University of California • Berkeley William Hammond Hall Papers Purchased from The Peter and Resell Harvey Memorial Fund c4 (^ 2f^A,*'Hvnvv^ EM JS Y S T E M OF '" FOR THE USE OF SCHOOLS, BY THE REV, J. JOYCE. ASAPTU) TO THE COMMERCE OF THE UNITED STATES, BY J. WALKER. BALTIMORE : PUBLISHED BY N. G. MAXWELL, 140, M ARKET^TREE 1 J. Robinson, Printer. 1819. DtSTRlCT OP MAnTLAITD, SS. BE IT liEMKMBERED, That on the Twenty-First day of June ,*:Mai**» »" the Forty-third year of the Independence of the United IskalI ?t^tes ot America, X (i. Maxwell ofthesaid District hath %„^^^ deposited m thisofllce the title of a Book, the right where- « A c * '^^ he Claims as Proprietor in the words followinP to wit: « l^et 7 i^"" ""^ iVactical Arithrnetick. for the Use of Schools, by the " J \Vaike?''^ Adapted to the Commerce of the United Stites, by In conformity to an act of the Congress of the United States, enU- tied. An act for the encouragement of learning, by securing the cop,esof nxaps, charts, and books, to the authors and ' proprSrs of such cop.es during the times therein mentioned." And ilso to the act em.tled, "An act supplementary to an act, entitled, «Anacrfor?he encouragementof learning, by securing the copies of maps, ch^t and books, to the authors and proprietors of such copies during the times therein mentioned," and extending the benefits 'thereof t? the arts of designing, engraying, and etching historical ami other printe " PHILIP MOURE, •Clerk of the District of Maiyljind. PREFACE TO THE ENGLISH EDITION. IN" presenting a new System of Arithmetick to the publick, some account of its plan and execution will be expected. It is hoped, that the title of the present Work will briefly explain the views of the Author, who, from his own experience, in the business of education, has long since been convinced, that, among the excellent introductory books to this useful science, no one, that he has met witii, is sufficiently adapted to the occasions of Gominon life : some are too abstruse for novices, while others are defective in such examples, as point out the application of the several rules to transactions of real business. If the Author of this System of Arithmetick has not deceived himself, he has completely supplied these de- ficiencies, and he appeals without apprehension to that publick, whose candour and liberality he lias already and often experienced, to decide upon this attempt to rendei the elementary rules of arithmetick practical and po- pular. There are few children who do not experience some disgust in passing through the first four rules ; occasion- ed, without doubt, by the paucity of examples, and by the want of interest in those that are given. The Au- thor has therefore filled a large portion of his work with the early rules, and has illustrated them by miscellane- ous questions, in which will be found much useful infor- mation, applicable in the advancing stages of life. The modes of treating the Rule of Three, of illustra- ting Vulgar and Decimal Fractions, Practice, &c. &c., will best speal^ for themselves. But a reason may be de- manded for the introduction of Logarithms, and for the particular method adopted in those parts in whici: the doctrine of Annuities, Reversions, Leases, &c. is illus- trated. ARITHMETICK, ARITHMETIC K. is the science which explains the va.- riou^ methods of computing by numbers. All its operations are performed by Addition, Sub- traction, Multiplication and Division. OF NUMERATION OR NOTATION. When two or more figures are placed together, the first or right hand figure is taken for its simple value : the second to the left signifies so many tens : the third so many hundreds ,- and the fourth so many thousands / and soon, according to the following Tabie : ^^ ^ ;§ jg^ H:g ^ 53 ^ t) 5 431 26978 Thus figures, besides their common value, have one which depends upon the place in which they stand when joined to others; 6 and 5 are read six and five; but if they stand together, 65, they are read sixty-five. The figure 5 on the right-hand denotes its simple value only, hut the 6, from its situation, becomes ten times greater than its simple value, or sixty, therefore the two together are called sixty -five. 10 NUMERATION, If there be three figures, as 978, the first figure to the rijfht-hand denotes its simple value, as eij^ht ; the second a value ten times j^reater than its sinple va- lue, as seventy ; and the third is a hundred times 5;reater than its simple value, as nine hundred : the figures to- gether are read nine hundred and seventy-eight. In this manner, the value of each figure to the left is always ten times oreater than it would be if it stood in the next place on the right ; thus 6666, the first figure 6 is simply six, the next is sixty, and the third six hun- dred, and the fourth six thousand ; the whole number is read, Six thousand six hundred and sixty -six. The first six figures in the table, are read, One hundred twenty-six thousand, nine hundred and seven- ty-eight. The whole period of nine figures is thus read. Five hundred and forty-three millions, one hundred and twenty -six thousand, nine hundred and seventy -eight. The enumeration of figures may be carried much fur- ther, according to the following Table : IE o 5 fi t^ .£ . ^f .2 « . -S ■^^S-5<«o -^-^s-o^.S -acrs-Oci;* « .~4)_n3a.:^ 3a)-C3a;i-^ 3a).c3a;S KHHXHCQ Er-r-XH^ KHr^EHlD 123456 4-87951 4627 5 3 In large numbers it is common to divide them into periods of six figures each, and half periods of three fi- gures. The foregoing three periods are read — One hun- dred twenty-three thousand, four hundred and fifty-six billions, four hundred eighty -seven thousand, nine hun- NUMERATOIN". n dred and fifty-one millions, four hundred sixty-two thou- sand, seven hundred and fifty-three.* Hence the following Rule. To the sim/ile value of each figure, join the the name of its place according to the situation in the se- ries, as hundreds, thousands, millions, billions, trillions^ EXAMPLES IN NUMERATION AND NOTATION. Read, or write down in words, the value of the fol- owing Numbers Sx. 1. 19 Ex. 11. 40005 Ex. 21. 340 2. 244 12. 324060 22. 436i:0l 3. 3045 13. 4(>0569 23. 36945 4. 45060 U. 765 24. 9874000 5. 69305 15. 564001 25. 6.54328 6. 93614 16. 439762 26. 4328764 7. 564875 17. 9300044 27. 856540 8. 4500342 18. 70000021 28. 4376(5 4.; 652 20502 g* i« ADDITION. :. 22. 12345 Ex. ,23. 12349 Ex. £4. 99887 54321 56789 44556 6'854 48672 17280 58108 '24 69776 49328 51403 43509 98765 46795 49312 432U0 31274 56418 87219 45670 43004 Ex. ^. 764329 Ex. 26. 527648 Ex. 27. 397648 S97643 476239 473465 249764 765473 247396 354673 620728 4789i3 576894 437649 862759 35^649 276354 380475 476392 762938 928/64 734629 476849 387649 563:93 327649 258763 Ex. 28. 476293 Ex. 29. 587649 Ex. 30. 53''649 547689 356.-43 827649 536754 873649 567937 645'64 786492 326-53 4736*9 5673ifr 478943 6 '4^59 S8'.745 47 3659 7684y2 764:>32 476.S43 324768 976439 267568 374689 567834 743687 Ex. ADDITION • 31. S27638 Ex. 32. 432999 4 619.7 7fl3427 5S86T4, 632'-42 7e^!i27 763487 48"634. 629-64. 927865 394276 73t2i86 839467 47^288 364237 367495 - 648^.76 19 MISCELLANEOUS EXAMPLES IN ADDITION. Ex. I. Add to^fther the following sums; 98764, 397652, 876. 459321, f I, 80, and 76942, Ex 2. Add 39:64, 47652, 34291, 225, 48, 764871, and 10000 together. Ex. 3. What is the sum of thirty-five thousand and four ; five hundred and forty thousand, three hundred and nine ; four hundred and twenty-seven ; fifty thou- sand nine hundred and eighty; two millions and five; and seven hundred and seventy-seven ? Ex. 4. When will a child, horn in 1806, be forty- nine years old ? Ex. 5. How many days are there in the first eight months of the year, ^hen it is not leap year ? Ex. 6. How old is the world this year, 1808, sup- posing it was created 4004 vears before the bnth of Christ ? Ex. 7. A person at his death left 3287/. to his wi- dow ? to his eld'st son he bequeathed 5250/ and to each of five other children, he left a thousand pounds less than to the eldest son : he left also to a nephew 105/., and the same sum to be divided auiong four dis- tant relations : How much money did he leave behind him ? Ex. 8. The lease of my house was granted me in the year 1793. for ninety-nine years; when will it expire ? Ex. 9. How many days v\ill there be between January the first and November the 20th, 1808, being leap year, both days inclusive ? 20 ADDITION. Fx 10. What do the following; sums amount to, 1268 + 86l2-f 10018+275+919+8+550099 ? Ivx. 11. How manv chapters are there in the several books of the New Testament ? Kx 12. How many chapters are there in the several books of the Old Testament? Kx. 13. How manv chapters are there in the Bible, which consists of the Did and New Testaments ? Ex. 14. In travelling from London to Bath in a post- chaise, for how many miles shall 1 have to pay ? The distance from London to Hounslow is 10 miles, from Hounslow to Maidenhead is 16 miles, from Maidenhead to Reading: 15 miles, from Reading to Spleenhamland 16 miles, from Spleenhamland to Marlborough is 19 miles, from Marlborough to Chippenham is 19 miles, and from < hippenhao) to Bath is 13 miles. Ex. 15. How far is it from London to Harwick ? To Romford are 1 1 miles, from thence to Ini>atestone 12 miles, from Ingatestone to Chelmsford 6 miles, from Chelmsford to Colcliester are 21 miles, and from Col- chester to Harwick 20 mile^^. Ex. l6. In travelling post to Margate I pay a shilling a mile : How many shillings shall 1 have paid at the end of the journey ? The distance from London to Dartford is 15 miles, from thence to Rochester is 14 miles, from Rochester to Sittinghourne is 11 miles, from Sittinghourne to ('anterbury i» 15 miles, and from Can- terbury to Margate is 17 miles. ( 21 ) SUBTRACTION. By Subtraction we find the difference between two numbers. Rule ( 1 .) Place the lesser number under the greater, so that units may stand under units, tens under tens, Sfc. / begin at the right hand, and take each figure in the lower line from the figure above it, and set down the remainder. (2.) If the figure in the lower line be the greater, add ten to the upper one, and then take the lower one from the sum, set down the remainder and carry one to the next lower figure, with which proceed as before. (3.) When the figure in the lower line is equal to that above it, the difference is nothing, for which a cypher must be set down. EXAMPLES. From - - - 874698 Take - - - 561436 Remainder 313262 765087 425436 339651 762134 597082 165052 Proof. Add the remainder to the last line, and if the sum be equal to the first, the work is right." 22 SUBTRAi From - - - 658742 Take - - - 346121 OTION. 390076 184193 431267 280795 Remainder 312621 205883 150472 Proof - - $587 42 390076 431267 EXAMPLES FOR PRACTICE. Ex. 1. 4867434 2.6789491 3.5876486 4.3390761 25S4213 5468354 3564214 1478490 Ex.5. 7052673 6.9276807 7.7231607 8.9104008 3860749 4859434 5587465 9031618 Ex. 9. 6734078 10. i201832 11. 6000342 12. lOOO^OO 5943769 4b76543 5999343 99999(> Ex. 13. 4002103 14. 3874205 15. 9000123 16. 5301864 3987654 1796432 8123456 99 Ex. 17. 7962038 18.91111118 19. 468103520. 8302697 6498100 80000009 93006 2912934 Ex. 21.60001234 22.71216003 23.30061217 24.26013032 49993490 59876543 19996642 19125346 SUBTRACTION. f3 Ex. 25. '98743205 26- 50257480 27. 49764321 9^99999 41926321 1587549!? Ex. 28. 93816('86 29. 94286730 30. 92370800 927908 32199739 4812719 Ex. 31. 42601304 32. 27000019 53. 76253922 22500894 4102094 344939 Ex. 34. 33861400 35. 94681039 S6. 6901090 23713509 3041316 1860018 Ex. 37. 591040029 38. 271216904 39. 97348098 490300019 28391767 9290412 Ex. 40. 974689019 41. 593902742 42. 913062138 31689247 312003717 44823165 Ex. 43. 797260839 44. 170909009 45. 99326104 62310079 24710905 2128 i 299 24 SUBTRACTION. Ex. 46. 19390P09 47. 3.'921090 48. 1115677333 2109109 1937(;99 38 103475 MISCELLANEOUS EXAMPLES IN SUBTRACTION. Ex. 1. The invention of gunpowder was discovered in the year 1302 : How long is it since to the present year, 181 1 ? 2. What is the difference between thirty-five thou- sand three hundred and nine, and nine thousand and ninety-nine. 3. How much does seven hundred six thousand and four, exceed fourteen thousand nine hundred and thir- ty-seven ? 4. How much does fifteen thousand and five want of twenty-three thousand ? 5. The art of printing was discovered in the year one thousand four hundred forty-nine. How long is it since 1808 ? 6. Coaches were first used in England in the year 1580 : How many years is it to 1808 P 6. Needle making was introduced into England from India in the year 1545: How many years was that be- fore the present king came to his throne, which was in 1760. 8. Required the answers of the three following sums ; 18045—999; 2059—928; and 258764 — 49876. 10. How many more chapters are there in the Old Testament than in the New ? (25 ) MULTIPLICATION. Multiplication is a short method of Addition, and it teaches us to tind what a number will amount to, when it is repeated a certain number of times. Rule. The number to he multiplied is called the J^Iuttiplicand : and the number muttiptied is called the JUuUijplier, The number found is called the Product. 1 ' MULTIPLICATION TABLE. 2 times or twice i are 2 2 4 6 8 10 12 14 16 20 22 24 8 times 1 are 8 2 lo times are 3 6 9 n 18 21 24 27 30 3C\ 3 4 5 6 7 8 9 10 11 12 4 times 1 are 4 2 8 times are 5 10 15 2u 25 3u 35 40 45 50 55 60 6 times 1 are 6 1 9 times 10 tin les Ill til nes 1 are 9 lar 2 10 lar e 11 1 2 lu t 2 20 2 2z 3 27 3 3v 3 33 4 36 4 40 4 44 5 45 5 50 5 55 6 54 6 60 6 66 7 63 7 rv> 7 7. 8 72 8 80 8 8« 9 81 9 90 9 99 10 90 10 loo 10 110 11 0^ 11 110 11 1^1 12 108 12 12. 12 132 7 times 1 are 7 12 1! 24 30 o6 42 48 54 60 66 72 y^ 12 times 1 are 12 2 24 3 36 48 60 72 84 96 108 120 133 144 y 26 MULTIPLICATION. 1. When the Multiplier does not exceed 13. Rule. Multiply every figure in the multiplicand from ri^ht to left, consider how many tens there are in each product, the remaining units set down under the figure multiplied, and carry the, tens as so many ones to the next product. The last product is to be wholly set] down. Ex. 1. 420847 8 EXAMPLES. Ex. 2. 94564875 5 Ex. 3. 3476819 12 3366776 472824375 41721828 Thus in the first exatnple, I say 8 times 7 are 56, in which there are five tens and six over, I put down the ;8ix, and say 8 times 4 are 32, adding the 5 from the last product, 1 have 37 ; 1 put down the 7, and carry the 3 for the three tens ; I then say 8 times 8 are 64, and 3 are 67, 7 and carry 6 ; 8 times is 0, but put down the 6 brought from the last product; 8 times 2 are 16, put down the 6, and then 8 times 4 are 32, and the one brought forward are 33, which as being the last pro- duct, must be set down. EXAMPLES FOR PRACTICE. Ex. 1. 4653245 2 Ex. 2. 8756894 3 Ex. 3. 4986587 4 Ex. 4. S39076S Ex. 5. 705id673 Ex. 6. 9276807 5 6 7 Ex. 7. 7231607 Ex. 8. 9134908 Ex. Q. 6734078 8 9 10 MULTIPLICATION, Si7 Ex. 10.5201832 Ex. U. 6393476 Ex.12. 8874025 11 12 11 Ex. 13. 83022697 Ex. 14. 5391864 Ex. 15. 4681953 12 11 12 Ex. 16. 9874.3205 Ex. 17. 50947496 Ex. 18. 49764329 9 8 7 Ex. 19. 5972834 Ex. 20. 5097648 Ex. 21. 5875496 5 6 4f Ex. 22. 5439027 Ex. 23. 9999999 Ex. 24. 8888888 7 8 7 Ex. 25. 9734895 Ex. 26. 9237085 Ex. 27. 5942867 9 8* 9 This character Xi wliich is cal]«^d St. Andrew's cross, is us'mI to denote Multiplication, and when it stands be- tween two numbers, it signifies that those numbers are to be m'lltinl'ed into one another: thus 9 x 6 "Z: 54, is re«d, nine niuUip! od by six "s equal to fifty-four. Asrain 12 ' 11 —132, that is 12 multiplied by U is equal te 132. 28 i MULTIPLICATION. EXAMPLES. Ex. 1. 528318769 X 5 Ex. 2. 9567283 14 X S Ex. 3. 825934685 X 7 Ex. 4.486875294 x 9 Ex. 5. 496745 8 3S X 9 Ex. 6. 683637544 X 8 Ex. 7. 578940245 X 2 Ex. 8-759654318 X 11 Ex. 9. 987234617 X 6 Ex. 10. 867122436 x 12 Ex. 11. 716432978 X 9 Ex. 12. 6876493^1 X 7 Ex. 13. 795483206 X 11 Ex. 14.779368245 X 9 Ex. 15. 91872648 x 12 Ex. 16. 986G49005 x 5 Ex. 17. 85678654 X 4 Ex. 18.390057864 X 6 Ex. 19. 894.367542 X 8 Ex. 20. 765438958 X 4 II. To multiply by 10, add an to the multiplicand : thus 567 X 10 is 5670 ; and 567 x 100 is 56700 ; and 6489 X 1 0000 = 64890000. Therefore, to multiply a given number of one denomination, by a number whose significant figures do not exceed 12, having a cypher or cyphers joined to it : Rule. Write down the cypher or cyphers for the first part of the product towards the right hand, and then multiply every figure in the multiplicand by the significant figures of the multiplier, as in the preceding case. Thus, S469456 X 50 ZT 173472800, and 98765432 X 8000 IT 7901 23 456000, for 3469456 98765432 50 8000 173472800 790123456000 EXAMPLES. Ex. 1. 6754328 X 70 Ex. 2. 987654329 x 800 Ex.3. 8329674 X 1 10 Ex.4. 56780943 X 120 Ex. 5. 6470078 X 9000 Ex. 6. 9237654 X 1100 Ex.7. 7856423 X lOOO Ex.8. 7490434 X 600 III. When the multiplier consists of several figures. MULTiriCATlON, ^9 Rule. The multiplicmd must be multiplied by each fl2;ure of the multiplier separately be^inmn<»; with the rij^ht hand tii^ure, and the first fij^u re of every product must Hta:jd exactly under the figure tnultiplied by. Add these products together for the whole product. To multiply by any number between 13 and 19 in one line. Rule. Multiply the unirN figure of the multiplicand, by the right-hand «ligit of the multiplier ; set down the uint's fiiiure of the product, and remember what is to be carried. Multiply the second figure of the multiplicand ; to the product, idd what was to be carried, and also the first fiiture of t »e multiplicand. Then set down the unit's ti;;ure, and retain in your mind the number to be Curried, as before. Multiply the third figure of the mul- tiplicand : add the number to be carried, and also the second figure ol the multiplicand, and so on 5 thus 74365487596 17 1C642I3:89132 Here I say 7 times 6 are 42 ; I put down the 2 and carry 4, and say 7 times 9 are 63, apd 4 are 67, then add the b, which makes 73 ; put down the 3, and say 7 times 5 are 35, and 7 are 42, to which add the 9, which make 51, put down 1 and carry 5, and so on, till the last figure, when I say 7 times 7 are 49, and 3 to be carried are 52, take in the 4, which make 56, put down 6, and add 7 to the 5, and set down \2. To multiply by 21, 31, 41, &c. to 91 in one line. Rule. Bringdown the unit's figure of the multipli- cand for the unit's figure of the product ; multiply the same figure by the left hand digit of the multiplier, to which add the next figure on the left hand of the multi- phcandi set down the unit's figure and carry the tens, 3f MULtlPLICATION. multiplj the next figure of the multiplicamd by the same Multiplier, and so on, always observing to add the num- ber you carry and also the first figure on the left hand of that which you multiply. EXAMPLE. 3760942 21 78979782 Bring down the 2 then say twice 2 are 4, and 4- are 8, put down 8 and say twice 4 are 8, and 9 are 17, put down 7, and carry 1 ; then say twice 9 are 18 and 1 are 19, put down 9 and carry 1 ; next twice 0, will be 0, but the 1 you carried, and 6 make 7, put down 7 ; twice 6 are 12 and 7 are 19, put down 9 and carry 1 ; then say twice 7 are 14 and 1 are 15 and 3 are 18 ; put down 8 and car- ry 1, and lastly twice 3 are 6, and 1 are 7. EXAMPLES. 57864329 579 35964827 846 520778961 405050303 289321645 2157889C2 143859308 287718616 33503446491 3042624 3642 Proof. The readiest way of proving the truth of sums in Multiplication is, by casting out the ninesi Rule. Make a cross like that which is used to de- note Multiplication : add toj^ether the figures in the •multiplicand, casting out all the nines in the sum as of- ten as they amount to 9, and put the remainder down on one side of the cross 5 do the same with the fnultiplier. jvruLTiPLiCATiosr. 3X and put down the remainder on the other side of the cross Multiply the two remainders to}j;ether, and cast- ing out the nines of their product, will leave the same remainder as the nines cast out of the answer, when the work is right. EXAMPLES. 459S267 7628954 1 568 OXl 857 5Xt — I 56746136 534026:'8 27559602 38 44*70 22966335 61031632 2e.089'^5656 653801^578 To prove the second example, I saj 7 and 6 are 13 5 4 ahove nine, (omit the 9) : 4 and 2 are 6 and 8 are 14; 5 ahove nine, (omit the 9) : 5 and 5 are 10, 1 ahove 9, 1 and 4 are 5 : I place the 5 on the left hand of the cross, and say 8 and 5 are 13, 4 ahove 9 ; 4 and 7 are 11,2 ahove 9 : the 2 I put on the right hand of the cioss; Kovv 5Xfi gives 10, which is 1 above 9, I put the 1 at the top of the cross, and then cast out the 9's of the whole product, and I find the remainder is 1, which an- swering to the 1 at the top of the cross, leads me to con- clude that the operation is right. IV. When cyphers are intermixed with the figures In the multiplier. Rule. Omit the cyphers, and let the first figure of each product be placed under its multiplier* 32 MULTll^XieATlOW. EXAMPLES. Ex. 1. 76503^.9 Ex. 2. 4465348 6'i0j09 I 7()0< 608 3 5X2 7 X 8 6P^<5 961 1 357 -784 . 8 3S-i5i645 26:9:^0B8 459' l'r4 31257436 45^4091417461 31 2601,1093 i 584 Ex. 3 849275 X 706 Ex. 4. 973648 X 8005 Ex. 5. 59:384 X 830004 Ex. 6. 364*59 X 2709 Ex. 7. 245918 X 70.S006 Ex. 8. 609483 X 95007 V. When the multiplier is the product of two or more numbers in the table. Rule. Multiply the multiplicand by one of the com- ponent parts, and that product by ti»e other, and so or? : thus if I have to multiply a triven sum by 64, I find 8 X 8 = 64; instead, therefore, ofmultiplyinwhy 6 and 4 in the usual way, I multiply first by 8, aiid then that product by 8 again. 864392 8. X 64 5 5 X 5 EXAMPLES. 39746285 - 7 X 168 3 8X6 S 6915135 8 I . 278223995 6 55321088 1669343970 4 667? 375880 t i MULTIPLICATION. 33 EXAMPLES IN ALL THE CAS£:S. :. 1. 99365497 X 13 2. 54962874 X 26 3. 36729876 X 56 4. 47893062 X 48 5. 73167482 X 77 6. 8274386 X 96 7. 39745371 X 86 8. 5487962 X 357 9, 72983456 X 99 10. 3891307 X 464 11. 737394 X 4567 ]2. 35846 X 4682 13. S29357 X 2839 14.. , 58427 X 3957 15. 462875 X «874 16. 47683 X 3456 17. 594326 X 5936 18. 87491 X 7892 19. 486752 X 4608 20. 29687 X 3579 21. 8739690279 X 3978i^9 32. 7936820056 X 500634 23. 25764.'>2874 X 613487 24. 9167 4032^8 X 653000 25. 872694325 :>< 2900008 26 715970032 X 350706 27. 52673'it69 X 590734 28. 37 M5687 X 999999 29. 74714323 X 3^5627 30. 46382719 x 50000092 MISCELLANEOUS EXAMPLES. Ex. 1. ^^ultiply three millions thirty-nine thousand and three, bj thirty-five thousand and twenty-eight. 2. Multiply six billions, six hundred thousand and sixty-five, by eight thousand and thirty-nine. 3. There are eleven hundred hackney coaches in Lon- don } suppose, on the average, esich coach earns thir- 34 DlYlSIQNk teen shillings adaj^ how many shillings will be expend- ed in the hire of these carriages in a year of 365 days, Sundays being excepted ? 4. In Jamaica only there were imported, annually, not less than ten thousand eight hundred negroes from the coast of Africa: How many slaves had free-born Englishmen made in that island, between the year 1799 and the year 1807, in which the infamous traflick was abolished. 5. A boy can point sixtee»thousand pins in an hour; How many will he do in six days, supposing he works eleven clear hours in a day? See Blair*s Universal Free ptor. f>. What is the continual product of f5, 19, 703, and 999 ? 7. How many changes can be rung on twelve bells ? 8. Multiply the difference between 50487 and 30056, by the sum of 850. 9067, and 800 ? 9. The sum of two numbers is 30 ;55, and the greater nunjber is 25S51 ; What is their product ? 10 The sum of two numbers is 45 o 4, and the less is 1876; What is their product ? 11. What is the difference between twelve times fifty- seven, and twelve times seven and fifty ? 12- How many miles will a person walk in sixty-six years, supposinij he travels, one day with another, six mil*--, and there are .^65 liays in a year ? 13 How maov cn; c foet does this room contain, which i> fifteen feet lonj^, fourteen feet wide, and thir- teen feet high ? DIVISION, Bt Division, we find how often one number is con- taineil in another of the same denoiuination ; this is a short method of performing Subtraction. »ivisroN. 35 The sum to be divided is called the dividend ; the fi- gure, or figures by wliich we divide, is called the divisor ; and the result is called the quotient. In this Rule, as ia Multiplication, there are several distinct cases. I. When the divisor does not exceed 1 2. Rule. Write the divisor on the left hand side of the dividend, make a curve, and consider how often the di- visor is contained in the iirst figure, or in the first two or three fia;nres, and set tlie quotient under it ; and for every unit remaining after subtraction, carry ten to the next figure of the dividend. EXAMPLES. Ex.1. 4)78654328 Ex.2. 9)85674327 •19663582 95iy3o9— 6 Ex.3. 11)10876341 Ex.4. 12)11272459 988776—5 939371—7 In the second example, I saj there are 9 nines in 85 and 4 over ; I put down the nine and carry the 4, as 40 to the 6, and the 9*s in the 46. ,5 times and 1 over 5 put down the 5 and carry 1, as JO. and say the 9's in 17, once and 8 over; put down the 1 and carry 8 as80 j 9's in 83, 9 times and two over, and so on : at>* the last figure there are 6 remaining, put down this be- yond a small line. It is usual, in giving the answer, to make a short line under the remainder, and place under it the divisor ; tlius the answer to the secori3'2640 -T- 11 15. 327-3742 -t- 12 16. 333u333 ~ 12 17. 44444444 -— H 18. 5598764 -^ 12 19. 98897bU3 -~ 9 20. 933U048 -r- 8 Proof — I'lie method of Division, IS to multiply the take in the remainder, the dividend. proving the truth of sums in answer by the divisor, and result will be equal to the Ex. 7959467b'?4 ~ 7 7)7959467834 Quotient- 1137066833—3 7 8 7X2 8 Proof - - 7959467834 WIVISION. Sf Another method is bj casting out the nines, as in Multiplication. — Rule. Cast away the nines in the divisor, and put the remainder on one side of the cross ; then for the top figure multiply these two numbers to- gether, cast away the nines, and add the excess of nines in tJie remainder after division, and the excess of nines in this sum will be equal to the excess of nines in the dividend, if the work is right. See the preceding exam- pie, where I put down the 7 on one side of the cross ; do the same with the quotient, for the other side of the cross : the excess of nines in the quotient is 2, which I put ow the other side of the cross, then I say 7 times 2 are 14, and the remainder 3 make 17, which is 8 above nine, this I put at the top of the cross, and T find that 8 is the excess above the nines in the dividend, therefore I con- elude the operation is right. II. To divide a number of one denomination, by another number whose significant figures do not exceed 12, having a cypher or cyphers joined to the right hand. Rule. Cut off the cyphers from the divisor, and the same number of figures from the right-hand of the divi-. dend ; then divide the remaining figures of the dividend by the remaining part of the divisor, and the result is the anstver. To the remainder, if any, join those figures of the dividend, which were first cut off, and the whole will be the true remainder. Divide 46R5321 by 800 ; and 326441 by 1200. 8.00)46853 21 12.00)3264.41 ^856—521 2:2—41 Of course the true answers to these sums are 5856|U, an4 272^1^ S8 BIVISION, EXAMPLES Ex. 1. 3476521 -7- 60 Ex. 2. 8543009 -^ 700 ■ 3. 2y.'i7()48 -f- 8o0 4. 9C034 6 -T- 9000 5. 5620042 -T- noo 6. 7641121 -^ 500 7. 40 2079 -^ 1200 8. 84^6531 -r- 1»000 9. 7021164 -f- 90 10. 993' 216 -r- 80f)0 11. 46201 132 -i- 700 12. 1234567 -r- 120 IIT. To divide a given number of one denomination, by a divifior which is compounded of two or more num- bers in the Multiplication Table. Rule. Divide the given number bj one of those parts, and th^* quotient by the other component part, and so on till each of the component parts has been used as a divisor; thus 46875815777^ 105 is performed as fol- lows : the divisor 105 is equal to 7 x 5 X 3 ; I there- fon divide the dividend first by 7, and the quotient by 5, and this second quotient by 3. Ex 7)fl6875815777 5)6696545111 3); 339309022. -. r I Answer - - 446436340 • --2_ \ examples • . 1. 84596543 -r- 36 Ex. 2. 545069549 -r- 42 3. 45897642 -r- 56 4. 94^960542 — 99 5. 39200761 -T-66 6. 87932874 -r- 768 7. 38426587 -r- 550 8. 44444444-7- 121 9. 28476974 -^ 720 10. 55555555 H- 37S 11. 56342b72 -J- 132 12. 33992288 -f- 288 13. 34765982 -T- 144 14. 9845S392 -r- 432 35. 24853274 -f 512 16. 83547552 -r- 99 17. 43.-33999 -h 343 18. 54954535 -^ 720 19. 5555556 -^ 729 20. 25574538 -r- 343 DIVISION. 3i> IV. To divide by a number cansistinj^ of two or more dij^its, which Dumber is not compounded of those in the table. Rule (I.) Draw a curved line on the right and left of the dividend, and write the divisor on the left. (2.) Find how many times the divisor is contained in as many figures of the dividend as are just necessary, and place the number on the right for a quotient. (3.) Multiply the divisor by the quotient figure, and place the product under the above-mentioned figures of the dividend, subtract this product from that part of tha dividend under which it stands, and bring down the next figure in the dividend, or more if necessary, to the right hand of the remainder, and proceed as before, till the whole is finished. This is called Long Division. Ex. 5537049 H- 954 954)5537049(5804 Quotient. 4770... 7670 7632 3849 3816 • 33 Remainder. Answer 5804-^^j\, Here the divisor not being contained in the first three figures, I consider how often it is contained in the first four, and find it to be 5 times, the 5 I put in the quotient, and multiply the divisor by it, setting the product under the dividend. I now subtract this product, and to the remainder 767, 1 bring down the 0, and find that the di- visor is contained 8 times in 7670, the 8 I place in the quotient, and proceed to multiply the divisor by it ; the product subtractetl leaves only 38.; I now bring down the 4, but the divisor not being contained in 384, I put down in the quotient, and bring down the 9, the remaining figure in the dividend, and proceed ^s before 40 DIVISION. EXAMPLES. Ex. 1. 78654321 -J- 76 Ex. 2. 56943278 -~ 97' 3. 68742164 -i- 87 4. 84365487 ~ 69 5. 77755502 -J- 654 6. 45687403 h- 187 7. 53-430432 ~ 7654 8. 56943286 -r- 429 9. 57078443 -r- 8439 10. 58456942 -^ 327? 11.564320376-7-3976 12. 92876487 -i- 7392 13,677744032-^-5189 14. 468592 10 -j- 1437 15.627432871-7-4967 16. 35555555-7-7777 17. 44444444 -r- 5555 18. 888000999 -r- 999 19. 33333333-7-999 20.111111111-7-7777 Ex. 21. 487264325876 ~- 56780909 22. 87(3842'>8762J -t- 9095;.843 23. 948318296542-4-56400032 24. 5678432 '•i549 -r- 64785321 25. 877896543210 -i- 92836058 26. 444444*44444 -■- 750000564 27. 222000333^046 -7- 708385032 28. 5409»i53 :876i -7- 5406057 29. 32899438654 -^ 10010433 30. 784363254871 -7- 99834369 MISCELLANEOUS EXAMPLES. Ex. 1. Divide fifty millions by four thousand and seventy -nine. 2. The planet Mercury goes round the sun in 88 days, which is the length of her year, how many years of Mercury wauhl make 50 of our years, supposing each year contained exactly 365 days ? 3. It is estimated that there are a thousand millions of inhabitants in the known world : if one thirty-third of this number die annually, how many deaths are there in a year ? 4. The national, debt at present, cannot be less than five hundred millions sterling : how long would that be in piyiii.1 off, at the rate of two millions and twenty-five pounds per annum ? DIVISION. 41 5. The taxes annually collected amount to full thir- ty-three millions of pounds : how many poor families of six persons each would that sum supporl, supposing the annual expenses of the father and mother to be 20/., and of each child 7l. ? 6. My friend is to set sail to Jamaica on the first of March, 1812, the distance is reckoned to be 3984 miles from England, at what rate will he go, supposing he reaches the Island on the 10th day of April, that is, in 41 days ? 7. What is the diiference between the 12th part of 20,100 and the 5th part of 9110 ? 8. The prize of 30,000/. of the last Lottery became the property of 15 persons : how much was each per- son's share, after they had allowed 750/. to the office- keeper for prompt payment ? 9. The sum of two numbers is 1440, the lesser is 48 : what is their difference, product, and quotient ? 10. The crew of a ship, amounting to 124 men, have to receive, as prize-money, 1890/. ; but as they are' to be paid oft', they determined to make their com- mander and boatswain a present, the one of a piece of plate, value 25/. ; the other of a whistle, which is to cost 5/. : how much will each receive after these deduc- tions are made ? 11. In all parts of the world a cubical foot of water weighs 1000 ounces: how many pounds are there, sup- posing 16 ounces make a pound ? 12. A cubical foot of air weighs one ounce and a quarter, how many pounds avoirdupois of air does a room contain, which is 10 feet high, 14 feet wide, and 16 feet long? 13. Hydrogen gas, or, as it was formerly called, in- flammable air, that is, the gas vvith which balloons art filled, is full nine times lighter than the common air which we breathe : how much less would a balloon, con- taining 27,000 cubical feet, weigh if tilled with hydrogen gas, than if filled with common air ? 14. At what rate per hour and per minute does a place on the equator move, supposing the great circle of the earth to be 25,000 miles, and the earth to turn on its axis exactly in 24 hours ^ 4* ( 42) COMPOUND ADDITION, ADDITION OF MONEY. PENCE AND SHILLING fABLES. Pence s. d. Pence s. d. Shill. L. s. d: 20 - are 1 8 12 are 1 20 1 «5 2 I 18 1 6 25 1 5 30 2 6 24 2 30 1 10 35 2 11 30 2 6 35 1 15 40 3 4 36 3 40 2 45 3 9 42 3 6 .^0 2 10 ,50 4 2 48 4 60 3 55 4 7 34 4 6 70 3 10. 60 5 60 5 80 4 65 5 5 66 5 6 90 4 10 70 5 10 72 6 100 5 75 6 3 78 . 6 6 110 5 10 80 6 8 84 7 120 6 85 7 1 90 7 6 130 6 10 90 7 6 96 8 110 7 95 7 11 102 b 6 150 7 10 100 - 8 4 108 9 |60 8 i05 . 8 9 114 9 6 170 8 10 110 . 9 2 120 10 180 9 115 . 9 7 1S2 1 1 (90 - 9 10 150 - JO 144 12 20O - 10 UNITED STATES, OR FEDERAL MONEY^ 10 Mills (m.) make 1 Cent, c. 10 Cents — 1 Dime, d. 10 Dimes — — 1 Dollar, D. or g 10 Dollars 1 Ka-le, E. lOOCts.— — »1 COMPOUND ADDITION. 4»5 ENGI/ISH MONEY* 4. Farthings (qrs.) make 1 Penny, d. 12 Pence I Shilling, s. 20 Shillings 1 Pound, £, Compound Addition is a method of collecting seve- ral numbers of the different denominations into one sum. Rule (1.) Arrange the numbers so that those of the same denomination may stand directly under each otheri and draw a line under them. (2 J Add the numbers in the lowest denomination to- gether, and find how many units of the next higher de- nomination are contained in their sum. f3.J Write down the remainder, and carry the units to the next higher denomination, and proceed so to the end. L, s. d, I first add together the farthings, Ex, 4.68 19 Al which I find to be 14, but 14 farthings J 23 J 6 !l4 make 2>ld» I put down the 3 and carry 987 12 9 the 3 to the column of pence, which I 654 i3 7l then add together, and find the sum to 123 17 4a be 58, but by the table, 55 pence are 45t> 18 \0\ 4s. 7rf., therefore 58 pence are 4i». lorf., 439 4 Oa 1 put down the 10 and carry the 4 592 12 4j to the column of shillings ; i now add ■ the •'hillings together, and find the sum 3847 15 lU^ to be 115, but U5 shillings make 5l, ^ 15s., 1 put down the 15^ and cary the 5 to the pounds, and proce«d as in simple addition. EXAMPLES OF money. L. s. d. L. s. d, L, s. d. L. s. d. Ex. 1. 55 3 8 2. 67 2 8 3. 95 2 9 4. 49 9 II 62 6 3 24 9 9 89 7 8 33 8 7 90 2 1 38 2 5 72 4 3 96 12 9 31 8 4 42 5 9 67 9 2 75 3 4 43 7 5 78 6 6 51 8 9 51 8 9 10 9 8 ti4 d 9 45 5 4 12 19 7 44 COMPOUND ADDITION. X. S. d, L. S. d, Ex. 5. 58 15 9 6. 42 16 9 7. 92 13 41 79 5 5 37 15 11 84 14 9 61 7 10 73 9 9 73 18 4* 64 16 3 62 10 6 69 17 10 32 15 10 29 4 4 48 15 7 19 12 8 19 17 11 35 14 11* L. s, d: L. s. d, L. s. d. Ex. 8. 50 19 8l 9.54 17 6| 10.67 16 8^ 97 16 n 93 12 8 71 13 9 35 14 2 31 6 9i 84 . 11 81 46 16 8^. 25 10 11 32 19 3 07 16 2 76 13 10 48 10 4i 24 15 9\ 44 6 61 55 18 7i ___ 33 8 3 21 1 12 4 L. s. d. L. s. d. Ex. 11. 18 14 81 12. 41 15 9^ 93 15 10^ 56 10 9 37 6 11 62 16 31 78 16 5l 87 4 69 12 y\ 78 13 n 43 8 11 92 19 01 12 17 SI 13 16 7 B. cts. mis i. D. cts. mis. " — Z>. cfs. Wis/. L 73 14 5 14. 84 13 8 15. 69 17 4 27 37 4 79 57 3 37 16 2 46 18 3 99 14 7 48 27 6 74 29 9 37 74 5 62 74 3 38 17 4 29 18 6 73 65 7 85 63 7 47 13 2 18 11 1 COMPOUND ADDITIOJJ. 45 Ea, D. rf. c. »n. Ea. n. d. c. w. 16. 34 4 7 6 3 17.174 3 4 2 4 29 3 2 7 6 27 4 2 6 3 13 4 1 2 149 7 3 2 8 ir 6 2 7 76 4 2 9 7 39 4 2 1 8 37 5 6 4 7 ■ 4S 9 1 2 7 59 7 4 2 6 L. s. d Z. s. d. X. s. rf. 18. 46 2 H 19. 45 19 94 20. 43 17 10| 65 10 H 63 17 111 50 14 6* 6 4 74 10 79 13 51 72 81 17 H 46 10 9| 65 19 7», 39 15 10 35 8 7 91 5 1^ 23 10 8| 47 19 10} 38 19 10 19 14 ^i 19 14 6 29 12 9^* L. s. e£. L. s, d. L. s, d. «1. 52 18 10 22. 77 15 4\ 23. , 57 15 91 67 12 2i 69 10 9i 64 9 2 77 14 9 41 lOi 76 17 \0l 82 13 JO^ 57 13 8 97 IS 9 98 12 11^ 87 9 lOi 39 18 111 21 17 ■71 91 16 U| 45 10 10 45 12 9 76 i4 8 59 17 9 L. s. rf. L. s. d. L. s. d. 24. 446 19j^l 25. 48 14 10^ 26 ;. 92 19 91 152 1^ ^10| 36 13 10 56 10 9 695 1-2 0^ 74 15 71 64 18 71 7o8 3 5 23 18 2i 38 16 3 S3H 14 3^ 48 9 6 49 15 lU 166 19 11 81 16 4| 64 19 SJ 279 12 91 77 '11 M 92 ir 8^ 46 COMPOUND ADDITION. L. s. d. L. s. d. L. s. A 27. 1^2 14 91 28. 54 11 10 29. 414 19 9 93 16 10^ 22 19 &\ 627 17 111 17 12 U 61 16 % 741 f) 41 56 13 7\ 14 17 01 865 14 8 91 19 11 58 12 11.^ 917 6 101 76 14 5\ 72 10 6 347 14 101 14 11 3 r^i 14 11 44^ 13 4 30. 427 18 10* 31. 548 11 6 32. 493 2 SI 941 17 9 932 18 41 347 14 31 712 19 6 379 61 729 19 5 6:25 12 7 J 414 17 0^ 672 5 85 511 11 10 573 4 31 548 10 3 46^2 10 6^ 697 13 % 217 12 SI 383 11 9^ 551 6 11 974 16 7\ 146 5 OJ 33. 412 9 Hi 34. 152 15 21 35. 504 3 92 924 19 65 255 18 61 636 15 5 750 11 35 348 12 % 421 2 71 627 IQ na Atf\ f\ t rt ^A7 Iv 10 01 438 10 4i 566 13 otf 12s 383 7 363 2 105 631 6 45 848 15 2i 221 15 8 781 3 10 710 81 147 1 5 949 16 7 483 10 45 123 15 11 426 19 7 36. 576 14 9 37. 827 18 111 38. 792 19 31 613 12 U^ 550 11 81 437 14 9| 719 13 41 938 9 4 354 10 lOl 914 14 Q\ 344 3 516 18 4 9,n 10 9 615 16 U 209 13 105 759 8 5\ 471 2 7 524 17 2| 433 15 35 214 15 loi 739 6 10 918 11 4| 745 19 2 365 2 61 564 7 2 90 9 9 147 17 9 _, irOMPOUND ADDITION. 47 L, s. d. L» s. d. L. s. ,d 39, 88 16 1 m 40. 28 9 41 41. 60 15 55 26 11. 51 54 17 9 48 13 |3 »4 9 7 2^ 6 11 93 18 6 36 13 4| 28 13 51 7 7 105 41 18 3 65 18 71 35 19 41 27 3 81 92 6 41 73 6 91 54 15 ] u^ 7 16 01 31 17 3 12 1 J 6 14 5 10 59 14 1(^ 20 10 40 9 60 10 42. 94 1 9i 43. 53 11 4^ 44. 68 19 51 88 2 6*a 6 2 8 84 7 35 46 5 Hi 18 5 31 8 6 51 29 16 35 26 10 75 25 11 95 48 '2 42 41 9 13 7 '5 17 7 64 2 2 47 15 61 61 13 3i 71 18 icl 32 1 3 7, 14 101 3 14 115 1 18 05 12 18 •^1 80 61 2 16 4 ~ 45. 76 12 8^ 46. 39 14 45 ^ 47. 78 12 5 40 61 97 12 21 17 14 85 8 17 4 73 15 105 35 6 24 19 51 6 10 111 28 16 iOl 59 15 21 30 2 9 11 8 35 82 6 51 16 12 51 49 15 71 7 18 H 58 16 15 6 11 41 S3 2 91 2 13 7 62 15 8 8 10 01 82 35 5 18 41 10 10 10 90 10 48 eOMPOUND ADDITION, L. s. d. L, s. d. L. s. d. 48.127 10 \ol 49.515 14 9| 50. 657 16 \^\ 356 14 91 043 17 Si 734 17 4^ 483 9 < 623 15 11* 879 14 ^l 849 7 11 417 19 S* 919 12 10« 680 18 lU 338 14 10 131 19 11 774 19 n 385 18 \M 235 7 6J 114 6 2^ 764 13 6 496 18 SI 251 18 9^ 453 19 91 587 9 5 428 15 6 5m 18 51 673 U 10 567 16 2 223 14 2 820 19 4 .4;n 16 9 52. 7tf2 10 9i 53.477 16 41 272 15 61 966 4 81 395 15 ^ 889 17 10^ 899 13 6 736 5 11 647 19 21 248 16 101 692 14 91 398 16 7 53*2 14 9 565 13 •51 563 16 loi 476 19 '^i 937 17 770 51 744 12 9^ 441 16^ 41 945 17 7 6f^9 15 71 7ro m 01 420 13 9i 593 15 1 1* 672 11 11 150 10 150 10 O 40 10 54. 494274 12 91 765502 6 4 S00089 2 21 402193 If 9 375451 3 10 269440 18 61 123428 15 10 567865 11 910649 10 6 55, 901442 16 10| 23497 1 5 91 567 S 52 14 71 912261 19 21 34551^ 17 9S 678830 12 6 912887 19 10 456713 10 31 891391 17 81 COMPOUND ADDITION. 4^ L, s. d: L. s. d L. s. d. 5'&, 4567 14 11^ 57. 3^256 19 6* 53. 3567 12 9\ 4934 15 9 4397 10 111 7960 17 10 2765 16 10* 1974 12 9i 1234 15 71 9876 19 lU 7-246 8 4 5678 12 8* 34'J7 9 5 S94-2 15 lOl 9123 14 10 1234 10 8^ 4567 8 91 4567 13 lU 5678 16 10 4567 17 111 8912 17 9 4376 8 9 9376 12 8* 1456 9 6 £794 15 4| 4623 2 5 7891 10 41 7921 12 lOl 5932 5 4 2845 ^ 3 59. 1764 13 91 60. 2487 7 3 61. , 6789 12 5] 1805 17 4 5764 16 11* 2345 13 11 1704. 12 7 12:?4 J 8 2 6789 16 9l 3459 15 11 5678 19 91 4972 15 10 2946 16 10^ 9012 17 10 3456 19 5* 1796 14 10 3456 2 2 7891 16 71 4325 16. 8 7890 14 5 2345 14 11^ 5678 12 111 1234 13 10 6782 12 9 4932 14 6 5678 15 7 4315 11 71 2005 9 51 9123 13 4 2105 8 6 50 COMPOUND ADDITIOX. EXAMPLES OF fVEIGHTS AND MEASURES. TROY WEIGHT. 24- Grains {s,\\) mako 1 Penny wt. pwt 20 Penny wt. — 1 Ounce, 6z. 12 Ounces — 1 Pound, lb. Note. — By this weight are weighed Gold, Silver, Je'wels, Liquois, &c. In adding up the column of grains I find the sum to be 122, which I divide by 24 to bring it into pennyweights; and 122 £,rains make 5 pennyweights and 2 grains over ; the 2 I put down, and carry the 5 to the column of penny- weights ; [ then add these togetfier, and find the sum to be 101, which i divide by 20 to bring to ounces, I put down the 1 and carry 5 to the coluiiin of ounces ; then adding the ounces, I find the sum 79. which, by dividing by 12, give 6 lb. 7 oz. the 7 I put down, and carry the 6 to the pounds, and proceed as in simple Addition. lb. oz. dwtj ^.gr. 7684 9 16 22 1234 11 5 19 9876 8 11 22 14^)3 9 19 12 3587 10 10 3 2345 7 6 15 678P 9 14 21 3257 11 15 8 3627 1 7 1 2 lb. oz. dwt. lb. oz. dwt. S''- lb. oz. dwt. i.414 9 14 2. 410 9 12 19 3. 526 10 19 617 5 13 342 11 16 12 712 9 17 713 10 9 912 3 14 14 944 6 14 322 7 15 751 6 10 22 633 10 11 413 2 10 626 10 17 16 319 4 10 5\4- 11 15 427 4 11 23 247 9 12 976 8 7 - 123 11 17 12 123 10 17 oz. lb. oz. dwt. Rr. [■:. OZ. dwt. dwt, •?;r. k 9'R) 10 ]9 15 5. 174 11 19 6. 174 19 23 738 6 4 23 7 4 10 13 714 11 14 6J4 3 17 13 944. 9 14 714 19 546 7 16 19 7 1 11 19 74 1 22 321 10 5 22 944 10 13 948 o 21 230 9 15 15 74 11 3 74 2 12 946 11 19 i23 12 4 6 301 14 4 COMPOUND ADDITION. 51 lb. oz. (Iwt oz. dwt. s;r. 7.71 11 19 8.74 19 23 dl. 8 14 64 14 17 77 74 19 Jl 1 + 3 11 66 13 9 64 2 9 74 11 11 7 + 6 14, 14 10 3 71 2 13 ]9 11 14 105 9 12 13 17 5 AVOIRDUPOIS WEIGHT. 16 Drains (dr.) make 1 Ounce, oz. 16 Ounces — - 1 Pound, lb. oQ Pftiinfia . - ^ of a htinc 1. qr. . Cwt. 4 Quarters T Hundrt'd 20 Hundred 1 Ton, T. Note. — By this weight are vveighevi all kinds of coarse and heavy Goods, except Gold, Silver, &c. !b. oz. dr. tons. cwt. qr. lb. lb. oz. dr. 1.318" 10 10 2.416 19 2 26 3, .539 13 15 436 9 8 313 10 20 316 14 13 62A 14 6 2 1 11 3 1& 223 \2 7 419 6 15 725 19 ^ 18 811 9 6 245 9 7 357 14 2 25 700 6 14 853 11 10 429 17 3 22 414 12 12 145 9 8 235 15 2 19 toJis. cwt. qr. lb. tons. cut. qr. cwt. qr. lb. 4. 305 14 2 11 5. 174 19 3 ( 5. 174 3 27 418 18 74 14 2 724 2 24 336 2 1 14 714 13 1 149 1 14 119 13 3 27 718 16 2 719 2 16 767 16 8 734 15 2 407 1 23 782 9 1 16 714 14 1 149 2 17 421 15 3 19 155 3 7ti 3 15 ^2 COMPOUND DIVISION. qr. lb. oz. lb. oz. drs. 7- 44 27 15 8. 17 15 15 74 26 14 27 14 11 19 14 13 16 13 9 74 12 14 74 14 14 66 27 13 70 74 19 10 04 13 10 13 17 5 13 4 5 APOTHECARIES' WEIGHT. 20 Grains (gr.) make 1 Scruple, 9. 3 Scruples 1 Dram, 5* 8 Drams — 1 Ounce, ^. 12 Ounces 1 Pound, j^ lb. oz. dr. oz. dr. so. gr. lb. oz. dr. sc. gr. 1.314 8 4 2. 22 3 2 19 3. 646 U 4 1 19 210 11 4 56 1 13 715 3 7 1 14 766 10 2 43 2 2 11 934 3 4 12 555 9 6 54 7 17 373 10 5 2 9 417 8 1 76 5 2 14 216 5 1 2 16 324 7 3 45 6 1 159 2 5 14 lb. oz. dr. oz. dr. sc. dr. sc. gr. lb. oz. 1 dr. 4. 47 11 7 5 . 149 7 2 6 .749 2 19 7. 84 11 7 94 10 6 714 3 607 1 18 74 10 6 74 10 4 619 2 1 714 2 17 37 5 4 75 9 3 74 6 2 400 19 4 3 69 2 162 5 2 74 1 13 74 1 2 57 1 2 74 1 2 715 2 14 79 2 6 18 2 1 779 6 1 64 1 18 19 2 4 19 3 5 146 4 16 10 13 4 8 , COMPOUXD ADDITION'. 53 CLOTH MEASURE. 2J. Inches (In.) make 1 Nail, ni. 4 Nails. I of a yard, qr. 4 Quarters 1 Yard, yd. 3 Qufirters 1 Ell Flemish, E. Fl. 5 Quarters I Ell English, E. E. 6 Quarters 1 Ell French, E. Fr. yd. qr. nl. K.e. qr. nl. E.e. qr. nl. yd.qr. nl, 1. 4:i4 3 2 2. ,511 4 2 3.565 4 4. .543 3 2 527 1 ? 660 2 626 2 I 836 2 2 613 2 3 439 4 2 724 1 754 2 3 758 3 1 337 I 2 882 2 3 217 1 3 840 1 3 854 2 3 933 3 725 3 2 925 2 2 766 2 227 1 1 438 2 2 E.e. qr. nl. E.e. qr. nl. yd. qr. nl. E e. qr. nl 5, 120 2 2 6. . 537 2 7. 74 3 S 8. . 77' 4 3 3P4 4 1 916 3 1 64 2 1 14 3 2 110 2 328 3 3 74 1 3 74 2 I 481 1 2 457 I 2 49 2 I 49 1 2 556 4 3 646 3 2 74* 1 2 74 2 I 664 3 I 2^7 4 2 44 3 / 44 I 2 779 2 3 561 2 2 16 2 3 94 2 LONG MEASURE. 3 Barley Corns(bc.)n] lake 1 Inch, in. 12 Inrhes - 1 Foot, ft. 1G» Feet -i 1 Rod,r. 40 Rods - 1 Furlong. , fur. 8 Furlonc;s 1 Mile, m. 69^ Statute Miles — - 1 Degree, Deg. 5 i* ^4 eOMPOUND ADDITIOI?. ALSO, 4 Inches make 1 Hand, 3 Feet I Yard, 5f Yards 1 Hod, Pole, or Perch. 6 Feet 1 Fathom. 66 Feet — .— 1 Gunter's Chain. S Miles EXAMPLES. miles, fur. p. yds. yds. ft. in. be. lea. mi. fur. p. 1. 427 6 23 3 2. 214 2 9 3. 320 1 6 IS 689 5 26 5 183 2 11 2 623 1 7 27 322 7 30 2 597 8 1 721 4 I6 510 2 38 4 649 2 7 2 826 1 3 32 777 4 3 725 16 1 932 2 G 1 888 3 10 4 930 13 315 1 2 28 ]26 ^^4 492 1 4 1 409 1 5 39 412 7 39 4 291 2 10 2 376 2 7 27 lea. m. fur. fur . p. yds p. yds. ft. feet in. b.e 4. 17 2 7 5. 147 39 5 6. 177 5 2 7. 174 11 2 14 1 6 614 37 4 714 4 I 49 10 1 74 1 7 714 193 714 12 74 11 2 68 2 4 674 17 1 615 I 64 9 I 74 1 719 27 2 714 1 2 74 10 I 69 2 1 197 19 I 719 1 1 64 11 2 74 1 2 724 14 3 437 2 1 74 10 96 2 4 604 29 5 610 4 94 n 2 LAND, OR SQUARE MEASURE. 144 Square Inches make 1 Square Foot. 40 Rods 1 Rood. 4 Roods I — — Acrtfi COMPOUND ADDITION. ^5' ALSO, 9 square Feet make 1 square Yard. sol Yarda — ■ I Rod. 160 Rods I ^cre. 640 Acres 1 Mile. EXAMPLES. ac. r. P- ac. r. P- ac. r. P- 1. 452 2 38 2. 982 2 24 3. 921 1 29 114 1 35 618 3 14 604 3 32 715 2 16 100 1 27 736 2 29 430 2 35 474 2 10 559 3 28 529 3 7 363 1 31 265 1 17 2 '1-6 1 23 755 3 38 427 30 6fil 3 11 647 6 883 I 39 314 2 35 234 2 29 P- 291 3 25 ac. r. P ac. r. p. ac. r. ac. r • P- t. 17 3 39 5. 714 3 39 6. 14 3 39 7. 174 3 39 64 2 37 619 1 36 74 1 19 714 1 27 74 1 U 714 2 27 64 2 14 618 2 12 64 2 19 619 1 34 74 1 18 719 1 14 74 1 18 719 2 37 4*7 2 24 734 2 II 64 2 17 719 1 24 18 1 14 715 1 24 14 1 13 615 2 14 74 2 19 639 2 24 94 3 54 174 3 38 74 2 24 714 1 34 '_ LIQUID MEASURE. 4 Gills (gl.), make 1 Pint, pt. 2 Pints I Quart, qt 4 Quarts I Gallon, gal. 63 Gallons 1 Hogshead, hhd. 2 Hogsheads I Pipe, P. or Butt. B. 2 Pipes or Butts — ^ 1 Tun, T, 56 e-^MPOUND ADDITION. hhd.^al. pt./ tuns, h. g. qt tuns, h. g. q. 1. 626 U 7 2. 522 I 39 3 3. 148 2 25 3 753 17 1 257' S 34 2 513 42 3 4:8 5^ 6 763 2 58 3 614 1 S6 1 217 1.^ 7 611 3 43 1 349 3 43 2 1.33 1^5 937 1 16 3 416 2 56 1 A97 56 2 23S 31 2 952 3 26 J^312 11 3 749 3 7 567 1 19 3 W 256 <> SI9 2 59 3 792 3 46 2 • tun^.hlid.s:. hh«l ^al qt. p:. q- p- 4. 7 1 4 3 62 5. 74 41 3 6. 14 3 1 614- 2 61 64 40 3 74 I 174 1 39 74 19 1 39 3 1 164 2 47 64 39 2 17 1 274 1 49 74 40 I 19 2 175 -2 37 69 16 I 77 1 1 375 I i9 17 39 2 39 3 1 704 n 64 28 44 3 24 ^2 V DRY MEASURE. 2 Pints (pt.) make I Quart, qt. 4 Quarts 1 Gallon, g;al. 2 Gftllons >- ' I Peck, pk. 4 Pecks 1 Bushel, bu. 40 Bushels 1 Load, Lo. bu. pk. gal. bu. pk. 8:al. bu. pk. gjal. 1.73 3 1 2.29 2 3.36 2 1 46 2 57 1 99 3 I 39 3 1 38 3 1 36 3 1 48 2 26 2 •27 2 1 37 2 48 1 46 3 46 1 I 28 1 27 2 1 27 Q 1 76 3 I 36 1 1 39 I 1 24 2 1 57 2 1 COMPOUND ADDITION, 57 gal. qts. [)ts. pks.gal. qts. bu. pk. gal. qts. 4. 56 3 1 5. 76 1 3 6. 58 3 1 3 77 2 1 39 2 74 2 1 3 64 1 92 1 3 63 3 2 76 1 1 47 3 49 1 1 3 67 2 1 36 1 2 48 2 1 3 74. 3 27 1 3 63 3 1 3 62 1 1 64 15 ^9 2 1 77 I 2 3o 3 2 TIME. 60 Seconds (Sec.) make 1 Minute, ra. 60 Minutes 1 Hour, li. 24 rtours 1 Day, d 3651 l)ays 1 Year. yr. 100 Years 1 Century, Cen. mo. w. d. h* w. d. h. "mi. d. h. mi. sec. 1. 19 2 6 19 2.57 4 23 38 2 .62 7 47 38 46 1 4 21 64 6 13 47 18 12 54 56 22 3 5 S 9 15 3 21 18 19 15 76 21 16 34 9 ^ZO 49 57 2 21 36 2 31 62 1 ii 12 78 6 9 59 bO 23 3i 45 17 3 2 14 49 20 * 6 52 22 28 32 11 3 4 16 71 5 14 48 15 4 5S 23 29 I 3 21 23 3 7 24 64 I6 13 16 yrs. mo. w. uio. w. d . d avs, h. m. hrs. min. se. 4.737 12 3 5. 64 3 6 6. 714 23 59 7. 647 59 59 347 11 2 74 1 5 74 14 54 137 51. 54 618 10 1 34 2 8 94 21 55 375 56 56 374 9 2 74 1 4 74 13 53 714 17 19 175 I 1 63 2 1 69 12 14 615 54 54 714 12 3 74 1 2 74 12 19 714 17 13 615 10 I 64 2 1 37 11 17 613 34 56 314 9 3 94 2 6 46 22 49 626 47 49 58 COMPOUND ADDITION. A8 rR0N(3xMY. 60 Seconds (") make 1 Prime Minute,'. 60 Minutes 1 Degree, °. 30 Dei^rees 1 Siun, S. 12 Sij2;nsor> __ C The jjjreat circle 360 Degrees^ I oftheZodiack 11 24 37 41 5 3 26 25 6 9 54 35 7 12 5- 21 9 5 37 56 3 29 59 7 3 25 13 17 8 24 42 59 11 26 21 19 4 29 18 29 3 9 12 15 9 24 50 40 5 16 52 43 4 8 17 41 11 18 29 27 3 19 47 51 S2(^ 9 8 5 13 51 46 11 29 51 36 5 16 8 27 6 7 1 9 9 18 30 SO 11 20 40 50 10 12 24 35 3 4 44 44 10 9 55 • >7 7 21 42 .'=6 7 25 36 51 4 21 4-4 56 5 ^23 51 46 MISCELLANEOUS EXAMPLES IN ADDITION. 1. What is the sum total, in shillinu;s, ot 54 guineas, 29 pounds, 36 guineas, and 48 pounds? Answer, 3430 shillings. 2. Add together 16/. 12s. 2(1. ; 156/. 9s. 9l(l.; 20395/ 12s.: 24/. 19s. lljd. ; 37/. 6s. Id.; 327/. 18s.; and 100 iTuineas. Ans. 21063/. 18s. 6d. 3. In collecting an account of debts owing to «ne, I find Mr. A. owes me 74D. 16cts. ; Mr. B. 69 D. 50cts. ; Mr. C. 73!>. 4cts. ; Mr. D 38!3. 37^cts. ; Vlr. E. 14D. e^cts. ; what is the whole sum due to me } Ans. 26? ). 13lcts. 4. A gentleman ordered a service of plate from his silversmith, anil on receiving his bill, be fin Is that he had dishes and covers weighing 45 lb. 9 oz. 12 dwts. ; plates weighing 70 lb 7 oz! 16 dwts. ; spoons of difter- ent sizes, and ladles, 24 !b. 9 ;>z. 12 Iwts : waiters, 15 lb. 10 oz. ; salts and castors, 4 lb. 4 oz. 3 dwts.; candlesticks, 19 lb. 11 oz. 17 dwts. ; and sundry snial- COMPOUND ADDITION. 59 ler articles 5 lb. 3 oz. 10 clwts. ; what is the weight of silvtT he will hiv • to pay for ? Ans. l86ib. Soz. iOitwt: 5. A carrier hritie;s goods to a shop k«^eper, viz. 8 h;i<;s of hops wei^hioi; 19 cwt 3 qrs. 14 lb. : cheeses wei}>;hing J 5 rwt. 1 qr. 21 lb. ; butter wei^jhino; 12 cwt. 2 qra. ; two chests of tea weighing; ]l cwt. each ; and a sack of salt weighing 8 cwt. 2 qr. 12 lb. ; how n>uch weight wdl the earner have to cliarge t' Ans. 59 cwt. 1 qr. 19 lb. 6 The lent of my house is 30/. per annum : the house taxis three pound fifteen shillings ; land tax 51. ; vvin- dous 15/. I2.S'. Oc?. ; poor's rates 10/.; lightincr, watch- ing, and street rates 3/. 9s. 3ld. : how much therefore do my house and taxes stand n»e in per annum ? Ans. 87/. lijs. 3ld, 7. 'I'he following is an estimate of the repairs wanting to my house; how much is the whole sum ? ('arpenter*s bill 27/. 9s. 9(11, ; bricklayer's and plasterer's 17/. 7s. ad. ; mason's 5/. 5s. ; painter's, glazier's, and plum- ber's, fourteen gu.neas ; smith's, for new rails, 12/.; and the slater's 9/. 18.s. Ans. 86/. 14.>-. 3^. 8. A man purchased some goods for the country ; the first parcel contained 25 y«ls. 2 qr- 2 nl of broad cloth ; the second \26 yds. 2 qrs. of serge ; the tisird a thousand yards of green baisc ; and the fourth 19 yds. 3 qrs. 2 nl. of shadoon ; what was the whole quantity ? Ans. 1172 yds. qr. nl. 9. A wine merchant, retiring from business, takes an account ot the stock of wines in his cellar, and finds 6 pipes and 50 gallons of port wine, four pipes of sherry ; ten pipes of Lisbon ; 2 pipes of claret ; of Madeira he hail 36 gallons ; of brandy 50 gallons ; of rum two hogs- heads ; and of Holland, I hhd. and 12 gallons; what quantity of liquor dia his cellar contain ? A us. 23 pipefe, 1 hhd. 22. gal. 10. \ friend in Essex desired me to measure his farm, which he holds on a lease ; t[»e three fields at the back of the house measured 59 ac. 2 r. 20 p. ; the large piece of ground in the valley measures 74 acres, three others measure each on an average I \ ac. 1 r. 36 p.; the field laid down in clover contai^is 7 ac. 3 r. 2 p. one sosvn with caraways, 1 find to be 3* acres ; and the ground be- 60 COMPOUND SUBTRACTION. longing to the garden, out-houses, &c. makes about li acres ; how many acres ought he to pay for ? Ans. 180 ac. 2 rd. 10 per, 1 1. A merchant sends to his banker on the 2d day of the month, in money and bills, to the amount of two thousand guineas ; on the fifth he sends him 900/. I9s, 4-d, ; on the eleventh he sends 500/. ; and in the course of the remaining days of the month he sends 1515/. I2s. Hid.; how much therefore may he draw as occasion requires ? Ans. 6016/. I2.s. 3ld. 12. A gentleman's steward received the following sums of money for rents ; what was the gentleman's in- come ? Of farmer A he received 394/. 12s. 6d., of B 971. Us. 9rf., of C 175/. 10s., of D 99/. 4s. and of K 139/. 12s. 46/. Ans. 906/. I3s.7rf. 13. A person borrows of several friends the following suujsof money : of the first 500/. ; of the second 225/. 12s. ; of the third fifty guineas ; of the fourth seventy guineas and 22 crowns ; of the fifth he had 150/. 7s. 6d.i how much will he have to pay interest for ? Ans. 1007/. 9s. 6d, 14. A man borrowed a sum of money, and paid at dif- ferent times 87 dollars, but he still owed 64 D. 372 cts., what was the original debt ? ' Ans. 15 ID. 37aCts. COMPOUND SUBTRACTION, Is the method of finding the difference between two given compound numbers. Rule. I. Having arranged the numbers so that the smaller may stand under the greater, subtract each num- ber in the lower line fiom that which stands above il;, and write down the remainders. 2. \\ ben any of the lower denominations are greater than the .;pper. increase the upper numuer by as ujany as make one of the next superior denomination, from which take the figure in the lower line, set down bOMPOUNP SUBTRACTION. 61 the difference, and carry one to the next number in the lower line, and subtract as before. Ex. Subtract 595/. ITs^^ld. from 600/. lOs, 1\d. Here I say 2 farthings from 1, I cannot, but I add 4 to the 1, be- cause 4 farthings inak*^ a penny, and 2 from 5, and there reinains \ ; 1 carry one to the 9 ; 10 fro^ 7 I cannot, but I add 12 to 7, be- Proof 600 10 1\ cause J2 pence make a shilling, and 10 from 19 and there remain 9 ; I carry 1 to 17 ; and 18 from 10, T cannot, but 1 add 20 to the 10, because 20 shillings make a pound, and 18 from 30 and there remain 12; I now carry one to the five, and go on as in simple sub- traction. The method of proof is the same as in simple Sub- traction. 600 595 10 17 d. 9^ 4 12 ^\ 600 10 7i EXAMPLES. D. cts. rals. D. cts. mis. D. cts. mis. Ex. 1.39 44 3 2.76 29 4 3.18 7e» 5 27 7G 2 49 13 6 * 9 47 6 yJU rsrwT D. cts. mis. D. cts. mis, D. cts. mis. Ex. 4. 57 13 7 5.62 13 7 6.48 30 1 37 4 1 • 24 97 % 49 76 9 fwrr TjtTT Ea. D. D. cts. mis. Ex. r. 67 S 7 4 6 39 4 2 9 7 /7ir"F TSJL Ea. D. D. cts. mis. 8. 79 4 I 6 7 37 6 7 4 9 {rrcTri: 6£ COMPOUND SUBTRACTION. L. s. d. L. s. d. L. s. d» Ex. 9. 145 19 9* Ex. 10.370 17 H Ex. 11.450 12 61 130 17 6^ 369 12 4^ 371 10 4 Answer Proof L. s, d. L. s. d. L. s. d. Bit. 12. 594 10 91 Ex. 13.465 12 7^Ex. 14.564 12-2* 374 19 51 349 17 91 37d 18 4^ L. s, d. Ex. 15.371 19 2| 199 17 1]^ L. s. Ex. 16. 700 376 16 d, L. s. d, Ex. ir. 47d 19 4 374 12 9 L. s. Ex. 18.473 18 291 12 d. L. s. d, Ex. 19.24j^ 9 9^ 159 19 Hi L. s. Ex. 20. 376 17 299 14 d. 7 4| L. s. d. ' Ex. 21. 594 593 19 9^ L. s. Ex. S2. 796 12 ] ^69 8 d. '^ r- • j'\' ../ %f^f L. s. d, Ex. 23.476 17 7 • 3^9 19 111 Ex. 24. 399 2 177 12 d. 2 n \ \ r ' « \i COMPOUND SUBTRACTION. 63 L. Ex. 25. 209 159 s. d. 18 8 19 9^ Ex. Ex. Ex. Ex. Ex. Ex Ex.: Ex.4 L. 26. 500 499 19 11 L. Ex. 27. 422 371 s. d. 3 6* 15 7^ 28. 224 156 s. d. 2 6| 6 6^ L. Ex. 29. 794 367 15 ()l 16 4^ L. , 30. 999 [800 19 111 _ L. Ex. 31, 704 398 .s. d, 15 4^ 12 11 L. S2. 074 249 s. d, 6 02 19 91 ■" " " L, Ex. 33. 372 U9 s, d. 10 61 6 4i L. 34. 649 597 s. d. 12 91 19 81 L. Ex.35. 341 230 s. d. 5 llj 9 4^ , 36. 846 375 s. d. 9 81 9 9^ L. Ex. 37. 124 109 ^. J.. 9 10^ 10 3* L. J8. 90441 67217 5 91 13 10 L. 0. 12i27 7618 L, Ex. 39. 418 399 s. rf. 7 10 16 9^ s. d. 16 lU 14 91 64 COMPOUND SUBTRACTION. L, s. a. L. s. d. Ex. 41.1654 12 7 Ex. 42. 14476 5 6* 585 9 10^ 7iJ4 13 8| L. s. d. L. s. d. Ex. 43.222 18 9^ Ex. 44. 96481 16 9 i42 7 10^ 37(>8 10 91 X. s. d, L. s. d. Ex.45. 164 17 8^ Ex.46. 18149 14 0* 29 2 91 ]7'2\6 4j L. s, d. L s. d. Ex. 47. 417 4 10^ Ex. 48. 20412 13 9| 319 11 7\ 19911- 14 tl L, s. d, L, s. d. Ex. 49.425 18 9 Ex. 50. 22425 14 9* 139 10 9| 21018* 8 Hi L. s. d. L. s. d- EX.5U183 9 IJ Ex;52. 24463 13 HI 24 14 10* 177S2 iQ o» L. s. d. L. s. d. Ex. 63. 42116 91 Ex.* 54. 86476 6 9^ 326 19 56117 13 10 L. s. d. L. s. d, Ex. 55. 433 17 2| Ex. 56. 28446 17 9 311 19 41 19994 14 8| COMPOUND SUBTRACTION. 6^ L. s. d, L. s. d. ES. 57. 194 12 8i Ex. 58. 80490 9 9 117 12 9 2^089 15 \0l L s. d. . L. .s. d, Ex. 59. 474 19 4^ Ex. 60. 26475 13 9 3t>2 ]3 ll 247 10 18 Ml L; s. d. L. s. rf. Ex.61. 4559 16 91 Ex. 62. 3U87 15 11^ 3228 9 5i 31767 19 lO I. s. d. L. s. tZ. Ex. 63. 2139 7 10 Ex. 64. 56492 7 5^ 1914 13 10* L, si d. L. a d, Ex. 65. 3471 19 9i Ex. 66. 38410 14 9 293 19 9^ 28019 19 10^ if. Ex. Ex. Ex; 67. L. 4557 3945 s. 18 17 d. 68. 601273 462104 >. d, 11 7 1 5 8] • E3^ 69. L. 5:34. 559 .s. 11 12 d, 3 7 L. 70. 424156 379 1 2* i s. d. 11 6i 10 9^ • L. s. rf. L. s. d. Ex. ri. 7 860 Ex. 72. 44I.J91 (3 0* 32. I 4. 7 ' 38909 i 9 8^ 66 COMPOUND SUBTRACTION. L. s. d. Ex. 72>. 6234 6 6 309 12 lOi Ex. 74. 1414 9 91 7c9 12 111 L. 'S. d, L. s. d, *Ex.r5. 1173 14 9| Ex. 76.484760 10 9 437 18 11* 329189 19 9l Ex. Ex. Ex Ex. 84 Ex. Ex. 88 ' ^ L. s. d. Ex. 77. 791 5 11^ 261 19 11* L. s. d. 78. 14112 0| 4612 19 1 L. s. d. Ex. 79. 1345 19 9^ 34.5 17 91 L. s. d, 80. 4621 15 9l S94 1 J 0| L. s. d. . fix. 81.396 19 9\ 29 19 9^ • L. s. d. :. 82. 254 14 9^ 244 19 10| . ^ L. s. d. Ex. 83. 1214 5 880 8^ L s. d. . 564121 10 I0| 379178 16 lOi L. s. d, Ex. 85. 4465 10 9* 304 \\\ L s. d. 86 453 2 13 9l 4319 15 n| L. s. d. Ex. 87.408 19 4* 254 I 10^ L. s. d. .60985-14 4-1 1427 19 91 COMPOUND SUBTRACTION. 6t L. s. d. L. s. d. Borrowed SOO o Borrowed ] i 1000 ( \9 1.5 '577 16 7* Paid at \ S9 7 71 Paid at ^ 105 3 different J 76 8 1 different < ' 52 10 11 times j 4d 15 10 times i 2i6 9 9* (^105 . ( Paid -~ ..'JOG 9 Paid - 283 6 6| 881 18 4 Unpaid. 16 13 51 Unpaid - M8 1 8 Suppose a person is debtor And is creditor, by book- to sundry persons in the debts from different peo- following sums. ple, in the foilowiny; sums. L. s. d. L. s. cf. r678 14 9^ 764 14 9| 2* 17 4i 39 14 4 5 5 500 1054 12 91 8V9 5 95 26 5 2500 7 7 5503 5 n 95 19 9* 30(Kt ^J , 3'J 11 S Br. CK Dr. Balance in favo ur of Cr. Required the balance of Required the ha'anceof this account ? tills account ? Dr. . Cr. Dr. Cr. L. s. d, L. s. d. Z. s. d. L. s. rf. 764 <4 9 39r 14 111 769 19 \0l 49 12 11 397 10^ 2()7 Ij 9 643 4 4 1000 17 91 .2M> 19 9^ 72i) 13 8^ 24s 11 7 iro6 5 5 467 16 7^ 464 Id 591 8 4 4 4 O 371 14 9 215 12 6. < 19 6 250 12 8| 564 12 6| 345 Q \L)\ 3^0 i; 175? 17 &S COMPOUND SUBTRACTIOIS'. EXAMPLES OF WEIGHTS AND MEASURES, TROY WEIGHT. lb. oz.dwt. gr. Ih. oz.dwt.sr. lb. oz.dwt.gr Ex. 1. 187 9 12 20 2.256 6 22 3.567 4 169 6 14 17 199 9 3 20 379 i 1 9 9 lb. oz. (Iwt.i!;r. lb. oz dwt. gr. lb. oz. dwt.gr 4.254 5.673 3 9 6 423 5 15 14 25.1 11 19 20 567 9 17 16 246 '1 18 23 lb. oz dwt. oz.dwt.gr. lb. oz. dwt. oz.dwt.gr. 7.14 II 9 8. 74 12 18 9.175 3 10 10.17 10 20 11 10 14 71 14 17 159 11 14 14 n 33 AVOIRDUPOIS WEIGHT. tons; cwt. qr. lb. oz. dr. tons. cwt. qr. lb. oz. dr.^ 1.72 10 3 14 10 12 2. 64 15 2 15 JO 9 • 9 16 1 25 14 6 46 15 3 5 12 14 tons. cwt. qr. II). oz. dr. tons, cwt. qr. lb. (\z dr. 3. 25 4. 67 2 1 4 14 2 24 2 15 29 14 3 -2 14 tons, cwt. qr. lb. t)Z. dr; tons. cwt. qr. II;. oz. dr. 5. 36 7 1 1 1 1 6. 76 3 4 30 3 2 5 5 5 67 12 2 14 4 tons, cut. qr. cwt.qr. lb. qr. lb. oz. lb. oz. dr. 7. 14 12 2 8. 17 I 25 9. 143 22 12 10. 174 II 10 1 14 3 14 2 27 . 74 19 14 39 12 13 COMPOUND SUBTRACTION. Q9L APOTHECARIES WEIGHT. Ih. oz. dr. scr. lb., oi. dr. scr. lb. oz. dr. scr. I. 4-56 9 4 2. 269 8 3 2 3. 987 4-4 399 4 7. 2 178 11 3 1 379 10 5 I lb. oz. dr. scr. lb. oz. dr. scr. lb. oz.dr.scr. 4.564 5, 375 7 7 I 6. 394 2 2 469 3 3 2 369 4 7 2 299 11 7 2 lb. oz. dr. oz. dr. scr. dr. scr. gr. lb. oz.dr 7. 144 10 6 8. 27 4 1 9. 27 1 14 10. 74 10 5 64 11 7 14 7 2 14 19 65 1 1 6 CLOTH MEASURE. yds. qr. n. E.e. qr. n. yds. dr. n. yds. qr.n Bx. 1.218 2 2.46 3.567 I 1 4.459 12 167 1 3 23 2 2 469 2 399 3 3 yds. qr. n. E.e. qr.n. E.e. ^qr, n. E.e. qr. n. 5. 174 2 1 6. 174 3 1 ' 7. 171 1 3 8. J2 1 I 39 3 2 49 4 2 74 4 2 10 4 3 LONG MEASURE. yds. ft. in. b.c. yds. ft in. b.c. yds.ft.in.be. Ex. 1.456 2 10 1 2.669 3.267 1 1 1 379 1 11 2 599 1 I 1 199 2 2 2 lea. n^. fur. p. lea. m. fur. p. lea. m. fur. p. 4.470 1 4 19 4. 367 6. 225 1 11 279 2 7 23 179 2 5 23 167 2 4 4 * 70 COMPOUND SUBTRACTIOK. lea. m. fur. fur. p. yds. p. yd. ft. ft. in. b.c. 7.21 2 4 8.14 34 5. 9.14 3 1 10.17 11 2 3 2 6 12 39 5 9 4 2 14 11 1 LANF) MEASURE. ac. r. p. ac. r. p. ac. r. p. ac. r. p. Ex. 1.456 2 25 2.4.57 1 29 3.356 39 4.594 I 1 Syy 29 2b4 3 39 279 3 39 259 3 17 ac. r. p. ac. r. p. ac. r. p. ac. r. p. 5. 12 32 6. 112 I 31 7. 12 I 25 8. 19 1 20 1 3 14 74 2 37 10 3 39 14 2 21 WINE MEASURE. tuns lihd. gal. qt. pt. tuns, hhd. gal. qt. pt, Ex. 1. 456 2 24 1 2. 257 3 10 1 I 399 3 46 3 1 199 50 3 1 tuns, hhd. gai. qt. pt, tuns, hhd. gal. 3. 467 2 4. 27 2 54 299 3 32 2 I 19 3 62 hhd. gal. qt. hhd. gal. qt. gal. qt. pt. 5. 147 14 2 6. 14 1 2 7. 24 2 79 3 3 12 41 3 17 1 DRY MEASURE. bu. pks. gal. bu. pks. gal. bu. pks. gal 1. 86 3 i 2. 59 1 3. ()2 46 1 39 3 I 24 1 1 COMPOUND SUBTRACTION. 71 pks. gal. qts. pks. gal.qts. pks. gal. qts. pts. Ex. 4. 67 2 5. 28 I 6. 74 1 1 I 32 1 1 12 1 3 27 I 3 1 TIME. cL hr. mim. d. hr. min. sec. mo. w. d, hr. Ex. 1. 37 2 39 2.74 3 12 14 S. 46 I 1 4 29 21 49 " 47 21 54 36 • 29 3 6 Z\ w, d. hr. m. s. ysr. m. w. m. w. d. 4. 36 5. 17 10 2 6.147 2 3 35 6 23 50 59 14 12 3 19 2 4 d. hrs. 111. hrs. min. sec. 7. 167 21 50 8. 174 50 51 19 23 54 94 59 57 MISCELLANEOUS EXAMPLES IN SUBTRACTION. Ex, 1. I borrowed of a friend five hundred guineas, and hav^ paid at different tim.es, three hundred and ninety pounds six shillings and seven pence three far- things : what have I still to pay ? Answer, 134^. ISs. 44^. 2. A horse and his harness are worth 175 dol., but the harness is worth 47 I). 37^ cts. I demand the value of the horse ? Ans. 127 D. 623^ cts. 3. What sum added to 150 guineas, will make up 1 99^ 9.S. 9^ ? Ans. 41 /. I9s. 9ld. 4. At an eclipse of the sun, the moon is situated be- tween the earth and sun ; how far distant is the moon from the sun, supposing the distance between the earth and the sun 95 millions of miles, and that between the earth and moon 240 thousand ? Ans. 94760000 miles. 7*2 COMPOUND SUBTRACTIOK. 5. The great bell at Oxford weighs 7 tons, 1 1 cwt, 3qrs. 41b. ; that at St. Paul's 5 tons, 2 cwt. Iqr. 22lb.; and the great Tom of Lincoln weighs 4 tons. 16 cwt. 3 q.rs. 16 lb. : how much heavier than these together is the great bell at Moscow, .which is 198 tons ? A'iS> 180 tons, 8 c»vi. 3 qrs. 14- lb. 6. The Royal Exchange cost 80 thousand pounds in building; ; the Mansion-house 40 thousand ; Blackfnars- bridyje, 153 thousand; Westminster-bridge, 389 thou- sand ; and the Monument, 13 thousand! pounds; but the Cathedral of 8t. Paul's cost »00 thousand : how much did this cost more than all the resr ? Ans. 125000^. 7. If my income is 3G7/. 8.s A-jd. and my expenditure be 340 guineas : how much can 1 lay by ? Ans. 10/ 8s 4]i. 8. A person, by great losses, was oblij^ed to call his creditors together : lie found his whole property amount to .'527/. 12s. 8ld. ; but he owed to one man 150/. ; to another 300 guineas ; to a third 20 crowns; to a fourth 55/. 8s. 9ld. ; and to a fifth 200 guineas ; how much will they be losers ? Ans. 207/. 1 6s. Old. 9. A gentleman leaves between his two children 50,000 dollars ; to the younger he leaves 17478 dollars : what was the fortune of the elder ?• Ans. 32522 Dollars. 10. An apprentice has served of his term of seven years, three years, two months, three weeks, four days, seventeen hours ; how much longer has he to serve ? Ans. 3 yrs. 10 m. w. 2 da. 7 ho. 1 1. From a field of 6| acres, I take out two gardens, one measuring ia roods, and the other 24 roods, and a piece of ground for coach-house and stables, that mea- sures 1 rood and 12 perches : what will be the size of tht field after these pieces are taken away .? Ans. 4 ac. 1 r. 38 poles. 12. A plumber puts lead upon the diiTerent parts of my house that weighs 5 cwt. 3 qr. ; and he takes avuy, in return, old lead weighing 2 cwt. 24 lb. : what is the difterence in the vreight between the new an'i the old lead ? Ans. 3 cwt. 2 qrs. 4 lb. (") COMPOUND MULTIPLICATION Is the method of finding the amount of any given num- ber of different denominations, by repeating it any num- ber of times : I. When the given multiplier does not exceed 13. RuLK. Write the multiplier under the lowest denomi- nation of the multiplicand, multiply every number of the multiplicand by the multiplier, and bring the several products as they occur, to the next higher denomination. Write down the remainders, and carry the integers to the next product. Ex. Multiply Z..768 14s. 9\d, by 9. Ex. Dlls. cts. L. 768 s. 14 H 9 4 Dlls .cts 691B 13 i! ] Ex. 1. 79 14 Ex. 3. 67 37| X3 X4 Ex.2. Ex.4. 84 79 62| 25 X6 X5 Ea, . D. d. cts }. Ea.D. d. cts. mis. 5. 7 7 5 4X7 Ex. 6. 6 3 4 7 6X9 Ea. D. cts. Ea. D. cts. Ex. 7. 74 3 50 X 8 Ex. 8. 29 4 43^ X 12 ?4 COMPOUND MULTIPLICATION. L. s. d. L. s. d, Ex. 9. 3987 4 6| X 2 11, 2987 3 95 >^ 5 13. 3487 12 8 X 6 15. 5094 10 11^ X 8 17. •37(-4. 12 h\ X 10 19. 4610 15 4 X 12 21. 1456 16 10 X 12 23. 3420 13 5* X 10 25 2675 19 31 X 9 27. 467i) 17 8* X 1 1 II. VMien the multiplier is a composite number, and can he resolved into tuo or more component parts. Rule. JMultipIv bv its component parts successively, and the last product will be the answer. Ex. Multiply L.374 10s. ll^c?. by 63. L. s, d. 374 10 11^ X 63ZI9 X 7 9 10. 3564 10 n X 3 12. 2648 16 81 X 5 14.3498 2 6i X 7 16. 2691 18 lU X 9 18. 3465 15 10^ X 11 20. 359 1 19 9^ X 4 22.2761 14 4 X 6 24. 4694 12 7 X 8 26. 3476 17 81 X 5 28. 4900 92 X 7 3370 1 8 9l 7 Ans. 23596 ] [1 8* EXAMPLES. L s. d. L. s. d. Ex. 1. 456 12 % X 15 Ex. 2. 436 14 31 X 16 3. 784 15 4 X 18 4. 397 16 10 X 21 5. 674 18 lOl X 22 6. 487 16 9^ X 24 7. 245 10 3 X 30 8. 376 15 11 X 30 9. 246 19 91 X 35 10 489 18 81 X 42 11. 397 13 3 X 48 12- 369 10 2 X 54 13. 384 15 10* X 56 14. 565 i2 91 X 63 15 592 12 9 X 66 16. 80U 9 8 X 72 17. 911 13 21 X 84 18 914 16 4 X 77 19. 397 4 41 X 96 20. S74 12 51 X 108 21 459 9 91 X 100 22. 279 13 3 X 120 23 S76 15 4 X 121 24 347 3 9 X 132 25. 376 4 91 X 144 26. 5b7 14 7 X 45 27. 897 16 X 108 28. 675 13 31 X 88 29. 487 19 111 X 121 30. hb<^ 12 2 X 132 COMPOUND MULTIPLICATIOST. 75 III. When the multiplier is not a composite number. Rule. Take the composite number which is nearest to It, and multiply by the component parts, as before : then add or subtract as many times the first line, as the composite number is less or greater than the given mul- tiplier. (1) Multiply L.324 12*-. 6^ by 394. L. s. d 3^4 12 6* X 394 IZ 8 X 7 X 7 + 2. 8 The nearest composite number is 392 =8X7X7; I accordingly multiply by these three figures, and to the product 1 add twice the ori- ginal sum, which gives the true answer. EXAMPLES. 2597 18179 2 649 5 127903 1 5 L s. d. L. s. d. Ex. 1.574 12 6* X 38 Ex. 2. 387 18 71 X 46 3. 325 8 4 X 58 4.' 222 12 8^ X 68 5. 2 d 18 91 X 78 6 136 14 5 X 94 7. 300 3* X 273 8. 246 12 0^ X 359 9. S'-IS 16 Ol X 412 10. 326 18 3 X 687 11.239 9 9 X 740 12. 560 ^21 X 388 13. 660 15 4,1 X 1004 14. 407 13 1 X 1325 15.700 01 X 1450 16.110 10 11 X 1208 EXAMPLES OF IVEIGHfS AND MEASURES, TROY WEIGHT. lb. oz. dwt. gr. lb. oz. dwt.gr. Ex. 1. 187 9 12 20 X 4 Ex. 2. 256 6 252 X 5 3. 169 6 14 17 X 6 4. J79 1 1 9 y X 7 5. 254 3 3 3X9 6. 253 11 4 20 X 8 7. 675 4 15 10 X 11 8. 375 17 X 12 76 COMPOUND MULTIPLICATrON. AVOIRDUPOIS WEIGHT. tons.cwt.qr.lb. oz. dr. tons,cwt.qr. lb. oz.dr. i5xJ.l2 10 3 14 10 12X2 Ex. 2, 6i 13 2 15 6 8X4 3.25 2 8 4 4X3 4.46 15 5 12 4 4X6 5.75 13 18 6 10X8 6. 39 12 2 16 10 8X9 APOTHECARIES' WEIGHT. lb. oz. dr. sc. lb. oz. dr. sc. Ex. 1. 456 3 4 1 X 5 Ex. 2. 748 5 2 2 X 8 3. 534 7 6 2 X 12 4. 378 10 1 Xll 5. 321 5 4 IX 10 6. 491 5 7 2x9 CI.OTH MEASURE. yds. qr. nl. E.e. qr. nl. yds. qr. nl. Ex. 1. 210 2 1 X 4 2. 378 4 3 X 7 3. 596 3 1 X 12 4. 357 1 3 X 6 5. 738 3 2 X 9 6. 876 3 X 10 LONG MEASURE. yds, ft. in. be. lea. Ex. 1.556 2 10 1X5 Ex. 2. 379 3. 369 1 9 2X8 4. 376 5. 241 2 11 1 X 10 6.674 LAND MEASURE. m. 1 2 2 fur. p. 6 20 X 7 5 37 X 9 7 18X 6 r. 3 2 P- 12 X 12 25 X 10 12 X 8 ac. r. p. ac. Ex. 1. 456 O 25 X 11 Ex. 2. 597 3. 371 2 18 X 4 4. 271 5. 189 3 32 X 8 6. 430 LIQUID MEASURE. tuns,hhd. gal. qts. p. tuns, hhd.gal.qts. Ex. 1.456 3 28 2 1x4 2.456 3 46 2x6 3.374 2 60 3 1X8 4.350 2 25 1X2 5. 221 1 4 1 0X5 6.124 3 50 3X10 DRY MEASURE. bu. pks. gal. bu. pks. gal. Ex. 1.29 2 1X3 Ex. 2. 29 3 1 X 5 3. 76 3 X 4 4. 27 2 1 X 6 pks. gal. qts. pts. bu. pks.gal.qts. 5. r.4 I 3 1 X 7 6. 64 2 1 2 X 8 7. 76 1 2 X 9 8-37 1 13x11 9. 6^ 3 1 X 12 10. 64 3 1 2 X 12 COMPOUND MULTIPLICATION 77 TIME. w. d. hrs-m. s. yrs. mo. w.d.' Ex. 1. 73 6 10 40 30 X 5 Ex. S. 594 12 3 4 X 7 3. 36 4 1-2 15 20 X 9 4. 364 8 2 6 X 8 5. 98 5 17 13 53 X iZ 6. 443 lO 3 3 X 11 MISCELLANEOUS EXAMPLES. Ex. 1. What cost 12 lb. of tea at 1 dol- 50 cts. per lb. ? Ansv\'er. IS.'dolls. 2. What cost 16^ lb. of suj^ar, at Is, \ld. per. lb- ? Ans. 1 «.s. nld. 3. What is the value of 24 yards of Irish linen, at 3s. 6^. per yard ? Ans, 4/. 5s. 4. What will 79 bibles come to, at Idol. 12^ cts. each? Ans. 88 dol. 87^ cts. 5. What is the value of 85 gallons of brandy, at l9s. Old. per gallon ? Ans. 84^. 2s. 3ld. 6. What is the weight of 28 ingots of gold, each weigh- ing 6 lb. 7 oz. 15 dwts. 20 gr. s' Ans. I 86 lb. 2 oz. 3 dwt. 8 gr. 7. What will 157 oxen cost at 15/. 5s. 9^. each ? Ans. 2400/. 2.S-. 9d. 8. What is the value of 576 sheep, at 1/. 6s. sU. each ? ' Ans. 756/ Oa.Od, 9. How much must I pay for 759 chaldrons of coals, at 58s. 6rf. per chaldron ? Ans. 2220/. Is. 6d. 10. What is the value of 199 firkins of ale, at I2s 6d. per firkin :' Ans. 124/. 7s. 6d. 11. What is the value of 245 yards of broad cloth, at 19s. Id. per yard ? Ans. 239/ 17.^. l\d. 12. What is the worth of a stack of hay, containing 75 loads, at 3/. 19s. 9d. per load ? Ans. 299/. Is. 3 J. 13. What is the worth of 12^ lb. of cofiee, at 25 cts. per lb ? Ans. 3 dol. l-ig cts. 14. How many pounds sterling are there in 28 purses, each containing 15 guineas, 15 half-guineas, 15 seven- shilling pieces, and tliree crowns ? Ans. 829/. 10s. Od, 15. What is the weight of 1 000 guineas, each guinea weighing 5 dwts. 9| gr. ? Ans. 22 lb. 3 oz. 15 dwt. 20 gr. 7* 78 COMPOUND MULTIPLICATION, 16. I bought at a sale 47a dozen of port wine, at 2l. 5s, 6d. per dozen, how much money must 1 send to pay for it? Ans. 108/. Is. 3d. 17. What is the value of 83 tons of iron at 18^. 17s. ^^. per ton ? Ans. 1605/. 12s. 3^. 18. What do 79 packages of good* weigh, supposing that each package weighs 3 cwt, 3 qrs. 15 lb. ? Ans. 15 tons. 6 cwt. 3 qr. 9 lb. 19. If one ounce of gold cost 3/. 16s. 8(/ , what is the value of 43C4 ounces ? Ans. 1673^. 5h. Od. 20. What shall I pay annually for 459 acres of land, at 2 dol 37^ cts. per acre ? Ans. 1090 dol. 12J cts. 21. What is the price of 185 gallons of rum, at l3s. &ld per gal. } Ans. 125/. 5s. 2*. 22. If a man spend 1 dol. 62a cts. per day, how much does he expend in a year ? Ans. 593 dol. I2a cts. 23. How much federal money in 49/. sterling, allow- ing 4 dol. 44 cts. to a pound sterling ? Ans. 217 dol. 56 cts. BILLS OF PARCELS. A mercer's bill. L s. d. L. s, 12 yards of silk, at - 15 ? peryard| 114 Do. of flowered silk at 18 7^ of velvet, at - 1 2 4 of satin, at - 13 9 of brocade, at - 15 7 of lustring, at - 6 3 16 Do. 12 Do. 27 Do. 14 Do. I I A stationer's bill. L 250 Reams of paper, at - 1 112 D». do. at - 2 34 Do of imperial brown at 1 500 Dutch quills, at - 2500 Do. common, at - s.d L. s. 2 6 per ream 4 6 15 3 9 per hun. 2 3 d. OOMPOUND MULTIPLIOATION. 79 A carpenter's bill. 65 cubick feet of oak, at 125 Do. wrou;^lit and framed, at 176 Do. fir frained aad mould- ed, at - - - 15 square shed roofing, at 8 Do. hip aad valley roof- 70 feet water trunk, at - € 3C>4 feet ovolo wainscot sashes, at - - - 124 Do. do. mahogany, at 1 10 men's labour, for 25 days, at 4 d. 3 per foot 8 L. s. 6 6 per square 3 lo per foot 9 - 4 8 per day 1 A BRICKLAYER S BILL. 39 rod of grey -stock brick- work, at 7 Do. in party wall, at 105 feet of J 8 inch drain, at 1050 Do. of pointing old work at - - - 1500 grey stocks, at 125 pan-tiles, at 45 hods of mortar, at 13 Do. of tarras, at - - 15 bricklayers, 25 days, at 12 labourers, ditto, at - 66 load of rubbish carted away, at L. s. d. L. s. d. 13 per rod 15 3 per foot (\5l - 4 6 per bun. 1^ each 7 4 2 4 6 per day 3 2 6prload. ao COMPOUND MULTIPLICATION. A slater's bill. L. s. d. 9 square of Westmore- land slatinu;, at - 2 19 6 per square 7 do. of Welsh ladies, at 1 IT 4 - 5 Do. of Welsh coun- tess, at 1 18 3 35 Do. of ripped and rub- L. s. d. bish cleared, at 2 7 12 slaters 7 days, at 4 5 per day tJ labourers, do. at 2 9 5050 clout nails, at 4 per hun. painter's adn glazier's bill J *• 1035 yards of painting 3 times u. in oil, at 7^ per yard 56.5 Do. do. and sand, at 1 3 3(5 sash frames, at 11 each 432 sash squares, at 8a per doz. 12()5 feet of best Newcastle j^lass, at 1 71 per foot 356 Do. Iar«;e size, at 2 1.^ iOOO Do. in lead work, at 1 05 L. s. d. COMPOUND DIVISION, Is the method of finding how often one ^iven number is contained in another of different denominations; or, to divide a given compound number into any proposed number of equal parts. I. When the given divisor does not exceed 12. Rule. Place the divisor to tlie left-hand of the divi- dend. Divide the highest denomination of the dividend by the divisor, and write down the quotient ; reduce the remainder, if any, into the next lowerdenomination, adding to it the aumber which stands in that place oC COMPOUND DIVISION. 6l the dividend, and divide as before, and so proceed to the end. Ex. 1695?. 14s. 4ld. H- 8. L. s. d. 8)1695 14 4^ 211 19 3^—2 • 8 Proof 1695 14 ^l EXAMPLES. D. cts. D. cts; Ex. 1. 74 50 -T- S Ex. 2, 62 25 -i- 4 3. 56 371 -^ 5 4. 79 18i -T- 6 5. 49 49 ^7 6. 63 25 -T- 8 Ea. I), cts. Ea. D.cts. 7. 43 7 37^ -7- 9 8. 56 6 25 -r- 10 9. 17 4 50 -^ n 10. 13 7 75 H- 12 L. s, d. L. s. rf. Ex.tl. 457 8 9^ ^ 3 Ex . 12.579 18 4*H- 2 13. 3;^6 18 7i -=- 4 ' 14. 768 2 6^ -i- 5 15. 474 12 10 -r- 6 I6. 93t 14 5 H- 7 17. 897 16 4 -j- 8 18. 2.56 17 10^ -r- 10 19. 759 0-^9 20 694 19 6-7-12 21. 101 15 9l~- 1( 22.496 -i- 12 23. 900 0-^8 24. 500 5 5 -f- 4 25. 800 10 2 -i- 7 26. 270 17 7| -T- 6 2r. 464 3 91 -V- 6 28. 901 1 1 -r- 9 II. When the divisor is a composite numher. Rule. Divide bj the component parts of the divisor successively, and the last quotient will be the answer* Ex. Xl48 8s. 8^. -7- 27 = 3 -^ 9. L. s. d. 3)148 8 8^ 9H9 9 6i— 61-n 111—6 J 5 9 '^ The answer is 5/. 9s. 1 1]^' ^^. 82 COMPOUND DIVISION. ;x. L. s. d. L. s. (i. 1 167 12 Gl^ _ 14 E5 :. 2. 769 9 81- - 20 3. 33^ 15 8^ - 15 4. 594 7 6 - - 25 5. 4.86 9 9 - - 16 6. 333 10 lOi- - 28 T.^67 ol- - 18 8. 498 9 9^- - 32 9. 4:<9 5 61 -i - 24 10. 596 12 ri- - 36 11. 37^ 18 7 - - 27 12. 465 11 11 - - 44 13 487 9 9^- •- 30 14. 564 13 5^ - 49 15. 596 4 6 - - 33 16. 678 6 5 - 7- 54 17.834 3 6^H - 42 18. 999 9 8 - r- 56 19. 32: U 4 H - 48 20. 564 4 6 - - 63 21 387 12 11 H - 72 22. 248 3 - - 84 23. 565 II 8 - - 88 24.' 505 5 51- - 99 25. 674 18 8H - 108 26. 564 2 2 - - i20 27.465 3 3 ^ - 132 28. 888 8 8 - - 144 When there are three component parts. Ex. L. 1350 10s. llrf .-r- 240 =5 X6 X 8 , L. s. d. 5)1350 10 u 6)270 2 2- — 4 r 8) 45 4^ ~n.- i Air Ex. 1.L.5527 10s. 6^.-J-243. 2. 18568/. 12s. 1^^-4-1296. III. When the divisor is greater than 12, and not a composite number ? Rule. The several quotients must be found by the method of Lon^ Division, (see pp. 28 and 29), reducing the remainders to the next lower denomination, and tak- ing in tho!^e numbers of the dividend which are of the same denomination. COMPOUND DIVISION. 83 Ex. Divide L.l 350 lOs. lie?, bv 242. L. s. d. 242) J 350 10 11(5 1210 140 20 242)2810(1 2662 148 12 242)1787(7- 16'J4 93 4 242)37:^(1 242 D. cts. Ex. I. 234 50 -4- 3. 427 C)2l — Ea. I), cts, 5. 17 3 8 ^ 7. r23 4 J5 L. s. d. ISO 17 37 9. 985 18 11 405 16 4^-7- 13. 565 13 3- 15. 800 8 17. 987 14 19. 598 12 21. 483 6 23. 98ci 5 25. U«5 19 27. 2690 12 9 -7- 8^ 4 6 6 2 3 ■59 - 74 19 29 37 41 46 67 73 89 107 166 ^ D. cts. Ex. 2.* 627 25 -J- 26 4. 317 75 ^ 43 Ea. D. ct-. 6. 127 7 12^ 68 8. 319 4 50 -T- 89 /.. s. d, 10. IO(M 12 111 12. 2468 13 Zl 14. .5746 9 6 -T- 16. 6321 3 3 -r- 18. 4'^68 12 8 20. 4«21 9 71 22. 5943 16 6 24. 3618 4 6 — 26. 4683 15 5^, -^ 376 28. 5649 9 9 23 39 59 61 69 87 97 97 439 84 gOMPOUND DIVISION. L. s, d. L. s. d. 29. 6259 11 6 H- 215 30. 3604 10 -T- 509 Si. P654 7 71-7- 649 32. 6534 16 3^ -^ 606 S3. 5942 17 ^\~ 757 34. 4.593 12 4 -^ 1585 35. 4628 .5 9-7- 1001 36. 5349 0-7-4786 37. 1456 16 7 -T- 3761 38. 9504 1 1* -=- 8078 IV, When the divisor consists of a numher not exceed- ing 12, with one or more cyphers. Rule. Cut off, by a line, as many places in the pounds as there are cyphers in the divisor, and divide by short division ; then i educe the remainder to the next lower denomination, as in the last rule. Ex. Divide L 5645 14s. 4d. by 1200. 12.00-56 45 14 4 L. 4 — 845 20 12.00; 169. 14 s. 14— 114 12 12.00J13.72 d. 1—172 EXAMPLES OF WEIGHTS AND MEASURES. TROY WEIGHT. lb. oz.dwt;. gr. lb. oz.dwt.gr. Ex. 1. 287 9 12 20 -T- 4 Ex. 2. 356 6 22 -^ 5 3.269 6 14 7 -=- 6 4. 379 11 9 -r- 7 5. 854 3 3 3 -r- 9 6 3j5 11 4 20 -^ 8 7. 675 4 15 10 -f- II 8. 775 17 -i- 12 AVOIRDUPOIb Wti:iGHT. tons, cwt. qr. Ih. oz. dr. ton*, cvvt.qr.lb. oz. dr. 1, 412 10 3 14 10 12 -^ 2 2. 664 13 1 12 6 8 -r- 4 3. 526 18 6 6 -~ 3 4 464 3 27 S -~ Q 5.678 2 2 2 8 2 ~- 8 6.591 5 4-3 12 -i- 9 COMPOUND DIVISION. 83 APOTHECARIES WEIGHT. lb. oz. dr. scr. lb. oz. dr. ser. Er. 1. 591 8 4 1 -~ 5Ex. 2. 748 5 7 -=- 8 3. 639 1 1 2 H- 12 4. 392 10 6 ~ II 5. 487 2 -7- 10 6. 421 4 5 1-7-9 CLOTH MEASURE. yds. qr. n. E.e. qr. n. Ex. 1. 5210 2 1-5-4 Ex. 2. 596* 3 I -r- 1 1 3. 3976 1 2 -i- 6 4. 7645 4 2-^12 5. 4721 -r- 8 6. 3492 3 -r- 9 LONG MEASURE. yds. ft. in. b.c. lea. m. fur. p. Ex. 1. 5946 2 10 1 -T- 5 Ex. 2. 3795 2 7 30-7-7 3.4736 1 8 2 H- 8 4-4965 1 3 18 -f- 5.2005 11 2 H- 10 6.6743 2 6 4 -i- 6 LAND MEASURE. ac. r. p. Ex. 1. 654 2 24-7-11 3. 371 18 -H 4 5. 891 3 32-7- 8 ac. r. p. Ex.2. 958 3 12-7-12 4. 379 25-7-10 6. 496 1 1-^8 Ex. 1. 2. 3. 4. 5. 6. LIQUID MEASURE. tuns, hbd. gai. qts. pt. 456 3 27 2 1 -t- 656 3 31 2 -r- 594 30 3 -j- 391 2 25 1 0-7- 271 O 2 0-7- 421 3 50 3 -T- 10 DRY MEASURE. bu. pks. gjal. Ex. 2. 87 3 1-7-5 bu. pks.gal. Ex. 1. 16 2 1-7-3 pks. gal. qts. pts. pks. gal qts. pts. Ex. 3. 327 1 3 -i- 7 Ex. 4. 219 ^0 2 -r- 9 5. 12.) 2 1-7-11 6. 99 1 3 0-7-12 8 86 COMPOUND DIVISION. TIME. w. d. hrs.m. sec. yrs. mo. w.d. Ex. 1. 779 6 20 40 25 -4- 5 Ex. 2. 594 12 2 4 -f- 7 3. S91 4 12 16 12 -i- 9 4.954 6 3 5-4-6 5.913 4 5-4-J2 6.348 10 3 3-^11 MISCELLANEOUS EXAMPLES. Ex. 1. If 17 yards of doth cost 19/. 3s. 9rf., what is it per yard ? Answer. I/. 2s, tiliL-^-^. 2. V\ hat is the price of one pound of sugar, if 8Jb. cost nine shillings ? Ans. Is. lid. 3. I he expenses of a journey amounting to 97/.'9s. 6rf. are to be defrayed by six persons : how much will each have topay ? Ans, 16/. 4*-. lid. 4. I have bought 12 gallons of wine for 32 dollars 50 cts. ; how much is that per gallon ? Ans. 2 dolls. 70 cts 5. Twelve boys are to have a guinea and a half divi- ded among them : what will be each boy's share ? Ans. 2s. 7ld. 6. A hundred and twenty -five sailors have taken 8465/. prize money : how much will each man be entitled to ? Ans. 67/. U>.4,ld, ^. 7. I have bought 144 pair of stockings for 27/. ; at what rate can 1 sell them so as to gain by each pair one shil- ling .►* Ans. 4s. yd, 8. What did I pay a piece for sheep, having bought 75 for 135/. r 9. Cheese at 3/. 12s. dd, per cwt. : per 1 .'. ? 10. If 81 oxen cost 1781/. 12s. 6d, : of one } An.%. 2 1 11. Ifapipeof wine cost 95/. : how dozen, which contains three gallons ? Ans. 12. Bought 50 dozen of wine for a hundretl guineas : how much is thai per bottle ? Ans. 3s. (jd, 13. Divide a tliousand guineas bitweei. 5i3 |jeoj)h', and see how much it is tor each ."^ Ans. 4d/. 13s. O^d, 4* Ans. 1/. 16s. how much is that Ans. 7ld. 8 iia» w hat is , tht value /. 19s. Wld. I / much is that a . 2L 5s, 2rf. 108 136' MISCELLANEOUS QUESTIONS. 87 14. If 12 pieces of linen cloth contain 250 yards, what is the length of a single piece ? Ans. 20 yds. 3 qr. 1*9 nail. 15. How much can I afford to spend a day, a week, and a month, if my income Ire 5(iO/. per annum, allow- ing 52 weeks, or 13 months to a year ? Ans. 1/. 7.S-. 4>ld. per day. 9/. 12s. 3^. per week. • S8/. 9s. 2ld. per month. 16. If 12 tea-spoons weigh 9 oz. 17 dwt. 12 gr. ; what is the weight of each spoon ? Ans. 16 dwts. 1 1 gr. MISCELLANEOUS qUESTIONS. Ex. 1 . It is said that Syrius, or the Dog Star, is the nearest of all the fixed stars, and that its distance is com- puted at 2,200.000,000,000 miles ; how many years, (each containing 365 days, 6 hours exactly,) would a cannon ball be in passing from the earth to 8irius, supposing it travelled at the rate of 480 miles per hour ? Ans. 522853S. Ex. 2. The Planet Mercury is about thirty -seven mil- lions of miles from the Sun ; Venus sixty-eight millions ; the Earth ninety-five millions ; Mars a hundred and forty five millions ; Jupiter four hundred and ninety- three millions ; Saturn nine hundred and eight, and the Herschel one thousand eight hundred millions of miles from the Sun : put these several distances down in figures, and add them together as a sum in Addition. Ans. 3546.000.000 Ex. 3. How much nearer the Sun, is Mercury than Mars; and how much farther is the Herschel than the Earth ? See Ex. 2. Ans. Mercury 108 millions nearer the sun than Mars, and Herschel ^705 millions further from the Sun than the Earth. Ex. 4. The beautiful planet Venus travels, in her an- nual journey round the Sun, at the rate of 75,000 miles in an hour : how many miles does sh« travel in one of her years, or in 22bi days ? Ans. 410.850.000 Ex. 5. The Earth travels, in her annual course, at the rate of 68.400 miles in an hour ; how many mUes there- fore do we move in a second ? Ans. ly S8 MISCELLANEOUS QUESTIONS. Ex. 6. There are in the Old Testamftnt 89 hooks, and 929 chapters, and in the New there are 27 books, and 260 chapters: how many books and chapters are there in the Bible ? Ans. 66 books, and 1 189 chapters. Ex. 7. There are 23214 verses in the Old Testament, and 7959 in the New i how much therefore do the verses in the former exceed those in the latter ? Ans. 15255 Ex. 8, There are 592439 words in the Old Testament, and 181253 in the New ; how many words are there in the Bible ? Ans. 773692 Ex. 9. In the Old Testament there are 2,728,100 let- ters, and in the New there are 838.380 : what are the sum and difference of these two numbers ? Ans. 3.566.480 sum, 1.889.720 difference. Ex. 10. There are in the Bible 3.566.480 letters : how }ong would a person be in counting tJiem, supposing he could count 200 in a minute } Ans. 297 hrs. 12 minutes. Ex. 11. A printer charges 54^. for every J 000 letters that he sets up : how many thousand must he set up to earn IL 15s. per week ? Ans. 80,000 Ex. 12. If a printer set up 8500 per day, how long would he be in composing the Old Testament, and how long in composing the whole Bible ? See Ex. 9 and 10. Ans. 321 da^s Old Test, and 419^ Bible, nearly. Ex. 13. If a printer be desired to set up the Bible in Latin, how much would he earn in the business, at the rate o^ 5ldr per lOOO letters, supposing there are as ma- ny letters in the Latin as there are in the English ? Ans. 85/. 8s. Wld. Ex. 14. If there be as many letters in the Greek Tes- tament as there are in the English, how much would a printer earn in setting it up at Sid. per thousand ? Ans. SOL 1 1 s. 3ld. ^o* Ex. 15. The name of Jehovah occurs 6855 times in the Old Testament : what proportion therefore does this word bear to all the other words in that book ? Ans. 862 nearly. Ex. 16. The word and occurs in the Bible 46227 times : what proportion does that bear to the other words? See Answer to Ex. 8. Ans. 17 nearly. MISCELLANEOUS QUESTIONS. 89 Ex, 17. There. are in tlie northern side of London 126 houses newly built, and unlet, the average rent of which is 85^. ; and 75 houses at 50L each, and 68 at 30 guineas each : what is the total annual loss of these empty houses to the proprietors ? Ans. 16602/. Ex. 18. There are 1100 hackney coaches in London, each of which earns on an average 18s. per day : how much is expended weekly, daily, and annually, on these vehicles, Sundays excepted ? Ans. 990^ per day. 5940/» per week. 308880/. per annum. Ex. 19. What are 256 reams of paper worth, at 33s. 6d. per ream ? Ans. 428/. 16s. Ex. 20. Fifty thousand larks have been sold in a sin- gle season in London : what did they fetch, supposing they were bought at lid. each ? Ans. 26oZ. 8s. 4>d. Ex. 21. The circumference of the Earth, in the lati- tude of London, is 15,120 miles, which is the space we pass over in 24 hours, by the diurnal motion of the earth : how much space do we pass over in a minute ? Ans, lOa miles. Ex. 22. Three thousand ounces of gold are imported into England annually : how many pounds and grains are imported in 50 years, at this rate, and what is the value of it at 3/. 18s. per ounce ? Ans. 12.500 pounds, 72.000,000 grains, and 585,000/. value. Ex. 23. To work the silver mine^ in South America, 40,000 negroes are imported annually : how many of these poor creatures have perished in this work during the last century ? Ans. 4.000,000 Ex. 24. The duty on hops amounted, at lid. per lb. in a certain year, to 26,357/. 9s. 9c/. : how many hops were grown that season } Ans. 1882 tons, 13 cwt. 2 qrs. 6 lb. Ex. 25. The battering ram employed by Titus to de- molish the walls of Jerusalem, weighed 100,000 lbs. : how many tons did it contain ? Ans. 44 tons, 12cwt. 3 qrs. 12. lb. Ex. 26. The copper mines in the island of Anglesey produce 1500 tons annually, and those in Cornwall 4000 tons : what is the value of the whole at 9ld. per lb. ? Ans. 487,666/. 13s. 4rf. 8* 90 MISCELLANEOUS QUESTIONS. Ex. 27. Mr. Bolton coined 40,000,000 penny pieces, each weighing an ounce : how many pounds of copper were used for them : how much was the value of these in pounds sterling ; and what was gained by this coinage, supposing the copper and expense of coining to be esti- mated at li2ld. per pound ? Ans. 2.500.000 lbs. 130,208/. ds. Sd. 36,458Z. 6s. 8.d Ex. 28. In the year 1794, 43,259,746 yards of Irish linen were exported from Ireland : how many packages did they make, each package containing 20 pieces, and each piece 26^ yards ? How many shirts would this linen make, at the rate of 31 yards per shirt ? Ans. 81.622 pack. 86 yds. 11.535.932i*6. Ex. 29. The circumference of the earth is estimated at 24,912 miles: how many barley-corns, (three of which make an inch,) would fill up this space ? Ans. 4.735.272.^60 Ex. 30. The territory of the United States of America contains a million of square miles, or 640 millions of square acres : of these, about 56 millions are water : what number of acres, roods, and perches of land, do the United States contain, and how many inhabitants will they support, allowing to each 4a acres ? Ans. 129,777.777. Ex. 31. There are now in England, Scotland, and AVales, 23 millious of acres of waste land : how many farms might these be divided into, allowing to each 75 acres : — and allowing 5 persons to each farm, how many souls would these waste acres support ? Ans. 306.606 farms 50 acres. 15.333.333 inha. Ex. 32. Between the 5th of July, 1810, and the same day, 1811, there were brewed, by 12 brewers only, 939, 900 barrels of porter : how much would this quantity sell for when retailed out at 5d. per qt. allowing 36 gals, to the barrel ? Ans. 2.819.700/. Ex. 33. How many hours, minutes, and seconds have elapsed since the birth of Christ, which is 1808 years, supposing 365^ days in a year ? Ans. 15.848.928 ho. 950,935,680 rain. 57,056,140,800 sec. MISCELLANEOUS QUESTIONS. 91 Ex. Si. It is said the Small-pox carries off in London, by death, 50 persons in a week : how many (if the dis- ease is not checked) will it destroy in ten years ? Ans.26.000 Ex. 35. There are about 10,540 tons of cheese import- ed into London annually : how much do they sell for at the average price of Tgfl^. per lb. .? Ans. 7.^7.800/. Ex. 39. It is computed that there are 50,000 tons of butter annually consumed in London : what is the ex- pense, supposing the average price lOld. per lb. '.^ Ans. 5.016.656^. 13s. 4-d, Ex. 37. About 120,000 persons are employed in the cotton trade ; if of these one-fourth are men, who earn 3s. 6d. a day, and one-fourth women, who earn Is. \d, a day, and the rest children, who earn, each, 3s. per week, how much is earned by manual l^our in the cot- ton manufacture every year ? Ans. 2,613,000 Ex. 38. There have been 20,000,000 lbs. of tea im- ported in a single year from China ; what was the value of it, supposing the average price 4s. 9t/. per lb. Ans. 4.750,000/. Ex. 39. The consumption of tobacco in this country is about 169,000 cwt. ; how much is expended on this arti- cle at \ld. per oz. ? Ans. 1.577,333/. 6s. 8d. Ex. 41. The consumption of milk is not less than- 6,980,000 gallons annually in London ; how much is ex» pended on this article at Sets, per pint ? Ans. 1,675,200 dollars. Ex. 42. The iron rails round St Paul's cost 11,202/. Os. 6c?., and they weighed 200 tons and 81 lbs. ; what was the iron charged per lb. .^ Ans. 6d. per lb. Ex. 43. Westminster-bridge cost 389,500/. in build- ing ; how soon would it have been paid for by foot pas- sengers, at a halfpenny each, supposing 2420 went over each day ? Ans. 21 1 years, 241 ^ days. 92 REDUCTION. Reduction is the method of converting numbers from one nauie, or denomination, to another of the same va- lue; and it is divided into Reduction descending^ and Reduction ascending. When numbers of a higher denomination are to be brought to a lower, it is called Reduction descending^ and it is pei formed bv Multiplication. When numbers of a lower denomination are to be brought to a higher denomination, it is called Reduction ascending, and is performed by Division. REDUCTION DESCENDING, OR CONVERTrXG GREAT INTO SMALL. Rule. Multiply the given number by as many of the lower denomination as make one of the higher. Thus, in reducing 55l. into shillings, I multiply the 55 by 20, and the answer is 1 100 shillings ; in both ca- ses the value is the same, that is, 551. is equal to 11 00 shillinss. REDUCTION ASCENDING, OR CONVERTING SMALL INTO GREAT, Rule. Divide by as many of the lower denomination as make one of the next higher. Thus, in bringing 890 pence into shillings, I divide the number by 12, and the answer is 74 shillings and two pence over. heduction, 53 EXAMPLES. L, s, d. Ex. 1. Reduce 29 6 8^ into farthings. 20 586 shillings 12 7040 pence 4 Answer 28 1 Q3 farthings. Ix. 2. In 28163 farthings how many pounds sterling ? 4)28163 12)7040--| 2,0)58,6--8c?. Ans.L.29 6 8| Ex. 3. Reduce 37 Dimes to mills. Ans. 3700 Mills. 4. Reduce 53 dollars to cents. Ans. 5300 cents. 5* Reduce 163 eagles to dollars. Ans. 1630 dollars. 6. Reduce 74 dollars to dimes. ^Ans. 740 dimes. 7. Reduce 217 dollars to mills. Ans. 217000 mills. 8. Reduce 35 eaj2;les to mills. Ans. 350000 9. Reduce 28 shillings to pence. Ans. 336 Pence. JO. Bring56 pounds into shillings. Ans. 1120 shills, 11. Reduce 672 pence into farthings. Ans. 2688 farthings. 12. How many pence are there in 105^ ? Ans. 25200 pence. 13. In 1000 guineas how many shillings ? Ans. 210OO shillings. 14. In 4704^how many pence ? Ans 1 1 23960 Pence. 15. In 3995^. how many farthings ? Ans. 3835200 Far. 16. In 7968 guineas, how many farthings ? Answer, 8031744 farthings. 17. How many farthings are there in 75 guineas } Ans. 75600 farthings. 94 TROY WEIGHT. 18. Reduce 576Z. into farthings. Ans. 552960 far, 19. In 99/. hoM n)an)' shillings, pence, and farthings ? Ans. 1980 shilings, 23760 pence, and 95040 farthings. 20. Reduce 567/ &s. 9^. into farthings. Answer. 544790 farthings. 21 How many halfpence are there in 157/. 7s, 7^. An,«. 75543 halfpence. 22. fn 1084890 pence, how many pounds ? Answer. ^b2(jt. Is. bcL 23. in 84^0896 pence, how many guineas ? Answer. 33376 guineas l2>. 24^. in 4808764 farthings, how many pounds ? Ans. 5009/. 2s. 7rf. 25. How many seven-shilling pieces are there in a thousand guineas ^ Ans. 3000. 26. How many groats are there in a hundred guineas ? Ans. 6300 groats. 27. Bring 3 110456 pence into groats. Ans. 777614 groats. 28. How many crown-pieces are there in 79/. 15s. ? Ans. 319 crowns. 29. How many half-crowns are there in 85/. \2s\ 6d, ? Ans. 685 half-crowns. 30. In 769 guineas, how many sixpences I Answer. 32298 sixpences ? TROY, OR, GOLD SMITHS' WEIGHT, lb. oz. dwt. gr. Ex. 1. Reduce 3 9 6 18 to grains, 12 45 20 906 24 I 3632 1813 21763 TROY WEIGHT. 95 Ex. 2. How many pounds Troy are there in a million of graius ? 4)1,000000 6)250000 2.0;4-U6 .6 — 4 ZI 16 grains. 12V20^3— 6 175—7 Answer 173 lbs. 7 oz. 6 dwts. 16 grs. Ex. 3. In 36 lb. 10 oz. 12 dwts. 16 grs. how many grains? Ans. 212464 j^rains. 4. How many pounds troy are there in 5987 penny- weights ? Ans. 24 lb. 1 1 oz. 7 dwts. 5. In 1434 lb. oz. dwts. 19 grs. how many a;rains ? Ans. 8259859 grs. 6. How many pounds are there in 45065 grains ? Ans. 7 lb. 9 oz. 17 dwts 17 grs. 7. Reduce 105 lbs. troy into grains. Ans. 604800 8. In 495 spo(ms, weighing 103 lbs. 1 oz. lu dwts., how many grains ? * Ans. 594000 Si 96 AVOIRDUPOIS WEIGHT. AVOIRDUPOIS, OR GROCERS' WEIGHT, Ex. 1. How many drams are there in 225 tons, 17 cwt. 3qrs. 241b. 12 oz. 8 dr. ? tons, cwt, qr« lb. oz. dr. 225 17. 3 24. 12 8 20 4517 4 18071 28 144-572 3r)l44 506012 16 SOS 607 4 506(1 13 8096204 16 48577232 8096204 Answer, 129539272 drams. • 129539272 AVOllRSUPOIS WEI6H9*. S7 Ex. 3. How many tons are there in 259078544 drams ?• 4)259078544 v4>64769636 4)16192409 4)4048102 — 1 7)1012025 — 1 7 oz. 4)144575 lb. 4)36143 — 3=21 2.0)903.5—3 451 15 3 21 ^ Ans. 451 tons, 15 cwt. 3 qrs. 21 lb. 9 oz. Ex. 3. In 179 cwt, how many pounds ? Ans. 20048 lb. 4. Reduce 8345 tons into quarters. Ans. 607600 qrs. 5. How many ounces are there in 4 tnns, 15 cwt. 2qrg. 12lb. ? Ans. 171328 ounces. 6. In 233076 ounces of sugar, how many cwt. ? Ans. 130 cwt. Oqr. 71b. 4 oz. 7. How many drams are there in 53 tons, 14 cwt. 1 qr. 4^ lb. 14 oz. 8 dr.? Ans. 3()804200 drams. 8. In 3-2384818 drams, how many tons weight ? Ans. 56 tons, 9 cwt, 1 qu 27 lb, 3 ©z. 2 dr* 98 apothecaries' weight. APOTHECARIES* WEIGHT. Ex. I. How many grains are there in 2 lb. 5 oz. 4 dr* 1 scr. 17 gr. ? lb. oz. dr. scr. gr. t 5 4 1 17 29 8 236 3 709 20 14i97 Answer 14197 grains. £^» 2. In 42591 grains, how many pounds \^ 2.0)4259.1 3)2129 — 11 8)7(:9— .2 22)88 — 5 7 4 5 2 11 Answer - 71b. 4 oz. 5 dr. 2 scr. 11 gv. Ex. 3. In 51 lb. 2 oz. of rhubarb, how many scruplesi Aiis. 14736 scrup. 4. Id 234876 grains, how many pounds ? Ans. 40 lb. 9 oz. 2 dr. 1 scr..l6gr. ,5. How many pounds are there in 1000 6^. of -opium ? Ans. 83 lb. 4 < z. 6. In 239 lb. 9 oz. 2 dr. 2 scr. 14 gr., how many gis. .>^ Ans. lS8n64grs. 7. How nany scruples are there in or.e hundred and three ounces of Peruvian bark ? Ans. 2472 scrup. 8. In J20794 grains, how many pounds? Ans. 22 lb. oz. 1 scr. dr. 14 gr. LONG MEASURE, LONG MEASURE. Ex. 1. How many yards are there between London and Bath, the distance of which is JOS mile&? JOS 8 864 40 345^0 5| 172800 17280 190080 Answer 190080 yds, Ex. 2. In f 60S29 feet, how many leagues ? 3)760329 253443 2 11)506886 4.0)4608.0 — . 6 IZ 3 8)1151^ — 3)144 — 43 Ans. 48 lea. m. fur. p; 3 yards. Ex. 3. How many inches are there in 1009 miles ^ ^ Ans. 63930240 inches. 4. Reduce 57 ra. 4 fur. 38 p. 3 yds. 2 ft. 3 in. 1 b.c- into barley -curns ? Ans. 10952378 b. corns. 5. In 100004 poles, how many inches r Ans. 19800792 inches. 180 JELOTH MEASURER Ex. 6. In 40y683 feet, how many furlongs ? Ans. 620 fur. 161 yards. 7. How often will the wheel of a coach turn round in going from London to Sheffield, or in 160 miles, suppos- ing the circiunference of the wheel to be J 6 feet } Ans. 32800 times. 8. Suppose on an average I step two feet and a half; how many steps shall I take in walking from London to Richmond, a distance of 10 miles ? Ans. 21 120 steps. CLOTH MEASURE. Ex. I. How many inches in length are there in 15^ ^8 English of cambrick ? 15& 780 4 SI 20 7020 Answer 7026 inches; Ex. 2. In 1000 inches of cotton, how many yards arc there ? 27 3 1 Ans. 27 yds. 3 qr. On. 1 in; Ex. 3. How many English ells are therein three thou- sand and fifty -five nails ? Ans. 152 E.e. 3 qrs. 3 n. 4. In 15 yds. 2 qr* 3 n. 1 in., how many half inches ? Ans. 1131^ half inches. 6, How many inches are there in 10056 yards ? Ans. 362016 inches. 6. Reduce 546 English ells to nails. Ans. 109:^0 n\sh SQUARE) OR LAND MEASURE. 101 SQUARE, OR LAND MEASURE. Ex. 1. How many yards are there in 5604 acres ? 5604 4 224- Id 40 896640 50| 26899200 224160 27123360 Ans. 27123360 yds, Ex. 2. In 6534 square feet, how many perches ? 9)6.534 50^ 4 7xi6 4 121 121)2004(24 242 484 484 Answer 24 perches. Ex. 3. How many roods are there in 382 perches ? Ans. 9 roods. 22 perches* 4. In 561 acres of ground how iiarvy (lerch. and vds. ? Ans. 89760 ot-r. 271 5240 yds. 5. In 2967 4f)0 inches how m iiy acres ? Ans. Not quite a an acre, l)i*.*ng ordy 2289 yds. 5*^. 6. How many perches are there in 997 acr. 3 rd. I0j>.? Ans. lo9610 perches. 9* 102 OUBIC, OR SOLID MEASURE. CUBIC, OR SOLID MEASURE:. £^. 1. In 36 solid ^ards, how many inches E 36 27 72 972 1728 7775 1944 6804 972 Answer - 1 6796 1 6 inches. '3. In 1259712 solid inches, how many yards ? Ans. 27 yardat LIQUID MEASURE. •Ex. J. How many gallons j o 2 10 63 Answer . 630 ire there in gallons. 5 pipes of' wine? Ex. 2. In 700C. pints, 2;70Q6 how many gallons ? 4)3503 876 3 Ans. 875 gal. S qts. DRY MEASURE. 103 Ex. 3. In 31490 pnts, how many gallons ? Ans. 3936 gal. 1 qt. 4. In 3 tuns, I hhd. 49 gallons of claret, how many quarts ? Ans. 347^1 quarts 5. Mow many tuns of port wine are therein 46088 gallons.^ Ans. 183 tuns, 3 hhd. 65 gal. miv MEASURE. Ba. 1. In 79 pks. how many pts. ? Ans. 12^64 pts. 2. How many bushels are there in 7649 pints ? Ans. 119 bush. 2 pks. 1 pt. 3. How many pts. are there in 23 bush. 3 pks. 2 qts.? Ans. 1324 pts. 4. In 3 pks. and 1 gal how many qts. ? Ans. 28 qts. 5. How many pks. are there in 187406 quarts ? Ans. 23425 pks. 1 gal. 2 qts. COMMERCIAL NUxMBERS, OR ARTICLES SOLD BY TALE. 1 2 articles of any kind - 13 ditto 12 dozen - - - 20 articles of any kind - 5 score 6 score 12 score 6 dozen skins of parchnnint - 72 words in Common l/ SO _ in the Kxcheq* 9o i.i Chancery 34 -ihe"t> of paper 520 quires 21a quires, or 3l6 sheets 2 reams hmen 3m: 1 dozen 1 lon^ dozen 1 gross I score 1 hundred 1 great hundred 1 pack of wool 1 roll I sheet 1 ditto 1 ditto 1 fiuire 1 ream 1 do. printer's 1 bundle 104 ARTICLES BT TAtE. Folio is the largest size of books, of which, 2 leaves, or 4 pages, make a sheet. Quarto, 4io. - 4 leaves, or 8 pases, make a sheet^ Octavo, 8vo. - 8 h-aves, or 16 pages, ditto. Duodecimo,12no. 12 leaves, or 24 pages, ditto. Octodecimo,! Brno. 18 leaves, or 36 pages, ditto* EXAMPLES. Ex. 1. How many long dozeu ^^..^ there in ten thou- sand oranges ? Ans. 769 doz» ^ ^^anq-es. Ex. 2. How many gross are there in one hundred ana fifty thousand corks ? Ans. 1041 gross, 8 doz. corks. Ex. 3. In seventy thousmd quills, how many great lundref's are there ? Ans. 583 hundreds 40 quills. Kx. 4. I have a deed containing 4 skins of parchirtent^ and each skin contains* 850 words ; for how many sheets shall I have to pay the person who copies it, reckoning according to the common law charge ? Ans. 47 sheets and 16 words. Ex. 5. The writing of an Kxchequer cause occupies 315 sheets ; for how many, words shall I have to pay the clerk who copies it for me ? Ans. 25200 words. Ex. 6. A suit has been four years in chancery, and I wish to have a copy of all the proceedings ; for how ma- ny sheets shall I pay. supposing it occupies 1264 skins of parchment, and each skin 6Q0 words ? Ans. 9690 sheets and 60 words. Ex. 7. How many sheets are there in 4o reams of paper? Ans. 19200 sheets. Ex 8. How many common reams of paper are there in ten thousand printer's reams ? Ans. t0750 reams. tCx. 9. W hat number of sheets less are tliere in 500 common rea?MS of paper, thaik there are in the same num- ber of printer's reaios ? Ans. 18000 sheets. Ex. 10. What number of pages are there in a folio containing 2!l sheets ? Ans. 844 pages.. Kx II. What will be the difference in the numher Of pages, whether 1 print in l2mo. or iSno., supposing my work^wiil make lourteeo sheets ? Aus. ids pages. TIME. 1Q5 Ex. 12. What numbers of words are there in Dr. Gre- gory's Dictionary of Arts an\l Sciences, which contains 240 sheets 4to., and each page contains 1848 words ? Ans. 3548160 words. Ex. 13 How many reams of paper ivere used in print- in jj that Dictionary, six thousand copies having been taken off? Ans. 3000 reams. Kx. 14. How many pens were used in writing the said Dictionary, supposing each pen to write 840 words ? Ans. 4224 pens. TIME. Ex. 1. In 4199 days, how many months of 28 days each, and years of 365 days each ? Ans." 149 ms. 27 days; or ii yrs. and i nearly. 2. Ucduco ISO days to hours and minutes ? Ans. 3600 hours, 216000 minutes. 3. In 70 years how many days, supposii>g each year to consist of 365^ da^s ? Ans. 25567 days and a. 4. How many minutes, hours, and days are there ia 5960034 seconds ? Ans. 99333 min. 1655 ho. or 68d. 23ho. 33min. 54s. 5. How many minutes are there in 1808 years, allow- ing 365^ Ans. r lb. 4 oz. 4 dwts. Ex. 7. In 78 bags of hops, each weighing 3 cwt. how many pounds ? Ans. 26208 lb. Ex. 8. How many pounds and cwts. of tobacco are there in 73 hogsheads, each containing 3 cwt. 1 qr. 14 lb.? Ans. 28350 lb. 253 cwt. qr. 14 lb. Ex. 9. In 98465 inches of broad cloth, how many yds. and ells ? ^Ans. 2735 yds. 5 in. ; 2188 ells, 5 in. Ex. 10. In five thousana yards of cloth, how many nails ? Ans. 80000 nails. Ex. 11. How many inches are there between London and Bristol, a distance of 120 miles ? Ans. 7603200 in. Ex. 12. How many barley-corns will reach round the earth, which is a great circle of 360 detjrees, and each degree contains 69a miles. And how many quarters Cff MISCELLANEOUS S1CAMPLES. 107 barley would be necessary to perform this, supposing 9200 barley-corns to fill a pint measure ? Ans. 4"53^Ol600 b.c; 1009 qrs. 5 bush. 6 pts. 8800 b.c. Ex. 13. How often will a wheel turn ingoing trom London to York, a ra one of 56 acres, how many square yards will remain ? Ans. 246840 sq. yards. Ex. 16. How many pints anxi gallons* are there in 39 hogsheads of cyder ? Ans. 19666 pts. ; 2457 gal. Ex. 17. How many minutes have elapsed suice the creation of the world to the present time, 1808 inclusive, supposing the world to have been created 1004 years be- fore the birth of Christ, and each year to consist of SQ5l days ? Ans. 3056879520 minutes, AMERICAN COIN. TABLE. Currency s, d. Fed.money, In Maryland Pennsylvania, > , g make 1 DoL Delaware and Mew Jersey, y ISew England and Virginia. 6 ditto. New York and North Carolina. 8 ditto. South Carolina and Georgia. 4 8 ditto. Canada and Nova tScotia. ^^ 5 0' ditto. 1. To reduce Maryland, Pennsylvania, Delaware and New Jersey currencies to Fe Ans. 1794 dols. 82 cts. 5. Redfuce 71. 6s. 8d. New Jersey currency to Fede- ral money ? Ans l9 dols. 55 cts. 6. Reduce 39/. 7s 6d, Maryland currency to Fede* ral money ? Ans. 105 dollars. 7. Reduce 48/. 9s. 5d. Pennsylvania currency to Fe- deral money ? Ans. 129 dols. 25 cts. a 2. To change Federal money to Maryland, Pennsyl- vania, Uelauare and New Jersey currencies. KvLE. It the given sum be doijars only, multiply by 90 and the result wilkibe pence, but if there should be cents in the given sum, multiply by 90 and cut oft two fij^ures on the right hand, the result will be in pence also^ which reduce to shillings and pounds. KOTE, IF there should be half-pt-ticc or farthings jn the given sUrt, retiiice it to the lowest denomination mentioned, and reduce also the number ut pi nee in ont dollar to the same denomination, and divide by iiii^ tor the answer. REDUCTION. 109 EXAMPLES. Ex. How much Maryland currency in 76 dols. 50 cts.? duls, cts. 76 50 90 12)6885,00 20)57,3 9 Answer - - - 2Sl. I3s. 9d, 2. Change 744 dols. into Pennsylvania currency ? Answer 279^ 3. In 365 dols. 25cts. how much New-Jersey cur- rency ? Ans. 136/. 19*-. 4*c/. 4. In 7493 dollars 50 cents, how much Delaware cur- rency ? Ans. 2810/. Is. 3d. 5. In 627 dollars 75 cents, how much Pennsylvania currency ? Ans. 235/. 8s. 1^. 6. In 134 dollars 60 cents, how much Maryland cur- rency? Ans. 50/. 9s. 6d. 7. in 1216 dollars 80 cents, how much Pennsylvania currency ? Ans. 456/. 6s. 3. To change New England and Virg;inia currencies to Federal iin)ney, the value of the dollar being 6 snil- lings or 72 pence. Rule.— If there be pounds and shillings only; re- duce the given sum to shillings, and divide by 6 : but if there be peiic also, retfuce the given sum to pence ; then divide bv 72, and the quotient will be dollars, an- nvx two cyphers to the dividend^ and cuntiuue the operatLun for ceuts. 10 110 REDUCTION. EXAMPLES. 1. In 74?. 6s. 8d. New England currency, how much Federal money ? L. s. d, 74 6 8 20 U86 12 dols. cts. 72)17840.00(247 77 Answer. 144 344 or thus, «IOO0 288 f8)l7! n] — 560 (_ 9)2 3U00 8 247,: 7 cts. 504 560 504 560 504 56 2. Tn 64i. 15s. Virginia currency how much Federal money? Ans. 215 dollars «3 cents. 3. Tn 327?. l6s. 4rf. Virginia currency, how much Federal money ? Ans. 1092 dollars 72 cents, 4. In 463/. 12s. 9d. Virginia currency, how much Federal money ? Ans. 1545 dollars 45 cents, 5. In 579?. 18s. 2d, New England currencv, how murh Federal money ? Ans. 1933 dollars 2 cents. 6. In 6214?. 12s.. 9d. Virginia currency, how much Federal iN oney ? Ans. 2''7 15 dollars 45 cents. 7. In 7 >09?. 13s. 7d. Virgitiia currency, how much Federal money ? Ans. 24365 dollars 59 cents. REDUCTION. H) 4. To chann;e Federal money to New England and Vir2;inia currencres. Rule. — Multiply the given number of dollars by 6, and divide by 20 for pounds, or if ther*^ be cents in the question, multiply the number of cents by 72, and divide by 100, the quotient will be pence, which reduce to shillings and pounds. EXAMPLES. Ex. I. Change 273 dollars 23 cents to New England currency ? 27325 cts. 72 54630 loo) 191275 12)19674,00 2,0)1(53-9 d Answer - - - 8H. 19s. 6i. 2. Change 49G dollars to New England currency ? Ans. 148/. rSjf. 3. Change TO dollars 5o cents to Virginia currency ?^ Ans. 231. 17 s. 4. Change 673 dollars 60 cents to Virginia currency ? Ans. 202/. Is. 7(1. 5. Change 762 dollars 15 cents to Virginia currency } Ans. 228/. 12^. lOi. 6. Change 847 dollars 75 cents to Virginia currency ? Ans. 254/. OS. 6d. 7. Change 1740 dols. 30 cents to Virginia currency ? Ans. 323/. 17 s. 9d. 5. To change New York and North Carolina currencies to Federal money, the value of the dollar being 8 shil- lings or 96 pence. KvLE — If tiie lowest denomination mentioned in the given sum be shillings, reduce it to that denomination ^12 REDUCTION. and divide bj 8 ; but if the lowest denomination be pence, reduce the given sum to pence and divide by 96, the quotient will be dollars ; bring down two cyphers and continue the operation for cents. EXAMPLES. Ex. 1. In 74/. 16s. New York currencj how much Federal money ? L, s. 74. 16 20 8)1496 • 187 dollars. Ans. 2. In 29/. 17s. New- York currency how much Federal money ? Ans. 74 dolls. 62^ cts. 3. In 365/. 7s. 4rf. New- York currency how much Federal money ? Ans. 913 dols. 41 cts. 4. In 497/. 16s. 10^. North Carolina currency how much Federal money ? Ans. 1244 dols. 60 cts. 5. In 563/. 12s. 6d, New-York currency how much Federal money ? Ans. 1409 dols. 6 cts, 6. In 728/. ]3s» 9rf. New-York currency how much Federal money ? Ans, 1821 dols. 71 cts. 7. In 3674/. 8s. 7d. North Carolina currency how much Federal money ? Ans. 9186 dols. 7 cts. ' «. To change Federal money to New-York and North Carolina currencies. Rule— Multiply the given number of cents by 96. and divide the product by 100, the quotient will be pence, which reduce to shillings and pounds, or if there be dol- lars only in the question, multiply thorn by 8 and. divide by 20 for pounds, the remainder, if any, will be shillings. \ REDUCTION. 113 EXAMPLES. Ex. 1. Reduce 49 dols. 50 cts. to New York currency. cts. 4950 96 207 OO 1,00) 44550 J2)475?.00 2,0)39.6 Ans. L19 16 2. Reduce 246 dols. to North Carolina currency. Ans. 987.8s 3. Reduce 418 dols. 75 cts. to New-York currency. Ans. 167/. 10s. 4. Reduce 672 dols. 25 cts. to North Carolina curren- cy. Ans. 263^. ISs. 5. Reduce 847 dols. 60 cts. to North Carolina curren- cy. Ans. 339/. Os. [)d, 6. Reduce U84 dols. 40 cts. to North Carolina curren- cy. -Ans. 473/ 15s\ 2/. 7. Reduce 2756 dols. 50 cts. to New -York curr, In 92"'/. I6s. 9 Ans. 478 dols. 26 cts. 7. In 5106/. 17s. 4c?» Georgia currencv, how much Fe- deral money ? Ans. 13315 dols. 14 cts. V 8. To change Federal money to South Carolina and Georgia currencies. i^«bi^. HEDUOTION. 115 Rule. — Multinly the gjiven nimber of cents hv 56, and divide by 100. the quotient will be pence, which re- duce to shillings and pounds, t. EXAMPLES, Ex. I. How much Georgia currency, in 216 dols. SOcts. ? cts. 21650 56 12P900 1,00) 1 OK 250 12)12124.,00 20)101,0 4 Ans. L.50 10 4 2. How much South Carolina currency in 467 dols. 25 rts. ? Ans. 109/. Os. 6d, 3. How much South Carolina currency in 762 dols. 30 cts. .? Ans. 177/. 1 7s 4^^. 4. How much Georgia currency in 939 dols. 70 cts. ? Ans. !^19/. 5>. 3d. 5. How much Georgia currency in 1000 dols. P Ans. 233' 6^, 8. 6. How much Georgia currency in 2 172 dols. .50 cts. P Anv 506/ 18>. 4rf. 7. How much Georgia currency in 9999 dols 99 cts. ? Ans. 2333/. 6^. id. '' 9. To change Canada and Nova Scotia currencies to F'deral money, the value of the dollar being 5 shillings or 60 pence. Rule — Reduce' the given sum to pence and divide by 60, the quotient will be dolUrs ; or if the lowest denouii- 116 REDUCTION. nation be shillings, reduce to shillinj^s and divide by 5, the quotient will be dollars ; annex two cyphers and con- tinue the operation for cents. EXAMPLES. Ex. 1. Reduce 87/. 16. 4rd. Canada currency to Fede- ral money ^ L. s. d, 87 16 * 20 «,0)2 107,60 Ans. g33I 26 cts. 2. Reduce 24 14 to Federal money h 20 6)494.00 Ans. 898 80 cts. 3. Reduce 827/. 15s. Nova Scotia currency, to Federal jnonpy ? Ans S3] I dols. 4. Ueduce 268/. I2s. 3d. Canada currency, to Federal money ? Ans. 1074 dols 45 cts. 5. Reduce 719/. 9s. 2d, Canada curr»'ncy, to Federal money ? Ans. 2877 dols. 83 cts. 6 Reduce 672/. iOs. lOrf. Canada currency, to Fede- ral tvoney f Ans. 2ti90 dols 16 cts. 7 Reduce 926/ 1 Is. I \d, Canada currency, to Fede- ral nu>ney ? Ans 3706 dols. 38 cts. . 8 Reduce 5119/. VOs. id, Canada curr' ncv,io Federal money ? Ans. 20479 dols. 91 cts. V'^.' REDUCTION. 117 10. To change Federal money to Canada and Nova Scotia currencies. Role— Multiply the given number of cents by 60, and divide the product by 100, the quotient wdl be pence, which reduce to shillingis and pounds; or it there be dol- lars only in the question multiply them by 5, divide by 20 for pounds, and the remainder will be shillings. EXAMPLES. Ex. 1. In 68 dels. 50 cts. how much Nova Scotia cur- rency ? cts. 6850 (jO l^oo) 12)4110,00 2,0)34,2 6 Ans. L.17 2 6 2. In 124 dols. 25 cts. how much Canada currency ? Ans. 31/. Is. 3d. 3. In 7648 dols. how much Canada currency ? Ans. 1912Z. 4. In 867 dols. 35 cts. how imich Canada currency ? Ans 216/. I6s.9d, 5. In 1714 dols, 75 cts. how much Nova Scotia cur- rency ? Ans. 428/. 13s. 9d. 6. In 6179 dols. 20 cts. how much Canada currency? Ans. 1541./. 16s. 7. In 4444 dols. 44 cts. how much Canada currency ? Ans. llll/.2i. 2rf. (118) PROPORTION, OR THE RULE OF THREE, This rule is called the Rule of Three, because by three numbers being givcMi we find a fourthj and it is either the Rule of ilirce Direct or Inverse. THE RULE OF THREE DIRECT teaches, from three given numbers to find a fourth, which slidll have the same proportion to the second, as thetliird has to the first ; that is, if the first be greater than the third, the second will lie greater tlian ihc fourth ; and, if the first be less than the third, the second will be less than the fourth. Rule I. State the question: that is, place the given numbers so that the first and third may be of the same kind, and the second the same as the number re- quired. 2. Bring the first and third numbers into the same de- no;nination, and the second into the lowest denouiina- tion mentioned. 3. Multiply the second and third numbers together, and divide the product by the first, and the quotient will be the answer, in the same denomination as that in w bich the second number was left. nULE OF THREE DIRECT. Il9 Ex. 1. What is the value of a pipe of wine, if 5 gal- lons cost 4/. 178. gal. L. s, pipe. 5 : 4 17:: 1 20 2 97 2 63 « 126 97 882 1134 5)12222 3.0)244 4 — 2 12 124.4 5)Z4 4 — 4 4 ^ 5,16 — 1 120 B.ULE OF THREE DIRECT. Ex. 2. If T can buy 27lb. of sugar for iL 13s. how much can 1 purchase for thirty guineas ? L, s lb. guineas. 1 13 : 27 : : 30 20 21 33 6J0 27 4410 1260 Ibsi 33)17010(515 165 51 33 180 165 15 16 ■ oz. 33)240^7 231 9 Ex. 5. What is the value of 28 ells of cloth, if 4 ells cost 18s. ? ells, shill. ells. ' 4 : 18 : : 28 18 nULE OF THREE DIRECT. l2l Ex. 4. If six yards of cloth cost 24 shillings, what will SI yards cost? Ans. 16/. 4s. Kx- 5. If 8 gallons of wine cost 17 dollars 25 cents,^ what is the value of 35 gallons ? Ans. 73 dols. 4b cts, ▼ Kx. 6. If 5lb. of Dotatoes cost 4ii., what is the worth of 1 cwt. on the samfe terms ? Ans. 7.s. 5ld. Kx. 7. If 5lb. of old iron cost 3rf., how many can I buy for 40>. ? Ans. 7 cwt. qrs. 16 lb. Kx. 8. If 10 English ells of cloth cost 11 dols. 11 cts., what is the value of 5 pieces, each containing 26 yards ? Ans. 116 dollars 54 cents. Ex. 9 If 16 yards of muslin cost 10 guineas, how many ells can I buy for 45^ Ans. 54 ells 4|S)q»'S. % Ex. 10. If I can purchase 25 books for 30 dollars, how many can I have for 75 dols. 60 cts. ? .4ns. 63 books. Ex. 11. If a servant's wages' be 25 guineas a year, how much has he to receive for 87 days' service ? Ans. 6(, 5s. lid. ^. Ex. 12. If a servant receive three guineas and a half for 20 weeks service : how long ought he to remain in his pliice for 12 guineas.^ Ans. 68 weeks 4day8. tEx. }J. If I pay half a crown for 4Un of cheese : how much can i have for three crowns and nine-pence ? "^ Ans. 25lb. SSi oz. Ex. IJ^^ If 2 lb. 4 oz. of honey cost 3>. 9rf. : what is the value of 28 lb. ? ^ \ns. 21. 6-. 8^, Ex. yf. If 3 lb. of sugar cost 37a cents, what will ^ 1 cwt. amount to .? Ans. 14 dollars. Ex. 1^ If a dozen of wine glasses cost lOs. 6d., what is the value of dOO.P Ans. 21/ \7s. 6d. •y Ex. \f. If I can buy S pair of shoes for 7 dollars 50 cts. what must I pay for 17 pair ? Ans. 42 dols. 50 cts. Ex. ^. If a cwt. of tobacco cost 8 guineas; what is the value of 7,000,000 of ll)S. ? Ans. 525,000/. 1^ Ex. (9. II 6 lb. of saop cost 1 dollar 37a cents, what is t!»e value of 1 cwt. Ans. 25 dols. 66 cts. Ex. 10. If I pay 39 shillings per cwt. for lead ; how much will it cost to cover the roof of a building with lead that weighs 5505 lb. > Ans. 961, \6s. lid. itz- Kx. t|. I warit to know how much 1 have to pay for a cistern 990 lbs. at the rate of 2l. 2s, per cwt., the plum- 11 Itt RULE OF THREE DIRECT. ber agreeing to allow me at the rate of \l. 14s. per cwt: for the old lead, which weighs 458 lb. ? ^ Ans. 11/. I2s. ^Id. ^ Ex. 22. If a journeyman can earn 9d*»llars 50 cents, in 6 days, how much will he earn in 3^)5 days ? Ans. 4ro dollars 9 1 cents. Ex. 23. The brazen statue of Apollo, that was erect- ed by Chares, at Rhodes, weighed 720,000 lbs. : how much did the old brass sell for at four guineas per cwt. I Ans. 27,000/. Ex. 24 If I pay \L 7t. the sack of five bushels ? Ans. 19,379/. 5s. Ex. 59. What is the value of 6 casks of raisins, each weigliing 3 cwt. 2 qrs. 14 lb. at 31. \0s 6(1. per cwt. ? Ans. 1 201. Ss. 4*rf. Ex. 60. How much must I jrive for a gold snutt-box that weighs S oz. 9 dwts., at the rate of 4/. 3s. 9ci. per oz } Ans. 35/. 7.s. S^c/. Ex. 61. How much tax must I pay for 586 dollars, at 61 cts. per dollar ? Ans. 36 dols. 62^ cts. Ex. 6'2. A Bankrupt has but 1020 dollars to pay debts to the amount of 3235 dollars ; how much can he pay in the dollar } Ans. 31 cts. 7 mills nearly. Ex. 63. A merchant failing, his assignees find eflfects and good debts to the aniount of 3335/. ; but he owes 4225 1. ; the expences attending his bankruptcy will be 212/. 9s.: how much, therefore, will he pay in the pound.? Ans. l4i»/9c?.S Kx. 64. An honest tradesman, tlirough unfore-eei> misfortunes, is obli^jed to call his creditors togetlier ; he ,fin«ls his debts to be 432b/. and he can pay J4.s. 6d. \n the pound : how much has he still left ? Ans 3136/ 7s. Kx. 65. Hops are remarkably cheap, and I have 100/. to spare : how many can [ purchase at 3/. 15s 6//. per cwt. Ans. 26 cwt. 2 qrs. nearly. Ex.66. If 10 lbs. of tea are worth 10 dols. 25 cts.; how much of the same sort can 1 purchase for 38 dols. 431 cts. ? An.«. 37 lb. 8 oz. Ex. 67. What must I pay for the carrisig** by th«^'. ca- nal, froui Manchester to Etruri;i, 705 tons* 5 cwt. of goods, at 15s. per ton ; and what is the diHeren- e be- tween tlus and the ia'id caniage, at "Zl. I5s. per ton ? Ans. I iiust pay 62s/, I8>. 9^/.. for carriage by ('anal^ and 14-10/, 10s. oittV'FtM.ce in t :e price of carriage. Ex 68. \\h.iit weight of goods can be r;i'7;'"d on the the canal l»etv\ern Manchester and Hjrniiitjjf^^-:i f(»r 85/- at the rate of I/. 10s. per ton : and liow lo ^.p fan be earned the same distance, by land-fHrria;^e, at 5/. per toa i Ana* 6G| tons by water, and 17 tons by land. 126 RULE OF THREE DIRECT.. Ex. 69. The clothino; of a regiment of 760 men cornea to 3050/ how much is that per man ? Ans. 4/. 7^. Ex. 70. What may a man spend per week, whose in- come is 2000/. per annum, supposing 52 weeks in a year ? Ans. 38/. 9.s. 2i;/.^a. Ex. 71. If hy selling fine Irish cloth at 5 dollars per ell, I gain 108 dollars ; how much shall I gain if I sell it at 6 dols. 25 cts. per ell ? Ans. 135 ollars. Ex. 72. If sugar that cost 9 cts. per lb. be sold at 3 lb. for 37a cts. what will be the gain by selling 1 cwt> ? Ans 3 dols. P2 cts. Ex. 73. I purchase 5 pieces of Holland, each contain- ing 30 yards, at 4s. 9c?. per yard : how much shall I gain by selling it at 6s. 2rf. per ell English ? .Ans. I/. 13s. profit. Ex 74. Two persons part at the same time for the same place, tlie one travels north 24 miles a iVdj, and the other 21 miles a day south : when w\\\ they be 1000 miles asunder ? Ans. 224 days nearly. Fix. 76. If a pack of wool weighs 3 cwt. 2 qrs. 7 lb., what is it worth at 2 Is. bcl, per tod of 14 lbs. ? Ans. 30/. \2s.9d, Ex. 76. The rents of a parish amount to 1750/., and a rate for the poor is wanted of 65/. 7s. 6d* : wliat is that per pound? Ans. 9rf. nearly. ( 127 ) THE RULE OF THREE INVERSE. This rulp, like the last, teaches, from three given numbers, to find a fourth, which fourth number shall bear the same proportion to the second, as the first has to the third. 'I'hus, if the question be, .If 10 men can nmw a certain field in 6 days, how soon can it be done by 2U n»en ? The answer will evidently be in 3 days, because do ibie the number of men will certainly do the same work in half the time; the proportion will therefore stand, 10 men : days : ; 20 men : 3 days ; and 3 bears the same proportion to &. that 10 does to 30 ; that is, the fourth number bears the same pri>portion to the second, that the first does to the third. Rule. — State the question, and when necessary, re- duce the terms as before. Multiply the first and s*eco)id terms together, and divide the product by the thud term ; the quotient is the answer in the same denomina- tion as the second term 5 thus in the foregoing example> JOxt) IZ3 days. 20 Ex. 1. If 15 reapers can cut down a field of corn in 4 days, in how long time will the same work be performed by 40 men ? 15 : 4 : : 40 4- 4.0)0.0 128 RULE ©F THREE INVERSE* Ex. 2. If the penny loaf weighs 4 ounces when flour is 4.S, per peck, how much must it weigh when flour is 5s. 4>d. per peck ? Ans. 3 ounces. Ex. 3. A person lent me 240 dollars for 8 months ; in return for his kindness, how much ought I to lend him for 18 months ? Ans. 106 dollars 66 cents. Ex. 4 How many men must be employed to finish a canal in 12 days, which 5 could perform in six weeks, or 36 days ? Ans. 15 «nen. Ex. 5. If 24- pioneers can make a trench in 12 days, what length of time would the same work employ 9 men P Ans. 3-2 days. Ex. 6. The. floor of a chapel 96 feet in length, and 7Q feet in breadth, is to be covered with matting 2 feet six inches broad : how many yards will it require ? Ans. 2688 feet Ex. 7. If a person travel 12 hours a day, and finish his journey in three weeks ; how hmg would the same journey take bin), if he travelled only 9 hours a day at the same rate ? Ans. 4 weeks. Ex. 8. If the town and garrison of Bhurtport, con- taining 22.400 persons, have provisions to last three weeks, how many tnhaiiitants must tiolkar send away, so as to make the provisions last 7 weeks, which is as long as General Lake can carry on the siege ? Ans. 12,8oo persons. Ex. 9 If a besieged garrison have 4 months provisions, at the rateot 18 ounces per man per day ; how long wdl they be able to hold out, if each man is allowed only \Z ounces per day ? Ans. 6 months. Ex. 10. If there arc in a garrison provisions sufficient for 150(/ men 10 weeks, which, on account of the rains, is seven weeks longer tiian the siege can last; how many sohhers may be brought to defenti the place for three weeks, without lessening the quantity of food tt> anj individual ? ^ Ans 330J soldiers. Ex 11 If 9 plasterers can finish the uiside ol a cha- pel in 10 days: how lonjj wdl it take 4 rt>en. supoo-in* the other 3 sent away '-n a i»ew job P Ans. 22a *'•*>*• Ex. 12 it S3 yards of broadcloth, if wide will usake a suit of clotlies ; how xuuch will be nece^sai v of cloth onlj- I wide ? Ans. 8 jards 0|. ^ RULE OF THREF: INVERSE. 1 Kx. 13. If 32 clerks in the bank are sufficient to make up the books in a certain office in 15 days, how many clerks would be required to do the same work in 6 days ? Ans. 130 clerks. Ex 14 If the carriage of 15a cwt. for 60 miles, came to 7s. 9d. ; how far can [ have carried 3^ cwt. for the same sum } Ans. 248 miles- F^x. 15. If 24 men can finish a piece of w>)rk in \d hours; how manv men will it require to do the same work in 12 hours? Ans. 32 men. F.x. ^6. If 12 inches in length, and 1^ inches in breadth, njake a square foot ; what length of board, 8 inches broad, will be equal to the same measure. Ans. 18 inches. Ex. 17. If 220 yards in length, and 22 in breadth, make an acre ; what must be the breadth when the length is 120 yards } Ans. 40 yards. Ex. 18. If 5 horses can he maintained when oj minutes ; how many such cocks will empty it in 42 ndnutes .'' Ans. 8 cocks. Ex. 22. The sides of a room are found to measure J 38 feet in lenu;th. a')d thf height is 14 feet 6 inches ; how much paper, 2 feet 3 inches wide, will cover it ; and what is the valun of it at 9rf. per yard } Ans. 296 yds. I ft. 4 in. \\l. 9s 4d. Ex. 23 If 50 cows can be kept in a field 17 days; how Ion"; will the same pasture feed 70 cows ? Ans. 125. ( no ) THE DOUBLE RULE OF THREE. THE Double Rule of Three teaches, from five given nuuihers to find a sixth. Three of the numbers^ontain - the suppositions, and the remaining two are terms of de- mand. Rule (1) Put tlie terms of supposition one above another in tlie first place, except that which is of the same nature with the term sought, which put in the se- a cond place. (2.) Place the terms of demand one above another in the third place, in the same order as the terms of the supposition were put in the first place. (3.) The first and third term in every row will be of the same nature, and must be reduced to one denomina- tion ; and the middle term must be brought to the low- est denomination mentioned. (4.) Examine each stating separately, using the mid- w die term as common to both, in order to know if the proportion be direct or inverse. When it is direct mark the first term with an asterisk, and when it is in- verse, mark the third term with an asterisk. {5.) Multiply the numbers together which are marked for a 22 da vs. in how many days will six persons coR- sume 20 pecks ? pecks. 15 : days. 22 : pecks. ; 20 persons 9 : persons. : : 6 9 : or, ^ 21 X 20 — ^ZZ 44 days' 15 X 6 132 DOUBLE RULE OF THREE. Ex. 4. If 6 pioneers can dig a ditch 34 yards long in 10 davs ; how many yards may be dug by 20 men in 15 days ? Ans. 170 yards. Ex. 5. If 1050 soldiers consume 230 quarters of corn in 6 months ; how many soldiers will 960 quarters serve 4 months ? ' Ans 6048 men. Ex. 6. If a cask of beer last 8 persons 14 days : how man)' casks will serve 2 persons 3()5 days ? Ans. 6^ casks. Ex 7. If 10 men in 6 weeks earn 500 dollars ; how many weeks must 15 men work to earn lOOO dols. Answer, 8 weeks. Ex 8. Suppose I walk 66 miles in 4 days, of eight hours each day : how n)any days, of 14 hours each, shall I be in going from London to York, or 196 mites. Ans. 6^. almost 7 days. Ex.9. If three boats take 6000 herrings in 8 days : how long will (iOO boats be in taking 20,000 barrels, each containintj; 700 herrings ^ Ans. ^Sg Ex. 10. If, against a general mourning, 6 tailors can make JO suits of clothes in 4 days : how many suits can 600 men make in the 7 days which occur before the mourning is wanted ? Ans. 1750 suits, Ex. 11. If 12 niantua-makers can make 27 mourning dresses in 4 days : how many persons would be required to make 189 dresses in 8 days ? Ans. 42 mantua-makers. Ex. 12 If 3000 copies of a History of America, each containing U sheets, require 66 reams of paper : how much paper will 5000 take, if the work be extended t» 123 sheets. Ans. 125 reams Ex. 13. As 12 inches in length, 12 in breadth, and 12 in thickness, make a solid foot : what length of plank, which is 7 inches broad and 3 inches thick will make the same? Ans. 827 inches. Ex 14. If 430 tiles, each 12 inches square, will pave my cellar : how many tiles must 1 have, if the tiles are 9 inches long and 8 broad ? Ans. 900 tiles. fcx. 15. If the expence of 3 persons on a tour for 5 months be 133/. 8s. : what will 2 persons spend in 9 months ? Ans. 148^ Is, Id. MISCELLANKOUS QUESTIONS. IS3 Ex. 16. If 12 ounces of wool make 2l yards of very tine cloth, 6 quarters wide : how much wool would be required to loO yards, 4 quarters broad ? Ans. 480 ounces. Ex. 17. If 300 dollars gain 15 dollars interest in a year, in what time will 900 dollars gain 180 dollars. Ans. 4 years. Ex. 18. If an iron bar 4feetlong, 3 inches broad, and IJinch thick, weigh 36 lbs. : how much will a bar weigh that ts SIX feet long, 4 inches broad, and 2 inches thick > Ans. ll5Hbs. MISCELLANEOUS QUESTIONS ON ALL THE FOUEGOING RULES. Ex. 1. What three numbers are those, the first of which is 105, the second ^ds of the first, and the third 67 less than the first and second together I Ans. first, 105, second 70, third, 108. Ex. 2. A gentleman left his eldest daughter 1000 gui- neas more than the youngest, and to tliree other daugh- ters he left 7000 guineas between them, which was equal to the sum left to the youngest and eldest together : what was each child's fortune. Answer, Eldest, 4000, youngest, 3000, three other daughters, 7000. Ex. 3. W hat is the difference In value between five times five and twenty guineas, and five times twenty- five guineas ? Ans. 80 guineas. Kx. 4. What was the value of a prize taken by 25 sai- lors, besides officers, so that each sailor received 19^. 9s, 96/., and the officers received as much as the sailors ? Ans. 974/. 7s. 6d, Ex. 5, A prize valued at 13,177/. 10s., after the offi- cers have had their share, is to be divided among 525 sailors : what would each man have to take ? Ans. 25/. 2s. each man's share. Ex. 6. What is a fourth proportional to the numbers 6, 9, and 24 ? Ans. 36 fourth proportional. Ex. 7. What is the value of 4 packs of cloth, eack pack coutainiug 4 parcels, each parcel 10 pieces, and 12 134 MISCELLANEOUS QUESTIONS. each piece £6 yards, at the rate of 12 dols. 50 cts. for 3 vards ? Ans. 17333 dols. 33 cts. Ex. 8- How many yards of paper, 3 quarters wide, will be sufficient for a room 48 yards round, arid four yards high : and what is the value of the paper, at the rate of 18s. per piece of 24 yards ? Answer. 2o6 yards, worth 9/. 12s. Ex. 9. If 100 dollars jiain 5 dollars in J 2 months, what will 75 dollars gain in 9 months ^ Ans, 2 dols. 81^ cts. Ex. 10. If 48 cannon consume, in 3 da; s, 288 barrels of powder, how much will be spent in 15 days, when 144 cannon are to be supplied ? Ans. 4320. Ex. ll. Fifteen people joined to purchase a lottery ticket, for which they gave three sshillings less than eighteen guineas: if it came up a prize of 30,000 gui- neas, what did each n an receive, and what was his gain } Ans. 2100/. each man's share, and 2098/. 15s. each man's gain. Ex. 12. A tobacconist bought two parcels of tobacco, which weighed 9 cwt, 2 qrs , for a hundred guineas, the difference of the parcels in weight was 3 qrs. 12 Id., and in value eight guint'as : what was their weight and va- lues ? Ans. one parcel. 5 cwt. Oqr. 20 lb. the other par- cel 4 cwt. I qr. 8 lb., cost 54 guineas, and 36 guineas. Ex. 13. 'J he clothing of 100 charity children came to 211/., of which 135/. was expended on 60 boys: what was paid for the 40 girls, and how much did the clothes of each child cost? Ans gris clothing 76/., price of each boy's clothes, 2/. 5s. ditto girl's clothes 1/. 18s. Ex. 14. A great graz-ier left to his four sons 220 oxen and 1200 ^heep : 1 demand the value of each son's lega- cy, supposing the oxen worth 18 guineas each, and the sheep 39 shillings each ? Ans. 1624/. 10s. each sou's legacy. 15. What number is that which, multiplied by 384, will give a product of 3.013,2+8 ? ' Answer, 7847 Ex. 16. What is gained by the sale of 456 yi rds of cli)th. that was bought at the rate of 7 dols. 25 cts. per yard, and sold at the rate of 11 dols. 50 ct^. per yard ? Ans. 1853 dols. MISCELLANEOUS qUESTrONS. 135 Ex. 17. ir 9 printers can set up the New Testament in %% (lays, in what time could it be done if 15 were em- ployed ?" Answer, 1 Sg day^s. Kx. 18. If 8 men will earn on an average 84 dols. in 6 days, how much can 16 men earn in 27 days ? Ans. 708 dols. 75 cts. Ex. 19. When the quotient is 1083, and the divisor 555, what is the dividend, if there be a remainder of 79? Answer, 601144. Ex. 20. The silk mil! at Derby winds oft' 7 3.726 yards of silk every time the great wheel goes round, which is thrice in a minute : how many yards wdl it wind in a year, allowing that it works every day, except Sunday, 15 hours, and how many skeins will be made, supposing 960 yards gO to the skein i* Ans. 62,305,842,(iOO yards made in a year, and 64901919s the number of skeins. Ex. 21. In tlie partition of some waste lands in the west of England, A had 59^ acres, B 761 acres, C 110 acr. 2r. 12 per., D. 15 acres, and E 39 acr. Or. 12 n-r.. but these, taken together, were but one-filth of the whole : how many acres were divided, and what was the value of the whole, supposing each acre worth 16^. 9s. 6rf. } Ans. 1502 acr. .Or. Op. land divided, 23243/. 9s. Oc/. va- lue of the land. Ex. 22. An Island in the West Indies contains 42 parishes, and every parish 7Q houses, and each house at thtj rate of b\ white persohs ; besides these, there were 65 negroes to each of 54 plantations : how many people were there on the whole island ? Ans. 21066 persons. Ex. 23. \n the club mentioned in the Spectator (No. 9.), there were 15 persons, weighing together 3 tons : how many pounds, ounces, and diains. Avoirdu- pois, did each man weigh .'^ Ans. 448 lb. 7168 oz. ll468Bdr. Ex, 24. The British possessions in Hindostan contain 212,406 square miles, and the population is estimated at fourteen miilions : how many inhabitants are tiiere to a square mile } Ans. 66 persons nearly, Ex. 25. If 9 lb. of tea cost 7 dollars 20 cts., what is the worth of 4 chests each weighing l cwt. 2 qi s. ? Ans. 537 dollars 60 cents. * 365—52 equal 313 the number of working days in a year. 136 mSOELLANEOVS QUESTIONS. Ex. 26. What shall I give for a farm containing: 256 acres, for which I am to pay at the rate of 95 dollarii for 4 a^res ? Ans. 6080 dollars. Ex. 27. What will it cost a young man to come into a farm, for the lease of which he is to pay lOOO guineas; fur 22 horses he is to pay at the rate of 18 guineas each 5 for crops in t' e ground 354/. ; for 210 hushels of wheat he is to pay 4l. los. per 8 bushels ; the household furni- ture is appraised to him at 298 guineas, and for farming, utensils of all kinds he is to pay 196/. ? Ans. 2i46/. 16.s. 6d. ^ Ex. 28 The revenue collected in Hindostan by the Bri- tish, is reckoned at 3,400,000/., how much is that from each inhabitant, supposing they amount to 14- millions? Ans". 4s. \Old. Ex. 29. The number of negroes in Jamaica is estimat- ed at 250,000, and of whites 20,000, how many slaves are there to a single white man, and what do the planters :-.-.'I...;. their nronprfv i.nr+u \n the article of slaves only, supposing each to be worth 93 guineas r Ans. 125 slaves, 24,412,500/. Ex. 30. The population of the United States is esti- mated at six millions and a half, and the number of slaves still existing in that free country is reckoned to be 697,097, how many free people are there to one slave ? Ans. 93 nearly. Ex. 31. The extent of China Proper is equal* to 1,397,999 square miles, and the population is estimated at 333,000,000, how many inhabitants are there to a square mile .^ Ans. 238 nearly. Ex. 32. In Spain each person pays 10 shillings to go- vernment for protection ; in France, under tlie old go- vernment, each paid 20s. for protection ; and in England we pay full three guineas each for the same advantages, how much is the revenue of the three governments, suppo- sing the population of Spain to be 10^ millions ; of France, at the period referred to, 25 millions ; and of England and Wides 9,343,1 73.? Ans. 59,555,994/. 19s. Ex. 33. The population of London, Westminster, and fcouthwark, is 8()4,8d5, that of Paris 547,75t), how MISCELLANEOUS qUESTlONS. 137 much does the population of London exceed that of Paris? Ans. 317.109. Kx. 34. How many minutes and seconds have elaps- ed since the birth of Christ, or 1808 vears* ? Ans. 95'),Q35,6RO min. 57,056,140,800 sec. Ex. 35. How Ions; would it require to count five hun- dred tnillions sterling, supposins; a person were to reck- on 150L in a minute, and were to he employed 10 hours ea h day, and six days a week, till he had finished the job? * Ans. 9^2(> weeks, nearly. Kx. 3d. How many barley-corns will reach round the earth, supposing the length to be 25,200 miles ? Ans. 4,790,016,000 Ex. 37. How many seven-slnHinw pieces are there in a thousiind pounds ? Ans. 28.57 seven -shil. 1 shil. Ex. 38. A French franc is worth \0d., how many francs are there in 100^ Ans. 2400 Francs. Ex. 39. If 8 men can mow 18 acres in 4 days, how many men will be required to mow 50 acres in six days f Ans. 14f^ kx. 40. A balloon has moved at the rate of 6492 feet in a minute, how long would it have been sailing round the earth at the same rate, supposing the circumference of the earth to be ^5,^^200 miles } Ans. 14 days 5 hours 35 min, 22 sec. Ex. 41. How much oftener will the small wheel of a coach turn than the large «>ne, between London and Bris- tol, or 120 miles, if the former be 10 feet 8 inches in circumference, and the latter 18 feet 4 inches ? Ans. 24840. Ex. 42. If my income be 250^ per annum, and I have fo(»lishIy expended 15.s. per day, how much »'ofit to the proprietor, what would he clear hy the night, al- lowing that the incidental expenses were 250/. ? Ans. 1258/. 17s. Ex. 45. If 3l yards of cloth will make a shirt, how much of the same stuft' will he wanted to make two shirts for each man of a regiment, consisting of 855 men ? Ans. 555/3 yards. Ex. 46. In November, 1800, 276,334 five-pound hank notes were issued ; in l)ece;nhcr 2,626,700 ; and in the January following 2,769, JOO; what was the nominal va- lue of the notes issued in these three months ; and what was the cost of white rags, from which they were made, supposing each ounce of rag might be manufactured into twenty five-pound notes, and the raij;s to be worth 8c?. per lb..? Ans. 28,360.970 nominal, 590/. 17s. 0*^7. Ex. 47. Two persons depart from London to York on the same day ; the one walks 19 miles a day, the other only 15a miles; '^'^^ far distant will they be from one another after ten days travelling, and when will each get to York, which is 197 miles from London ? Ans. 35 miles distant, Jie who goes l9 miles a day will complete his journey on the IJth. day, whde the other will not complete his journey till the 13th. day. Ex, 48, The population of the world is estimated at a thousand millions of human beings ; if the face of the earth be repeopled every 33 years, how many persons are born and die in a year, week, day, and minute ? Ans. 30.303,030 year, 582,750^^ week, 832oOday. 34118 nearly in an hour, 58 nearly a nun. Ex. 49. The field opposite my house will serve 50 cows forty days ; how long will it afford 220 with equal feed ? ' And. 9 days and a fraction. Ex. 50. If 10 persons expend 250 dols. in 4 months; how much ought 3 persons to expend in 12 months ? Ans. 225 dollars. ( 1^9 ) FRACTIONS. A Fraction is the part, or parts of a whole, or of any wh.>l;* quantity expressed by unity, and is expressed by two fifftjies, with a line drawn between them, as |, J, |. The upper fijjure of a fraction is called the numerator, and the under one the denominator. The denominator sliews how many parts the unit is divided i ito, and the numerator, how many of these parts are to be taken : thus |, or three-fourths, shews that the wliole is divided into four parts, and that three of those i>arts are to be taken : and f , or five-ei«:hths, shew that the whole is divided into eh^ht parts, and that five of these parts are taken. There are four sorts of fractions, simple and com- pounv the word of placed between them, as | of 6: or I of | of ^%, A proper fraction is, when the numerator is less than the denominator. An improper fraction is, when the numerator is equal to, or greater than, the denominator. A mixed number is formed from an integer and a frac- tion joined together, as 8^. A complex fraction is one that has a fraction or a mix- ed number tor Us nuaieratur) or deuomioator^ or both. ( no ) REDUCTION OF FRACTIONS. The method of diangincr fractions from one form to another, without altering: their value, is called Reduc- tion ',^ZZIIii:[qZI:^ZZ.I. Reduction serves to prepare frac- tions for Addition, Subtraction, Multiplication, and Di- vision. Cask 1. To reduce fractions to their least terms. Rule. Divide the terms of the given fraction by any Dumber, which will divide them both without a remain- der, and the quotients will be the terms of a new frac- tion, equal in value to the given fraction. Repeat the operation, till the terms of the reduced fraction are di- visible only by 1. ^ Ex. 1. Reduce |ii2 to its lowest terms. )3136 392 \392 49 \49 7 IT , and 8 ] — — , and 7 j — zr— 3584 448 /448 56 J 5Q 8 Keduce the following fractions to their lowest terms. Kx. 1 Ex, 2. Ex. 3. Ex. 4. Ex. 5. 32 4 208 52 136 17 156 If- S60 30 120 15 684 171 72 9 356 28 708 59 Ex. 6. Ex. 7. Ex. 8. Fx. h. Ex. 10. 384 1 5184 432 24-5 55 4032 224 3105 69 1152 3 6012 ,01 2880 64 '4806 267 15705 267 2X3X4X5 Reduc? — to its lowest terms. 3X4X7X8 2X3X4X5 -10 5 3 X 4 X 7 X 8 65 28 -o DEDUCTION OF FRACTIONS. 141 3X8X9X2 Reduce — to the lowest terms, 4 X 3 X 14 X 36 3X8X9X2 3x2x4x9x2 2 1 4X3X14X30 4X3X2X7X4X9 28 14 3X4X15X41 Ex. U. ZZ- 5X6X24X3 2 10 X 27 X 30 X 12 24 Ex. 12. — — 15 X 9 X 55 X 30 55 Case TI. To find the greatest common measure of a fraction. Rule. Divide the greater term by the less, and this divisor by the remainder, then the fast divisor will be the greatest common measure of both terms of the frac- tion. ■■4X,. laaj issj isa* 144 nenuoTioN OF fractions; Ex. 1. Reduce I, ^. and l to a common denominator. A,,e 33 m 88 2. Reduce J, g, *, and I, to a common denominator. Anau/or 2160 1400 2016 2305 y\ijswci , 2«zo^ agaoi aeaot sfiso* 3. Reduce 7, g, 7, and 3, to a common denominuior. Anewpr 7" "7 ^lO 738 miJ)Wer, 24S? 245* 246* 4. Reduce^, io» 8, and llg. to a common denomina- fnv A neu'Ar ** '"> 2400 s4fio Lui. rviidwri) 300*300' 300« 300' 5. Reduce 11, 2, 7, 4, and 2^, to;i common denomina- •*Or. A neurpr ^^° ^^ 220 3080 16M lui. Aribwer, 770' 770- 770 770* 77o» 6. Reduce I, % 7, and ^, to a common denominsitor. Ancu'Pr *'»0 *8s 80 310 /\nhuer» gso^ 23o» sso? 280» 7. Reduce L 9> «? « and 7, to a con^mon denor- i.'-i or. AncwPr 1260 800 360 576 10080 ^IJHWtrr, 1410' 1410» 1440. 1440> 1+40 * 660» 660* 9. Reduce e, i, 7, and *j, of 9, to a common dinonnna- inr Anuwpr B^^ 1388 623 l?098 lor. Answer, sggo, gegsj 3696> sege* ^^ (2.) To find the least common denominator. Set down the denominators ot the given fraction? in a line, and divide as ninny of them as possible, by any number v\bich will leave no remainder, and set down the quotients, and the undivided numbers below. Repeat the operation till there be no two numbers which can be divided without a remainder. Then the product of all the divisors, and the quotients in the last lines will give the least common denominator. Divide this least com- mon denominator by each of the given denominators se- parately, and nmltiply the quotients by their several nu- mnators, their products will be the new numerators. Reduce % I, ",|, to the least common denominator. 3y5. 9, 3. 1 , then 3X5X3X1X1=45, is the common 5, 3, 1, I non denominator, and 45 divided by the given denomi- nators, 5, 9, 3, 1, give 9, 5, 15, 45; these multiplied by the given nunierators, give 27, 35, 165, 135, for new numerators, and the fractions will stanU S» H, ^^» *^« REDUCTION OF FRACTIONS. 1^8 Reduce I, 1, « ti and 1, to the least common denomi- nator. 3)3, 4, 5, 6, 8 The least denominator is, accordingly, ————— 3 X4'X2x5== 120 • 4)1, 4, 5, 2, 8 120-J-3, 4, 5, 6, 8=40,' 30, 24, 20, 15 40X^5 30X3; 24x2; 20X4; 15x3, 2)1,1,5,2,2 for new numerators; therefore the fractions required are ^o Z, ^y Z^ i^. 1,1.5,1,1 ^ Case VII. To reduce a fraction of one denomination to the fraction of another denomination of equal value. Rule. ( 1 .) When it is from the less to a greater deno- mination, '• Multiply the denominator by all the deno- minations from that given to the one sought." Thus, to reduce 4 of a penny to a fraction of a pound, 3 3 the answer will be — -= - — . 4 X 12 X 20 960 (2.) When it is from a greater to a less denomination, '' Multiply the numerator by all the denominations, from that given to the one sought.'* Thus, to reduce ^ of a pound to the fraction of a farthing, 6X20X12X4 5760 7 7 Ex. 1. Reduce ^ of a farthing to the fraction of a pound. Answer, 8^. 2. Reduce 9 of a penny to the fraction of a shilling. Answer, i^* 3. Reduce g of a pound to the fraction of a farthing. Answer, ^. 4. Reduce ^ of a pound to the fraction of a penny. Answer, ^. 5. Reduce l\ of a pound to the fraction of a farthing. Answer, *T- 6* Reduce 3 shillings to the fraction of a pound. Answer, ^ 13 140 REDUCTION OF FRACTIONS. 7. Reduce 9 of a dwt. to the fraction of a lb. Troy; Answer, g^o* 8. Reduce 5 of a cwt. to the fraction of an ounce. Answer, 1280 oz.=80lb. 9. Reduce ^ of a week to the fraction of an hour. Answer, "J^ 10. Reduce 4 of a mile to the fraction of a yard. Answer, 1320 yards. 1 1 . Reduce s of a pipe to the fraction of a gallon. Answer, ^*. 12. Reduce * cent to the fraction of a dollar Ans. 200 dollar. Caes VIII. To find the value of a fraction in numbers of inferior denomination. Rule. Multiply the integer, or its value in the next lower denomination, by the numerator, and divide by the denominator : 3 X 20 Thus, the value of i of a pound is equal to =12 £ X 12 5 shillings, and § of a shilling equal to = 8 pence. • 3 ^*" Ex. 1 . What is the value of 9 of a pound ? Ans. llsAld. . 2. What is the value of ^ of a shilling ? Ans. lojc?. 3. What is the value of ^ of half a crown ? Ans. 18c?, 4.. What is the value of ^ of a lb. Troy ? Ans. 9 ounces. 5. What is the value of ^^ of a cwt. ? Ans. 72 lb. 6. What is the value of 9 of a mile ? Ans. 977^ yards. 7. What is the value of f of a cwt. ? Ans. 48 lbs. 8. What is the value of ^ of a dollar? Ans. 4l|cts. 9. What is the value of 7 of a hogshead of wine ? Ans. 54 gallons. Case IX. To reduce a complex fraction to an equivalent simple fraction. REDUCTION OF FRACTIONS. . H7 Rule. If the numerator or denominator, or both, be whole or mixed numbers, reduce them to improper frac- tions ; and multiply the denominator of the lower frac- tion into the numerator of the upper, for a new numera- tor, and the denominator, of the upper fraction into the numerator of the lower for a new denominator. ^ 4 * 4X8 52 *o ^0 4 ' I I 7X1 1 5 I 50 . 51 'i 4r. 9 ? G3 51 And — = — = — . And — ~ — = — . And again — 51— 8 ? 64 ■ 3^ f 23 3^ !^ 147 — = . No. other varieties can occur. '^ 96 31 Ex. i. Reduce — to a simple fraction. Ans. ie 4 3 4 2. Reduce — to simple fraction. Ans. ^^. 2, Reduce to a simple fraction. An. ^. 1% ^ 4. Reduce to a simple fraction. Ans. ^. DO Ex. 5. Reduce — to a simple fraction. Ans. j^ 6. Reduce — to a simple fraction. Ans. i^i 5 7. Reduce — to a simple fraction. Ans, ^l 7 148 ADDITION OF FRACTIONS. 4 8. Reduce to a simple fraction. Ans. i^ • 19^ ADDITION OF FRACTIONS. Rule. Reduce mixed numbers to improper fractions , and compound or complex fractions to simple ones, and bring them all to a common denominator. Add all the numerators together, and write the sum over the com- mon denominator. Ex. Add I, 3, 5*, and I together ; which is thus per- formed : I I, ", I 3X3X2X4= 72' 2X5X2X4= 80/ Therefore ^^ + ^ + Ifo ■I ] I X 5 X 3 .X 4 = 6601 -^ Z == f^l = 7,^ = 7^, 1X5X3X2= 30 r which is the answer. 5X3X2X4= 120 ; This may be performed by bringing the given fractions to the least common denominator. 2)5, 3, 2, 4, Thus, % I, ") « then , and the new deno- 5, 3, 1, 2, minator zi 60 ; the fractions will beiS + 62+M+w=? 60 = «60' Ex. 1. Add^, ?, and ; together. Ans. l=r« S. Add t, I, and I together. Ans. 5^^^ S. What is the sum of i, *, and 4>l? Ans. 5% 4. Add together 3^, 41, and I. Ans. 8',^ 5. Add ^, «, 21 and 5l together. Ans. 9^ 6. What is the sum of 7i, 3*, and |. Ans. 1 1|', 7. What is the sum of 7 of a guinea, g of a shiU ling, and g of a penny ? Ans. 0/. 9s. 5Jrf. 8. What is the sum^ of g of a pound, 7 of a shilling, and ^ of a penny ? Ans. 0/. 5s ^ Q^d. |i I SUBTRACTION OF FRACTIONS. 149 Ex. 9. What is the sum of 4 of a guinea, 3 of a shil- ling, and 10 \ ? Ans. ]3^\l. 10. How m:ny pounds are there in 83 cheeses, each containinor £5^ lb, Ans. 2 18^. DIVISION OF FRACTIONS. RuLv. Reduce the fractions, as in MuUiplication ^ tht >n ID ert the divisor, and proceed as in Multiplication : thus, I to be divided by 9. 3 . 3 3w9 27 9 ^"^ » - — ft ^ 3 \B fi« Ex. Divide^ of 4,^hy^of 1. 3 23 3 1 3 XS.'i 3* 8 5 7 48X5 4X7 =s>^;;,'t= i6io the answer. EX.VMPLES. tx. I. Divide If of 12 by I. Ans. 7lf, 2. Divide A of 8 by ^. Ans. 22^. 3. Diviile f by ^,J. Ans. 5^. 4. Divide I of 64 by J. Ans. i^iO. f- ^ivide|of 12byS.?. Ans. jf?. ^":':;^l«f 36by3i. Ans. 8^. Ans. fg. 7. Divide ^^ 4bvlof 2. M|i^i^%»'|b^lof^ Ans 9. Divide ^ of ^ of 5bv?of L 10. Divide /, of ^ by ,Vof 5. 11. What nu .Tiber multiplied by f will give 9* ? Ans. 15 An&. «^7: 16* 1100* Ans. 2. What number multiplied by ^ of 3 will give 56 ? An^ 44|» 152 PRACTICE. 13. What number multiplied by ^ of f of 15 will produce! of 4? Ans. i^g. 1 4. From 5 subtract ^ of f of 4| and divide the re- mainder by 4. Ans. l^^. 13. What is a person's share of a prize of jL.2O,0uO gths of which is to be divided among; 13 per- sons? Ans. 1230/. 15s.~4|rf. ^^ PRxiCTICE. Practick is a method of finding the value of any quantity of goods, from the p'rice of an integer being given. ALiquoT PARTS of any number or quantity, are such as will exactly' divide it without leaving a remainder: thus 7 and 4 are aliquot parts of 28, 4 pence is an ali- quot part of a shilling, and 5 shillings is an aliquot part of a pound. TABLES OF ALIQUOT PARTS. IMOt parts of a L. Parts of a shil. Parts of 3 pence. S. d. d. q- 10 ==i 6 = i 8 1 4 '4 6 8 =3^ 4 = I 1 r °. 5 4 3 2 = I = 1 4 = S 6 = i 3 =. 1 2 = I 1 =r 13 4 — la , Parts of a penny. \ = t =/o Farts 4 *= 4.- 8 =i of a sixpence. 4 -.*» * = 1 3 = ?« i = i =i PRACTICE. 163 I. When the price is less than a penny. Rule. Divide the quantity hy the aliquot parts in a penny, and the quotient by 12 and 20. ^ Ex. What is the value of 7853 yards of tape, at I per yard ? 7853 392')* 1963^ 12)58891 2,0)49.0 91 Answer, X.24 10 9^ ■ EXAMPLES • L, s. d. Ex7l.456rat^peryd. Answe r, 4 15 11 2. 6784 at i per lb. 14 2 3. 3976 at i 12 8 6 - 4 7655 at ^ per yd. 15 18 m ^5. 7486 at 1 per lb. S3 7 101 6. 9984 at 1 20 16 7. 6327 atiper yd. 19 15 51 8. 5934 at i per lb..' 6 3 n 9. 7585 att ./ 15 16 01 10. 4767 at 1 per yd. / 4 19 31 11. 6493 at 1 per lb. 20 5 91 12. 5388 at i 16 16 9 II. When the price is an aliquot part of a shilling. Rule. Divide the s:iven number hy the aliquot part, aad tliis quotient by 20 : the answer will be in pounds. 154 PRACTICE. Ex. What is the value of 2785 lbs. of salt at 4i. perjb.?- W. 2785 2.0)92.8 4 Answer, LA6 8 4 Ex. i. s. d. 1. 3764at2(£. Answer, 31 7 4 2. 5943 at 3d, 74 5 9 3. 4953 at V^d. 30 19 n 4. 5943 at 4d. ' 99, 1 5. 3987 at 3d, 49 16 9 6. 5964 at \d. 24 17 7, 5684 at Ad, 94 14 8 8. 2105 ^tQd, 22 10 10 9. 3456 at 2i. 28 16 10. 3924 at I|(i. 37 6 11. 5904 at 2(/. ^ 49 14 12. 5215 at4f/. 86 18 4 JII. "When the price is pence and farthings, and no aliquot part of a shilling. RuLK. (1.) Find what aliquot part of a shilling is nearest to the given price, and divide the proposed num- ber by it. (2) Consider what part the remainder is of this aliquot part of the given price, and divide the for- mer quotient by it, &c. (3) Add the several quotients together, and the answer will be in shillings, which di- vide by 20 to bring into pounds. Ex. What is the value of 4277 yds., at i02^. per yd.?- 6 .1 4277 3 1. 2138 6 34 |\l069 5 i i 534 7i 89 H 2,0^ 38:vl 5| Answer, L. lyi 11 ^i ^^^■V FRACl :icE. 1 d. L. s. d. Ex. 1. 4784 at U , Answer, 24 18 '4 . 2. 5964 at \% 43 9 9 3. 4659 at 2* 43 13 62 4. J765at 2\ 16 10 ill 5. 4305 at 21 49 6 62 6. 3694 at 35 50 Si 7. 7641 at 2^ 79 11 \ol 8. 9875 at 6^ 267 8 11* 9. 5476 at 10^ 245 5 7 10. 3592 at 3^ 52 7 8 11. 3046 at 62 85 13 4* 12. 32l4at 111 154 1 13. 8764 at 3^ 136 18 9 14. 5921 at 1\ 178 17 3* 15. 5178 at 91 204 19 3 16. 9714 at 41 182 2 9 17. 5643 at 8^ 199 17 1* 18. 4932 at 10* 210 12 9 19. 8934 at 5\ 195 8 71 20. 2458 at 9i 99 17 1* 21. 8764 at 11| 429 1 5 22. 5687 at 51 136 5 Oi 23. 1435 at 10* 62 15 71 24. 5842 at 7i 176 9 6* 25. 5943 at 9| 235 4 10* 26. 187 6 at 2| 21 9 11 27. 43 16 at 7^ 139 7 5 28. 1956 at 82 71 6 3 29. 4235 at 5\ 97 1 01 30. 1327 at 9^ 52 10 6* 31. 2748 at 11 125 19 32. 9374 at 7 J 283 3 5* 33. 4285 at ll^ 200 17 2* 34. 1594 at 3* 23 4 11 35. 5632 at 5' 117 6 8 36. U14at 5* 25 10 ? 155 156 PRACTICE. ^ ' IV. When the price is more than one shilling, and Iqss than two. Rule. Let the given number stand f©r shillings, and work for the pence and farthings as bt?fore. ^ _, Ex. What is the value of 1187 quartern loaves, at Is' lid, each ? Ex H i 11187 i I 6" 1 J48 4^ 24 al 2 ,0)136.0 \l Answer L.68 l\ s. d. L. s. d. '. 1. 3456 at 1 21 Answer, 208 1 6 2. 487«> at 1 51 355 10 10 3. 5792 at 1 8^ 494 14 8 4. 2632 at 1 35 172 14 6 5. 4092 at 1 71 328 4 3 6. 2.-96 at 1 ]0 237 19 4 7. 4735 at I n 325 10 7| 8. 3724 at I 91 333 12 2 9. 3451 at 1 61 269 12 21 10. 7321 at I 72 602 9 12 11. 5928 at 1 11 568 2 12. 6542 at 1 81 S6r, 12 21 13. 8465 at 1 9^ 758 6 5^ 14. 4371 at 1 31 282 5 10^ 15. 8937 at I SI 586 9 92 16. 1234 at 1 11 lis 5 2 17. 5629 at I n 322 9 102 18. 4516 at 1 2 263 8 8 19. 5678 at 1 2^ 348 19 21 20. 9272 at 1 42 647 9 J 21. 5461 at 1 7 432 6 7 2S. 8234 at 1 51 600 7 11 23. 5i)28 at 1 lOl 555 15 24, 8750 at 1 5 619 15 10 r PEACTIOE. 1^7 When the price is any number of shillings under 20. Rule. (1.) If the price is an even number, multiply the given quantity by half the said number, doubling the first figure to the right hand for shilling^;, and the rest are pounds. (2.) If the price is an odd number, find for the greatest even number, as before, to which add the j^th •:, of the given number for the odd shUling, and the sum is the answer. / Ex. What is the value of 3456 yards of cloth, at 18s, per yard ? 34.56 9 Ans. L. 3110 8 Ex. What is the value of 2592 yards of second cloth? at lis. per yard? l = ^\ 2592 5 1296 129 12 Answer L. 1425 12 EXAMPLES. Ex. I. S'^rs at 2 Answer s. I. 6'^75 at 2 2. 4374 at 3 3. 5916 at 4 4. 7691 at 5 5. 6743 at 6 6. 9430 at 8 7. 6734 at 10 8. 594f) at 11 9. 3004 at 7 10. 2yc^5 at 13 11. 4392 at 14 12. 5931 at 19 L. S. 597 10 6J6 2 1183 4 1897 15 2022 J8 3772 2867 3270 6 1051 8 1907 15 3074 8 5634 14 PRACTICE. s. L. s. 13. 491 r at 18 Answer 44.25 6 14. 3271 at 9 1471 19 15- 9515 at 17 7917 15 16 2514 at 16 £011 4 17. 1392 at 10 696 18. 54S2 at 19 5160 8 VI. AVhen the price is shillings and pence. Rule. (1.) If they are an aliquot part of a pound, divide the quantity by that part, and the quotient is the answer. (2.) If they are not an aliquot part, multiply by the shillings, and take parts for the pence. Ex. What is the value of 2769 yards of Irish, at 3s. 4d. per yard ? 3s. 4d. I 2769 Answer L. 461 10s. Ex. "What is the value of 3758 yards of muslin^ at I2s« 9d, per yard ? 6 3 1 1 3756 12 * 4-072 1878 J 938 Answer 2.0)47889 L. 2394 9s. EXAMPLES. %. S 1. 8943 at S 2, 3532 at 4 S. «67l at 7 4. 2524 at 3 5. 5971 at 5 6. 5460 at 7 . d. 5 Answer 6 91 10 6 L. 894. 706 3251 478 1741 2047 s. 6 8 12 10 10 10 d. 6 10 ^^B' PRACTICE. s» d. L. s. d. Ex. 7. S764 at 10 Answer 1 882 8. 5638 at 8 n 2513 12 2 9. S7+5 at 9^ 11 1856 17 U 10. 8756 at 15 10 6931 16 8 11. 31)42 at 4 5 870 10 6 12. 2475 at 16 8 2062 10 16. 5642 at 18 4i 5183 11 9 14. 1764 at 5 8 499 16 15. 5931 at 17 6 5189 12 6 19. 9J43 at 6 8 304-7 13 4 17. 7 189 at 3 7 1288 7 18. 4004 at 19 6 4488 18 159 VII. When the price is pounds and shillings, or pounds; shillini^s, pence, and farthings. Rule. Multiply the quantity by the pounds, and work the rest by the foregoing rules. Ex. What is the value of 5428 hogsheads of ale, at 4>l, 12s, per hogshead ? 5438 4 12 21712 3256 16 Answer, L. 24968 16 Ex. What is the value of 2714cwt. of sugar, at 5U 12s. v9^. per cwt. ? 10s. 1 2 2714 S 8142 25. 6d. 1 4 1357 3d, 1 .10 339 5 t a 1 33 18 6 5 13 1 Answer, L, 9877 16 7 / leo PRAGTICE. L. s. d. L. s. d'. Ex. 1. 5674 at 5 17 6 33334 15 2. 6431 at 4 8 4 28403 11 8 3. 3416 at 5 11 6^ 19U54 17 6 4. 4^31 at 9 4 45365 4 5. S146 at 10 12 9 334'>5 11 6 6. 4316 at 10 19 Gl 4737 7 1 10 7. 5648 at 12 13 71447 4 8. 1436 at 10 10 6 15113 18 9. 1346 at 3 13 4 4935 6 8 10. 2714 at 18 9 500." 3 6 11. 9614 at 4 14 6 45426 3 12» 5789 at 7 7 7 42717 19 11 13. 1590 at 12 12 20034 14. 6341 at 8 18 6 56593 8 6 15. 4S03 at 9 9 91 45583 9 5i 16. 3465 at 8 15 50.3 18 15 17. 7<8£ at 11 12 10 83610 9 18. 1604 at 4 11 10 7S65 8 VIII. If there be a fraction in the given quantity* Rule. Work for the whole number, according to the preceding rules, to which add 1,2,1,85 &c. of the price, according to the nature of the question. Ex. .What is the value of 5354| cwt. of soap, at U. 4^. Sd. per cwt. ? 4s. Bd. 5354 I 4 21416 1070 16 178 9 3 3 4 6 4 4 8 2 2 4 1 2 2 L.3 3 6 Answer, L.22668 8 10 Ex. L. s, d. L. s. d, 1. 4562i at 3 15 9^ Ans. 17289 6l— 3 2. 6744^ at 9 9 lU^ 64030 11 II* 3. 26541 at 7 15 4 20618 11 2 4. 73941 at 12 8 SJ 91949 1 10* 5. 465 U at 5 12 10 26240 16 0^ §. 3749^ at 16 9 5 61757 7 9^ PHAeTICE. 161 Z. s, d. Ex. 7. 3875 at 8 18 a^ 8. 43()5l at ll 11 11 9 9724i at 6 16 4l 10. 36t8i at 4 4. 6^ L. , s. Ans. 34596 9 50624. 10 66307 4 15426 6 7 3^ Aliquot part of Aliquot parts a ton. ot a cwt. cwt. qr. lb. qrs . lb. 10 ozz I 2 0- I 5 0=1 1 - I 4 z: I 10 - ,* 2 3 12 — ^ 14-^ 2 2 — 1 8- A 2 ::= /o 7-^ 1 0— i Aliquot parts of a lb. oz. TABLES OF ALIQUOT PARTS. Aliquot parts of a qr. of cwt. lb. 14 -1 7 —1 * =7^ ol 1 •>9 8 * 14 13 1 *4 — 16 1 -^ IX. When the given quantity is of several denominations* Rule. Multiply the ^iven price by the highest deno- mination, as in Compound Multiplication, and take parts of the price for the inferior denominations of the given quantity. Kx. What is the value ^23 cwt. 3 qr. 21 lb. of hops^ at 4/. \8s. Qd. per cwt. ? 2 qr. 1 qr. I4lb . 71b. l\ 18 d. 6 11 54 3 Here, for the 22 cwt, I multi- ply by 1 1 and by 2 ; then 1 take parts lor the 3 qi-s. 21 lb., accord- ing to the preceding table, and by case Vlll. 108 7 ZZ value of 22 cwt ^ 2 9 3 :ZI ditto 2 qrs. 1 4 n ZZ ditto 1 qr. 12 Si Z: ditto 14 lb. 6 \i- irz ditto 7 lb. iUis. L. U:^ 19 4 — 1 14* 162 PRACTICE, cw4:. qrs. lb. dolls, cts. Ex, I. 8 '2 14 at 20 50 per cwt. Ans. 176 ('oils. Slfcfs. 2. 16 1 21 at 14 80 per cvt. Answ«r, 243 «Jo11s 27^ cts. 3. 37 3 22 at 12 ll 7 per cut. Answer, L. 477 6 8. 4. 73 2 10^ at 3 16 9 per cut. Answer, L, 282 8 S], 5. 38 1 16 at 2 12 6 per cwt. Answer, L 100 15 7^. 6. ^i 2 8 at 39 3 8 per cwt. Anf^wer. L. 1315 8 % 7. 84 3 14 at 12 11 8 por cwt. Aoiwer, L. 106S 2j. />. s. d. 6. 56 tons, 4 cwt. S qrs. Olb. at58 7 6 per ton. Answer, L. 3282 2 8^, 9. 39tons, 12 c^S^t. 1 qr. 14lb. at23 12 8 per ton. Answer, L. 10J5 1 1 2. 10. 124 tons, 16 cwt. 2 qr. 16lb. ail2 18 7 per tun. Answer, L. 1613 19 6 nearlv, 11. 16 lb. 8 oz, 12tlr. - - at 4 3 6 per lb. Answer, L. m 1 7\, 12. 25 lb, 12 oz. 4 dr. - - st 8 12 6 per lb. AHlswer, L. 222 4 6i. IS. 35 lb. 4 oz, 12dwt.- - at 1 1 9 9 per lb. Answer, L. 4.06 9 3^ 14. 48 lb. 8 '>z. 16dwt,- - at 14 4 4 per lb. Answer;,/., 692 16 5|. 15. 25 lb. 6oz. 5dwt. - - at 15 3 9 per lb. Answer, L. 387 I J Hi, 16. 18 yds. 2qr. Snails- - at 16 b per yd. Answer, L, \5 11 5*. 17. 55yds. 3 qr. Snails- - at 1 3 9 r# yd. Answer, L. 66 7 Oi. 18. l5acF. 3rd. 24 per. - - at 38 3 6|jeraiT« Answer, L. 606 19 7^ §, 25 acr, 1 rd. 4 per. - - at 22 50 per acr. Answer, 568 dulU, 68| ts. 20, 59 acr 3rd. 18 per. • - at 36 25 per acr. Answer, 1317 Uuii&. lliaCtS* ( 1G3 TARE AND TRET. Tare and Tret are a set of practical rules for de- ducting; c»'rtuin allowances, made by wholesale dealers in selling their goods hy weight. Guo'^s Weight is the whole wei2;ht of goods, includ- ing package, or whatever contains them. Neat Weight is what remains after all allowances are made. Take is an allowance to th#buyer, for the weight of the package, and is either at so much per barrel, chest, &c , or at so mucii per cwt., or at so much for the whole. Tret is an allowance of 41b. in every i04lb. for waste, vlust, &c., or the i^ part of the whole. Cloff is an allowanci, after Tare and Tret are de- ducted, of21h. upon every 3 cwt. that the weiglit may hold good when sold by tiie retail. Suttle is when only part of the allowance is deduct- ed from the gross. I'hus, after the tare is deducted from the gross, the remainder is calhd tare suttle. Case 1. When the tar-e is so much for the whole. KuLE. From the gross weight subtract the tare, and ihe reinaiuder will be tiie neat weight required. . 164 TARE AND TRET. Ex. What is the neat wei^'it of 25 barrels af in(Ti^(iy weiuihin;^ 116 cvvt. 2 qr. 14 lb., allowing 2 cvvt. 3 qr. 12 lb. tare ? Gwt. qr. lb. IJ6 2 14 2 3 12 Answer - 113 3 2 neat weio;ht. Fx. 1. What is the neat weit;ht of 55 barrels of figs^ weighing 35 cwt. 2 qr. !J lb., tare beiny; allowed at I cwt. I qr. 24 lb.? Ans. 34 cwt. qr. 19 lb. Ex. 2. What is the neat vvei;^ht of 20 casks of Ilussiair tallow, weighing 74 cwt, tare be*ng allowed at 2 cvvt. 2qr. 3 lb.? Ans- 71 cwt. 1 qr. 23 lb. Case II. When the tare is at so much per barrel, chest, &c. Rule. (\.) Multiply the tare by the number of hogs- heads*' barrels, chests, &,c. subtract the product from the )rvnss, and the remalider will be the neat weight re- quired : or (2.) Subtract the tare of each parcel fron^. the given weight, and multiply by the number of parcels. Ex. What is the neat weight of 8 hhds. of tobacco, each weighing 4 cwt. 2 qr. 24 lb. gross, tare being allowed at 2 qrs. 4 lb. per hhd. ? cvvt. qr. lb. qr. lb. 4. 2 24. 2 4 8 8 Gross weight 37 2 24 4 1 4 Tare. 4 1 4 a Answer - 33 1 20 rieat weight* Ex. 1. What is the neat weight of tS^frails of Mala- ga raisins, each weij>hing 2 cwt. 3 qrs. 12 lb., when the Ure upon each trail is 17 ib.? Aua. 07 cwt. 2 qr. 15 ib« TARE AND TRET. 16^ Ex. In 79 barrels of fi2;s, each weiojhinj; ] cwt. 12 lb. and tare 91b. per barref, what is the neat weight ? Ans. 81 cwt. C qr. 13 lb. neat weight* Ex. 3. What is the neat weight of 24 hhtls. of tobac- co, the weight of each being 4.^ cwt., and tare 67 lb. per hhtU? Ans. 93 cwt. 2 qr. iO lb. neat weight. Ex. 4, In 18 casks of currants, each weighing 6 cwt. 1 qr. 12 lb. and tare 61 lb per cask, what is the neat weight ? Ans. 104 cwt. 2 qr. 14 lb. neat weight. Case. III. When the tare is at so much per cwt. Rule. Take the aliquot part or parts of the whole gross weight that the tare is of a cwt , as in Practice, and subtract the result from the gross weight. Ex. What is the neat weight of 24 barrels of figs, each weighing 3 cwt. 2 qrs, 12 lb. and tare 12 lb. per cwt. B cwt. qr. lb. X 24 = 6 3 2 Tare, Ans 77 21 2 16 4 .86 2 8 oz. , 9 1 . 2 13^ X 4 lb. 8 cwt. 86 qr. 2 , lb. 8 — oz. 20 9 10 4i 9 1 2 5 2\ nt. wt. 13i [tare. Ex. 1. What is the neat weight of 21 barrels of pot- ash, each barrel weighinu; 1 cwt. 3 qr. 8 lb., tare being 10 lb. per cwt.? Ans. 34 cwt. 3 qr. 9^ lb. neat wt, Ex. 2. What is the neat vveight of 35 barrels of ancho- vies, each weighing 1 qr. 12 lb., tare at 14 lb. per cwt.? Ans. 10 cwt. 3 qr. 21 lb. neat weight. Ex. 3. Requrttd the neat weiu,ht of 15 hhds. of tobac- co, each weighing 4 cwt. 2 qrs. 12 lb, tare at 20 lb. per cwt. ? Ans. 56 cwt. 3 qr. 2 lb. neat weiglit nearly. 166 TARE AND TRET. Ex. Jn^What is the value of 26 hogsheads of tobacco,^ at 8/. 5|s. per cwt. each hogshead weighing 4^1 cwt.^ and the allowance for tare being J 3 lb. per cwt, ? Ans, 8 68/. 14.S. 6d, Case IV. Wlien there is an allowance both of tare»and tret. HuLE. Find the tare by the last rule, subtract it from the i;n)ss weight, the remainder or suttle, divided by 26, gives the tret, which being subtracted from the suttle^ gives the answer. ^ Ex. What is the neat weight of 15 casks of tallow, each weighing cwt. 2 qr. 12 lb., tare being 12 lb. per cwt. and tret as usual ? cwt. qr. lb. 6 2 12 X 15 = 5 X 3. 5 33 4 3 -6 lb. owl* qr. lb. 8 1 A 99 12 4 1 ^ 7 8 — 12 1 3 2 4— 6 Gross wt. 99 Tare - 10 2 12 12 3 2 1 17- 10 2 12—18 Answer 85 10 neat weight. Ex. I. In 18 cwt. 1 qr. 6 lb. gross, tare 63 lb., and tret as usual, how much neat ? Ans. 17 cwt. Oqr. 7 lb. neat weight. Ex. 2. In 14 casks of raisins, each 2 cwt. 14 lb. gross, tare 18 lb. per cwt., and tret as usuaL what is the neat weight ? Ans. 24 cwt. qr. 1 lb. neat weight. Ex. 3. In 9 chests of sugar, each weighing 8 cwt. 2 qr. 10 lb., tare 14 lb. per cwt, and tret as usual, what is the neat weight ? Aus. 64 cwt. 3 qr 24 lb. nt. wt.- TAllE AND TRET. 167 Case V. When cloftis allowed, Pule. Subtract the tare from the r,ross,^Rj(] the tret frOiii the taro suttle ; then liiviHc the tre." suttle by 168, and the result will he the iJlotf, which l>eirjir sub- tracte'l from the last suttle, gives the neat weigiit re- quired. Ex. What is the neat weij;ht of 19 cwt. 1 qr. 2 Jh. gross, tare 3 cwt. 3 qr. 2-1 lb., and tret and cloft* at the usual rate ? cwt. qr. lb. cvvt. qr. lb. Gross -19 12 4)14 2 26-T-168— 4x6x7 Tare - 3 3 22 26)15 18 6)3 -Z 20 8oz. Tret - 2 10 7)2 12 12 Tret suttle 14 2 26 9 13^ Cloff - 9 13 oz. Ans. cwt. 14 2 16 3 neat weight. Ex. 1. What is the neat weight of 224 cwt. 3 qr. 20lb. of tobacco, tare beinji; 25 cwt. 3 qr., tret and cloff' as usual. Ans. cwt. 190 I 14 neat wei•^!lt. Ex. 2. Inl4hhds. of tobacco, each weighing 5 cwt. 3 qr. 17 lb. gross, tare 11 lb. per cwt., and tret and cloll* as usual, what is th€f/nrtit weij^ht ? Ans. cwt. 70 2 2 neat weight. Ex. 3. What is the neat weight of 15 casks of cur- rants, each weighing Si cwt. gross, tare 35 lb. per cask, tret and cloff usual? Ans. cwt. 74 J 14 neat weight. F.K. 4. In 9 chests of sugar, each containing 7 cwU 2qr. 12 lb. gross, tare 13 lb. per cwt.. tret and cloii as usual, what is the neat weight, and what is the value of it at 9^. per lb. ? Aus. cwt 67 3 14 nt. wt 256/. lis. 7d. ( 108 ) DECIMAL FRACTIONS. 1. Decimal, or Decimatfd Fpaotions, are such as always h.ivt I cith one or nsor^ (jf-he's for their deuo- iT)i'fi.*^ors. The denoiti';r»ator« are n«ner express«?(l. being uinlerstood to b^ 10, 100,1000, ifec, acrorditit: a^ the nu!>iprators consist ot" 1, (J comma before then>, as 2: 24; 211. 2. Ha (iecin»al consists of only t.ne fimirts one is sup- p. - d o be divided into ten equai partsi and the decimal rep! -sents as many ot those parts as fhe decimal fijiure expresses ; thus, J nu ans seventi-nths of an unit : If it copsist of two fi^;ures, one is supposed to be divided into 1XX> equal parts, of which the tlecir.tai represents as manjr as tl e figur** expresses : thus, .65 means sixty-five hun- dredths of an unit. 3. (Jypiiers to the right-hand of decimals cause no diffierence in their valut, for .5 ; .50 ; 500, are decimals of the saa.e vahie. ben.g each equal to \ ; that is, .5 — io5 .50 = 1^; .500 = ,^ ; hut if Ae cypliers are placed on the left-hand of aecid.als, they dinnnish their value in a ten-fold proportion, thus .3 ; .03 x .003, are 3-tenths, 3- hundredths ; S-rhousaudths ; and answer to tht vuigar fractious I,, i^, i^o, respectively. 4. A whole iiufi'l'er and decimal is thus expressed^ 85 74 which is 8.^>74 8504 equal to 85ioo = and 45.04 =» 85^^ = — - ^-c. 100 100 ( 169 > REDUCTION OF DECIMALS. Case I. To reduce a vulgar fraction to Q^ecimal of an equal value. ''^^* Rule Divide the numerator of the fractrtwEi'increased by a cypher, or cyphers, by the denominator, and the quotient will be the decimal sought. Keduce |, ^, ^^ jV» to decimals of the same value. 1=10 =,.5. 1^1:00^.25. 1=^1.000^,125. tV= ^?°- = .0625. The cyphers avlded to the numerators are separated from the original figures by a dot, to shew that they are borrowed for the sake of forming the decimal. Ex. 1. What decimal expre^^sioas answer to the fol- lowing vulgar fractions, |, |, |, f , || ? Ans. |-ooo = 375. f ^°^ = .623. |--ooo = .875. f-^o = .222, &c- 11-000 = .733, &c. Ex. 2. Required the equivalent decimals of the frac- finnc 5 9 3 7 9 Ans. If = .2. -I 0000 ^ .5625. |-oo = .75. -^'Oo* = a363,&c. T?^ = 1 =.5. Ex. 3. What is the decin\al that answers to -^^ ? Ans. T>^ = 10^0000 ^ .015625. Ex. 4. What are the decimals answering to the frac- tions T-l^, aVV;* and ^^^ ? An^ {■^■s: = .0390625. |fg- = .05859375. ^^|^ = .019531, &c. Ex. 5. What decimal expressions answer to ^, -^ and /^Jj ? All!,. -} = .333, &c. -^ = .020202, &c. ^Vt- = .123123123, &c. Case 11. To reduce numbers of different denominations to their equivalent decimal values. Rule. (1) Write the given numbers under each other for dividends, proceeding from the least to the greatest 15 /: 4 12 20 179 REDUCTION OF DE«IMALS* (2) Place on the left side of each dividend, for a divisor, the number that will bring it to the next superior deno- mination. (3) Begin with the uppermost number, and set down the quotient of each division, as decimal parts, on the right hand of the dividend next below it, and so proceed to the last quotient, which is the decimal re- quired. Ex. Reduce 12s. 3^d. to the decimal of a pound. 3 qrs, I divide the | by 4, supplying cy- Sd.,75 phers to the 3 by the imagination ; 12s.. 3 125 the quotient is .75, which is placed _ by the side of the 3d.f and then di- .615625 > vide the 3SS by 12; the quotient, decimal of a L. J .3125, 1 set by the side of the 12s, «— and divide by 20, which ^ives .615625 for the answer : that is, if a pound were divided into 1,000,000 parts, the 12s. 3|rf. would be 615625 such parts, in the same manner as if a penny were divided into 100 parts, | would be equal to 75 such parts. Ex. 1. Reduce 8s. 4|d. to the decimal of a pound. Answer, .41875. 2. What decimal of a pound is 15s. 5|(i. .♦^ Answer, .77395833, &c. 3. What decimal of a pound is 4s. 6if/. ? Answer, .22604166, &c. 4. Reduce 18s. Od., 8s. 2d,, and 5s. to decimals of a pound. 1st. Ans. .925. 2d. Ans. .40833, &c. 3d. Ans. .25. 5. Reduce 5 oz. 6 dwts. 8 gr. troy, to the decimal of a pound. Answer, .443055. 6. Reduce 3 qrs. 7 lb. 8 oz. avoirdupois, to the decimal of a cwt. Answer, .816964. 7. Reduce 2 qrs. 1 n. to the decimal of a yard. Answer, .o625. 8. Reduce 3 pks. 1 gal. 2 qts. to the decimal of a bushel ? Answer, .9375. Case III. To find the value of any given decimal in ierms ©f the integer. This is the reverse of the last case. ADDITION OF DECIMALS' 171 Rule. Multiply the decimal by the number of parts, in the next less denomination, and cut off as many places to the ri2;ht-hand, as there are places in the given deci* mal, and so proceed through each denomination. Ex. What is the value of .615625 of a pound ? .615625 It may be observed, that as cyphers to the right do not alter the value in deci- mals, they are omitted in each step «f the operation. 20 12.312500 12 V^- 3.7500 4 3.00 Ex. Answer, 12s. 3|d. 1. What is the value of .625 of a shilling ? Answer, 7^ penc'^ 2. What is the value of .1275 of a pound ? Answer, 2s. 6|d. 4. 3. What is the value of .575 of a cwt. Answer, 2 qr. 8 lb. 6 oz. 6 dr 4. What is the value of .875 of a bushel ? Answer, 3 pks. 1 gal. 4. ADDITION OF DECIMALS. Rule. (1.) Arrange the numbers under each other, accordmg to their several values. (2.) Find the sum as in Addition of whole numbers, and cut off, for decimals) as many figures to the right as there are decimals in any on^ of the given numbers. \A 472 SUBTRACTION OF DECIMALS. Ex. What is the sum of 23.45, 7.849, 543.2, 8.6234 and 253.004 ? 23.45 7.849 543.2 8.6234 253 004 Answer, 836.1264 Ex. 1. What is the sum of 37.035, 4.26, 598.034, 9.3076, 4.321, and 5 ? AnswiM-, (J57.9o76. 2. Find the value of 39.33, 4.2056, .987 35, 46.x:87, 3,7491, and 8.004. Answer, 97.56305; SUBTRACTION OF DECIMALS. Rule. Arrange the numbers according to their value ; subtract as in whole numbers, and cut off for decimals, a^ in Addition. Ex. Subtract 35.87043 from 132.005. 132.005 35.87043 Answer, 96.13457 Ex. 1. What is the difference between I04.3'?6 and 74.05 ? Answer, 30 ^276 Ex. 2 Find the difference between 394.832 and 14H.0076. Answer, 246.8244 Ex. 3. From 3r2.97 1 take 270.30041 . Answer, 102.6705'9 ( 173 ) MULTIPLICATION OF DECIMALS. Rule. Multiply as in whole numbers, and cut off as many figures from the product as there are decimals in the multiplier and multiplicand, Ex. Multiply .025 by .045 : also 4.82 by 3.53. 'Ji „t„ In the first instance, there • _[ being but four figures in the -g^ ~ .^.g product, and six decimals in * 10 ^-'^^ multiplier and multipli- \±Afi cand, two cyphers must be * ***^ added to the left hand of the 100 2.410 .001125 17.0146 P'^^"^^- Ex. 1. Multiply 76.43 bv. 875: also .897 by .452. Answers, 66.87625— -.405444 Ex. 2. Multiply 324.004 by .7 872 Answer, 255.0559488 Ex. 3. What is the product of 9.57 and .074 ? Answer, .70818 Ex. 4. Multiply .643 Ijy .389 Answer, .250127 When the number of decimals in the multiplicand is large, and it is not wished to carry the operation to more than a certain number of decimals in the product, it is done by the following Rule, which I shall illustrate by an example. Rule. Having arranged the multiplicand, count as many figures from the decimal point, as you intend to keepdecirnals in the product, and make. a * over the last of these, under which, after you have inverted the multi- plier, place the units figure of the multiplier thus invert-?|^ ed, and the others in their proper order. Then multiply V^;- each figure of the inverted multiplier, beginning, as usual M 174 MULTIPLICATION OF DECIMALS. at the right hanH and set down the respective products s© that the right hand figures may fall in a straight line under one another. In multiplying, no attention is to be paid to the figures on the right hand of that which you multi- ply by, unless it be with the two preceding figures, td find what number should he carried. Ex. Required the product of 1.570796, multiplied by 26.3719, with four places of decimals in the product. This, in the usual method, would yield ten places of de- cimals ; by contraction it is thus performed. 1.570796 9.17562 314159 31 product with 2 regard being had to 2 x 6 94247 rz 6 — ' 6X9 4712ZI 3 3x7 1(,99 :zz 7 41.4246 263719 We will now work the example in the common way. 1.570796 From this it will appear plain, why in the contracted form the multiplier is inverted : the last product here being the first there. In the contracted form, the units place is 6 ; it would however be 8, if the 2 were carried from the 27, obtained in the next line bj Addition. Ex, 2. Multiply 128.678, by 33.24 sous to have but one |}lace ^f decimals. 14 137164 15 ro796 1099 557« 4712 388 94247 § 314159 2 41.4248 75 >324 DIVISION OF DECIMALS. Common method. Contracted method. 17S 128.678 38.24 128 678 42.83 514712 25^306 1029421. 386034 38603 10294 257 51 4920.64672 4920.5 DIVISION OF DECIMALS.* Rule. (1 ) Divide, as in whole numbers, and cut off as many figures in the quotient, as the decimal places in tiie dividend exceed those of the divisor. (2.) If there be notfij^ures enouj^hin the quotient, the deficiency must be supplied by prefixing cyphers, (3.) If there be a re- mainder, or there i)e more decim il places in the divisor than in the dividend, cyphers may be affixed to the divi- dend, and the quotient carried on to any extent. Divide 1.71 54 Hy 1.5 ; and .37046 by 16. 1.5)1.7134 16) i7046 In the first example, by ■ — _-^— - supplying a single cypher 1.1436 .02315375 there is no remainder left; but in the second I must sunplv three cyphers to obtain an even answer ; and I find the quotient has one figure less than there are deci- mals in the dividend *(» supplied. I must therefore prefix a cypher to the quotient found. KO'lE. * The Contracted method of Divison maybe thus performed. Rule. Having determined how many places of whole ly here will be ^n the quotient, if any, which is easily know section ; if there are nooe^ then consider of what v^ 176 Ex. DIVISION OF DECIMALS. Ans. 7.0343 nearly. Ans. 4936.7ii||. Ans. 5.64i77 nearly. Ans. .01992, &c. 1. Divide 25.64 by 3.645. 2. Divide 4732 by .9587. 3. Divide .865439 by .156. 4. Divide 79 by 3965. 5. Divide 3.3.64472 by 882. Ans. .038146, nearly, 6. Divide .218 by 7.435. Ans. .0293, &c. 7. Divide 76.42 by 58. Ans. 1.317, &C' 8. Divide 88 by .88. Ans. lOO. first figure in the quotient will be, and proceed as in common Division, only omitting one fij^ure of the divisor at each opera- tion ; viz. for every figure of the quotient dot off one in the divisor, remembering to carry for the increase of the figures cut off, as was done in Multiplication. Ex. Let it be required to divide 23.41 by 7.9863. Contracted method. i'ommon method. 7.9863)23 4100(^2.9312 | 7.9863)23 410U(2 9312 15.9726 15.9726 Here it must .74374 be observed, that 71876 in each of the subtractions ex- .2497 cept the first, .2395 unit mustbe car- -— ried to the first .101 figure, as would 79 be the case in — — the usual course. .21 15 w5- .74374 71876 .2497,30 239 189 .101 79 .21 15 410 865 .470 t'726 .5 5744 ( 17/) REDUCTION OF DECIMALS, To chang;e the currencies of the different states to "Federal money, and Federal money to currency by de- cimals. 1. — To reduce Maryland, Pennsylvania, Delaware, and New Jersey currencies to Federal Money. * Rule. Reduce the given sum to the decimal of a pound, and divide by .375 the quotient will be the answer. EXAMPLES. Fx. 1. Reduce 76i. Hs. 6d* Maryland currency to Federal money ? 12)6 2,0)14.5 .375)76.725(204.6 or 204 dols. 60 cts. 75 1725 1500 — \ 2250 22j0 Ex. 2. Reduce 237/. 17s. 4i. Pennsylvania currency to Federal money ? Ans. dols. 634.3111 &c. Kx. 3. Reduce 673^ Is. 2d, New Jersey currency to Federal money .? Ans. dols. 1794)8222 &c. Ex. 4. Reduce 7/. 6s, 8rf. New Jersey currency to Federal money ? Ans. dols. 19.5555 &c. * NOTE. As 7s 6rf. of this currency make a dollar, reduce it to the decimal of a pound, and it will be .375/. the divisor ghren in this ruler. 178 HEDUCTION OF DECIMALS. 2-— To change Federal money to Maryland, Penn- sylvania, Delaware, and New Jersey currencies. Rule. Multiply the given sum by .375 and the pro- duct will be pounds, which reduce to shillings and pence. EXAMPLES. Ex. 1. How much Maryland currency in S76.50 ? 76.50 •375 38250 53550 22950 L. 28.68750 20 13.75000 12 9.00000 Ans. 28i. 13s. 9d, Ex. 2. Change 744 dols. into Pennsylvania currency ^ Ans. 279Z. 3. Change 365.25 dols. into Pennsylvania cur- rency ? Ans. 136^ 19s. ^d. Ex. 4. Change 627.75 dols. into Mairvland currency ? Ans.'^235^ 8s. \\d. 8. — To change New England and Virginia currencies to Federal money. * Rule. Reduce the given sum to the decimal of a pound, and divide by .3, the quotient will be the answer. * As 6 shillings of this currency make one dollar, reduce 6 shillings to the decimal of a pound, and it will be .3, the di- visor given in the rule. KEDUCTION OF DECIMALS. 179 EXAMPLES. Ex. I. imU, 68. 8rf. New England currency, how much Federal money ? 1 2)8 2,0)6.6' ,3)74.333' &c. g 247.77' &c. Ex. 2. In 64Z. J 5S' Virginia currency,' how much Fe- deral money ? Ans. dols. 215.833 &c. Ex. 3. In 3271. l6s. ^d. Virginia currency, how much Federal money ? Ans. dols. 1092.722 &c. Ex. 4. In 463^ 12s. 9d. Virginia currency, how much Federal money .^ Ans. dols. 1545.45833 &c. 4 To change Federal money to New England and Virginia currencies. Rule. Multiply the given sum by .3, and the pro- duct will be pounds, which reduce to shillings and pence. examples. Ex. 1. Change 273.35 dols. to New England currency ? 273.25 .3 b 1.975 20 19,500 12 6,000 Ans. 81/. 19s. 6d. Fix. 2. Change 496 dols. to New England currency? Ans. 148/. 16s, 3. Change 79.50 dols. to Virginia currency .? Ans. 2.d. 17 s, 4k Change 673.60 dols. to Virginia currency ? Ans. 202/. Is. 7.2rf. 180 REDUCTION OF DECIMALS. 5. To change New- York and North-Carolina curren- cies to Federal money. * Rule. Reduce the given sum to the decimal of a pound, and divide by .4-, the quotient will be the answer. Examples. Ex. 1. In 74L l6s. New- York currency how much Fe^ deral money ? 2,0) 1 6 • .4)74.8 gl87 Ans. Ex. 2. In 29^ 17s. New-York currency, how much Federal money ? Ans. g74.625 Ex. 3. In 365/. 7s. 4d. New-York currency, how much Federal money? Ans. g9l3.4l()6. &c. Ex. 4 4y7/. 16s. \0d. North Carolina currencv, how much Federal money ? Ans. gl244 604166, &c. 6. — To change Federal money to New-York and North Carolina currencies. Rule. Multiply the given sum by .4 and the product will be pounds, the decimal parts of which reduce to shillings and pence. Examples. Ex. 1. Reduce S49 50 to New-York currency ? 49.50 .4 19.800 20 I6.U00 Ans. 19?. 165. ♦ Note. As 8 shillings of this currency make one dollar, re- duce 8 shillings to the decimal of a pound, and it will give .4 the divisor given in the rule. REDUCTION OF DECIMALS.' 181 Ex. 2. Reduce g246 to North-Carolina currency ? Ans. 98/. 8s. Ex. S. Reduce S418.75 to New-York currency ? Ans. 167/. 105. Ex. Reduce &847.60 to New-York currency ? Ans, 339/. 09, 9.6d. 7. — To change South-Carolina and Georgia currencies to Federal money. * Rule. Reduce the given sum to the decimal of a pound, multiply by 30 and divide the product by 7 ; the quotient will oe the answer. Examples. Ex. 1. In 69/. I5s.6d, South-Carolina currency, how much Federal money ? 12)6 2,0^15.5 69.775 30 ) 2093.250 299.03571428' Ex. 2. In 864/. l7s. 2d. South Carolina currency, how much Federal money ? , Ans. 23706.535714285'. • Note. — As reducing the currency of these states to the decimal of a pound, would produce a circulating de- cimal, I have formed this rule on the principle of Vul- gar Fractions. 4s. Srizz^V or 1^ of a pound, consequently the proportion will stand thus -^qL : \doll. :: pounds : dollars, or as 7 is to 30 so are pounds to dollars, agreeably to tht rule. 16 182 REDUCTION OF DECIMALS. Ex. 3. In 9271. 16s. 9d. Georgia currency, how much Federal money ? ^ , Ans. S3976. 4.4642857 1 Ex. 4. In 6.73^. 12s. 8d. Georgia currency, how much Federal money ? , Ans. 2886.99 or 2887 dollars. 8. — To change Federal money to South-Carolina and Georgia currencies. Rule. Multiply the given sum by 7, and divide that product by 30, the quotient will be the answer in pounds ; the decimal parts of which reduce to shillings and pence. Examples. Ex. 1. How much Georgia currency in %2i6.50 ? g216.50 7 3,0)151,5.50 50.516' circulates 20 10.33' &c. 12 3.99' &c, Ans. 50/. lOs. 4<^. Ex. 2. How much South-Carolina currency in 8467.25 ? Ans. 109^. Os. 6d. Example 3. How much South-Carolina currency ing762.S? Ans. 177/. 17s. 4.8rf. Example 4. How much Georgia currency ing939.7? Ans. 219/. 5s. 3.19'd: 9. To change Canada and Nova«Scotia currencies to Federal molfiey. REDUCTION OF DECIMALS. 183 * Rule. Reduce the given sum to the decimal of a pound; and divide by .25 the quotient will be the answer ? Examples. Ex. 1. Reduce STL 16s. 4rf. Canada currency to Fe- deral money .^ 12)4. 2,0)1,6.3 .26)87. 8l6'(S351.26'&c. 75 128 or thus, 125 {.5)87.816' .25 (.5)1' 31 t,5)n56i 25 g35l.26' &c. 66 50 166 150 16 Ex. 2. Reduce 827^ 15s. Nova-Scotia currency to Federal money .^ Ans. g33ll. Ex. 3. Reduce 268^ 12s, Sd. Canada currency to Fe- deral money .^ g 1074.45. Ex. 4. Reduce 71 9^ 9s. 2d. Canada currency to Fe- deral money ? Ans. gC877.83' 10. To change Federal money to Canada and Nova- Scotia currencies. ♦ Note.— 5 shillings of this currency make one dollar. The divisor in this rule is obtained by reducing" this sum to the de- cimal of a pound. 184 REDUCTION OF DECIMALS. Rule. Multiply the given sum by .25, and the pro- duct will be pounds, the decimal part of which reduce to shilliDgs and pence. EXAMPLES. Ex. 1, In 68.5 dols. how much Nova Scotia currency } 68.5 .25 3425 1370 17.125 20 £.500 \t C.OOO Ans. 17/. 2s. 6c?. Ex. 2. In g 124.25 how much Canada currency ? Ans. 3U. Is. Sd, Ex. 3. In g7648 how much Nova Scotia cur- rency ? Ans. 19I2Z. Ex. 4. In 8867.35 ho much Nova Scotia cur- rency? Ans. 216/. 16s. 9d. Note. — The shortest method of working the exam- ples in this currency, is to multiply the given number of pounds by 4 for dollars, and to reduce dollars to pounds divide by 4. — This number is used because 5ihillingsare ^^ or 1 of a pound. ( 185 ) INVOLUTION, Involution is a method of raising numbers to higher powers. A power is the product arising from multiplying any- given number into itself once, or oftener : thus, 3x3 ZZ 9 is the second power of 3, and it is denoted in thi» manner 3*. The number denoting the power is called the index^ or exponent of that power : thus, in 3^ the 2 is the in- dex or exponent. The third power of 4- is 4^ :^ 4 X 4 X 4 ZI 64 The fourth power of 3 is S*iz3x3X3X3:^8l The sixth power of5is5«lz5x5x5x5X5x5 [ZI 15625. The third power of i is | ^ — i X ^ X J ZI ^V- The fifth power of .03 is .03* ZZ .03 X .03 X .03 X .03 X .03 = .O0OOCUO243. EXAMPLES. Ex. 1. What is the sixth power of 6 ? Ans. 46656 2. What is the eighth power of 7 ? Ans. 5764801 3. What is the fourth power of | ? Ans. /^V 4. What is the fifth power of J.? Ans. i||gi 5. What is the third power of .25 ? Ans. ,015625 0. What is the fourth power of .05 ? Ans. .00000625 186 INVOLUTION. 7. What is the third power of .305 ? Ans. .028372625 8. What is the ninth power of 9 ? Ans. S87420489 9. What are the squares of 3 and 6 ; 5 and 10 6 and 12; 2, 4, 8, and 16? Ans. 3* = 9 52 = 25 6* = 36 6* = 36 102 ^ 100 122 ^ 144 22 = 4. 42 = 16 82 = 64 162 = 256 10. What are the cubes of 3 and 6 ; 5 and 10; 6 and 12 ; 2, 4, 8, and 16? Ans. 33- = . 27 63 = 216 c= 8 X 27 53 = 125 103 = 1000 = 8 X 125 - 63 = 216 123 = 1728 = 8 X 216 23 = 8 43 = 64 = 8 X 8 83 = 512 = 8 X 64 163 -_ 409a = 8 X 512 (187 ) EVOLUTION. Evolution is the method of extracting root9. The root of any number, or power, is such a number, as being multiplied into itself once, or oftener, produces that power : thus 3 is the square root of 9, because 3 multiplied into itself gives 9 : 4 is the cube root of 64, because 4 multiplied into itself twice, gives 64. The roots are denoted by indices, or exponents, in this man- ner : The cube root of 125 is -^^mIs ~ 5. The square root of 81 is ^ Slzz9, The fifth root of 243 is ^/Hs zz 3. Ex. 1. What are the square roots of 49 and 64 ? Answer, 7. 8. 2. What are the cube roots of 216, 343, 512, and 729 ? " Answer, 6. 7. 8. 9. 3. What are the fourth roots of 625, 2401 , and 4096 ? Answer, 5. 7. 8. 4. What are the fifth roots of 3125 and 32768 ? Answer, 5. 8 To extract the square root. Rule. (1.) Divide the given jumber into periods of two figures each, by placing a dot over units, another over hundreds, and so on. {2.) Find the greatest square in the first period, and set its root on the right-hand, as a quotient figure in division. (3 ) Subtract the square thus found, and to the remainder annex the succeeding period for a new dividend. (4.) Double the root for a divisor, and examine how often it is contained in the dividend, exclusive of the place of units, and.put the ■«*. 188 EVOLUTION. result into the quotient and in the units place of the divi- sor. (5 ) Multiply the divisor thus increased hy the new quotient fig^ure, and subtract the product from the dividend. (t>.) Bringdown the next period, find a divi- sor as before, by doubling the figures already in the root, and proceed as before. The rule will be rendered clear by the follcvin^ examples: What are the square roots of 16777216 and 43040321 ? 167771^10(4096 16 809). .7772 7281 8186)49:16 49116 43046721(6561 36 125)704 6-25 1306):"967 7836 13121)13121 13121 EXAMPLES. Ex. 1 . What is the square root of 1 1 7649 } ' Answer, 343. 2. What is the square root of 262144 ? Answer, 512. 3. What is the square root of 531441 ? Answer, 729. 4. What is the square root of 1679616 ? Answer, 1296, EVOLUTION. 18^ Ex. 5. What is the square root of 5764801 ? Answer, 2401. 6. What is the square root of J 073741 824. ? A;)9wer, 32768. 7. What is the square root of 1 195506G9121 ? Answer, 345761. 8. What is the square root of 20 ? Answer, 4.4721, &c. 9. What is the square root of 300 ? Answer, 17.3205, &c. 10. What is tlie square ro©t of 1000 ? Answer, S1.622, &c. 1 1 . What is the square root of | : -^^ ; -^^ ? ♦ Answer, .7071, &c. .4082, &c. .4166. 12. What is the square root of .25 ? Answer, .5. MISCELLANEOUS EXAMPLES. Ex. 1. A gentleman desirous of making his kitchen garden, which is to contain 4 acres, a complete square, I demand what will be the length of the side of the garden ? Ans. 139 yards. Ex. 2. Six acres of ground are to be allotted to a square garden ; but for the sake of more wall for fruit, there is to be a smaller square within the lar;>;er, which is to contain 3 acreSj 1 demand the length of the sides of each square ? Ans. ouJ;er 170.41 yds. inner 120^nearlj. Ex. 3. What is the mean proportional between 12 and 75 ? Ans. 30. Kx. 4. flow long must a ladder be to reach a winilow 30 feet high, when the bottom stands. 12 feet from the house ? Ans. 22.31 feet. To extract the cube root, I. UuLE. (1) Find, by trials, the nearest cube to the given number, and call it the assumed cube. (2) Say as twice the assuuied cube added to the given num- ber, is to twice tiie number added to the assumed cube, 80 is the root of the assumed cube to the root required nearly. 190 EVOLUTION. What is the cube root of 27455 ? Here the nearest root that is a whole nnmber is 3<^, the cube of which is 27000 : therefore I say, As 27000 X "2 + 2745.5 : 27456 X 2 + 27OO0 : : 30 or 81455 : 8l9lO :: 30 : 30.1675. It is evident that the true root, omitting the last two figures, is somewhere between 30.16 and 30.17, the for- mer being too little, the latter something too large. By taking the root thus found 30.16, as the assumed eube, and repeating the operation, the root will be had to a still greater degree of exactness, Ex. 1. What is the cube root of 15625 ? Ans. 25 2. What is the cube root of 140608 ? Ans. 52 3. What is the cube root of 444194947 ? Ans. 763 4. What is the cube root of the difference be- tween 140608 and 14625 ? Ans. 50.13 nearly. II. Rule. (1) Separate the given number into periods of three figures each, beginning from units place ; then from the first period subtract the greatest cube it con- tains, put the root as a quotient, and to the remainder bring down the next period for a dividend. (2) Find a divisor by multiplying the square of the root b}"^ 300, see how often it is contained in the dividend, and the answer gives the next figure in the root. (3) Multiply the divisor by the last figure in the root. Multiply all the figures in the root by 30, except the last, and that product by the square of the last. Cube the last figure in the root. Add these three last found numbers to- gether, and subtract this sum from the 20 94 . 12i767 30 ,0027397 45 ,0061643 70 ,00958i^0 95 ,0130137 21 ,0028767 46 ,0063013 71 ,0097i>60 96 ,0131506 22 ,00.30137 47 ,0064383 72 ,009863> 97 ,0132876 23 ,0031506 48 ,0065753 73 ,01000CO 98 ,0134246 24 ,0032876 49 ,0067121 74 ,0101369 99 ,0 35616 25 ,0014246 50 ,0068493 75 ,0102739 100 ,0136986 204 COMMISSION AND BROKERAGE. Rule. Multiply the figures corresponding with the number of days by the sum : Thus, if the interest of 75l. for 61 days be required : I find opposite to 61, the number .0083561, wliich multi- plied by 75, gives .6267075 of a pound, which reduced, is iQs.aid. Ex. I . What is the interest of 155Z. for 49 days ? Ans. 1^ Os. 9|rf. nearly COMMISSION AND BROKERAGE. Commission is an allowance of a certain sum per cent, to a correspondent or agent, for buying and selling goods for his employer, or to a banker {or drawing bills and managing accounts. Broker A«E, though of a difierent name, is of the same nature as Commission. Ex. 1. A salesman a^ Smithfield, in the course of a year, sells for his correspondents 1 120 loads of hf.y, at the average price of 5/. lOs. per load ; and 620 loads of straw, at 55s. per load : 1 wish to know t!.e c .mmission money, at ^i per cent .^ Answer, 176/. 1 9s. 3d, COMMISSION AND BROKERAGE. 205 1120 620 H = what the hay sold for« m 5600 560 L.6l60= 1705 L.7865 2i 1240 310 155 1705 = : what the straw [sold foi;, 15730 1966.25 176.9625 20 19.25 12 Answer, 176^. i9s. Sd. 3.00 Ex. 2. A Manchester manufacturer allows his agent in London i-} per cent, for goods sold by him ; in the course of the year 1807 he sold to the amount of 15,400/., what was his commission for that year, an^ how much was the agent's clear gains, supposing his losses on the year's account, by bad debts, amounted to 225/. 10s. 6d, ? Ans. 654/. 10s. Od. Com. 428/..l9s. brf. clear gains Ex. 3. A Liverpool merchant sells goods in a year, for his American correspondents to the amount of 144,454/. lOs., on which he reckons his clear gains at the rate ot | per cent., what is his income on this one concern? Answer, 54 iZ. 14s. Id Ex. 4. What is the commission of g 1026.50, at 3 J per cent ? Ans. 38 dolls. 49 cts. 3.75 mis. Ex. 5. A bookseller in London allows his agent in America 5 per cent, commission ; what does he pay him for the remittance of 8540i, 15s 9d, ? Answer, 427/. Os. 9ld, 18 20d BISCOUNT. Ex. 6. What is the brokerage of gl210, at ^ per cent. ? Answer, 3 dolls. 2 cts. 5 mis. Ex. 7. What is the claim of a broker at 3| per cent. on gl550.50. ? Ans. 52 dolls. 32 cts. 9.375 mis. Ex. 8. What is the commission on glOOO atf per cent. ? Answer, 6 dolls. 25 cts. Ex. 9. What have I to pay my broker for the sale of goods to the amount of 9950/. 95., at li per cent.? Answer, 124/. 7s. 7^d. Ex. 10. What will the commission of a country banker amount to on 12314i. 8s. 9d., at | per cent. ? Answer, 15/. 7s. lO^dr. Ex. II. What is the brokerage of 1526/. iSs, 6d,, at 1^ per cent. ? Answer, 22/. I8s. DISCOUNT. Discount is an allowance made for advancing mo- ney on securities before they are due. The presen^t worth of any sum, due sometime hence, is such, as if put to interest for that time at the rate per cent, given, would amount to the given sum. Rule.— As the amount of 100/. or dollars, at the rate and time given is to KfO : so is the given sum to the present worth. The present worth taken from the given sum will be the rebate or discount. or thus, for the discount ; As the amount of 100/. or dollars, at the rate and time given, is to the interest of the same sum at the same rate and time, so is the given sum to the discount required. I DISCOUNT. 207 Kx. 1. What is the present worth and discount of 620 dollars, due 4 years hence at 6 per cent, per annum discount ? 6 4 ____ • 24 Interest of S 100 at 6 per cent, for 4 years. 100 124 Amount of glOO for 4 years at 6 per cent, 124 ; 100 :: 620 100 g 124)62000(600 620 00 g620 500 present worth. §120 discount. Proof. or thus ; 500 124 ; 24 : : 620 6 24 30.00 2480 * 1240 120.00 124)14880(120 500 124 2620 248 248 620 120 discount. 500 present worth. 20S DISCOUNT. Ex. 2. What is the discount of g7 18.75 for 5 years at 5 per cent, per annum ? Ans, 143 dols. 75 cts. Ex. 3. What is the present worth of 1092^. 13s. due 5 years henc6 at 6 per cent, per annum ? Answer, 846/. 10«. Ex. 4. What is the present worth of 284 dols. 28 cts. due 8 months hence at 4V per cent, per annum ? Answer, ii76 dollars. . Ex. 5. What is the discount of 250/. 10s. 6d. due 2 years and 4 months hence, at 6^^ per cent per annum ? Answer, 31/. 17s. 8|:/. Ex. 6. What is the present w^orth of 1000/. due 3 years and 7 months hence, at 5| per cent, per annum ? Answer, 8S9/. 3s. 2d. /j\% Ex. 7. What is the present worth of G40/. l0>. due 10 years and 2 months hence, at 4^ per cent, per an- num discount ? Answer, 43Q/. 9s. 0^/. i|| Ex. 8. What is the discount of 740 dols. 30 cts. due 7^ years hence, at 6j per cent, per annum ? Answer, 242 d«ls. 68 cts. -/-j^g- Ex. 9. What is the present worth of 500 dollars, one half payable in 6 months, and the other half in 8 months, discount at 6 per cent, per annum ? Answer, 483 dols. 10 cts. -^\W Ex. lO. AVhat diflference is there between the interest of 600 dollars for 1 year and 9 months at 6 per cent, per annum, and the discount of the same sum at the same rate and for the sametime ? Ans. 5 dols. 98 cts. |^|f Discount in business is generally reckoned in the same man- ner a» common interest. . 1 j-r When the sum is not very large, and the time short, the dit- ference between the discount and the interest is a mere trifle ; but when the sum is large and the time considerable, their dif- ference then becomes essential, and the sum should be calculat- ed on correct discount principles. (209 ) PROFIT AND LOSS Is a rule that discovers what is gained or lost on thifc prime cost in the purchase and sale of goods, and it tea- ches how to to fix the price of their goods so as to gain so much per cent. Questions in this rule are performed bj the Rule of Three Direct, upon this principle, that quantities, or sums of money, which gain or lose at the same rate, are to one another as their gains or losses. Ex. 1. A tallow chandler has this day purchased mot- tled soap, at 102s. 6d. per cwt.,at how much per lb. must he retail it out to gain 10 per cent, profit ? L. s. d. : 110 :: 102 6 -i- 112 102 6 2000 2£0 1 100 .- 5^ 2.000)11.275 5.6375 L. 5.6315 and =ls.^=:is. 0|rf. nearly, 112 Ex. 2. How much per cent, is gained at the rate of 2d. in a shilling ? Answer 16Z 13b. 4rf. Ex. 3. If 3 dollars he gained in selling at 21 dollars, at what rate per cent is that ? Answer I6f per cer* 18* £10 PAUTKERSHIP. Ex. 4. Three pounds of tobacco are bought'at 5*. 9(1. and sold for 7s. 6d., what is the gain upon the sale of what cost loo/. Answer 30/. 8s. S^c?. Ex. 5. Bought cheese at 3/. 3s. per cut., and sold it again at lO^d. per lb. : what is the gain per cwt. suppos- ing the loss in weight to be 4lb. per cwt. Answer, L. 1 11 6 gain per cwt. Ex, 6. Bought silk stockin^^s at ^4 25 per pair, what must thej be sold for to gain 20 per cent profit ? Answer, gS.lO. Est: r. If 375 yards of cloth be sold for 290i. and there be 20 per cent, profit, what did it cost per yard ? Answer, 12s. lOfo?. Ex. 8. If90 English Ells of Cambric cost 120 dolls, for how much must 1 sell it per yard to gain 18 per cent ? Answer, gl 25i|. Ex. 9. A plumber sold 5 fother of lead, for \02l2s.6d. (the fother being 19| cwt.), and gained after the rate of 12/. 10s. percent. : what did it cost him per cwt. ? Ans-vver, l8s. 7^^ Ex. 10. Bought 218 yards of cloth, at the rate of 8s. 6d. per yard, and sold it for lOs. ^d. per yard : what was the gain of the whole ^ Answer, 19/. 19s. 8d. Ex. 1 1. Paid 69/. for one ton of steel, which is retailed at Sd, per lb., what is the profit or loss by the sale of 12 tons? Answer X. 68 gain. PARTNERSHIP Is a general rule, by which merchants, &c., trading in company with a joint stock, are enabled to ascertain each person's particular share of the gain or loss, in proportion to his share in the stock. This rule divides itself into two parts, vi«. 1. Part- -irship without regard to time : and 2. Partnership with me. PARTNEKSHIP. 2U I. Partnership without Time. Rule, " As the whole stock is to the whole gain or loss, so is each man's share in the stock to his share of the gain or loss." Ex. 1. Two merchants embark in business, the one puts in as capital X.5550, a id the other L.34i20, and they gain in the first year i.l 260, what is each man's gain ? L.5550 3420 8970 —joint stock. 8970/. : 1260/. : : 5550/. : 779/. 12s. nearly ; of course the profits of the other are 1260/— 779/. 12s.=480/. 8s. Ex. 2. Three persons trade together : A puts in li/0/. ; B 150. ; C 200/. ; and they gain 900/.: what is each man's gam ? Ans. A 200--B 300 — C 400. Ex. 3. A, B, and C, enter into partnership ; A puts in S640/., B 4S20/., and C 5000/., and they gained 8670/.; what is each man's share in proportion to his stock P Ans. A 2344/. 13s. nearly — B 3104/. i 4s. — C 3220/.l3s. Ex. 4. Four merchants. B,C, D, and E, nmke a stock ; B put in 2270/., C 3490., D 11 50/. and E 4390 ; in trad- ing they gained 4280/. I demand each merchant's share ot the gain ? Ans. JB 859/. 16s. nearlv—C 1321/. 175. 6d D 435/. lis. 6d, nearly— E 1662/. \5s. Ex 5. Three persons, D, E, and F, join in company ; D's stock was 3750/., E's 2800/., .md F's 250O/., and at the end of 12 months they gained 3420/. ; what is each man's particular share ot the gain ? Ans. D 1417/. 2s. 6^fl/.-E 1058/. 2s. 5c/.— F 944/. 15s. |d!. J/. Partnership with Time. Rum. As the sum of the product of each man's mo- ney i.nd time is to the whole gain or loss, so is each man's product to the share of the gain and loss. 21S PARTNERSHIP WITH TIME. Ex. 1. Two persons lay out 1500^. in trade, m the proportion of 3 to 2 : that is, A put in 900L, and B 600l. ; A leaves his money in the concern, 9 months, and B docs not want his for 12 months : what profits belong to each, supposing they gain 250^ ? X..900 X 9 = 8100 600 X 12 = 7200 15300 15300 : 250 : : 8100 250 153.00)20250.00(132^. 7s. Ans. A*s share of profit L. 1 32 7 B's - - - - lir 13 Z<.250 Ex. 3. A puts into a concern 208O/. for 2 months, B 970/. for 5 months, and C 400^ for 15 months ; they gain among them 650/. ; what must each receive for his share of profit ? Ans. 180/. 3s. A*s profit nealy. ; 210/. B's profit; 259/. 17 s. C's profit.^ Ex. 3. Three merchants join in company for 18 months : D put in 500/. and at 5 months' end took out 200/. ; at 10 months' end put in 500/., and at the end of 14 months takes out 130/. ; E puts in 400/, and at the end of 3 months 270/. more ; at 9 months he takes out 140/., but puts in 100/. at the end of 12 months, and withdraws 99/., at the end of 15 months. F put in 900/., and at 6 months took out 200/.; at the end of 11 months puts in 600/., but takes out that and 100/. more at the end of 13 months. They gained 200/. I desire to know each man's share of the gain ? Ans. 57/. nearly D's gun ; 59/. 7s. 5d, E's gain } 83/. 12s, Id, = F's gain. ( 213 ) ALLIGATION Teaches to mix things of different values, so as to as- certain the price of the mixture. There are two cases in this rule. T. To find the mean value of a mixture composed of several quantities of ditierent values. Rule. Multiply each quantity by its respective value, and divide the sum of the products by the sum of the quantities. Ex. 1. A tea-dealer mixes 3i cwt. of tea, at 9s. per lb., with 2 cwt., at 7s. and 4-i cwt at 5s. 6d., at how much per lb. can he sell the whole mixture ? 3^X112 = 392") r392x9 ==3528 2 X 112 = 224. land-} 224 X 7 =1568 4J X 1 12 = 4-76 J i4>7Q X 5i = 2618 1092 )lTU{7s.0^d,f^ 7644 Answer - 7s. O^d, . .70 Ex. 2. What is a lb. of sugar worth which is com- pounded of 3 cwt. at 46s. : 2 cwt. at 59s. ; l| cwt. at 84^. ; and 56#. at 60s. ? Answer, 6ifl?. -fg-f. Ex. 3. What is the average earnings of workmen, 4 of whom earn 10 dollars each per week ; 8 earn 9 dollars each ; and 12 will get only 6 dolls. 50 cts. each ? Answer, 7 dolls. 91f cts. Ex. 4. A tobacconist mixes 80 lb. of tobacco at SOJ. per lb. ; 150 lb. at 2s. 3d. per lb. ; and 40 lb. at 3s. lOd. per lb. ; wiiat will be the value of the mixture per oz. ? Answer, I|d. nearly. 214 ALLIGATION. 11. To find how much of difterent things of different values, must be taken, in order to make a mixture of a certain mean value. Rule (1). Set down the names of the things to be mixed, together with their prices; then, finding the dif- ference between each of these, and the proposed price of the mixture ; place these differences in an alternate or- der, and they will shew the proportion of the ingredients. Ex. 1. Orange wine, at 9s. per gallon, is to be mixed with raisin wine at 6s. per gallon ; what will be the pro- portions, so as to sell the mixture at 7s, per gallon ? Proposed Orange - 9s, C price, r 1 "J A mixture therefore of these J 7 s. J f wines in the proportion of one j J r orange to two raisin, will be Raisin - (is.' (_ 2) the answer. Ex. 2. A spirit at 10 shillings, and another at 12 shil- lings per gallon, are to be mixed with low wines at 6 and 5 shillings, in order to produce a mixture worth 9 shil- lings per gallon ; what must the quantities of each be ? 3 Theansweris, 3 gallons at 16s., 4 4 at 1 2s. : 7 at 6s. ; and 3 at 5s. ; 7 will make a mixture that may be 3 sold for 9 shillings per gallon : for Spirit, - 16 Ditto, - ^1 Wine, - Ditto, - ^ ^ 3 X 16 = 48 4 X 12 =48 7 X 6 = 42 3 X 5 = 15 — • — — T53 17 I53and-— =9s, Pro«f, 17 Ex. 3. A tea-dealer would mix four sorts of tea toge- ther, viz. at 4s., 4s. 6d., 5s. 6rf., 6s., and 7s. per lb. ; in order that he maj sell the whole mixture at 5s. 6rf. per lb., what proportion of each will he use ? Ans. lA lb. at 4s. ; | lb. at 4s. 6d. 1 lb. at 6s, ; and 1 J lb, at 7s, ; and as much as you please at 5s. 6d. ^ at 48 X — 42 1 — 27 H — 24 lb. c^s. 1 at 48 1 — 42 3 — 27 2 — 24* ALLIGATION. 215 Ex. 4. How much cofifee at 48 cts., 42 cts., 27 cts. and 24 cts. per lb. will compose a mixture worth 30 cts. per lb. lb. cts. lb. cts. Ans. 6 at 48 3 — 42 or 12 — 27 24 — 18 OR THUS. lb. cts. Ans. 1 at 48 2 — 42 6—27 4 — 24 III. When the prices of all the things to be mixed are given, likewise where the quantity of one, and the mean rate are also given, to find the several quantities of the others. Rule. (1). Take the difference between each price and the mean rate as before. (2). As the difference of that thing, whose quantity is given, is to the rest of the ditferences severally ; so is the quantity given to the several quantities required. Ex. 1. A rectifier of compounds has 200 gallons of spirit that he can sell for 12s. 6d, per gallon, but he means to mix it with three other kinds of spirit at 1 3s. 4c?., at I5s., and 18s. 4i., per gallon, in order that he may sell the whole at 14s. 2i. per gallon ; how much must he use of each ? I reduce the several prices to pence, which stand as follows : 170 150--. 50 160.1 10 50; 10: : 200 ; 40 180^ 10 50 : 10 : ; 200: 40 220—' 20 50 : 20 : : 200 : 80 * Note.— A variety of answers can be obtained to these ques- tions, by linking tliem different ways, they may also be made infinite by multiplying or dividing any result by one common number. 40 at 13 4 = 26 40 at 15 = 30 80 at 18 4 = 73 216 ALLIGATION. The answer is ; to 200 gallons, at 12s. 6d., must be added 40 at l3s. 4d., 40 at l5s., and 80 at 18s. Ad. ; the truth of which is proved thus ; 200 at 1:2 6 = 125 13 4 6 8 360 255 and = 14i,. 2d. Proof. 360 Ex. 2. A grocer has 100 lb. of tea worth 4s. per lb. which he means to mix with others at 12s. 3d., 10s., and 6ii. per lb. ; in order to sell the whole at 8s. how much of each must be used ? Ans. 100 lb. at 4s. ; 100 lb. ^t 12s. 3d. ; 200 lb. at 10s. ; and 212^ lb. at 6s. IV. When the price of each thing is given, also the quantity and the mean rate, to find how much of each sort will make that quantity. Rule. (1). Take the difference between each price and the mean rate as before : then (2). As the sum of the diflferences is to each particular difference, so is the quantity given to the quantity required. Ex. 1. A wine merchant means to mix 860 gallons of wine to sell for 8s. a gallon, out of other wines that he already sells for 12s., 9s. 6s., and 5s. per gallon, how much must he take of each ? . . 8 1 12-^ 3 10: 3 :: 860: 258 2 10 :2 ::860: 172 10: J ::860: 8S 10:4;: 860 : 344 2- 6 9 2 6^ 1 Sum of differences = <0 The answer is 258 gallons at 1 2s. ; 172 at 9s. ; 86 at 6s. } and 344 ai 5s. per gallon, /iiay be mixed and sold at 8s. per gallon. rosiTioN". 217 Ex. 2. A ^oKlsmit'n has (our sorts of goh],viz. ofS4, 10, IS, and 15 carats fine, vvis!ies~ 125 oz. of the. fineness of 17 carats, how much will he want of each sort ? Ans. 14. 02. 16 dwt. 1 1 /^ e;r. of 24.. ; 7 oz. 8 dwt. 5j-f gr. oflO. ;5l oz. 17 dwt. 15^\ gv. oi IS. ', 51 oz. 17 dwC l5,Vgr.ofl5.? Ex. 3. A drug grinder has hark worth l6.s. per lb., some at lOs., and some at 4s. ; but he is desirous of making up two parcels, viz. one containing a cwt. at 9s., a«d the other 84- lb. at 12s.; what proportions of each must be used ? Ans. 43J.J lb. at 16s. ; 8-^^^ lb. at 10s. ; 60^\ lb. at 4s. for 112 lb. 5 48 lb. ; 12 lb. ; 24 ib. for 84 lb. ? POSITION Position, or as it is sometimes called, the Rule of False, is a rule, that by means of any supposed num- bers, others thart are true, and that answer to the terms of the question, are found. There are two kinds of Po- sition, viz. Single and Double. Single Position is performed, by using a supposed number, and working with it as the true one, till the real number is found. Rule. Take any number and perform the work with it, as if it were the right number : then say, As the re- sult of this work is to the position, so is the result in the question to tlie number required. Ex. 1. A person counting some guineas, being asked how many he had, repli-ed : " If you had as many, and as many more, and half as many, and one quarter as many, you would have 26 k" How many had the per- son who was counting his gold ? 19 SIS- POSITION. By way of supposition, I take 80 as the number; then, by the terms of the question, it will be 80 96 As many more, 80 220 : 264 : : 80 96 Half as "many, 40 80 48 ith. as many, 20 24 220)211 20(96 Ans. 220 264 Proof. Ex. 2. A person after spending A, ^, and -}, of his money, finds he had 500/. left, wiiat was his original propel ty ? I take a number divisible by 2, 4, and 6, for the sup- position, viz. 60. Suppose 60 60 — 55 ZI 5, tiierefore Proof. — As 5 : 60 : : 300 ^ == 3000 4- 30 60 A = 1500 15 10 5)30.000 55O0 55 Answer, L.6.O0O 500 rem. Ex. 3. Three persons bought goods at Baltimore, which cost 600 dollars. The first person was to have a third part more than the second, and the third a fourth part nK)re than the first ; what was each man's share .? Ans. g200 first person's share, Si 50 second share, . g250 third share. Ex. 4. In a leaky vessel there were three pumps of ilitferent capacities ; the first would enipty the hold of the ship in 20 minutes, the second vvould require double that time, and the third would not perform the business in less than an hour ; how long would all three together take in doing it ? Answer, 1 1 minutes nearly. ( '^19 ) DOUBLE POSITION QuKSTiONS in this rule are resolved by rnakino:supp<>- sitio?is of two nusiibers, whicli maif both prove false ; ill that case the errors are made to correct each other. Rule. (1.) Place each error against its respective po- sition, and multiply them cross ways. (2.) If the errors are alike, that is, both greater or both less than the given number, take their difference for a divisor, and the dif- ference of their products for a dividend. But if unlike, take their sum for a divisor, and the sum of their pro- ducts for a dividend, the quotient will l)e the answer. Ex. 1. Three persons have obtained the 2O,000Z. prize in the lottery, and it is to be so divided, that the second is to have 600^. more than the first, and the third 800/. morg than the second, what is each person's share ? Suppose the first had 5000 Suppose the first had 5600 Then the second had 5600 The second had 0200- ajid the third had 6400 The third had 7O0O 27000 too little by 300O 1 8800 [too little by 1200. 3000 5000") fSOOO X 5600 ZZ 16800000 X Vthat is,-< 1200 5600j (1200 X 50OO n GOOOOOO Difference of Products - lObOOOOO = [dividend. 3000 ^ 1200 = 18(0 (diff of errors) for a divisor. 10.800.{)00 GOOO Therefore, . — = L.COOO dtiOO 1800 7400 L. 20.000 Proof 220 COMPOUND INTEREST AN1> ANNUITIES. Ex. 1. A genfleman at Christmas, wished to give se- veral poor families 5 shillings each, but he found he had 16s. 8(7. too little; he then gave them 3s. 6c?. each, and found he had 4s. 4i. left, how many families we-re there? Answer/ 14 families. Ex. 3. A person purchased a house and land, togeth- er with a carriage and horses, for 13 000 dollars; he paid 4 times the price of the carriage and horses for the land, and 3 times the price of the land for the house, Vhatwas the value of each separately ? Ans. gCOO carriage and horses, 825400 land, g 12,000 house. COMPOUND INTE^^EST AND ANNUITIES. Compound Interest, or interest «pon interest, is that which is paid not only for the use of the money lent, but also for the use of the interest as it becomci: due. There are two methods of working Problems in this. Rule, viz. by Common Arithmetic; and by Decimals j 1 shall give examples under each. 1. liy Common Arithmetic, Rule. 1. Find the amount of the given principal for the time of the first payment by simple interest. (2.) Consider this amount as the principal for the second pay- ment, the amount of which is to be calculated as before, and so on through all the payments to the last, stdl reckotiing the last amount as the principal for the next payment. COMPOUND INTERES-n 221 Ex. 1. What is the amount of 550L for three years, at 5 per cent, compound interest ? 20)550 iiven principal. 27 10 first year's interest. 20)577 10 second year's principal. 28 17 second pear's interest. 20)606 7 6 third year's principal 30 6 4-| third j'ear's i..terest. Answer - 63(> 13 10| V.x, 2. What is the amount of 400^ for four years, at 5 per cent, compound interest ? Ans. 48(1^ 4s. O^d, Ex. 3. What is the compound interest of COO dols. for 5 years at 5 per cent per annum .^ Ans. Si 65.7689375 II. By Decimals, Rule. 1. Find the amount of \L for a year, at the given rate per cent. 2. involve the amount thus found, to such a power as is denoted by the nun»ber of years, o. Multiply this power by the pri;icipai or i^iven sum, and the product Mi!! be the amount required. 4. Sub- tract the princip.il from the amount, and tlie remainder will be the interest. Ex. 1. What is tite compound interest of 550l, for 3 years, at 5 per cent, per annum ? 1.05 = amount of l7. for a year, at 5 per cent. ; Then 1.05 X 1.05 X 1.05 = 1.157625, and 1.157025 X 550 == f)35 69375 ^ amount^ 636.69375—550 = 86.693:5 — 86/. ISs. lO-J-f^. Ex. 3. Wliat is the amount of ! 00 dols. for 4 yearSj at 6 per cent, per annum, compound interest ? Answer, gl26 247696. Ex. S. AVhat is the compound interest of 620/. for 6 years, at 5 per cent...^ Answer, 171L 58. lOd^ 222 COMPOUND INTEREST AND ANNUITIES. A TABLE. Shewing the Sum to which J^ or gl Principal will in- crease at 5 per cent. Compound Interest, in any num- ber of years not exceeding a hundred. Yrs Amount, Yrs. 26 Amount. Yrs. 51 Amount. Yrs. Amount, 1 1.05 3.555572 12.040769 76 40.774320 2 I 10:5 27 3.733456 52 12.642808 77 4-813036 3 1.157625 28 3.920129 53 13.274948 78 44953688 4 1.215506 29 4 11 6135 54 13.938696 79 47.201372 5 1.276^281 30 4.321942 55 14 635630 80 49.561441 6 1.340095 31 4.53803iy 06 15.367412 81 .52 039513 7 1.407100 32 4 764941 57 16.135783 82 54.641488 8 I 477455 33 5.U031S8 58 16.942572 83 S7:373563 9 1.551328 34 5.253347 59 17.789700 84 60 242241 10 1.62^894 35 5.516013 60 18 079185 85 63.254353 11 1.710339 •36 5.791816 61 19.613143 86 66.417071 12 1.795856 37 6 081 406 62 JO 593802 87 69 737924 13 1.885649 38 6.3S5i77 63 21.623492 88 73.224820 14 1.979931 39 6.704751 64 22 704667 89 76 886061 15 2.0789?8 40 7.03998^ 65 23.839900 90 80.730365 16 2.182874 41 7.391988 66 25.031395 91 84 766883 17 2.292018 42 7.761587 67 26.283190 92 89.005227 18 2.406619 43 8.149666 6S 27.597C-64 93 9.5 455438 19 2.526950 44 8..'} 57 150 69 28.97754^ 94 98.128268 2U 2-623297 45 8.985037 70 :0.425425 95 103.034676 21 2.785962 46 9 434258 71 .1.947746 96 » 08- 1864 10 22 2.925260 47 9 9u597] 72 33 545134 97 11 3. 59.0730 23 3.071523 48 10.4012g1 73 35.222390 98 119.275517 24 3.225099 49 10.92 i3.>;i 74 36 983510 99 125.239293 25 '3386354 50 11.467399 75 ^8.832685 100 131.5012.57 1. To find by means oi' tlie table what any sum will amount to in a given number of years. RuLB. Multiply the number in the table, opposite to the term of years, by the sum, and the product will be the answer. Ex. 1- To what sum will 500/. amount to in 44 years, at 5 per cent, compound interest ? Opposite to 44. in the table I find 8.5.07 150, this I mul- tiply by 500, and the answer is 4278/. 11*-. 6(i. COMPOUND INTEREST AND ANNUITIES. 22o Ex. 2. What will 350^ araount to in 25 years, at 5 per cent, compound interest ? Answer, 1185/. 4s. 5^d. nearly. Ex. 3. A prudent young man marries at the ajre of 22 ; the fortune which he has with his wife is 2500/., half of which he readily gives into the hands of trus- tees to be accumulated at 5 per cent compound inte- rest ; what will it araount to, supposing he lives 32 years, which he may reasonably expect ? Answer, 5956/. 3s. 6^d, Ex. 4. The year 1808, is tliat in which the late Mr. Pitt calculated there would be four millions surplus to be applied to the payment of the national dehi of En- gland : 1 demand how much this single four millions will accumulate in half a centurj', at 5 per cent, com- pound interest ? Answer, 45,8oQ,596/- (See other questions on this subject after the next table.) II. To find the number of years in which a given sum will increase to another given sum, in consequence of being improved at Compound Interest. Rule. Divide the latter sum by the former, and the sum in the table which is nearest to the quotient will shew the terms required. Ex. 1. In what time will 200?. increase to 1500/., if improved at 5 per cent, compound interest ? 1500 7.5. The nearest number in table I. to 7.5 is 2fO 7.391988, opposite to which is 41, the number of years. Of cuuise 2(i0/. in a little more than forty-one years wouhl, by beini; accumulated at compound interest, at 5 per cent., amount to 1500/. Ex. 2. In what time will 100/. increase to 600/ sanie rate of interest ^ Ans. 33 year^ Ex. 2. In what time will 8G0/. increase to K Ans. between 50 an ,^24 COMPOUND INTEREST AND ANNUIIIES Ex. 4. In how long would five millions be in pajino; ti.e national debt, which m January, 1806, was upwards of 580 millions ? Ans. between 97 and 98 years. Ex. 5. Admiral Rainier left, in 1808, 25,000/. to- wards paying off the national d(^bt, \. hen will if have accumulated to a million at 5 per cent compound in- terest i Ans. 76 years, nearly. TABLE II. Shewing the sum to winch 1/ per annum will increase at 5 per cent. Compound Interest, in any number of years not exceeding a hundred. YrsT Amount. Yib Amount Yis. 1 Vinouut. Yrs. 76 Amount. X 1,0000 26 51,1135 51 220,8154 795,4864 2 2,0500 27 54,6691 52 232,8562 77 836,2607 3 J,l>25 28 58,4026 53 245.4990! 78 879,0738 4 4,U01 29 6 \ 1227 ^4 r58,7739, 79 9-^4,0274 .5 5, 5256 iO 66,4388 55 272,7126 m 971,2288 6 6,8019 31 70,7608 56 287,318 81 1020,790''i 7 8.1420 ''o 7.S^^'98b 57 302,7j57 82 1072,.-.293 8 9,5491- oo 8','.u6.iB .^8 318,851^ 83 1127,4713 9 11,0 66 34 85, u 670 59 335, .■940 84 1184,8448 10 12,577^ ;.-; 90,3203 60 353,583;' 85 1245,0871 11 '.4,-:?068 jG 95,8363 61 372,2629 86 1308,34J4 12 15,9171 3? 101 ^281 62 391,8760' 87 1374,7585 13 17,7130 ;8 ; 07,7095 63 412,4698 88 1441, 4964 » 14 19,5986 39 114,095' 64 434,0933 89 1517,7212 15 21.3786 40 120,7998 65 456,7980 9(' 1 594,6073 16 23,6575 41 I27.83^h 66 480.6379 91 \675,'i377 17 25,,-'40l \42 135,2317 67 505,6698 92 1760,1045 18 :8,1328 43 142,993; 68 5.1 9583 93 1849.1098 19 30,53^/0 44 r 1,14-^0 69 559,5510 94 1942,5653 20 33,06^9 45 1 5-^,7(02 70 588.5285 95 2040,6935 ^1 33,7192 4f> 1-38.685^ 71 618,9549 96 2143,7282 22 38,505S 47 178,119'", 72 6'-.0,902r| 97 2251,9146 23 41.430.' .18 188,0254 73 684,4478 98 2365,6x03 24 44,5020 49 19.^\4267 74 719,6702 99 2484,7859 2.. 4. .7271 -0 2US.34.S 1 7.5 756,6h37 100 2610,0232 COMPOUND INTEREST AND ANNUITIES. 2S^5 I. To find in what time a given annuity will amount te a given sum at compound niterest. Rule. Divide the given sum hy the given annuity, and the number in the table nearest to the quotient will be the answer. Ex. 1. A person owes lOOO/. and resolves to appro- priate 201. per annum, to be accumulated at 5 per cent, per ann. compound interest, in how many years will the debt be paid ? ICOO ZI50. Tlie nearest number m table II. preceding 20 page, to 50 found, is 51.1135, and the number answering to this is 26, so that in less than 26 years a debt of 1000/. would be extinguished by laying by, and accunmlating, at compound interest, annually 20/ per annuiii. If the rate of interest had been 6 per cent 24 years would have paid the debt, but at 4 per cent, it would have taken be- tween 28 and 29 years. Ex. 2. How long will 75 guineas a year be in accu- mulating to 2000/. at the same rate ? Ans. in somewhat less than 17 years'. Ex. 3. In what tim^e will an annuity of 25/. amount to 3575/., at the same rate } ^ Ans. in little more than 43 years. Ex. 4. How long will the national debt, left at the time of Mr. Pitt's death, viz. 581 millions, be in paying off, supposing five millions annually be appropriated for that purpose, and the rate of compound interest 5 per cent. ? Ans. in less than 40 year? , Ex. 5. The national debt was, at Midsummer 1807, 756 millions of pounds, out of which the commissioners had redeemed \\7 millions and a half, how long would the remainder take in paying olf, if eight millions be ap plied annually, at the rate of 5 per cent, compound inte- rest for the purpose ? Ans. 33 years, 226 AND ANNUITIES. II. To find how much a given annuity will amount to in a given term, at 5 per cent, compound interest. Rule. Multiply the given annuity by the number in the table standing opposite to the given term of years. Ex. 1. I can lay. by 30/. per annum with its interest; that is, i can appropriate 50/. a year to be accumulated at 5 per cent, compound interest, how much shall 1 have saved if 1 live 21 years ? Opposite to 21 years I find S5.7192, which multiplied by 60, gives 1785.9600. Answer, 1783/. 19s. 2d. Ex. 2. How much will an annuity of 35/. amount to in 83 years ? Answer, 39461/. 9s. lOjc/. Ex. 3» To what sum will an annuity of 100 guineas amount in 19 years, at 5 per cent, compound interest ? Answer, 3206/. 12s. Ex. 4! To what sum will 60 dollars per annum amount to in 25 years, at 5 per cent, compound interest? Answer, g2863, 62 cts. 6 mis. III. The PRESENT VALUE of an annuity is that sum which, if improved at compound interest, would be suf- ficient to pay the annuity. — For this the following table is adapted. ANNUITIES. 227 TABLE III. Shewing; the present Value of an Annuity of ll for any number of Years not exceeding 100, at 5 per cent, per annum, Compound laterest. Yrs Value. Yrs. Value. Yrs. Valine. jYrs. | Value. 1 ,952381 26 14,375183 51 18,338977 76 19,509495 o 1,859410 27 14,643034 52 18,418073 77 19,532853 3 2,723248 :^S 14,89-14/' 53 18,493403 78 19,555093 4 3,545950 29 1-), 14 1074 54 18,565146 79 19,576284 5 4,329477 30 15,372451 55 18,633472 80 19,596460 6 5,075692 31 15,592810 56 18,698543 81 19,615677 7 5,78637S 32 l.;,80. 677 57 18,7605 i 9 82 19,638978 8 6,453213 33 16,002549 58 18,819542 83 I9,65l4u7 9 7,107822 34 16,192901 59 18,875754 84 19,668007 10 7J21735 15 16,374194 60 18,9292&0 85 19,683816 11 8,306^14 36 16,546852 61 18,1^80276 86 19,698873 12 8,863252 37 .6.711287 62 19,028834 87 19,713212 13 9,393373 38 16.86789;] 63 1j 075030 83 19,726869 14 9,898641 39 17,01701-1 64 19,119124 89 19,739875 15, 10,379638 40 17,139086 65 19,161070 90 19,752262 16 10,8:37770 41 17,29456 b6 19,24019 91 19,764039 17 11,274066 42 17,42320'^ 67 19 239066 92 19,775294 18 11,689587 4J 17.54591: 68 19,275301 93 19,785994 19 12,085321 44 17,662773 69 19,309810 94 19,796185 20 12,462210 45 17,774070 1 70 19,342677 95 19,805891 21 12,821153 46 17,880066 71 19,37397S 96 19,815134 22 1 13, 16 "^003 47 17,981016 72 19,403788 97 19,823937 23 13,488574 48 18,077158 73 19,4321.-9 98 19,832321 24 13,798643 49 18,168722 74 19,459318 99 19,840306 25 4(.> 93945| 50 18,253925 75 19,484-70 100 19,847910 To find the present value of an annuity for a term of years. Rule. Multiply the number in the table opposite to the given term of years, by the sum, and the product is the answer. Ex. I. What is the present value of an annuity of 126^. for 21 years ? In the table opposite 21 is 12.8!2ll53; this multiplied by 126, gives 1615.465278 ZZ 1615/. 9s. Si. 228 OHAXCES. Ex. 2. WItat is the present value of an annuity of 75 dollars for 12 3 ears, at 5 percent. ? Answer, 8664:74.39. Ex. 3. What present sum is equivalent to a nett rent ©f 45/. per annum for 84 years, allowing interest of money at 5 per cent. ? Answer, 883^ nearly. CHANCES.* Q_uestlon /.—Suppose a counter, having a black and a white face, be thrown up, to see which will be upper- most, after the counter has fallen to the ground^ and if the whit^' face appear uppermost, a person is to have 5 shillings, what is the chance, or probability, that he will be entitled to the five shillings ? Solution. Since either the black or the white face must be uppermost, there is an equal chance for the ap- pearance of either face, of course the chance, or the pro- bability, may be expressed by ~, or a bystander ought to give him 2s. 6d. for his chance of getting the five shil- lings. (Question II. — Suppose there are three counters put into a bag, one red, another white, and a third black; out of whicli.if aperson blindfolded take the red he is to have 5 shillings, 1 demand the value of t.ie chance, or what is the probability of his drawing the red counter } * It is meant only to give so much of the doctrine of chances, as shall enable the pupil to understand upon vvliat ground the doctrine of Anmiities, 8ic. depends. To illustrate this part of the subject, recourse will be had to some familiar instances, vjrhich may seem, at first sight, to lead to gaming ; but it is be- lieved, that the facts adduced must, if properly considered, de- ter young persons from tins pernicious and destructive vice, which IS too much encouraged by the almost perpetual drawing of state lotteries. CHANCES. 229 'Solution. lie has evidently one chance out of three, and therefore the probability may be valued at •}, and another person inclining to purchase his chance, ought to give for it the ^d of 5 shillings, or Is. Sd. In the former case, the chances for the event's hap- pening and failing are equal, and each being equal to |, the certainty is reckoned as 1, or unity. In this last ease, there is one chance for the event's happening, and two for its failing : in other words the chance for its happening is -J, and for its failing -|: here, again, the chances for the happening and failing are equal to unity, because ^ + f iz | z: 1. Question 111. — Suppose there are five counters, two white and three black, out of which, when mixed, a per- son blindfolded is to draw one of the white, and in that case is to be entitled to 5s., what is his chance for so doing, and what is his expectation worth ? Solution. It is plain here are five chances in the whole, of which there are two only out of five for taking a white counter, and the other three for taking a black one ; therefore the probability of winning may be expressed by the fraction f, and of missing -^, ai:d he might sell his expectation of the five shillings for fths of that bum, that is, for two shillings. Ex. J. At the conclusion of the last state lottery, when there were only five tickets left in the wheel, there were two prizes oli 50^ each, and three blanks, what was the value of one of those tickets ? ' Answer, £0/. Ex. 2. What is the value of one ticket when only five are left in one wheel, and in the other there is one prize of 100/. and four blanks .^^ Answer, 201. Ex. S. AVhat chance has the holder of a single lottery ticket of a prize, when there are three blanks to a prize ? Answer, 4 to 1. 20 230 CHANCES. . Question IV. — What is the prohability of throwing an ace with a single die, in one trial ? Solution. There are six faces to a die, of which one only is the ace, therefore the probability of throwing an ace with a single die in one trial is expressed by \ ; and the probability of not throwing an ace is | : here, as before, the chances for not throwing the ace, and that for throwing, are together equal to unity. Question V. — AVhat is the probability of throwing an ace in four throws ? Solution. "We must consider the probability of fail- ing in the four tlirovvs. 'J he probability of missing the first lime will be -^ ; so it is the second, third, and fourth times ; therefore the probability of missi.ig in all four 5 5 5 5 625 throws will be— X— X — X— . = ; which sub- 6 6 6 6 J 296 12^6—625 67 \ tracted from unity or J, gives = , which 1296 1^96 is the probability of throwing it once or oftener in four turns ; therefore the odds of throwing an ace in four times, is as 67 J to 025, or rather more than an even chance* The probability in three throws will be, 5 5 5 125 216—125 91 - 6^6 d £1() 216 216 Here the odds is against throwing the ace in three throws, as 91 is less tlian 125. question VI. In two heaps of cards, one containing the 13 diamonds, the other the 13 spades, placed promis- cuously, what is the probability that, takiiig one card at a venture, out of each heap, 1 shall take out the two aces ? Solution. The probability of taking the ace out of the first heap is ^V ? ^^® probability of taking the ace out of E5i:PECTATI0N OF LIFE. 231 le second heap is also -Jj, therefore the probability of iking out both aces is -j\ x -jV' ^^ » vvhich sub- 169 168 tracted from I, gives , of course the chances against 109 me are as 168 to 1 : in ojier words, I may eji'pect to do this once in 169 attempts. On similar principles the evpectntinn of life is found. It is known by accurate observation, tliat of 46 per- sons aged 40 years, one will die every year, till they are all dead in 46 years ; therefore half 46, or 23 years, will be the expectation of life of a person 40 years of ao;e. That is, the number of years enjoyed by them all, will be just the same as if every one of them had lived 23 years, and then died. The same reasoning ap- plies to all other aijes, which leads us to a more particu- lar consideration of the subject. EXPECTATION OF LIFE. From the Bills of Mortality in different places, tables have been constructed which shew how many persons, upon an average, out of a certain number born, are left at the end of each year to the extremity of life. From such tables, wliich, as we have seen, are founded on the doctrine of Chances, the probability of tiiexontinuance of a life, of any proposed a^e is known. 232 EXPECTATION OF LUTE. TABLE I. Shewing the Probabilities of tbe Duration of Hmnaii Life, deduced from the Register of Mortality at Northampton. Persons Dec. iVrsoiiB IXc. PevsoJis 1 Dec Age living-. ' of Life Age living. of Life. 75 1 Age. 06 living of Life. il6iu 4J-'0 1552 80 1 8650 13(S7 ' 34 4083 75 67 1472 80 2 723; 502 .15 .'AO 75 68 1392 80 3 67 SI 335 36 3935 75 69 1312 89 4 6U6 197 37 3860 75 70 1232 80 5 6249 184 38 3785 75 71 1152 80 6 &'65 1925 140 ..9 3710 75 72 1072 80 7 110 40 3685 76 73 992 80 8 58:5 80 41 3559 77 74 912 80 9 r>7r>5 60 42 3^82 73 75 832 80 10 5G75 52 43 3404 7S 76 752 77 11 562.) 50 44 3326 78 77 675 73 12 5f,7S 5U 45 3248 78 78 602 68 13 5523 50 46 3170 78 79 534 65 14 5473 50 47 3092 78 ^^o 469 63 15 5423 50 48 3014 78 81 406 60 16 5373 5:i 49 2936 79 8.> 346 57 17 5320 58 50 n57 81 83 289 55 18 5262 C3 51 2776 82 84 234 48 19 5199 67 52 2694 82 S5 186 41 20 5132 72 53 2612 82 86 145 34 21 5.6'J 75 54 2530 82 87 lit 28 22 49 -,55 75 -55 2448 82 88 83 21 2li 4910 73 ' 56 2366 82 89 62 M> 24 4835 75 57 2281 82 9,) 40 12 25 - 4760 75 58 2202 82 91 3i 10 26 4685 75 5? 2120 82 92 24 8 27 4610 75 e^j 2038 82 93 16 7 23 45J5 75 61 1956 82 94 9 5 29 4460 75 62 1874 81 95 4 3 30 43S5 75 63 1793 81 96 1 1 Si 4310 75 64 1712 80 3i 4235 75 65 1632 80 Case L To find, by this Table, the expectation of any single life. .Rule. Divide tbe sum of all the living in the table, at the age whose ex[>ectution is required, and at all great* EXPECTATION OF LIFE. 23S cr ages, bv the sum of all that die annually at that age, and ab(>ve it, or. wiiicli is the same thinj^, by the number in tlje table of the living at that a^^e, and half unity, or .5 subtracted from the quotient will be the expectation re- quired. Ex. 1. What is the expectation of a life at 60 ? The sum of the living at the age of 60 and upwards, by the table, is 27947, which divided by 2038, the num- ber of lining at that age, gives 13.71, from which subtract .5, and the expectation of a life at 60 is equal to 13.2 J, or 13 years 1 1 weeks nearly. Ex. 2. What is the expectatin of a life 70 years of age, one of 80, and one of 90 ? Ans. life of 70 is 8 years 31 weeks — Hfe of 80 is 4|. years — life of 90 is 2 years^ 20 weeks, and 5 days. Case II. To find the probability that a given life shall continue any number of years, or attain a given age. Rule. Make the number in the table, opposite to the proposed age, the numerator of the fraction, and for the denominator take the number opposite the present age. Ex. I. What is the probability that I, who am 45, shall live to 60 .^ The number against 60ii:2038 "| Therefore the chances in Lmy favour are 20 : 12 The number against 45IZ3248 J nearly, or as - 5 : 3. 2038 For, since the probability of living is equal to , the 3248 chance of dying during that period is 2038 3248—2038 1210 1 -zz — iz . The denominators being 3248 3248 3248 the aiiie, tho chance of life is to the probability of dying as 2038 to 1210, or as 20 to 12, or as 5 to 3 nearly. go* 23 i EXPEcrATION OF LIFE. Ex. 2. What is the probability that a peiison aged 21, as lattain to 54 ? 2530 Ans. chance of living: 5060 Ex. 3. What is the probability that a person aged 15 should live till 70 ? 1232 Ans. chance of living. 5423 Ex. 4. What chance has a person aged 70 of living 10 years longer ? 469 Ans. — ^ chance of living. 1232 From the foregoing table is formed TABLE H. Shewing the expectation of Human Life at every Age according to the Probabilities found bv Table I. Age. Expectation. Age. Expectation, Age. 50 Epoctation. Ag.-. Expectatioi». 25,18 25 o0.85 17,99 75 6,54 1 32,74 26 30,33 51 17,50 76 6,18 2 37.79 27 29,82 52 17,02 77 n,S3 3 39,55 28 29.3:; 53 16,54 78 5,48 4 40,58 29 28,79 54 16,06 79 5,11 5 40,84 30 28 27 53 15,58 80 4,75 6 43,07 31 27,76 55 15,10 81 4,41 7 41,08 32 27,24 57 14,63 82 4,09 8 40,79 33 2e,72 58 14,15 83 3,80 9 40,36 34 26,20 b9 13,68 H4 3,58 10 39,78 35 25.6^ 60 l3,2l 85 3,37 11 39,14 36 25,16 6] 12,75 86 3,19 1$ 3a. 49 37 24,64 62 12,28 87 3,01 13 37,83 38 24, 12 63 11,81 88 '2,S6 14 37,17 39 23,.0 64 11,35 89 2,66 15 36,51 40 23,03 65 10,88 90 2,41 16 35,85 41 22,56 66 10,42 9". 2,09 17 35,20 42 22,! 4 67 9,96 92 1,75 18 34 58 43 21,54 68 9,50 93 1,37 19 33,99 4i 21,03 69 9,05 94 1,05 20 33,43 45 20,52 70 8,60 95 0,75 31 32,90 IG 20.02 71 8,17 96 0,50 22 32,39 47 19,51 72 7,74 23* 31,88 48 19,00 73 7,^3 ^k 31,36 49 18,49 r4 6,92 LIFE ANNUITIES. 233 To find the expectation of any given life. Rule. Seek in the table the given age, and oppo- site to it is the expectation. Thus, the chance of life to an infant just horn is 25.18, or rather more than 25 years ; to a person of 45 years of aire 20,52, as we have found before, and to a person of 69, just 9 years. Upon these tables is found^^d the doctrine of LIFE ANNUITIES. Life Aknuities are annual payments to continue dufins: any life or lives. The^^e are generally purchased or s>!ld for a present sum of money. ** The present value of a life annuity** is the sum that would be sufficient, (allowins: for the chance of life failing, which has been considered in the preceding pages) to pay the annuity without loss. If money bore no interest, the value of an annuity of 1/ would be equal to the expectation of life- Thus, Table [I. the value, of an annuity for a life of 20 years of age, if money bore no interest would be equal to nearly S3 years and a half purchase ; that is 3S/. los. in hand for each life, would be sufficient to pay to any number of suc!^ lives iL per annum. If money is capable of being improved bv being put out to interest, tlie sum just mentioned would be more th.in the value, because it would be more than sufficient to pay the annuity ; and it will be as much more than sufficient as the interest is greater. As an example. If money can be improved at 5 per" cent, compound iviterest, the half of 33', 10s., or 16/ 15s., will, as we have seen, in little more than 14 years, produce the 33^ lO-. required. It must not however he supposed, that 16^ I5s is the true value of an annuity of il. during a life of 20. The 236 LIFE ANNUITIES. value of an annuity certain for a term equal to the ex- pectation, always exceeds the true value, because, In a number of life annuities, many of the payments would not be to be made till a much more remote period than the term equal to the expectation. Upon this principle the following table is computed, from which it appears that the present value of an annui- ty of 1/. on a life of 20 years of age, is equal to 14/. and a small fraction only ; that is, 14/. in hand for each life, improved at compound interest, will be sufficient to pay to any number of such lives 1/. per annum. TABLE I. Shewing the Value of an Annuity of U, on a Single Life, at every Age, according to the probabilities of the Duration of Human Life at Northampton, reckoning interest at 5 per cent. Age. Value. Age 25 1 Value. Age. ' 50 Value. Age. 75 Value. Birth. 8 863 13.567 10,269 4,744 1 year 11.563 26 13473 1 ^1 1C,097 76 4,511 ^2 13.420 2? 13.377 52 9,925 77 4,277 3 14.135 2« 13.278 53 9,748 78 4,035 4 1k613 29 1.3.177 54 9,567 79 5,776 5 14 827 30 13.072 55 9,382 80 3,515 6 15.041 31 12.965 56 9,193 81 3,263 7 15166 32 12.354 57 8,999 82 3,020 8 15.226 33 12 740 58 8,801 83 2,797 9 15.210 34 12.623 59 8,599 84 2,627 10 15139 35 12.502 60 8,392 85 2,471 11 15.043 36 12.377 61 8,181 86 2,328 .12 14.937 37 12.249 62 7,966 87 2,193 13 14.826 38 12.116 63 7,742 88 2,080 14 14.710 39 11.979 64 7,514 89 1,924 15 14.588 40 11,837 ' 65 7,276 90 1,7^3 16 14.460 41 11,695 ' 6b 7,034 91 1,447 17 14.334 1 42 11,551 67 6.787 92 1,153 18 li.2J7 • 43 11,407 68 6,536 93 0,ii\6 19 14.108 44 11,258 69 6,281 94 0,524 20 14.007 45 11,105 70 6,023 95 0,238 21 13917 46 10,94? n 5,764 96 0,000 22 13.83 i 47 10,784 72 5,504 23 13.746 48 10,616 73 6,245 24 1 13.658 4y 10,44 -. 74 4,990 4-IFE ANNUITIES. 237 I To find the value of an annuity for a person of any- given age. Rule. Multiply the number in the table against the given age, by tlie sum, and the product is the answer. Ex. 1. What shouhl a person, aged 45, give to pur- chase an annuity of 60^ per antium during life, interest being reckoned 3 per cent ? The value in the table against 45 years is U. 105, and this multiplied by 00 gives the answer, 066/. 6s. Ex. 2. A person aged 69 years would purchase an annuity of 200/. for life, what must he pay for it in ready money at the same rate of interest ? Answer, 1256^ 4s. Ex. 3. A merchant marries a lady aged 28, whose for- tune for life is 300/. per annum, being desirous of con- verting the same into money, what ought he to have for itj allowing interest 5 per cent. ? Answer, 3983/. 8s. Ex. 4. What is the value of an annuity of 200 dollars during the life of a person aged 25 years ? Answer, 552713 40 cts. Ex. 5. What is the value of 50/. per annum, payable during the life of a person aged 41 years ? Answer, 584/. 15s. Ex. 6. What is the value of a clear annuity of 75/. during the life of an old man aged 76 ? Answer, 338/. 6s. Qd. Ex. 7. What is the value of a landed estate during the life of a person aged 38, produing nett 3o/. 9s. per annum? Answer, 3(J8/. IBs. 7^fi/. Ex. 8. What is the life interest of a person aged 53, in 1250/. 3 per cent. Consols worth .^ Answer, 3G5/. lis. Ex. 9. A gentleman aged 60, who receives an annuity of 150/. per annum, for life, out of a freehold estate^ 238 LIFE ANNUITIES. wishes to exchangie his life for that of his wife, aged 32 ; what ought to be required of him for so doing ? Answer, 669^, 6s. Ex. lO. A person having an annuity of lOOl. during a life of 37 years, agrees to exchange it for an equivalent annuity during a life of 45 ; what annuity should be granted him ? Answer, ilOZ. 6s. Ex. 11. What annuity will \00l. purchase during the life of a person aged 28 .^ Answer, 7^.. 10s. 7c?. Ex. 12. A parish means to raise a sum of money for building a workhouse, by life annuities ; at what ages should they grant 7, 8 and 9 per cent. ?* Ans. To persons of 18, 35 and 45 years of age. Ex. 13. What is the difference in value between an annuity of 40/. during a lifeof36, and an annuity certain for 20 years Pf Answer, 3l. 8s. Qd. nearly. Ex. 14. What annuity should be granted to a person aged 57 during his life, for 2,000Z. five pes cent, stock, which is now at 99| ? Answer, 22U. 8s. NOTES. * Questions of this sort are answered by dividing \00l. by the rates per cent., and opposite to the numbers in the table that are nearest the quotient, are the required ages : thus, to find at what age a life annuity of 9 per KG cent, should be granted, — = 11.111 : the nearest 9 number in the table is 1 1.105, by the side of which is 45, hence, to ages of 45, an annuity of 9 per cent, may be granted. LIFE ANNUITIES. 239 TABLE II. Shewing the Value of an Annuity during the joint con- tinuance of Two Lives, accorciing to the probabilities of Life at Northampton, reckoning interest at 5 per cent. Ages, Value. Ages. Value. Ages. Value. Ages. Value. 5-5 11,984 15-35 10,655 30^0 10,255 45-70 5,195 5-10 ; 2,315 15-40 10,205 30-35 9,954 45-75 4,206 5-15 11,954 15-45 9,690 30-4U 9,576 45-80 .:>,197 5-20 ri,5f)l 15-50 9;076 30-45 9,135 50-50 7,522 5-25 11,281 . 15-55 8,403 ' 30-50 8,596 50-55 7,098 5-30 10,959 15-60 7,022 30-55 7,999 50-60 6,568 5-35 10,5/2 i 5-65 6,705 30-60 7,292 50-65 5,89r 5-40 10,H)2 15-70 5,r)31 30-65 6,447 60-70 5,054 5-45 9,571 15-75 4,495 30-70 5,4^^2 50-75 4,112 5-50 «,94l 15-80 3,372 3U-75 4,365 50-80 3,140 5-55 8,256 20-20 11,232 30-80 3,290 ,55-55 6,735 5-60 7 406 20- .^5 10,989 ( 35 35 9.680 55-60 6,272 5-u5 6.546 20-30 10,707 ' 35-40 9,331 55-65 5,671 5-70 5,472 20-35 10,363 : 35-45 8.l"zl 55-70 4,893 5-75 4,362 20-40 9,9j7 ' 35-50 8415 55-75 4,006 5-80 3,238 20-45 9,4+8 35-55 7,849 55-80 3,076 10-10 12,665 20-5'J 8,801 35-60 7,174 60-60 5,888 10-15 12,302 20-55 S,216 35 65 6,360 60 65 5,372 10-20 11,906 20-60 7,463 35-70 5,382 60-70 4,680 10-25 11,627 20-65 6,576 35-75 4,3^7 60-75 3,866 l'y-50 11,304 20-70 5,532 35-80 3,268 60-80 2,992 10-35 10,916 20-75 4,4.-4 40-40 9,016 65-65 4,960 10-40 10,442 20-80 3,325 40 45 8,643 65-70 4,378 10-45 9,900 25-25 10;764 40-50 8,171 65-75 3,iJ65 10-50 1 9,2C0 26-30 10,499 40-55 7,654 65-80 2,873 10-55 8.5(30 25-35 10,175 40 60 7,015 70-70 3,930 lo-fiu 7,750 25-40 9,771 40-65 6,240 70-75 3,347 10-05 6,803 25-45 9,301 40-70 5,298 70-80 2,675 10-70 5,700 25-50 8,73^ 40-75 4,272 75-75 2,917 10-75 4,522 25-55 8,116 40-80 3,236 75-80 2,381 10-80 3,395 25-60 7,383 45-45 8,512 80-80 2 01!| 15-15 11,960 25 65 6,515 45-50 7,891 85-85 1,256 1 5-20 11,^85 25-70 5,489 45-55 7. ill 90-90 0,909 15-25 11,324 1 25^75 4,390 45-60 6,822 15-30 111,020 25-80 3,308 1 45-65 6,094 Case I. To find the value of an annuity on the longest of two single lives. Rule. From the sum of the values of the single lives, subtract the value of their joint continuance, and the re- mainder will give the value of the longest of the lives. 240 LIFE annuities: Ex. 1 What is the value of the longest of two lives aged 10 and 15 ? ri' 11 T S The value of a life at - - 10=15.159 lablel.^ . . . 15 = 14.588 29.727 ♦ Table II, The value of the joint continuance of two lives of - - 10 and 15 = 12302 Value of Ihe longest of the two lives 17.425 Therefore an annuity ot iCO/. a year upon the longest of two lives, one 10 and the other 15, would be worth near- ly 17 years and a iialf purchase, or more accurately, 1742^. 10s.' Ex. 2. What is the value of an annuity on ihi longest of two lives whose ages are thirty and forty ? Answer, 1533/. 6s. Case II. To find the value of an annuity on three joint lives. Rule. Take the value of the two elder, and find the age of a single life equal to that ; then find the value of the joint lives of this now found) and the youngest Ex. I. Let the three lives be 20, 30, and 40. The value of the joint continuance of the two eldest ; viz. of SO and 40 (by Table II.) is equal to 9.576, which answers to a single life (by Table 1.) of 54. Now, the value of the joint lives of 20 and 54 by Table IF., or the ages which come nearest, viz. 20 and 55, is 8 210* for the value sought : hence an annuity of 40/. on three joint lives would be worth about 328/. 12s. Ex. 2. To find the value of 3 joint lives of the ages 15, 30, and 45. Answer, 8.403. NOTE. * The numbers 9.576 and 8.216, are not quite ac- curate, because the limits of this book do not admit of a table giving the combinations of all ages. LIFE ANNUITIES. 241 Ex. 3- \Miatis the value of an annuity of 150/. on the joint continuance of three lives of the ages 50, 60, and 70 ? Answer, 587/. 14s. Case III. To find the value of the longest of any three lives. Rule. From the sum of the values of all the single lives subtract the sum of the values of all the joint lives, combined two and two. To the remainder add the value of the three joint lives, and the sum will be the value of the longest of the three lives. Ex. 1. What is the value of the longest of three lives, whose Bges are 20, 30, and 40 ? rvalue of a life of 20 = 14.007 Table 1.4 _ — — 30 = 13.072 (^ '— — — 40= 11.837' 38.916 Value of two joint lives of 20 and 30 = i0.ro7 — — — 20 and 40 = 9 937 — — — — — — 30 and 40 = ^9.376 38.916 ■ S0.22O 30.220 8.096-f8.216 (the value of the joint lives found in Ex. 1, Case II.) = 16.912 = the value of the longest of the three lives. Ex. 2. What is the value of the longest of three lives, ■whose ages are 15, 30, and 45 ? Ans. 17.126. Ex. 3. What is the value of an annuity on the longest of three lives, whose ages are 50, 60, and 70 ? Answer, 12.300. EXAMPLES FOR PRACTICE. Ex. 1. What is the present value of an annuity of 50^, on the joint lives of two persons, each 30 years of age > Answer, 61 2^» Ids'. 242 LIFE ANNUITIES. Ex, 2. What is the present value of an annuity of 65^., during the joint lives and the life of the survivor, of a man aged 45, and his wife aged 35 ? Answer, 954/. 12s. nearly. Ex. 3. What is the value of a lease producing 27Z. 13s. per annum, on the longest of two lives aged 60 and 45 ? An3wer, 3501. 9s, Ex* 4. What is the value of an annuity of 4-0/, on two joint lives of 70 and 5 years ? Ans. 218/. I7s. 7d, Ex. 5. What is the value of an annuity of 50/. on the longest of two lives of 70 and 5 years ? Answer, 768/. 18s. Case IV. To find the value of an annuity on a given life for any number of years. Rule. Find the value of a life as many years older than the given life as are equal to the term for which the annuity is proposed. Multiply this value by ll. payable at the end of this term, and also by the probability that the life will continue so long. Subtract the product from the present value of the given life, and the remainder multiplied by the annuity will be the answer. Ex. 1. What IS- the value of an annuity of 50/. per annum, for 14 years, on a life of 35 ? 35 + 14 = 49. The value of a life of 49 (14years older than the given life, by Table 1.) - - - - = 10.443 The value of 1/. payable at the end of 14 vears (Table ) = .505068 TH^e probability that a life of 35 will con- 1 2936 tinue 14 years (Fable and the 2d >. ^ Case.) J 4010 10.443 X .505068 X (?^^ .7322 =3.861, which, sub-. \40i0/ ' tracted from 12.502, the value of a life of 35, Table I. gives 8.641 ; and 8.641 X 50 = 432/. Is. Ex. 2. What is the value of an annuity of 80/. per annum for 20 years provided ^a person aged 45 live so long ? ' Answer, 785/. 13s. 7d, LIFE ANNUITIES. TABLE. Shewing the present Value of H. to be received at the end of any number of years, not exceeding 100 ; dis- counting at 5 per Cent. Compound Interest. Yrs. Value Yrs. 26 Value. Yrs. Value. Yrs. Value. 1 .95i38J .281241 5i .083051 76 .024525 2 .907029 27 .267848 52 .079096 77 .023357 3 .863838 28 .255094 53 .075330 78 .022245 4 .822702 29 .242946 54 .075743 79 .021186 5 .783526 30 .231377 53 .068326 80 .020177 6 .746215 31 .?20359 56 .065073 81 .019216 ^. 7 .710681 32 .209866 57 .061974 82 .018301 Wk 8 .676839 33 .199873 58 .0:>9023 83 .0174^^0 Wt 9 .644609 34 .190355 59 .056212 84 .016600 10 .613913 35 .181290 60 .053536 85 .013809 11 .584679 36 .172657 61 .050986 86 .015056 12 .556837 37 .164436 62 .048558 87 .014339 13 .530321 38 .156605 63 .046246 88 .013657 1* .505068 39 .149148 64 .0440t4 89 .013006 15 .481017 40 .142046 65 .041946 90 .012387 16 .45M112 41 .135282 66 .039949 91 .011797 17 .436297 42 .128840 67 .038047 92 .011235 18 .415521 43 .122704 68 .036235 93 .010700 19 .395734 44 .116861 69 .034509 94 .010191 20 .376889 45 .111297 70 .032366 95 .009705 21 .358942 46 .105997 71 .031301 96 .0'< 9243 22 .341850 47 .100949 72 .029811 97 .008803 23 .325571 48 .096142 73 .028391 98 .008384 24 .310068 49 .091564 74 .027039 99 .007985 25 295303 50 .087204 75 .02 5753 100 .007604 In order to find the present worth of any sum which is to be received at the end of a certain number of years. — Multiply the number in the table opposite to the term of years, by the sum, and the product will be the answer, Ex. 1. What is the present value of 750^ to be re ceived at the expiration of 9 years ? 244 LIFE ANNUITIES. The number in the table even with 9 years is .644.609. which IS to be multiplud by 750. .644609 750 3223045 4512263 483.45675 20 y.l350 12 1.620 4 Answer, 483^ 9s. l^i. 2.48 Ex. 2. What is the present value of 574^. 10 = 2l. lOs. 6d» 13 072 nearly, in annual payments continued during life. If tlie interest of money be supposed 4 per cent., then the value of a life of 30 is equal 14.68,* and the perper- 100 tuity is equal = 25. Therefore 25 — 14.68 = 4 4123 10.32. This multiplied bv 400/. = 4128. And = 104 391' 1 4s. 39/. 14s. nearly ; and = 2/. 14s. 14.63 • This is taken from a table not in this book. See Price'* Reversionary Payments, and Morg^an's Doctrine of Annuities, &tt SI* 246 LIFE ANNUITIES.- Hence it appears, that when the values are required in a single payment, the difference in the rate per cent. is considerable, though but trifling when made in annual payments during life. In this question, if money be im- proved at 5 per cent., tiie value of the single payment would be 33/. ; but at + per cent, it would be 39/. 14s., which is one fifth more in the latter case than in the for- mer : l)ut, when the value is pai". 6c?., and at 4 percent, it is 2/. 14,';., making a difference ot 3s. 6'/. per annum, being an increase of less than one fourteenth. If the first of the annual payments is to be mado. im- mediately, then the single payment is to be divided by the value of the life, with unity aided to it, so that at 33 5 per cewt. it will be = 2/. 6s. lid, nearly 5 and 14-.0r2 39/. 14s. at 4 per cent, it will be — = 21. 9s. 4-^d. 15. OH Ex. 3. Let the life be 25, the sum lOOO/., and the rate 5 per cent. Answer, 21/. annually. Ex. 4. Let the life be 60, the sum 1 000/., and the rate 5 per cent. Answer, 39/. nearly. Case VL To determine the value of an annuity cer- tain on a given life for any number of years. Rule. Find the value of a life as many years older than the given life as are equal to the term for which the annuitv is proposed. Multiply this value by 1/. payable at the end of this term, and also by the probability that this life will continue so long. Subtract the product from the present value of the given life, and the re- mainder multiplied by the annuity will be the answer. Ex. I. Let the annuity be 50^ the age of the given life 30 years, and the term proposed 15 years ; interest 5 per cent. LIFE ANNUITIES. 24^7 The value of a life of 45, or 15 years older than the given life, by Table, pa^^e 2)6, = 11.105. The value of 1/. payable at the end of 15 years is, by table, page 243, IZ .481 : and the probability that the life of 30 will 3^48 exist so long, is by Table, page 232 = = .74 near- 4385 ly. Therefore 11.105 X .481 X .74 = 3 953. And th^ present value of the given life, by the Table, page 236, = 1.3.072: therefore "13.072 — 3.953 =9.119, and this multiplied by 50 = 4,35/. 195. Had the interest been only 4 per cent, the value would have been about 490/.: that is, in the one case 455/. 19s., and in the other 40O/., by a person who would insure an annuity of 50'. per annum for 15 years certain, which depends on the contingency of the life of a person aged 30. Ex. 2. Let the annuity be 40?. the age of the given life 40, and' the term proposed 20 years. Answer, 402/. IGs. \0d. Case VII. To find the value of a given sum paya- ble at the decease of a person, should that happen with- in a given term. In other words : What ought a per- son to give for haviitg his life assured to him for a cer- tain term ? Rule. From the value of an annuity certain for the given term, subtract the value of the life for the same term, and reserve the remainer. Multiply the value of l/. due at the end of the given term, by the perpe- tuity, and also by the probability that the given life shall fail in the given term. The product is to be added to the reserved remainder, and the sum multiplied by the given sum : this last product divided by the perpetuity increased by unity, gives the value in one present pay- meat. 24S LIFE ANNUITIES. Ex. I . A merchant at Liverpool, aged 30, expects to realize a considerable property in the next 15 years ; but as he may die before he can accomplish his views, he is willing; to insure on his life, during that period, the sum of 5000/., what must he pay for the same f The value of annuity certain for 15 years, by Table, p. 227, is equal to 10.379 ; and by example, page 247, the value of an annuity certain for 15 years on a life of SOiz 9.119; therefore 10.379— 9.11 2izl.26zireserved re- mainder. The value of 1/. to be received at the end of 15 years? by Table, page 243, ZI.481 ; and the probability that a life of 30 shall fail in 1 5 years, is 1142 lOp < •z:.26 :* and the perpetuity is Z220. There- 4385 5 fore, .481 X .26x20 =2.5, and this added to the reserv- ed remainder 1.26=3.76, which multiplied by 5000, the given sum, and divided by 21 (the perpetuity increased by unity) is equal 895/. 5^. nearly, the value required in a single payment. That is, a person of 30 must give 895/. 5S. to secure to his heirs 5000/. supposing he dies within 15 years. Or he must pay annually during the 15 years, if he live so long, 985/. 5s. divided by 9.119, or 98/. 3s, 4c?.t for *^he same security. NOTES. * The probability of life's failing, is always equal to the probability of its continuing, subtracted from unity. Thus the probability of a life of SO continuing 15 years, 3S48 is by table, p. 232, zz IZ .74, and the probahili- 4385 3^48 4485— S248 1137 ty of its failing 1 zz: — — :^ ^^ ^^ •2^' 4385 4385 4385 See Chances, p. 230. t The payments are supposed to be made at the end of every year. But in all assurances, the first premium is paid immediately, and the remaining ones at the be- LIFE ANNUITIES. S4& If money can be improved at 4 per cent. onl>', then the sum to be paid at once will be 929/. 4s. 2(1., and the annual payments will be \0\L nearly. Ex. 2. If I live 7 years, I shall receive 20OOZ. ; what must I give to insure my life for that period, being now 46 years of age ? Ans. b\L for each annual payment for 7 years, if he live so long. Case VIII. To explain, by examples, the mode of grant- ing annuities by the British Government established in the year 1808. [The following examples are deduced from the tables printed and circulated by -Government, and which may be had, gratis, at the Office, Bank Buildings, Royal Ex- change, London.] Ex. 1. By the tables it appears, that for every 100^ stock in 3 the per cent, consolidated annuities, will be given annually for life, to a person of 46 years, 5t. lis.* If, therefore, a person of that age transfer \OQOL stock, he will receive an annuity for life of 55l. 10s. But he will receive interest 30/. and keep his capital ^ and to insure 660/. at the Equitable, or Royal Exchange Offices, he must pay rather more than 4 per cent, j that is, he must NOTES. ginning of ever year after ; hence the proper divisor will be the value of the life for one year less than the given term added to unity, or, in this case, the value of a life for 14 years. And generally : the divisor for determin- ing the annual payments must he increased by unity, whenever it is proposed that the first payment should be made immediately. See p. 246. * Supposing stocks to be at 66, which they are at present* 250 REVERSIONS. pay between 26 and 271. annually, during life, to insure to his heirs at his death the 660/. which he transfers to Gi)vernment : he will ©f course be a loser by the transfer, of between one and two pounds per annum. It is there- fore obvious, that no one, when stocks are at 66, can join in the plan held out by Government, who is not willing to give up his capital. Ex. 2. When stocks are at GO, he will receive for lOOO^. stock, 59.1. lOs. ; and to insure 000/. must pay more than 24/. to insure his life, and will of course be a loser of 1/. lOs. per annum. Ex. 3. When stocks are at 80, as they may be, he will receive for the 1 GOO/, stock 62/. ; but, to insure 800/. he niu^t pay annually rather more than 32/. ; in this case there will be his interest left, and he will he neither gainer nor loser. These examples will suffice for the whole. REVERSIONS. Reversions, or Reversionary Annuities, are those which do not commence till after a certain number of years, or till the decease of a person, or some other future event has happened. Case I. To find the present value of an annuity for a term of years, which is not to commence till the expi- ration of a certain period. Rule. Subtract from the value of an annuity for the whole period, the value of an annuity to the time when the reversionary annuity is to commence. HEVERSIONS 251 Ex. 1- What is the present value, at 5 per cent, com- lund interest, of 80/. per annum for 24 years, com- encing at the end of 8 years ? ^4i-\~S=32, pound The present value of an annuity (Table, p. 227,) for 32 years, is 15.802677, and the value of one for 8 years is 0.4632 1 3, therefore, 15.802677 6.463213 9.339464 X80=74r. 157 12=747^. 3s. l|d/. Ex. 2. "What is the present value of an annuity of 55?, for 15 years, to commence at the end of 15 years f Answer, 274/, 12s. Ex. 3. What is the present value of an annuity for 49 years, to commence at the end of 47 years ? Answer, Something more than a year and half's purchase. Case II. To find the value of an annuity certain for a given term, after the extinction of any life or lives. Rule. Subtract the value of the life or lives from the perpetuity,* and reserve the remainder. Then say, as the perpetuity is to the present value of the annuity cer- tain, so is the reserved remainder, to the number of years purchase required. NOTES. * Perpetuity, is the number of years purchase to be given for an annuity which is to continue for ever ; and It is foujid by dividing 100/. by the rate of interest j thus, 252 HEYERSIONS. Ex. L What is the value of an annuity certain for 14 y^ars, to commence at the death of a person aged 35, al- lowing 5 per cent. ? The value of a life of 35 (Table, p. 236) «= I2.50g; this subtracted from 20, the perpetuity, leaves 7.498 = reserved remainder. Then, as 20; 9.898t : : 7.498: ^.7107 = number of years purchase. NOTES. allowing 5 per cent., the perpetuity is 20 years, or — = 20 ; and at the rates most usually adopted, the perpetuity is as follows : At 3 per cent— r- = 33.33, &c. 3i ditto ^/°=28.5r,&c. 4 ditto T='^- 4} ditto —-= 22.22, &c. 5 ditto 100 5 =2^- 6 ditto 100 ,^ ^^ „ —T-= 16.66, &c, 7 ditto 7=14.2S,&C, 8 ditto --,2.5. These are the number of years purchase to be given for a perpetual annuity, on the supposition that it is re- ceivable yearly : but, as annuities are more commonly received half-yearly, and the interest of money likewise paid half-yearly ; in this case the perpetuity will be somewhat greater or less than the above, as the periods at which the annuity is payable are more or less frequent than those at which the rate of interest is here supposed payable. t The value of an annuity certain for 14 years. Table- REVERSIONS. 253 Ex. 2. A and his heirs are entitled to an annuity of Zr.lOO certain for 25 years, to commence at the death of a cousin aged 43 years ; what can A sell his interest in this annuity for ? ^ Answer, 6261. I6s. Case III- To find the value of an annuity for a term certain ; and also for what may happen to remain of a given life after the expiration of this term: Rule. Find the value of a life as many years older than the given life, as are equal to the term for which the annuity certain is proposed. Multiply this value by il. payable at the end of the given term, and also by tho probability that the given life will continue so long. Add the product to the value of the annuity certain for the given term, and the sum will be the answer. Ex» 1. What is the value of an annuity of 60/. for 14 years, and also for the remainder of a life now aged 35, after the expiration of that term ? 35 + 14 = 49. The value of a life aged 49 ( I'able 1, page 236.) =10.443 The value of ll. payable at the end of 14 years ( Fable, page 243 ) - - == .50.5068 The probability that the life wdl exist > _2936 so long, (Table, page 232.) - J "ioib 2936 Therefore, 10,443 x .505063 X =386 1 ; this added 4010 to 9.898, the value of an annuity certain for 14 year?, (see Table, page 227.) :z: 13 759, the number vS years purchase ; and 13.739 X 60 — 825/. 10.>-. 9-^. Ex. 2. What is the value of an annuity of 73/. for 10 years, and also the remainder of a life now aged 24, After the expiration of that term ? Ans. 1070/. 5s. Case IV. To find what annuity can be purchased for a given sum, during the joint lives of two persons of given ages, and also during the life of the survivor, on condition that the annuity shall be reduced one-iialf at the extinction of the joint lives. 22 254 REVERSIONS. Rulp:. Divide twice the given sum by the sum of the value of the two single lives, and the quotient will give the annuity to be paid during the joint lives, one-half of which is therefore the annuity to be paid during the remainder of the surviving life. Ex. 1. A man and his wife, aged G5 and 27, are de- sirous of sinking 2000/. in order to receive an annuity during their joint lives, and also another annuity of half the value during the remainder of the surviving life : what annuities ought to be granted them ^ The value of a life of 27 > .j, , j . 335 S ='3.377 The 35 S ^^^^ ^' P- ^"^^ ? =12.502 ^5.879 4000 (twice the sum) Therefore, « 1=154^. lis. Sc/. =: 25.879 annuity during their joint lives: and 77^ 5s. l-^d, an- nuity during the Hie of the survivor. Ex. 2. A single man, aged 60, possessed of 1500/. is tiesirous of purchasing with it an annuity for himself and his sister, aged 40, during their joint lives, with one of half the value, during the remainder of the life of the survivor, at the death of either : what will be the value of the annuities ? Answer, 148/. 6s. annuity during joint lives, and 74/, Ss, do for the survivor. Ex. 3. A man possessed of lOOO/. which he will sink in the same way, and for the same purposes, during the joint lives of himself and father ; the age of the one is 65, of the other 80 : what annuities can be given for it ? Answer, 155/. annuity during joint lives, and 77/. 10s. do for thesurvivor. V. To find the value of the expectation of a perpe- tual annuity, provided one person of a given age sur- vives another of a given age. r REVEUSIONS. S55 (l.) If the Expectant be the elder* Rule. Find the value of an annuity on two equal joint lives, whose common a^^e is equal to the age of the oldest of the two proposed lives ; subtract this value from the perpetuity, and take half the remainder : tlieii say, As the expectation of the duration of life of the younger, Is to that of the elder : So is the iialf remainder to a fourth proportional : which will be the number of years purchase, if the ex- pectant is the older. (2.) If the Expectant be the younger. Add the value found, as above, to that of the joint lives, and let the sum be subtracted from the perpetuity, and the remainder is the answer. Ex. 1. What is the value of B's expectation, (aged SO,) of an estate 50/. per annum, provided he survive A aged 20 ? Value of two joint lives, aged 30, (Table TI. p. 239) = 10.255, the dirterence between which and 20, (the perpetuity,) is 9.745, the half of which is 4.872 : there- fore, : : 4 872 : 4.119 ZZ 205/. I9s. Ex. 2. What is tlie value as above, when B is 20^ and A 30 .^ Then, to 4.1 1 9, just found, add [p. 239) 10.707, value of the joint lives (Table 11. 14.826; this subtracted from 20, the per- petuity and the remainder, 5-174 x 50 = 258/, 14^. is the uae answer. ^^^ ~ REVERSIONS. ^ EXAMPLES FOR PRACTICE. Ex. 1. What is the difference in the value of an an- nuitj of 20/. certain for 30 years, and an annuity of the same amount on the longest of two lives, ao;ed 25 and 40 ? Answer, L.5 4 4| difference. Ex. 2. What is the valu6 of an estate of 150/. per annum held on the longest of two lives, ased 40 and 50, suhject to the payment of an annuity of 14/. to a life of 62, and another annuity of 18/. to a life of €5 } Answer, 1847/. 16.«. value Ex. .S. What is the present worth of 2000/. to be received at the decease of a person aged 65 } Answer, 1272/. 8^-. present worth. Ex. 4. W^hat is the present value of 36/. a year, being the third part of a farm in Essex, after the death of a person aged 54 years .? Answer, 375/. 11. «J. 9^^. present value. Ex. 5. What is the present value of a reversionary- annuity of 252/. S.«. 8c/. during the life of a person aged 24, in oase he survives his brother, a2;ed 34 ? Answer, 1539/. 5s. 9d. present value. Ex 6. What shouJd be the consideration to be paid at the death of a person aged 85, for 1000/. now advanced to a person aged 25, in case the latter survives the for- mer ? Answer, 1193/. tis. Ex. 7. What is the value of the reversion of PI/, per annum forever, after the death of a person aged 53 ? Answer, 932/, 18.v. Id, value; Ex. 8. A person aged 52, is entitled to 800/. at the death of another aged 76, provided the former survives the latter ; what is its present worth ? Answer, 522/. Os. 9d, Ex. 9. What is the present value of an annuity on the longest of two lives, now aged 25 and 30, the an- nuity not to commence till 14 years hence ? Answer, 854/. 19s. Id. LEASES. LEASES. 257 A Lease is a conveyance of any lands and tenements, made, in consideration of rent, or of a present sum of money, for life, or for a term of years. The purchaser of a Lease may be considered as the p^irchaser of an annuity equal to the rack-rent of the estate; its value must therefore be calculated on the same principles as that of an annuity. The sum paid down for the grant of a lease is so much, as being put out to interest will enable the landlord to repay himself the rack-rent of the estate, or the year- ly'value of his interest tiierein. The value of the lease depends on the length of the term, and the rate of interest which the landlord can make of his money. The value of leases at 5 per cent, compound interest may be found in the Table page 227. Thus, the value of a lease for 14 years, of a farm worth 150^ per annum, is by that table, 9.898641X150 = 1484^ \5s> l\d. - Kx. I. What ought to be given for a lease of 26 years of an estate of 18^ per annum clear annual rent, in-or- 576 7,' 037 63 16,2424 14,0844 12,4020 14 9 2919 8,7454 8,2442 64 16,2664 14,09/6 12,4092 15 9.7122 9,l(»79 8,5594 65 16,2891 i 4. 1099 12,4159 16 UK 105 8 9,44436 1 8,8513 66 16 3104 14,1214 12,4222 17 10 4772 9.76.32 9,1216 67 16,3306 14,; 321 12,4279 18 10.8276 10,0590 9,3718 68 16,3496 14,1422 12,4333 ll> 11 1581 10,3.S55 9,6035 69 16,3676 14,1516 12,4382 fiO 1 1 4099 10,5940 9,8181 70 16,3845 14,1603 12,4i28 iil 11764(^ 10,8355 10,0168 71 16,4005 14,1685 12,4470 22 12.0415 11,0612 10,2007 72 16,4155 14,1762 12,4509 i.l 12 3033 11,2721 10,3710 73 16,4297 14,1834 12,454r» '24 12 5503 11,469.) 10,5287 74 16,4431 14,1901 12,4579 25 127S.n3 11,6535 10,6747 1 75 16,45 5 S 14,1963 12,4610 ■26 13 00,31 11,8257 10,8099 1 76 16,4677 14,2022 12,4639 '27 13.2W3 11,9867 10,9351 • 77 16,4790 14,2076 12,4666 28 13 44)61 12.1371 11,0510 78 16,4896 14,2127 12,4691 39 13.59U7 12.2776 11,T584 79 16,4996 14 2175 12,4713 30 13 76 i 8 12,40110 11,2577 80 16,5091 14,2220 12,4735 31 1 3 9290 12,5318 11,3497 81 16,5180 14,2261 12,4754 3/ 14 0810 12/i465 11,4349 82 16,5264 14,2300 12,4772 33- 14 2.302 12,7537 11,5138 83 16,5343 14.2337 12,4789 ;i4 r • .3681 .12,8540 11,5869 84 16,5418 14,2371 12,48(5 55 144982 12,9476 11,6545 85 16,5489 14,2402 12,4819 36 14 6209 13,0352 11,7171 86 16,5556 14,243: 12.4833 ;37 147367 13,1170 a. 7751 87 16,5618 14,2460 12,4845 ^8 14 8460 13,1935 ll,8J88 88 16,5678 14,2486 12,4856 ->9 14 y UiO 13,2649 11,8785 89 16,5734 14,2510 12,4867 4o 15.0 Mri 13.3317 11,9246 QO 16,5787 14,2533 12,4877 4! 15.1.38'' 13,3941 11,9672 91 16,5856 14,2554 » 2,4886 4^i 15.2245 1.).4524 12,00r.6 92 16,5883 14,2574 12,4894 4.S 15 3061 13,5069 12,0432 93 16.5928 14,2.S92 12,4902 44 15 3831 13,5579 12,0770 94 16,5969 14,2610 12,4905 45 15 4558 13,6055 12,1084 95 16,6009 1 4,2626 12 411 G 46 15.524.3 i 13,6500 1 -•,1.374 96 16,6046 14,2641 12,4922 47 15 5890 13,6916 12.1642 97 lft,6081 14,2655 12,4928 ' 48 15.6500 13,7304 12,1891 98 16,6114 14,2668 12,4933 49 15 7075 13,7667 12,2121 99 16,6145 14,2680 12,4938 50 15 7618 13,8007 12,2334 100 16,6175 14,2692 'r'.4943 XEASES. 259 Case 1. To find the sum that oiuht to be given fo lease. Rule. Look in the table an;ainst the number of years for which th»! leu^e is to continue, and on the line even with it, under the given rate of interest, is the number of years purchase that ou^ht to be given for the same. Ex. What sum ought to be given for the lease of an estite of 17 years, of the clear annual rent of 75i. al- lowing the purchaser to make 7 per cent, interest of his money ? Answer, 9.7632x75— 73 2.24=732^. 4s. 9|^, T*Tx. 2. What must he given for a lease of 21 years, at the clear annual rent of 50 guineas, allowing 8 per cent, for money ? Answer, 525L n."-. 9d, Ex. 3. What is the worth of a lease of 83 years of aH estate of 78/. per annum, interest being 6 per cent..^ Answer, 1239/. 13s. Ex. 3. What sum ought to be given for a lease of 69 years, of aTarm of 150^ per annum, the purcliaser being alioAed 6 percent, for his money ? Ans. 24i55L \0s. Ex. 5. What sum ought to be given for the lease of 46 years, of an estate estimated at 200^, but which is charged with the payment of a reserved rent of 70/. i5s. besides taxv^s and incidental expenses to the amount of 49/. l^s. annually; allowing tl»e purchaser 6 per cent, interest for his money .^ Answer, 1236/. 9s. 9rf. Ex. 6. What sum ought to be given for the ground rent of a house of 15/. per annum, for 18 years, allow- ing the purchaser 8 per cent .'^ An«. 140/. lis. 6c?. «. Cask II. To find the annual rent correispondirig to any- given sum paid for a lease. Rule. Divide the aura paid for the lease by the num- ber of years purchase that are found against' the given term, and under the rate of interest intended to be made of the purchase money, the quotient will be the anauai rent required". 260 LEASES. Kx. I. I a Answer, 4.932 year's rent 9.64f RENEWAL OF LEASES. Ex. S. I have a house for a lease of 48 years, but I wish to extend the lease to 97 years : how much must 1 pay for it, supposing the house worth 50^. per annum, and the interest 8 per cent.? Ans. 15/. 4s. It will be seen by working Ey. 2. of Case 1, by this rule, that the answer will be precisely the same by both metliods : for the whole term for which the new lease is granted is 21 years : the value of a lease for this term is, by Table, 1 1.764, and the value of the 14 years' lease yet to come is 9.295; this subtracted from the other, gives 2.4G9, as before, which, multiplied by 60, and the an- swer is \4SL 2s. did. The following table will comprehend the cases that most frequently occur at ti>e rate of 5 and 6 per cent. RENEWAL OF LEASES. 265 TABLE, For Renewing, with one Life, the Lease of an Estate held on Three Lives. » Life Age of 1 1 Life Age of put lives in 5pr.Ct. 6pr.Ct.| put lives in 5pr. Ct 6pr.Ct, in. [)ossession. •1,741 in. possession. 30-30 1,305 40—75 3,943 3.076 30-40 2,035 1,521 50-50 3,289 2,536 30-50 2,431 1,832 50—60 \ 3,910 3,059 30—60 2,838 2,160 50-70 4,546 3,579 30—70 3,277 2,535 15 50-75 4.816 3,819 30—75 3,402 2,571 60-60 4,692 3,678 40-40 V,397 1,792 60-70 5,780 4,327 40-50 2,916 2,204 60-75 6,054 4,849 40_60 3,451 2,637 70-70 7,125 5,805 10 40_70 40_ 75 50_50 50_60 50—70 50-75 60—60 60_70 60—75 7C-70 3,914 4,264 3,563 4,206 4,873 5,174 5,023 6,161 6,452 7,556 3,03 J 3,273 2,723 3,242 3,819 4,062 3,911 4,917 5,U2 6,124 1 30-30 30-40 30—50 30-60 30—70 30-75 40-40 40-50 40-60 1,404 1,673 2019 2,363 2,813 2,845 2,027 2,467 2,943 1,079 1,284 1,557 1,831 2,213 2,241 1,558 1,908 2,293 " 20 40—70 3,358 2,641 30-30 1,572 1,191 40-75 3,615 2,873 30-40 1,857 1,407 50-50 3,010 2,841 30—50 2,227 1,699 50—60 9,607 2,828 30-60 2,600 1,996 50-70 4,208 3,337 15 30-70 3,052 2,381 50—75 4,474 3,576 30-75 3,127 1 2,403 60-60 4,347 3,433 40—40 2,224 1,687 60-70 5,386 4,338 40 50 2,701 2,067 60-75 5,636 4,558 40-60 3,205 2,474 70-70 6,695 5,489 40-70 3,641 2,839 Rule. The years' purchase in the tablej multiplied by the improved annual value of the estate, beyond the rent payable under the lease, gives the fine to be paid for put- ting in the new life. S3 266 PERMUTATIONS AND COMBINATIONS. Ex. What must be given to prt in a life of 10 years, when the ages of those in possession are 40 and 50, al-- lowing 6 per cent, for money ? Ans. 2.204, or not quite 2i years* purchase. If the life to be added be 15 years, the answer would be 2.067, or very little more than 2 years* purchase. And, If the life to be added be 20 years, the answer would be 1.908, or less than 2 years* purchase. PERMUTATIONS AND CONBINATIONS. The Permutation of quantities is the changing or varying the order of things. The Combination of quantities is the shewing how often a less number of things can be taken out of a greater, and combined together, without considering their places, or the order in wiiich they stand. Gase I. To find the number of changes that can be made of any given number of things, all different from each other. Rule. Multiply all the terms one into another, and the last product will be the number of changes required. Ex. 1. How many changes can be rung on 12 bells ? 1X2X3X4X5X6X7X8X9X10X11X121= 479,001,600. Ex. 2. How many days can eight persons be placed in a different position at a dinner table ? Answer, 40320. PERMUTATION AND COMBINATIONS. 267 Case TI. Any number of different things being given, to find how many changes may be made out of them, by taking a given number of quantities at a time. Rule. Multiply the number of things given, by itself less 1, and that product by the same number less 2, dimin- isliing each succeeding multiplier by an unit, till there are as many products, except one, as there are things taken at a time the last product will br the answer. Ex. 1. How muny changes can be rung with 4 bells out of 12 ? 12X 12—1 X 12 —2 Xl2— 3 = 1^X11X10X9 = 11880. Ex. 2. How many changes can be rung with 5 bells out of 10? Answer, 30240. Ex. 3. What number of words, containing each 6 let- ters, can be formed out of the 24 letters in the alphabet, supposing any 6 to form a word ? Answer, 96909120. Case HI. To find the combinations of a less number of things out of a greater, all different. Rule. Take the series I, 2, 3, 4, &c. up to the less number of things, and multiply them continually to- gether for a divisor : then take a series of as many terms, decreasing each by an unity, from the greater number of things, and multiply them continually together for a dividend. Divide the latter product by the former, and the quotient will be the answer. Ex. 1. How many combinations can be made of 10 things out of 100 .^^ 1X3X3X4X5X6X7X8X9X10 (the number to be taken at a time)=»,628,800 100X99 X98X97'X96X95X94X93X92X9 I (the same number of terms taken from 100) =62,815,650,955,529,472,000. 6281565095529472000 and ^ = 17310309456440. 3628800 268 EXCHANGE. Ex. 2. How many combinations can be made of 3 letters out of the 24 letters in the alphabet? Answer, 2024 combinations required. Ex. 3* A club of 21 persons aj^reed to meet weekly, five at a time, so long- as they could, without the same live persons meeting together, how long would the club t'xist? Answer, 391 years. t'ASE IV. To find the compositions of any' number, in sets of equal number:^, the things or persons them- selves being different. Rule. Multiply the number of things in every set continually together, and the product is the answer. Ex. 1. There are three parties of cricketters, in each eleven men, in how many ways can 11 of them be cho- sen, one out of each ? Answer, 11X11x11 = 1331. Ex. 2. In how many ways can the four suits of cards b'e taken, four at a time ? Ans. 28561. Ex. 3. There are four parties of whist players ; in one there are G, in the second, 5, in the third 4, and in the fourth 3 persons^ how often can the set differ with these persons? Ans; S6o. EXCHANGE. By Exchangf^. is meant the bartering, or exchanging,^ the money of one place for that of anotlier, by means of an instrument in writing, called a bill of exchange. Exchanges are carried on by merchants and bunkers all over Europe, and are transacted on the Royal Ex- change of London, the ilovai Exchange of Dublin, the Exchange of Amsterdam, and those of the principal ci- ties of this country and the continent. When an exchange is mentioned between two places, one place gives a determined price, to receive an unde- termined one. EXCIIA'U.E. '269 The deternuncd price is called certain : thus, London i^iv6s a pound sterling, which is a certain price, lo receive from Paris a number of francs, more or less, to he paid or received there. As.i,ain London gives 100'. which is a certain price, to Duhiin and other parts of Ireland, for an uncertain number of pounds, shillings anil pence Irish, to he paid or received there, "viz. frora 105'. to 115/. Irish, as the exchange may he. The undetermined price is called uncertain, because it is alvvays 8-ubject to variation : for instance, London pays an uncertain price to Spain, as a num- ber of pence sterling, to receive a dollar which is cer- tain in exchange. Thereat money of a state signifies one piece or more, of any kind of metal coined, and made current by pub- lic authority, as guineas, shilliiiiis, &.c. of .^^ngiand. The imaginary moneif is chiefly used in keeping ac- counts, as pounds steiling, for which there is no coin to nsTver. The yar of exchange is the quantity of the money, whether real or imaginary, of otie countrv, which is equal in value to a certain quantity of the money ot another ; thus^ 200/. sterling is equal in value to 108/. 6 S3* 270 EXCHANGE. months* date. Half usance is 15 daySj be the month- what it may. Bays of Grace are a certain number of days allowed for the payment of bills of exchange, after the expira- tion of the term specified in such bills, and are variable in diflferent countries. In England three days are al- lowed. Rules for finding what quantity of the money of one country will be equal to a given quantity of the money of another according to a given course of exchange. Case I. When the course of exchange is given, how much money of one country answers to a certain sum of another, as of Great Britain ? Rule. As the given course of exchange, is to one pound sterling, so is the given sum in foreign money, to its corresponding value in sterling money. Ex. 1. How much sterling money can I have for 2035 Flemish shillings, when the course of exchange is 37 shillings for 1/. ? Here 1 say, As 37 : 1 : : 2C35 ; 55 «= pounds sterl. Ex. 2. How much sterling money can 1 get for 4086 florins, 4 stivers, 6 penings banco, supposing 1^. is worth 38 schillings and 2 grotes r* schil.gr. X. florins st. p. 38 2 1 : : 4086 4 6 12 40 458 163440 grotes 8 grotes = 4 stivers I of a grote=6 penings 458)1 634481(356^. I7s. 6d. Ans. NOTE* S. d. * 8 penings make 1 grote, or penny = 54 2 grotes — — 1 stiver - - - =r 1.09 12 grotes - 1 L'chilling - - = 6 56 20 schillings 1 pound Flemish = 10 11.18 ^ 40 grotes ^-— 1 guilder or Florin = 1 9.8» EXCHANGE* 271 Ex. 3. What sterling money will 293?. 10s. 6i. Irish fetch, when the exchange is 1 14/. Irish for 100/. sterling ? 114/- : lOOl, : : 293/. 10s. Qd. : 257/- 9s. C^d. Ex. 4. Dublin remits to London 826/. I3s., what must be received there, exchange being HO/, per cent? Answer, 751/. 10s. Ex. 5. Jamaica remits to London 287/. Os- lO^d, currency, what must be received for it, exchange being 135/. per cent. ? Answer, 212/. ]2s. 5d, Case II, Given the course of exchange, to bring any quantity of sterling money into the money of another country. Rule. As 11. sterling is to the course of exchange, so is the given sum, in sterling money, to its correspond- ing value in foreign money. Ex. 1. How much Flemish money will 233/. 6s. 8d» sterling be worth, when the exchange is 34s. per l/. sterling ? 1/. ; 34s. : : 233/. 6s. Si. : 396/. 13s. 4i. Answer. Ex. 2. How much Flemish money must be given for 628/. 10s. sterling when the exchange is 33s. 8c/. per L, sterling. Answer, 1057/. I9s* 6d, Case III. To reduce the currency of any state into bank or exchange money. Rule. As 100, with the agio added to it, is to 100, so is any given sum current to its value in bank money. Ex. 1. How much bank money can a merchant in Amsterdam have for 5550 guilders, when the agio is 4^ per cent. ? 1044 : loo : : 5550 : 6311 • Answer. • 104.5 . 272 EXCHANGE. Ex. 2. How many floiins bank will 3000 currency purchase, agio being 6i per cent. ? Answer, 2H23 florins, 21 grotes, I penning. Case IV. To reduce bank money in4 per cent, what has he to pay me ? In this cast the regular interest is 55 dollars, which at 4s. 6d. each, v^t^n exchange is at par, or at l66l. 13s. 4i., would be 12L ns. 6d., but the exchange is 164 5 therefore I say. As 164 : 166^. 13s. 4rf. : > 12^ 7s. 6d. : l2l. lis. 6^^. Ans. Or by Aecimals, 164 : 166.66, &c. : : 53 : 55.^^9 dollars ZZ 12Z. lis. 6.d The following is a Table of the Course of Exchange^ taken with slight variations from Hie Monthly Maga- zine for the J St of May, 1808. COURSE OF EXCHANGE. April 5; gives 34.5 34.7 35.5.2U.- 34.9 23.13 - Hamburgh Altona - gives Amsterdam gives Ditto, sight gives Paris, l.d. gives Leghorn receives Naples ditto Genoa ditto Lisbon ditto Oporto ditto ^ladrid ditto, Palermo ditto Dublin ditto Agio of Bank \ of Holland $ 49J pence- 42 ditto - 45 ditto - 60 ditto 60 ; 65 ditto 65$ SH| doEff- April 12, 34 6 for U. 34 7 for do 35.4.2.U. do 34 8 for do l.d. 24.0 for do[rials 49| 42 for 1 pezza of 8 for 1 ducat for 1 pezza for 1 milrea — - for 1 dollar 92 per oz. — 11 01/. — 6| per cent . 92 .110 per oz. tor 100 6| percent. 276 e:xckangk. Tliis table, in addition to what is gone before, will afford an opportunity of explaining everything that a mail of business vvill wish to be aciiuainted with. On the 5th of April, the exchange between Hamburgh and London was at the rate of 34 schillings, 5 grotes for a pound sterling; that is, if a merchant in London sell a bill on Hamburgh for 500/., he would be paid for /t 34.5 X 500 zz 17208 schillings, 4 grotes; but on che 12th, such bill would have fetched 34.6 X 500 = J/250 schillings. Here, the higher the exchange the greater the advantage to England ; for the merchant, in this in- stance, gains 41 schillings, 8 grotes, by the ^ise in the exchange. For Altona, the course of exchang-'^ ^s the same on both days, viz. the L. is worth 34 schillings, 7 grotes ; and for Amsterdam, the course o^ exchange falling, the merchant in London would be a loser, who put off his market from the oth to the i-^th. In this case S5.5 2U. means, that a pound sterling is worth, on the 5th 35 ^^chillings, 5'grotes, allowing it to be payable at two^ionths' date : but if it is payable at sight, it is then n^orth only 34 schillings 9 grotes. This difference, w/iich on a bill of 100/. is equal to 34 schil- lings 4 grotes, is instead of the interest of money for the interval. The course of exchange rose between London and Paris from the 5th to the 12th of April. On the first of these days 1/. uas at l.d., that is, at one day's sight, worth 23.13, or 23 francs, and 13 cents.; but on the 12th its value was 24 francs. Leghorn receives 49j pence for 1 pezza of 8 rials, that is, a bill of exchange of 5O0O pezza would be worth 4s. ^d. multiplied by 5000, or 1036/. 9s. 2d, A Na- ples ducat was worth 3s. 6d. : a Genoa pezza 3s, 9di : a milrea of Lisbon 5 shillings, one of Oporto 5s. 5d. EXCHANGES. 277 Madrid receives SS^d. Eff. for 1 piastre of 8 rials,* that is, a Spanish piastre of exchange was worth 3s. 2|i. A species of paper money, denominated tvt/es rials, is circulated in Spain, the value of vvhich, independently of interest on them, is this: — Vales rials for 000 dollars are worth 9035 rials, 10 maravedies of ve^/on,t that is, as 34 maravedies is equal to one rial, 1 dollar payable in this sort of paper is worth 15 ria4s, 2 maravedies. The paper is transferable by indorsement ; and, by lawi should be received in payment according to the nominal value; but as it experiences depreciation, it is necessa- ry in drawing on Spain for effective money, to insert the words '' payable in effective" in the body of the bill, which might otherwise be payable in vales rials : hence the word Eff. in the table, which is an abridgment of " in effective'^*. NOTES. * In some parts of Spain they reckon by silver money, which is of two kinds, viz. old and new plate, the former is the most valuable : thus the piastre of exchange con- sists of 8 rials old plate, or of 10 rials new plate, the rial being at the par of exchange worth little more than o\d, t The copper money of Spain is called vellon. In Madrid, and theprincipal places ot Spain, accounts s^re kept in piastres (called also dollars) rials, and mara- vedies 5 and sometimes in ducats. TABLE. s. d. 51- 34 maravedies 1 C \ rial li: 5 8 rials I make ■< 1 piastre ZZ 3 7 375 maravedies J (^ 1 ducate zz 4 Ijf Hence the piastre at par is 3s. Id., and the ducate at par 4s. Iji. ; but the course of exchange of the piastre varies from 35 to 45 pence. 278 EXCHANGES/ Palermo 92 pence per cz. In Sicily exchanges are made per onza by the ounce of Silver, for which on the day referred to Palernao, received 92 pence, or 7s. Sd.* Dublin 1 10^- for 100/, that is, at the date of the table there would have been given on the exchange of London a bill on Dublin for 110/. 5s. for 100/. sterling, fcee page 275. By the agio of the Bank of Holland is meant, as we have seen, page 269, the difference between cash and bank money, which, by the table, is on the 5th of April, 6|, or 6/. 10s. percent. ; that is, 106/ lOs. currency must be given for 100/. bank, and so in proportion. Exchange between London and other Ftaces in this Country, The several cities, towns, &c. in Great Britain, ex- change with London for a small premium in favour of London, as from ^ to 1 or |A percent. The premium is more or less according to the greater or less distance, and according to the demand for hills. Ex. York draws on London for 560/. 10s. exchange being -J per cent. ; how much money must be paid at York for the bill ? i ?w 560 10 -^ i 2 16 «i 1 8 oi L.564 14 0| To avoid paying the premium, which in some cases, would not be just, it is the usual practice to take the bill payable a certain number of days after date. On this principle, interest being 5 per cent, 73. days are equiva- 365 lent to \L per cent because — =73. 5 NOTE. * The Sicilian ounce is 600 grains, and the monies are regulated by the following Table i 10 grains - make - 1 carlin, 2 carlins - make - 1 tarin, 30 tarins - (600 gr.) - 1 ounce. A crown (seudo) is equal 240 grs., therefore 5 crowns —2 ounces. EXCHANGE. 279 Ex. A friend at Exeter has received for me OS sjuineus, in which lie is no ways interesteil, and haviuir no means of sending the money liut by a bill of exchange, he a<^rees with iiis banker to draw it 30 days after date, rather than pay the premium of | per cent., is my friend, or the banker, tlie gainer, allowing 5 per cent. ? Answer, the banker loses Is. 2d. of his usual profit. EXAMPLES FOR PRACTICE. Ex. 1. How much currency will 6330 {guilders, bank- money, be worth iu Holland, agio, being 8^ per cent. ? Answer, 7176 guilders, 39 grotes/ Ex. 2. What is the agio of 3310 guilders, 6i per cent.? Answer, 206 guilders, S5 grotes. Ex. 3. A London merchant draws on Amsterdam for 1564/. sterling ; how many pounds Flemish, and how many guilders will that amount to, exchanyje being 3 1 schil. 8 gro. per L. sterling. See table, page 270. Answer, 2710 18 8=pounds Flemish — 1G265 24 guilders. Ex. 4. How much sterlinu: money will pay a Portu- guese bill of exchange of 1654^=^372 'millreas ; that is, of 1654 niillreas and 372 reas, exchange being 65|^ pence sterling per millrea ?* Answer, 451/. lOs. l^i. ■^^^. NOTE. *In Portugal accounts are kept in reas and millreas, the latter being equal to 1000 of the former ; and they are distinguished from each other by some such mark as that in the question. The millrea, in exchange with this country, is at par 67 1 sterling or 5s. 7| sterling, and the course usually runs from 5s. 3d. to 5s. Si. TABLE— Par in sterling. s. d.f. 1 rea IZ 0.27 400 reas > l S ^ crusade zz 2 3 1000 reas 5 ^^^^ I I millrea IZ 5 7 ^ The reas being the thousandth parts of the millreas, are annexed to the integer, and the work proceeds as in de- cimals. 52,30 EXeHANOE. Ex. 5. How many Portuguese reas will 750?, sterling amount to, exchange being 64| per millrea ? Answer, ^2785 milr. 299 reasfff||. Ex. 6. A Spanish merchant imports from Seville, gomls to the value of 1031 piastres, 6 rials : how much srerii Jig money will this amount to, exchange being, on the day of payment, 41^ pence per piastre ?" See Ta- ble, page 277. ? Ans. 187/. Is, a|^. Ex. 7. I want to purchase goods at Cadiz, and for this purpose pay into a Spanish house lOOOZ. : how muek value, in piastres, may I expect, exchange being 3s. 6^0?. per piastre ? Answer, 5647^2/t piastre^, ARBITRATION OF EXCHANGES, The coarse of exchange, between nation and nation, naturally rises or falls, as we have seen, according as the circumstances and balance of trade may happen to vary. To draw upon, and ta remit money to foreign pla- ces, in this fluctuating state of exchange, in the way that will turn out mo&t profitable is the design of arbitration. Arbitration of Exchange, then, is a method of finding such a rate of exchange between any two places, as shall be in proportion with the rates assigned between each of them and a third place. By comparing the par of exchange thus found, with the present course of exchange, a person is enabled to find which way to draw bills or remit the same to most advantage. Arbitration of exchange, is eitheF simple or compound. In simple arbitration, the rates of exchange from one place to two others are given, by which is found the correspondent price between the said two places, called the arbitrated price. An example or two will make the subject clear. EXCHANGl^. 2^t Ex. J. If exrhan;5e between Ti)nilon ami Amster- dam be 34 schil. 9 §rotes per L. sterlini^. and if excban<;e between London and Genoa be 45 pence per pezza what is the par of arbitration between Amsterdam and Genoa : Here IZ. =24.0 pence: therefore, as 240fl?. : 34.S. 9 gr. : : 45i, : 78-A^ g)\ Answer, 78 Flemish grotto, or pence per pezza Genoa Kx. 2. If exchange from London to Amsterdam 3Ss. 9:/. per L an;e between P.iris and Amsterdam is fallen to 52 pence Flemish per crown : what may be trained per cent., i)y drawing oa Paris and remitting to Amsterdam } 24^ 282 COMPOUND ARBITRATION. By Ex. 2, the par of arbitration between Paris and Amsterdam is 54id. Flemish per crown ; then d. cr. L. cr. 32 : 1 : : loo : 750 drawn at Paris. cr. d.Fl. cr. ^.Fi. 1 : 52 : : 750 : 39000 credit at Amsterdam. rf.Fl. /.. rf.FI. L. s. d, 405 : I :: 39000 : 96 5 11 to be remitted. therefore 100^.— 96/. 5s. lid. =3/. lis. lfi.=gain per cent. to r- If the course of exchange between Paris and Amster- H#am he at 56 Flemish per crown, instead of 52 ; and iff would gain by the negociation, I must draw on Amster- dam and remit to Paps ; thus, X. d.lX Z. f/.Fl. 1 : 405 :: lOO : 40500 drawn at Amsterdam. d.¥\, cr. rf.FI. cr. 56 : 1 : : 405OO ; 723 credit at Paris, cr. d. cr. L. s. 1 : 32 : : 723 : 96 .8 fherefore 100/.— 96/. 8s. zz 3/. 12s. gain per ceijl, COMPOUND ARBITRATION. In Compound Arbitration, the rate of Exchange between three or more places is given, to find how much a remittance passing; through them all will amount to at the last place : or to find the arbitrated price, or par of arbitration, between the first and last place. Examples of this kind may be worked by several suc- cessive stagings in the Rule of Three, or according to tlie following Rules i OOMPOttJND ARBITRATION. 25» {l) Distinguish the given rates, or prices, into ante- cedents and consequents, placing the antecedents in one column, and the consequents in another, with the sign of equality between them. (2) The first antecedent, and the last consequent to which an antecedent is required, must be of the same kind. (3) The second antecedent must be of the same kind with the first consequent, and the third antecedent of the same kind with the second consequent, &c, (4) Multiply the antecedents together for a divisorj and the consequents together for a dividend, and the quotient will be the answer required. Kx. If a merchant in. liondon remit 500/. sterling to Spain by way of Holland, at 35 shillings Flemish per* pound sterling, thence^o France at 58 pence per crown, thence to Venice at 10 crowns for 6 ducats, and thence to Spain at 360 mervadies per ducat ; how many pias- tres of 272 mervadies will the 500/. amount to in Spain ? H. = 35s. or 4,Q0d. FL 5Sd. = 1 crown 10 cr. ZZ 6 ducats 1 due. = 360 mervadies 272 mer. =3 I piastre How many piastres =500/. Omitting the units, we have by the rule^ 420 X 6 X 360 X 500 . and this fraction reduced to it^ , 58 X 10 X 272 21X3X45X500 ] 417500 lowest terms, gives = =s= 29 X 17 41)3 2875| piastres, \^hich is the answAff. 1284 DUODECIMALS. By the Rule of Three we should hav6 said, 1/. : 420rf. : : 5i>0L : 2IO000df. 58^/. : 1 cr. : ; 210()00c?. ; 3620 cr.* 10 cr. : 6 due. ": : 3')20 or. ; 2172 due. 1 due.: 360 mer., : : 2172 due : 781Q20 mer. 272 mer.: 1 pias. : : 78l920mer: 2875» pias. If the course of direct exchange to Spain were 42| pence sterling, then 500/. remitted would only p.nount to 2823-^- piastres, of course 2875|— 2823^, gives 52, which is the number of piastres gained by the negotiation.. DUODECIMALS Duodecimals, or Crnsa jyiiiltlplicatian, is made u^e of by artificers in measuring theif several works, and is performeil by means of the following table ; 12"" fourths - make 1 third. 12'" thirds - - 1 second. 12" seconds - - 1 inch. 12' inches - - 1 foot. Glaziers, Masons, and others, measure by the square foot. — Painters, PavU.rs, Plasterers, &e , by the square yard — elating, tiling, flooring, &c. by the square of 100 feet. — Brickwork is n^easured by the rod of 16| feet, the square of which is 27-21. Rule, (l) Arrange the terms of the multiplier under the same denomination of the multiplicand. (2) Multiply each term in the multiplicand, beginning at the lowest, by the (eet in the multiplier, aud write the result of each under its respective term, observing to carry one for every twelve. (3) Multiply, in the same manner, by the inches, and set the result of each term one place re- NOTE. • The fractions are omitted, and on that account the answer by this method will not be quite accurate. DUODECIMAIS. 2^5 moved to the right-hand of those in the multiplicand.* (4) Multiply then by the seconds, setting the result of each term two places removed to the right-hand of those in the multiplicand. Multiply 9 ft. 4 in. 8 sec. by 5 ft. 8 in. 6 sec. 9 4 8 5 8 6 46 11 4 6 *3 1 4'" '4840' 53 7 1 8 Ex. 1. How much must I pay for a slab of marble 7 ft. 4 in. long, and 2 ft. 1 in. 6 sec. broad, at the rate of 7s. per square foot ? Answer, 5L 9s. id, Ex. 2. What will ^e the expence of glass for a win- dow that measures, in the clear 10 ft. 6^ in. in height^ and 4 ft. 9 in. in width, at is. 9d. per toot ? Answer, 4?. 7s. 6d* Ex. 3. How much will a room cost in painting, at Old. per yard ; the sides are 18 ft. 10 in. by 10 ft. 3 in. and the two ends are 16 ft. 6 in. by 10 ft. 3 in. ? Answer, 3/. 3s. 8|-c?. Ex. 4. What shall I have to pay for statuary marble about my fire-place, at 14s. per foot ; the hearth mea- sures 6 ft. 4 in. by 2 (t. 3 in., the three fronts are each 4 ft. 2 in. by 8 in., and the mantle-piece slab is 6 ft by 9 in. ? Answer, 18/ 19s. Ex. 5. What will the paving of a court-yard come to, at Is. 2d, per foot, the yard being 74 feet long, and 56ft. Sin. wide? Answer, 244^. 12s. ^d. * Feet multiplied into feet give feet. Feet multiplied into inches give inches Feet multiplied into seconds give seconds. Inches multiplied into inches give seconds. Inches multiplied into seconds give thirds. Seconds multiplied into seconds give foufths. 586 DUODECIMALS. Ex. 6. How much shall T have to pay for slatins; a house, consistina; of two slopinj^ sides, each nieasurinj^ 24 ft. 5 in. hy 15 ft. 9 in. at the rate of 41s. per square of 100 feet? Answer, loL 18s. 7d, Ex. 7. What will the tilinj^ of 10 houses come to, the roof of each house consistin*^ of two sides, each 18 feet by 14, and the price of tiling at 28s. per square ? Answer, 701. lis. 2ld. Ex. 8. How many square rods are there in a brick wall 44 ft. 6 in. long, and 7 ft! 4 in. high, and 2^ bricks thick ?* Answer, 2 rods nearly. Ex. p. If an oblong garden be 254 ft. Bin. long, and 184 ft. 8 in. wide, what will a wall cost 10 ft. 6 in. high^ and 2J bricks thick, at 15/. 15s. per square rod ? Answer, 888/. 6s. Ex. 10. How much shall I have to pay for the plate- glass of four windows; each window consists of 16 panes, and each pane measures 20|^ inches by 15 J inches at 9s. 6d, per foot i Answer, t\Bl. 3s. Sd. KOTE. * Bricklayers value their work at the rate of a brick and a half, or three half bricks thick ; and if the wall be more or less than this, it must be reduced to that thick- ness by the following; rule : — '< Multiply the measure found by the number of half bricks, and divide by three :*' thus, if the wall be 2^ bricks thick, 1 multiply by 5, and divide the product by 3. Ex. If the wall be 50 feet long, and f high, and 2 4 600 bricks thick, it will be 50x9x— =600 feet 5 and 3 272^ = 2^ square rods nearly. ■^ if.' %^