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Un dm symboles suivants apparaftra sur la dernlAre image de cheque microfiche, selon le cas: le symbols — »> signifie "A SUIVRE", le symbols V signifie "FIN". IMaps, plates, charts, etc., may be filmed at different reduction ratios. Those too large to be entirely included in one exposure are filmed beginning in the upper left hand corner, left to right and top to bottom, as many frames as required. The following diagrams illustrate the method: Les cartes, planches, tableaux, etc., peuvent Atre filmAs A des taux de rAduction diff Arents. Lorsque le document est trop grand pour Atre reproduit en un seul clichA, il est f limA A partir de Tangle supArieur gauche, de gauche A droite, et de haut en has, en prenant le nombre d'images nAcessaire. Les diagrammes suivants illustrent la mAthode. 1 2 3 1 2 3 4 5 6 TL1 COM. Arithmeti being a br its Rules, concise Me published ; toeachRul variety of siness, with ed. II. Vulgar 1 treated witi ness and pe III. Decimal tion of tiie quadrate 1 plain and \i'li!cb »»re s easy calciih nuities, at ( Tiie whole b« School9{ or Knowledge 77*1* Work i and Accom ¥tibliithed,j FROM PR Vk J. f . ■ : I TLTOx^s ASSIST Ai^ BEING A COMPENDIUM OF ARITHMETIC, AND A COMPLETE QUESTION-BOOK. i ! CONTAI Arithmetic in whole Numbers ; being a brief Explanation of all its Rules, in a new and more concise Method than any hitlierto published ; \nth an Application to each Rule, consistinja; of a. large variety of Questions in veal Bu- siness, with their Answer:! annex- ed. II. Vulgar Fractions, which are treated with a great deal of plain- ness and perspicuity. III. Decimals, with the Extrac- tion of the Square, Cube and Bi- quadrate Roots, after a very plain and familiar manner ; in which »*re set down Rules for the easy calculation of Interest, An- nuities, and Pensions in Ar- ning: rears, the present worth of Au' nuities, AN APPENDIX ON CIRCULATING DECIMALS. The whole being adapted either as a Question-Book for the Use of Schools^ or as a Remembrancer and Instructor to such as have some Knowledge therein. 77ii.« fVork having been perused by several eminent Mathematicians and Accomptants, is recommended as the best Compendium hitherto published, for the Use of Schools, or for Private Persons. BY FRANCIS WALKINGAME, WniTlNG-MASTER AND ACCOMPTANT. FROM THE FlFTy-FIRST LONDON EDITION. Montreal: printed by nakum mo*kh, 9j, st, pacl steet, 1818, ../fe' J ■■■A !l i •t t| i -!■ .uijlaHMETICAi. w NrHIKRATlON. =: 5 o - ■ =r. ■" c = 2 g . M §• v: • S3 : 1 1 2 2 3 : I 1 4 2 3 3 4 4 & e " E- 0) • • 1 : I ^ 1 2 3 2 3 4 4 5 6 7 3 4 6 5 6 7- 8 18 '-iJO.. i^ .. 1 30 40 m so PO 100 1 10 2 2 10 3 3 4 4 S 10 10 PENCE. d. ^'^ is ;4 .. 10 .. 33 .. 40 t. 48 .. .50 .. tin .. 70 .. 72 .. SO .. S4 .. 00 .. 90 .. iOO .. 108 .. 110 .. i20 .. 1 30 .. 132 .. 140 .. 144 .. 1.50 .. 100 .. jr. d. 1 8 2 2 3 3 4 4 4 2 5 5 10 6 6 8 7 7 6 8 8 4 9 9 2 10 10 JO 11 T) 11 8 12 12 13 4 MULTIPLICATION. COINS. VALVE. £. s. d. A 5Ioidore 1 7 (• Half ditto 13 « A Guinea 1 JO Half ditto 10 6 Eighteen Siiilling .0 18 (. Half ditto *.0 Ol: A Pistole 1/ I Halfditto 8fi A Mark 13 ■< All Angel 10 4 Noble 6 8 WEIGHT. dwt 6 22 3 11 5 9 2 4 1«* U 2 4 8 2 a .s Gold is .and each* i a a 1 i ,-. •4 .■) (i 7 8 « 10 n 4, 4 1 ft \n ~ ^^ "^~ 1>! ^\f •"• ~ •1 ~~> >■■> i.H 27 77 «: 3 -• T i'l. Id 20 £4 is 3*, 30 4»; .n ,, TT io i!- ■J,'> ■J'J .'.,0 10 4.5 .50 .5.) b 17 "ii 24 3(. M< 4-.' . *, »4 ■•u o*. "TiUlIT i>. 3,1 -iJ ^^"^ ii) (,;.,» ".0 7. 8 lu!24 3c -'«> -I*! J'; .\-V \i\ ^^{i i". . 9! 18 2 2 .'ii- TT u\->:' f2i 81 ; '.^..i )."• IC M iO 4t> .>i> ;>'■' "!■; H< ~^l' ah' ; p Ti ~"J 33 44 .5» * Aw' lTION. • lid 4.5 >4 fJ.» \i H\ ^'■' 10 ■''. • >• -**; .50 ■•u 10 '.0 .\tv in ■' ■ 3: ' '■» 'i ■ ■ , ' * IT" )■"• ill ; i = . Ui '■ Hi i4 \BLES. 1 Q/" i:(.. of Out 11. ly. ^'^ Pouiitls ... 1 Dm ».)f \ir\v Hay. \6 Tru.«;se.s ... I L(Wil. TONG MiiASUPiE. li liitinfs ... I lAt'rt. ^ r\('t I Yuif.i. ■'^ Yitr.ls ... 1 P.ilr'. 4'J Poles . , . . » I'a:l;in!?. S Purlon^s . J AJ.ie. LAN^D .VJKASUaE. IVet .... 1 Yard. .Ji)-t Yards ... I I';,l«. 40 Poles ... I Kord. 4 liooti." . . : 1 Acre. 'CT.OTH ,vi{*;ASURE. .i^ Ti:clie« . . . . i Nail. 4 NuUs . . . . J (iuartvr. i QiiarUM's . . ! Fie;ii!.s Mliiu.'es .... I Hour. 4i Hours 1 Dav. 7 IXivs i Wc'ek. 4 Week.s .... 1 MrntU. 3(1,5 Day.s, 6 Hours I Year. DPiY mp:asurr. i Quarts .... I Pottle. 4 Pottles . . . . [ Gall(,;i. 4 Gallons ... 1 Peck. 4 Pecks 1 Busli*!. i Bushels .... I Strike, n Bushels .... 1 Quartar, •5 Quarters ... 1 Wey. i Weys 1 Last. SOLID MRA.SURK. 174^ Inches ... I Solid Foot. '41 Feet .... 1 Yard or Load. I ADVERTISEMENT. TWALKINGAME's Assistant having been so long held in high and deserved estimation hy the Public, it cannot be necessary for the Proprietors or Editor of the present edition (the 51st J to enter upon any discussion of its merits ; were they inclined so to do, they should undoubt- edly refer to tJie numerous republications of the "work itself, as forming its most powerful reconu mendations. Nevertheless, it cannot have esca- ped the observation of those who are engaged in studying or teaching the higher bra.xhes ofArith* melic, that hating hitherto coyitained no rules up- en the management of Circulating Decimak, this xvork has not only been incomplete in regard to the Theory and Practice of Decimal Fractions, but also incofTCct in many of the answers to the questions contained therein^ from want of atten- tion to the practice of Circulating Decimals in their solutions ; an oversight which has been a frequent source of trouble and anxiety to every Teacher, when the Pupil has been workiiig the ea ample s in Decimal Interest, Purchase of An- nuities, c^c. since the approximations which have been hitherto uniformly substituted in the place of the true results, are insufficient for their solution^ when t der an\ this in wantin^ and en ded Of Decimi and tht ring L to a w precede wholej cheapen which As to have in Dec repeten ces, a therefo commei found i gar Ft most rt n' g been so estimation for tJie tion (the s merits ; undoubt. IS of the ul recom- ive esca- gaged in ofArith^ rules up- lak, this 'gard to raciionSy s to the if at ten- limals in been a '0 every king the of An' ich have place of wlution. ADVERTISEMENT. V when the same questions have again occurred un- der another rule, with other data. To rei^iedj/ this inconvenience, and that nothing might be wanting in regard to the perfection of this new and enlarged Edition, an Appendix has been ad* ded on Circulating Decimals, the examples in Decimal Interest, S^c, carejully wrought afresh, and their corrected anszvers inserted ; the recur- ring Decimals pointed, and the errors incident to a work of this nature, which have crept into preceding editions, every^ where corrected; the whole forming, in its present improved slate, the cheapest and most practical work on Arithmetic which has hitherto appeared. As it would have been frequently inconvenient to have given the remainders to many examples in Decimal Interest, (§c. in a decimal form, the repetend consisting of too great a number qfpla-^ ces, a vulgar fractional expression has been, therefore, introduced, and the Editor would re* commend, in all instances where the repetend is found to consist of many places, the use of VuU gar Fractions in preference to Decimal . as the most ready mode of calculation. [\ 'y.!. .. .•:•', 'i.'j i "■"«'/ - »!t v» PREFACE. THE Public, no doubt, will be surprised to find there is another attempt made to publish a book of Arith- METic, vhen there are such numbers already extant on the »ame subject, and several of them that have so lately made 2^rg^r5, any that is already published ; as likewise an Interest Table, calculated for the easier finding the Interest of any sum of money, at any Rate pe}- cent, by MultipliCfition and Additioi only ; it is also useful in calculating Rates, Incomes, and Servants Wages, for any Number of Months, Weeks, or Days ; and I may venture to say, I have gone through the whole with so much plainness and perspicuity, that there ii none better extant. I have nothing further to add, but a return of my sincere thanks to all those Gentlemen, Schoolmasters, and others, ^ills of F^ whose kind approbation and encouragement have now ^< deduction tablished the use of this Book in almost every School of em' inence tj^roughout the Kingdom : But I think my gratitude more especially due to those who have favoured me with their remarks ; though I must still beg of every «(fandid and judicious Reader, that if he should, by chance, find a trans 'ahles .. Uldition minatio ShVUlS^ilO. )onble Ih ^tacticc h-e and position of a Letter, or a false Figure, to excuse it ; for, not- i'nple Ini withstanding there has been great care taken in correcting, ' yet errors of the press will inevitably creep in ; and some may also have slipped my observation ; in either of which caaes the admonition of a good-natured Reader will be verj acceptable to his Much obliged ^ > • and most obedient 4 humble Servant, F. WALKING AMIS, ■L ^: >t: 3'i; 1 — J ingle Rid eduction ddition . nbtractioi hltipUcai ■8 in liitegen ms ; in order be perform- irdiny^ to the dotting. In nd the use of 18 that occur jduction sov- eul business, led Tables to ties, &c, and •d lew CONTENTS, FART L ARITHMETIC IN WHOLE NUMBERS. ss, AC. and of making)!? w Rule lor" tpr way thw integers, Addition 3 terest Table, f any sum oi, and Addition ncomes, and Weeks, or tddilion of several Deno- through the minations 16 that there i«J— — — Suost.-aclion 22 MvltipUcation ... 25 af my sincercl- Divhion 30 i, and others, Mils of Parcels 32 have now eS' lednction 35 School of eni< my gratitude ired me with y Candid and find a trans it ; for, not in correcting, ; and some ler of which will be verj ingle Rule of Th ree Direct 4 1 Inverse 4'4' )rAihle Rule f^f Three .... 46 NGAMJS. NTROCTION 1 f'meralion 2 — Subtraction 4 Multiplication ... 5 — — — Division 7 'ahlcs 9 Purchasing of St ocic 59 Brokerage ib. Compound Interest 62 Rebate, or Discount Gf* Equation of Payments ... 64 Harter ...'. 65 Profit and Loss 67 Fellcwship 68 ith Time 70 Alli(:;ation, Medial 71 Alternate 73 Position^ or Rule of False 76 — Double 7« rogresston, ■tactice 48 V:?e and Tret 55 iinple Interest 5SlPermutation 'ammisdon Exchange « ,.... 79 Comparison of Weights S^ Measure^* 83 Conjoined Profortion Arithmetical Geometrical 591 81. 8.> 92 PART IL VULGAR FRACTIONS. Pajre eduction 94 ddition 100 ubfraction ib. Mtiplicatibn , 101 Division 102 The Rule oj Three Direct ib. ■ ■ Inverse 1 03 The Double Rule of Three lOi ! .1 1.1 I i : CONTENTS. fi PART III. DECIMALS. i THE ] Page Numeration 105 Addition 106 Subtraction 107 Multiplication ib. Contracted Muliiplicat ion 108 Division 109 Contracted 110 Reduction , Ill Decidual Tables of Coins, Weights and Measures 114 The Rule of Three 117 Extraction of the Square Root lis Vulgar Franctions 119 '■ ' Mixed Nu mbers 1 20 Extract, of the Cube Root 122 — — — Vulgar Fractions 124 — — Mixed Numbers ib. -^-«— — Biquadrate Root 126 A General Rule for ex- tracting the Roots of all Powers Simple Interest for Dai/.s Pa Annuities and Pensims, S(c. in Arrears Prcsentxjoorth of Annuiiki Annuities, ^c. in revenloii Rebate, or Discount Equation of Pat/ments .. Compound Intere'it Annuities, 8^-0. in Arrears Preseht ivujih (jf Annuities Annuities, S^c. in reversion Purchasing Freeh Id, or Real Estates r:; '-1 IS u 14: 14: 14; J 4! I Si 15; CoUectii {,et down frr the g thejbreg tn reversion Rebate or Disccunt Multiplication of Feet and Inches 157 Measuring by the Foot Square 159 Measuring by the Yard Square ib* i PART IF. DUODECIMALS. Page Paj: Measuring bif the Square of 100 i^e'et Measuring by the Rod , Multiplying several Fi' gurcs by several, and the Operations in one line only 16( ^nitiom eduction ^^ ddition „ 16 CONTENT* XI I THE MENSURATION OF CIRCLES, &c. 16i Pil:; 'i? /or ej:- Roois of 11 • • • • • ••••• - i'j* PensimSi >-s 13 ^ An null I'd ^ 1j irevenim llrl 02(n/ 14| ymenU .. 14^ ^f?^ i4'i ?» Arrears 143 'Annuities \^i f. reversion I5'f eh lci\ or 153 nion 151 mil •<•«#••• iQt I i ■ I ; Collection of Questions \},et down prcmiscuously f^rr the greater trial of PART ri. QUESTIONS. Page A creneral Tnhle for calculating Interests^ Pentsy Incomes, and Fag© thejbregoing Ruks.,, iTOl Servants Wages,,,,,, 176 , APPENDIX. % Paj: j. . ON CIRCULATING DECIMALS. \f'n Page Subtraction 184> 185 Multiplication, c Square 1 - . . ^^^ l^efinttwns 179 eduction 181 ■e Rod . IG^ddition J82 {DivUton 189 eral Fi' ral, and | i s in one I ..I ' *' ''v. .:■:•' ; ;• * ■ i '■.'{ >^ # xu 11 i \ EXPLANATION OF THE CHARACTERS MADH USE OF IN THIS COMPENDIUM. issEqual. '—Minns, or less. The sign of Equality ; as 4 grs.=sl civt, signiiies, that 4< qrs. are equal to 1 ad, The Sign of Subtraction; as, 8—2=3, that is 8 lessened by 2 is ecjual to 6. •^PluSf or more. The Sign of Addition ; as, 44-1=8, that is^ 4 added to 4 more, is equal to 8. The Sign of Multiiilication ; as, 4X6=24-, that is, 4 multiplied by 6 is equal to 24. The Sign of Division ; as, 8-5-2=1?, that is, 8 divided by 2, is equal to 4. Numbers placed like a fraction do like- wise denote Division ; the upper num ber being the Dividend, and the lower the Divisor. The Sign of Proportion ; as 2 : 4 : : 8 : 16, that is, as 2 is to 4, so is 8 to 16. Shews that the Difference between 2 and 7, added to 5, is equal to 10. Signifies that the Sum of 2 and 5, takeu from 9, is equal to 2. XMuUipUed by. •i-Divided by, . 2357^ : : So is. 7—2+5=10. 9—2+5=2 10—3+6=51. 8 I. r. 1 ARi RIT on whic NOTAT ON, Mu ™. EAC diffe ber. Over any number of quantities, denote that they must be taken together which are under it, thus 10 less the sum of 3 and 6 is equal to i ; without this char- acter the preceding expression would be ambiguous, and might be read thus, 10 less 3 and 6 added to the difference, is equal to 1. Prefixed to any number, signifies the Square Root of that number is required. Signifies the Cube or Third Pdwer. Denotes the Biquadrate, er the Fourtli Power, &c. id est, that is. THB ?.S MAD I' VM. . : qrs.=s\ civt,' ual to 1 cn-L as, 8—2=5, qual to 6. 14-4=8, that qual to 8. as, 4X6=24, equal to 24, ?.i-2=l, that 1 to 4. :tion do like- upper num-j md the lower! ! : 4 : : 8 : 16,| J to 16. etween 2 and| 10. and 5, takeul ities, denote qether which le sum of 3 It this char- ssion would )e read thus, le difference, signifies the r is required TUTOR'S ASSISTANT: BEING A COMPENDIUM OF ARITHMETIC. PART I. ARITHMETIC IN WHOLE NUMBERS. THE INTRODUCTION. RITHMETIC is the Art or Science of computing by Numbers, and has five principal or fundamental Rules on which all its operations depend, viz. Notation or Numeration, Addition, Subtrac- ON, Multiplication, and Division. NUMERATION lEACHETH the different Value of Figures by their different Places, and to read and write any ^uni or iber. THE table. Pdwer. the Fourtli ■*^ . B ■^ (' C Millions. X Millions. Millions. C Thousands X Thousands Thousands. Hundreds. Tens. Unite. 9 8 7 .654^. 3 2 1 9 .000. OOP 8 .000. 7 .000. 6 0. £ . fc^ •_ ' 4- . 3 2 i V !• { ii i'-: IH 1? ' II Vi } 2 Numeration* THE tutor's V I Rule. There are three Periods ; the first on the Right Hand, Units ; the second Thousands ; and the third Mil- lions ; each consisting of three Figures, or Places. Reckon the first Figure of each from the left Hand as «o many Hun- dreds, the next as Tens, and the third as so many single Ones of what is written over them : As the first period on the Left Hand is read thus, Nine hundred eighty-seven MilHons ; and so on for any of the rest. THE APFLCATION. Write down in proper figures tkejbllomng Numbers t Twenty-three. Two hundred and fifty-four. Three thousand, two hundred and four. Twenty- five thousand, eight hundred and fifty-six. One hundred, thirty-tw6 thousand two hundred forty-five,| Four millions, nine hundred forty-one thousand, fourhun* dred. Twenty -seven millions, one hundred fifly-seven thousand,| eight hun icd thirty-two. Seven iindred twenty- two millions, two hundred thirty-] one thousand, five hundred and four. Six hundred two millions, twx) hundred ten thousand, five hundred. Write down in Words at Length the following Numbers ; 35 2017 519007 5207054 65700047 59 5201 754058 2071909 900061057 172 20760 5900030 70054008 221900790 NOTATION BY ] ROMAN LETTERS. I One XI Eleven II Two XII Twelve HI Three XIII Thirteen IV Four XIV Fourteea V Five XV Fifteen VI Six XVI Sixteen VII Seven XVII Seventeen VIII Eight XVIII Eeighteeft IX Nine XIX Nineteen X Tea XX. Twenty tn the Right i e third Mil- • s. Reckon 5 many Hun- many single period on eighty-sevem ASSISTANT. XXX Thirty XL Forty L Fifty LX Sixty LXX Seventy LXXX Eighty XC Ninety .C Hundred CC Two Hundred CCC Three Hundred Addition. 3 CCCC Four hundred D Five Hundred DC Six Hundred DCC Seven Hundred DCCC Eight Hundred DCCCC Nine Hundred M One Thousand MDCCCXHI One thousand eight hundred & thirteen. :i >i it lumbers 2 y-six. ed forty-five, nd, four hun- en thousand, mdred thirty* ihousand, five • Numbers .* 65700047 100061057 121900790 to n em. INTEGERS. ADDITION. rW'^EACHETH to add two or more sums together, £ make one whole or total sum. ituLE. There must be due regard had in placing the Figures one under the other, i. e. Units under Units, Tens under Tens, &c. then beginning with the first row of Units, add them up to the top ; when done, set down the Units, and carry tim Tens to the next, and so on ; continuing to the last Row, at which set down the total amount. Proof, iiegia at the top of the Sum, and reckon tha Figures downwards, the same as you added them up, and, if the same as ihe tirst, the Sum is supposed to be right. , Vearif, 271048 S'25476 107584^ 625608 754^087 279726 Qrs, Months, £ ^15 12-54. 75245 110 7098 37502 4.73 3:314. 91474. 354? 6732 32145 271 2546 47258 352 / 0709 214-76 What is the sum of 43, 401, 9747, 3464, 2263, ^14, 974. Arts, 17206. Add 246,034, 298,765, 47,321, 58,653, 64,218, 5/J76, 9,821, and 640 together, Ans, 730,828. If you give A.;^.56, B.;^.104, C.;^.274, D.>f.391, E. £. 703, how much is given in all ? Ans. £. 1528. How many days are there in the twelve Calendar months ? Ans, 365, B2 t 1 1 I 4\ i* Subtraction, TKH TVTOB'8 - SUBTRACTION ' ^ ! ,^ ri^EACHETH to take a less sum from a greater^ andi I shews the Remainder, or Difference. 1 ituLE. This being the Reverse of Addition, you must] borrow here (if it require) what you stopped at there, alwaysl remembering to pay it to the next. Proof. Add the Remainder and less Line together, andj if the same as the greater, it is right. 37502051 3150874 From Take 271 154. 117 271 4754. 42087 452705 271508 2723 34096 327616 152471 Rem. froof. \i . . MULTIPLCATION TE ACHETH how to increase the greater of two Num- bers given as often as there are Units in the less ; and | compendiously performs the office of many additions : To this Rule belong these principal Members ; viz 1, The Multiplicand, or Number to be multiplied; 2, The Multiplier, or Number by which you multiply ; 3, The Product or Number produced by multiplying. Rule. Begin with that Figure that stands in the Unit's Place of the Multiplier, and with it multiply the first Figure of the Unit's Place of the Multiplicand. Set down the L nits and carry the Tens in Mind, till you have multiplied the next Figure in the MultipKcand by the same Figure in the Multiplier ; to the Product of which add the Tens you kept in Mind, setting down the Units, and proceed as before, till the v> hole Line is multiplied. Pkoof. By casting out the Nines : or make the former Multiplicand the Multiplier, and the Multiplier the Multi- plicand ; and if the Product of this Operation be the same Mt> bei'ore, the Work is right. ^ tutor's ASSISTANT. Multiplketion of Integers, 5 ? .*-« MULTIPLICATION TABLE. a greater, and ion, you roust j \t there, alwa}?) j *. together, and] 508 3750205 m 3150874 of two Num- the less ; and itions : )ers; viz plied ; multiply ; iltiplying. in the Unit's e first Figure wn the Lnits iultiplied the •"igure in the ens you kept s before, till the former the Multi- be the same 28 32 36 40 44. 48 25 30 35 40 45 50 55 60 *^ 11 9 10 is 63 ^ 70 — 77 — 84 8 8 9 10 11 12 64 72 80 88 96 36 42 48 54 60 66 72 r7 - 4 18 — 5 — 49 6 — 100 — 110 — li!0 — 121 — 132 12 Multiplication of Integers, the tutor's When the Multiplier is more than 12, and less than 'u multiply by the Unit Figure in the Multiplier, adding to l Product the back Figure to that you multiplied. 5710593 51072.-2 7653210 92057165 13 U 15 16 6251721 17 9215324 18 2571^41 19 3592104 20 When the Multiplier consists of several Figures, tlid nmst be as many Products as tliere are Figures in the IVl| tiplitr, ohservin^^ to put the first Figure of every Prodi( unJer that Figure you multiply by. Add the several Pi* ducts together, and their sum will be the total Product. Multiply 271041071 by 5147. Multiply 62310047 by 1608. ; ' - Multiply 170925164 by 7419. 1 Multiply 9500985742 by 61879. |. •' Multiply 1701495868567 by 470S756. "When Cyphers are phced between the significant Figi in the Multiplier, they may be omitted ; but great care nil be taken that the next Figure must be put one place ni| to the left hand ; i. e. under the Figure you multiply by. ., , , Multiply 571204 * \ , ', By 27009 ^ : 51408S6 3998428 - 1142408 Product 1-5427648836 •' - ' Multiply 7561210325 by 57002. * Multiply 562710934 by 590030. Wlien tjiere are Cyphers at the end of the MultipHci i»r Multiplier, they may be omitted, by only multiplying the rest of the Figures, and setting down on the right-hi of the total Product as many Cyphers as were omitted. iimm'iiii'.-'"** E tutor's ASSISTANT* Division, id less than a r, adding tQ i6d. 92057165 16 3592104 20 Figures, tli^ iircs in the Ml every Prodil tlie several Pr* tal Product. Multiply 1379500 By 3400 55180 41385 4690300C00 Multiply 7271000 by 52600. Multiply 74837000 by 975000* When the Multiplier is a composite Number, i. e. if any. ^wo Figures, being multiplied together, will make th|it uinbcr, then multiply by one^of those Figures, and the *rodact by the crhor will give ^he answer. Multiply 771039 by 35, or 7 times 5. 7 5397^273 5 26986365 ■ &756. piificant Figi great care ml one place ni| multiply by. Multiply 921563 by 32.. Multiply 715241 by 56. Multiply 7984956 by 144* he Multiplic! y multiplying B the rigbt-hi re omitted. DIVISION TEACHETH to find how often one Number is con- tained in another; or to divide any Number into Parts you please. In this liule there are three Numbers real, and a fourth accidental : viz, 1, The Dividend or Number to be divided : 2, The Divisor or Number by which you divide : 3, The Quotient, or Number that sliews how often the Divisor is cont;tined in the Dividend : 4, Or accidental Number, is what remains when the Work is finished, and is of the same Name as tlie Dividend. Rule. When the Divisor does not exceed 12, find how often it is contained in the first Fif;ure of the Dividend ; set it down under the Figure you divided, and carry the over- plus (if any) to the next iathe dividend, as so many Tens i. ■! ,1 ' ^^' " / i i 8 Division of Integers. the tutor's then find how often the Divisor is contained tlierein, set d down, and contir.ae the s&iiie till you hnvf. gone througj the Line : but whon thu uivisor is more than 1*2, rauUiplj it by tlie Quotient J'^iguie. ihe ]Yo'^-2t subtract from thj Dividend, and to the rwn.rtlndci* bring dov n jhe nixt FiguM in the Dividend, Uiid \ rccc'^d as bcf-^ie, till tht: Figures ar| all broagiU.uvivn. PiiooB'. "Tuitip'y *}ie DivisoF and Quotient togethei] adding <-he " . om i* i . Ji* (ii an"), and the Product will be tiJ same 09 tbe D:viu':jd. DivitK-".U. Htm.; Divisor (2)726107 ( Quotient 362J53| 2 3)^2i0472( 4)72104161 Proof 725107 S)7203287( 6)5231037 ( 7)2532701 ( 8)2547325( 9)25047306 Divisor. Dividend. Quotent. 29)4172377(143875 29 29 127 116 1294875. 287750 2 Rem. U2 87 4172377 Proof .253 232 ' .217 203 - ' / .147 145 Rem. . S 'Divide 7210473 by 57 Ans. 194877,'" Divide 42749467 by 347 Divide 734097143 by 57431 Divide 1610478407 by 54711 Divide 4973401891 by 51081 Divide 51704567874 by 476501 Divide 174537989461237^ ; by 3147948 'i ;:. ffl TUTOR S ASSISTANT. Tables qfMonei/. 9 ^en there are Cypliers at the end of the Divisor, thev be cut off, and as many places from off the Dividend, must be annexed to the Remainder at last. !00)254732|iil(9S9 5721|00)7253472|16(1267 1000)7524731729(2756 2151000)63251041997(29419 hen the Divisor is a composite Number, t. e. if any two ures, being multiplied together, will make that Number, , by dividing the Dividend by one of those Figures, and Quotient by the other, it will give the Quotient requir- But as it sometimes happens that there is a Remainder ach of the Quotients, ana neither of them the true one, ay be found by this 4)721041 6||luLE. Multiply the first Divisor into the last Remain- to that Product add the first Remainder, which will give true one. ov^9j^|Q<^-/B 3210473 by 27 7210473 by 35 6251043 by 42 5761034 by 54 118906. 11 Hm. 206013. 18 Rm. 148834. 15 Rm. 106685.44 Rm. tlierein, set gone throug' n 12, rauUipl tract from thi he mxt Figur he Figures ar| tient togethei luct will be til 9)25047306 4 Farthings make 1 Penny 1 Shilling 73 by S7 Ans. 194877^ )467 by 347 )7143 by 57431 78407 by 54711 01891 by 5108JI 567874 by 476501 379«94612S7 by 814794( MONEY, Iked. \Farihpng, \Halfpenny. 12 Pence Yl'hree Farthings, 20 Shillings — larthinvs. 4= 1 Penny, 1-8= 12= 1 Shilling., )0f= 240= 20= 1 Pound, Marked. . d, • s, 1 Pound - /. SHILLINGS. 1^. ^0 is JO — to — 50 — 50 — . ^0 ^ JO — )0 — 10 — 20 — JO — /. 1 1 2 2 3 3 4 4 5 5 6 6 s. 10 10 10 10 10 10 d. 20 24 SO 36 40 48 50 60 70 72 80 84 PENCE TABLE. is s, 1 2 2 3 3 4 4 5 5 6 6 7 d. 8 6 4 2 10 8 d. 90 96 iOO 108 HO 120 130 132 140 144 150 160 is 8, 7 8 8 9 9 10 10 U d, 6 4 2 10 11 : 8 12 : 12 : 6 13 : 4t \ )l m % t » ! .» ■' : ti :/l Ij i' Hi 1 1 t ' ■ 10 Tables qf Weight. the tutor's mmsi • TROY WEIGHT, 24* Q rains make 1 20 Pennyweights — 12 Ounces — Markfii Pennyweight ^^J^J 1 Ounce oz. 1 Pound /(^. Graing. 24=x 1 Pennyweight, 480= 20= 1 Ounce. 5760= 240= 12=1 Pound. By tiiis wtight are weighed Gold, Silver, Jewels, E!( ories, and all Liquors. N. S. The standard for Gold Coin is 22 Carats of Gold, and 2 Carats of Copper, melted together. For Sil is 1 1 oz. 2 divLs. of fine Silver, and 1 8 dwts. of ( 25 lb. is a quarter of a 100 lb. 1 ctvt. 20 cwt. 1 Ton of Gold or Silver. Copper. . : : AVOIRDUPOI&E WEIGHT. 16 Drams make 1 Ounce \ 1 Pound tb. 1 Quarter qrs. 1 Hundred Weight cwt. 1 Tou Ton,\ 16 Ounces — 28 Pounds -- 4 Quarters or Wl^ Ih. 20 Hundred Vveli^Iit Draiiv. !(;= 1 Ounce. 2,5f)^ 16=1 Pound. 7i(i^= 44'8=28=l Quarter. 28t>r2" 17y2= 112= 4=1 Hundred Weight. 573 iiO=35b40=2240=80— 20=1 Ton. There are several other Denominations in this Wei that are used in some particular Goods, viz, lb. 'ey m Clovi Jlove... ftone... rod [y this Mature all Me kte. O \rs. i 1 I Grains iScruph iDrams [Ounces I'lS. ro= lo= 1 A Firkin of Butter 56 Soap 64 A Barrel of Anchovies 30 Soap 256 ' I Raisins... 112 A Puncheon of Prunes 1120 A Pother of Lead, 19 ewt. . 2 qrs.. A Stone of Iron Shot! or Horseman's wt. J Butchers Meat A Gallon of Train Oil. A Truss of Straw ■ ' New Hay.... Old Hay 36 Trusses a Load. 3 [0= 24 |0=288 'ote. e, but ght. he Ap ceTr( ided. ^aijs Juarteri Quarters Quarters ^uarten kSSISTAN / Tables qf might 1 1 CHEESE AND fii;lTER. A Clove or Half Stone, 8 M. fey in Suffolk, 1 lb, A Wt7 in Essex, 7 16' Clov€8, or 3 256 42 Cloves, or J 336 wooi:« r, Jewels, Eld M. A Wey is 6 Tod and I lb. {love 7 1 Stone, or J 182 (tone H A Sack is 2 Weys, or 364 rod 28 A Last is 12 Sacks, or 4368 ky this Weight is weighed any thing of a coarse or dros- Jature ; as all Grocery and Chandlery Wares, Bread, all Metals, but Silver and Gold. lote. One Pound Avoirdupoise is equal to 14 o;e. 11 dwt, rrs. i Troy. make APOTHECARIES WEIGHT, Marked 1 Scruple 9 1 Dram 3 1 Ounce 5 1 Pound » Grains jScruples prams I Ounces 'IS. [0= 1 Sruple. [0= 3= 1 Dram. 10= 24= 8= 1 Ounce. - |0=288=96=12=1 Pound. ^ote. The Apothecaries mix their Medicines by this le, but buy and sell their Commodities by Avoirdupoise ^ight. le Apothecaries* Pound and Ounce, and the Pound and ice Troy, are the same, only differently divided and sub- Kded. ma . Quarters Quarters Quarters Quarters CLOTH MEASURE, make Matkei 1 Quarter of a Yard I"' Iqrs. 1 Flemish Ell FLE. 1 Yard i/d» 1 English Ell E, E, 1 French Ell F, E, I i: ■(!' n; n !1 !:i 1 .V It Tabks qf Measures* the tutor's Inches. 2^= 1 Nail. 9 = 4=sl Quarter. . 36 =16=4=1 Yard. 27 =12=3=1 Flemish Ell. 45 =20=5=1 English £11. 54 s=24=6=l French £11. LONG MEASURE. Marki 3 Barley Corns make 1 Inch J*"*"* 12 Inches — 1 Foot ^ ,feet. 3 Feet — 1 Yard i,j^d. 6 Feet — 1 Fathom /tk. S^Yards — 1 Rod, Pole, or Perch...roc?,) 40 Poles — 1 Furlong .Jur. 8 Furlongs — 1 Mile ^.mile. 3 Miles — 1 League lea. 60 Miles -.- 1 Degree deg. Barley Corns. 3= 1 Inch. 36= 12=1 Foot. 108= 36= 3 = 1 Yard. 594= 198= 16^= 5^=1 Pole. 23760=17920= 660 = 220 = 40=1 Furlong. 190080=63360=5280 =1760 =320=8=1 Mile. N. B. A Degree is 69 Miles, 4 Furlongs, nearly, thoug, commonly reckoned but 60 Miles. This measure is used to measure distance of Places, any thing else that hath length only. WJNE MEASURE. ' Marked 2 Pints make 1 Quart j^J** 4 Quarts -^ 1 Gallon gal. 10 Gallons — 1 Anchor of Brandy anc. 18 Gallons •— 1 Rundlet run. 31^ Gallons — Half a Hogshead ^hhd, 42 Gallons — 1 Tierce tierce. 63 Gallons — 1 Hogshead hhd. 2 Hogsheads — 1 Pipe or Butt P. or buil 2 Pipes or 4 Hogsheads 1 Tun tun. IASSI2 Pints mm c tmrnim''. m i ' -v itUt' '»*> i* » ■ lASSISTANT. [nclies. Tables of Measures. 1 3 Pints make 1 Quart. 28f= 1 Pint. [571= 2=: 1 Quart. 531 = 8= 4<= 1 Gallon. 02 = 336= 168= 42=1 Tierce. 53 =» 504= 252= 63=4=1 Hogshead. 04 = 672= 336= 84=2 =H=1 Puncheon. C6 =1008= 504=126=3 =^2 =4=1 Pipe. 12 =2016=1008=252=6 =4 =3 =2=1 Tun. 11 brandies, spirits, perry, cyder, mead, vinegar, honey, •il, are measured by this measure ; as also milk, not by , but custom only. ALE AND BEER MEASURE. Markei SPts. iQis, 1 Gallon Gal. 1 Firkin of Ale J^r, 1 Firkin of Beer B.^r, 1 Kilderkin Kil, 1 Barrel Bar. 1 Hogshead of Beer.... Hhd. 1 Puncheon Pun. Barrels, or 2 Hogsheads 1 Butt BuiL EER, ic incl.es, 35^= 1 Pint. ' ' 70^= 2= 1 Quart. 282 = 8= 4= 1 Gallon. 538 = 72= 36= 9= 1 Firkin. )76 =144= 72= 18= 2=1 Kilderkin. 152 =288=144= 36= 4=2=1 Barrel. 228 =432=216= 54= 6=3=ljr.^l Hogshead. Oil =576=288= 72= 8=1=2 =1 Puncheon. t56 =864=432=108=12=6=3 =2=1 Butt. ALE, )ic Indies. 35^= 1 Pint. •'-;' . 70|= 2= 1 Quart. >82 = 8= 4= 1 Gallon. 256 = 6i= 32= 8=1 Firkin. 512 =128= 64=16=2=1 Kilderkin. •- 324 =256=128=32=1=2=1 Barrel. 536 =381:^:1 92= 18 =6=3= 11-= 1 HoffslieacJ. C Quarts — Gallons — Gallons — Firkins — Firkins, or 2 Kilderkins Barrel and^, or 54 Gal. Barrels — 5Vi.1& 1 1 !! 'i ! j. ■ ' 1 ' } i! .!i' ii I ;. 'A- i!, •:i til H II i ! ii 14 Tables of Measures, the tut( In London they compute but 8 gallons to the firkin of and 32 to the barrel ; but in all other parts of England ale, strong b«er, and small, 34< gallons to the barrel, S gallons I to the firkin. N, B. A barrel of salmon or eels is 4-2 gallons^ A barrel of herrings ----- 32 gallons, ^ A keg of sturgeon ------ 4 or 5 gallons, A firkin of soap -- 8 gallons, DRY MEASURE, Mai 2 Pints make I Quart f^f \qts 2 Quarts — 1 Pottle ^oi 2 Pottles — 1 Gallon goi 2 Gallons — 1 Peck pk. 4* Pecks — 1 Bushel- - - - - bu, 2 Bushes — 1 Strike ----- str 4* Bushels — ' 1 Coomb- - - - -» cot 2 Coombs or 8 Bushels - - - 1 Quarter - - - - qr 4" Quarters — 1 Chaldron - - - - ch( 5 Quarters — ' ' 1 Wey we 2 Weys — 1 Last lai ^- In London 36 Bushels make a Chaldron. Solid Inches. t 268-t= 1 Gallon, 537|= 2= 1 Peck. 215(H'= 8= 4=1 Bushel, • r 43004= 16= 8= 2= 1 Strike. 86011= 32= 16= 4= 2= 1 Coomb. 17203|= 64= 32= 8= 4= 2= 1 Quarter. 86016 =320=160=40=20=10= 5=1 Wey. 172032 =640=320=80=40=20=10=2=1 Last. The bushel in Water Measure is 5 pecks. A score of coals is 21 chaldrons. A sack of coals — 3 bushels. A chaldron of coals — 12 sacks. < A load of corn — - 5 bushels. A cart of ditto — 40 bushels. This measure is applied to all dry goods. The standard bushel is 18 inches and ^ wide, and 8 1 es deep. ■- 1 1 ^ v THE TUTOr'I gallons to the firkin of ak ather parts of England, fJ gallons to the barrel, aD| iels is 42 gallons. - - - - 32 gallons. - - . - 4. or 5 gallons. - - - - 8 gallons. SURE, Quart Pottle - - - - Gallon - - - - Peck - - - - - Bushel - - - - Strike - - - - Coomb- - - - Quarter - - - Chaldron- - - Wey Last - - - - - make a Chaldron Markedi pot. gal. pk. bu. striked coorm chal. V3ey, last. [1,. ■ . ■' ■- -" trike. 1 Coomb. ^•^■.- ; 2= 1 Quarter, v 10= 5=1 Wey. 20=10=2=1 Last. easure is 5 pecks. 21 chaldrons. 3 bushels. 12 sacks. 5 bushels. 40 bushels, to all dry goods. tes and ^ wide, and 8 ini ASSISTANT. Seconds mako Minutes — Hours — Days — Weeks — Months, 1 day, 6 hours 2'ables of Measures, 1 : TIME, Marlcea \m, hour. dull xvei'r.- via. 1 Minute - - - 1 Hour - - - - 1 Day 1 Week - - . 1 i^ionth - - - 1 Julian yoi.r - ijy. S"Conis. 60= 1 Minute. 3600= 60.^ 1 Hour. 86400= 1440= 21= 1 Day. 601800=10080= 168= 7 r=l Week. !4l920O=403i^O= 672=28=4=1 Montlj. d. h. n». d.h, L557600:==.52j960=S766=365.6==.5i'J.6=l Julian yr-.r. ' '".A .\ ^ . d. h. 711, s. 1556937«.525£'1S=.87G5==;365.5.18.,57=1 Solar year. To know the days in each month, observe : Thirty days hath September, April, June, and November. February hath twenty-eight alone. All the rest have thirty and one. Except in Leap Yei'.r, and then ft lua time. February's days are ivoenfy and nine. SQUARE MEASURE. 44 Inches make 1 Foot. 9 Feet . — • 1 Yard. 00 Feet •.mmm ■ 1 Square or floor:;.g. 72^ Feet . mm> 1 Rod. 40 Rods ___ X Rood. 4 Roods, or 1€0 rods, or 4840 yar, 1 Acre of land. 40 Acres m.... 1 Square mile. 30 Acres mmm 1 Yar I ■■• 1/ APOTHECARIES WEIGHT. 17..10..7..1 7" 2..1..0..13 27..11..1..2 9..10..2..0..U 9.. 5..6..1 . 7.. 5..7..1..15 37..10..5..2 3.. 9..5..2..13 49 7..0 7.. 1..4..1..18 ! FE. qr. n. i27..2..l 53..1..3 .'<76..2..1 197..1,.3 CLOTH MEASURE. yd. qr, n. 135..3..3 .: ." I ^ 70..2..2 95..S..O 176..1..3 26..0..1 279..2..1 EE.qr.n. 572..2..1 152..1..2 79..0..1 156..2..0 79..3..1 154..2..1 m ! LONG MEASURE. y yd.feet in. bar* lea.m^ur.p, *i25..1.. 9..1 ^ 72..2..1..19 171..0.. 3..2 27..1..7..22 52..2.. 3..2 S5..2..5..31 V 397..0..10..1 79.,0..6..12 J54.,2.. 7..2 S1..1..6..17 137..1.. 4..1 72..0..5..21 LAND MEASURE. a. r, p. 726,.!.. »1 219..2..17 U55..3..U. 879..1..21 1195.,2..H «. r, p, 1232..1..14 327..0..19 131..2..15 1219..1..18 459..2..17 HE TUTOR'J BSISTANT. • d gr, :\ hhds, gal, qts» .0..13 Si •• 57 •• 1 ..1..17 97.. 18 ..2 ..0..14. 76.. 13.. 1 ..1..15 55 .. 46 .. 2 ..2.. 13 o7 .. oo •• >5 ..1..18 55 .. 17 .. 1 Addition of Measures. 19 WINE MEASURE. 7\ hhdx. gal, qis, 14 ..3.. 27 ..2 19.. 2. .56 ..3 17. .0.. 39 ..2 75..2.. 16.. 1 54.. 1 ..19 ..2 97 .. 3 .. 54 .. 3 EE.qr.n* S72..2..1 152..1..2 79..0..1 1S6..2..0 79..3..1 154..2..1 fur.p, 1..19 7..22 5..31 6..12 6..17 5..21 r. p, .1.14 .0..19 .2..15 .1..18 .2..17 * ' alTb and beer measure. A,B,Jir, gal, 25 .. 2 .. 7 17..3..5 ' 96 ..2.. 6 75 .. 1 .. 4 96 .. 3 .. 7 75 ..0.. 5 ;S3> B,B.Jir, gal, 37 .. ^ .. 8 54 .. 1 .. 7 97 .. 3 .. 8 78 .. 2 .. 5 47 ..0.. 7 35 ..2. .5 hhd, gal, ji/.*. 76 ..51 ..2 97 ..27 ..3 22 .. 17 .. 2 55 .. 38 .. 3 ch, bu, pks. 75.. 2..1 41 .. 24 .. 1 92 .. 16 .. 1 70.. 13. .2 54.. 17 ..3 79..25..1 t w. d, h^ V 71 ..3..U 51 .. 2 .. 9 I -■•• 76..0..21 95 .. 3 .. 21 79..!.. 15 i - -• . ■ DRY MEASURE. lasts, lueys, gts, bu, pJcs,. 38 ..1 .. 4 .. 5 .. 3 47 .. 1 .. 3 .. 6 .. 2 62 .. .. 2 .. 4 .. ^ 44f ..1 .. 4 .. 3 .. 3 To a. 1 •• 1 •> ^ •« ^ 29 ..1 .. 3 .. 6 .. 2 TIME. .. .» iv, d. h. m, sec. 57 ..2.. 15 ..42 ..41 95 .. 3 .. 21 .. 27 .. 51 76 ..0» 15.. 37 ..28 i 98..2.. 18..47.. 38 , I\\ w I I ll i •J* iV. i:l| ;l i. i 20 Addition. n;'V5i» THE TUTOR*S.issrSTA i cher £122..11..5 ; to his baker je37..9..5 ; and to his s( 1^^142.. vants for wages £53.. 18. I dtsire to know what menej had to raise in the whole, when we add to the above su i£100 which he wished to take with him ? Ans. £1032..17..3. V A ho weighec the 1 thef ed two weight ^ Ac >d in Ja id in Fe 1..17 ; g to th hhede e to kn E tutor's) .ssistant. XX will he be Ans. 1797. chase of a qu and a cro^ii aid out in all ins. £l3..6..3, es these seve 19..14.? Ins, £20l..lo. 215, the seco: Addition. 21 ). A father ^ras 24 years of age (allowing 1$ months year, and 28 days to a month, ) when his first child born ; between the eldest and next born was 1 year, lonths, 14 days ; between the second and third were 2 s, 1 month and 15 days; between the third and fourth 2 years, 10 months and 25 days; when the fourth 27 years, 9 months and 12 days old ; how old was the r ? Ans. 58 years ^ 7 months j 10 days, u.XOy and foil. A banker's clerk having been out with bills, brings and a shillii e an account, that A paid^ him i£7«5..2. B 15..18..6^- is, ;C236..8..4. 150..13..2^. ~ " D £l7..6-.8. E 5 guineas, 2 crown pieces, when the teKftf crowns, and 4^. and 2d, F paid him only 20 groats. 76..15..9g. and H £121..12..4. I desire to know now h the whole amounted to that he bad to pay ? Ans, je396..7..C|. o. What is t I. A nobleman had a service of plate, which consisted Ans, 1468. I paid £54..1 spent about t d me in ? ns, £57..10..3. ireof is 40, thi k'enty dishes, weighing 203 oz. 8 dixit, ; 36 plates woigli 08 oz. 9 dvot. ; 5 dozen of spoons, weighing 112 cz. 8 6 talts, and 6 pepper-boxes, weighing 71 o^. 7 dxit ; ES and forks, weighing 73 oz. 5 dwts, ; two large cups, a aid and a mug, weighing 121 oz. 4 dwts.; a tea*ktttle lamp, weighing 131 oz, 7 dtvts. ; together with sundry greater numbi r small articles, weighing 105 oz. 5 diuts, I desire to mber, 94 sum. Cl 500 more tl >rtune, and wii it sister's foi !.j them ^£25722. wn, was desiio enquiry he fou s wine-merch 4 ; to his drap > his coachmaK .9 ; to his cor what meney > the above su . £i032..17..3. V the weight of the whole ? Ans. 102 lb, 2oz, 13 dtvts. and, 1 1 hundn . A hop-merchant buys 5 bags of hops, of which the weighed 2 cut. 3 qrs. 13 lb.; the sacond 2 ctvt. 2 qrs, the third 2 cut. 3 qrs. 5lb. ; the fourth 2 c*ot. 3 qrs. the fifth 2 cwt. 3 qrs, 15 lb. Besides these he pur- ed two pockets, each weighing 84 lb, I desire to know weight of the whole ? ^n^. 15 cwt, 2 qrs, . A of Vienna, owes to B of Liverpool, for goods re- d in January, the sum of jfil03..12..2 ; for 'goods re- d in February, i£93..3..4 ; for goods received m March I1..17 ; for goods received in April jS142..15..4 ; for ^.0 ; to his h\ s received in May £171«15..10; for goods received in and to his sf i;^142..12..6; but the latter six months of the year, g to the falling off in the demands for the articles m h he dealt, amounted to the sum only of j£205..7*'2. I e to know the amount of the whole-dear's bill ? Ans. £9Sl.,S..i!, 1 ■' 1 r {2S Subtraction. THE TUTd ^TANT. '. M, I fO li 'I' ' I !: « I .1, SUBTRACTION OF MONEY, WEIGHTS, Ai MEASURES. ,,. .- RUJfiE. Subtract as in Integers only when any o lower denominations are greater than the upper, row as many of that as make one of the next superior, ing it to the upper* from \vhich take the less ; set (low ililFerence, anu carry one to the next higher denomini for what you borrowed. {'roof. As in Integers. MONEY. J^» s. d. Bough Sbl< Unsol t . \ Borrowed 715.. 2.. 7+ Paid 476.. 3.. s| Remains to pay 238 .. 18 .. 10^ Proof 715.. 2.. 7^ r j^. s, d. £. 5. d. ^, s, d, 87 .. 2 .. 10 3 .. 15 „ li 25 .. 2 .. 5k 79..3..7i I. .14 -7 17. .9.. 8 J Lent 316 .. 3 .. Received 21 8.. 2.. f' f* 25 .. 5 .. £, s. d, £. s. d, 321.. 17.. H 59.. 15 ..3^ 257.. 14. ..7 36.. 17. .2 £» s. d. 71 ..2. .4 £' *• 527 ..3.. 1~ 19.. 13. .71 139 ..5.. Borrowed 25107 .. 15 .. 7 Lent 250156.. 1 .. 375.. 5^5^ Paid 259.. 2..7f at359.. 13.o4| different 523 .. 17 *. 3 271 .. 13 .. .•Received 359 .. 16 - at 475 ..r^.. several 527 .. 15 .. .. ^ •• ! .. 5 .. E, qrs, 55 .. 2 7 .. 2 idt. ft 07 .. 2 78 ..2 twes 274 .. 15 .. 7^: •♦ * . payments 272 ..16 325 .. 13 .. 5 .Paid-in all •s 150 .. ^— .. ■ hJ '■ '"■-,.——— — Remains to pay « i ■ l 'i » a, r W .. 1 59 .. C ITANT* Substractim* T THE TUTd TROY WEIGHT. lb', oz. dxvt.gr. :IGHTS, AM Bbuglit 52 .. 1 .. 7.-2 SbldS9..0.. 15..7 lb. oz.dtui.gr, 7..2..2..7 S..7..1..S y when any o|j an the upper, j cxt superior, less ; set dowil ^her dcnominj Unsold (,: ' t. c , AVOIRDUFOISB WEIGHT. OZ. dr, ctut. grs. lb, T, cvot, qrs, lb, \5 .. 10 .. 5 S5 .. 1 .. 21 21 .. 1 .. 2 .. 7 b •• 12 .. 7 25 .. 1 .. 10 9 .. 1 .. 3 .. 5 ent 316.. 3. . ved218..2.. f' !' o ( •• o •• 25.. 5 J 527.. 3.. ^ 7| 139.. 5. . 250156.. 1.. b ! 3 9 ) •• ^ •• X •• i/ ! .. 5 .. 2 .. 1 APOTHECARIES WEiGHT. '^E. qrs. n, \5 .. 2 .. 2 7 .. 2 .. 1 9^ •• I „ £ „ \ „ *J^ 6=.. 7 .. 3 .. I .. 1* CLOTH MEASURE. yds, qrs, n, EE, grs, n. 71 .. 1 .. 2 35 .. 2 .. 1 * 3 .. 2 .. 1 14f .. 3 .. 2 n,' .a» 271 .. 13 .. ed 359 .. 15 .. at 4t75.,Vi .. ral 527 .. 15 .. Its 272.. 16.. 150 . "Vr ^ LONG MEASURE* ' ' fdt. ft, in, bar, ^ t ■ leag, mi, Jjtif, po. 07 .. 2 .. 10 .. 1 V 1*7 .. 2 .. 6 " 29 78 .. 2 .. II .. 2 58 .. 2 .. 7 •• 33 r:\f' LAND MEASURE. a. r, p, &75 .. 1 ,. 27 59 .. .. 27 a, r, p, 325 .. 2 .. 1 ^f«7 •• ZV •• O :1 ill ii I ■ .1 ! ,11 ?i: !;' .'! 1 ! ? ■ ( ■' ■ 24 SuhtracUon. the tutcl _JITANT. WINE MIESURI. hhd. gal qts. pi* tun hhd. gal, qts. #. A mei 47 .. 47 .. 2 .. 1 42 .. 2 .. 37 .. 2 Jimd in c 28 .. 59 .. 3 .. 17 .. 3 .. 49 .. 3 'fijlO..? ; ,t is the ALB AB,Jir, gal, «5 •• 1 •• ^ 21 .. 1 .. 5 AND BEER MEASURE. . A gen tlie y) BB» Jir» gal, hhd, gal. qts. twice ;^ 87 25 27 12 27 50 1 2 ? u, 2 , 35 hu. 1 . r 3 DRY qu, 65 57 MEASURE. 6u. p, •• A (t 1 •• ^ •• o 79 54 . 3 . .7 . 1 \ 79 23 mo. • o • . 9 . TIME. toe. da. 2 .. 4 3 .. 5 v.' ho. 34 IS mm. 42 . 63 . S€C, 45 47 THE APPLICATION. 1. A man born in the year 1723, what was his age in year 1781 ? Ans. 58. 2. What is the difference between the age of a man In 171C, and another born in 1766 ? Ans, 56- 3. A merchant had 5 debtors, A 6. C. D. and E. vi together owed him ;^1156. B. C. D. and E. owed j^737 ; what was A's debt ? Ans. £^\9. 4. When an estate of ;^300, per annum is reduced payment of taxes, to 12 score and ;^14 .* 6, what is the I Ans, £4:5 .. 14. 5. W^hat is the difference between ;^9154, and tli mount of ;^754, added to ;^305 ? Ans. 8095. 6. A horse in his furniture is worth £^"7 ..5, ; out 14 guineas; how much does the price of the furniture •eed that of the horse ? Ans, ;^7 •• 17. . A trail crediforj 1] £i05 k.l.%.H; litors Ibi ; he had i\.8£'n us fllec f were k 0. My c wing ac (Is sent 1 L15..4.; cloth £ the same to iny 01 value t Spanish liow or? Ans. OF ULE. n o J e one the quo the gi\ bers, w beri bi THE TUTCl IITANT. Mulliplicatton, ^IS '. gal. qts. . gal, qts. .. 27 .. 1 .. 50 .. 2 A merchant at his out-setting in trade, owed jC7!iO ; d in cush, cominoditiuH, the 8tucks, and good dihts, 15\0..7 ; ho cloared the lirstyear by commerce /'15'J.;i..j litors found the Vidue of his stock to oe /.'Jf2..G, and : he had owing to him in good hook (lebts^^liy .8..!{, dv.'Sj^'ii..H)..5, mont.'y in hand. As Ins crtiiitortj took us effects into tlijir hands, I desire to know whctlicr f were losers or gainers, and how much 'f Ans. tlie Cii'ditor.K lost £14.6..]}.. \0. 0. My correspondent, at Seville, in Spain, sends me the 3wing account of money received at different sales for (Is sent him by me, viz. liees-wax to the value of '..15..4-; stockings £\)7Jj..7 \ tobacco ;^'i25..i 1..6 ; Iin» cloth £\ i2..14'..S ; tin ^'1 15..10..5. My correspond! nt he same time informs me, that lie has s-hipped, agrcea- to my order, wines to the value of ;^!;,^^()..15 : fiuit to value of /'51..12..(>; figs ;^19..17..0; oil £[9..VA..li wag his age inl Spanish wool to the value of ;^"115..15..G. 1 desire u> Ans. 58. l-v Iiow the account stands between us, and who is the \. hu. p. ) .. 3 .. !< .. 7 .. 1 mn. stc, 12 .. 45 53 .. 47 ge of a man Ans. 56' . D. and E. w md E. owed Ans, £^\.9. tm is reducecl I, what is the t ns, £4:5 .. 14. 9154, and tli Ans. 8095. 17 ..5, ; out ( r the furniture 4ns, £l .. 17. or.'' Ans, Due to my Spardsh correspondent £2S„\4!..^, MULTIPLICATION OF SEVERAL DENOMINATIONS, IILE. Multiply the first Denomination by the quanti- j ty given, dividing the product by as many of that as e one of the next, setting d.)Wii the remainder, and the quotient to the next superior, after it is nmltiplied. the given quantity is above 12, multiply by any two bers, which multiplied together, will make the same ber J but if no two numbers multiplied t(>^ellier will D i ( 1 1 i \ 1 fj ■' t i I i; ! » t ! I » < ''( Oil ' i I ! ! l\!- i26 Compound Mi'Mplkation. the tuj^oBistant make the exact number, then multiply the top line by as i 25§ ells ny as is wanting, adding it to the last product. Pkoof. by Division. £, s. d, £. s. d, £. s. d. £. s. d 35..12..7^ 75..13..1^ 62..5..4i 57..2..4 8 4f 5 n id 71 ^ ^> \- 1.18 yds. of cloth at 9s. 6d, 2. 26lb. of tea, at £i .. 2 per yard 9X2=18 per lb 4f " 5 " G 2 8.. 11 ..0 9..0 27 ..0 Top line X 2 2. .5 ^^ •• 3. 21 ells of Holland, at 7?. Si^/.^prr ell. Facit £H..l 4. 35 firkins of butter, at 155. 3\d. per firkin. Facit £26.-15 5. 7.)lb of nutmegs, at 7*. 2|rf. ;?i?;" lb. lacit £21. .1 6. 37 yards of tabby, at9»'. 7d. per yard. Facit iei7..1 7. 97 cwt. of cheese, at £l,. 5..S per cv>t. Facit £122.. 8. 43 dozen of candles, at 6s, ^d. per dozen. Facit jel3..1 9. 1271b. of bohea tea, at 125. 3rf. /)^ ells of diaper, at l.v. Sd. per ell. Facit /,'^i .. U .. ^ 19^ ells of damask, at 45. Sc?. ^^r ell. JFflczV £4! .. 2 .. 10.J 35| ells of dowlas, at I5. 4f/. /)- cwt. /rtc^Y ^166.. 4..7-i 96^ cwt. of currants, at £2 .. 15 .. ^ per cwt. Facit £267 ..15. .9 45| lb. Belladine silk, at 18£. 6d. per lb. Facit £4>2..6.»i^ y H ;;„ I '^ ' i i- i;, 1(1 ;i f I ■'I' Hi i ! in in i ,. !' • t i28 Compound Multiplication 29. 871 bushels of wheat, 45. 3 per THE TUTOl bushel. Facit 18 .. 12 30. I20| cwt. of hops, at £{< .. 7 •• 6 per cwt. i^ac^V jC528 .. 5 'U. 407 yards of cloth, at S.f. 9|(/. /jcr yard 32. 729 ells of cloth, at Is, 1\d. per ell Facit £11 .. 3 , Facit £211 .. 3 . 51 ST A J Jo. In la mac Thn trs, 10 ^ how 1. A spend ual inc 2. A t utoirt e six ivn pie( to her 3. Adi a quar s, wha ;h in 7 A i. A r ment ^\ :3S6..1.' haml( 19 33. 206S yards of lace, at 9$. 5\d. per yart\. Facit £911.. 19 . THE APPLICATION. 1 . What sum of money must be divided amongst 18 n so that each man may receive £11: .. 6 .. 8^ ? Am. £258. .0.. 2. A Privateer of 250 men took a prize, which amoui to £125.. 15 "6 to each man, what was the value of prize? Ans. £31443.. 15.. *. What is the difference between six dozen dozen half a dozen dozen : and what is their sum and product A7i;i. TJ2 Dijf. Sum 936. Product 6220f 4. What difference is there between twice eight fifty, and twice fifty eight, and what is their product ? Ans. 50 Diff. 7656 Produc 5. There are two numbers, the greater of them is times 1-5, and their difference 19 times 4 ; their sum product arc required ? Ans. 3254 Sum 26i-5685 Produci 6. The sura of two numbers is 360, the less of them 1 what is their product and the square of their difference Ans. 31104 Product, 513 1 Square of their Different 7. In an army consisting of 187 squadrons of hoise, e 157 men, 207 battalions, each 560 men, how m «ny effec soldiers, supposing that in seven hospitals there are sick? ^ ^ Am. 144806 8. What sum did that gentleman receive in dowry v his wife, whose fortune was her wedding suit : her pettic juoriiMi having two rows of fur inflows, each furbelow 87 quills, n was I in each quill 21 guineas ? 4ns, £^836 .. 14 .. 9. A merchant had £19118 to begin trade with: f(l r.x, yours together he cleared £1086 a year ; the next 4 y< Muiri{i he made good ^^271.) .. 10 ..<> a year ; but the lastSyi Miiitij he was in trade, had the misfortune to lose, one year « Muluj^ another, £475 .. 4 .. 6 a year ; what was his real fortun MuUip 12 years end ? ' Ans, ^£33984 .. 8 .. 6 Multip Multip !. A g< biic ch )JVVG £ poor is fXec Hi liiue A dm THE TUTOl shel. ^acit 18 .. 12 ..) cwt. acit £528 .. 5 \ ird. Facit £11 .. 3 i 'acit £211 .. 3 .| ard. cit £911 .. 19 . nSTANT. Compound Multiplication, 29 jo. In some parts of the kingdom they weigh their coals la machine, in the nature ot* a steti-yard, waggon and Three of these draughts together ainount to 1 'J7 *^*'-'^ trs. 10 lb. and the tare or weight of the waggon 13 avt. . how many coals had the customer in 12 such ih\'iu<.).tt5? Ans. 391 ctvi. 1 yr, }-Z l.'j. 1. A certain gentleman lays up every y^ar £29i.,l2..6, spends daily £1 ..12 .. 6. I desire to know m hat is h's ual income? " Atis. .9^887. 3 5..0 2. A tradesman gave his daughter as a mnrriage portirm 1 amongst 18 n ;ii.i(oire, in wliich tiiere were 11* drawi^TS, in each driivver e six divisions, in each division there were £50. four wn pieces, and eight half crown pieces, how much hud to her fortune ? Ans. £ >744. 3. Admitting that I pay eight guineas and half-a-crown a quarter's rent, and am allowed quurtcily 155. for re- s, what does my apartment cost me annually, and hovr ;h in 7 years ? 85. £258. .0.. iy which amoun 5 the value of £J14.43.. IS.. : dozen dozen n and product Product 6220? twice eight leir product ? 7656 Produc er of them is 4- ; their sum ^5685 Produci less of them 1 eir difference ? their Differena ons of hoise, eBo poor housekeepers ten guineas eacli, aiul 150 j^uiacus low m *ny effecijs rvtcjor. Wliai: sum -."'t-t iic fi-ive been pn-sessed of Ills there are Ae tiave of ]]i;j death to aiiower ail tliej.;> ]eg^cit.\s ? Am. 144806 An ■■. £ > U'o .. 1 .. 0. ive in dowry V ;. Admit '?0 to be tlie rensimsKr nt' a (^vision PU;n, 4^3 >uit : her pettic juotient, the lii victor th;.- jjiv.u of both, and 19 more, low 87 quills, |it was the nunbtr of the dividend ? Ans. 195H6. rXAMTLES OF WTTCHTS ANll MEASURES. Mukipiy ylh. 10 cz. M {l\v[s. 19 qv. by 9. , V> Multij.iy 13 tons, 19 vwt. 3 qr>. 18 lb." by 7. -" '•' Multiply 107 v.-ii-ds, S nrs. 2 nails, by 10. ' 'V.* Multiply 33 ale bar. 2 frk. 3 gii). by U. Multiply 27 beer bar. 2 ink. 4 gal. *3 qts. by 12. Multiply 110 miles, 6 far. 2(3 polcS; bv 12. D2 Arts. In one year £^l ..2s. in seven £2\1 .. \is. 4. A robbery being committed on the highway, an as- ment was made on a neighbouring hundred for the siim !386..15..6, of which four parishes paid each £o7..\i..'2\ hamlets £il .. 4 .. 2 each, and the four townships .. 12 .. 6 each : how much v.as the deficiency ? An.s. £-36.. 12. .2 . A gentleman at his decease left his widow £\-5f^0 ; to b!ic ch;iraty he bequeatliod £572..] ; to each of his four lews £7lO.. 10; to each of his four nieces £:i75..\iL.(); £'j8S6.. 14..0 trade with : f( 1 the next 4 y It the last 3 y ise, one year \ his real fortur jt^33yb4 .. 8 .. 6 s m i 1 ' ^^ \t .» 30 T> iiuion* i':- Hl|: , ; M' 1^ ! I i ! THE TUTClsrSTA^ DIVISION V OF SEVERAL DENOMINATIONS, ULE. Divide the first Denomination on the left hi 11. A ( X\j and, if any remains, multiply them by as many oi next less as make one of that, which add to tje next, divide as before. Proof. By Multiplication. 5)2J..2..4.( 3)37.-7.. 7( 4)57 ..5.. 7 5)52 ..7 12.. 11. .2 Divide £ 1407. 17*. 7d. by 243. Divide £ 700791. 14<*. 4>d. by 1794. Divide 2* 4<90y81. 35. I^d. bv 31715. Diride £ 1974-3062. 5s. I^d' by 214723. each .. 14 .. An 0. A J 'ds, the for a Ic £4> thua : I ■ 19. Ar ered a c hole bei 20. M Vhat die 21. A d 12crt'/ omt of > d. I di Ans. THE TUTClsiSTANT. each ITIONS, n on the left h by as many o to ilie next, .7 5)52 ..7 723. Divhion, Si man's share, supposing ev«ry bullock worth ).. 14..6? Ans. 5 bullocks each marty £4-8 .. 12 ..6 each share, 10. A gentleman has a garden walled in, containing 9625 ds, the breadth vtras '65 }ards, what was the lergth ? Ans, 275. 11. A club in London, consisting of f 5 gentlemen, join- for a lottery ticket of afflO value, which came up a prize f't'JOO. laesire to know what each man contributed, d what each man's share came to ? Am. each contributed 8^. each share jf 160. 12. A trader cleared £I1S6 equally in 17 years, how - uch did he lay by in a year ? Ans. 68/. 13. Another cleared ^62805 in 7^ years, what was the larly increase of his fortune ? Ami. £37^, H. What number added to the 43d part of £4429 will ise it to ^'240 ? Ans. aei37. 15. Divide 20*. between A. B. and C. in such sort that . may have 2^. less than B. and C. 2«. more than B. Ans. A 4.V. Sd. B 6s. 8sl. C 8*. 8d. 16. If there are 1000 men to a regiment, and but 50 offi- !rs, how many private men are there to one officer ? Ans. 19. 17. What number is that which multiplied by 7847 will ake the product 3013248 ? Ans. 384. IH. The quotient is 1083, the divisor 28604, what was 4ns. £l ..12, (ces of cloth, /Ins. je4 .. 2 .. !7 .. 5 .. 4i, at Ans. 1 .. 7 . of beer, who Am. £1 .. 5 .. 6.. 0, I de, Ans. £i ..:^ months time, ns. £2i ..8 .. ame to £57 ..lie dividend if the remainder came out 1788 ? /4w5. 30979920. 19. An army consisting of 20,000 men, took and plun- ered a ciry of jC 1 2 000. What was each man's share, the hole being equally divided among them ? Ans. I2s, 20. My purse and money, said Dick to Harry, are worth '2s. 8d. ; but the money is worth ieven times the purse, V'hat did toe purse contain ? Ans. lis. Id. *^1. A mercliant bought two lots of tobacco, which weigh- d 126rtJ^. Sqrs. I5lb. for igll4. 15s. 6d. Iheir difference in omt of weight was 1 cwi. "iqrs. 13 lb. and of price £7 I5s. d. ~ ' ■ cloth ; what \ns. £l5," 5 ,• d. I desire to know their respective weights and value? be equally ( Ans. Leaser voeight 5 cvot. 2 qrs. 15 lb. Price £53 lO*". 5 share ? Grealer tvcight 7 cixt. I qr. Price £6 1 . 5s. 6d. IS. £14 ..10 .. 22. Divide lOOO crowns in such a manner between A, B, ded among 50 nd C\ that A may receive 12i^ njore than B, and B 178 had, and th« Js& than C ? Ahs, A 3G0, B 231, C 4t9. I h : ! i ,■ y ■ .'I I ." -if M! t ; i : 't 52 Bills of Parcels. THE TUT( '! I ;■' I EXAMPLES OF WEIGHTS AND MEASURES. Diviflc 8S lb. 5 oz. 10 dwts. 17 gr. by 8. Divide '29 tons, 17 ctvt. qrs. 18 lb. by 9. Divide ]\^ yardsy 3 yr^. 2 wr///^, by 10. Divide 1017 milesy 6 furl. 38 />o/^6', by 11. , . . :. Divi»le ii019 acresy 3 r■. ^ -i V .7/ rs. Brig - - ' HOSIER S. 'Mr. John Thomas Marcli 7, 180S Bought of Samuel Green, •; 5. d. ■ 8 Pair of worsted stockings at 4 .. 6 /7^r pair £ 5 Pair of thread ditto at 3. .2 3 Pair of black silk ditto at 14 ^O , 6 Pair milled hose at 4 .. 2 _ 4 'Pair of cottoii ditto at 7 •• 6 2 Yar.ds of tine, flannel at 1 •• 8 Tarda of air of fii •'ans of I •"ine lace )o9en Ir iets of k /k.'> ._.i .... r;. v---ivif .' . 1 j:.„.i- --Xi £7'.. 12 iTr. Isaac Grant March 7, ISOal'i ^ ' ^ Bought of John Sims, '' i . , '■■-'. ..... s. a. 15 Ynr'ls of satin at 9 .. 6/)cr j'ard aff ■' ■ ' 38 Yards of flowered silk at 17. .4 ' -^ ' ;■ ' 12 Yards of rich brocade...at W..H _ ' •^['" 16 Yards of sarsenet at 3 .. 2 , ' ' ■ 13 Yards of Gei-oa vtlvet at 27 ..'6 ' ' / *^3 Yards of kites u-ing at 6 » 3 ^'^^ L:'.. i(i-~f: -rt Sftjj, £62.. 2 ttV^> ;!: .* ^' r. Thor ards of I'ards of Tds. of s Tards of Tards of Isivdi of STANT. THE TUIC A.o ;asur£s. J. 9. 11. . I 26. 5 days* 11 ii jv-^ >■ . ,^ . i ^ Iarcli7, 180!j . ,1,;' ,|_;vTr pair £ J5///5 of Parcels. 35 LINEM draper's. . Simon Surety 27th March, 1809. Bought of Josiah Short, s* d. ards of cambric at 12 ..6 per yard £ ards of muslin at 8 .. 3 ards of printed linen at 5.. 4< ,.• iQzen of napkins at 2.. 3 each illls of diaper at I „7 per e\\ Dllg of dowlas at 1 » 1| o aei7..4...6} ••••«••! ^'- £7'.. 12 milliner's. rs. Bright April 25, 1809. Bought of Lucy Brown, /. s> dt fards of fine lace at .. 12 .. 3 per yard £ ^air of fine kid gloves... at C. 2.. 2 per pair... - •ans of French mounts at 0.. S .. 6 each W laced tippets at 3 .. 3.. •'- ^ )ozen Irish lamb at 0^. 1 .. 3 />er pair. „ • • Sets of knots at C 2 ..677^}' set..... £23.. U.. 4 WOOLLEN DRAPER S. r. Thomas Sage April 7, 1809. Bought of Ellis Smith, /. S. dm ' ' • Ir.rch 7, ISO^ards of fine serge at .. 3 .. 9 per yd, £ I'ards of drugget... .....at C 9..0 fds. of superfine scarlet at 1 .. 2..0 Tards of black at 0.. 18.. Tards of shalloon at 0.. 1 ..9 fardsofdrab atO.. 17..6 ird £ ^'''- * . . » • » <:?U"* ■ .... .-i ^ f • • t • £59.. 5..0 £62.. 2 ; II r I) I' ■ t S 1; ,;• '* i ; I m M I'..! : i I ' t f 1 111 j ' !' '^' ii -1 i i '■>:' •i; ;t i ! ■ r, ill il'l I'l'i IHI II! 1% |:(!iiii!j ill'! • III '1:1 31 Bills of Parcels, THE TUTOlsTANT LEATHER-SELLER S. Mr. Giles Harris April 12, 180S Bought of Abel Smith, s. d, 27 Calfskins at ^ ..Q per skin £ 75 Sheep ditto at 1 ..7 36 Coloured ditto at 1 .. 8 15 Buck ditto at 11. .6 , 17 Russia hides at 10.. 7 each 120 Lamb Skins at 1..2^ j£90 tall grocer's. Mr. Richard Grove* April 21, 480i Bought of Francis Elliot, c. d, 25 lb. of lump sugar at 0.. ^per lb. £ 2 loaves of double re- 1 at 11' fined, weight ISlb.j "" " ^ 14 1b. of rice at 0.. 3 28 lb. of Malaga raisins at .. 5 15 lb. of currants at 0.. 5\ 7 lb. of black pepper at 1 .. 10^ r. Abra s, 19 bi e, 18 bL , 7 qua 8, 15 lb , 6 quai IS, 12 bi je3..2. CHEESEMONGER S. Mr. Charles Cross April 23, 180 Bought of Samuel Grant, s. .d, 8 lb. of Cambridge butter at .. 6 pfr lb. jf 17 lb. of new cheese at .. 4 1 Fir. of butter, wt. 28 lb. at ..5^ m 5 Cheshire cheeses, 7 «. n a wt. 1271b. 3 at O..* 2 Warwickshire do. 1 * /> o wt, 151b. } at 0..3 12 lb. of cream cheese at .. 6 • ••« £'6..\ the brir into othe auie vail nt, All g with s concllu, g with S( BL£ OF Gu Ha Oni """• Crc Ha Shi )TE. 7 '■: such ( 7, half-j) InjCs 20 160 12 1920 THE TUT0|3TANT. pril 12, 18 Reduction. 55 in £ • • • • I • • •« »• ••• Iff t 4f CORN CHANDLERS r. Abraham Doyh-y April 29/ 1909. Bought of Isaac Jones, s, d. s, 19 bushels at 1 .- ^ par bushel... ,18 bushels at 3.. 91 , 7 quarters at ^H ,. )o^r quarter , 15 lb at 1.. 5per\h , 6 quarters at 2.. 4 ;?rr bushel... s, 12 bushels at 4< .. 8 £23 .. 7 .. 4. Lpril21, 180 \h.£ REDUCTION J&S..2. April 23, 180 t, lb. Ji the bringing or reducing numbers of one denomination nto other numbers of another denomination, retaining ame value, and is performed by multiplication and divi* rati All great names are brought into small by raulti- g with 60 many of the less as make one of thj greater. condlu, All small naiiies are brought into great by di- g with so many of the less as make one of the greater. lBL£ of such coins as are CUUREMT IK ENGLAND. /. S' d, Guinea 1 .. 1 ..0 Half ditto 0.. 10. .6 One tkird ditto .. 7 •• Crown 0.. 5.. Half ditto... 0.. 2. .6 Shilling .. 1 .. [)TE. There are several pieces tvhich speak their oia^ '■ : such as six-pence. Jour-pcncei ihree-pencef tivO'pemef /, half-penyiTj, farthing In jC 8 how many shillings and pence ? 20 160 shillings, 12 £'6 .. 1 1920 penc€* ^, ^ -■, I '|- !, •■ i! \\\ ', f li: ' )■ I < i 56 Reduction* THE TUWSTANl i if!' 2. In jC 12 how many shillings, pence, and farthingj Ans. 2405 2880^. 11520/j S. In 311520 farthings, how many pounds ? Am. 324./. 105. 4. How many farthings are there in 2 1 guineas ? Am. 211(1 5. In \vns ivni shill ts, £55^ .. 18 . into ihe mi' they have in ii guineas f 7s. oi Ans. 8 ingots. 10.32 V. -t 102..! ;hed 50 oz. 8 diot. and ordered hmi to make it into itid nic.i J'Hf s. ns, each to weigh 2 oz, 16 dwt, how many had he? 2s. 76 ii;oidor ■ ^ Ans, 18». and cro\vn> !. A gentleman delivered to a goldsmith 137 oz. 6 dwt, . of silver, and ordered him to makt it into tankards of z. 15 dwt, 10 sr, each ; spoons of 21 oz. 11 dwt, 13 gr, doz; salts of ^S oz. 10 dwt. each, and fo^ks of 21 ox, ivt. 13 gr. per doz. ; and for every tankard to have one a dozen of spoons, and a dozen of forks, what is the ber of each he mMst have ? Am, 2 wo of each .fort^ 8 ox. 9 dwt, 9 gr, over. I 'i I I - Hi ! 1 i I ■VJ .11 lit 1 i' I 1 I!! if ( . I. !!ii I n r ( I I ir ill r 38 Reduction* THE TUTO AVOIJIDVPOISE WEIGHT, Not 15. TVjrrff are: several sorts of silk xn^hich an* t I 5-2iN CLOTH MEASURE. jids. 1.1C\ '. In 2T yards, how mmy nails? y. In 75 I'inglish ells, how many yards? j^.7v. 93 yr/rV', Syr. }. In 93J yards, how many English ells? ' An.s. ";!), c 10 tji ^' In 2i pieces, each containing 32 Flemish ells, how z. of pur. 12 SI ,y jv^gij^j, ^..j,j,P ^^,^ ^g() ^^^^^ 4. ^^^ '"® ^ ^^11. In 17 pieces of cloth, each 27 Flemish ells, how y yards? jIm, 9>i^ ynrdSf }qr. L\ Bought 27 pieces of Englis^h stulfe, euch 27 elb, many yards ? y/z/v. 91 1 yardsy 1 qr, 3. In 9t 1^ yards, how many English elU ? Ans. 7-9. 4. In 12 bales of cloth, each 25 pieces, each 1.5 English how many yards ? ^, Ans, 5625. LO^G MEASURE. V. celsof 18M. ea 6. In 7 miles, how many feet, inclies, and barley corns? jins. 16 iw many pound . Aas. 1078 wt. I qr. 15 lb. 1 the whole ? I cio^ 2 yr. 10 Is of tobacco, 5. In 57 miles, how many furlongs and poh^s ? -«^ji5. 'iStJ'in longs, 1S2'^0 poles. Alls. S6960feeL 443.520 inc/ies, l^SO^OO hmiey cnrns. 7. In 1824<0 poles, how many furlon}i;8 and niiJes? Arm. i 56 J)ir longs, 57 miles. 8. In 72 leagues, how many yards ? >5rt.f. 3801(i0. 9. In 380160 yards, how many miles and leagues? Ans. 216 miles ^ 72 leagues 0. If from London to York be accounted 50 leagues, Ans. 264(i Jmand how many miles, yards, feet, inches, and harley , how many j as? Ans. 150 miles, 264000 i/ard,; 792000 /c-e^ Ans. 3( 9504000 inchesy 28512000 barley cor?? v. 3 lb. 2 oz. are I 1. How often will the wheel of a coach, that is 17 feet 2 lb. 14 oz. ovfircumference, turn in 100 miles? / Ans. 31058] :) times round. I 1 I ' I' t n r; in IV ;■ i 11 *-,i I ir ' i .si % < \r m:; .4 r ill ;i| [|§ i 'I- i! P f 'I'll ! i .; 1 f i i\ \ 40 Reduction, THE TWTO 62. How many barley corns will reach ronud the wo which is 360 degrees, each degree 69| miles. Ans. 4755801600 barlei/ cor 5 LAND MEjiSURE. 7. In 2 8. In 2 63. In 27 acres, how many roods and perches ? Ans, 108 roods^ ^S20 perches 64. In 4S20 perches, how many acres ? Ans. T, 65. A person having a piece of ground containing acres, 1 pole, has a mind to dispose of 15 acres to A. I sire to know how many perches he will have left. Ans. 352] 66. There are 4 fields to be divided into shares of perches each ; the first field containing 5 acre? ; the sec 4 acres 2 poles ; the third 7 acres 3 roods ; and the fo^o^ jjoi 2 acres 1 rood ; I desire to know how many shares are c tiiined therein ? Ans. 40 sharesy 42 perches WINE MEASURE. TSTAN' . Inl ons, an 4ns. 96 6. In 3 ^9. In ^ Christw }1. Sto ■ Savioi )4? n. Fro P ALE AND BEER MEASURE. r 7 1 . In 46 barrels of beer, how many pints ? *?ri : * . i Ans. 1324 72. In 10 barrels of ale, how many gallons and quart Ans. 320 gal. 1280 q> 73. In 72 hogshead of beer, how many barrels ? Ans. 108|e 74. In 108 barrels of beer, how many hogsheads? Am* 73|ird ai's and 67. Bought 5 tun of Port wine, how many gallons pints? y/«.?. 1260 ^«//ow*, 10080 ;?/«/5fc3. Fro 68. In 10080 pints, how many tuns ? Ans. 5 ir, how 69. In 5896 gallons of Canary, how many pipes i 84. Fro hagshead, and of each a like number? Ans. 31 of each, 37 gallons over 70. A Gentleman ordered his butler to bottle off | a pipe of French wine into quarts, and the rest into pii I desirqi to know hov/ many dozen of each he had? , Ans. 2S dozen of each EA _ foL U tllO f IIULK. such c e socoi first a to the I nuir THE TWTOJ ronud the wol lies. 00 barley corn\ erches ? Itstant. Single Rule of Three Direct 4 1 DRY MEASURE. u In 120 quarters of wheat, how many bushels, pecks, |ons, and quarts ? IrtS. 960 bushelsf 3840 pecks, 7680 gallon's. 5J072O qt^, \Q. In 30720 quarts of corn, how many quarters ? Ans, 120. 43^0 verchem^' ^^ ^^ chaldrans of coals, how many pecks ? jilns. 27 Ans. 87360. TIME. A'ls. 2880. nd conVahiingl^* ^" ^^^ **^** *^ *^'^"' ^^^^ "'•'^"^ P^^^^® •'* acres to A. I T ve left. yins. 35211 !"!! ®'li?l^f§9. In 72015 hours, how many weeks? " """ ulns. 428 Keeks, 4 ^/rt;y5, 15 hours. 0. How many days is it since the birth of our Saviour, Christmas, 1794 ? Ans. 655258 J. Jl. Stowe writes, London was built 1108 years before r Saviour's birth, how many hours is it since to Christmas, n ? Ans. 25438932 hours. 32. From Nov. 17, 1738, to Sept. 12, 1739, how many many gallons ys ? Ans. 299. nst IQO^O pint i3. From July 18, 1749, to December 27 of the same ir, how many days ? Ans. 162. 84. From July 18, 1723, to April 18, 1750, how many lu's and days ? Ans. 26 years y 9770 ^ daySi reckoning 3C5 days, 6 hours a year. acre? ; the seo 5 ; and the fo ly shares are iresy 4i2perchei Ans. 5 many pipes 37 gallons over to bottle off I e rest into pii I he had ? S dozen of each URE, nts? Ans. 1324 ons and quart 10 gal. 12S0 qi barrels ? THE ogsheads ? SINGLE RULE OF THREE DIRECT EACHETH by three numbers given to find out a fourth, in sucj proportion to the third as the second tlio first.. UuLK. First state .the question, that is, place the numbers such order, that the first and third be of one kind, and 3 second the same as the number required ; then bring Ans. 108 e first and third numbers into one nanit, aiui tiie second r t) t to the lowest term mentioned. Multiply the second and Ans, 73 ird numbers together, and divide tlie product by the firsts If h. i' 'l\ I i I i i : !^ I 1 'i I 1 N 1 :\U W illi V ]■) ! 1 |l j !!'!! 42 Single Rule of Three Direct, the tuto stant the quotient will be the answer to the question in the si denomination you left the second number in. EXAMPLES. 1, If 1 lb. of sugar cost 4^ rf, what cost 54t Ih ' 1 :4i;: 54. ^ 18 18 4)972 Ans.£ 1»0..3 12)S:43 20^ 3rf. . If a J ds IP . s end? ). If lb ish ells 1 ^ If 50^ t rate m ?. Gave J9 lb. 4 . ). If 1 I ards ^ ( 0. If 1 h of on 1. If 14 2, If a gallon of beer cost lOrf. what is thutper barrei ^I^ase of Ans. 4*l..lO..0 S. If a pair of shoes cost 4j. Qd, what will 12 do come to ? Ans. £32..8i.fl 4. If 1 yard of cloth cost i5c. Qd. what will S2 yards at the saiuc rate ? Ans. jr24!.A6..(B" l>oag 5. If :}2 yards of cloth cost i024..16..O. whatis the vi n for 5 •fa yard? Am. ios. 6.! 6. if I give £4..18..0. for 1 ctot. of sugar, at what i 5. If 7 did I b'ty [t;>e?- /i. '^ Ans. 10.vl>f ^4- /i 7. If i buy 'JO pieces of cloth, each 20 ells, for 12a\ per ell, wli.it is the value of the whole ? Ans. £'ZrA 8. V/hat will 25 cu'^. 3 yr.?. 14 lb, of tobacco couie to 1 5 \d. per lb. '^ Am: s£'I87..3..!^ 9. Bought 27 yards } of muslin, at 6s. 9\d. per y^ what docs it amount to ? Ans. i£9..5..(>|- 2 revt 10. Bought 17 cxjot 1 qr. 14 /^. of iron, at S^d. pcv 2. If 27 Lsl oan a. j: - .1. (k. what iloes it come to ? /6. 1. A dr of 14.V the who J. Age ^. 3 oz. )ay for //;?.?. €2o..7..0fi. A gj 6 t'lo/. t is the 9. A d Ans. 27 t/urd.y, 3 qrx. 1 yi(7?7. S4 /rw; irccls, 13. If 1 c:e^ of Cheshire cheese cost 3ei..ll'..8. w must I give tor S^ 10. ? A7is. Is. Id 14. Bought" 1 c'Mf. 24 lb. 3 o«. of old lead, at 9s. per c 11. If cofi'ee is sold for 5ld. per ounce, what must given for 2 cwt^ ' Ans. .£b2..2..8 12. How many yards of cloth may be bouglit l'21..ii..i^v, when 3.V coi^t :l& .:o..3. THE TUTo JsTANT. Singk Ruk of Three Direct 4S 5tion in the s^ ,^ *i . • • o^r^ j i. p. Ira gentleman s income is t500. a year, and he (Is IP . 4rf. /jer day, how much does he lay b^ at the 'send? Aiin. £IVI .."i.A. 1. If I buy 14- yards of cloth for 10 guineas, how many lish ells can 1 buy for £283..17..6. at the saue rate ? Am. 504) Fl. elh, *2 grs, J. If504< Flemish ells, 2 quarters, cost £283..17..6. at t rate must 1 give for 14 yards? u4ns. flei0..10. u Gave jei..l..8. for 3 /6. of coffee, what must be given 19 lb. 4 oz. ? jins. ^€10..! I ..3. ). If 1 English ell, 2 quarters, cost 4*. Id, what will ards ^ cost at the same rate ? jtlns. £5..S..5^^. D. If 1 ounce of gold is worth je5..4..2. what is the th of one grain ? yins. 2id. 20 rem. . If 14 yards of broad cloth cost £9..12. what is the :Iiase of 7rj yards? j4ns. jG51..H..6|-. 6 rem. 2. If 27 yards of Holland cost £5..12..6. how many ells ILsI, oan 1 buy for £'11*0? j^ns. 3b4. '. cost £12..12..6. wliat must I give for 14 civi, _ ^ 1. If,. ' J j^ns. igl82'..0..11|. 8 rem. ;^"24..16..C t' I>oaght 7 yards of cToth for 17s. Sd. what must be . what is the va n for 5 pieces, each containing 27 yards I ? ylns. jei7..7..0-^. 2 rem. , at what il5. If 7 oz. 1 c^u?^ of gold be woith 3^35. what is the val- Ans. 10.'^#f 14 lb. 9 oz.\2 dxuts. 16. gr. at the same rate? Ans. £82:?..9. 3|. 55'1 rem. I. A draper bought 420 yards of broad cloth, at the of 14.V. 10^ d. per ell English, how much did he pay the whole? .. ' ' ,'. ■ Ans. £250. .5. J. A gentleman bought a wedge of gold, which weighed L 3 oz. 8 divt. for the sum of ;^'514..4. at what rate did )ay for it per ounce ? ^ns. £3. S. A grocer bought 4 hogsheads of sugar, each '.veighing 6 avt. 2 qrs. 14 lb. which cost him a£'2..8..6. per cvot, t is the value of the 4 hogsheads ? Am. £6\!..5..3. 9. A draper bought 8 packs cf cloth, each outuining I nail. SI renfjircels, each parct;! 10 pieces, and each piece 20 \ardis, gave after the rate of /Ji^..l6. for 6 yards ; I desire to w what the 8 packs stood him in? Ans. £tJ65G. }. If 24 lb. of raisins cost 6s. Qd. what will 18 frails cost, 1 weighing neat 3 qrs, 18 lb* ? jitis. £2h.l7^S, hut per barrel Ans. £l..lO..O It will 12 do Ans. £32..8..0 >. <" will 32 yards ( ns ar I ells, for 12s. Am. £'IrA] >acco come to ins. s£'If^.7..3..;< ?. d},d. per y ^9...0..(>|- 2 rem , at 3]^/. pc/ ■his. £2o..7..0 i, what nmst Arts. £h2..2..H he bouglit 3ei..lt..8. w Ans. \s. \d id, at 9s. per c il\d, WZrm in n ji, il ' I, Mi 4 I i i.: id- \ n ■,'i 1 1 ! ;!t ill' ] !rt^ I i,' ' in liij w'-n M i'l ''i ■ m i! ; m'i .d'i Ml \'i :l H 44 Rule qf Three Inverse. THE TUT »' If 8 in caa 16 SI* If 1 ounce of silver be worth 5s» what is tlie prii ]4 ingotS} each weighing 7 lb. 5 oz 10 dtct. ? Ms. ^313., S2. What is the price of a pack of wool weigiiing 2 i ^. 19 lb. at 85. 6rf. per stone ? i4«.?. j^S..^. 6^. 10 ^3. Bought 59 cwt. 2 grs. 24 /A. of tobacco, al^2..1 per ctvt. what does it come to ? 34. Bought 171 tons of lead, at ^ii'per ton, paid riage and other incident charges £4..l0. I require the ue of tlie Tead, and what it stands me in per lb. ? Ans. £2:^98.. iO. value 1^. 432 rem. per S5. If a pair of stockings cost 10 groiits, how many d may I buy for £i;S.^5.? Am. 21 doz. 7| jj" 36. Bought 27 dozen 5 lb. of candles, after the rati Yld.per 3lb. wliat did they cost me ? j4ns. ^7..\5..4!i 1 37. If an ounce of fine gold is sold for ;^':j..10..0. come 7 ingots to, each weighing 3 lb. 7 oz. 14 dwt 21 the same price ? ^?w../ri071..14..' 38. If my horse stands me in 9^rl. per day ke< ping, will be the charge of 1 1 horses for the year ? 39. A factor bought 86 pieces of slutf", which cos! j^517..19..4. at 4^. lOd. per ya.\d, I demand how many j there were, and how many ells Englisli in a piece ? Ans. 214J^ yards, B^ rem. and 19 ellsy^ quarters, 2/ 64 rem. in a piece: 40. A gentleman hath an annuity of ;^896..17..0. annum, I desire to know how nmch he may spend daily, at the year's end he may lay up 200 guineas, and give t poor quarterly 10 nioidores? A7ii. £l..H.S, 176 n n TANl e the )rtioa t THE RUL*E OF THREE INVERSE. N VERSE Proportion is, w!«»en more requires am.1 less requires more. More requires loss, is whe third term is greater than the fii'st, and requires the U;rm to be less than the second. And less requires is when the third term is less than the first, and requiri fiouith term to be greater than tiie seconds I If54i can do Ifwh« :hs8o2 h but 1 How 1 pieces How sure to If He If for nds can le If 108 ly are si An ai with arted, v 0. If ;^ 00 men . wort! I. A c< ut 12 h journ VI THE TUTd what is the pri(j wt,? 3ol weighing 21 .;^8..4.6^. lol i)acco, at^2..]| n,.S..1\. 80 n per ton, paid I require the per lb. ? 432 rem, per ts, how many d 21 doz. 7 J }>n after the rat JTANT. Rule of Three Inverse. 45 ?7..15..4i 1 H }r ;^':>..10..0. •2. 14 dmt 21 , s../ri07l..l4..j day kec ping, ear? iff, which CO md how many J a a piece? , 4 qMurters, 2 f ;^8%..17..0. ay spend daily, teas, and give t L..14.J3. 176 r( NVERSE. rLK. Multiply the first and second terms together, and the product by the third, the quotient will bear such )rtio» to the secondTas the first does to tlie third. EXAMPLES. If 8 men do a piece of work in 12 days, how many caa 16 men perform the same in ? Am. 6 days^ 8 : 12 : : 16 : 6 8 l6)9e{ 6 daj/s. If 54 men can build a house in 90 days, how many can do the same in 50 days? Ans,91 men^. If when a peck of wheat is sold for 2s. the penny loaf hs 8 oz, how much must it weigh when the peck is h but Is. ed. ? Ans. 10 ox. lOrfr.^- How many pieces of monej of 205 value, are equal t& pieces of 12«. each? Ans. 144. llow many yards of 3 quarters wide, are equal ia sure to SO yards of 5 qnarters wide ? Ans. 50. /•158 18 ( If I lend my friend jS200. for 12 months, how long jtf whicli rnsi '^ ^® '® ^®°^ "*® £lSO. to requite my kindness? Ans. 16 months. If for 24*. I have 1200 lb. carried 36 miles, how many nds can I have carried 24 miles for die same money ? Ans. 1800 lb. . If 108 workmen finish a piece of work in 12 days, how ty are sufficient to finish it in 3 days ? Ans. 432. . An army besieging a town, in which were 1000 sol- 8, with provisions for three months, how many soldiers arted, when the piovision lasted them 6 months ? Ans, 500. 0. If ;^20. worth of wine is sufficient to serve an ordinary 00 men, when the tun is sold for JI^'oQ. how many will ). worth sufiice, when the tun is sold but for dC24. ? Ans, I'-ZSmen. 1. A courier makes a journey in 24 days, when the day ut 12 hours long, how many days will he be going the more requires res less, is whe requires the ti ess requires St, and requiri '^* jo»^rn<^y? ^^h^n the days are 16 hours long ? dns, 18 da^ i'l Ji '■■ ;! I ■;ii l!'l .!l ^ Double Rule of Three, the tui rANT. liil I ■ ^1' : ill 12. How much plush is sufScient for a cloak, whicll in it 4 yards of 7 quarters wide of stuff for the liniiigj plush being but 3 quarters wide? Ans, 9 yardf 13. If 14 pioneers make a trench in 18 days, how days will 34 men take to do the same ? j4ns. 7 dayft. 4 hours, 56 min, j\ at 12 hours for a dl 14. Borrowed of my friend £64. for 8 months, arj hath occasion another time to borrow of me for 12 mcl how much must I lend him to requite his former kinl to me? A us. £"12.. 13, 15. A regiment of soldiers consisting of lOOi) nien,| have new coats, each coat to contain y^ yards of 5 quarters wide, and to be lined with shalloon of 3 quJ wide ; I demand how many yards of ghalloon will line til Ans. 4166 ya, ds, 2 quatters, 2 nails, 2 rJ THE DOUBLE RULE OF THREE IS so called, because it is composed of 5 numbers to find a 6th, which, if the proportion is direct, ir such proportion to the 4th and 5th, as the third to the 1st and 2d. But if inverse, the 6th number bear such proportion to the 4th ismd 5th, as the 1st bei the 2d and 3d. The three first terms are a supposition two last a demand. [fl4ho lusheis \two sin^ \i)r. bu. i^ :56 bn. 16:80 X 8 ne» Inien mi \s.daysM 112:U: I?. rM?w. i ■iBi) : 8 If iSltX '5 gai !lf a car lilcs, he 3 qrs lU" a re^ 1 351 qu£ leat will If40a( acres ( Rule. 1. Let the principal cause of loss or gain, int or decrease, action or passion, be put in the first place 2. Let that which betokeneth time, distance, or p and the like, be in the second place, and the remainin in the third. 3. Place the other two terms under their like in the position 4. If the blank falls under the third term, multipl] first and second terms for a divisor, and the other three a dividend. But, 5. If the blank falls under the first or second term, n ply the third and. fourth terms for a divisor, and the three for the dividend, and the quotient will be the ans| weigh rate o 'jCroof. By two single rules of thr^e If 40s. 12 men lf;^-.(> rill gain If a rej ers of ' 1404 i In a 2 kilde there 1 ys? lfth( I THE TUl rANT. Double Bute of Three. 47 a cloak, whid for the liniii ■^ns, 9 yard 8 days, how I hours for a ( 8 months, aii me for 12 m US former kin of lOODnien, t?5 yards of alioon of 3 qu; loon will line tl 'St 2 naihf 2 n EXAMPLES. f 14 horses eat 56 bushels of oats in 16 days, how bushels will be sufRcicnt for 20 horses for 24 days ? tvio single rules 1 or in one stating^ uoorked thus : tr. bu. hor. bu. hor. day. bu. i*:.'>6::20::8()> 14 : 16 : 56 J bu. hor, bu. 20 : 24 : — 16: 80:: 24: l20^ i'8 men in 14 . ^s »,. ^ow 112 acres o^ ■» ^s, how men must there be to liiow 2000 acres in Iv/ days ? s.day&.acren.dnys. "^ men. days, acres. 12:U::2000:250(8 : : 14 : 112 8X14X2000 56X20X24 =120 14X16 days. "^ ):250 ( 9. m^n. days. men. ( =200 150 : 8 :: 10 : 200 j — : : 10 : 2000 1 12X 10 f jt'lOO. in 12 months gain £6. interest, how much T15. gain in 9 mouths ? Ans, £3..7..6. If a carrier receive £2..2..0. for the carriage of 3 cwt, lilts, how much ought he to receive for the carriage of 3 qrs. 14 lb. for 50 miles ? jdns. £\„{6..9. If a regiment of soldiers^ consisting of 136 men, con- 351 quarters of wheat in 108 days, how many quarters leat will 1 1 232 soldiers consume in 56 days ? Ans. 15031. If 40 acres of grass be mowed by 8 men in 7 days, how acres can be mowed by 24 men in 28 days ? Ans. 480. If 405. will pay 8 men for 5 days work, how much will 12 men for 24 days work ? Ans. £'ZS..%, ]f;^'iOO. in 12 months gain ;^(). interest, what princi- ill gainj^"3..7..6. in 9 months? Jins. £15. If a regiment, consisting of 939 soldiers, consume 351 ers of wheat in 168 days, how many soldiers will coii- J404 in 56 days ? Ans, 11 268. In a family consisting of 7 persons, there are diank 2 kilderkins of beer in 12 days, how many kilderkins there be drank out by anoUier family of 14 persons in ys? Ans 2 /lil. 12 gal. . If the carriage of 60 ctyf. 20 niil^s cost ;^14..10..0. Iff 11 be the ans ; weight can 1 have carried 30 miks fur ^5..8..9. at the : rate of carriage ? Ans. 15 c*ot. THREE i 5 numbers on is direct, as the third 3 6th number , as the 1 St bea ; a supposition OSS or gain, int the first place. iistance, or p the remainiDj 3ir like in the term, multiplj he other three econd term, ni or, and the n I |i i'fi 1 1 i.l '■' : ; Ml 4S Practice. the tu 12. If 2 horses eat 8 bushels of oats in 16 day uany horses vr ill eat up 3000 quarters in 2'! days ? 13. If;^100. in 12 months gain^^?. interest, what interest ot ;^"571 . for 6 years ? ^;is. /*^'i9.. 1 t'v I*. If I pay 105. for the carriage of 2 tuns 6 niiLs,Bjr»«j g^ must I pay for the carriage of 12 tuns, H cwf. 17 mill ... . '^'**' ^^••-•13751 ai ANT. 7547 ai )62|8..1 ■m PRACTICE 312..7 78..1I I S so called from the general use thereof by all pA)39|0..8: concerned in trade and business. ■ ^ AH questions i« this rule are performed by taking all9..10..J or even parts, by which means many tedious reduiT are avoided ; ihe table of which is as follows : m25 at 1 Of a Pound. \Of a Shilliv^ d. 6 - h * - ^ 3 - i 2 - i 1 - tV d, 10..0 is I 6..8 - i S..0 - I 4..0 - i 3..4^ - ^ 2.6 -X 2..0 -,v 1..8-J- Ofa Ton. Hit, IC is i 5 - i * - i 21- I 2 - Of a Hundred!^ *> ^»^°* /rs. /6. 2 or 56 is 1 - 28 - 1* - Of a Quarter. 5 at In £i5..l 14. 7.. 4 . 3i 10 at 2i Rule 1 . When the price is less than a penny, divic cit 3&'28 the aliquot parts that are in a penny ; then by 12 and will be the answer. i is ^ 5704 lb. at \ 12)1426 2;0)11I8..10 ie5..18..10 7695 at i Faa7;^'16..0..7^ 5740 at ^ Facit /,'i I. .19..2 6547 a FacU £20. 51 at 2fl 't £19..1 10 at 2c £61..\ m at 2^ '.it £37.. 4573 Facit£li. )S2 at 3 47 at 3( Rule 2. When the price is less than a shilling, talc jt jgsi. ftliquot part or parts that are in a shillincr, ^^^ them to er, and divide by 20, as before. ooo at i it j^lOl 715 at ; it £33.. s in 16 days 24 days? 17547 at W. iterest, wlvit|jNg2|8..11 !tuiis6mils,|£3i,.8..11 \7 cwf. 17 m Ans, ;^9..ti.p75i at Id.i 312..7 78..1| ireof by all pl)39l0..8f ■ > ?d by taking ali9..10..8| tedious redur S257 at 'iff. FacU £54!..5..S 2056 at 4rf.i Facit je36..8..2 3752 at 4^4 Facit £70..7..0 1325 at h/4 Hows : Of a Hundredi ;^'339..10..7 ' ^ lb. or 56 is - 28 - 14. - 2107 at 4rf.| FflCiV;^'41..14..0i Fractice, V.i 3714 at "Id.l FacU £\\9..\h,.'Tl 2710 at 8rf. i'V/cif JC90..6..8 35l4at8rf.i F<7««£l20..15..10i 3210 at 5d. Facit £66..17"6 Of a Quarter. 4, 7.. 4 . 3i 6547 a FacU £20. 5 at Id.^ 51 at 2rt?. ^£19..11..10 10 at 2d.\ £67..11..10rl 2715 iit5d.\ Facit jC59..7..9| 2759 at Hd.j Facit €97..l4..3rl 9872 at 8^.1. FacU i£'359..18..* 3120 at 5d.^ Facit £71. .10.(0 7521 at 5d.l Facif£lSO..'3..9l 10 at 24/.^ a penny, divic cit 36,'28..i..7 hen by 12 and !50 at 2d.r^ '.it £37..4..9l 4573 4-573 a ZZ — 7], * It £95..12..7^ a shilling, talc f; jg31..6..2^ ^. add them to 715 at Sd. it £3S..18..9 3271 at Gd. Facit £'81..15..G 5272 at 9^. Faa7;^'197..14..0 G325 at 9d.\ Facit ;^243..15..6} 7921 at 9d.} Facit £'3]3..hi..'2 2150 at 9d.f Facit £H7..6..iQ^ 7914 mod.}; Facit £20G.A.A0l 3250 at 6d.^ Facit £SS..0''..5 2708 at ed.l Facit ;^76..3..3 47 at 3d,j 000 at 3r/.| :t £109..7..6 3271 at Id. Facii £95..8..l 3254 at 7d.i Facit £98..5..11;; G i25 at lOrf. Facit £263..10..10 5724 at Wd.^ Facit ;^'244..9..3 6327 at lOr/.i Facit £270..4..3? 3254 at 10^4 Facit £H2..7.,$ 7291 at lOd.^ Facit £^V26..l\..G\ 2701 at 7d.l Facit ^^84..3.-U F 3256 at lie?. i^ffciY;^'l49..4..3 i I '■ ■ i , ■J ^>'lf i IN ll' ^.ll M ( i - 1 1. in ii - irt! 11 jt i| I I MM !|:f 'H 50 Praclice. 72.) 1 at lh/.| THE TUTOI fC.'lat I hi'.'. '^O'T f)72at llrf.r^ Rule 3. Wlicn the price is more than one sliilhng, al less than two, take the part or parts, with so much of tl given price as is more tlian a shiUing, which add to the giv quantity and divide by 20, it will give the answer. SISTA 5251- a it91n a \ ^\2n)6 at 12^.;- 43..10i 210)21 l|9..10i- /'J07..9..I0.\ 3215 at I*. ld.\ ^"7C7 Note. JVhcn the price is 5s. divide the quantity by 1 ij" any remains it is 5s. llui.E 6. When the price is shillings and pence, and the aliquot part of a pound, div livide by the aliquot part j^J ^ : but if they are not an al „„t. ' i it will j?ive the answer at once : t)ut it tney part, then multiply the quantity by the sliillings, and parts' for the rest, add them together, and divide by 20, d. It. 3 2 2710 at 6s. 8d. Facit /■903..6..8 S150at 36\4rf. Facit £b25.,0..0 2715 at 2.5. 69..1.S 7150 at 1,?. 8r/. Facit £595..\Q..^ 3215 at \s.\d. Facit £'214..G . 7211 at \s.?>d. Facit /;i50..l"...d 2710 at 3s. 2d. 8130 45 1. .8 85811. .8 ;^429..l..8 7514. at As.' Facit /j: 21. 25U at 5s. Facit £660.. 2517 at 7s. /•rt(7/^928..l 3271 at 55. Facit ;^'94S.. 2103 at 15j facit ;£1616.. 7152 at 17s. Facit £62m. 2510 at li,y. /''ac//;^'1832..1 3715 at 9s. Facit £\7-^\ 2572 at 13.?. Facit ^1752 7251 at 145, Facit £5mi by the 7 Uy, dly, Wi pound, tiply the them to ote, W> proceec K Fai \Fa\ niU TUT IUi2 at i:| BISTANT. 3210 at 15*. U.} Facit £25\l..3..H Practice, 53 2710 at I9s. 2d.h Facit xC!2602..li..7 - ^- XVr v<2 ff ^^" '^' ^^^' ^'^'^^ ^^* price is pounds and ejhiliingg, utat £zOi^'p^jp|y jjjg quantity by the pounds, and proceed with the — ;;; — ~ ""Blinds if they are even, as in the 4 When the given quantity is no 'more than three Jig- * proceed as in Compound Multiplication. 25 V7 at 5s, Facit /"(JGCl 2517 at Is, r«c.7/^928..lj S271 at 55.1 Facit £9^S..]\ 2103 at 1 5s Facit £lQi6.M t 7152 at 17s. Facit ie6280, 2510 at \U. Facit£lS32.. 371 5 at 9s. Facit £m\ 2572 at 13.?. Facit £11521 7251 at 145 \Facit £on^. 7215 at £7..i..O 7 50505 1443 51948 jS 2104 at IS5..3..0 5 10520 263 52.. 12 10835..12 U i 2lO 2107 at;^*2..8..0 Facit £5056..16..0 71^6 at ;^'5..6..0 Ffl«V £37926..! 6..0 2710 at ie2..3..7| 43 116530 1355 338..9 11822I3..9 f5911..3..9 '*''» 3215 at£l..l7..0 Facit JC5947..15..0 2107 ai £l.A'6..0 Facit £'M76..U..O 3215 at £4..6..8 FaaY;^'13931..1i>..4 F2 2154 at ;^7..1..3 |FaaV;^15212..12..6 h -i> f , li '' I t : I i. ll 1,1 '-I i I i1 11 fill ="i .1 l\' -\ !:if m .1 fill,. :iii >i) IJ! .( I i I':':;! !■ 1 I; I I ii r. Hi 54 Practice, i Ji70l at je2..3..4 F«a^ je5852..3..4 2715at;^1..17..2i Facit £5051. .0..7^ 21,37 at ;^3..15..2J Ffl6/^£8108..19..5] 3210at;^'1..18..6^ /flCiY £6189..5..7i THE TXJTObBsTAVT. 142 at £l..l5.m' o 1 J Pflc/^ £250..2.fXt do ^. .. ,». H Tobacc 95 at £15..17.M . ,K // FrtCiV £l4-94..7..rAt /"^ .13/R( >. Bougl 5 cwt, 1 . At -«4 ». of tob oap i ? 37at£l..l9..5 Fflc/^ £73..0..8| 217Satf2..15..| Facit £6022..0.. 2150at£l7..16..1 Facit £38283..8.. 2157 at £2..7..4f Fffc/^£5109..7»10^ Rule 8. When t! e price and quantity given are of eral denominations, multiply the price by the integer, t;ike parts with the parts of the integers for the rest. 1. At /,3.,17..6 per cwt. what is the value of 25 c, ^ qrs. 14 lb' of tobacco ? lb. 14 I 4 ;^":i..l7.. 6 5 19.. 7.. 6 5 96..17.. 6 1..18.. 9 9.. 8^ %j%^»% •y«*XJ.'*f 5+j=25 2. At;^1..14..9. jo^r cw^ what comes 17 cwt,^ i qr. 17 cheese to ? An& £2\ ..10..8 3. Sold 85 ciKit. 1 yr. 10 lb. of cheese, atj^l..7..8pe/c what does it come to ? Ans, ;^'l 18..1..0^ 4. Hops at j^4..5..8. per avt. what must be given 72 cxvt. 1 qr. 18 lb 'j? ^«4". ;^310..3./2 5. At;^'l..I..4. ^er ctof. what is the value of 27 c 2 grs. 15 lb of Malaga raisins ? -..l4i..6» the ctc^ what is the value of 37 caU AS to. of double refined sugar? Ansi £l77..lif..S^ ). Bought sugar at j^3..14..6 the cwt. what did I give \5 cwt. 1 qr. 10 lb? Ms. £57..2..9. "m. At -t4..15..4'. the civt. the value of 172 ctvt. 3 qrs. 75 at £2..15.M. of tobacco is, required ? yfns. ^823..19..0j. oap at i3..11..6. the cut. what is the value of 53 ctvt. I? Am. Ii90..0..i!, \he Allowances usually/ made in this Weight are Tare, Trett, and Cloff. :o xit£6022..0, at £17..16.J ii£38283..8. jiven are of si le integers, the rest. Rare is an alloin^nce made to the buyer for the weight alue of 25 c. ^^ ^^^> barrel, bag, &c. which contains the goods ht and is either ^ ;.. 'vi- t so much per box, barrel, &c. . ' ^' t so much per cent, or tso much in the gross weight. * ••' " ' ' - ^ - RETT is an allowance of iflb. in every r04;lb. for waste, &c. made by the merchant to the buyer. lOff is an allowance of 2/6. to the citizens of London, very draft above Scxut. on some sort of goods. ROSS Weight is the whole weight of any sort of Is, and that which contains it. UTTLE is when part of the allowance is deducted from jross. EAT is the pure weight, when all allowances are de- ed. ULE 1st. When the tare is at so much per bajr, barrel, multiply the number of bags, barrels, Sfc. by the tare, subtract the product, from the gross, tlie remainder is ! cuit. i qr. 17 Ins £2l,.lO..S i£i..7..iiperc s.£il8..l..0l USt be given us. £S10..3.,'2 value of 27 c !ns. ;^'29..9..6i tits, at £2..n ns. ;f 227..14.A DTE. To reduce pounds into gallonst multiply hif 2, dl- hy \.5. ♦ ' In 7 frails of raisins, each weighing 5 cwt. 2 qrs. 5 lb. ?i tare, at 23/6» per frail, how much neat weight ? ^ Ans, 37 cwt. I qr. i'^ lb. r. : .,: !i I 'I I; i; !' ki . ii 1 ' h » ■ i' \ lit / if"' IS i . ! .<|l*!l;]:i. i t I I ! k \^ \ n\ \ '< \< !■ i ! I 56 Reduction, THE tutI 23 7 A) 5. .2. .5 7 i .vn: li 28)161(5 140 1 .. 1 21 38.. 3.. 7=gross, 1 .. 1 .»2ltB=iare, or, £.. 5..1 37 .. 1 .. 14 neat. IsTANT. What is weighin LE 4. ds suttU the suti In one 2. What is the neat weight of 25 hogsheads of tob^' ^^^^ ^ weighing gross 163 ctvt, 2 qrs, 15 lb, tare 100 lb, per ^^ ^^^^ head? yins. 141 cwt, 1 qr, 7 3. In 16 bags of pepper/ each 85 lb, 4 oz. gross, tar bag 3 lb, 5 oz. how many pounds neat ? jfns. 131 Rule 2. When the tare is at so much in the whole weight, subtract the given tare from the gross, the remali is neat. 4. What is the neat weight of 5 hogsheads of toba weighing gross 75 cwt, 1 yr. 14 lb, tare in the whole 752 jfns, 68 cwt. 2 qrs. 18 / 5. In 75 barrels oi figs, each 2 qrs, 27 lb. gross, tar the whole 597 lb, how much neat weight ? - A'is. SOCivt. 1 qj Rule 3. When the tare is at so much per ciut. divide gross weight by tke aliquot parts of a cwt, which subti from the gross, the remainder is neat. Note, 7 lb. is j\, 8 lb,i8j\, 14 lb. is |, leM.is^ 6. What is the neat weight of 18 butts of currants, ei 8 cwt, 2 qrs, 5 lb, tare at 14 lb. per cut, f cwt. qr. lb. ' : 8. .2.. 5 9X2=18 76 ..3.. 17 2 In 7 b. how \5'2 c m lb. ii LE 5. divide which I Wha hing 1; per 10 14=5 153 .. 3 19..0..25^ 134 ..2.. 8f 7. In 25 barrelis of figs, each 2 cwt. 1 qr. gross, tare cwt, 16 lb* how much, neat weight ? ^n^. 48 cwt. qr, 24 /^ ; ^;| 111 ' or. THE TUT r, 5.. 5..1 37 ..1 STANT. Reduction. 57 What is the neat weight of $ hogsheads of natmegs, weighing gross S ctvt, 3 qrs. M lb. tare 16 lb. per civt ? jins. 68 cwt. 1 qr. 24.< lb. [jLE 4f. When tret is allowed with tare, divide the ds suttle by 26, the quotient is the tret, which subtract the suttle, the remainder is neat. In one butt of currants, weighing 12 cwt. 2 qrs. 24 /^. heads of tobl' ^^^^ ^^ ^^' P^^ ^^^'^' ^^^^ '^ ^^' P^^' ^^^ ^^' ^'^^ many 100 lb. per li ctvt. I qr.l I ipz. gross, tare jfns. 131 in the whole g OSS, the remaii leads of toba the whole 752 xvt. 2 qrs, 18 L lb. gross, tar IS. 50 ctvt. 1 q\ ler cwt. divide ds neat ? 12 ..2. .24. 50 28 i, 16/A.isi of currants, e 8 r. gross, tar« ^ tiot, qr, 24 /^ 14 I 1424 ^row. 178 ^arc. 2Q)\2\Q suttle, 47 ^rtf^* 1199 M^a^ )^. which subti ** ^" *^ ^^'^* ^ 5^'''^* ^^ ^^* ^^^^^t ^^'^^ 36 /5. tret 4 lb. per lb. how many pounds neat ? ^/w. 826 lb. 152 ctu^. 1 qr, 3 /A. gross, tare 10 lb, per ctvt. tret 4/5. 04 /6. how much neat weight ? ^n.?. 1 33 cut, 1 yr. 1 1 lb, ULE 5. When cloff is allowed, multiply the cwt. suttle divide the product by 3, the quotient will be the pounds which subtract from the suttle, the remainder will he neat* . What is the neat weight of 3 hogsheads of tobacco, liing 15 ctvt. 3 qrs, 20 lb, gross, tare 7 lb, per ctvt, tret per 101' lb, cloff 2 lb. for 3 avi If Ans, 14 cwt, I qr, 3 /6. 7=J„ 15.. 3 .. 20 ^ro^y. 3 .. 27i tare. 14 .. 3 .. 20^ 5««/e. 2.. 8 tret. 14.. 1.. 12^ suttle, 9^ c/(#. 14 .. 1 .. 3 neat. '! t I n It ! ■I'tii (::' •{ • V I n iil'iiil ■li' i V 1 i ' \ I I" t i ' ■f •a, ■ii^i I? u 1 ' ' ' i ijijli ri jl' ' i iij i '■•r 58 Interest, THE TU 13. In 7 hogsheads 6f tobacco, each weighing gross* ^^'^ 2 ^r*. 7 lb. tare 8 /^. jper cwt. tret 4 /i. j!)er lOi lb. cloft.. fer 3 cvot, how much neat weight ? ■vliat is t ^M5. 34 cid.^ 2 ^r.«. SIMPLE INTEREST IS the Profit allowed in lending or forbearance sum of money, for a determined space of time. The Principal is the money lent for which Inf to be received. The Rate per Cent, is a certain sum agreed tween the Borrower and the Lender, to be paid for ^100. for the use of the principal 12 months. The Amount is the Principal and Interest add gether. Interest is also applied to Commission, Brokagc, chasing of Stocks, and Insurance, and are calculated •anie rules. urn? Jhoi is t t. per an 'hat is t for thn /hat is tl . per an hat is tl for 5 J y corres |o the anr 5 commii I allow demand To find the Interest of' any Sum of Money for a Yt noj^pe ea Stoci eofiClV Rule 1. Multiply the Principal by the Rate;7eT ceni 1 96| pt product, divided by 100, will give the interest require For several Years, 2. Multiply the interest of one year by the num years given in the question^ and the product will be tli swer. 3. If there be parts of a year, as months, weeks, or work for the months by the aliquot parts of a year, an the weeks and days by the Rule of Three Direct. EXAMPLES. 1. What is the interest of i375 for a year at 5 pe\ per annum 9 5 18175 20 15K}0 allowar )ersons :. Divid ient wi I cmp J.. 17 •5175 .. 20 15117 12 2110 Ans, il8..l5 ^"cn 5.. 10. OS. 6d. nnuities t 124f a Stoc THE TUi ;]iing gross,] 101; lb, clof :ant. Interest, 59 cv:t. 2 (p's. hat is the interest of 268/. for one year at 4- per cent, um? ^/w. ilO.. 14.. 42. hat is the interest of X945 .. 10 .. 0. for a year at 4- /. per annum ? ^ns. £37 .. 16 .. 43. hat is the interest of £54!7 ..15 .. 0. at 5 per cent, per for three years ? Ans, £S2 .. 3 .. 3. /hat is the interest of £25^ .. 17 •• 6. for 5 years, at 4 per annum ? Ans. £50 .. 19 .. 6. f'hat is the interest of £556 •• 13 .. 4. at 5 per cent, per for 5 years ? Ans. /CI39 .. 3 .. 4. 3' correspondrnt writes me word, that he has bought |o the amount of jC754 .. 16 .. 0. on my account, what commission come to at 2^ per cent ? Ans. £18.. 17 "^. I allow my factor 3f per cent, for commission, what „ 1 I demand on the laying out jC876 .. 5 .. 10 ? ion. Brokflgci ^ o .*„. rqo.. forbearance ;e of time, which Intf um agreed be paid for iiths. nterest adil re calo^llated 1 Jonei^ for a 1 J Ans. iC32 .. 17 .. 2^. 1 10^ per cent, what is the purchase of ^2054 .. 16 .. 0. ea Stock ? Ans. £2265 .. 8 .. 4. t 104v' per cent. South-sea Annuities, what is the e of i:i797 .. 1.5 .. ? Ans. £1876 .. 6 .. 1 1 f . B Rate per cewAt 96f per cent, what is the purchase of ^577 .» 19 •. 0. iterest requirelnnuitie!} ? Ans. 559 .. 3 .. 5^. 1 124f percent, what is the purchase of ^758 .. 17 •• ia Stock ? Ans, 945 .. 15 .. 4^. by the numn BROKAGE iuct will be thl allowance to Brokers, for helping merchants or fac- )ersons to buy or sell their goods, ths weeks ori' divide the sum given by 100, and take parts from s of a year, an ree Direct. a year at 5 pei ient with the rate per cent. I employ a broker to sell goods for me, to the value 5 .. 17 .. 6. what is the brokage at 4.v./>cr cent? !5!75.. 17..6. 20 46. i 25.. 15.. « 15117 12 2iiO Am. to .. 3 .• 0[ Ans, <18..15 ^"^'f^ ^ broker soil? goods to the amount of 5.. 10. what may he demand for broknge, if he ii \os. 6d. per cent ''■ Ans, <19 •• 10 .. 9{,. \ 'i' 1: JM Mi! ■J! Ii' ■iTF 'i i ill ,!f. ' J: '1 w V' 1.1 ii 11 f [■ •■ < I' ' ,i\ -Hi'Vlj/ I in i; :;i ':! i I 4 I i ; li! !ii I ' I i •'i.i i r if: iJ i >■ hh 1;!'!' ! I 11 ^:'!| Hi , li wn CO Interest, THE TlWOl ISTAN^ What unt to i |7. What 15. If a broker is employed to buy a quantity of g to the value of i975..6..4. what is the brokage at 6s,\ per cent ? Ans, '£3..3..4L 16. What is the interest of /C547..2..4.. for 5 years a J^^^^'*^/'' half, at 4; per cent, per annum f Ans. ^120 .. 7 .. 3 17. What is the interest of i257 .. 5 .. 1. at 4 per C€nt\ a year and three quarters ? Ans. ;C18 .. .. 1 18. What is the interest of ""' *® £734<..8..0 at 4 per cent per annum ? Ans.£54>0. 7. What principal being put to interest for 7 years, at ;r cent per annumt will amount to ii'334..16..0 ? Ans, j£248. en the principal Rate per cent, and the amount are given, tojind the time. luLE. As the interest of the principal for 1 year : is to ear : : so is the whole interest : to the time required. at 5 ' ner ^' ^" ^^^' ^''"^ ^'^^ ^^^^ amount to £^Q2..\Q..O. at * ^-^ ' t J er cent, per annum? f. *ai .. / .. ii ^ ,^g jy .jQ . . g2,.io : 5 mherafWec^^ 20 20 of the givei interest requ I. for 20 week a.. 19.. 10 . for 12 week . ;C3T9 .. 4 .. C 10 3 50 20 oo t of the given erest require! is the intere ?. /:289 .. 15 . . 4. at 4;;^ per JC4.19 .. 15 .. 210 21l0)105|0(5i/ea«. ^«M02. 10 105 ' 350.. o 2..10 19. In what time will £540. amount to ;^734..8..0. at er cent, per annum ? Ans. 9 years. 0. In what time will je248. amount to ;^334..*1G..0. at er cent, per annum ? - Ans. 7 i/ears. \en the Principal, Amount, and Time are given, tojind the Bate per cent. luLE. As the principal : is to the interest for the whole e : : so is ;^ 100 to the interest of the same time. Divide t interest by the time, and the quotient will be the rate • cent. en: to the pi 51" At what rate per cent, will ;^350 amount to 02..1O..O. in 5 yeari time ? st will amouf 50 As ;^350 : ;^'52..10 : : ^ 100 : jTlS. 20 cut. are give ;he rate and r annum ? 12 .. 10 20 8050 100 00) i'^BO Alts 52..10..0. 1050 100 35|OU05000!0(300.^^*15-r-.:=.Wr cent. 12. At what rate per cent. will.;(^248 amount to £ .S34..1t». 7 years time .^ . Anu 5 per cent^ ) m ;f m 4 il i ; m v\ Mi\ w\ m ■\i "j nil 62 Interest. ■ ■ I , 'i Jil THE TUT(B|STANT 33. At what rate ;)• cent, will £.T10 aniouuj £lii'i!..S. in 9 vcars time ': A US. 1 per cci What i b, i) nion COMPOUND INTEREST IS that which arises both from the principal and intci that is, when the interest on money becomes dm.', not paid, the same interest is allowed on that interest paid, as was on the principal before. Rule 1. Find the first year's interest, which add ti principal: then find the interest of that sum, which ai before, and so on for the number of years. 2. Subtract the given sum fi am the last amount, a will give the compounc^ 'nterest required. \ ' EXAMPLES. 1. What is the compound interest of £500* fori S ye|fe at 5 per cent, per annum ? i'- ■-; 5c3K500 525 - 5 25 26..5 - . 25100 5':5 1st year. 55U.5 2d year £6125 20 5100 27I56..5 20 551..5..0 27.11. ..3 H 11125 12 3 loo 578..16..3 3c?^c«r. 500.. 0..0 prin. sub. £'JS..l6..S='hiterest. for 3 yei tlie ab I before it in the lent uione |ar to cor [ULE, As i?.r, intert led. jb tract tl n\\ be th What is I months, — : K K What 2. What is the amount of ;^400. forborne 3^ years fer cent, per annum, componnd interest ? Jno\'£490..l3..1 3. Wliat will £Q50. amount to in 5 years, at 5 pei per annum, compound interest ? ^ns. £829..! I.. 4. What is the amount of ;^550..10..0. for 3 years igreed t months, at 6 per cent, per annum, CQmpound interest? im ? , j^ns. £6'75..6 What I 8. What is the compound interest of «e764. for 4 " cent pel «nd 9 months, at 6 per cent, per annum ? ji!is.£'2i:3.. Boua-htl 6. What is the compound interest of £57..10..6. months,] years, 7 months, 15 days, at 5 per cent per annum ? im alio Ans. jei8..3 487. 14.1 J.;^473J 'ITE TUT* STAN T. " Rebate or Discount, CS 10 amouiiJ US. 4; per cc\ What is the compound interest of ;^2o9..10..0. for S [s, t) months, and 10 days, at 4^ })er cent, per annum '^ pal and intc: comc8 due, hat interest wWiTilUklM I'hich add t un, which ai REBATE OR DLSCOUN^r tlie abating so much money on a debt to h« receivrd before it is due, as that money, if put to iuturest, woald in the same lime, and at the sanio rate. As /tlOO. ent money would discharge^ a debt of /.'I l^. to be paid ar to come, rebate being made at o per cent. St amount, ^^^^^^ As ;^'I0O. with th'e interest for the time given : is u interest : : so is the c:un givjn to the rebate re- bel. btract the rebate from the given suni, and the remain- ill be the present worth. . ;^500. ford ;..o V )..3 Sc? year. )..Q prin. suk ^..S=*interest. for 3 yei erne 3^ years ?.je490..l3..1 [fears, at 5 pc) lis, £829.. n. K for 3 years mnd interest ? ■ £764-. for 4 J,is.£'2iS. if £57..l0..6. EXAMPLES. What is the discount and present worth of ;^4'37..12. ) months, at 6 per cent, per annum ? 6?H.';() As 103 : 3 : : 487..12. — ~ 20 m 3 100 103 487..12 14.. 4 2060 9752 3 £. s. 20610)292516 ( U.,ii, rebaler ^06 oer annum 9 Atts, jei8..f J. ;^473.. 8..0 present worth. 865 !5 824 416=i^. What is the present payment of £357.- 1 CO. which agreed to be paid nine months hence, at 5 per cent, per mi? Ans. ie344..n..7. What is the discount of ie275..iO..O for 7 months, at • cent per annum ? Ans. £7..16..1 f . Bouffht goods to the value of /109..10..0. to be paid months, what present money will discharge the same, im allowed 6 per cent, per annum discount ? ^m. £l04..15..Si-. Hi .1? s I m ii !:'fct 15^ '1. . I ! '! ; ! I V, til' I *u I iii ) ' I.' * fl Il . 'ri m L^ if M Ml 'i 1.1 i ■ i( !■!: I ini I' ' ill ill; i I ill i I 64t liquation of Payments* ^ ' the tuto| 5. What is the present worth of £.527. .9.1.. payabl months hence, at \'l per cftnt ? Ans. ;^.514'..13..10jJ (>. What is the discount of £85. .10, due Septemberf ^th, this being July the 4th, rebate at 5 per cent per anni Ans. \5s..'ii 7. Sold ijoodr, for £f^75..5..6. to be paid 5 months hen what is the present worth at 4 J per ccnt.f Ans. £S59..S..^} 8. What is the present wortli of ^£500. payable ini months, at 5 pet cent, per annum ? Ans. £48(j 9. How much ready money can I receive for a note of j due 15 months hence, at 5 per cent, Ans. ;^70..11..91 10. Wliat will be the present worth of £\S^. payabl 3 four months, i. e. one-third at four months, one-third months, and one-third at 12 months, at 5 per cent, discc Ans. £U5..'i..%\ 11. Sold goods to the value of ;^575.. 10. to be pail two 3 months, what must be discounted for present payinl 9X5 per cent. Ans. £\0.»\.\.a\ 11. What is the present worth of ;^500. at 4 per j^-100. being to be paid down, and the rest at two 6 moni Ans. £488..7..8| EQUATION OF PAYMENTS IS wheiv several sums are due at different times to fil mean time for paying the whole debt ; to do whichf is the common lluLE. Multiply each term by its time, and divide| sum of the products by the whole debt, the quotient counted the mean time. EXAMPLES. 1 . A owes B ^200. whereof £^0. is to be paid at months, £GQ. at 5 months, and £100. at 10 montbl what time may the whole debt be paid together, wil prejudice to either ? 40 X 3= 120 60 X 5= 300 --' 100 X 10=1000 ^ 2|00) 14|20 7 monihsy -^ I't •HE TUTOl )i3TA>rr., Barter, 65 ., B ovTcs C ;^800. whereof ^^200. is to be paid at 3 iriths, £10U. at 4 months, /JiOO. at 5 mjuths, and £200. 5 months ; but they agreoiag to make but one payment the whole, I demand wiiat timj that must be ? ji IS. 4 months^ 18 days, . I bought of K a quantity of goods to the value of Ji). which was to have been paid as follows: affl20. at iionths, ;^200. at 4 months, and the rest at 5 months ; we afterwards agreed to have it paid at one mean time, time is demanded? Ans. 3 month.'}, 13 dai/i, . A merchant bought goods to the value of £500. to pay ()(). at the end of 3 months, £150. at the end oT > months^ I .£250, at the end of 12 monlhs ; but afterwards they t'jd to discharge tlie debt at one payment ; at what time this payment made? Ans. 8 months, 11 da^s. . H is indebted to L a certain sum, which is to be paid 6 different payments, that is, ^ at two months, -J at 3 inths, ^ at 4 months, ~ at 5 months, 4 at G months, and I rest at 7 mont'is ; but they agree that the whole shall be d at one equated time, what is that time ? Ans. 4 months, 1 quarter. I. A is indebted to B ;^I20 whereof | is to be paid at 3 nths, -[ at 6 mouths, and the rest at 9 months, what is the equated time of the whole naymcnt ? Ans, 5 months, 7 dai/s. BARTER S the exchanging one commodity for another, and- in- forms the traders so to proportionate their goods, thiit ther may sustain los«. / ?ULE 1st. Find tlie value of that cotrimodlty whose quan- is given : then find what quantity of the other, at the c proposed, you may have for the same money. Jdly. When one has goods at a certain pr-ce, rea-Ji/ mon- hut in bartering advances it to sometning more, find It the other ought to rate his goods at, in proportion to. t advance, ai)4 tl^en proceed as before,. ' G 2. w ' r ;■! !!; I i '** ii'i ki\ |:!lr:' 'I {111 V i' i >' r i I n i;i I'l:.; I i ! ill ii!:P !' 1 66 Barter. \ TIXE TUTOBISTANT BXAMPLAK. 1 . What quantity of cho- colate at 4:s, per lb. must be delivered in barter for 2 cwt. of tea at 9.V. jupy lb. ? 2cxvt, 9 4-j'2016 the value of the tea. 504- lb, chocolate. rest in CI on B ga^ If B h{ A for 1 2. A and B barter ; A 10 cwt. of prubes, at 4 * 8. A ajid B barter ; A hath 320 dozen of candles, ai ; Af a y 6i/. per dozen ; for wJiich B, giveth hixii £iQ, in monj&yi. An, K 1 lb le gain ; I6f . vfhi I2£ TUTOAlSTANT. >arter; A es, at 4' ve the differei he qf-Aj^S.. pper at I3{d< h ginger mus s. 3 lb. 1 oz. . 2d. per do: ;. 16 lb. of tal *. 26 doz, 3 h t.v. per yard, oney and 85 1 is. what di ight for 115. and sold for . (id, what is the gain per f As 11 12 13 1m6 100 2C 2000 18 1 ^ ) 36000 12)5272 20)272..8 Jns, £l3..l2,.nj\. As 100 : 18 108 cost j^ 18. what must 1 ell be sold for to gain 8 per cent f : 108 12x5=60 12)19..8..9i- Sll..l2..4i 1100)19144 20 8180 12 9!60 4 2140 6..5J jfns, €s, 5f . If.l lb. of tobacco cost I6d. and is sold for 20^. what »e gain per cent ? yf,ts. 25, «,, TT««v « • If a parcel of cloth be sold for £560. and at jei2 per Am, £b..\[ ' g*'"» ^^^'^^ ^^ '^fi prime cost ? Ans. £500. . If a yard of cloth is bought for I3s, ^, and sold agaia. I6r. what' b the gain per cwt f Ans. 20.. of candles, at 10. in moneji. r M I I' I I, \\l 'i \ (' »;| ■" 11 I ! !lf / Pii. m rH I , :: \\\ \v ! r, 'i ( . 1 ii- I I ,;| ! I :{ I . Ill (!; 1 -• lis FtUowshlp^ ™^ ^^^' ISTANT. 6. If 1 12 M. of iron cost 27« 6fl?. what must 1 act. be for to gain 15 per cent ? y4us. £1, 11."™ , ^r. 7. If 375 yards of broad cloth be sold for;^;490. an Zu^^ll} per cent, profit, what did it cost per yard? Ans. /.'l..].m 8. Sold 1 curt, of hops at /''i..l5. at the rate of '25 tent, profit, what would have been the gain per cent, if I ■old them for £8. per cwt. ? Ans. jfi8./2..11 9. If 90 ells of cambrick cost j^60. how muft I sell ii yard to gain 18 per cent ? /his. I2s Id^ 10. A plumber sold 10 fother of lead for ^'-iO^.. 1.5. fother being If/ ao/. \) and gained after the rate of _^'1'J f)er vent, what did it cost him per cwt. Ans. 18,^. 8 1 1 . Bous^ht ^Mi yards oi^ cloth, at the rate of Ss. 6d. yard, and sold it for 10^. Ad. per yard, wl:at was the ga the whole? Ans. £39.. 1 9., 12. Paid £o9 for one ton of .^teel, which is retailed al ■fier lb. what is the profit or loss by the sale of 14 tons ? 13. Bought 124 yards of linen for;^32. how should •ame be retailed ^^'^r yard to gain £15. per cent f Ana.Ss.Ud.fl H. Bought 249 yards of cloth, at 2>s. Ad. per yard, tailed the same at 4*. '■2d. per yard, what is the profit in whole, aud how much per cent ? Ans. £l0..7..6. profity and 25 per cci FELLOWSHIP IS when two or more join their s^or. and trade togel so to determine each person's pai ticular share of gam or loss, in proportion to his principal in joint stoc Bi/ this ?'ule a bnn/cnipt's estate mat/ be divided amon^ creditors ; a& also legacies may be adjusted volien there dejicieucy of assets or effects. Fjellowship is either toith or toithout Timb. FELLOWSHIP WITHOUT TIME. Rule. As the whole stock : is to the whole gain or :.: so is each manVt share, ia stock :. to his share of the crlosa,. ' Four m :227. C led jS42J A ROOF. A il to the j whole ga 's sliare c , Two m< B jC40. t eof? ^ 60 : 50 2C 6!0)100i £16..13, Three n :20. B ^: I's part of • A, B, a tm. and j's share I Ans. A i . Three ;k was months re of the A men .to D estate is livided Ans. B' D's )h Four Ahal THE TUTd ISTANT, Fellowship. 69" [roof. Add all the shares together, and the sum will be to the given gain or loss : — but the surest way is, as kvhole gab or loss : is to the whole stock : : 90 is each ['s share of the gain or loss to his share in stock. EXAMPLES. Two merchants trade together ; A put in stock £20. I B £4:0. they gained £50. what is each person's share bof? 20+40=60 60 : 50 : : 20 As 60 : 50 : : 40 33.. 6..8 B's share. 20 40 16..13..4 A's. 6!o)ioo;o £16..13..4 6|0>200|0 £50.. 0..0 4ns.£lS.£ /oM Three merchants trade together, A, B, and C ; A puti how should|r-20. B ;^30. and C ;^40. they gained ;^.180. what is each. 's part of the gain ? Ans. A £\0. B £Q0. C £S0. • A, B, and C, enter into partnership ; A puts in £364/*^ ^482. and C £500, and tl^ey gained £867. what is each 's share fn proportion to his htock ? Ans. A £234..9..3i— rm. 70. B J&310..9..5— rem. 248.. C. £322..!.. 3^— rem. 1028. . Four merchants, B, C, D, and E, make a stock ; B put "227. C;£*349. D;^il5. and E ;^439. in trading they icd jC428. I demand each merchant's share of the gain? Ans. B £85-.19..6|— 690. C jei32..3..9— 1^20. X) £43..11..1|— 250. E jfil66..5..6|— 70. i. Three persons, D, E, and F, join in company ; D'a ;k was £750. E's £460. and Fs je500 and at the end of months thev gained ^^684. what is each man's particular re of the gain ? Ans. D jg300. JS £l84. and F jg200. !. A merchant is indebted to B je275..14..0. to C £i504.. . to D £152. and to L 4^ 104..6..O. but upon his decease, estate is found to be worth but d^G75..15..0. how must it livided among his creditors ? Ans. B's ie2i^2..15..2— 6584. Cs £245..18. 1|— 15750. Ds £122..16..2|— 12227. and Ks £8i..5.. 5-^1 5620. . Four persons trading together in a joint stock, of ;ii A has ^, B ^^ C }, and D the remainderf and at the TF ■ .' 1 '.■ i '] C 1 : , 1 ' S H ,1 vh I ¥ ;• 'il 1^ r •i 1 '1 ! ^^;! t; Il IS; : ^ ' ' 1 ■ h 1- 'I- ■ ■ :r ' ■' 1 jii. ■ r 1 ^ il ' ' 1 i 1 ' 3'' '1 1 i J r' II' ; %\ Il i 1 11 ' if SIM , , n - 1 1 11' h I. iJ ( -r hi f f; m 'M' I * ilr'i ■u \ lli hW i I n liiilii i . 'If I! :■ I.. I If :iP '1 li i s J : ii'i ii! 'ill';!' i .k \{ Tto- FellcxsUp, THE TUTO €nd of six montlis they gain ^100. what is each mail's ji of the said gain ? , Jns. A £f}3..6..8. B £25..0..C. C £20..0..0. D £li..U,A, 8. Two persons purchased an estate of £\100 per an freehold for £27,'200. when money was at 6 per cent, i: est, and 45. per pound hind tax, whereof D paid;^'15, and E the rest ; some time after the intercf;t of the ir^c ■falling- to 5 per cenl. and '2s. per pound kind tax, llu y the said estate for 2% ycqrs purchase. 1 desire to know ( person's share ? Am D £22,500. E £iH,20 9. D, E, and F join their stocks in tratlc, the amoun their stock is £647, and are in proportion as !•, G, and K, to one another, and the amount of their gain is equal to Stock, what is each mans stock aud gain? Jfs stock, £i'V.}..l5.ASh" gain, £31..19..0A?. Ks - ^ 2ir>..13..4.*" 4-7..18..6^4. F's - - 287..11..14§ 63..\S..0/^., 10. D, E, and F join stocks in trade ; the amount oft stocks was ;^i 00. D's gain /^.i. E's £5. and F's ^8, v was each man's stock ? Jim. D's stock ;^'i8..15..0. E's £3l..5..0. ct7id F's i FELLOWSHIP ivrrn time. Rule, As the sum of the product of each man's nso and time : is to the whole gain or loss : : so is each iiii product : to his share of the gain or loss. Proof. As in Fellowship without time. EXAMPLES. 1. D. and E enter into partnership ; D puts in £iQ, three months, and E £l5. fur four months, and they gai jfe70. What is each man's share of the gain ? Ans. D £20. E £b[ 40 x3=al20 As 4^20 : 70 : : 120 As *20 : 70 : : *?5><4=»300 120 • 300 420 4210)84010(20 4210)210010(50 I^TANT. . Three i .5..14..0. 59..H'..i ich \x\mi Jns. D' . Three i in 500/. iths end ) £130. J 0. more ; D. at the of 15 mc £200; ar 3S out tha y gained gain ? Ans, D D, E, ch they a s; E21 ( ach man Ans. ALLIGAl > wncn t given t( U1.E. A any pn ROOF. and it Uities a *IIE TUTO ach man's si £20..0..0. 1700 j^fr an\ per cev.t. v.. ) paid £\ij.\ f^t of the ii.c d tax, tli'v V re to know ( . E £\'6fi{. , the amour 1, G, and K n is equal to J amount of tl ul F's ^8, V ..0. rt/zc? F's IME. ach man's nioi so is each in; I^TANT. AUi!:ation, 74 . Three merchants join in company ; D puts in stock .l-.l^.-O. for '] months, V, t'I69..18..'J. for 5 months, and 59..H'..10. for II months, they gained ;^'iJ6t..l8..0. what ich man's pju't of the gain ? Ans. D's £im..G.A—50m. Ks i€14-8..1..1^— 482803. and Fs ;^'l 14'.. 10..6;— 14707. , Three merchants join in company for 18 months: D in 500/. and a;t 5 months end took out 200/. : at lU iths end put in £'iO{). and at the end of 14 months takes £130. E puts in XLOO. and at the end of 3 months 0. more ; at 9 months he takes out /;140. but puts in 0. at the end of 12 months ; and withdraws ld9. at the of 15 months. F put in ^900. and at 6 months took iC200; and at the end of 11 months put in ^500. but !S out that and ;C100. more at the end of 13 months. y gained jC200. I desire to know each man's sliare of gain ? Ans. D >C50..T..6— 21720. E ^62..12..5];— 29S59, and F ^S7..0..0^— 14167. . D, E, and F, hold a piece of ground in common, for ch they are to pay if J6..10..6 1) puts in 23 oxen 27 E 21 oxen 35 (lays ; arid F 16 oxen 23 days. What ach man to pay of the said rent ? Ans. D ^'l3..3..1i— 624. /; /:15..11..5-- 1683. and > .. /'-<:7..l5..11— ll:i6. puts in £iQ. > and they gai lin ? ) ;^20. E £b{ i -^20 : 70 : : soo 210010(50 ALLIGATION. ALLIGATION IS EITHER MEDIAL OR ALT'2P.>TATa. ALLIGATION MEDIAL ■ S when the price and qnnnciries of several simples ar.:; given to be mixed, to iind the mean price of that mix- !uLE. As the whole comp:;sition : is to its total value : : s any pnrt of the composition : to its mean price. -• 'roof. Find the value of the whole mixture at the mean , and if it agrees with the total value of the sevpral ntities at their respective prices, the work is right. Ji. /i( I Ih ;:i- , I I'i l;;}| ' I .' ' ' ; ;'1 ■ .i' iii ■ ! , .1 'iHi 1 I'll .if 1; ! Ill tiH :r ' ■ :\ I'll!! 7^ Jllifatien, THE tutI 5SISTAN SXAMPLBS. 1. A farmer mixed 20 bushels of wheat at 5s. per U and 36 bushels of rye, at 3*. per bushel, with 40 bushij barley, at 2«. ;}cr bushel. I desire to know the worth! bushel of this mixture? As 96 :288 :; 1 : 3. 20 X 5 =. 100 36 X 3 — 108 40 X 2 = 80 Ans. Ss, 96 288 lis when ch quan t price pre In orde |1. Place lid the pi Ite at the 2. Link Iserving t Is. Agair 2. A vintner mingles 15 gallons of Canary at JSslean and i gallon, with 20 gallons at 7*. ^d. per oralion: io ^^WMWhen th sherry, at Qs. Sd. per gallon, and 24 gallons of^whitt; at 45. per gallon. What is the worth of a gallon oi mixture ? Aris. Gs. 2\d, 3. A grocer mingled 4 ao^ of sugar, at 56^. per c^ cuit. at 43.S. per civt. and 5 ctvt. at 375. per ctvt. I dei the price of 2 civt. of this mixture? Ans. iCi..8 4. A malster mingles 30 quarters of brown malt, ai per quarter, with 46 quarters of pale, at 30a> per qui and 24 quarters f high-dried ditto, at 255. per q "What is the value t ' 8 bushels of this mixture ? M^:. i:i..8..2| 5. If I mix 27 bushels of wheat, at Ss. 6d. per b with the same quantity of rye, at 45. per bushel, bushels of barley at 25. Sd. 2)er bushel, wliat is the wo a a bushel of this mixture ? Ans. 45. ?>ld P 6. A vintner mixes 20 gallons of port, at 5s. 4 per qw ; 25^. per qu xture '^ ^».y.Tl..8..2i. 55. Qd. per b' oer bushel, a fhat is the wo Ans. 45. 3|/i , at 5s. 4d. pi per gallon, 3 10 gallons of lion of this 111 Atis. OS. 3|r! ullion, of 6 a \ 10 ib. oi' a o: ture? oz. 18 dcvt. }'. tobucco at 1 EXAMPLES. 1. A vintner ivould mix four sorts of wine together, of Id. 20d. 24fd, and ^Z8d. per quart, what quantity of each ust he have to sell the mixture at 22d. ;pcr quart ? Answer. Proof. 2 of I8d.^ md. 6of20rf.=120 1 of 2trf.= 96 2of28c/.= 5Q 14 ixture be wor Ans. 16|c/ 22d. 18 — 2.720- ^-2t - 28 - r thus, Proof. 6 of \M.z=\0id' 2 of 20d.= 40 i of 24r/.= 48 4 of 28rf.= !12 14 )'J08 22^. Note. Questions in this Ride admit of a great Variety of tistve7's, according to the manner of linking thrm. 2. A grocer would mix sugar at 4rf. 6^dperlb. Whit quantity IvV/m. hn wni 'each must he take? Ans. 2to, at^d.2lb. at 6d. aruf b. at 10a'. H ■ li il $ Ml' u ' :' ' ,? ;i.f ■ ; '!lii .',*: m j ^ij- Ill' 'l! ■^1 1 5 It' \1 t I', ■ 1 t m Ml ' ''; 11 i(i''. k •( 'if 1 74f Alternation Partial, THE TUTOliljisxAN: 2. A farr shel, witI shel. H B composi 3. A disti J. per gal Ion. W or 8.V. pt' .///j,s. 4( '. A gro( 3. I desire to know how much tea, at 16.?. 145. 9^. %s, peril), will compose a mixture worth \0s. per lb ? Ans. Mb. at \Gs. 'J.lb> at iAs. 6lb. at 9s. and 4:1b. at 8.?.| 4. A farmer would mix as much barley at 3^. 6(1. bushel, rye at 4a-. y?t?r bushel, and oats at 2s. yn^T bushel, al make a mixture worth 25. Gd. per bushel. How much is ti of each sort ? Ans, 6 ofbarlnj, 6 of rye., and 30 oyow/jj 5. A grocer would mix raisins of the sun at ^i. perl with Malagas at 6d, and Smyrnas at 4fl?. per lb. I desire! know what quantity of each sort he must take to sell tli at 5d. per lb ? Ans. lib. of raisins of the sun, 1/ MnlognSt and 'Mb. of SmijrnmX 6. A tobacconist would mix tobacco at 2?. l5.6r/. andl ^d. per lb. so as tiie compound may bear a price of I'f'B,"^ 'f* ^^'^'^ per id. What tpii^iitity of rach sort must he take? M V o'"^ Ans. lib. at 2.y, 4/i. at is. 6d. and 4/^. at ls.§ ^''*'- ^^' A win( Inary at ti ALTERNATION PARTIAL i!"il,n^^ V Clin on* IS wheri the prices of all the simples, the quantity of y4«^ o;\e of them, and the mean rate, are given, to find aeveral quantities of the rest in proportion to that given Rule. Take the difference between each price, and mean rate, as before. Then, As the diiference of tliat simple, whose quantity is giv is to the rest of the differences severally ; so is the quan '^ fer gallon ; and Canary at 10^. per gallon, would nialiB™*"y "™o mixture of 60 gallons, to be worth 5s, per gallon. W quantity of each must he take ? Ans. 45 gallons qfiuhite 'voine^ 5 gallons ofFlem 5 gallons of Malaga, and 5 gallons of Ca 4. A silversmith hath four sorts of gold, viz. of 24 ca iine, of 22, 20, and 15 carets fine, would mix as mud «?ach sort together, so as to have 42o«. of 17 carets i How much must he take of each ? 2, a peri^ Ans. 4 0/ 24, 4 of 22, 4 of 20, and 30 o/*15 carets j jal pieces] 5. A druggist having some drugs of 8*. 5^. and 4s. pei Xii3.de them into 2 parcels ; one of 28/6. at 65. per Ih. other of 42/6. at Is, per lb. How much of each sort dii take for each parcel ? Ans,\2lb.o^8s. Ms. 30 of 8s. tppose he as many., half as m, \ as mam IK added wmany 3. A ger Sib. of OS, Sib. of 4s. 6 of 55. 6 of 4s. 28/6. at 6s. per lb, 4'2lb. at 7s. per lb. POSITIO:^, OR THE RULE OF FALSE iO. the he e chaise hat did Ans. Ho\ A, B,) wUvS, whif ves that ourth paj istpay.^f 6. A pe^ own, to num, sii JS a rule that by false or supposed numbers, talci . pleasure, discovers the true one required. It is ( P''*'*^*P| ea into two parts ; Single and Double. : IE tutorIsistant. Position, or the Rule qf False. 77 144 :48 114; 21 . 6s. Qs. and SINGLE POSITION Js, by using one supposed number, and workine with it [the true one, you find the real number required, by the jlowing LuLE. As the total of the errors : is to the true total : : so Ithe supposed number : to the true one required. IProof. Add the several parts of the sum together, andl irth 7s*. vtrW *^^*^* ^'^^ '^*^ *""* *' ^^ "^'^^ \ EXAMPLES. ndUllh.qfi hite winefr* ^ schoolmaster beJhg asked how many scholars he Malaga atf' '^*'**' if I had as many, half as many, and one quarter „r^iii§ ninll"'*"y tnore, I should have 88. How many had he ? would mal'na^y gallon. lions qfFlem\ lions of Cam viz. of 24 caJ mix as mucll f 17 carets of 15 carets^ ippose he had, 40 la; }nan^.. ...... 40 ihalfas many : 20 1^ as many.,.* 10 110 AsllO:88::40 40 ll|0)352l0(3a Ans. 32. 16 M 88 proofs 3. A person having about him a certain number of Por- al pieces, said, if the third, fourth, and sixth of them 55. and 4s. p^^ added together they would make 64. I desire to know It 6s. pcf l^^\ f each sort dii Is, per lb OF FALSE, numbers, talc( uired. w many he had ? Ans. 72. 3. A gentleman bought a chaise, horse, and harness for iO. the horse came to twice the price of the harness, and e chaise to twice the price of the horse and harness, hat did he give for each ? Ans. Horse £l3..6..8. Harness £6..13..4', Chaise £\0, 4-. A, B, and C, being determined to buy a quantity of oQvS, which would cost them £l20. agreed among thera- ves that B should have a third part more than A, and C burth part more than B. I desire to know what each man St pay ? Ans. /4 £ 30. B £ 40. and C £ 60. 5. A person delivered to another a fiutn of money un- own, to receive interest for the same, at 6 per cent, per nuniy simple interest, and at the end often years received It'iTi " P'"*''*cipal and interest £ii00. What was the sum lent ? ^ wf/is. £187.,10.*0. \\ ,11 -!!ti mm ; ! fli 1 1 iNii: * llii iiliji II, ; i I ■ 78 Position, orths Rule ifjt" False, the TuroMssisTA? DOUBLE POSITION i. Thre( I, 1 r.m [i IL : and igj of eaci ^. l>, E; isputing i uuld ; 1> J lore ; and ether. I Is, by making use of two supposed numbers, and if prove false (as it generally happens) they are, with tli| errors, to be thus ordered : Rur.E 1. Place each error agalas its respective positii 2. Multiply them cross ways, S. If the errors are alike, /. e. both greater or both than the given number, take their difference f'-r a divli and the difference of their product for a divuu > '. B unlike, take their sum for a div or, and the sum ofi!^ . product for a dividend, the qujt:c..t will be the answJ ^' S^! *^ ^ ■dies, and EXAMPLES. I"'./""" nut Ji we w 1. A, B, and C, would divide £200 betweon themj ""^"y »! that li may have £Q. more than A, and C £S. movel^^y * ii, how much must each have ? Suppose A had 40 I'/ien suppose A had 50 then B had 46 then B must have 56 Mnd C 64 mid C 6h sup. UO too little by 60 errors, 40 60 50*30 60 80 110 too little i 60 A 66 B 74 C "S recei paid ir The Par fie infrinsi ng; t)u |iou£ occa& They kej ivres, soJsl SOOO 1200 — riOO 30 divisor. fij0)l80|0 -' 2Q0proo/: GO Ans./or A. ^ 2. A man had 2 silver cups of unequal weight, h _ «ne cover to both, of 5 oz. now if the cover is put oi ^, ^t par^ lesser cup, it w ill double the weight of the geater and set on the greater cup, it will be thrice as heavy a lesser cup. What is the weight of each cup ? Ans, 3 ounces lesser, 4 greai t 3. A gentleman bought a house with a garden, i horse in the stable, for £500. now he paid 4 times the of the horse for the garden, and 5 times the price garden for the house. What was the value of the garden and horse, separately ? , . . .., Ans, Horse £20. Garden £80. House £1 Rule. 'reach sul Rule. the sterl :HE TUrOBliSSISTANT, JEwchange, 79 ?r8, and if 1 are, with till tectivc posltij iter or both ce f ' r a divii he sum of il e the answer! tweon thewJ £8. morel r?50 )e 56 70 too liVk I 60 A 66 B 74. C 200 proof. 4. Three persons discoursing concerning their ages ; say* h, I *vn ;K) years ol" age ; «ays K, 1 am as old as 11, and ^ [t'L : and says L, I ivm as old as you both. What was the Ige of each pv rson ? An^. II 'M, K 50, and L 80. . 1 ;:. U, E, and F played tit cards, ataked 324' crowns ; but lisputing abijut^ the tiick.^, each man took as many as he Luld; l> got a certuiii lui.uber ; E as many as D, and 15 lore ; and V got a fifth part of both their suras added to- luther. How many did each get ? Am. D 127 ^, F. \\2 |, and F 54. 6. A gentleman going into a garden, meets with some klies, and says to them Good morning to you '0 fair maids 1 kir, you mistake, answered one of them, "^ c are not 10 ;. lut if we were twice as maviy more as we wc should be |s many above 10 as we are now under. 1 many were key ? Ans, 5. EXCHANGE S receiving money in one country for the same value paid in another. The Par of Exchange is always fixed and certain, it being e intrinsic value of foreign mone}', compared with ster- ng ; t)ut the' Course of Exchange rises and falls upon yj|» ■ious occaibions. J. FRANCE. il weight, ha over is put « the geater ice as heavy a cup? lesser f 4 greal h a garden, d 4 times the-, les the price f'^^'^cn alue of the They keep their accounts at Paris, Lyons, and Rouen, m vres, sols, end deniers, and exchange by thj crown==4«. d, at par. Note. 12 deniers make 1 soT. 20 sols \ livre, 2 livres I croiin. To change French into Sterling. Rule. As 1 crown : is to the given rate : : so is the sum : to the sterling required. ITo change SierHng into French. Rule. As the rate of exchange : is to one crown : i le „^ . the sterling supn; to the Fpendi required* ii: t' J •4? ■ '"1 '4 'I' Mi, 1 !Ti \ ^i hi i!i .^; I' m .>^> ^% v<^.^^-^^ IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I ■^■2.8 12.5 ■^ i^ 12.2 Ui lis u 140 1^ t p^ll'-^l'-^ .- . < 6" ► \ ^^ J^ 0> /. ^'^*' ^ '# gross 1 ducat. Rule. The same as before. 9. How much - sterling money may a person receive i» ndon, ifhepaysin Genoa 976 dollars, at53e^.j7(7r dollar? Ms. je215..10..8. 10. A factor has sold goods at Florence, for 250 duca- 118, at 54ed each, what is the value in pounds sterling ? jihs. JC56..5..0. 11. If 275 ducats, at 4«. 5d. each, be remitted from Ve- ;e to Loudon, what is the value in pounds sterling ? jins, £G0,.U..7, 12. A gentleman travelling would exchange ^60..14k.7« rling for Venice ducats at 4$. 5d, each, how many must receive? Ans, 27 5, IV, PORTUGAL. They keep .their accounts at Oporto and Lisbon, in reai» d exchange on the milrea=s6«. Sj^d, at par. Note. 1000 reas make 1 milrea. Rule. The same as with France. j^ , EXAMPLES. 13. A gentleman being desirous to remit to his correspon* pi nt in London 2750 milreas, exchange at 6s, 5d, ^ermiU 6i I, how much sterling will he be the creditor for m Lon* n? Ans. £SS2,.5,>10, 14. A merchant in Operto remits to London 4366 mil- is, and 183 reas, at 5s. 5d, f exchange per milrea, how yfich sterling must be paid in London for this remittance ? ^«*.;fn93..17..6i, 0375. It' 5'<1 fi i|j :i i I i IS ! ii I I! I 1 i l' I I ' ! i ill, • U 11' i t jli 'I; I m ii 82 Ejcchangc, THE TUTOU's ilSTAN- 15. If I pay a bill in London of ;^'ll9P>..17../)J,()3li8792g what must 1 draw for on my correspondent at Lisbon e gi'oats/ To am [OTB. 1 dtfferei \railyfr change at 5s. Sd. | 'per milrea ? Am. 4366 mitreas, 183 na;. F. HOLLAND, FLANDERS, AND GEllMANY. They keep their accounts at Antwerp, Amsterdam, Bn sels, Rotterdam, and Hamburgh ; some in pounds, shilli and pe? :e, as in England ; others in guilders, stivers, #ijle. A pennings ; and exchange with us in our pound, at 33i. ed: :so flemish, at par. Note. 8 pennings make 1 groat. ^2. groats, or \G pennings \ stiver. .: 20 stivers \ guilder or florin ALSO }2 groats or 6 siix)ers make 1 sehelling, 20 schellings or 6 guilders... 1 pound. To change Flemish into Sterling. KuLE. As the given rate : is to 1 poimd : : s3 is the I mish sum : to the sterling required. ULE. A is the 0. Chan^ !■: florin . Chan iiey, agi 2. A g< will >ie •^ To chanst^ Sterling into Flemishm Rule. As £1 sterling : is to the given rate : : so is I' ^^^\ sterling given : to the Flemish sought. EXAMPLES. 16, Remitted from London to Amsterdam a bill £75if..\0. sterling, how many pounds Flemish is the the exchange at 33*. 6d. Flemish per ^^ouid sterling ? Jns. £i ':63 .. l.» .. -'). FlemiM If 50 E 17. A merchant at Rotterdam remits £^ \A5..9\ y Dutch mish to be {^id in London, how much sterii ^ money n he draw for, the exchange being at 33*. 6d. Flemish pound sterling ? Ans. £75^..lO. 18, If 1 pay in London :e852..12..6. sterling how ni l^SOlb. guilders must I draw for at Amsterdam, exchange a schel. 4^ groats Flemish per pound sterling? Ans, 8792 guild. 13 stiv. 14>1 pennini^ 19. Wh^tmust I draw for at London, if I pay at An^ r/? t 3..6..6 I MPJRj Ifl2yi at Paris y lb. at If 95/5, isli are ( HE TU Toil's i9f5..17..G|,03; nt at Lisbon f (reaSj 183 nu; lEHMANY. mstcrdam, Bn pounds, shill 8792 guild. 13 stiv. 14-^ pennings, exchange at 34 scliel. rroatsyvtr pound sterling? Ans, jC852.. 12.. 6. Toconxwrt Bank Money into current, and the contrary, oTB. The Bank M' ikcy u ivorih morethnn the Current. difference between one and the other i- called ?iglo, and is 'rally Jrom 3 to 6 per cent, in favour q/the bank. ,To change Dank into Current Money. iers, stivers, a ule. As 100 guilders Bank: is to ICO with the agio ound, at 33i. )at. ver, llder or florin, chelling. found. ling. d : : S3 is the lish, rate : : so is tSTANT, Excharifre* 83 'to" ed: : so is the Bank given : to the cuirent required. To change Current Money into Bank, ULE. As 100 with the agio added: is to 100 Bank is the current money given : to the Bank required. 0. Change 794- guilders, lo stivers, current money, into il; florins, agio 4 1 per cent'. Am. 761 gadders, 8 stivers, \i\^^ pennings, . Change 761 guilders, 9 stivers Bank, into Current ley, agio 4| fer cent. Ans, 794- guilders, 15 stivers, 4^^^ pennings, VI, IRELAND. 1. A gentleman remits to Ireland £575.. 15. sterling, twilDie receive there, the exchange being at 10;7er ^ Ans. £633..C..6. . What must be paid in London for a remittance o ^^..^..6 Irish, exchange at iO per cent? Ans. 575,15, EXAMPLES. .„ MPARISONOF WEIGHTS AND MEASURES. terdam a bill emish is the m id sterling ? * . , ) .. '). Flemhl If 50 Dutch pence be worth Q5 French pence, now y Dutch pence are equal to 350 French pence? Ans, 269if . If 12 yards at London make 8 ells at Paris, how many at Paris will make 64 yards at London ? Ans. 42^2 • P \.15..9 rh ^ money: Qd, Flemish is./*754..10.. , exchange a .g? V. \^\penmn[ I pay at An^ terling how ni If 30/i. at London make 28/c». at Amsterdam, how y lb. at London will be equal to 350/6. at Amsterdam ? Ans. 375. 11*95/6. Flemish make 100/6. English, how mapy lb, iish are equal to 275 lb, Flemisli ? Aits, 289i^f . i\ 1 ' 1' ' 1 ' 1 l' : i 1 ■til if |/l |H| i i 1 1 . •:li S \: II I'l H V. i I il 'f'll i^ . il i: : 1 ^■'j m>iU n ti •4 Proportion, the tut#istan CONJOINED PROPORTION IS when the coin, weight, or measures of several coui» jf jo are compared in the same question : or it is ^ii^^^inglKrnsterd Iher a variety of proportions. When.it is r; quired tu find how many of the first 8i coin, weight, or measure, mentioned in the questioi equal to a given quantity of the last. Rule. Place the numbers alternately, beginning left hand, and let the last number stand on the left then multiply the first row continually for a dividem the second for a divisor. Proof. By as many single Rules of three as the qi requires. e arc ei If 10 'i at A tzick aj EXAMPLES. 1. If 20/&. at London make 23/d. at Antwerp, and at Antwerp make 180/6. at Leghorn, how many /&. all don are equal to 72/6. at Leghorn ? Right 23 180 20 X 155 X 72=223200 23 X 180 = 4U0)223200(53| Left 20 155 72 2. If 12/6. at London make 10/6. Amsterdam, at Amsterdam 120/6. at Thoulouse, how, many lb, ; don is equal to 40/6. at Thoulouse? Ans. 3. If 140 braces at Venice are equal to 156 braces horn, and 7 braces at Leghorn equal to 4 ells Engli many braces at Venice are equal to 16 ells English Ans. 2 'I 4. If 40/6. at London make 36/6. at Amsterdi 90/6. at Amsterdam make 1 16/6. at Dantzick, how at London is equal to 130/6. at Dantzick ? Ans. 112 When it is required to find how many of the lasi coin, weight, or measure, mentioned in the quci equal to the quantity of the first. Rule, t'lace the numbers alternately, beginni left hand, and let the last number stand on the rig then multiply the first row for a divisor, and th for a dividend. ' ARITI when th py by th£ pers: As [by the C( 1. by th TE. iV '(il Progr ttvo niici the ox vo midMt If!', the wn the m mil be eg f>/ 'utant 'mble nf] lAritijinel 1. The 2. The 3. Th( 4. The] 5. The three oj THE TUTOf isTANT. ^ Progression. 85 ON I ;,v.,,^, EXAMPLES. -.,.,.'. ..iy ■ '^ :.* •several cou*, if 12 /5. at London make 10 Uj. at Amsterdam, 100 Ih, it is linking tj^msterdam, 1 20 /6. at Thoulouse, how many lb. at Thou* ,e are equal to 40/6. at London ? Ans. 40 //n jf the first 8(B; if iq /6. at London make 36 /6. at Amsterdam, and the questionjj. at Amst.Tdam 1 lb /6. at Dantzick, how many lb. at Uick are equal to 122 lb, at London? , beginning a| , . Ans. 141 ^fi§. i on the left 1 or a dividend^ "^^i""*— • .-.:* ^ee as the qui PROGRESSION i ' CONSISTS OP TWO PART8» ' '"' ARITHMETICAL AND GEOMETRICAL. ;wmany/6^'il ARITHMETICAL PROGRESSION when the rank of numbers increase or decrease regular- ly by the continual adding or subtracting of the equal jbers: As 1, 2, 3. 4, 5, 6, are in Arithmetical Progres- =223200 Iby the continual increasing or adding of one; 11, 9, 7, 0)2232OO(531 j^ ^ty the continual decreasing or subtraciing of two. ■[)TE. IVhen any even number qflernu dlff": r by Arilh- Amsterdam, J^j^ progression^ the sum of the two extnmius will be equal )W,many lb. Mgtvoo middle numbers, or any tvoo nii ans equally di-int ^^^'mthe extremes: as 2, 4 6, ^, 10, i2, xvbere ^H-H, o 156 braces ■jijj „j/flf,y/g numbers flr. =12+J=I t, the txao ext,eines 4 ells Englil^.!,, the tw mean,= 14. Ans. 2W"'" '^^ number of terms are odd, th^ double of the middle at Amsterdiil"''*''' ^"^ equal to the two extremes; r-r ffiny twomeHrij^ Intzick how eK/ '^' ''^'*''. /'O^ ^^'^ "> ddli'term ; as 1, 2, ii, t, !},.uifhere ck? * fibteof',=5+l='2+i'=3. i4n5. Il2l-^'''*t^'^"^'^^cai Progression five things arc to be observ- ,nv of the la8ll^* din the quel 1* The first term; better expressed thusy F. ^ ' 2. The last term, L. , ely beginnini 3. The number of terms, N. d on the r\Jlk ^' ^ ^^^ equal ditidrence, -- D. isor and th« ^' ^'^*^ ^^^^ of all the terms ------ S. * ■■ three of vohich being given the other tvoo m ly be found. ■ I ■'. 5ll.,f , Ifff ^■ I! ' ii ' I; ; I ' \; 1 . 1 t\ ^> I I 86 Progression. the tutr _!ISTAN The first, second, and third terms given, to find the fi KuLE. Multiply the sum of the two txtrenics by halftiXhe fifBl ruDiber ofti.'rnis, or multiply half the sum of the twotlRuLR. ] tremes by the whole number of terms, the product is the tal of all the terms: or thuSj 1. F. L. N. arc given to find S. *• N r divide third : m.F.i L EXAMPLES. I. How many strokes does the hammer of a clock sti « )^ ^ei Li 1 2 hours ? / 1 9, A man buys 17 yartls of cloth, and gave for thef yard 2«. and for the last 10«. what did the 1 < yards aiuo to? ... -^n*' i^5u.^>..0 3. If 100 e^gs were placed in a right line, exactly ^. y asuad r from one another, and the first a yard from a ba et, w at length of ground does that man go wh« gathers gpcond the^e ' CO eggs singly, returning with every egg to the b: «t " put it in? ylus, 5 miles, IHOO ^ad 'i he first secono^and third teim given to find the foi Rule. From the second subtract the first, the remain dlvic^ed by the third less one, gives th« fourth : orthuSi II. F. L. N. are given to find D.. ■ L-F . , •: '..'■■ 5S 7. A ma e younge neased o Rule. A i product IV. L. N K-1 4-;^lO— 1 i: flXAMPLCS. 4. A man had eight sons, the youngest was 4 year tnd the eldest 32, tbey increase in Arithmetic^ Piogt •n ; what was the common difierence of their ages ? A n * 32—4=28, then 2^^^-;— 1= . common diferem m* differ' 5. A man is to travel from London to a certain plai , ^^^^r 12 dsiys, and to go but 3 miles the first, day increasinge ^j^^ ^^^^ 4ay by an equal excess, so that the last day's journey Rule I be 68 miles, what is the daily increase, and how many i ^j subtra distant is that place from London ? Ans. 5 daily inert , ^j^j^^ , Therefore, as three miles is ihe first dat^^sjournej/, y, j^ D 3+5=8 the second day, ' a 8 + A = 1 3 the third day, S^c. ^^ The tvhore distance it 366 miles, ^ i. A mai the coun and the STANT. Progressian, S7 to find the filL ... nits by half tlThe firtit, second, and fourth terms giren, to find the third. of the twntlRuLR. From the second subtract the firbt, the remaitf roduct is their divide by the fourth^ and to the quotient ftdd 1, gives third : or thus, '/n- of a clock stij gave for the 1 , yards amo Atts* .it>&»«-"' e, exactly ^t y ^ard from a b o wh« gathers f egg to the b: es, 1300 ^.ardl o find the fou ^j** rst, the remain urth^ orihus, lUt F. L. D. are given to fin L— F IXAMPLES. Id, A person travelling into the country, went 3 miles thr. St day, and increased every day 5 miles, till at last list int 68 miles in one day, how many days did he travel ? 58— 3=*55. then 55-5-5=1 1. 11+1=^2 /Ac Jna, 7. A man being asked how many sons he had, said thut youngest was 4 years old and the oldest 32 ; and that he eased one in his family every 4 years, how may had he ? Afis, 8. le second, third, and fourth terms given, to find the first. Rule. Multiply the fourth by iha third made less by f, e product subtracted from the second givet tlie imi ; or iV. L. N. D. are given ta find F. L— D+N— 1=F. > ■■6. EXAMPLES. $, A man in 1$ days went from London to a certain town - the country, every day's journey increasing the former by and the last he went was 46 miles, what was the first ? t was 4 year Ans.iO miles. meticfd Frogi ^^^lO— 1=36, then 46— 36=10 M^/ri daysjourneif, leir ages • A\ 5^ ^ ^^^^ ta'.■% gives the second : or thus, VI. F. N. D. are given to find Lr N D— DtF=L. . EXAMPLE. f r-jtv* in Arith th an un mber ofh cA Geomt der thenu 12 3 11. What is the last number of an Anttinrctical Pro^g* ^* g* feion, beginning at 6, and continuing by the increase ol * * ' to 20 places? jlns, 15^ hi if the 20X8—8=152, then 152+6=158 the last number, %m4he rat $SIITAN1 ■.A Kote As COi 'I t'.- tT*» r «■; -^-^ ■.Jb. V . h »,> •.,'-#♦ fri (tiff GEOMETRICAL PROGRESSION f-. ■ IS the increasing or decreasing of any rank of numbers some common ratio ; that is, by the coHtinual multi cation or division of some equal number : as 2, 4, 8, increase by the multiplier 2, and 16) 8, 4, % decrease, die divisor 2. NcfTE. When any number of terms is continued in geo trical Progression^ the product of the two extremes tc;/l equal to any two means, equally distant from the extren us 2 : 4, 8, J 6, 32, 64, Vihere 64X2 «re=4X32, 8X16=128. 0, 1, 2, 1, 2» 4, i When th lices mad (f terms y the sea Add any reemth tl As in i Geome M TUTO«|„„^N^. --I* Progression. 89 « WA«i the number qfter0u are odd ; the middle term mulii" ^ied into itself will be equal to the two extremes, «r ant/ two leans, equaUy distant from the mean, as 2, 4, 8, 16, 32, , J ^' ^* ^» *» ^» ^» ^«^«^^*' '' metical rro^2, 4, 8, 16, 32, 64. Numbers in geometrical proportion, e mcrease oH ./Ins* loStoy^ iflhejp,rst term in geometrical proportion be different 'ast number* mm^e ratio, the indices, must begin with a cypher. \'.< : V'"'' im what it Ans, £8,' sf payment. the second, tuct oi the thill ided to the f j'-i*>! r u iiP:^' ' 0, 1, 2, 3, 4, 5, 6, Indices. 1, 2» 4, 8, 16, 32, 64. Numbers in geometrical proportion. iSION When the indices begin with a cypher, the sum of the lices made choice of must always be one less than the num' . tf terms given in the question ; for I in the indices is nk of numbers ^ ^^^ ^^^^^^ ^ ^^^ 2 over the third, &c. o»tmual multt as 2, 4, 8, ^jj gj^^ ^^^ g^ f^g indices together j and that sum will 2, decrease, reewith the product of their respective terms, ntinued in geo ^^ ,•„ the first table of indices 2+5 extremes t<;?fl geometrical proportion 4 X 32=^128 om the extrenf «rc=4X32, li T/^f n Me second 12 2 + 4=6 4 X t6«6i A • 11 ii' liJ If 4 ill I ii 1 i' ■t 1 ' 1 M M;l ,i ' 1' I t 111. *■ Am! ffS^Porgression. THS TUTOfilflnSlAif In any geometrical progression proceeding from unii tlie ratio being known, to find itny remote term, withi producing all the intermediate terms. liuLB. Find what figure of the indices added togei would give the exponent of the term wanted ; then tiply the numbers standing under such exponent into ei other, and it will given the term required. Note. When ihe exponent 1 stands over the second tei the number of exponent must be 1 less than the numbei terms. EXAMPLES. nii 4. A gei xecutors, 1. A man agrees for 12 peaches, to pay only the pi '^ '?'*' ^*^ ef the last, reckoning a farthing for the first, and a hi 'i»<^h more penny for the second, &c. doubling the price to thel 8*tyoun| what must he give for them ? • ■ Ans, £2..*2..8 "ill**"; 16= 4 * The firsi 0, 1, 2, 3, 4, Exponents, '' 16=4* le sum of 1, 2, 4, 8, 16, iVo. of terms ' «^^^- J 256= 8 «* ^^^^ » ) the quo! 8=3 For4H+3=U, No, of terms less, 1 5. A sui It to hav in; what 60, 1 !t3740 quired. f !•. 4)2048=11 No, ol 12)512 . 2!0) l|2..8 Jb^a.^oO 5. A sei ) serye hi farthing icond, 4ti 0. I, 1, 4, 1^ +4 + 3= 2. A country gentleman going to a fair to buy some o meets with a person who had 23 ; he demanded the ] of them, was answered j^ 16 a piece: the gentleman him;^15 apiece, and he would buy all ; the other tells it could not be taken; but if he would give what the ox would come to, at a farthing for the first and doul farthing it tcTthe last, he should have all. What was the prii ere four the oxen ? Am. £^369..l In any geometrical progression not proceediLf; unity, the ratio been given, to find any remidte term, lay, and g out producing all the intennediale terms. . ing to dc RiTLE. Proceed as in the last, only observe that < ear j whi prorluct must be divideil by the trist term. 6. A mi HE TUTOalflflSlANT.^ ing from unit] ) term, witho added togetl ited; then lonent into eil he lecond teil n the number] Progression, 91 5. A sum of money is to be divided among 8 persons, the 'nt to have jfi20, the second £60, and so in triple propor- on ; what will the last have ? , Ans. 1^43740. 1, 2, S, 540 xSiO ^^^gan ^^^^ 14510X60 0,60, 180,540, 20 , ., . i *^ 20 Bi8740 ,- ' 3+S+l— 7, o»tf /tf« than the number qf terms. = 4 = 4' = 8 = 3 4. A gentleman dying left nine sons, to whom and to his ixecutors, he bequeathed his estate in manner following : only the piV*^ ^^^ executors iC50. his youngest ;Son was to have as Lst and an '^^^ more as the executors, and each son to exceed the Drice to the I ^^^ younger by as much more ; what was the eldest son's ns. £2..*2..8 "'*'**" ** ''''*• ^25600. The first term, ratio, and number of terms given, to find le sum of all the terms. Rule. Find the last term as before, then subtract the rst from it, and divide the remainder by the ratio, less 1 ; > the quotent of which add tlie greater, gives the sum cqoired. '" -v ;=n No,0l .:'.>!'"" EXAMPLES. .,,!.,;p, ^■'.-'i,.:. • ^ 5. A sentnt skilled in numbers agreed with a gentleman ) serye him twelve months, provided he would give him farthing for his first month's service, a penny for the icond,.4(/. for the third, Sfc, what did his wages amount )? • Ans. £5S25S-'5\. 256x256=^5536 then 65536x64=4194304 0, 1, 2, 8, 4, 4194304—1 ,„_, 1, 4, 16, 64, 256. IZj- =1398 191, then i- 4, 4-3=3:11. No. of terms less 1, 2..8 ^..8 to buy some o manded thef gentleman ;he other tells me what the first and doul IS. 4'4369,.l proceeding eaibte term, •bserve that 1. '-... 1 398101 +4194304-=5592405/ar/A/«^*. 6. A man bought a horse, and by agreement was to give farthing for the first nail, three for the second, &c. there it was the prifere four shoes, .and in each shoe 8 nails; what was the orth of the horse ? y^ws. 169651 14681693..13..4. 7. A certain person married his daughter on New-year's ay, and gave her husband Is. towards her portion, prom- ing to double it on the first day of every month for 1 ear ; what was her portion? Ms* iS204..15«. )1 1 .! I ; V i* ;< rl I i'l 1^ I '1 H: i>, 1 1 in : . H'':' ' I : \H' tin iH I 2 • i! I 'I I' fjt J^ermuktUott* TUB TUTOR SISTANTi 8. A laceman, well Tersed in> numbers, agreed \rith gentleman to sell him 22 yards of rich gold brocaded lac for 2 pins the first yard, 6 pips the second, &c, in treb proportion ; I desire to know what he sold the lace for^ the pins were valued at 100 for a farthing ; also what tl laceman got or lost by the sale thereof, supposing the ta ftood him in £7» per yard ? Ans, The lace sold for j€326886..0.^. Gaiw £326732..0«9. *%:i "^' '.'> ■ . , \ -ii; '. li TU •r' ,-.■! PERMUTATION -w. a^^^ FRACI ten wit IS the changing or varying the order of things. Rule. Multiply all the given terms one into anotli aitd the last product will be the number of changes reqi «d» . .' 1. How many changes may be rung upon 12 bells; ai liow long would they be ringing but once over, supposi 40 changes might be rung in 1 minute, and the year to co tain 36S days, 6 hours? , % v - ^ «« 1 X 2x 3 x4.»< 5x6x7x8x9x10x11x12=479001600 . ^y, . changes^ '03hich^\0=^l9O0\60 minutes ; ad he figure ref/«ce';'.v agreed with )rocaded lac . ■■'[ 'V::.i.r &c. in treb * * ; a: ;: ■■■^'- i\ 'i^ ^^" ^''i- ,- ■''■■'\''' '* le lace forj also what t ..'.■■: 1.,. )Osing the !ai ■ ' 'THE f^-'- ^ f '<=•.'•'' ^_'^ »26886..0.A ••; ';*.'■;• > .. ■ . ; i\ } ■■♦> >»ii' . ./'■ '•' ••'• ^^"^t-' ■••*'■ I6732..0-9, • \r' ; I , t . . 1 • :• M 1 . 1 >■ vi.-i ! ' •:.•*' ' ! f; TUTOR'S ASSISTANT. ,1. i ' ."' *\' "« . - * » " * flings. le into atiotli! changes requl . V » ;■■( i >i >■* PART II. ?).■ .; f, 9^ • '^'t ' j-i > .i, , I, , VULGAR FHACTIONS. ■Kr I n 12 bells ; al }ver, supposif the year to c« FRACTION is a part or parts of an unit, and writ- »»;-.?. iL ten with two figures, with a line between them as =4790016001' f» *c. minutes • and '^^ figure above the line is called the numerator, and the 5 dm/s 6 hou er one the denominator ; which shews how many parts ' unit is divided into ; and the numerator shews ho.T f()r the conve jy of those parts are meant by the fraction. ar who toldl ^^^^ ^^^ ^^^^ ^^^^^ of valgar fractions : proper f improper^ ttime he shoi Pow»«^» and mixed, viz, ,^^ , . >v ^ , .» .,ii . or so long as . A proper fraction is when the numerator is less ionsliesideshi a the denominator, as |, f, ^, /p, 4t7» ^c* ^ ^ ^^Vk ^ w' ^^ IMPROPER FRACTION is when the numeiator 11 £ 5, to wnjq^^l j^ ^^ greater than the denominator, as f , I, \\, iholar stay v|^^^ » » i 3»4»i8* M. 5040 rffl^ i . A COMPOUND FRACTION is the fraction of a fraction, known by the word of as |, o/f, of^, ofj\y ofj%i &c. A MIXED NUMBER or FRACTJdt^^f IS composcdofa le number and fraction, 8f , ITj, 8J4, &c. ii/: i^i I ;«'■, '. ir i < ! J i 1 yJC 4. Reduce j^t f » |, |, to a common denominator. Viirit IflAO.. JULQ- 3 4_o. _c„4 J 4. Reduce |» |, f , and | to a common denominator, 8" tuLE. 1V1 he fractio ' numeral Note. Ti deHomino Facit f p- 5(? » 8 4 (T> no 8"! '•> J ft 8-J 6. Reduce |, f r f» and | to a common denominator 2. To reduce a vulgar fraction to its lowest terms. Rui E. Find a common measure by dividing the 1 term by the upper, and that divisor by tlie remainder foil tng, till nothing remain ; the Iftst div;sor is the conmion n 9* Redu sure ; then divide both parts of the fraction by the comi measure, and the quotient will give the fraction cequl !0. Redu Note. If the common measure happens to be one, thej \rf' o^j Hon is already in its lowest term ; and when a fraction i lo* i? g,i,. cyphers at the right handy it may be abbreviated by cut them o^, a« ill* IX AM PL KS. 7. Reduce f | to its lowest terms.. 24)32(1 — then 8)|^)=| Faeit^ 9$m, measure 8)24(3 ;»■ 3. Reduc lb i. Redu< 5. Redu( 6. Redu( Ik Redu( 8. Redu( 4, Tore Rule. 4. Redu 5, Toi RvLE. : and all tcduce t PHE TUTOS CTIONS. nominator. i the denomiij and all thed he several g'«| net by the ew numeratd naton it ^f , and it) sisTANT. Reduction of Vulgar Fractions. ^5 Reduce ,Vs Reduce f{ff Reduce ^of Reduce ||f to its lowest terms, to its lowest terms, to its lowest terms, to its lowest terms. Reduce ^i\^ to its lowest terms. FacU /g" Facit f^f Facit A. Facit f|, Facit |. To reduce a mixed number to an improper fraction, luLE. Multiply the whole number by the denominator [he fraction, and to the product add the numerator for • ' numeratori which place over the denominator. Note. To express "whole number^fracUon-iuaifit set 1 fir \dmoriiinotQr given, El^AMPLES. rM.=if, and orainator. trit ^t. i«. -'" I denominator » s I C 2.ajo '■» T. Tff I' ' 3 3 6 enominator. 3. Reduce ISftoan improper fraction. Facit ^*, lbX7+3=129 netu numeta^or,=^^^, 4). Reduce 56^ | to an improper fraction. Faeit *l\', 5. Reduce 183/^ to an improper fraction. Facit 'f f % 6. Reduce 13f to an improper fraction. Facit ''-f, i>» iVA» iV/rl?* Reduce 27f to an improper fraction. /Wsf a is, denominator 8. Reduce 514? y\ to an improper fraction. Facit oaf", 4, To reduce an improper fraction to its proper terms, RuLS. Divide the upper term by tlie lower. i (5 nun J PS 14 (7> 8 I '» «•» denominator 5 4J) 13 9 _ 1 ■» 2 I 6 » a I B • vest terms. en a fraction aviated by cut EXAMPLES. ividing the lo remainder foil :he conmion n 9. Reduce ^^* t« its proper terms, n by the comi 12y-r7 18| fractiim r^qui tO. Reduce ^H* to its proper terms. , tl. Reduce ^sls to its proper terms. beone,theJ f^. Reduce ej iq its proper terras. 13. Reduce a^' to its proper terms, ii. Reduce »£|» to its proper terms. Facri 18f* FacU .56 if, j'ncit 18:^3*^, .Fac// I3f, Facit 27f , /"««> 5l4-tV. Mit^ 5, 7\> reduce a compound fraction to « swgt^ o«^. RvLE. Multiply all the numerators for a new numera- and all the denominators for a new denominator. Icduce the new fraction to its lowest term, by rule 2, :!t: C 'I 1 1 :;H :{.! .; L;;! i ■: \t i! il" , , ^5, ReduM 25. Reduce » of | of | to a single fraction. |20X12=1 Facit l^ . ^= , 20 ^^'^w^^^ '<^ '^^ ^''«'''*' *m»i =*• Reduce 26. Reduce | of ^ of f-^ to a single fraction. m. Reduce FaciV ^J!|=%.weight. 27. Reduce H o^ if o^li *o ^ '^^"g^® traction. §. Reduce '. Facit Un^l 28. Reduce | of | of ,?„ to a single fraction. Wo reduce! Facit H|= iVl'"^ yalue, 29. Reduce | of | off to a single fraction. maction. Fneit if|=TVl"i'E. As 1 SO. Reduce f of | of j% to a single fraction. l)niinator : Facit ^\%^^\mAeviomin .6. To reduce fractions of one denomination to thefradr ' ^ o/* anothery but greater, retaining the same value. i . Reduce RuLS. Reduce the given fraction to a compound oi tor shall I by comparing it with all the denominations between it, ' "- ^ that dei'Oinination which you would reduce it to ; then duce that compound fraction to a single one. ' ' ' 'i-'U :3? ^#r. EXAMPLES. V V.-.'i-'^ >\ 31. Reduce } of a penny to the fraction of a pound. . Reduce tor shall . Reduce tor shall 1 reduce Facit ;, of ,'. of , = \ ne Value, 32. Reduce ] of a penny to the fraction of a pound. Facit 10' S3. Reduce f of a dwt. to the fraction of a lb. troy. Faoit ,- ,*T7r 34'. Reduce 4 of a lb. avoirdupoise to the fraction of a c Facit ,ii-^. 7. To reduce fractions of one denomination to the fract of another, but less, retaining the same^value. Rule. Muhiply the numerator hy the parts contain in the several denominations bctwem it, and tliai you w rediAcc it to, for a new nuuierator, aud place it over giv(ui denomi»iator. t v. Reduce the new fraction to its lowest terms. /i ^ mired, LE. As 1 imerator ' to its ni (It Reduce lator shaJ ' Reduce lator sh« Reduce lator sha HE TUToiisTANT. Reducliott of Vulgar Fractions, OL EXAMPLES. - • V |5, Reduce ,-|5^ of a pound to the fraction of a penny. Facit |. 20X12=1680 1^1^ reducecHo its lowest term— I, west terms Jf' ^^^^ce 9^^ of a pound to the fraction of a pen:iy. ~ Facit {. . Reduce j^^j df a pound troy, to the fraction of a 'acit Hl=*y-weight. Facit |. ction. n. Reduce yf ^ of a etot. to the fraction of a lb. icit Uil—M . ^acit *^. on. Wo reduce Fractions of one Denomination to another of the icit 111= iVb'"^ Valusy having the Numerator given of the required n. vaction. rteit HI=tV1''^^' As the numerator of the given fraction : is to its ;ion. minator : : so is the numerator of its intended fraction; ^cit H''A=fi\# denominator. ^30 6 S i t EXAMPLES. n to thefrac\\ mine value. . Reduce ^ to a fraction of the same ralue, whose nu- , compound oAtor shall be 12. ^a* 2 : 3 : : 12 : 18. Facit ||. between it, al. Reduce 4 to a fraction of the same value, whose nu- s it to ; then §t9r shall be 25. Facit ^f . . Reduce -f to a fraction of the same value, whose nur itor shall be 4<7. • ^ , 47 '■'•-•■" -.V--: ••- Facit 65f of a pound. Wo reduce Frattionsofone Denomination to another of the ,'., of /--==: *ie Valuey having the Denominator given of the Fractions of a pound, lairefl?. Facit i 5. Ille. As the denominator of the given fraction : is t* 'a lb. troy, pnaerator : : so is the denominator of the intended frac- traction ot a c1 Facit .-^^^i. I EXAMPLES. I ' to the fractm Reduce ^ to a fraction of the same value, whose de- ator shall be 18. ^.s 3 : 2 : : 18 : 12. Facit \l. Reduce 4 to a fraction of the same value, whose de- ator shall be 35. Facit |;. lace"^ it over i Reduce 4 to a fraction of the same value \\hose de- ator shall be 65 4. 47 ?* -?-■ .: V:- Facit to Its numerator. m tme,value. purts cuutaitil il thai you w( inns. ! '.I if!' ;i ■I £1 N : . !' 'i I : t I ■ ■ Si ! il .'I' . i 4 1 iiljifl !:::.;■, Ull;!: 'I r II '.I ' 9S Reduction of Fttlga?^ Fractions, t. 10. To reduce a mixed Fraction to a single one. SSISTAI ./■»». 1 'I 36| 45. Reduce ,1. i'' 36X3+2=110 numerator, 48X3 =14'4 denominator, 23f , n: Rule. When the numerator is the integral part, mulil^^' ^^' ply it by the denominator oi" the fractional part, adding! tlie numerator of the fractional part for a new numeraM^^* ^^^ then multiply the denominator of the fraction by the denC minator of the fractional part for a new denominator. W^' R^*(l £XAMP|,E8. miQ' Red a simple fraction. Facit JT|««fi.l^^' ^^^ v' |J8. Red J S9. Redi 46. Reduce to a simple fraction. Facit -^Hzme^^^Mj' 38 PO. Redii When the denominator is the integral part, multiplJ by the denominator of the fractional part, adding in the m^, J'q y. meratOi' ^i the fractional part for a new daiominator ; vikreater D .inulti[)!y the numerator of the fraction by the detiomin«p •f thi fractional pwt for a new numerator. IKule. I *^ ■■-''- iDtionedfc ft (reduce 4,7 i &ye the 47. Reduce to a simple fraction. Facit y|: 65* 1*9* - . w 11. Redui 48. Reduce to a simple fraction. Fac'U ira^L 44 » i^. Redu^ 1 1 . To find the proper Quantity of a Fraction in the h\ Parts of an Integer, W^' Redu^ Rule. Multiply the numerator by the common partf^* the integer, and divide by the denominator. P* Redm . EXAMPLES. p -jjgj^^ 49. Reduce J of a pound sterling to its proper quaiiT 3 X 20=30^4= \5s, Facit ll6. Reduc 50. Reduce § of a shilling to its proper quantity. Facit 4r. EXAMPLES. il. Reduce 15s. to the fraction of a pound sterling. Facit )^\=^l£, 2. Reduce ^Id. \ to the fraction of a shilling. Facit f . \i. Reduce 9 oz. 2 dr. f to the fraction of a lb. avoirdu- e. Facit ^, \. Reduce 3 qrs, 3 lb, \ oz,l2 dr, J to the fraction o£ t, ' Facit I, 5. Reduce 7 oz, 4 dvots, to the fraction of a lb, troy. s proper quani Facit ?. Facit 1S|6' Reduce 2 qrs, 3 nails J to the fraction of an English Facit I, Reduce 6 furlongs 16 poles to the fraction of a mile. _ Facit %, it 9 oz. 2 dr. \ S* Reduce 2 roods 20 poles to the fraction of an acre, uantity. Facit |. . 1 oz 12 rfr.f • Reduce 5^ gallons to the fraction of a hogshead of - i- . ■'■;■::> .^■" ' Facit {, quantity. icit 4rf. S qrs. >ise to its pi 1:1 |,l Hi !„ ft: J fir lilt. l/'\ I !' : :! I '^i;-! :^h I.- s l':;ipi il i '■\ ?v i I u r.i t f i i i jii m III -1 I .' ■1 i !: I : .. , • I- 'f ' I i], i ' J" i'i- li.M, L'»* % 100 Reduction qf Vulgar FraQllons. the tutoi ■T- ♦.-,-.- T' 70. Reduce 12 gallons to the fraction ofa barrel of b( Facit 71. Reduce 15 bushels to the fraction ofa chaldroff* V?k" coals. Facit I '^^'"^ ^''^ 7'2. Reduce 2 weeks, 2 days, 19 hours, 12 minutes, the fraction of a month. Facit ^- 5ISTANT 1. Wher ninator, or, carrjri nber. •'. ADDITION OF VUI^GAR FRACTIONS. RULE. Ruduce the given fraction to a common nominator, ihen add all the numerators together, Aer nhich place the common denominator. EXAMPLES. I. From \ 4X7=^^ . From I I. From i . From ; . From , I. From ( ■i- Multii - i SUBTRACTION OF VULGAR FRACTIONS. RULE. Ruduce the given fractions to a comnio '* Multij nomin ator, then subtract the less numerator froi • Multij greater, and place the remainder over the common ^' Multij oiiuator. I '• Multij i* Multij II. Whei ce them t( 1. Add f and 4 together. Facit 4-J+H 2. Add f, f,and f together. ^--'^^ Facit 1 jj;'. '6. Add I, 4 -^ and f together. Facit iff/ 4. Add 7 I and f together. Facit 8 Vj. 5. Add f, and |.of f together. Facit }i. 6. Add 51, 6 I, and 4 i together. Facit 17 ^^-. 11. When the fractions are of several denominations, ducfc thto to their proper quantities and add as befot ^' "om 7. Add f of a pound tof of a shilling. Facit los, 8. Add \ ofa penny to | ofa pound. if'aaV IS*. 9. Add 2 of a pound troy to j of an ounce. Facit 9 oar. 3 f^uf. 10. Add I of a ton to « ofa llu Facit 10 ciKt, qr. /^. 13o2. 5 : H. Add \ ofa chaldron to ^ ofa bushel. Facit 2 1 husheUy 3 ffc 12. Add I of a yard \o \ of an inch. ~ r ,. , ,, Facit 6 inch. 2 bar, ', From I !. From I >. From I 1. From 2. From M ULE. ^ by the ors toget for a DC THE TUTOI 'a barrel of b( Facit Facit f 12 minutes, Facit \ CTIONS. to a common 5ISTANT. MullipUcation ofViilg» Fractions, 101 I. When the lower fraction is gi'eater jthan the upper, ot a chaldron ^^^^^^ ^^g numerator of the lower fraction from the ile ninator, and to that diiference add the upper numc- Dr, carrying one to the unit's place of the lower whole nber. EXAMPLES. 1. From ? take j. 3x7=21 fx 4=20.21— 20= 1 nnw, B. From I take | off .,.Facitl\, o^r^ together! '• F*"^"* ^ I *^^e^ Facit '^i, * ' i. From ^ take | Facit^^, 5. From ^^ take | of i^ Facit |^». 5. From 64- \ take § of | Facit, 63^ .9 0_> ! 1 T^2 I 1 14') 5 If. 17 ^. enominations, . add as befot , Facit \os, Facit IS*'. unce. it 9 vz. 3 • Multiply 48 | by 13 | Factt612i,. the commor. ^' Multiply 430^ by 18 ^... Facit 7935fA. >. Multiply ^« by | of 4 off Facit^^\=l^, I. Multiply |4 by 2 off | Facit ^» K2 I . I 1 1 ■ -,11 i r :i^^ Hi If It; ■V f I' ! '> It if . ■ i\! il Mi ■ !l ,? l1 1... 1 i'/jl i::,niU! 'lis ! *m ill i % ■ I ii i! I i I ■ hi ■!im:'. Mi ' |l f! '■ ■1 ! I ^i. 102 Division of Fulmar Fractions, the tutoI 7. Multiply I of a by * of -V Facit^ 8. Multiply ^ of if by | Fee// ;\ t). Multiply 5 ^ by a Faciti^ 10. Multiply 24- by ?- Fac// 1{ 1 1. Multiply ^ of 9 by I Facit 5ll li'. Multiply 9 J by f Fa«< 3/ R DIVISION OF VULGAR FRACTIONS. ULE. Prepare the given numbers (if they requir by the rules of reduction, and divcit the div it that rat ilieu precced as in Multiplication. tXA.MPLES. t rj. 4. 5. Divide /y by | 5x0-^'i5 num. 5x20=.6O den.^ DividoA^bya ' ' Facit i. Divide 672 T^^ by 13} . . Facit iS Divide 7935 fj- by 18 ^ " Facii O Divide f by f of | off Fac/V ^Sj, Divide ^ of 16 by 4 of | /Ti/c// 191 tSSISTAI 1. Iff i*me to a 2. Iff 3. If a, tost? -t. If^i 5.1f|< 6. If 12 It the sair ,7. If ,% ne same i 8. If J 3 fards f CO 7. Divide ^ of | by | of ,- 8. Divide 9 j^ by ^ of 7 9. Divide A by 4^ 10. Divide 16 by 2i 11. Divide 5205 j% by ^ of 91 ri. Divide 3^ by 9^- Fucit 11= Facii 2^,. Tacit J-. Faeit %. Facit 111 Facit J . - 9.1f I . ish come 10. Ifi Jome to ? 11. If 1 fost, e£;.c! 12. liou 1 6^..03 p 08t? THE SINGLE RULE OF THREE DIRECT, VULGAR FRACTIONS. TJ ULE. Reduce the numbers as before directed i -t*^ ducfcion. State the question as in the Rule of ' in whole numbers, jynd invert tlie first terra of the pi tion, then multiply the three terms continually tog and the product will b« the answer. sing: F48ra can d 2. If 2.5 , how fai [ 3. If 3^ ) make a iyard :he tutoIassistant. Single Rule qf Three Direct. lOo FflCJ^ll ..." EXAMPLES. ' ''* '^ ' ,...Fccit .}K Facitim 1. If | of a yard cost | of a;£'. what will ,*y of a yard Facit iflieme to at that rate ? Ans, ^]^l5s. Facit sl i yard : f £ -r : ^\ yd, \\ £. Facit m forixSX 9=180 «uwi. ,„__,.,.,,, i, '- aW3X8Xl0=2t0r/^fis-ni)VA2. 2. If f of a yard cost | £, what will j^ of a yard cost ? i4«5. 1 \s. Sd. 3. If a of a yard of lawn cost 7*. Sd. what will 10 yards } ''TIONS. i^^'*''* Ans. £4f..i 9.. 19^ f,. 4). \f^lb» cost 3j. how many pounds will ^ of h. buy ? f they requirj 5. If f ell of Holland cost ^ ^. what will 12 ells ^ cost vcit the divp that rate ? >4w.v. ;^'7..0..8 J J4. 6. If 12^ yards of cloth cost I5s, dd. what will 48| cost it the same rate ? /f/js. ;^3..0. .9^ yVV* 7. If |*i^ of of an cxvt. cost 2845. what will 7 ca/. ^ cost at e same rate ? i4«5. ;^ 1 1 8^6.. 8. 0--60f/e«.:l 8. If J yards of broad cloth cost;^2..f, what will 10 r ., 7 w^'^'^* ^ ^°®*- Ans,£9.A9, r- IqX ^' '^ * °^* y^^^ ^^^^ I 0* a j^- what will f of an ell Eng- ■'1^^^{ 48j|ijh come to at the same rate ? ^«5. j^"2. Facit i.M 10. If 1 /^. of cochineal cost jC1..5, what will 3G lb..j\ ^" •/ VqF'"® to ? ^«.s. £^. -..n.-e. ,, "• oij II- If 1 yard of broad cloth cost 15*.^, what will 4 pieces rt |«f|<>st' ei;c'.i containing 27 yards ^ ? ^«.«. ;^85..l4.. S^f. /^^c» JjiJ 12. bought 3 pieces a of silk, each containing 'J4 ells f, ^^^* C P 65"0^ »^r ell, I desire to know what the whole quantity FacHn^''' • • ^"*- ^25..17..2i H- F«a^i. I SINGLE RULE OF THREE INVERSE, IN . VULGAR FRACTIONS. : DIRECT, I EXAMPLES F 48 men can build a wall in 24 days f , how many men ^S. 11^ can do tlie same in 192 days? Arts. 6 men^W. , . 1 •■ ^* If 2.5.S. ^ will pay for the carriage of 1 cxvt, H^ mile* ore directed il j^^^ f^^ may 6 cut, ^ be carried for the same money ? the Rule ot J ^„^^ 22 niilcsrPe, eira of the pi 3^ jf 31 y^rds of cloth, that is l}yard wide, be sufficient ntinually tog|j n^^ke a cloak, how much must I have of that sort whicli } yard wide, to make another of the same bigness ? , Ms, 4^ yardst ifrl l-''1 :fi; t i ii' til I I |n;ii ,11 I., JiSISTAI 104 The double Rule of Three, the tutoh 4 If 3 men can do i piece of work in 4 hours \^ in hi ' ' * many hours will 10 men do the same work ? Am, I hour 5'.. 5* If a penny white loaf weigh 7 ot* when a bushtl wheat cost 5u 6d. what is the busht'l worth when a pent white loaf weighs but 2^ oz. ? Ans. \5s. 4r/. i,. 6. What quantity of shalloon that fs -^ yard wide »i line 7 ^ yards of cloth that is 1|| ynrd wide? Ans, \5ifd DOUBLE KULE OF THREE IN VULGAR FRACTIONS. xxAMPtEs. bN Dec J pound F a carrier receives £2 r\ for the carriage of 3 crt^ 1 ued into miles, how much ought he to receive for the carria 7 etc/. 3 ^rj. "i 50 miles ? Ans.£\,.\^..^, on with So that 1 2. If £106 in' 12 montlis gain J.6 interest, what pif consist cipal will gain j^3f in 9 months ? Ans. £75. 3. If 9 students spend ;^'10J in 18 days, how muchi ily distin 20 fetudents spend in 30 days ? Ans, £S9.,\B,Aj\%% 4, A man and his wife having laboured one day, earn ^ tA,, 4s, f , how much must they have for 10 days \, when th two 8on» helped them? Ans. ;^4'..17..1l 5* If ;6'50 in 5 months gain £2j\^ what time \ ;^11 A require to gain fly's? Arts, lO^f month 6. If the carriage of 60 ctvt. 20 miles cost £lif\, v weight can 1 liave carried 30 miles for £5y^ ? Ans, 15 axi r has pla efixed : I But the Ilowiiig t ^>^l •^ ^»,.;- ^- - ' s- t ;i\!r . ^:* From w crease in rt« decri !" - ♦;.-,, i. • "■- ■>»■(>. .Jim HE TUTOH tours \t ill hi I f, I hour 5'v ten a builul I when a penii 5. \5s, w. ^. yard wide u] Ans, \5i/ds, VULGAR iSISTANT. • ,/ ' f • • » TH& TUTOR'S ASSISTANT, a PART III. ( r t . r . I ' '■ *1 >» DICIMAL FRACTIONS. ge ol' 3 cTc#. 1 for the carri fns. £1..16..9. N Decinuil Fractions the integer or whole thing, as on« pound, one yard, one gallon, ike. is supposed to be di- ed into ten equal parts, and those parts into tenths, and on without end. _ So that the denominator of a decimal being always known jrest, what pr consist of an unit, with as many cyphers us the numera- Ans. £lb> ' ^^* places, therefore is never set down; the parts be:r,5 liowmuchi ily distinguished from tht; whole numbers by a comnta '39..18..4fV'ii' ^fi''®^ • t^i'^s ,5 which .stands for jV* |25 for f/o, ,123 one day, earn 'tz'»'>' .„. , ^ lys 4, when tli B"' ^^^ different value of figures appears plainer by the ns.£^.,n..n llowingtab'le: what time i lO^if month cost £iH Am.\5 cui < . '^'.f- . i-;«-. whole numbers. Decimal parts 76543 2 1234567 § Kj .^ r «^ a -**=*=»** =» "^^ "fit' ii So 5 8 3 eg M-" a s S ai5^k- a s From which it plainly appears, that as whole number* crease in a ten-told proportion to the lef^ hand, decimal rtfi decrease in a tenfold proportion to the right hand : I I • Vi I in lit m '" , I i '•. 1 : !'■'' Ml;! ■' i ii ii ",■' ■■^]' i ii i t (1. r,: , ! 106 AddiUon of Decimals. the tutoi f f% ,50 ,500, fo that cyphers placed before decimal parts decrease tli value, by removing them father from the comma, or un place ; tnu8 ,5 is 5 parts of ten, or f\ ; ,05 is 5 parts iOO, or y^^;,005 is 5 parts of 1000, or ,jffnj ,000.5 5 parts of 10000, or ^o.rao' ^"* cyphers, after decii parts, do not alter their value. For, 5, •re each but -,*„ of the unit. A FINITE DECIMAL is that which ends at a certain m bcr of places; but an infinite is that which nowhere en A RECURRING DECIMAL is that wherein one or m figures are continually repeated, as 2,75222 And 52,275275275 is called a compound recurri DECIMAL. s Note, r finite decimal may be considered as infuiite making cyphers to recur ; for they do not alter the value the decimal. In all operations^ if the result consists of several hi reject them^ and make the next superior place an unit mo thus for 26,25999 write 26,26. In all circulating numberSj, dash the last Jigure, as 86,54666. EULE who lU be ca msm R ULE. ADDITION OF DECIMALS. In setting down the proposed numbers to SfSTAN From ,2 From 2, From 2 From a: t\ added, great care must be taken in placu)g figure directly underneath those of the same value, wlitt they be mixed numbers, or purt decimal parts ; and to form which there must be a due regard had to the com or separating points, which ought always to stand in a rect line, one under, another, and to the right hand of tl if ^|tin|y carefully place the decimal parts, according to thiirKr.,i«^-i!i„ spective value ; thea add them as in whole numbers. » * :. EXAMPLES. 1. Add 72,5+32,071+2,1574+371,4+2,75. Facit 480,878|v\hen an 2. Add 30,07+2,007 H-59,432+07,1. 3. Add 3,5+47,25+927,01+2,0070+1.5. 1. Add 52,75+47,21+724+3 1 ,4V>j+,3075. 5. Add 3275+27,514+1,005+725+7,^2. 6. Add 27 ,5+52+3,267£+,5741+2720. Ml )ULE. who It hand, )Oth fact ces n th left hani Multiply Multiply Multiply Multiply ilaltiply 1000, multiplii e are cy ><100= THE TUTO rts decrease tli comma, or un ; ,05 is 5 parts r ,irf«? .0005 rs, after decii 5, ,50 ,500, at a certain m ch no where en L-ein one or ii 22. JND RECURRI ed as injlnite, alter the value of several i ace an unit mo last Jigtire, as sisTANT. Multiplication of Decimals, 107 -'f';' SUBTRACTION OF DECIMALS. , ... \ CLE. Subtraction of Decimals differs but little fron ^ whole numbers, only in placing the numbers, which '\ii be carefully observed, as m Additiou. SXAMPLES. ':.i>:,l^.. .„', ">;! '\ ;.r From ,2754 take ,2371 From 2,37 take 1,76 From 271 take 215,7 From 270,2 take 75,4075 5. From 571 take 54,72 6. From 625 take 76,91 7. From 23,4 15 take ,3742 8. From ,107 take ,0007 MULTIPLICATION OF DECIMALS. \ ULE. Place the factors, and multiply them, as ia 1/ whole numbers, and from the product towards the It hand, cut off as many places for decimals as there are oth factors together ; but if there should not be so many ces HI the product, supply the defect with cyphers te left hand. I ding to thfcit e numbers. f2,75. Facit 480,878 75. ALS. d numbers to in placing t\ ne value, wlitt sarts ; and to ad to the com s to stand in a ^f^itipiy ^2365 by ,2435 ■ght hand ot tl Multiply 2.071 by 2,27 Multiply 27,15 by 25,3 iIultiply72347by23,L Multiply 17 105 by, 32 o7 ilultiplyl7105by,0237 EXAMPLES. Facit ,05758775. 7. Multiply 27 ,35 by 7 ,70071 8. Multiply 57 ,21 by ,0075 9. Multiply ,007 by ,007 10. Multiply 20, 15 by ,2705 11. Multiply ,907 by ,0025 ^^hen any number of decimals is to be multiplied by 10, , 1000, &c. it is only removing the separating point in multiplicand so many places towards the right-hand as e are cyphers in the multiplier ; thus ,578 x 10=5,78. jx 100=57,8. ,578X1000=578. 578 X 10000= :780. I 1i:l ii;. I if L ; S ril ■m I ■';!i ^M I : I ' ' tr. * n ^09 Contracted Multiplication, the tutor'I CONTRACTED MUTIPLICATION OF DECIMALS. SULE. Put the unit's place of the multiplier undj that place of the multiplicand that is intended to in the product, then invert the order of all the oth| figures, i. e. write them all the contrary way ; then in tiplying begin at tHe figure in the multiplicand, which stanij over the figure you are then multiplying with, and set do? the first figures of each particular product directly one undJ the other, and have a due regard to the increase aiisio from the figures on the right hand of that figure you gin to multiply at in the multiplicand. Note. That in miiitiplt/in^ the figure left out every tin next the right-hand in the multij>licandi ifthe product he 3, ujmardSi to \5 carry IfiflStOr uptvardSf to 25^ carry'. and if 25, or uptoards, io 36, carry 3, &c* EXAMPLES ' 12. Multiply S84,S72l')8 by 36,8345, and let there only four places of decimals ia the product. Facit U'l69y20ij^ kllSTAN' Contracted Way, 384,672158 5J38,63 115401647 23080329 ; 3077377 115402 15387 1923 14169,2065' n J! vu 1i Conwion Way. . 384.6721 5S V ,.;. 36,8345 ;•;•. 1923;360790 16386 88632 1154016474 3077377264 23080329 48 115401647 IHIS the< 5one by tULE 1. le value [stands o The Ices, as t INote 1. fher qfc Ifih in ike of, ! dividenc \n be atK But bianyfg mjcyphe) Divide i Divide ^ Divide i 14169,20650 38510 FacH ,1166|Diy|\|g \ 13. Multiply 3,141592 by 52,7438, and leave oniylDiyjjjj . places of decimals. Facit 165,699 i.lDjyj^g ^ 14. Multiply 2,386 15, by 8,2175, and leave only 4 plac of decimals. iwiaV 19,6107. 15. Multiply 375, 13758 by 16,7324, and let there be oiLf.^ 1 place of decimals. Facit 6276,9. K.- • ; 16. Multiply 375,13758 bv 16,7324, and leave oiil^ placQB of decimals " Facit e'21G,95''2i 17. Multiply 395,3756 by ,75642, and let there be oi ,, .^^ 4 places of decimals. iW^ 299,0(;99 ^"^' ^:^ i •',' '1 ».«=• » ■ hen rii )0, &c. he divid re are c} r> '^".. [E TUTOU'I )N OF IlISTANT. Division of Decimals, 10» iltiplier undd intended to i i'all the oth^ ; then in d, which stanij I, and set do\( sctly one undJ icrease aiisiol figure you ; out every t'd 'producthe 5j io 2i>f carry\ ' DIVISION OF DECIMALS. |HIS Rule is also worked as in whole numbers ; th» the only difficulty 13 in valuing the quotient, whick fone by any of the following rules : luLE 1. The first figure in the quotient is always of the ne value with that figure of the dividend, which answcri [stands over the place of unites in the divisor. 2. The quotient must always have so many decimal kces, as the dividend has more than the di visor. ote 1. If the divisor and dividend hnxie both the samr nber of decimal partSt the quotitnt voill be a tohole number, • nd let there " if the dividend has not so many places ofdecimnh a:; It 14169/206.11? fi the divisor, then so viany cyphers must be annexed fo imon Waif, ^dividend as mil wake them equals and the quotient iviU n be a tvhole number* ) ;8*.67215S 36,8345 923,360790 38688632 40164.74 377 264 32948 64714. 50 38510 But if, when the division is done, the quotient has no i htxany Jifrnres as it should have places of decimals, then mj cyphers must be projixed as there are places Viantiug. EXAMPLES. Divide 85643,^25 by 6,321. Facit 1351.9,09429 ^•. Divide 48 by 144. Divide 217,75 by Go. Facit ,nG6.|Divide 12riby ,1045. d leave oniylDivide 709 by 2,574. xit 165,699 l.lDivide 5^714 by 8275. |ave only 4 plaf ^flcii 19,6U)71|^^j^gj^ numbers are to be divided by 10, 100, 100«, P ^^^^^Q'C t|l^^^' ^^'' '^ '"^ performed by placing the separating point I ^als, am 'arthinfrs 'ys remei e numbet f be inert Reduce Reduc( Reduc( '*► M^"v :hii TUToAgigxANT. ReducttoTi of Decimals. Ill XIMALS. x^' •V REDUCTION OF DECIMALS. ;he value oft nowing the I nay be redud ift-hand figul to reduce a vulgar fraction to a ducimxl in dividing or operation, mr^ ULE. Add cyphers to the numerator, nnd divide ! v [e/% out in ihemV the denominator, the quotient is the decimal f'laciioii dsjo 15; j/luirtd. irdsj to 35, cflf EXAMPLES. let there be I t. tnon Way. Jl, 17562(319,1 72296 4.394'60l2 2257432 21S717 20316888 10554-8 9029728 Reduce \ .to a decimal. RecUure| to a deciniaK Reduce I to a decimal. Reduce ^ i..to a deciniaj. Reduce -g^ to a decimal. Reduce \\ of |5 to a decimal. 4)l,CO/2J Jr/t/f. Facit ,5. Facit ,75. Facit ,375. Fncil 192r.0TG+. Facit ,604-31)j(>+. 1518 1354 163 158 8920 4592 »iote. If the civen parts be of several denGminafionr> may be reduced either by so many distinct opera! ions «? k are different parts, or by the first reducing them into their est denominations, and then divide as before ; or, [dly, Bring the lo'Joest into declniah of f,he next superior fminution, and on the right hand of the decimal found, ]re the parts given of the next superior denomination ; so feeding till you bring out the decimal pirts of the highest :er required, by still dividing the pre duct by the next SU' lor denominator ; or, 9328l|(lly, Tq render pence, shillings, and Jitrthings. If the QQXHmber of shillings be even, take half for the frst place of Imals* and let the second and third places bejilled up xcith urthings contained in the remaining pence and farthings, •ys remembering to add 1 , tohen it is or exceeds 25. But e number of shillings be odd, the second jdace of decimals be increased by 5. Reduce 58. to the decimal of a £. Facit ,25. Reduce 9s. to the decimal of a £. Facit ,45. Reduce 168. to the decimal of a £. Facit ,8. I. 1- lia, J|'!:i 1 Decim ,75 ,5 ,^0 6 5 4 3 2 1 ,003348 ,002790 ,002232 ,001674 ,001116 ,000558 Decimals ,000418 ,000279 ,000139 TABLE V, VOIRDUP. WT. lb. the Integer. 5 Lbs. 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Dec ill' ,12") ,116( ,1071 ,098 ,089 ,080 ,07 ,oe^j ,05:1 ,041 ,OSo ,026 ,01 ,00s unces. 8 7 6 5 4 3 2 1 ?r. ic. rams. 8 7 6 5 4 3 2 1 Decimals ,5 ,4375 ,375 ,3125 ,25 ,1875 ,125 ,0625 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 1 ,317465 27 ,23809.'> ,198412 ,158730 ,119047 ,079365 ,039682 ,035714 ,031746 ,027 ,023809 ,019841 ,015873 ,011904 ,007936 ,003968 Pints, 3 2 1 Decimals ,005952 ,003968 ,001984 TABLE VIL MEASURE. Liquid. Dry. Gallon, 1 Quarter. Integer. Decimals ,03125 ,027343 ,023137 .019531 ,015625 ,011718 ,007812 ,t)03906 Pints. 4 3 2 1 Decimals ,001984 ,001488 ,000992 ,000496 Pints. 4 3 2 1 Decim. ,5 ,375 ,26 ,125 Bush. 4 3 2 1 A Hogshead the Integer. TABLE VL IQUID MEAS. Tun the Integ. 02. Deci 8 ,00 7 I ,00 allons 00 90 Decimals ,396825 ,357141 Gallons 30 20 10 9 8 7 6 5 4 3 2 1 3 2 1 Decim. ,09375 ,0625 ,03125 3 2 1 Decimals. ,0234375 ,015625 ,0078125 Q. Pks. 3 2 1 Decimals. ,005859 Dmwia/il ,003906 ,476190 ,001953 ,317^60 ,15873C ,142857 ,126984 ,111111 ,095238 ,079365 ,063492 ,047619 ,031746 .015878 Pints. 3 2 1 TABLE VIIL LONG MEASURE. 1 Mile the Integer. Yards. 1000 900 800 700 600 Decimals ,568182 ,511361 ,4.54545 ,397727 ,340909 HIlHl'! Ill' \v .ill i fli' 'jif ill t Nlt:^ , 11; 1 :■ ;i; ^[H II ;i! n ; I 1 1 -; ; l;!i, h. ' I' :=f Hi: M :ri'. ( 116 ) THE TUTO «STAN: Decimal Tables of Coi>h IVeight, and Measure. 500 400 300 200 100 90 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 Feet. 2 I ,284091 ,227272 ,170454 ,113636 ,056818 ,051136 ,045454 ,039773 ,034091 ,028409 ,022727 ,017045 ,011864 ,005682 ,005114 ,004545 ,003977 ,003409 ,002841 ,002273 ,001704 ,001136 ,000568 Inches. 9 3 1 Decimah 0003787 0001894 80 70 60 50 40 SO 20 10 9 8 7 6 5 4 3 2 1 ,21917P ,19178} ,164381 ,136986 ,10958G ,08219ii ,054794 ,027397 ,024657 ,021918 ,019178 ,016438 ,013698 ,010959 ,008219 ,005479 ,002739 TABLE X. CLOTH MEASURE 1 Yard the Integ Qrs. the same (u Table 4. 1 Day the Integ. Decimals ,0000947 ,0000474 ,0000158 TABLE IX. TIME. 1 Year the Integ. Months the sti'me\ as Pence in ihe\ second Table. Hours, 12 11 10 9 8 7 6 5 4 3 2 1 Decimals ,5 ,4583 ,416 ,375 ,3 ,2916 I ,2083 ,16 ,125 ,083 ,0416 Nails. 2 i Deciim ,125 ,C625 k.««ii>'4k.-.i •«• TABLE XL LEAD WEIGHT. A Fother the Inte| j What man? THE F 26J yi to? Days. 365 300 200 100 90 Decimals. 1,C00000 ,821918 ,647345 ,273^2 ,246575 Min 30 20 10 9 8 7 6 5 4 3 2 I Hund. 10 9 8 7 6 5 4 3 2 1 drs. 2 1 Decima ,51285 \ ^9f . ■us or the Ifacl .12..9. 6, ,4615- ,4102; ,3589 ,2564 5. A gro ,2051! *^cco for ,1538 ,1025! 6. WiiHt r\^\q\ sold lor '"^ '■ What ,0256 ,0128 Decimals ,02083 ,013883 ,006944 ,00625 ,005555 ,004861 ,004166 ,003472 ,002777 ,00208s ,001388 ,000694 Poundi. 14 13 12 11 10 9 8 7 6 5 4 3 2 1 7. 8. VVhu r yard ? Decimi t). if i Je ,00641 ght he t( ,0059; ,0054! ,0050; ,0C45 ,0041 ,00361 ,0032 ,0027 ,0022 ,0018 ,001 S ,00C9 ,0004 fto. If ;arment, same ? 1. If or an7;ard t 2. If 1 ghing t( THE TUTO 8ISTANT. Rule oj Three in Decimals, \Vt \d Measure. 'ABLE X. TH MEASURE rd the Integf 's. the same tu Table 4. VABLE XL BAD WEIGHT iher the Intei THE RULE OF RHREE IN DECIMALS. EXAMPLES. . ■ ': F 26 J yards c«8t j^3..,16...S, what will 32J yards come to f An.s ;i'4f...l2...94. Yds, £. Yds. 26,5 : 3,8125 : ; 32,25 : * 32,25 26,6)I22,953125(4,63974=je4...12...9J-. . What will the pay of 540 men come to at;^1...5...(SL Decimat *"*" ** '^"*' ^^88... 10. 5128ff' '^^f y^'**^s ot cloth cost £2. ..12. ..9. what will 140^ '4.QI5jds of the same cost? Atis. £ 47...16...3,2yr5. *4102j'' ^^* chest of sugar, weighing 7 caY. 2 qrs. 14 lb. cost ,05121 ,2051! >acco 1538 ^''*' *"• 'l025 ^' Wiurt will 3*26 /i. 1 (/r. of tobacco come to, when l^i(&. sold lor ?)S. 6d. Ms. jess.. 1.. 3. .7. What is tlia worth of 19 oz. 3 dxvt. 5 gr. of gold, at oxa '"^^' P^^ ^-' '• -^"•'- £56..10..5..2,3 qrs, ntoc ^' ^^^'^^ ^^ ^^^^ ^°'^'^* of 827| yards of painting, at lOJrf. ,0128|. yg^j p ^^,^^ £36..4..3..1,5 yr«. Decim [). If 1 lent my friend je34. for | of a year, how much ,00641 ght he to lend me y^j of a year to requite my kindness? ,0059; Ans. £51. ,0054! 10. If p of a yard of cloth, that is 2 yards \ broad, make ,0050; ;arment, how much that is | of a yard wide will make ,0045 ! same ? Ans. 2,109375 j/ards, ,0041 U , If one ounce of silver costs 5s. 6d. what is the price of ,00S6 an!iard that weighs 1 lb. 10 oz. 10 dwt. 4 gr. ? ,0032 Ms. £6..S..9 2,2. ars. ,0027 12. If 1 lb. of tobacco cost l^d. what cost 3 hogsneadg ighing together 15 cwt I qr, 19 lb„ ? i4n«.^107..18..9. 4< ! I Si': ;i:i ! i !'■! i < i ■ / iJ :i|?:'l i m K 3} ; '( •: ^ 1' iiii: !(r 118 Extraction of the Square Root, tiie tutorPI^tant ! ; (H i-ili^ '1:1 i; 1 ^i n W !| 13. If 1 ctiot, of currants cost ;^2..9..6 what will 45 nj 9 qrs, H Ih. cost at the same rate ? ^/)J. iC113..l0..9..3 J 14. Bought 6 chests of sugar, each 6 cvot. 3 qrs. at^i 16*. per cwt. what do they come to ? jfns. I'liSM 15. Bought a tankard for £10.. 12. at the r€ite of 5s, J per ounce, what was the weight? /tns. 89 oz. 15 dut] 16. Gave ;^187..3..8. for 25 cm/. 3 qrs. 14 M. of toba CO. at what rate did I buy it per Ih. ? Arts. 15(1. 2 qrs,\ 17. Bought. 29 lb. 4 oz. of coffee for tflCll-S. whati the value of 3 lb.? Ans. £l..l..8.J 18. If I gave Is. Id. for 3j/i. of cheese, wh >♦ will be t| ▼alue of 1 avt. ? Ans. k . ..14.,8.l What » c 6 EXTRACTION OF THE SQUARE ROOT. I. What i ). What i \. What ii 5. What ii I. What it hen the EXTRACTING the Square Root is to find out suclt'"*'® *?f number as being nmltiplied into itself, the prodil"?^ cyph c equal to the given number. wF^ ""**' * Rule. Firsts Point the given number, beginning at ■' ^JJ^^V wnit's place, tlien to the hundreds, and so upon every secol' jXlj**'? figure throughout. I* What ii 1. What ii Secondly y Seek the greatest square number in the fil What ii point towards the left liand, placing the square number il What ii der the first point, and the root thereof in the quotient ; s tract the square number from the first point, and to the maindcr bring down the next point, and call that the sol vend. Thirdhf^ Double the quotient, and place it for a divi on'the left hand of the resolvend ; se<;k how often the li ^' , ^ sor is contained in the resolvend (preserving always ' *«^ ^^ unit's place, ) and put the answer in the quotient, and a '' "^ ^^^^ on the right hand side of the (Jivisor ; then multiply by '^^^ f"'^'* figure last put in the quotient, and subtract the product ir the resolvend ; bring down the next point to the remain (if there be any more) and proceed as before. Roots. . 1. 2. 3. 4. .s. 6. 7. 8; 9. Squares. 1. 4. 9^ 16. 25. 36. 49. 64. 81. extrnc\ ULE. Ri t the squ the squa Whati What What U 1ST ANT. Extraction of the Square Root, llf BXAMPLR8. Wh^t is the square root of »11902i ? Ant* 3i£* 119025(345 9 d\ 64)290 685)3425 3425 tl -• U' TIIE TUTOR liat will 45 cw 13.JO..9..39 vt. 3 qrs, at 2' 4ns.£\\i.». i rdte of 5i, 59 02. IS divt. Ulb. oftoba I,?. I5d. 2 0|n. CM. 3. what 4ns. £l..l..8. wb '♦ will be tP Ins. k . ..14.,8.i What is the square root of 10692^? Anf. 32?. ' What is the square root of 2268741 ? Ans, 1506,23+. . What is the square root of 7596796? i4»j. 2756,228+. . What is the square root of 36372961 ? Ans. 6031 . I, What is the square ro9t of 2207 1 204 ? Ans. 4698, hen the given number consists of a whole number, and mals together, make the number of decimals even br >!<•" ♦v,«"«3!ling cyphers to them ; so that there may be a point fall .It, tne proui ^^ ^^r^,^ pj^^^ ^^ ^^^ ^^^^^ number. . What is the square root of 327 1 ,4007 ? Ans. 57, 1 9+. . What is the square root of 4795,25731 ? .^ws. 69,247+. . What is the square root of 4,372594? Ans. 2,091+. . What is the square root of 2,2710957 ? Ans. 1,50701 f. . What is the square root of ,00032754 ? Ans. ,01809+. uare number 1 . What is the square root of 1,270054 ? Ans. 1,12G9+. tie quotient ; si b extract the Square Root 0/ a Vulgar Fraction. ULE. Reduce the fraction to its lowest terms, then cx- t the square root ot the numerator for a new numerator, the square root of tiie denominatorTTor a new denomi- )r. f the fraction be a surd (i. e.) a number "where a root can r be exactly founds reduce it to a decimal^ and extract root from it. IE ROOT. 3 find out suclj beginning at Jon every seco nber in the It, and to the call that the ce it for a divi )w often the ci rving always [uotient, and ti multiply by the product ir to the remain fore. . 8; 9. , 64. 81. IXAMPLES. What is the square root of | ?|.| ? What is the square root of *|^?^^ What is the square root^f ^igL«T^ Ans. %. Ans. J. Ans. i^f, W- "U.'l : •" lil' .!!■! I ': \t ' 'iv^> If fill' H 1 1 : i it (1) f m I ' 1 il'l •1 i'l; ■I : !': 'ii If Extraction of the Square Root the tutoIistant SURDS. 16. What is the square root of f Jf ? Ans. ,89802+ 17. What is the square root of \\l ? Ans, ,86602+ 18. What is the square root of |jj ? Ans, ,933099+ To extract the Squai'e Boot of a Mixed Number. EcJLE. 1. Reduce the fractional part of the mixed ni ber to its lowest term, and then the mixed number to improper fraction. 2. Extract the root of the numerator and denominft''^ ibr a new numerator and denominator. If the mixed number given be a surdy reduce thejrnctu part to a decimal^ annex it to the whole number^ and exl the sqiigre root therefrom, i.«i' 19. What is the square root of 51|| ? liiji;' Ans. T 20. What is the square root ofSTr^ ? ? '^-■ii-ijins. 5 81. What is the square root of 9Jf ? : 'n ?« Ans. 8 EXAtUPLES. • 7'-» • .-1 •j« >' SURDS. 82. What is the square root of 85 if I, u Ans, 9, '2 23. What is the square root of 8f ? Jns. 2,9519 24. What is the square root of 6f ? jins. 2,5819 Tojind a mean proportional between any ttvo given num Rule, f he square root of the product of the given bers is the mean proportional sought. EXAMPLES. U', Jv*-U |t S7. What is the mean proportional between 3 and \:.!- Ans. 3 ^12=36 thenVS6=s6 the mean proportiom 28. What is the mean proportional between 41276 and Ans. 1897,1 To Jlnd the side cf a square equal in area to any give perficies,- Rule. The square root of the content oCany give perficiesy is the square equal sought. 9. If the of the 8(; If the ire equal The Ar ule. a to the root G be diame 1. What other enc ng an aci tail to be Area qfi w ule. a le square of the ai ce. !. When . When ny twos lliird sidt he Base i ule. and per The ti surrounc THE TUTopsTANT. Extroclton of tfic Squctre Root. Iti EXAMPLES. Ans, ,89802+19. If the content of a given circle be 160, what Is the Ans, ,86602+1 of the square ? Ans. 1 2,6t9 ' 1 . ins, ,933099+io. If the area of a circle is 750, ivhat is the side of the e equal ? ^ns, 27,38612. ED Number^ . The Area of a Circle given iojind the Diameter* ed number tj^'*'^* As 355 : 4 2, or, as 1 : 1,273239 : t so is the : to the square of the diameter : — or, multiply the and denominl""® ''®°'' o^ ^^^ *'*ca> hy 1,12837, and the product will he diameter. , - . EXAMPLES. iuce thejrnctu imbeff and ext :^^iihpAns.'l n ^'. ii i)Ans. i) . . I ♦ Ans, 9,27 Jns. 2,9519 jfns. 2,5819 ttoo given num of thegiven t i :■'. . ,!% 'm' ■ s'fc fc 1. What length of cord will be fit to tie to a cow's tail, other end fixed in the ground, to let her have liberty of ng an acre of grass and no more, supposing the cow tail to be 5 yards | ? Ans, 6,136 perches. Area of a Circle given tojind the Periphery (w Circum* Jerence, :^'ti y. Ans. SluLE. As 113 : 1420, or, as 1 : 12,56637 : : the area le square of the periphery, — or, multiply the square of the area by 3,5449, and the product is the circuni- nce. ,:'-;-^. .,,-,, EXAMPLES, . When the area is 12, what is the circumference ? Ans, 12,2798. ' i. When the area is 160, what is the periphery ? Ans. 4'i,S^9, ny two sides of a right angled triangle given to find hird side. 'tween 3 and ! in pvoportiona ireen 4276 and Ans. 1897,4 rea to any give lit or any gi he Base and Perpendicular given tojind the Hypothenuse,, ULE. The sijuare root of ihe sum of the squares of the and perpendicular is the length of tlie h^ pothcnuse. EXAxMPLES. . The top of a castle from the around is 45 yards higk, surrounded with a ditch 60 yards brotid ; nliat Icngtii M I rmm' l;i' "hi M ■ Mil ;i v < ' • 1- 11 :. ,ii J4 ■ ■( ¥ rs ! . *,r, ii f ' .'Ms J i V 1 ' ■' ' m ;if,! U\ ^ I'M :. i '! .1 i,|i 122 Etxr action qfthe Square Root the tuto of the ditcl *'V f«» ^-f -5^ *-«^»^ 1 must a ladder be to reach from the outside the top of the castle ? 0) U en Cm O t, Ditch. '3 S l^> y/«5. 75 ^«/rf f4 «ij!«?* •'.I J -v» , « •« «> ISTANl point, f itient, an reniaind . Find J Itient by end, rej< he quotl To fini he quoti( excep 3. Ml iiicts toj n the res It, and p OCXS. U3ES. ; Wliat i 9 6 Divisor arc o/'ix llase 60 yards. 3'j The wall of a town is 25 feet high, which is rounded by a moat of 30 feet in breadth : I desire to k th« length of a ladder that will reach from the outsid( the moat to the top of the wall ? i !♦} t' Ans. 39,05/« The Hi/pothenuse and Perpendicular given tojind the B Rule. The square root of the difference of the squari the hypothenuse and perpendicular is the lengthpf theb The Base and Hypothenuse givffn *o find the Perpendia Rule, The square root of the difference of tlie square the hypothenuse and base is the hight of the perpendicu N. B. The two last Questions muy be varied Jbr Exan to the tivo last Proposititns. n* >>» Any number of men being given to form them in square battle, or to find the number of ranks and files. Rule. The square root of the number of men given tlie number of men either in ranK or ile. 86. An army consisting of 3S1776 men, I desire tol ''^ ^'^6 how many rank and file. Ans. i 37. A certain square pavement contains 48841 sq stones, all of the same size, I demand how many are tained in one of the sides ? Ans* 22 EXTRACTION OF THE CUBE ROOT. extract the Cube Root is to find out a num What i! I'-hich being multiplied into itself, and then into What i: product, produceth the given number. What i; Rule. 1. Point every third figure of the cube given. What if ginuing at the unit's place ; seek the neatest cube to What n mm h Deviso I wl product, / ,. - » THE TUTG^ ISTANT. Extraction of the Cube Root, 1^3 . of the ditcll ,, .,0. -t. Ans, 75 ?/«/dPP^'"'» ^^*^ subtract it therefrom ; put the root in the *tient, and bring down the figures in the next point^to remainder for a Resolvend. I. Find a Divisor by multiplying th« square of the tient by 3. See how often it is contained in the re- end, rejecting the units and tens, and put the answer ;lie quotient. ;. To find the Subtrahend. 1. Cube the last figure |hc quotient. 2. Multiply all the figures in the quotient except the last, and that product by the square of the . , >i* h ■ ■ ^* ^^u^^'P^y the devisor by the last figure. Add these ign, w 10 1 J^^^jg jp^g^jgj.^ gjygg jl^g subtrahend, which subtract lu^''f..4?:j Jn the resolvend; to the remainder bring down the next t, and proceed as before. ooTS. 1. 2. 3. 4. 5. P. 7. 8. 9. U3ES. 1. 8. 27. 6K 12.). 216. 31-3. 512. 729. EXAMPLES. What is the cube root of 99252817 ? 9925-284.7. (463 .> - ^ v Divisor, ' ^i ._> - "^ ' V" m the outsidfj Ans, SOJObfet n tojind the ce of the squad length Qf the bj the Perpendic ;e of the square the perpendicu ariedjbr Exam form them int| anks and files, of men give«| n, I desire to Ans, lins 48841 sqi Dw many are( Ans, 22 BE ROOT. ire 0/4x3=48 ( 35252 resohend, -^ - • ■* 2l6=cubeof6. 432 =4X3X bt/ square of 6. 288 -devisor X b^ 6. 33336 subtrahend. U » '3 Devisor. re()f46 X 3=6348)1916847 mo/vewt/. - .4' * *■ *^ » .' ':-'A 27 —cube of S, 1 242 = t6 X Sxbij sqr, ofi, 19044 ^divisor 'x by 3. 1916847 substrahend. nd out a nuiu What is the cube root of 389017 ? and then into Ans, 73. What is the cube root of 5735339 ? ^ns, 179. What is the cube root of 32461759 ? /ins. 319. the cube givenJ What is the cube root of 84604519 ? Ans. 439. iatest cube to|What is the cube root of 259694072 ? , jins. 638. Wliat is tlie cube root of 48228541. ? Ans, 364. ill m^ w» ! ,,l: • ii'j' i m'^ % ■■1 ur h .111 '^y. .1' alt I ' A I 'i' I' Hi I; ■ ' 154 Exixaction of the Cuhe RooU the tuto 8. What is the cube root of 27051036008 ? J«i. 30 9. What is the cube root of 220G9S1012.5 ? Ans. % 10. What is the cube root of 12?«f. 5327232 ? y^ns. 49 11. What is the cube root of 219365327791 ? Ans. 60 12. What is the cube root of 673373097125 ? Ans. 8; fr/t-r=., - -^ « If the s 'fa cub ■J* so that theii lie number. ) Alls. 2^, Ans. 34. Ans. 7|i THE TUTo,^^^^^^^ Effraction of the Cube Root. HJ ? Ans. 3(1 ? Ans. 28| ' *•■; EXAMPLES. 2 ? y^««. 49| ^|,at is the cube root of 12^^ ? H .-» ^«s. 601 ^i,jjt ig tj^e cube root of 31 jW ? Ans. 8,1 ^hat is the cube root of 'iOSpVa ? /n?/Vj6«>'fl»'i| SURDS, t , I2 . per 'I I. ^^1 t ii ill 1 •II I IH V' mi 'l.;!' # .i'M' i I '; •'! ti.i ::i'r ! '(. "'i ■Ci ! imp .,' i 1 5 ji ih ■ ii '!■, !:/' 1S6 Extracting the Roots of Powers, thetutoi The side of the Cube being ffiven, to find the Side of the C\ that shall be double, treble, &c. in Quant Hi/ to the C\ given' Rule. Cube the side given, and multiply it by 2, 3, 1 the cube root of the product is the side sought. 5 > ' ^ EXAMPLE. 7. There is a cubical vessel, whose side is 12 inches, a It is required *o find the side of another vessel, that is contain three times as much.^ i^ Ans, 17,306 [SSISTA 5. Fim to the ; wer, ai 6. Sub iven poi lac^ an le next i 7. Fine efore. 1. Whj ol divii - EXTRACTING OF THE BIQUADRATE ROOT. TO extract the Biqwadrate Root is to find out a nunil j^376(3 which being involved four times into itself, will \ iluce the given number. Rule. First extract the square root of the given niT ber, and then extract the square root of that square and it will give the biquadrate root required. EXAMPLES. 1 . What is the biquadrate of 37 ? Am. 5S\ % What is the biquadrate of 76 ? Ans, 33362 9. What is the biqulidrate of 275 ? Ans, 571914C)( 4i. What is the biquadrate root of SSHil ? Ans, &, What is the biquadrate reot of 3bS62176 ? Jns, 6. What is the biquadrate root of 5719140625 ? Ans.'. A GENERAL RULE FOR EXTRACTING ROOTS OF ALL POWERS. PREPARE the number given for extraction, pointing off from the unit's place as the root ^uireU directs. 2. Find the first figure in the root by the table of po which subtract from the given number. S. Bring down the first figure in the next point to remainder, and call it tlie dividend. . 4. Involve the root into the next inferior power to \htch is given, multiply it by the given power, and c tke divisor. 2. Wha 27 50( 4107 5S X X X X . thetutobIssistant. Eatracliag the Boois qf Po^^ers. 127 Stdeoftue LM ^ YinAa quotient figure by common division, and annex liy to the ^<§ to the root ; then involve « he whole root into the given lower, and call that thSsubtrahend. 6. Subtract that number from as many points of the^ iven power, as is brought down, beginning at the lower lac^ and to the remainder bring down the first figure o( le next point for a new dividend. 7. Find a new divisor, and proceed in all respects as efore. - ' ^ -.- . - ,. ly it by 2, 3,i ught. is 12 inches, a vessel, that Ans, 17,306 rADRATE find out a numl to itself, will J EXAMPLES. 1. What is the square root of 141376 ? H376(376 ' the given n\ that square n red. ]b\ dividend, 1369 subtrahend. ' S X 2=6 divisor. S7 X 37=1369 subtrahend, ' 37 X ^==1^ divisor, 876 X 376=141376 *u676 subtrahend, Ans. 333b21 Ans. 571914C^ I ? Ans, 176? Ans. M)625? Ans.\ 2'. What is the cube root of 6315737G? lACTING T ERS. for extraction ce as the root 531573(376 27 .V . .J 1 * ^ . 27)261 dividend. : t 50653 subtrahend. ••It. „ u: the table of po)i next point to[ ierior power to|7 X power, and c 4107)25043 dividend. ^J% *< 53157376 subtrahend. 3 X S=2T divisor, 37 X Sl^mSS subtrahend, 7 X 57 X 3=4107 divisor. 6 X 376 X 376=53157376 subtrahend. 1 1, m: ;; i: % ll ■ ,1 '|:ii 11 II 'i! I'. :! . -1, . Ml lii 'it:, f pm ti^' r m tl I ■! ■ 128 Simple Interest. the tutou'! 3» What is the biquadrate root of 19987173376 ? f. '.)>. fcf. 19987173376(376 .81 .fii'sD 108)1188 dividend, 1874161 subtrahend. •202612)1245563 dividend. 19987173376 subtrahend. 'I ■ 5 X 3 X 3 X 4=108 divisor, 37 X 87 X 37 X 37=1874161 5tt/&^;a^f«rf, 37 X 37 X 37 X 4=202612 divisor, 376 i^ 376 X 376 x 376=19987173376 sw^^^raAe/u/. SIMPLE INTEREST. T HGRE are five letters to be observed in Simple Iniouths, a terest, viz. P the Principal* T the Time. R tlie Ratio, or per cent. Jii., I the Interest. X 3=14.1,825, or £l 11..16..6. 2. What is the interest of;^54'7"l -..O. at 4' per cent, pet mum^ for 6 years ? ^ns, ;^'131..8..l i .'2 qrs. ,08 3. What is the interest of je796..1 "^..0. at 4 J per cetil. per mum for 5 year a? Ans. £l7 9.. 5. A. 2 (/r.'i. 4'. What is the interest of ;^397..9...5. for 2 years and , at 3^ per cent, per annum ? Ans. ^34?.. 1 3. .6. S,55(/rs. 5. VVhat is the interest of £.'>^4'..17..6. for 3 years » onths, at 4-^ per cent, per annum ? Ans. ^€91 ..11 ..I — 22. 6. What is the interest of jfe"236..18..8. for 3 years & louths, at 5.^ per cent, per annum ? Ans. a^4'7..15..7| ,293. SSt ,08 ,085 ,09 ,095 ,1 IFhen the Interest is for any Number of Days only. Rule. Multiply the interest of ;^'l. for a day, at the ven rate, by the principle and number of days, it wilf ve the ans>ver. '-' INTEREST OF £1. FOR ONE DAY. ^er Cent, 3 H 4 4i 5 cf £1. for found thus .* .5 :: 1,035 0! Decimals, ,00008219178 ,00009589011 ,00010958904 ,00012328767 ,00013698630 ,00015068493 ,00016438356 Per Cent. 6.V 7* I' I' Decimals. ,00017808219 ,00019178082 ,00020547945 ,00021917808 ,00023287671 ,0002i657534 ,00026027397 Note, The above Table is thus found s J 365 : ,03 : : 1 : ,00008219178. AndT as 365 « ,035 : 1 :, 00009589041, &c. Hi? ;, ! !„(!.;: Y\ ■ \ ,;r iii I m it: |i,.i rill i \ flit |s{ 3!' ! 'i I ' I i !! i I '!■ ! / i; !i 'I'! W ' ','.*,( «'-'' 13' 'ijll if. I Ifli 1 ;S' -.1 '!i t M '- , I: I. I 1 i ,;,!l|i'!' 8' 1 ! M 1-30 Simple Interest THE tutor' ►^ISTAN ♦■'l IXAMPLKS. 1. ,^'*..i2..0. amount to in £ years ^, at per cent, per annum ? yins. £1 130..9..0.1,92 qrs. 14?. What will j^'273..18..0. amount to in 4; years, K da^s, at 3 per cent, per aunum ? Ans. £310..14..1| ,35080064 qr^ III. JVhen A, R, T, are giverif to Jind P. a 19. At w f)7..1S..5 5. 367,6 88,0 p. At \ 6..19..1 II. At w 130,.9..0 a- CLE p rt^l ' T \ i EXAMPLES. 5. In wh (]rs. at i9,6 X ,( \. In whs ITS. at 3 . In wh qrs, at . In whj 0064 qr 15. What principal, being put to interest, will amount C367..13..5.3,04 qrs. in 7 years at 4^ per cent, per annu) Ans. ,045 X 7+1 -=1^315, then 367,674-i-l ,3lNUITI =£279..12i.0. 16. What principal, being put to interest, will amount j€376"19.er <-JWf. pel EXAMPLES. ^3..1()..U^ 19. At what rate ;jer c^»^ will £2'79..12..0. amount t« )7.. 1 3. ,5.3 ,04 yr,?. in 7 years ? u ,i-:\. |. 367,674— 279,6=88,074, 275,6 x7=-1957,2, then i .. , V? I 88,074-i-1957,2=,45, or4i^frmi< 10. At what rate per cent will £320,. 17..0. amount to 7 years, at 4li6..19..11. 2,8 qrs. in o years? Anx. 3h per cent. 3..5.'},')1' <7r;. |1. At what rate percent, will ;^926..12..6. amount to ^6^=1^67 ,671. Il30,.9..0. 1,92. qrs. in 6 years i ? -4ws. 4/>t?r cewf. I 5 years, at Sp. At what rate percent, will jC273..18 .0. amount to )..ll. 2,8 qrs,mO..U'..ll ,35080064 qrh, in 4 years, 175 days ? xvhole years, rM Ans, 3 per cent. , he decimal parljl' When A, P, R, aregiven^ to Jind T. a — -j? :; v ^ '. ■■•: . j ■'" -•>. \ £ years ^, atiCLE =:T. .. - v,^..^ yNUa | «t t, .9..0.1,92V''-| P^ n 4 years, V\ examples, .In what time will 1^272.. 12..0. amfltmt to ^367..1S..5 J5080064 qr^ I ^^^^ ^t 4} per cent. ?' Jw5. 367,674—279,6=88,074. 1:9,6 X ,045i=l 2,5820, Me» 88,074-M 2,5820=7 ^ear^ . In what time will je32O..17..0.araount tO;^376..19..1 1. rs. at 3 'y per cent ? Ans. 5 years. ^ B. In what time will £926..12..0.amount to ;^1 130..9..0. qrs, at 4 /)er c r, (i«::| UJ. ;. J •'^1 i:!*;'"^. 132 Simple JvtercsU ' the TUToii'i Note. iJ njpre^dntji the annuity, pension, or yearly rentj T, R, A/flTA' hefure, I, U, K, T, are given to Jind A. Rule.. Xr^ f /u»A VSI9T. S3. If 1925 x^ 3*. If thei XXAMPLESU 27. If a salary of £150. b* forbom 5 yean, at 5percen what would it amount to ? yf n*. ^ 825. |iat is t «^o -35. If 3000 b x5xl50—5x 130=3000 then X,05 + 5x 150= i/. win 36. Su <> at 28. If 1^250. yearly pension be forbom 7 years, whCJJ^jj^ will it amount to in that time at 6 per cent. ? Ans, d'20^ ^gif 2). There is a house let upon lease for 5 years ^ at ^''J^rterfy per annum, what will be the amount of the whole time, at ff. w' , percent.? Ans. £i63.. 8..^. 30. Suppose an annual pension of £2S. remain unpa for 8 years, what would it amount to at 5 per cent.? Ans. /'263..4..0. , Note. fVhen the annuities^ Sfc. are to be paid half yea #r quarterly y then ■ For half yearly payment, take half of the ratio y h,aJfoJ^y,w ^^ mnnuity, Sec, and twice the number of years — andy I ' For quarterly payments, take ajbutth part of the ratimn fourth part of the annuity y &c and four titnes the number yearsy and work as before* 37. If rs, an 33. If ( EXAMPLES. 59. If J SI. If a salary of £ 150. payable every half-year, remawears, v ynpaid for 5 years, what would it amount to in that tim»;- — S per cent. ? Ann. leSS*..?..^^ X -?- 32. If a salary of /*! 50. payable every quarter, was! uupaid for 5 years, wliat would it amount to in that tim J40. If •< 5 .per cent. ? Ans. £839..! ..sf r.mtuii ■ Note. // may be observed by. comparing these last exm^^l is th plesy the amount of the half-yearly payments are more adx^J^ . If t tag^ous than the yearly, and the quarterly more than half-yearly. Ill When A, R, T, are given to fnd U. 2rt HuLE. ■ «*U. ttt^tr-{-2t . i ycara l3i.A..{ N^ora, a Uixii ITHE TUTOU' , or yearly rent ISISTANT. ISimplc Inicrest, 135 EXAMPLES. an, at T) per ced 5S. If a salary amounted to ifc'825 in five yearn, at 5 per i/. what was the salary ? Ans, £ 1 50. 125 x2*» 1650, 5x5x,05— x,05+5x2=^ll then 1650 -M1=«jC150. 34. If a house is to be let upon a lease for 5 years ;, d the amount for that time be £'MS'^..S,.li, at l^ per cent* Arts, £S25. |iat is the yearly rent ? Am. £lH). 'i5. If a pension amounted to ;^2065 in 7 years, at (J per nt. what IS the pension ? Ans. £250. 5 + 5x 156= Ijg. Suppose tlie amount of a pension be jC'263.. 1..0 in S law, at 5 per cent, what is the pension ? j4hs.£'2S. "7 years, wn fJoTE. /f/iew the pn/jments are halfyearlij^ then take 4 a .? ^;i|. rf206 i half of the ratio, and twice the number of i/ears: and if 5 years J at U j,.^gy/y^ ^^^„ ^^^.^ 8 a, one fourth of the ratio, and/our whole time, at ^^^ ^^^^ number of years, and proceed as be/ore. W. igi63..8..3. main unpa ^^^^ j^^jj ^^ ^ -,^ c«rn^ be £834V.7..6, what is the'salary ? MiercentJ Am. i:\50. 37. If tho amount of a salary, payable half yearly, for J. remam ""ni" . . - . . /^/^ .* _•' -. ,..•'.•'. '**• /)^"*^-'** 33. If tlie amount of an annuity, payable quarterly, i)a paid half year J39 1^,3 f^r 5 ytjai-g^ at b per cent, what is the annuity ? .. 17^/., -4/i'S- JCI 50. e»-a/«o, Aa/^o/|jjf^ fF/ifw U, A, T, ftr^^/rc^ /rj/wrf R. 2a- oart of the ratio «,,,„*' ?> nes the number •t»4.i} i« . fiXAMPLfi'^. S9. If a salary'or_^1.50. per annum amount t0;^S2jIa half-year, rema years, what is the rate per cent. ? Ans. 3 per cent. t to in that timi 150 im. *e83*..7..6 5x2— 1jO;xj>;-2^ 150 then =. r , ■■ ; ■: ===r^,OJ quarter, was 1 1 jl x5 Xj — 150x5. i to in that tim^ 40. If a house be lei upon lease for 5 years \, at ^c;o 4n.«. £839..i..JI '■^"'"'"'j a»itl t'la amount for tlut time be i£3G3..S..5. tr fAwe /ffs^ ^Jf' '^t *^ ^^'^ '■^^^ Z'^'' '^^'^^' ^ ^''■^' "^h per cent. 41. If a pension of £'250 per annum amounts to £'ii06S 7 years, v, hut is the rate per cent, i Ans. 6 per cent. 42. Suppose the aaioant of a yearly pension of ;^'t:?8. be !6i..4..0 I.i 8 years, \ihat is the rate per cent, f Ans. 3 per cent. lioris. IVhen the pa//ments are half-yearly, take 4 u — 4 iit a Uevideud, and uork with httlf the annutfy, and U&itL'.e N its are more adv ■ly more than i \ rj.i h ,i!< :,',)! •1 ! ' I I t' .1 i t: !:hU.;:i': <:l'- •i< I ',.!!, 134* Simple Interest. the tutoIistan' the numler of yean for a divisor; if quarterly, ■ pRl S a — 8 ut, and luork mth a fourth of the annuittfy andj hjE. titnes the number of years, 43. If a salary of £\50 per annum, payable half-yea , Wher^ amounts to £B3^,<1.S. in 6 years, what is the rate cent? , Ans^Spercen ttii 44-. If an annuity of ;^ 150 p«r annum, payable quaiJuLE. ly, amounU to^839..I..3 in 5 years, what is the rate cent? Ans,5^rccn\ 2/ IV. When U, A, R, are given tojind T, 1. Wha kinue 5 2 ' ■• ' ' ^a XX X BuLE. First, — I =x : /7*^825 at 5 per cent ? Ans, 5 year 2 82: X2 30X39 ^ V 1=39 —220^ —380,25 J . ,05 150H,05 -39 .i^flfO' ^220+380,25=24,5 =5 years. 46. If a house is let upon lease for a certain timelifnff £60 per annum, and the amount to ;^363..8..3, atHj cent, what time was it let for ? Ans. 5\ year n 5 yea 47. If a pension of cf 250 per annum , being forbor jte. By certain time, amounts to £2065, at 6 per cent, what ' the time of forbearance ? i 5| yea !. What inue 7 j \. What ey, at 5 ote. Th muities 1. What \he presi Ans. 7 year wcous ti When 48. In what time will a yearly pension of ^'28 amoiu ;^263..4..0, at 5 per cent ? Ans* 8 yeat Note. If the payments are half-yearly, take halj '^** — ratio and half the annuitu ; if quarterly, one Jburih cj ratio and one fourth of tne annuity ; and T mil be equ those half-yearly or quarterly paj/merifs. 49. If an annuity of ;^iSO per annum, payable Yearly, amounts to ^834..7..6, at 5 per cent, what time the payment forborne ? Ans. 5 yeai 50. If a yearly pension of ;^150, payable quarti amounts to ^ 839..!. .3. at 5 per cent, what was the tin forbearanee ? Ans, 5 yeai V, ti . If the years al . it \ THE TUTdlSTANT. Simple Interest, 155 ' quarterly^ tnnuitt/i andj yoble half-yej i is the rate I An&,5 per ceni PRESENT WORTH OF ANNUITIES, TE. P represents the present worth ; U, T, 11, as be- When U, T, R, aregiven, to find P. payable quarluLE.- ttr-^tr^2t. .: X tt=P. It is the rate! Ans, 5 per cc«| T. ■! ' '': 1, 2 50 per annu 2/r+2 EXAMPLES. 1. What is the present worth of £150 per annum, t« Itinue 5 years at 5 per c ent f Aas. £6(jO. ars^ X 5X ,05— 5X ,03+5X 2=11, 5X ,05X1 2+2=>2,5. then ll-f-2,5X 150=^;^660. ^ _/. What is the yearly rent of ft house of jC60, to con- Ans, ^^^<""ig5| years, worth in ready money, at 4^ per ceni^ .What is the present worth of /''ioO^jrr fl;;/2>/w, to nue 7 years, at 6 per cent? An6. _;^i454..4..6/, . . What is a pension of £QS per annum worth in icady y, at 5 per cent, for 8 years ? -^ * Ans. £\^%. [ote. The same thing is to be chserved as in thejint rule muities in arrears^ concerning half-yearly and quarterly a certain time|len^f 63..8..3, atHjl. What is the present worth of j^l50. pa} able quart^r- Ans, 5\ yeamx 5 years, at 5 per cent ? Ans. £67l'.5..0. , being for borlote. By comparing the last examples it will he found er cent. whatlfAe present worth of half -yearly payments is mort ad- Ans. 7 yeamgcous than yearly : and quarterly than half-yearly. of j^'28araoui| When P, T, R, are given to find U. tr-i-i LE. — — :X 2p=s\J, . * «r— ili i| "1. ' ;^li HI' 'ii:. ^ j'l 1'^ a tt'Hi' 3l t.l> ■ ■ ': ■ ■ ;» lifeH'^ ! - "'/ 136 Eimple InieresL THE TUTOElgsisx; I i .' :1 'i-!;' ■'( :i ■■ 1- ' ; i'l i' ,;'* ' I V .,! 57. There is a house lei upon lease for .5^ years to con| 65, jf I desire to know the yearly rent, when the present woiEa^g^ pr at 4'.', per cent, is j^^^iOl ..e..3 j'/tt. -^'" £60.1;. cent f />8. What annuity is that which for 7 years continuai-.a at 6 percent, produces, jf 1464'..4..6W present worth? I Note. Ans. £250Al/year 50. What annuity is that which for 8 years continuani produces iCl88 for the present wwth, at b per cotif l-^or ha \ Vfthen Note. When lhe payments are Jiolf -yearly ^ tahhalf ratio, twice the number of years^ and multiply by 4: p; uhiut quarterly^ take nne fourth of the ration four times tmmher of years, and multiply by 8 p. (3v). There is an tumuity payable half-yearly, for 5 yel66. If s For qui andf ratio 1 come, what is the yearly rent, when the present woi Ans. jglSO crate;»< at oper cent, is 4'667..10..0? Gl. There is an annuity payable quarterly, for 5 year come, I desire to know the yearly income, when the p y]' fe-nt worth, at 5 per cent, is £d71..o'.0? Ans, £15% III, When U. P. T. are given tofnd R» » Vt — ^PX2 :" HULE. s=R. .7 2pt\ut — utt vl^^ r ;.';, having 67, If a ing5y e per ce y. m PwULE. v« EXAMPLES. J». If ar present €9. At what rate 775r C(?«^ will an annuity fcf j^l5C aii7iunt, to continue 5 years, produce the present worl lance ? *fe'660? Ans. 5 per cei 15ui< 5— 6(50>^ 2=im,2X 660X 5+i50X 5—150X ^ =3600 then 180^^5GOO=,05pe'r cent. C)^. If a yearly rent of £60 per annum^ to contini 2X 30,2 years, produce A)29l..6..Hj'y'*^ for the present worth, — — IS the rate per cent ? Ans, i,'. per ct 4 b^, If an annuity of j^^.^O per annum, to contii . go^ years, produce Z'1454..4'..6/,- for the present worth, w " — ^ the r&tc per cent ? jins. 6 per. ce 9. For 'J..6..3,, < I ?,HE TUTOalssisTANT. H^T ri'L Simple Interest 137 years to coni 65. If a pension of jC2vS per annum, to continue eight present woiEars, produce £188 for the present worth, what is the rate Atis £.^^Mtr cent? v . '► jins. 5 per cent, \x% continuaT.fl .nt worth? I Note. When the annuities y or rents, &c. are to be paid Ans. i^^h^wf yearly y or quarterly y then, ^^^ )ii* \For half-yearly pa)rments, talce half of the annuity y &c. ^^^ ^Ans f'i^M^ '^"^'^^ '^^ number of years, the quotent mil be the ratio uf ^ ** IZ/TMe ro/e per cent and, . • . :- ^ Z hu 4 P ; l'^"''' *l"^*''<2'^^y payments tale a fourth part of the annuity, ipi/ If .^'^gjlc, and Jour times the number of years, the quotient mil be 0, jou ■ ^^^^^ of the fourth part of the rate per cent. ^ ■' arlv for 5 yl^S. If an annuity of £\50 per annum, payable I. 'ilfyear- \ present wof having 5 years to come, is sold for;^667..10..0, what is Ans, jglSOr ''**® f^*^ ^^^^ ^ ■^^^' ^ V^^ ^^^* rly, for 5 yearlgy. If an annuity oi £\oQ per annum, paya'^ - ^-^larterly, le, when the viving 5 years to come, is sold for £Gri\..5.' v' at is the Ans, ^ISjie per cent, f ■ Am,^ o per cent* IV. IVhen U, P, R, are given tofnd T. R. lVI.£. i— ^_l=xat anni I. To Jind the present Worth of an Annuity, Sjc. faliceP, at in Reversion. "e of itfi 4r^2t IifLE. 1. Find the present worth of., llie yearly sum at the given rate and — - for the time of its continuance, thus: *'^+2 '2. Change P into A, and find whr.t principal being put to interest " will amount to A at the same rate," r.nd for the time, to come before the^'T^ uiliiiuity, ^c. eommence, thus : .:Xw: -=P. vs. EXAMPLES. 74. What is the present worth of an annuity of £1 per flwwww,'^^'continue 5 years, but not to commence till rnd of 4 years, allowii;i|f'^5 per cent, to the purchaser ? Ans. £550 CGO ^\-05x2+;S 4X,W0T 1 78. A I ich doe it for £ s the yt ^0X4 I136SG r9. The commei Be for jl. t to th( ^0. A irs, whii posed o ing 5 p HE TUTOR*|sSIgTANT. Simple Interest. 139 annum, be pul "js. What is the present worth of a lease of j€50 per an- An5.7 i^fars.Kjfi^ to continue 4 years, but is not to connmenec till the ' (innuin bepuMjd of 5 years, allowing 4 per cent, to the purchaser ? Ans. 8 years ', then u ixiill alio, and T < nil be equal to qftht-ratiot ai wable half-yea 76. A person having the promise of a pension of ^10 per urn, for 8 years, b»U not to commence till the e d of 4. ars, is willing to dispose of the same at 5 per cent, t^hat ill be the present worth ? Am. *11 1 ..I8..I4. 77. A legacy of £40 being left for 6 3'earp to a person of I years of age, but is not to commence till he is 21 ; he,, anting money, is desirous of selling the same at 4 per cent. tlcsire to knc hat is the present worth ? ome? yiB' writs, 5 years yahle quarter »t is the numb Ans. ^'171..1S^..11 iVr *- jftnd the Yearly Income of an Annuity, S^q. in Reversion*. Rule. 1. Find the amount of the * + _a ! '*''^ aymentSf 5 yea) esent worth at the given rate, and P ^ ^~~ " rthe time before the reversion, - . AVERSION, us. 2. Change A into P; and find lat annuity being sold will pro- , nuity, ^c. id ice P, at the same rate, and for the:^L_i. :X2p—'U.' rie of its continuance, Ma5, ttr — //+2^ 2tf\-2 -:Xm=: -=P. EXAMPLES'. 78. A person having an annuity left him for 5 y cars, lich does not commence till the end of 4 years, disposed it for jC5oO, allowing 5 per cent, to the purchaser, what IS the yearly income ? Ans. £150. 650X4X,05 + 5.50=660 5X,05+1. 11S6S6X 660X2=^150. 5X5\ ,06— 5 X ,05 + 5X2 19. There is a lease of a house taken for 4 years, but not r /» Jcommence till the end of 5 years, the lessee would sel] the ^y .Mile for;^152..5..11^. 'ij, present payment, allowirg4 per annul ^ commence till purchaser ? Ans. £550 CGO =50 t. to the purchaser, what is the yearly rent? yfns.£50, 0. A person having the promise of a pension for 8 irs, which does notccmn.ence till the end of 4 years, has posed of the same for jCl 11.. 18.. Inf, present money, al- ing 5 per cent, to the purchaser, what was the pension ? Ans. £2Qi, ri: r M\'\: r>^ iHi^iil ■V .. ,, ;; Kil' Mil t 1 hS {11 Pr^ h '! M\ ■? ''1' lit Cli,, ; .'■■: ?■!),! t I, f I'' li ,i t 'i' \y 1 '■\Ik ■' ■:! 140 R§hate or Discount THE tutor'! 'S^*"^ 81, There is a certain legacy left to a person of 15 year of age, which is to be continued for 6 years, but not ti commence till he arrives at the age of 21 ; he wanting ; sum of money, sells it for jC171..1S..I1.7Vt> allowing^ jper cent, to the buyer^ what was the annuity left him. Ans, £40. •y. % N REBATE OR DISCOUNT. OTE. S represents the sum to be discounted. *■ F the present worth. T the time. , ,. r '^ :-'\ R the ratio. ' "' ^ * < I. When S, T, R, are given ioJindV,: ^ -^ • * ■ ;v' ■■• ■ - ' Rule =P. EXAMPLES. , -' ^' ' ''■■ '^'' 1. What is the present worth of j^57..lO, to be paid months hence, at 5 per cent* ? ■ — ??!>! =£'344.,5783 >4««. dC344..11..6f, 168. du< pre! 8. A lence, uch »i HI. RULI 1 12. A ence, p IV. I Rule ,75 X, 05 + 1 ^j..». jw^ •^..* ...^J|4, .w^. ■ 1 2. What is the present worth of ;^'275..10. due 7 montff""^* hence, at 5 per cent, ? ie267..13..10/4«y. 3. What is the present worth of £S1 5.S„6 due 5 month hence, at 4^ per cent, Ans. ;^'859..3..3 j rl?* 4. How much ready money can I receive for a note ;^'75, due 15 months hence, at 5 per eent.f ^n5.;^;70..I1..9iV II. When P, T, R, are giveyi^ tojind S. Rule. 7)ir+/>'=*S. - ~^ • *- • EXAMPLES. 5. If the prssent worth of a sum of money due 9 mont hence, allowing 5 per c^w^ be jCj^^.-ILiS. 3,168 qrs. W was the sum first due ? Ans. i'S51.A0. n 344,5783X,75X05+344.,)783=;C3.')7..10. 6. A person owing a certain sum, payable 7 montlig hen agrees with his creditor to pay him down jC267..13..10t;V allowing 5 per cent, for present payment, what is the deb Ans. iC275..10..0. 7. A person receives jSS59..3..3?,|3, for a sum of z 9. A nonths eot pa} 10. A nonths laymeni 11. A 13. T rae to hat tir ate? 14w Tl THE TUTOR*©'^^*'^^^'''* Rebate or Diseomt 14.1 ^rson of ISyearl ar«, but not t ; he wanting t,Vt, allowing J left him. Arts, £iO. r. iscouated. to, to be paid .6 due 5 montii ve for a note 9 ;^.70..11..9f. V ey due 9 mon 3,168 qrs. w! Ans. iC^57..10. 783=;f3'>7..10. 5 7 months hem :267..13..10^V' vhat is the deb ns. iC275..10..a >r a sum of £ due 5 months hence, allowing the flebter 4| per cent, i)f present payment, what was the sum due? Ans. je875..5..6 8. A person paid £70..11..9,^y for a debt due 15 months eoce, he being allowed 5 per cent, for tlie discount, how uch was the debt ? Ans, £1$. J II. UrhenSt^.TyaregivenytoJindVi, Rule.— -»=R. tp EXAMPLES. 9. At what rate per cent, will ;^357..10, payable 9 nonths hence, produce £344'.rll..6 3,168 qrs, for pre- ent payment ? ^ns, 3 per cent, 357,5— 84>,5783 — . ==r,05i=5 per cent, 344.,5783X,75 10. At what rate per cent, will /'275..10, payable 7. nonths hence, produce £267..13..105Vt ^^r the present lay menfe ? ', v ; J^/ij. 6 per cent, 14 11 6^ 168 11* '^'^ ^^^' ^^t6 77fr ce»f. will je875..S.,6, payables 10* due Vmontl months hence, produce the present payment of £859..3..^^ Ij jins. 4|, percent. 12. At what rate per cent, will ^75, payable 15 months, ence, produce the present payment of ;^70..11..9,\? Ans, 5 per cent* IV. When S, P, R^ are given to jind T. s — p Rule =T. rp EXAMPLES; 13. The present worth of ;^S57..10» due for a certain rae to come, is jf34'4<..11..6 3,108 qrs. ^X. b per cent /wk. hat time should the sum have been paid without any re- ate ? ^ Ans, 9 months^ 357,5—344,5783 =s:,76=9 months. <^: 844,5783X ,05 14. The present worth of ;^275..10. due for acertaift. li.' '^il'ilil ,!S! 'Mi u <'M III ,,i £; m. kl ifi. > m i r r J'P 'i' I i r I ■ 'f rti 142 Equation qf Payment ■ ihe tutor^s time to come, is £267"lS..\0^j%fi at 5 per cent, in what time should the sum have been paid without any rebate ? Arts. 7- months. 15. A person receives ie859..3..3|,|Tj, for £875..S..6 due at a certain time to come, allowing 4A per cent, dis- count, I desire to know in what time the debt should have been discharged without any rebate ? jIm. 5 months. 16. I have received £70.. 11..9,4r for a debt of £15, iJlowing the person 5 per cent, for prompt paymnct, I de- sire to know when the debt would have been payable with* out the rebate ? Ans. IS months. EQUATION OF PAYMENTS. T-O FIND THC EQUATED TIME FOR, THE PAYMENT 01 A SUM OF MONEY DUE AT SEVERAL TIMES. RULE. Eind the present worth of each s payment for its respective time, thuSf — -.s=P. Add all th« present worths together, therii s — p=D* d and'—^-==Ei» EX^AMPLES.. 1. D owes E jfiSOO, whereof £40 is to be at threi months, ^60 at 6 months, and £ 100 at 9 months ; at what time may the whole debt be paid together, rebate being made at. 5 per cent. ? Ans. 6 mouths 26 dai/s. 40 eo 100 — =39,5061 =i58,5365 =96,3855. • :200 > >a J,0125 1,025 1,0375 then 200— &9,5061 +58,5365t 96,3855=5, 5719 5,5719 J .=,573 15 =6 months, 26 dai/s» 194,4281 X ,05 2. D owes F 2,01219 2,13?92 2,26090 2,39665 5 Rates, 6 2,18287 2,29201 2,40662 2,52695 2,65329 2,78596 2,92526 :3,07152 3,22510 3,38635' %5556 3,73345 3,95J01f M16U 4,32194 2,54035 2,69277 2,85434 3,02559 3,20713 3,39956 3,60353 3,81975 4,04893 4,29187 4,54938 l',82234 5,11168 5,4183« 5,74340 n ^;< ,t ill H '?P .1': i' A i 1 m ;f; (I til'! IC; '.■mn li'i] ! I SSIS 144 Compound Interest. THE TUTOI Note. The preceding tahU isthuimadp : AnKXi'AOA 1 ,05 for thejirst year : then, As 100 : 105 :: 1^,05 : 1 > lol ^ second year, &c. ■ * ^ I. Whm P, T, R, are given, Ujind A. }^rs. m EXAMPLES. llO. . 1. What will jC225 amount to in 3 years time, at 5 1 j|'^ * cent, per annum ? Ans. 1 ,05 X 1 ,05X 1 ,05= 1 , 1 51&^^>7 ' A then 1,1 57625 x225=iC260..9..3..3 qrWiQ^' 2. What will ;C200 amount to in 4 years, at 5 per W19..3 per annum f Ans. i24f3..2,02j| jyj 3. What will j£i50 amount to in 5 years, at 4« per per annum f Ans. ^547..9..IC. '2,0538368 i 4. What will /C500 amount to in 4 years, at 5^1 $ent. per annum? Ans. i6 !9»8..2. 3,8323 q^ II. fVhen A, R, T, are given, tojind P. {a Rule.— = P. [r. * ;'>.^ rt ' : • . '■'-• !> •■ ■ i ^. ^V\ 13. U per\ 0,465e "^^ EXAMTLES, — •- 5. What principal being put to interest will amoiJ 260..9..3. 3 qrs. m 3 years, at 5 ver cent, per annum A 22. i^QO ^60,465625 1 ,05X 1 ,05X 1 ,C5=1,1 52765-j-|5^go-5 =>£225. it I, the U. h per ceti^ i5. h 0J383( 6. What principal being put to interest will amoi >C24!3;.2,0255. in 4 years, at 5 per cent, per annum f J\' j^*^y ^j '•% ,.M... __._.._.,_.„ . ■ .;^'^*^J8323^ 7. What principal will amount to -C547.. 9.. 10. 2,05: qr9. in 5 years, at 4 ;jer cen^ per annum ? • 8. What principal will amount to £619..8..2. 3,83 in 4 years, at 5^ per cent f Ans. £: V III. When P, A, T, are given tojind R. ? . l*r ■ a iohich being extracted by tf^e rutesi" **^> ■»] Rule.— =/f, traction (the time given in the qi \ ...... ^-. f shcxing the power. J toill give S. ^323 ai TVxvd Notej SSISTANl Compound Interest, Hi THE TUTOI I r ^ EXAMPLES. • As 100 : lOSr 5 :; 1*05 : l|l^| 9. At what rate per cent, will ^2*25 amount to iC260..9..3. ^rs, in 3 years ? ^«* 5 /jer cf «/. 260,465625 »1, 157625, ihe cube root qf whitifi 225 fit being the Sd power J s^lyOSsxo per cent. 10. At what rate per cent, will 1^200 amount to i2 13. ,0i55. in i years ? Ans. 5 per c*>nt, ,rs tinne, at 5|ii. At what rate per cent, will C^HO amount t« 1,05= l»1576'2J5i7..9..10. 2,0538368 qrs. in 5 years ? Am. 4- yjer cew^ i{260..9..3"S 9''l 12. At what rate per cent, will /;5')0 amount to irs, at 6 J»^^ J619..8..2. 3,8323 qrs. la 4 years? Am. 5^ ;xt cne^ ns, iC243..2,0'2j| IV. When P, A, 11, are ^Iven, tofnd T. ars, at 4 /J^*" \ a which being continually divided bj/Ti, till ?. ^,0538368 ^vRoLE, — --r/, nothing remains^ the number uf inose di' ' vcars, at 5^1 p visions will be equal to T, J..2, 3,8323 qf ' ■ ■ * EXAMPLE. ■ I srest will amoi! ,nt. per anriMn'^ 13. In what time will -£225 amount to £260..^^. 3 o'-i'. t per cent, ? )0,46562j • 1,157625 1,1025 1,05 «1,157625 =1,1025 =1,5 — 225 1,05 1,05 1,05 125 1, the number of divisions heiiig three times sought. U. Inwfiat time will ^200 amount to >C 243. 2.025 s\ at per cent. ? Ans. 4 years. _15. In what time will £ 50 amount to jC517..9..iO. rest will ft'^°|0J38368 qrs. at 4 per cent.'^ Ans. 5 ^can. per annum • l^ll^ ^^\^^ ^i^^jg ^yuj ^-qq amount t5 ^619.8..^'. -^"i" *:l8323 at 5L per cent. ? An,, \ years. V ;i9 ^^*t%ANNVlTIES, OR PENSIONS, IN ARREARS. .8..2. 3,832 Ans, £' hy the rules riven in <^« ^ ' will give B. Note. U represents the annuity, pension, or yearly rent : , R, T, as be/are. 1; . I ■.I. i-'ii i ■ /';' \''\ ,! :< in i!i \tl ( ■■■! 1 ,•! I I n: i.T ! I 11 G Compound Interest THE TUTOR* I8SII jf Table shexving the amount o/'>Cl , jtnnuityyor any numb of Yeays under 31, at 5 and 6 per cent, per annum. Year» 5 /^a tes, 6 Years 1 1,00000 1,00000 16 2 2,05000 2,06000 17 3 JJ,15250 3,18360 18 4 4,31012 4,37461 19 5 S,52")6^ 5,63706 20 ti 6,80191 6,97532 21 7 8,14200 8,39383 22 8 9,54910 9,89746 23 9 11,02656 11,49131 24 10 12,57789 13,18079 25 11 14,2067b| 11,97164 26 VI 15,91712 16,86994 27 15 17,71298 18,88213 28 14. 19,59863 21,01506 29 15 21,57856 23,27597 30 5 Ra^es. 6 23,65749 25,84036 28,13238 30,5390(» 33,06595 35,71925 38,50521 41,43047 t4,50199 t7,72709 •1,11345 54,56912 58,40258 62,32271 166,43884 2 5,672 2 28,21288 30,90565 33,76999 .^6,78559 39,99272 43,39229 46,99582 50,81557 54,86451 59,15688 63,70576 68,52811 73,63970 79,03818 19. )rbori 20. e omii lie ami II. I Rul; 21. \ ^ns. 22. V\ U'248. 23. V\ lount I 24. If ars, a r cent, III h 1,05—1 hythetahle th 5, 4,31012X10— >C215..10..1 1,76 qrs. IS. What will a pension oi iC45 per an»Km, payi 6. Ir yearly, amount to in 5 years, at 5 per sent. ? Rule. Note. The above table in made thus: take ihejird yea umotint^ lahich is Ht multiply it by i,0>)rl='^y05=S€coi years dmcuntf which also multiply by 1,05+1=3,1523 thit d year's amount. I. Wften U, T, R, arc given tojind A, urt — u Rule. =A, or by the table thus .* r— 1 Multiply thsj amount of iCl for the number of yearp, a at the rate per cent, given in the question, by the annui pensicii, &c. and it will give the answer. * EXAMPLES. ^ f 5. In 17. What will an annuity of iC50 per annumj paya 15.. 10, yearly, amount to in 4 years, at 6 per cent, A71S, 1,05X 1,05X 1,05 XI, 05X 50=60,77531 25( 60,7753125—5 *• then — — =/C215..10..1. 2 grs. or, 215 ch beii ons tui 8.,13.. Am i£248..13..0 3,27 qr went? rHE TUTOR*ii88i8TANT. Compoufid Interest, 117 for anjf numb per annum. \9\1 5,072 i' 36128,21288 i3bl30,9056S I0()|33,7&999 ;95b6,78559 )*23l39,99272 521^3,39229 047*6,99582 199 50,81557 I709l5*,86*51 34.5 59,15688 >912103,70576 3258 68,52811 227173,63979 19. If a talary of i\[0 per annum, to be paid yt'aily, Le orborne 6 years, at 6 per cent, what ix the anioui.t ? Arts. Jt:279..0..3 ..V.'*.. V- 20. If an annuity of ;^'76 per annum, pnyable } '.'aily, ^e omitted to be puid lor \i) yuATgy at 6 ncr len*. \^iat 'i» he amount ? ins. iC938. 1 1..^^^. i..^i5851424i>4(ii 12. II. fV/ien A, R, T, are gheut to Jtud U. ar — a RuLl. =U. r/— 1. IXAMPLf.S. 21. What annuity, being forborne 4 years, M'ill amount 4ii;i5..10..1. 2 yn. at 5 ;;<^r a/'/.f* Ans. *215,50625>< 1,05— 215,5062, .-=,jC50. 1,05X1,05X1,05X1,05-1 3884i79 ,Oj818H 22. What pension, being forborne 5 years, will aim)m;t l'24'8..13..0. 3,27 y>*. at 5 j^tr cent.? Ans. £^o. ake ihejirft yMf ^g. what salary bcin^' o»iiitt*id to be paid o years, will tl^Tx, _rSl"0""' *o £'279..0..3. -^^VV.'6» at 6 per cent. ? Am. /^ti). l,Oo+l--3»w^|24.. If the payment of an annuity being forborne 10 trs, amount to je988..1 1 ..2^ 23585 U243461 12. at o rcentf what is the annuity ? Ans. £15. III. When U, A, 11, are given y to find T. ar^ii — a tvhich being continuallij divided hy EuLE, ■ =srt 11, till notning remains, the number u of those divisions will be equal to '£ IS : iber of year*, a n, by the annul annunif pays 775312i • EXAMPLES, !5. In what time will ;^50 per annum amount to :15..10..1. 2 qrs. at 5 per cent, for non-payment? 10=60 215,50625X 1,00+50—215,50625 \s. .1. 2 qrs. or, |0..1 1,76 qrs. zr annum, pay' ?nt.? .13..0 5,27 qr\ 60 ■=^1,21550625 ich being continually divided by R, the number of the di» [ons toill Ae = 4 years, [6. In what time will j€45 per anpum amount to 18.,13..0 3,27 qrs* allowing 5 /}er cent, forbearance o lent? Ans, 5 years. Ur. ' lit. I- m 'V hil. ms :, I a i. 148 Compound Interest '27. In what time will iC^O ^'279..0..3./,-sV2iJ at 6 per rcw^? 28. In \vliat time will £13 /;988.. 1 1 ..2|. 23o8Sl 4243461 12, i'urbearance of payment ? THE tutor's |3 ••") Rates. 6 0.9i:238(0.94339 !,859!1 2,72324 3, "4.59.5 4,32917 5,07769 5.78617 fi,4n32l 7,1()7S'2 7,72 17'^ 8,30641 S,8G32.5 9,39357 9.S9S6 . !0,3796r) l,S33t^9 2,67301 ■^,46510 S21236 1,917.32 >..oS238 ),20979 (;,^;0!69 7,3600.S 7,88687 8,38384 8,85268 9,29498 9.71225 \ears, 1% 17 18 19 20 21 22 23 21. 2) 26 27 28 29 30 5 Rates, 6 10,83777 11.27406 11,68958 12,08532 12,46221 12,82115 13,16.300 13,48857 13.7986 14,09394 14,37518 14,64303 14,89812 15,14107 15,37245 10,1 C5S9 10,47726 10,82760 11,15811 11,46992 11,76407 12,04158 12,30338 12,55035 12,78365 13,00316 13,21053 13,40616 13,59072 13,76483 30 l,5036i =167,4: 30. W munif t 31. W intinue 32. W BTth in ] Noi l:. The above table is thus made: divide £1. by 1 :■" 952.-?8 the present worth of the first ycar^ xuhich -r- =-_:.9070J. addtd to the first year s present xt:orth=\,hr)\) the second year s ptesent xvorth : then yo70:i-irl,05 and tjuotic/d added to 1,85941=2,72324, third year's prcit ivortn. _ ■_. . - ^ I. JVhen V, T, R, are given, to find P. u rt Rule. r— 1 or, by the table, thus. er cent. 15. If tl • be req i. what 16. If tl £216..! uity? .it \ HE TUTOR- ll^SISTANT. Compound Interest, 149 m amount t^ ins, 6 years tm amount t ns Multiply the present woith of ;^1 annuity for the time id rate per cent, given by the annuity, pension,^ &€, it 6 p^rccw). folill give the answer. 10 years, I * EXAMPLES. 29. What is the present worth of an annuity of £30 per mumf to continue 7 years, at 6 per cent, ^ Ans.£i 7..9..5..184f/. SO 10,0483 =19,9517. 30—19,9517=10,0483 1,50363 1,06—1 167,47 16- 5y the table 5,58238X30=167,4716. JITIES, any Number fcr annum. Rates, 6 77110,10539 ,0610,47726 )5810,82760| 532 11,15811 221 11,46992 115 n,76407 300 30. What is the present werth of a pension of £40 ^er mum, to continue 8 years, at 5 per cent, ? Ans. 258..10..63 HflHIMf?- 31. What is the present worth of a salary of £35, to ntinue 7 years, at 6 per cent, ? ^ Ans. >;-195..7..8 ^i|ifMH}|i^. 32. What IS the yearly rent of ^50, to continue 5 years, th in ready money, at 5 per cent. ? ^ „o«^ «i I ^«*- ^216. 9..5. 23i|| Jf ft qrs, 12,783fi5 |ii, jyken P, T, R, are given tofnd U. 13,00316 1 prtXr-.prt 13,21053 ■" '^ ^ 13,40616 13,59072 1 3,76488] 2,04158 12,30338 12,55035, 857 86 394 518 13,00316' 30'J i812 107 1245 I Rule. '=U. rt-^l EXAMPLES. hide £\.hy 1. 33. If an annuity be purchased for ;^167..9..5..184r/. to continued 7 years, at 6 per c ent, what is the annuity ? 167',4716X 1,50363X1,06—167,4716 X 1,50363 _ "ar^ which -r- K ti'orfA=:l,^50 |ori-i-l,05 amliP' Id year's V^^^\q^ 1,50363—1 J4. If the present payment of ;^258..10«6f. MtfiHf T 9^"^' he made for a salary 8 years to come, at ier cent, what is the salary f Ans £40. )5. If the present payment of;^J95..7..8^ifAa.2|a£^«i_. be required for a pemion for 7 years to come, at 6 per i. what is the pension ? Ans, £'i5, \Q, If the present worth of an annuity, 5 years to come, £216..9..5. ^UHiVi9^^' a* 5 per cent, what is th« kuity ? Ans, £50» ' 02 , .,r, .11 i :! i^i<' vi- I m ill sV 1^ "li !i' i ill: fi- ll' f >t in 50 Compound Interest THE TUTOr'IassIST !l i': 'I II .1.^* I i ,1 III n» I''. ■■\i III. JVhen, U, P, R, are given, to JindT. u ivhich being continually divided h\ KxjLE. I — --s=r^ R, /«'// nothing remainSf themimh p\ u — pr of those divisions tvill be equal to EXAMPLES. S7. How long may a lease of £30 yearly rent be hadfo /,'167..9..5. ,181(/. allowing 6 per cent, to the purchaser? tuhich bciyg couth 30 ually divided, th Ans. ■ :g=l ,50363 number of tJm 167,4716+30—177,5198 divisions 'will a= ^0 T = years. 1,41852 =175,0 42i \ !lf;^60 mence i ^urchas 43. 1 i yet in i lease ease shj he said baser ? ^ 38. If ^2.58..10..6. 3 iflfflHilT 9"- " paid dow Cor a lease of£W per annunij at 5 per cent, how long is tli lease purchased for ? jins, 8 ^ean. S9. If a bouse is let upon lease for £3^' per annm imd the lessee makes present payment of ;^195. 7. 3HfHf?HI?r he being allowed 6.per cent. 1 demand hol^^ ^^^ long the lease is purchased for ? ^ns, 7 years. 40. For what time may a lease of £50 per annum be pu cl'iated when present payment is made of j£2l6,.9.. *2 iHlxo^ ^t 5 p^r cent, f Ans. 5 years. ANNUITIES, LEASES, S^c. taken in REVERSIOl ^^^^J^ To find the present worth of Annuities,. Leases, Sfc. taken Beversion.. Rui E. Find the present worth of the annuity, &c. at the given rate, u and for the time of its continuance ; thus, — u rt =P. r— 1 2. Change P into A, and find what principal being put to interest will amount to Pat the same rate, and for a the time to come, before the annuity — — commences, which will be the present rt vrntUx of th« annuity, &c. J*- >•■'■■ *• , .=p.. 1 V, 41. ) of £40 mence i purcbaa 40 Rule orth at cnt beij ame rat ance, v uired ? 44. W id thei 5..1. 'W The HE TUTOR'liSSISTANT. Compound Interest 151 a% divided h\ inSf the nnmbt U be equal to rent behadfo e purchaser ? vit • n EXA^MPLES. 41 . What Is the present worth of a reversion of a lease lof £40 per annunit to continue for 6 years, but not to com- Imcnce till the end of two years, allowing 6 per cent, to the Ipurchaser? ^, Ans, £n5.A.A. '2,0^& qrs. 40 ' ' 40—28,1984 199,6933 =28,1984, =196,1933. 1,06—1 1,1236 1-, . ,.,1,41852 f^ *ST;i=l 75,0163. ^ 42i What is the present worth of a reverwon of a leas« f ;^60 per annum, to continue 7 years, but not to com- iience till the end of 3 years, allowing S per cent-, to the urchaser? ^„5. £299.J8..2 -.WV'^JAVtViVt: 43. There is a lease of a house at £30 per annum, which % yet in being for 4 years, and the lessee is desirous to take lease in reversion for 7 years, to begin when the old ease shall be expired, what will ba the present worth of he said lease in reversion, allowing 5 per cent, to the purr Ihaser ? Ans. ;f 142..16..3 -.V^ViVtWiViVo^ ^rs. \y divided, th ruber of th, isions will : to T - ars. s. is paid dow !, how long is tli Ans. 8 years £3' per o««M« t of ;^195.7. 1. 1 demand ho p^ ^„^ ^^g Yearlg income of an Annuitt/i Sfc. taken in Ans. 7 years. Iteverdon. }er annum bepul of ae2l6,.9.l Rule. Find the amount of the present Ans. 5 y^«^*'^orth at the given rate, and for the time T> T jrr T> c rnl^^'^'"^ '^® annuity commences ; Mm?, prtt^iA. REVLRbiW\ Change A into P, and find what yearly nt being sold will produce P. at the ime rate, and for the time of its cottin- ance, which will be the yearly sum re- r<>< r — prt uired ? ......... thus. =U. lasesy Sfc. taken] u rt .=P. .ni5 r;— L r— 1 a Wt .=p. .K ■< '.J n EXAMPLES. 44. What annuity to be entered upon 2 years hence, id then to continue 6 years, may be purchased for It75..1..1 2,058 qrs. at 6 per cent. ? i4/is. 175,0515:3X1,1236=196,6933 '-*';- Then 196,6933X 1,41852X 1,06—279,01337 141862—1 mi it' '^i \i ■■iV-. '^; . 'II HJ !' M' ii I I'l ■'f an house lie reversion fo d of 3 years, sJ at is the yearl Ans. /eO. reversion 7 vean ^** ^^ ^ freehold estate of £l4iO per nnnum be bought ired paid dow "" *^3^^» ^^^^ ^^ *^6 rule per cent, allowed ? t was the vearl '^"*- * P^^ '^'^^'* lowed 5 per een ^^'. ^^ *" estate of .•€75 per annum is sold for 1^1250, Ans rSO ^^^^ ^^ ^^^ ^^^^ P^^ ^^'^^ allowed ? yfns. 6 per cent. Es • such as ar f*^^^^' per cent, t Ans, £3500: rent was to b r 6 per cent. ? Ans, £\250, nd 4 per cent, a nt? Ans.£\^ ns. 1,05—1 1,2155 57. What is an estate of 3^200, to continue for ever, It not to commence til! the end of 2 years, worth in ready onty, allowing the purchaser 4- per cent. ? ^ns. £4622..15..7i. Ilf. 58. What is an estate of £240 per annum worth in ready oney, to contmue for ever, but not to commence till the 1000, and the a d of 3 years, allowance being made at 6 per cent. lyer, what is tl ^/w. ;^S358..9..0. 2,|||4^. 1000^ ,05=£5 find the Yearly rent of an Estate taken in Reversion. Rule. Find the amount of the worth the estate, at the given rate and time '' thus, tur/=A. find what 250 present mi fore it commencos ; B to the buyer " " yearly Ans, £75 fit being sold will produce U, at the lurXr — tcr le rate th us. ^J, iigh will be the yearly rent required. .;, \M ii I 'i ■ i''i ii l;!l !!: !■ !r' A ' I :1'^ '.' :,.! 1.': 1: liii ■i.!! II I i ii; ■•' i4 i> Jl K m '"j'l 1 1 hs' : 5i Rebate or Discatmt, THE TUTOR*! jsiSTi XXAMPLES. 59. If a freehold estate, to commenee 4 yews hence, i sold for ;^822..14>..l .2 ^rs. allowing the purchaser 5 ft etnU what is the yearly income ? jins. 822,70625 XI ,2155=1 000. then 1C00X1,05>< 1,05—1050 =^^50. 1,05 60. A freehold estate is bought for £4622..! 5 .7^. j^? which does not commence till the end of 2 years, the buyt being allowed 4 per cent, for his money ; I desire to knoi the yearly income ? Am, £200. 61. There is a freehold estate sold for £3338. .9..6 ns, — ^tHIt? ^^s. but not to commence till the expiration o 3 years, allowing 6 per cent, for present payment ; what - the yearly income ? ' • Jwc. j€2^0. I. Wl Rule 1. Wl years h 31J 1, REBATB, OR DISCOUNT. ' /I TABLE thettiing the present worth of £\ due an number of year Sy to commence under 31 > rebate at *tnd 6 per cent. jrf 1-" Years, 1 2 S 4 5 6 7 8 9 10 11 12 13 14 15 5 Mates, 6 ,952381 .907030 ,863838 ,82270^; ,783526 ,746215 ,710682 ,676SS9 1,644'609 ,61391:5 ,584679 ,558837 ,30-21 ,505068 ,481017 ,943396 .88999C ,839619 ,79209 ,74725> ,70496(' ,165057 ,6^74 U: ,591898 ,558394 ,5ti6787 ,496969 ,46883i:' ,442301 ,417265 Years. m 17 18 19 20 21 22 23 24 25 26 27 28 29 30 5 Hates, 6 ^58lin,393647 5f. a i ne, wh r cent. \ 3. The payabh esent \t te beini II. Wh li ULX. ,436296 ,415520 ,395734 ,376889 ,358942 ,341849 ,325571 ,310067 ,2953(/^ ,28124( ,2678 S ,25509f- ,24294t .231377 ,371864 ,350313 ,330513 ,31180 ,2941 5-i ,27750.5 ,261797 ,246976 ,232998 ,219810 ,207368 ,1956^0 ,18 556 ,174llf ^lOTE. first vear'i eonayear The above table is thus made, 1-r-l ,05 =,952-3! present toorth ; anrf ,95238 14-1 ,05 =,90703 s ; and ,90703-rl,05=, 863838 third year^ Sfc *. If a the pri at was 1 an all rment, . Thei ra, bui aaent, : II. m VLl. ■ HE TUTOR*lssiSTANT. Rdbate or Discount. 155 .1 1. When S, T, R, are given to Jind P. rears hence, m g archaser 5 jiM fiyi^^:, =P. ft ' V ■ , . Ill Mt! EXAMPLES. 50 I i. What is tlie present worth of JS315..12..4, 2c?. payable 22..I5 -7^' ill years hence, at 6 per cent.f ears, the buye| desire to kno Am. ;^200 •or £3338..9.J e expiration lyinent; whati| Ans, 1,06X 1,06X 1,06X 1,06=1,26247. then by the table 315,6175 315,6175 ,. =.;f 250. ,792093 siW 1,26247 249,9984124275 s:. ir £344..14..9 2,01940875 qrs, be payable in 7 years e, what is the present worth, rebate being made at 5 cent.? Ans. £'24f5. 3. There is a debt of 1^441. .17..4. ,06464 qrs. which payable 4 years hence, but it is agreed to be paid \m ^ Jesent money ; what sum must the creditor receive, re- 31 > rebate ai ^e being maie at 6 per cent, f Ans. jfiSdO. 11. When ?, T, 11, are given, to Jind S. IT. I of£l due a 5 Hates. 6 58111 1,3936471 .36296,371364 tiULE. EXAMPLES. a 5520 1957-^4 176889 S58942 541849 125571 10067 5953(// 58124( ,3503131 .3305131 ,31180 ,2941 551 ,2775051 ,261797 ,2469761 ,232998| ,21981( . If a sum of money due 4 years hence, produce £2i0 the present pa3rment, rebate being made at 6 per cent. lat was the sum first due ? Ans. £205 X l,26247«=je315..12..4 2. j2d. rebate OtGper cent, in what time w tae debt payable ? 3 1 5|6I75 %uhkh being continually diviit Ans, =1,26247 those divisions will be equal 250 ^ssthe number of years. 8. A person receives if245 now for a debt ie344<..14..9. 2,01940875 qrs, rebate being madeat5;j( eent, I demand in what time the debt was payable ? Am. 7 years. 9. There is a debt of £A^\,M..A>, ,06464.. due at certain time to come, but 6 per cent, being allowed to t debtor for the present payment of £350, I desire to km in what time the sum should have been made without a rebate ? Ans. 4 yearh 'i IV. When S, P, T, are given, tojind Vi, s tohich being extf tided by the rules of -KVLB. —~^=srt traction (the time givm in the quesii p shelving the poller J vjill be equal to R EXAMPLES. 10 A debt of 315..12..4. ,2c/. is due 4 years hen but it is agreed to take £250 now, what is the rate cent, that the rebate is made at ? 315,6175 r . Ans ■ =1 ,26247 ; V 1 ,26247 =1 ,06=6 /7#r c 250 „ OB loss 01 hi; u N Mul the lo\ jlt und very erior. ' 11. The present worth of 3^844..] 4..9. 2,01940875 ' }^\^^ payable 7 years hence, is £245, at what rate per cen ^^^ '" rebate made ? Ans, 5 per cen P^^^® 12. There is a debt of ;^4U..17..'1. ,CGl«. payabl ^* 4 years time, but it is agieed to take £3bO present ment, I desire to know at what rate per cent, reba f .J*.^'" made at ? - . , Ans. 6 per ce) 't»t>hei| right ds, fou [•HE TUTOR* SISTANT. C 157 J de for a debt what time wd niinuallydivide^ s will be equal i r of years, for a debt ng made at 5 ;) payable ? Am. 7 .yffl^y 06464'. due at ng allowed to t , I desire to km made without a Am. 4f yearh THE ^s TtTTOR's ASSISTANT. ';/ :\ PART IV. f* n( I. hy the rules of }m in the questi Ul be equal to B DUODECIMALS; OR, WHAT IS GENERALLY CALLED toss MULTIPLICATION AND SQUARING OF DIMENSIONS BY ARTIFICERS AND WORKMEN, ; RULE FOR MULTIPLYING DUODBCIMALLY. ae 4* years hat is the rate hrTND^R the Multiplicand write the corresponding . II. J denominations of the INIultiplier, 7 =1,06 =6 prrc ,9. 2,0194.0875 . Multiply each term in the Multiplicand (beginning he lowtst) by the feet in the Multiplier; write each lit under it3 respective term, observing to carry an unit every 12, from each lower denomination to its next erior. . In the same manner multiply the Multiplicand by the ni6s in tfe Multip''''r, and write the result of each term lat rate /Jtfr cei ,^^ ^^^^^ ^^ ^^^^ right hand of those in the Multipli Ans. 5 per cen r » r . ,CGl^. payabl ^JjU prest ^^ .Work in the same mapner with the* seconds in the per ^^" • ^^' iti^Uer, sietting the result of each terrn two places to >^«*' ^ P right hand of those in tho Multiplicand, and so on for ds, fourths, &c. . '» ' . ^ w '■i 4 ':;; m '( I "li! if ill! ' !;i.r:'ii '■\* I r i m I !i -M l! I," l^f 158 Duodecimals, THE tutor's EXAMPLBS. ISSISl M Jl in. f, in. 1. Multiply 7 . 9 by 3 . 6 Cross ^luUipll, Practice 6^ 7 . 9 3 . 6 7^9 3^6 DuodecimaU* Decimah. 21.0.0=7x3 2.3.0::i^-9X3 3.6.0=7 H 6 0.4.6=9X6 27.1.6 23 . 3 3 . 10.6 27 . 1 . 6 7 . 9 3 . 6 23 . 3 X3 3 . 10.6X6 ■ 7.75 • 3,5 3875 2326 27 . 1 . 6 27125 2. Multiply 3. Multiply 4. Multiply 5. Multiply ./ «'«• fAn. f, in. 8.5 by 4. 7. Facit 38.6.11 ' :) 9.8 by 7. 6. Fncit 72.6. 8.1 by 3. o. i^rtc/V 27.7.5. .^ 7.6 by 5. 9. Facit 4^3. 1. 6. C. Mult.ply 4.7 by 3.10. Facit 11.6 W. ^' 7. Multiply 7.5.9** by 3.5.3^' i^aciV 2S.*.6.2.3. 8. Multiply 10. J.5 by 7.8.6. Facit 79.11.0.6.6. 9. Multiply 75.7 by 9.8. /W/ 730.7.8. 10. Multiply 97.8 by 8.9. Facit H54.7. '' 11. Multiply .57.9 by 9.:>. Facit 54f?.9.9. 12. Multiply 75.9 by 17-7. Facit 133i.I1.3. . -.• 13. Multiply 87.5 by 35.8. Facit 3U7.10.4. , ! 14. Multiply 179.3 by 38.10. Facit 6960.10.6. 15. Multiply 259.2 by 18.11. F,tcit 12677.6.1Q. 16. Multiply 257.9 by 39.11. Facii 10288.6.3. 17. Multiply :^1 1.4.7 by 36.7.5 Facit 11402.2.4.11.11. 18. Multiply 321.7.3 by 9.'i.6. Facit 2988.2.10.4.6. ' '' THE APPLICATION. 3 20. \V ng 4 f i- per 21. Tl bot 6 i y com Artificer's work is computed by different measures, vii eet 7 i , 1. Gidzing a.id mason's Hat-work by the foot. 2. Painting, plastering, paving, &c. by the yard. 3. Partitioning, flooring, roofing, tyling, &e. by tli square of 100 feet. 4:. Brick woi V, &c. by the rod, or 16 feet ^, whose squar -.v7 - - . ^ 19. 1 iir, tilt feet 8 t' each 1 1 id. hodec. 7 . 1 6 . 5 . 4 9 . b i? ME. As: IE tutor'sI ASSISTANT* Duodecimals, 1.59 MEASURING BY THE FOOT SQ.VARE, As Glaziers and Masons Flat-work, EXAMPLES, X3 GX6 1 . 6 2712 tn. .6.11 .'- .6. .7.5. .1.6. .6 10. M.6.2.S, .11.0.6.6. ,7.8. 3>^| 19. TTiere Is a house with 3 tier of windows, 3 in a iir, the lieight of the first tier 7 feet 10 inciifs, the second 3^7^ feet 8 inches, and the tliird 5 feet 4 inehes, the bi'.adth 232o Ij- ^,^,ch is 3 feet 1 1 inches, what will the glazing come to Hid. per fool? ,, - * . uodecimah. 7 . 10 :he 6 . 8 height's 15.4 added »' \\d .10 'i=xvindGXvs, Jeet in. .pts. 233..0..6 at lid. per foot 2d. ^ 233 =1.?. 3S..10 = 2d. ^ = 6 parts. p9 . 6 in a tier. 2I0)27:1..10}. If! ?.9.9. Ji.11.3. 7.10.4. ^0.10.6. 77.6.10. 88.6.3» 02.2.4.11.11. b.2.l0.4.6. 8 3 . 11 in breadth. £13..11..10^ Ans, 6 6 . 6 '•yr p-i 3 . 0. What IS the worth of 8 squares of glass, each meas- ng 4 feet 10 incheti long, and 2 feet 11 inches broad, at .jperfoot? Ans, £l..l8..9. 1. There is 8 windows to be glazed, each measures [oot 6 inchds wide, and 3 feet in height, how much will y come to at 7^d. per foot? A.s. £l..3..3. 2. What is the price of a marble slab, whose length is t measures, vileet 7 inches, and the breadth 1 foot 10 inches, at 6^. per foot. |t? Ans. £3..l..d. Ithe yard ,g, &c. by thi MEASURING BY THE YARD SQUARE. It i whose squarl -^^ Paviours, Painters, Plasterers, and Joiners, ' loTE. Divide the square Jbot by df and it will give the :h ,i?iIJvi i^f^er of square yards. lii '^M ■i'.! !!,! V!''' i^: iHi ii:!! r it t' /'.' .•l !■ 160 Duodecimals. THE TUTOI V iSSIS F.XAMPLK8. '«:3. A room k to be ceiled, whose length is 74 foci 32. inches, and width 1 1 feet 6 inches, what wijl it come tJvas tc V^s. I0\(l. per yard? Atis. jl,'\ii..U)..\maem\i 24-. What will the paving of a court-yard come to,|7 fire j A^d. per yard, the length being 58 feet 6 inches, 6 inchi breadth 54 feet 9 inches ? Jus, 4'7..0. 1( Iwo of 2^. A room painted 97 feet 8 inches about, and 9 f5 ih iO inches high, what does it come to at 2,?. 8Jrf. per ya » iO ft j4n'!.£H..\\..] tome t £6. What is the content of a piece of wainscottiiij 3.3. yards square, that is 8 feet 3 inches long, and G feet 6 im n long broad, and wliat will it come to at 6s. Ijid. per yard ? 'I tri Ans.£l..\9.l quare: 27. What will the paving a court-yard come to at Si. Not ):cr yard, if the length be 27 feet 10 inches, and ttkpH t i)readth 1 4 feet 9 inches ? Ans. d*?. A. Me then 28. A person has paved a court-yard 42 feet 9 incli«o(/' is front, and CS fett (3 inches in depth, and in this he Imread/h tuurvtiy tlio urpth of the court, of 5 feet 6 inchSke irue li! eacUh ; the footv»ay is laid with purbcclc Stone, at SsMod or s fo yard, und the rest wiih pebbles, at 3^. 2;r'r yard, J S4i, \ V. iiltlie whole come to? j4ns. ae49..17.. qunre ; 29. What will the plastering a ceiling, at lOd. per j 7 feet come to, suppoi?ing the length 21 feet 8 inches, and iches o breadth 14 feet 10 inches? Ans. £l..9.> SO. W'hat will the wainscotting a room come to a per square yard, supposing the height of the room (ta in the corHice and moulding) is 12 feet 6 inches, am j^q^^ compass S:< feet 8 inches, the three window shutters ^^^-^^ ^ 7 feet 8 inches by 3 feet 6 inches, and the door 7 fef ^yg ^^ .'^ feet 6 inches ? The shutters and door being worke jjule both sides, is reckoned work and half work ? ' ^ajf Ans. i,36..12..I yj^j^j MEASURIXG BY THE SQUARE OF 100 Fl ■ Js Flooring^ Partilioningy Roojingy Ti/Iing, S^c EXAMPLES, 31. In 173 feet 10 inches in length, and 10 feet 7 i in height of partitioning, how many squares ? AnSyl^ sguares, ^^J'^et, 8 inches. in *^-tiirs\ii ^r^^ 35. If S3 two ? 36. 1( dies Jj ntaiu? lUl THE TUTOI iSSISTANT. Duodecimals, l6l jrth is 74 ft'cl 32. If a house of three stories, besides the ground floor, will it come tJwM to be floored at J[^Vy,.\O.A) per ^quare, and the house Ins. /'I8..10..llwcaaured 20 feet S inches, by IfJ feet 9 inehcs: there are ard come to,p fi*^ places, whose measures are two of 6 f^et, by 4 feet ^et 6 inches, m iaches each, two of 6 feet, by 5 feet -t inehcs eaeh, and Ans* JL*7««0. ic|t^<* o^5 ^'^**^' ® inches, by 4 feet 8 iaches, and iho seventh About and 9 ip^5 feet '2 inches, by 4 feet, and the well-hole for the stairs U S^d. per yaV ^^ ^^^^ 6 inclics, by 8 feet 9 inches, whatwiU the whole ii/js. Vu-.l 1 ..!|on»e to ? A)is. 4'5 }.. I Ji. :i:):. f wmnscottinJ 3i5. If a house measures within the walls d'-l feet 8 inches and 6 feet6 in«tt length, atid 30 feet 6 inches in breadth, and the roof be / «5i, A71S. iei..9.. . ■ 00m come to a MEASURING BY THE ROD.' ^yj -^ of the room (tt 2t 6 inches, an( 'fifoTE. Bricklayers always xmlue their tvnrk at the rate of indow shutters j^/^^ q/kJ a half thick; and if the thickness of the wall is the door 7 fel(„.g or less, it must be reduced to that thickness bijihis )or being "workeljIuLE. Multiply the area of the wall by tlie number work ? ' half bricks the thickness of the wall is of; the product, Ans. £36.'I2"|vided by 3, gives the area. EXAMPLES, IE OF 100 Fl . ^yt^-^ ngy TifUngySiC ; |35. If the area of a wall be 4085 feet, and the thick- ss two bricks and a half, how many rods doth it con- and 10 feet 7 i '" ? Ans. 25 rods. luares? '^^* ^^ * garden wall be 254 feet round, and 12 feet 7 feet d incheSf K '^hes high, and 3 bricks thick, liow many rods doth it ' %yxi ntaiu? Ans.2S rodsyiaefegt, . ^ 1:11 f '!: i I 'M I f !:'i Uh ii ;1 '^if: M,' :,i! r 1 162 Dmdechnals, THE TUTOU 37. How many square rods are there in a wall 62 J ft long, 14* feet 8 inches high, and 2i bricks thick? Ans,S tods^ \&1 feet. S8. If the side walls of an house be 28 feet 10 inches length, and the height of the roof from the ground 55 f( 8 inches, and the gable (or triangular part at top) to r 42 course of bricks, reckoning 4 course to afoot. Ni 20 feet high is 2i bricks thick, 20 feet more, at 2 bri( thick, 15 feet 8 inches more, at 1| brick thick, and t gable at 1 brick thick, what will the whole work come at ;f 5..i6«0 per rod ? Ans, /'48..13..5i m^ '-<4t /a^fft \m&m^^r^ -m «;&* ^^t- (»*A^1^*' Multiplying several ^guj-es hy several, and the prodijifit to - * ^ ■ "' produced on the line only. ■ ■■im^ Hule; Multiply the units of the multiplicand by units of the multiplier, setting dqwn the units qf the p duct, and carry the tens ; next multiply the tens in the m tiplicand *by the units of the multiplier, to which add product of the units of the multiplicand multiplied by tens in the multiplier and the tens carried ; then multi the hundreds in the multiplicand by the units of the mi plier, adding the product of the tens in the multiplic multiplied by the tens in the multiplier, and the units of multiplicand by the hundreds in the multiplier ; and so } ceed till you have multiplied the multiplicand all throii by every figure in the muitiplierrf^-v "s . V'- ■J-^ * ■ EXAMPLES. Multiply - - S5234 hy - - 52J24 1847107216 tv> '\ 362S4 ' 5J* *242» '•;-• ■.f:,\ ai, k.-.. ^^kv^::^\ 140936 70468 140936 70468 176170 i' ■ " ; t > 184710721& ti f . •*'*( THE TtJTOtt n a y/9\\ 62]^ f( thick? todSi . 167 feet feet 10 inches le .ground 55 fi irt at top y tor to a foot. ^ more, at 2 bri( :k thick, and ole work come d the prodiift U ultiplicand by B units qf l^he p :he tens in the n , to which add multiplied by Led ; then niulti units of the mi n the multiplici and the units of iplier ; and so \ licand all throu •> .\ * * '.^ v.- ■!-*-,- ' ' '' "' . 352S4 S242S ASSISTANT. i ■^■y.. Jkiodeckiialif 163. IKPtANATION, „ First, 4X4=:16, that is 6 and carry 1, Secondly, 3X4? Hr.X!2 and I that is carried is. 21, set down 1 and carry 2. Thirdly, 2IX4+3X2+.4-K4+ 2 carried =32; that is 2 and carry 3. Fourthly, 5><4+2X243X4+.4X2+3 carried =47; set down 7 and carry 4.. Fifthly^ 3X4+5X2+2 X4+3X2+4X5f4. carried=s60; set down and carry 6., Sixthly, 3X2+ X 4+2X2+3X5+6 carried = 51; set down 1 and carry 5* Seventhly^ 3X4+5X2+2X5+5 carried=r)7, that is 7 and carry 3. Eighthly, 3X2+5X5 *3 carried=34; set down 4 and cany 3. Lastly, 3X5 +3 carriedaeis ; which being multiplied by the last figure in the multiplier set ths whole down, and the work is iiiiished. tfc,j(w,^ , ,-. -<*t».;;.^*w-^-* w»»-^» '% t 'vi 'ts''\. •>. t. f^nz^' *• t' *y )♦» !'?» t'\ ♦! 140936 70468 10936 )468 70 710721a ' u-} ' ■ ........ I . _]^^_ , ; : ^ - , ■ ■ ^' ' ■■■' 1 '■■ i . 5 ■."■■■-'. MV'rvo*- --t- *.■ v.1-;*^-A- ^i r;'*^ I : 1 ■' )■'} v'j. ., i ri.''i !'♦* i-i "t r ■ i. >!:('■'*■> J:!f; 1 iij ill I -■■^>i; ) - ijli! lift i V t! iliiilj !l V. ...I" 11.,' f'i^i il ■i .' • il -r, I ti '.■I, ■,*; I III! i' I ut.. 'A C 164 ] .1 " . THE tutor's -:''■■* > 1 - THE I'-i: '{ ■??:-,* TUTOR'S ASSISTANT. <\u '■•cS >"( tiO' At- ,..,-1. PART V. . > ( thje; meats c7/e^ no n of circles, Sfc-. A CIRCLE is a plain figure, contained under one line which is called a cixcumfcrence, unto which all line! drawn from a point in the middle of the figure, called thi centre, and falling upon the circumference, are equal thf one to the other. The circle contains more space than an] plaia figure of equal compass. The proportion of the diameter of a circle to the eircuir. lerencewas never yet exactly found, notwithstanding man eminent and learned men have laboured very far therein ^rnong whom the excellent Van Ceulenhas hitherto outdon al', in his having calculated the said proportii>n to thirty-si places of decimals, which are engraven upon his tomb-ston in St. Peter's church in Leyden. Let it be required to find the area of a circle, whose di{ meter is an unit. By the proportion of Van Ceulen, it'tl diameter be one, the circumference will be 3.1 4-1 59265, &( of which 3.1 416 is sufficient in most cases. Then the ru teaches, to multiply half the circumference, by half the d.! meter, and the product is the area : that is, niuiti))ly 1.67( by .5, {viz. hali' 3.1116 by half 1 ) and the proJact is ,7b5' ^I'.ich is the area of the circle, whose diameter is 1, ASSISTANT. Memuratwn. 165 HE tutor's '•• it' > .. . . •r 1 lNT. CLES, Sfc: I under one line to which all lines gure, called th e, are equal thi re space than an] le to the clrcuir, thstanding man very far therein hitherto outdon tion to thi»ty-si )ri his tomb-ston circle, whose dif an Ceuien, it' tli 3.14-1 59265, &( . Then the rul c, by half the d.i 5, njuitiply 1.67(' pro J act is ylbS' iieter is U Again if the area be required when the circumference ii 1, first find what the diafneter will be, thus: 3. 1416: to 1 : : 1 to 318r.09, which is the diaweter when the circum- ference is 1. Ihen multiply half .318309 by half 1, that is .15.9154 by 5, and the product is .079577, which is the area of a circle whose circuniferencels 1 . If the area be given to find the side of the square equal, you need but extract the square root of the area given, and it is done. So that the square root of .7854 is 8f^62, which is the side of a square cquiil when the diameter is 1 . And if you extract the square root of .079577 it will be 2821, which is the side of the square equal to the circle whose circumference is 1. If the side of a square vv'ithin a circle be required, if you square the semi-di;uneter, and double that square, and out of that sum extract the squtu-e root, that shall be the side of the .square, which may be inscribed in that circle : so if the diameter of the circle be 1 , then thu half is .5, which squar- ed is .2.5, and this doubled is .5, whose square root is .7071 > the side of the square inscribed. From what has been here said, the ingenious scholar will easily perceive liow all other proportional numbers iire found, and may examine them at pleasure. We shall now proceed to the different problems. - -» Problem 1. Having the diameter and circumference to find the area. Every circle is equal to a parallelogram, whose length is equal to half the circumference, and the breadth equal to half the diameter ; therefore multiply the circumference by half the diameter, and the product is the area of the circle. Thus, if the diameter of a circle, that is, the line drawn cross the circle through the centre, be 22.6 ; and if the circumference be 71, the half of 71 is 35.5, and the half of 22.6 is 11.3, which multiplied together, the product is 401.15, which is the arcct of the circle. Problem 2. Having the diameter of a circle to find the circumference. As 7 to 22, BO is the diameter to the circumference. Or, as 113 to 355, so is the diameter to the circumference. Or, as 1 to 3«14'1593» so is the diameter to the circumference. Let the diameter, as in the first problem, be 22.6. This multiplied by 22, and jives^ I it*' : !h;l / f I i :■) (U> < ^4 ^ ; 166 Mensuration. THE TUTOR 8 71 .028 for the circumference ; but the other two proportions are more exact, as appear by the following work. 22.6. 355 , ' ' 22.6 1884-9,758 6283186 6283186 : ; ' 2)30 >,.,,;>■: 1 710 , • i^ • 710 710000018 118)80230 (Ti- lls '■ty^- • • »- ASSIS' c.'idin forpiu square square the ar« Prti the arc Beca circum its circ its circi nother circumf 07^»SS tlie oil ore It w Pro/}/ quare c If the Problem 3i Having the clrcunifecenty? oi"a circle, to find tlie diametor. As 1 is to. .3 18309, so is the circumference to the dia meter. Or, as 356 to 113, so is- the circumfertnce to thelifrcto dia.Tieter. Or, as 22 to 7, so is the circuBiference to the d'ametvr. , V 1-y.t tlie circumference be 71, and then proceed wkh. either of the three proportions, as follow ; 318:^09 71 J; 18 309 2^28163 22.5j9')39 <»» ;.* 113 71 113 791 355)8023(22.6 . . 923 2130 71 ". 7 ^^--• 22)497(2?I9 57 130 200 tlie dia ProCt\ f the sn If th( liuare w '^e circ ProO/e uare, m If the scribed .98016 Jmtter, ubif wj quare \ tf tJie Thus, by the second proportion, the diameter is 22.6 ; jbed w but by the other two it falls something short. (),«2I t '^ro/jlen Problem 4-. Having the diameter of a circle, to find the t' the a area. ,J732. All circles are in proportion one to another, as the square! square of their diameters, (by Euclid, lib. 12. prop. 2 ) Now thcf ''^^^'^J area of a circle, whose dimneter i» 1, will be .785398, ae ASSISTANT. M 6)1 sura lion. 1^ E TUTOR SI ) proportions! irk. 5 5 1 '11 a circle, to find to lite C'.rding to Van Ceulen's proportion beforenientioned ; but for pructice .TS.'jJ will be sufiiciei t. Therefore, as 1 (the square ol tilt; djameter I) is t-t* ''>.&\, so is 510,76 (the isqUiire of 3 .\3 the diamttcr of the given circle) to 4-1,15 the ar« a of tht: |?iven circle. Pr»blan 5. Having the circumference of a circle to find ithe area. Because the diameters of circles are proportional to their Icircumfer- nces ; that is, as the diai'Vjtkr of one circle is to ts circamfireuce, so is the diameter of another cix le to its circumference : therefore the areas tf circles art to one inother, as the squares of the circumferences. And if the ircumfercnce of a circle be 1, the area of that circle vi'ill be |071^»5S ; tlien the squre of 1 is 1, and the square of 71, the circumft rence ol' the former circle) is 5iiil. There- lore It will be, as 1 : 079S8 :•: 5041 :-t01, 16278. Problem 6. Having the diameter, to find the side of a uare equal in area to that circle. it the diameter of a circle be 1 . the side of a square equal liereto will be .8S62 Therefore as 1 : 8862 : : 22.6 the diameter) : 20,02812, the side of the square. ProL'tm 7. By having the circumference, to find the side the square equal thereto. If the circumference of a circle be 1, the side of the uare equal will be .2821. Therefore as 1 : 2f21 : ; 71 'ie circumference) : 20,0291? the side of the square. Problem K Having the diameter, to find the side cf a uare, which may be inscribed in that circle. If the diameter of a circle be 1, the siele ot the square cribcd will be .7071. Therefore, as 1 : 707' : : 21^6 : 98016, the side inscribed. Or, if you squa.e thesemi- mtter, and double that square, the square root of the ubie sfjLiare will be the side of the square inscribed. '^robtetn 9. Havi»g the circumference to find the side of whch may be inscribed, f the circumference he I, the side of the square in- iameter is 22.6 f bed will be .2251. Therefore, as 1 : 2251 : : 7l ; 9821, the side of the square. i- ' rable'm 10. Having the area to find the diameter. ■ the area of a circle be 1 , the square of the diameter J732. Therefore as 1 : 1.2732 : : 401.15 : v)10,744-180, ice to the aference iference to th a proceed w nm 71 2)4.97(2?S9 57 130 200 2 as the g-yjjfgjsquare root of which is 22.599, the diameter. ^ow thel ''^^^^^^ ^^* Having the area, to find the circumference. 2 ) Now e .785398, ac^ .l')'-l,y '.}.ifJM 1»'->J S-i i-4l»lv.»-!. ~--- 81 1 HI 111 , vM iii'i il 1^1 m" 'i 'it 16S Mensuration, If Ml' -^.i .')i THE TlTTOH*S tssii ;«' land t e22 he] eter irde to ircle, e ei| by the e qu£ Thes le wh< rcle, g If the area of a circle be 1, the square of the circumfei- rence will be 12.566S7. Thtreibre. as 1 : l2.5i>:VX7 : • 40'. 15 : 504.0.99932530, the square raot of which J* 70.9999. Problem 12. Having the area/ to find the sid<- ora square mscribed. If the area of a circle be 1, the area of a squiv-o inscribed within that circle will be .6306. Fhcrefore, as i • 10] i5 : : .6366 *. '.'55. .72090, the root of which is 15.980, the side of the square sought. :, -i4{v: Problem 13. Having the side of A square, to find the dia- meter of the circnniiicrihing circle. If the side of r square be I , the diameter of a circle that will circumscribe ^"sat square, will be 1.4<142. Iherefore, as : \.4iW2 : : L 98 : 22.598916, the diameter souuht. Problem 1 1 Having the side of a square to find the dia> meter of sGri'i^i cqcai to it. If the Pide c ' -,: square be 1, the diameter of a circle equal to it will h.- 1.128. Therefore, as 1 : 1 128 : : 20.0291 : 22..^925w48 the diameter required Prohiem 15. Having the side of a square to find the cir- ^ ^P' cumference of a circumscribing circle. In* j • If the side of a square be 1 , the circumference of a circlej*,. '*^ that will encompass that square will be 4.44 H. Therefore, as J*^3Jiie 1 : 4.44:i : : 15.98 : 70.99914, the circumfcrenc- rt^quired, "1® su Prohiem 16. Having the side of a square, to find the cir< *"^ ^^ cumference of a circle that will be equal to it. If the side of a square be 1, the circumftrence of a cir cle that will be equal to it is 3..545 Then, as 1 : 3.545 : 20.0291 : 71.0031595, the circumference. Note. In several of the foregoing problems, tvhere th diameter and circumference are required^ the aniwers are no exactly the same as the diameter and circuniference oj th given circle, but are sometimes too mv(h. am! sometimes P little, as in the tvoo last problems, ivhere ,he answers in enc should be 71) the one being too much, c^n'. the other too ^'th The reason of this is, the f^mnll defect luat happens to be i the decimal Jradions, thetj being sometimes too great, an sometimes too little ; yet ilie defect i:> so smjll, that it is nea less to calculate them to more exactness. Of the Semicircle. To find the area of a semicircle, multiply the ftMirth p£ of the circumference of the whole circle ky the semi-Uiamet f ill tutor's liSSlSTANT. j^fiensuration, 169 e circumfei- of v^hiclV !• k; of a square inre inscribed IS I : 101 ;S s 1S.9.S0, tlie land the product will be the area. Suppose the diameter Ibe 22,6 and the half circumference, or arch line, is 35,5. [he half of it is 17.75, which multiplied by the serai-dia- jeter 11,3, the product is 200.575, the area of the aenii- kircle. , > , ''*■ .V ''■'■"'. " • ''' X' Of the Quadrant, To find the area of a quadrant, or the fourth part of a lircle, multiply half the arch line of the qliadrant, that is, e eighth part of the circumference of the whole circle by the semi -diameter) aid the product will be the area of )f a circle that |e quadrant. . . _ . ,. . These are the rules commonly given for findmg the ea of a semi-circle and quaelrant ; or find the area of le whole circle, and then take half the area for the semi- rcle, and the fourth part for the quadrant. o find the dia- l. Ihcrefore, meter aoujiiit. to find the dia- ler of a circle 8 1 : 1128 : : d itoftndthecir To find the Solidity of a Sphere or Globe. rence of a circle Therefore, as A sphere or globe is a round solid boiy, every part of surface being equally distant from a point within iff lied its cratre. To find its solidity, multiply the axis, diameter, into the circumference, the product of which ^^^■^aui'r'ed ^^^ superficial content. This multiplied hy u sixth parr, crenc • It^q . ' ^j^^ axis, thp nrndnnh is the KoHilifv. to find the cir^ it. ftrence of a cir as 1 : 8»54?5 J the axis, the product is the solidity. blems, where th e an-wers are m ourtijerence of th ind sometimes P e answers in enc the other too 'M \ happens to be i ,es too great, ai\ 4ly thai a is nee ,ly the fi^^'vth pj the senii-diamet »vV«fc?f • . u m.. ■a H /■ ' \ ■II i.f V f! ,1 I'l.l, I i ^ .» :4*' •| K .1 , !. ri 1 . :. ••-* >tl T' t 170 3 THE THE TUTOR* TUTOR'S ASSISTANT- PART VI. A COLLECTION OF QUESTIONS. 1. \757'^^'^ ^ ^^ ^'^® value of H barrels of soap Y V 4^flf. ;9er /6. each barrel containing 254- //a ? , LSS] 10 \lb cc II ■cost 12 land { (come 13. vn- '■ pany fnodit |(>r uhi Ifiuall 1.5. li'lSU 16. ortli , y^WA-. £66..\?j..6. 2. A and B tra-Ic together : A puts in £^'>2{) for months, B £160 for a months, and they gained ;^'IOi .What must each man receive ? jins. A £5">..l.S..9-:^f anrf B ^+6..6..2jVh- S. How many yards of cloth, at lis. 6d. per yard, c I ' avc for 13 cnot. 2 qrs. of w«ol at 14^/. per lb? Ans. 100 yards, S qrs.. }. 4. If I buy 1000 ells of Flemish linen for 4*50, wh in;-y I sell it per ell in London, to gain ;^*10 by the whole jins. 3.V. id. per ell. [^ ^'^ 5. A has 64-8 yards of cloth, at Hv, per yard, ready mip^' "^ ney, but in barter will have \6s. B has wine at ;^4-2 /j tun, ready money : the question is, how much wine mi be given for the cloth, and what is the price of a tun wine iivteter ? . Ans.£\^ the tun, and 10 tun 3 h/i " \2'l ^KiOf ivine mut be given for the clol\ 6. A ^Jeweller sold jewels to the value of £1200, which he received in part 876 French pistoles, at I65. each, what sum remains unpaid ? Ans. £117. .6, 7. An oilman bought itll cvot. 1 qr. \5 lb. gross weigl of train oil, tare 20 lb. per 1 12 lb. how many neat gallo "^^squ were there, allowing 7^ to a gallon ? Ans.. ."'120 gallons, ^f' 8. If I buy a yard of cloth for 14.,s. 6d. and sell it for 1( ^"'e; \l. what do I gain per cent ? Ans. £15.. 1 0..4< ni'le f} 'I 9. Bought 27 bags of ginger, each weighing gross 84";^ tav< ]^./(i,. pgr bag, trett ilb. per lOUb, what do they co to at^i,/. p^r lb i' Ans. ^76..13..2§.| / fence, ij 17. ersons uch f hare i 18. 19. feet 20. lonths low m uch t 21. each THE TUTOR LssisT.iNT. A Collecllon of QitcsUons, 171 10. If *^ of an ounce cost Z of a shillln'r, what will f of a \\h cost ? Ans. 17.?. ()f/. (>. 11. If J of a gallon cost f of a^*. what will I of a t un Icost ? Ans. £\' \NT. 'IONS. arrels of soap a aininjT 254- ^/''' ? ' *" ed /"10( ''■orth_;^13 apiece, how much will make good the al;fe ^y o *» ,^QQ in case thev interchange their said drove of cattle ? 12. A gentleman spends one day with another £1 ..7..!(^|. and at the year's end layeth ui)^".''J40, what is his worlj in- come? Jus. £US.Jl'..1i. 13. A has 13 fother of lead to send abroad, each hiln^j^ 19^ times 112. . B has 39 casks of tin, each S^^ld. how many ounces difference is there in the weight of tlicpe com- iiiodities ? Ans. t! 1 2 1 (JO r,-. ii. A captain and 160 sailors took a prize worth €l;;(jO. i»r which the captain had ^ for his share, and the re.*t wa-r Lf^ually divided among the sailors, what was each man's part? A71S. the captain had £272. and each sailor £\^...\Q, 1.5. At what rate fcr cent, wil) £'^)56 amount to ^ISl^-lO. in 7}j years, at simple interest? Ans. 5 per cent, 16. A hath 21< cows worth 7-«". each, and B 7 1 ovse.i good the al;fer- they Ans. £i..V?., 17. A man dies and leaves £120 to be given to ilirce )crsons, viz. A, B, C : to A, a share unknown; li twice as nuch as A, and C as much as A and B; what was the ihar^ of each ? Jns. A £lO, B jrM\ and C /;60. 18. There is a sum of 3^1000 to be divided among 3 men, n such manner, that if A has £'^, B shall have iC5, and C ^8, how much must each man have ? Ans.. A /;i87..U), B,;^312..10, andC£oOO. u ,.,;.. ^ r.,„ 19. A piece of wainscot is 8 feet 6 inches h long, and much wuie nui „ ^ r^ - ^ - ■ , i i. • .i r ■ } ' ^ 1-j .:_„ ..c -> f.,r. f reet 9 inches ^ broad, what is the superhcia! content .'' Ans. 2\' feet 0..3..^..G. 20. If 360 men be in garrison, and have provisions for 6 Qonths, but hearing of no relief at the end of 5 month?, iow many men must depart, that the provisions may last so nuch the longer ? Ans. 'i8S men. 21. The less of two numbers is 187, their difference 34, i £i6..6..^-' 2!t n' i&, .''120 gallons and sell it for 1( ighing gross Si' diat do they coi [«5.^76..13..2^ \d. per yard, ca per lb F fards, 3 qrs.. \. n for £S0, will 10 by the whole , 3.S-. 4rf. per ell r yard, ready m( wine at £\?. p wi price of a tun uid 10 f«n 3 hh (riven for the clol Fueof £1200, " igtoles, at 165. (5 /#«*. ;^'l-77..6. 5 lb. gross weig ^^ ^^^^^^.^ ^^ ^^^ product is required ? u-Ins. 1 707920929. "^"^.o" -_//.., o 22. A butcher sends his man with ^'216 to a fair to buy attle; oxen at £ll, cows at 40^. colts at £l..5, and hogs t £1..15. per piece, and of each a like number, how many if each sort did he buy ? jins. 13 of each sort, a}2dS o''-cr. irf.-- li 'II h\ :" ji . '::il ■ili' :'t| •|h m j:f[|N /.'/ i ■•If ;.( 172 A Collection of Questions* the tutor's '23. What number, added to II4 will produce 36f|^? Ans. 24^;§. 24 What number, multiplied by ^, will producell y\. Am. 36 if. 25. What is the value of 179 hogsheads of tobacco, each weigianf* 13 cxvL at £2.-7. .1 per crvt ? Am. ;^54f78..2..ll. 26. M} factor sends ine word he has bought goods to the value of .300.. 13. .6. upon my account, what will his com- mission come to at 3 J per cent¥ ^ns. £17..10..5, ^grs-j*^^^. 27. If ^ of 6 be three, what will ^ of 20 be ? Ans. 7 J. 28. What is the decimal of 3 qrs. Wb. of an cwt ? Alls. ,875. 'J9. How many lb. of sugar, ^tA^d.per lb, must l.c given in bart«.r for 60 gross of incle, at 85. Sd. per gross ? Ans. 1386|. SO. If 1 buy yarn for 9d. the lb. and sell it again for \^d. ■per lb. what is the gain per cent f Ans. 50. 31. A tobai -"onist would mix 20/^. of tobacco at 9^^. per Ih. with ijQlb, at \9.d per lb. 4>0lb» at 18d. per lb. and with V2lb. at 2s. per lb. what is alb. of this mixture worth ? Ans. \s. 2\d.f^. 32. What is the difference between twice eight and twenty and twice twenty-eight ; as also between twice five and fifty and twice fifty-five ? Ans. 20 and 50. 33. Whereas a nob'p and a m-'jrk just 15 yards did buy ; How many ells of the sat* i cloth ibr £50 had I ^ Ans.60Q, :J1. A broker bought /or bis principal in the year 1720, j^" iOO capital stock in the South Sea, at £650 per cent, and 'Ul it again when it was worth but ;^130 per cent, how • was lost in the whole ? Ans. £2080. .. t, C! hath candles at 6a'. per dozen ready money, but in •''' ,vill have 6s. 6d. per dozen ; A hath cotton at 9d. per lb. ready money ; I demand at what price the cotton must be at in barter ; also how much cotton must be bar- tered for 100 dozen of candles? Ans. the cotton at 9d. 3 qrs, per lb. and 7 civt. qrs. 16/6. of cotton must be given Jbr 100 dozen of candles, S6. If a ckrk's salary be ^^"73 a year, what is that per day ? Ans. 4*. 37. B hath an estate of £53 per annum, and payeth 5>. lOd. to the subsidy, what must C pay whose estate is wottii £i 00 per annum f ^ Ans. lis. Od.-^. ASSn 38. ling, J the rn and h 39. remaii 40. biishe of eac 41. the wl 42. were i 43. ofioi je220( contin he foi the CO first? 44. them ii seconc is;^"28 45. per cei 46. 4 mon ill teres 47. tred cover 1 48. une o ivas bu r'tarly 49. n- a d von J an lomcn ach ? HE TUTOR** luce 36J|t ? jducell/if. Am. 36|f. r tobacco, each .;f54.78..2..ll. lit goods to the ,t will his com- »e ? -^ns. 7i« an cu>t ? Ans. ,875. , must he given ' grbsa ? Ans, 1386|. again for \^^d, Ans. 50. acco at 9rf. ^er pr /^. and with ire worth ? vice eight and i^een twice five s. 20 and 50. rards did buy ; id 1 3 Ans.mo. the year 1720, ;0 per cent, and per cent. Kow ^»s. £2080. f money, but in cotton at 9d. rice the cotton m must be bar- Wot.O qrs. 16/6. » of cartdles. hat is that per Ans. if. im, and payeth whose estate k s, J Is. Orf. j%. ASSISTANT. yf Collection of Quehiions. 173 38. If I buy 100 yards of ribband, at 3 yards for a shil- ling, and 1()( more at ? yards for a shilling;, and fA\ it at the rate of 5 yards for 2 shilling, whether do I git or lose, and how much ? Ans. lose :;,v'. \d. 39. W iia number is that, from which if von take '{, the remainder will be J ? * yi;/.y. f J*. 40. A farmer is willing to make a mixture of rye at 4-?. a bushel, barley at 3s. and o..ts at 2s. how much mufet he take of each to sell it at 2^. Gd. tlie bushel ? Ans. 6 ()f rye, 6 of bnrleif^ and 24* of oats. 41. If ;? of a ship be worth j6'3740, wliat is the worth of the whole ? Ans. ;^9973..6..S. 42. Bought a cask of wine for £62..8, how many gallons ^vere in the same, when a gallon was valued at 5s. 4af. ? Ans. 231-. 43. A merry young fellow in a small time got the better of \ of his fortune ; by advice of his frienus he gave j£220O for an exempt's place in the guards ; his profusion continued till he had no moje than 880 guineas left, which he found by computation was 7^^^- part of his money alter the commission was bought ; pray what was his Ibrtunc at first ? Aus. £10150. 44. Four men have a sum of money to be divided amongst, them in such a manner, that the first shall have -!, of it, the second \, the tliird |, and the fourth the remainder, which is ;^'28, what is the sum ? Ans. £i\2. 45. What is the amount of jSIOOO for 5 years i, at l^ lier cent, simple interest? Ans. ^'126 1. .5. 46. Sold goods amounting to the value of £lOO for two 4 months, what is the present worth, at 5 per cent, simple iatercst ? Ans. £M'1..\ 9..o\ ,1^7 . 47. A room 30 feet long, and 18 feet wide, is to he cov- ered with painted cioth, how many yards of } wide vvill cover it ? » " . An^:. SO yards. 48. Hetty told her hrcthcr George, that though' licr for- tune on her marriage took £i9.'>12 out of htr faniily, it ivas but ? of two years rent, Heaven be praised I of his f'carly income ; pray what was that? Ans. ;^*1G093. 6. .8 a year. 49. A gentleman having Z'ds. to pay among his labourers -K a day's work, would give to eviry b« \ 61, £,4 part; divide this equitably among them, ac $ci:ding to tlieir father's intention. Am. A £\52m, B £\ 1414^ C ^91||j I> ^76H!> E £65i|t, $9. When first the marriage knot wiw ty'd Between my wife and me. My age did her's as far exceed As three times three does three ; But when tMi years, and half ten yearo^ We man and wife had been, Her age came then as near to mine, As eight is to sixteen. H'tfif, What "was each of our ages when we married ': Af)S» jt^j years th^ manf \o ihc iKoman, f :7' FHU TUTOR^S 1 A3SI8TANT% A'COlt^Hotl of ^tSt'tmS, 17S ,i..'iii IS longi 2 feet 9 low many solid feet, 3 inches. f war's crew of ce, each man's Ans je23040. ch crowns at \s, must pay ahalf- nust he receive ? sd ;ClOO; Bput i receive ^60 of [srest arise in one f An. £4-00. e^h, how many Ans. 76|?. , 2 qn. 5 lb., how It, each weighing Ans. 356t^|f . md 72 taken fron e leaves 12, wliai lave taken 48, 72 Ans. 158. rdered £500 to b A. says he, J, B ^ among them, ac WM, C £91ii| £65i||, 60. If 12 oxen wiJl mt 3^ acres of gras* in four weekg and 21 oxen will vrv JO ncres in 9 w pVp bow many oxen will eat 24- acres in 18 weeks, the grass oting allowed to grow uniformly. If f?^ acres r 12 oxen : : 10 acres 36 oxen, which 10 acres will keep in 4 weeks. Inversely, as 4 weeks : 36 oxen : : 9 weeks : 16 oxen, to be kept in 9 weeks. The growth of the grass on 1# acres in 5 weeks, will be so much IS alone would feed 5 oxeji 9 weeks ; that is, 21-— 16=5 oxen. Inversely, as 9 weeks : 5 oxen : : 18 weeks : 21 oxen in 18 weeks. 18 weeks-r^ weeks == 14' weeks, 9 weeks — 4 ==5 weeks. Inversely, As 14 weeks : 2\ oxen : : 5 weeks : 7 oxen^ 7 oxen+8 oxen =15 oxen, which 10 acres will keep or feed in 18 weeks. La^tly^ As 10 acres : 15 oxen : : 94 acres : 36 osiien. . ty'd e; ) yearsj^ ne. hen we married r If?, 15 the •woman, C ■4 "'ii !( !; 1 mI Mi 1ii A TABLE for findipc the Interest of any Sum number of Months, fVeekSy or Days at any rate of Money for any per cent. Year. Cul«n. Mon. Week. Day. £. £. s. d. £. s. d. £. a, d. 1 1 8 4§ 1 2 3 4 9 l| S S 1 l| 2 4 « 8 1 6i 2{ 5 8 4 1 11 3i 6 10 2 Sl - Ai*f- 7 11 8 2 84 : n 8 13 4 3 1 5| 9 IS • 3 5| 6* 10 16 8 3 loi 6| 20 1 13 4 7 8i 1 7| 30 2 10 11 6^ 40 3 6 8 15 4* 19 sf 2 2^ 50 4 3 4 2 9 60 5 1 3 ci- 3 si 70 5 16 8 1 6 11 3 10 80 6 13 4 1 10 9l 4 4i 90 7 10 1 14 74 4 Hi 5 5i 100 8 6 8 1 18 5t 200 16 3 4 3 16 11. 10 llf 300 2S e 5 15 41 16 24 400 33 6 8 7 13 10 1 1 11 500 41 13 4 9 12 3i 1 7 41 h 600 50 11 10 9 1 12 10| 700 58 6 8 13 9 si 1 18 4| 800 66 13 4 15 7 84 2 3 10 900 75 17 « 1^ 2 9 5| 1000 83 6 8 19 4 7f 38 9 2I 2 14 of 2000 166 13 4 5 9 7 5000 250 57 l3 10 8 4 4i 4000 333 6 8 76 18 5\ 10 19 2 5000 416 13 4 96 3 o| 13 13 111 6000 500 115 7 84 134 12 3» 16 8 9 ril 7000 583 6 8 19 3 6| 1 8000 666 13 4 153 16 11 21 18 4i le 9000 750 1 73 1 6i 24 13 11 10000 833 6 8 192- 6 V 27 7 Hi "J^ 20000 -166G 13 4 384 12 i. 54 15 10| ^ 30000 2500 576 18 si- 82 3 10 f«K(<'LE. Multiply the prinoiial hy the rate per cent, and the number of months, vecfifi, or j^, ivhick are required, cut off ti'w figures Oii the ri'jht hmi side of :h^' product, and collect frcni the tahlc the st*;- eral sums agniust the different n.imbcra as when addt-d will tn ike the number renaining. And the several sums together will give the in- terest required. N. B. For erery 10 that is cut off in months, add 2d ; for crcry 10 cut 9ffin weeks, aid an hal/'pemvj ; and for c\iery 40 in the rf«i'v, Ifarthiuj Money for amj \ r cfiiit. . Day. 1 1 £. s . d. ll 2 24 si 4 'n Si • 4 6 • 6^ 1 U 1 7| 2 2i 2 9 3 5i 3 10 4 44 4 Hi 5 si 10 llf 16 24 1 1 11 1 7 41 1 12 10^ 1 18 4i 2 3 10 2 9 51 2 14 9§ 5 9 7 8 4 4f 10 19 2 13 13 lli 16 8 9 19 3 6i 21 18 4i 24 13 1| 27 7 Hi 54 15 I04 82 3 10 f 177 3: EXAMPLES. ' 1. What i& the interest of ;^2467m10..0 for 10 mflfntb% at 4 J9ffr cent, per annum f 2870..10 900 75.. 0..0 4 80 6..13..4 :h 7=«0..11..8 9870.. a 10 987«82., 6..Q 98710O 2. What is the interest of ;f246?..lQ..0 for 12 weeks, al 5 percent? '....• i 2467..10 5 12337..ia 12 1000=19.. 4.. 7i 400= 7..13i.lO 80= 50: \^ l.ilO.. 9\ 1= 0.. 0.. 2| 1480|50=28.. 9.. 5 1102150.,— 3. What is the interest o£ £2467..10..0 for 50 days, at. )per cent, ? 2467..10 7000=19..S 6^ 6 400= 1..1..11 -.— 2= 0..f.. U 14805,. 50= 0"0.. Oj 50 7402|50te20..5.. 7 KUt. and the niunber t off twofigvres oa iTii tUe lalilc the sta- addvd mil mike the er will give the in- 7402150 To find what an estate, from 1 to i€60,000;7^ annum^ rill come to for 1 day. Rule 1. Collect the anntial rent oi* ineeme from the ta- le for 1 year, against which take the several sums for 1. ay, add them toother, it will give the answer. An estate of j^76 jtet attnumt what is that per day t 300=0..16.. 6J 76=0.. 3..10 6=0., 0.. 4^ td; forcncrylOciit thediiy'h^MtlUnj. 376=1.. 0..7| To find the amount of any income, salary, or serv«Dt^i:. ages, for any number of months, weeks, or days. ':ii I 178 jl Rule. Multiply the yearly income or salary by the mvroi- ber of months, weeks, or days, and collect the product from the table. What will ;^270 per annum come to at 1 1 months, for 3 weeks, and for 6 days ? For 11 Months, 270 2000-^lS6..13..4. Jl 2970 900= 75.. 0..0 70= 5..16..8 2970=2n,.10 For 3 tveeh. 270 800=15.. 7.. 8^ 3 10= 0.. 3..10 810 — 15..11.. 6i .•( .; :;■; v.^ . |M4l! For 6 days. 270 1000=2.. 14... 9i 6 600=1. .12.. 10^ 20i=0.. 1.. 1 1620 1620^4.. 8. 9 ,., For the 'whole time. 247..10..0 15..11..6^ 4.. 8..9 267..10..3f Jns. ^ Table shelving the Number of Dar/s Jiom any "Day in the Months to the same Day in any other Mmith through the Year. To • c "'1 *- u < rJan. S6.5 31 59 90 Feb. 334. 365 28 59 . Mar. 306 337 365 31 Apr. 275 ^06 334 365 - May 24-5 276 30+ 'i55 2 June 214 245 27.^ 304 3 July 181 215 243 274 Aug. 153 184 212243 Sep. 122 153 181212 Oct- 92 12.^ 151,182 Nov. 61 92 120,151 .Dec. 31 62 90121 120 89 61 30 365 331 304 27.-; 242 212 181 151 3 151 18 120150 92122 61' 9! 31 61 365| 30 3351 :>65 304; 334 2731303 243^273304 212:242,273 182,212(243 < 212 181 153 122 92 61 31 365 334 CM or 24.'-; 212 184 153 123 91 62 31 365 O 273 24'J 214 183 153 122 92 61 30 > 1 w O I u 12; Q 304S34 273:303 335 365 304'334 274 '304 245 214 184 153 123 92 61 31 335 275 244 214 183 153 1221 911 G\ 3 36; ..!• 1. , ^i- *:u. ;' - ' iry by the nura*' le product from I months, for 3 tveeh. 15.. 7.. 8i 0.. 3..10 15..11.. 6i iole time' I..0 1..9 3..33 Jn$. >om any Vny in Month ihroiigh 1 335'365 3304^334 3l274<>304| ^ C 179 3 ■»i • • ! ' I APPENDIX, #- -- ••#■• .J- CIRCULATING DECIMALS. DEFINITIONS, 61 91 31 Gl 335^6 = 1. '\']|THEN the denominator of a vulgar fraction, is W ilo aliquot part of its numerator, th« lattt r being increatsed vrith any necessary number of cypliers, the de- cimal fraction equivalent thereto is called a repctend, or circulating decimal, from the continual repetition of a cer- tain figure or series of figures, circulating alternately. 2. A single repetend is a decimal having one figure con- stantly repeating, ai ^=3333, &c. «==6666, &c. ex- pressed by either drawing a stroke through the rrpeating figure, thus 8, or more neatly, hy putting a dot over it, 35^3=.666=6 ; by which contrivance the series is pointed out, and any repetition of the circulating figure rendered unnecessary. \ 3. A compound repetend consists of two or more figures circulating alternately, the first and last of which are distinguished as in single repetends thus, 636383=63. 21496^21496=21496. 4. In a compound recurring decimal, ^either of the re- peating figures may be made the first in the repetend, pro- vided the nev series be so far continued that it shall con- tain as many figures as the original repetend ; thus H2b57 may be expressed in either of the ithcwn^ way 1428571 = l4iHi714=142^57U2 =1428571*28=14285714285, &c i 11 ( - I I '' ! I r i V I, 180 Appends. So, likewise, the ries may be repeated any number of times before the repetend be supposed to begin, the fig* ures between the first of the repetend and the decimal point being considered and treated as terminate numbers: the truth uf this proposition may be proved by converting the above decimals into their least equivalent vulgar frac- tions, by Rule 2, when they will respectively be found equal to each other =s I ; hence is derived the method of making several repetends begin at the same distance from the decimal point, when they are then said to be similar^ EXAMPLE. • • • • • 1,1213,4,019, and 24,92 are disshnilar, because the re- pretends begin at different distances from the decimal point, but expressed thus; 1,123, 4,01201, and 24,9292, they become similar. 'Nqt£. Terminate decimals may be considered luad man. aged as repetends by the addition of cyphers. S. Any circulating decimal may be transformed into an- •ther containing some multiple of the nuinber of places in 4he original repetend. EXAMPLE. Tof RUL niani this f requi 1. iSand 2. ] §216 3. 1375 fV S14=:S14314«314314314, Ac. If any number be compound repetends be continued till they are equal to the least common multiple of their seve* ral places, they will all necessarily end at the same place, and are then eddied counterminoui. The examples following Def. 4. thus carried out, stand ae follow: 1,21*333333, 4,01201201, and 24,92929292*. for the number of places in the several repetends being 1,3, and 2, the least com- mon multiple will be 6 by the 3d of the following Rules. 6. similar and conterminoi^s repetends are such as begin an^ end at the same plaee. i Rui rfinit raai ght th(^ )mmc nns. 1. R 7, ar Fi: 59 La Appendix* ISl .y numbeYfl REDUCTION. the decimal I CASE I. ate '™'^"* jjjj' 1 STo reduce a pure repetend to its equivalent vulgar /taction, t vulgar frac-lRuLE. TT ET the given decimal be made the numerator irely be found I § j of the vulgar fraction, and its denominator as the method ot Ijnany nmes as th^re are figures in the repetend. Ileduce distance from Ithis fraction to its lowest terms, and it will give the answer ) be similar^ |required. EXAMPLES. 1. Required the least equivalent vulgar fractions t# ^ and 135.* because the.re. r p^^^^^ ^^,_, ^^^ 13S=HI=3't. e decimal pemt» I g. Required the least equivalent vulgar fractions to 14, 24,9292, they J^gie, 413, and 0091, 3. Required the least equivalent vulgar fractions te dered and man- 11375, 47002, 0125, and*2743'4. irs. 1 CASE II. gformed into an- 1 j,^ reduce a mixed repetend io a vuharjr action, nber of places in 1 * ' Rule. From the given decimal subtract the terminate finite part for a numerator, and fur a denominator annex many cyphers as there are terminate numbers to the ht hand of the same number of nines as there are figures. ,;^- I the repetend. This fraction, divided by its greatest « ^'\\ Imnion measure, will give the required answer in its lowest ,le of their sevt- 1 examples. . the same pi»ce, ample* foWow^^gjl. Required the least equivalent vulgar fractions to 18, 1,2<3333S3,|y^ and 5925, and 1,209. dumber of places I the least com- 1 [liowing Rules, ire such as begip First. ,18 5925 = 18—1 17 . 027—2 = =— ; ,027= = 90 90 900 1925—5 5920 16 025 1 900 36 9990 ■J Lastly. 1,209 = »990 1209—12 27 1197 990 990 133 110 23 a—, no. ft !!li ^1, f' 182 AppcndiiV, Vi ;•» J.' 3 f< .-i ,:,•■ 2. Ilcquircd the least equivalent vulgar fractions tt 11850, 74,048, lie, 142857, 3,518, 42175, and 12^75. To find tJie least common multiple of several numbers, 1. Ir the numbers given be incommensurable, that is, if no number can be found which is an aliiiuot part of both the given numbers, the product of the said numbers will be the multiple requned ; thus, *2 and 5 being incommensura- ble, the multiple is 2><5.-=l0 If a number can be found which is an aliquot part of both, let either of them be divided by it ; and this quotient multiplied into the remaining number, will give the multi- ple sought. Til us, if the numbers 4- and G be given, being commensurable by 2, cither number divided by it, and the quotient multiplied by the other, the product 12 will be the multiple required. If the multiple of 3 or more numbers are required, proceed to find the least common multiple of any 2 of the said numbers, with this multiple and either of the remaining numbers, proceed as before, &c. : for in stance, let the numbers given be 2, 3, 4, and 6, then will the multiple of 2 and 3 bo 6, the multiple of 6 and 4 be J 2, and the multiple of 12 and 6 be 12, the least common multiple of all the numbers 2, 3, 4, and 6, as required. EXAMPLES. \ 1. T''c least common multiple of the numbers 2, 4, 5, and 7 it quired. Ans, 140. 2. The least common multiple of the numbers 3, 7, 21 4, aiid 8 is required. ni\ ADDITION. CASE I. When the decimals contain single repetends, KuLE. "T\/¥ AKE them all similar and conterminot _|^^ then add as in common numbers, only the last, or right hand figure, or add as many units as th are nines in the sum of the row standing over it, and t figure, if tiot a cypher, will be a repetend / r fractions t% and 12^75. •ral numbers, ruble, that is, if \)t part of both lumbers will be incommcnsuia- aliquot part of ndthis quotient I give the multi- be given, beins d by it, and the ct 12 will be the r more numbers imon multiple of pie and either ol; .re, &c. : for in- and 6, then will pie of 6 and 4 be the least common 1, as required. numbers 2, 4, 5, Ans. 140. numbers 3, 7, 21 epetends. e ri^ and conterminoi I numbers, only many units as the ag over *■ '"^ * snd it, and t ri Appendh. EXAMPLES, 29,IC(> 6.347 2,0'0 ,333 1,700 1 8S 39,.'>97 Ada 1,727083'; 2,583; ,002081J ; 9,02916 ; 4-.0V, 25 ; and 17,C35756 together. Am', !>7,'I0^>oTS. % Add ,083; 12,5; ,7,60806'; ,75; ai-1 4,006 lii to- gether. ... 3. Add 74,617; 40,013; 1,25, ,6 and O'JT trgcther, 4. Add 41,15 J ,10086; ,27; 4,62; and 9,6 togoilicr. CASE IL When the decimals contain compovad rcpekmh^ Rule. Make them similar and cor.vcniunoiis, and adfV as in common numbers,, witli this difference, to the sum (if the right hand repetcnd, or first rev/ of figures, Jitld «s many units as must be carried by the comnion vui:. of addi- tion to the next row of figures beyond the lel'trhand repc- tends or place where all the repel-inds bejrin together. The figures under the left and right-hand row of repetends will be the first and last of the repetend of the sum. EXAMPLES. Add ,6 ; ,027 ; ,73 ; 5,125 Dissimilar^ Made similar. ,6 ,027 ,73 5,125 ,127647 y6Qm ,0277 ,73737 5,1250 ,127647127 Ans* and ,127647 together. Himilur and Contcrnunous. ,666666666 ,027777777 ,737*373737 5,125000000 ,127647127 6,684465309 181 Appendi ii/» 'i\i 2. Ada IG'2,162; 134,09; 2,93; 97,26; 3,709230, 99,083 ; !,'> ; and i'SU togetlier. jins. 501,62651077. 3. Add ,29543; ,io4; ,37; ,466582iB ; and ,4731 to- fjotlitr. .. .. i4«5. 1,6^30109431099. 4. Add ,7045 and ,795 1 together. Ms» 1,5, Add ,2161 3+50,0(33+00025+ 1 ,34,703. \ Add \ ,1^5061 ^'li^ll-^ i*M)5^ ,6^ ,05. *' Add 2,93726 t],2H,0003+, 7 12+ 3076. SUBTRACTION. PtULE. li yyAKE the decimals, tvhether thej contam J3rjL ^"^fJ^*? o*" conjpound repetends, similar and conterniinoiis, as in Addition : then subtract as in whole numbers with this dift'erence, when the repetend of the number to be subtracted is greater than the repetend of the subtrahend, the right-hand iigure of the remamdermuBt be one less than it would be in common numbers. EXAMPLES. 1. From 39,2178 take 17,68. 39,2*178178 17,6868686 21,5309491 Prom 1,2 take 1,0072a 1,20000 1,01723 ,18276 # h. 3. From. 10,0413 take ,264. 4. From 9,17386 take 4,20013. 5. From 1, take ,3. 6. From 4,oi23 take 2,703. 7. From 14.047 take 12,36/ Ans. 9,7766948» # ( 44 Jppendiv, 185 5; 3,TG92fJ0, 1,62651077. and ,473*1 to- )109431099. /ins. IfS, r they contain ds, similar and ict as in whole !petend of the repetend of the naindermuBtbe its. i* IS. 9,7766948. MULTIPLICATION. CASE I. When the multiplicand contaias a y^nglc. icnclendy the muhi" plicr being terminate nwnhcrs. Rule. TJROCEED as in whole numbeis, only obscrv- X/ ing to increase the product of the ii<,'ht-hanJ figure cfthe multiplicand with eacii of the several figures in tile multiplier, in every line by as many units as the re are nines contained therein ; make the several products conterminous, and add them together by Case I. tlie right- hand figures of the sum will be a ciicuiatc or u cypher. EXAMPLES. , 1. 21,GrJ3 2. 1G,1IG '' G 10,82 1S0,03R0 32293 1291733 01586666 659,10693 S. ^Multiply 91,6167 by 4.26,8. '4. Multiply 40613,52 by 2,0068. CASE 11. JVli.cn iJie midtiplicand contains a compound rrpcl^nd, and the multiplier conmts ()f terminate numhtrs, RuL.':. Increase the products of tlio riglit-Iiand circu- late arising from the muitiplication of the sevtral iignrcs »;f the rnuitiplier, with as many units as jire curried Irom tlie product of the left-hand circulate to the product ol' tiio next figure to tlie left hand, then inuitij)ly as in consmon numbers, observing each pre. duct as wci! as tliC sum of tJie prodiicis contains a repetend of t!ie same nu'nher of f'^ur- s aji the repetend of the inuitkjjLcand liiukc tiie icvci'iu pro- 11 2 I .1 m t'iJ 186 AppendiJ^. ducts conterminous, as in tlie last case, and add them«U' gether by Case II. in Addition. EXAMPLES. 1. ,9437 7 2. 3,246 28,6 6,6062 19478 2597 i7 64-9292 92,8488 3. Multiply 5,1637 by 2,84. Ans, 14,665089.- 4. Multiply 56,0042941 by 461,2. Ans, 25829,18045872. 5. Multiply 8,42543 by 1001.,8. g. Multiply 37,603 by ♦)1,C2. CASE IIL /^i. When the, multiplier has a single repetend, Rui,E. Multiply by the repetend, as in common num bers, unless the multiplicand contain a single or compound repetend, in w'.:irh case, increase the product b^ the pre- ceding rules, divide the repetend product by nin«, and con- tinue the division till it terminate or end in a single or com- pound repetend, proceed with the remaining figures as usu- ill ; in adding the several products together, the repetend product must be considered as containing the same numbc tt" tii"U'cs as before the divi^ion^ ^ ij» i , y^ add themiU' \^ . 14,665089. 59,18045872. 'pelend. common num ;le or compound iuct by the pre- ly nii)«', and con a single or com g figures as usu er, the repetend he same numbci * w Appendix, SZAMPLES. 187 I. S1.64 '^ # -f". 2,3 * 9)6492 7213 4328 Am* 60,493 2. 14,013 ,146 i# ■ 9)84080 93422 5605*33 1401333 2,055288 3. 2,9l5 8,46 9)17477 194194 194 . 1165165 165 23303303 303 24,662662 662, &c. 4. Multiply 14861,6 by 40,73^ 5. Multiply ,1.19637 by 15,7. 6. Multiply 21464,3 by 12,6.. CASEir. I When the multiplier contains a^compound repetend. Rule. If the multiplier contains finite numbers let, tliem first be subtracted from it for a new multiplier ; if it be a pure repetend it undergoes no alteration. Then mul- tiply as in whole numbers if the multiplicand be terminate numbers ; if it contain a single repetend by Case I. and if a compound repetend, by Case II. Lastly, add the total product to itself in the following manrier : — Set the left- hand figure of it so many places forward, or to the right hand, as exceeds the number of places in the repttcnd of the multiplier by one, the remaining figures in order after it ; repeat this addition till the product last added fuU be- yond the first, and if the multiplicand consists of tenuinate I l^f - : f "> *'-''*' * jli^' - *^ i llWJ*. m i f^Ji > ' "''i 188 Appendix. numbers, the repetend of the product will consist of the same number of places as that of the *niultipiier ; should there be repetends in the multiplicand, the repetend of the product will be most readily determined by continuing and repeating the first product. ^ « EXAMPLES. ! ;. V ** ' - *- t* 1. 11,7505 ■^ 826 705030 235010 352515 38306630 38306630 38306630 "f ■M '•:# # m Ru repi peai I. S,834'i9749, &ci by repeating the additions, the sssiSSSSmmSBss series will be readily seen. 2. Multiply 225,6 by ,1225. 3. Mult. 8,594 by 12,581 1225 1 1224 225,6 ,1224 V 9026 45133 451333 2256666 2762160 2762 2 27,6492 Ans, ( 8,594 12,458 51567 4297*29 3437837 17i89189 125 12458 * 859459545 10705 t27027O 107054-2702 10705427 107054 1070-^ Z**^-; 108,1?5()265S Ms ■% , 4. Multiply 49,273 by 6,14902. 5. Multiply 7,0046 by 00413. 6. Multiply 4,12643 by 5,127:J. 7. Multiply 9,24685 by 46. 8. ]Multiply ,012 i 643 by 24,3721 4. 5. 6. 7. 8. Ru ei ras yph consist of the plier ; should jpetend of the ontinuing and Appendix:, Division. CASE I. 1S9 s When the dividend contains a single or contpound repetendt ' the divisor beins terminate numbers* Rule. "JJROCEED as in terminate numbers, only ob- ]lj serving to bring down instead of cyphers the repeating figure, or if it be a compound repetend, the re- peating figures in their proper order. EXAMPLES. i^- ^ e additions, the be readily seen. $,594. by 12,581 ,4 125 58 L_ 124-56 S7 , = 29 ^ ' . 37 89 5 70270 .-2702 05427 07054. 1070 . 10 \26f->S Jus 1. 8)146,158333S 2. 12)96,*317317, &c. 18,2697916 8,026443109776 3. 32,6)167,41519(5JL35i 1630 441 326 1155 978 1771 1630 1419 1304 115 ad infinitum,. ♦. Divide 461,17527 by 7. 5. Divide 51,6*4328 by 11. 6. Divide ,414 by ,3048. 7. Divide 24,6i4368 by 8,4461. 8. Divide 4,14 by 8,64. ^^^ CASKIL ^n the divisor contains either a single or l;ompound repe* tendf the dividend being terminate numbers. Rule. Annex to the right hand of the dividend as many /phers as there are places in the repetend of the divisor, )r a subtrahend; from which subtract the dividend, the re- ■% 190 Appendix. mainder will be a new dividend, with which proceed as in ter- minate numbers. Should there be any terminate numbers in the divis«r, they must be first subtracted from it, but ii' the divisor be a pure repetcnd it undergoes no alteration.^ but is to be used in all respects as terminate numbers. ■I -Yi- ^ ■f^ EXAMPLES. 1. Divide 12,487 by 8. 124870 12^87 ii 8)112383 U^. /#.. f> f- ■' Si, 1,4017875 Ans. 2. Divide 428,364 by 2.-43. 243 428,3640 24 428364 2,19) 385,5276 ( 1 76^04 terminate. 2l9 1665 1533 . *^: ^J^ 1322 1314 876 876 S. Divide 3 by ,462. 3000 3 .M 462)2997 6,4870129 4. Divide 214,160 by 1,476, 5. Divide 921,4 by 83. 6. Divide 1000 l^y 516r ?i Divide 754,03 .by 7'^ r m ceed as in ter- nate numbers from it, but ii' no alteration!.^ numbers. #' ■r. linate. !000 3 2997 6,4870129 Afpendix* CASE III. 191 When there are either single or compound repetends in the divisor and dividend. Rule. Should the divisor and dividend be both pure re- petends, and each containing the same number of places, proceed as in common division, and continue the quotient by bringing down cyphers till it either terminate, repeat, or is sufficiently ' exact. If the divisor and dividend are pure repetends, but dissimilar, not consisting of the same number of places in each, make them conterminous wad proceed as above. If the divisor and dividend be dissimilar mixed repetends, mPvke tlicm similar and conterminous, and subtract the ter- minate numbers from each for a new divisor and dividend,, which proceed as above. ZXAMFLES. 1. Divide 47 by 26 2. Divide 27,64-92 l^ 225,6. 27,6492 276 Quotient. ,18076923. 225,6666 2256 225441)27,62160 New dividend. 2254410 Netio divis. ,1225 Quotient. 3. Divide 8,68363 by 3,53*7. 3,53777 3537 3,5024 4. Divide 4,193 by ,1417. ,14171717 14 $,68363 8683 . 8,5968 2,45 quotient. 4,19319319 419 „ Vf J m ,14171703) ^' 4193189 29,588t % ^ I 199 Appendix, 6. Divide 108.13562S5 by 12,581. 12,i818181 108,1356265 126 1081 U ill' 12,5818056 1084355184' n- ^M \\ .„■».■ V, i,594 Quotient* S' m 111 m «. Divide ,017t31 16449 by 1,003, 100300300300 01783116449 100 01 «j. ill''' 5 ** i|^ % 100300300200) 7. Divide 1,831 by ,042. 18318318 18 01783116448 ,017 Quttient ■^ ,0424242)18318300 :. 43,l7889t f «. Divide 406,3 by 1,61456. 9. Divide 914,00014 by '417. 10. Divide 3201,40338 by 73,2586. 11. Divide 13,5169533 by 4,297. 12. Divide 46,0431712 by ,*42168. N. B. If the Student should be at any loss respecting the sertainty ojfhis operations in the Rules of Multiplication and Divisiony he wilt Jind considerable advantage tn turning the repetends into theit equivalent VI fractions, and proceed' ing with them by the rules of vulgar fractions* dgarfr 1 i w ^k uV mm- sEk li^Br,; 'IR Wm fll Ifli' FINIS. % 265 081 594< Quotient* 16U9 01 16448 ,017 Quotient #- w respecting the dti plication and e tn turning the ^s, and proceed' IS*