPLES OF ADDITION. Edd cm DC DC 25,6,4,8,2 5675,25 53258,75* 24.7.6.2.4 2386,63 93620,33^ 63.8.1.3.5 3972,80 30176,56$ 92.2.3.4.6 7285,75 . 27532,35 206,4,5,8,7 SUBTRACTION. E d d c m DC D 83,6,5,3,5 32,9,3,7,5 50,7,1,0,0 FEDERAL MONEY. 23 MULTIPLICATION. Edd cm DC DC 23,6,3,5,7 2637,25 6378,75 369 70,9,0,7,1 DIVISION. Edd cm DC DC 2)63,3,8,6,2 5)3632,75 8)82750,33 31,6,9,3,1 Promiscuous Questions. 1. Add 25 eagles, 62 dollars, 8 dimes, 75 cents, and 5 mills. Ans. 3.13d 55c 5m. 2. A person deposited at bank 1055 dollars in notes, 260 dollars in gold, 3650 dollars in silver, and 2,50 cents : how much is the amount? Ans. 4967d 50c. 3. Bought a barrel of sugar for 39 dollars 87 cents, a bag of coffee for 22 dollars 18J cents, and a pound of tea for 2 dollars 12^ cents ; how much do they all cost? Ans. 64d 18jc. 4. Bought goods to the amount of 645 dollars 95 1 cents, and paid at the time of purchase 350 dollars ; how much re- mains to be paid ? Ans. 295d 95 f. 5. A man lent his friend 1000 dollars, and received at sundry payments, first 160 dollars 25 cents, second 285 dollars 66-J cents, third 300 dollars 28| cents ; what remains yet to be paid? Ans. 253d 79fc. 6. What is the product of 102 dollars 19 cents, multiplied by 120? Ans. 12262d 80c. 7. What will 16 barrels of flour amount to, at 4 dollars 50 cents per barrel ? Ans. 72d. 8. How much will 132 pieces of calico come to, at 11 wllars 37 i cents a piece? Ans. 2293d 50c. 9. What is the quotient of 6022 dollars 50 cents, divided by 5? Ans. 1204d 50c. 10. A butcher bought 18 beef cattle for 252 dollars 90 cents; how much did he pay for each? Ans. 14d 05c. 11. Bought 45 yards of linen for 22 dollars 50 cents, what was the price of one yard? Ans. 50cU- 24 COMPOUND ADDITION. 12. If 25 men expend 15555 dollars 50 cents in the erec- tion of a bridge, how much has each one to pay, if the shares are equal ? Ans. 622d 22o Having treated of federal money separately, inasmuch as it requires to be well understood, seeing it is the general currency in the United States; we now proceed to the other parts of mixed numbers, or as they are frequently termed, divers denominations. SECTION 2. OP COMPOUND ADDITION. COMPOUND Addition is the collecting together, and thereby ascertaining the amount of several quantities, of divers de- nominations. RULE. Place the numbers in such a manner, that all of the same denomination may stand directly under each other, then be- ginning with the lowest denomination, add as in whole num- bers, carry at that number which will make one of the next greater; set down the remainder (if any) and so proceed till nil are added. PTCOOF as in simple addition. ENGLISH MONEY. The denominations are, pounds, shillings, pence and far- things, and are Thus valued : 4 farthings (marked qr.) make 1 penny (marked) d. 12 pence 1 shilling *. 20 shillings 1 pound . TABLE OF SHILLINGS. s. . s. 20 shillings make 1 00 30 .... 1 10 40 .... 2 00 50 .... 2 10 60 .... 3 00 70 .... 3 10 80 .... 4 00 90 . . . 4 10 100 . 5 00 PENCE TABLE. d. 20 pence make 30 . . . 40 ... 50 ... no . . . 70 ... 90 ... 00 ... 100 ... *. d. 1 8 2 6 3 4 4 2 5 5 10 6 8 7 6 a 4 COMPOUN D A DDITIO3N . 25 EXAMPLE* . s. d. . s. d. qr. . s. d. qr. 12-*6 11 8 35678 11 9 \ 2368 17 5 4 9462 8 4 37562 18 7 $ 3969 19 11 | 3215 10 6 63497 15 10 i 9386 14 6 $ .13934 10 6 raor WEIGHT. This weight is used for jewels, gold, silver, and liquors. The denominations are, pounds, ounces, pennyweights, and grains. Thus valued. 24 grains (gr.) make . 1 pennyweight dwt. 20 pennyweights . . 1 ounce oz. 12 ounces .... 1 pound ft. EXAMPLES* ib.oz.dwt.gr. lb. oz. dwt. gr. lb.oz.dwt.gr. 4 10 15 16 5 8 11 16 45 17 11 8 6 10 11 9 10 15 21 9 6 12 9 6 9 14 23 6 11 18 17 18 11 19 23 20 3 1 2 AVOIRDUPOIS WEIGHT This weight is used for heavy articles generally, and all metals but gold and silver. Tlie denominations are, tons, hundreds, quarters, pounds 1 ounces, and drams. Thus valued. 16 drams (dr.) make ... 1 ounce oz. 16 ounces 1 pound lb. 28 pounds 1 quarter qr. 4 quarters, (or 112 lb.) . . 1 hundred cwt. 20 hundreds 1 ton T. C 2f> COMPOUND ADDITION EXAMPLES. T. cwt. qr. Ih. oz. dr. T. cwt. " qr. Ib. oz. dr. 10 16 2 24 9 12 856 12 3 19 11 10 15 11 1 15 12 9 537 19 1 23 8 9 85 8 3 19 13 13 638 10 2 21 12 6 18 15 1 14 10 8 897 19 3 27 15 15 APOTHECARIES' WEIGHT. This weight is used by apothecaries in mixing medicines, .' "t they buy and sell by avoirdupois. The denominations are, pounds, ounces, drams, scruples, and grains. Thus valued. 20 grains (gr.) make ... 1 scruple sc. or 9 3 scruples 1 dram dr. 3 8 drams , . 1 ounce oz. 3 12 ounces 1 pound Ib. ft EXAMPLES. Ib. oz. dr. sc. gr. Ib. oz. dr. sc. gr. * 5 2 1 16 17 5 7 2 14 3 11 7 2 19 80 3 2 1 16 6 8 6 1 12 85 10 3 2 5 5 2 4 2 9 36 6 2 1 15 CLOTH MEASURE. By this measure, cloths, ribands, &c. are measured. The denominations are, Ells French, Ells English, Elh Flemish, yards, quarters, and nails. Thus valued. 4 nails (na.) make ... 1 quarter qr. 4 quarters I yard yd. 3 quarters i Kll Flemish E. FL 5 quarters 1 Mil Kn^iish E. En. quarters i Kll French E. Fr. EXAMPLES. Yd. qr. na. E. 1?l. qr. nn. E. Fr. qr. na. E. En. qr. na 56 2 2 HO 2 3 16 4 2 53 4 3 sG 13 18 1 2 17 5 1 53 3 ? 33 3 2 36 2.1 80 2 2 32 2 1 1*8 2 1 :W 21 13 3 3 81 COMPOUND ADDITION. 27 LONG MEASURE. By this, lengths and distances are measured. The denominations are, degrees, leagues, miles, furlongs, poles, rods or perches, yards, feet, inches, and barley- corns. *y Thus estimated. 3 barleycorns (be.) make 1 inch in. 12 inches 1 foot ft- 3 feet 1 yard yd. 5 yards (or 16J feet) . 1 rod, pole, or perch P. 40 poles rods, or perches, > fo , /wn (or 220 yards) $ 8 furlongs (or 320 poles, ? -, -, a/ r ?-r^n i \ t 1 mi l e * or 17 GO yds.) $ 3 miles 1 league L. 60 geographic, or ) ., , flft-f * * miles 1 degree deg. 69i statute ^ 360 degrees make a circle, or the circumference of the earth. A hand is a measure of 4 inches, and a fathom of 6 feel. EXA3fPLES. deg. m. fur. po. yds. ft. in. be. L.' M. fur. yds. ft. in. 50 30 5 15 2 2 9 2 5 2 6 75 2 11 60 25 7 12 4 1 10 1 3 1 4 95 1 9 75 35 2 92281 2 1 3 15 2 8 20 55 6 8 1 1 11 2 125 200 1 6 LAND MEASURE. By this the quantity of land is estimated. The denominations are, acres, roods, perches, yards, anl feet. Thus rated. 9 feet (ft.) make . . . . . 1 yard yd. 30^ yards ..._....! perch P. 40 perches 1 rood R. 4 roods (or 160 perches) . . 1 acre A. 28 COMPOUND ADDITION. EXAMPLES. A. R. P. A. R. P. A. R. P. 25 3 20 265 2 15 246 3 29 33 1 16 375 1 29 762 1 12 33 2 34 860 3 39 632 2 11 68 1 39 632 2 20, 357 3 20 CUBIC, OR SOLID MEASURE. By this, wood and other solid bodies are estimated. The denominations are, cords, tons, yards, feet, and inches Thus rated : 1728 inches (in.) make ... 1 foot ft. 27 feet ....... 1 yard yd. 40 feet of round timber, or > ^ tQQ y 50 feet of hewn timber $ 128 feet 1 cord cor. EXAMPLES. Co. ft. in. T. ft. in. T. ft. in. 4 112 1260 6 39 1384 23 *12 1400 6 84 1500 2 26 526 68 45 1600 8 127 1700 8 18 260 82 49 1700 5 63 403 3 12 1100 96 18 50 TIME. This relates to duration. The denominations arc, years, months, iceeks, days, hours, minutes, and seconds. The relative differences are these. 60 seconds (sec.) make ... 1 minute mi. (>0 minutes 1 hour h. 24 hours 1 day d. 7 d.'iys 1 week w. 4 weeks 1 month M. 12 months, 5'^ weeks, or 365 > . . vr days and 6 hours \ Note. The solar year, according to the most .xnrt observation, con tains 3G5dav.s, 5 JuMirs, 4ft minutes, 57 seconds. COMPOUND ADDITION 21) TTie number of days in the 12 calendar months, is thus found: Thirty days are in September, April, June, and November ; February hath twenty-eight alone, And all the rest have thirty-one. JNote. Every fourth year is called bissextile, or leap-year, in which February has twenty-nine <days. EXAMPLES. Y. M. da. h. mi. sec. Y. da,, h. mi. sec. 22 10 25 16 34 55 4 350 15 19 5 34 6 16 20 48 33 2 268 13 54 38 46 9 13 23 59 59 6 350 22 50 50 MOTION. This relates to the measure of circles. The denominations are, circles, (or revolutions) signs, de- grees, minutes, and, seconds. The relative differences are, 60 seconds (sec.) make - 1 minute mi. or ' 60 minutes ..... 1 degree deg. or 30 degrees - - - . - 1 sign sig. 12 signs (or 360 degrees) 1 circle EXAMPLES. sig. deg. mi. sec. sig. ' " 2 24 48 58 3 20 30 40 2 29 59 59 2 25 35 45 3 21 20 20 3 2d 38 58 LIQUID MEASURE. This is used for measuring wine, spirits, cider, beer,*&c. The denominations are, tuns, pipes or butts, hogsheads, bar- rels, gallons, quarts, pints, and gills. Thus estimated. 4 gills (gi.) make - - 1 pint pt. 2 pints - - - - 1 quart qt. 4 quarts 1 gallon gal. 63 gallons - - - - 1 hogshead hhd. 2 hogsheads 1 pipe, or butt pi. bt 2 pipes (or four hogsheads) 1 tun T. Note. By a law of Pennsylvania, 32^ gallons make a barrel, and 16 gallons make a half barrel. C2 30 COMPOUND SUBTRACTION. EXAMPLES. T.hhd.gal. qt. pt. 4 3 53 2 I 6 2 25 3 1 8 1 62 1 1 T. hhd. gal. 24 2 33 19 3 54 34 1 50 DRY MEASURE. This is used for measuring grain, salt, fruit, &c. Tht denominations are, bushels, pecks, quarts, and pints. Thus estimated. 2 pints (pt.) make - - 1 quart qt. 8 quarts - - - 1 peck P. 4 pecks (or 32 quarts) 1 bushel bu. bu. P. qt. 25 2 4 36 3 6 34 1 2 78 2 7 EXAMPLES. bu. P. qt. 256 3 6 243 1 6 468 3 1 584 2 7 bu. P. qt. 34156 3 7 2003 1 2 90 3 6 4809 SECTION 3. OF COMPOUND SUBTRACTION. COMPOUND SUBTRACTION is the taking a lesser numher from a greater, of divers denominations, and thereby finding l he difference. RULE. Set down the lesser number under the greater, as in com- pound addition. Then beginning with the lowest number, subtract as in subtraction of whole numbers : when the lower number is greater than the upper, take it from as many of that denomination as will make one of the greater, and to the remainder add the upper number, set down the amount and carry one to the next, and so proceed till all are subtracted. COMPOLLN 1> SUBTRACTION PROOF. Add the remainder to the lower line. EXAMPLES. . s. d-. From 25(T 15 6 Take 129 12 8 $ 31 T. cwt. qr. Ib. oz. dr. From 246 15 2 18 11 5 Take 89 16 1 24 8 15 Rem. 127 2 9 } Proof 256 15 6 mi. fur. P. ft. in. be. From 250 4 24 10 61 Take 125 6 30 5 10 2 D. h. mi. sec. From 325 18 30 24 Take 236 20 45 50 bn. , P. qt. pt. From 204 2 6 1 Take 150 3 2 sig. deg. mi. sec. From 6 16 32 29 Take 3 . 24 16 48 T. hhd. gal. qt. pt. From 50 2 45 2 1 Take 20 3 60 3 A. R. P. From 1658 2 Vj Take 1249 3 ?4 Promiscuous Questions in Compound Addition and Subtraction. 1. A merchant bought five pieces of linen, containing as follows: No. 1, 36 yards 3 quarters 2 nails; No. 2, 45 yards 1 quarter 3 nails ; No. 3, 48 yards 2 quarters 1 nail ; No. 4, 52 yards 3 nails ; No. 5, 64 yards 2 quarters ; how many yards were in all ? Ans.. 247 yds. 2qr. Ina. 2. Sold 5 head of beef cattle, at the following prices, viz : the first far 67. 2s. 4d. the second for 5Z. 10s. 9%d. /he third for 11. the fourth for SI. 10s. 6d. the 5th for Ql. 2s. 6d. and received. 2.21. 10s. 6d. in ready payment, and a note for the remainder ; how much did the cattle cost, and for how much was the note given ? Ans. The cattle cost 367. 6s. l^d. and the note wastfor 13/. 15*. l%d. 32 COMPOUND MULTIPLICATION. 3. A silversmith bought 26lb. 9oz. lOdwt. of silver, and . wrought up ISlb. \6dwt. Wgr. how much has he left ? Ans. Sib. Soz. ISdwt. llgr. 4. A physician bought Qlb. Woz. 6dr. 2sc. (apothecaries' weight) of medicine, and has used 4Z&. Soz. 4dr. Isc. I7gr. what quantity has he yet remaining ? Ans. 2lu. 5oz. 2dr. Osc. 3gr. 5. William was born on the 15th day of January, 1816, at 6 o'clock in the morning, and Charles was born on the 20th of March, 1817, at 9 in the evening ; how much older .6 William than Charles 1 Ans. 1 year, 2mo. 5d. 15h. ft An innkeeper bought four loads of hay, weighing as lollowing, viz. first load, 19 hundred 2 quarters and 14 Ib. second load, 16 hundred 3 quarters 18 Ib. third load, 22 hundred and 24 Ib. fourth load, 24 hundred and 1 quarter how much hay in all ? Ans. 4 tons 2 hundred. 7. From a piece of broadcloth which at first measured 55 yds. I sold to A 5J yds. to B 6|, to C 7|, to D a quan tity not recollected, and to E just half as much as to D ; on Pleasuring the remainder, I found there was 20J yds. left ; how many yards did D and E each receive 1 Ans. D 10, and E 5 yds. 8. A wine merchant bought 1 pipe 2 hhds. and 3 qr. casks of wine, each 26 galls. ; of these he sold 1 hhd. and -jr. casks; he also found that the pipe had leaked 17 galls, the remaining hogshead 11, and the cask 5-J ; how many gallons did he buy, and how many had he left ? Ans. Bought 330 galls, left 18l galls. 9. Bought 4 pieces of cloth, the two first measured 9 Ells Fr. 3 qr. 2 na. each, and the two last 8 Ells Fr. 2 qr. 3 na. each, of these I sold 40 5 yards ; how much have I left ? Ans. 13y. 2 qr. 2 na. SECTION 4. OF COMPOUND MULTIPLICATION. COMPOUND MULTIPLICATION teaches to multiply any given quantities or numbers of divers denominations. Case 1. When the multiplier does not excer<i 12. COMPOUND MULTIPLICATION RULE. 33 Begin by multiplying the lowest number first, as in inte- gers ; divide the product by that number which will make one of the greater. Set down the remainder, if any, and carry the quotient to the next number. KXAMPLES. Ib. 14 . 5. d. 24 10 6 cjr. T. cwt. qr. Ib. oz. dr. \ 48 14 1 4 12 11 2 3 ft. 1 11 2 ) 49 1 1 24 10 6 4 Proof. oz. dwt. gr. 4 11 11 5 bu. pc. qt. hhd. gal. qt. 24 3 7 25 48 3 8 * 8 mi. fur. p. 24 6 34 8 yds. y/. in. be. A. R. 24 282 89 3 6 P. 26 9 to/. 48 y?c. #. 3 6 11 d. h. mi. sec. Y. _m. w* d. 84 19 38 15 125 8 3 4 9 12 Application. I. 5 yards of cloth at .2. 9 ' do. at 3. 11 bushels of flax-seed at 4, 12 do. clover-seed at . s.d. 264 1 2 \>\ 12 9j 2 4 2| Ans. . 11 10 7 s. d. 11 8 2, 84 8i 26 10 6 Case 2. When the multiplier exceeds 12, uut is the exact produd of any two factors in the table- 1 . 34 COMPOUND MULTIPLICATION. RULE. Multiply the given sum by any one of these, in the same nanner as above, and the product by the oilier. EXAMPLES. . s. d. qr. Multiply 12 8 6 by 18=3x6 3 37 5 6 | 6 223 13 4 Application. 1. Multiply 4T. Scwt. Iqr. 16ZZ>. Soz. Wdr. by 36. Arts. 150 T. 2cwt. Iqr. lib. 6oz. 8dr. 2. 120Z. (5s. 9d. by 24. A/is. 238SZ. 2s. Od. 3. 24 T. 4e-7/tf. 2<p% lib. by 48. Aws. 1162T. Wcwt. Oqr. OZ/;. Case 3. When the multiplier is not the exact product of any two factors. RULE. Multiply as in the last case, by any two factors that will come the nearest to the multiplier, but less ; and add for the deficiency. EXAMPLES. bll. pC. qt. Multiply 12 2 4 by 17 4x4+1 = 17 50 2 4 202 = 16 12 2 4= 1 214 ^ 4 17 3. Multiply H/>. 4/>. I2?ni. f>v. by 2JK AH*. 237 />. I/?. 507/w. COMPOUND DIVISION. 35 Case 4. When the multiplier exceeds the product of any two fac- tors in the table. RULE. Multiply by the units figure, as in case 1, and set down the amount; again multiply the given sum b}* the figure of tens, and that product by 10, and place tins amount under the first ; again multiply by the figure of hundreds, and the product by 10 and 10, which set down under the other pro- ducts- in the same way for thousands, by three tens, &c. EXAMPLES. - s. d. 2626 Multiply 2 6 by 245 4 2 5 10 50 12 6 first, or units product 10 10 500 second, or tens do. 25 third, or hundreds do. 500 2100 10 Ans. 30 12 6 total. 25 . s. d. . s. d. 2. Multiply 14 6 by 240 Ans. 174 3. 123 117 130 3 3 4. 126 275 309 7 6 SECTION 5. OF COMPOUND DIVISION. COMPOUND DIVISION, teaches to divide any sum or quan- tity of divers denominations. Case 1. When the divisor does not exceed 12. RULE. Divide the highest, or left-hand denomination; if any re- mains, multiply by that number which will reduce it to the next highest, add this product to the second, then divide as Df.'ibre, and so proceed till all are divided. 30 COMPOUND DIVISION. PROOF By compound multiplication. EXAMPLES. . s. d. qr. . s. d. qr 2)465 10 6 i 3)563 15 4 Quotient 232 15 3 2 Proof 465 10 6 T. cwtf. f/r. Ib. yds. ft. in. 6)91 16 1 14 5)960 1 9 T. Mid. gal. qt. . w. d. h. mi. sec. 8)468 1 48 3 10)30 6 18 48 50 Case 2. When the divisor is the exact product of any two factors in the table. RULE. Divide first by one as above, and the quotient by the other. EXAMPLES. 6)224 12 6 by 30=6x5 5) 37 8 9 Quotient 7 9 9 . s. d. . s. d. 2. Divide 134 18 S by 44 Ans. 3 1 4 3. 9H4 144 6 16 8 4. 474 72 6 11 8 Case 3. When the divisor is not the exact product of any two (ac- tors in the table, or eA,:ee.ds them. RULE. Divide the highest denomination in the ^iv<-n sum, in the .s.-iuic manner as in case 2, of whole numbers; reduce I lit remainder, if any, to the next lower denomination, adding it COMPOUND DIVISION. 37 to the number of the same denomination in the given sum ; di- vide this in the same manner, and so proceed till all are divided. EXAMPLES. 1. Divide 264Z. 10s. 7$d. by 25. 25)264/. 10s. l^d. (Wl.lls.l^d. Ans. 25 20 25)290(11 25 ~40 25 15 12 25 ) 187 ( 7 175 Is 4 25)50(2 50 . s. d. . s. d. 2. Divide 409 13 9 by 345 Ans. 139 3. 232 4 9 524 8 l(5i 4. 3236 12 4J 654 4 l3 llj 5. 132 8 68 1 18 10 Promiscuous Questions, for exercise, in the foregoing rules of Compound Addition, Subtraction, Multiplication and Division. 1. What is the value of 672 yards of linen at 2s. 5d. per yard? Ans. 81Z. 45. 2. A goldsmith bought 11 ingots of silver, each of which weighed 4db. loz. I5dwt. 22gr. how much do they all weigh? Ans. 45Z&. loz. 15dwt. 2gr. 3. Bought 8 loads of hay, each weighing 1 ton 2 hundred 3 quarters 16 pounds; how much hay in all? Ans. 9 ton 3 hun. 161b. 4. Divide 9 ton 3 hundred 161b. into 8 shares. Ans. 1 ton 2 hun. 3qr. 16lb. 5. Bought 15 tracts of land, each containing 300 acres 2 roods and 20 perches ; what is the amount of the whole ? Ans. 4509 acres 1 qr. 20 rods. D 38 COMPOUND DIVISION. 6. Divide a tract of land containing 4509 acres 1 rood and 20 perches equally among 15 persons; what is each one's share? Ans. 300 acres 2 roods 20 perches. 7. Bought 179 bushels of wheat for 201 dollars 37^ cts. what is it per bushel? Ans. I doll. 12^ cents. 8. If a man spends 7 pence per day, how much will it amount to in a year? Ans. 10Z. 12s. lid. 9. What is the value of 1000 bushels of coal at 10| cents per bushel? Ans. 105 dolls. 10. Bought 135 gallons of brandy at 1 dollar arid 62 cents per gallon, which was sold for 2 dollars and 5 cents per gallon ; required the prime cost, what it was sold for, and the gain? Ans. prime cost 219 dolls. 37^ cts. sold for 276 dolls. 75 cts. gain 57 dolls. 37 cts. 11. If 27 cwt. of sugar cost 47/. 12s. W$d. what cost 1 cwt. ? * Ans. ll. 15s. 3^d. 12. Suppose a man has an estate of 9708 dollars, which he divides among his four sons : to the eldest he gives f , and to the other three an equal share of the remainder ; what is the share of each? Ans. eldest son, 3883 dolls. 20 cents, other sons, each 1941 dollars 60 cents. 13. A dollar weighs lldwt. Sgr. what will 45 dollars weigh at that rate? ^Ans. 39oz. 14. An eagle of American gold coin should weigh lldwt. tigr. now 150 were found to weigh 84oz. 7dwt. 20gr. how much was this over or under the just weight ? Ans. &gr. over. 15. What cost 2^- cwt. of sugar, at 13 cents 3 mills per pound ? Ans. 37 dolls. 24 cts. 16. A merchant deposited in bank 35 twenty-dollar notes, 63 eagles, 284 dolls. 642 half dolls. 368 qr. dolls. 256 twelve and a half cent pieces ; he afterwards gave a check to A for 560 dolls, and another to B for 820 dolls. ; what sum has he still remaining in bank? Ans. 679 dolls. 17. A merchant bought a, piece of broad cloth containing ,o yards, at 4 dolls. 66 cts. per yard ; of this he found 4 yards were so damaged that he sold them at half price ; 8 yards he sold at 5 dolls. 50 cents per yard : on the whole piece he gained IM) dolls. 56 cents ; at what rate did he sell the remainder? Ans. (> dollars per yard. 18. Five travellers, upon leaving a tavern in I ho morning, icund they were charged 12 cents eaeh HM- their beds, 4 limes flint sum for their spnpor rind break (hst. 75 eenls fnr RKDIJCTJOJV. 39 liquor among them all, 25 cents each ibr hay ; the remain- der of their bill, which amounted to 6 dollars, was for oats at 2 J cents per quart; how many gallons of oats had they, and how much had each man to pay ? Ans. 8| gallons ; each paid 120 cents. 19. A laborer engaged to work for 75 cents per day, working 8 hours each day, or 8 hours for a day's work ; but being industrious he worked 12 hours 25 minutes each day for five days, and then 11 hours 30 minutes for 9 days more ; what sum is he entitled to receive for his services ? Ans. 15 dollars 52 cents 3 mills-f 20. If 25 hhds. contain 1534 galls. 1 qt. and 1 pt. of brandy, each an equal quantity, how much is there in each hogshead? Ans. 61 galls. 1 qt. 1 pt. 21. If a man do 114 hours 45 minutes' work in 9 days, how long did he work each day? Ans. I2h. 45mi. 22. Divide 180 dollars among 3 persons A, B, and C; give B twice as much as A, and C three times as much as B. C A 20 dolls. Ans. I B 40 f C 120 SECTION 6. OF REDUCTION. REDUCTION is the changing of numbers from one denomi- nation to another, without altering their real value. Thus. 1 -dollar, if reduced to cents, will be 100 cents, which in their real value are equal to 1 dollar: or, 3 feet reduced to yards, is one yard, which is still the same length as the 3 feet. RULE. If the reduction is from a higher to a lower denomina- tion, multiply ; but if from a lower to a higher denomina- tion, divide by as many of the next less as make one of the greater ; adding the parts of the same denomination to the product as it descends ; and setting down the remainders as il ascends. Reduction ascending, and descending, mutually prove each t.lher. 40 REDUCTION. >>CQd ^ 3 3 ^ & >>>>>>>> 1 N. b S" hrl Ifli <? S". GCtJo tr CD ^ Iff ^Si^li f ^l-o js|o> s r ra ^* fa o s* ~3 a 5 30 i^-p I*. . >? . ^ S .p p ^ >-t 3 p "p 3 * " p eta. * PT* 5' o ... ^ p^ - ,. *~ ft P- XV'^ i. ^ H- t i R* ^ 5* COCO <l CO ^^O'e^'OiOiOODS. Standard C* 1 -^' jo H-i H* I-' tO rr rre^AL ^ ^, o - OD GO o ^ 05 en 05 GO H-* J 3 o >gr l ~*3 __ ' a * 000 OOl 1 t-- 1 i- u COH- 'iob^ J 55 IS ^ O 1 > 4- Cn I 1 H* 1 > ^ i 1 " O5 O5t 'H->v!O5O5tO-:o j JJ , g 1 ^ St' Cft 1 H-* 00000000?- ? ^ If ir! 1 000 ^H-H-^H-^tO^^. ?^^^ ^',0 P a s |]1 5 Cfe^ 1 i t h- O5 05 tsw-siaoScDooS* |.|.^l| 5" o s 6C tr^ O 00 oooooooo a. ^*I|| ! r^^' 3 ff 1 O O O H- H- 00 o 00 t H- ( 'I * tO Oi i'5 O5 ^ 1 ' I H- 1 ||1^ ., |L 1" ^1 CO CO oooooooo?-- 1 : ^j^ | OO o H.M^MW-^W^^ ^ b3'^ t'l 2 t-t )t -> 00 t H-* 1 * ^ooi^aiCflto-oo? 9 -2 5 a"^ ^, ! ^^ r*. o 1 O5 GO O5 ^ ^ ^ a ~" ^ o 2 "^ as 'CO C? C^ r ' OO OOl H-'H-OOtO^p ? S 5 > ^ H-* o O H-* ^ en ^JQDI 'h-'OOOOO;* Illl ^ j^ t0 00 ^ O5OOCOOOOO .^ ? a <Ht do o N PP r* H- w^^^c^I^o^L Ktiff/e.} | /)o/r#. \ % /> v 2. - "^ ^ ~~ * *~* -"'' i 2 <a o to o o sa ^i o J5 S M o p .V,Vs. ) e ;* o to o o i| Gfi ' ** O O C. ? ? .REDUCTION. 41 A TABLE of other Foreign Coins, &c. with their value in Federal Money , as established by a late act of Congress. E. d. d. c. m. \ E. d. d. c. m. Pound Sterling' . 4, 4 4 4 t The Guilder of the Pound of Ireland . 4, 1 Pagoda of India . 1, 9 4 Tale of China . 1, 2 4 Millree of Portugal 1, 2 4 Ruble of Russia . 0, 6 6 Rupee of Bengal . 0, 5 5 5 U. Netherlands 0, 3 9 Mark Banco of Ham- burg 0, 3 3 5 Livre Turnois of France 0, 1 3 5 Real Plate of Spain 0, 1 MONEY. Cents are reduced to pence by subtracting one-tenth of their number. Pence are reduced to cents by adding one- ninth of their number. Pence are to cents as 9 is to 10, and to mills as 9 to 100. This only applies where the dollar passes at 7$. Qd. or 90 pence. 1. Reduce 100 cents to pence. 10) 100 cents 10 90 pence. Arts. 2. Reduce 90 pence to cents. 9 ) 90 pence 10 100 cents. Ans. 3. Reduce 1251. 10s. 6%d. to farthings. 125Z. 105. 20 2510 shillings 12 30126 pence 4 120506 farthings. Ans. 4. Reduce 120506 farthings to pounds. Ans. 1251. 105. 5. Reduce 260 cents to pence. Ans. 234d. D2 42 REDUCTION. 6. Reduce 480Z. 1 9s. 9d. to cents. Ans. 128263 cents. 7. Reduce 4658 pence to pounds. Ans. 191. 8s. 2d. S. Reduce 648 pence to cents. Ans. 720 cents. 9. Reduce 720 cents to pence. Ans. 648 pence. 10. Reduce 24235 half- pence to pounds. Ans. 507. 9s. 9%d. 11. How many pounds, Pennsylvania currency, in 216 French crowns ? Ans. 897. 2s. 12. In 29Z. 17s. how many cents and dollars? A?i. 7960 cents 79 dolls. 60 cts. 13. In 375Z. Pennsylvania currency, how many dollars ! Ans. 1000 dolls. Note. To bring pounds (Penn. currency) to dollars, multiply by 8 and divide by 3 ; and dollars to pounds, multiply by 3 and divide by 8. TROY WEIGHT. 1. Reduce 115200 grains to pounds. Ans. 201. 2. Reduce 30Z&. to grains. Ans. 172800r. 3. Reduce 45648 pennyweights to ounces. Ans. 22S2oz. Sdwt. 4. Reduce 4.lb. Soz. I5dwt. 20g?\ to grains. Ans. 2726Qgr 5. Reduce 27260 grains to pounds. Ans. 4Z6. Soz. 15dwt. 20gr. 6. In 24 spoons, each weighing Sdwt. 6gr. how many grains ? Ans. 4752gr. AVOIRDUPOIS WEIGHT. 1. Reduce 3 tons to pounds. Ans. 6720Z6. 2. Reduce 2867200 drams to tons. Ans. 5 tons. 3. Reduce 5 tons to drams. Ans. 2867200dfr. 4. In 6 barrels of flour, each weighing Icwt. 3qr. Low many pounds? Ans. 1176/6. 5. In \ficwt. 2qr. 14Z&. how many pounds? Ans. 18627A. 6. In a load of hay weighing 2876Z&. how many hun- dreds? Ans. 25cwt. 2qr. 207ft. APOTHECARIES 1 WEIGHT. 1. Reduce 15Z&. to scruples. Ans. 4320*?. REDUCTION. 43 2. In a bottle containing 3lb. of calomel, how many grains? Ans. IT2$Qgr. 3. In 2%lb. of drugs, how many parcels, each 16 drams ? Ans. 15 parcels. 4. In 576000 grains, how many pounds ? Ans. 100Z&. CLOTH MEASURE. 1. Reduce 250 yards to nails. Ans. 4000 nails. 2. In 8642 nails, how many Ells English ? Ans. 432 Ells E. 2 nails. 3. In 324 Ells French, how many yards ? Ans. 486 yards. 4. In 16 bales of cloth, each measuring 36 Ells Flemish, how many yards.? Ans. 432 yards. LONG MEASURE. 1. Reduce 260 miles to inches. Ans. 16473600 inches. 2. Reduce 11 miles 7 furlongs 38 perches 2 yards 2 feet, to barley-corns. Ans. 2280060&C. 3. Reduce 1267200 feet to geographical degrees. Ans. 4 degrees. 4. Reduce 3 leagues 2 furlongs 110 yards 1 foot 5 inches, to inches. ' Ans. 590057 inches. 5. How many inches will reach round the world, at 60 miles to a degree? Ans. 1368576000 inches. o LAND MEASURE. 1. Reduce 25 acres to perches. Ans. 4000 perches. 2. Reduce 176000 perches to acres. Ans. 1100 acres. 3. A tract of land containing 640000 perches is to be iivided into 400 equal shares ; how many acres will be in rach share ? Ans. 1 acres. 4. In 10 acres, how many square inches ? Ans. 62726400 inches. CUBIC, OR SOLID MEASURE. 1. Reduce 3200 feet of wood to cords. Ans. 25 cords. 44 REDUCTION. 2. In. 20 tons of square timber, how many feet? Ans. 1000 feet. 3. In 30 tons of round timber, how many inches ? Ans. 2073600 inches. Note. The cubic feet of any circular body, such as grindstones, &c is found in the following manner. Add halt' the diameter to the whole diameter ; multiply the amount by the aforesaid half^ and this product by the thickness; this will give the contents in cubic inches; divide these by 1728, and the quotient will be the cubic feet. 4. In a grindstone 48 inches diameter and 6 inches thick, how many feet ? 48 diameter 24 half do. 72 24 288 144 1728 6 1728 ) 10368 ( 6 cubic feet. Ans. 10368 5. In a millstone 4 feet 6 inches diameter, and averaging 18 inches in thickness, how many cubic feet ? 54 27 81 27 567 162 2187 18 17406 2187 1 728 ) 39366 ( 22 feet 1350 inches. Ans. 3456 4806 3456 1350 DECIMAL ARITHMETIC. 46 TIME. 1. Reduce 8 weeks 2 days 6 hours 20 minutes, to minutes. Ans. 83900 minutes. 2. Reduce ten years to seconds. Ans. 315576000 sec. 3. How many days since the commencement of the Christian era to the present time, 1823 '? Ans. 665850 days 18 hours. 4. How many seconds in a week ? Ans. 604800 sec. LIQUID MEASURE. 1. Reduce 4 tuns to pints. Ans. 8064 pints. 2. Reduce 4032 pints to hogsheads. Ans, 8 hhds. 3. Reduce 38 hogsheads to pints. Ans. 19152 pints. DRY MEASURE. 1. Reduce 78 bushe's 3 pecks 7 quarts to pints. Ans. 5054 pints. 2. Reduce 2196 pints to bushels. Ans. 346w. Ipc. 2qt. PART III. DECIMAL ARITHMETIC. DECIMAL ARITHMETIC is a plain and easy method of dis- covering the value of an unit, or one, divided into any given number of parts. Thus, if 1 dollar is divided into 10 equal parts, any one of these parts will be one-tenth, 2 will be two-tenths, 3, three-tenths, &c. Again, if 1 dollar is divided into a hundred equal parts, any one of these will be one- hundredth, 2, two-hundredths, &c. The number of parts into which the unit is divided is called the denominator, and any number of these parts less than the whole is called the numerator, and which always stands over the denominator ; thus, 2 numerator 10 denominator is read two-tenths ; and these two so placed constitute what is termed a fraction. In decimal fractions, the de- nominator can only be an unit, with one or more ciphers 46 ADDITION OF DECIMALS. added thereto ; as y 5 -, y 2 ^, T 2 / F V The numerators of these are usually written without their denominators, and are dis- tinguished from whole numbers, by prefixing a point called the separatrix, as ,5 ,25 ,'225. Ciphers placed to the right hand of decimals, make no change in their value, for ,5 ,50 ,500, &c. are decimals of the same value, each being equal to J. But when prefixed to the decimal, they decrease the value in a tenfold propor- tion. Thus, ,5 ,05, ,005, have the same proportion to each other a,s 5, 50, 500, have in whole numbers. This is made plain by the following TABLE. Integers. Decimals. 5 5 ,5 500 ,05 5000 ,005 50000 ,0005 500000 ,00005 5000000 ,000005 all's 11 .2 13 w 3 "2 < ,2 =3 5 G Q 5 C -^ SECTION I. ADDITION OF DECIMALS. RULE. SET down the numbers according to their value, viz. units under units, tenths under tenths, &c. Then add as in addition of whole numbers, and place the point in the amount exactly und^i those in the given sum. SUBTRACTION OF DECIMALS 47 EXAMPLES. 2468,5036 3460000,0000643 521,0428 460000,000643 32,0004 3400,3680005 3021,5468 3. Add 283,604 + 490006,003275 -f 21,05 -f 1,2 + 6200,3476. Ans. 496512,204875. 4. Add ,246 + ,012 + ,02 + ,6 + ,41 3 + ,5. Ans. 1,791. 5. Add 25,52 + 225,005 + ,0035 + 844 -f2,2 + 300,825 + ,00005. Ans. 1397,55355. 6. Add one hundred and twenty-five, and five-tenths, + ten thousand, and five millionths, + fifteen, and seventy-two thousandths, + two, and one hundredth. A.ns. 10142,582005. 7. Add five, and four-tenths, + fifteen and four hundredths, + one hundred, and four thousandths, + six thousand and four hundred thousandths, + ninety-three thousand eight hundred and eighty, and four ten thousandths. Ans. 100000,44444. SECTION 2. SUBTRACTION OF DECIMALS. RULE. SET the less under the greater, with the points as in addi- tion, and place the point in the remainder, in the same manner. EXAMPLES. Prom 6432,50437 From 848,045 From 15,6547 Take 369,95429 Take 162,54936^ Take 7,35 Kern. 6062,55008 4. From 45,005 take 23,65^82. Ans. 21,35018. 5. From six hundred and twenty, and two-tenths, take wo hundred and two thousandths. Ans. 420,li/ k rt 48 MULTlPLICATIGiN OF DECIMALS. 6. From 5 take ,10438. Ans. 4,89562. 7. From 2 take ,00002. Ans. 1,99998. 8. From sixteen take sixteen thousandths parts. Ans. 15,984. SECTION 3. MULTIPLICATION OF DECIMALS. RULE. MULTIPLY as in whole numbers, and from the product point ofF as many on the right-hand for decimals, as there are in both the factors. If the whole product should be too few, then must ciphers be added on the left of the product till an equal number is had. EXAMPLES. Multiply 29,831 by ,952 59662 149155 268479 Product 28,399112 24,021 4,23 22,2043 ,12345 72063 48042 96084 1110215 888172 666129 444086 222043 101,60883 4. Multiply ,385746 by ,00463 5. 158,694 23,15 6. ,024653 ,00022 2,741120835 Ans. ,00178600398 3673,7661 ,00000542360. 7. Multiply twenty-five and four hundredths, by two thou- sandths. Ans. ,05008. 8. Multiply six hundred and forty-five, and three thou- sandths, by five millionths. Ans. ,003225015. Note. The product of any number when multiplied by a decimal only, will be less than the multiplicand, in llrt PUIIW proportion as the multiplier is les than one. MULTIPLICATION OF DECIMALS. 49 CONTRACTION IN MULTIPLICATION OF DECIMALS. If only a limited number of decimals is sought for, instead of retaining the whole product, obtained by the foregoing method, work by the following RULE. 1. Set the multiplier, in an inverted order, under thn multiplicand, placing the units figure of the multip'ier under the lowest decimal place in the multiplicand, that is wished to be retained. 2. In multiplying, omit those figures in the multiplicand which are on the right of the multiplying figure, but to the first figure in each line of the product, add the carriage which would arise from the multiplication of the omitted figures, carrying one from 5 to 15, 2 from 15 to 25, 3 fron 25 to 35, &c. Place the first figures in each product di- rectly under each other, and add as in addition. Note. If you would be absolutely certain that the last figure retained is the nearest to the truth, work for one place more than you wish to retain* EXAMPLES. 1. Multiply 34,6733 by 3,1416, retaining four decimal places in the product. 34,6733 61413 inverted 1040199 34673 13869 347 209 108,9296 Arts. 2. Multiply ,78543 by ,346787, retaining five decimal places in the product. 34G787 Or thus, ,78543 34587 inverted 787643,0 24275 235G3 2774 3142 173 471 14 55 1 6 .27237 4ns 427237 50 DIVIS[OIN OF DECIMALS. 3. Multiply 23,463 by 2,34, retaining three decimals. Ans. 54,903. 4. Multiply 234,216 by 2,345, retaining two decimals. Ans. 549,23. 5. Multiply 3,141592 by 52,7438, retaining four deci mals. Ans. 165,6995, SECTION 4 DIVISION OF DECIMALS. RULE. DIVIDE in the same manner as in whole numbers, and point off on the right of the quotient as many figures for de- cimals, as the decimals in the dividend exceed those in the divisor. When the decimals in the divisor exceed those in the dividend, let ciphers be added to the dividend, till they equal those in the divisor. And if there be a remainder, let ciphers be annexed thereto, and the quotient carried on to any degree of exactness. EXAMPLES. 29,831 ) 2' ,d991 12 ( ,952 24,021 ) 101,60883 ( 4,28 ' 268479 96084 155121 55248 149155 48042 59662 72063 59662 72063 Nate. When the divisor is 10, 100, 1000, &c. the division is perform- ed by pointing off as many figures in the dividend for decimals, aa there are ciphers in the divisor. 10 ) I 685,6 Thus, 6856 divided by 100 > is < 68,56 1000 > ( 6,856 3. Divide 65321 by 23,7 Ans 2756,16 + 4. 234,70525 64,25 3,653 5. 10 3 3,3333 + 6. 9 ,9 10 7. ,00178600398 ,00463 ,385746 DIViSIOiN OF DECIMALS. fj 1 8. Divide ,2327898 by 2,46 Ans. ,09463 9. ' ,2327898 ,09463 2,46 10. * ,000162 ,018 ,009 CONTRACTION IN DIVISION OF DECIMALS. When only a limited number of decimals in the quotient is sought for, work by the following RULE. 1. Take as many figures only on the left hand side of the divisor, as the whole number of figures sought for in the quotient, and cut off the rest. 2. Make each remainder a new dividend, and for a new divisor, point off one figure continually from the right hand of the former divisor, taking care to bring in the increase, or carriage of the figures so cut off, as in multiplication. Note. When the whole divisor does not contain as many figures as are sought for in the quotient, proceed as in common division, without cutting off a figure, till the figures in the divisor shall equal the re- maining figures required in the quotient, and then begin to cut off as above directed. EXAMPLES. 1. Divide 14169,206623851 by 384,672258, retaining four decimal places in the quotient, or in all six quotient figures. 3.8.4,6.7.2|258 ) 14169,206623851 ( 36,8345. Ans. 1154017 262903 230803 32100 30774 1326 1154 172 153 19 19 52 REDUCTION OF DECIMALS. 2. Divide ,07567 by 2,32467, true to four decimal places, or three significant figures, the first being a cipher. 2,3.2|467 ) ,07567 ( ,0326. Ans. 697 59 46 13 14 3. Divide 5,37341 by 3,74, true to four decimal places. 3,74 ) 5,37341 ( 1 ,4367. Ans. 374 1633 1496 1374 1122 252 224 28 26 4. Divide 74,33373 by 1,346787, true to three decimal places. Ans. 55,193. 5. Divide 87,076326 by 9,365407, true to three decimal places. Ans. 9,297. 6. Divide 32,68744231 by 2,45, true to two decimal olaces. Ans. 13,34. 7. Divide ,0046872345 by 6,24, true to five decima places. Ans. ,00075 SECTION 5. REDUCTION OF DECIMALS. Case 1. To reduce a vulgar fraction to a decimal. REDUCTION OF DECIMALS. 53 RULE. Annex one or more ciphers to the numerator, and divide by the denominator; the quotient will be the answer in decimals. EXAMPLE. 1. Reduce to a decimal. 4)1,00 ,25 Ans. 2. Reduce to a decimal. Ans. ,5 3. - J to a decimal. ,75 4. - -J to a decimal. ,875 5. - -Jj to a decimal. ,04 6. - |j to a decimal. ,95 7. - T 6 3 of a dollar to cents. ,40 cts. Case 2. To reduce numbers of different denominations to a deci- mal of equal value. RULE. Set down the given numbers in a perpendicular column, having the least denomination first, and divide each of them by such a number as will reduce it to the next name, annex- ing the quotient to the succeeding number ; the last quotient will be the required decimal. EXAMPLE. 1. Reduce 17s. 8%d. to the decimal of a pound. 4 12 20 3 8,75 17,720166 ,8864583 + Ans. 2. Reduce 195. to the decimal of a pound. Ans. ,95 3. - 3d. to the decimal of a shilling. ,25 4. - 3d. to the decimal of a pound. ,0125 5. - kcwt. 2qr. to the decimal of a ton. ,225 6. - 2qr. 14/6. to the decimal of a cwt. ,625 7. - 3qr. 3na. to the decimal of a yard. ,9375 E2 54 REDUCTION OF DECIMALS. Case 3. To reduce a decimal to its equal value in integers. RULE. Multiply the decimal by the known parts of the integer EXAMPLE. 1. Reduce ,8864583 of a pound to its equivalent value in integers. ,8864583 20 s. 17,7291660 12 d. 8,7499920 4 qr. 2,9999680 It is usual when the left hand figure in the remaining decimal exceeds five, to expunge the remainder, and add one to the lowest integer. Thus, instead of 17s. 8d. 2,999, die. we may say 17s. 8%d. Ans. 2. What is the value of ,75 of a pound ? Ans. 15s. 3. What is the value of ,7 of a pound troy? Ans. Soz. Sdwt. 4. What is. the value of ,617 of a cwt. ? Ans. 2qr. 13Z&. 1 oz. W + dr. 5. What is the value of ,3375 of an acre? Ans. I rood, 14per. 6. What is the value of ,258 of a tun of wine ? Ans* Ihhd. 2 -{-gals. 7. What is the proper quantity of ,761 of a day? Ans. ISh. 15?ni. 50,4src. i. What is the proper quantity of ,7 of a Ib. of silver? Ans. Soz. Sdwt. : What is the proper quantity of ,3 of a year? Ans. lQ9d. 13ft 10. What is the difference between ,41 of a day and ,16 of an hour ? Ans. 97*. 40wi. 48sec. 11. What is the sum of ,17T. 19^. ,l7qr. and lib. 1 Ans. Zcwt. 2qr. DECIMAL KHACT1O.NS. 55 Promiscuous Questions in Decimal Fractions. 1. Multiply ,09 by ,000. Ans. ,00081. 2. In ,36 of a ton (avoirdupois) how many ounces ? Ans. 12902,402. 3. What is the value of ,9125 of an ounce troy? Ans. 18dwt. 6gr. 4. Reduce ^j to a decimal. Ans. ,0127 nearly. . 5. Reduce 2oz. I6dwt. 20gr. to the decimal of a pound troy. Ans. ,2368 -f nearly. 6. What is the length of ,1392 of a mile? Ans. 1 fur. 4 per. 3 yds nearly. 7. What multiplier will produce the same result, as mul- tiplying by 3, and dividing the product by 4 ? Ans. ,75. 8. What decimal of Icwt. is 6lb. Ans. ,0535714. 9. What part of a year is 109 days 12 hours? Ans. ,3. 10. In ,04 of a ton of hewn timber, how many cubic inches? Ans. 3456. 11. What is the value of T 3 T of a dollar divided by 3? Ans. 6f cents. 12. What is the value of ,875 of a hhd. of wine? Ans. 55 gal. qt. 1 pt. 13. What divisor, true to six decimal places, will produce the same result as multiplying by 222 ? Ans. ,004504. 14. In ,05 of a year, how many seconds, at 365 days 6 hours to the year? Ans. 1577880. 15. What number as a multiplier will produce the same result as multiplying by ,73 and dividing first by 3, and the quotient by ,25 ?" Ans. ,973^. 16. What is the difference between ,05 of a year, and ,5 of an hour ? Ans. 2i. 2d~ ISh. 42m. 17. In ,4 of a ton, ,3 of a hhd. and ,8 of a gallon, how many pints ? Ans. 964. 18. How many perches in ,6 of an acre; multiplied by X)2? Ans. 1,92.* 19. What part of a cord of timber is 1 cubic inch? Ans. ,000004 -f 20. What part of a circle is 28 deg. 48 minutes ? ATU. ,08. 66 SINGLE RULE OF THREE DIRECT PART IV. PROPORTIONS. Tins part of arithmetic which treats of proportions is very extensive and important. By it an almost innumerable variety of questions are solved. It is usually divided into three parts, viz. Direct, Inverse, and Compound. The first of these is called the Single Rule of Three Direct, and sometimes by way of eminence the Golden Ride. The sec- ond is called the Single Rule of Three Inverse: and the last is called the Double Rule of Three. In all these, cer- tain numbers are always given, called data* by the multipli- cation and division of which, the answer in an exact ratio of proportion to the other terms is discovered. SECTION 1. SINGLE RULE OF THREE DIRECT IN this rule three numbers are given to find a fourth, that shall ha\e the same proportion to the third, as the second has to the first. If by the terms of the question, more requires more, or less requires less y it is then said to be direct, and belongs to this rule. In stating questions in this rule, the middle term must always DC of the same name with the answer required; the last term is that which asks the question, and that which is of the same name as the demand, the first. When the ques- tion is thus stated, reduce the first and third terms to the lowest denomination in either ; and the middle term (if com- pound) to its lowest, and proceed according to the following RULE. Multiply the second and third terms together, and divide the product by the first ; the quotient will be the fourth term, or answer, in the same name with the second. SINGLE RULE OF THREE DIRECT. 5? PROOF. Invert the question, making the answer the first term ; the result will be, the first term in the original question. Note. 1. After division if there be any remainder, and the quotient be not in the lowest denomination, it must be reduced to the next less denomination, dividing as before, till it is brought, to the lowest denomi- nation, or till nothing" remains. 2. When any of the terms are in federal money, the operation is con- ducted in all respects as in simple numbers, taking care to place the separatrix bet'ween dollars and cents, according to what has already been laid down in federal money and decimal fractions. EXAMPLE. 1. If 8 yarcls of cloth cost 32 dollars, what will 24 yards cost ? Yds. D. Yds. D. Yds. D. As 8 : 32 :: 24 Proof. As 96 : 24 :: 32 24 32 128 48 64 72 8)768 96)768(8 768 96 Ars. 2. When sugar is sold at 12 dollars 32 cts. per cwt. whni will Wlb. cost? Ans. I doll. 76 cts. 3. What is the amount of 3 cwt. of coffee at 36 cents per pound? Ans. 120 dolls. 96 cts. 4. What will 4 pieces of linen come to, containing 23, 24, 25, and 27 yards, at 72 cents per yard ? Ans. 71 dolls. 28 cts. 5. What will bcwt. 2qr. Sib. of iron come to at 48 cents for 4/6. ? Ans. 61 dolls. 44 cts. 6. What will V2Slb. of pork come to at 8 cts. per pound? Ans. 10 dolls. 24 cts. 7. If 9 dozen pair of stockings, cost 68 dollars 40 cent? what will 3 pair cost ? Ans. 1 doll. 80 cts. 8. If 20 bushels of oats cost 9 dollars 60 cents, what will three bushels come to? Ans. 1 doll. 44 cts. 9. A merchant bought a piece of cloth for 16 dollars 50 cents, at 75 cents per yard ; how many yards were there iii the piece ? Ans. 22 yds. 10. If 17 cwt. 3qr. 17 Ib. of sugar cost 320 dollars 80 cts. what must be paid for 6oz. ? An*. 6 o*nts. 58 SINGLE RULE OF THREE DIRECT 11. If 9,7 Ib. of silver is worth 97 dollars, what is the value of l,5oz. 1 Ans. 1 doll. 25 cts. 12. If 125,5 acres are sold for 627,5 dollars, what will 4,75 acres cost ? Ans. 23 dolls. 75 cts. 13. If 1,5 gallons of wine cost 4 dollars 50 cents, what will 1,5 tuns cost? Ans. 1134 dolls. 14. How many reams of paper at 1 dollar 66 cents, 1 dollar 97 cents, and 2 dollars 31 cents per ream may be purchased for 528 dollars 66 cents, of each an equal num- ber ? Ans. 89 reams of each sort. 15. When iron is sold for 224 dollars per ton, what will Iqr. 1Mb. cost? Ans. 4* dolls. 20 cts._ 16. A merchant paid 1402 dollars 50 cents for flour,- at 5 dollars 50 cents per barrel ; how many barrels must he re- ceive? Ans. 255 barrels. 17. A man has a yearly salary of 1186 dollars 25 cents, how much is it per day ? Ans. 3 dolls. 25 cts. 18. A man spends 2 dollars 25 cents per day, and saves 378 dollars 75 cents at the end of the year, what is his yearly salary? Ans. 1200 dolls. 19. What will 4T. Wcwt. Iqr. I2lb. of hay come to at 1 dollar 12 cents per cwt.? Ans. 101 dolls. 20 cts. 20. How much will a grindstone 4 feet 6 inches diameter, stfid 9 inches thick, come to at 1 dollar 10 cents per cubic foot? Ans. 12 dolls. 53 cts. 21. \\hat will a grindstone 28 inches diameter, and 3,5 inches thick, come to at 1 dollar 90 cents per cubic foot ? Ans. 2 dolls. 26 cts. 22. At 221. Ss. per ton, what will 203 T. Scwt. 3qr. Mb. of tobacco come to ? Ans. 4558Z. 3s. 23. If 850 dolls. 50 cents is paid for 18 pieces of cloth at the rate of 11 dollars 25 cents for 5 yards, how many yards were in each piece, allowing an equal number to each piece? Ans. 21 yds. 24. If 124 yards of muslin cost 1Z. 17$. 6rf. what is it pe> yard? Ans. 3s. 25. If a staff 4 feet long cast a shadow (on level ground) 7 feet long, what is the height of a steeple whose shade at the same time, is 218 feet 9 inches? A?is. 125 feet. 26. If 4292 dollars 32 cents are paid for 476 acres 3 roods 28 perches of land, how much is it per acre ? Ans. 9 dollars. 27. If a man's annual income be 1333 dollars, and he SLN(iLK lUiJJ-: OK TUkiaO INVERSK. 59 expend daily 2 dollars 1-4 cents, how much will lie save at 'the end of the year 1 Ans. 551 dolls. 90 cts. 28. If 321 bushels of wheat cost 240 dollars 75 cents, what is it per bushel ? Ans. 75 cts. 29. If l yard of cloth cost 2 dollars 50 cents, what will 1 quarter 2 nails come to ? Ans. 62^ cts. 30. Bought 3 pipes of wine, containing 120-|, 124, and 126| gallons, at 5s. 6d. per gallon ; what do they cost ? Ans. 102Z. Is. lOicZ. 31. A sets out from a certain place and goes 12 miles a day ; 5 days after, B sets out from the same place, the same way, and goes 16 miles a day; in how many days will he overtake A? Ans. 15 days. 32. If I have owing to me 1000Z. and compound with my debtor, at, 12s. Qd. per pound, how much must I receive? Ans. 625Z. 33. If 365 men consume 75 barrels of pork in 9 months, how many will 500 men consume in the same time ? Ans. 10241 barrels. 34. How much land at 2 dollars 50 cents per acre, musl f>e given in exchange for 360 acres at 3 dollars 75 cents 1 Ans. 540 acres. 35. If the earth, which is 360 degrees in circumference, turns round on its axis in 24 hours, how far are the inhabit- ants at the equator carried in 1 minute, a degree there being 69! miles ? Ans. 17 miles 3 fur. SECTION 2. SINGLE RULE OF THREE INVERSE. IF in any given question, more requires less, or less re- quires more, the proportion is inverse, and belongs to this rule. Having stated the question, as in the rule of three direct, proceed according to the following RULE. Multiply the first and second terms together, and divide the product by the third ; the quotient will be the answer, in the same name as the second. 60 SINGLE RULE OF THREE INVERSE EXAMPLE. 1. If 20 men can build a wall in 12 days, how long will it require 40 men to build the ^ame ? M. d. M. d. M. d. As 20 : 12 :: 40 Proo/ As 6 : 40 :: 12 12 40 40 ) 240 ( 6 days. Ans. 12 ) 240 ( '20 men. 240 24 2. If 60 men can build a bridge in 100 days, how long will it require 20 men to build it ? Ans. 300 days. 3. If a wall 100 yards long requires 65 men 4 days, in what time would 5 men complete it ? Ans. 52 days. 4. If a barrel of flour will last a family of six persons 24 days, how long would it last if 3 more were added to the family? Ans. 16 days. 5. If 5 dollars is paid for the carriage of \cwt. weight, 150 miles, how far may Sent* weight be carried for th^ same money ? Ans. 25 miles. 6. If a street 80 feet wide and 300 yards long, can be paved by 40 men in 20 days, what length will one of 60 feet wide be paved by the same men in the same time ? Ans. 400 yards. 7. If a field that is 30 rods wide and 80 in length, con- tain 15 acres, how wide must one be to contain the same quantity, that is but 70 rods long ? Ans. 3412. 4/7. 8&iri. S. If a board be ,75 of a foot wide, what length must it be to measure 12 square feet? Ans. 16 feet. 9. How much cloth 1,25 yards wide, can be lined by 42,5 yards of silk that is ,75 of a yard wide ? Ans. 25,5 yards. 10. If 10 men could complete a building in 4,5 months, what time would it require if 5 more were employed ? Ans. 3 months. 11. In what time will 600 dollars gain 50 dollars, when 80 dollars would gain it in 15 years? fins. 2 years. 12. If a traveller can perform a journey in 4 days, when the days are 12 hours long, what time will lw require when thf! days arn 1 (> hours long ? _ Ans. 3 days. I ;*. Suppose 100 nvn in a garrison are supplied SINGLE RULE Oi TliUKE INVERSE. 61 provisions for 30 days, how many men must be sent out if they would have the provisions last 50 days ? Ans. 160 men. 14. Lent a friend 292 dollars for six months ; afterwards J borrow from him 806 dollars ; how long may I keep it to balance the favor? Ans. 2 months 5 days. 15. 1200 men stationed in a garrison, have provisions for 9 months, at the rate of 14 ounces per day; how long at the same allowance will the same provisions last if they are reinforced by 400 men ? And also what diminution must be made on each ration, that the provisions may last for the same time ? Ans. 6| mo. at the same allowance 3J oz. deduction to last for the same time. 16. If a piece of land 40 rods in length and 4 in breadth, make an acre, how wide must it be if it is but 25 rods long? Ans. 6| rods. 17. How much in length that is 3 inches broad, will make a square foot ? Ans. 48 inches. 18. If a pasture field will feed 6 cows 91 days, how long will it feed 21 cows ? Ans. 26 days. 19. There is a cistern having 1 pipe, which will empty it in 10 hours ; how many pipes of the same capacity will empty it in 24 minutes ? Ans. 25 pipes. 20. How many yards cf carpeting that is half a yard wide, will cover a floor that is 30 feet long and 18 feet wide? Ans. 120 yards. 21. What is the weight of a pea to a steelyard, which being suspended 39 inches from the centre of motion, will l Ans. 4Z6. 22. A and B depart from the same place, and travel the same road ; but A goes 5 days before B at the rate of 20 miles a day, B follows at the rate of 25 miles a day ; in what time, and at what distance, will he overtake A? Ans. 20 days, and 500 miles. The following- rule, if adopted, will suit for the stating of all questions in single proportion, whether direct or inverse. GENERAL RULE, Place that number for the third term, which signifies the same kind, or thing, as that which is sought ; arid consider whether the number sought will be greater or less ; if great- F equipoise 2QSlb. suspended at the draught end f of an inch ? f>2 SLXGLK RULF. OF THRRK. er, place the least of the other terms for the first, but if less place the greater for the first term, and the remaining one for the second. Multiply the second and third terms together, and divide the product by the first ; the quotient will be the answer required. EXAMPLES. 1. If 30 horses plow 12 acres, how many will 40 horses plow in the same time ? k. h. acr. Direct Proportion. 30 : 40 : : 12 12 30)480(16 acres. A us. 2. If 30 horses plow 12 acres in 10 days, in how many days will 40 horses plow the same quantity ? h. h. D. Inverse Proportion. 40 : 30 : : 10 10 40)300(7,5 days. Ans. 3. If 800 soldiers in a garrison have provisions sufficient for 2 months ; how many must depart that the provisions may last them for 5 months ? Ans. 480. 1. Bought a hogshead of Madeira wine for 119 dollars, nine gallons of which leaked out ; what was the remainder sold at per gallon, to gain 12 dollars on the whole 1 -Ans. 2 dolls. 42-J-fcts. 5. If 225 pounds be carried 51*3 miles for 20 dollars, how many pounds may be carried 64 miles for the same money? Ans. 1800Z6. 6. If 87 dolls. 50 cents be assessed on 1750 dolls, what is the tax of 10 dolls, at the same rate? Ans. 50 cts. Promiscuous Questions in Direct and Inverse Proportion. 1. Suppose a man tnivels to market with his wagon loaded, at the rate of 2- miles .n hour, and returns with \\ empty at the rate of 3 miles an hour ; how long will he be in performing a journey, going and returning, to a place 123 miles distant? Ana. 84J-J hoars. SINGLE HULK OF TlfREK. (>3 2. A lent B 1000 dollars fur IS!) days, without interest how long should B 'end A iif>0 dollars to requite the favor? Ans. 290fg days. 3. Bought 14 casks of butter, each weighing Icwt. Iqr. 4/6. at 12 dollars 60 cents per cwt. ; what did they come to, and how much per Ib. ? Ans. 226 dolls. 80 cts. whole cost ; 1 1 cts 2 \ m. per Ib. 4. Sold 4 chests of tea, each weighing Icwt. Qqr. 14//>. the first for 80 cents per Ib. the second for 90 cents, the third for 1 doll. 5 cents, and the fourth for 1 doll. 25 cents ; how many pounds of tea were there, what was the average price, and what did the whole come to ? Ans. 504//?. average 1 doll. come to 504 dolls. 5. When flour is sold at 2 dolls. 24 cents per cwt. what will be the first cost of one dozen of rolls, each weighing 5oz. allowing the bread to be in proportion to the flour, as five is co four ? Ans. 6 cents. 6. If a merchant bought 270 barrels of cider for 780 dol- lars, and paid for freight 37 dolls. 70 cents, and for other charges and duties- 30 dolls. 60 cents ; at what must he sell it per barrel to gain 143 dolls. Ans. 3 dolls. 67^ 4 T cts. 7. If half a ton of hay was equally divided among 80 horses, how much must be given to 7 ? Ans. 3qr. 14Z&. 8. Suppose the circumference of one of the larger wheels of a wagon to be 12 feet, and that of one of the smaller wheels 9 feet 3 inches ; in how many miles will the smaller wheel make 1000 revolutions more than the larger? Ans. 7 m. 5 fur. 34 yds. 1 ft. 7 T \ in. 9. If a man perform a journey in 18 days, when the days are 15 hours long, how many days will it require to per- form the same journey when the days are only 12 hours long? Ans. 22^ days. 10. A merchant bought a piece of broadcloth measuring 42 yds. for 191 dolls. 25 cents; 15 yards of this being damaged, he sells it at two-thirds of its cost ; the residue he is willing to sell so as to gain 1 doll, per yard on the whole piece ; at what rate must he sell the remainder ? Ans. 6 dolls. 86 T 4 T cents per yd. 11. If 60 yards of carpeting will cover a floor that is 30 feet long and 18 broad, what is the width of the" carpeting ? Ans. 3 feet. 12. ff a piece of -land be 40 rods in length, how wide must it be to contain 4 arres ? Ans. 1 6 rods. 64 SINGLE RULE OF THREE. 13. Suppose a large wheel, in mill work, to contain 70 cogs, and a smaller wheel, working in it, to contain 52 cogs; in how many revolutions of the greater wheel, will the les- ser one gain 100 revolutions? Ans. 2'88. 14. The number of pulsations in a healthy person is, say 70 in a minute, and the velocity of sound through the air is found to be 1142 feet in a second : now I counted 20 pulsa- tions between the time of observing a flash of lightning from a thunder cloud, and hearing the explosion of the thunder; what was the distance of the cloud ? Ans. 3 m. 5 fur. 145 yds. 2j ft. 15. A merchant bought 5 pieces of cloth, of different qualities, but of equal lengths, at the rate of 5, 4, 3, 2, and 1 doll, per yd. for the different pieces ; the whole came to 532 dolls. 50 cents ; how many yards did each piece contain ? Ans. 35 yds. 16. What principal will gain as much in 1 month, as 127 dollars would gain in 12 months? Ans. 1524 dolls. 17. If a pair of steelyards be 36 inches in length to the centre of motion, the pea 5 Ib. and the draught end \ inch in length, what weight, will they draw? Ans. 360 Ib. 18. Supposing the nbovp steelyards would only Hrow 00 Jb., what is the length of the draught end ? Ans. 2 inches. 19. If 1 yard of cloth cost 2 dolls. 71 cts. 1J mills, what will 67i yards come to at the same rate? Ans. 183 dolls. 4 c. 6 m. 20. If a man's income be 16s. 5d. l-J^qr. per day, what is it per annum ? Ans. 300Z. -21. How many pieces of wall paper that is 3 qrs. wide. and 11 yards long, will it require to paper the walls of a room that is 25 feet long, 15 wide, and 10- high, allowing a reduction of j\ for doors and windows? Ans. lQ-f T . 22. The length of a wall being tried by a measuring line, appears to be 1287 feet 4 inches; but on examination the line is found to be 50 feet 10 inches in length, instead of 50 feet its supposed length; rrquirod the true length of the wall? Ans. 1309 feet 10} J inch. 23. If a dealer in liquors use, instead of a gallon, a mea- sure which is deficient by half a pint, what will be the true measure of 100 of these false gallons? Ans. 93j galls. DOUBLE RULK OF THREK DIRECT. 66 SECTION 3. THE DOUBLE RULE OF THREE. THE DOUBLE RULE OF THREE, or as it is often called, Compound Proportion, is used for solving such questions as have five terms given to find a sixth. In all questions be- longing to this rule, the three first terms must be a supposi- tion, the two last a demand. RULE FOR STATING. 1. Set the two terms of the supposition, one under the other. 2. Place the term of the same kind with the answer sought in the second place. 3. Set the terms of the demand, in the third place, ob- serving to place the correspondent terms of the supposition and demand in the same line. Consider the upper and lower extremes, with the middle terms, separately, as in the single rule of three ; if both lines are direct, then the question will be in direct proportion ; but if either lines are inverse, then will the question be in inverse proportion. When the question is in direct proportion, multiply the product of the two last terms by the middle term for a divi- dend, and multiply the two first terms for a divisor; the quo- tient will be the answer in the same name with the middle term. But if the proportion be inverse, transpose the inverse terms and proceed in the same manner as in direct proportion. DIRECT PROPORTION. EXAMPLE. 1. It' 6 men in 8 days earn 100 dollars, how much will 12 men earn in 24 days ? dolls. 6 men ) , mn 5 12 men >t \ f ' 8 days f ' J ) 24 days 48 288 100 48) 28800 (600 dolls. AM. 288 00 F2 06 DOUBLE RULE OF THREE INVERSE 2. If 10 bushels of oats suffice 18 horses for 20 days, how many bushels will serve 60 horses 36 days ? Ans. 60 bushels. 3. If 56 pounds of bread will suffice 7 men 14 days, how much bread will serve 21 men 3 days ? Ans. 36 pounds. 4. If 8 students spend 384 dollars in 6 months, how much will maintain 12 students 10 months? Ans. 960 dollars. 5. If 20 hundred weight is carried 50 miles for 25 dol- lars, how much must be given for the carriage of 40 hun- dred weight 100 miles? Ans. 100 dollars. 6. If 14 dollars interest is gained by 700 dollars in t> months, what will be the interest of 400 dollars for 5 years '.' Ans. 80 dollars. 7. If 4 men can do 12 rods cf ditching in 6 days, how many rods may be done by 8 men in 24 days ? Ans. 96 rods. INVERSE PROPORTION. EXAMPLE. 1. If 4 dollars pay 8 men for 3 days, how many days must 20 men work for 40 dollars ? days. As 4 dolls. ( q J 40 dolls. 8 men. \ ) 20 men. Here the lower line is inverse, which transposed will stand thus : <t days. As 4 dolls. ( o ) 40 dolls. 20 men. { ) 8 men. 80 320 3 80 ) 960 ( 12 days. Ans. 80 160 160 2. If 4 men are paid 24 dollars for 3 days work, how many days may 16 men be employed for 3^4 dollars? Ans. 12 days. JJ. If 4 men arr {-aid '24 dollars f<;r 3 days work, hou many mon may !/o rmployrfl 16 days for 96 dollar? Ana. 3 men. DOUBLE KULK OF THRICE. (57 4. If 7 men can reap 84 acres of grain in 12 days, bow- many men can reap 100 acres in 5 days ? Ans. 20 men. 5. If 7 men can reap 84 acres of grain in 12 days, how many days will it require 20 men to reap 100 acres ? Ans. 5 days. 6. If 40 cents are paid for the carriage of 200 pounds for 40 miles, how far may 20200 pounds be carried for 60 dollars 60 cents ? Ans. 60 miles. 7. If 5 men spend 200 dollars in 22 weeks and 6 days, now long will 300 dollars support 12 men ? Ans. 14 weeks 2 days. Promiscuous Questions. 1. If 12 oxen in 8 days eat 10 acres of clover, how many acres will serve 24 oxen 48 days? Ans. 120 acres. 2. A person having engaged to remove 8000 weight 15 miles in 9 days ; with 18 horses, in 6 days, he removed 4500 weight ; how many horses will be necessary to re- move the rest, in the remaining 3 days ? Ans. 28 horses. 3. If the carriage of 9 hogsheads of sugar, each weighing 12 cwt., for 60 miles, cost 100 dollars, what must be paid for the carriage of 50 barrels of sugar, each weighing 2,5 cwt., 300 miles? Ans. 578 dolls. 70 + cents. 4. If 1 pound of thread make 3 yards of linen, 5 quar- ters wide, how many pounds of thread will it require to make a piece of linen 45 yards long and 1 yard wide ? Ans. 12 Ib. 5. If a footman travels 240 miles in 12 days, when the days are 12 hours long ; in how many days will he travel 720 miles, when the days are 16 hours long? Ans. 27 days. 6. A perch of stone measures 16^ feet long, 1^ foot broad, and 1 foot high ; at 1 doll. 25 cts. per perch,. what will a pile of stone *come to, which measures 30 feet long, 26 feet broad, and 4 feet high ? . Ans. 177 dolls. 27 -f cts. 7. How many cords are there in a pile of wood 200 feet long, 10 feet high, and 36 feet broad ; the cord measuring, according to law, 8 feet in length, 4 in breadth, and 4 in height ? Ans. 562J cords, 8. If 3 pounds of cotton make 10 yards of cloth, 6 qr. wide, how many pounds will it take to make a piece 100 yards long and 3, qr. wide ? Am. 15 Ib 68 PRACTICE 9. If 24 men buiid a wail 200 feet long, 8 feet high, and 6 feet thick, in 80 days, in what time will 6 men build one 20 ft. long, ft. high, and 4 ft. thick ? Ans. 16 days. 10. If a family of 9 persons spend 450 dolls, in 5 months, how much would they spend in 8 months, if 5 more were added to the family'? Ans. 1120 dollars. 11. If a baker's bill for a family of 8 persons amounts to 11^: dolls, in a month, when flour is at 10 dolls, per barrel, what will it amount to in 6 months if 4 more are added tc the family, and flour is at 11 dollars per barrel ? Ans. Ill dolls. 37 cts. 12. If a cellar which is 22,5 feet long, 17,3 feet wide, and 10,25 deep, is dug by 6 men in 2,5 days, working 12,3 hours each day ; how many days of 8,2 hours will it re- quire 9 men to dig one which is 45 feet long, 34,6 wide, and 12,3 feet deep ? Ans. 12 days. PART V. MERCANTILE ARITHMETIC. SECTION 1. OF PRACTICE. PRACTICE, so called from its frequent use in business, is only a contraction of the preceding rules of proportion. By it a compendious way is given of finding the price of any given quantity of goods or other articles of trade, when the price of 1 is known. Case 1. When the price consists of dollars, cents, and mills. Reduce the given quantity by multiplication, as in whole numbers, and point off from the right of the product for mills and cents, according to the rules in federal money. Or, multiply by the dollars only, and take aliquot or frac- tional parts for the cents and mills. PRACTICE TABLE. 50 cents is of a dollar. 144 cents is 33J . . . , 124 . 25 . i . . 11J . 20 . i . 10 ief . ; t 5 EXAMPLE. 1. W^hat will 175 pounds of tea come to at 1 dollar 30 cents and 5 mills per Ib. ? T'O 130,5 175 6525 9135 1305 228375 cts. Or, 25 5 5 m. i J TV 175 4375 875 875 228375 2. 3. 4. 5. 6. 7. 8. 228 dolls. 87 cts. 5 mills. Z). c. m. D. c. 7 250 yards at 1,75 Ans. 437,50 201 do. 4,20 884,20 2210 do. 1,10 2431,00 421 do. 2,41,5 1016,71,5 625 do. 25 156,25 8275 do. 4,4 864,10 8275 do. 5 41,37,5 Case 2. When the price is the fractional part of a dollar, or cent, such as J of a dollar, f of a cent, multiply the quantity by the numerator, and divide the product by the denominator ; the quotient will be the answer. EXAMPLE. 1. What will 375 yards of muslin cost at } of a dollar per yard ? 375 at | 3 4 ) 1125 cwt. qr 2. 4 1 8. 12 2 4. 14 2 281,25 Ant. Ib. D. c. m. 14 of sugar at of a doll, per Ib. Ans. 122,50 13 of spice at f do. do. 942 7 of lead at J do. for Mb. 285,42,5 70 PRACTICE. Application. 1. Bought 6 hogsheads of tobacco, each weighing 12,5 cwt. at f of a doll, per pound ; what did it cost ? Ans. 3150 dolls. 2. A gentleman bought a vessel of 60 tons burden, and gave at the rate of 2J eagles per ton ; what did the vessel cost 1 Ans. 1560 dolls. 3. A carpenter bought 12650 feet of boards at 10 dol- lars per thousand ; what did they cost him ? Ans. 137 dolls. 56 cts. 8 m. Case 3. When the price and quantity given are of several denom- inations, multiply the price by the integers, or whole num- bers, and take aliquot parts for the rest. Table of a hundred weight. 56 Ib. is of a Cwt. 28 4 16 . . 141b. is 4 of a Cwt. A EXAMPLE. 1. Bought Wcwt. Iqr. IQlb. of tobacco at 12 dollars 44 cents per hundred weight ; what did it cost ? 12,44 16 Iqr. is 1 4 I 19904 1Mb. I 4 I 311 1777} 2039274 Ans. 203 dolls. 92 cts. 7| ms. 2. ITcwt. 3qr. I9lb. of sugar at 10 dollars 94 cents per nundred weight. Ans. 196 dolls. 4cts. 3. 5c*ot. Iqr. Qlb. of tobacco at 13 dollars 41 cents per hundred weight. Ans. 70 dolls. 40 cts. 2 m. 4. 7&vt. Qqr. I9lb. of sugar at 15 dolls. 5 mills per hun- dred weight. Ana. 107 dolls. 58 cts. PRACTICE 71 Case 4. When the price consists of pounds, shillings, pence, and farthings. 1. Reduce the given price to dollars and cents, (see re- duction of money, page 41) and then proceed according to the foregoing cases. Or, 2. Multiply by the integers, and take aliquot parts for the remainder. of a shilling. TABLE. ,?. d. J. 10 is i of a pound. 6 68 J 4 5 3 4 f 2 34 i 26 i 2 T V 1 8 Jy EXAMPLES. 1. What will 4548 yards come to at Is. fid. per yard ? Is. 6d.= 18d. 20 cents= } of a dollar. cts. 20 4548 909,60 dollars. Ans. Or, 6d. | i | 4548 at 1 shilling, will be the same num- 2274 oer of shillings. 2|0 ) 682,2 shilling* .341 2 = 909 dolls. 60 cents. 473 yards at 6 397 do. 3 \5\lb. of coffee at 1 658/6. of tea at 12 745 yds. of cloth at 16 969* do. 19 3715 do. 9 4. 5. 6. 7. 8. 9. 4567 d. . s. d. D. c. 8 Arc*. 157 13 4=420,44$ 4 66 3 4 = 8 13 5 5r= 394 16 596 0= 11 964 19 a 4i 1741 8 ij do. 19 114 4557 9 i = 72 PRACTICE. Case 5. When both the price of the integer and the quantity are of different denominations, multiply the price by the inte- gers of the quantity, and take parts of the price for those of the integer. EXAMPLES. 1. What will 45cu?i. 2qr. 14Z&. of sugar come to at 37. 7 s. 9d. per hundred weight? 5s. is 2 6d. is 3 t TV i i i i 45 2 14 379 135 11 5 4 10 1 2 6 11 3 1 13 104 8 54 2qr. is 1Mb. Or thus, . 3 d. 9 5X9=45 16 18 152 8 9 2qr. is 4 1 13 104 14Z6. i 8 54 .154 11 1 Ans. .154 11 1 2. 37 T. llcwt. 2qr. 14Z6. of hemp at 89Z. 65. Sd. per ton. Ans. 3370Z. 13s. 2d. 3. 3T. I2cwt. Sqr. 27lb. of sugar at bZ. 11s. 5d. per cwt. Ans. 6251. 11s. Wd. 4. IT. Icwt. 2qr. 2llb. of rice at 3Z. 17s. 6d. per cwt. Ans. 5601. 13s. 3Jd. 5. 476 acres 3 roods 28 perches of land at 3Z. 7s. lid. per acre. Ans. 1619Z. 11s. l$d. 6. 640 acres 2 roods 20 perches at 10 dollars 55 cents per acre. Ans. 6758 dolls. 594 ct s- 7. 229 acres 3 roods 18 perches at 18 dollars 50 cents per acre. Ans. 4252 dolls. 454 cts. 9. I2cwt. Qqr. lib. at 6 dollars 34 cents per cwt. Ans. 76 dolls. 47 cts. 6 m. 9. llcwt. 3qr. 24JZ&. at 14 dollars per cwt. Ans. 251 dolls. 56 cts. 24 m. 10. 16 acres 3 roods 25 perches, at 125 dollars 50 cents per acre. Ans. 2121 dolls. 73 cts. 3-f m. 11. 25ctrf. 3gr. 14Z/>. at 3Z. 17*. 6d. per cwt. AIM. 1007. 5*. 3j<f. TARE AND TRET. 73 SECTION 2. Allowance on the weight of goods, called Tare and Tret. Tare, is an allowance made for the weight of the barrel, box, trunk, &c. in which goods are packed. Tret, is an allowance to retailers, for waste in the sales of their commodities. Gross, is the whole weight of the .goods with the barrel, box, &c. in which they are put up. Neat, is the weight of the goods after all allowances are deducted. RULE. Find the amount of the tare, and subtract it from the grosi weight, the remainder will be the neat. EXAMPLE. 1. What is the neat weight of a hogshead of tobacco, weighing gross I2cwt. 3qr. I2lb. tare 14Z&. per cwt. Ib. cwt. qr. ib. 14 12 3 12 gross 12 1 2 12 tare 168= the tare of I2cwt. Am. 11 1 neat 10 8oz. = the tare of 3qr. 1 Soz.~ the tare of I2lb. 180= Irw*. 2$r. 12Z&. 2. What will 3 barrels of sugar come to, weighing as fol- <ows: viz. No. 1, 2cwt. Iqr. 25/6. No. 2, 2cwt. 2qr. No. 3, 2cwt. 2llb. tare 2llb. per barrel ; at 12 dollars 50 cents per cwt.? An,s. 82 dolls. 47 cts. 7 m. 3. At 45 cents per pound, wnat wil 14 barrels of indigo come to, weighing as follows : cwt. qr. Ib. Ib. No. 1, 3 3 2 Tare 29 No. 2, 4 1 10 38 \^ 3, 4 o ]9 32 No. 4, 4 00 35 Am. 760 dolls. 95 cts. 74 INTEREST. 4. Bought 2 hogsheads of sugar, weighing as follows viz. No. 1, llcict. Iqr. lllb. tare 112Z6. No. 2, I2cwt. 2qr rare 74ZZ>. at 16 dollars 80 cents per cwt. neat; for which I gave 18 barrels of flour at 4 dollars 50 cents per bbl. and I % ton of iron at 120 dollars per ton; what was the balance still due? Ans. 112 dolls. 65 cts. 5. What is the neat weight of 12 barrels of potash, each weighing ^cwt. 2qr. 26lb. tare I2lb. per cwt. ; and what will it come to at 9 dollars per cwt. ? Ans. SQcwt. 2qr. 23lb. and comes to 456 dolls. 34| cts. 6. Sold a hogshead of sugar, weighing 6cwt. gross, tare 100Z&. tret 4Z&. per 104, for 82 dolls. 50 cents ; what was it sold for per pound? Ans. 15 cents. 7. In I20cwt. 3qr. gross, whole tare llllb. tret 4Z6. per 104, how much is the neat weight in pounds, and what the amount at 73 cents per pound? Ans. 12833,6Z&. and comes to 9368 dolls. 53 cts. 8. Bought 9 hogsheads of sugar, each weighing 6cwt. '2qr. I2lb gross ; tare lllb. per cwt. what is the neat weight, and wh?' Joes it amount to at 16 dollars per cwt. ? Ans. 5Qcwt. Iqr. 22lb. amounts to 807 dolls. 14 cts. 9. Sold 27 bags of coffee, each 2cwt. 3qr. lllb. gross; ea re l'3ib. per cwt. tret 4Z&. per 104; what is the neat weight, and what will it co*ne to at 32 cents per pound ? Ans. 66cwt 2qr. lllb. and comes to 2386 dolls. 88 cts. SECTION 3. OF INTEREST. I NT FOREST is a compensation allowed for the use of money, ibi a givim time; and is generally throughout the United States fixed by law at the rate of 6 dollars for every 100, ;-K>;- annum. 1. Thr sum of w < .ney at interest, is called the Principal. ^. The sum per -nt. agreed on, is called the Rate. 3. Tho principal id interest added together, is called the A/Hf*itn.t. * Interest is firher. nfflc or compound. SIMPLE INTEREST. 75 SIMPLE INTEREST. SIMPLE INTEREST is a compensation arising from the principal only. Case 1. When the given time is one or more years, and the prin- cipal dollars only. RULE. Multiply the given sum by the rate per cent. ; the product will be the interest for one year in cents, which multiplied by the number of years, will be the answer required. EXAMPLES. 1. What is the interest of 454 dollars for one year, at 6 per cent. ? 454 6 2724 cents. Ans. 27 dolls. 24 cts. 2. Required the interest of the same sum for 5 years, at the same rate ? 454 2724 5 13620 cents. Ans. 136 dolls. 20 cts. 3. Required the amount of the same sum for 5 years, at he same rate ? 454 6 2724 5 13620 interest. 4.5400 principal. 59020 amount. Ans. 590 dofts. 20 cts. 4. What is the interest of 200 dollars for 2 years, at 6 per cent ? Ans. 24 dolls. 76 SIMPLE INTEREST. 5. What is the interest of 1260 dolls, for 4 years, at 7 per cent. ? Ans. 352 dolls. 80 cts. 6. What is the amount of a note for 560 dollars for 3 years, at 8 per cent. 1 Ans. 694 dolls. 40 cts. 7. W T hat sum must be given to discharge a bond given for 4520 dollars, on which there is 6 years interest at 5 per cent. ? Ans. 5876 dolls. Note. When the rate per cent, contains a fraction, such as , ^, f the principal must be multiplied by the fraction, as well as the whole number : this may be done either by adding- the parts of \, i, &.c. of the principal to the product of the whole number; or reduce the frac- tion to a decimal. See case 1, in reduction of decimals. 8. W r hat is the amount of 400 dollars for 2 years, at 6| per cent. 1 Ans. 452 dolls. 9. What is the interest of 4925 dollars lor y years, at 7^ per cent. ? Ans. 3324 dolls. 37 cts. 5 m. 10. What is the amount of 2500 dollars for 1 year, at 7$ per cent. ? Ans. 2693 dolls. 75 cts. Case 2. When the principal is dollars and cents, or dollars, cents, and mills, and the time years only. RULE. Multiply the given sum by the rate per cent, and divide the product by 100 ; or, what is the same, point off two figures on the right of the product ; .the quotient, or remain- ing figures, will be the answer, in the same name with the lowest denomination in the principal. EXAMPLES. 1. What is the interest of 264 dollars 50 cents for 1 year, at 6 per cent. ? 264,50 6 cents 1587,00 Ans. 15 dolls. 87 cts. 2. What is the interest of 468 dollars 22 cents and 5 mills for 1 year, M 8 per cent. 468,225 mills 37458,00 Ans. 37 d.|ls. 45 cts. 8 m. SIMPLE INTEREST. ^ 77 3. What is the interest of 364 dollars 50 cents for 5 years, at 6 per cent, per annum ? 36450 Or, 36450 6 30= product of the rate and time. 218700 cents 10935,00 5 cents 10935,00 Ans. 109 dollars 35 cents. 4. What is the amount of a note for 1260 dollars 50 cents and 5 mills for 3 years, at 7^ per cent, per annum? Ans. 1544 dolls. 11 cts. 8-f ms. 5. What sum will discharge a bond given for 630 dollars 50 cents, on which there is 5 years interest at 8 per cent, per annum ? Ans. 882 dolls. 70 cts. 6. What is the difference between the interest of 1274 dollars 64 cents 6 mills for 3 years, at 7J per cent, per an- num, and the interest of 3462 dollars 84 cents, for 4 years, at 3^: per cent, per annum ? Ans. The lattep is 163 dolls. 37 c. 3,85 m. the greater. 7. A gave B his bond for 3422 dolls. 25 cents, to be paid in the following manner, viz. one-third at the end of one year, one-third at the end of two years, and the remainder at the end of three years, with interest from the date, at 6 per cent, per annum ; what will be the annual payments, and what the whole amount ? Ans. 1st payment 1209 dolls. 19,5 cts. 2d, 1277 dolls. 64 cts. 3d, f346 dolls. 8,5 c. ; whole amount 3832 d. 92 c. Case 3. . When the principal is dollars, cents, &c. and the time is years and months, or months only. RULE. Multiply by half the number of months in the given time, when the rate is 6 per cent, per annum : but if the rate per cent, be more or less than 6 per cent., multiply the given number of months by the rate, and divide the product by 12 ; the quotient will be the rate for the time; the principal multiplied by this rate, will give the interest required. G2 78 ., SIMPLE INTEREST EXAMPLES. 1. What is the interest of 650 dollars for 8 months, at 6 per cent, per annum ? 650 4 half the months cents 2500 Ans. 26 dollars. 2. What 13 the interest of 860 dollars for 1 year and 6 months, at 6 per cent, per annum ? 860 9 half the months cents 7740 Ans. 77 dolls. 40 cents. 3. What is the interest of 420 dollars for 9 months, at 8 per cent, per annum ? 9 months 420 8 per cent. 6 12 ) 72 2520 Ans. 25 dolls. 20c. 6 the rate for the time. 4. What is the amount of a note for 724 dollars, with 18 months interest due thereon, at 4 per cent, per annum ? Ans. 767 dolls. 44 cts. 5. What is the interest' of 240 dollars for 15 months, at 7 2 per cent, per annum ? A.ns. 22 dolls. 50 cts. 6. What is the interest of 1260 dollars for 4 months, at G per cent, per annum ? Ans. 27 dolls. 30 cts. Case 4. When the principal is dollars, cents, &c. and the time is months and days, or days only. RULE. Find the interest for the given months by the last case, find take aliquot parts for the days. Note. In calculation of interest, 30 days make a month. Or, multiply the given sum when the rate is 6 per cent. i)\ the number of days, and divide the product by 60 ; the <]ii<>;ient is the interest required. Note. Though both the foregoing methods are considered sufficiently exart for common business, by merchants and jiccomptants generally, SIMPLK INTKRKST. 7<J yet as this is only allowing 360 days in the year, and not 365, the true time ; if, therefore, the principal is large, on which interest is due, and greater exactness is required, then find the interest of the given sum tor 1 year, and proceed according to the single rule of three. As 365 days : to the interest for one year :: the given number of days : the an swer. Or by the double rule of three, find the fixed divisors, which fbi 5 per cent, is 7300, fer 6 per cent, is 6083, for 7 per cent. ^214 ; multi- ply the principal by the days, and divide by these divisors according to the rate per cent, required. EXAMPLES. 1. What is the interest of 260 dollars for 5 months aid 20 days, at 6 per cent, per annum ? 260 2 \ month I i |5,2G interest ibr 4 months 15 days [ i UfSO interest for 1 do- 5 j |4 I 65 interest for 15 days 2 1,6 interest for 5 do- 7,36,6 Ans. 7 dolls. 36 cts. 6 m. 2. What is the interest of 450 dollars for 36 days, at ti per cent, per annum ? 450 6,0 ) 1620,0 36 2,70 Ans. 2 dolls. 70 cts. 2700 1350 16200 3. What is the interest of 564 dollars for 44 days, at 6 per cent, per annum ? 564 6 days days Ai 365 : 3384 :: 44 44 13536 13536 365 ) 148896 ( 4,079, Ans. -i dolls. 07 cts. 9 -f m. 1460 2896 2555 3410 3285 125 80 SIMPLE INTEREST. 4. What is the interest of 960 dollars for 70 days, at 6 per cent, per annum ? 960 " 70 6083 ) 67200 ( 1 1 + dolls. Ans. 6083 6370 6083 287 5. What is the interest of 12000 dollars for 40 days, at 7 per cent, per annum 1 Ans. 92 dolls. 6 cts. 6. What is the interest of 8400 dollars for 20 days, at 5 ,,er cent, per annum ? Ans. 23 dolls. D. c. days D.c.m. 7. 517,90 for 84 at 6 per cent, per annun*. Ans. 7,15,1 e. 73,41 27 33 9. 225,24 40 .... 1,48,1 10. 1200,00 80 - 15,78,1 11. ^62,19 254 123,68,8 12. 1733,97 102 - 29,07,5 A TABLE, knowing the number of day s+ from any day in any mojtth, to the same day in any other month through the year. From jJa.|Fb.|Mr.|Ap.|Ma. Ju. |Jly.|Au.| Se.|Oc.|No.|De. ToJan.|365|334|306|275|245|214|184|153|122| 92| 61| 31 Feb. | 31j365|337|306[276|245|215|184|153|123| 92| 62 59| 28|365|334|304|273|243|212!lftl|151|120| 90, April | 90| 59| 31|365|335|304|274',243|212|182|151|121 May |120| 89| 61| 30|365J335|304|273|242|212|181|151 |151|120|' 92 1 61| 31|365J885|304!273|243|212|182 July |181|150|122| 91| 61| 30|365|334|303|273|242|212 Au ? ust|2121181|153|122| 92 1 61 1 31|365|334|304|273(243 sept. |243|212|184|153|123| 92| 62| 31|365|335|304|274 Oct. |273|242|214|183|153|122| 92| 61 1 30|365|334|3Q4 |304|273|245|214|184|153|123| 92| 61 1 31;365|335 |334|303|275|244|214|183|153|122| 91 1 61| 30|365 Suppose the number of days between the 10th of April and l()th of Ortohpr wpre rerjuirorl ; under the column of SlMl'LK 1NTKRKST. 8) April at the top of the table, look for October, and you find =188, the number required. Ii the days in the given months be different, their differ- ence must be added or subtracted, to or from the tabular number. Thus, from the 10th of April to the 20th of Oc- tober, is 183 + 10 = 198 days. And from the 20th of April to the 10th of October, 18310=173 days. If the time exceed a year, 365 days must be added for each year. Case 5. When the amount, rate, and time are given to find the principal. RULE. As the amount of 100 dollars, at the rate and time given, is to 100 dollars, so is the amount given to the principal re- quired. EXAMPLE. 1. What principal being put to interest for 9 years, at 5 per cent, per annum, will amount to 725 dollars ? 9 45 100 As 145 : 100 :: 725 : 500 Ans, 2. What principal being put to interest for 12 years, at ft per cent, per annum, will amount to 2752 dollars? Ans. 1600 dolls. 3. Received 728 dollars as payment in full for a note with 5 years interest thereon, at. 6 per cent, per annum ; for how much was the note given? Ans. 560 dolls. 4. What sum put to interest for 4 years, at 7 5 per cent. per annum, will amount to 1 638 dollars ? Am. 1260 dolls. 5. Received 2000 dollars as payment in full for a bond, with 5 years interest thereon, at 5| per cent, per annum ; what principal did the bond contain ? Ans. 1553 dolls. 39 cts. 8 T Jm. 82 SIMPLE INTEREST Case 6. When the amount, time, and principal are given to find the rat3. RULE. 1. As the principal is to the interest for the whole time so is 100 dollars to its interest for the same time. 2. Divide the interest so found by the time, and the quo- tient will give the rate per cent. EXAMPLE. 1. At what rate of interest per cent, will 500 dollars amount to 725 dollars in 9 years ? 725 500 D. D. D. D. 225 As 500 : 225 :: 100 : 45 9)45 5 per cent. Ans. 2. Paid 858 dollars in full for a note given for 650 dollars, with 4 years interest due thereon; what was the rate per cent, per annum charged on said note ? Ans. S per cent. 3. At what rate per cent, will 1600 dollars amount to 275*2 dollars in 12 years? Ans. 6 per cent. 4. At what rate per cent, will 640 dollars amount to 860 dolls. 80 cents in 6 years? Ans. 5f per cent. 5. At what rate per cent, will 12000 dollars amount to 50100 dollars in 15 years? Ans. 4 percent. Case 7. When the principal, amount, and rate are given to find the time. RULE. Find the interest of the principal for one year. And then s as the interest for 1 year, is to 1 year, so is the whole inter- est to the time required. KX AMPLE. 1. In what time will 500 dollars amount to 725 dollars at 5 per cent, per annum? 500 5 D. Y. D. Y. !, acres! for 1 year 25,00 As 25 : 1 :: 22,) : <) Ans. Amount 725 Principal 500 Whole interest 22o Sl^lPLK INTKRKST. 83 2. ln what time will 050 dollars amount to 910 dollars, at S per cent, per annum 1 Ans. 5 years. 3. In what time will 1600 dollars amount to 2080 dollars at 6 per cent, per annum ? Ans. 5 years. Case 8. When the principal is in English money, viz. pounds-, shillings, and pence, and the interest required either in Fed- eral or English money. RULE. Reduce the English to Federal money, and find the in terest by the~preceding rules. EXAMPLE. 1. What is the interest in Federal money, of 325Z. 105. English money, for 5 years, at 6 per cent, per annum ? . s D. c. 325 10=1445,22 30=the time multiplied by the rate. 4335660 Ans. 433 dolls. 56 cts. 6m 2. What is the amount of a note for 640/. 3s. 6d. with 3 years interest due thereon, at 5 per cent, per annum, in Fed- eral money? Ans. 3268 dolls. 73 cts. 3|i m. 3. What is the interest of 1374Z. ls.-9d. for U ycar : . aJ 5f per cent, per annum? Ans. 1157. 185. Or/. Case 9. Computing interest on bonds, notes, &c. on which differ- ent payments have been made. RULE I. Find the interest of the principal from the time the inter- est first commenced, to the time of the first payment made ; add that interest to the principal, and subtract from the amount the payment made ; the remainder forms a new principal ; on which proceed in the same manner, till all the payments are brought in. Note 1. When a payment alone, or in conjunction with any pre- ceding- payment, is. less than the interest due at the time, then no cal- culation must be made ; but these lesser payments added to the next 2. By this rule, no part of the interest ever forms a part of the prin- cipal carrying interest, the payments being first applied to discharge the interest. 84 SIMPLE INTEREST. EXAMPLES. * 1. A has B's note for 1000 dollars, dated 1st January 1816, payable in 18 months, with interest from the date, at 6 per cent, per annum. On which the following payments are endorsed, viz. 1816. July 1. Rec'ci on the within note 230 dollars 1817. Jan. 1. Rec'd - 300 March 1. Rec'd ... 4 April 1. Rec'd - 250 What was the balance due on the 1st of July, 1817, when the whole note is payable ? D. c. Principal at interest from January 1, 1816, 1000 00 1816, July 1. Interest (6 months) - 30 1030 Paid same date 230 Remainder for a new principal - 800 1817, January 1. Interest (6 months) '24 824 Paid same date ..... 300 Remainder for a new principal - ^ 524 March 1 . Paid 4 dollars less than the interest and not to be calculated. April 1. Interest (3 months) - 7 86 531 86 Paid same date 250 + 4= - . . 254 277 86 July 1. Interest (3 months) - 4 17 Balance due Ans. 282 03 RULE II. Multiply the principal by the number of days, till the first payment is made ; the remaining principal by the number of days, between the first and second payment, &c. till aU the payments are made ; divide the whole amount by 60 ; the quotient will give the interest required. Note. By this method the interest is generally calculated among merchants. INTEREST. 85 1. A bond was given by B to C, for 2400 dollars, payable in 2 years, with interest from the date. Dated July 1, 1815. On this bond the following payments are eridgrsed ; viz. May 1, 1316, 900 dollars ; October 1, 1816, 450 dollars , January 1, 1817, 620 dollars. Required the amount due on the 1st of May, 1817? 1815. July 1. Principal 2400 dollars. 1816. May 1. 2400 multiplied by 304 is 729600 Paid 900 Oct. 1. 1500 .... 153 . 229500 Paid 450 1917. Jan. 1. 1050 . .-. . 92 . 96600 Paid 620 May 1. 430 .... 120 . 51600 Interest 184 55 . Divide by 6,0 ) 110730,0 Balance 614 55 Ans. Interest 184,55 2. A note was given by A to B, for 1800 dollars, dated 1st January, 1820, with interest from the date. On which the following payments are endorsed, viz. April 1, 1821, 700 dollars; January 1, 1822, 400 dollars; July 1, 1822, 500 dollars. Required the amount due on the 1st of Janu- ary, 1823 ? Ans. 414 dolls. 16 cts. 6 m. COMPOUND INTEREST Is a compensation allowed not only for the principal, but also for the interest as it becomes due. RULE. Add the simple interest of the given sum for one year to the principal. This amount forms a new principal for the second year, and so on for any number of years required. Subtract the first principal from the last amount, the re- mainder will be the compound interest required. H COMPOUND INTEREST. EXAMPLE. 1. What is the compound interest of 500 dollars for 3 years, at 6 per cent, per annum I dolls. 500 1st principal 6 30,00 interest 500 530,00 2d principal 6 31,80,00 530 561,80 3d principal 6 33,70,80 561,80 595,50,8 last amount 500 95,50,8 Ans. 95 dolls. 50 cents, 8 mills. Compound Interest may be more expeditiously calcufctted by the follow- ing Table, in which the amount of one dollar for any number of years under 30 is shown, at the rates of 5 and 6 per cent, per annum, com- pound interest. Years 5 Ra tes 6 Years 5 Ri ites 6 1 1.05000 1.06000 16 2.18287 2.54035 2 1.10250 1.12360 17 229201 2.69277 3 1.15762 1.19101 18 2.40662 2.85434 4 1.21550 1.26247 19 2.52695 3.02559 5 1.27628 1.33822 20 2.65329 3.207 1.5 , 6 1.34009 1.41852 21 2.78596 3.39956 7 .40710 1.50363 22 2.92526 3.60353 8 .47745 1.59384 23 3.07152 3.81i>75 9 .55132 1.68948 33 3.22510 4.04898 10 .62889 1.79064 25 3.38635 4.29187 11 .71034 1.89829 26 3.55567 l.r,ii)38 12 .79585 2.01219 27 .T73345 1234 13 1.88565 2.l32;j^ 28 3.920 J 3 5.11168 14 1.97993 2.26090 29 4.11613 5.41838 15 2.07892 2.39655 30 4.32194 5.74349 To find the compound interest of any sum by this labic, multiply the figures opposite the number of years, under the rate percent, b r 'n prim-ipal ; the product will be tbt: amount required ; from this subtract the principal, tht. remainder will b'' th< INSURANCE COMMISSION. AND BROKAGK. 87 EXAMPLE. 2. What is the compound interest of 1000 crbllars for W years, at 6 per cent, per annum ? 1,59384 the tabular number for the time 1,000 the principal 1593,84000 1000 593,84 the interest. Ans. 593 dolls. 84 cts. 3. What is the amount of 1500 dollars for 5 years, at 5 pej* cent, per annum? Ans. 1914 dolls. 42 cts. 4. What is the compound interest of 4500 dollars for 16 years, at 6 per cent, per annum ? Ans. 6931 dolls. 57 cts. 5 m. 5. A has B's note for 650 dollars, payable at the end of 20 years, at 6 per cent, per annum, compound interest ; what sum will it require to discharge the note, at the expira- tion of the given time ? Ans. 2084 dolls. 63 cts. 4 m. 6. A father left a legacy of 8000 dollars at compound in- terest, 6 per cent, per annum, to be equally divided among his three sons, when the youngest, who was 4 years old, should arrive at the age of 21 ; what will be each one's share? Ans. 7180 dolls. 72 cts. each share. SECTION 4. Insurance, Commission, and Brokage. INSURANCE is a premium given for insuring the owners of property against the dangers and losses to which it is liable, or indemnifying for its loss. The instrument of agreement by which this indemnity is secured is termed the policy of insurance. Commission is a compensation allowed to merchants and others for buying, selling, and transporting goods, wares, 88 INSURANCE A\V> COMMISSION. Brokage is an allowance given to brokers for exchanging money, buying and selling stock, &c. The method of operation in all these is the same as in simple interest, EXAMPLES, INSURANCE. 1. What is the premium of insuring 1260 dollars, at 5 per cent ? 1260 5 63,00 Ans. 63 dolls. D. c. 2. 1650 dollars at 15 per cent. - Ans. 255 75 3. 4500 25 - - - 1125 00 4. What sum must a policy be taken out for, to cover 900 dollars, when the premium is 10 per cent? 100 policy 10 premium 90 sum covered As 90 : 100 :: 900 : 1000 dolls. Ans. 5. What sum will it require to cover a policy of insurance for 4500 dolls, at 25 per cent 1 Am. 6000 dolls. 6. What sum witl.it require to cover a policy of insurance for 560 dollars, at 9 per cent 7 Ans. 615 dolls, 38$ cts, COMMISSION. 1. What is the commission on 850 dolls, at 5 per cent. 850 5 42,50 Ans. 42 dolls. 50 cts. 2. What is the commission on 1260 dollars, at 6 per rent? Ans. 75 dolls. 60 cts. D. c. 3. 2550 dollars at 4 per cent. - - Ans. 102 00 4. 26342 H 790 26 5. 6422 } 48 10 J BROKAGE Bl'YINU AND SELLING STOCKS. 89 6. A commission merchant receives 1260 dollars to fill an order, from which he is instructed to deduct his own commission of 5 per cent, how much will remain to satisfy the order? 100 5 per cent. As 105 : 100 :: 1260 : 1200 dolls. Ans. 7. A commission merchant has received 4120 dollars \vith instructions to vest it in salt at 8 dollars per harrel ; de- ducting from it his commission of 3 per cent, how many barrels of salt can he purchase ? Ans. 500 barrels. BROKAGE. 1. What is the brokage on 1000 dollars, at lj per cent? 1000 H 1000 500 15,00 Ans. 15 dolls. 2. What is the brokage on 1625 dollars 50 cents, at 3^ per cent? Ans. 54 dolls. 18 cts. 3. 1868 dollars at 2 per cent. Ans. 46 dolls. 70 cts. 4. 560 6 38 60 SECTION 5. BUYING AND SELLING STOCKS. Stock is a fund vested by government, or individuals in a corporate capacity, in banks, turnpike roads, bridges, &c. the value of which is subject to rise and fall. RULE. Multiply the given sum by the rate per cent, and divide the product by 100 H2 90 REBATE OR DISCOUNT. EXAMPLES. 1. What is the amount of 1650 dollars, United States bank stock, at 125 per cent, or 25 per cent, above par? 1650 125 6250 3300 1650 1,00 ) 2062,50 Ans. 2062 dolls. 50 cts. Or thus, 25 is i)1650 412 50 2062 50 AM. D. D. c. 1500 bank stock at 110 per cent. Ans. 1650 00 1686 128 . 2158 08 4. 25000 108 - - 27000 00 5. 1260 90 - - 1134 00 6. 9254 84 - 7773 36 7. 1518 83| 1271 32$ SECTION 6. REBATE OR DISCOUNT, Is a reduction made for the payment of money before il tecomes due. It is estimated in such a manner, as that the ready payment, if put to interest at the same rate and time, vvould amount to the first sum. Thus, 6 dollars is the discount on 106 dollars for 12 months, at 6 per cent, leaving 100 dollars the ready payment, which, if put to interest for the same rate and time, would regain the 6 dollars discount. RULE. As 100 dollars and the interest for the given time, is to 100 dollars, so is the given sum to its present worth. Sub tract the present worth from tho given sum, and the remain dor is the discount. REBATE OR DISCOUNT. 91 ^ EXAMPLE. 1. What s the discount of 1696 dollars, due 12 months hence, at 6 per cent, per annum ? As 106 : 100 :: 1696 : 1600 1600 96 dolls. Ans. 2. What is the present worth of 2464 dollars, due 1 year and 6 months hence, discounting at the rate of 8 per cent. nor annum] Ans. 2200 dolls. 3. A has B's note for 1857 dollars 50 cents, payable 8 months after date ; what is the present worth of said note, discounting at the rate of 5J per cent, per annum? Ans. 1791 dolls. 80 cts. 4. What reduction must be made for prompt payment of a note for 650 dollars, due 2 years hence, 7 per cent, per annum being allowed for discount ? AILS. 79 dolls. 83 + cts. 5. What is the present worth of 5150 dollars, due in 4^ months, discounting at the rate of 8 per cent, per annum, and allowing 1 per cent, for prompt payment ? Ans. 4950 dolls. Note. Discount and interest are often supposed to be one and the same thing; and in business, the interest for the time is frequently taken for the discount, and it is presumed neither party sustains any loss. This however is not true, for the interest of 100 dollars for 12 months, at 6 per cent, is 6 dollars, whereas the discount for the same sum, at the same rate and ti'ne, is only 5 dollars 66 cents, making a difference of 34 cents for every 100 dollars for 1 year at 6 per cent. The following 1 examples will show the difference. EXAMPLES. 1. What is the discount of 1272 dollars, due in 12 months, discounting at 6 per cent, per annum ? .As 106 : 100 :: 1272 : 1200 discount 72 dolls. 2. What is the interest of the same sum, for the same time and rate ? 1272 D. c. 6 . Interest 76 32 Discount 72 76,32 interest, Difference 4 32 92 BANK DISCOUNT. 8. What is the difference between the interest and dis- count on 7280 dollars, for 18 months, at 8 per cent, per an- num? Ans. 93 dolls. 60 cts. difference. Note. But when discount is made for present payment, without re- gard to time, the interest of the sum as calculated for a year, is the discount. EXAMPLE. 1. How much is the discount of 260 dollars at 5 per cent? 260 5 13,00 Ans. 13 dollars. 2. What is the discount on 1650 dollars, at 3 per cent! Ans. 49 dolls. 50 cte. 3. What sum will discharge a bond for 2464 dollars, on which a discount of 8 per cent, is given ? Ans. 2266 dolls. 88 cts. SECTION 7. BANK DISCOUNT. BANK discount is the interest which banks receive for the use of money loaned by them for short periods. And as banks from long established custom, give three days over and above the time limited by the words of the note, called days of grace ; and as the day of the date, and the day of payment are both calculated, which makes the time 4 days longer than expressed in the note, so interest must be calcu lated on these days in addition to the regular interest on the given sum, for the specified time. RULE. Add 4 to the number of days specified in the note, multi- ply the given sum by this number, and divide the product by 60. Or, Multiply the given sum by half t)ie number of days, and divide bv 30. EQUATION OF PAYMENTS. f 98 Note. When the cents in the given sum are less than 50, the bank loses the interest on them, but when they are more than 50 they charge interest for one dollar. EXAMPLES. 1. Required the discount of 1500 dollars for 60 days. 1500 Or, 1500 64 32= half the days 6000 3000 9000 4500 6,0)9600,0 3,0)4800,0 16,00 Ans. 16 dolls. 16,00 2. What is the discount of 250 dollars for 30 days ? Ans. 1 doll. 41 1 cts. 3. What is the discount of 600 dollars for 90 days ? Am. 9 dolls. 40 cts. 4. What is the discount of 1260 dollars 40 cents for 60 days 1 Ans. 13 dolls. 44 cts. 5. What is the discount of 2649 dolls. 75 cents for 60 days ? Ans. 28 dolls. 26 cts. 4 m. Form of a note offered for discount. Pittsburgh, July , 1832. Dollars Sixty days after date, I promise to pay A. B. or order, at the bank of , the sum of dollars, without de- falcation ; value received. J. P. SECTION 8. EQUATION OF PAYMENTS. EQUATION of payments is the finding the mean time, for the payment of two or more sums of money payable at dif- ferv.~t times. RULE. Multiply each sum by its own time. Add the products 94 FELLOWSHIP. into one sum and divide this amount by the whole debt ; t/ie quotient will be the mean time. EXAMPLE. 1. A owes B 600 dollars, of which 200 is to be paid at 4 months, 200 at 8 months, and 200 at 12 months; but they agree to make but one payment ; when must that paymetit be made? 200 X 4= 800 200 X 8=1600 200X12 = 2400 600 ) 4800 ( 8 months. Ans. 4800 2. A merchant has owing to him from his friend, the sum of 3000 dollars, to be paid as follows, viz. 500 dollars at 2 months, 1000 dollars at 5 months, and t le rest at 8 months ; but they agree to make one payment of the whole ; whit will be the mean time of payment? Ans. 6 months. 3. A buys of B 50 acres of land, for which he agrees to pay 1000 dollars at the following times, viz. 200 dollars at 5 months, 300 dollars at 8 months, and the rest at 10 months, but an equation of payments is afterwards agreed upon ; when must the payment be made? Ans. 8 months 12 days. 4. C owes D 1400 dollars, to be paid in 3 months, but D being in want of money, C pays him 1000 dollars at the ex- piration of 2 months ; how much longer than 3 months may he in justice defer the payment of the rest ? Ans. 2% months SECTION 9. FELLOWSHIP. 'SHH' loaches to find tho profit or loss arising f o different partners in trade, in proportion to the cap: J or stock each has advanced. Fellowship is either nngle or compound. SINGLE FELLOWSHIP. 95 SINGLE FELLOWSHIP, Is when the stocks employed are different, but the time alike. RULE. Find the amount of the whole stock employed ; and then (by proporfton) as the whole stock is to the whole gain or loss, so is each partner's stock to his share of the gain or loss. EXAMPLE. 1 . Two merchants join their stock in trade ; A puts in eOO dollars, and B puts in 400 dollars, and they gain 250 dollars ; what part belongs to each ? A 600 A 1 nnn - n 600 to A's share 150 ^ B 400 ; 400 to B's 100 ! ^^ 1000 250 J 2. Three merchants enter into partnership in trade ; A advanced 7500 dollars, B 6000, and C 4500, with this they gained 5400 dollars ; what was each partner's share ? C A 2250 dolls, Ans. I B 1800 I C 1350 3. A bankrupt is indebted to A 1291 dollars 23 cents, to B 500 dollars 37 cents, to C 709 dollars 40 cents, to D 228 dollars ; and his estate is worth but 2046 dollars 75 cents ; how much does he pay per cent, and how much is each creditor to receive ? D. c. (A receives 968 42$ B 375 27 | C 532 05 D 171 4. Three men, A, B, and C, rent a farm containing 585 acres 2 roods and 34 perches, at 600 dollars per year, oJ which A pays 180 dollars, B 195, and C 225, and they agree that the larm shall be divided in proportion to the rents; how many acres must each man have? A. R. P. ; share is 175 2 34j Ans. { B's 190 1 17^ 219 2 22] i A's share is 175 , { B's 190 f C's 219 96 S1JNGLE FELLOWSHIP. 5. Three merchants freighted a ship with 2160 barrels of flour, of which 960 barrels belonged to A, 720 barrels to B, and 480 barrels to C ; but on account of stormy weather they were obliged to throw 900 barrels overboard ; how many barrels did each man lose? i A lost 400 barrels. Ans. IE $00 (C 200 6. Three merchants join stock in trade ; A put in 1260 dollars, B 840 dollars, and C a certain sum ; and they gained 825 dollars, of which C took for his part 275 dollars ; re- quired A and B's part of the gain, and how much stock C put in? ( A gained 330 dolls. < Ans. <B 220 ( C's stock was 1050 7. Four men traded with a stock of 800 dollars, and they gained in two years time twice as much, and 40 dollars over ; A's stock was 140 dollars, B's 260, C's 300 ; required D's stock, and what each gained ? (D's stock was 100 dolls. A's gain was 287 B's 533 C's 615 D's 205 8. Three butchers lease a pasture field for 96 dollars, into which they put 300 beef cattle ; of these 80 belonged to A , 100 to B, and 120 to C ; how much had each to pay ? Ans. A 25 dolls. 60 cts. B 32 dolls. C 38 dolls. 40 cts. 9. A father left an estate of 5000 dollars to his three sons, in such a manner that for every 2 dolls, that A gets, B shall have 3, and C 5 ; how much did each son receive ? Ans. A gets 1000 dolls. B 1500, C 2500. 10. A, B and C put in money together, A put in 20 dolls B and C together put in 85 dollars ; they gained 63 dolls, of which B got 21 dollars ; what did A and C gain, and how much did B and C separately put in ? {A gained 12 dolls. C 30 B put in 35 C 50 COMPOUND FELLOWSHIP. 97 COMPOUND FELLOWSHIP. COMPOUND FELLOWSHIP is when both the stocks 'and times are different. RULE. Multiply each partner's stock by the time it is employed, add all the products into one sum; then say, as the sum of the products is to the whole gain or loss, so is each part- ner's stock multiplied by the time, to his share of the gain or loss. EXAMPLE. 1. Three merchants entered into trade; A put in 2500 dollars for 4 months, B 3000 dollars fur 6 months, and C 4000 dollars for 8 months, and they gained 1200 dollars; what is each man's share of the gain ? D. M. A 2500x4=10000 B 3000X6=18000 04000X8=32000 Sum 60000 10000 : 200 A's share ) As 60000 : 1200 :: < 18000 : 360 B's V Ans. 32000 : 640 C's > 1200 proof. 2. Three merchants enter into partnership for 16 months; A put into stock at first 600 dollars, and at the end of 8 months, 200 dollars more ; B put in at first 1200 dollars, but at the end of 10 months, was obliged to take out 600 dollars; C put in at first 1000 dollars, and at the end of 12 months put in 800 more ; with this stock they gained 2300 dollars ; what was each man's share ? C A's share is 560 dolls. Ans. 1 B's 780 ( C's 960 3. A and B join stock in trade ; A put in 600 dollars or. the first of January ; B advanced on the first of April a sum 98 COMPOUND FELLOWSHIP. which entitled him to an equal share of the profit at the end of the year ; required the sum B put in ? Ans. 800 dollars. 4. D put in stock 1800 dollars ; E at the end of 4 months agrees to advance such a sum as at the end of the year wili entitle him to an equal share of .the profits; what sum must E advance? Ans. 2700 dollars. 5. Two gentlemen, A and B, hired a carriage in Pitts- burgh to go to Philadelphia, and return, for 160 dollars, with liberty to take in two others by the way. When at Philadelphia they took in C, and afterwards, 100 miles from Pittsburgh, they took in D. Now allowing it to be 300 miles from Pittsburgh to Philadelphia, and also that each man pays in proportion to the distance he rode ; it is required to tell how much each must pay ? !A pays 60 dolls. B 60 C 30 D 10 160 proof. 6. Three graziers hired a piece of pasture ground for 145 dolls. 20 cents ; A put in 5 oxen for 4 months, B put in 8 oxen for 5 months, and C put in 9 oxen for 6 months ; how much must each pay ? Ans. A pays 27 dolls. B 48 dolls, and C 70 dolls. 20 cts. 7. A, B, and C have received 665 dollars interest; A put in 4000 dolls, for 12 months, B 3000 for 15 months, and C 5000 for 8 months ; how much is each man's part of the in- terest ? ( A 240 dolls. Ans. I B 225 f C 200 HI. Three merchants lost by somo dealings 263 dollars 90 -outs ; A's stock was 580 dolls, for 6 months, B's 580 dolls, for 9J months, and C's 870 dolls, frr 8| months ; how much is each man's part of this loss ? A's loss 59 dolls. 15 cts. Ans. B's 86 45 C's 118 30 PROFIT AND LOSS. 99 SECTION 10. PROFIT AND LOSS. BY this rule we discover what has been gained or lost on the purchase and sale of goods, and merchandise of every kind. RULE. Prepare the question by reduction when necessary, and then work by the Rule of Three or Practice, as the nature of the question may require. EXAMPLE. 1. Bought 360 barrels of flour for 6 dollars 25 cents per barrel, and sold it for 7 dollars 50 cents per barrel; what is the profit on the whole ? D. c. 7 50 6 25 1 25 gain per barrel. B. D.c. B. D. As 1 : 1 25 :: 360 : 450 Ajis. 2. Bought a piece of cloth for 1 doll, and 20 cents pei yard, and sold it again for 1 dollar 50 cents a yard ; what is the gain per cent 1 Ans. 25 per cent. 3. Bought a piece of linen containing 42 yards for 21 dollars, and sold it at 66 cents per yard ; what is the gain or loss on the whole piece ? Ans. 6 dolls. 72 cents gain. 4. A merchant bought 6 barrels of whiskey containing 32 gallons each, for 96 dollars ; while in his possession he l ost 12 gallons by leakage, the residue he sold for such a sum as gained him 12 dollars on the whole; how much per gallon did he buy and sell for ? Ans. Bought for 50 cents, and sold for 60 cents per gall. 5. Bought 120 doz. of knives for 20 cents each knife, and sold them again for 17 cents each, what was the loss on the whole? Ans. 43 dolls. 20 cts. 100 PROFIT AND LOSS 6. A merchant gave 149 dollars for 100 yards of cloth ; at how much per yard must he sell it to gain 51 dollars on the whole 1 Arts. 2 dollars. 7. Bought a chest of tea at 1 dollar and 25 cents per pound, but finding it to be of an inferior quality, I am will- ing to lose 18 per cent, by it ; how must I sell it per pound ? Ans. 1 doll. 24 cents per Ib. 8. A merchant bought 20 dozen of wool hats at 90 cents per hat ; at what rate must he sell them again to gain 20 per cent, and how much does he gain on the whole 1 Ans. he must sell at 1 dollar 8 cents per hat, and gains 48 dollars 20 cents. 9. A trader bought a hogshead of rum of a certain proof, containing 115 gallons, at 1 dollar 10 cents per gallon ; how many gallons of water must he put into it to gain 5 dollars, by selling it at 1 dollar per gallon 1 Ans. 16i gallons. 10. A merchant bought 4 hundred weight of coffee for 134 dollars 40 cents, and was afterwards obliged to sell it at 25 cents per pound ; what was his loss on the whole, and how much on each pound ? Ans. 5 cents, loss on each pound, and 22 dollars 40 cents on the whole. 11. If by selling 360 yards of broadcloth for 1728 dollars, there is gained 20 per cent, profit, what did it cost per yard ? Ans. 4 dollars. 12. A merchant laid out 1000 dollars on cloth, at 4 dol- lars per yard, and sold it again at 4 dollars 90 cents per yard ; what was his \yhole gain ? Ans. 225 dolls. 13. A sells a quantity of wheat at 1 dollar per bushel, and gains 20 per cent. ; shortly after he sold of the same to the amount of 37 dollars 50 cents, and gained 50 per cent. ; how many bushels were there in the last parcel, and at what rate did he sell it per bu >hel ? Ans. 30 bushels, at 1 doll. 25 cents per bushel. 14. A trader is about purchasing 5000 galls, of whiskey, which he can have at 48 cents per gallon in ready money, or 50 cents with two months credit ; which will be the most profitable, either to buy it on credit, or by borrowing the money at 8 per cent, per annum, to pay the cash price? Ans. he will gain 68 dollars by paying the cash. 15. A butcher bought 12 head of beef cattle of equal BARTER. 101 weight, for 240 dollars, which he sells again for 4 cents per pound ; what ought each one to weigh, that the butcher may have the hides and tallow as clear gain ? Ans. 4:cwt. Iqr. 24Z&. SECTION 11. BARTER. BARTER is the exchanging of one commodity for another at the rates agreed upon by their owners. $ RULE. Proceed by the rules of reduction and proportion, as the nature of the question may require. EXAMPLE. 1. How many yards of linen at 50 cents per yard must be given for 6^ yards of broadcloth, at 4 dollars 50 cents per yard ? 4,50 dollars 6* 2700 112J 28,12$ c. yd. D. c. yds. As 50 : 1 :: 28,12* : 56% Ans. 2. A has 320 bushels of salt at 1 dollar 20 cents per bushel, for which B agrees to pay him 160 dollars in cash and the rest in coffee at 20 cents per pound ; how much cof- fee must A receive? Ans. 1120 Ib. 3. How much rye at 70 cents per bushel must be given for 28 bushels of wheat, at 1 dollar 25 cents per bushel ? Ans. 50 bushels. 4. A barters 319 Ib. of coffee at 23* cents per pound, with B for 250 yards of muslin ; what does the muslin cost A per yard 1 Ans. 30 cents nearly. 5. C has flour at 5 dollars per barrel, which he barters 12 102 EXCHANGE. to D at a profit of 20 per cent, for tea which cost 1 dollar 25 cents per pound ; at what rate must D sell the tea to make the barter equal? Ans. I doll, 50 cts. per Ib. 6. A has cloth which cost him 2 dollars 50 cents per yard, but in trade he must have 2 dollars 80 cents ; B has wheat at 1 dollar 20 cents per bushel ; at how much per bushel should he sell to A, to make the barter equal ? Ans. I doll. 34f cents. 7. P has 240 bushels of rye which cost him 90 cents per oushel ; this he barters with Q at 95 cents per bushel for wheat which stands Q 99 cents per bushel ; how many bushels of wheat is he to receive in barter, and at what price, that their gains may be equal ? Ans. 218/y bushels, at 1 doll. 4| cts. per bushel. 8. A gives B in barter 26lb. 4oz. of cinnamon, at 1 dol- lar 28 cents per pound, for rice at 6 cents per pound ; how much rice must A receive ? Ans. 5 cwt. 9. C and D barter ; C has muslin that cost him 22 cents per yard, and he puts it at 25 cents ; D's cost him 28 cents per yard ; at what price must he put it to gain 10 per cent, more than C ? Ans. 34|| cents per yard. 10. A buys 250 barrels of flour from B, at 6 dollars 25 cents per barrel, in payment B takes 4 cwt. of coffee at 30 cents per pound, 64 pounds of tea at 1 dollar 75 cents per Ib. 25 yards of broadcloth at 6 dollars per yard, 206 dollars 10 cents in cash, and the balance in salt, -at 8 dollars per barrel ; how many barrels of salt must B receive ? Ans. 120 barrels. SECTION 12. EXCHANGE. EXCHANGE is the reducing the money, coin, &c. of one tate or country to its equivalent in another. Par is equality in value ; but the course of exchange is often above or below pcu . Agiu is a term sometimes used, to express the difference . between bank and current money. Case 1. To change the currency of one state into that of another. EXCHANGE RULE. Work by the Rule of Three ; or by the theorems in the following table : * The New England and Maine. Note. In some part* Louisiana, Mississippi, I GQ O f ? . - ~ p p r OKI 5 2 CL ^ ^i 3 o' co !> "~i P CD 3 R?' ^1 II 5* P~* - 02 ri 3 p: Cta CD [TABLE, exhibiting Theorems fo p P CD .0- sr CD 02 i ,-*JL^ K >l * Is 3 i l^ 3 CD o- O 02 P C CD cT t |f|.^ s ^** 11 B o '->-' 01 ^T co r ^ 2 s^ s ^^ S P CD p * P cr jj* Q P^ O ^. re a ^ Ct5 ^5 0> " a <-^ p. ^ s" 3" | 1" 3' 50 ^^f f! o 4 S -<i S ' || 1 ?^^ | < o CD t> ^&^J a^&'S s ** S 5 j| 1 P t * J^ O5 p Sf *m Ril- ii *-"* JO 0- ^ c^ o> i sr C^ "S _ > . 53 S* s'S ; ' ll I BQ 0^ rj- (0 QO O^ D H-* p- CJ P- O g > New York and Vorth Carotin Z. f? ^ a- si s> ^ "* "i ^H =!- CC |; &1 1 1 E^ ^2- .j. ^ ? g" fifi 3 ^ i^ - oi' 0|- 8 S ^'^S, 4. r T MH-P- s' S The value of a dollar in any state is found, either opposite to that state, or under it in the table. 104 EXCHAN^K. EXAMPLE. 1. What is the value of 480Z. Pennsylvania currency in North Carolina? s. d. s. . . As 7 6 : 8 :: 480 : 512 Ans. . Or, 480 Add V t = 32 512 Ans. 2. What is the value of 256Z. New York currency in Pennsylvania? Ans. 240Z. How much South Carolina currency is equal to 1500Z. of New Jersey? Ans. 933Z. 6s. 3d. 4. What sum New York currency is equal to 180Z. in Massachusetts? Ans. 240Z. 1. How much Virginia currency will purchase a bill for 280Z. South Carolina ? Ans. 360Z. 6. A bill of exchangj being remitted from Rhode Island __ Kjouth Carolina for 304Z., what is its value in the currency of the latter? Ans. 236Z. 85. Case 2. To change the currency of the different states to Federal money. RULE. Divide the given sum, reduced to shillings, six-pences, or pence in a dollar, as it passes in each state. EXAMPLES. 1. Change 127Z. 12s. New England money to dollars and cents. 127/. 12s.=2552 shillings. The dollar, New England, is 6s. ) 2552 425,33 Ans. 425 dolls. 33 cts. FOREIGN KXCHAIS'GK. I0i 2. Change 37/. 10s. Pennsylvania currency, to dollars. 37 L 10s. 1500 six-pences. 7s. 6d. or 15 six-pences make a dollar Pennsylvania cur- rency ; hence 1500 ~ 15= 100 dolls. Ans. Or, 37Z. 10s. = 9000 pence= 100,00 cents. Ans. 3. Change 2251. 12s. New York currency to Federal money. 225L 12s. 4512 shillings -h 8 the dollar New York cur- rency=564 dollars. Ans. 4. A bill of exchange for 468Z. 9s. 6d. Virginia curren- cy, is remitted to Philadelphia ; what is its value in Federal money? Ans. 1563 dollars 25 cts. 5. A merchant deposited in the United States branch bank at Pittsburgh, the sum of 750Z. 10s. Pennsylvania cur- rency, for what sum may he draw for in Federal money ? Ans. 2001 dollars 33^ cents. Note. Federal money being now generally introduced into mrrc;;r tile business throughout, the United States, has nearly superseded ti": use of the above rules of exchange between the different States. Case 3, FOREIGN EXCHANGE, Accounts are kept in Kngland, Ireland, and the West India Islands, in pounds, shillings, pence, and farthings though their intrinsic value in these places is different. A TABLE Of different Moneys^ as they are denominated and valued in different countries. REAT BRITAIN, IRELAND, AND THE WEST INDIES. 4 farrhings =1 penny 12 pence 1 shillino 20 shillings I pound 106 FOREIGN EXCHANGE. FRANCE. 12 Deniers = 1 Sol 20 Sols 1 Livre 3 Livres - 1 Crown - SPAIN. 4 Marvadies Vellon, or Marvadies of Plate = 1 Quarta 8* 34 Quartas, or Marvadies Vellon 1 Rial Vellon 16 34 Quartas, or Marvadies of Plate 1 Rial of Plate 8 Rials of Plate 1 Piaster, Pezo, or Dollar 5 Piasters - 1 Spanish Pistole 2 Spanish Pistoles - 1 Doubloon ITALY. 12 Deniers = 1 Sol 20 Sols 1 Livre 5 Livres 1 Piece of Eight at Genoa 6 Livres 1 Ditto at Leghorn 6 Solidi - 1 Gross 24 Grosses - 1 Ducat PORTUGAL. 400 Reas = 1 Crusadoe 1000 Reas - 1 Millrea HOLLAND. 8 Penning - = 1 Groat 2 Groats - 1 Stiver =2d. 6 Stivers I Shilling 20 Stivers 1 Florin, or Guilder 3i Florins - 1 Rix Dollar 6 Florins - 1 . Flemish 5 Guilders - 1 Ducat DENMARK. 16 Shillings - = 1 Mark 6 Marks ~ - 1 Rix Dollar 32 Rustics - 1 Copper Dollar 6 Copper Dollars - - 1 Rix Dollar RUSSIA. 18 Pennins - = 1 Gros 30 Gros - 1 Florin 3 Florins - 1 Rfx Dollar 2 Rix Dollars - 1 Gold Puoai FOREIGN EXCHANGE ,10? RULE, In exchanging of foreign moneys, work by the R lie of Three, or by Practice ; and for exchanging foreign moneys to Federal, work by the table in page 40. EXAMPLES. 1. Philadelphia is indebted to London 1749/. currency, what sum sterling must be remitted, when the exchange is 65 per cent. ? . . . . As 165 : 100 :: 1749 : 1060 sterling. Ans. 2. London is indebted to Philadelphia 1060Z. sterling; what sum Pennsylvania currency must be remitted, the ex- change being 65 per cent, as above ? . . . . As 100 : 165 :: 1060 : 1749 Ans. Or, 50 J 1060 10 i 530 5 | 106 53 1749Z. Ans. 3. Baltimore, Oct. 1, 1817. Exchange for 1260Z. 10s. sterling. Thirty days after sight of this my first of exchange, sec- ond and third of like tenor and date not being paid, pay to A. B. or order, twelve hundred and sixty pounds ten shil- lings sterling, value received, and place the same to account, as per advice from P S n. W. L. merchant, London. What is the value of this bill in Federal money ? 1260Z. 1 Os. = 1260,5.x by 444 cents=5596 dolls. Oa cents. Ans. 108 FOREIGN EXCHANGE. 4. London, January 1, 1818. Exchange for 5596 dolls. 62 cts. Federal money. Thirty days after sight of this my second of exchange, first and third of the same tenor and date not paid, pay to J. B. or order, five thousand five hundred and ninety -six dollars sixty-two cents, value received, and place the same to account, as per advice from S.'S. Mr. T. L. merchant, Baltimore. How much sterling is the above bill, 4,44 cents to the pound? 444)5596,62(1260 444 1156 888 2686 2664 222 20 4440(10 Ans. 1260Z. 10*. 4440 5. A merchant of Philadelphia receives from his cor- respondent in Dublin, a bill of exchange for 540Z. 15s. Irish currency ; what is its value in Federal money ? Ans. 2217 dolls. 7 cts. 6. A merchant in Philadelphia draws on his correspond- ent in Dublin for the balance of an account amounting to 2217 dolls. 7 cents ; what sum Irish currency mast be re- mitted to satisfy the draft? Ans. 540/. 15s. Note. In tiPrie last examples tha course of exchange is considered ;IH 1 cin{r at jKir: when the exch.-mge is nbove or below par, the per cent, must be added or subtracted, as the case requires. 7. In a settlement between A of London and B of Phila- delphia, B is indebted to A in the sum of 3207. sterling, what sum must be remitted by B to A to settle the balance, the uxv hange being 12J per cent, from the United States to (iivat Britain? Ans. 1598 dolls. 40 cts. ALLIGATION. 109 8. C of New Yo] ;v remits 3259 dollars to his correspond- ent in Dublin, to be placed to his account ; for what sum Irish currency, must he receive credit, the course of ex- change being 8 per cent, in favor of Ireland ? Ans. 7361. nearly. Note. The par ?f exchange between the United States of America and most other trading- countries, may be found by the table in page 40. SECTION 13. ALLIGATION. ALLIGATION is a rule for finding the prices, and quantity of simples in any mixture compounded of those things.* Case 1. To find the mean price of any part of the composition, when the several quantities and prices are given. RULE. As the sum of the whole quantity, is to its total value, so is any part of the composition, to its value. EXAMPLE. 1. A merchant mixed 2 gallons of wine at 2 dollars per gallon, 2 at 2 dollars 50 cents, and 2 at 3 dollars ; what is one gallon of this mixture worth ? gal. 2 at 2,00^400 2 at 2, 50 = 500 2 at 3,00 = 600 6 1500 G. D. c. G. D.c. As 6 : 15,00 :: 1 : 2,50 Ans. 2. A grocer mixed 20 Ib. of sugar at 10 cents per Ib. 30 Ib. at 15 cents, and 40 Ib. at 25 cents; what is one pound of this mixture worth? Ans. 18-1- cts. 3. A trader mixes 10 bushels of salt at 150 cents, 20 at K 110 ALLIGATION. 160 cts. and 30 at 170 cts. per bushel; at what rate can he afford to sell one bushel of this mixture? Ans. 1G3-J cts. 4. If 4 ounces of silver at 75 cents per ounce, be melted with 8 ounces at 60 cents per ounce, what is the value of one ounce of this mixture ? Ans. 65 cents. Case 2. To find what quantity of several simples must be taken at their respective rates, to make a mixture worth a given price. RULE. Place the rates of the simples under each other, and link each rate which is less than the mean rate, with one or more that is greater. The difference between each rate and the mean price set opposite to the respective rates with which it is linked, will be the several quantities required. Note. 1. If all the given prices be greater or less than the mean rate, they must be linked to a cipher. 2. Different modes of linking will produce different answers. EXAMPLES. 1. How many pounds of tea at 150, 160, and 200 cents per pound, must be mixed together, that 1 pound may be sold for 180 cents? t 150x 20 at 150 cents } Mean rate 180 2 160\) 20 at 160 V Ans. ( 200^ 30 + 20=50 at 200 ) 2. How many gallons of wine at 3, 5, and 6 dollars per gallon, must be mixed together, that one gallon may be worth 4 dollars? Ans. 3 gallons at 3 dolls. 1 gallon at 5 dolls, and 1 gallon at 6 dollars. 3. How many bushels of rye at 40 cents per bushel, and corn at 30 cents, must be mixed with oats, at 20 cents, to make a mixture worth 25 cents per bushel ? t 20^rx 15 f 5 C 6 bushels of rye 1. Ans. 25 ? 30-J ) 5 2. Ans. < 6 do. corn ( 40-^ 5 (24 do. oats 4. A grocer has four several sorts of tea, viz. one kind at 120 cents, another at 110 cents, another at 90 cents ALLIGATION. Ill and another at 80 cents per pound, how much of each sort must be taken to fnake a mixture worth 1 dollar per pound ? ' 2 at 120 cents. ( 3 at 120 cents. 110 2 ^ ns >2 110 (3 80 f 1 at 120 cents. ( 1 80 f 2 at 120 cents. 6. Ans. ) j* *g0 (3 80 Note. From this last example it is manifest that a great many dif- ferent answers may result to the same question, according to the va- rious modes of linking the numbers together. Case 3. When the rate of all the simples, the quantity of one of them, and the compound rate of the whole mixture are given, to find the several quantities of the rest. RULE. Arrange the mean rate, and the several prices, linked to- gether as in case 2, and take their difference. Then, as the difference of the same name with the quan- tity given, Is to the rest of the differences respectively : So is the quantity given, To the several quantities required. EXAMPLE. 1. A grocer would mix 40 pounds of sugar at 22 cents per pound, with some at 20, 14, and 12 cents per pound ; how much of each sort must he take to mix with the 40 pounds, that he may sell the mixture at 1 8 cents per pound ? f!2 N 41b. is) 14 V ~ 2 ^ 20_ / / 4 . 22 6 against the price of the given quantity. As 6 : 40 :: 4 : 26,66 Ib. at 12 cents.) 6 : 40 :: 2 : 13,33 do. 14 } Ans. and 26,66 do. 20 112 ALLIGATION. 2. How much wheat at 48 cents, rye at 06 cents, and barley at 30 cents per bushel, must be mixed with 24 bush- els of oats at 18 cents per bushel, that the whole may rate at 22 cents per bushel ? Ans. 2 bushels of each. 3. How much gold at 16, 20, and 24 carats fine, and how much alloy must be mixed with 10 ounces of 18 carats fine, that the composition may be 22 carats fine ? Ans. 10 oz. of 16 carats fine, 10 of 20, 170 of 24, and 10 of alloy. Case 4. When the price of all the simples, the quantity to be mix- ed, and the mean price are given, to find the quantity of each simple. RULE. Find their differences by linking as before : Then, as the sum of the differences, Is to the quantity to be compounded ; So is the difference opposite to each price, To the quantity required. EXAMPLE. 1. How much sugar at 10, 12, and 15 cents per pound, will be required to make a mixture of 40 pounds, worth 1 3 cents per pound ? 8 sum of the different simples. As 8 : 40 :: 2 : 10 lb. at 10 els. ) 8 : 40 :: 4 : 20 do. 15 \ Ans. and 10 do. 12 > 2. How much golJ of 15, of 17, of 18, and of 22 carats fine, must be mixed together to form a mixture of 40 ounces of 20 carats fine? Ans. 5 oz. of 15, of 17, and of 1 8, and 25 oz. of 22. 3. How many gallons of water must be mixed wkh wine at (> dollars per gallon, to fill a vessel of 70 #,i lions, so that it may he sold without loss at 5 dollars per gallon? 4/7A-. 1 1 07) lions of water. VULGAR FRACTIONS. 1 1 3 PART VI. VULGAR FRACTIONS. A VULGAR FRACTION is any supposed part or parts of an unit, and is represented by two numbers placed one above the other, with a separating line between them ; thus, | one- fifth, four-ninths. The number above the line is called the numerator, and that below the line the denominator. Thus, 4 numerator 6 &c. 9denomina. 10 The denominator shows how many parts the unikpr inte- ger is divided into, and the numerator shows how many of those parts are contained in the fraction. Vulgar fractions are either proper, improper, compound, or mixed. A proper fraction is when the numerator is less than the denominator, as -J, f, j, |f , &c. An improper fraction, is when the numerator is eitner equal to, or greater than the denominator, as f , f, 2 T , &c. A compound fraction is a fraction of a fraction, as f of $, f of/ of jf,&c. A mixed fraction is a whole number and fraction united, as 8|, 4J, 120f , &c. SECTION 1. Reduction of Vulgar Fractions. Case 1. To Reduce a Vulgar Fraction to its lowest terms. RULE. Divide the greater term by the less, and that divisor by the remainder, till nothing be left, the last divisor is the K2 114 REDUCTION OF VULGAR FRACTIONS. common measure, by which divide both parts of the frac- tion : the quotient will be the answer. Or, Take aliquot parts of both terms continually, till the frac- tion is in its lowest terms. Note. 1. 1.' the common measure when found is 1, the fraction ia already in its lowest terms. 2. Ciphers to the right of both the terms may be cut off thus, w=f EXAMPLES. 1. Reduce J| to its lowest terms. 36 ) 48 ( i 36 Common measure 1 2 ) 36 ( 3 36 12)36(3 36 3 2 div. 6 div. Ans. 36 18 3 12)48(4 Or, = =- Ans. 48 48 24 4 2.. Reduce ^f to its lowest terms. Ans. f 3. Reduce T 7 ^ to its lowest terms. | 4. Reduce fVVVV to i ts lowest terms. f 5. Reduce T y T to its lowest terms. | 6. Reduce gW/T to its lowest terms. J Case 2. To reduce a mixed number to an improper fraction. RULE. Multiply the whole number, by the denominator of the fraction, and add the numerator to the product, for a new numerator, under which place the given denominator. EXAMPLE. 1. Reduce 8$ to an improper fraction. 8 4 3-35 - Ans. 4 REDUCTION OF VULGAR FRACTIONS. 115 2. Reduce 12 T f to an improper fraction. Ans. 2 j~? 3. Reduce 183 -/ T to an improper fraction. *ff 8 4. Reduce 514 T \ to an improper fraction. 8 T | 9 5. Reduce 68425? to an improper fraction. a 7 3 T 7 3 Case 3. To reduce an improper fraction to a whole or mixed number. RULE. Divide the numerator by the denominator ; the quotient will be the answer required. Note. This case and case 2, prove each other. - EXAMPLE. 1 . Reduce \ 5 to its proper terms. 4)35(8| Ans. 32 3 2. Reduce 3 f T 8 to its proper terms. Ans. 183^ 3. Reduce 2 \ 6 5 to its proper terms. 352i 4. Reduce 3 T 6 y to i ts proper terms. 56y 5. Reduce 8 y| 9 to its proper terms. 514 T 5 Case 4. To reduce several fractions to others that shall have one common denominator, and still retain the same value. RULE. Reduce the given fractions to their lowest terms, then multiply each numerator into all the denominators, but its own. for a new numerator ; and all the denominators into each other for a common denominator. EXAMPLE. 1. Reduce t, |, and |, to a common denominator. 1X3X412V 2X2X4 16> numerators. 3X2X318) 2X3X424 common denominator. 116 REDUCTION OF VULGAR FRACTIONS. 2. Reduce f , |, and -, to a common denominator. Ana. ?&, V&, || 8. Reduce i, -|, T 4 j, and f , to a common denominator. Case 5. To reduce several fractions to others, retaining the same value, and that shall have the least common denominator. RULE. Divide the given denominators by any number that will divide two or more of them without a remainder ; set the quotients and undivided numbers underneath ; divide these numbers in the same manner, and continue the operation, till no two numbers are left capable of being lessened ; the product of these remaining numbers, together with the di- visor or jdivisors, will give the least common denominator. Divide the common denominator, so found, by each par- ticular denominator, and multiply the quotient by its own numerator for a new numerator, under which place the com- mon denominator. EXAMPLE. 1. Reduce |, J, j-, and f , to the least common denom- inator. 3)2 368 2)2 128 111 4 X~2x 3=24 common denominator. '2)24 Divisors 12X1 = 12 3 8x2=16 6 4X5=20 ^8 3X7=21 Then, if, ||, ff, f i Ans. 2. Reduce f , f , T \, and vV, to the least common donom. mator. Ans. A n -, rV 5 o> T V REDUCTION OF VULGAR FRACTIONS. _ 117 3. Reduce |, f , T 4 5 -, and |, to the least common denomi- nator. Ans. J-f , H, H> If Case 6. To reduce a compound traction to a single one. RULE. Multiply all the numerators together for a new numerator, and all the denominators for a new denominator. Note. Such figures as are alike in the numerators and denominators may be cancelled. EXAMPLE. 1. Reduce f of J of to a single fraction. 2 X 3 X 4 24 2 Or cancelled. = - Ans. 2 3 4 2 3X4X5=60 5 _ _ _ = _ Ans. 3 4 5 5 2. Reduce J of of T \ to a single fraction. Ans. fi 3. Reduce of f of J to a single fraction. Ans. ^ 4. Reduce f of ^ of f J to a single fraction. Ans. T Vo Case 7. To reduce a fraction of one denomination to the fraction .of another, but greater, retaining the same value. RULE. Make it a compound fraction, by comparing it with all the denominations between it and that to which it is to be reduced ; reduce this fraction to a single one. EXAMPLE. 1 . Reduce f of a penny to the fraction of a pound. 5 X 1 X 1 5 6 X 12 X 20 1440 2. Reduce of a pennyweight to the fraction of a pound troy Ans. J T % Reduce T 9 :i - of a pint of wine to the fraction of a hogs- head. Ans. ^fa 118 REDUCTION OF VULGAR FRACTIONS. 4. Reduce TT of a minute to the fraction of a day. TJ84 Case 8. To reduce tne 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 the denom- inator, between it and that to which it is to be reduced, for a new numerator, and place it over the given denominate,,-, which reduce to its lowest terms. 1. Reduce TT 5 T -g- of a pound to the fraction of a penny. 5 X20X12 1200 5 1440X1 X 1 H40 6 2. Reduce -$~ of a pound troy to the fraction of a pen- nyweight. Ans. j 3. Reduce 7 ~ of a hogshead to the fraction of a pint. Ans. ^ 4. Reduce yj-g-4 f a day to tne fraction of a minute. Ans. VT Case 9. To- find the value of a fraction in the known paits of an integer. RULE. Multiply the numerator by the known .parts of the inte- ger, and diviilo by the denominator. EXAMPLE. 1. What is the value of f of a pound sterling? 20 shillings = 1 pound. 2 3 ) 40 13 4 Ans. 13*. 4ct t. Red are J of a pound troy tc iu proper quantity. Ans. 7oz. 4dwt. REDUCTION OF VULGAR FRACTIONS. 11D 3. Reduce f of a mile to its proper quantity. Ans. 6fur*. Wp. 4. Reduce T 3 F of a day to its proper time. Ans. 7h. I2min. 5. What is the value of f of a dollar. Ans. 80 cts. Case 10. To reduce any given quantity, to the fraction of a greater Munomination of the same kind. RULE. Reduce the given quantity to the lowest denomination mentioned for a new numerator, under which set the integral part (reduced to the same name) for a denominator. EXAMPLES. 1. Reduce 6s. Sd. to the fraction of a pound. s d. s. 68 20 12 12 80 1 240 =- Ans. 240 3 2. Reduce 25 cents to the fraction of a dollar. 25 1 100 4 3. Reduce 31 gallons 2 quarts to the fraction of a hogs- head. Ans. i. 4. Reduce 6 hundred weight 2 quarters 18| pounds to the fraction of a ton. Ans. J. Case 11. To reduce a vulgar fraction to a decimal of the same value. RULE. Add ciphers to the right-hand of the numerator, and cfivide by the denominator. 120 ADDITION OF VULGAR FRACTIONS EXAMPLE. ' 1. Reduce f o a decimal fraction of the same value. 4 ) 300 ,75 Ans. 2. Reduce ^J to a decimal fraction. Ans. ,85 SECTION 2. ADDITION OF VULGAR FRACTIONS. Case 1. To add fractions that have the same common denomi- nator. RULE. Add all the numerators together, and divide the amount by the common denominator. EXAMPLE. 1. Add T ^, r 5 2, T 7 2> T 9 2 and |J together. numerators. 1 5 7 9 11 common denominator 12 ) 33 ( 2| An*. 24 9 3 12 4 2. Add /y, T 8 T , i j, {, and if together. Anj. 2$ 8. Add Jf, H, AJ,and JJ together. 3*. ADDITION OF VULGAR FRACTIONS. 121 Case 2. To add fractions having different denominators. RULE. Reduce the given fractions to a common denominator, by case 5, and proceed as in the foregoing case. EXAMPLE. 1. Add $, J, and T 9 j together. 12 1359 12 18 15 18 18 ---- = -- --- 15 I 4 8 12 24 24 24 *4 18 24)63(2} 48 15 2. Add i, i, and together. Ans* 1 & 3. Add f , f ,3, f and r 8 5 together. 3^ Case 3. To add mixed numbers. RULE. Add the fractions as in the foregoing cases, and the inte- gers as in addition of whole numbers. EXAMPLES. I. Add 13^, 9 7 <y and 3/j together. 2. Add 5|, 6 j and 4 together. 5|-rr5if common denominator 6J = 6U 15 122 SUBTRACTION OF VULGAR FRACTIONS. 3. Add 1J, j of J, and 9^ together. Ans. 4, Add I fa 6, of i, and 7i together. Case 4. To add fractions of several denominations. RULE. Reduce them to their proper quantities by case 10 in re- duction,, and add them as before. EXAMPLE. 1 . Add J of a . and y 3 ^ of a shilling. s. d. f of a . = 15 6|=|| common denom. ft of a ,.-0 3j = A 9 Shillings 15 10/3- Ans. 2. Add i of a yard to f of a foot. Ans. 2 feet 2 inches. 3. Add i of a day to of an hour. Ans. 8 hours 30 minutes. 4. Add J of a week, J of a day. and \ of an hour to- gether. . Ans. 2 days 14 hours 30 minutes. 5. Add J of a miie, f of a yard, and J of a foot together Ans. 1540 yards 2 feet 9 inches. SECTION 3. Subtraction of Vulgar Fractions. RULE. PREPARE the fractions as in addition, and sub:ract the rower numerator from the upper, and place the rfrilerence over the common denominator. Note. 1. When the lower numerator is greater than the upper, sub- tract it from the common denominator, adding the .nipper numerator to *he difference, and carry 1 tcf the units place of t^e integer. 2. When the fractions are of different integers, find their values, separately t and subtract as in compound subtraction of whole tium MULTIPLICATION OF VULGAR FRACTIONS. 123 EXAMPLES. From | Take f From f Take From 1= Take i= Rem. f =i A/is. Rem. From ji Take J Rem. i From Take Rem. From f Take j Rem. TjV From 13 Take 6 Ans. Rem. From |?f Rem. : From Take Rem. fc Ans. - From }| Take |i Rem. T V From 19 T \ Take T 7 Rem. From Jof a .=15 6| Take ft of a s. = 3J Rem. 15 From 7 weeks Take 9 T V days Rem. . 7h. 12m. SECTION 4. Multiplication of \ ulgar Fractions. RULE. REDUCE the compound fractions to simple ones, and mixed numbers to improper fractions, then multiply the nu- merators together for a new numerator, and the denomina- tors for a new denominator. 1. Multiply J by EXAMPLES. 2X1=2 1 - - =- An*. 3X4=12 124 DIVISION 7 OF VTLHAR FRACTIONS. 2. Multiply 4i by | 2 9X1= 9 - - An*. 2x8=16 3. Multiply | by f Ans. ^ 4. ' J of | by . . . - & 5. H by i - . - 1 j T. 48} by 13| - . . . 672y\ SECTION 5. Division of Vulgar Fractions. RULE. PREPARE the fractions, if necessary, by reduction ; inrert the divisor, and proceed as in multiplication. EXAMPLES. 1. Divide f by j| 3x3 9 - -= An*. 8x2 16 2. Divide 4i by 1| *J H 2 3 9X3 27 5 Then - -==2,7, 2X5 10 2 a RULE OF THREE IN VULGAR FRACTIONS. 125 3. Divide f by f Ans. f } 4. Divide if by f Iff 5. -Divide li by 4 T \ ft 6. Divide J by 4 /, 7. Divide 9| by i of 7 2if 8. Divide 5205} by f of 91 71* SECTION 6. The Rule of Three in Vulgar Fractions. THE operation of tbe Rule of Three in Vulgar Fractions, whether direct, inverse, or compound, is performed <in the same manner and agreeably to the principles laid down in whole numbers under these rules. When the question is in direct proportion, prepare the terms by reduction, and invert the first term ; then proceed as in multiplication of fractions. EXAMPLE. 1. If i of a yard of cloth cost f of a dollar, what will I of a yard come to ? yd. D. yd. As i : 1 :: | 4X2X7 = 56 D.c. Inverted - - - =2J dollars. Ans. 2 33 J 1X3X8 = 24 2. If f of a ton of iron cost 164J dollars, what will ^ of a ton come to? Ans. 211 dolls. 28| cts. 3. A person having f of a coal mine, sells | of his share for 171 dollars, what is the value of the whole mine at the same rate? Ans. 380 dollars. 4. At f of a dollar per yard, what will 42 yards come to ? Ans. 35 dollars. 5. A gentleman owning | of a vessel, sells f of his share for 312 dollars, what is the whole vessel worth ? Ans. 11 70 dollars. L2 126 INVERSE PROPORTION. 6. If li bushel of apples cost 79 J cents, what will 3f bushels cost at the same rate ? - Ans. 202 T \ cents. 7. If J of a ship be worth 175 dollars 35 cents, what part of her may be purchased for 601 dollars 20 cents? Ans. 3j SECTION 7. INVERSE PROPORTION. RULE. PREPARE the question as in direct proportion, invert the third term, and proceed as in multiplication of fractions. EXAMPLES. 1. How much shalloon f yard wide, will lino 4J yards of cloth l yard wide? H = % Then, as I : f :: f 4i=4 Or, inverted f : f : : | = LO 9 Ans. 2. If 6 hundred weight be carried 22 ^ miles for 25f dollars, how far may 1 hundred weight be carried for the same money? Ans. 145J miles. 3. If 12 men can finish a piece of work in 37 f days; how long will it take 16 men to do the same work ? Ans. 281 days. 4. A lends to B lOOf dollars for 6| months; what sum should B lend to A for 3|- years, to requite his kindness ? Ans. 14|f f dollars. 5. Mow many feet long must a board be, that is | of a root wide, to equal one that is 20 feet long, and f of a foot wide? Ans. 17 i lert long. 6. In exchanging 20 yards of cloth of 1 i yard wide, for some of the same quality of | yard wide, what quantity of the latter makes an equal barter ? Ans. 34 yards. INVOLUTION. 12? PART VII. EXTRACTION OF THE ROOTS, AND COMPARATIVE ARITHMETIC. SECTION 1. Involution, or the Raising of Powers. INVOLUTION is the multiplying of a given number by itself continually, any certain 'number of times. The product of any number so multiplied into itself, is termed the power of that number. Thus 2x2 = 4= the second power or square of 2* 2x2x2 = 8= the third power or cube of 2. 2x2x2x2 = 16= the fourth power of 2, &c. The number denoting the power to which any give n sum is raised, is called the index or exponent of that powe r. If two or more powers are multiplied together, ttu ir pro- duct will be that power, whose index is the sum of the expo- nents of the factors. Thus 2x 2=4, the 2d powei of 2 , 4X4=16, the 4th power of 2; 16x16 = 256, tlie 8th power of 2, &c. TABLE Of the first nine powers. 1 CU rt Jf i o o 1 1 1 S 1 1 ft o 1 00 5 1 1! 11 11 1 1 11 1| 1 2 4, 8| 16! 32 64 128 256| 512 3| yi 27| 81 243 729| 2187) 6561! 19683 4|I6| 64 j 256| 1024 4096| 16384J 65536) 262144 5|25|125| 625 3125| 15625) 78125J' 3J0625| 1953125 6|36j 216|12:)6| 7776, 46656) 279936| 167961 6j 10077696 7|4i)|343|2401|16807|11764<)| 823543| 5764801J 40353607 8|64|512)4096!32768 262114|2097152| 16777216J i 34217728 [ 9(81 72:)|6561I5904 ; ): 531441 14782)6:) 43046721)3874204891 123 EVOLUTION. SQUARE ROOT. EXAMPLE. 1. What is the 3d power of 15 ? 15x15x15=3375 Ans. 2 What is the 4th power of 35 ? Ans. 1500625. 3. What is the third power of 1,03 1 Ans. 1,092727. 4. What is the 5th power of ,029 ? Ans. ,000000707281. 5. What is the 4th power of J 1 Am. -fa SECTION 2. Of Evolution, or the Extracting of Roots. EVOLUTION js the reverse of involution. For as 3x 3=9 y 3 = 27, the power ; so 27-^-3=93=3, the root of that power. Hence the root of any number, or power, is such a number as being multiplied into itself a certain number of times, will produce that power. Thus, 4 is the square root of 16, for 4X4=16; and 5 is the cube root of 125, for 5x 5x 5=125. SECTION 3. THE SQUARE ROOT. number multiplied once into itself is called the square of that number. Hence, to extract the square root <;f any number, is to find such a number as being multiplied f >y itself will be equal to the ^iven number. RUf.R. 1. Point off the given sum into periods of two figures a_, beginning at the right hand. SQUARE ROOT. 12!) 2. Subtract from the first period on the left, the greatesi square contained therein ; setting the root, so found, for the first quotient figure. 3. Double the quotient for a new divisor, and bring down trte next period to the remainder lor a new dividual. Try how often the divisor is contained in the dividual, omitting the units. figure, and place the {lumber, so found, in the quo- tient, and on the right of the divisor ; multiply and subtract as in division. 4. Double the quotient for a new divisor ; bring down the next period, and proceed as before, till all the periods are brought down. When a remainder occurs, add ciphers for a new period, the quotient figure of which will be a decimal, which may be extended to any required degree of exactness. PROOF. * Square the root, adding the remainder (if any) to the product, which will equal the given number. EXAMPLE, 1. What is the square root of 531441 ? 53144J ( 729 An*. 49 I. double the quotient 14,2 ) 414 284 2. double do. 144,9) 13041 13041 729 729 6561 1458 5103 Ans. 327 2187. 6561. 19683. 4698. 6031. 1506,23 -t 2756,22^4 531441 proof. 2. What is the square root of 106929 ? 3. What is the square root of 4782969 ? 4. What is the square root of 43046721 ? 5. What is the square root of 387420489 ? 6. What is the square root of 22071204 ? 7. What is the square root of 36372961 ? 8, What is the square root of 2268741 ? 9. What is fhe square roof of 7596796 ' 130 SQUARK ROOT. When there are decimals joined to the whole numbers in /he given sum, make the number of decimals even by adding ciphers, and point off both ways, beginning at the 'decimal point. 10. What is the sqiuire root of 9712,718051 ? Ans. 98,553 + 11. What is the square root of 3,1721812 ? Ans. 1,78106 + 12. What is the square root of 4795,25731 ? Ans. 69,247 13. What is the square root of ,00008836 ? Ans. ,0094 To extract the square root of a vulgar fraction. RULE. Reduce the fraction to its lowest term ; then extract the "Kjuare root of the numerator for a new numerator, and the scjuare root of the denominator for a new denominator. Nute. When the fraction is a surd* that is, a number whose exact ff>ot cannot be found, reduce it to u decimal and extract the root there- from. EXAMPLES, 1 . What is the square root of Jf A 7 Ans. J "2. What is th<; square root of |Jf A ! f 3. What is the square root of iff f ? ^f Surds. 4. What is the square root of ffj '! Ans. ,86602 + 5. What is the square root of f Jf ? ,93309 + 6. What is ihr square root of f|J ? ,72414 + To ertraci thr vjimre root of a mired number. RUTJv 1. Reduce the fractional part of the mixed number to its lowest term, and thn mixed number to an improper fraction. SQT T ARK ROOT. 131 2. Extract the roots of the numerator and denominator, for a new numerator and denominator. If the mixed number given be a surd, reduce the frac- tional part to a decimal, annex it to the whole number, and extract the square root therefrom. EXAMPLES. 1. What is the square root of 37 ^f? Ans. 6-J 2. What is the square root of 27 r 9 F ? 5i 3. What is the square root of 51|}? 7} 4. What is the square root of 9ff ? 3i Surds. 5. What is the square root of 7 T 9 T ? Ans. 2,7961 + 6. What is the square root of 8f ? 2,951 9 + 7. What is the square root of 85-ff ? 9,27 -f Any two sides of a right angled triangle given to find the third side. RULE. As the square of the hypothenuse or longest side, is al- ways equal to the square of the base and perpendicular, the other sides added together ; then it is plain if the length of the two shortest sides are given, the square root of both these squared and added together, will be the length of the third or longest side. Again, when the hypothenuse, or longest side, and one of the others are given 5 the square root of the difference of the squares of these two given sides will be the len ;th of the remaining side. 132 SQUARE ROOT. EXAMPLE. 1. The wall of a fortress is 3fi feet high, and the ditch before it is 27 feet wide: it is required to find the length of n ladder that will reach to the top of the wall from tho op- posite side of the ditch ? Ans. 45 feet. 27 ft. ditch 2. The top of a castle" from the ground is 45 yards high, and is surrounded with a ditch 60 yards broad, what length must a cable be to reach from the outside of the ditch to the top of the castb? Ans. 75 yards. 8. In a right angled triangle, ABC, the hypothenuse line A C is 45 feet, the base A B 27 feet; required the length of the perpendicular line B C? Ans. 36 feet. B 4. In a right angled triangle, A B C, the line A C is 75 feet, B C 45 feet; required the length of the line A B? Ans. 60 feet. To find the side of a square equal in area to any given ./ RULE. Extract the square root of the content of the given su |H;rfices ; the quotient will give the side of the equal square sought. SQUARK ROOT 133 EXAMPLES. 1. If the content oi a given circle be ICO, what is the side of the square equal? Ans. 12,(>4911-{- 2. If the area of a circle he '2025, what is the side of the square equal ? Ans. 45. 3. It* the area of a circle be 750, what is the side of the square equal? Ans. 27,38612 + To find tht diameter of a circle of a given proportion larger or less than a given one. RULE. Square the diameter of the given circle, and multiply (if greater) or divide (if less) the product, by the number of times the required circle is greater or less than t.\c given EXAMPLES. 1. There is a circle whose diameter is 4 feet; what is the diameter of one 4 times as large ? Ans. 8 feet. 2. A has a circular yard of 100 feet diameter, but wishes to enlarge it to one of 3 times that area; what wall the diameter of the enlarged one measure? Ans. 173,2 -f 3. If the diameter of a circle be 12 inches, what will be the diameter of another circle of half the size ? Ans. 8,48 -f inches. The area of a circle given to find the diameter. RULE. Multiply the square root of the area by 1,12837, and the produce will be the diameter. EXAMPLES. 1. When the area is 160, what is the diameter? Ans. 1 4,272947 -f 2. What length of a halter will be sufficient to fasten a horse from a post in the centre, so that he may be able to graze upon an acre of grass, and no more? Ans. 7,1364 perches, or 117 ft. 9 inches* M 134 SQUARE ROOT. * Application. 1. If an army of 20736 men is formed into a square column ; how many men will each front contain? Ans. 144 men. 2. How many feet of boards will it require to lay the floor of a room that is 25 feet square? Ans. 625 feet. 3. A certain square pavement contains 191736 square stones, all of the same size ; how many are contained in one of its sides ? Ans. 444. 4. In a triangular piece of ground containing 600 perches, one of the shortest sides measures 40 perches, and the other 30 ; what is the length of the longest side ? Ans. 50 perches. 5. Two gentlemen set out from Pittsburgh at the same time ; one of them travels 84 miles due north, and the other 50 miles due west ; what distance are they asunder ? Ans. 97^ -{- miles. ft. What is the square root of 964,5192360241 ? Ans. 31,05671. 7. What is the square root of 1030892198,4001 ? Ans. 32107,51. As it is probable many teachers rind it difficult to explain satisfactorily the reasons and principles upon which the rules for the extraction of the roots are founded, I have subjoined the following demonstration of the rule for extracting the square root ; and which will also serve to show the reason of the rules for extracting the roots of the higher powers. From what has been already said on this rule, it is sufficiently evident that the extraction of the square root has always this operation on numbers, viz. to arrange the number of which the root is extracted into a square form. Thus, if a car- penter should have 625 feet of dressed boards for laying a floor ; if he extracts the square root of this number, (625) he will have the exact length of one side of a square floor, which these boards will be sufficient to make. Let this then be the question : Required to find the length of one side of a square room, of which 625 square feet of boards will be sufficient to lay the floor. The first step, according to the rule given, is to point off the numbers into periods of two figures each, beginning at the unit's place. This ascertains the number of figures of which the root will consist, from this principle, that the product of any two numbers can have, at most, but so many . places of figures, as there are places in both the factors, and at least, but one less. The number (625) will then have two periods, and con- sequently the root will consist of two figures. Operation. The last, or left-hand period in this . . number is 6, in which 4 is the greatest 625 ( 2 square, and 2 the root ; hence 2 is the 4 first figure in the root, and as one fig- ure more is yet to be found, we may 225 for the present supply the place of that figure with a cipher (20) ; then 20 Figure 1. will express the just value of that part of the root now obtained. But a root is the side of a square, of equal sides. A 20 20 Hence, figure 1 exhibits a square, each side of which is 20 feet, and the a| 400 |b area 400, of which 20 is the root now 20 obtained. As the rule requires, we next subtract the greatest square contained in the first period, and to the remainder bring down the next period. 4 is the greatest square contained in the first period (6), and as it falls in the place of hundreds, is in reality 400, as may be seen by filling up the places to the right-hand with ciphers ; this subtracted from the (i leaves 2, as a remainder, to which if the next period is brought down, the remainder will be 225 ; and the original number of feet (625) has been diminished by the deduction of 400 feet, a number equal to the superficial content of the square A. Figure 1, therefore, exhibits the exact progress of the operation, and shows plainly how 400 feet of the boards have been "disposed in the operation thus far, and also that 225 feet yet remains to be added to this square, by en- larging it in such a manner as not to destroy its quadrate form, or its continuing a complete and perfect square. Should the addition be made to one side, only, the figure would lose its square. The addition must be made to two sides: accordingly the rule directs to lt double the quotient 136 SQUARE ROOT. (viz. the root already found) for a new divisor ;" the double of the root is equal to two sides of the square A, for the double of 2 is 4, and as this 4 falls in the place of tens, since the next figure in the root, according to the rule, is to be placed before it in the place of units, it is in reality 40, and equal to a b and b c which are 20 each. Operation continued. Again, as the rule di- . . rects, try how often the di~ 625 ( 25 visor is contained in the 4 dividual, omitting the units figure. 45)225 The divisor is here 4, 225 which, as has already been shown, is 4 tens, or 40 ; 000 this is to be divided into the remainder 225 ; omit- n 5 ting the last figure=220. But 40 the sum of the two sides b c and c d / to which the remaining 225 is to be added, and the square A enlarged, which omitting the last figure (5) gives 5 for the last quo- tient figure ; 5 is the breadth of the two parallelograms B B, the area of each (5 X 20 b 5 20) is 100. The rule requires us to omit the last figure in the divid- ual^ and also, to place the quotient figure, when found, on the right of the divisor ', the reasons for which are, that ad- ditions of the two parallelograms B B ta the sides of the square A (fig. 2) do not leave it a perfect square, a deficiency remaining at the corner D ; the right-hand figure is omitted to leave something of the dividend for this deficiency. And as this deficiency is limited by the two parallelograms B B, and the quotient figure (5) is the breadth of these, consequently the quotient 5= the length of each of the sides of the small square D ; this quotient then being placed on the right of the divisor and multiplied into itself gives the area of the square I) ; which being added to the 20 5 B 5 D 5 en d 100 25 c A B 25 20 20 20 5 a 400 100 X CUR!-: ROOT. 137 contents of the two parallelograms B B each (100) 200, shows that the remaining 225 feet of boards have kasji dis- posed of, in these three additions (B B D) made to the T^st square A ; whilst the figure is seen to be continued a corrt* plete square. Q. E. D. PROOF. The square A =400 feet The parallelograms B B =200 The square D = 25 625 feet. SECTION 4. THE CUBE ROOT. THE cube is the third power of any number, and is found by multiplying that number twice into itself. As 2 X 2 x 2^. To extract the cube root, therefore, of any number, is to find another number, the cube of which will equal the given number. Thus 4 is the cube root of 64 ; for 4X 4x 4=64. RULE. 1. Point off the given number into periods of three figures each, beginning at the units place, or decimal point. These periods will show the number of figures contained in the required root. 2. Find the greatest cube contained in the first period, and subtract it therefrom ; put the root of this cube in the quotient, and bring down the next period to the remainder for a new dividual. 3. Square the quotient and multiply it by 3 for a defective divisor; 2x2x3 = 12. Find how often this is contained in the dividual, rejecting the units and tens therein, and place the result in the quotient, and its square to the right of the divisor. 4 x 4=16 put to the divisor 12 = M2 138 CUBE TtOOT. 4. Multiply the last tigure in the quotient by the rest, and the product by 30 ; add this to the defective divisor, ana multiply this sum by the last figure in the quotient, subtract that product from the dividual, bring down the next period, and proceed as before. Note. When the quotient is 1, 2, or 3, put a cipher in the place of tens in filling up the square on the right of the divisor. EXAMPLE. U What is the cube root of 48228544 1 Greatest cube in 48 is Operation. 48228544(364 Ans. 27 21228 Square of 3 X by 3=27. 1 def. divis. ' Square of 6 put to 27 =2736 6 last -quo. fig. X by the rest and 30 = 540 Complete divisor 3276 J 19656 Square of 36 X 3=3888. 2 def. divis. Square of 4 put to 3888 = 388816 4 last quo. fig. X by the rest and 30 = 4320 Complete divisor 393136 1572544 1572544 2. What is the cube root of 13824 T 3. What is the cube root of 373248 ? 4. What is the cube root of 5735339 ? 5. What is the cube root of 84604519 ? 6. What is the cube root of 27054036008 I 7. What is the cube root of 122615327232 ? 8. What is the cube root of 22069810125 ? 9. What is the cube root of 219365327791 ? 10. What is the cube root of 673373097125 ? 11. What is the cube root of 12,977875 ? 12. What is the cube root of 15926,972504 ? 13. What is the cube root of 36155,027576 ? Ans. 24 72 179 439 3002 4968 2805 6031 8765 2,35 ?Mfi f 33,06 -f CUBE ROOT. 139 14. What is the cube root of ,053258279 ? Ans. ,376 + 15. What is the cube root of ,001906624 ? ,124 16. What is the cube root of ,000000729 ? ,009 17. What is the cube root of 2 ? 1,25 + To extract the cube root of a vulgar fraction. RULE. Reduce the fraction to its lowest terms ; then extract the cube root of the numerator for a new numerator, and the cube root of the denominator for a new denominator ; but if the fraction be a surd, reduce it to a decimal, and extract the root from it for the answer. EXAMPLES. 1. What is the cube root of f f-f 1 Ans. 4 2. What is the cube root of T 3 /^ ? f 3. What is the cube root of 4-f f ? f Surds. 4. What is the cube root of 4 ? Ans. ,829 -f- 5. What is the cube root of f ? ,873 -f 6. What is the cube root of f ? ,822 -f To extract the cube root of a mixed number. RULE. Reduce the fractional part to its lowest terms, and the mixed number to an improper fraction ; extract the cube roots of the numerator and denominator for a new numerator and denominator ; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. ^ EXAMPLES. 1. What is the cube root of 31 J/j ? Ans. 3| 2. What is the cube root of 12if ? 2J 3. What is the cube root of 405 r 2 ^ ? 7f 140 CUBE ROOT. Surds. 4. What is the cube root of 7} 1 Ans. 1,93-f- 5. What is the cube root of 8f ? 2,057 + 6. What is the cube root of 9 ? 2,092 + To find the side of a cube that shall be equal to any given solid, as a globe, a cone, <fyc. RULE. Extract the cube root of the solid content of any solid body, for the side of the cube of equal solidity. EXAMPLES. * 1. If the solid content of a globe is 10648, what is the side of a cube of equal solidity ? Ans. 22. 2. If the solid content of a globe is 389017, wha't is the side of a cube of equal solidity? Ans. 73. Note. The relative size of different cubical vessels is found by mul- tiplying the cube oi' the side of the given vessel, by the proportional number, and taking the cube root of the product for the answer sought. EXAMPLES. 1, There is a cubical vessel whose side is two feet I de- mand the size of another vessel which shall contain three times as much 1 Ans. 2 feet 10 inches and } nearly. 2. There is a cubical vessel whose side is 1 foot ; re- quired the side of another vessel that shall contain three times as much ? Ans. 17,306 inches Application. 1. If a ball of 6 inches diameter weigh 32 lb., what will one of the same metal weigh, whose diameter is 3 inches ? Ans. 4 lb. 2. What is the side of a cubical mound equal to one 28H feet long, 216 broad, and 48 high? An*. 144 feet. 3. There is a stone of cubic form, which contains 389017 solid feet ; what is the superficial content of one of its sides ? AIM. 5329 feet. PROGRESSION. 141 4. What is the difference between half a solid foot, and a solid half foot ? Ans. 3 half feet. 5. In a cubical foot, how many cubes of 6 inches, and how many of 4 are contained therein ? Ans. 8 of 6 inches, and 27 of 4 inches. SECTION 5. OF PROGRESSION. PROGRESSION is of two kinds, arithmetical and geometrical. Arithmetical progression is when any series of numbers increase or decrease regularly by a common difference. As 1, 2, 3, 4, 5, 6, &c. are in arithmetical progression by the continual adding of one ; and 9, 7, 5, 3, 1, by the continual subtracting of two. Note. In any series of even numbers in arithmetical progression, the sum of the two extremes will be equal to the sum of any two terms, equally distant therefrom ; as 2. 4. 6. 8. 10. 12., where 2 -f- 12 = 14, so 4 -f 10 = 14, and 6 -f 8 = 14. But if the number of terms is odd, the double of the middle term will be equal to any two of the terms equally distant therefrom ; as 3. 6. 9. 12. 15. where the double of 9 the middle term = 18, and 3 -f 15 = 18, or 64-12 = 18. In arithmetical progression five things must be carefully observed, viz. 1. The first term, 2. The last term, 8. The number of terms, 4. The equal difference, 5, The sum of all the terms." Casel. The first term, common difference, and number of terms, given to find the last term, and sum of all the terms. RULE. 1 . Multiply the number of terms, less 1 , by the common 142 PROGRESSION. difference, and to the product add the first term, the sum will be the last term. 2. Multiply the sum of the two extremes by the number of terms, and half the product will be the sum of all the terms. 1. A merchant bought 50 yards of linen, at 2 cents for the first yard, 4 for the second, 6 for the third, &c. increas- ing two cents every yard ; what was the price of the last yard, how much the whole amount, and what the average price per yard ? 50 number of terms 1 49 number of terms less one Multiply by 2 common difference 98 Add 2 first term 100 last term 2 + 100=102 sum of the two extremes Multiply by 50 number of terms 2)5100 50 ) 25,50 sum of all the terms 51 cents C 100 cents the last yard Ans. < 25,50 do. the whole amount f 51 do. the average price per yd. 2. Bought 20 yards of calico at 3 cents for the first yard, 6 for the second, 9 for the third, &c. ; what did the whole cost? Ans. 6 dolls. 30 cents. 3. If 100 apples were laid two yards distant from each "other, in a right line, and a basket placed two yards distant from the first apple, what distance must a person travel to gather them singly into the basket ? Ans. 11 miles, 3 furlongs, 180 vardau PROGRESSION. 1 13 4. A agreed to serve B 10 years, at the rate of 20 dol- lars for the first year, 30 for the second, 40 for the third, &c. ; what had he the last year, how much for the whole time, and what per annum 'I Ans. 110 dolls, for the last year, 650 dolls, the whole amount, and 65 dolls, per^xnnum. 5. A sold to B 1000 acres of land, at 10 cents for the first acre, 20 for the second, 30 for the third, &c. ; what was the^price of the last acre, and what did the whole come to? j $ 100 dolls, the last acre, ' I 50050 do. whole cost. Case 2. When the two extremes, and number of terms are given, td find the common difference. RULE. Divide the difference of the extremes, by the number of terms, less one ; the quotient will be the common difference. EXAMPLE. 1 . A is to receive from B a certain sum to be paid in 1 1 several payments in arithmetical progression ; the first pay- ment to be 20 dollars, and the last to be 100 dollars ; what is the common difference, what was each payment, and how much the whole debt ? Operation, 100 last term 20 first term No. of terms 11 1 = 10)80 the difference 8 common difference 20 -f 100X 5i=660 whole debt 20 first payment 20 + 8=28 second do. 28 -j- 8 =36 third do, &c. 2. There are 21 persons whose ages are equally distant from each other, in arithmetical progression ; the youngea! 1 44 GEOMETRIC A L PROGRESSION. ts 20 years old, and the eldest 60 ; \\ hat is the common dif- ference of their ages, and the age of each man ? C 2 common difference Ans. < 20 + 2 22 the second (22 4-2 = 24 the third, &c. 3. A man is to travel from Pittsburgh to a certain place, tn 12 days, and to go but three miles the first day, increas- ing each day's journey in arithmetical progression, making the last day's travelling 58 miles ; what is the daily increase, and what the whole distance I * ( 5 miles daily increase * < 366 miles whole distance. SECTION 6, GEOMETRICAL PROGRESSION. ANY series of numbers increasing or decreasing by one continual multiplier, or divisor, called the ratio, is termed geometrical progression; as 2, 4, 8, 16, 32, &c. increase by the multiplier 2 ; and 32, 16, 8, 4, 2, decrease, continu- ally, by the divisor 2. In geometrical progression there are five things to bo carefully observed. 1. The first term, 2. The last term, 3. The number of terms, 4. The ratio, 5. The sum of all the terms. To find the last term, and sum of all the series in gcf>. netricnl progression, worh by the folloiring RULE. 1 . R iise the ratio in the given sum, to that power whose index 5 hall always bo one less than the number of term? give n ; multiply the number so found by the first term, and Tho jrviduct will be the last term, or greater extreme. GEOMETRICAL PROGRESSION. 145 2. Multiply the last term by the ratio, from that product subtract the first term, and divide the remainder by the ratio, less one ; the amount will be the sum of the series, or of all the terms. EXAMPLE. 1. Suppose 20 yards of broadcloth was sold at 4 mills for the first yard, 12 for the second, 36 for the third, &c. what did the cloth come to, and what was gained by the sale, supposing the prime cost to have been $15 per yard ? Note. In this question observe, the first term is 4, the ratio 3, arid the number of terms 20, consequently the ratio 3 must be raised to 20 1= 19th power. Thus, g S S 1 ! I 1 | & 3 8 I Indices 1234 Ratio 3 9 27 81 . 81 81 648 6561= the 8th power 6561 6561 39366 32805 39366 43046721= 16th power 27= 3d power 301327047 86093442 1162261467=19th power X 4 first term 4649045868=20th or last term X3 13947137604 4 first term Ratio 31=2)13947137600 69 73568,80,0= sum of the series, or number of First cost of the cloth 300,00 mills for which the cloth was sold 0=gam, N 146 GEOMETRICAL PROGRESSION. 2. A father gave his daughter who was married on the first day of January, one dollar towards her portion, prom- ising to double it on the first day of every month for one year ; what was the amount of her whole portion 'I Ans. 4095 dollars. 3. A merchant sold 15 yards of satin ; the first yard for is. the second for 2s. the third for 4s. &c. in geometrical progression ; what was the price of the 15 yards? Ans. 1638Z. 7s. 4. A goldsmith sold 1 pound of gold at 1 cent for the first ounce, 4 for the second, 16 for the third, &c. ; what did it come to, and what did he gain, supposing he gave 20 dollars per ounce ? Ans. He sold it for 55924 dollars 5 cents, and gained 55684 dollars 5 cents. 5. What sum would purchase a horse with 4 shoes and 8 nails in each shoe, at one mill for the first nail, 2 mills foi the second, 4 for the third, &c. doubling in geometrical pro- gression to the last ? Ans. 4294967 dollars 29 cents 5 mills. t>. What sum would purchase the same horse, with the same number of shoes, and nails, at 1 mill for the first nail, 3 for the second, 9 for the third, &c., in a triple ratio of geometrical progression to the last ? Ans. 926510094425 dollars 92 cents. 7. What sum would purchase the same horse, with the same number of shoes, and nails, at 1 mill for the first nail, 4 for the second, 16 for the 3d, &c., in a quadruple ratio of geometrical progression to the last ? Ans. 6148914691236517 dollars 20 cents 5 mills. S. Sold 30 yards of silk velvet, at 2 pins for the first yard, 6 for the second, 18 for the third, &c., and these dis- posed of at 100U for a farthing ; what did the velvet amount to, and what was gained by the sale, supposing the prime cost to hav<3 IKJCII 100/. per yard '! Amount 214469929/. 5s. 3 An *' * Gained 214466929/. 5*. Sjd. SINGLE POSITION. 147 SECTION 7. OF POSITION. POSITION is a rule for finding the true number, by one or more false or supposed numbers, taken at pleasure. It is of two kinds, viz. Single and Double. Single position teaches to resolve such questions as require but one supposed number. RULE. 1. Suppose any number whatever, and work in the same manner with it as is required to be performed in the given question. 2. Then, as the amount of the errors, is to theN sum, so is the given number to the one required. PROOF. Add the several parts of the result together, and if k agrees with the given sum, it is right. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, if I had as many, half as many, and one quarter as many more, I should have 1 32 ; how many had he ? Suppose he had 40 As 110: 40 :: 132 : 48 Ana. as many 40 p roof ^ as many 20 40 \ as many 10 g. US J? 132 2. It is required to divide a certain sum of money among 4 persons, in such a manner that the first shall have , the second ^, the third -J, and the fourth the remainder, which is 28 dollars ; what was the sum ? Suppose 72 As 18 : 72 :: 28 : 112 dolls. AIM. i is 24 Proof 112 i is 18 , . rzr 1 i 19 * ls d ' * 1S _ i is 28 .54 J- is 18f Rem. 18 84 28 last share. 148 DOUBLE POSITION. 3. A, B, and C, buy a carriage for 340 dollars, of which A pays three times as much as B, and B four times as much as C ; what did each pay ? f A paid 240 dolls. Ans. IE 80 (C 20 4. What is the sum of which 1, ]-, and , make 148 dol- lars? Ans. 240 dollars. 5. A person having spent ^, and J of his money : had 26f dollars left; what had he at first? Ans. 100 dollars. 6. A, B, and C, talking of their ages, B said his age was once and a half the age of A ; C said his was twice and T \ the age of both, and that the sum of their ages was 93 ; what was the age of * each? i A's age 12 years. Ans. < B's 18 ( C's 63 7. Seven-eighths of a certain number exceeds four-fifths by 6; what is that number? Ans. 80. 8. A gentleman bought a chaise, horse, and harness, for 360 dollars ; the horse came to twice the price of the har- ness, and the chaise to twice the price of the horse and har- ness together ; what did he give for each ? ( 80 dollars for the horse Ans. < 40 harness ( 240 chaise 9. A gentleman being asked the price of his carriage, an- swered that i, i, i, and of its price was 228 dollars ; what was the price of the carriage ? Ans. 240 dolls. 10. A saves of his wages, but B, who has the same sal- ary? by spending twice as much as A, sinks 50 dollars a year; what is their annual salary? Ans. 150 dolls, each. SECTION 8. DOUBLE POSITION. DOUBLE POSITION is making use of two supposed num- bers, to find the true one. RULE. 1. Take any two numbers, and proceed with them ac- cording to the conditions of the question, noting the errors DOUBLE POSITION. 149 of the results ; multiply these errors cross-wise, viz. the first position by the last error, and the last position by the first error. 2. If the errors be alike, that is, both greater, or both less than the given number, take their difference for a divisor, and the difference of the products for a dividend : but if they are unlike, take their sum for a divisor, and the sum of the pro- ducts for a dividend, the quotient will be the answer required EXAMPLE. 1 . A father leaves his estate to be divided among his three sons, A, B, and C, in the following manner, viz. A is to have one-half wanting 50 dollars, B one-third, and C 10 dol- lars less than B ; what was the sum left, and what was each son's share? Operation. 1st. Suppose 240 dollars. Then 240 -f- 250=70 A's part 240-+- 3= 80 B's part B's share 80 10= 70 C's part Sum of all their parts 220 20 er. too little, 2d. Suppose 300 dollars. Then300-f- 2 50=100 A's part 300-:- 3= 100 B's part B's share 10010= 90 C's part Sum of all their parts 290 10 er. too little, errors. 1st. sup. 240^20=6000 2d. sup. 300-^-10=2400 10)3600(360 An*. Proof 360-;- 2 50=130 360-^ 3= 120 120 10= 110 360 N2 150 DOUBLE POSITION. 2. A and B have the same income ; A saves the of his, but B by spending 30 dollars per annum more than A, at the end of 8 years finds himself 40 dollars in debt ; what is their income, and how much does each spend per annum ? ( Their income is 200 dolls, per annum Ans. < A spends 175 ( B spends 205 3. A, B, and C, would divide 100 dollars between them, so as B may have 3 dollars more than A, and C 4 dollars more than B ; how many dollars must each have ? ( A 30 dollars. Ans. I B 33 (C37 4. A, B, and C, built a house which cost 10,000 dollars; A paid a certain sum, B paid 1000 dollars more than A, and C paid as much as both A and B ; how much did each one pay? ( A paid 2000 dolls. Ans. 1 B 3000 I C 5000 5. A gentleman has 2 horses and a saddle worth 50 dol- lars, which saddle if he put on the back of the first horse, will make his value double that of the second; but if he put it on the second horse, it will make his value triple that of the first ; what is the value of each horse ? A $ First horse 30 dolls. s ' I Second do. 40 6. The head of a fish is 9 inches long, and its tail is as long as its head and half its body, and its body is as long as its head and tail together ; what is its whole length ? Ans. 6 feet. 7. A laborer hired 40 days upon this condition, that he should. receive 20 cents for every day he wrought, and for- l<'it 10 cents for every day he was idle; at settlement he re- ceived 5 dollars ; how many days did he work, and how many was he idle? Ans. Wrought 30 days, idle 10 8. A and B vested equal sums in trade ; A gained a sum equal to J of his stock, and B lost 225 dollars ; then A's money was double that of B's ; what sum had each vested ? Ans. 600 dollars. 9. Divide 15 into two such parts, so that when the greater PERMUTATION. 151 is multiplied by 4, and the less by 16, the products will be equal? Ans. Greater 12, less 3. 10. A person being asked in the afternoon what o'clock it was, answered, that the time past from noon was equal to fa of the time to midnight ; what o'clock was it ? Ans. 36 minutes past .1 o'clock. SECTION 9. PERMUTATION. PERMUTATION is the finding how many different ways any given number of things may be varied in position, or ar- rangement. Thus, 123 are six different arrange- 132 ments made upon the three 213 figures 1 2 3. 2 3 1 3 1 2 321 RULE. Multiply all the terms from an unit up to the given num- ber into one another, and the last product will be the num- ber of changes required. EXAMPLE. 1. In how many different positions may 5 persons be placed at a table. 1X2X3X4X5^120 Ans. 2 How many changes may be rung on 12 bells, and how Jong would they be ringing but once over, allowing 10 changes to be rung in 1 minute, and the year to contain 365 days 6 hours ? Ans. 479001600 changes, and would require 91 years, 3 weeks, 5 days, and 6 hours. 3- Seven men not agreeing with the owner of a boarding house about the price of boarding, offer to give 100 dollars each, for as long time as they can seat themselves every day differently at dinner ; this offer being accepted, how long may they stay ? Ans. 5040 days, or 13 years, 295 days. 152 COMBINATION. 4. What number of variations will the 9 digits admit of? Ans. 362880. 5. How many changes may be made on the 26 letters of the alphabet ? Ans. 403,291,461,126,605,635,584,000,000. Quatril. Trilns. Billions. Millions. Units. Note. From the answer to this last question, which amounts to a number, of which we cannot form any conceivable idea, we may dis- cover the surprising power of numbers, and also the endless variety of ideas that may be distinctly communicated by these 26 simple charac- ters. It will also be evident from the method of notation here used, that a row of figures of any given length whatever, may be numerated, though we may be entirely unable to comprehend the amount. SECTION 10. COMBINATION. COMBINATION of quantities, is the showing how often a less number of things can be taken out of a greater, and combined or joined tdgether differently. RULE. Take a series of 1, 2, 3, &c. up to the number to be com- bined ; take another series of as many places, decreasing by unity from the number out of which the combinations are to be made ; multiply the first continually for a divisor, and the latter fcr a dividend, and the quotient will be the answer. EXAMPLE. 1. How many combinations may be made of 7 dollars out of 12? 1x2x3, &c. up to 7 =5040 divisor. Again, 12 the whole number of terms less 7 = 5 Hence 12 x 11 X 10, &c. down to 5 = 3991680 dividend. And 5040 ) 3991680 ( 792 Ans. 2. How many combinations can be made of 6 letters out of 24 of the alphabet ? Ans. 134596. 3. In how many different ways may an officer select 8 men out of 30, so as not to make the same selection twice Ans. 5H52925 ADDITION OF DUODECIMALS. 153 PART VIII. MENSURATION. SECTION L Duodecimals, or Cross Multiplication. DUODECIMALS are fractions of a foot, or of an inch, or parts of an inch, having 12 for their denominator. Inches and parts are sometimes called primes ('), seconds ("), thirds ('"), &c. The denominations are, 12 Fourths ("") make 1 Third 12 Thirds - 1 Second 12 Seconds 1 Inch in. 12 Inches 1 Foot ft. Nole. This rule is much used in measuring 1 and computing the di- mensions of the several parts of buildings ; it is likewise used to find the tonnage of ships, and the contents of bales, cases, boxes, &c. ADDITION OF DUODECIMALS. RULE. Add as in compound addition, carrying 1 for each 12 u. the next denomination. EXAMPLE. Ft. in. "" '" "" Ft. in. " 25 9 3 5 8 244 6 3 10 5 34 3 9 2 7 355 9851 28 10 4 8 4 559 10 9 5 8 64 11 9 7 2 129 5569 82 7 5 6 8 895 1 10 5 11 15 3 7 9 10 651 1759 44 6 11 2 8 555 9 8 5- 5 22 3 6 1 5 388 11 10 10 9 318 8 9 8 4 154 SUBTRACTION OF DUODECIMALS. SUBTRACTION OF DUODECIMALS. RULE. Work as in compound subtraction, borrowing 12 when accessary* EXAMPLE. Ft. in. ' Ft. in. " '" "" From 125 4 3 8 2 2756 5780 Take 68 9 2 10 1 1839 9 5 11 10 Rem. 56 7 10 1 3. From a board measuring 35 feet, 9 inches, 2 seconds, cut 24 feet, 10 inches, 5 seconds, and 4 thirds; what is left ? Ans. Wft. lOin. Ssec. 8'" 4. A joiner having lined several rooms very curiously with costly materials, finds the amount to be, in square measure, 803 feet, Z inches, 4 seconds ; but several deduc- tions being to be made for windows, arches, &c. those de- ductions amounted to 70 feet, 3 inches, 7 seconds, 10 thirds, 5 fourths ; how many feet of workmanship must he be paid for? Ans. 132ft. llin. 8" I"' 7"" MULTIPLICATION OF DUODECIMALS. Case 1. When the feet of the multiplier do not exceed 12. RULE. Set the feet, or the highest denomination of the multiplier under the lowest denomination of the multiplicand, and mul- tiply as in compound numbers, carrying 1 for every 12 from ;m<i denomination to another, and place the result of the ;owcst denomination in the multiplicand under its multiplier. TABLE. ________ 2. Feet multiplied by inches, give inches, 1. Feet multiplied by feet, give feet. 2. Feet multiplied by inches, give inc^>. 3. Feet multiplied by seconds, give seconds, &c. 4 Inches multiplied by inches, give seconds. MULTIPLICATION OF DUODECIMALS. 155 5. Inches multiplied by seconds, give thirds, &c. 6. Seconds multiplied by seconds, give fourths. 7. Seconds multiplied by thirds, give fifths, &c. PROOF. Reduce the given sum to a decimal, or work by the rules of practice. 1. Multiply EXAMPLES. Ft. in. " Ft. in. 8 6 9 by 7 3 7 3 Ans. Ft 59 2 11 3 1 8 3 . 62 11 3 Or practice. 3 is |i| 8 6 9 7 3 59 2 11 1 3 8 3 62 11 3 A?is. 7 3 Proof decimally. 6 9 = 8,5625 = 7,25 Ft. in. 2. Multiply 9 5 3. 7 10 4. 846 428125 171250 599375 62,078125 12 0,937500 12 11,250000 12 3,000000 Ft. in. " Ans. 36 10 7 69 10 2 21 10 5 Ft. in. by 3 11 by 8 11 by 2 74 5. What is the price of a marble slab, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches, at 1 dollar and I 50 cents per foot? Ans. 15 dolls. 35^ cts. 6. There is a house with three tiers of windows, 3 in a tier, the height of the first tier is 7 feet 10 inches, of the second 6 feet 8 inches, and of the third 5 feet 4 inches, and the breadth of each window is ^ ft. 11 inches. ; what will the glazing come to at 14 cts. per ft. ? Ans. 32 dolls. 62 cts. . 156 MULTIPLICATION OF DUODECIMALS. Case 2. When the feet of the multiplier exceeds 12. RULE. Multiply by the feet in the multiplier, and take parts for the inches. EXAMPLES. 1. Multiply 84 feet 6 inches, by 36 feet 7 inches and 6 seconds. Operation. 84 6 1 6*cc. 507 6 3042 42 3 3 6 Ans. 3094 9 Ft. in. Ft. in. 2. Multiply 76 7 by 19 10 3. 127 6 by 92 4 4. 184 8 by 127 6 Ft. in. sec. 5. Multiply 311 4 7 by 36 7 6x6=36 Ft. in. sec. Ans. 1518 10 10 11772 6 23545 Ft. in. sec. 5 (tin. is II 1868 3 6 11209 9 " 155 8 3 6 25 9 8 7 7 ' 10 4 21 987 11402 (I 1 I MULTIPLICATION OF DUODECIMALS. 157 6. A floor is 70 feet 8 inches, by 38 feet 11 inches ; how many square feet are therein 1 Arts. 2750ft. lin. 4sec. 7. If a ceiling be 59 feet 9 inches long, and 24 feet 6 inches broad, how many yards does it contain ? Arts. I62yds. 5ft. W$in. Note. Divide the square feet by 9, and the quotient will be square yards. 8. What will the paving of a court yard come to at 15 cents per yard, the length being 58 feet 6 inches, and the breadth 54 feet, 9 inches ? Arts. 53 dolls. 38 + cts. 9. What is the solid content of a bale of goods, measur- ing in length 7 feet 6 inches, breadth 3 feet 3 inches, and depth 1 foot 10 inches? Ans. 44/fc. 8in. Ssec. Note, To find the cubic feet, or solid content of bales, cases, boxes, &c. multiply the length by the breadth, and that product by the thick- ness. 10. A merchant imports from London six bales of the fol- lowing dimensions, viz. Length. Breadth. Depth. Ft. in. Ft. in. Ft. in No. 1. 2 10 24 19 2. 2 10 26 13 3. 36 22 18 4. 2 10 28 19 5. 2 10 26 19 6. 2 11 28 18 What are the solid contents, and how much will the freight amount to, at 20 dollars per ton of 40 feet ? Ans. lift. lin. and freight 35 dollars 79 cts. To find a ship's tonnage by Carpenter's measure. For single decked vessels, multiply the length, breadth at the main beam, and depth of the hold together, and di- vide the product by 95 ; but if the vessel be double decked, take half the breadth of the main beam for the depth of the hold, and work as for a single decked vessel. O 158 MULTIPLICATION OF DUODECIMALS . EXAMPLES. 1. The length of a single decked vessel is 60' feet, the breadth 20, and depth 10; what is the tonnage? Then 60x20x10 = 12000 And 1 2000 ~ 95 = 1 26 fy tons. An*. Or, as 95 : 20x10 :: 60 : 126 r <V Ans. 2. Required the tonnage of a double decked vessel, whose 'ength is 90, and breadth 30. Then 90 X 30 X 15 (half breadth) =40500. And 40500-^-95=426-,^ tons. Ans. Or, as 95 : 30 X 15 :: 90 : 426 T \ Ans. 3. A single decked vessel is 64 feet long, 22 feet broad, and 10 feet deep; what is its tonnage? Ans. 148 T 4 y- tons. 4. What will be the tonnage of a double decked vessel whose length is 80 feet, and breadth 26 feet ? Ans. 284}| tons - To find the Government tonnage. " If the vessel be double decked, take the length thereof from the fore part of the main stem, to the after part of the stern post, above the upper deck ; the breadth thereof at the broadest part above the main wales, half of which breadth shall be accounted the depth of such vessel, and then deduct from the length three-fifths of the breadth ; multiply the remainder by the breadth, and the product by the depth, and divide this last product by 95, for the tonnage. But if it be a single decked vessel, take the length and broadth, as directed above; deduct from the said length three-fifths of the breadth, and take the depth from under the side of the deck plank to the ceiling in the hold, then multiply and divide as aforesaid, and the quotient shall be deemed the tonnage." CARPENTERS' OR SLIDING RULE. 159 SECTION 2. The Carpenters' or Sliding Rule. THIS rule is not only useful in measuring timber, artifi- cers' work, and taking dimensions, but also in ascertaining the contents of such work ; it is therefore a rule which all mechanics, having any thing to do with mensuration, ought to possess and understand. It consists of two equal pieces of box, each one foot in length, connected together by a folding joint ; in one of these equal pieces, there is a slider, and four lines marked at the right-hand with the letters A, B, C, D ; two of these lines are upon the slider, and the other two upon the rule. Three of these lines, viz. A, B, C, are called double lines, because they proceed from 1 to 10 twice over ; these three lines are alike both in number and division. They are numbered from the left-hand towards the right with the figures 1, 2, 3, 5, 6, 7, 8, 9, then 1, which stands in the middle; the numbers then proceed, 2, 3, 4, 5, 6, 7, 8, 9, and 10, which stands at the right-hand end of the rule. These numbers have no determinate value of their own, but depend upon the value you set upon the unit at the left-hand of this part of the rule ; thus, if you call it 1, the 1 in the middle will be 10, the other figures which follow, will be 20, 30, &c. and the 10 at the right-hand will be 100, If you call the first or left-hand unit. 10, the middle 1 will be 100, and the following figures will' be 200, 300, &c. and the 10 at the right-hand end will be 1000. Or if you call the first or left-hand unit 100, the middle 1 will be 1000, the following figures 2000, 3000, &c. and the 10 at the right-hand 10,000. Lastly, as you alter, or number the large divisions, so you must alter the small divisions in the same proportion. The fourth line D is a single line, proceeding from 4 to 40 ; it is also called the girt line, from its use in casting up the contents of timber. Upon it are marked W G at 17, 15, and A G at 18, 95, the wine and gauge points, to make it serve the purpose of a gauging rule. Tne use of the double lines, A and B, is for working the rule of proportion, and finding the areas of plane figures. On the other part of this side of the rule there is a table to 160 CARPENTERS' OR SLIDING RULE. ascertain the value of a ton, or 50 cubic feel of timber at all prices from 6 pence to 24 pence per foot. On the other side of the rule are several plain scales divided into 12th parts, marked mrA, ^> i ?, &c. signifying that the inch, 5 inch, &c. are divided into 1:2 parts. These scales are useful for planning dimensions, that are taken in feet and inches. Again, the edge of the rule is divided into inches, and each of these into eight parts, representing half inches, quarter inches, and half quarters. In this description the rule is folded ; but when it is opened and the slider drawn out, the hack part will be ibund divided like the edge of the rule, so that all together will measure 3 feet or one yard. USE OF THE CARPENTERS' RULE. 1st. To multiply numbers together. EXAMPLE. 1. Suppose the two numbers 13 and 24. Set 1 on B to 13 on A ; then against 24 on B, stands 312 on A, which is the required product of the two given num- bers 13 anu 24. Note. In any operation when a number runs beyond the end of the line, seek it on the other radius, or other part of the line ; that is, take the 10th part of it, or the 100th part of it, &c. and increase the product of it proportionally 10 fold, or 100 fold, <fec. 2. Multiply 12 by 16 Ans. 192. 3. 35 19 665. 4. 270 54 14580. 2d. Division of numbers by the Carpenters' Rule. EXAMPLE. 1. Required to divide 360 by 12. Set the divisor 12 on B to the dividend 360 on A ; then against 1 on B stands 30, the quotient on A. 2. Divide 665 by 19. Quotient 35. 3. 396 " 27. 14,6. 4. 741 42. 17,6. 5. 76HO 24. 320. Note. In this last example, because 7G80 is not contained on A, one- lenth of the number, viz. 768 is taken, to make it fall within the com- pass of the scale The quotient of this sum is 32, but as the dividond w;is diminished by a division of 1(1, so the juotieiit nmst be multiplied uy the Rane number, arid 32 10320. CARPENTERS' OR SLIDING RULE. 161 3d. To square numbers by the Carpenters' Rule. EXAMPLE. 1. Required to square the number 25. Set 1 or 100 on B to the 10 on D ; then against every number on D stands its square on the line C. Thus against 25 on D, stands 625, its square on C. 2. Required the square of 30. Ans. 900. 3. Required the square of 35. 1225. 4. Required the square of 40. 1600. Note. If the given number be hundreds, &c. reckon the 1 on D for 100, or 1000, &c. then the corresponding one on C is 10000, or 100000, &c. , thus the square of 230 is found to be 52900. 4th. To find a fourth proportional to three numbers : or to perform the Rule of Three by the Carpenters' Rule. EXAMPLE. 1. Required to find a fourth proportional to 12, 28, and 114. * Set the first term 12 on B to the second term 28 on A ; then against the third term 114 on B, stands 266 on A, which is the fourth proportional sought. 2. Required the fourth proportional term to the numbers 25 : 75 : : 100. Ans. 300. 3. Required the fourth proportional term to the numbers 27 : 20 : : 73. Ans. 54 2 T 5th. To extract the Square Root of any number by the Carpenters 1 Rule. EXAMPLE. 1. Required the square root of 400, Set 1 upon C to 10 upon D ; then against the number 400 on C, stands its root 20 on D. 2. Required the square root of 529. Ans. 23. 3. What is the square root of 900 ? 30. 4. What is the square root of 300 ? 17,3 -f O 2 162 MEASURfNG OF BOARDS AND TIMBKR. SECTION 3. Measuring of Boards and Timber. 1st To find ike superficial content of a board or platik. RULE. Multiply the length by the mean breadth. When the board is broader at one end than the other, add the brewlih of the two ends together, and take half the sum for tin..- mean breadth. By the Carpenters' Rule. Set 12 on B to the breadth in inches on A; then against the length in feet on B, you will find the superficies on A, in feet. EXAMPLES. 1. How many feet are there in a board that is 13 {(-el long and 16 inches broad ? Operation. By duodecimals. By decimals. 'Ft. in. 13, 13 1,33 1 4 39 13 39 440 13 17 4 Ans. 17,29 Ans. By the Carpenters' Rule As 12 on B : 16 on A : : 13 on B : 17 on A. Ans. 2. Required the superficies of a board, whose mean breadth is 1 foot 2 inches, and length 12 feet 6 inches? Ans. 14 feet 7 inches. 3. Required the value of 5 oaken planks, at 3 cents pei foot ; each of them being 17 feet long, and their several breadths as follows, viz. ; two of 13 inches in the middle, one of 14 inches in the middle, and the two remaining ones, car-h 18 inches at the broader end, and 11:1 inHvs at the narrower 1 Ans. 3 dolls. 9i cts. MEASURING OF BOARDS AND TIMBER. 163 2d. Having the breadth of a board or plank in inches, tojind how much in length will make afoot^ or any other assigned quantity. RULE. Divide 144, or the area to be cut off, by%ie breadth' in inches, and the quotient will be the length in inches. EXAMPLE. 1. How many inches in length will it require to make one foot, of a board that is 9 inches broad? Operation. 144-1-9=16 inches, the length required. 2. How many inches in length, of a board that is 23 inches wide, will make 1 foot? Ans. 6,26 -f inches. 3. From a mahogany plank 26 inches broad, a yard and a half (or 13 feet 6 inches) is required to be cut off*; what distance from the end must the line be struck ? Ans. 74,7692 inches, or 6,23 feet. 3d. To find the solid content of squared or four sided timber. RULE. Multiply the mean breadth by the mean thickness, and that product by the length ; the last product will give the solid content. Note. 1. If the tree taper regularly from one end to the other, take the mean breadth and thickness in the middle ; or take half the sum of the dimensions at the two ends, for the mean dimension. 2. If the piece does riot taper regularly, take several different di- mensions, add them all together, and divide their sum by their num- ber, for the mean dimension. 3. The quarter girt is an arithmetical mean, proportional between the mean breadth and thickness, that is the square root of the pro- duct. EXAMPLES- 1. If a piece of timber be 2 feet 9 inches deep, and 1 foot 7 inches broad, and the length 16 feet 9 inches,, (or which 164 MEASURING OF BOARDS AND TIMBER. is the same thing,* if the quarter girt be 26 inches, and the length 16 feet 9 inches,) how many solid feet are contained therein ? Operation. 26 inches quarter girt 16,75 16 feet 9 inches, the length. 26 676 156 10050 52 11725 10050 676 square 144) 11323,00(78,63 + feet. Arts. 1008 1243 1152 910 864 460 432 28 rem. By the Carpenter's Rule As 12 on D : 16f on C : : 26 on D : 78 on C. Ans. 2. The quarter girt of a piece of squared timber is 15 inches, and the length 18 feet; required the solidity? Ans. 28 J feet. 3. If a piece of squared timber be 25 inches square at the greater end, and 9 inches square at the less, and the length be 20 feet ; what is the solid content ? Ans. 40,13 feet. 4. Suppose a piece of squared timber to measure 32 by 20 inches at the greater end, and 10 by 6 inches ai the less, and the length 18 feet; how many feet of timber are con- tained therein? Ans. 34,12 -f feet. * This is making- use of an arithmetical mean, instead of a geomet- rical one, which is not exactly true; it is, however, sufficiently exact for common purposes, when the timber is nearly square; the error in- creases, the more the breadth and depth differ from each other. When greater exact ness is required, multiply the breadth by the depth in the middle, and that, product by the length, ibr the true con- tent. MEASURING OF BOARDS AND TIMBER. 165 4:th. To find the solid content of round timber. RULE. Multiply the square of the quarter girt, or of ^ of the mean circumference, by the length, for the content. Note. 1. To find the quarter girt of round timber, measure round the middle with a line, one-fourth part of this is reckoned the quarter girt 2. When the tree is tapering 1 , take either the mean dimension, as in squared timber, or girt it at both ends, and take half the sum. If the tree is very irregular, divide it into several lengths, add all the girts to- gether, and divide the amount by the sum of them, for tho mean girt. 3. The buyer is allowed to take the girt anywhere between the greater end and the middle, if it taper ; an allowance must also be made for bark ; one-tenth for oak, but less for ash, beech, &c. EXAMPLES. 1. A piece of round timber being 9 feet 6 inches long,, and its mean quarter girt 42 inches, what is the content? Operation. Decimals. Duodecimals. 3,542 inches, quarter girt. Ft. in. 3,5 3 6=42 inches. 3 6 175 105 190 10 6 12,25 9,5 length 12 3 9 6 6125 11025 6 1 110 3 116,375 content. 116 4 6 By the Carpenters' Rule As 9,5 on C : 10 on D : : 3,5 on D : 116J on C ( 4 Or,9,5 on C : 12 on D : : 42 on D : 116i on C $ 2. The length of a tree is 24 feet, its girt at the thicker end is 14 feet, and at the smaller end 2 feet; what is its content? Ans. 96 fert. 166 CARPENTERS' AND JOINERS' WORK- 3. If a piece of round timber 18 feet long, measure 00 inches in circumference, or the quarter girt 24 inches ; how many feet of timber does it contain? Ans. 72 feet. 4. If a piece of round timber measure 11 feet 4 inches at the larger end, 2 feet 8 inches at the less, and its length 21 feet, how many feet of timber are contained therein? Ans. 64,31 feet. 5. Required the amount of three pieces of round timb< r measuring as follows, viz. The first 24 feet long and mean girt 8 feet, The second 14* do. do. 3,15 The third 17i do. do. 6,28 Ans 147 ft 4. feet. SECTION 4. OF CARPENTERS' AND JOINERS' WORK. To this branch belongs all the wood-work of a house, such as framing, flooring, partitioning, roofing, &c. Carpenters usually measure their work by the square, (consisting of 100 superficial feet) the yard or foot; but enriched mouldings, cornices, &c. are estimated by run- ning or lineal measure, and some things are rated by the piece. In measuring of Carpenter's work, the string is made to ply close to every part of the work over which it passes. Partitions are measured from wall to wall for one dimen- sion, and from floor to floor, as far as they extend, for the other. In framing, no deductions are made for door-ways, fire- piacrs, or other vacancies, on account of the additional trouble of framing arising from them. For stair-case's, take the breadth of all the steps, by milking a line ply close over them, from the top to the bot- inm, and multiply the length of this line, by the length of thr step, for the whole axea. T>y Iho length of a step is CARPENTERS' AND JOINERS' WORK. 167 meant the length of the front, and the returns at the two ends ; and by the breadth, is to be understood the girt of its two outer surfaces, or the tread and rise. The rail of a stair-case is taken at so much per foot in length, according to the diameter of the well-hole ; archi- trave string boards, by the foot superficial ; brackets and strings at so much per piece, according to the workmanship. Wainscoting is measured by the yard square, consisting of 9 feet. Door cases, frame doors, modillion cornices, eaves, frontis- pieces, <fec. are generally measured by the foot superficial. Joists are measured by multiplying their breadth by their depth, and that product by their length. They receive va- rious names, according to the place in which they are laid to form a floor ; such as trimming joists, girders, binding joists, bridging joists, ceiling joists, &c. In boarded flooring, the dimensions must be taken to the extreme parts ; and the number of squares of 100 feet, are to be . calculated from these dimensions. Deductions must be made for chimneys, stair-cases, &c. In roofing, take the whole length of the timber, for the length of the framing, and gird over the ridge from wall to wall with a line, for the breadth. This length and breadth multiplied together give the content. In measuring of roofing for work and materials, all holes for chimney-shafts, sky-lights, &c. are included in the mea- surement, on account of their trouble and waste of materials ; but for workmanship alone, they are generally deducted. It is a common rule among carpenters, that the flat of any hou.se, and half the flat thereof taken within the walls, ks equal to the measure of the roof of the same house ; this is, however, only when the roof is the true pitch where the length of the rafters are 5 of the breadth of the build- ing. The pitch of roofs varies according to the materials with whio.h they are covered, and fancy of the builder. Weather- boarding, like flooring, is measured by the square, arid sometimes by the yard. 168 CARPENTERS' AND JOINERS' WORK. EXAMPLES. 1. If a floor be 57 feet 3 inches long, and 28 feet 6 inches broad, how many squares of flooring does it contain ? Operation. By decimals. By duodecimals. Ft. in. Ft. in. 57 3=57,25 57 3 28 6= 28,5 28 6 28625 456 45800 114 11450 28 7 6 100)1631,625 16,31,625 700 100)1631 7 6 16,31 7 6 Ans. 16 squares 31 feet 7 in. 6' 2. Let a floor be 53 feet 6 inches long, and 47 feet 9 inches broad, how many squares does it contain ? Ans. 25 squares 54 feet. 3. A floor being 36 feet 3 inches long, and 16 feet 6 inches broad, what will it cost at 4 dollars and 50 cents per square? Ans. 26 dolls. 91 cents. 4. A room is 35 feet long and 30 feet wide ; there is in it a fire-place which measures 6 feet by 4 feet 6 inches, and a well-hole for the stairs measures 10 leet 6 inches by 8 feet ; what will the flooring come to at 3 dollars and 75 cents per square? Ans. 35 dolls. 21 -f cts,. 5. How many squares are contained in a partition that i* 82 feet 6 inches long, and 12 feet 3 inches high? Ans. 10 squares and 10-ffeet. 6. If a partition between rooms be in length 91 feet 9 inches, and its height 11 feet 3 inches; how many squares are contained in it, and how much does it come to at 4 dol lars and 50 cents per square ? * Ans. 10 squares 32 feet, and costs 46 dolls. 44 cts. CARPENTERS AND JOINERS WORK- 169 7. If a house within the walls be 44 feet 6 inches long, and 18 feet 3 inches broad ; how many squares of roofing will it contain, allowing the roof to be the true pitch ? Operation. By decimals. By duodecimals. Ft. Ft. in. 18 3 18,25=18 3 the breadth. 44 6 44,5=44 6 the length. 72 9125 72 7300 7300 792 11 0" Flat 812,125 9 1 6 Half 406,062 Flat 812 1 -100)1218,187 Half 406 + Sum 12,18+ + 100)1218 1 6 Sum 12,18 Ans. 12 sq. 18 ft. 8. What cost the roofing of a house at 1 dollar and 40 cents per square ; the length within the walls being 52 feet 8 inches, and the breadth 3-0 feet 6 inches ; the roof being of a true pitch ? Ans. 33 dollars 73 cents. 9. Suppose a house measures, within the walls, 40 feet 6 inches in length, and 20 feet 6 inches in breadth, and the roof being a true pitch ; how many squares of roofing does it contain, and how much will it cost at 2 dollars 25 cents per square ? Ans. 12,45375 squares, and costs 28 dolls. 2 + cts. Note. All timbers in a roof are measured in the same manner as in floors, except king-posts, which are measured by taking their breadth and depth at the widest place, arid multiplying these together., and the product by the length. 10. If a room or wainscot, being girt downwards over !?(> BIUCKLAYEKS' WORK. * the mouldings, be 15 feet 9 inches high, and 126 feet 3 in* -lies in compass ; how. many yards does that room contain '/ Operation. By decimals* By duodecimals. 12V5 ' Ft - in - 15,75 Ig 3 63125 63Q 88375 126 63125 12655 189 63 1 6 31 6 9 9)1988,4375 390 Sum 220,8 Sum 220 8 Ans. 220 yds. 8 feet. 11. If a room of wainscot be 16 feet 3 inches high, and the compass of the room 137 feet 6 inches ; how many yards are contained in it ? Ans. 248 yards 2 -f feet. 12. If the window-shutters about a room be 69 feet 9 inches broad, and 6 feet 3 inches high ; how many yards are contained therein, at work and half? Ans. 72,656 yards. 13. What will the wainscoting of a room come to at 80 cents per square yard, supposing the height of the room, in- cluding the cornice and moulding, be 12 feet 6 inches, and the compass 83 feet 8 inches ; three window-shutters, each 7 feet 8 inches by two feet 6 inches, and the door 7 feet by 3 feet 6 inches ; the shutters and door being worked on both sides, are reckoned work and half? Ans. 96 dollars 60J cents. SECTION 5. OF BRICKLAYERS' WORK- BKICK WOKK is measure. I ;ind rsfimatr'd in various ways. !i somn places walls aiv mrasim-d !>y fho rod square of 1 i frc'i ; so th.-.t on'- rod in length, and one in breadth BRICKLAYERS' WORK. 171 contain 272,25 square feet ; in other places the custom is to allow 18 feet to the rod, that is, 324 square feet. In other places they measure by the rod of 21 feet long, and 3 feet high, that is, 62 square feet. Again, in other places they account 16^ feet long and 1 foot high, that is, 16^ square feet, a rod or perch; or again, by the yard of 9 square feet ; and oftentimes the work is estimated at so much per thousand bricks. When brick work is measured by the rod, or perch, it must be estimated at the rate of a brick and a half thick; so that if a wall be more or less than this standard thickness, it must be reduced to it by the following RULE. Multiply the superficial content of the wall by the num- ber of half bricks in the thickness, and divide the product by 3 for the superficial feet in standard thickness. EXAMPLES. If a wall be 72 feet 6 inches long, and 19 feet 3 inches high, and 5 bricks and a half thick ; how many rods of brick work are contained therein, when reduced to the standard thickness ? Operation. By decimals. By duodecimals. 19,25 = the height Ft. in. 72,5 = the length. 72 6 "9625 19 3 3850 648 13475 - 72 139^625 1368"" 11= the thickness. 18 1 6' -f-3)15351,875 9 6 272,25) 5117,291 (18 rods. 1395 7 ^6 239479 i- 68,06) 216/79( 3 quarters. -3)15351 10 6 lalT 272)511^ (18 rote. 2397 68) "22f ( 3 quarters Tffeet. Note. Observe that 68,06 is the one- fourth part of 272,25, and 68 is only the one-fourth part of 272. As the number 272^ is an inconve- nient number to divide* by, the \ is usually ornitfed, and the content in feet divided only by 272 ; the difference being too trifling to be con- sidercd in practice. 172 BRICKLAYERS' WORK. To find fixed divisors for bringing the answer into feet or rods of a standard thickness, without multiplying the su- perficies by the number of half bricks, <$*c. RULE. Divide three, the number of half bricks in l, by the number of half bricks in the thickness, the quotient will be a divisor, which will give the answer in feet. Or if a divi- sor is sought for, that will bring the answer in rods at once, multiply 272 by the divisor found for feet, and the product will be a divisor for rods ; as in the following TABLE. 1 The thickness of the wall. 2 Divisors for the ansiver in feet. 3 Divisors for the ansiver in rods. 1 brick 1, 408 l brick 1 472 2 bricks ,75 204 2J bricks ,6 163,2 3 bricks ,5 136 3i bricks ,4285 116,6 4 bricks ,375 102 4^ bricks ,3333 90,6 5 bricks ,3 81,6 5J bricks ,2727 74,18 Application of the above Table. Multiply the length of the given wall bv the breadth , o serve the number of half bricks it is in thickness ; and op- [>rsite thereto will be found in the second column the di- visor to reduce it to feet ana in the third column the divi- sor for rods. Thus in the above example 72,5 X by 19,25 1395,625. Ana 1395,620 2727 = 5117+ the number of feet in standard nn-asure. And i:W5,(>-jr>-74,18^18,8-f th<> num l >cr of rods BRICKLAYERS' WORK. 173 Or, by the Carpenters' Rule As the tabular divisor, against the thickness of the wall : is to the length of the wall : : so is the breadth : to the content. As 74,18 on B : 72,5 on A :: 19,25 on B : 18 j on A. Ans. To find the dimensions of a building, measure half around on the outside, and half on the inside, for the whole length of the wall ; this length being multiplied by the height gives the superficies. All the vacuities, such as doors, windows, window -backs, &c. must be deducted, for materials ; but for workmanship alone no deductions are to be made, and the measurement is usually taken altogether on the outside. This is done in consideration of the trouble of the returns or angles. There are also some other allowances, such as double measure for feathered gable ends, &c. 2. How many yards and rods of standard thickness are contained in a brick wall, wWbse length is 57 feet 3 inches, and height 24 feet 6 inches ; the wall being 2J bricks thick ? Ans. 259,74 yards, or 8,58 + rods. 3. If a wall be 245 feet 9 inches long, 16 feet 6 inches high, and 2| bricks thick; how many rods "of brick work are contained therein, when reduced to standard thickness ? Ans. 24 rods 3 quarters 24 feet. 4. A triangle gable end is raised to the hei^ of 15 feet above the wall of a house, whose width is 45 feet and the thickness of the wall 2 bricks ; required the contei^ in rods at standard thickness? Ans. 2 rods 18 feet. Chimneys by some are measured as if they were solid, de- ducting only the vacuity from the hearth to the mantle, on account of their trouble. But by others, they are girt or measured round for their breadth, and the height of their story, taking the depth of the jambs for their thickness. And in this case no deduc- tion is made for the vacuity from the floor to the mantle-tree, because of the gathering of the breast and wings, to make -oom for the hearth in the next story. P2 174 MASONS' WORK. If the chimney back be a party wall, and the wall be measured by itself, then the depth of the two jambs and length of the breast is to be taken for the length, and the height of the story for the breadth, at the same thickness with the jambs. Those parts of the chimney-shaft which appear above the roof are to be girt with a line round about the least place of them for the length, and take the height for the breadth ; and if they are 4 inches thick, they are to be accounted as one brick work, and if they are 9 inches thick, they are to be taken for l brick work, on account of the trouble of plastering and scaffolding. It is customary in most places to allow double measure for chimneys. SECTION 6. OF MASONS' WORK. MASONS' work is measured sometimes by the foot solid, sometimes by the foot superficial, and sometimes by the foot in length. It is also measured by the yard, and mostly by the rod or perch, which is 16 feet in length, 18 inches in breadth, and 12 inches in depth. Walls are measured by the perch; columns, blocks of stone, or marble, &c. by the cubic foot; and pavements, slabs, chimney-pieces, &c. by the superficial or square foot. Cubic, or solid measure, is always used for materials, but square measure generally for workmanship. In solid measure, the true length, breadth and thickness are taken and multiplied into each other for the content. !n superficial measure, the length and breadth of every part of the projection, which is seen without the general upright face of the building, is taken for the content. MASONS' WOHK. 175 EXAMPLES. 1. If a wall be 97 leet 5 inches long, 18 feet 3 inches high, and 2 feet 3 inches thick, how many solid feel, and perches, are contained therein ? Operation. By decimals. By uuodecimals. 97,417 length Ft. in. 18,25 breadth 97 5 18 3 4870S5 194834 776 779336 97 97417 24 4 3" 600 1777,86025 superficies 1 60 2,25 thickness 1777 10 3 888930125 2 355572050 355572050 3555 8 6 444 5 6 4000,1855625 solidity. in cubic ft. 4000 2094 u. 4000^-24,75^161,616-f feet. Ans. 2 How many solid feet and perches are contained in a wall 53 feet 6 inches long, 12 feet 3 inches high, and 2 feet thick? Ans. 1310,75 feet, and 52,9595 rods. 3. If a wail be 107 feet 9 inches long, and 20 feet 6 inches high, how many superficial feet are contained therein ? Ans. 2208 feet 10 inches. 4. If a wall be 112 feet 3 inches long, and 16 feet 6 inches high, how many superficial rods, each 63 square feet, are contained therein ? Ans. 29 rods 25 feet. 5. What is a marble slab worth, whose length is 5 feet 7 inches, and breadth 1 foot 10 inches, at 80 cents per fool superficial 1 Ans. 8 dolls. 19 els* 176 PLASTERERS WOixK SECTION 7. OF PLASTERERS' WORK. PLASTERERS' WORK is principally of two kinds, viz. first, plastering upon laths, called ceiling ; and second, plas- tering upon walls, or partitions made of framed timber, called rendering, which are measured separately. Plasterers' work is usually measured by the yard square, consisting of 9 square feet ; sometimes it is measured by the square foot, and sometimes by the square of 100 feet. Enriched mouldings, cornices, &c. are rated by running, or lineal measure. In arches, the girt round them multiplied by the length, is taken for the superficies. Deductions are to be made for doors, chimneys, windows, and other large vacuities. But when the windows, or other openings, are small, they are seldom deducted, as the plas- tered returns at the top and sides are allowed to compensate for the vacuity. Whitewashing and coloring are measured in the same manner as plastering. EXAMPLES. 1. If a ceiling be 59 feet 9 inches long, and 24 feet 6 inches broad, how many superficial yards of 9 square feet does it contain ? Operation. By decimals. By dtfodecimals. Ft. in. Ft. in. 59 9=59,75 feet 59 9 24 6= 24,5 do. 24 6 29875 236 23900 118 11950 29 10 6" 18 -7-9)1463,875 feet Ans. 1 62,65 -f yards -r 9) 1463 10 6 162 5 10 6 An*. \VKHJS WORK 177 2 If the plastered partitions between rooms be 141 feet 6 inches about, and 1 1 feet 3 inches high, how many yards (Jo tiiey contain? Ans. 176,87 yards. 3. What will the plastering of a ceiling come to at 15 cents per yard, allowing it to be 22 feet 7 inches long, and 13 feet 11 inches Lroad ? Ans. 5 dolls. 20 cts. 4; The length of a room being 20 feet, its breadth 14 feet 6 inches, and height 1 feet 4 inches ; how many yards of plastering does it contain, deducting a fire-place of 4 feet by 4 feet 4 inches, and two windows, each 6 feet by 3 feet 2 inches ? Ans. 73^ T yards. 5. The length of a room is 14 feet 5 inches, breadth 13 feet 2 inches, and height 9 feet 3 inches, to the under side of the cornice, which projects 5 inches from the wall, on the upper part next the ceiling ; required the quantity of render- ing and plastering ; there being no deductions but for one door, the size whereof is 7 by 4 feet ? Ans. 53 yds. 5 ft. of rendering, and 18 yds. 5 ft. ceiling. 6. The circular vaulted roof of a church measures 105 feet 6 inches in the arch, and 275 feet 5 inches in length ; what will the plastering come to at 12 cents per yard ? Ans. 387 dolls. 42 cts. 7. What will the whitewashing of a room come to at 2 cents per yard, allowing it to be 30 feet 6 inches long, 24 feet 9 inches broad, and 10 feet high ; no deductions being made for vacuities? Ans. 4 dolls. 13 cts. SECTION 8. OF PAVERS' WORK. PAVERS WORK is measured by the square yard, consist- ing of 9 square feet. The superficies is found by multiply- ing the length by the I readth. 178 FA1JNTKRS' WORK. EXAMPLES. 1. What cost the paving of a street 225 feet 6 inches long. and 60 feet 6 inches wide, at 30 cents per square yard ? By decimals. By duodecimals. Ft. in. ^ in. 225 6=225,5 feet 60 6= 60,5 do. 13500 11275 US 13530 9)13642 9 1515 7 30 -7-9 ) 13642,75 superficial ft. 1515,86 yards. 30 26 =the price of 7 ft. 9 in. Ans. 454,7580 ~454jeT Ans. 454 dolls. 76 cts. 2. What will the paving of a foot-path come to at 28 cents per yard, the length being 35 feet 4 inches, and the breadth 8 feet 3 inches ? Ans. 9 dolls. 33 cts. 3. What cost the paving of a court-yard at 38 cents per yard, the length being 27 feet 10 inches, and the breadth 14 feet 9 inches ? Ans. ?7 dolls. 33 cts. 4. What will be the expense of paving a rectangular yard, whose length is 63 feet, and breadth 45 feet, in which there is laid a foot-path 5 feet 3 inches broad, running the whole length, with broad stones, at 36 cents a yard ; the rest being paved with pebbles, at 30 cents a yard ? Ans. 96 dolls. 70 cts. SECTION 9. OF PAINTERS' WORK. PAINTERS' WORK is computed in square yards of 9 feet Kvery part is measured where the color lies ; and the mea suring line is pressed close into all the mouldings, corners, &ic. over which it passes. PAINTERS' WORK. 179 Windows, casements, &c. are estimated at so much a piece ; and it is usual to allow double measure for carved mouldings, &c. The value of painting is rated by the number of coats ; or whether once, twice, or thrice colored over, and the dif- ferent qualities and costliness of the colors. EXAMPLES. 1, How many yards of painting will a room contain which (being girt over the mouldings) is 16 feet 6 inches, and the compass of the room 97 feet 6 inches ? Operation. By decimals. By duodecimals. Ft. in. 97 6 97 6=97,5 16 6 16 6=16,5 4875 5850 975 -r 9 ) 1608,75 feet -r-9 ) 1608 Yards 178,6,75 178,6,9 Ans. 178J yards. 2. A gentleman had a room painted at 8J cents per yard, the measure whereof is as follows, viz. the height 11 feet 7 inches, the compass 74 feet 10 inches, the door 7 feet 6 inches by 3 feet 9 inches ; five window shutters, each 6 feet 8 inches by 3 feet 4 inches; the breaks in the windows 14 inches deep, and 8 feet high ; the opening for the chimney 6 feel - inches by 5 feet, to be deducted, the shutters and doors are painted on both sides ; what will the whole come to ? Ans. 10 dolls. 43 cts. 3. How many yards of painting are there in a room, the length whereof is 20 feet, its breadth 14 feet 6 inches, and height 10 feet 4 inches; deducting a fire-place of 4 feet by 4 feet 4 inches, and two windows, - each 6 feet by 3 feet 2 inches? Arts. 73^ T yards. 180 GLAZIERS' WORK. 4. What cost the painting of a room at 6 cents per yard , its length being 24 feet 6 inches, its breadth 16 feet 3 inches, and height 12 feet 9 inches ; also the door is 7 feet by 3 feet 6 inches, and the window shutters of two windows, each 7 feet 9 inches by 3 feet *6 inches, but the breaks of the windows themselves, are 8 feet 6 inches high, and 1 foot 3 inches deep ; deducting a fire-place cf 5 feet by 5 feet 6 inches. A?is. 7 dolls. 66 cts. 9 m. SECTION 10. OF GLAZIERS' WORK. GLAZIEKS compute their work in square feet ; and the di- mensions are taken either in feet, inches, and seconds, &c. or in feet, tenths, hundredths. Windows are sometimes measured by taking the dimen sions of one pane, and multiplying its .superficies by the number of panes. But more generally they measure the length and breadth of the window over all the panes, and their frames for the length and breadth of the glazing. And oftentimes the work is estimated at so much per pane ac- cording to the size. Circular, or ovtil windows, as fan-lights, &c. are measured as if they were square, taking for their dimensions the greatest length and breadth, as a compensation for the waste of glass, and labor in cutting it to the proper forms. EXAMPLES. 1. I low many square feet are contained in a window, which is 4 feet 3 inches long, and 2 feet 9 inches broad? Hv decimals. By duodecimals. Ft. in." Ft. in. 4 = 4, 25 the length 4 3 2 9 -=2,75 the breadth 2 9 2125 8 6 2975 3 2 sf,0 11/H75 foot. 11 S 3 Ans. MEASUREMENT OF GROUJND 181 2. If a window be 7 feet 3 inches high, and 3 feet 5 inches broad, how many square feet of glazing are contained therein ? . An*. 24 feet 9 inches. 3. There is a house with three tiers of windows, 7 in a tier; the height of the first tier is 6 feet 11 inches, of the second, 5 feet 4 inches, and of the third, 4 feet 3 inches ; the breadth of each window is 3 feet 6 inches : what will the glazing come to at 14^ cents per foot? Ans. 58 dolls. 61 cts. 4. What will the glazing of a triangular sky-light come to at 10 cents per foot, the base being 12 feet 6 inches long, and the perpendicular height 16 feet 9 inches? Ans. 10 dolls. 465 cts. 5. What is the area of an elliptical fan-light of 14 feet inches in length, and 4 feet 9 inches in breadth ? Ans. 68 feet 10 inches. 6. There is a house with three tiers of windows, and & in each tier; the height of the first tier is 7 feet 10 inches, of the second, 6 feet 8 inches, of the third fret 4 inches and the common breadth 3 feet 11 inche? , what will the glazing come to at 14 cents per foot? Ans. &' <Kis. 87 j cts. SECTION 11 MEASUREMENT OF GKOUND. 1st. To find the content of a square piece of ground. RULE. MULTIPLY the base in perches, yards or feet, as the case may be, by the perpendicular, and the product will be the answer required. Note. 1. Any area, or content in perches, being divided by 160, will give the content in acres ; the remaining- perches, if more than 40, being divided by 40, will give the roods, or quarter acres, and the last remainder, if any, will be perches. 2. Ground is generally measured by chains, of two poles or rods in length ; the two pole chain measures 33 feet. Chains of 4 poles are sometimes used, and sometimes chains or poles of one rod in length only. Q 182 MEASUREMENT OF GROUND EXAMPLE. 1. In a square field, A, B, C, D, each side of which measures 40 rods, or poles, how many acres ? Operation. 40 DC 40 4,0) 160,0 10 Acres. 4 ) 40 A 40 B 10 acres. Ans. 2. In a square field, each side of which measures 35 two pole chains, how many acres? Ans. 30 acres 2 roods 20 perches. 8. A piece of square ground measures 16^ perches on each side ; what is the content in acres ? Ans. 1 acre 2 roods 32| perches. 2d. Tojind the content of an oblong square piece of ground, called a parallelogram. RULE. Multiply the length by the breadth, and the product will be the answer. EXAMPLE. 1. There is an oblong square piece of ground, A, B, C, D, the longest sides of which measure 64 perches, and the shortest sides, or ends, measure 40 ; how many acres does it contain ? Operation. D C 64= the length 40= the breadth 4,0 ) 256,0 perches 4)64 16 Acres. 64 16 acres. An*. MEASUREMENT oV GROUND. 183 2. In a piece of ground lying in the form of an oblong square, the length measures 120 perches, and the breadth 84; what is its content in acres? Ans. 63 acres. 3. A lot of ground lying in the form of an oblong square, measures 240 feet in length, and 120 in breadth; what is its content in acres '.' Ans. acres 2 quarters 25 perches 21 3J feet. 4. There is an oblong piece of ground, whose length is 14 two pole chains 25 links, and breadth 8 chains 37 links ; how many acres does it contain ? Ch. L. Perches. 8 37 = 17,48 breadth 14 25= 29 length 15732 3496 4,0)50^,92 4)12^26" 3 26,92 Ans. 3 acres quarters 27 perches nearly. Note. The English statute perch is 5 yards, the two pole chain is 11 yards, or 33 feet, and is divided into 50 links; the four pole chain is 22 yards, or 66 feet, and contains 100 links ; hence the length of a link in a statute chain is 7,92 inches, and 25 links make 1 rod. And consequently, if the links be multiplied by 4, carrying 1 to the chains for every 25 links, and the chains by 2, the product will be perches, and decimals of a perch. 5. An oblong piece of ground measures 17 two pole chains and 21 links in length, and 15 chains 38 links in breadth ; how many acres are contained therein ? Ch. L. Ch. L. 17 21 15 38 24 24 34 84 perches, the length. 31 52 p. the breadth. Then 34,84x31,52=1098,1568 perches=6 acres 3 qr. 18,1 5 -f- perches. 3d. To find the content of a triangular piece of ground. RULE. Multiply the base by half the perpendicular, or the per- pendicular by half the base, or take half the product of the base into the perpendicular. 184 MEASUREMENT OF GROUND B D C 1. Let A, B, C, be a triangular piece of ground, the ongest sio or base B, C, is 24 chains 38 links, and perpen- dicular, A D, 13 chains 29 links; how many acres does it contain ? Operation. Ch. L. 24 38=49,52 perches 13 28=27,12 9904 4952 34664 9904 1342,9824 Half the sum is 4,0)67,1,4912 perches 4 ) 16,31 4 31,4 Ans. 4 acres roods 31,4 perches. 2. In a triangular piece of ground, the base or longest side measures 75 perches, and the perpendicular 50 ; how many acres does it contain ? Ans. 11 acres 2 qrs. 35 perches. 3. How much will a triangular piece of ground come to at 45 dollars per acre, the longest side or base of which measures 120 perches, and the perpendicular 84 perches. Ans. 1417 dolls. 50 cents. 4. How many superficial yards are contained in a trian- gular piece of ground, fho bavc of vhirh measures 140 feet and the perpendicular 70 feet'! Ans 544 vards 4 feet. MEASUREMENT OF GROUND. 185 4ith. To find the content of a piece of ground, in the form of an oblique parallelogram. RULE. Multiply the base into the perpendicular height for the content. EXAMPLE. D C A E B . Let A, B, C, D, be a piece of ground in the form of t~n oblique parallelogram, the base of which, A, B, measures 44 perches, and the perpendicular, D, E, 40 perches ; how many acres does it contain 1 44 length -40 breadth 4,0)176,0 perches 4) 44 11 acres. Ans. 11 acres. 2. A piece of ground lying in the form of an oblique par- allelogram, is found to measure 80 perches along its base, and its perpendicular height 24 perches; how many acres does it contain? Ans. 12 acres. 5th. To find the content of a piece of ground bounded by four sides, none of which are parallel or equal. RULE. Find the length of a diagonal line between the two most distant corners, and multiply this line by the - sum of the two perpendiculars falling from the other corners to that di- agonal line, and half the product will be the area. Q2 186 MEASUREMENT OF GROUND. 1. Let A, B, C, D, be a field with four irregular and un- equal sides, the diagonal line of which, A, C, measures 80 perches, the perpendicular, B, m, measures 25 perches, and the other perpendicular, D, n, 35 perches; how many acres does it contain ? 80 the length of the diagonal line. 25 -f 35=60 the sum of the two perpendiculars. 2 ) 4800 4,0)240,0 perches 4) 60 15 acres. An*. 2. In a field of four unequal sides, the diagonal line be- tween the two most distant corners measures 120 rods, and the perpendiculars measure, the one 48, and the other 24 rods ; required the number of acres it contains? Ans. 27 acres. 6/7i. To find the area of a pitce of ground lying in a cir- cle y or ellipsis. RULE. Multiply the square of the circle's diameter, or the pro- duet of the longest and shortest diameters of the ellipsis, by the derimal number ,7854, the product, will give the area. Note. In any circle, the Diameter multiplied Circumference flivid'-il MEASUREMKiVr OF GROUND. 187 EXAMPLE. 1. How many acres are contained in a circular piece of ground, whose diameter measures 320 perches, or 1 mile ? 320X320=102400 ,7854 409600 512000 819200 716800 4,0 ) 80424,9600 perches 4)2010,24,9 502 2 24,9 Ans. 502 acres 2 qr. 24,9 perches. 2. A gentleman has an elliptical -yard in front of his house, the longest diameter of which measures 30 perches, and the shortest 20 ; how much ground is contained therein ? Ans. 2 acres 3 qr. 31,2 perches. 3. How many square yards are contained in a circular piece of ground, the diameter of which measures 160 feet? Ans. 22 34 + yards. From the foregoing simple methods of finding the con- tents of ground lying in different forms, it will readily be seen, that the content of fields and small pieces ' of land, lying in any shape whatever, and bounded by any number of sides, may be calculated, without having recourse to the more expensive and troublesome practice of employing a regular surveyor. No other apparatus than a common rod- pole, or line of a known length, is requisite. Pieces of land having more than four boundary lines, may be easily divided into squares, parallelograms, triangles, (fee. and each calcu iated separately by some of the foregoing rules, and then the whole amount added into one sum for the content. It is of great importance to every practical farmer to know the size of the different fields which he cultivates. Besides the satis- faction thereof, this knowledge is necessary to enable him to regulate the quantity of seed which he should sow, as well the price for clearing^ plowing, planting, reaping, &c. 188 GAUGING. SECTION 12. OF GAUGING. GAUGING is taking the dimensions of a cask in inches, to find its content in gallons. RULE. 1. Find the mean diameter, between the head and bung diameters, by adding two-thirds of the difference between them to the head diameter. If the staves be but little curv- ing from the head to the bung, add only six-tenths of this difference. 2. Square the mean diameter, so found, and multiply the product by the length of the cask in inches, for the content thereof in cubic inches. 3. Divide the cubic inches, so found, by 294, for wine or spirits, and by 359 for ale ; the quot'ent will be the answer in gallons. EXAMPLE. 1. How many gallons of wine will a cask contain whose bung diameter is 31 inches, head diameter 25 inches, and whose length is 3 feet, or 36 inches 1 Operation. 31 bung diam. 25 head diameter 25 head diam. f of 6= 4 two-thirds difference 6 difference. 29 29 261 58 841 square of the mean diam. 36 the length 5046 2523 30276 cubic inches. Then 80276 -f- 294 =102ffJ gals. Or, 102 gals. 3 qt. GAUGING. 189 2. The diameter of a barrel at the bung measures 24 inches, and at the head 18 inches, and its length is 24 inches ; what is its content in wine measure 1 Ans. 39f f gals. 3. How many gallons of spirits will a cask contain, whose bung diameter is 36 inches, head diameter 28 inches, and whose length is 3 feet 4 inches ? Ant. ISlfN, gals. 4. What is the content, in ale measure, of a barrel whose bung diameter measures 18 inches, head diameter 15 inches, and whose length is 2 feet 5 inches ? Ans. 23i||. gals. 5. Bought a barrel of ale of the following dimensions, viz. bung diameter 22 inches, head diameter 18 inches, and length 3 feet ; how many gallons does it contain ? Ans. 42|f| gals. Of the Gauging or Diagonal Rod. The diagonal rod is a square rule, having four faces, be- ing commonly four feet long, and folding together by joints. This instrument is used for gauging, or measuring casks, and computing their contents, and that from one dimension only, namely, the diagonal of the cask ; that is, from the middle of the bung- hole, to the meeting of the head of the cask, with the stave opposite to the bung ; being the longest line that can be drawn within the cask from the middle of the bung-hole. Am! accordingly one face of the rule is a scale of inches, for measuring this diagonal, to which are placed the areas in ale gallons, of circles to the correspond- ing diameters, in like manner as the lines on the under sides of the three slides, in the sliding rule. On the opposite face are two scales of ale and wine gallons, expressing the con- tents of casks having the corresponding diagonals. And these are the lines which chiefly form the difference between this instrument and the sliding rule. EXAMPLE. The rod being applied within the cask at the bung-hole. 190 MECHANICAL POWERS. the diagonal was found to be 34,4 inches ; required the con- tent in gallons. Now, to 34,4 inches, will be found corresponding on the rod, 90 1 ale gallons, and 111 wine gallons, the content re- quired. Not e. In taking the length of a cask to find the cubic inches, an allowance must be made for the thickness of both the heads of 1 inch, of 1^ inch, or 2 inches, according to the size of the cask ; and the head diameter must always be taken close to the chime. The contents ex- hibited by the rod, answer only to casks of the common form. SECTION 13. OF MECHANICAL POWERS. 1st. OF THE LEVER. To find what weight may be raised or balanced by any given power. RULE. As the distance between the body to be raised, and ful- crum, or prop, Is to the distance between the prop, and the point where the power is applied, So is the power to the weight which it will raise. EXAMPLE. 1. If a man weighing 150 Ib. rest on the end of a levei 12 feet long ; what weight will he bafemce on the other end, supposing the prop 1 foot from the weight ? Operation. 12= the length of the lever l,5=distance of the weight from the prop 10,5= the distance from the prop to the man. Then, as 1,5 : 10,5 : : 150 : 1050. Ans. 2. The pea of a pair of steelyards weighing 5 Ib. is re- moved 20 inches back from the fulcrum ; what weight will MECHANICAL TOWERS. 1^1 it balance, suspended at I irvh distance on the opposite si le? Arts 100 lo. *2d. OF THE WHEEL AND AXLE. To find what power must be applied at the wheel, to r< \ise a given weight suspended to the axle ; or what weight ai the axle will be raised by a given power at the wheel. RULE. As the diameter of the axle : is to the diameter of th< wheel : : so is the power applied to the wheel : to the weight suspended to the axle. EXAMPLE. 1. It is required to make a windlass in such a manner, that 1 Ib. applied to the wheel, shall be equal to 12 lb. sus- pended to the axle; now allowing the axle to be 4 inches diameter, what must be the diameter of the wheel ? lb. in. lb. in. As 1 : 4 : : 12 : 48 = 4 feet the diameter of the wheel. Ans. 2. Suppose the diameter of an axle to be 6 inches, and that of the wheel 5 feet; what power at the wheel will bal- ance 10 lb. at the axle? Ans. 1 lb. 3d. OF THE SCREW. In the screw there are four things to be considered : viz. the power, the weight, the distance between the threads, and the circumference. To find any one of these, the other three being given, observe the following proportional RULE. As the distance between the threads of the screw : Is to the circumference : : So is the power : To the weight. Note. 1. To find the circumference of the circle described by the end oi the lever ; multiply the double of the lever by 3,14150, and the product will be the circumference. 2 It is usual to abate of the effect of the machine for the friction 192 PROMISCUOUS QUESTIONS. EXAMPLE. There is a screw whose threads are an inch asunder ; the lever by which it is turned is 36 inches long, and the weight to be raised a ton, or 2240 Ib. ; what power or force must be applied to the end of the lever sufficient to turn the screw that is to raise this weight ? Thus, the lever 36 X 2=72,and72 X 3,14159^226,194 + the circumference. Circum. in. Ib. Ib. Then, as 226,194 : 1 :: 2240 : 9,903 the power. Ans. A COLLECTION OF PROMISCUOUS QUESTIONS, TO EXERCISE THE SCHOLAR ON THE FOREGOING RULES. 1. What is the sum of 2578, added to itself? Ans. 5156. 2. What is the difference between 14676, and the fourth of itself! Ans. 11007. 3. There is the sum of 1468 dollars in three bags; the first contains 461, the second 581, how many are in the third bag? Ans. 426. 4. What is the sum of the third and half third of 1 dol- lar ? Ans. 50 cts. 5. What number is that which being multiplied by 45 the product will be 1080? Ans. 24. 6. Required the quotient of the square of 476, divided by the half of itself, or its single power? Ans. 952. 7. A general drawing up his army into a solid square, found he had 231 over and above, but increasing each side with one soldier, he wanted 44 to complete the square ; how many men did his army consist of? Ans. 19000. 8. What number added to the cube of 21, will make tin- sum equal to 113 times 147 ? Ans. 7350. 9. A person possessed of $ of a ship, sold f of his share for 1260 dollars; what was the value of the whole ship at Th" sani" rate? Ans. 5040 dolls. 10. A jruJirdinn paid his ward 3500 dollars for 2500 dol- lars, uhieh he had in his hands for 8 years; what rate of fiU'rest did lie allow him 7 .!//*. 5 per cent. PROMISCUOUS QUESTIONS. 193 11. A young man received 210 dollars, which was f of his elder brother's portion ; now three times the elder bro- ther's portion was half of the father's estate ; how much was the estate worth? Ans. 1890 dolls. 12. A broker bought for his principal in the year 1720, the sum of 400 dollars capital stock, in the south sea, at 650 per cent, and sold it again when it was worth but 130 dollars per cent. ; how much was lost upon the whole ? Ans. 2080 dolls. 13. A gentleman went to sea at 17 years of age; 8 years after he had a son born, who lived 40 years, and died before his fether ; after whom the father lived twice 20 years, and then died also ; I demand the age of the father when he died? Ans. Ill years. 14. A, B, and C, entered into partnership in trade, A put in a sum unknown, B put in 20 pieces of cloth, and C put in 500 dollars ; at the end of one year they had gained 1000 dollars, whereof A received 350 dollars for his share, and B 400 dollars ; required C's share, how much A put in, and the value of B's cloth? Ans. C's share 250 dollars, A put in 700 dollars, B's cloth was worth 800 dollars. 15. A captain and 160 sailors took a prize worth 2720 dollars, of which the captain gets J for his share, and the rest is equally divided among the sailors ; what was each one's part ? Ans. The captain gets 544 dollars, and each sailor 13 dollars 60 cents. 16. A lady tells her husband, upon her marriage, that her fortune, the interest of which for one year at 6 per cent, was 972 dollars, was but the f of the interest of her father's estate for three years, at the same rate per cent. ; what was the lady's fortune, and what was the value of her father's es- tate?" Ans. Her fortune was 16,200 dollars, and her father' estate was 150,000 dollars. 17. A stone measures 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches deep ; how many cubic feet does it contain ? Ans. 41 feet 3 inches. 18. Suppose I of a' mast or pole stands in the ground, 12 feet in tne water, and f of its length above the water; what is its whole length? Ans. 216 feet. R 194 PROMISCUOUS QUESTIONS. 19. A gentleman being asked his age, answered, my grandfather is 112 years old, and my father 4 o f his age, whilst mine is but ^ of my father's ; what was his age ? Ans. 21 J years. 20. A person who was possessed of f share of a copper mine, sold 3 of his interest therein for 1710 dollars; what was the value of the property at the same rate ? Ans. 3800 dollars. 21. There are two numbers, the one 63, the other half as much ; required the product of their squares, and the differ- ence of their product and sum 1 ' A $ Product of the squares 3938240,25. 5 * J Difference 1890. 22. Two men set out at the same time from the same place, but go contrary ways, and each of them travels 34 miles a day ; required the time in which they will have travelled 2000 miles ? Ans. 29 days 9 hours 52ff mi. 23. If a cannon may be discharged twice with 6 Ib. of powder, how many times will 7Cwt. 3qr. I7lb. discharge the same piece ? Ans. 295 times. 24. What number is that, to which if you add f of itself, the sum will be 20? Ans. 12. 25. What number is that, which being divided by j}, the quotient will be 21 ? A?is. 15J. 26. What number is that, which being multiplied by 15, the product will be \ 1 Ans. -^\ 27. W T hat number is that, from which if you take f , the remainder will be J? Ans. ff 28. A gentleman wishing to distribute some money among a number of children, found he wanted 8 cents to give them :J cents a piece, he therefore gave each 2 cents, and had three cents left; how many children were there? Ans. 11. 29. In what time will 500 dollars amount to 1000, at 6 per cent, per annum? Ans. 16 years 8 months. 30. When \ of the members of congress were assembled -t 15, there were J-flO absent; how many members were in all? Ans. 150. 31. If the earth be 360 degrees round, each 69 miles, how long would it take a man to travel orce round, at 20 miles a day, admitting there were no obstacles in the way, and reckoning 365 i days in the year. Ans. 3 years 155J days. PROMISCUOUS QUESTIONS. 195 32. -What is the mean time for paying 100 dollars a; 3| months, 150 dollars at 4^ months, and 204 dollars at 5if months ? Ans. 4 months 23Jf A days. 33. If A can do a piece of work alone in 7 days, and B do the same in 12, how long will it require them both to- gether ? Ans. 4 T V days. 34. A minor of 14 years of age, had an annuity left him of 400 dollars ; this sum his guardian agreed to receive yearly, and allow him compound interest at 5 per cent, thereon, till he should arrive at 21 years of age ; how much must he then receive ? Ans. 3256 dolls. 80 -f cents. 35. Sold goods to the amount of 700 dollars for four months ; what was the present worth, at 5 per cent, simple interest? Ans. 688 dolls. 52 -f cents. 36. Three persons, A, B, and C, purchased a lot in part- nership, for which A advanced f , B -f , and C 140 dollars ; what sum did A and B pay, and what part of the lot be- longed to C ? C A paid 267 dolls. 27 + cts. Ans. < B paid 305 45J ( and C had J-J parts. 37. A gentleman finding several beggars at his door, gave to each four cents, and had sixteen left ; but if he had given to each six cents, he would have wanted 12 ; how many beggars were there ? Ans. 14. 38. B and C can build a wall in 18 days, but with the assistance of A they can do it in 11 days; in what time can A do it alone 1 Suppose the work to consist of 198 parts. Then 198-^18=11 parts performed by B and C, in one day. Again, 198-^11 = 18, performed by A, B, and C, in one day. But 18 11=7 parts performed by A alone. P. D. P. D. h. m. And as 7 : 1 : : 198 : 28 3 25f Ans. 39. Twenty members of congress, 30 merchants, 24 law- yers, and 24 citizens, spent at a dinner 192 dollars ; which sum was divided among them in such a manner, that 4 members of congress paid as much as 5 merchants, 10 merchants as much as 16 lawyers, and 8 lawyers as much 196 PROMISCUOUS QUESTIONS. as 12 citizens; the question is to know the sum of. money paid by all the members of congress ; also, by the mer- chants, lawyers, and citizens ? Ans. The 20 members of congress paid 60 dollars the 30 merchants paid 72, the 24 lawyers paid 36 and the 24 citizens paid 24. 40. What difference is there between a piece of ground 28 perches long, by 20 broad, and two others each of half those dimensions ? Ans. 1 acre 3 qrs. 41. Required the dimensions of a parallelogram, con- taining 200 acres, which is 40 perches longer than wide ? Ans. 200 perches by 160, 42. How many acres are contained in a square field, the diagonal of which is 20 perches more than either of its sides? Ans. 14 acres 2 qrs. 11 per. 43. The paving of a triangular yard, at I8d. per foot, came to 100Z. ; the longest of the three sides was 88 feet ; what then was the sum of the other two equal sides 1 Ans. 106,85 feet. 44. Required the length of a line by which a circle that shail contain just half an acre may be laid off? Ans. 21% yards. 45. A ceiling contains 114 yards 6 feet of plastering, and the room is 28 feet broad ; what is its length ? Ans. 36f feet. 46. A common joist is 7 inches deep, and 2 thick, but I want another just as big again, that shall be three inches thick ; what must be its^ether dimensions ? Ans. llf inches. 47. If 20 feet of iron railing weigh half a ton, when the oars are an inch and a quarter square, what will 50 feet come to at 3 \d. per pound, the bars being but of an inch square ? Ans. 20Z. Os. 2d. 43. A may-pole whose top being broke off by a blast of \vind, struck the ground at 15 feet distance from the foot of the pole ; what was its whole height, supposing the length of the broken piece to be 39 feet ? Ans. 75 ft. 49. Required a number, from which if 7 be subtracted, arid the remainder be divided by 8, and the quotient be multiplied by 5, and 4 added to the product, the square root of the sum extracted, and three-fourths of that root cubed, the cube di- vided by 9, the last quotient will be 24 ? Ans. 103. , PROMISCUOUS QUESTIONS. 197 50. A vintner has a cask of wine containing 500 galls, of which he draws 50 galls, and fills it up with water, and re- peats the same thing five times ; I demand what quantity of wine, and also of water, is then in the cask ? Ans. 295 galls. 1 qt. of wine, and 204 galls. 3 qts. of water nearly. 51. Since a pile of wood 4 feet long, 4 feet high, and 8 feet broad, makes a cord, what part of a cord will be in a pile of half the dimensions each way ? Ans. % part. The answers to the following questions are designedly omitted, that the scholar may be induced to apply to the resources of his own mind alone for the solution thereof. Without habits of reflection and inves- tigation are acquired, by which he can compare, examine and apply the various rules and directions that are contained in this treatise, he never can have any good claim to be considered a proficient in arithmetic 52. A owed B 1864 dollars, for which he gave his note, on interest, bearing date April 1st, 1817. On the back of the note are the following endorsements*, viz. Oct. 15th, 1817. Received in cash 225 dolls. 50 cts. Jaiv 10th, 1818. Received in cash 150 Same date, one bag of coffee ; weight 1 Cwt. 22lb. at 29 cents per pound. May 16th. Received 3 ton of iron at 195 dolls, per ton. What is the sum due from A to B, on the 1st of August 1818? 53. How many cords are there in a pile of wood 36 feet long, 6| feet wide, and 8J feet high? 54. If a man spends 356 dollars 34 cgnts per year, how much will it be per day ? 55. A bankrupt, whose whole property is worth 2564 dollars 95 1 cents, can pay his creditors but 18 J cents on a dollar ; how much does he owe ? 56. If 8 men spend 20 dollars 50 cents in 30 days, how long will 64 men be in spending 100 dollars at the same rate ? 57. A bridge built over a stream in 6 months by 34 men, being washed away by a flood, how long time will it take 86 men to build another in its place, of twice as much work ? 58. Three gardeners, A, B, and C, having bought a piece of ground, find the profits of it to amount to 240 dollars a year ; now the sum of money which they gave, was in such R 2 198 PROMISCUOUS QUESTIONS. proportion, that as often as A paid 5 dolls. B paid 7, and as often as B paid 4 dolls. C paid 6 ; how much must each man receive for his share of the profits per annum? 59. If a county tax of 7 cents and 3 mills per cent, is as- sessed on property, how much must that man pay, whose property is valued at 8564 dollars 20 cents ? 60. Suppose a cistern having a pipe which conveys 4 gal- lons 2 quarts into it in an hour, and has another that lets out 2 gallons 2 quarts and 1 pint in an hour ; in what time will it be filled, allowing it to contain 84^ gallons? 61. What is the length of a lane, which, being 36 fcul wide, will contain just one acre of ground ? 62. If 50 men consume 12 bushels of grain in 30 days, how much will 40 men consume in 90 days ? 63. A gentleman had 18 dollars 90 cents to pay among his laborers ; to every boy he gave 6 cents, to every woman S cents, and to every man 16 cents; now there were three women for every boy, and two men for every woman ; re- quired the number of each ? 64. Two men depart from the same place, and travel the same way ; the one travels at the rate of 3 miles an hour, for 8 hours every day ; the other goes at the rate of 4J miles, for 7 hours each day ; how far are they apart at the end of 13 days? 65. A began to trade on the 1st of January, with a capi- tal of 962 dollars ; on the 15th of April following, he took in B as a partner, with 1635 dollars ; on the 1st of July, A put in 320 dollars more, and 1 month after B drew out i of his capital ; on the last day of December, on settling their .accounts, they found a gain of 486 dollars 64 cents'; what was each partner's share ? 66. Suppose the Ohio river to be 2500 feet wide, 6 feet deep, and runs at the rate of 3 miles an hour ; in what time will it fill a cistern of two miles in length, breadth, and depth, the mile being 5280 feet? 67. A sloth was observed climbing a tree at the rate of 9^ inches every day, but during the night slipped down 6J inches ; how long will it be in reaching a limb 45 feet 6 inches from the ground! 68. In an orchard of fruit trees, of them bear applet, | peaches, cherries, plums, and 46 are pears; how many trees does the orchard contain ? 69. An old soldier lately received a sum of money as PROMISCUOUS QUESTIONS. 199 pension from government : of this sum he paid 94 dollars in the payment of debts which he then owed, half of what remained he lent to a friend, and the fifth he gave for a suit of clothes ; he then found that nine-tenths of his money was gone ; what sum did he at first receive ? 70. What number is that, of which the difference between its third and fourth parts is 84 ? 71. In turning a chaise within a circle of a certain diameter, it was discovered that the outer wheel turned thrice, while the inner turned twice; now supposing the axle-tree 4 feet long, and the wheels of an equal size, the length of the circumference described by each wheel is re- quired ? 72. The sum of the sides of an equilateral triangle is 125 feet ; required the area thereof'/ A, in a scuffle, seized on f of a parcel of sugar-plums; B catched three- eighths of it out of his hands, and C laid hold on three-tenths more, D ran off with all that A had left except one-seventh, which E afterwards secured slily for himself: then A and C jointly set upon B, who in the conflict let fall he had, which was equally picked up by D and E : B then kicked down C's hat, and to work they went anew for what it contained ; of which A got i, B A, D two-sevenths, and C and E equal shares of what was left of that stock: D then struck f of what A and B last acquired out of their hands ; they with difficulty recoved five-eighths of it in equal shares again, but the other three carried off one -eighth apiece of the same. Upon this they called a truce, and agreed, that the J- of the whole, left by A at first, should be equally divided among them How much of the prize, after this distribution, remained with each of the competitors ? Ans. A got 2863, B 6335, C 2438, D 10294, and E 4950. Solution. First, f of f = J B's ^ First acquisition And y\of |=i C's \ ^ their sum Then -| ^=H, or ^ left I of if J^- E's first acquisition. Also, T V -o = 'iTo D' s * Thus ended the 1st heat. Again, i of J = J B's "j Retained 1 C's ! Part, at the end of the And T Vo+TV=Ht I) ' s | second scuffle. Also -V3_. + T i_ = _i_5_L. E > s J Proceeding, i of 1 = ^ A's " f of !=*'! Then ^V + rV + A= T^V to be taken from C's. Thus, ^-^ and i of T u_ '=j&C 9 a And '>* Their situation at the end of the 3d attack. 200 QUESTIONS FOR EXAMINATIONS, A. B. Further, ^ + ^=J^ and f of &=& lost by A and B. Then, t V of ft + iof & ^^i^A's^ p f ' ' " Also, A of /, + i of And * of W + riV =T*tt* C's i of A+W* * Of T'O +TWo * Of * = T ' ? _.__l_l_ _ _S_4_3J o l~3*0 Tl 5 26880 ^ f n ff ot j n of r A got 2863 ^ So that if the } 6335 sugar-plums > then <j C 2438 I Arts. were 26880, ) B 10294 IE 4950 J QUESTIONS FOR EXAMINATION. THIS collection of questions is designed to assist the teacher in l.ie ex- amination of his scholars. It will contribute very much to the progress of scholars, to assign them a certain number of these questions as lessons, to be answered correctly and with facility. Many similar questions will no doubt, from time to time, occur to the mind of the teacher, on thf different sections, as the scholar proceeds. By accustoming his pupils to answer such with ease, not only will his, own burden in teaching be lei^ened, but the parents of children, who have been intrusted to his care, will find that neither their trouble or expense has been in vain. PART I. What is Arithmetic ? How many parts does it consist of? What are the characters used in arithmetic ? \Vhat is numeration ? How are the digits divided ? What is the rule for writing numbers? What is simple addition ? How do you place numbers to be added ? How is the sum or amount of each column to be set down ? Why do you carry at 10, rather than for any other number? How is addition proved? What is simple subtrac- tion? How must the given numbers be placed ? How is subtraction performed ? How is subtraction proved ? What is simple multiplication? What are the numbers called? In what order are the numbers in multiplication to be placed ? IIoxv many cases are there in multiplication? How is the operation to be performed in the first and second cases? How is multiplication proved ? When there are ciphers on the right-hand of either of the factors, how do you proceed ? What is simple division ? Wnat are the given numbers called ? How are they to be placed ? QUESTIONS FOR EXAMINATION. 201 How many cases are there in division? How is division performed in each case ? When the multiplier is the exact product of any two factors, how do you proceed ? When there are ciphers on the right of the divisor, how do you proceed ? When the divisor is 10, 100, 1000, &c. how do you proceed? How is division proved ? PART II. Federal money, why so called ? What are its denominations, and standard weights ? How is addition, subtraction, multiplication, and division severally per- formed in federal money ? What is compound addition? How is compound addition performed \ How is it proved ? What are the denominations of English money, and how are they valued ? \Vhat articles is troy weight used for? What are its denominations and how valued ? What articles is avoirdupois weight used for? What are its denominations, and how valued ? What is apothecaries' weight used for? What are its denominations and how valued? What are the denominations of cloth measure, and how valued ? What are the denominations of long measure, and how estimated ? What are the denominations of land measure, and how rated ? What is cubic measure, what are its denominations, and relative difference ? What are the denominations of time, and what their relative differences ? What is the exact length of the solar year? What are the denominations of motion, and the relative difference? For what is liquid measure used, what its denominations, and relative di ference ? What are the denominations of dry measure, w 7 hat used for and how estimated ? What is compound subtraction, and how performed ? What is compound multiplication, how many cases, and how performed? What is compound division, how many cases, and how performed ? What is reduction, and how performed ? How is reduction proved ? How are pence reduced to cents, Penn. currency ? How are pounds, shil- lings, and pence, reduced to dollars, Pennsylvania currency? PART III. What is decimal arithmetic, and how distinguished from whole numbers? What is the decimal point called? What effect has ciphers, placed on the right-hand of the integer, and what effect when placed on the left-hand ? llow is addition of decimals per- formed ? How is subtraction of decimals performed ? How is multiplication of decimals performed ? How is division of decimals * performed ? How many cases in reduction of decimals? How is a vulgar fraction reduced to a decimal ? How are numbers of different denominations reduced to a decimal of equal value? How are decimals reduced to their equal value in in- tegers ? PART IV. What is proportion ? Into how many parts is it divided ? What are the given terms in proportion called ? Wha f is required in the single rule of three direct ? How may you know when the question is in direct proportion? What is the rule for stating questions in the single rule of three direct? How is the operation performed ? How do you prove questions in the single rule of three direct ? What is the single rule of three inverse ? How may you know when the question is in the single rule of three inverse ? How is the operation performed in the single rule of three inverse ? How are questions proved in this rule ? What is the double rule of three ? How many, and which terms must be a supposition, and how many, and which must be a demand ? What is the rule for stating questions in the double rule of three ? [low is the operation performed in the double rule of three direct? 202 QUESTIONS FOR EXAMINATION. How do you know when the question is in direct proportion, and when in inverse? How is the operation performed in inverse proportion ? PART V. What is practice, and why so called ? How many cases are there in practice ? When the price consists of dollars, cents, ana mills, how is the operation performed ? When the price is the fractional part of a dollar or cent, how is the operation performed ? When the price and quantity given are of several denominations, how is the operation performed ? When the price consists of pounds, shillings, pence, and farthings, how do you proceed ? What is meant by aliquot parts? When both the price of the integer and the quantity are of different de- nominations, how do you proceed ? What is tare and tret, and what is gross and neat? How do you work questions in tare and tret? What is interest? What is the general rate of interest? What is the sum of money loaned, called? What do you understand by the amount? How many kinds of interest are there ? What is simple interest ? How many cases are there in simple interest? When the given time is years and the principal dollars, how is the interest found ? When there are cents and mills in the principal, how do you proceed ? When the lime is years and months, or months *only, how is the interest found ? When the time is months and days, or days only, how is the ope- ration performed? How is the interest computed on bonds, notes, &C;? What is compound interest, and how is it performed ? What is insurance ? What is the instrument of agreement termed ? How are the questions in insurance performed ? What is commission, and how performed ? What is brokage, and how per- formed ? What is stock, and how bought and sold ? What is rebate or discount, and what the rule to work questions therein ? What is the difference between discount and interest? What is bank discount, and how is the discount calculated ? What do you mean by the equation of payments? How is the mean time found in the equation of payments ? What is fellowship, and how many kinds are there ? What is single fellowship, and how is the operation performed ? What is compound fellowship, and what is the rule for working questions therein ? What is profit and loss, and what is the rule of operation therein ? What is barter, and how performed ? What is exchange, ami of how many kinds ? What do you understand by par in exchange, and what by agio? How do you reduce the currency of different states to federal money? How do you reduce the currency of one state to another, where it is differ- ent in them? How are accounts kept in England, Ireland, and how in France, Spain, &c. ? What is alligation, and how many cases are therein ? How are the operations performed in the first case, second case, &c. ? PART VI. What is a vulgar fraction, and how many kinds are there ? What is a proper fraction, what is an improper fraction, what is a com- pound fraction, and what is a mixed fraction? What are the numbers above the fine called, and also those below? How do you reduce vulgar fractions to their lowest terms ? EIow are mixed numbers reduced to an improper fraction ? How is an improper fraction reduced to a whole or mixed number? How do you reduce fractions to others that shall have a common denomina- tor? How do you find the least common denominator? How do you find the value of a fraction, in the known parts of an integer? How are given quantities reduced to the fraction of a greater denomination ? How are vulgar fractions reduced to decimals of the same value ? How do you reduce a compound fraction to a single one? How are vulgar fractions added, subtracted, multiplied, and divided? QUESTIONS FOR EXAMINATION. 203 How do you perform the single rule of three in vulgar fractions, direct, and inverse ? PART VII. What is involution ? What do you understand by the power of a number ? ilow is involution performed ? What is the number denoting the power, termed ? What is evolution ? What do you understand by a root ? How do you extract the square root ? If there be decimals in the given number, how must it be pointed? How do you extract the square root of a vulgar fraction? How is the square root of a mixed number extracted ? Hew do you find the side of a right angled triangle, the other two being given f How do you find the side of a square, in any given area ? How do you find the diameter of a circle, when the area is given ? How do you prove the square root? What is a cube ? How do you extract the cube root in whole numbers? Ilow do you extract the cube of a vulgar fraction ? How do you point off in decimal numbers? How is the cube root of a mixed number extracted ? How is the cube root proved ? What is progression, and how many kinds ? What is principally to be ob- served in arithmetical progression ? How 7 do you find the last term ; and sum of all the terms ? How do you find the common difference ? vVhat is geometrical progression, and how does it differ from arithmetical ? What is principally to be observed in geometrical progression ? How do you find the last term, and sum of all the series ? What is position, and of how many kinds ? How do you resolve questions in singlo position? How is single position proved ? What is double po- sition ? What is the rule for working questions in double position ? What do you understand by permutation ? How is the number of varia- tions found in this rule? What do we learn from the results of this rule ? What is combination? How do you find the greatest possible number of combinations in any given number? PART Vlli. VVhat are duodecimals? What are the denominations in duodecimals, and what are they termed? How are duodecimals added, subtracted, and multiplied ? How do you prove multiplication of duodecimals ? How is the solid content of bales, &c. found by duodecimals? How do you find a ship's tonnage ? What is the carpenters' rule, and what its use ? How do you find the superficial content of boards, &c. ? How do you find the solid content of squared timber? How do you find the solid content of round timber? What things belong to carpenters' work? By what numbers do carpenters usually measure their work ? How is brick- work estimated ? What is the standard thickness of a brick wall? How do you reduce a wall of a different thickness to a standard one ? How is masons' work measured ? What kind of measure is used for ma- terials? How is the solid content of walls calculated ? What is superficial measure ? How is plasterers' work divided ? In what manner is plasterers' work measured ? How is white-washing and coloring estimated ? How is pavers' work calculated ? How do painters compute their work ? In what manner is glaziers' work estimated ? How are the contents of squares calculated? In what way is the area of an oblong piece of ground ascertained ? How do you calculate triangular pieces of ground ? How do you calculate a piece of ground lying in the shape of an obiiquo parallelogram? How are pieces of ground, oounded by four irregular sides, calculated,? How do you calculate the area of a circle? What is gauging? How are the contents of casks, &c. calculated? How do you calculate the power of the lever? of the axle and wheel ? VVhat things are to be considered in finding the power of the screw? How do you find the proportion between the weight and the power? CONTENTS. PART I. P^?e Numeration 7 Addition 9 Subtraction 11 Multiplication 13 Division 17 PART II. Federal Money 21 Compound Addition 24 Compound Subtraction 30 Compound Multiplication .... 32 Compound Division 35 Reduction 39 PART III. Addition of Decimals 46 Subtraction of Decimals 47 Multiplication of Decimals ... 48 Division of Decimals 50 Reduction of Decimals 52 PART IV. Single Rule of Three Direct . . 56 Single Rule of Three Inverse . 59 Double Rule of Three 65 PART V. Practice 68 Tare and Tret 73 Simple Interest 75 Compound Interest 85 Insurance, Commission, and Brokage 87 Buying and Selling Stocks . . 89 Rebate or Discount 90 Bank Discount 92 Equation of Payments 93 Single Fellowship 95 Compound Fellowship 97 Profit and Loss 99 Barter 101 Exchange 102 Alligation 109 PART VI. },-.?< Vulgar Fractions 113 Reduction of Vulgar Frac- tions '. ib. Addition of Vulgar Fractions 120 Subtraction of Vulgar Fr<ic- tions ..; . 122 Multiplication of Vulgar Frac- tions 123 Division of Vulgar Frac- tions 124 The Rule of Three in Vulgar Fractions 125 Inverse Proportion 12G PART VII. Involution 127 Evolution 128 The Square Root ib. The Cube Root 137 Arithmetical Progression .... 141 Geometrical Progression .... 144 Single Position 147 Double Position 148 Permutation 151 Combination 152 PART VIII. Duodecimals 153 The Carpenters' Rule 159 Measuring of Boards and Timber 162 Carpenters' and Joiners' Work 166 Bricklayers' Work 170 Masons' Work 174 Plasterers' Work 176 Pavers' Work 177 Painters' Work 178 Glaziers' Work 180 Measurement of Ground 181 Gauging 188 Mechanical Powers 190 Promiscuous Questions 192 Questions for Examination . . 200 14 DAY USE RETURN TO DESK FROM WHICH BORROWED LOAN DEPT. This book is due on the last date stamped below, or on the date to which renewed. Renewed. books are subject tn imnnpHmtp recall. ICLf JAN 2 01968 6 5 -b$ TH LD 21A-60ra-3,'65 (F2336slO)476B General University of v Berkeley YA 02433 >J i // z. vQf+4- frr<S?t~ - iT[ n