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UC-NRU- IN MEMORIAM FLORIAN CAJOR1 THE JUVENILE ARITHMETICK AND SCHOLARS' GUIDE, [LLUSTRATED WITH FAMILIAR QUESTIONS. AND CONTAINING NUMEROUS EXAMPLES IN FEDERAL, MONEY* TO WHICH IB ADDED, A 8FHORT 8TUTEM OP BOOK KEEPING: BY MARTIN RTTTER, President of Augusta College. REVISED AND ENLARGED By JVathan Guilford. CINCINNATI: FTBLI8HED AND SOLD BY N. 4& G. GTTILFORD. efcereotyped at the Cincinnati gtereotype foundry: 1831. DISTRICT OF OHIO, TO WIT : BE IT REMEMBERED, That on the twenty-second day of Vpril, in the year of our Lord one thousand eight hundred and tweii- y-seven, and in the fifty-first year of the American Independence, VIARTIN RUTER, of said District, hath deposited in said office the itle of a book, the right whereof he claims as author and proprietor n the words following, to wit: THE JUVENILE ARITHMETICK AND SCHOLAR'S GUIDE: wherein theory and practice are combined and adapted to the capacities of young beginners; containing a due proportion of examples in Federal Monty, and the whole being- illustrated by nu- merous questions similar to those of PESTALOZZI, by J\lAR- TLVRUTER.A.M;" In conformity to the Act of the Congress of the United State?, en Jtled " An Act for the encouragement of Learning, bv securing the copies of Maps, Charts, and Books, to the Authors and Proprietors of such copies during the times therein mentioned ; and, pl-o, ot of the Act entitled " An Act supplementary to an Act entitled an Act for the encouragement of Learning, by securing the copies of Maps, Charts, and Books, to the Authors and Proprietors of such Copies during the times therein mentioned, and extending the benefits there- of to the Arts of designing, engraving, and etching historical and other Prints. \VM. KEY BOND, Clerk of the District ofOkio. RECOMMENDATIONS. The following have been selected from the recommenda- tions bestowed upon this work. Messrs. Guilfords, I have examined hastily the "Juvenile Arith- metick," which you sent me, and am of opinion that it possesses scne advantages over those generally in use: I particularly refer to the part intended to cultivate in the learner, the habit of going through the solutions mentally. Very respectfully yours, JOHN E. ANNAN, Professor of Mathematicks and Natural Philosophy in the Miami University. Oxford, June 5, 1827. From a hasty review of Dr. Ruter's Artihmetick, I am inclined to think well of it. The attempt to introduce a rational method of in- struction in any department of education, is laudible and especially in common schools. This I think the Juvenile Arithmetick is well calculated to do, in that branch of study to which it belongs. The plan of Pestalozzi is excellent, and Dr. Ruter has perhaps imitated it more successfully (by comprizing more in less space) than Mr. Col- burn, between whose Arithmetick and this there is however a consid- erable resemblance. Your's &c. WM. H. M'GUFFY, une28,m7. Professor of Languages, &c., in the Miami University. We have examined your ' Juvenile Arithmetick/ and feel a pleas- ure in recommending it to the schools of our country. We think the general arrangement good, and have no hesitation in saying, that the questions prefixed and appended to the rules, give it superior ad- vantages. Respectfully yours, JOSEPH S.'TOMLIN SON, JOHN P. DURBIN, March 12, 1828. Professors in Augusta College. The Juvenile Arithmetick, from the cursory examination which I have given it, appears to be a manuel of value for the introduction of youth into the science of numbers. In furnishing a second edition, I wish you success. ELIJAH SLACK. I concur most cheerfully in the above opinion. Cincinnati, April 2, 1828. S. JOHNSON. I have used thy compilation of Arithmetick during the last year; and do not hesitate in recommending it to the publick. The ques- tions preceeding the rules, the particular attention to fractions, and the sketch of mensuration give it a decided preference to any other here in use. JOHN L. TALBERT. Cincinnati, Fourth mo. 5, 1828. Having examined the above Arithmetick, I cheerfully concur in the foregoing opinion of its merits. ARNOLD TRUESDELL. i RECOMMENDATIONS. I have carefully inspected the "Juvenile Arithmetick and Schol lar's Guide," by Dr. Ruter, and am of the opinion, it is well calcula- ted and arranged, to conduct the pupil by an easy gradation to a per- spicuous conception of the science of numbers. I therefore recom- mend it to the publick use, particularly in common schools. SAMUEL BURR, September 2, 1827. Professor of Mathematick*. A cursory examination of Dr. Ruter'a Arithmetick, has convinced me, that the simple and familiar manner in which the learned author unfolds the principles of this science, and adapts them to the under- standing of the young learner, can not fail to give his work a decided preference, for practical purposes, over those arithmeticks in common uge. In my opinion, teachers who adopt it, as well as pupils who stu- dy it. will realize satisfactory and highly beneficial results. S. KIRKHAM, Author of Grammar io Familiar Lectures. Pittiburgh, Afril 2, 1828. I have examined the system of Arithmetick compiled by Dr. Ru- ter, and am of opinion that it is well calculated tor conveying to youth, a general knowledge of that science in a shorter time, man any I have seen. G. GARDNER, March 29, 1828. Teacher of Mathematicks, Mill-Creek Township. Having examined the Juvenile Jlrithmetick, I have no hesitation n pronouncing it an excellent elementary School Book, The rules are judiciously arranged, and peculiarly well adapted to juvenile com- prehension: The work contains multum in parvo, and I think its publication will be conducive to publick utility. Hoping its merits will be duly appreciated, 1 take great pleasure in 'Commending it to the publick patronage. Yours respectfully, RICHARD MORECRAFT. Cincinnati, January 2, 1828. Teacher. From my acquaintance with Ruter'* Arithmetick, I am convinced that it is well calculated to encourage the student, improve his mind, and prepare him for business. JOHN LOCKE, May 16, 1828. Principal of Cincinnati Female Academy. Gentlemen I have examined with some attention the Juvenile Arithmetick, &c. by the Rev. Dr. Ruter, and am decidely of opinion, that it is admirably calculated for conveying to youth with great fa- cility a general knowledge of that important science. The ingenious manner in which the compiler has given an elucidation of Vulgar Fractions, together with an exclusion of all extraneous matter, ren- ders it in my estimation a treatise of peculiar merit. Your obedient servant, JOHN WINRIGHT, Cincinnati, September 2, 1827. Teachar. PREFACE. Tins ARITHMETICK has been compiled with a view to facilitate the progress of pupils, and lessen the labour of teachers. The questions preceding and following the rules, are designed to lead young learners into habits of thinking and calculating; and thus, to prepare them for practical operations. Experience has demonstrated, that, in the instruction of children in any science, it is necessary to excite their entire attention to the subject before them. The latent energies of their minds must be roused up, and called forth into action. When this can be effectually done, success is rendered certain. To accomplish this important object, the best method has been found in the frequent use of well selected questions. Though it is a successful course in all ju- venile studies, it is particularly so in the science of numbers ; and the progress of pupils must be slow with out it. The questions in the following pages are thought to be sufficiently numerous for the purposes in- tended ; the rules have been arranged according to the plan of some of the best authors on this subject, ard the work is offered to the publick with the hope that it will be useful in the schools of our country. M.R. 1* EXPLANATION OF THE CHARACTERS USED IN ARITHMETICS -[* Signifies plus, or addition. Signifies minus, or subtraction. Denotes multiplication. Means division. : : : Signifies proportion. Denotes equality. Thus, 4-{-7 denotes that 7 is to be added to 4. 5 3, Denotes that 3 is to be taken from 5. 8X2, Signifies that 8 is to be multiplied by 2. 9-r-3, That 9 is to be divided by 3. 3:2: : 6: 4, Shows that 3 is to 2 as 6 is to 4. 7+9= 16, Shows that the sum of 7 and 9 is equal to 16 V or 3 ,/ Denotes the Square Root. *J Denotes the Cube Root. 4 J Denotes the Biquadrate Root. This mark, called a Vinculum, shows that the several figures over which it is drawn are to to be taken together as a simple quantity. ARITHMETIC!*. ARITHMETICK is the science which treats of the nature and properties of numbers : and its operations are con- ducted chiefly by five principal rules. These are, Nu- meration, Addition, Subtraction, Multiplication, and Di- vision. Numhers in Arithmetick are expressed by the fol lowing ten digits or characters, namely : 1 one, 2 two, 3 three, 4 four, 5 five, 6 six, 7 seven, 8 eight, 9 nine, cypher. An Integer signifies a whole number, or certain quan tity of units, as one, three, ten. A Fraction is a broken number, or part of a number, as i one half, two-thirds, 4 one-fourth, | three-fourths, I five-eigths, &c. Numeration teaches the different value of figures by their different places, and to express any proposed num bers either by words or characters ; or to read and write any sum or number. miMERATION TABLE. r-HO?'NO}COTj< Units. ojO' iooTt<co Tens. Hundreds. Thousands. CO^COTJ.IO.XN Tens of thousands. 01 co ^ ifl ^ x eo Hundreds of thousands. ^* Millions, co i- woo 10 Tens of millions. 00 ^ *- Hundreds of millions. ' Thousands of millions. 10 Ten thousands of millions. * Hundred thousands of millions. 8 NUMERATION. Here any figure in the place of units, reckoning from right to left, denotes only its simple value ; but that in the second place denotes ten times its simple value ; and that in the third place, one hundred times its simple value ; and so on, the value of any figure in each suc- cessive place, being always ten times its former value. Thus in the number 6543, the 3 in the first place denotes only three ; but 4 in the second place signifies four tens or 40 ; 5 in the third place, five hundred ; and six in the fourth place, six thousand ; which makes the whole number read thus six thousand five hundred and forty- three. The cypher stands for nothing when alone, or I when on the left hand side of an integer ; but being joined on the right hand side of other figures, it increas- es their value in the same ten fold proportion : thus, 50 denotes Jive tens ; and 500 is read jfoe hundred. Though the preceding numeration table contains only twelve places, which render it sufficiently large for [young students, yet it may be extended to more places I at pleasure. EXAMPLE. Quatrillions. Trillions. Billions. Millions. Units. 987,654 ; 321,234 ; 567,898 ; 765,432 ; 123,456 Here note, that Billions is substituted for millions of millions : Trillions, frr millions of millions of millions : Quatrillions, for millions of millions of millions of mil- lions. From millions, to billions, trillions, quatrillions, and other degrees of numeration, the same intermediate denominations, of tens, hundreds, thousands, fyc. are 'ised, as from units to millions. And thus, in ascertain- ing the amount of very high numbers, we proceed from Millions to Billions, Trillions, Quatrillions, Quintillions, SextillioRS, Septillions, Octillions, Nonillions, Decillions, Undecillions, Duodecillions, Tredecillions, Quatuorde- cillions, Quindecillions, Sexdeciliions, Septendecillions, Octodecil lions., Novemdecillions, Vigintillions, &c. all |! r )f which answer to millions so often repeated, as their |j indices respectively require, according to the above pro- f j portion. THE APPLICATION. Write down in figures the following numbers Ten. . )0 Twenty-one. .... Thirty-five. - - - 35 Four hundred and sixty-seven. 467 Two thousand three hundred and eighty-nine. - 23S9 Thirty -four thousand live hundred and seventy 04570 Six hundred and three thousand four hundred. 6034 )C Seven millions eight hundred end four thou- ) -c n/l qr>q sand three hundred and twenty-nine. Fifty -eight millions seven hundred and thir- ty-two thousand one hundred and five. Eight hundred and ten millions nine him- dred and two thousand five hundred 610902512 and twelve. Three thousand two hundred and three millions six hundred and eight thou- 3203608999 sand nine hundred and ninety -nine. Question 1. What is Arithmetic!*. ? 2. What are the ten digits by which numbers are expressed ? 3. What is an integer ? 4. What is a fraction ? 5. What are the principal rules by which the operations in Arithmetick are conducted? 6.What does Numeration teach ? SIMPLE ADDITION. Simple Addition teaches to put together numbers of ;he same denomination into one sum ; as 5 dollars, 4 dollars, and 3 dollars, make 12 dollars. Before the pupil enters upon Addition in the usual way, with figures, it would be useful for him to learn to perform easy operations in his mind. For this purpose et him be exercised in the following questions, or in others which are similar. to SIMPLE ADDITION. 1 . If you have two cents in one hand and two in the other, how many have you in both ? 2. If you have three cents in one hand, and two in he other, how many have you in both ? 3. If you have five cents in one hand, and two in the jther, how many have you in both ? 4. John has six cents,.- and Robert has three ; how many have they both together ? 5. Charles gave five cents for an orange, and two for in apple ; how many clid he give for both ? 6. Dick had four nuts, John had three, and David !iad two ; how many had they all together ? 7. Henry had five peaches, Joseph had three, and Tom had two, and they put them all into a basket ; how many were there in the basket ? 8. Three boys, Peter John and Oliver, gave some money to a beggar. Peter gave seven cents, John four, ind Oliver three. How many did they all give him ? 9. A man bought a sheep for eight dollars, and a calf for seven dollars ; what did he give for both ? 10. A boy gave to one of his companions eight peach- es to another six ; to another four ; and kept two him- self ; how many had he at first ? 11. How many are two and three ? two and five ? three and seven ? four and five ? 12. How many are two and four and one? 13. How many are three and two and one? 14. How many are four and three and two? 15. How many are five and four and three ? 16. How many are four arid five and two ? 17. How many are seven and three and one ? 18. How many are eight and four and two ? 19. How many are nine and five and one ? 20. How many are five and six and seven ? 21. How many are four and three and two and one ? 22. How many are two and three and one and four ? I 23. How many are five and three and two and one ?* *It is expected that many of these questions will be varied by the teacher, and rendered harder, or easier, or others substituted as the capacity of the pupil may require. SIMPLE ADDITION. ll RULE. Place the figures to be added, one under another, so that units will stand under units, tens under tens, hun- dreds under hundreds, &c. Draw a horizontal line un- der them, and beginning at the bottom of the first col- umn, on the right hand side, that is, at units, add togeth- er the figures in that column, proceeding from the bot- tom to the top. Consider how many tens are contained in their sum, and how many remain besides the even number of tens ; place the amount under the column of units, and carry so many as you have tens to the next column. Proceed in the same manner through every column, setting down under the last column ih full amount. PROOF. Begin at the top of the sum and add the Several rows of figures downwards as they were added upwards, and it the additions in both cases be correct, the sums will agree. EXAMPLES. I. II. III. IV. 12 321 4000 542210 21 123 3124 18540? 34 410 2345 350212 10 203 5234 201304 1057 14703 V. VI. VII. 2405670 50678 4 5 7 8 f 3540210 76543 876542 4321023 20134 450780 4065243 56787 876543 2123456 65432 2 3 4 7 9 g SIMPLE ADDITION. VIII. IX, X. 57898765 45 20000000 4321234 678 3000000 567898 9876 400000 76543 54321 50000 2123 234567 6000 212 987654 700 72866775 9287141 2345670C XL XII. XIII. 24681012 54321231 98765432 42130538 19000310 12345576 71021346 20304986 98765432 20324213 19876540 12345678 98765432 98755432 98765432 12345678 12000987 12345076 APPLICATION. 1. A boy owed one of his companions 6 cents ; he owed another 8, another 5, and another 9. How much did he owe in all ? Ans. 28 cents. 2'. A man received of one of his friends 7 dollars, of another 10, of another 19, and of another 50. How many dollars did he receive ? Ans. 86 dollars. 3. A person bought of one merchant ten barrels of flour, and paid 40 dollars ; of another 20 barrels of ider, for which he paid 60 dollars, and 20 barrels of sugar at 450 dollars ; and of another 95 barrels of salt at 570. How many barrels did he buy, and how much money did he pay for the whole ? Ans. 145 barrels, and paid 1120 dollars. 4. A had 250 dollars ; B had 375 ; C had 5423 ; D, 84320 ; E, 287432, and F, 4321567. How much would it all make, if put together ? Ans. $4679367. Question. 1. What does Simple Addition teach ? 2. How do you place the numbers to be added? Where do you begin the Addition? How do you prove a sum in Addition? SIMPLE SUBTRACTION. Simple Subtraction teaches to take a less number from a greater of the same denomination, and thus tc find the difference between them. Questions to prepare the learner for this rule. 1. If you have seven cents, and give away two; how many will you have left? 2. If you have eight cents, arid loee four of them; how many will you have left? 3. A boy having ten centa, gave away four of them; how 7 many had he iefe? 4. A man owing twelve dollars, paid four of it; how much did he then owe? 5. A man bought a firkin of butter for fifteen dollars, and sold it again for ten dollars ; how much did he lose ? 6. If a horse is worth ten dollars, and a cow is worth four; how much more is the horse worth than the cow? 7. Ahoy had eleven apples in a basket, and took out five; how many were left? 8. Susan had fourteen cherries, and ate four of them; how many had she left? 9. Thomas had twenty cents, and paid away five of them for some plums; how many had he left? 10. George is twelve years old, and William is seven; how much older is George than William? 11. Take four from eight; how many will remain? 12. Take three from nine; how many will remain? 18. Take five from ton; how many will remain? 14. Take six from ten ; how many will remain? 15. Take six from eleven; how many will remain? 16. Take five from twelve; how many will remain? 17. Take four from thirteen; how many will remain? 18. Take six from fourteen; how many will remain? 10. Take six from fifteen; how many will remain? 20. Take eight from sixteen; how many will remain? 21. Take nine from twelve; how many will remain? 22. Take nine from ftv.rtesn; how many will remain? 23. Take three from thirteen; how many will remain? 2 14 SIMPLE SUBTRACTION. 24. Take eight from seventeen ; how many will remain ? 25. Take nine from sixteen; how many will remain? 28. Take nine from eighteen ; how many will remain? RULE. Place the larger number uppermost, and the smaller one under it, so that units may stand under units ; tens under tens; hundreds under hundreds, &c. Draw a line underneath, and beginning with units, subtract the low- er from the upper figure, and set down the remainder. But when in any place the lower figure is larger than the upper, call the upper one ten more than it really is ; subtract the lower figure from the upper, considering it as having ten added to it, set down the remainder^ and add one to the next left figure of the lower line, and proceed thus through the whole. PROOF. Add the remainder and the less line together, and if the work be right, their sum will be equal to the greater line. EXAMPLES. I. II. III. IV. V. 23 457 54367 73214 84201 11 215 20154 54876 49983 12 242 34213 18338 34218 VI. VII. 9812030405321 700000000000 6054123456789 98765432123 VIII. IX. X. XI. 32016 98700 500612 65040032 12045 25290 499521 7000302 XII. XIII. XIV. 974865 400000 100000000 863757 7 1 SIMPLE ADDITION AND SUBTRACTION. 15 APPLICATION. 1. A ship's crew consisted of 75 men, 21 of whom died at sea. How many arrived safe in port? Ans. 54 men,. 2. A boy had 100 miles to travel, and went 33 miles the first day. How far had he still to go? Ans. 67 miles, 3. A tree had 647 appies on it, but 158 of them fell off. How many were there then remaining on the tree? Ans. 489 apples, 4. A boy put 1000 nuts into a basket and afterwards took out 650. How many were left in the basket? Ans. 350 nuts. 5. A general had an army of 43250 men, 15342 of them deserted. How many remained? Ans. 27G08 men. Question 1. What does Subtraction teach? 2. How do you place the larger and smaller numbers? 3. What do you do when the lower number is larger than the upper number? 4. How is a sum in subtraction proved? Exercises for the slate under the two preceding Rules. 1. I saw 15 ladies pass up street, and 8 down street. How many passed both ways? Ans. 23 ladies. 2. A boy who had 15 buttons upon his jacket, lost off 7 of them. How many were left on? Ans. 8 buttons. 3. A man bought a barrel of flour for 10 dollars, a barrel of molasses for 29 dollars, arid a barrel of rum for 36 dollars. How much did he pay for all the arti- cles? Ans. 75 dollars. 4. A man bought a chaise for 175 dollars, and to pay for it gave a wagon worth 37 dollars and the rest in money. How much money did he pay. Ans. 138 dollar? 5. James bought at one time 89 apples, at another 54, at another 60, and at another 75. Hovy many did he in all? Ans. 278 apples 16 SIMl'LE ADDITION AND SUBTKAC'TIOK'. 6. A merchant bought a piece of cloth containing 489 yards, and sold 355 yards. How many yards hue he left? Ans.* 124 yards 7. Suppose my neighbour should borrow cf me a one time 658 dollars, at another 50 dollars, at anothei 3655 dollars, and at another 5000 dollars; how much should I lend him in all? Ans. 9361 dollars 8. Charles has 42 marbles and John has 25. Hou many has Charles more than John? Ans. 17 marbles 9. If Charles give John 200 nuts, and James give him 56, and Joseph give him 195; how many will John have? Ans. 451 nuts 10. My friend owed me 150 dollars, but has paid me 90 dollars. How much does he still owe me? Ans. 60 dollars, 11. If you buy 20 peaches for 40 cents, and sell 15 for 35 cents, how many peaches will you have left, and bow much will they cost you? Ans. 5 peaches, & will cost 5 cents 12. A person went to collect money, and received of one man 90 dollars; of another 140 dollars; of an other 101 dollars, and of another 29 dollars. How much did he collect in all? Ans. 360 dollars. 13. A man deposited in bank 8752 dollars, and drew out at one time 4234 dollars, at another 1700 dollars, it another 962 dollars, and at another 49 dollars. How much had he remaining in bank? Ans. 1807 dollars. 14. Gen. Washington was born 1732, and died in 1799. How old was he when he died ? Ans. 67 years. 15. A man owed 11,989 dollars. He paid at one ime 2875 dollars; at another 4243; at another 3000 dollars. How much did he still owe? Ans. 1871 dollars. 16. A man travelled till he found himself 1300 miles rom home. On his return, he travelled in one week 235 miles; in the next 275; in the next 325, and in he next 290. How far had he still to go before he vould reach home? Ans. 175 miles. SIMPLE MULTIPLICATION. Simple Multiplication toadies a short method of find- ing what a number amounts to when repeated a given number of times, and thus performs Addition in a very expeditious manner. I. What will fjur apples cost at two cents a piece? 12. What must you give Ibr two oranges, at six cents a piece? 3. What are two barrels of flour worth, at five dol- lars a barrel? 4. What will three pounds of butter come to, at three cents a pound? 5. If you can walk four miles in one hour, how far can you walk in three hours? 0. If a cent will buy five nuts, how many nuts will four cents buv? 7. What are two barrels of cider worth, at three dollars a barrel? 8. If you give four cents for a yard of tape, how mnny cents will buv three yards? l\ If I pat in your pocket five r.pples at three differ- ent time?, how many apples will you have in your pocket? How many are three times five? 10. If four boys have each four apples, how many have they all? How many are four times four? II. What will six marbles cost at three cents a piece ? How many are six times three ? 12. A horse has four legs. How many legs have five horses? How many are four times five? 13. I gave six boys four peaches each. How many di I I give them all? How many are six times four? 14. How many cents will buy ten marbles if one cost three cents? How many are three times ten? 15. If I can walk three miles in one hour, how far can I walk in six hours? Before entering upon this Rule, let the pupil so learn the follow- ing table, as to answer with readiness any question implied in it; after which, he will be able to proceed with facility. IS SIMPLE MULTIPLICATION. ; MULTIPLICATION TABLE. Twice 3 times 4 times 5 times o ume, e / times I Imake 2 Imake 3 1 make 4 Imake 5 Imake Jniake 7 2 4 2 6 2 6 2 10 2 IS 2 14 3 6 3 3 12 3 15 3 It 3 21 4 8 4 12 4 16 4 20 4 24 4 28 5 10 5 15 5 20 5 25 5 3C 5 35 6 12 6 15 6 24 6 30 6 3 6 42 7 14 7 21 7 28 7 35 7 42 7 49 8 16 t 3 24 8 32 8 4C 8 r4fc 8 56 9 16 9 21 9 36 9 45 9 54 9 63 10 20 10 30 10 4C 10 5<T: '0 6C 10 70 11 22 11 33 11 44 11 55 11 66 11 77 12 24 12 36 12 # 12 6C 12 72 12 84 8 times 9 times 10 times 11 times IxJ times 1 make 8 1 make f. Imake 1C Imake 11 lmakel 2 V 16 2 18 2 2C 2 22 2 24 3 24 3 27 3 3G 3 33 3 36 ' 4 32 4 36 4 40 4 44 4 46 5 40 5 45 5 50 5 55 5 6C f 6 48 6 54 6 60 6 66 6 7 1 7 56 7 63 7 70 7 77 7 84 1 8 64 8 72 8 80 8 88 8 96 9 72 9 81 9 90 9 99 9 ioe 10 80 10 90 10 100 10 110 10 12C 11 88 11 99 11 110 11 121 11 13S |12 96 12 108 12 120 12 132 12 144 Though the foregoing table extends no farther than 12, it may be easily continued farther; and if pupils were to extend it, and commit it to memory, as far as 30 or 40, it would afford them great advantage in their progress. The number to be multiplied is called the multipli- cand. The number which multiplies is called the multiplier.* The number produced by the operation is called the product. *The multiplier and multiplicand are called Factors. SIMPLE MULTIPLICATION. 19 CASE I When the Multiplier is no more than 12. RULE. Place the greater number, or multiplicand, upper- most; set the multiplier under it, and beginning with units, multiply all the figures of the multiplicand in succession, carrying one to the next figure for every ten, and setting down the several products, as in Addition. The whole of the last product must be set down. PROOF. Multiply the sum by double the amount of the multi- plier, and if the work in both instances bo right, the product will be double the amount of the former pro- duct* i. 234 2 n. 3201 3 EXAMPLES. III. 51000 4 rv. 43201 5 v. 354610 6 168 9603 204000 216005 2127660 VI. 453210 7 VII. 3245013 8 VIII. 98765432 1 9 3 172470 IX. X. XI. 678987654 321234567 898765432 9 11 12 *Multi plication may be proved by Division; for if the product be divided by the multiplier, the quotient will be the same as the multi plicand. 1 2* 20 SI.MfLF MULTIPLICATION. CASE II. When the Multiplier is more than 12. RULE. Multiply each figure in the multiplicand by every figure in the multiplier, and place the first figure of each product exactly under its multiplier; then add the several products together, and their sum will be the answer. When cyphers occur at the right hand of either of the factors, omit them in multiplying, and annex them to the right hand of the product.* When the multiplier is the product of any two whole numbers, the multiplication may be performed by mul- tiplying the sum by one of them, and the product by the other. Thus, if 24 were to be multiplied by 18, (as 6 times 3 make 18,) let it be multiplied by 6, the product by 3, and the answer will be the same as if multiplied by 18. EXAMPLES. I. II. 43021678 8765432 C 432 543 86043356 26296296 C 129065034 350617280 1 720867 12 438271600 18585364896 4759629576C in. iv. v. 679100 26043 432000 32 34 4300 13582 104172 1296 20373 78129 1728 21731200 885462 1857600000 *Multiplying by 10, add a cypher to the right hand side of the sum, and it is done. Thus, let it be required to multiply 12 by 10, the product will be 120; but if a cypher be added, it will bring the ;ult. In multiplying by 100, add two cyphers: by 1000, same res three, <fcc. SIMPLE MULTIPLICATION. 2l Multiply 18450 by 35. vi. vn. vm. 18450 18450 18450 7 5 35 129150 92250 92250 5 7 55350 045 750 645750 645750 9. multiply 420 by 7 product 2940 10. "3240 9 29160 11. 54134 18 974412 12. 37990 24 911760 13. 84522 54 4564188 14. 90203 587 52949161 15. 370456 7854 2909561424 16. 7654876 8765 67094988140 APPLICATION. 1. A man had 29 cows, and his neighbour had five times as many. How many had his neighbour? Ans. 145. 2. There are twelve barrels of sugar, each contain- ing 256 pounds. How manv pounds do they all con- tain? Ans. 3072. 3. How far will a man travel in a year, allowing the year to contain 365 days, if he travel 40 miles per day? Ans. 14600 miles. 4. In one hogshead are 63 gallons ; how many gal- lons are there in 144 hogsheads? Ans. 9072. Q. 1. What does simple Multiplication teach? 2. What is the number to be multiplied, called? 3. What is the number called which is used in mul- tiplying another number? 4. Are the multiplicand and multiplier called by any other names? 5. How do you proceed when the multiplier is no more 'than 12? 6. When the multiplier is more than 12, how do you proceed? 22 SIMPLE DIVISION. 7. What do you do when cyphers occur at the right hand of either of the factors? 8. How do you proceed when the multiplier is the product of two other numbers? 9. How may sums in Multiplication be proved? SIMPLE DIVISION. Simple Division teaches to find how often one num- ber is contained in another, and is a concise way of performing several subtractions. Questions to prepare the learner for this rule. 1. James had 4 apples and John half as many; how many had John? 2. If two oranges cost 6 cents, what does one cost? 3. If you divide 8 apples equally between two boys, how many will each have? 4. What is one half of eight? 5. If you divide 6 nuts equally among 3 boys, how many will each have? 6. What is one third of six? 7. If 12 cherries cost 9 cents, what will 4 cost? 8. A third of 9 is how many? 9. If you divide 16 nuts equally among 4 boys, how many will each have? 10. A fourth of 16 is how many ? 11. How many times two are there in six? 12. How many times three in six? 13. How many times four in eight? 14. How many times two in twelve? 15. In nine, how many times three? 16. In eight, how many times two? 17. In ten, how many times five? 18. In twelve, how many times three? 19. In twelve', how many times four? 20. In twenty, how many times five? 21. In eighteen, how many times six? 22. In sixteen, how many times two? SIMPLE DIVISION. 23. In thirty, how many times five? 24. In thirty, how many times six? 25. In twenty-one how many times seven? 26. In twenty-eight, how many times seven? 27. In thirty-six, how many times twelve? 28. In forty-eight, how many times twelve? 29. In forty-eight, how many times sixteen? 30. In fifty-five, how many times eleven? 31. In sixty IK;W many times twenty? 3*2. In eighty, how many times twenty? 33. In one hundred, how many times twenty? 34. In one hundred and twenty, how many times thirty ? 35. In ten, how many times four ? Answer. Two times, and two remain. 3f>. In fourteen, how many times three ? Answer. Four times, and two remain. 37. In twenty-five, how many times four? Answer. Six, and one remains. There are in Division four principle parts, viz: The dividend, or number to be divided. The dwiwr, or number given to divide by. The quotient, or answer, which shows how mnn times the divisor is contained in the divr-err . The remainder, which is any overplus of figure, that may remain after the sum is done, and i; always less than the divisor. CASE I. RULE. First, find how many times the divisor if contained in as many figures on the left hand of the dividend as are necessary for the operation, and place the number in the quotient. Multiply the divisor l^ this number,, and set the product under the figures r ' the lef, hand of the dividend I ef >re mentioned. Sul trac this product from that part of the dividend under whicl it stands, and to the remainder bring down the nex ; figure of the dividend; but if this will not contain th divisor, place a cypher in the quotient, and I rin<r dcwi< another figure of the dividend, and so on, until! it wi! 24 SIMPLE DIVISION. contain the divisor. Divide this remainder (thus in creased) in the same manner as before ; and proceed in this manner until all the figures in the dividend are brought down and used. Multiply the quotient by the divisor, and to the pro- duct add the last remainder, if there be any; if the work is right, the sum will be equal to the dividend. EXAMPLES. i, Divisor. Dividend. Quotient. 3)143967182(47989060 12 3 2 3 Proof 1 43067182 2 1 In this example, I find 2 9 that 3, the divisor, can 2 7 not be contained in the first figure of the divi- 2 6 dend ; therefore I take 2 4 two figures, v iz : 14, and inquire how often 3 i 27 contained therein,which 27 I find to be 4 times, and put 4 in the quotient. 1 8 Then multiplying the 1 8 divisor by it, I set the: . product under the 14, Remainder. 2 in the dividend, r.nd find by subtracting that there is a remainder of two. To this 2, I bring down the next figure in the dividend, viz: 3, which increase? he remainder to 23. I then seek how often 3 is con- tained in 23, and proceed as before, When I bring down the one that is in the dividend, I find that three can not be contained in it, and therefore place a cypher in the quotient and bring down the 8, which makes 18. Find- ing that 3 is contained 6 times in 18, and that there is no SIMPLE DIVISION. ^ remainder, I bring down the 2; but as 3 can not be con- tained in it, I place a cypher in the quotient, and let 2 stand as the last remainder. In proving the sum by Multiplication, the 2 is added. This mode of operation is called LONG DIVISION u. in. 2)3456789(1788394 5)8789876(1357975 22 55 14 Proof 3456789 17 Pr 678987G 14 15 5 28 4 25 16 39 16 35 7 48 6 45 18 37 18 35 9 26 8 25 1 1 IV. V. 42)9870(235 320(12864016081(40200050 84 1280 147 640 126 640 210 1603 210 1600 61 SIMPLE DIVISION. VI. VII, 12 ) 301203 ( 25100 15 ) 218760 ( 14584 24 12 15 15 61 Pr,301203 68 72920 60 60 14584 12 87 218760 12 75 -03 126 120 60 60 VIII. 848)2468098(3808 1944 '3808 5240 Proof 648 5184 - 30464 5698 15232 5184 22848 514 514 2468098 1. Divide 87654 by 58 Quo. 1511 Rem. 16 2. 456789 679 672 501 3. 3875642 7898 490 5622 4. 98765432 1234 80036 1008 5. 12486240 87654 142 39372 6. 57289761 7569 7569 Note. When there is one cypher, or more, at the right hand of the divisor, it may be cut off ; but when this is done, the same number of figures must be cut off from the right hand of the dividend ; and the figures thus cut off, must be placed at the right hand of the re- mainder. SIMPLE DIVISION. 27, EXAMPLES. I. II. 6|00)567434| 10(94572 18|000)246864|593(13714 54 18 27 66 24 54 34 128 30 126 43 26 42 18 14 84 12 72 210 Remainder 12593 Note. In dividing by 10, 100, or 1000, &c. when you cut off as many figures from the dividend as there are cyphers in the divisor, the sum is done ; for the figures cut off at the right hand are the remainder, and those at the left are the quotient, as in the following sums : nr. iv. Quotient. Quotient. 1 10)98765(4 Rem. 1 100)123456(78 Rem. Quo. Quo. 1|000)56789(876 Rem. 1|0000)8765(4321 Rem. CASE II. When the divisor does not exceed 12, seek how often it is contained in the first figure or figures of the divi- dend, and place the result in the quotient. Then mul- tiply in your mind the divisot by the figure placed in the quotient, subtract the product from the figure under which it would properly stand in the former case of di- vision and conceive the remainder, if there be any, to be prefixed to the next figure. See how often the divi- sor is contained in these, and proceed, as before, till 3 28 SIMPLE DIVISION. ;he whole is divided. This operation is called SHORT DIVISION. EXAMPLES. I. II. 4) 987654321 8) 123456789 Quo. 24691358 01 Quo. 1543209 85 In the first example, I find that 4 is contained twice in 9, and that 1 remains. The 1, 1 conceive as prefixed to the next figure, which is 8, and they become 18. In 18, 1 find 4 is contained 4 times, and 2 remain. By prefixing the 2 to the following figure, which is 7, they make 27. In this manner I proceed, setting the result of each calculation in the lower line which is the quo tient. In the second example, as 8 can not be contained in 1, take two figures, and proceed as in the first. in. rv. 9)1023684200 12) 19 14678987 v. vi. 11)6789870062 12)1000001246 Note. When the divisor is of such a number that two figures being multiplied together will produce it, divide the dividend by one of those figures, the quotient thenc< I irising, by the other figure, and it will give the quotien required. As it sometimes happens that there is a re tnainder to each of the quotients, and neither of then I "he true one, it may be found thus : Multiply the firs divisor by the last remainder, and to the product adc the first remainder, which will give the true one. EXAMPLES. I. Divide 249738 by 56. 8] 249738 8 7 | 312 17 2 4 4459--* 32 2 34 Remainder. SIMPLE DIVISION. 29 The same done by Long Division. 561249738F4459 224 333 280 538 504 34 Remainder. ii. Divide 1847562324 by 84. 12] 1847562324 7] 1 847562324 7]153963527 12] 26393747 46 2199478 94 12 4 8 Rem, 2199478 96 7 - 42 6 Rem. 48 3. Divides 463 098 6 by 72. 4. " 6 7 8 6 1 2 1 by 63. 5. 124 5 6743 by 96. 6. 3 4 2 1 3 9 by 81. 7. 5 4 6 9 7 2 8 3 by 103. 8. 7 5 3 9 2 6 1 8 by 113. Note. In all cases in Division, when there is any re- mainder, the remainder and divisor form a Vulgar Fraction. Thus, if the divisor be 8 and the remainder 5, they make f or five eights ; or, as in one of the pre- ceding examples, the divisor is 56 and the remainder 34, which make . 30 SIMPLE DIVISION, APPLICATION. 1. A man bought 6 oxen for 318 dollars. How much did he pay a head. Ans. 53 dollars. 2. How much flour at 7 dollars per barrel can be bought for 1512 dollars? Ans. 216 barrels, 3. If 1600 bushels of corn are to be divided equally among 40 men, how much is that a piece? Ans. 40 bushels. 4. The salary of the President of the United States is 25000 dollars a year. How much is that a day, reckoning 365 days to the year? Ans. 68-iff 5. A regiment consisting of 5CO men are allowed 1000 pounds of pork per day. How much is each man's part? Ans. 2 pounds. 6. If a field of 32 acres produce 1920 bushels of corn, how much is that per acre? Ans. 60 bushels. 7. A prize of 25526 dollars is to be equally divided among 100 men. What will be each man's part? Ans. 255 T W dollars. 8. How many oxen at 30 dollars a head, may be bought for 38040 dollars? Ans. 1268 oxen. Question 1. What does Simple Division teach? 2. What are the four principal parts of Di vision? 3. How do you proceed when there is one cy pher or more on the right hand of the divisor? 4. How do you proceed in dividing by ten, or a hundred, or a thousand? 5. How do you proceed when the divisor does not exceed 12? 6. When you divide by nny number not ex- ceeding 12, what is the operation called? 7. When the divisor is of such a number that two figures multiplied together will pro- duce it? 8. What can be made by placing the remain- der of a sum over the divisor? Ans. A Vulgar Fraction. 9. How is a sum in Division proved? moniscrors QUESTIONS. 31 Promiscuous Exercises on the slate under all the fore- going Rules. 1. A man bought a cart for 25 dollars, a yoke of oxen for 69 dollars, and a plough for 7 dollars. What did Sie give for the whole? Ans. 101 dollars. 2. What will 315 bushels of rye cost at 42 cents a ushel? Ans. 13230 cents. 3. If my income be 1647 dollars, and I spend 1010 dollars of it, how much do I save? Ans. 637 dolla.rs. 4. Four boys had gathered 113 bushels of walnuts; in dividing them equally, how many will each have? Ans. 284 bushels. 5. There are 63 gallons in a hogshead. How mapy gallons are there in 25 hogsheads? Ans. 1575 gallons. 6. A merchant bought a stock of goods for 30250 dol- lars, and sold it again for 40000 dollars. How much did he gain? Ans. 9750 dollars. 7. A merchant bought at auction, broadcloth for 350 dollars, muslin 97 dollars, linen 1010 dollars, silk 874 dollars, and calico 8 dollars. To what did the whole amount? Ans. 2339 dollars. 8. There are 328 rows of corn in my field and each row has 169 hills. How many hills are there in the field? Ans. 55432 hills 9. John had in his desk 1000 dollars. lie took cut 120 dollars to pay a debt; he afterwards put in 75 dol- lars. How much was there in the desk? Ans. 955 dollars. 10. A farmer has a flock of 60 sheep; one third of them are black and the rest white. How many of them [are black? Ans. 20. 11. A merchant has 50 boxes of raisins with 17 pounds in each box. How many pounds are there in all? Ans. 850 pounds. 12. There are 4 quarts in a gallon. How many gal Ions are there in 760 quarts? .. Ans. 190 gallons. 13. Ann had a paper of pins which had 600 in it when she bought it, but she used 245 of them. How many are left? Ans. 355 pins. 14. If I divide 364 cents among 14 boys, how many will each have? Ans. 26 cents 32 PROMISCUOUS QUESTIONS. 15. There are 1681 nuts in a basket. James took out 150, Charles 272 and John 1005; after which, Jo- seph put in 95. How many were there in the basket? Ans. 349 nuts. 16. A cooper worked 115 days, and made 6 barrels each day. How many barrels did he make? Ans. 690 barrels. 17. A person has in money 5000 dollars; in bank stock 3500 dollars, and in merchandize 12500 dollars, i He intends to divide all this property equally among his 13 sons. What will be the share of each son? An?. 7000 dollars. 18. William had 372 pears; he kept 120 of them, and divided the rest between his two sisters. How ma- ny did each sister receive? Ans. 126 pears ID. There are 10 bags of coffee weighing each 120 pounds, and 12 bags weighing each 135 pounds. What is the weight of the whole? Ans. 2820 pounds. 20. There are 15 firkins of butter each weighing 49 pounds. The fh'kins which contain the butter wei/ each 7 pounds. How much would the butter weigh without the firkins? Ans. 630 pounds 21. A man died, leaving 12426 dollars in cash. He directed in his will that 1000 dollars should be given to his niece; and that the remainder should be equally divided between his two nephews? What is the share of each nephew? Ans. 5713 dollars 22. A farmer, who had a farm of 520 acres, bought an adjoining one of 375 acres, and divided the whole equally among his 5 sons. How many acres had each son? Ans. 179 acres 23. A merchant bought 5 pieces of linen containing 25 yards each, and 2 pieces containing 24 yards each and 1 piece containing 26 yards. How many yards were there in the whole? Ans. 199 yards, 24. Three boys bought 3 baskets, each contain- ing 150 apples, and 2 barrels, each containing 540 apples. They found 219 to be rotten, which they threw nway and divided the rest equally among themselves, How many had each for his share? Ans. 437 apples FEDERAL MONEY. The denominations of Federal money, or the money )f the UNITED STATES, are, Eagle, Dollar, Dime, Cent, lid Mill. TABLE. 10 Mills (m) make 1 Cent, c. 10 Cents - - 1 Dime, d. 10 Dimes - - 1 Dollar, /?, or 10 Dollars - - 1 Eagle, E. In writing Federal Money, it is customary to omit Uagles, Dimes, and Mills, and set down sums in dol- ars, cents, and parts of a cent. The parts of a cent generally used are, halves, thirds, and quarters. Thus, is a half; * a third; \ a quarter. Exercises for tfic learner. 1. How many mills make a cent? How many half a cent? How many a cent and a half? How many two cents? 2. How many halves of a cent make a cent? 3. How many thirds of a cent make a cent? 4. How many fourths of a cent make a half cent? 5. How many fourths make a cent? 6. How many cents make one fourth or quarter of a dollar. 7. How many cents make a half dollar? 8. How many cents make three-fourths of a dollar? 9. How many cents make a dollar? 10. How many dollars and cents in one hundred and ten cents? How many in two hundred and six cents? How many in three hundred and forty-eight cents? How many in five hundred and one cents? 11. If you give a dollar for a book; thirty cents for a slute, and one cent for a pencil, how many cents will you give for the whole? 12. Write down one dollar arid eight cents. TWO dol- lars and sixteen cents. Twenty dollars and five cents? 13. Write down three hundred dollars and forty cents 14. Five hundred eighty-four dollars and fifty cents. 4 O* FEDERAL MONEY. 15. Eight hundred sixty dollars and sixty-seven cents. 16. Four thousand eight hundred dollars and two cents. 17. Six hundred thirty-one dollars fifty-six and a fourth cents. 18. Nine hundred and eighty-seven dollars. 19. Thirty -two thousand five hundred dollars eighty seven and a half cents. 20. Ten dollars sixty-eight and three-fourth cents. 21. Twelve dollars ninety-three and three-fourth cents. 22. Twenty dollars thirty-seven and a half cents. 23. Thirty-three dollars thirty-three and a third cents. 24. Sixty dollars sixty -six and two third cents. 25. Read the following sums, viz. $3448.87* $3450.25 $47967.91 $7.10 $115.334 $170.931 $19.01 $85.06^ ADDITION OF FEDERAL MONEY. RULE. Begin at the right hand side of the sum, add one row of figures at a time, and carry one for every ten, from the lower denomination to the next higher, as in Simple Addition, until the whole is added. When you come to the hist row on the left hand, instead of setting down what remains over ten, twenty, or thirty, &c. set down the fall amount. Note. When there are parts of a cent in a sum, such is halves, &c. find the amount of them in fourths of a lent; consider how many cents these fourths will make, ind add them to the first row in the column of cents. When the parts of a cent are not sufficient to make a ^ent, place their amount at the right hand of the column -jf cents, as in the first example ; and when the parts of i cent make one cent or more, and some parts remain, but not enough for another cent, the parts thus remaining mast be set down in the same way, according to the se- cond example. The proof is the same as in Simple Ad- dition. FEDERAL MONEY. 35'' EXAMPLES. I. D. cts. 5432.12* 1234.564 7898.76 5432.12 3456.78 ii. D. cts. 324.87* 987.431 720.30 842.431 103.62* in. D. cts. 885.90 125.87* 440.40 867.12i 390.97 IV, D. cts. 987654.32 123456.78 87654.32 123000.45 678987.65 23454.341 2975.67* 2710.27 2900753.52 APPLICATION. 1. A man bought a farm in five parcels; for the first, he gave $250.75; for the second, $350; for the third, $475,87* : for the fourth, $550; and for the fifth, $600. What was paid for the farm? Ans. 2226.62* 2. A merchant, in buying gave for flour, $325.43! ; for sugar, $854.25; for molasses, $520.62* ; for coffee, $944.50 ; and for cotton, $6427.12*. What was the sum paid? Ans. $9071.931. 3. What is the amount of 10* cents; 93! cents; 87* cents; 50 cts.; 314 cts.: 43! cts.; and 11 dollars? Ans. $14.16! cents. 4. Gave for an Arithmetick 314 cents; for asiate, 37* cents; for quills, 50 cents; for an inkstand, 62* cents; for a Geography, 1 dollar, and for a History, 87* cents. How much do they amount to? Ans. $3.68! cents 5. Add $75212.50, $544225.75, $4587220.50, $90000, and $5876432.75. SUBTRACTION OF FEDERAL MONEY. RULE. Place the smaller sum under the larger, setting the dollars under dollars and cents under cents, and proceed as in Simple Substraction. When there is a fraction, or part of a cent in the upper line of figures, and none in the lower, set it down at the right of the remainder, a.-- 36 FEDERAL MONEY. a part of the answer. When there is a fraction in each line, and the upper one is the larger, subtract the lower one from it and set clown the difference ; but if the lower one is larger than the upper, subtract it from the num- ber that it takes of the fraction to make a cent add the difference to the upper one, and set down the amount. When there is a fraction in the lower line and none in the upper; subtract the fraction irom the number that it takes of it to make a cent, and sat down the remain- der. In this case, and likewise when the part or frac- tion I clow is larger than the upper one, it is necessary to cany one to the right hand figure of the lower row of cente^ KXAXPLF.3. IT. ni. iv. 7). c. D. c. D. c. 587.25 687.31 9000.43 292.50 500.81 8220.314 $277.15 $294.75 $37.50 $780.111 v. vr. vn. vin. I), c. D. c. D. c. J). r. 6U5.624 820.431 5078.314 9810000.12! 457.87* 790.37* 4689.031 1037654.68* $237.75 $30.064 $1288.37* $7822345.44-! 9. Subtract $ 387.20 from glOOO. 10. Subtract $'5871.31:1, from $5430.87*. 11. Take $44.874 from 300 dollars. 12. Take $11000, from $19876.87*. APPLICATION. 1. Bought goods amounting to $3875.62*, and hav- ing paid $2350.93*; how much remains due? Aw. 2024.681. 2. My account .ipiirtst my neighbour amounts to $759. 25; and his account against me is $346.87*. How | much does he owe me? " ADO. 212.37*. FEDERAL MONEY. 37 3. Having bought a quantity of goods at $5425, and sold them at $3932.681. How much did I make on the goods? Ans. $1507.6^. 4. A owes me $11587.50, but having failed in busi- ness, he is able to pay $9263.62 L How much do I lose? Ans. $2323.874 5. Subtract $3427.874, from $9000. Ans. $572.124. MULTIPLICATION OF FEDERAL MONEY. BULE. Set the multiplier under the sum, and proceed as in Simple Multiplication, carrying one for every ten from a lower to a higher denomination, until the whole is multiplied. After the sum is done, separate, by a pe- riod, the two right hand figures of the product for cents, and the figures at the left hand of the period will be dollars. Note. When the sum to be multiplied contains a frac- tion, or part of a cent, multiply it by the multipler, and consider how many cents are contained in its product. Then multiply the first figure of the cents and add to its product the cents contained in the product of the fraction, and proceed as directed above. In multiplying a fraction, if you find in the product one cent or more, and a remainder not large enough to make another cent, set down the fraction at. the right hand of the product, that is under the row of fractions or parts of a cent. When there is a fraction in the sum, and the multiplier exceeds 12, multiply the sum without the fraction, and afterwards multiply the fraction and add it to the sum. EXAMPLES. I. II. III. IV. V. D. c. D. c. D. . D, c. D. c. 124.10 830.12* 172.30 2451.624 275.431 23 4 5 12 248.20 2490.374 689.20 12258.12* 3305.25 FEDERAL MONEY. VII. D. C. 3120.17 24 1314600.88 1643251.1 2957851.98 12480.68 62403.4 74884.08 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. Multiply $420.50 519.75 99.62* 75.314 62.12* 750.25 330.12* 248.87* 95.931 24.17 37.50 58.931 9876.624 by by by by 3 4 5 by 6 7 Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. Ans. by by 8 by 9 by 12 by 10 by 28 by 36 by 208 Ans. 2054338.00 1542.17 6609.3 94 8151.564 $841.00 1559.25 398.50 376.564 372.75 5251.75 2641.00 2239.87* 1151.25 241.70* 1050.00 2121.75 APPLICATION. 1. How much will 18 barrels of flour cost, at 3 dol- lars per barrel ? Ans. 54 dollars. 2. What will 35 pounds of coffee cost, at 20 cents per pound? Ans. 7 dollars. 3. Sold 87 barrels of flour, at $3.12* per barrel. What was the amount? Ans. $271.87*. 4. Bought 160 acres of land, at $1.25 per acre. What did the whole cost? Ans. 200 dollars. 5. What will- 225 bushels of apples cost, at 62* cents per bushel? Ans. 140.62*. 6. What will 5SO bushels of salt cost, at $1.12* per btishe ? Ans. $652.50, *Io multiplying by 10, when there is no fraction in the sum, it is necessary to add a cypher to the rierht hand of the sum, placing the period that separates cents from dollars one iigure farther towaros the rlerht hand, anrl the sum is done. In multiplying by 100 t add two cy- phers: by 1000, thret:-, &c. FEDERAL BS.OKEY. 3J DIVISION OF FEDERAL MONEY. RULE. Proceed as in Simple Division. When the sum con- sists of dollars and cents, the two right hand figures of the quotient will be cents. When there is a remainder, multiply it by 4, adding the number of fourths that are in the fraction of the sum (if there be any) to its pro- duct; then divide this product by the divisor, nnd its quotient will be fourths 3 , which must be annexed to the quotient of the sum. When the sum consists of dollars only, if there he a remainder, add two cyphers to it; then divide by the divisor as before, and its quotient will be cents, which must be added to the quotient of the sum. When the sum is in dollars, and the divisor is larger than the dividend, add two cyphers to the divi- dend then divide, and the quotient will be in cents. EXAMPLES. I. II. III. IV. D. c. D. c. D. c. D. c. D. c. 2)120.50 4)SOOO.OO 5)580.75 7)34.4C(12.07 210.25 2000.00 HG.io 14 14 49 49 T. VI. VII. D. c. D. c. D. c. D. c. DC (27.81(3.09 12)144.60(12,05 38)162.36(4.51 27 12 144 81 24 . 183 81 24 180 60 36 60 36 40 FEDERAL MONEY* VIII. IX. D. c. D. c. D. c. D. c. 35)1234.72(34.291 44)87654.32(1992.144 108 44 154 436 144 396 107 405 72 396 352 94 324 88 28 63 4 44 36)112(3 192 108 176 4 16 . ' 4 44)64(1 44 20+ 10. Divide $840 by 12 11. 717.12J by 8 12. " 246.25 by 9 13. 687 . 20 by 12 14. 980 by 34 15. 87654 by 128 16. 1284 by 112 17. 40000 by 188 18. 976 . 87* by 225 19. " 1334.37* by 212 20. 9876 . 44 bv 345 21. " 89786.54 by 374 FEDERAL MONEY. 41 APPLICATION. 1. Divide 400 dollars, equally, among 20 persons. What will be the portion of each person ? Ans. $20. 2. Divide 1728 dollars, equally among 12 persons. What does each one of them share? Ans. $144. 3. If 240 bushels cost 420 dollars; what Is the cost of one bushel at the sams rate? Ans. $1.75. Promiscuous Examples. 1. What will the following sums amount to, when ad- ded together, namely r $124.62* ; $248.874 ; $342.40 ; $9850.25. and $20.314? Ans. $10588.464. 2. If my estate is worth 12870 dollars, and 1 meet with losses amounting to $4364.50, how much shall J have left? Ans. $8505.50. 3. A merchant enters into a trade by which he re- ceives $1324.624 per yeai\ for four years: how much is his whole gain? Ans. $5298.50. 4. An estate of 98740 dollars is to be divided, equally, between 8 heirs; what will each receive? Ans. $12342.50. A bought of B, 1 barrel of sugar at - - $24.50 1 chest of tea, - - - 60.00 1 hogshead of salt, - - 3.75 20 yards of cloth, - - - 15.00 1 barrel of flour, - 3.87 i Ans. $107.124. Q. 1. What are the denominations of Federal Money ? 2. How many mills make a cent? 3. How many cents make a dime? 4. How many dimes make a dollar? 5. How many dollars make an eagle? 6. How are the denominations generally used in writing Federal Money, and in reckoning? 7. Where is Federal Money used as a currency? Answer. In the United States of North America. 4 42 FEDERAL MOXEY. Exercises for the slate in Federal Money. 1. What is the sum of 50 cents and 5 dimes? Ans. 1 dollar 2. A man owed $35.46.5, and paid $27.69.6; how [much did he th.on owe? Ans. $8.76.9 3. A man laboured 6 months at 25 dollars 6 cents and 5 mills per month. How mr.ch did his wa amount to? Ans. $150.3$ 4. If you divide $35001.50 equally among 125 men how many dollars will each have? Ans. $280.01,2 5. Bought an umbrella for four dollars, a penknife for 37 cejts and 5 mills, a hat for 3 dollars and 6* cents, a cane for one dollar and 5 dimes, and a book for one dollar and a quarter. How much did he p for all these articles. Ans. $10.19 6. What will 1473 peaches come to at 5 mills u piece? Ans. 7 dollars 36 cents 5 mills, or $7.38.5 7. A lady bought 8 yards of silk for 10.37.6. How much is that a yard? Ans. $1.29.7 8. A boy sold a top for a dollar, and gave away 12 2 cents of the money. How much had he left? Ans. 87i cents 9. Add two mills, two cents two dimes, and two dol- ars together. Ans. $2.22.2 19. You borrow $535.15 and pay $236.18; how much remains unpaid. Ans. 299.97. 11. Going a journey I took two 50 dollar bills to oear my expenses, which were as follows, viz: Stage 'are eighteen dollars; board, nine dollars and fifty cents; carrying trunk, seventy-five cent$; private convey- ance at one time, six dollars and 37 i cents; and at an- ather, seven dollars. How much had I left, on my re- urn home? Ans. 58^37* . 12. From five dollars take one mill? Ans. $4.99.9. 13. From 4 dollars take 3 dollars 99 cents and 9 mills. Ans. $0.00.1. 14. What will 2640 bushels of oats cost at 12* cents [>er bushel? Ans. $330. 15. A man paid $fr.60 for 12 yards of cloth. How much was the cloth a yard? Ans. $2.30. FEDERAL MONEY. 43 16. If John spends 6 cents a day, how many would he spend in a year, or 365 days? Ans. $21.90. 17. I bought a wagon for 70 dollars, and paid $9.124 for repairing ir, and sold it again for $75.624. Did I make or lose by the bargain? and how much? Ans. Lost $3.50. 18. I bought at a store a pair of gloves for 75 cents, a vest pattern for $1.12* , two yards of cloth for a coat lat $2.874 per yard, and trimmings for 18! cents; and jto pay the bill, I gave the merchant an eagle. How much should he give me in change? Ans. $2.18!. 19. Three men sold their marketing for 10 dollars. What would be the share of each, if the money were divided equally among them? $3.334 20. A lady went a shopping with the sum of 55 dol- lars and her purchases amounted to 29 dollars and 624 |cents; how much had she remaining? Ans. $25.374. 21. If a person pay for board $3.50 a week, how much will he pay in 26 weeks? Ans. $91.00. 22. A person bought a box of tea weighing 14 pounds for $17.50; how much is it a pound? Ans. $1,25. 88. A gentleman rents a house for 650 dollars a year; how much is it a month? Ans. $54.16.6. 24. A person has due to him the following sums of money, viz: 574 dollars 64 cents; 425 dollars 25 cents; 175 dollars 68! cents; 341 dollars 874 cents; 1021 dol- lars 124 cents. How muclxis due to him in all? Ans. $2538.00. 25. Bought 35 yards of cloth at 2 dollars and 64 cents a yard; 51 pair of shoes at 874 cents a yard; 10 'dozen of buttons, at 16! cents a dozen, and 75 hats, at $3.25 a piece. To how much did the whole amount? Ans. 362.43!. 28. Four men about to descend the Ohio river in a boat, laid in the following provisions, viz: 26 loaves of bread, at 64 cents a loaf: 41 pounds of ham, at 124 cents a pound ; 17 pounds of coffee at 18! cents a pound, 15 pounds of cheese at 9 cents a pound, and paid for sundry other articles $6.434. How much did the whole amount to, and what was the share of each? Ans. $17.72. Each man's share $4.43. TABLE OF MONEY, WEIGHTS, MEASURES, &c^ ENGLISH MONEY. A table of Federal Money has already been given. The denominations of English Money are pound, shil- ling, penny, and farthing. 4 farthings (qr.) make 1 penny d. 12 pence - 1 shilling s. 20 shillings - 1 pound OirFarthings are written .as fractions, thus : 4 one farthing. h two farthings, or a half-psnny. 1 three farthings. PENCE TABLE. SHILLING TABLE. d. s. d. s. * 20 pence make 1 8 20 shillings make 1 30 " "-26 30 1 10 40 "-34 40 2 50 "-42 50 2 10 60 " "-50 60 3 70 " " - 5 10 70 3 10 80 " "-68 80 4 90 " "-76 90 4 10 100 " "-84 100 5 110 " "-92 110 5 10 120 " - 10 120 6 240 - 20 130 6 10 TABLE CP WEIGHTS AND MEASURES. 45 TROY WEIGHT. By this weight, jewels, gold, silver, and liquors are weighed. The denominations of Troy Weight are pound, ounce, pennyweight, and grain. 24 grains (gr.) make 1 pennyweight dwt. 20 pennyweights 1 ounce ox. 12 ounces - 1 pound lb, AVOIRDUPOIS WEIGHT. By this weight are weighed things of a coarse dros- sy nature, that are bought and sold by weight; and all metals but silver and gold. The denominations of Avoirdupois Weight are ton, hundred weight, quarter, pound, ounce, and dram. 16 drams, (dr.) make 1 ounce - - ox. 18 ounces - 1 pound - - lb. 28 pounds - 1 quarter of a cwt. qr. 4 quarters, or 112 lb. 1 hundred weight cwt. 20 hundred weight. 1 ton. - - T. APOTHECARIES WEIGHT. By this weight apothecaries mix their medicines, but buy and sell by Avoirdupois Weight. The denominations of Apothecaries Weight are pound, ounce, dram, scruple, and grain. 20 grains (gr.) make 1 scruple 9 3 scruples - 1 dram 3 8 drams - - - 1 ounce 3 12 ounces - 1 pound fc LONG MEASURE. Long measure is used for lengths and distances. The denominations of Long Measure are degree, league, mile, furlong, pole, yard, foot, and inch. 45 TABLE CF MEASURES. 12 inches (in.) make 1 foot - ft. 3 feet 1 yard - - yd. 5i yards, or 10 foot 1 rod, pole, or perch P. 40 poles (or 220 yds.) 1 furlong - - fur. 8 furlongs (or 1760 yds.) 1 mile - - M . 3 miles 1 league - L. 60 geographick | l , , _ . or 6fii statute \ Note. A hand is a measure of 4 inches, and used in measuring the height of horses. A fathom is 6 feet, and used chiefly in measuring the depth of water. CUBICK, OR SOLID MEASURE. By Cubick, or Solid Measure, are measured all things that have length, treadth and thickness. Its denominations are, inches, feet, ton, or load, and cord. 1728 inches make - 1 cubick foot. 27 feet - - -1 yard. 40 feet of round timber) or 50 feet of hewn> - 1 ton or load, timber. ) 128 solid feet, i. e. 8 in) length, 4 in breadth,) 1 cord of wood and 4 in height. ) LAND, OR SQUARE MEASURE. This measure shows the quantity of lands. The denominations of land Measure are acre, rood, square perch, square yard, and square foot. 144 square inches make 1 square foot ft. 9 square feet 1 square yard yd. 304 square yards - 1 square perch P. 40 square perches 1 rood - J?. 4 roods 1 acre - A. 640 acres 1 mile - m. TABLE OF MEASURES. 47 CLOTH MEASURE. By this measure cloth, tapes, &,c. are measured. The denominations of Cloth Measure are English ell, Flemish ell, yard, quarter of a yard, and nail. 4 nails (na.) make 1 quarter of a yard qr, 4 quarters 1 yard - yd. 3 quarters - 1 ell Flemish - E.F1. 5 quarters - - 1 ell English - E. E. 6 quarters - - 1 ell French - E.F. DRY MEASURE. This measure is used for grain, fruit, salt, &c. The denominations of Dry Measure are bushel, peck, quart, and pint. 2 pints (jpt.) make 1 quart - - qt. 8 quarts - 1 peck - pe. 4 pecks - 1 bushel bu. WINE MEASURE. By Wine Measure are measured Rum, Brandyy Per- ry, Cider, Mead, Vinegar and Oil. Its denominations are pint, quart, gallon, hogshead, pipe, &c. 2 pints (ptf.) make 1 quart - qt. 4 quarts - 1 gallon - gal, 42 gallons 1 tierce - tier. 63 gallons 1 hogshead - kfid. 2 hogsheads - 1 pipe or butt P. or J3. 2 pipes - 1 ton - T. ALE, OR BEER MEASURE. The denominations of this measure are pint, quart, gallon, barrel, &c. 4* [8 TABLE OF TIME AND MOTION. 2 pints (pt.) make 1 quart - qt. 4 quarts - - 1 gallon - - gal. 8 gallons - - - 1 firkin of ale - fir. 2 firkins - 1 kilderkin - kil. 2 kilderkins - 1 barrel - bar. 11 barrels, or 54 gallons 1 hogshead of beer hJid. 2 barrels - 1 puncheon - pun. 3 barrels, or 2 hogsheads 1 butt - butt. TIME. The denominations of Time are year, month, week, tay, hour, minute and second. 60 seconds (sec.) make 1 minute - min. 60 minutes - 1 hour H. 24 hours - 1 day ' - D. 7 days - 1 week - W. 52 weeks, 1 dav, and 6 hours, I 1 v or 365 days, and 6 hours, \ L >' es 12 months (mo.) 1 year - Y* Note. The si# hours in each year are not reckoned ill they amount to one day; hence, a common year con- sists of 365 days, and every fourth j r ear, called leap year, of 366 days. The following is a statement of the number of days in each of the twelve months, as they stand in tho cal- ender or almanack : The fourth, eleventh, ninth, and sixth, Have thirty days to each affix'd : And every other thirty-one, Except the second month alone, Which has but twenty-eight in fine, Till leap year gives it twenty-nine. MOTION. 60 seconds w make ] prime minute, 60 minutes - 1 degree - 30 degrees - 1 si^n 9. 12 signs, or 360 degrees > n ^ REDUCTION. Reduction teaches to change numbers of one denomi- nation into those of other denominations, retaining the same value. Its operations are performed by Multipli- cation and Division. When performed by Multiplica- tion, it is called Reduction Descending, when performed by Division, it is called Reduction Ascending. QUESTIONS. 1. How many farthings will it take to make two pence ? How many pence to make two shillings ? How many shillings to make two pounds? 2. How many gills to make three pints? How many pints to make three quarts? How many quarts to make three gallons? 3. How many quarts to make four pecks? How ma- ny pecks to make four bushels? 4. How many pence are there in eight farthings? How many shillings in twenty-four pence? How many pounds in forty shillings? 5. How many pints in twelve gills? How many quarts in six pints? How many gallons in twelve quarts? 6. How many pecks in thirty-two quarts? How ma- ny bushels in sixteen pecks? 7. How many pounds and shillings in thirty shillings? How many shillings and pence in thirty pence? REDUCTION DESCENDING. RULE. Multiply the numbers in the highest denomination given, by the number that it takes of the next less de- nomination to make one of that greater; and thus pro- ceed until you shall have multiplied each higher de- nomination by the number that it takes to form the next lower, until you come to the lowest of all. REDUCTION. PROOF. Descending and Ascending Reduction prove each other. SIMPLE EXAMPLES. I. Reduce 25 pounds to shillings 25 20 shillings in a pound. 500 shillings. Ans. 500 shillings. n. Reduce 50 shillings to pence. 50 12 pence in a shilling, 600 pence. Ans. 6CO pence, in. Reduce 15 pence to farthings. 15 4 farthings in a penny. 60 farthings. Ans. 60 farthings IV. Reduce 10 tons to hundred weights. 10 20 hundreds in a ton. 200 hundreds. Ans. 200 cwt REDUCTION. 51 V. Reduce 36 pounds to ounces. 36 16 ounces in a pound. 216 3G 576 ounces. Ans. 576 ounces. 6. Reduce 70 miles to furlongs Ans. 560 fur. 7. Bring 30 furlongs to rods. Ans. 1200 rods. 8. Bring 20 rods to feet. Ans. 330 feet. 9. Bring 24 feet to inches. Ans. 288 inches 10. Reduce 32 acres to roods. Ans. 128 roods. 11. Bring 24 square perches to square yards. Ans. 726 square yards. 12. Reduce 10 hogsheads to gallons. Ans. 630 gal. 13. Bring 25 gallons to pints. Ans. 200 pints. 14. Reduce 23 bushels to pecks. Ans. 92 pecks, 15. Bring 12 pecks to pints. Ans. 192 pints 16. Reduce 15 years to months. Ans. 180 months 17. Bring 75 days to hours. Ans. 1800 hours 18. Bring 24 hours to minutes. Ans. 1440 minutes 19. Bring 10 signs to degrees. Ans. 600 degrees COMPOUND EXAMPLES. I. . s. d. qrs. In 15 17 11 3 how many farthings ! 20 shillings in a pound. 317 shillings. 12 pence in a shilling. 3815 pence. 4 farthings in a penny. 15263 farthings. 52 REDUCTION. Note. In multiplying by 20, I added in the 17 shil- lings, by 12, the 1 1 pence ; and by 4, the 3 farthings ; and this must be observed in all similar cases. To prove this sum, let the order of it be changed, and it will stand thus: in 15263 farthings how many pounds ? 4)15263 12)3815+3 quarters. 2|0)31 |7-}-ll pence. 15 17s. 11 d. 3 qrs. Ans. In reducing Federal Money from a higher to a lower denomination, it is only necessary to annex as many cy- phers as there are places from the denomination given to that required ; or if the given sum be of different denominations, annex the figures of the several denom- inations in their order, and continue with cyphers, when the sum requires it, to the denomination intended. 2. Thus, in 7 eagles, 3 dollars, how many mills? Ans. 73000. 3. In 85 dollars, how many mills? Ans. 85000. 4. In 574 eagles, how many dollars? Ans. 5740. 5. In 469 dollars, how many cents? Ans. 46900. 6. In 844 dollars, 75 cents, how many mills? Ans. 844750. 7. In 1000 dollars, how many mills? Ans. 1000000. 8. In 25 dollars, 47 cents, 8 mills ; how many mills ? 9. In 29 guineas at 28s. each, how many pence ? Ans. 9744. 10. In 20 acres, 29 poles, or perches, how many square perches? Ans. 3229. 11. How many solid feet in 30 cords of wood? Ans. 3840. 12. How many grains in 100 Ibs. Troy Weight? Ans. 576000. 13. How many Ibs. in a ton: Avoirdupois Weight? Ans. 2240. REDUCTION. 53 14. In 27 Ibs. Apothecaries Weight; now many grains? Ans. 155520. 15. In 30 yards, how many nails? Ans. 480. 16. In 360 degrees, being the distance round the world, how many inches, allowing 69i miles to a de- gree? Ans. 1,587,267,200. 17. How many pints are there in one tun of wine? Ans. 2016. 18. How many half pints in one hogshead of beer? Ans. 864. 19 How many pints in 400 bushels? Ans. 256CO. 20. How many seconds in 80 years of 365 days each ? Ans. 2,522,880,600 21. How many yards in 4567 miles? Ans. 603 "A&O. 22. In 20 17s., how many pence and half pence? Ans. 5004 pence, and 10,008 half pence. REDUCTION ^ASCENDING. RULE. Divide the figure or figures in the lowest denomina- tion, by so many of that name as make one of the next higher; and continue the division until you have brought it into that denomination which your question requires. In reducing Federal Money from a lower to a higher denomination, nothing more is necessary than to cut off so many places on the right hand side of the sum, as there are denominations lower than the one required. Thus, 98765 mills are reduced to dollars, cents, and mills, by cutting off one figure for mills, two more for cents, and the remaining figures being dollars, the amount is $98|76|5 or ninety-eight dollars, seventy-six cents, five mills. SIMPLE EXAMPLES. 1. How many dollars are there in 8000 mills? 8|00|0 Ans. 8. 2. In 487525 cents, how many dollars and cents? 4875|25 Ans. $4875.25. 54 REDUCTION. 3. In 999888 mills, how many dollars, cents and mills? 999|8S|8 Ans. $999.88.8. 4. In 19200 farthings, how many pounds? 4)19200 12)4800 20)400 Ans. 20 pounds. 5. In 480 nails, how many yards? 4)480 4)120 30 Ans. COMPOUND EXAMPLES. 6. In 52300 farthings, how many pounds? 4)52300 12)13075 2|0)108|9+7 Ans. 54 9s. 7d. 7. In 8828 Ibs. Avoirdupois Weight, how many tons? Ans. 3 tons. 18cwt. 3qrs". 81bs. 8. In 524 Ibs. Avoirdupois Weight, how many cwt. &c. Ans. 4 cwt. 2 qrs. *201bs. 9. In 253440 grains, Troy Weight, how many Ibs? Ans. 44. 10. In 155520 grains, Apothecaries W T eight, how many pounds? Ans. 27. REDUCTION, 11. How many miles are therein 1,585,267,200 in- ches? Ans. 25020. 12. In 4000 nails, how many yards? Ans. 250. 13. In 8000 square rods, how many acres? Ans. 50. 14. In 2016 pints of wine, how many tuns? Ans. 1. 15. How many bushels are there in 80,000 quarts? Ans. 2500. 16. In 2,522.880,000 seconds, how many days? Ans. 29,2CO. 17. In 3840 solid feet, how many cords? Ans. 30. 18. In 1728 half pints of beer, how many hogsheads? Ans. 2. 19. Bring 240,000 pence to pounds. Ans. 1000. 20. Bring 112 quarters to cwt. Ans. 28 cwt. 21. Bring 120 miles into leagues. Ans. 40L. 22. Bring 1280 poles into furlongs. Ans. 32 fur. 23. Reduce 960 nails to quarters. Ans. 240 qrs. 24. Reduce 17280 cubick, or solid inches, to solid feet. Ans. 10 solid feet. 25. In 768 pints, how many bushels? Ans. 12. 26. In 1890 gallons, how many hogsheads? Ans. 30. Q. 1. What does Reduction teach? 2. By what rules are its operations performed ? 3. When performed by multiplication, what is it called? 4. What is your rule for Reduction Descending? 5. When performed by Division what is it called? 6. What is your rule for Reduction Ascending? 7. How do you reduce Federal Money frorn a, lower to a higher denomination? 8. How is Reduction proved' 56 COMPOUND ADDITION. Compound Addition teaches to add numbers which represent articles of different value, as pounds, shillings, pence; or yards, feet, inches, &c. called different de- nominations. The operations are to be regulated by the value of the articles, which must be learned from the foregoing table. RULE. Place the numbers to be added so that those of the same denomination may stand directly under each other. Add the figures of the first column or denomination to- gether, and divide the amount by the number which it takes of this denomination to make one of the next high er. Set down the remainder, and carry the quotient to the next denomination. Find the sum of the next col- umn or denomination, and proceed as before through the whole, until you come to the last column, which must be added by carrying one for every ten as in Simple Addition. EXAMPLES. I. ii. in. s. d. qrs. s. d. qrs. . s. d. qrs. 14 10 8 2 19 19 11 3 18 17 11 3 11 16 10 1 10 14 4 1 15 14 10 3 8 3 11 3 13 13 10 2 17 18 9 2 34 11 6 2 44 8 2 2 52 11 8 OAns In the first of the above examples, I begin with the right hand column, or that of farthings; and having ad ded it, find that it contains 6. Now, as 6 farthings con tain 1 penny and 2 over, I set the 2 farthings, under the column of farthings, and carry the penny to the column of pence. In the column of pence I find 29, which, with COMPOUND ADDITION. 57 the one carried from the farthings, make 30. In 30 pence I find there are 2 shillings and 6 pence over : set- ting the 6 pence under the column of pence, I add the 2 shillings to the column of shillings. In this column are 29, and the 2 added make 31. Thirty-one shillings con- tain 1 pound, and 11 shillings over. The 11 shillings are then placed under the column of shillings, a.rA the 1 is carried to the column of pounds. In that column are 33 pounds, which, with the 1 added, make 34. Thus the amount of the sum is 34 pounds, 11 shillings, 6 pence, and 2 farthings. In ail cases in Compound Addition, one must be car- ried for the number of times that the higher denomina- tion is contained in the column of the lower denomination. Thus, in Troy Weight : as 24 grains make one penny- weight, one from the column of grains is carried for every 24; in the column of pennyweights, one for every 20; and in every instance the learner must be guided by the foregoing table of "Money, Weights, Measures, &c. ir. v. s. d. qrs. . s. d. qrs. 487 16 11 3 9876 15 4 3 830 10 9 1 2123 14 5 500 11 4 2 6789 18 10 2 620 18 3 3 1234 15 II '-1 900 8 10 7876 493 Note. Sums in Compound Addition may be proved in ihe same manner as in Simple Addition. TROY WEIGHT. VI. VII. Ib. oz. dwt. gr. Ib. oz. dwt. gr. 487 10 18 22 6780 11 11 12 500 8 11 10 1100 9 18 22 234 11 10 16 3090 10 17 20 876 3 17 23 2468 8 13 19 58 COMPOUND ADDITION. AVOIRDUPOIS WEIGHT. VIII. IX. Ton. cwt. qr. Ib. oz. dr. Ton. cut. qr. Ib oz 16 18 2 25 11 14 27 17 3 27 8 97 12 3 17 9 11 98 19 2 11 <j 34 11 1 10 10 10 70 11 1 18 7 82 19 2 27 15 13 18 16 10 6 APOTHECARIES WEIGHT. x. xi. fc 3 3 gr. fc 3 3 gr. 74 9 7 1 13 20 10 7 1 18 18 11 6 2 17 37 11 5 2 17 91 10 3 10 28 9 3 1 15 17 9 5 1 19 14 8 4 11 LONG MEASURE. XII. XII. deg. mil. fur. po. ft. in. mil. fur. po. yd. ft. 118 36 7 19 13 3 976 2 13 4 2 921 15 4 16 10 10 867 6 10 3 1 671 10 6 27 11 11 500 1 11 643 26 5 15 8 8 123 4 15 3 2 123 14 5 16 7 8 345 6 17 1 CUBICK, OR SOLID MEASURE. XIV. XV. XVI. Ton ft. in. yd. ft. in. Cord ft. in. 17 10 1229 29 20 1092 48 120 1630 24 13 1460 11 11 1195 54 110 1500 98 25 1527 18 11 1000 75 88 1264 18 16 1079 27 9 1330 87 113 1128 COMPOUND ADDITION. 59 LAND OR SQUARE MEASURE. XVII. XVIII. XIX. acr. TOO. per. 987 2 23 acr. roo. per. 8423 1 38 acr. roo. per. 9432 3 24 798 3 .28 1234 10 4324 2 12 123 2 11 4821 3 11 5678 1 36 567 1 27 6789 2 30 5865 3 11 700 00 8000 1 13 8765 2 15 CLOTH MEASURE. XX. XXI. XXII. yd. qr. riL 175 3 3 El. Fr. qr. nl. 247 2 3 El. Fl. qr. nl. 9876 2 3 481 2 1 456 1 1 8765 1 2 234 1 2 345 3 3456 2 3 345 1 236 2 2 4000 234 1 2 567 1 7898 2 3 XXIII. XXIV. XXV. , El. E. qr. nl 87654 1 2 yd. qr. nl. 656547 1 1 yd. qr. nl. 987654321 3 3 56788 3 1 987654 2 234567876 87654 3 2 765432 1 3 543212345 3 2 12345 134545 3 2 900087654 1 3 84231 2 3 584050 1 384563200 3 DRY MEASURE XXVI. XXVII. XXVIII. buxh. pk. qt. 187 7 3 bush. pk. qt. 356 3 7 bush. pk. qt. 874 3 6 290 6 2 120 1 6 123 1 2 185 3 1 543 2 1 345 2 5 349 1 2 678 3 5 753 1 7 160 5 3 432 1 3 936 2 4 5* 60 COMPOUND ADDITION. WINE MEASURE. XXIX. XXX. Tun. khd. gal, qt. pt. Tun. hhd. gal. qt. pt 4820 1 16 2 1 987654 1 12 1 1 9765 3 18 3 1 321234 3 15 1 8645 2 19 1 125780 2 18 3 1 5432 1 22 3 1 876531 2 27 1 6787 1 10 1 248765 1 49 2 1 ALE OR BEER .MEASURE. XXXI. XXXII. khd. gal. qt. pt. hhd. gal. qt^ pt^ 4820 48 3 1 17819174 18 3 1 8765 34 1 1 21350000 27 1 1 9877 53 2 1 12168400 35 1234 12 1 21346870 15 S 1 5678 50 1 43212345 50 1 1 TIME. XXXIII. XXXIV. w. d. h. m. s. y. mo. w. d. h. m. s. 3 6 23 58 24 75 11 3 6 22 50 57 3 5 20 49 57 18 10 2 5 16 16 15 1 4 21 30 30 84 11 14 15 10 10 3 2 13 53 53 40 9 1 00 00 00 1 10 10 10 SO 10 1 1 11 11 11 MOTION. XXXV. XXXVI. XXXVII. 18 54* 44^ 26 19 X 15 n 10*. 24 53' 50" 20 25 30 19 26 20 90 19 31 87 30 10 50 15 19 39 23 42 00 11 11 33 10 11 8 17 44 45 27 29 34 12 34 31 7 10 20 10 COMPOUND SUBTRACTION. t)l Q. 1. What does Compound Addition teach? 2. How do you place the different denominations in Compound Addition? 3. How do you proceed after placing the denomina- tions under each other? 4. What do you observe, in carrying from one de- nomination to another, that is different from Simple Addition? 5. How is Compound Addition proved? COMPOUND SUBTRACTION. Compound Subtraction teaches to find the difference between any two sums of different denominations. RULE. Place those numbers under each other which are of the same denomination the less always being below the greater.* Begin with the least denomination, and if it be larger than the figure over it, consider the up- per one as having as many added to it as make one of the next greater denomination. Subtract the lower from the upper figure, thus increased, and set down the remainder, remembering, that whenever you thus make the upper figure larger, you must add one to the next superior denomination. PROOF. As in Simple Subtraction. EXAMPLES. ENGLISH MONEY, i. ii. s. d. qrs. s. d. qrs s. d. qr. 460 14 10 3 744 10 8 1 689 792 320 10 8 2 398 18 10 3 37218 4 3 140 4 2 1 345 11 9 S 316 943 *By this is meant that the lower line of. figure must always be a less sura than the upper 1'me, though sorae of its au^ller denominations may be larger man those immediately above theru, in the upper line 'JOMPOUND SUBTRACTION. The first example, is in itself, sufficiently plain. In the second, finding the upper figure smaller than the lower one, as it is in farthings, and ns four farthings make a penny, I suppose four added to the upper figure, which makes it 5. Then I say. 3 from 5, and 2 remain. Placing the 2 underneath, I add 1 to the next lower figure, name- ly, the 10, which thus becomes 1 1 ; and as the 8 standing above is less, I suppose 12 added to it, which makes it 20. Taking 11 f;om 20, remain. Setting the 9 un- derneath, and adding one to the 18, it becomes 19; and as the upper figure is smaller, I suppose 20 added to it, which makes it 30. I take 19 from 30, and 11 remain. Placing 4he 1 1 underneath, I carry one to the next figure, namely, 8; and then proceed as in Simple Subtraction. TROY WEIGHT. iv. v. $. oz. dwt. gr. Ib. oz. dirt. gr. 947 5 13 16 876543 7 16 11 123 10 18 20 549876 9 17 19 AVOIRDUPOIS WEIGHT. TI. VII. Ton. c\vt. qr. Ib. Ton. cu-t. qr. Ib. oz. dr. 5 13 1 12 8 16 24 11 11 1 11 3 16 6 18 2 26 12 13 APOTHECARIES WEIGHT. VIII. IX. fc 5 3 gr. ft 3 3 6 gr. 44 7 5 1 12 87 4 1 10 39 9 6 2 16 48 10 4 1 18 COMPOUND SU3TRACTION. C LONG MEASURE. X. XI. deg. mil, fur. po. ft. in. deg. mil. fur. po. 85 53 7 16 10 10 95 10 3 12 60 57 27 14 11 79 44 6 13 CUBICK, OR SOLID MEASURE, xn. xin. xiv. Ton ft. in. yd. ft. in. Cord ft. in. 18 17 1040 40 10 940 ' 874 110 1122 11 21 1485 32 16 1080 499 120 1699 LAND, OR SQUARE MEASURE. XV. XVI. XVII. acr. roo. per. acr. roo. per. acr. roo. per. 987 2 23 8423 1 36 9432 3 12 798 3 28 4123 10 7324 2 24 CLOTH MEASURE. xviir. xxix. xx. xxi. yd. qr. nl. E. E. qr. nl. E. Fl. qr. nl. E. Fr. qr. nl. 45 12 537 2 1 567 1 2 945 3 3 29 3 1 409 3 3 389 21 730 5 2 DRY MEASURE. XXII. XXIII. XXIV. bush. pk. qt. bush. pk. qt. bush. pk. qt. 74 1 1 230 a 56 1 1 42 3 2 199 2 1 28 3 3 64 COMPOUND SUBTRACTION, WINE MEASURE, xxv. xxvr. Tun. hhd. gal. qt. pt. Tun. khd. gal. qt. pt. 482 1 16 1 1 654 2 12 1 297 3 22 3 1 276 3 40 2 1 ALE OR BEER MEASURE. XXVII. XXVIII. hhd. gal. qt. pt. hhd. gal. qt. pt. 8240 12 1 1 11917400 10 1987 52 2 2 11654000 27 2 2 TIME. XXIX. XXX. w. d. h. m. s. y. mo. w. d. h. m. s. 8 2 12 42 30 20 10 1 4 10 27 37 1 1 16 54 40 11 11 35 17 40 54 MOTION. xxxi. xxxn. xxxm. 16 15' 35" 8s. 10 10' 10" 7s. 8 37' 47" 12 45 48 6 15 50 30 4 11 44 55 Application of the two preceding rules. 1. A B & C purchased goods in partnership. A paid 12 pound?, 10 shillings and 8 pence; B paid 124 pounds, 16 shillings; and C paid 8^ pounds and 11 pence. What was the whole amount paid? Ans. 224 7s. 7d. 2. A merchant has money due him : from one man, 587 pounds; from another, 420 pounds, 17 shillings and 6 pence; from a third, 200 pounds; and from a fourth, 978 pounds, 16 shillings and 8 pence. How much had he due in all? Ans. 2186 14s. 2d. COMPOUND MULTIPLICATION. 65 3. From 20 pounds, take 12 pounds, 19 shillings and 3 farthings. Ans. 7 Os. lid. Iqr. 4. From 22 pounds, take 19 shillings and 1 farthing. Ans. 41 Os. lid. 3 qrs. 5. From 17 pounds, take 9 pounds, 9 shillings and 9 pence. Ans. 7 10s. 3d. 6. A has paid B 7 2s. 3d. 19 lls. 4d. and 17 18s. Sid. on account of a debt of 30. How much remains unpaid? Ans. 15 7s. 7id. 7. A ropemaker received 3 tons, 4 cwt., 2 quarters, and 5 pounds of hemp; of which he made into cordage 2 tons, 9 cwt. and 1 quarter. How much had he left? Ans. 15cwt. Iqr. 51bs. Q. 1. What does Compound Subtraction teach? 2. H nv do you set down Compound Subtraction ? 3. What do you do when the lower denomination is larger than the one that is above it ? 4. How is Compound Subtraction proved? COMPOUND MULTIPLICATION. Compound Multiplication teaches how to find the value of any given number of different denominations, re- peated a certain number of times. It is of great use in finding t^e value of goods, which is generally done by multiplying the price by the quantity, CASE I. When the quantity or multiplier does not exceed 12. Set down the price of 1, and place the multiplier un- der the lowest denomination ; and in multiplying by it t observe the same rules for carrying from one denomina- tion to another as in Compound Addition. PROOF. Double the multiplicand, and multiply by half the multiplier : or, divide the product by the multiplier. 66 COMPOUND MULTIPLICATION* EXAMPLES. ENGLISH MONEY. 1. What will 7 yards of cloth cost, at 1 12s. per yard ? 11 10s. 3|d. In this example, 1 say 7 time 3 make 21 that is, 21 farthings, equal to five pence and one farthing. I set down the one farthing under the place of farthings, and carry the five pence to the place of pence saying, 7 times 10 are 70, and 5 make 75 pence equal to 6 shillings and 3 pence. I set down the 3 pence under the pence in the sum and carry the 6 shillings, saying 7 times 12 are 84, and 6 make SO shillings, equal to 4 pounds and 10 shil- lings. Setting down the 10 shillings under the shillings, I carry the 4 pounds, saying 7 times 1 are 7, and 4 make 11 pounds, making thefanswer to the question 1 1 pounds, 10 shillings, 3 pence and 1 farthing. II. III. IV. s. d. s. d. *. d. Multiply 4 14 101 7 12 9 14 15 94 by 2 4 8 9 9 91 V. VI. VII. *. d. *. d. *. d. Multiply 14 17 84 24 16 10* 50 15 5f by 9 7 12 TROY WEIGHT. VIII. IX. Z&. oz. diet. gr. Ib. oz. dwt. gr. Multiply 11 9 16 10 17 5 12 6 by 4 5 COMPOUND MULTIPLICATION. 67 AVOIRDUPOIS WEIGHT, x. xi. Ton. cwt. qr. Ib. oz. dr. Ton. cwt. qr. Ib. oz. dr. Mult. 3 11 3 10 5 4 6 17 3 13 2 15 by 6 8 APOTHECARIES WEIGHT. XII. XII. fc 3 3 6 gr. ft 3 3 6 gr. Mult. 43 10 6 2 10 4 10 7 2 13 by 5 6 LONG MEASURE. XIV. XV. dcg. m. fur. p. yd. ft. in. L. m. fur. p. Mult. 7 22 6 20 2 2 10 15 2 7 30 by 26 CUBICK. OR SOLID MEASURE. XVI. XVII. XVIII. Ton. ft. in. yd. ft. in. Cord ft. in. Mult. 10 16 15 14 2 19 24 13 18 by 2 4 LAND, OR SQUARE MEASURE. XIX. XX. XXI. ^ A. R. P. A. R. P. A. R. P. Mult. 20 3 12 37 2 15 47 1 18 bv 2 4 6 68 COMPOUND MULTIPLICATION. CLOTH MEASURE. XXII. XX1TI. XXIV. XXV. yd. qr. nl. ELFr. qr. nl. El. Fl. qr. nl. El. E. qr. nl MulU7 33 32 2 1 42 2 1 53 2 1 by 4 6 8 9 bush. pk. qt. Mult. 637 by 5 DRY MEASURE. XXVII. bush. pk. qt. 14 3 2 6 XXVIII. bush. pk. qt. 34 2 3 8 WINE MEASURE. XXIX. XXX. Tun. hhd. gal. qt. pt. Tun. hhd. gal. qt. pt, Mult. 1 2 12 by 3 1 4 40 3 1 10 ALE, OR BEER MEASURE. XXXI. XXXII. hhd. gal. qt. pt. hhd. gal. qt. pt. Mult. 3 12 2 1 4 15 3 1 by 5 8 XXXIII. y. mo. w. d, Mult. 7 735 by 9 XXXV. Mult. 24 19' 11" by 10 TIME. XXXIV. y. mo. w. d. h. m. s. 8 5 3 6 20 32 10 7 MOTION. XXXVI. 10s. 30 17' 101" 12 C03IPOUKD MULTIPLICATION. 69 CASE II. When the multiplier or quantity exceeds 12, and is the pro- duct of two factors in the Multiplication Table; that w, of two numbers which being multiplied together, amount to the same as the multiplier. Multiply the sum by one of the two numbers, and then multiply the product by the other. EXAMPLES. I. II. 8, d. S. d. Multiply 8 18 111 by 18. 1312 9* by 27. 6 9 53 13 10* 122 15 1* 3 3 161 1 74 368 5 44 s. d. s. d. 3. Multiply 10 10 10 by 14. Product 147 11 8 4. " 11 11 11 by 15. 173 18 9 5. 12 12 9 by 24. 303 6 6. " 5 13 44 by 28. 158 14 6 7. " 4 15 10 by 42. " 201 5 8. 7 17 71 by 64. 504 9 4 9. 6 10 3 by 72. " 468 18 10. " 9 19 Hi by 81. 809 16 74 11. 10 15 91 by 84. 906 8 3 12. 3 11 74 by 96. 343 16 CASE III. When the quantity, or multiplier, is such a number that no two numbers in the Multiplication Table will produce it. Multiply the sum by two numbers whose product will amount to nearly the same as the multiplier; then mul- tiply the sum by the number which will make the pro- duct of the two numbers equal to the multiplier, and add its product to the sum produced by the two num- bers. 70 COMPOUND MULTIPLICATION. EXAMPLES. I. . ,. d. . .. d. Multiply 7 10 5 7 10 5 by 62 10 2 75 4 2 15 10 6 451 5 He re note, I multiply by 10, then 15 10 by 6, because 10 times 6 make 60; then I multiply the same sum by 468 5 10 2, that I multiplied, first, by 10, and add its product to the other product, which makes the amount of the answer. s. d. s. d. 2. Multiply 2 10 10 by 31. Product 18 15 10 3. " 3 11 11 by 38. " 136 12 10 4. 4 11 24 by 68. 310 9 5. " 1 8 8 by 26. 37 5 4 6. " 1 3 3* by 47. " 54 14 Si 7. 124* by 83. 92 17 li CASE IV. When the multiplier is greater than the product of any two numbers in the Multiplication Table. Multiply the sum by 10, and thatproductby 10, which is equal to multiplying by 100; then multiply the pro du-^t by the number of hundreds in the multiplier, and il the sum be even hundreds, the product will t e the answer. If (here be odd numbers over even hundreds, as 70, 60, or 37, &c., multiply the amount or product of the first mul ^plication by 10, by the number of tens over 100; thus, if there be 70 over, multiply by 7. If, in additon to tens, there are smaller numbers, as 7, 8, 5, &c., the sum must be multiplied by such number; and the amount of all the multiplications being then added together, their sum will be the answer. COMPOUND 3HTJLTIILICATIOX. 71 EXAMPPLES. 1. s. d. Multiply 1 2 4 by 4300 10 11 3 4 amount of 10. 10 111 13 4 amount of 100. 10 1116 13 4 amount of 1000. 4 4468 13 4 amount of 4000. 335 00 amount of 300. 4801 13 4 Answer. In the foregoing example, I first multiply by 10, tbree imes, which gives the amount of the sum multiplied by lOOOf then by 4, which gives the amount of 00 \ The s\imis<yet to be multiplied by 300. To do this, 1 take the product of thssum multiplied by 100, viz. 111Z. 13s. 4d. and multiply it by 3, which gives the product of the sum by 300. The sum of these is the answer. s. d. s. d. 2. Multiply 1 4 by 190. Product 12 13 4 3. 1 2 3 by 430. " 478 7 6 4. " 76 by 506. 189 15 5. 8 8 by 684. 296 8 6. 139 by 375. 445 6 3 7. 1 2 by 3458. 201 12 APPLICATION. 1 . What do 84 pounds of sugar cost at 9d. per pound ? Ans. 3. 3s. 2. What do 18 yards of cloth cost at 19s per y^rd. Ans. 17. 2s. 3. Sold 7 tons of iron at 32 10s. per ton; hi/v much is the amount? Ans. 227. 10,?. COMPOUND MULTIPLICATION. 4. What is the weight of 4 hogsheads of sugar, each weighing 7 cwt. 3 qrs. 19 Ib? Ans. 31 cwt. 2 qrs. 20 rbs. 5. What is the weight of 6 chests of tea, each weigh- ing 3 cwt. 2 qrs. 9 Ibs? Ans. 21 cwt. 1 qr. 26 Ibs. 6. What is the value of 79 bushels of wheat, at 11s. 5}d. per bushel? Ans. 45 6s. 1Q4. 7. What is the value of 94 barrels of cider, at 12s. 2d. per barrel? Ans. 57 3s. 8d. 8. What is the value of 114 yards of cloth, at 15s. d. per yard? Ans. 87 5s. 7d. 9. What.is the value of 12 cwt. of sugar, at 3 7s. 4d. per c>vl.? Ans. 40 8s. 10. What is the worth of 63 gallons of oil at 2s. 3d. per gallon? Ans. 7 Is. 9d. 11. What is the amount of 120 days wages at 5s. 9d. per day? Ans. 34 10s. 12. What is the worth of 144 reams of paper at 13s. 4d. per ream? Ans. 96. 13. What will 1 cwt. of sugar cost, at ICfd. per Ib?* Ans. 5 Os. 4d. 14. If I have 9 fields, each containing 12 acres, 2 roods and 25 poles; how many acres have I in the whole? Ans. 113A. 3R. 25P. 15. What will 1 ton of lead cost, at 1 pence per pound? Ans. 37 6s. 8d. 16. If a man can travel 25 ms. 3 fur. 20 ps. 4 yds. in 1 day, how far can he travel in 6 days? Ans. 152ms. 5fur. 4ps. 2yds Q. 1. What t!c?* Compound Multiplication teach? 2. In what is it par'i'-nlarly useful? 3. Which is made the multipli^v the price, or the quantity ? 4. How r do you proceed when the multiplier does not exceed 12? 5. How do you proceed when the mul tiplier exceeds 12 ? 6. When the multiplier consists of no two component numbers, as in case third, how do you proceed? 7. How do you proceed in case fourth? 8. How is compound multiplication proved? *It must be recollected that Icwt. is U21h>. COMPOUND DIVISION. Compound Division teaches the manner of dividing numbers of different denominations. CASE I. When the divisor does not exceed 12. Begin at the highest denomination, and after dividing that, if any thing remain, reduce it to the next lower de- nomination, adding it to that denomination in the sum, and proceed in this manner until the whole is divided. If the number of either denomination should be too small to con- tain the divisor, reduce it to the next lower denomina- tion, and add it thereto, as directed in case of a remain- der. The denominations in the quotient must be kept separate. PROOF. Multiply the quotient by the divisor, and the product, if right, will be equal to the dividend. EXAMPLES. I. II. III. s. d. s. d. s. d. Divide 2)6 8 8 4)3 3 10 5)7 2 3 344 15 Hi 18 54+3 iv. v. vi. s. d. s. d. s. d. 5)6 17 11 6)9 9 9 12)21 16 lid 177 1 11 7i 1 16 4*+10 In doing the 6th sum, which is divided by 12, I find the divisor is contained once in 21; and setting down 1 I find 9 pounds remaining; which, reduced to shillings, and added to the 16 shillings in the sum, make 196 shillings. The divisor being contained 16 times in 196, with 4 remaining, I set clown 16, and reducing the 4 shillings to pence, and adding them to the 11 pence, in the sum, the amount is 59 pence. The divisor is con- 74 COMPOUND DIVISION. tained 4 times in 59, leaving 1 1 pence remaining. I set down 4, and the remaining 1 1 pence reduced to farthings and added to the half penny or 2 farthings in the sum, make 46 farthings ; and as the divisor is contained 3 times in 46, leaving a remainder of 10, 1 set down I and place the final remainder at the right hand of the sum. s. d. *. d. 7. Divide 12 10 10 by 5. Quotient 2 10 2 8. 13 13 9 by 4. 3 8 5* 9. 2 18 Hi by 3. 19 7+l 10. 7 7 7 by 4. 1 16 101 11. 177 19 111 by 12. 14 16 71+11 CASE II. When the divisor exceeds 12, and is the product of two numbers multiplied together. Divide by one of the numbers: then divide the quo- tient by the other. EXAMPLES. Divide 5 10s. 6d. by 48. ! d. 6)5 10 6 8) 18 5 2 3+5 Answer. Note. If there be any remainder in the first opera- tion, and not in the second, it is the true one. When there is a remainder in the second operation, multiply it by the first divisor, and add it to the first remainder, if there be any, and it forms the true remainder. s. d. s. d. 2. Divide 240 12 10 by 16. Quotient 15 9i+8 3. 88 13 11 by 21. 44 51+14 4. " 90 15 4i by 32. 2 16 81+2 5. 450 8 8 by 42. 10 14 51+26 " 789 19 9 by 64. " 12 6 10i+52 " 840 4 3* by 72. 11 13 4d+62 COMPOUND DIVISION. 75 CASE III. When the divisor is more than 12, and cannot be produced by multiplying any two numbers together. Divide after the manner of Long Division, reducing from higher to lower denominations, as in the following EXAMPLES. Divide 61 12s. by 23. Sfc i. *. 23)61 12(2 13s. 6d. 3qrs.+3 Ans. 46 Divide 14 10s. llfd. by 95. 15 ii. 30X *. d. 95)14 10 1 H(0 3s. Of d.4-2 Ans. 312 20 X 23 - 290 82 285 69 - 5 13 12X 12X - - 71 156 4x 138 - 18 28 4X 2 72 69 Note. In the second example, I find the divisor great- er than the number of pounds in the dividend. I there- fore set down a cypher in the place of pounds in the quotient, then reduce the 14 pounds in the sum, into shillings, at the same time adding the ten shillings in the 70 COMPOUND DIVISION sum, which thereby becomes 290. In 200 the divisor is contained 3 times, and 5 over. This 5 shillings I re- duce to pence, adding to it the 1 1 pence in the sum ; and the amount being still smaller than the divisor, I set down a cypher in the place of pence, in the quotient,* and reduce it to farthings, and proceed as before. Though this operation is longer, it is, perhaps, less liable to error than either of the preceding cases. s. d. ;*. d. 3. Divide 20 10 8 by 17. Quotient 1 4 l4-9 4. 27 18 7 by 29. 19 3 +4 5. 147 4 4 by 65. 25 3 -4-18 6. " 581 19 11 1 by 73. 7 19 5J4-49 7. 77 3 3J by 19. " 4 1 2'+17 8. 319 7 lOj by 29. , 11 3J+1 APPLICATION. 1 . If 42 cows cost 126 16s. 6d ; what was the price of each? Ans. 3 Os. 4'd. 3. Five men bought a quantity of hay, weighing 21 tons, 13 hundred, and 3 quarters; which they divided, equally among them. What was the share of each? Ans. 4 tons, 6cwt 3qrs. 4. A farmer had 3 sons, to which he gave a tract of land containing 520 acres, 3 roods, 29 perches; and the land was to be divided equally among them. What was the portion of each? Ans. 173A. 2R. 23P. 5. Divide 375 miles, 2 furlongs, 7 poles, 2 yards, 1 foot, 2 inches, by 39. Ans. 9M. 4fur. 39P. Oyd. 2ft. Sin. Q. 1 . What does Compound Division teach ? 2. How do you proceed when the divisor does not exceed 12? 3. How ? When the number of either denomination is too small to contain the divisor ? 4. How? When the divisor exceeds 12. and is the product of two numbers multiplied together? 5. How? When the divisor is more than 12, and cannot be produced by multiplying any two numbers together? 6. How is Compound Division proved? 77 EXCHANGE. Exchange teaches to change a sum of one kind of money to a given denomination of another kind. To reduce the currency of each of the United States tc dollars and cents, or Federal Money. RULE. Reduce the sum to pence; to the pence annex twc cyphers ; then divide by the number of pence in a dol lar, as it passes in each State, the quotient or answer will be in cents, which may be easily reduced to dollars Note. This rule applies to the currency of any other country, if its currency be in pounds, shillings, pence c. EXAMPLES. 1. Reduce 621 pounds, New England Virginia, anc Kentucky currency, to dollars and cents; a dollar being 72- Dence. 621 20 72)14904000($2070.00 144 504 504 000 2. Reduce 12 pounds, 3 shillings, and 9 pence to dollars and cents. Ans. $40.62*. 3. Reduce 30 pounds and 3 shillings to dollars and cents. Ans. $100.50. 4. In 763 how many dollars, cents and mills? Ans. $2543.33cts. 3 m. 5. Reduce 19 shillings and 10 pence to dollars and ents. Ans. $3.30cts. 5m. 78 EXCHANGE. 6. In 9 pounds and 16 shillings in New York and Vorth Carolina currenncy, how many dollars and cents, eckoning 96 pence to a dollar? Ans. $24.50. 7. In 30 pounds, how many dollars and cents, same currency? Ans. 75.00. 8. In 27 pounds, 2 shillings, how many dollars and cents, same currency ? Ans. 67.75. 9. In 942 pounds of New Jersey, Pennsylvania, Del- aware, and Maryland currency ; how many dollars and cents, a dollar being 90 pence? Ans. $2512.00. 10. In 12 pounds how many dollars and cents same currency? Ans. $32.00. 11. In 86 6s. 5d. how many dollars, cents and mills, same currency? Ans. $230.18 cts. 8m 12. In 21 pounds, South Carolina and Georgia cur- rency, how many dollars and cents, there being 56 pence in a dollar? Ans, $90.00 13. In 56 pounds, how many dollars, &c. same cur- rency? Ans. $240.00 14. 108 pounds, Canada and Nova Scotia currency, low many dollars, &,c. there being 60 pence in a dollar \ Ans. $432.00 15. In 460 pounds and 16 shillings sterling, or Eng- ish money, how many dollars, &c. there being 54 pence in a dollar? Ans. $2048.00 16. Reduce 16 pounds, 6 shillings, and 3 pence Eng- lish money, to dollars and cents. Ans. $72.5'~ To bring Federal Money into pounds shillings, fy pence RULE. Multiply the dollars, or dollars and cents, by the number of- pence in a dollar of the currency to which you wish to change the given sum ; the answer will be in pence, which can then be reduced to shillings and pounds. When there are cents in the sum to be re duced, two figures must be cut off from the right of the product, before bringing it into pounds. Note. This rule applies to the currency of any country whose currency is in pounds, shillings, &c. EXCHANGE. 79 EXAMPLES. 1. In $16.50 how many pounds and shillings, in ster- ling, or English money; a dollar being four shillings and six pence, or 54 pence? $16.50 9X6=54 9 12)891.00 2|0)7|4.+3 3 14s. 3d. Answer. 2. In 33 dollars, how many pounds, &c. ? Ans. 7. 8s. 6d. 3. In 1000000 dollars, how many pounds sterling? Ans. 225000.* 4. Reduce 432 dollars into the currency of Canada and Nova Scotia, a dollar being equal to five shillings; or 60 pence. Ans. 108. 5. In $490.50 how many pounds, shillings, &,c. same currency? Ans. 122 12s. 6d. 6. Bring $150.25, into the currency of New England, Virginia, and Kentucky, a dollar, being equal to 72 pence. Ans. 45. Is. 6d. Questions. 1. What does Exchange teach? 2. How do you reduce the currency of any one of the United States to Federal Money? 3. Does this rule apply to the currency of any other country? 4. How do you change Federal Money into pounds, shillings, and pence of any state or country? 5. Among the various kinds of money, what kind is the most easily reckoned ? *Federal Money may, also, be changed into English Money, by multiplying the dollars by 9, and dividing the product by 40. to PLATE TO BE USED IN STUDYING VULGAR FRACTIONS. i i Jn using the Fractional Plate, the student must count the white spaces, and not the black lines. The first row of squares, or white spaces, at the top, are whole numbers; the second row is divided into lalves; the third, into thirds, and so on from the top to the bottom. Thus it may be shown, at one glance, that 7 halves make three and a half, or that 8 thirds make 2 and 2 thirds, &c. VULGAR FRACTIONS. Fractions are broken numbers, expressing any as- signable part of an unit, or whole number. They are represented by two numbers placed one above another, with a line drawn between them; thus f , I, &,c. signi- fying two fifths and five eights. The figure above the line is called the numerator, and that below the line, the denominator. The denom- inator shows into how many equal parts the whole quantity is divided, and represents the divisor in divis- ion. The numerator shows how many of those parts are expressed by the fraction ; being the remainder af- ter division. Both these numbers are sometimes called the terms of the fraction. Questions to prepare the learner for this Rule. 1. If a pear be cut in two equal parts, what is one of those parts called? Ans. a half. 2. If you cut a pear into three equal parts, what is one of those parts called? Ans. one third. 3. How many thirds of a thing make the whole? 4. If a pear be cut into four equal parts, what is one of those parts called? Ans. one fourth. What are two of the parts called? Ans. two fourths. What are three of them called ? Ans. three fourths. 5. How many fourths of a thing make the whole? 6. If an orange be cut into five equal parts, what is one of the parts called? Ans. one fifth. What are two of the parts called? Ans. two fifths. What are hree of them called? Ans. three fifths. What are four of them called? Ans. four fifths. 7. How many fifths of a thing make the whole? 8. If you cut a pear into six equal parts, what is one of the parts called? What are two of them called? What are three of them called ? What are four of them called? 9. How many thirds are there in one ? How many fourths? If four fourths make the whole, what part are two fourths? What part of three is one? What part of four are two? What part of six are two? 82 VULGAR FRACTIONS. What part of eight are two ? What part of eight are six? What part of 9 are 6? What part of 10 are 2? I What part of 10 are 4? What part of 10 are 7? What part of 12 are 6? What part of 12 are 4? What part of 12 are 3? What part of 12 are 2? 10. How many are two fourths of 12? How many are three fourths of 12? Two thirds of 12, are how many? How many are 5 times 8? In one eighth of 40, how many? In three eights of 40, how many? Four eights of 40, are how many? Then $ of any p-;mber, or of any thing, amount to how many, or how much? How many are f of 30? How many are of 30? How many in i of 60? In * of 60, how many? In } of 60, how many In ^ of 60, how many T How many are -fa of 60? How many are f of 60?' How many are of 60? In 2 and J, how many fifths? In 5 and f how many fifths? In } of 100, how many? In of 100 cents, or 1 dollar,, how much? How much are I and i? How much are I and f ? How much are i and |? How much are f? How much are f and How much are J, |, and A? Iny, how many? In how many? If you take I from one dollar, how much will remain? If you take J from one dollar, how much will remain? If you take A from a pound, how much will remain? If you take -| from one, how much will remain ? How many fourths are 2 times I ? How many are 5 times f? How many are 3 times f? In how many ? In - 1 , how many ? 11. If one half, three fourths and a quarter, be add- ed, how much will be their amount? 12. If you take two eights from eleven eights, how much will remain? 13. What is a proper fraction? Ans. When the numerator is less than the denomi- nator, as i, or |, &c. 14. What is an improper fraction? Ans. It is that in which the numerator is equal,, or superior to the denominator; as, ^, or |, or J, &c. 15. What is a simple fraction? Ans. It is a fraction expressed in a simple form; as, 4, A, f VULGAR FRACTIONS 85 16. What is a compound fraction? Ans. It is the fraction of a fraction, or several frac- ions connected together with the word of between hem; as , of I of I; or I of T 7 T , &c. which are read hus, one half of two thirds; &,c. 17. What is a mixed number? Ans. It is composed of a whole number and a frac- ion; as 3i, or 12i. 18. What is the common measure of two or more lumbers? Ans, It is that number which will divide each of them without a remainder; thus 5 is the common meas- ure of 10, 20, and 30; and the greatest number that will do this, is called the greatest common measure. 19. What is meant by the common multiple? Ans. Any number which can be measured by two 9r more numbers, is called the common multiple of .hose numbers; and if it be the least number that can :>e so measured, it is called the least common multiple; hus 40, 60, 80, 100; are multiples of 4 and 5; but their least common multiple is 20, 20. When is a fraction said to be in its lowest terms? Ans, When it is expressed by the smallest numbers possible. 21. What is meant by a prime number? Ans. It is a number which can only be measured by itself, or an unit. 22. What is meant by a composite number? Ans. That number, which is produced by multiply- ing several numbers together, is called a composite number. 23. What is a perfect number? Ans. A perfect number is one that is equal to the sum of its aliquot parts.* *Tbe following perfect numbers are all which are, at present known 6 8589869056 28 137438691328 496 2305843008139952128 8128 24178521639228158837784575 33550336 99035203 142S297 1830448816 128 84 VULGAR FRACTIONS. REDUCTION OF VULGAR FRACTIONS. Reduction of Vulgar Fractions, is the bringing of them out of one form into another, in order to prepare them for Addition, Subtraction, Multiplication, &c. CASE I. To reduce a fraction to its lowest terms. RULE. Divide the greater term by the less, and that divisor by the remainder, and so continue till nothing be left; the last divisor will be the common measure ; then divide both parts of the fraction by the common measure, and the quotients will express the fraction required. Note. If the common measure happen to be 1, the fraction is already at its lowest term. Cyphers, on the right hand side of both terms, may be rejected; as-f |, ^. EXAMPLES. 1. Reduce iff to its lowest terms. 144)240(1 48)144(3 144 144 96)144(1 i Ans. 96 48)240(5 Greatest common 240 measure 48)36(2 96 Thus 48 is the greatest common measure, and the true answer is obtained by dividing the fraction by it. This reduction may be performed, also, by another rule, thus : Divide the numerator and denominator of the fraction by any number that will divide them both without a remainder ; divide the quotients in the same manner, and so on, till no number will divide them both, and the last quotients will express the fraction in its lowest terms. The same sum done by this method : 12)-ma 2 o 4)Jf(f Answer. 2. Reduce 9 T to its lowest terms. Ans. *. 3. Reduce iff to its lowest terms. Ans. * . 4. Reduce fff- to its lowest terms. Ans. ^. 5. Reduce } to its lowest terms. Ans. I. REDUCTION OF VULGAR FRACTIONS 85 CASE II. 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; then set that sum, namely, the whole product, above the de- nominator for the fraction required. EXAMPLES. Reduce 23 f to an improper fraction. 5 117 4 7 Answer. 2. Reduce 12 to an improper fraction. Ans. iJ 3. Reduce 14 T 7 T to an improper fraction. Ans. * T 4. Reduce 163^- to an improper fraction. Ans. 3 | CASE III. To reduce an improper fraction to a whole or mixed number. RULE. Divide the numerator by the denominator, and the quotient will be the whole or mixed number sought. EXAMPLES. 1. Reduce y to its equivalent number. 3)12(4 Answer. 12 2. Reduce V to its equivalent number. 7)15(2} Answer. 14 1 3. Reduce 7 r y to its equivalent number Ans. 44 T * T . 5. Reduce y to its equivalent fraction. Ans. 8. 5. Reduce ^f 3 to its equivalent fraction. Ans. 54|J. 6. Reduce 2 |} 8 to its equivalent number. Ans. 171||. CASE IV. To reduce a whole number to an equivalent fraction, hating a ghtn denojniniztor. Multiply the whole numt rr by the r ; ven flenomina- 36 REDUCTION OF VULGAR FRACTIONS. tor ; then set down the product above for a numerator, ind the given denominator below, and they will form the traction required. EXAMPLES. 1. Reduce 9 to a fraction whose denominator shall be 7. 9X7=63, then 6 T 3 is the answer. 2. Reduce 13 to a fraction whose denominator shall be 12. Ans. iff. 3. Reduce 27 to a fraction whose denominator shall bell. Ans.Yr 7 - CASE V. To reduce a compound fraction to a simple one. RULE. Multiply all the numerators together, for a new nu- merator, and all the denominators for a new denomina tor, then reduce the fraction to its lowest term. EXAMPLES. 1. Reduce * of ! of I to a single or simple fraction. 1X2X3 6 1 - _ - = - = - Answer. 2x3X4 24 4 2. Reduce f of f of |f to a single fraction. Ans. / T 3. Reduce ^ of % to a single fraction. Ans.'if. 4. Reduce f of | of }J- to a simple fraction. Ans.y/ T . CASE VI. To reduce fractions of different denominations to oilier s of the same value, and haying a common denominator. RULE. Multiply each numerator into all the denominators except its own, for a new numerator, and all th'e* denom inators into each other for a common denominator.* +The least common denominator, or multiple, of two or more numbers, may be found thus: Divide tlie given denominations by any number that will divide two or more of them without a remainder, and set the quotients and undivided numbers underneath. Divide these quotients by any number that will divide two or more of tliem as before, and thus continue, till no two numbers are left, capable of bein? lessened. The.n multiply the last quotients and the divisor, or divisors together, and the product will be the answer. What is the least common multiple of f ,f ,/y, and T ^ ? 8)9 8 15 16 3)9 1 15 2 3X1X5X2-30X3X8X720, Ans. REDUCTION OF VLLGAK Fit ACTIONS. 87 EXAMPLES. 1. Reduce 4, 5, and f to a common denominator. 1 X3X4=12 the numerator for *. 2x2X4=16 the numerator for 3. 3x2X3=18 the numerator for I. Denominator 2x3x4=24 the common denominator. Therefore the results are , , !=f , |f , . Or the multiplications may be performed mentally, and the results given 1, 3, !=, if, if. 2. Reduce f and to a common denominator. Ans J? and 3. Reduce , |, and I to a common denominator. 5. Reduce 4, f , and f to fractions of a common de. nominator Ans. T 4 /^, T \^ and }$ CASE VII. To reduce the fraction of one denomination to the frac- tion of another, but greater, retaining the same value. RULE. Make the fraction a compound one, by comparing it with all the denominations between it and that denom- ination to which you would reduce it; then reduce that compound fraction to a simple one. EXAMPLES. 1. Reduce | of a cent to the fraction of a dollar. By comparing it, it becomes - of ^ of yV which being re- duced by case five, will be 4xlXl =4 and 7x10x10 =700. Ans. T | J . D. 2. Reduce f of a mill to the fraction of an eagle. 3. Reduce f of a penny to the fraction of a pound. 3 X 1 Xl= 3 1 = - 5x12x20=1200 400 4. Reduce | of an ounce to the fraction of a pound, Avoirdupois Weight. Ans. -^Ib. 5. Reduce J of a dwt. to the fraction of a pound, Trow Weight. Ans. T / 7r lb'. 6. Reduce j J of a minute to the fraction of a (iny. i. cla. BO REDUCTION OF VULGAR FRACTION*. CASE VIII. To reduce the fraction of one denomination to the frac- tion of another, but less, retaining the same value. RULE. Multiply the given numerator by the parts in the denomination between it and that to which you would reduce it, and place the product over the given denom- inator. EXAMPLES. 1. Reduce T | T of a dollar to the fraction of a cent. The fraction is ^ T of y of l j ; then i xioxio 100 , A . . , , . and this reueced, is equal to 175x 1 X 1 175 An M 2. Reduce IT -J of an eagle to the fraction of a mill. Ans. J. 3. Reduce y J- 7 of a pound to the fraction of a penny. Ans. 1 4. Reduce ^ of a pound Avoirdupois, to the fraction >f an ounce. Ans. '$-. 5. Reduce r ^ T of a pound Troy, to the fraction of a pennyweight. Ans. Jdwt. 6. Reduce ,-jL^ of a day to the fraction of a minute Ans. }f of a min* CASE IX. Tojind the value of the fraction in the known part* of the integer; or, to reduce a fraction to improper value. RULE. Multiply the numerator by the known parts of the integer, and divide by the denominator. EXAMPLES. 1. What is the value of 1 of a pound? 2 thirds of a nound. 20 3)40 thiroi of a shilling. . 13 .-|-t third of ashiling. 3)12 thirds of a penny. ~ Am. 13i. 4d REDUCTION OF VULGAR FRACTIONS. 89 2. Reduce & of a shilling to its proper value. 2 fifths of a shilling. 12 5)24(4d. 20 4 fifths of a penny, 4 6)16 fifths of a farthing, 3 qr.+l fifth. Ans. 4d. 3qr. { 3- Reduce f of a Ib. Troy, to its proper quantity. Ans. 7 oz. 4dwt 4. Reduce of a mile to its proper quantity; Ans. 6 fur. 16 poles, 5. Reduce ^ of a cwt. to its proper quantity, Ans. 2 qrs, 6. Reduce f- of an acre to its proper value. Ans. 2R. 20R 7. Reduce ,^ of a day to its proper value. Ans. 7 hours 12 rain. r CASE X. To reduce any given quantity to the fraction of a great- er denomination of the same kind. RULE. Reduce the given quantity to the lowest denomina- tion mentioned for a numerator, and the integer into the same denomination for a denominator. EXAMPLES. 1. Reduce 16s. 8d. to the fraction of a pound. 16 8 Integer 1 12 2G Numerator 200 20 ==| Ans. 12 Denominator 240 240 Denominator. IK) ADDITION OF VULGAR FRACTIONS. 2. Reduce 6 furlongs and 16 poles to the fraction of a mile Ans. f . 3. Reduce | of a farthing to the fraction of a pound. 4. Reduce ^ dwt. to the fraction of a pound Troy. Ans. -3^ 5, Bring 80 cents to the fraction of a dollar. A dollar is 100 cents, then 80 cents are equal to -J of a dollar; which, being reduced, is equal to | Ans, 6) Bring 16 cents 1) mills to the fraction of an eagle. 16 cents 9 mills = 169 . : A nc 1 eagle = 10000 7. Bring 2 quarters 3 J nails to the fraction of an ell Fnglish.* 2 quarters 3i nails. 4 11 9 Numerator 100 Denominator 9 of 4- of 1 = *-$% = 4 Ans. ADDITION OF VULGAR FRACTIONS. CASE I. To add fractions having a common denominator. RULE. Add all the numerators together, and place the sum over the common denominator, which will give the sum of the fractions required. EXAMPLES. 1. Add I, A and together. ______ iXf Xi=f =1* Answer. *When the sum contains a fraction, as in the seventh example, multiply both parts of the sum by the denominator thereof, and to the numerator add the numerator of the given fraction. ADDITION Oi' VULUAIl FRACTION'S. 91 2. Add-^,2,?- and togctlier. ' -}~M- +?+?- = V = H Answer. CASE II. To add fractions having different denominators. RULE. Find the common denominator by Case VI. in Re- duction; then add, as in the preceding examples. . EXAMPLES. 1. Add J and together. 4 X 5=20> 3 X 9=27$ 47 sum. 4x^=36 com. denom. = ! Ans. 2. Add f and -fa together. Ans. T 9 T . CASE III. To add mixed numbers. RULE. Add the fractions as in Case I, in Addition, and the whole numbers as in Simple Addition; then add the fractions to the sum of the whole numbers. . If the fractions have different denominators, reduce them to a common denominator, and then add the fractions to the integers or whole numbers. EXAMPLES. 1. Add 13^, 9 T 4 T , 3^ together. 13+9+3=25 whole numbers. TV+T 4 5+7 7 7=if =f Thus, 25| Ans. 2. Add 5f , 6| and 4i together. 5-|_6-|-4=15 whole numbers. Then, 2x^X2=32 7x3x2=42 98 sum of the numerators. 3x8X2=48 common denominator. Then, ff=2^ . Thus, 15+2^= 17 Jj Answer. 3. Add If, 2| and 3-f togetner. Ans. 7}J ~ ADDITION OF VULGAR FRACTIONS. CASE IV. To add compound fractions. RLLE. Reduce them to simple ones, and proceed as before. EXAMPLES. 1. Add i of | of |, to f of f of |f. *-^* simple fraction * 3 = ! - 18X5X11=^ simp fractlon - Then find a common denominator. 4v 4 _ \Q forj, T 4 r tnus iCn- -11 numerator. 27 sum of the numerators. 4x11=44 common denominator. Therefore JJ is the answer. 2. Add J of J, and of i together. Ans. 3. Add | of -j^, and T ^ of f together. Ans. 4. Add f , 9-} and | of 1 together. Note. The mixed number of 9j = -y 5 ; the compound fraction | of i=|. Then the fractions are, ^, y and |> which must be 'reduced to the fractions of a common denominator and added. Ans. 5. Add 1^, 6|, | of J and 7j together. Ans. CASE V. When the given fractions are of several denominations. RULE. Reduce them to their proper values, or quantities, and add them according to the following examples. EXAMPLES. 1. Add S- of a pound to J of a shilling. Thus, | of a pound=13s. 4d. and f of a shilling= Os. 4d. 3iqr. 13s. 8d. 3lqr. Ans. 2. Add I of a pound and T 3 F of a shilling together. Ans. 15s. lO^d. 3. Add of a week, of a day, and 1 of an hour together. Ans. 2d. SUBTRACTION OF VULGAR FRACTIONS. 4. Add I of a yard, -J of a foot, and of a mile to- gether. Ans. llGOyds. 2ft. 7iru 5. Add of a dollar, f of a cent, -f^ of a cent, and of a mill together. Ans. 20c. 9m. 6. Add -J- of a pound, ^ of a shilling, and of a pen- ny together. Ans. 2s. ~ SUBTRACTION OF VULGAR FRACTIONS. CASE I. When the fractions have a common denominator. RULE. Subtract the less numerator from the greater, and set the remainder over the common denominator, which will show the difference of the given fractions. EXAMPLES. 1. Subtract f from 4. Ans. -f . 2. What is the difference between | and I? Ans. |=r 2. Take ^ from T ^. Ans. T 2 ? = ; 4. Take f from f Ans. |=i. CASE II. When fractions, or mired numbers, are to be subtracted from whole numbers. RULE. Subtract the numerator from its denominator, and under the remainder place the denominator; then carry one to be deducted from the whole number. EXAMPLES. 1. Take f from 12. Thus, 12. llf Answer. 2. Subtract 27f} from 32. Ans. 4 3. From 10, take T V. Ans. 9 4. From 9, take 5f Ans. 5. From 25, take 24^. Ans. T V 94 SUBTRACTION OF VULGAR FRACTIONS. CASE VI. To subtract fractions hating different denominators. RULE. Reduce the fractions to a common denominator, by Ca^o VI. in Reduction, and subtract the less numerator from the greater the difference will be the answer. EXAMPLES. 1. What is the difference between ij and JJ? Thus If and } f are equal to |JJ, # ff , And 88 from 171, leaves 83. Ans. 7 \ 3 2. From f take f . Ans. 3. Take from f . Ans. 4. Subtract -fe from ^-. Ans. J CASE VI. To distinguish the largest of any two fractions. RULE. Reduce them to a common denominator, and the one that has the larger numerator is the larger fraction. EXAMPLE. Which is the greater fraction, j-J or ||? Thus 192 common denominator. 12x15=180 numerator. = 176 numerator. 4 numerator = A . Therefore -J-f is the greater fraction by J^, Ans. CASE V. To subtract one mixed number from another, when the fraction to be subtracted is greater than that from which the subtraction is to be made. RULE. Reduce the fractions to a common denominator; sub- tract the numerator of the greater from the common denominator, and add to the remainder the less numera- tor; then set the sum of them over the common denom- inator, and carry one to the whole number, and sub- tract as in Simple Subtraction. MULTIPLICATION OF VULGAR FRACTIONS. 95 EXAMPLES. 1. From 12 J subtract 8}, Thus | reduced to a common denominator ,= JJ~, and }|. reduced to a common denomipator,=^ T 7 A. Then 72 taken from 114, leaves 42; which, added to 7, the less numerator, makes 99 for the numerator in he answer. Then carrying 1 to the whole number, namely, 8, makes it 9; and taking 9 from 12 leaves 3. Therefore the answer is 2 From 10 T \, take 1 T V Ans. CASE VI. When fractions are of different denominations. RULE. Reduce them to their proper values, or quantities^ and subtract as in Compound Subtraction. EXAMPLES. 1. From fof a pound, take 1 of a shilling. Thus, f of a pound =17s. 6d. And i of a shilling =0 4 17s. 2d. Answer. 2. From of a ton take / ff of a cwt. Ans. 14cwt. Oqr. lllb. 3oz. 3dr. 3. From J of a pound, take J of a shilling and what will be the remainder? Ans. 14s. 3d. 4. From of a pound, Troy Weight, take of an ounce. Ans. 8oz. 16dwt. 16gr. MULTIPLICATION OF VULGAR FRACTIONS. RULE. Reduce compound fractions to simple ones, ami mix ed numbers to equivalent fractions; then multiply all the numerators together for a numerator, and all the denominators together for a denominator which will give the product required. > MULTIPLICATION OF VULGAR FRACTIONS. EXAMPLES. 1. Multiply I by f. Here, J X=3% == o the answer. 2. Multiply f by }. Ans. ^ 4. " J*of7by|. Ans. 1}. 5. " t>J by 4-. Ans. 1 6. 4 by 3-5. Ans. 14< 7. " ^ibyi Ans -^ DIVISION OF VULGAR FRACTIONS. RULE. Reduce compound fractions to simple ones, and mix- ed numbers to equivalent fractions; thea multiply the numerator of the dividend by the denominator of the divisor, for a new numerator, and the denominator ol he dividend by the numerator of the divisor, for the de- nominator; the fractions thus formed will be the answer. EXAMPLES. 1. Divide 4 by J. Thus, 4 numerator of the dividend, 3 X denominator of the divisor. 12 numerator. Then 7 denominator of the dividend. 2 X numerator of the divisor. 11 denominator Therefore, J-J=-J is the answer 2. Divide % by }. . g 3ass g Answer. 3. Divide U by .*. Ans. 4 4. " T\byf. Ans. 5. I by V s - Ans. 6. ^J by f . Ans. 7. % by ->. Ans. 8. < 4 ^ by ^. Ans. T DECIMAL FRACTIONS. 97 9, Divide f by 2. Ans. T V 10, 7|by9f. Ans. fj. 11- " I of b Y 4 of 7 f Ans - rfr- 12, What part of 33 J T , is 26|i.? Ans. f Questions. 1. What are Vulgar Fractions? 2. How are they represented in figures ? 3. What is the upper figure called? 4. What is the lower figure called ? 5. What does the denominator show ? 6. What does the numerator show ? 7. What are the two numbers cf a fraction some- times called ? DECIMAL FRACTIONS. Decimal Fractions are parts of whole numbers, and nre separated from them by a point, thus, 8.5; which is read, eight and five tenths, or 8 T 5 ^. All the figures on the left of the point are whole numbers; those on the right arc fractions. An unit is supposed to be divided into ten equal parts, and the figure at the right of the point expresses the number of those parts. Decimals decrease in a tenfold proportion, as they depart from the separating point. Thus, .5 is 5 tenths, or one half; .57 is 57 hundredths; .05 is 5 hundredths; and .005 is 5 thousandths. Cyphers placed at the right hand of de- cimals do not alter their value; thus, .5 or T \; .50 or T $$ ; .500 or 1 J , are all of the same value, and equal to . The first place of decimals is called tenths ; the second, hundredths, &c. DECIMAL NUMERATION TABLE. 07 o? v> "^ tr> & "^ r3 3 J S 3 J . 2 T3 en "is "-* w "** w ill fj . ^ i ill 1 i 1 I J 1 I Illgll 7654321.654321 98 ADDITION OF DECIMALS. ADDITION OF DECIMALS. RULE. Place the figures according to their values units un- der units, tenths under tenths, &,c. and add as in Sim- ple Addition of whole numbers; observing to place the point in the sum under those in the given numbers. EXAMPLES. 1. Add together the following sums, viz: 252.25. 343.5, 17.85, 1244.75 and .425. Thus, 252.25 Note. The answer to this sum 343.5 is read thus: One thousand eight 17.85 hundred and fifty eight, and seven 1244.75 hundred and seventy-five thou- .425 sandths. 1898.775 Answer. ii. in. 87654.321 987654.3 23456.78 212345.67 98765.4 898765.432 209876.501 2098765.402 4. Add 420.4, 38.05, 54.9, 27.003 and 29.384. Ans. 569.737. 5. Add 376.25, 86.125, 6.5, 41.02 and 358-865. An?. 868.760. 6. Add .64, .840, .4, .04, .742, .86, .99 and .450. I Ans. 4.9H2. : 7. William expended for a gig $255^, for a wagon 7-flfr, fora bridle T y and for a saddle '$ilifffc; What did they amount to? Ans$804.455. JVbfe. Dimes, cents and mills are decimals of a dollar. A dime is one tenth, a cent is one hundredth, a mill is one thousandth; which shows that the addition of Federal Money is the addition of decimals. Thus 5 tenths of a dollar is the same as oO hundredths, or 50 cents; and 25 hundredths of a dollar is equal to 25 cents, fee. It may be likewise added, that .5, or .50, or .500 being- equal to one half, .25 equal to one quarter, and .75 equal to three quarters or three fourths, 'O .7, or .35, or any intermediate, fractious, have a proportionate SUBTRACTION OF DECIMALS. 8. James bought 2/> cwt, of sugar, 23.265 cwt. of hay and 4,2657 of rice. How much did he buy in all ? Ans. 30,0307cwt. 9, James is 14^ years old, John 15-^- and Thomas 16 T 7 ^. What is the sum of all their ages? Ans. 46.5 years. xO. What is the sum of one and five tenths; forty-five and three hundred and forty nine thousandths; and six- teen hundredth*? Ans, 47.009. SUBTRACTION OF DECIMALS. RULE. Write the larger number first, and the smaller one under it; then subtract as in Simple Subtraction; obser- ving, that the dividing point in the answer, or remain- der, must be placed under those in the sum. EXAMPLES. I. II. III. From 91.73 2.73 1.5 take 2.138 1.9185 .987654321 89.592 0.8115 0.512345679 4. Bought a hogshead of molasses, containing 60.72 gallons and sold 40.721 gallons. How much was left in the hogshead? Ans. 19.999 galls. 5. A merchant, owing $270.42, paid $ 191.626; how much did he then owe? Ans. 78.794. 6. I bought 20,25 yards of cloth and sold 5.75 yards. How much had I left? Ans. 14.50 yds. 7. From .650 of a barrel take .6 of a barrel. Ans. .050 barrel. 8. From $ of a bushel take -J^ of a bushel. Ans. .126 bushel. 9. A farmer was 42.075 years old when he came to the Western country, and is now 64.67 years of age. How long since he emigrated? Ans. 22.595 years. 10. Take one hundredth from on tenth. Ans. .09. 100 MULTIPLICATION AND DIVISION OF DECIMALS. MULTIPLICATION OF DECIMALS. RULE. Place the multiplier under the multiplicand, and mul- tiply as in Simple Multiplication; then point off as ma- ny places for decimals as there are decimals in the multiplicand and multiplier. If there be not so many figures in the product as there are decimals in both fac- tors, the deficiency must be supplied by prefixing cy- phers. EXAMPLES. in. .63478 .8994 253912 571302 571302 507824 .570921132 4. Multiply .63478 by .8204. Ans. .520772512. 5. Multiply .385746 by .00464, Ans. 00178986144. 6. What will 5.66 bushels of wheat cost at $1.08 a bushel? Ans. $6.1128, or $6.11c. 2 T yn. 7. What will 8,6 pounds of flour come to al $.04 a pound? Ans. $ .34cts, 4m. 8. At $ .25c. a bushel, what will 12.67 bushels of apples cost? Ans, 3.1675. 9. If I travel 30.75 miles a day, how far shall 1 travel in 8.325 days ? Ans. 255,99375 miles. Multiply 24.85 by 6.25 12425 4970 14910 228375 319725 365400 155.3125 3996.5625 DIVISION OF DECIMALS. RULE. Divide as in Simple Division, and point off as many figures from the right hand of the quotient, for decim- als, as the decimal figures in the dividend exceed in number those in the divisor. When there are not so many figures in the quotient as this rule requires, the DIVISION OF DECIMALS. 101 deficiency must be supplied by prefixing cyphers to the left of the quotient. When there are more decimal fig- ures in the divisor than in the dividend, place as many cyphers to the right of the dividend as will make them equal. When the number of decimals in the divisor, and the number in the dividend are equal, the quotient will always be in whole numbers, unless there should be a remainder after the dividend is all brought down. When there is a remainder, cyphers must be annexed to it and the division continued and the quotient thence arising will be decimals. EXAMPLES. I. II. .5).75(1.5 *" 324.8)9876.5(30.4079 5 9744 25 13250 cypher 25 12992 annexed. IV. 6.4)128.64(20.1 128 64 64 25800 22736 30640 29232 1408+ in. .48)65.88(137 48 178 144 348 336 12 rem, 5. Divide 234.70525 by 64.25. Ans. 3,653. 6. Divide 14 by .7854. Ans. 17.825. 7. Divide 2175.68 by 100. Ans. 21.7568. 8. If you divide 116.5 barrels of flour equally among 5 men, how many barrels will each have? Ans. 23.3 barrels. 9. At $ .25 a bushel, how many bushels of corn may be bought for $300.50. Ans. 1202 bushels. 10. At $ .12 or $ .125 a yard, how many yards of cloth may be bought for $16? Ans. 128 yds. 11. Bought 128 yards of tape, for $ .64; how much was that a yard? Ans. $ .005, or 5 mills. 102 REDUCTION OF DECIMALS. REDUCTION OF DECIMALS. CASE I. To reduce a vulgar fraction to a decimal. RULE. Place cyphers to the right of the numerator until you can divide it by the denominator; and divide till nothing remains; or, if it be a number th^t will not di- vide without a remainder, then divide until you get three or more figures for the quotient. The quotient will be the vulgar fraction expressed in decimals. EXAMPLES, 1. Reduce 1 to a decimal. Thus, 2)1.0(3 10 2 Reduce } and f to decimals. 4)1.00(55 4)3.00(.75 8 28 20 20 20 20 3. Reduce to a decimal. Ans. .375. 4. Change j, f , -fe an< * TJ to decimals. Ans. .125, .5, .75. 04, 5. What decimal is equal to ^1 Ans. .05. What is equal to J? Ans. .2. What is equal to ? Ans. .3333-f. 6. Change 64 T 3 bushels to its equivalent value. Ans. 64.25 bushels. CASE II. To reduce any sum, or quantity, to the decimal of any given denomination. RULE. Reduce the quantity to the lowest denomination, and reduce the proposed integer to the same denomination then divide the quantity by the amount of the integer and the quotient will be the answer. REDUCTION OF DECIMALS. 103 EXAMPLES. 1. Reduce 3s. 9d. to the fraction of a pound. One^pound reduced to pence makes 240; and 3s. 9d reduced to pence makes 45. Then, 240)45.0000(.1875 Answer. 240 2100 1920 1800 1680 1200 1200 The same sum may be done by writing the given numbers from the least to the greatest in a perpendicu- lar column, and dividing each of them by such number as will reduce it to the next denomination, annexing the quotient to the succeeding number. Thus, 12 9.00 2|0 3.750|0 .1875 Answer. 2. Reduce 7 drams to the decimal of a pound, Avoir- dupois Weight. Ans. .02734375. 3. Reduce 14 minutes to the decimal of a day. Ans. .009722. 4. Reduce 21 pints to the decimal of a peck. Ans. 1.3125. 5. Reduce 15s. 6d. to the decimal of a pound. Ans. .775. 6. Reduce 56 gallons 3 quarts 1 pint to the decimal of a hogshead. Ans. .9027. 7. Reduce 12 dwts. 16grs. to the decimal of a pound Troy Weight. Ans. .0527, 8. Reduce 4 mills to the decimal of a dollar. . Ans. .004 .- In doing sums in this rule, it will be necessary to keep in mind the tables of the different weights, measures, money, &c. 104 REDUCTION OF DECIMALS. CASE III. To find the value of any decimal fraction. RULE. Multiply the decimal by the number of parts in the next lower denomination ; point off as many figures for decimals as is required by the rule in multiplication of decimals; then multiply the decimal by the number of parts in the next lower denomination, and so on, to the last. The figures on the left of the points will show the value of the decimal in the different denominations. EXAMPLES. 1. What is the value of .775 of a pound? .775 20 ^.15.500 12 d.6.000 Answer. 2. What is the value of .625 of a cwt.? 4 2.500 28 4000 1000 14.000 Ans. 2 qr. 1416. 3. What is the value of .625 of a shilling? Ans. 7 4. What is the value of .4694 of a pound, Troy Weight? Ans. 5oz. 12dwts. 15.744grs 5. What is the value of .6875 of a yard? Ans. 2qrs. 3na. 6. What is the value of .3375 of an acre? Ans. 1R. 14P, 7. What is the value of .0008 of an Eagle? Ans. 8m DUODECIMALS. 105 Questions. 1. What are decimal Fractions? 2. How are they separated from whole numbers? 3. In what manner do they decrease as they depart from the separating point? 4. In the table of numeration, what is the first place called ? 5. What money, or currency, is reckoned after the manner of Decimal Fractions ? DUODECIMALS. Duodecimals are fractions of a foot or of an inch, or parts of an inch, and have 12 for their denominator. They are useful in measuring planes, or surfaces, and solids. In adding, subtracting, and multiplying by Du odecimals, it is necessary to carry one for twelve. The denominations are feet, inches, seconds, thirds and fourths. 12 fourths"" make 1 third 12 thirds - - 1 second ". 12 seconds - - 1 inch /. 12 inches 1 foot Ft. MULTIPLICATION OF DUODECI3IALS. RULE. Set down the different denominations, one under the other, so that feet stand under feet, inches under inches, seconds under seconds, &/c. Multiply each denomina tion in the sum, by the feet in the multiplier, and set the result of each under its corresponding term,observ ing to carry one for every 12 from one denomination to another. Then multiply the sum by the inches in the multiplier, and set the result of each term one place removed to the right of those in the sum ; and in like manner, multiply the sum or multiplicand by seconds, thirds, &,c. if there be any in the multiplier. Or, instead of multiplying by inches, &c. take such parts in the multiplicand, as these are of a foot. 106 DUODECIMALS. Add the amount of the multiplications together, and their sum will be the answer. EXAMPLES. i. IT. Ft. L Ft. 7. Multiply 47 14 9 by 6 4 .46 59 746 29 4 66 4 6 .\otc 1. Feet multiplied by feet, give feet. Feet mul- tiplied by inches give inches. Feet multiplied by se- conds, give seconds. Inches multiplied by inches, give seconds. Inches multiplied by seconds, give thirds. Seconds multiplied by seconds, give fourths. III. IV. Ft. /. " "' Ft. /. Multiply 8 4 2 10 11 10 by 4 2 10 9 33 4 11 4 118 4 14858 8 10 6 34 9 7 9 8"" 127 2 6" Note 2. 'In doing the third sum, I begin with 4, which stands under the 8, and multiply the sum, begin- ning with the right hand figure which is 10; saying 4 limes 10 are 40. In 40, 1 find there are 3 times 12 and 4 over. Setting down 4, I multiply the next figure, adding three to it, which makes 11. and thus multiply the whole sum. Then taking the 2 for the multiplier, I say 2 times ten are 20. In 20 1 find 12 is contained once, and 8 over. Setting down 8 one place farther to the right, I say 2 times 2 are 4, and one to carry makes 5; and after this manner multiply all the figures in the sum. Then adding the two rows of figures together, I obtain the answer DUODECIMALS. 107 Method of doing the same sum by taking the frac- tional parts. Ft. 7. " "' 2 inches = 842 10 4 2 33 1 4 11 4 8 4 5 8 34 9 9 8"" Answer* In this last example, I multiply the sum by 4, as in the former case. Then, as 2 inches make -J of a foot, I divide the sum by 6, which I hnd multiplied by 4, divi- ding it after the manner of Compound Division, multi- plying each remainder by 12, and adding it to the next lower denomination; and setting the result under the amount of the multiplication. Then 1 add the two sums as before. I. What jrre the solid contents of a cubick block that is 4 feet 4 inches in length, 3 feet 8 inches in breadth, and 2 feet 8 inches in thickness? Ans. 42JFV. 41. 5" 4"'. 6. What i? the product of 12 feet 9 inches, multiplied by 6 feet 4 inches. Ans. SO Ft. 97. 7. What is the product of 3 feet 2 inches 3" multi- plied by 3 feet 2 inches 3"? Ans. 10.FUJ. 11". 0'". 9"". 8. What is the price of a marble slab whose length is 5 feet 7 inches, and breadth 1 foot 10 inches, at one dollar per foot? Ans. $10.23. 9. How rnnnv square feet in a board 177V. 77. long and IFl. 57. wide? Ans. 24F*. 107. II". 10, How many solid feet in a load of wood 6Ft. 77. long and 3R 57. high and 3JFV. 87. wide? Ans. S2Ft. 57. 8". 4"', 11. What will be thr, expense of plastering the walls of a room S.f-1. 67. high arid each of the 4 sides IfiFt. 37. long, at 50 cents per square yard ? Ans. $30 .(590-}-. 108 SINGLE RULE OF THREE. Questions. 1. What are Duodecimals? 2. In what are they useful ? 3. In adding, subtracting, and multiplying Duodeci- mals, what do you observe in carrying from one denomination to another? 4. What are the denominations used in Duodecimals? 5. Repeat the rule for Multiplication of Duodecimals? SINGLE RULE OF THREE. The Rule of Three, which is sometimes called the Rule of Proportion, teaches how to find a fourth pro- portional to three numbers given. As it has three terms given to find a fourth, it is generally called the Rule of Three. Questions to prepare the learner for this rule. 1. If 2 apples cost 3 cents, how much will 4 apples cost at the same rate ? 2. If you give 2 cents for 4 nuts, how many cents must you give for 8 nuts? 3. If a pound of butter cost 8 cents, how much will pounds cost? 4. A boy has 20 melons to sell, and asks 10 cents for two, how much will they all come to at the same rate? 5. If 6 men can reap a field of wheat in 4 days, how long will it take 12 men to reap the same field? 6. If 4 yards of cloth cost 1 dollar, how much will 2 yards cost? 7. How much will a gallon of milk come to, at four cents a quart? 8. How much will a bushel of peaches come to at 25 cents a peck ? 9. If 3 cents will buy 2 apples, how many apples will 9 cents buy? 10. If a boy can run 2 miles in one hour, how far can e run in 4 hours? SINGLE RULE OF THREE. 109 RULE. Set the term in the first place, which is of the same kind with that in which the answer is required. Then determine whether the answer ought to be greater or less than the third term. If the answer ought to be greater than the third term, set the greater of the other two numbers on the left for a second or middle term; and 'hs L ss number on the left of the second term, for a first term. If the answer ought to be less than the third term, the less of the two other numbers must be the mi. Idle t2rm, and the greater must be the first term. After iluis stating the sum, proceed to do it in the fol- lowing manner, viz: Reduce the third term to the lowest denomination mentioned in it. Reduce, likewise, the first and second terms to the lowest denomination that either of them has. Then multiply the second and third terms together, and divide their product by the first term. The quotient thus obtained will be the answer. It will not be necessary to distinguish between direct and inverse proportion, because the foregoing rule is calculated for both. PROOF. By reversing the statement. EXAMPLES. 1. If 3 pounds of sugar cost 25 cents, what will 18 pounds cost at the same rate? Ibs. Ibs. cts. Thus, 3 : 18 : : 25 18 200 25 3)450 $1.50 Answer. 110 SINGLE RULE OP THREE. 2. If 7 pounds of coffee cost b?i cents, what must I pay for 244 pounds? Ibs. Ihs. cts. Thus, 7 : 244 : : 87J 1708 1952 122 7)21350 $30.50 Answer. 3. If 450 barrels of flour cost $1350, what will 8 bar- rels cost? Ibis. bbls. $ $ Thus As 450 : 8 : : 1350 : 24, Answer * 4. If 15 yards of cloth cost 5, what number of yards may be bought for 125? yds. yds. As 6 : 125 : : 15 : 3121 Answer. 5. If twelve men can do a piece of work in 20 days, in what time will 18 men do it? m. m. d. d. As 18 : 12 : : 20 : 13| Answer. 6. What will be the cost of 17 tons of lead, at 223 dollars 66 cents per ton? T. T. D. ctfs. D. cts As 1 : 17 : : 223.66 : 3802.22 Ans. 7. What will 72 yards of cloth cost at the rate of 9 yards for 5 12s. yds. yds. s. s. As 9 : 72 : : 5 12 : 44 16 Ans. 8. If 750 men require 22500 rations of bread for a month, what will a garrison of 12CO require? Ans. 360CO. I *The sum in the third example is read thus: As 450 is to 8 so is 1350 to the answer. This is the manner of reading all sunui when stated in the Rule of Three. SINGLE RULE OF THREE. Ill 9. What must be the length of a board that is 9 inches in width, to make a surface of 144 inches, or a square foot? Ans. 16 inches. 10. How many yards of matting 2 feet 6 inches broad, will cover a floor that is 27 feet long, and 20 broad? Ans. 72 yards. I 11. If a person's annual income be 520 dollars, what iis that per week? Ans. 10 dollars. 12. If a pasture be a sufficient for 3000 horses for 18 days, how long will it be sufficient for 2000? H. H. D. D. As 2000 : 3000 : : 18 : 27 Ans. 13. What must be the length of a piece of land 13J- rods in breadth, to contain one acre? Ans. 11 rods, 4yds, 2ft. 0-^in. 14. If 8 men can build a tower in 12 days, in what time can 12 build it? M. M. D. D. As 12 : 8 : . 12 : 8 Answer. 15. If a piece of land be 5 rods in width, what must I be its breadth to make an acre? R. R. R. R. As 5 : 160 : : 1 : 32 Answer. 16. How much carpeting that is lj yards in breadth, will cover a floor 7-J- yards in length, and five yards in breadth? By Decimal Practlons. yds. yds. yds. As 1.5 : 5 : : 7.5 : 25 Answer.. 17. What will one quart of wine cost at the rate of 12 dollars for 16 gallons? gals. qts. qt. D. cts. As 16 or 64 : 1 : : 12.00 : 18f Ans. 18. If 10 pieces of cloth, each piece containing 42 yards, cost 531 dollars 30 cents, what does it cost per yard? Ans. $1.2i3i 19. If a hogshead of brandy cost 78 dollars"? 5 csnts, what must be given for 5 gallons at the same rate? Ans. $6.25 112 SINGLE RULE OF THREE. 20. If a staff 4 feet in length, cast a shade on level ground, 8 feet in length, what is the height of a tower, whose shade, at the same time, measures 200 feet? ft- ft- ft- ft- As 8 : ^00 : : 4 : 100 Answer. 21. I lent my friend 350 dollars for 5 months, he pro- mising to do me the same favour; hut when requested, he could spare only 125 dollars. How long ought I to keep it to balance the favour? D. D. M. M. As 125 : 350 : : 5 : 14 Answer. 22. If 7 oxen he worth 10 cows, how many cows will 21 oxen be worth? Ox. Ox. C. C. As 7 : 21 : : 10 : 30 Answer. 23. If board for 1 year, or 52 weeks, amount to $182, what will 39 weeks come to? Ans. $133.50. 24. If 30 bushels of rye may be bought f>r 120 bushels of potatos, how many bushels of rye imy be bought for 300 bushels of potatos? An. 150 bushels rye. 25. A gentlemin bought a bag of coffee for the use of his family, weighing 127 Ibs., for which he gave $15.- 25 cts.; what was it a pound? Ans. 12-}- cts. 26. If a gentleman spends 4 dols. 62i cts. every day how much will that amount to in a year? Ans. 1688.124, 27. A lady bought for the use of her family, a piece of cloth, containing 16 yds. 3 qrs. 2 na., at $1.25 cts a yard; what was the amount of the whole? Ans. $21.09+ 28. A, failing in trade, owes the following sum=, viz to B $1600.60, to C $500, to D $750.20, to E $1000 to F $230; and his property, which is worth no more than $1020.20, he gives up to his creditors; how mucl does he pay on the dollar? arid what is the amount oi loss sustained by all? Ans. 25 cts. on $1 ; and the amount of loss is $3060.60 29. A farmer made from an orchard of apples, 146 barrels of cider, which he afterwards sold at $3.12j cts a barrel; what was the amount of the whole? Ans. $456.25. SINGLE RULE OF THREE. 113 30. A flour merchant sold 1574 bbls. of flour, at $5.12i cents a barrel, what was the amount? Ans. $8066.75. 31. A merchant failing, was able to pay his creditors only 62 J- cents in a dollar; how much wi.l a person re- ceive, whose claim is 746.25 cents? Ans. $466.40,6,25. 32. A lady purchased a set of silver, weighing in the whole 5 Ib. 6 oz. 5 dwt. at 1.50 cents an ounce; what was the cost of the whole? Ans. $93.37,5. 33. A lady, intending to make a bed quilt, containing 8-J- square yards, desired her daughter to inform her how much muslin l yard wide would be required to line the same; how many did it take? Ans. 6.8 yds. or 6-} yds. 34. A certain field will afford pasture for 10 oxen, for 60 days; how long will the same pasture suffice for 24 oxen? Ans. 25 days. 35. A pipe will drain off a cistern of water in 12 hours; how many pipes of the same size will empty it in 30 minutes? Ans. 24 pipes. 36. Two persons travel on the same road, and in the same direction. One sets out 5 days before the other, and travels 20 miles a day; the other travels 25 miles a day; how long before the latter will overtake the for- mer? Ans. 20 days. 37. If 48 men can build a fortification in 24 days, how many men can do the same in 192 days? Ans. 6 men. 38. A certain piece of work was done by 120 men in 8 months, how many men will it take to do another piece of work of the same magnitude in 2 months? Ans. 460. 39. A merchant failing in trade, owes 29475 dollars: he delivers up his property, which is worth 21894 dol lars 3 cents; how much does this sum pay on the dollar towards what he owes? Ans, 74cts. 2m.-{- 40. If property rated at 28 dollars, pay a tax of 21 dollars, how much is that on the dollar? Ans. 75 cents. 41. If i of a yard cost 62i cents, what will f of a yard cost? Ans. $2.187.+ 114 SINGLE RULE OF THfiEE 42. If a tax of 30,000 dollars be laid on a town in which the ratable property is estimated at 9,000,000 dollars, what will be the tax of one of the citizens whose ratable estate is reckoned at 750 dollars? D. D. D. D. cts. As 9,000,000 : 30,000 . : 750 : 2 : 50 .Ans* 43. How far are the inhabitants on the equator carried n a minute, allowing the earth to make one revolution .n 24 hours; and allowing a degree to contain 6J miles? The earth being divided into 360 degrees, allowing 69* miles to a degree, makes the distance round it to be 25020 miles; the number of minutes in 24 hours is 1440. Ans. 17 mile?, 3 fur. 44. There is a cistern having 4 spouts; the first will empty in 15 minutes, the second in thirty minutes, the third in 45 minute?, and the fourth in 60 minutes: in what time would the cistern be emptied, if they were all running together? As 15 : 1 : : CO : 6 30 : 1 : : : 3 45 : 1 : : 90 : 2 60 : 1 : : SO : 1-} cisterns, cist. mln. rnin. sec. Then, decimally, as 12.5 : 1 : : 90 : 7 . 12 Ans *In making taxes in a due proportion, according to the value of each man's ratable estate, proceed in the following manner. Make the amount of ratable property the first term ; make the sum to be raised the second term ; and one dollar the third term and the number arising from this operation will be the amount to be raised on the dollar, From thi-, make a tax table from one dollar to 30, or any amount necessary. In the same manner fine what is to be paid on a cent of ratable estate ; and from this make a table from 1 to 99 cents; then, from these tables, take each man's tax. Thus, if the tax were 75 cents on the dollar and you would know what a portion of property pays, that i rated at $28,80, the tables will show the amount to be $21, fo the dollar?, and 60 cts., for the cents. In estimating property for making taxes, it is customary to rate it much lower than its real value. DOUBLE RULE OF THREE. 45. If a ship^s company of 15 persons have a quantity of bread, sufficient t-j afford to each one 8 ounces per day, during a voyage at sea, what ought to be their al- lowance, under the same circumstances, if 5 persons be added to their number? Ans. 6 ounces. Note. As the Rule of Three in Vulgar and Decimal Fractions require the same statements as in whole num- bers, and is performed by multiplication and division after the same manner of other sums in the Rule of Three, it is deemed unnecessary to give any examples. When the pupil understands Fractions and the Rule of Three, he will find no difficulty with the Rule of Three in Fractions. Q. 1. What is the Rule of Three sometimes called? 2. What does it teach? 3. Which of the terms must be set in the third place? 4. How do you ascertain which ought to be the first term, and which is the second? 5. If the third term consist of different denomina- tions, what do you do with them? 6. What do you do if the first and second terms are of different denominations? 7. After stating the sum, and reducing, when neces- sary, the terms to similar denominations, how do you proceed to do the sum? 8. How are sums in the Single Rule of Three proved? DOUBLE RULE OF THREE. The Double Rule of Three is that in which five or more terms are given to find another term sought. RULE. Set the term which is of the same denomination aa :he term sought, in the third place; then consider each pair of similar terms separately, and this third one, as making the terms of a statement in the Single Rule of Three, setting the similar terms in the first or second places, according to the rule of the Single Rule of Three. After stating the question in this manner, and reducing, 116 DOUBLE RULE OP THREE. if necessary, the similar terms to similar denominations, then multiply the terms in the second and third places |together for a dividend, and the terms in the first place together fora divisor the quotient, after dividing, will be the term sought. Sums in this rule may also he done by two or more statements in the Single Rule of Three. PROOF. By inverting the statement, or, more easily, by two statements in the {Single Rule of Three. EXAMPLES. 1. If 8 men, in 16 days, can earn 96 dollars, how much can 12 men earn in 26 days? men 8 : 12 : : days 16 : 26 : : 128 312 1872 2808 D. 128)29952(234 Answer. 256 435 384 512 512 2. If $100 gain $8 in 12 months, what will $400 gain in 9 months? As 100 : 6 : J months 12 : 9 : : \ 1200 54 400 X 12|00)216|00 $18 Answer. DOUBLE RULE OF THREE. 117 3. If 16 men can dig a trench 54 yards in length in 6 days, how many men will be necessary to complete one 135 yards in length, in 8 days? By two statements in the Single Rule of Three. yds. yds. men. men. As 54 : 135 : : 16 : 40. days. ds. men. men. Then, as 8:6:: 40 : 30 Answer. 4. If $100 in one year gain $5 interest, what will be the interest of $750 for 7 years? Ans. $262.50. 5. If 9 persons expend $120 in 8 months, how much will 24 oersons spend in 16 months at the same rate? Ans. $340. 6. If 54 dollars be the wages of 8 men for 14 days, what mast be the wages of 28 men for 20 days at the same rate? Ans. $270. 7. If a horse travel 130 miles in 3 days, when the ;lays are 12 hours in length, in how many days of 10 hours each can he travel 360 mile^s? Ans. 9$ f days. 8. If 60 bushels of corn can serve 7 horses 28 days, how many days will 47 bushels serve 6 horses? Ans. 25f days. 9. If a barrel of beer serve 7 persons for 12 days, how many barrels will be sufficient for 14 persons for a year, or 365 days? Ans. 6C| barrels. 10. If 8 men spend 32 dollars in 13 weeks, what will 24 men spend in 52 weeks? Ans. $384. |Q. 1. How rmny terms are generally given in the Dou- ble Rule of Three? 2. Which of the terms must be set in the third place? 3. How do you ascertain which of the other terms should he placed in the first, and which in the second place? 4. Which of the terms do you multiply together for a dividend? 5. How do you form a divisor? 6. How do you proceed when the terms consist of dif- ferent denominations? 7. How is a sum in the Double Rule of Three proved ? i io DOUBLE RULE OF '1HKLE. Promiscuous questions in Simple and Compounl Pro- portion. 1. Wh^,t can you buy 15 tons of hay for, if 3 ton: cost $36? "Ans. $150. 2. William's income is $1500 a year, and his daily expenses are $2.50; how much will he have saved ; t the year's enci? Ans. $567.50. 3 Jf 7 men can reap 84 acres of wheat in 12 davs; how many can reap 100 acres in 5 days? Ans. 20 men. 4. If a horse will trot in a gig 8 miles in an hour, how far will he trot at the same rate, in 3i hours? Ans, 28 miles^ 5. A merchant bought 5 pieces of muslin, each con- taining 26 yards, at 11 cents a yard; what did they amount to? Ans. $14.30. 6. If a family of 9 persons, in 24 months, spend $480, how much would 16 persons spend in 8 months? Ans. $320. 7. A merchant, owning | of a vessel, sells T of his share for $500; what was the whole vessel worth? | O/"|=JL=J; then, as 2 . of the vessel is $500, -i is $250, and j, or the whole vessel, is 5x*250=$1250. Or thus; f o/f : 1 : : 5CO : $1250. Ans., as before. 8. If li Ib. indigo cost $3.84, what will 49,2 ll:s. cost |at the sariKx rate? Ans. $125,952. 9. If 1 12 acres of meadow be mowed over by 16 men. in 7 days; how many acres can -24 men mow over, in 19 days? Ans. 456 acres. 10. If 8 cwt. of iron can be carried 128 miles for $12.80, what will be the expense of carrying 4 cwt. 32 miles? Ans. $1.60. 11. A merchant bought a bale of cloth, containing 375 yds. at ^ 3.12^ a yard; what did the whole amount t? " Ans. $1171.87,5. 12. A mother allows her daughter, at a hoarding school, 3 cents a day for spending money; how much will that amount to in a year? Ans. $10.95. 13. Suppose the wages of 6 persons fjr 21 weeks be 288 dollars, what must 14 persons receive for 46 weeks \ Ans. $1472. PRACTICE. 11<J 14. If 1 hundred weight of sugar cost 13 dollars 50 cents, what must be paid for ITcwt. 3qrs. 141b.? Ans. $'241.31-} cents. 15. How many yards of paper, 2i feet wide, will be required to cover a wall, which is 12 feet long, and 9| fee high? Ans. 14yds. 1ft. Sin. 1 . If it take 5 men to make 150 pair of shoes in 20 days> how many men can make 1350 pair in 60 days? Ans. 15 men. 17. If a footman travel 240 miles in 12 days, when the days are 12 hours long; how many days will he require to travel 720 miles, when the days arc 16 hours long? Ans. 27 days. 18. If 4 men receive $24 for 6 days 1 work, how much will 8 men receive for 12 days' work? Ans. $96. 19. If 9 persons in a family spend $1512 in 1 year (or 12 mo.), how much will 3 of the samo persons spend in 4 months? Ans. $168. 20. A regiment of soldiers, consisting of 800 men, arc to be clothed, each suit containing 4|- yds. of cloth, which is If yd. wide, and lined with flannel J yd. wide; how many yards of flannel will be sufficient to line all the suits? Ans. 8633 yds. 1 qr. lj na. PRACTICE. Practice is a short method of doing all sums in the Single Rule of Thiee, that have one for their first term, and is of great use among merchants. It may be proved by Compound Multiplication, or by the Single Rule of Three. Questions to prepare the learner for this rule. 1. What will 50 yards of tape cost at i of a cent per yard? 2. What will 40 pounds of beef come to at 1 of a cent per pound? 3. What will 100 figs come to at J of a cent a piece? 4. How many pence will 40 peaches come to at one farthing a piece? 5. How many shillings and pence will 52 peaches come to at one farthing a piece? 120 PRACTICE. PRACTICE TABLE, OR TABLE OF ALIQUOT PARTS. ds. dots. *. A d. s. 50 =3 .1. * i 10 z= i ' } 1 1 "\ O 25 i c 68 | H~ " i 7 20 i 9 50 j 2. 2 * i s- 12* \ > 2~ 4 P 3 jf*! 6 1 A jr 3 4 "5 4 | j jf- 5 -j 2 f> J 6 v J ' 4 1 8 T V 5 ^r5. Ib. cwt. 771. c~ts. i o A . a or 56 = i ^ 5 2 1 *1 s> .A)? gr. d. d. 1 28 I 16 | 14 J 8 T V 1 CASE I. rV I When the price of one yard, pound, $c. is in farthings. BULK Divide by the aliquot parts of a penny, and the an- swer will be in pence, which reduce to shillings, pounds, EXAMPLES. 1. What is the value of 2. What is the value of 380 at one farthing each? 744 at 3 farthings each? 1 J 380 2 i 744 J 12) )5 1 i 372 - 186 v 7s. lid. Ans. f 12)358 3. What is the value of 460 at 2 farthings each? 20)46 G 2 i 460 2 6. 6d. Ans. 12)230 pence. 19s. 2d. Answer. 4. What is the worth of 298 at Jr !.? An?. 6s. 2id. 5. What is the worth of 586 at id .? Ans. 1. 4s. 5d. | 6. What is the worth of 964 at fd.? Ans. 3. Os, 3d. PRACTICE. 121 CASE II. When ike price is any number of pence less than 12. RULE. Divide by the aliquot parts of a shilling, and the an- swer will be in shillings, which may be reduced to pounds. EXAMPLES. I. II. 672 at Id. 20)56 2. 16s. Ans. 444at2d. 20)74 3. What is the value of 237 at 3d.? 4. What is the value of 594 at 4d.? 5. What is the value of 868 at 6d.? 6. What is the value of 988 at 5d.? 7. What is the value of 1049 at 8d.? 3 14s. Ans. s. d. 2 19 3 9 18 14 Ans. 20 11 8 Ans. 34 19 4 Ans. Ans. Ans. 21 8. What is the value of 1294 at 10d.? Ans. 53 18 4 CASE III. When the price in pence exceeds the number of 12. RULE. Consider the number given in the sum as containing so many shillings. Then divide by such aliquot parts as may be formed by the pence over a shilling, adding the product to the sum. The answer will be in shillings. EXAMPLES. 20 600 75 675 at 13jd. 33 15s. Answer. Note. In this example, I consider the sum as 600 shillings. Then, as the given price is 1 Jd. over a shil- ling, which makes J of a shilling, I divide the sum by 8, and add the quotient to the given sum j which makes 675 shillings, or 33 15s. 122 PRACTICE. 2. What is the worth of 450 at 14d.? Ans. 26. 5s. 3. What is the worth of 570 at 16d.? Ans. 38. Os. CASE IV. When the price is any number of shillings under 20. RULE. Divide by the aliquot parts of a pound, and the answer will be in pounds. Or, consider the sum as being so many shillings, then multiply the sum by the number of shillings in the price. The product will be the an- swer in shillings; which reduce to pounds. I. n. 5s. 1296 at 5s. 324 Ans. 723 at 12s 12 X 20)3676 433. 16s. Answer. The second example is done by the second method, which is thought by many to be the easier way. 3. What is the value of 1128 at 3s.? Ans. 169. 4 4. What is the value of 889 at 4s.? Ans. 177. 16 5. What is the value of 1616 at 9s.? Ans. 727. 4 6. What is the value of 2868 at 18s.? Ans. 2581. CASE V. When the price is in pounds, shillings and pence. RULE. Multiply the sum or quantity by the number of pounds in the price, then divide the aliquot parts of shillings and pence, and add the quotients to the product theii; sum will be the answer. I. EXAMPLES. II. 10 448at410s.6d.4 4 2027. 4s. Ans. 5678 at 7 4s. 9d 7 39746 1135 12 141 19 70 19 6 U094. 10s. 6d. Note. In the second example, after multiplying the sum by the number of pounds, as 4s. is i of a pound,! divide by 5, which gives 1135 pounds in the quotient; and leaving a remainder of 3 pounds, which reduced to shillings and divided by 5 give 12s. Then, as 6 and 3 make 9, the number of pence, and as 6d. is J of 4s., I divide the quotient by 8, which gives 141 pounds wish a remainder of 7; this being reduced to shillings, and the 12 shillings above added to it make 152, which di- vided still by the 8 give 19 shillings. And as 3 is 1 of 6, or its aliquot part, I divide the hist quotient bv 2 This gives 70 pounds and a remainder of I, which is 20 shillings; and adding it wilh 19 shillings above, the amount is 39 shillings. This divided by the 2 gives 11 shillings and a remainder of 1 shilling, or 12 pence; which divided still by the 2, makes 6d. And thus the answer is obtained. 3. What is the amount of 288 at 5. 10?. 4d.? Ans. 1588. 16s. 4. What is the amount of 642 at :>. 4s. 6d.? Ans. 51 22. Cs. T>. What is the amount of 734 at 12. *s. d.? Ans. 8905. 17s. 4d. CASE VI. When the quantity consists of different denominations, and the price is in pounds, shillings, 6$c. RULE. Multiply the price of the highest denomination given, by the whole of the highest denomination, then divide by aliquot parts of each of the lower denominations in the sum. Add the results together, and their sum will be the answer. EXAMPLES. i. s. d. 3 cwt. 2 qrs. 14 Ibs. 2 qrs. are ^ of a cwt.. 14 Ihs. are 1 of 2 qrs. at 4 6 2 per cwt 3 12 18 6 2 3 1 10 9i 15. 12s. 4id. Ans. 124 FELLOWSHIP. 3. 4 cwt. 3 qrs. 12 Ibs. at 8. 4s. 4d. per cwt. Ans. 39. 18s. 2d. 3. 5 cwt. 3 qrs. 4 Ibs. at 9. 6s. 8d. per cwt. Ans. 54. 0. 0. 4. 7 cwt. qr. 14 Ibs. at 2. 3s. 4d. per cwt. Ans. 15. 8s. 9d. 5. 8 cwt. 3 qrs> 24 Ibs. at 1. 2s. 3d. per cwt. Ans. 9. 19s. 5id. 6. 9 cwt. 1 qr, 18 Ibs. at 3. 10s. lOd. per cwt. Ans. 33. 6s. 7d. 7. 10 cwt. 2 qrs. 10 Ibs. at 4. 4s. 6d. per cwt. Ans. 44. 14s, &}d. 1 1. What is practice? 2* Wherein is it particularly useful? 3* Repeat the table of aliquot parts. 4. How many cases are there in poundsj shillings, &.C.! 5. Repeat the rule of each different case. 6. How are sums in practice proved? FELLOWSHIP. Fellowship is an easy rule by which merchants or other persons in company, are enabled to make a just division of the gain or loss in proportion to each person's share. Sums in Fellowship are generally done by the Rule of Three. CASE I. When the several shares are considered without regard to time. RULE. As the sum of all the stock is to each person's partic- ular share of the stock^ so is the sum of all the gain or loss, to the gain or loss of each person, PROOF, Add together all the shares of gain or loss, and if it be right, the sum will be equal to the whole gain or loss. FELLOWSHIP, 125 EXAMPLES. 1. A and B purchase certain gocc's amounting to |f 580, of which A pays $350 and B $230. They gain " what is each man's share of the gain? A $350 B $230 A's share, gain. $ cts. 580 : 350 : : 262 : 158.10 JJ- A's gain. B's share, gain. $ cts. Then, as 580 : 230 : : 262 : 103.8?) jf B's gain. 2. A, B and C formed a company. A put in $-10,B 60 and C 80. They gained $72: what was each man's share? Ans. A gained $16, B 24 and C 32. 3. A, B and C lose a quantity of property worth $2400; of which A owned J, B -J-, and the remainder to C; what does each lose? Ans. A loses $600, B 800 and C 1000. 4. Three persons entered into partnership in trade iThe first put in 250 dollars; the second put in 350 dol liars; and the third put in 500 dollars; and in 12 months, they found, by examining their hooks, that they had gain- ed 460 dollars; how must the gain be divided between them, so that each may have his due proportion? fA's share, $104.54,545+ Answer. \ B's share, 146.3G,363-f IC's share, 209.09,09 + 5. A and B purchase goods worth $80, of which A pays 30 and B 50. They gain $20; what is the gain of each? Ans. A's gain is $7.50 and B's 12.50. 6. Four men formed a capital of $3200. They gain ed in a certain time $6560. A's stock was $560, B's 1040, C's 1200 and D's 400. What did each gain? Ans. A gained $1148, B 2132, C 2460 and D 820. CASE II. When the different stocks in company are considered in \ relation to time. RULE. Multiply each man's stock by the time it has been a part of the whole stock; then, as the sum of the pro- lucts is to either single product, so is the whole sum of or loss to the gain or loss of each man. 12(1 TAKE AND THET. EXAMPLES. 1. A, Band'C hold a pasture in common, for which they pay $40 per annum. A put in 9 cows for five weeks; B, 12 cows for 7 weeks; and C, 8 cows for 16 weeks. What must each man pay for the rent? 9x 5= 45 12x 7= 84 8x16=128 As 257 - 45 : : 40 : 7^ T A's part. As 257 ; 84 : : 40 : 13 A, B's part. As 257 : 128 : : 40 : 19f f ? C's part. 2. A with a capital of 1000., entered into business on the first of January. On the first of March follow- ng he took in B as a partner, who brought with him a capital of 1500.; and three months after they are joined by C, with a capital of 2800. At the end of the year they find they have gained 1776. 10s. How must it be divided among them? Ans. A^s part will be 457. 9s. 4d. B's part will be 571. 16s. 8-jd. C's part will be 747. 3s. lljd. Q.I. What is Fellowship? 2. By what rule are sums in Fellowship usually done ? 3. How do you proceed when the shares are consider- ed without regard to time ? 4. How do you proceed when the shares are consider- ed in relation to time ? 5. How are sums in Fellowship proved? TARE AND TRET. Tare and Tret are certain allowances made by mer- chants in selling their goods by weight. Tare is an allowance made for the weight of the bar- el, bag, &c., that contains the article or commodity bought. Tret is an allowance of 4 Ibs. in every 104 Ibs. for vaste, dust, &c. TARE AND TRET. 127 Gross weight is the weight qf the goods, together with the barrel, box, or whatever contains them. When the tare is deducted from the gross, what remains is called suttle. Neat weight is the weight of articles after all allow- ances are deducted, CASE I. When iJie tare is so much in the whole gross weight. RULE. Subtract the tare from the quantity the remainder will be the neat weight. EXAMPLES. In Ghhds. of sugar, each weighing 9 cwt. 2 qrs. 10 Ibs. gross, tare 25 Ibs. per hhd. how much neat weight? cwt. qr. Ib. cwt. qrs. Ibs. 25x6=1 1 10 tare 9 2 10 6X 57 2 4 gross, 1 1 10 tare 50 22Ans. 2. What is the neat weight of 456 cwt. 1 qr. 19 Ibs. of tobacco, tare in the whole 15 cwt. 2 qrs. 13 Ibs? , Ans. 440 cwt. 3 qrs. 6 Ibs. 3 What is tho neat weight of 5 casks of sugar, the gross weight and tare as follows? cwt. qrs. Ibs. qrs. Ib. No. 1. Gross 4 2 14 Tare 1 5 2. 3 17 1 1 3. _ 5 3 10 2 11 4. 6 1 16 2 27 5. 3.2 18 * 1 3 ' Ans. 21 cwt. 2 qrs. CASE II. When the tare is at so much per cwt. RULE, Divide the gross weight by the aliquot parts of a cwt then subtract the quotient from the gross, and the re- mainder will be the neat weight... 128 TARE AND TRET. EXAMPLES. 1. In 129 cwt, 3 qrs. 16 Ibs. gross, tare 14 Ibs. per cwt, what neat weight? 14 Ibs 129 3 16 gross. 16 26i 113 2 171 Answer. 2. In 97 cwt. 1 qr. 7 Ibs. gross, tare 20 Ibs. per cwt. what neat weight? Ans. 79 cwt. 1 qr. 20J Ibs. 2. What is the neat weight of 35 kegs of raisins, gross weight 37 cwt. 1 qr. 20 lb.; tare per cwt. 14 Ibs.? Ans, 32 cwt. 3 qrs. 3. What is the neat weight of 6 hogsheads of sugar, each weighing 8 cwt. 2 qrs. 14 Ibs. gross; tare 16 Ibs. per cwt.? Ans. 44 cwt. 1 qr. 12 Ibs. Note. When the tare per cwt. is not an aliquot part, the tare may be found by the Rule of Three, thus As 112 is to the number of pounds gross, so is the rate per cwt., to the tare required. 4. What is the neat weight of 38 cwt. qr. 4 Ibs. tare, at 11 Ibs. per cwt. cwt. qr. Ibs. 38 4=4260 pounds. Ibs. Ibs. Ibs. Then, as 112 : 4260 : : 11 : 4 18^ Answer. 4260 r 4 A cwt. qr. Ibs. = 34 1 5 f 6 T 8 2 - Answer. CASE III. Wlien tare and tret are allowed. RULE. ' Find the tare according to the preceding rules, sub- tract it from the gross, and the remainder will be suttle; then divide the suttle by 26, and the product will be the tret, which subtract from the suttle the remainder will be the neat. Note. As 4 pounds on the 104 Ibs. is the customary I allowance for tret, we divide by 26, because 4 is % of j 104. TARE AND TRET. 129 EXAMPLES. 1. In 247 cwt. 2 qrs. 15 Ibs-. gross, tare 28 Ibs. per cwt. .nd tret 4 Ibs. for every 104 Ibs. how much neat? 28 lbs.= | J cwt. 41bs.= | ^ of 104 cwt. qr. Ib. ox. 247 2 15 61 3 17 12 tare subtract 185 2 25 4 7 16 Ans. 17S 29 4 neat. 2. In 9 cwt. 1 qr. 10 Ibs. gross, tare 28 Ibs. per cwt. and tret 4 Ibs. for every 104 Ibs. how much neat? Ans. 6 cwt. 2 qrs. 26 Ibs- 3. A merchant purchased 4 hhds. of tobacco, weigh- ng as follows : The first 5 cwt. 1 qr. 12 Ibs. gross, tare 35 Ibs. per hhd.; the 2d. 3 cwt. qr. 19 Ibs. gross, ;are 75 Ibs.; the 3d. 6 cwt. 3 qrs. gross, tare 49 Ibs.; the 4th 4 cwt. 2 qrs. 9 Ibs. gross, tare 35 Ibs. and allowing ret to each at the rate of 4 Ibs. for every 104 Ibs. What was the neat weight of the whole? Ans. 17 cwt. qr. 19 Ibs. 2 oz. Exercises under the foregoing rules. 1. There are 24 hogsheads of tobacco; each hogshead weighs 6 cwt. 2 qrs 17 Ibs. gross; tare in all, 17 cwt. 3 qrs. 27 Ibs. How much will the tobacco amount to at 1. 10s. 6d. per cwt. Ans. 216. Os. 4|d. 2. Bought 5 bags of coffee, each of which weighed 95 Ibs. gross; tare in the whole lOlbs. How much did it amount to, at 25 cents per pound? Ans. $116.25. 3. What is the value of 10 casks of alum; the whol weighing 33 cwt. 2 qrs. 15 Ibs. gross; tare 15 Ibs. per cask; price, 23s. 4d. per cwt.? Ans. 37. J3s.6jd. 4. A farmer sent a load of hay to market, which with the cart, weighed 29 cwt. 3 qrs. 16 Ibs.; the weight of the cart was 10 J cwt. ; what did the hay come to, at $" " a ton? Ans. $14.357+. 130 SIMPLE INTEREST. 5. A merchant bought sugar in a hogshead, both of which weighed 8 cwt. 15 Ibs.; the hogshead alone weigh- ed 1 cwt. 1 qr.; what was the cost of the sugar, at 111 cents a pound? Ans. $86.73f. Q. 1. What do you understand by Tare and Tret' 2. What is tare? 3 What is tret? 4. What is gross weight? 5. What is neat weight? 6. What is called suttle? SIMPLE INTEREST. Interest is a premium paid for the use of money In calculating interest on money, four things are necessary to be considered, viz. the principal, the time, rate per cent, and amount. The principal is the money lent for which interest is to be received. The rate percent, per annum (by the year) is the in- :erest for 100 dollars or 100 pounds for one year. The time is the number of years, months, or days, for which interest is to be calculated. The amount is the sum of the principal and interest, when added together. Questions to prepare the learner for this rule. 1. If you give $6 for the use of $100 for a year; how much must you give for the use of $50? 2. If you give $6 for the use of $100 for a year; how much must you give for the use of it for six months? How much for three months? -How much for 4 months? How much for 8 months? How much for 9 months? 3. If the interest of $200 be one dollar for a month; how much will it be for 15 days? How much for 10 days? How much for 20 days? SIMPLE INTEREST. 131 CASE I. When the time is one year, and the rate per cent, is any number of dollars, pounds, RULE. Multiply the principal by the rate per cent, divide the product by 100, and the quotient will be the interest for one year. EXAMPLES. 1. What is the interest of 328 dollars for one year at 6 per cent.? 328 In this example, as cutting off 6 the two right hand figures is the same as dividing by 100, the di Ans. $19.|68cts. vision is omitted. 2. What is the interest of $9876 for one year at 6 per cent.? 6 $592|56 cts. Answer. When the sum is in pounds, if there be a remainder after dividing, or after cutting off the two right hand figures, the remainder, or figures cut off must be reduced to shillings; and if there be still a remainder after di- viding the shillings, it must be reduced to pence, &c. 3: What is the interest of 573. 13s. 9oV^at-6-per cent, per annum? 573. 13s. 9|d. Note. When the interest is 6 for more than one year, mul- tiply the interest for one year 34|42 2 9 by the number of years. To 20 obtain the amount, the interest must be added to the princi- 8|42 pal. 12 5|13 Ans. 34. 8s. 5d. 4. What is the interest of 40. 19s. lid. 3 qrs. for one year, at 6 per cent, per annum? Ans. 2. 9s. 2d. Iqr. 132 SIMPLE INTEREST. 5. What is the interest of 87 dollars for one year, at [> percent, per annum? Ans. $5.22. 6. What is the interest of 143 dollars for one year, at 7 per cent, per annum? Ans. $10.01. When the rate per cent, consists of a whole number and a fraction, as 6j, 6J, or 6 j, multiply the principal >y the whole number, to the product add J, or |, as the case may be, of the principal and then divide by 100, or cut off the two right hand figures as before. 7. What is the interest of 228 dollars for one year, at per cent per annum? $228 $14|25cts Answer. When the principal consists of dollars and sents, mul tiply by the rate per cent, without any reference to the separating point; then from the product cut off the first right hand figure as a fraction or remainder, the next ftgure will be mills, the two next cents, and the other figures, that is, those on the left of the cents, will be dollars. 8. What is the interest of $98.79 for one year, at 6 per cent, per annum? 6 5|92|7|4 fraction Ans. $5,92c.7m. 9. What is the interest of 432 dollars 73 cents for 4 years, at 6 per cent, per annum? $432.73 6 rate per cent. 259638 4 number of years. 103|85|5|2 frac. Ans. $103.85c.5m 10. What is the interest of $8420-82 for three years at 8 per cent, per annum? Ans. $2020 99c 6m. SIMPLE INTEREST. 133 11. What is the interest and amount of $7462.13 for bur years, at 7 per cent per annum? Ans. Interest, $2089.39c. 7m. AmH. $9551.53c.2m. CASE II. Tojind the interest when the given time is months or days. RULE. Find the interest for one year, then say as one year s to the given time, so is the interest of the sum for one r ear, to the interest for the time required. Or, instead f the Rule of Three, it may be done by Practice, thus: ?or the number of months, take aliquot parts of a year; ind for days, the aliquot parts of 30.* EXAMPLES. 1. What is the interest of $98.50 for 9 months and 18 jays, at 6 per cent, per annum? $98.50 6 $5.91 100 for one year. year. mo. days. $ cts. $ cts. m. Then,as 1 : 9 18 : : 5 91 : 4 72 8 Ans. tn this sum, the year is reduced to 360 days, the months and 18 days to 288 days, and the third term stands as 591 cents. The same is done by Practice, thus $98.50 6 mo. 6, ofa year. 3, J of 6 mo. 15d.of3mo. 3,Jofl5ds. 5.91.0|0 2.95.5 1.47.7^ 24.6J Ans. $4.72.8 *ln these calculations, a year is reckoned at 360 days, and month at 30 days. 134 SIMPLE IXTEHEST. 2. What is the interest of $120.60 for one year and ihree months, at 6 per cent, per annum? Ans. $9.04c. 5m. 3. What is the interest on $187.06j for 10 months, at 6 per cent per annum? Ans. $9.35c 3rn. 4. What is the interest and amount of 640 dollars for 4 years and 7 months, at 5 pet cent, per annum? Ans. $146.66f interest. Arn't. $786.66j. 5. What is the interest of $300 for 4 years, 4 months, and 20 days, at 81 per cent, per annum? Ans. $111.91f 6. What is the interest of $5430 for 17 months, at 4 per cent per annum? Ans. $307.13i. 7. What is the interest of $7200 for 14 months, at 6 percent, per annum? Ans. $$04. 8. What is the interest of $8050.871 for 3 years and 11 months, at 6 per cent, per annum? Ans. $1891. 95c. 5m. 9. What is the interest of $948.621 f or 8 months, at 8 percent, per annum? Ans. $50.59c. 3m. 10. What is the interest of 421. 16s. 9d. for 2 years and 8 months, at 5 per cent, per annum? Ans. 56. 4s. lOfd. 11. What is the interest of 580 pounds for 5 years, 2 months and 10 days, at 7 per cent per annum? Ans. 210. 17s. 12. What is the interest of $36 for 1 month, at 8 per cent, per annum? Ans. 24 cents. When the rate is 6 per cent, another method ^of finding the interest for any number of months, is, to multiply the principal by half the number of months and divide the pro- duct by 100 (or cut off the two right hand figure* as before.' KLUIPLES. 1. What is the interest of $1500 for 4 months? $1500 2 half the number of months. - 30(00 Ans. $30. SIMPLE INTEREST. 135 2. What is the interest of $7656 for 3 years and 4 months? $7656 20 half the number of months. 1531J20 Ans. $1531.20. 3. What is the interest of $230.25 for 8 months? Ans. $9.21. 4. What is the interest of $750 for 9 months? Ans. $33.75. 5. What is the interest of $967.64 for 28 months? Ans. $135.47. The interest for any number of days* at 6 per cent, can be found by multiplying the dollars by the number of days, &id dividing' the product by 83: the answer will be in cents, if the principal consist (f dollars and cents , cut ojf the two right hand figures. \gj~Bank interest is reckoned by this rule. 1. What is the interest of $1542 for 90 days? And of $754.54 for 60 days? $1542 $754.54 90 60 6|0)13S7810 6|0)452724|0 2313 Ans. $23.13. 754|54 Ans. $7.54. 2. What is the interest of $3084 for 30 days at 6 per cent, per annum? Ans. $15.42. 3. What is the interest of $2324 for 54 days, at 6 per cent, per annum? Ans. $20.91. 4. What is the interest of $281.75 for 93 days, at 6 per cent, per annum ? Ans. $4.36. CASE III. The amount, time, and rate per cent, given to find the principal RULE. Find the amount of 100 dollars at the rate and time given;" then say, as the amount of 100 dollars, is to the amount given, so are 100 dollars to the principal re- quired.. SIMPLE INTEREST. EXAMPLES. . 1. What principal at interest for two years, at 6 per ;ent. per annum, will amount to $134.40? iifrl HA $100 6 12.00 100.00 $112 amount of 100 for two years. dolls. $ cts. dolls, dolls. Then, as 112 : 134.40 : ; 100 - 120 Ans. 2. What principal at interest for 5 years, at 6 per sent, will amount to $780 ? Ans. $600. 3. What principal at interest for 4 years and 3 months, it 6 per cent, will amount to $119235? Ans. $950. CASE IV. Tofnd the rate per cent, when ike amount^ time and prin- cipal are given. KULE. Take the principal from the amount, the remainder will be the interest for the given time; then,. as the prin cipal is to one hundred dollars, so is the interest of the principal for the given time, to the interest of 100 dollars for the same time. Divide the interest of 100 dollars thus found, by the time, and the quotient will be the rate per cent. EXAMPLES. 1. At what rate per cent, will $500 amount to $650 in three years ? 650 Amount. 500 Principal. 150 Interest for the time. D. D. D. D. As 500 : 100 : : 150 . 30 Interest of 100. Then divide by the time 3)30(10 Ans. per cent, 30 COMPOUND INTEHK3T, 137 2-. At what rate per cent, per annum will $1850 dou- ble in 5 years? Ans. 20 per cent. 3. At what rate per cent, per annum, will 600 dol- lars amount to $856.50 in 9 years and 6 months? Ans. 4 per cent. CASE V. To find the time when the principal, amount, and rale per cent, are given. RULE. Find the interest of the principal for one year; find the interest of the principal for the whole time, by sub- tracting the principal from the amount; then divide the whole interest by the interest for one year the quotient will show the time required. EXAMPLES 1. In what time will $800 amount to $1000 at 5 per cent. per annum? 800 1000 Then, 4|0)20|0 5 800 5 $40|00 200 Whole In't. Ans. 5 years. 2. In what time will $80 amount to $182.40 at 8 per cent, per annum? Ans. 16 years. 3. In what time will $600 amount to $798 at 6 pei cent, yer annum? Ans. 5 years. ' COMPOUND INTEREST. Compound Interest is that which arises from the in terest being added to the principal, and becoming a par of the principal, at each time of payment. RULE. Find the amount of the principal, for the time of the first payment, by Simple Interest; this amount, contain 38 COMPOUND INTEREST. EXAMPLES. 1. What is the compound interest of $8000 for two -ears, at 6 per cent per annum ? Interest for the first year 480|00 Principal 8000 Amount 8480 6 In't. for the second year 508.|80 Principal 8480.00 8988.80 Subtract 8000.00 $988.80c. Answer. 2. What is the compound interest of $554 for 3 years, at 8 per cent, per annum? Ans. $143.88. 3. What is the compound interest of $744 for 2 years, at 7 percent, per annum? Ans. $l07.80c. 5m 4. What is the compound interest of $50 for 8 years, at 8 per cent, per annum? Ans. $42.54c. 6m 5. What is the compound interest of 48. 5s. for 3 years, at 6 per cent, per annum? Ans. 9. 4s. 3|xl. In computing Interest on Notes. When a settlement is made within a short time from the date or commencement of interest, it is generally the custom to pro- ceed according to the following RULE. Find the amount of the principal, from the time the inte rett commenced to the time of settlement, and likewise the amount of each payment, from the time it was paid to the time of settlement; then deduct the amount of the sever a payments from the amount of the principal. Exercises for the Slate. 1. For ralue received, I promise to pay Rufus Stanly, or order Three Hundred Dollars, with interest. April 1, 1825. $300. PETER MOSELY. INTEREST. 139 On tnis note were the following endorsements : October 1, 1825, received $100 > April 16, 1826, . . . $ 50V December 1, 1827, . . $120} tf hat was due April 1, 1828? Ans. $60.73, CALCULATION, The first principal on interest from April 1, 1825, - $300.00 Interest to April 1, 1828, (36 mo.), - - - 54-00 Amount of principal ?irst payment, Oct. 1, 1825> - nterest to April 1, 1828", (30 mo.) Second payment, April 16, 1826, Interest to April 1, 1828, (23* mo.) Third payment, Dec. 1, 1827, - Interest to April 1, 1828, (4 mo.} $100.00 15.00 50.00 5.87 120.00 2.40 - $354.00 Amount of payments deducted - - $293.27 Remain* due, April 1, 1828, - - $60.73 2. For value received, I promise to pay Peter Trusty, or order, Five Hundred Dollars, with interest. July 1, 1825. $500. JAMES CARELESS. ENDORSEMENTS. July 16, 1826, received $200) Jan. 1,1827, - - - $ 40V March 16, 1827, - - $230) What remained due July 16, 1828? Ans. $75.15. RULE IN SOME OF THE UNITED STATES. Compute the interest on the principal sum to the first time when a payment was made, which, either alone, or together with the preceding payments (if any,) exceeds the interest then due; add that interest to the principal, and from the sum subtract the payment, or the sum of the payments, made that time, and the remainder will be a new principal, with which proceed as with the Jirst principal, and so on, to the time of settlement. 1. For value received, I promise to pay Jason Park, or order Six Hundred Dollars, with interest. March 1, 1822. $600. STEPHEN STIMPSON. ENDORSEMENTS. May 1, 1823, received $2001 June 16, 1824, - $ 80 Sept. 17, 1825, - $ 12 | Dec. 19, 1825, - $ 15 I March 1,1826, - $100 Oct. 16, 1827, - $150 J What was there due August 31, 1828? Ans. $194.41. 140 INTEREST. The principal, $600, on interest from March 1, 1822, $600.00 Interest to May 1, 1823, (14 mo.) - Amount, $642.00 Payment, May 1, 1823, a sum greater than the interest, 200.00 Due May 1, 1823, forming anew principal, - - $442.00 Interest on $442, from May 1, 1823, to June 16, 1824, (13* mo.) - - - - - - - 29.83 Amount, $471.83 Payment, June 16, 1824, a sum greater tlian the interest then due, 80.00 Due June 16, 182$, forming a new principal, - - $391.83 Interest on $391.83, from June 16, 1824, to March 1, 1826, (20ft mo.) 40.16 Amount, $4ol.99 Payment, a sum less than the interest then due, $ 12 Payment, a sum less than the interest then due, $ 15 Payment, a sum greater than the interest then due, $100 $127.00 Due March 1, 1B26, forming a new principal, - $304.99 Interest on $304.99,/rom March 1, 1826?fo Oct. 16, 1827, (19 i mo.) - - - _29.73 Amount, $334.7 Payment, Oct. 16, 1827, a sum greater th&n the interest then due, - - - ' - - - - - 150.00 Due Oct. 16, 1827, jorming a new principal, - - $184.72 Interest on $184.72, from Oct. 16, 1627, to August 31, 1828, being the time of settlement, (10i mo.) - - 9.69 Balance due Aug. 31, 1828, - $194.41 2. For value received, I promise to pay Asher L. Smith, or order, Nine Hundred Dollars, with interest. June 16, 1820. $900. WILLIAM MoRRre. ENDORSEMENTS. July 1, 1821, received $150 Sept. 16, 1822, - - - $ 90 - $10 | - $20) - fee Dec. 10, 1824, - - June 1, 1825, - - Aug. 16, 1825, - - March 1, 1827, - - What remained due Sept. 1, 1828? Ans. '$483.07. Q. 1. What is Interest? 2. What are the four things considered in calculating interest? o. What is the principal ? What is the rate per cent What is the time? What is the amount? COMMISSION AND BliOKEKAGK. 14 j 4. How do you proceed in the first case? 5. How do you proceed in pounds, shillings, &c.? 6. How do you proceed when the rate per cent, con- sists of a whole number and a fraction f 7. How do you proceed when the principal is in dol- lars and cents? 8. How do you calculate interest for more than a year? How, when the time is in mouths? 9. What other method is there for calculating interest, besides the method of multiplying the sum hy the rate per cent.? 10. How is bank interest reckoned? What is the rule for casting it? 11. Do you understand all the cases and rules of in- terest? 12. What is Compound Interest? 13. Repeat the rule for calculating Compound Interest? INSURANCE, COMMISSION AND BROKERAGE. Insurance, Commission and Brokerage, are premiums allowed to insurers, factors and brokers at a certain rate per cent.; and is obtained after the manner of the first case in Simple Interest. EXAMPLES. 1. What is the insurance of $4500, at 2i per cent.? -J? 9000 2250 $112i50c. Answer, 2, What is the commission on a sale of goods amount- ing to $1184 at 5 per cent.? Ans. 59.20. 3, What is the brokerage of $987 at 3 per cent.? Aris. $-29.61. 4. What is tho commission on a sale of goods amount- ing to 4820 at 4J per cent,? $216.90. 142 DISCOUNT. DISCOUNT. Discount is an allowance made for the payment of any sum of money before it becomes due, and is the dif- ference between that sum, due some lime hence, and its present worth. As the amount of $100 at the given rate and time is to $100, so is the given sum or debt to the present worth. Subtract the present worth from the given sum, and the remainder will be the discount. EXAMPLES. 1. What is the present worth of $500 due in 3 years, it 6 per cent, per annum? 8(2 <2 tp w $100 113 : ICO : : 500 6 100 6|00 118)50000(423.72. 3 472 18 280 100 236 118 amount of $100. 440 354 S6.00|72c. 826 340 236 104 remainder. 2. What is the present worth of $350 payable in 6 months, discounting at 6 per cent, per annum? Ans. $339.80c. 5m. 3. What is the discount on 01000 due in one year, at 6 per cent, per annum? Ans. $56.60c. 4m. 4. What is the present worth of 65 due in 15 months at 6 per cent, per annum? Ans. 60. 9s. 3d. EQUATION. 143 5. What sum will discharge a debt of $1595 due af- ter 5 months and 20 days at 6 percent, per annum? Ans. $1541.32. 6m. 6. What is the present worth of $426.55 at 6 per cent, per annum, due in 8 months? Ans. $410. 14c. 5m. Note. When discount is made without regard to time, it is found as the interest of the sum would be for one year. EaUATION. Equation is the method for finding a time to pay at once, several debts due at different times. RULE. Multiply each payment by the time at which it is due, and divide the sum of the products by the sum of all the payments the- quotient will be the time required. EXAMPLES. 1. A owes B $480 to be paid in the following man- ner, viz: $100 in 6 months, $120 in 7 months, and $260 in 10 months; what is the equated time for pay- ment of the whole debt? 100 X 6= 600 120 X 7= 840 260x10=2600 480 )4040(8,V months, Ans. 3840 200 2. A owes B $1100, of which 200 is to be paid in 3 [months, 400 in 5 months, and 500 in 8 months what jis the equated time for payment of all? Ans. 6 months. 3. C is indebted to a merchant to the amount of $2500; of which $1000 is payable at the end of 4 months, $800 in 8 months, and 700 in 12 months when ought pay- ment to be made, if all are paid together? Ans. 7 months, 153 days. 144 LOSS AND GAIN, LOSS AND GAIN. Loss and Gain is a rule by which persons in trade are able to discover their profit or loss ; and to increase or lessen the prices of their goods so as to gain or lose on them to any given amount. Questions in Loss and Grain are solved by the Rule of Three, or by Practice. EXAMPLES. 1. A merchant bought 100 yards of silk at 75 cents per yard, what will be his gain in the sale, if he sell it for 90 cents y er yard ? 75 cents. yard, yards. ctx. dolls. 15 gain per yard. As 1 : 100 : ; 15 ; 15 Ans. 2. If a grocer buy 250 Ibs. of tea, at $225, and sell the whole at $1.25 per Ib. what will be his gain by the transaction? Ans. $87.50. 3. If a yard of calico cost 28 cents, and is sold for 31 cents, what is the gain on 293 yards? Ans. $8.79. 4. Bought 420 bushels of corn at 25 cents per bushel, and sold the same at 38 cents per bushel; what was the amount gained? Ans. $54.60. 5. A merchant bought 12 cwt. of coffee at 26 cents per Ib. and afterwards was obliged to sell it at 20 cents per Ib. what was his loss? Ans. $80.64, 6. If a merchant gain $80 on $560, what is that per cent.? Ans. 14^ per cent. 7. If a yard of velvet be bought for 16s. and sold again for 12s. what is the loss per cent.? Ans. 25 per cent. 8. A merchant bought 2 hhds. of wine, containing 126 gals., at $1.75 a gal. and retailed the same at $2.12 J a gal.: what did he gain in the whole? Ans. $47.25. 9. A merchant bought 2 pieces of broad-cloth, con- taining 56 yds., at $4.75 a yard ; but upon examination, found them damaged. He was, therefore, obliged to sell them for $4.12 J a yard; how much did he lose by the bargain? Ans. $35. INVOLUTION 145 10. A gentleman purchased 1500 Ibs. of coffee for $172.50, how must he sell the same to gain $32 b\ his bargain? Ans. 13 cts. 6 in. 33. 11. A merchant bought 250 hbls. of flour ;M g : bbl.; how must he sell the same to gain $55 b: bargain? An-. : . . 12. A lady purchased a quantity of milliner} , r'-r u hich she gnve $184; and sold the same for $2!0; how m.srn did she gain per cent.? Ans. 14.13-f-per cent. INVOLUTION, OR THE RAISING OF POWERS. The product of any number multiplied by itself any given number of times, is called its power, as in the f >I- lowing example. Thus, 2x2=4 the square, or second power of 2. 2x2x2=S the cube, or third power of 2. 2x2x2x2 = 16* the liqundrate, or fourth power of 2.* Hence, 3 r used to the 4th power mnkes 81. The number which denotes a power is called the index, or exponent of that power. When a power of a vulgar fraction is required, it is only necessary to raise, first the numerator, and then the denominator to the given power, and place the product of the one over the product of the other, thus, the 3d power of J <jv3 y 3=^V EXAMPLES. 1. What is the square of 4567? Ans. 20857489. 2. What is the cube of 567? Ans. 182284263. 3. What is the biquadrate of 67' Ans 20151121. 4. What is the ninth power of 2? Ans. 512. 5. What, is the cube of J? Ans. ?|. 6. What is the cube or third power of ,13? Ans. 002197 7. What is the sixth. power of 5.03? Ans. 16196.005304471)720. *Any eriven number is co- siderod the first power of itself, and when multiplied by itself th product is the second power, &c : 14(3 EVOLUTION. EVOLUTION, OR THE EXTRACTION OF ROOTS. The root of a number, or power, is any number, which being multiplied by itself a certain number of times, will produce that power; and is called the square, cube, biquadrate root, &.c. according to the power to which it belongs. Thus, 3 is the square root of 9, because when multiplied by itself, it produces 9; and 4 is the cube root of (34, because 4x4x4=64* and so of any other num- ber. THE SQUARE ROOT. Extracting the square root of a number, is the taking a smaller number from a larger, and such as will, being multiplied by itself, produce the larger number. RULE. 1. Separate the sum into periods of two figures each beginning at the right hand figure. 5. Seek the greatest square number in the left liand period; place the square, thus found, under that period, and the root of it in the quotient. Subtract the square number from the first period; to the remainder bring down the next period, and call that the resoivend. 3. Djubie the quotient, and place it on the left hand of the resolverid for a divisor. Seek how often the divi- sor is contained ki the resoivend, omitting the units fig- ure, and set the answer in the quotient, and also on the right hand side of the divisor. Then multiply the divi- sor, including the last added figure, by that figure, that is, by the figure last placed in the quotient; place the product under the resoivend, subtract it, and to the re- mainder bring down the next period, if there be any more, and proceed as alreidy directed. If there be a remain- der after the periods are all brought down, annex cyphers, |t.wo at a time, tor decimals, and proieed till the root is ;obtained with sufficient exactness. Note. When a sum in the Square Root consists of whole numbers and decimals, point off the whole num- bers as above directed, then point the decimal part, EVOLUTION. 147 Commencing at the decimal point and forming periods of two figures each towards the right, observing when there is only one figure left for the last period, to add u cypher to the right of it, to make an even period. When the sum consists entirely of decimals, separate the periods after the same manner. If it le required to extract the square root of a vulgar fraction, reduce it to its lowest terms; then extract the root of the numerator [for the numerator of the answer, and the root of the de- nominator for the denominator of the answer. If the fraction be a surd, that is, a number u hose root can never be exactly found, reduce it to a decimal, and then ex- tract the root from it; and if the sum 1 e a mixed number, the root may be obtained in tho same way. PROOF. Square the root, adding the remainder, (if any,) and the result will equal the given number. EXAMPLES. 1. What is the square root of 20857489? .... Root. 20857481(1567 Answer. 16 divisor 85)185 resolvend. 425 divisor 906)3074 resolvend. 5436 divisor 9127) >3S89 resolvend. 63889 2. What is the square root of 294849? Ans. 543 3. What is the square root of 41242084? A n?. 6422 4. Wh t is the sq-uire root of 17.3056? AriF. 4.H : 5. What is the sq lare root of .00072k? Ans. .027 6. Whet is the sq lare root of 5? Ans. 2.23600 7. What is the square r:>ot of T \ 7 T ? An.. if 8. What is the square root of 1 /? Ans. 4.168333 148 EVOLUTION. 9. A general has an army of 7056 men; how many must he place on a side to form them into a compact quare? Ana. 84. 10. If the area of a circle be 184.125, what is the side >f a square that shall contain the same area? Thus, /184.125=13.569+Answer. 11. If a square piece of land contain 61 acres and 41 square poles, what is the length of one of its sides? A. P. Thus, 61 41=9801 square poles. Then, ,J 380 1=99 rods, or poles in length, Answer^ 12. There is a circle whose diameter is 4 inches; what is the diameter of a circle 3 times as large? Thus, 4X4=16; and 16X3=48 and -[-inches. Ans. 13. There is a circle whose diameter is 8 inches; what is the diameter of a circle which is only one fourth as arge. 8x8=64; and 64-^-4= 16; and ,yi6=4inches. Aas. 4 inches. The square of the longest side of a right aug'edtriarigle, is equal to the sum of the squares of the other two sides; therefore, the difference of the squares of the. longest side, and either of the other sides , is the square of the remaining side. 14. The wall of a certain city is 20 feet in height, it is surrounded by a ditch 20 feet in breadth; what must be the length of a ladder, to reach from the outside of the ditch to the top of the wall? Ans. 28 feet. 15. On the margin of a river 24 yards wide, stands a tree; from the top of which a line 36 yards long, will reach to the other side of the stream; what is the height jf the tree ? Ans. 26.83-j- yards. 16. Two ships sail from the same port; one sails due e;ist 50 miles, and the other due south 84 miles; how far are they from each other? Ans. 97.75 miles. 17. A ladder or pole, 40 feet long, placed in the middle of a street, will reach a window of a house on each side of the street 24 feet from the pavement; what is the width of the street? Ans. 64 feet wide. THE CUBE KOOT. 149 THE CUBE ROOT. The cube root of a given number, is such a number as being multiplied by itself, and then into that product, produces the given number. RULE, 1 . Point off the sum into periods of three figures each, [beginning with units. 2. Find the greatest cube in the left hand period, place the root of it in the quotient, subtract the cube [from the left hand period, and to the remainder bring own the next period for a resolvend. 3. Square the quotient and multiply the square by 3 for a defective divisor. 4. Seek how often the defective divisor is contained the resolvend, omitting the units and tens, or two right hand figures. Place the result in the quotient, and its square to the rightof the divisor, supplying the place of tens with a cypher, whenever the square is less than ten. 5. Multiply the last figure of the quotient or root by all the figures in it previously ascertained; multiply that product by 30, and add their product to the divisor, to complete it. 6. Multiply and subtract as in Simple Division, and to the remainder bring down the next period, for a new resolvend. Find a divisor as before, and thus proceed until all the periods are brought down. Note. The cube root of a vulgar fraction is found by jreducing it to its lowest terms, and extracting, as in the square root; and if the fraction be a surd, reduce it to a (decimal, and then extract the root. In extracting the cube root, if the sum be in part de- |cimals, or if the whole be decimals, point the figures in the square root, observing to have three figures in a period instead of t\vo; and in all cases in the cul e root, when there is a remainder, if it be required to obtain decimal figures to the root, proceed as directed in the qna~e root, only add three cyphers, in place of two to ' t he remainder. 150 THE CUBE ROOT. PROOF. Involve the root to the third power, adding the remain der, (if any,) to the result. EXAMPLES. 1. What is the cuoe root of 182284263? . . . Root. Uefec.divJ , , 753G 182284263(567 &sq.of6.j 5X5X t3 - 7jl 125 6X5X30= 900 Complete divisor 6436 )57284 rcsolv. 50616 56X56X3 =940849 7x56x30= 11760 Complete divisor 952609" )6668263 6668263 2. What is the cube root of 48228.544? Ans. 36.4. 3. What is the cube root (or 3d root) of 2? Ans. 1.259921. 4. What is the cube root of 132651? Ans. 5J. 5. What is the cube root of 4173281 ? Ans. 161. 6. What is the cube root of .008649? Ans .2052-}-. 7. What is the cube root of iff? Ans. 8 What will be the cube root of 160, the decimal be- ing continued to three places? Ans. 5.428-}-. 9. If the contents of a globe amount to 5832 cubick inches, what are the dimensions of the side of a cubick block containing the same quantity? Ans. 18 in. square. 10. If the diameter of the planet Jupiter is 12 times as I much as the diameter of the earth, how many globes of the earth would it take to make one as large as Jupiter? Ans. 1728. 11. If the suu is 1000000 times as large as the earth, I and the earth is 8000 miles in diameter, what is the di- ameter of the sun? Ans. 8DOOOO miles. Note. The roots of most powers may be found by the square and cube roots only; thus the square root of the square root is the biquadrate, or 4th root, and the sixth root is the cube of this square root. ALLIGATION. 151 Questions concerning the powers and roots. 1. What is called a power? 2. What power is the square? Ans. The 2d power. 3. What is the cube of a number called? 4. How do you raise the power of a vulgar fraction? 5. What is the root of a power? 6. What is meant by extracting the square root? 7. Repeat the rule for doing it. 8. How do you proceed when the sum consists in part, or altogether, of decimals? 9. How do you extract the root of a vulgar fraction ? 10. How do you proceed when the fraction is a surd? 11. What dc you understand by the cube root? 12. Repeat the rule for extracting it. 13. How are sums in the square root proved? 14. How are sums in the cube root proved? ALLIGATION. Alligation is a rule for mixing simples of different qualities, in such a manner that the composition may be of a meaner middle quality. CASE I. To fold the mean price of any part of the mixture, wJien the quantities and prices of several things are given. RULE. As the sum of the quantities is to any part of the com- position, so is the price of the quantities to the price of any particular part. EXAMPLES. 1. A trader mixes 60 gallons of wine at 100 cents per gallon ; 40 gallons, at 80 cents,- and 30 gallons of water. What should be the price per gallon? gals. cte. $ Wine 60 at 100=60,00 Wine 40 at 80=32.00 Water 30 gals. gal. $ 130 : 1 : : 92.00. Ans. 70 . 152 ALLIGATION. 2. A trader mixes a quantity of tea as follows, viz: 4 Ibs. of tea at 42 cents per lb.; 6 ibs. at 33 cents; 12 Ibs. at 75 cents, and 15 ibs. at 80 ceMts. What can he sell it for per lb.? Ans. 6b'JA cents. 3. A farmer mixes 20 bushels of wheat at 5s. per mshel, with 33 bushels of rye at 3s , and 40 bushels of >arley at 2s, per r/ushel ; how much is a bushel of the nixture worth? Ans. 3s. CASE II. WJicn the prices of several simples are given to find what quantity ofeach^ at their respective pr ices <must be taken to maJce a compound at a proposed price. HULE. Set the prices of the simples in a column under each >ther. Connect with a continued line, the rate of each iinple which is less than that of the compound, with one r any number of those that are greater than the com pound, and each greater rate, with one or more of the ess. Place the difference between the mixture rate, md that of each of the simples, opposite to the rates, .viih which they are linked. Then, if only one differ ince stand against any rate, it will be the quantity be- ,onging to that rate; but if there be several, their sum will be the quantity. Different modes of linking, will produce different answers. EXAMPLES. 1. A merchant would mix wines at 17s. 18s. and 22s. per gallon, so that the mixture may be worth 20s. per gallon: what quantity of each must be taken? ^ 2 at 17s. ras rate 20s. 3+2==5at22s Ans. 2 gallons at 17s., 2 at 18s., and 5 at 22s. 2. H^w much barley at 40 cents, corn at 60, and |wheat at 80 cents per bushel, must he mixed together, that the compound may be worth 62 J- cents per bushel? Ins. 17i bush, of barley, 17of corn, and 25 of wheat. ALLIGATION. 133 CASE III. When the price of all the simples, the quantity of one of them, and the mean price of the mixture, are given, to Jind the quantities of the other simples. RULE. Find an answer as before, by connecting; then, as the lifferenceof the same denomination with the giveriqunu- ity, is to the differences respectively, so is the given luantity, to the different quantities required. EXAMPLES. 1. How much gold of 15, 17, 18, and 22 carats fine must be mixed together to form a composition of 40 oz. >f 20 carats fine? fl5 ^ ... 2 Mean or Mixture J 17X ... 3 rate. 20 1 18M - - 2 t22,yj 5+3+2=10 Then as 16 : 2 : : 40 : 5J A 10 and as 16 : 1 : 40 : 25 { Auswer ' Ans. 5 oz. of 15, 17 and 18 carats fine, and 25 oz. of 22 carats fine. 2. A grocer has currents at 4d., 6d., 9d., and lid., per Ih. and he would make a mixture of 240 Ibs. that migh: be sold at 8d. per ib.j how much of each kind must he take? Ans. 72 Ibs. at 4d., 24 at 6d., 48 at 9d. and 96 at 1 Id. CASE IV. When the prices of the simples, the quantity to be mixed, and the mean price are given, to Jind the quantity of each simple. RULE. Connect the several prices, and place their differences as before; then, as the s:im of the differences thus given, is to the difference of e.ich rate, so is the quantity to be compounded, to the quantity required. 154 POSITION. EXAMPLES. How much sugar at 9 cents, 11 cents, and 14 cents per Ib. will be necessary to form a mixture of 20 Ibs. worth 12 cents per ib.? (9 1 2 12 IHV 2 14 J\ 6 Then, as 8 : 2 : : 20 : 5 Ibs. 9 cents.) 8 : 2 : : 20 : 5 Ibs. 11 cents.l Answer. 8 : 4 : : 20 : 10 ll.s. 14 cents.) 2. A grocer has sugar at 24 cents per Ib. and at 13 cents per Ib.; and he wishes so to mix 2cwt. of it, that he may sell it at 16 cents per Ib.; how much of each Ikind must he take? Ans. 162 f Its. of that at 13 cents, land 61-j*,- Ibs. of that at 24 cents. 3. How many gallons of water must be mixed with wine worth 60 cents per gallon, so as to fill a vessel of 80 gallons, that may be sold at41J cents per gallon? Ans. &> gallons of water, and 55 of wine POSITION. Position is a rule for solving questions, by one or mor supposed numbers. It is divided into two parts, namel} single and double. SINGLE POSITION. Single position teaches to solve questions which re quire but one supposition. RULE. Suppose a number, and proceed with it as if it were the real one, setting down the result Then, as the re suit of that operation, is to the number given, so is the supposed number, to the number sought. EXAMPLES. 1. What number is that, which being multiplied by 7 nd the product divided by 6, will give 14 for the quo tient? Suppose 18 6)726 Then, as 21 : 14 : : 18 : 12 Answer. POSITION 155 2. What number is that, of which one half exceeds nc third by 15? " Suppose 60 Then i | 60 | J J80 30 *0 Subtract 20 To Then, as 10 : 15 : : 60 : 90 Answer. 3. What number is that, which being increased by 1, and i of itself, the sum will be 125? Ans. 60. 4. A schoolmaster being asked how many scholars he ,ad, answered, that if | of his number were multiplied >y 7, and J of the same number added to the product, he sum would be 292. What was his number? Ans. 60. 5. A schoolmaster being asked what number of schol- ars he had, said, if I had as many, half as many, and >ne fourth as many more, I should have 99. What was number? Ans. 36. 6. A person, after spending 1 and J of his money, hao $30 left; what had he at first? Ans. $180. 7. Seven eighths of a certain number exceed four fifths by 6. What is that number? Ans. 80. 8. A certain sum of money is to be divided among < >ersons, in such a manner that the first shall have 1 ol t,the second J, the third , and the fourth the remain der, which is $28; what is the sum? Ans. $112. 9. What sum, at 6 per cent, per annum, will amoun 3 860, in 12 years? Ans. 500. 10. A person having about him a certain num 1 er oi crown*, said, if a third, a fourth and a sixth of therr were added together, the sum would be 45; how man\ crowns had he? Ans. 60. 11. What is the age of a person who says, that if -^ ~. the years he has lived be multiplied by 7, and 2 of them be a<dded to the product, the sum would be29 ? Ans. 60 years. 12. What number is that, which being multiplied by and product divided by 6, the quotient will be 14? Ans. 12. I5t> POSITION. DOUBLE POSITION. Double Position teaches to resolve questions by means of two supposed numbers. RULE. Suppose two convenient numbers, and proceed with each according to the condition of the question, and set down the errours of the results. Multiply the errours 'into their supposed numbers, crosswise; that is, multi- ply the first supposed number by the last errour, and the last supposed number by the first errour. If the errours be alike, that is, both too much, or both too little, divide the'difference of their products by the Difference gf the errours the quotient will be the an- swer. B it if the errours be unlike, that is, one too large d the other too small, divide the sum of the products [bV the sum of the errours. EXAMPLES. 1. What number is that, whose \ part exceeds the part by 16? Suppose 24; and us of 24 is 8, and -| of it is 6, it is (evident tint the third part exceeds the fourth part by 2 linstead of 16; and therefore the errour is 14 too small. Again, suppose 48; and 1 of 48 being 16, and being 12, it is manifest that the third part exceeds the fourth by 4, instead of 16; hence the errour is 12 too small. Then, the errours being alike, proceed thus er. 1. supposition 24 \ /\ too small. /\ er. 2. supposition 48 / \ 12 too small. 14 672 product. 288 product. 12 288 2 dif. of er.2)384 difference of the products. 192 Answer. 2. A son asking his father how old he was, received this answer: Your age is now j- of mine; but 5 years ago, your age was]- of mine. What arc their ages? Ans. 20 and 80. ARITHMETICAL PROGRESSION 157 3. Two persons, A and B, have each the same income. A stives ] of his; but B, by spending 50 dollars per an- nnn more than A, finds himself at the end of 4 years 3ne hundred dollars in debt. What was their income, md what did each spend? Ans. Their income was $125 per annum for each* A spends $100 and B spends $150 per annum. 4r. What number, added to the sixty-second part of 7628, will make the sum of 200? Ans. 77, 5. A man being asked how many sheep he had in his irove, said, if I had as many more, half as many more, us fourth as many more, and 12 J, I should have 40. How many had he? Ans. 10. 6. An officer had a division, J- of which consisted of nglish soldiers, } of Irish, J- of Canadians, and 50 of idians. How many were there in the whole? Ans. 600. 7. A servant being hired for 30 days, agreed to re- ceive 2s. 6d. for every day he laboured, and to forfeit Is. or every day he played. At the end of the term his >ay amounted to 2. 14s. How many of the days did labour? Ans. 24. 8. What number is that, which heing multiplied by 6, the product increased by adding 18 to it, and the sum divided by 9, the quotient will be 20? Ans. 27, ARITHMETICAL PROGRESSION. Arithmetical Progression is a series of numbers in- creasing or decreasing by a common difference; as, 1, 2, 3, 4, 5; 1, 3, 5, 7, 9; 5, 4, 3, 2, 1 ; 9, 7, 5, 3, 1, &c. The numbers in a series are called terms the first and last terms are called extremes, and the common differ- ence is the number by which the terms in a series differ from each other, as in 2, 5, 8. 11, &,c. the common dif- ference is 3. In any series in Arithmetical Progression, the sum of the two extremes is equal to the sum of any two terms, equally distant from them, or equal to double the mid- dle term when there is an uneven number o terms in ARITHMETICAL PROGRESSION. the series. Thus, in the series 2, 4, 6, 8 10, 12, the extremes are 2 and 12, equal to 14, .md if \ou add 10 and 4, or 8 and 6, the result will Lethe same; and in the series 2, 4, 6, 8, 10, the extremes are 10 and 2, and the number of terms is uneven is the middleone, which, when doubled makes 12, and the extremes when added together make the same amount. CASE I. The first term, common difference, and number of terms, being given, to find the last term and sum of all the terms. RULE. Multiply the common difference by one less than the number of terms, and to the product add the first term, the sum will be the last. Add the first and last term* together, multiply their sum by the number of terms, and half the product will be the sum of all terms. EXAMPLES. 1. The first term in a certain series is 3, the common difference 2, and the number of terms 9; to find the last term, and the sum of all the terms. One less than the number of terrne is 8. 2 common difference. 8 number of terms less one. 16 product. 3-j- first term. 10 last term. 3-f- first term. 22 9X number of terms. 2)198 Answer 90 sum of all the terms. 2. A person so!d 80 yards of cloth at 3 cents for the first yard, 6 for'the second, and "thus increasing 3 cents every yard ; what was the whole amount ? Ans. $97.20 ARITHMETICAL PROGRESSION. 1 3. How many times docs a clock usually strike in hours? Ans. <8. 4. A man on Ji journey travelled 20 miles the tirst !,j\,^4 the second, and continued tu increase the nun- . er <>f iniitis by every day i*>r 10 days 11 w tar diu he tr.ivc ? A:is. 3&0 miies. .">. A firmer I'uiight 20 cows, and gave 2 dollars f-r flu lirsf,4 fr ihe?e^ud, and soon, giving in tlu same jr tpoiti'in irm the tirst to the last. Wh.it did he give fr the whole? Ans. CASE II. When the tiro extremes and Ike number of terms are given to fold t'te co.nmon difference. RFLR. S i! tract the less extreme from the greater, and divide ha remainder hy one less dun the mimi.er of terms - he <j io;i;:it will I e the common difference, 1. Th> extremes i ei -\ 3 ,.n<l ll v , a d the number of ';', \vh it is the comni >n difference? 9 1<> II 1 3 13 nutn.of -e ms. ' _^ 20 fl* 1 ^ 9 ' 12 _J ' Ans. 2 12)10 (in*, of extremes Common difference 5 Answer. 3. If the extremes le 10 and ")0, and the numlerof 'erms 21, what is the common difference, and the sum f the series? Ans. c< in. diff. 3, and the sum, 40. 4. A certain del t can 1 e p-sid in one yenr, or 52 veeks, by weekly payments in Arithmetical Progression, he first payment leing 1 dollar, and the las f K)3 dol- lars. What is the common difference of the '(-.us? Ans. $2. A de'^t is to he discharge! at 1G several payments i't A .'-i^hmetio'.il Progression; thetl^t payme -t tole20 1 .rs, and the last 110 dollars. What is the common liffererice? Ans. $6. 60 GEOMETRICAL PROGRESSION GEOMETRICAL PROGRESSION. Geometrical Progression is the increase of any series >f numbers by a common multiplier, or the decrease of anv series by a common divisor; as 3, 0, 12, 24, 48; ind 48, 24, 12, 6, 3. T[\Q multiplier or divisor by which any series is increased or decreased, is called the ratio. CASE I. To find the last term and sum of the scries. RULE Raise the ratio to a power whose index is one less ban the number of terms given in the sum. Multiply the product by the first term, and the product of that nulliplication will be the last term: then multiply the last term by the ratio, subtract the first term from the jroduct, and divide the remainder by a nmnl er that ii >ne less than the ratio the quotient will be the sum ot the series. EXAMPLES, 1. Bought 12 yards of calico, r,t 2 cents for the firs yard, 4 cents for the second, 8 for -the third, &c.: wha was the whole cost? NOTE. The number of terms 12, and the ratio 2. 1st. 1024 10th. power, term 2 1st. power. 4 2d. power. 2 F 3d. power. 2 It! 4th. power. J2 32 5th. power. 32 64 06 2048 llth power, or one less 2 2st term (than the : m- 4096 2 the ratio. 8192 2 .-subtract the 1st. term. -190 1 is one less than the [ratio. 24 10th. power. $Sl.l)0 Answer. 2. B-.MigUt 10 Ibs. often, and paid 2 cents for tlio fir? pound, 6 for the second, 18 for the third, &c. Wh;;t cm the whole cost? Ans. 590.48. G I-:O?,IETI! 1C AL PR G UESSIOX. 1 1) i 3. The first term in a sum is 1, 'he who'e nurn'erof 4 erms 10, and the ratio 2; wh.it is 'he neatest term, ;ul the sum of all the tern>? Ans. The greatest term is 512, and the sum of the terms 102'$. 4. Whit debt may be disch ;rged in 12 months, by >iviijr 1 dollar the first month, 2 dollars the second m >nth, 4 the third month, and so on, each succeeding .vivnrmt being double the last; and what will be the im >'int of the last payment? AM*. The debt is ,1095, and the last payment 2048. 5. A father whose daughter was married on a. nevv- veir's day, gave her one cent, promising to triple it on the first day of each month in the year: what was the irnoint of her portion? Ans. $2657.20. 0. Oae Sossti, an Indian, having invented the g-^me of rhess, shewed it to his prince, who was so delighted with it, that he promised him any reward he should ask; ;ip >n whih Sessa requested that he might be allowed me urn in of wheat for the fir<t square on the chess board, '2 f >r the second, 4 f >r the third, and soon, doubling 1 con- inuallv, to 64, the whole numl er of square?. Now, supposing a pint to contain 7080 of these grains, arul me quarter or 8 bushels to be worth 27s. 6d., it is requi- red to compute the value of all the wheat? Ans. 54481488206. 7. VVh-U sum would purchase a horse wi'h 4 shoes, -UK 1 eight nails in each shoe, atone farlhing f>r the first nail, i halfpenny for the second, a pennv for the third, &c., loubling to'the last? Ans. !47)24. 5s. 3^d. 8. A merchant sold 15 yards of satin, the first yard f>r Is. the second for 2s. the third for 4s. the fourth for Ss. &>o.; what was the price of the 15 yards? Ans. 1038. 7s. 9. Bought 30 bushels of wheat, at 2d. for the first Hishel, 4d. for the second, 8d. for the third, &c.; what Iocs the whole amotint to, and what is the price per bushel on an average? A ( Q 947848. 10s. 6d. Amount. ' } 298231. 12s. 4d. per bushel. 2 PERMUTATION. PERMUTATION. Permutation is used to show how many ways things may be varied in place or succession. RULE. Multiply all the terms of the series continually, from 1 to the given number inclusive; and the last product will be the answer required. EXAMPLES. 1. How many changes can be made with 8 letters of the alphabet? 1X2X3X4X5X6X7X8=40320 Answer. 2. In how many different positions can 12 persons place themselves round a tatle? 1X2X3X4X5X0X7X8X0X10X11X12= 479001600 Ans. 3. How many changes may be made with the alpha- bet? Ans. 620448401733239489360000. SKETCH OF MENSURATION, OR PLANES AND SOLIDS.* Planes, surfaces, or superficies, are measured by the inch, foot, yard, &,c., according to the measures used by different artists. A superficial foot is a plane or surface of one foot in length or breadth, without reference to thickness. Solids are measured by the solid inch, foot, yard, &,c.; thus, 1728 solid inches, that is 12x12x12 make one cubick or solid f>ot. Solids includeall bodies which have length, breadth and thickness. ARTICLE I. To measure a square having equal sides. RULE. Multiply any one side of the square by itself, and the product will be the area, or superficial contents, in feet, yards, or any other men sure, according to the measure used in measuring the sides. *Planes are the same as superficies, or surfaces. SKETCH OF MENSURATION EXAMPLES. Let A, B, C and D represent a square, having equal sides each measuring 20 feet . Multiply the length of one side by itself, thus 20 20 feet A 20 feet 20 C i Ans. 4CO square feet. B 20 D 20 2. How many square rods are there in a field 90 rods square? Ans. 8100 square rods. ARTICLE II. To measure iJie plane or surface of a parallelogram. RULE. Multiply the length by the breadth the product will be the superficial contents. EXAMPLE. Let A, B, C and D represent a parallelogram whose length is 40 yards, and breadth 15 yards. 40 Breadth 15 yards A B Length 40 yards 15 Il5 An?. 600 square yards. 40 2. How many square feet are there in the floor of a room 36 feet long and 16 feet wide? Ans. 576 square ft. 3. I engage to give a plasterer 15 cents per square! ard, for plastering the walls and ceiling of a room 30| eet long, 15 feet wide, *pd 9 feet high. How much vill his work come to? Ans. $21.00. Note. The contents of boards and other articles vhich are measured by feet, &c., may be easily found iy Duodecimal Fractions. ARTICLE III. To measure the plane or surface of a triangle. RULE. Multiply the base by half the perpendicular, if it be a ight angled triangle, and the product will be the area, r superficial contents; or multiply the base and perpen-i icular together, and half the product will be the area. 1 ,- SKETCH OF MENSURATION. But if it be an oblique angled triangle, multiply half the length of the base by a perpendicular let fall on the base from the angle opposite to it, and the product will be the area. EXAMPLES. 1. Let C, II and G represent a right angled triangle, having the right angle at G ; the base C G being 40 feet, and the perpendicular II G, 28 feet. No. 1. 14 feet, or half the perpendicular. . H 40 feet, or the base, ^'^ 5oO feet the area. vS ! rise. 2. Let B, C and D represent an oblique angled tri- ingle; the length of the base B D being 60 feet, and the perpendicular C E, 28 feet. No. 2. 28 the perpendicular C 40 half the base. 1120 Answer SO R 3. How many square rods are there in a triangular field, one of the corners of which is a right angle, and one of the shorter sides of which is 3d rods, and the oth*r 24 rods? Ans 432 square rods. NOTE. Right angled triangles arc tuch as hare one angle like the corner of a xquarc, and which is called the rig/it angle, containing 90 degrees; as the angle G in the triangle, No. 1 . Oblique angled triangles arc sw-h as have each of the angles, either more or less than IK) degrees* as in the triangle, No. 2. ARTICLE IV. To me as re a circle. Note. Circles are round figures, 1 ounderl every where by a circular line, called the* periphery, and aUo the circumference. A line passing through the cei.fpa SKETCH OF MENSURATION. 105 is called the diameter. Half the length of the diameter is called the radius. The diameter may be found by the circumference, thus As '& is to 7 so is the circumference to the diame- ter; and in like manner may the circumference be found by the diameter; for, as 7 is to 22,, so is the diameter to ihe circumference. 1. What is the diameter of a wheel, or circle, whose circumference is 16 feel? Ans. 5 feet, nearly. 2. What is the circumference of a circle, whose diam- eter is 2i) feet? Ans. 63 feet nearh . i>. if the distance through the earth be b>000 miles, how many miles around it? Ans. 25143 miles nearly. ARTICLE V. To find tlie superficial contents, or area, of a c ircle. JiULE. Multiply half the circumference by half the diameter, rid the product will I e the answer. Or, multiply the qii'ire of the .'iameter by .7&54; or multiply the square of the circumference by .07958, and in either case the product will be the answer. EXAMPLE. How many square feet are contained in a circle whose circumference is 44 feet, and whose diameter is 14 feet? 22 half the circumference. 7 half the diameter. 154 square feet. Answer. The same may be done by multiplying the diameter ind circumference together, and dividing the product y 4, thus, 44x14=616-7-4=: 154. Answer. 2. H >w many square 'feet are there in the area of a ircle whose circumference is 16 feet? Ans. 20 square feet. 3. Hv>w many sq-iare feet, are there in the area of a ircle whose diameter is 20 feat? Ans. 315 sqiriro ft. ARTICLE VI. To measure the surface of a globe or sphere. RULK. Multiply the circumference by the diameter, the juo- duct will be the surface, or arc. 16t> SKETCH OF MENSURATION. EXAMPLES. 1. What are the superficial contents of a globe whose circumference is 220 feet, and v. h;.-se diameter is "70 feel? 220 X H; = 1,3400 square feet. Answer. 2. How n> ny Mjuare nii.es ; re conuiLied on the sur- face of the vshole earth, or globe, vvhi< h we iiihal i-? The circumference of the earth is estimated to le 25020 miles, and the diameter, 7V.64, nearly. Then, 25020x79u^ = 191259280 square miles. Ans. ARTICLE VII. To find the &olid coidtnts of a cube* RULK. Multiply the length of one side 1 y itself, and muUi ply the product 1 y ihe sjane length, th.it i, by the same multiplier- the last product will Le ihe solid contents of the cube. EXAMPLES. 1. How many solid feet are contained in a cul e, cr solid block oft> equal sidos, each side Icing 3 feet in length, and 3 in breadth? 3X3X ; >=27 solid or cu 1 icfcfeet. Ans. When the contents are required of right angled solids J [whose length, breadth, etc., jsre not equal ; multiply th<i liength ly the 1 readth, and that product by ihe thickness, I; he pn dir-t will be the nswer. 2. Req-i'r d the con ents of a 1 ;ad of wood, whoi-e length is 8 feet, breadth 4 foet, a 1 d height or thickness [ feet. 8x1 X 1= 128 solid f ;e% or 1 cor'!. Ans. 3. Require 1 the contents of a j-torie L*>J Let in leng:b, 'A in bre ic'th, and 1 fxt in Lhickne.-s. 16.5X^.5X1=24^75 solid feet, or 1 perch. Am-. Note. S )lids whose climeriFions Jire in feet or inches, ire more easily measured by Duodecimals. AUTICLE VIII. To find the content* of a privm. A prism is an angular figure, generally of three eqi^l >i'Jes, whose ends are in the f.rm of trinr-glcs. It rt-- jsembles a file of three sijes, whose whole length is <;t equal bigness. *A cubo is a solU boily of equal sides, each of which is an cxactsqunrc. KKN31IKATION. 157 RULE. Find the area or superficial contents of one end as of diw other triangle, then multiply the area by the length jf the prism, and" the product will Le the soadity. EXAMPLE. What are the solid contents of a prism, the sides of he triangles of which measure 13 inches, the perpendic- ilar extending from one of its angles to its opposite side, [2 inches, and its length 18 inches? 13 X l^=li>t)-r~2 ' a X !&= 140-1 cubick inches. Ans. To find the coiittfittj of a cylinder. A cylinder is a lo.ig ro.uul body, ail its length being of equal bigness, like a round ruler. RULE. Find the area of one end, t>v the rule f >r finding the area of a circle, then multiply it by the length, and the product will be the answer. EXAMPLE. What is the solidity of a cylinder, the area of one end >f which contains ^.40 square feet and its length being 12.5 feet? 2.40 X 1^.-~>=30 solid feet. Ans. ARTICLE X. To find the solid content* of a round &tick of timber, which wr of a true taper from the larger to the smaller end. RULE. Find the area of both en-Is; add the two areas together md reserve the sum; multiply the area of the larger end ;>y the area of the smaller end, extract the square root f the product, add the root to the reserved sum, then multiply this sum by one third the length of the stick, md the product will be the solidity. NOTE. As this method requires considerable labour, the following has been preferred for common use, though not julle sj accurate. RULE. Girt the stick near the middle, hut a little nearer to Uie larger than to the smaller end ; this will give the cir oumference at that place. Fi id the diameter by the cir- cumference; m i.tiply the circumference and the diam- eter together; then multiply one fourth of the product by the length and the a nsiver will i.enearlv the solid contents. & PRACTICAL QUESTIONS. EXAMPLES. What is the solidity of a round stick of timber that is 10 feet long, and its circumference near the middle is 2.61 feet? As 22 : 7 : : 2.61 : .83 diameter, cir. diam. length, feet. 2.61 X.83=2.1663-r4=.5415x 10=5.4150. Ans. 5.4150 solid feet. ARTICLE XI. To find the solid contents of a globe, RULE. Multiply the cube of the diameter by .5236, the pro- duct will be the solid contents. Or, multiply the super- ficial contents, or surface, by one sixth part of (he sur- face. Or, multiply the cube of the diameter by 11, and divide the product by 21 in either case the product will be the solidity. EXAMPLES. 1. What are the solid contents of a globe whose di- ameter is 14 inches? 14X14XM=2744X.5236=1436.7584 cubick inches. Ans 2. How many solid miles are contained in the earth, or globe, which we inhabit? Suppose the diameter to be 7954 miles; then, 7954 X 7954x7^4=503218686664 the cube of the earth's axis, or diameter; then, 503218686664 X .5236=263485304337 cubick miles. Ans. Note. The solidity of a globe may be found by the circumference, thus Multiply *he cube of the circum- ferance by .016887 the product will be the contents. PRACTICAL QUESTIONS. 1. A cannon ball goes about 1500 feet in a second of time. Moving at that rate, what time would it take in going from the earth to the sun; admitting the distance to be 100 millions of miles, and the year to contain 365 days, 6 hours? . Ans. lOy^yW years. PRACTICAL QUESTIONS. 109 2. A young man spent of his fortune in 8 months, ^ of the remainder in 12 months more, after which he had 410. left. What was the amount of his tVtune? Ans. 956. 13s. 4d. 3, What number is that, from which if you take | of |, and to the remainder add T 7 ? of J^, the sum will be 10? Ans. lO^u 4. What part of 3, is a third part of 2? Ans". f . 5, If 20 men can perform a piece of work in 12 days, how many will accomplish another thrice as large, in :me fifth of the time? Ans. 300. (J. A person making his will, gave to one child ij of his estate, and the rest to another. When these lega- cies were paid, the one proved to be 600 more than the other. What was the worth of the whole estate? Ans. 2000. 7. The clocks of Italy go on to 24 hours ; how many strokes do they strike in one complete revolution of the index? Ans. 300. 8. What quantity of water must be added to a pipe of wine, valued at 33, to bring the first cost to 4s. 6d. per gallon? Ans. 20| gallons. 9. A younger brother received 6300, which was 7 f his elder brother's portion. What was the whole es- tate? Ans. 14400. 10. What number is that which being divided by 2, or 3, 4, 5, or 6, will leave 1 remainder, but which if divi- ded by 7 will leave no remainder? Ans. 721. 11. What is the least number that can be divided by the nine digits without a remainder? Ans. 2520. 12. How many bushels of wheat, at $1.12 per bushel, can I have for $81.76? Ans. 73. 13. What will 27 cwt. of iron come to, at $4.56 per cwt.? Ans. $123.12. 14. When a man^s yearly income is 949 dollars, how much is it per day? Ans. $2.60. 15. My factor sends me word he has bought goods to the v.; hie of 500. 13s. 6d. upon my account; what will his commission come to at 3 l per cent.? Ans. 17. 10s. 5^d. 170 PRACTICAL QUESTIONS. 16. How mnny yards of cloth, at 17s. 6d per yard, can I have for 13 cwt. 2 qrs. of woo!, at 14d. per 1! .? Ans. 1 00 yards, 8 qrs. 17. There is a cellar dug (hat is 12 feet every way, in length, I mdth,a d depth; how many solid feet of earth were taken out of i ? Ans. 172R. 18. If 2. of ari ounce cost J of a shilling, what will of a 11). cost? Ans. 17s. 6d. 19. If of a gallon cost of a . what will of a tun cost? Acs. '105. 20. Iff of a ship be worth 3740, whnt is the worth of the hole? Ans. .1973. 6s. 8d. 21. What is the commission on $2176.50, at per cent? Ans. $54.41J. 22. In a certain orchard < f th'^ trees 1 e tr nj pb>s, pears, J- plums, 60 of tl e '.i pearlvs, arid 40 cherries; how many trees are in the orchard? Ans. 12(0. 23. If A travel by m *il at the rate of 8 miles an h' ur, and when he is 50 miles on his way, B start from the wo place that A di; ? , and travel on horseback the s; me road at 10 miles an hour, how long and how far will B travel to come up with A? Ans. 25 hour?, and 250 miles. 24. Bright a quantity r.f cloth f r 750 dollar?, ^ of vvhi v h 1 found to be inferior which I Ind tosell at 1 dol- lar 25 cents per yard, and by this I lost 100 dollars: what must I sell the rest at per yard that I shall lose nothing by the whole? Ans. $3.15^f.. 25. If the ear h goes rrund the sun once in 365 days, 5 hours, 48 minutes, 49 seconds, and its distance from the sun 95000000 miles, what must be the distance of the planet Mercury from the Sun, admitting the time of its revolution round the Sun to be 87 days, 23 hours, 15 ninutes, 40 seconds? Note. The planets describe equal areas in equal f imes therefore, as the square of the time of the revo- lution of one planet, round the Sun, is to the squ re of he time of the revolution of any other planet, so is the u! e of the distance of one planet from the S-in, to the ^\ } o "f th .''ist -nee of nnv other from the S 'n. 171 A SHORT SYSTEM OP BOOK-KEEPING* FOR FARMERS AND MECHANICS. is the method of recording bus ness transactions. It is of two kinds single and double entry ; but we shall on.y lotice the former. Single entry is the simplest form of Book-Keeping and isem- )lojed by retailers, mechanics, farmers, occ. It requires a Day- JOJK, Leger, and where money is frequently received and pai,. out, a Cash-Book. A few examples only are here given, barely sufficient to givi the learner a view of the manner of Keeping books; it being in- tended that the pupil should be required to compose similar one*, ind insert them in a book adapted to tnis purpose. The Daj-B > jk contains entries of tne several articles in the successive order of iheir dates. Each person must be mule UY. for what he receives, and (Jr. by wnat is received of him o.. account. Every month, or oftener, the Day-Book should be copied 01 posted into the -.eger, as hereafter diiectcd. The crjsse v>n tin ieft hand column, show that the charge or credit, agam-t wmc,. they st ind, is ported, and the ligures show tlie page of the i^egei where the account is posted. Some use the tig-are* only as poo* Tne Leger is the grand book of account?, in which every pcr- f^s account is collected from different j>arts oi' the Day Book, and inserted in one place; the Dr. and v.-r. fronting each >thei ou opposite pages or on opposite sides of the same page, whici shows the w'hole state of the account at once. i^ost or transfer the entries from the Day Book to the Leger. thus: Open an account in the Leger for the first person vvt stands JJr. or CV. iu the Day Book, i. e. write his name with D m the left hand page of the folio, aud Cr.vii the ri c ht. 172 DAYBOOK. January 18th. 1831. IX IX Peter Simpson of Cin* Dr. To 15 yards of fine Broad-cloth, a 5.00 $75.00 " 24 do. superfine do. a 7.75 . . . 186.00 R. Fulton^ of Newport, Dn To 1 gall. Molasses $ 50 " 6 ibs. Coffee, a 37* cts. . . . 2.25 " 20 Jbs. Sugar, a 10 cts 2.00 I j - - 19 X IX IX John CatJiell, Carpenter, Dr. To 16 vards Calico, a I2i cts. . . $2.00 " 10"" Muslin, a 15 cts. . , . 2.50 * 1 Vest pattern, ........ 75 *' 1 pair Gloves, ......... 62 14 25 Ibs. Nails, a 8 cts ....... 2.00 Charles H. Glover, Dr. To 25 Reams post paper, a $3.00, . . $75.00 18 " foolsca > do. a 3.25, . . 56.50 George Whipple,Jr. Dr. To 300 Ibs. Pork, a 5 cts $15.00 50 bu. Corn, a 20 cts 10.00 _20 William Jones, Dr. To 35 Ibs. Iron, a 7 cts $J.45 4i Cash paid his order to John Bnker, 1.35 James L. Rowan, Dr. To 50 bis. Flour, a $3.50, .... $175.00 " 25 bu. Potato*, a 30 cts 7.50 " 4 bis. Cider, a 1.50, 6,00 * 75 Ibs. Beef, a 4 cts 3.00 Peter Simpson, Cr. ;By 30 cords Wood, a $2.25, .... $67.50 " 90 bu. Oats, a 12i cts 11.25 " 5 tons Hay, a 15.00, 75.00 1*1 DAY BOOK. I 7^ i January "2 1 at. le# 1. ix Burchel J. Barney Dr. Po 6 galls. Port u me. $J.DJ, . . . $15. M) " -JlGibs. Sugar, a v Jcts. .... 1:J.3.'I; " 1 ib Tea, 1. J5 1 $ C. 35! 60 ix Cumtims C. Williams, Dr. To 17 > bis. Whisky 5;9 J gulls, a >{) cts. $1J38.00 44 Paitl liis onlcvin j'avor of 'J^honias and miite, ibr salt, 234. ')0 ! | 1 C/. By 750 Ib?. Feathers, a 25 cts. . . . $187. )0 'i Cah, 5 IJ ] ^72 ,)0 1 6S7 50 I ix John Pkares, Cr. K sundries Tor which 1 .i^Jivc rny note at 60 <lny c , 184 50 2X Peter SitnpWH, Dr. 'o order ou ()l)crt F*ult<Hi . . 1 37 2X ttowrt l^uitoff, Cr. i [5v i"'t( r -^iinnson's order on him, 1 37 bx William JoneS) Dr. Po 14 Ib-. Veal, 4 cts g .5* i " ^00 Ibs. Flour, 2.5 Or. By his bill of blacksmith work, 3 f .)() 03 9 2X John Cathcll, Cr. By 4 days carpenter work, a $1.25, . , $5.0 " ^00 feet Poplar boards, a 31 d cts. . . 1.7" i 75 8X George Whipple, Jr. Cr. !>y Cash on account, . 30 Oi; ! 2X Claudius C. Williams, Cr. By 10 bis Mackerel, No. 1, a $12.00, $1'20.0:> u 1-20 galls. Cog. Brandy, a 1.50,. . 180.00 1 ! 00 oo ! : 2X John Phares, Dr. To cash on account of my note at 60 day*, 174 DAY BOOK. Pi) January 24th, 1831. IX 1 Robert Fulton, Dr. Fo 1 piece Broadcloth containing 25 yards, a $6 per yd. ; 90 days' credit, .... * < 150 ( )0 1 ix , Peter Simpson, Cr, 3y Cash in full, 08 32 1 2X, Win. Jones, Blacksmith, Dr. Fo 217 Ibs Iron, a S cts . 17 36 11 Cr. i 25 II O 1 ^ IX Charles H. Grover, Dr. o rent of my house 3 months, a $6 per month, Cr. 18 1 " Cash, 100.00 112 ' II IX George Whipple,Jr. Dr. 'o 1 Keg nails, weighing 215 Ibs. a 8 ts. . . Ofi 17 20 1 2X James L. Rowan, Cr. By use of his horse 20 days, A 50 eta. . $10.00 " 1500 Ibs. Rags, a 4 cts 60.00 " 160 bu. Salt, a 50 cts 80.00 150 00 2X Burchel J. Barney, Dr. To 1 Bag of coffee weighing 216 Ibs. a 16 cts. 34 56 1 3X C. Smith, of Lexington, Dr. To 15 Ibs. Wool, a 25 cts $37 " 20 Ibs. Flax, a 9 cts 18 II 55 I *X A. Dunn, of Columbus, Dr. To 800 ft. Pine boards, a $1.25, . . $10.0 " 84 ft. Scantling, a 3 ct? 2.5 IX Robert Fulton, Cr. By 1200 Ibs. Pork, a 4 cts 1 4 52 (\(\ i] IX William Jones, Cr. ,By Cash, 1 30 I 43 DAY BOOK. 175 |j January 21th. 1831. x John Cathell, o 350 lb c Nails a 8 cts . . . . Dr. $ . *28.00 a U 1 17 Door locks, a 1.^5, . . . . *dl.-^5 4 1 ) 25 1 IX Cftarles H. Graver, y Cash on account, Cr. . . i 37 2X Claudius C. Williams, To 856 Ibs. Tobacco, a 5 cts. . . . Dr. ... 4 280 I IX George Whipple, Jr. By 1700 Ibs. Cheese, a 7 cts. . . . Cr. . . . u 900 2X James L. Rowan, To 24 yds. Linen shirting, a 87 cts. Dr. >1 00 I IX John Cathell, By Cash in full, .... Cr. =iO 37 A 1 8 )U 0< 1 3X Calvin Smith, By 41 Ibs. Coffee, a 18 cts. . . . Cr. . . $7.38 " 36 Ibs. Sugar, a 12i cts. . . . . . 4.50 11 88 1 IX George Whipple,Jr. By 12 days labour of self, a 62* cts. u 7 days do. of his horse, a 25 cts. Cr. :: ^ 925 1 2X Burchel J. Barney By cash in full, Cr. 70 25 I 2X Andrew Dunn, To mending Wagon, ..... Dr. 5.75 " Timber and materials for do. . . . 1.25 700 I 2X James L. Rowan, By Cash in full, Cr. 62 50 || 1 2X John Phares, To 730 Ibs. Sea Island salt, a 15 cts. of my note Dr. in full 1HQ en II 3X Calvin Smith, To 75 yds. Domestic cloth, a 50 cts. Dr. 3 50 |! 170 FOUM OF A LEGEI?. [1 Dr. Peter Simpson, Cr. 1 1831.1 ;Jan.l8l " 222 11 To Sundri.-.-, Order on R. F. . $ \ < I'l ' 1831.- - Jan. 20 ? u 24 i>y sundries, 1 3 " Cash, H !'>( r c. 375 862 np Dr. Robert Fulton 9 - Cr Ian. 181 243 To Sundries, i 475 ,1 " Broadcloth J150 00 [ 154 75 " Balance, . UO^, 1831. an. 22 2 F^y Order P. S. tk 2tiS *'* Pork, . . ' " Balance, l( 18 )0 )5 8 Dr. John CatJiel'l, Cr. 1831. I Jan. 1911 " 274 To Sundries, [ " do. 57 l2 Jan. 22 . " 27 4 By Sundries, " Cash, 5 5 i- ^ c. G75 J37i 7 12* Dr. Charles H. Grocer, Cr. Jan. 19 " 25 To Paper, 3 Rent, . " Balance, 133 5*1 15 : -k )Q ^ 1 . 1831. ) Jan. 25 , " 27 > 3 Bv Sundries, 1 1 < ; Cash, " Balance, i i 10;37 28|51 5 1 "> .) Dr. George Wkippic,Jr. Cr. 1831. Jan. 19 25 1 1 To Sun hies 3, Xai *, " Balance i i*7 I 10605 !4S 25 } 1831. Jan. 23 l - 27 -J " 28^1 i \ By Cash, " Cheese, I " Sundries, 1 " Balance, ; 1 20 00 1900 9:25 4a 25 06 05 |2] FORM OF A LEGFR. l77 Dr. William Jones, 1831. $ c. 1831. c f.-\ Jan. 20 " 22! 1 TQ Sundries, 2 do. 370 306 Jan. 22 2 " 243 By Bill work, " do. 1 57 25 * 24 3 " Iron, 1736 263 44 Cash, 17 30 2412 in 24 12 Dr. James L. Rowan, Cr. l&U. $ < . 183 i. $ c. Jan. 20 i To Sundries, 1915 Jan. 26 3 By Sundries, JO 37 i " Linen, 210 " 28 4 " Cash, 62 50 2125 212 50 Dr. Bureliel J. Barney, Cr. l63l. $ c . 1831. .f e. Jan. 21 2 To Sundries, 356 3 Jan. 28 4 By Cash, 70 25 " 26 3 " Coffee, 345 6 70 2 5 ... Dr. Claudius C. Williams, Cr. 1831. $ c 1831. $ c. Jan. 21 27 2 To Sundries, i 4 Tobacco, 272 OC 42 8f Jan. 21 " 23 2 By Sundries, 300 5\ 00 u Balance, 327 30 I 31481 1 314 80 " Balance, 327 3( Dr. John Fhares, Cr. 1831. $ c- II 1831. $ c. Jan. 2? 2 To Cash, 7500 Jan. 2^ By Sundries, 184 50 " 28 1 Cotton, 10950 ] 18450 . Dr. Andrew Dunn, (Jr. 1831. Jan. 26J; * To Sundries, a Balance, I f 1831. Jan. 28J4 By Sundries, " Balance, $ 7 e. f)0 i 12 ii 178 CASii-LOOIi. [3 Dr. Ctdmn Smith, Cr. 1831. | $ c. ' Jan. :2G'3To Sundries, 5 55 44 28 J4 " Cloth, 37 50 43 05 1 " Balanco, 31 15 1631. $ c Jan. & 4 By Sundries, 1 1 $8 " Balance, 31 15 43 05 INDEX TO LEGER. R Pace: Barney, Burchel J. . . 2 P F*gt. Vhare?, John .... 2 C Cathell, John .... 1 Rowan, James L. . . 2 Dunn, Andrew ... 2 s Simpson, Peter ... 1 Smith, C akin . . 3 F Fulton, Robert ... 1 1^ Grover, Charles H. . . 1 w Whipple, Georee . . 1 Williams, Claudius C. . 2 j 91 BOOK. its and receiptt of cash. r. to cash on hand and what is paid out. ^ek, as may best snit the nature 1 is counted, and entered on the ake the sum of the Dr. equal to en struck, and the cash on hand i J .lories, \Villiuin . . . 2 pi CASH- This book record* the paymei It is kept by makinsr cash D received, and Cr. by whatever is At the end of every day or w< of the business, the cash on han Cr. side. If there i= no error, this will m that of the Cr. A balance is th carried again upon the Dr. side. Dr. OF A CASH-liOUK. CASH. 831: Jan. 1 To cash on hand 2 J.Bali 44 44 E. Jennison S 44 D. Roe paid) ** 44 H. Austin on) his note j 4 " 8. Ball on ace 5 44 A. Higby 6 44 Sales of Aer-) \ chandise ) I 3 t! * is 3H idol., Jan. 1 By ain't. | aid repairs epj 41 Paid note to, A. Y\ estou ' 44 Paid work nu 44 Family ex- pense's 44 Merchandise) bought at auction j (.'ash on hand 8 Cash on hand ^MiMMB JMB^MB .Form of a Bill of parcel* jrom the preceding Work. Mr. John Catliell To Solomon Thrifty, Dr. 1831. I ~ Jan. 16. To16y(KCalicf>, al2J cts. . . . $3.00 u a "10yds. Mnslin, a 15 cts 1.50 | a i v est pattern, 75 I 44 " 44 1 pair (ilove?, , 6^4' 44 44 25 Ibs. Nails a 8 cts 2.00 44 27 44 350 Ibs. Nails a 8 cts 28.00 44 " " 17 Door locks a $1.25 . .... 21.25 49 Cr. 56 124 44 5 By 4 days Carpenter worker $1.25 . . .$5.00 44 " " 200 ft. Poplar boards a 874 cts. . . 1.75 44 27 " Cash in full, 43.374 Errors excepttd. !56 124 CINCINNATI, Tan. 27th, 1831. SOLOMON THRIFTY. 2rf Form. Claudius C. Williams, To Solomon Thrifty, Dr. 1831. $ c . Jan. 21. To 172 bis. Whiskey, 5190 galls, a 20 cts $1038.00 44 44 44 Paid his order favour of Thomas White, for Salt 234.00 137? 00 27 856 Ibs. Tobacco, a 5 cts 4780 CONTENTS PAGE, Numeration - 7 Simple Addition * .9 Simple Subsiraction - 13 Simple Multiplication 17 Simple Divison 22 Federal Money 33 Table of Money, Weights, Measures &c. 44 Reduction - 49 Compound Additon - 56 Compound Substraction - 61 Compound Multiplication - 65 Compound Division ... 73 Exchange - 77 Vulgar Fractions 80 Decimal Fractions - 97 Duodecimals * 105 Single Rule of Three - - 108 Double Rule of Three - 115 Practice - - - 119 Fellowship -> ... 124 Tare and Tret - - 126 Simple Interest ... 130 Compound Interest - 137 Insurance, Commission and Brokerage 141 Discount - - - 142 Equation . . 143 Loss and Gain 144 Involution --..:. 145 Evolution - 146 Square Root - 146 Cube Root - - 149 Alligation - 151 Single Position - . 154 Double Position . - .156 Arethmetical Progression * - 157 Geometrical Progreesion - 160 Permutation - - - 162 Mensuration . . 162 Practical Questions - 168 Book-Keeping - - - 171 (*<* YA 02427 THE UNIVERSITY OF CALIFORNIA LIBRARY