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Un des symboles suivants apparattra sur la dernlAre image de cheque microfiche, selon le cas: le symbole — ► signifie "A SUIVRE", le symbols V signifie "FIN". Maps, 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 fr^ **. A THE TUTOE^S ASSISTA%T; BKINO A COMPENDIUM OF ARITHMETIC, AND COMPLETE QUESTION-BOOK; CONTAINING, V I. Arithmetic in whole pumbers ; being a brief explanation of all its Rules, in a new and more concise method than any hitherto published ; with an Application to each Rule, consisting of a great variety of questions in real Business, with their answers annexed. II. Vulgar Fractions, which ars treated with a great deal of plainness and perspicuity III. Decimals, with the extraction of the Square, Cube, and Biquadrate Roots, after a very plain and familiar manner ; to which are added, Rules for the easy calculation of Interest, Annuities, and Pensions', in arrears, &o., either by Simple or Compound Interest. IV. Duodecimals, or Multiplication of Feet and Inches, with Examples applied to measuring and working by Multiplication, Practice, and Decimals. y. A Oa^Sfl^jOn of Questions, promiscuously arranged, for the exercise of t^ .' c cholar in the foreg^l^ules. TO WHICH ARE ADDED, A new and very short method of extracting the Cube Root, and a General Table for readily calculating the Interest of any sum of money, at any rate per cent. ; Rents, Salaries, &c. 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 of Accounts. This Work having been perused by several eminent Mathematicians and Accountants, is recommended as the best Compendium hitherto published, for the use of Schools, or for private persons. BY FRANCIS WALKINGAME, WEITINO-MASTKR AND ACCOUNTANT. TO WHICH IS ADDED, A COMPENDIUM OF BOOK-KEEPING, BY ISAAC FISHER. NEW-YORKt PUBLISHED BY D. & J. SADLIER & CO., 164 WiLLIAM-STRiSiSi. BOSTON :-128 FEDERAL STREET. AND 179 NOTRE- DAME STREET, MONTREAL: C. E.* 1851. \ « M ?.^ ) i i i.^ H / A '? "^ '/'..-/' -^ f ./ j# w^'- ■ ~.„^im^SSf^^ // ^ ! / /^■/t'^'l PREFACE. The public, no doubt, will be surprised to find there is another attempt made to publish a book of Arithmetic, when there are Ruch numbers already extant on the same subject, an* several of them that have so lately made their appearance in the world ; but I flatter myself, that the following reasons which induced roe to compile it, the method, and the conciseness of the rules, which are laid down in so plain and famiUar a manner, will have some weight towards its having a favourable reception. Having some time ago drawn up a set of rules and proper ques- tions, with their answers annexed, for the use of my own school, and divided them into several books, as well for more ease to myself, as the readier improvement of my scholars, I found them by experience, of infinite use; for when a master takes upon him that laborious, (though unnecessary,) method of writing out the rules and questions in the children's books, he must either be toiling and slaving himself after the fatigue of the school is over, to get ready the books for the next day, or else must lose that time which would be much better spent in instructing and opening the minds of his pupils. There was, however, still an inconvenience ^hich hindered them from giving me the satis- faction I at first expected ; i. e. where there are several boys in a class, some one or other must wait till the boy who firet has the book, finishes the writing out of those rules or questions he wants, which detains the others from making that progress they other- wise might, had they a proper book of rules and examples for each ; to remedy which, I was prompted to compile one in order to have it printed, that might not only be of uso to my own school, but to such others as would have their scholars make a uuick progress. It will also be of great use to such gentlemen 9^ ^ A3 t>^^(^^ ^ - IV w PREFACE. have acquired some knowledge of numbers at school to make them the more perfect; hkewise to such as have completed themselves therein, it will prove, after an impartial perusal, on account of ite great variety and brevity, a most agreeable and entertaming exer- L-book. I shall not presume to say any thing more m favour of this work, but beg leave to refer the unprejudiced reader to the remark of a certain author,* concerning compositions of this nature. His words are as follows : — « And now, after all, it is possible that some who like best to tread the old beaten path, and to sweat at their business, when they may do it with pleasure, may start an objection, agamst^e use of this well-intended Assistant, because the course of anth- metic is always the same ; and therefore say, that some boys lazily inclined, when they see another at work upon the ^ame ques on will be apt to make his operation pass for their own But ih^e Uttle forgeries are soon detected by the diligence of the tutor thereforeras different questions to different boys do not in the least promote their improvement, so neither do the questions hinder it tfeither is it in the power of any master (in the course of his busi- ness) how full of spirits soever he be, to frame new questions at pleasure in any rule : but the same question will frequently oc- cur in the same rule, notwithstanding his greatest care and skill to ^'irma7also be further objected, that to teach by a printed book is an argument of ignorance and incapacity ; which is iio less triflin- than the former. He, indeed, (if any such there be,) who is afraid his scholars will improve too fast, will, undoubtedly, decry tJiis method : but that master's igorance can never be brought in question, who can begin and end it readily; and, most certainly, that scholar's non-improvement can be as little questioned, who makes a much greater progress by this, than by the common method. To enter into a long detail of every rule, would tire the reader, and swell the preface to an unusual length; I shall, therefore, only give a general idea of the method of proceeding, and leave the rest to speak for itself; which I bopc the kind reader will find to answer the title, and the recommendation given it. As to the Dilworth. '- «"« i^iiii 11' m PBEFACB. ' rules, they follow in the same manner as the table of coni^nts specifies, and in much the same order as they are generally taught in schools. I have gone through the four fundamental rules in In- tegers first, before those of the several denominations; in order that thev being well understood, the latter will be performed with much more ease and dispatch, according to the rules shown, then by the customary method of dotting. In multiphcation I have shown both the beauty and use of that excellent rule, in resolvmg most questions that occur in merchandising; and have prefixed l)efore Reduction, several Bills of Parcels, -which are applicable to real business. In working Interest by Decimals, I have added tables to the rules, for the readier calculating of Annuities, &c. and have not only shown the use, but the method of making them : as like- wise an Interest Table, calculated for the easier finding of the Inte- rest of any sum of money at any rate per cent, by Multiplication and Addition 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 pei-spicuity, that there is none better extant. I have nothing further to add, but a return of my sincere thanks to all those gentlemen, schoolmasters, and others, whose kind ap- probation and encouragement have now established the use of this book in almost every school of eminence throughout the kmgdom : but I think my gratitude more especially due to those who have favoured me with their remarks ; though I must still beg of every candid and judicious reader, that if he should, by chance, find a transposition of a letter, or a false figiu-e, to excuse it; for, not- withstanding there has been great care taken in correcting, yet errors of the press will inevitably creep in ; and some may also have shp- ped my observation; in either of which cases the admomtion of a good-natured reader will be very acceptable to his much obliged, and most obedient humble servant, „r a t Tr^xrn a mt? _ ARITHMETICAL TABLES NUMERATION. Units 1 I X. of Thousands 12,345 Tens 12 C. of Thousands 123,450 Hundreds'.'.'. 123 I Millions 1,234,507 Thousands 1,234 X. of Millions 12,345,673 Note.— This Table may be applied to Division by reversing it; as the \''i4 in 4 are 2, and 23 in 6 are 3, &c. fl^l»»« ARITHMKTICAL TABLES. Vll 20d 24 30 36 40 48 50 60 70 T2 PENCE. are Is. 8d ..2 6 4 2 10 2 3 3 4 4 5 5 6 TABLES OF MONEY. 2()3, are £1 80d, are 81 .. 90 .. U6 .. 100 .. 108 .. 110 .. 120 . . 130 .. 1 140 .. Ga. 7 7 8 8 9 9 10 10 11 8d. 6 4 2 10 8' 30 40 50 00 70 80 90 100 110 1 2 2 3 3 4 4 5 5 Os. 10 10 10 10 10 hHILLINCS. 1208. are £6 130 140 150 160 170 ISO 190 200 210 6 7 7 8 8 9 9 10 10 Os. 10 10 10 10 10 OF A POUND. lOs, 6 5 4 3 2 2 1 1 Od 8.. 0.. 0.. 4.. is 1 ..1 ..1 ..1 .1 6. . . .1 0. 8. 8. 8. 6. .'.1 ..1 ..1 I • • -L ...1 half third fourth fifth sixth eighth tenth twelfth twentieth thirtieth fortieth PRACTICE TABLES. OF A SHII^LING. 6d. is 1 half 4 1 third 3 1 fourth 2 1 sixth ll 1 eighth 1 1 twelfth OF A TON. 10 cwt. 1 half 5 1 fourth 4 1 fifth 2i 1 eighth 2 1 tenth qrs. 2 1. 0. 0. or OF A CWT. lb. 56 is 1 ..28....1 . .16. . . .1 . .14. . . .1 half fourth seventh eighth OF A QUARTER. 14lbs 1 half 7 1 fourth 4 1 seventh| 3i 1 eighth CUSTOMARY WEIGHT OF GOODS. Firkin of Butter is 56 Ibs.iA Stone of Glass. -5 lb.s.| * - "• A Stone of Iron oj bhot 14 A Barrel of Anchovies ..30 A Barrel of Pot Ashes 200 A Seam of Glass, 24 Stone, or 120 A Firkin of Soap 64 A Barrel of Soap 256 A Barrel of Butter 224 A Barrel of Candles A Faggot of Steel . . 120 120 TABLES TBOY WKIGHT. 24 gr..inake 1 dwt. 20dvvt 1 ounce 12 oz 1 poullfd apothecaries' 20 gr. make 1 scruple, 3 bcr 1 dram. 8 dr 1 ounce 12 oz 1 pound OF WEIGHTS AND WOOL WEIGHT. 7 lbs. make 1 clove 2 cloves....! stone 2 stone 1 tod 6i tods 1 wey 2 weys. ..... 1 sack 12 sacks 1 last MEASURES. land measure. 9 feet make 1 yard 30 yards .> . . 1 pole 40 poles 1 rood 4 roods i . . .1 acre I AVOIRDUPOIS. 16 dr. make 1 oz. 16 oz 1 lb. 14 1b 1 stone 28 lb 1 quarter 4 qrs 1 cwt. 20 cwt 1 ton, CLOTH MEASURE. 2i inch make 4 nails quar.; quar nnav. . . . . . quar 3 4 5 6 nail quar. Fl. ell yard En. ell Fr. ell LONG MEASURE. 3 bar. corn 1 inch inches . . 1 foot feet 1 yard feef 1 fathom (SOLID MEASURE. 172S in. make 1 sol. ft. 27 feet I ynrd 12 3 6 5^ yards . 40 poles. . 8 funn . . . 3 miles . 69^ miles. t i- pole • 1 furlon; mile • k. lenjrne • X degree m vtu AKITIIMHTK AL TAKLES. OLD STANDARD. Gals. 8 n 35 53 70 10() Q. 3 3 2 1 3 P. Gills. 3.i)3 I 3.8C 1 3.46 3.17 1 2.34 1 0.69 3,03 1.3S 1 2.06 1 12 21 50 75 100 151 302 1 2 1 2 3 1 1 1 1 1 1.00 2.41 0.10 3.28 1.22 3.S3 2.44 3.06 3.33 B. P. G. 1 1 1 2 Q. 1 1 2 P. Gills. 0.25 1.01 2.02 0.07 0.14 4 1 0.28 8 1 0.56 33 2.24 82 2 1 1.63 ALE AND BEER. NKW STANDARD. gills make pints ([uarts. ... Ijallons. . . firkins. .. 1 I , 1 .1 .1 pint quart gal. tir. kild. bar. kilderkins. 1 li ban-el 1 hhd 2 barrels 1 pun. 3 barrels 1 butt WINE MEASURE. 2 pints. . . 1 quart 4 quarts..,! gallon 10 gallons.. 1 anker IS gallons..! runlet 42 gallons..! tierce 63 gallons..! hogshead 84 gallons..! puncheon 2 hhds...l pipe 2 pipes...! tun DRY MEASURE. 3 37 1 3 0.21 2.52 2 pints make 1 quart 4 quarts 1 gallon 2 gallons .... 1 peck 4 pecks 1 bushel 2 bushels.... 1 strike 4 bushels. ...1 sack 8 bushels 1 quarter 4 quarters... 1 chald. 10 quarters... 1 last COAL MEASURE. 31 77 2 "W 3 bushels..! sack 2 3 36 bushels..! chaldron] 34 3 0.52 2.34 n CONTENTS. PART L— ARITHMETIC IN WHOLE NUMBERS Page. Introduction ^^ Numeration J3 Integers, Addition 15 . Subtraction 16 Multiplication 16 D^ision 19 Tables .• • Jl Addition of several denominationsSS Subtraction 34 Multiplication 37 Division 42 Bills of Parcels 44 Reduction • 47 Single Rufe of Three Direct. . . 53 ~~ -Inverse.. 56 Double Rule of Thre«, - 58 Practice 23 Tare and Tret 6/ Simi)le Interest i^ Commission Purchasing of Stocks 71 71 PART II.— VULGAR FRACTIONS iOG Reduction Addition Jl;^ Subtraction H- Multiplication 113 Page. Brokerage "'^ Compound Interest "j^ Rebate or Discount 75 Equation of Payments 70 Barter H Profit and Loss ^y Fellowship °y vsithoutTime »0 _ —with Time 82 Alligation Medial 83 Alternate o*^ Position, or Rule of False 88 Double 90 Exchange • • • ; ; • • Comparison of Weights and Mea- sures • Conjoined Proportion Progression, Ariir.hmetical 97 . Geometrical 100 Permutation 104 95 96 Division. The Rule of Three Direct.... 114 Inverse.... 115 The Double Rule of Three. . . HJ .- X ■^ CONTENTS. t^A PART III.— DECIMALS. 11^ 119 120 Page. Numeration '■'■' Addition ]\° Subtraction Multiplication Contracted Multiplication.. Division ]^\ Contracted l'*'* Reduction ; • • • • •.• • • 1^3 Decimal Tables of CoinjWeights, and Measures 126 The Rule of Three -129 Extraction of the Square Root. . Vulgar Fractions -Mixed Numbers. Extract of the Cube Root. . . . Vulgar Fractions . . . Mixed Numbers. . . . ^ ^Biquadrate Root. . . . 130 131 132 134 136 136 138 Page. A general Rule for extracting the Roots of all powers 138 Simple Interest 140 . . for days 141 Annuities and Pensions, Hcc. in Arrears 1^3 Present worth of Annuities. . . 147 Annuities, &c. in Reversion.. . 150 Rebate or Discount l'^>2 Equation of Payments l-)* Compound Interest l'^--> Annuities, &c. in Arrears IT)? Present worth of Annuities.. . 160 Annuities, &c. in Reversion. . . 102 Purchasing Freehold or Real Es- tates i'5J in Reversion ....... l*^)) Rebate or Discount 160 PART IV.r-DUODECIMALS. Multiplication of Feet & Inches, 169 Measuring by the Foot Square, 171 Measuring by the Yai-d Square, 171 Measuring by the Square of 100 F^et. 173 Measuring by the Rod 1^3 Multiplying several Fijipres by several, and the operation in one line only 1'''^ PART V.-QUESTIONS. A Collection of Questions, set down promiscuously for the greater trial of the foregoing Rules 1'^^ A COMPENDIUM OF BOOK-KEEPING A general Table for calculating Interests, Rents, Incomes and Servants' Wages 181 184 63 ^ u e. EXPLANATION OF THE CHARACTBB8. EXPLANATION = Equal. — ^Minus, or Less. 4- Plus, or More. X Multiplied by. ^Divided by. 235Y 63 ; : So is. 7^2 +5 = 10. 9__24-5=2. ^/ -y « OF THE CHARACTERS MADE USE OF IN THIS COiMPENDIUM. The Sign of Equality ; as, 4 qrs. = 1 cwt signifies thaj. 4 qrs. are equal to 1 cwt. The Sign of Subtraction; as, 8—2=6, that is, 8 lessened by 2 is eqpl to 6. Tbe Sign of Additional 1»s, 4+4=8, that is, 4 added to 4 more, is equal to 8. The Sign of Multiplication; as, 4X6 =2i, that is, 4 multiplied by 6 is equal to 24. The Sign of Division ; as, 8+2=4, that is, 8 divided by 2 is equal to 4. Numbers placed like a fra<5tion do likewise denote Division ; the upper number bemg the dividend, and the l^er %e divisor. The Sign of Proportion ; as, 2 : 4 : : 8 : 16, that is, as 2 is to 4, so is 8 to 16. Shows tlg^the difference between 2 and 7 added tJf, is equal to 10. Signifil^f the sum of 2 and 5 taken fi-om 9, is equ4 to 2. Prefixed * any number, signifies the Square Boot of that number is required. Signifies the Cube, or Third Power. T^ +^0 fhA Tiinnadrate. or Fourth Power, 1 i I id est, that is. y / /' TT] I ARITH^ bers, and all its ope Notat: riPLIOATIi i# Teadieth and to rei » \ THB TUTOR'S ASSISTANT; BEING A COMPENDIUM OF,,ARITHMETIC. ARITMETI TH ERS. INtRdOTC^ Aritkmetio is the Art or Scier^bf computing by Num- bers, and has five principal or fundamantal Rules, upon whicU all its operations depend, viz : — Notation, or Numeration Addition, Subtraction, Mui** riPLicATiON, and D^ision. ^ NUMERATION ^^^ Teacheth the diflferent value of Figures^Hu^^^rent Places, and to read and write any Sum oi|lium| *THE#A'BLE. #^ n .«._. 14 NUMERATION. 2ATI0N. • following Numbers. ^ THE fiftl Write down^ (») Twenty-thi H or&enSilbirVtwo Thousand, Two Hundred "t'f FoAuons, Nine Hundred and Forty^ne^ Thousand, ^7^ Tweniiiie Millions, One Hundred and Fifty^ven -^iWrtv-nS^^^^M^ive Hundred and ^oiir. Thousand, Five Hurilre Write down in Wo^d^ n 35 O 2017' ri 59 Jl 5?01 V 172# n 20766 (") 61700047 C*) Notatvm I One. II Two. III Three. IV Four. ^* V Jbive. VI Six. VII Seven. \ vm %ht. .<•> . .^ ig following Numbers. 7 ("^ 5207054 158 (") 2071909 10030 (") Jj|g4008 (") 22lfPl90 iters. IX Nine. X T^. XI Eleven. XII fclve. ;Yii| yji.irt.^.on. :l^en. XV Fifteen. XVI Sixteen. '■>x ADDITION OF IlfTEOERS. 16 % XVIII Eighteen. XIX Nineteen. XX Twenty. XXX Thirty. XL Forty. L Fifty. • LX Sixty. LXX Seventy. LXXX Eighty. XC Ninety. c Hundred 00 Two Hundred ccc cccc D DC DCC JDCCC DCCCC M MDcccx;n. MDCCCXXXVil Three Hundred. Four Hundred. Five Hundred. Six Hundred. Seven Hundred. Eight Hundred. Nine Hundred. One Thousand. One Thousand Eight Hundred and Twelve. Thousand 'Eight dred and Thirty -^■ iven. INTEGERS. • ADDITION Teacheth to add two or more Suras together, to make one whole or total Sura. j|^ ' Rule. Thei^e must be due regard lia4 in pjglg the Figures one under the other, i. e. Units underiUnit^Pns under Tens, &«. ; then beginning with the first row of4tTnfCs, add them up to the top; when done, set down the Units, and carry the Tens to the next, and so on; continuing to the last row, under which set down the Total amount. , . , Proof. Begin at the* top o downwards, the same as yo the first, the Sum is suppose Months. (•) 275 110 473 354 271 352 ^_ Sum, and reckon the Figures J them up, and, il the same as right. C) 1234 7098 3314 6732 2546 6709 W- £ (•) 75245 37502 91474 32145 47258 21476 Years (*) 271048 325476 107584 625608 754087 279736 ^ ^ / Ans. U7206. n x\dd 246034, 298765, 47321, 58653, 64218, 53t«, 9821,^' and 640 together. ^W5. 730828. t2 16 StTBTRACTION OP INTEGERS. If you give A. £56, B. £104, C. £274, D. £391, and E. £703, how much is given all ? , ^ , , ^,^'^,f- }^^^' (*) How many days are in the twelve Calendar Months i ^^ ^ - AnB. 365. SUBTRACTION Teacheth to take a less Number from a greater, and shows the remainder or difference. Rule This being the reverse of Addition, you must borrow here (if it require) What you stopped at there, always remember- ing to pay it to the next. Proof. Add lhe*remainder and the less Line tog^her, and if the same as the greater, it is right. i^\ (n (*) (') O From 271 4754 42087 452705 271508 3750215 Take 164 2725 34096 327616 ' 152471 3150874 Rem. 117 Proof 271 -'*:, -# H 2 3 4 5 6 7 8 9 10 11 12 MULTIPLICATION Teacheth how to increase the gpter of two Numbers given as dften as there are Units in the less; and compendiously performs the office of many additions. ^| To this Rule belong three fpncipal Members, viz. 1. The Multiplicand, or Number to be multiplied. '2. The Multiplier, or Number by which you multiply. 8. The Product, or Number produced by multiplying. Rule. Begin with that Figure which stands in the Unifs place of the Multiplier, and with it multiply the first figure in the Unit's Tens in mind, till you have multiplied the next Figure in the Multiplicand by the same Figure in the Multiplier; to the pro- fluot of which add the Tens you kept in mind, setting down tha Unite, and proceed as before, till the whole line is nmltiplied. MULTIPLICATION OP INTEOBBS. 19 Proof. By casting out the Nines; or «^f^\>^^,^. ^^^'"^ .^"^- tiplicand the^Multipliir, and the Mult^lier the Multiphcaiid' and it" the Product of this operation be the same as before, the work is right. MULTIPLICATION TABLE. 4 6 ■I? '' 8 ul2 4 . 6 8 10 12 6 9 12 15 18 8 12 16 20 24 10 16 20 25 30 12 W 24 30 36 14 21 28 35 42 16 24 32 40 48 18 27 36 45 54 20 30 40 50 60 22 33 44 55 66 24 36 48 60 72 Multiplicand MultipUer Q 25104736 2 • Product 50209472 (*) 27104107 6 (*) 231047 6 8 9 10 11 12 14 16 21 24 28 32 35 40 42 48 49 56 56 64 63 72 70 80 77 88 84 96 18 20 22 27 30 33 36 40 44 45 60 55 54 60 66 63 70 77 72 80 88 81 90 99 90 100 110 99 110 121 108 120 132 24 36 48 60 72 84 96 108 120 132 144 i I A A A A A A A A A A C) 52471021 7925437521 n 7092516 7 -A, (') 3725104 8 M -wt n 4215466 (•) 2701057 {") 31040171 9 10 11 When the Multiplier is more than 12, and less than 20, mulU- ply by the Unit Figure in the Multiplier, adding to tho Produc* the back figure to that you multiphed. b3 y 18 MULTIPLICATION OP INTEGERS. (») 5710592 (") 6107252 (") 7653210 13 14 15 (") 92067166 16 (") 6251721 - ('«) 9215324 (") 2571341 ('') 3592104 IV 18 19 20 When the Multiplier consists of several Figures, there must be as many products as there are Figures in the Multiplier, ob- serving to put the first figure of every Product under that Figure you multiply by. Add the several Products together, and their Sum will be the total Product. (") Multiply 271041071 by 5147. O Multiply 62310047 by 16G8. (") Multiply 170925164 by 7419. n Multiply 9500985742 by 61879. n Multiply 1701495868567 by 4768756. When Ciphers are placed between the significant Figures in flie Multiplier, they may be omitted ; but great care must be taken that the next Figure must be put one place more to t£e left hand, L e. under the Figure you multiply by. (") Multiply 571204 By 27009 5140836 • « 3998428 1142408 Product 15427648836 P') Multiply 7561240325 by 57002. (") Multiply 562710934 by 590030. yhen there are Ciphers at the end of the Muitiijlicand or Aul- fijdier, they may be omitted, by only multiplying by the rest of Ike Figures, and setting down on the right hand of the total Product as many Ciphers as virere omitted. n M Whei Figures multiply plied by Teachei or, to d In t] dental : 1. T 2. T 3. T is contj 4. C finishe( RUL contaiij the Fi, next ii Diviso] *..ii*v.,(».^^_^^ blVIBION OF IMTE0ES8. (*») Multiply 131^500 3400 55180 41385 4690300000 'n Multiply 7271000 by 52600. (") Multiply V4837000 by 975000. When the Multiplier is a composite Number, *. e. if any two Figures being multiplied together, will make that Number, then multiply by one of those figures, and that Product bemg multi- plied by the other will give the answer. (») Multiply 771039 by 35, or 7 times 5. "^ ' *^*' 7X5=25 5397273 5 26986365 Multiply 921563 by 32. * Multiply 715241 by 56. Multiply 7984956 by 144. DIVISION % Teacheth to find how often one Number is contained in another; or, to divide any Number into what parts you please. In this Rule there are three numbers real, and a fourth acci- dental: viz. 1. The Dividend, or Number to be divided : 2. The Divisor, or Number by which you divide : 3. The Quotient, or Number that shows how often the Divisor is contained in the Dividend : i. i • 4. Or accidental Number, is what remains when the work is finished, and is of the same name as the Dividend. Rule. When the Divisor is less than 12, find how often it is contained in the firat Figure of the Dividend; set it down under the Figure you divided, and carry the Overplus (if any) to the next in»the Dividend, as so many Ter- • then find how often the Divisor is contained therein, set it dov..., and continue the same 90 DIVISION OF INTEGERS. till you have gone through the Lino; but when the Divisor i« more than 12, multiply it by the Quotient Figure ; the Product subtract from the Dividend, and to the Remainder bring down the next Figure in the Dividend and proceed aa before, till the Figures are all brought down. Proof. Multiply the Divisor and Quotient together, adding the Remainder, (if any,) and the Product will be the same as tlw Dividend. Dividend. Rem. (*) Divi3or.2)725l07(l Quotient 362553 2 Proof 725107 7)2532701 ( ., 0) 3)7210472( (») 4)7210416( O 5)7203287( (•) 6)5231037( O 8)2547325( (") 9)25047306( ^^■| Divisor. Dividend. Quotient. ^H (') 29)4172377(143875 (") Divide (") Divide (") Divide (») Divide (") Divide ^B 127 ^H 116 ^^H 112 1294875 287750 2 rem. ^R ' 4172377 Proof. (») Divide ^H 232 (") Divide ^■i 217 ^W 203 ^H Rem. • 7210473 by 37. Arts. 194877|4 42749467 by 347. 734097J43by 5743. 1610478407 by 54716, 4973401891 by 510834. 51704567S74 by 4765043. 17453798946123741 by 31479461, When there are Ciphers at the end of the Divisor, they may be cut off, and as many places from off the Dividend, but they must be annexed to the Remainder at last. 1'**. TABLES OF MONfiY ai (") 271100)254732121(939 (») 3731000)7324731729(2017 (»") 5721100)7253472116(1267 («•) 215I0001G3251041997( 29419 When the Divisor is a composite number, i. e. if any two Fi- ssures, being multipHed together, will make that number, then, by dividing tlie Dividend by one of those Figures, and that Quotient by the other, it will give the Quotient required. But as it some- times happens, that there Is a Remainder to each of the Quotients, and neither of them the true one, it may be found by this Rule. Multiply the first Divisor into the last Remainder, to that Product add the first Remainder, which will give the true one. (") • (•») n n Div. 3210473 by 27. 7210473 by 35. 6251043 by 42. 5761034 by 54. 118906 11 rem. 20G013 18 rem. 148834 15 rem. 1066S5 44rem. Marked i Farthing i Halfpenny f Three Farthings Farthings 4 = MONEY. .M 4 Farthings make 1 Penny d. 12 Pence 1 Shilling. . . .*. 20 Shilhngs 1 Pound £ 1 Penny 48 = 12 = 1 Shilling 960 = 240 = 20 = 1 Pound. SHILLINGS. PENCE TABLE. a. £ s. d. s. d. d. s. d. 20 •.. 1 : 20 .. 1 : 8 90 .. 7 : 6 30 .. 1 : 10 24 .. 2 : 96 .. 8 : 40 .. 2^ : 30 .. 2 : 6 100 .. 8 : 4 50 .. 2 : ro 36 .. 3 : 108 .. 9 : 60 .. . 3 : 40 .. 3 : 4 110 .. 9 : 2 70 .. 3 : 10 48 .. 4 : 120 . 10 : 80 .. 4 : 50 .. 4 : 2 130 .. 10 : 10 90 .. 4 : 10 60 .. 5 : 132 . . 11 : 100 .. 5-: 70 .. 5 : 10 140 . . 11 : 8 110 .. 5 : 10 72 .. 6 ! 144 .. , 12 : 120 ..6:0 80 .. 6 : 8 150 .. 12 : 6 130 .. 6 : 10 84 .. 7 i . . 160 . ■ 13 '' K '* I 22 TABLES OF WBI0HT8. TROY WEIGHT. Marked 2^4 Grains. ..... .make. 1 Pennyweight > \v/t. 20 Pennyweights 1 Ounce oz. 12 Ounces 1 Pound lb. Grains 24 = 1 Pennyweight 480 r= 20 = 1 Ounce 5760 = 240 = 12 = 1 Pound By this Weight are weighed Gold, Silver, Jewels, Electuariea and all Liquors. N.B. The Standard for Gold Coin is 22 Carats of fine Gold, ind 2 Carats of Copper, melted together. For Silver, is 11 oz. i dwts. of fine Silver, and 10 dvvts. of Copper. 25 lb. is a quarter of 100 lb. 1 cwt. 20 cwt. 1 Ton of Gold or Silver. AVOIRDUPOIS WEIGHT. 16 Drams make. . . .1 Ounce Marked )dr. j oz. 16 Ounces 1 Pound lb. 28 Pounds 1 Quarter qr. 4 Quarters or 112 lb 1 Hundred Weight cwt 20 Hundred Weight 1 Ton ton. Drams 1 Ounce 16 = 1 Pound 448 == 28 = 1 Quarter 1792 = 112 = 4 = 1 Hundredweight A Wei 'Si AClov A Ston A To^*^- ^^'^'''- " ...iPottle P«*- 2Quart8 ^ Gallon. ^p«^i\!!^ •;;:::;:::::i....ipeck. .gal .pk. 2Gall6n8 J Bushel ^"•., 4 Pecks ;\|"4e '^''^ 2 Bushels Icoom ^°°°*- 4 Bushels ••*••; i Quarter %\ 2 Cooms. or 8 Bushels J C haldi-on . . -. ''^''^ 4 Quarters J Vey way. OQuarters \ V^I last. aWeys In London, 36 bushels make a chaldron. Solid Inches 268|= 1 Gallon 6371=: 2= 1 Peck 2150*= 8= 4= 1 Bushel 43004= 16= 8= 2= 1 Stnke 86011= 32= 16= 4= 2= l^^^ ,wof,oi— 64= 32= 8= 4= 2= 1 V^uarwT 860?6'=320=160=40=20=10= 5 = 1 Wey m032 =640=320=80=40=20=10=2=1 Last. The Bushel in Water Measure is 5 Pecks. » , Je 21 chaldrons. A score of coals is 3 ^hels. 12 sacks. n^^lr ^%.4> rk/'vn Its N !"**., A chaldron of coals t bushels Aloadofcorn i S!k A cart of ditto ..... .40 bushels. This measure is applied to a U dry goods. The standard bushel is 18^ inches wide, and 8 inches deep. 40 4 640 30 100 i^"-*'^' ■-•■1!|. TABLES OF MEASURES. 87 TIME. 60 Seconds make....l Minute. 60 Minutes i Hour... 24 Hours i Day.... 7 J)«>s 1 Week.*." 4 Weeks i Month. Marked \" ) m. ..hour. ..day. . . week. 13 Months, 1 day, 6 hours. . 1 Julian Year yr mo. Seconf^s 60 = 1 Minute 3600 = 60 = 1 Hour 86400 = 1440 = 24 = 1 604800 = 10080 = 168 = 1 2419200 = 40320 = 672 = 28 31657600=525960=8766=365 : 6= 31556937=525948=8765=365 : 5 ; Day = 1 Week = 4 = 1 Month, w. d. h. =52: 1 :6=] Julian Year, m. " 48 : 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, And all the rest have thirtv and one ; Except in Leap- Year, and then's the timo February's days are twenty and nine. SQUARE MEASURE. Foot. 144 Inches make i 9 Feet ^ y^^.^ l?oilHl 1 Squaie of flooring. '"'i^y 1 K'Hi. Acres' "'' "'" ^^^^ ^"^^ ^ ^^-''^ "^ ^''"'^ 40 4 640 30 100 '?^'*'^ I Square Mile ^^''^^ 1 Yard of Acres C3 Yard of land. Hide of land. 28 ADDITION OF MONEY. Inches , ^:^ * 144= 1 Fo^t . ^ , 39204= 272^=30^- i ^ ^ ^^^^ 1568160 = 10890 =1210 -- 4U ^ ^^re. 6272640=43560 =4840 -160 4 oil tl^inffs that have length and By this measure - V^^/^^^^ flooring, thatching, breadth ; such as land, painting, i plumbing, glazmg, S^ .. 14 LAND MEASURE. O a. 1232 321 131 1219 459 ', P 1 .. 14 .. l'-> 2 .. !•> 1 .. 18 o .. 11 ussaam ADDITION OF MEASURES. k 13 18 hhds. gala qts. 31 .. 57 .. 1 97 .. 18 .. 2 76 .. .13 .. 1 55 .. 46 .. 2 87 .. 38 .. 3 55 .. 17 .. 1 WINE MEASURE. (*) , t. hhds gals. qts 14 .. 3 .. 27 .. 2 19 .. 2 .. 56 .. 3 17 .. .. 39 .. 3 79 .. 2 .. 16 .. 1 54 .. 1 .. 19 .. 2 97 .. 3 .. 54 .. 3 2 1 1 1 ALE AND BEER MEASURE. A.B. fir. gal. 25 .. 2 .. 7 17 .. 3 .. 5 90 .. 2 .. 6 75 .. I .. 't 96 .. 3 .. 7 75 .. .. 5 0) B.B. fir. gal. 37 .. 2 .. 8 54 .. 1 .. 7 97 .. 3 .. 8 78 .. 2 .. 5 47 .. .. 7 35 .. 2 .. 5 u£j» ■] 1* O 1 'M hds. gals. qts. | 76 .. 51 .. 2 W 57 .. 3 .. 3 1 97 .. 27 .. 3 1 22 .. 17 .. 2 1 32 .. 19 .. 3 1 55 .. 38 .. 3 1 19 ; 22 . 31 . 12 . n . 21 ch. bu. pUs. 75 .. 2 .. 1 41 .. 24 .. 1 29 .. 16 .. I 70 .. 13 .. 2 54 .. 17 .. 3 79 .. 25 .. 1 DRY MEASURE. A last. wey. q. bu. pks. 38 .. 1 .. 4 .. 5 .. 3 47 .. 1 .. 3 .. 6 .. 2 62 .. .. 2 .. 4 .. 3 45 .. 1 •• 4 .. 3 .. 3 78 .. 1 .. 1 •• 2 .. 2 29 .. 1 .. 3 .. 6 .. 2 P- 1 .. 14 .. iv> 2 .. !•> 1 .. 18 o .. 17 {') w. d. 71 . . 3 . 51 «. 2 e 76 95 79 h. 11 9 21 21 15 TIME. (') w. d. h. 07 .. 2 .. 15 . j5 • . o . . 21 . 7(> . . . . 1 r. 53 98 m. r» t M t • 21 18 . 42 .. 41 . 27 .. 51 . 37 .. 28 . 42 . 47 A\ a. f-/ aii*«P9ft**r ) 32 ADDITION. THE APPLICATION. 1. A man was bom in the year 1750, when will I'^te f^ y,^*" "^ Tl B C and D, went partners in the purchase of a quanU- tity'of goods A laid out £7, half-a-g«inea and a crown; B. Tic, 54s;ca.-, and D, 87d. What w>. >a>d^o"y-l'» ._ 3. * 2er r^ds^talt^a-gnin'o. and a ^^J^^ t" •''fwhat i;.the estate worth per annum when U.e t^xes are 21 guineas, the neat income 8 score, £19 : U .^^ ^^^^ _ ^^ K Thpre are three numbers; the first is 215, the second 519, and tSdl t'much as the other two. What .^the ..mot "T Bolht a parcel of goods, for which I paid £54 : 17, for packing ll'sd/carriage ll : 5 : 4, and spent about the bargain 14s. 3d. What do these goods stand mo m !^^^ £5,, . jo : 3. 7 There are two numbers, the least whereof is 40, tlieir Sii- ference 14! I desire to know what is the greater nmnber, and the sum of both ? ^^^ ^^ ^^^^^ ^^^^^^ g^ ^„^ 8. A gentleman left his elder daughter £1500 more than the vouno-er and her fortune was 11 thousand, 11 hnudred and £11. 'mafw'rtt elder sister's fortune and what d d the father leave ., ,, • ^ns. Eldest sister's fortune, £13611. ""''" ' Father loft them £25722. ' 9. A nobleman, before he went out of town, ^va.s teirous of paying all his tradesmen's bills, and upon mquny, he found that he o^vid 82 guineas for rent; to Ins wme merchant £72 o . to his confecSonc.r, £12 : 13 : 4; to h,s draper f«^ 13 2 to his tailor, £110 ; 15 : 6; to Ins coach-maker, £157 .80 to his tallow-chandler, £8 : 17 =.9; to Ins corn-chandler, £170 . 6 8 . , to his brewer, £52 : 17 : 0; to ms Duiciier, ^...^.y^. '- _--■ ■" mXHJ^^t.:^^^:^ le^hf t^f oie'wJn f^^ to the above suL £100, whicW'J -^•'f,,*^ 3^' 10. A year, and between days; bel 15 days; and 25 c days old, .11. A 811 accou £150 : li half-crowi 15 : 9^, J whole air 12. A twenty di ing 408 dwts. ; si knives ai tankard, lamp, w( small art weight oi 13. A weighed the thirc the fifth, pockets, the whol ' 14. A in JanuE bruary, for good May, £1 but tiio the dem only £2' vear's bi ^i 3 qrs. 12 lb.; the fifth, 2 cwt. 3 qrs. 15 lb. Besides these, he purchased two pockets, each weighing 84 lb. I desire to know the weight of the whole ? Ans. 15 cwt. 2 qre. 14. A, of Vienna, owes to B, of Liverpool, for goods received in January, the sum of £103 : 12 : 2; for goods received in Fe- bruary, £93 : 3 : 4; for goods received in March, .£121 : 17: for goods received in April, £142 : 15 : 4 ; for goods received in May, £171 : 15 : 10 ; for goods received in June, £142 : 12 : 6 ; hilt, tiirt iHttPT' siY mnn V d £ s. d. £ «. d. 15 .. 3i 71 .. 2 .. 4 527 .. 3 .. 5i 17 .. 2 19 .. 13 .. 75 139 .. 5 .. H Borrowed 25107 s. 15 d. 7 375 .. 5 .. 54 Paid 259 . . 2 .. nk at 359 . . 13 .. u different 523 .. 17 .. 3 times 274 . . 15 .. 74 325 .. 13 .. a £ 8. d. Lent 250156 .. 1 .. 6 271 .. 13 .. 74 Received 359 .. 15 .. 3 at 475 . . 13 . . 9i several 527 . . 15 . . 34 Davments 272 .. 16 .. 5 Paid in all ^^^ns to pay '%7 (") Bongh Sol 29 .. lb. («) 5 2 Fl.E. (») 35 . 17 . yds. (») 107 . 78 . a. (») 175 . 59 . SUBTRACTION. TROY WEIGHT. lb. 02. dwt. gr. lb. oz. dwt. gr. (") Bought 52 . . 1 . . 7 . . 2 (*) 7 . . 2 . . 2 . . 7 Sold 39 . . . . 15 . . 7 5 . . 7 . . 1 . . 5 Unsold AVOIRDUPOIS WEIGHT. '' lb. oz. dr. cwt. qrs. lb. t cwt. qrs. lb (») 35 .. 10 .. 5 («) 35 .. 1 .. 21 (») 21 .. 1 .. 2 .. 7 29 .. 12 .. 7 25 .. I .. 10 9 .. 1 .. 3 .. 5 APOTHECARIES WEIGHT. lb. oz. dr. scr. -' lb. oz. dr. scr. gr (») 5 ,. 2 ,. J .« (•) 9 ,. 7 .. 2 .. 1 .. IS 2..5..2..1 6..7..3..1..1g Fl.E. qr. n. (>) 35 .. 2 .. 2 17 .. 2 .. 1 CLOTH MEASURE. yd. qr. n. («) 71 .. 1 .. 2 E.E. qr, n. (») 35 .. 2 .. 1 14 .. 3 .. 2 LONG MEASURE. yds. ft. in. bar. (>) 107 .. 2 .. 10 .. 1 78 .. 2 .. 11 .. 2 lea. mi. fur. po. 0) 147 .. 2 .. 6 .. 29 58 .. 2 .. 7 .. 33 a. r. T). (») 175 .. 1 .. 27 59 .. .. 27 LAND MEASURE. a. r. (•) 325 ,. 2 279 .. 3 .,-^,,, ■^"xm^A 36 SUBTRACTION. WINE MEASURE. hhd. gal, qts. pt. (') 47 .. 47 .. 2 .. 1 28 .. 59 .. 3 .. tun. hhd. gal. qt. O 42 .. 2 .. 37 .. 2 17 .. 3 .. 49 .. 3 5. Wha to £305 ? 6. A he much doe i, my i ALE AND BEER MEASURE. A.B. fir. gal. C) 25 .. 1 .. 2 21 .. 1 .. 5 B.B. fir. gal. O 37 .. 2 .. 1 25 .. 1 .. 7 hhd. gal. qt. C) 27 .. 27 .. 1 12 .. 50 .. 2 DRY MEASURE. qu. bu. p. (») 72 .. 1 .. 2 ou • t 2 « • 3 qu. bu. p. (') 65 .. 2 .. 1 57 .. 2 .. 3 • ch. bu. p. O 79 .. 3 .. 54 .. 7 .. 1 TIME. yr8. mo. w. ds. (*) 79 .. 8 .. 2 .. 4 23 .. 9 .. 3 .. 5 ho. min. " (») 24 .. 42 .. 45 19 .. 53 .. 47 THE APPLICATION. . 1. A man was born in the year 1723, what was his age in the year 1781 ? * Ans. 58. 2. What is the difference between the age of a man born in 1710, and another born in 1766 ? w?n». 56. 3. A Merchant had five debtors, A, B, C, D, and E, who together owed ip £1156 ; B, C, D, and E, owed him £737. What was A's debt > Xt Ana. £419, estate of £300 per annum, is reduced, on the paying of taxes £14:6. What is the tax ? 'SSCi^.SKrSl Ans. £45 : 14. 7. A m commodit year, by c months' ei 8. A ge who was t elder siste 9. A tri gether, an 5:2; to £143 : 12 and that h £21 : 10 : hands, I < much i ir 10. My count of rr Beeswax, 11 :6; lin the same i wines to t £19 : 17 : 15 : 6. I the debtor MULTI Rule.— the produ( iiiainder, It the g; multiplied multipliefl line by as COMPOUND MULTIPLICATION, f 87 3. What is the difference between £9154, and the amount of X754 added to £305 ? ^m. £8095. 0. A horse in his furniture is worth £37 : 5 ; out of it, 11 guineas ; how much does the price of the furniture exceed that of the horse ? ^ns. £7 : 17. 7. A merchant at his out-setting in trade, owed £7r)0; he had in cash, commodities, the stocks, and good debts, £12510 : 7; he cleared, the first year, by commerce, £452 : 3 : ; wliat is the neat balance at the twelve months' end ? Ans. £12212 : 10 : 6. 8. A gentleman dying, left £45247 between two daughters, the younger who was to have 15 thousand, 15 hundred, and twice £15. What was the elder sister's fortune ? jI/w. £28717. 9. A tradesman happening to fail in business, called all his creditors to* gether, and found he owed to A, £63 : 7 : 6 ; to B, £105 : 10 ; to C, £34 : 5 : 2 ; to D, £28 : 16 : 5 ; to E, £14 i 15 : 8 ; to F, £112 : U ; and to G, £143 : 12 : 9. His creditors found the value of his stock to be £212 : 6, and that he had owing to him, in good book debts, £112 : 8 : 3, besides £21 : 10 : 5 money in hand. As his creditors took all his effects into their hands, I desire to know whether they were losers or gainers, and how much ? it. Arts, The creditors lost £146 : 11 : 10. 10. My correspondent at Seville, in Spain, sends me the following ac- count of money received, at different sales, for goods sent him by me, viz : Beeswax, to the value of £37 : 15 : 4; stockings, £37 : G : 7; tobacco, £125: 11:6; linen cloth, £112:14:8; tin, £115 : 10 : 5. My correspondent, at the same time, informs me, that he has shipped, agreea<)ly to my order, wines to the value of £250 : 15; fruit to the value of £51 : 12 : 6; figs, £19 : 17 : 6; oil, £19 : 12 : 4; and Spanish wool, to the value of £115 : 15 : 6. I desire to know how the account stands between us, and who is the debtor ? Ans. Due to my Spanish correspondent, £23 : 14 ; 4. MULTIPLICATION OF SEVERAL DENOMINATIONS. RuLK. — Multiply the first Denomination by the quantity given, divide the product by as many of that as make one of the next, set down the re- mainder, and add the quotient to the next superior, after it is multiplied. If the given quantity is above 12, multiply by any two numbers, which multiplied together will make the same nuinbpr ; but if no two numbaf multiplied together will make the exact nvnnber, then multiply thtt|^^ f^ line by as many as ia wanting, adding it to the last product, yi ^[r^ D -V-A.. 38 / 7 Proof. By Division. (') O £. s. d. £ ». d. 35 :12: 2 •75: 13 : H 3 COMPOUND MULTIPLICATiaiJf^ je s. d. 62 : 5 : 4i 4 £ «. <2. 57 : 2 : 44 5 71 : 5 : 2i • 1. 18 yards of clot per yard. li, at 98. 6d. 9 4: 5:6 2 2. 26 lb, of tea, at £1 : 2 : 6 per lb. 8 8X2=18 8X3X — 'SO 9:0:0 3 8 : 11 : 27 : : Top line X2=2 : 5 : 29 : 5 : 3. 21 ells of Holland, at la. S^d. per ell. Faxiit, £8:1: IQi 4. 35 firkins of butter, at 15s. 3|d. per firkin. Facit, £2Q : 16 : 2^. 5. 15 lb. of nutmegs, at Is. 2|d. per lb. Facit, £27 : 2 : 2^. 6. 37 yards of tabby, at 9s. 7d. per yard. ^ Facit, £17 : 14 : 7. 7. 97 cwt. of cheese, at £l : 5 : 3 per cwt. Facit, £122 : 9 : 3. 8. 43 dozen of candles, at 6s. 4d. per dozen. Facit, £13 : 12 : 4. 9. 127 lb. of Bohea tea, at 12s. 3d. per lb. Facit, £77 : 15 : 9. 10. 135 gallons of rum, at 7s. 5d. per gallon. • ^ Faxjit, £50 : 1 : 3. 11. 74 ells of diaper, at Is. 4^d. per ell. ^ . Facit, £5 : 1 : 9. 12. 6 dozen pair of gloves, at Is. lOd. per pair. ^ Facit, £6 : 12. the given quantity consists oi f , §, yr 4. Divide the given price (or the price of one) by 4 for i, by 2 fpr IT 4, first divide by 2 for h, then divide that quotient by 2 for i, add >e product^ and their sum will be the- answer required. J 13. :* 14. 16. 16. ; 17. \ , 18. ( 30. 1 21. 1 00 o Wa4« O 23. 2 24. 1 25. 3' 26. 5( 27. 9( W»iMwI»^>pl»Mi'-n^ :2i 6 8 :0 .0 3 :0 :5 :0 :0 :5 :0 # . ,i. ^ J. ***** 5j|| COMPOUND MULTIPLICATIOIf. 13. 25^ ells of holland, at 3 : 4|d. per ell. 5 5X5=25 16:10^ ■ 6 39 4:4:4^ = 25 4:6:01 = 251 14. I5i ells of diaper, at Is. 3d. per ell. 15. m ells of damask, at 4s, k per ell. ^''^*' ^' '' '' '''^' 16. 35 J ells of dowlas, at Is. 4d. per ell. ^'"*' ^' ' ' = ''^• 17. n cwt. of Malaga raisins, at £l : 1 : 6 pe!l\^' ' ^ ^ ^' . 18. 6f .barrels of herrings, at £3 : 15 : 7 Sar^Ii; '' '' ''^' ^ 19. 35^ cwt. doubled refined sugar, at £4^^^:^ peV cU'^' ^0. 15H cwt. of tobacco, at £4:17: lO^f cwt '' '''''''' Qi n'7i. «.„n i? , Facit, £755 : 15 : 3. il. 117i gallons of arrack, at 128. 6d. per gallon. oo rks ^^4. i? 1 Facit, £73 : 5 : 7*. -2. 85f cwt. of cheese, at £1 : 7 : 8 per cwt. ^'.• ~n 40 . COMPOWB MWTIPUCATIOH. i 30. 120J cwt. of hops, at £4 . T • » > ^^^j^^ £528 : 6 : 1i- 31. 401 yards of cloth, at 3s. 9|d. per yard^^ ^^^ . ^ . 24. 32. 729 ells of cloth, at Is. H^- pe' «"• j,^;^^ £217 : 3 : 6*. 33. 2068 yards of lace, at 9s. Sid. per yard. ^^^^ _ ^^ . ^^ THE APPLICATION. ..Whats„.ofn,oney.J^..e..deda.oo^tl8.enso tiiat each man may receive £U . b . «t • ^^^ ^^SS : 0:9. * V ^ iirize which amounted tc 2 A prWateer of 250 men to^\ M^^^^' ^ ^ ,^, ize 1 £125 15 : 6 to each man; w^^t.^^^%;7£3l443 : 15 : 0.^ . Wtnt difference is there between t^v,ce eight ana y, Joe5-gHand«hatisthe.rproduct^^^.^_,^^,p auot. X. tW 'oreater of them is 37 times 5 nere are two nutnhers, *e greater ^^^ ,,oduct are «% their difference 19 Sr3254tum,%e«a Jproduc. Tth sum of two numbers ^^^^ X^ what is their product and the 'q"^'-^^ ^^^^ ^f their difference. Ani 31104 Product, f ^^^^ons of horse, each 157 1. In an army consistmg ot 187 »V"«' effective sol- J„, and 207 ba^t^Uon. -l^.^XthCa^ *" sill diers, supposing that in i nost^^** ^^s^ 144800. * •,«,;« finwrv with his 8. What sum did that geutW r^ive J do J^^ wife whyje fortune was her wedamg smt ,J 1 .^ ^^^^ q„,„ IJ^o'rows of furbelows, each tarbelow 87 qmlls, ^^^_^^ _ ^^ . ,^^ "rr*:Lanthad £19118 to begin trade with -^f^J^y;* .^^etter he deHred £1086 » J-^j-Jfa^T^^^* ^was'^in trade, he ■ y<''«;^»^'■^^'!!'riwnother,£475:4:6/ t ne to 'lose, one year with another, end? ds realfortune at 12 ycars^ena^^^^^^ . g f /% :/) ). 1, so 9. d tc \ 0. lalf a 08. ^, and I uct. timeB ict are luct. , 144; ence. icb 15^ Ave sol- 4806. ^ith his , having ich quill L4 : 0. • 5 yeapa ade 2.:o6d trade, he , : 4 *. 6/ -f ; 8 :/ ; .y COMPOUND MULTIPi^ICATION. 10. In some parts of the kingdom, they weigh their coals\ machine in the nature of a steel-yard, waggon and all. Ti, of these draughts together amount to 137 cwt. 2 qrs. 10 lb., \ the tare or weight of the waggon is 13 cwt. 1 qr.; how mai coals had the customer in 12 such draughts ? Ans. 391 cwt. 1 qr. 12 lb. 11. A certain gentleman lays up every year £294 : 12 : 6, and spends daily £l : 12 : 6. I desire to know what is his an- nual income ? •^^«- ^887 : 15 : 0. 12. A tradesman gave his daughter, as a marriage portion, a •scrutoire, in wl:ich there were twelve drawei-s, in each drawer were six divisions, in each division there were £50, four crown pieces, and eight half-crown pieces; how much had she to her fortune? Jns. £3744. 13. Admitting that I pay eight guineas and half-a-crown for a quarter's rent, and am allowed quarterly 15s. for repairs, what does my apartment cost me annually, and' how much in seven years? Ans. In 1 year, £31 : 2. In 7, £217 : 14. 14. A robbery being committed on the highway, an assessment was made on a neighbouring Hundred.for the suni of £380 : 15 : e, of which four parishes paid each £37 : 14 : 2, four hamlets £31 : 4 : 2 each, and the four townships £18:12:6 each ; how much was the deficiency? ■^^^•' ^^^ • ^2 • 2. 15. A gentleman, at his decease, left his widow £4560; to a public charity he bequeathed £572 : 10; to each of his four ne- phews, £750 : 10; to each of his four nieces, £375 : 12 : 6; to thirty poor housekeepers, ten guineas each, and 150 guineas to his executor. What sum must he have been possessed of at the time of his death, to answer all these legacies ? - Ans. £10109 : 10 : 0. 16. Admit 20 to be the remainder of a division sum, 423 tho quotient, tho divisor the sura of both, and 1^ more, what was the number of the dividend? ^»«- 195446. EXAMPLES OF WEIGHTS AND MEASURES. (*) Multiply 9 lb. 10 oz. 15 dwts. 19 grs. by 9. ,, n Multiply 23 tons, cwt. 3 qvs. 18 lb. by 7. ' . M Multiply 107 yards, 3 qrs. 2 nails, by 10. // I F*y 'J t> _1- I. ^ iliiV. -J Lilt. 1 1 Multiply 27 beer bar. 2 firk. 4 gal. 3 qts. by 12. J, Multiply miles, 6 fur. 26 poles, by 12, u3 ^.P^ /' c*.-»t*«« «MiltlMt«L~' ^.. Z^^m *^ L \: DIVISION. DIVISION OF SEVERAL DENOMINATIONS. Rule. Divide the first Denomination on the left hand, and if any remains, multiply it by as many of the next less as make one of that, which add to the next, and divide as before. Proof. By Multiphcation. (■) (') . £ s. d. £ s. a. 2)25 : 2 : 4( 3)37 :1 :1( 12 : 11 : 2 £ s. a. 4)57 : 5 : 1( £ s. d. 6)52 : 1 : 0( (') Divide £140Y : 11 : 1 by 243. •) Divide £700791 : 14 : 4 by I79i '') Divide £490981 : 3 : 7^ by 31715, ') Divide £19743052 : 5 : H by 214723. THE APPLICATION. 1. If a man spends £257 : 2 : 5 in twelve months' time, what IB that per month ? ^^s- ^21 : 8 : GJ. 2. The clothing of 35 charity boys came to £57 : 3 : 4, what ia the expense of each? -^^*^- ^I *• ^^ : 8. 3. If I gave £37 : 6 : 4| for nine pieces of cloth, what did 1 give per piece ? Ans. £4. : 2 : 11. 4. If 20 cwt. of tobacco came to £27 : 5 : 4^ at what rate is that per cwt. ? ^w«- ^^ • '^ ' ^• 6. What is the value of one hogshead of beer, when 120 are aold for £164 : 17 : 10 ? ^ns. £l : 5 : 9i 6. Bought 72 yards of cloth for £85 : 6 : 0. I desire to know at what rate per yard ? -4ws. £1:3: 8^. 7. Gave £275 : 3 : 4 for 36 bales of cloth, what [b that for 2 bajes? ^*''*- ^^AL^^- '\^8. A prize of £7257 : 3 : 6 is to be equally diVidSakWongst sailoi-s, what is each man's share ? ^^ Ans. £14 : 10 : 3^ ik\^*?hf>re ?,Tn 2545 bullocks to be divided amongst 509 men, 1 u_ \. ^^know how many each man had, and the value of e»«^;^ ' ^S^ 'share, supposing every bullock worth £9 : 14 : 6. f'J-' 4eL. 5 bullocks each man, £48 : 12 : 6 each share. 1^ :"!Hf>«,*.»^,*.,.>tj^ 'wKJf .'■^^>' 1 i men, I jj^ of eJiei^;^ DIVISION. 10. A gentleman has a garden walled in, containing 9625 yards, the breadth was 35 yards, what was the length ? ^ .. ' Aus. 215. 11. A club in London, consisting of 25 gentlemen, joined for a lottery ticket of £10 value, which came up a prize of £4000. I desire to know what each man contributed, and what each man's share came to ? Ans. Each contributed 8s., each share £160. 12. A trader cleared £1156, equally, in 17 years, how much did he lay by in a year ? -^w^- £6 8. 13. Another cleared £2805 in 1^ years, what was his yearly increase of fortune ? Ans. £374. 14. What number added to the 43d part of 4429, will raise it U/'.i40? ^ Ans. 131. 16. Divide 20s. between A, B, and C, in such sort that A may have 2s. .less thai B, and C 2s. more than B ? %^,,, . ,; Ans. A 4s. 8d., B 6s. 8d., C. 8s. 8d. tk If there are* 1000 men to a regiment, and but 50 officers, how many privateimen are there to one officer ? Ans. 19. 17. What number is that, which multiplied by 7847, wil make the product 3013248 ? Ans. 384. / 18. The quotient is 1083, the divisor 28604, what was the di- vidend if the remainder came out 1788 ? t Ans. 30979920. 19. An army, consisting of 20,000 men, took and plundered >a city of £12,000. What was each man's share, the wholo being equally divided among them ? Ans. 12s. 20. My purse and money, said Dick to Harry, are worth 12s. 8d., but the money is worth seven times the purse. What did the purse contain? ^^*; Us. Id. 21. A merchant bought two lots of tobacco, which weiglied 12 cwt. 3 qrs. 15 lb., for £114 ; 15 : 6. Their difference in point of weight, was 1 cwt. 2 qi-s. 13 lb., and of price, £7 : 15 ; 0. I desire to know their respective weights and value ? Ans. Less weight, 5 cwt. 2 qrs. 15 lb. Price, £o3 : 10. Greater weight, 7 cwt. 1 qr. Price, £61:5:6 / '. jrnce, juoi ; o . u- y ^ Ji;i, i/iViUU iUUU tJlUtt'iia in su-wn « Lin-^iiti-.i f.-. v -- — - -1-7 f j ( . I /v<^ -r^'-.iJ^ >^< 0. that A may receive 129 more than B, and B 178 le**' 'Mn 1/ Ans. A 300, F* £^r f iP"^.^ Ml^.^^^ HWP'- /I kK' BILLS OF PAKCEL8. EXAMPLES OP WEXOHTB A.. MEAB.BBS. 2. Divide 29 to^^' ^%'7,; Vnails, by 10. 3. Divide 114 yards, 3 qrs 2 "''^;'^'^ J^ n. 4 Divide 1017 miles, 6 ^^^'^- ^^ ^^^^ 26. t Divide 2019 acres, 3 -o^/^3 P^^,^^^^^ days, 11 bours, 27 6. Divide n1 years, 7 montns, o minutes, by 37. BILLS OF PARCELS. hosiers'. S?> ' ^.)Mr.Joh„«^^g^^^j^^^^„. s. «?. May 1, 18 4- 4. * 6 per pair ** 8 Pair of worsted stockings ^^"y., oj...... 5 Pair of thread ditto.. U:0....... 3 Pair of black silk ditto ^^ . ^ 6 Pair of milled bose -^ ^ . g 0, Pair of cotton ditto i • 8 per yard ^>^Yards of fine flannel ^^2:2 \ MERCERS . n Mr. Isaac Grant, May 3, 18 ^ ^ Bought of John Sims, ^^ ^ at . 9 ■• 6 per yard £ 15 Yards of satin . . . • • • • ' * * .17 : 4 '• 18 Yards of flowered sik ^^ . g 12 Yards of rich brocade ^ 3:2.... 16 Yards of sarsenet ^ g^ ^ . , .Q Y.vds of Genoa velvet a- a..., 3 Yards of lutestring "' £62 : 2 : 6 h yt«A« Jk If tp''^' * "%■, ■•/ ;#-^ BILLS OF PARCELS. LINEN DRAPERS . (') Mr. Simon Surety, Boufilit of Josiah Short. s. d. June 4, 18 4 Yards of cambric at. . . 12 : 6 per yard £ 12 Yards of muslin 8 : 3 1 5 Yards of printed linen 5 : 4 2 Dozen of napkins 2 : 3 eacli . . . 14 Ells of diaper 1 : 7 per ell . . 86 Ells of dowlas 1 : 1^ £17 : 4 : 6^ milliners'. (*) Mrs. Bright, Bought of Lucy Brown. £ s. d. 18 Yards of fine lace at. . .0 : 12 : 3 per yard £ 5 Pair of fine kid gloves : 2 : 2 per pair 12 Fans of French mounts : 3:6 each . . . 2 Fine lace tippets 3 : 3:0 4 Dozen Irish lamb : 1 : 3 per pair 6 Seta of knots. : 2 : 6 per set. . June 14, 18 £22 : 4 : 4 woollen drapers', (*) Mr. Thomas Sage. Bought of Ellis Smith. £ s. 17 Yards of fine serge. . . .at. . .0 : 3 1 8 Yards of drugget : 9 15 Yards of superfine scarlet . . .1 : 2 16 ^ardsof black : 18 : 25 i 'ards of shalloon June 20, 18 9 per yard £ 0... ...... If i ';ards of Urab \ i 1 :9 .0 : 17 :6 *Kr ..c y A iS"^ BILLS OF PAKCELS. leather-sellers'. (•) Mr. G-^les H^rris^^^ ^^ ^^^^ ^^,^^, July 1,^8 d. W ,,...3:9perBbu^ 27 Calf Skins 1 : T ^' cheepditto .... 1:8 I'e Co uved ditto •;.V.;..U:6 • 15 Buck ditto 10 :^. ; 17 Russia Sides 1:2^ _ 120 Lamb Skins £38 : 1*^ • ^ I Mr. Bic^-^^yof Francis Effiot.^ ^ July 5, 18 «igWl5lb. __'. 0-. 3.._ _^ July 6, 18 M [{' CHEESEMONGERS . • at ..0--6perlb.;fi p. . K. ^f htttter, «t. 28 ID. ; 4 "•■wA-e cheeses, 1'^ ; '"• • •;;■.... .0 : 3 ; ^ XUsWre ditto, 15 \b. • • • 0:6 ^ ■■ \ f\r<^xc. cheese "^ .\4 1 y 1 1 wit witll '. i 7:5 18 : 2 : 3i iy 6, 18 M \\ y REDUCTION. \ n Mr. Abraham Doyley, j | 20, 18 '* ^ ^ Bought of Isaac Jones. Juiy , 1 /» at .1 : 10 per bushel * Tares, 19 bushels ^^3. g^ Pease, 18 bushels '/.*.. 25 : per quarter Malt, 7 quarters /.'.'//. .1 : 5 per lb. . . • Hops, 15 lb ! ! [2 : 4 per bushel )at8, 6 qrs..... V/.'.4 : 8 : ^eans, 12 bushels . ^ £23 : 7 : 4 i • , — - \ REDUCTION PP y} \ hi. l: 1 :0 I IS inea 0:10:6 \ viisalf ditto '.'/.. .1 : 0:0 \\ \ Sovereign '. .0 : 10 : (Wie^ Half ditto . . . • ^ ^ : 5:0 ' U 3\CJrown 0: 2:6 Vvt8.\Talf ditto ^^ . i:0 J^i.^^^;:^^ T^.^^enr haM v.igh Crsixp^^^^ fourpence, threepence, twopence, penny, / i 33. A^ ^^^any shillmgs and pence ? .\ \ y. of silv' 1 j \ dwts. A ' ^, A if\*'^' 'M^hillings. • 4^p\ A|^ jiul^doS J^^^ ■»•« -T*V»». '***''^*^ ij i »y? ii«iii i-v REDUCTION. In £12, how many shillings, pence, and farthings? 3. In 311520 farthings, how many Ans. 2408. 2880d. ? . „ ^ , . Ans. £324 : 10. 4. How many farthings are there in 21 guineas ? ^ r n,^ , , ■^^^- 21168. 6. m i.17^: 6 : 3^, how many farthings ? ^n*. lGo73. / 6. In £25 : 14 : 1, how many shillings and pence? ^ws. 514s. 6169d. /. m 17940 pence, how many crowns? Am. 299. [| f ^- I" 15 crowns, how many shillings and sixpences? n T rH •. ,^ , ^W5. 75s. 150 sixpences. ' 9. In 57 half-crowns, how many pence and farthings ? ^ . - ^W5. 1 7 1 Od. 6840 farthings. 10. In 52 crowns, as many half-crowns, shillings, and pencvin how many farthings ? ^^ ^ 21424./ i^ri^' P'^r "^^°y P^'^^^' shillings, and pounds, ^re there)'^ Tn^TT "^^- ^^«- 4320d. 360s. £18. ' 12. How many guiijeas in 21168 farthino-s ? / 10 T 1 H i. 1 . ^4w5. 21 guineas, ■) 13. In 16573 farthings, how many pounds? v** Ans. £l7 * 5 * 3' 14. In 6169 pence, how many shillings and pounds ?' 15. m 6840 farthings, how many pence and half-crowns ? ^ — siiij- i« T „ , » , ^«. IVlOd. 57 lmlf-cro\)2 52 16. in 21424 farthings, how many crowns, halt-crowni+^ vis « hngs, and pence, and of each an equal number ? Anf^; Iqs 17. How many shillings, crowns, and pounds, in 60 gui^ & ' lo tj J H . -4ws. 12608. 252 crowns^ V» 18. Keduce 76 moidores into shillings and pounds? 4y ^v$[ ■.o T^ 1 ^ ^««. 2052s. £11, , U:c 19. Reduce £102.: 12 into shillings and n^oidores? # Vk TT "^'^^*' 2052s. 76 17 20. How many shillings, half-crowns, and crowns^' lu i^556, and ot each an equal number ? 1 o, T„ ,. -^^^6M308 each, an|. 1 ir*u5 nair-crowns, as ounds ? many crowns and si AnsM. vi men brought £15 : 10 each into the/^ X>, how many must they have in'^0 ',, Ans. 103 ffu--^ * i ?\ V 1 1 -( REDUCTION. 24. A certain person hnr) ok ^ , . •^^''' ^15 : 10. "eas a crown, and a 1 i. ? P"''''' ^"^ ^" ^^^^^ P"'^^ 12 ir„i. .^ ^ . ^ vn, and a moidore, how many pounds sterling had he 25. A gentleman, in his will loff -P^n . n ^^*- '^^^^• tlmt ^ should bo o. ven to Inripnf "^ *i'^ P^^^' '^'^^ ^^^^^^ed poor women, each to 14 2TTd ^t"' '"'^ l" ^''''' ''-^ to 1«.--| to poor girls, each to l.^vro^ T'^ ^^^'' ^^^^ ^^ ^^'^ve person wh^ distfibu ed it I demld T^'^'' ^'^"^'"^^"^^^ *^ ^^^ there were, and what tlm\J.nnCv''-J ""^"^ ^^ ^^^^^^ sort Wilis trouble? Pei^^onwho distributed the money had Ans. 66 men, 100 women 200 boys, 222 girls, *^ . 13 . 6 for the person's trouble, • TROY WEIGHT. 26. In 27 ounces of gold, how many grains ? 27. In 12960 grains of gold, how many ounces?^''' '''''• 28. In 3 lb. 10 oz. 7 dwt.. 5 gr. how many grains ? ^"'' ''' 29. In 8 ino-ots of ^llvn^ ^o^i, • i • ^ ^^*- 22253. 131. Bouofht 7 intrntc r.f c•,^ 1 -^''^'^'' 8 mjyots. |wf3. ho,v many S ''™'' ''••'<^'' ~"''»°"ff 23 ib. s'o^ 7 ' 32. A gentleman s^nf n +nr.i.„ 4^ i • , ^^*- 945336. «0 .... 8 Lfe. n^^" o?cl rod ± 1'" ''T ^f .^™'«'' «'«' --ish-l ! d«b. 10 .^r. eacl": Ll rf ""'it ", ,'"'? -^"'f-* -' " "^ _ I'Or do;v5^ *'|"3oz. 10dwts:e;ich; and 1 doz. and ror every tank.-ird to h forks of 21 i. u\Yt3. ly o-r. 'Jozen of forks ; what^is the 'ive one salt, a d oz. 11 dwts. 1 r. ifj Alls. 2 of each sort, 8 number of euch i">f'5 ', a Qozen .>f:^r;,. ^fg/ v-i oz. 9 d\vt' z^- u * ^ /*yL REDUCTION V 4 AVOIRDUPOIS WEIGHT. Wi Note.— There are several sorts of silk which are weighed by a great ppund of 2 1 oz. others by the common pound of IG oz. ; therefore, To bring great pounds into common, multiply by 3, and divide by 2, ot add one half To bring small pounds into great, multiply by 2, and divide by 3, or sub- tract one third. Things bought and sold by the Tale. 12 Pieces or things make 1 Dozen 12 Dozen 1 Gross. 12 Gross, or 144 doz 1 Great Gross. 24 Sheets 1 Quire. 20 Quires 1 Ream. 2 Reams 1 Bundle. 1 Dozen of Parchment. 12 Skins. 12 Skins 1 Roll. 34. In 147G9 ounces how many cwt. ? Ans. 8 cwt. qr. 27 lb. 1 oz. 35. Reduce 8 cwt. qra. 27 lb. 1 oz. into quarters, pounds, and ouncea. ./Jn*. 32 qrs. 923 lb. 14709 oz. 36. Bought 32 bags of hops, each 2 cwt. 1 qr. 14 lb. and another of 150 lb. how many cwt. in the whole ? Ans. 77 cwt. 1 qr. 10 lb. ^--^'I. In 34 ton, 17 cwt. 1 qr. 19 lb. how many pounds ? \ A71S. 78111 lb. 38.' In 547 great pounds, how many common pounds ? Ans. 820 lb. 8 oz. 39. In 27 cwt. of raisins how many parcels of 18 lb. each ? > Am. 168. ; 40. In 9 cwt. 2 qrs. 14 lb. of indigo, how many pounds ? \ Ans. 1078 lb. I 41. Bought 27 bags of hops, each 2 cwt. 1 qr. 15 lb. and one bag of 137 » ' lb., ho\\many cwt. in the whole ? \ Ans. 65 cwt. 2 qrs. 10 lb. 42. Hdw many pounds in 27 hogsheads of tobacco, each weighing neat 6S cwt. ? ^ Ans. 26460 ■j» ^ In 552 common pounds of silk, how many great pounds ? j ^, V vw ^ Ans. 36f C^ »^N^ '^7 parcels of sugar of 16 lb. 2 oz. are there in 16 cwt. l/ . j^ vS^ ■•,'■■ '{'") ^11^ ,^ '■c , . J^.=^^5. I ■^A i\ How often will the wheel of a coach, >hat ference, turn in 100 miles ? e2 Am. b\j:-2 : 1/ : i i.er ^--'^f,^., Jifis. £111 : 3 : '/.; ^^^ r^ P^ ^^m^ mi SSti S^mi. Ifow many barley rmference of \,hicli is REDUCTIOIV. * •corns will reach round the world, 360 degrees, each degree 69 miles Ans. 4755801600 barley-corni LAND MEASURE. th and A 63. In 27 acres, how many roods and perches? Ans. 108 roods, 4320 perches, 64. In 4320 perches, how many acres ? Ans. 27. 66. A person having a piece of ground, containing 37 acres. 1 pole, has a mind to dispose of 15 acres to A. I desire to know how many perches he will have left ? [. ^ Ans. 3521. ' 66. There are four fields to be divided into shares of 75 perches f each ; the fii-st field containing 5 acres ; the second, 4 acres, 2 poles ; the third, 7 acres, 3 roods ; and the fourth, 2 acres, 1 rood, I desire to know how many shares are contained therein ? |^ • Ans. 40 shares, 42 perches rem, f WINE MEASURE. } ' 67. Bouglit 5 tuns of port wine, how many gallons and pints? i ^ Ans. 1260 gallons, 10080 pints. 68. In 10080 pints, how many tuns ? ' Ans. 5 tuns. \ 69. In 5896 gallons of Canary, how many pipes and hogs- heads, and of each an equal number? -. ^ Ans. 31 of each, 37 gallons over. 70. A gentleman ordered his butler to bottle oflf f of a pipe of French wine into quarts, and the rest into pints. I desire to know how many dozen of each he had ? Ans. 28 dozen of each. ALE AND BEER MEASURE 71. In 46 barrels df beer, how many pints. Ans. 1.3248. / L^ lb , barrels of ale, how many gallons and quarts? * r f, . -^***- 320 gals. 1280 qta. / 42. I^»-»^ ' 2 hogslioads of ale, how many barrels"? . ' » common -j^f ale, how many hog-sheads ? Ans xin^f %^parcel3 of u Afuu.., ,■#■ s. mi \ SINGLE BULE OF THREE DIRECT. ^ 15. If a genflfeman's income is £500 a year, and he spends 19^. 4d. per day, how much does he lay by at the year's end ? \dtis. £147 : 3 : 4. 16. If I buy 14 yards of cloth for 10 guineas, how many Flemish ella can I buv for £283 : 17 : 6 at the same rate ? ...j^^Jj^^ ^-: Ans. a|KPit»W?> 2 qrs. f 17. If 504 Flemish ells, 2 quarters, cost £283 : 17 : %mfc^at rate did I pay for 14 yards .' Ans. IDs. lOd. 18. Gave £1 : 1 : 8 for 3 lb. of coffee, what must be given for 29 lb. 4 oz. ? Ans. £10 : 1 1 : 3. W. If one English ell 2 qrs. cost 43. 7d. what will 39i yards cost at th« •ame rate .' Ans. £5:3: 54, 5 rem. 20. If one ounce of gold is worth £5:4:2, what is the worth of one grain .' -Ans. 2h\. 20 rem. 21. If 14 yards of broad cloth cost £9 : 12, what is the purchase of 75 yards ? ' -^ws. 51 : 8 : G|, G rem. 22. If 27 yards of Holland cost £5 : 12 : 6, how many ells English can I buy for £100 ? -/ins. 384. 23. If 1 cwt. cost £12 : 12 : 6, what must I give for 14 cwt 1 qr. 19 lb. Ans, £182 : 0: lU, 8 rem. 24. Bought 7 yards of cloth for 1 7s. 8d. what must be given for 5 pieces, each containing 27 i yards. Ans. £17 : 7 : 04, 2 rem. 25. If 7 oz. 11 dwts. of gold bo worth £35, whut is the value of 14 lb. oz. 12 dwt. 10 gr. at the same rate ? Ans. £823 : 9 : 3|, 552 rem. 26. A draper bought 420 yards of broad cloth, at the rate of i4s. lO^d per ell English, how much did he pay for the whole ? ^ ^ • jfr«.». 250 : 5. 27. A gentleman bftught a wedge of gold, which weighed 14lb..3oz 8 dwts. for the sum of £514 : 4, at what rate did he pay fur it per oz? Ans £3. 28. A grocer bought 4 hogsheads of sugar, «ach weighing nt at cwt. 2 qrs. 14 lb. which cost him £2 : 8 : G ptr cwt. ; what is the value of the 4 hogsheads ? „ . . ^ ' Ans. £04 : 5 : 3. 29. A draper bought 8 packs of clotli, each containing 4 parcels, euch parcel 10 pieces, and each f)iece 26 yards, and gave after the rote of £4 % 16 for 6 vards ; I desire to linow what tlie 8 packs stood him to ? ■^ Ans. £jG56. 30. If 21 lb of raisins cost Gs. Gd. what will IS frails CDst, encb weigh- ing neat 3 qrs. 1 S lb. ? ^ ^;- *^ , .. Ans. £24 : \fi%. > 9J'/ JM oz. of stiver be worth 5.3. what is the price of 14 irtgot^s eachS weighing 7 1b, 5 oz. 10, dwts. <^ '' * " . Ans. £:jiy : 5, a-_,^. VTU^tL i3 CttCpti'-- '•' " |-"--< 5 D - --■-- - -•- ^ \-^^ Old. pKr titwie? %. Boa.^i!t 50 cwt. 2 qrs. 24 lb. of tobacco il^ioa Uconie to i \ yimin -i^^ifr ' ' BULE OF THREE INVERSE. 34. Bought l7l tons of lead, at £14 per ton; paid carriage and other incident charges, £4 : 10. I require the value of the lead, and what it stands me in per lb. ? ,- . ' Ans. £2398 : 10 value ; H5ilk32 rem. per lb. 35. If a pair of stockings cost 10 groats, how many dozen may I buy for £43 : 5 ? Ans. 21 dozen, 7^ pair. 36. Bought 27 dozen 6 lb. of candles, after the rate of 17d. per 3 lb. what did they cost me ? Ans, £7 : 15 : 4^, 1 rem. 87. If an ounce of fine, gold is sold for £3 : 10, what come 7 ingots to, each weighing 3 lb. 7 oz. 14 dwts. 21 gr., at the same pnce Ans. £1071 : 14 : 5^. 38. If my horse stands me in 9^d. per day keeping, what will be the charge of 1 1 horses for the year f Ans. £158 : 18 : 6^. 39. A factor bought 86 pieces of stuff, which cost him £517 : 19 : 4, at 4s. lOd. per yard ; I demand how many yards ther« were, and how many ells English in a piece t • Ans, 2143^ yards, 56 rem. and 19 ells, 4 quartei-s, 2 nails, 64 rem. in a piece. 40. A gentleman hath an annuity of £896 : 17 per annum. I deoire to know how much he may spend daily, that at the year's ^^ end he may lay up 200^ guineas, and give to the poor quarterly 40 moidores? Ans. £1 : 14 : 8, 44 rem. THE RULE OF THREE INVERSE. Inverse Proportion is, when more requires less, and less re- quires more. More requii-es less, is when the third term is great- ^er than the first, and requires the fourth term to be less than the ^cond. And less requires more, is when the third term is le^ than the first, and requires the fourth term to be greater than second. e first and second terms ¥y togeth \o, product by the third, the quotient will bear such propof- M second as the first does to the third. (- RX7LE OF THREE INVERSE. 67 EXAMPLES. r. it- • f 1. If 8 men can do a piece of work in 12 days, how many days can IG men perform the same in ? Ans. C days. 8 . 12 . . 16 . 6 8 16)90(6 days. 2. If 54 men can build a house in 90 days, how many can do the same in 50 days ? Ans. 974 men. 3. If, when a peck of wheat is sold for 2s., the p«?nny loaf weighs 8 oz., how much must it weigh when the peck is worth but Is. Gd. ?. Ans. 10| oz. 4. How many pieces of money, of 20s. value, are equal to 240 pieces of 12s. each?te Ans. 144. 6. How many yards, oi three quarters wide, are equal in .mea- iure to 30 yards, of 5 quartei*s wide ? Ans, 60. 6. If I lend my friend £200 for 12 months, how long ought he to lend me £160, to requite my kindness? Ans. 16 months^ 1. If* for 24s. I have 1200 lb. carried 36 miles, how many pounds can I have carried 24 miles for the same money ? Ans. 1800 lb. 8. If 108 workmen finish a piece of work in 12 days, how many arc sufficient to finish it in 3 days I Ans. 432. 9. An army besieging a town, in which were 1000 soldiers, with provisions for 3 months, how many soldiers departed, when the provisions lasted them 6 months ? Ans. 500. 10. If £20 worth of wine is sufficient to serve an ordinary of 100 men, when the tun is sold for £30, how many will £20 wortljr suffice, when the tun is sold but for £24 ? Ans. 125. Jf' ii.il Conner makes a journey m i;4: uuya, wiji;ij uio but 12 houi-s long, how many days will he be going tli journey, when the day is 16 hours lon^? Ans, Ij ' %if?f^-.ijU^f''^f\ ■ ■ 69 DOUBLE RULE OE THREE. 12. How much plush is sufficient for a cloak, which has in it 4 yards, of 1 quarters wide, of stuff, for the lining, the plush being but 3 quarters wide? Ans. 9^ yards. 13. If 14 pioneers make a trench in 18 days, how many days will 34 men take to do the same ? Ans. 1 days, 4 hours, 56 min. y\, at 12 hours for a day. 14. Borrowed of my friend £64 for 8 months, and he had oc- casion another time to borrow of me for 12 months, how much must I lend him to requite his former kindness to me ? Ans. £42 : 13 : 4. 15. A regiment of soldiers, consisting of 1000 men, are to hav* new coats, each coat to contain 2^ yards of cloth, 5 quarters wide, and to be lined with shalloon of 3 quarters wide ; I demand how many yards of shalloon will line them ? Ans. 4166 yards, 2 qrs. 2 nails, 2 rem. THE DOUBLE RULE OF THREE, Is so called because it is composed of 5 numbers given to find a 6th, which, if the proportion is direct, must bear such a proportion to the 4th and 5th, as the third bears to the 1st and 2d. Btit if in- verse, the 6th number must bear such proportion to the 4th and 6th, as the 1st bears to the 2d and 3d. The three first terms are a supposition ; the two last, a demand. Rule 1. Let the principal cause of loss or gain, interest or decrease, action or passion, be put in the first place. 2. Let that which betokeneth time, distance of place, and the like, be in the second place, and tiie remaining one in tlie third. 3. Place the other two terms under their like in the su}>posi« tion. 4. If tlie blank falls under the third term, multiply the first and second terms for a divisor, and the other three for a dividend. But, iail3 UHUCT OtltJ Hint or OCUVilUb Ut u. 11 Liro 1-1 i_ UUUIIv. ]yhxs third and fourth terms for a divisor, and the other three for jy^S^ivideiid, and the quotient will be tlie HLIt'^oof. By two single rules of three. answer. ^. *»iA-j( -».-.«•» *■- M,.. L;f *'ii / II iili I I ;)M B|ll > ili DOUBLE RULE OP THREE. ^jnSfcKL^ b^ EXAMPLES. ^J' U- • "^ I'^'^L^ u* ^° ^"^^^^^ °^ °^*« i» 16 days, how many bushels will be sufficient lor 20 horses for 24 days ? / '» «»" By two single rules, hor. bu. hor. bu. 1. As 14 . 50 .. 20 . 80 days bu. days. bu. 2. As 10 . 80 . . 24 . 120 or in one stating, worked thus : hor. 'days bu. 14 . Id . 50 50X20X24 20 . 24 . ^ _=120 14X10 #h?: ^l ^.""'" i*14 days can mow 112 acres of grass, how many men must there be to mow 2000 acres in 10 days ? ' acres, days, acres, days. ; men. days, acres. .1. As 112 . 14 .. 2000 . 250 r 8 . 14 . 112.8X14X2000 days. As 250 rnen. davs. 8 .. 10 men. 200 10 . 2000 112X10 -=200 3. If^lOO in 12 months gain £6 interest, how much will £75 eain in -^ '"''"^^s- ^m. £3:7:6. much inthf hil'''^'''''%^\* ^ ^"'-'^^ ''"'"""S^ «^ 3 '^^•*- 150 miles, how much ought he to receive for the carnage of 7 cwt. 3 qrs. 14 lb. for 50 miles ? r ,, . • -^ns- £1:10: 9. tP.^;nf whS"^^?no^/''^^'r'' ^^nsisting of 136 men, consume 351 quar- rnn.nmrr„ 4"; ^^^^y^' ^°^ "^^^^ quarters of wheat will 11232 soldiers consume m 56 days ? % -^ns. 15031 qrs. 804 rem. 6. II 40 acres of grass be mowed by 8 men in 7 days, how many acres can be mowed by 24 men in 28 days ? ,/ins 480 for'^oi d«v,'' wnll r^ ^ ""'" ^°' ^ ^^y^' ™^^' ^°^ "^"^h vvi)rpay'32 men lor -4 aa}s woilc f ^^^^ ^33 . ^ 8 If £100 in 12 months gain £0 interest, what principal will gain £3 : 7 : in 9 months ? JJnl £15. ■ '^ /ioV^^''^''"*' consisting of 939 soldiers, consume 351 qrs. of wheat m a 108 days, how many soldiers will consume 1404 qrs. in 56 days ? .ins fl2G8. Ul If a family consisting of 7 persons, drink out 2 kilderkins of beer in 12 days, how many kilderkins will another family of 14 persons drink out '" ^ "*^^ • ^ns. 2 kil. 12 gal n 11. If the carriage of 60 cwt. 20 miles, cost £14 : 10, what weii?ht can I i have carried 30 miles for £5:8: 9, at the same rate of carriage f f ^ . ^dns. 15 cwt. / J^'Jnr? ^'°T^ *•'*' ^ ^"^^^^'^ °^°^^^ '" 16 ^ays, how many horses will e^S up JOOO quarters in 24 days ? . 'M Ans. 4Qr' 13. If £100 in 12 months gain £7 interest, what is the interest of^ for 6 years ? mt'- _e'^:'^'^S:~^ZT- ... ^»M* ^839 : 16 : 4ifA „„«•— «r!rv 60 PBACTICB. ^f 9 tons 6 miles, wbat must U. If I pay lOs. f- ^^^C'^V vvlTxnUesS . I pay for the carriage of 12 tons, n ^ ^^ ^9 . 2 -. 0^ \ PRACTICE ed in trade and business. rf^^ed by taking alW"?*' «'. Of a Hundred. qrs. lb. 2 or 56 is ^ 1 or 28... l 14. ..i Of a Quarter. 14 lb I 7 i 4 ■ 4 3i the answer. ,. /i\ i;ai^^'704lb. att 1 ^1 ^♦^ 6547 at S Vv 12)1*'-"' I Faot, £lb_^-n_ '_ . >\ - \- ~„ ,, (n 4573 at i ^ 1€>; ? ;t.£5:18:10 tban a sWlUng, take the all "by 20, as before. O '»*'*%- IJ"^' ust cem- )t, or ided; ndred. 6i9 i Quarter. » • • • • 2 i • • • • •f i ! by tho , will bo TV , %-• ., ■ to tbe all jetber, and (») is tV 7547 at Id. 210)6218 : 11 Facit, £31 : 8 : 11. («)lis_U375latUd i is i 312 : V 78 : H 2i0)39l0 : 8| Facit, £19 : 10 : Si (") ,5432r/at 1-H- Facit, £339 : 10 : 7|. (*) 6254 at Ud. Facit, £45 : 12 : 0^ C*) 2351 at 2d. Facit, £19 : 11 : 10. («) 7210 at 2 id. Facit, £67:11 : 10^. n 2710 at2^d. Facit, £28 : 4 : 7. PRACTICE. (") 3257 at 4d. Facit, £54 : 6 : 8. n 2056 at 4id. Facit, £36 : 8 : 2. =* ("•) 3752 at 4id. Facit, £70 : 7 : 0. (") 2107 at 4td. Facit, £41: 14: O-i (")3210at5d. Facit, £66 : 17 : 6. n 2715 at5|d. Facit, £59 : 7 : ('*) 3120 at 5id. Facit, £71 : 10 : 0. (") 7521 at 6fd. Facit, £180 : 3 : 9|. {^') 3271 at 6d. T Facit, £81 : 15 : '6. C) 3250 at 2H Facit, £37 : 4 : 9^. 2715 at 3d. Facit, £33 : 18 : 9. '{>') 7062 at aid. Facit, £95 : 12 : 7f {'') 2147 at 3|d. Facit, £31:6: 2|. n 7000 at 3H Facit, £109 : 7 : 6. n 7914 at 6id. Facit, £206 : 1 : lOf (") 3250 at 6^d. Facit, £88 : : 5. (") 2708 at 6H. Facit, £76 : 3 : 3. {'") 3271 at 7d. Faeit, £95 : 8 : 1. {'°) 3254 at lid. Facit, £98 :5: Hi (") 2701 at 7id. Facit, £84 : 8 : H. 61 (") 3714 at 7|d. Facit, £ll9:18:7f (") 2710' at 8d. Facit, £90 : 6 : 8. C") 3514 at 8id. Facit. £120: 15 :10J. (") 2759 at 8^d. Facit, £97 : 14 : 3,f (''*) 9872 at 8^d. Facit, £369 : 18 : 4. ('') 5272 at 9d. Facit, £197 : 14 : 0. / ^^__ '("*) 6325 at 9id. Facit, £243 : 15 : 6^. {^') 7924 at 9^d. Facit, £313 : 13 : 2. ('") 2150at 9|d. Facit, £87 : 6 : lOf (") 6325 at lOd. Facit, £263:10:10. n 5724 at lO^d. Facit, £244 : 9 : 3. n 6827 at lOid. Facit, £270 : 4 : S^. •(") 3254 at lO^d. Facit, £142 : 7 : 3. {*') 7291 at 10^ Facit, £326:11:6' 1 0. y .'»**; ^left'V .\ (") 7254 at lUd. Facit, £340 : : 7i PRACTICE. (**) 3754 at ll^d. Facit, £179 : 17 : 7. (") 7972 at ll|d. Facit, £390 : 5:11. Rule 3. When the price is more than one «^^ili«S' ^"^ .|f;^ than two, take the part or parts, with so much of the given priTe n more than a shilling, which add to the given quantity, and divide by 20, it will give the answer. ('UxV2106atl2id. ^^ * 43 : lOi 210)21419 : 10^ Facit, £107 : 9 : 10^. {") 3215 at Is. lid. Facit, £177:9 :10i Oi2i37l5atl2H 154 : 9^ (•) 2790 at Is. l^d. Facit, £15G : 18 : 9. (') 7904 at Is. Ifd. Facit, £452 : 16 : 8. («) 3750 at Is. 2d. Facit, £218 : 15 : 0. (») 3291 at Is. 2id. Facit, £195 : 8 : Of. 210)38619 : 9^. Facit, £193 : 9 : 9i (") 3254 at Is. 3|d. Facit,£213:10:10i ('«) 2915 at Is. 4d. Facit, £194:6 : 8. (=>) 2712 at 125d. Facit, £144 : 1 : 6. (*) 2107 at Is. Id. Facit, £114 : 2 : 7. (") 9254 at Is. 2|d. Facit, £559 : 1 : 11. (") 7250 at Is. 2fd. Facit, £445:ll:5i (") 7591 at Is. 3d. Facit, £474 : 8 : 9. (") G325 at Is. 3id. •^^ . fi. O^rvi.io.nl racii, ^^ivi . io . u^ '(") 5271 at Is. 3^d. ^Kacit, £340:8 :4^d. (") 3270 at Is. Hd. Facit, £221 : 8 : li (»') 7059 at Is. 4^d. Facit, £485 : 6 : li ('") 2750 at Is. 4fd. Facit, £191:18:6|. (»°) 3725 at Is. 5d. Facit, £2G3 : 17 : 1. {^') 7250 at Is. 5id. Facit, £521 : 1 : 10^ ("^) 2597 at Is. 5^d. Facit, £189 : 7 : 3i (") 7210 at Is. 5fd. n 7524 at Is. 6d. Facit, £564 : 6 : 0. (") 7103 at Is. 6id. Facit, £540 : 2 : 5|. (") 3254 at Is. G^d. Facit, £250 : 1^': 7. (") 7925 at Is. 6|d. Facit, £619 : 2 : "' (") 9271 at Is. 7d. Facit, £733 : 19 : 1. (") 7210 at Is. 7id. Facit, £578 : 6 : 0^. n 2310 at Is. 7^. Facit, ^187 : 13 : 9. (") 2504 at Is. 7^. Facit, £206 : 1 : 2. C^) 7152 at Is. 8d. Facit, £590 : : 0. {^^) 2905 at Is. 8id. Facit, £245 : 2 : 2\. (3*) 7104 at Is. 8H Facit, £606 : 16 : 0. PRACTICE. k^ (") 1004 at Is. 8^d. Facit, £86 : 16 : 1. (") 2104 at Is. 9d. Facit, £184 : 2 : 0. (") 2571 at Is. 9icl. Facit, £227 : 12 : 9i n 2104 at Is. Old. Facit, £188 : 9 : 8. ♦ n 7506 at Is. 9^d. Facit, £680 : 4 : 7^ n 1071 at Is. lOd. Facit, £98 : 3 : 6. (") 5200 at Is. lO^d Facit, £482 : 1 : 8. (") 2117 at Is. lO^d. Facit, £198 : 9 : 4^ (") 1007 at Is. lOi Facit, £95 : 9 : 1^. (**) 5000 at Is. lid. Facit, £479 : 3 : 4. (") 2105atls. ll^d. Facit, £203 : 18 : 6^. \") 1006atl8. ll^d. Facit, £98 : 10 : 1. (") 2705atls. ll|d. Facit, £267 r^-a^-ai (") 5000 at Is. 11 -^d. J Facit, £48P : 11 : 8. n 4000atls. llH Facit, £395 : 16 : 8. RuLE 4. When the price consists of any even number^ of shillings under 20, multiply the given quantity by half, the price, doubling the first figure of the product for shillings, and the rest of the product will be pounds. (») 2750 at 2s. Facit, £275 : : 0. (") 3254 at 4s. Facit, £650 : 16 : 0. (=■) 2710 at 6s. Facit, £813 : : 0. (*) 1572 at 8s. Facit, £628 : 16 : 0. (') 2102 at 10s. Facit, £1051 : : 0. («) 2101 at 12s. Facit, £1260 : 12 : 0; {') 5271 at 14s. Facit, £3689 : 14 : 0. 3123 at 163. .. „ Facit, £2498 : 8 : 0. | 10s. (») 1075 at 16s. Facit, £860 : : 0. (>») 1621 at 18s. Facit, £1458:18:0. Note. When the price is 10s. take half of the quantity, and if any remains, it ii ■ Rule 5. When the price consists of odd shillings, multiply the given quantity by the price, and divide by 20, the quotient - will be the answer. (*) 2703 at Is. Facit, £135 : 3 : 0. (') 3270 at 3s. '", 210)98110 Facit, £490 : 10 : 0. i2 3271 at 5s. Facit, £817 : 15 : 0. J" Ki^mr '^t ■ '•^J«**^lM*'^ ['- iji/iiir G4 («) 2715 cat 7s. Facit, £950 : 5 : 0. (') 3214 at 9s. Facit, £1446 : 6 : 0. (•) 2710 at lis. Facit, £1490 : 10 : 0. rRACTICB. n 3179 at 13s. Facit, £2066 : 7 : 0. C) 2150 at 15s. Facit, £1612 : 10 : 0, (") 2150 at 199. Facit, £2042 : 10 : 0. {'') 7157 at 19s. Facit, £6799 -.3:0. (») 3142 at 17s. c,t *i4yu-..u.v. Facit, £2670 : 14 : 0. . , , . tol When the price is 5s. divide the quantity by 4, and if any remain, it is 5s. . T> p Whon the m-ice is shillings and- pence, and they the Rule 6. ^^^^ "'^ P'^ide by the aliquot part, and it will aliquot P'^'-t o^ ^ Pf""^' ^'V,,t /thc^ aliquot part, r '^t^lHL QuS by sSlh^ and takeSxvrts for then inultq)ly the quanuiy uy ,. . , , oo thd^rest, add thei* together, and dn ide by ^0. ' C) 7514 at 4s. 7d. Facit, £1721 : 19 : 2. (^) 2710 at 6s. 8d. Facit, £903 : 6 : 8. (') 3150 at 3s. 4d. Facit, £525 : : 0. (^) 2715 at 2s. 6d. Facit £339 : 7 : 6. (*) 7150 at Is. 8d. Pacit, £595 : 16 : 8. [") 3215 at Is. 4d. Facit, £21 4 : 6 : 8. C) 7211 at Is. 3d. Facit, £450 : 13 : 9. [') 2710 at 33. 2d. 3 8130 451 : 8 85811 : 8 Facit, £429 : 1 : 8. (») 2517 at 5s. 3d. Facit, £660 : 14 : 3. (»") 2547 at 78. 3|d. Facit, £928: 11: 10 J. {'') 3271 at 5s. 9^d. Facit, £943: 16 : 4i . ('") 2103 at 15s. 44d. Facit,£1616:13:7i ('") 7152 atl7s, 6H Facit, £6280 : 7 : 0. ('*) 2510 at 14: 7^d. Facit, £1832 :J 6 :5i. ('") 3715 at 9s. 4^d. Facit, £1741: 8 : 1^. ('«) 2572 at 13 : 7^d. Facit, £1752 : 3 : 6. (") 7251 at 14s. 8\d. Facit,£5324:19:0|. PRACTICE. 65 C. (") 3210 at 15s. 7|d. Facit, £2511 .3.1^. l(")2YlO at 19s. 2icL Facit, £2602. 14. 7. RuLK 1: Ut, When the price is pounds and shillings, multiply the quantity by the pounds, and proceed with the shillings, if they are even, as the fourth rule ; if odd, take the aliquot parts, add them together, the sum will be the answer. 2dly, When pounds, shillings, and pence, and the shillings and pence the aliquot parts of a pound, multiply the quantity by the pounds, and take parts for the rest. 3dly, When the price is pounds, shillings, pence, and far- things, and the shillings and pence are not the aliquot parts of a pound reduce the pounds and shillings into shillings, multiply the quantity by the shillings, take parts for the rest, add them together, and divide by 20. Note. When the given quantity consists of no more than three figures, proceed as in Compound Multiplication. s. d. 2:6 6 1 6 {') 7215 at £7. 4.0. Y 1 5 50505 1443 £51948 ft,- 6 H i 2104 at £5.3.0 5 (■*) 27lOat£2.3.7i. 43 210 10520 263 52.12 £10835.12 (3\ Oin'7 of -P9, R .0. Facit, £5056.16.0. (') 7156 at £5 .6.0. Facit, £37926. 16.0. F3 V 116530 1355 338.9 1182213.9 Facit, £5911 .3.9. ('')3215at£l .17.0. Facit, £5947. 15.0. (') 2107 at £1.1 3.0. Facit, £3476.11 . 0. («^ 3215 at £4.6.8. Facit, £1393 1.1 3. 4. (») 2154 at £7.1 .3^ Facit. £15212. ;5>'" ,K.^ ««<«»#'*'** < f 66 PRACTICE. ('») 2701 at £2. 3. 4. Facit, £5852 .3.4. (")27l5iit£l.l7.2i. Fncit, £5051 .0.7^. ('^)2157at£3.15.2i. Kiicit,£8l08.19.5i. ('=)3210at£1.18.6i Facit, £G189.5.7i ('*)2157at£2.7.4i Facit, £5109. 7. loi ("^)142at£l.l5.2i Facit, £250 . 2 . 6i. ("')95at£l5.14.7i. Facit, £1494. 7. 4i (")37at£1.19.5i Facit, £73 . . 8$. n 2175 at £2.15.4^. Facit, £G022,0.7|.' ('»)2150at£l7.l6.U. Facit, £38283 .8.9. Rule 8. When the price and quantity given are of several denominations, multiply the pri by the integers, and take parca with the parts of the integers fo .ne rest. ^ 1. At £3 . 17 . G per cwt., what is the value of 25 cwt. 2 qrs. 14 lb. of tob^co ? h lb. 14 £3.17. 6 5X5=25. 19. 7. 6 6 i 96.17. 6 1.18. 9 9. 8i 99. 5.1U y'^' o At £1 . 4 . 9 per cwt, what comes 17 cwt. 1 qr. 17 lb. of , ^ . 2 Ans. £21 . 10 . 0. S^'sold 85 cwt. 1 qr. 10 lb. of cheese, at ^l;.^^^^ p^^ cwt., what does it come to? , ^f • ^}^^ '^ ' ^,^- . 4. Hops at £4 . 5 , 8 per cwt., what must bo g'v^n for 72 cwt. " ^" At £1 . 1 . 4 per cwt. what is the value of 27 c^vt. 2^ qrs. 15 lb. of Malaga raisins Ans. £29 . 9 . C^. 6 Bouo-ht IS cwt. 3 qrs. 12 lb. of currants, at £2 17 . 9 per cwt.* what" did I give for the whole? Ans. £227 . 14. w SIJ Q9 .2i 6^. .4i . 5i 8i 5.4^. » 7-1. . i6.1i 8.9. ovcral parts rs. 14 lb. of ) . 8. cr cwt., . oi 72 cwt. ? . 2. 2 qra. . 9 per .14. 1' TARE AND TRET * J- Sold 56 cwt. 1 nr ni) . ' ^^ ^hat docs it come to?^ * '^ ^^' '^ '^^S^^> at £2 : 15 : the cwt c^...,U ;•"• ^° «^-'vt.. w,.at,-s the worth f^r ^« ib. of do„b,„^;|, Ans. £82 .3.3. 5. What is the interest of £254^ 17 . 6, for 5 years, at 4 per cent, pe? annum .> Ans. £50 . 19 . 6. 6. What is the interest of £556 . 13 . 4, at 5 per cent, per annum, for 6 years ? Ans. £139 . 3 . 4. 7. My correspondent writes me word, that l)e has bought goods to the amount of £754 . 16 on my account, what does his commission come to at 2i per cent. ? , Ans. £18 . 17 . 4|. 8. If I allow my factor 3| per cent, for commission, what may he de- mand on the laying out £876 . 5 . 10 .' Ans. £32 . 17 . 2i. 9 At llOi per cent., what is the purchase of £2054 . 16. South Sea Stock ? Ans. £2205 .8.4. 10. At 104g per cent. South Sea annuities, what is the purchase of 179TF . 14? - Ans. £IS1(S . Q . m. 11. At 96| per cent., what is the purchase of £577 . 19, Bank annuities ? Ans. £559 . 3 . 3|. 12. At £124§ per cent., what is the purchase of £758 . 17 . 10, India Stock ? £945 . 15 . 4i. BROKAGE, Is an allowance to brokers, for helping merchants or factors to persons, to buy or sell them goods. ^ RuLK. Divide the sum given by 100, and take parts from the quotient with the rate per cent. 13. If I employ a broker to sell goods for me, to the value of £2575 . 17 . 6, what is the biokage at 4s. per cent. ? 25175.17.6 20 4s.=:}25.'l5.2 Ans. £5 . 3.04 •Si#' r annum . 14 . 41 perannu . 16 . 4i 15117 12 ■ 2110 • ;- 14. When a broker sells goods to the amount of £7105 . 5 . 10, what may he demand for brokage, if he is allowed 5s. 6d. per cent. ? Ans. £19 . 10 . 9|. 15. If a broker is employed to buy a quantity of goods to the value of £975 .6.4, what is the brokage, at 63. 6d. per cent. ? wins. £3 . 3 . 4i 16. What is the interest of £547 .2.4, for 5i years, at 4 per cent, per annum .^ -• Ana. £120 . 7 , 3i 17= What is the interest of £257 ,5 = 1, at 4 per eent=s for a vear and three quarters ? Ans. £1$ . 6 . li. 18. What is the interest of £479 . 5 for 54 years, at 5 per cent, per an- num .' Ans. £125 . 16 . 0| 73 INTEREST. Itp 19. What is the interest of £576 : 2 : 6 for 1^ years, a.| 4| per cent, per annum. Jns. £18'7 : 19 : 2^. 20. What is the interest of £279 : 13 : 8 at 5^ per cent, per annum, for 3^ years ? Ans. £51:1 : 10. When the interest is required for any nuiuhcr of Weeks. Rule. As 52 weeks are to the Interest of the_ given sura for a year, so are the weeks given for the interest i-equired. 21. What is the interest of £259 : 13 : 5 for 20 weeks, at 5 per cent, per annum ? * ^ Ans. £4:19: 10^. 22. What is the amount of £375 : 6 : 1 for 12 weeks, at 4^ per cent, per annum? ■^^^^^' £379 : 4 : O^. When the Interest is for any number of days. Rule. As 365 days are to the interest of the given sum for a year, so are the .days given to the interest required., 23. At 5»j per cent, per annum, what is the interest of £985 . 2 . 7 for 5 years, 127 days ? '' Ans. £289 . 15 . 3. 24. What is the interest of £2726 . 1 . 4 at 4^ per cent, per annum, for three years, 154 days ? A71S. £419 . 15 . 6i. When the Amount, Time, and Bate per cent, are given to find the PrincijMl. Rule. As the amount of £100 at the rate ,nd time given : is to £100 :: so is the amount given : to the principal required. 25. What princit)al being put to interest, will amount to £402 . 10 in 5 years, at 3 per cent, per annum ? 3X5 + 100=£115 . 100 . . 402 . 10 20 20 2300 8050 100 23l00)a050l00(£350 Ans, I 4| pe lent, per : 10. cks. sura for ks, at 5 : 10^. ts, at 4 1 t : Oi. am for a f £985 . 15 . 3. cent, per 5 . 6i. to find given : is ired. ;o £402 . Ans^ INTEREST. 78 26. What principal being put to interest for 9 years, will amount to £734 : 8, at 4 per cent, per annum ? Ans. £540. 27. What principal being put to interest for 7 years, at 6 per cent per annum, will amount to £334 : 16 ? ^ Ans. £248. When the principal^ Rate per cent.^ aid Amount are given^ tofivd the Time. Rule. As the interest of the principal for 1 year : is to 1 year : : 80 is the whole interest : to the time required. 28. In what time will £350 amount to £402 . 10, at 3 per cent per annum ? * 360 As 10 . 10 : 1 : : 52 . 10 : 6 8 20 20 10)60 20 210 2110)10510(5 years. Ans. 402 . 10 106 350.10 lOlOO 52.10 29. In what time will £540 amount to £734 : 8, at 4 per cent per annum? uItw. 9 years. 30. In what time will £248 amount to £334 : 16, at 6 per cent per annum ? -^''^' ' years. Whm the Principal, Amount, and Time, are given, to find the Bate per cent. Rule. As the principal : is to the interest for the whole time : : 8i> is £100 : to the interest for the same time. Divide that in- terest by the time, and the quotient will be the rate per cent. 31. At what rate per cent will 350 amount to £402 : 10 in 5 years' time ? .350 As 350 :52 . 10:: 100: £15 20 52.10 1050 100 36i0)i0500»O(00O8. = £l5-7-6^3 per ctMit. 32. At what rate per cent will £248 amount to £334 : 10 in 7 years' time? -'^"*- '' per cui-.t. ; H 74 INTEKEST. » ; ■^ 33. At what ra^te per cent, will £540 amount to £784 : 8 in 9 years' time? ' ^»w. 4 per cent. COMPOUND INTEREST, Is that which arises both from the principal and interest; that is, when the interest on money becomes due, and not paid, the same interest is allowed on that interest unpaid, as was on the principal before. Rule 1. Find the first years' interest, which add to the princi- pal ; then find the interest of that sum, which add as before, and so on for the number of years. 2. Subtract the given sum from the last amount, and it will give the compound interest required. EXAMPLES. 1. What is the compound interest of £500 forborne 3 years, at 5 per cent, per annum ? 500 500 525 6 25 26 . . 5 « 25100 525 = 1st year. 55 1 . . 5 = 2d year. 5 6 651.. 5 26125 27156.. 5 2Y.11..3 20 20 5100 11125 12 3100 578.16..3=3dyear. 600 prin. sub 78 . 16 . . 3=inter.for3years. 2. What is the amount of £400 forborne 3^ years, at 6 per cent, per annum, compound interest? Ans. £490 : 13 : 11^. 3. What will £650 amount to in 5 years, at 5 per cent, per annum, compound interest? Ans. £829 : 11 : 7^. 4. What IS the amount of £550 : 10 for 3 years and 6 juionths, at 6 per cent, per annum, compound interest ? ^w5. £f>"5 :6j,5. 5. What is the compound interest of £764 for ■ years 'and 9 months, at 6 per cent, per annum? Am, £24.- 18 : 8. 6. What is the compound interest of £67 : 10 : a ft? a years, 7 months, and 15 days, at 6 per cent, per annum? Ans, £18 ; o : Sjf, BEBATE OR DISCOUNT. ^ :8m 9 r cent. 7. What is tlie compound interest of £259 : 10 for 3 years, 9 months, and 10 days, at 4^ per cent, per annum ? Ans. £46 : 19 : 10^. est; that paid, the s on the 16 pnnca- sfore, and id it will 3 years, REBATE OR DISCOUNT, Is the abating of so much money on a debt, to be received be- fore it is due, as that money, if put to interest, would gain in the same time, and at the same rate. As £100 present money would discharge a debt of £105, to be paid a year to come, rebate bemg made at 6 per cent. Rule. As £100 with the interest for the time given : is to Oiat interest : : so is the sum given : to the rebate required. Subtract the rebate from the given sum, and the remainder ^dll be the present worth. EXAMPLES. P. ar 3 years. at 6 per I : Hi. cent, per 1 :7i. 6 months, : 6/. 5. ars'and 9 18 : 8. y a years, 3 ;8i 1. 'WTiat is the discount and present worth of £487 : 12 for 6 months, at 3 per cent, per annum ? fm=i6 3 100 103 As 103:0::487:12 20 20 2060 9752 3 -£ 8. 487 : 12 principal. 14 : 4 rebate. 20610)292516(14 . 4 rebate. 206 Ans. £4:1 Z'. 8 present worth. 865 824 416==4s. 2 What is the present payment of £357': 10, which \y&& agreed to be paid 9 months hence, at 5 per cent, per aniiiim ? ^ jins\ ^o=it . IX 3. What is the discount of £275 : 10 for 7 months, at 6 per cent per annum ? Ans. £7 : 16 : If. g2 76 EQUATION OF PAYMENTS. I '' i i i- i . 4. Bought goods to the value of £109 : 10, to be paid at nine months, what present money will discharge the same, if I am al- lowed 6 per cent, per annum discount 1 Ans. £104 : 15 : 8^. 5. What is the present worth of £521 : 9 : 1, payable 1 months hence, at 4^ per cent. ? Am. £514 : 13 : lOf. 6. What is the discount of £85 : 10, due September the 8th, this being July the 4th, rebate at 5 per cent, per annum ? Ans. 15s.. 3fd. I. Sold goods for £875 : 5 : 6, to be paid 6 months hence, what is the present worth at 4^ per cent. ? Ans. £859 : 3 : 4. 8. What is the present worth of £500, payable in 10 months, at 5 per cent, per annum ? Ans. £480. 9. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent. ? Ans. £70 : 11 : 9^. 10. What will be the present worth of £150, payable at 3 four months, i. e. one third at four months, one third at 8 months, and one third at 12 months, at 5 per cent, discount? Ans. £145 : 3 : 8^. II. Sold goods to the value of £575 : 10, to be paid at 2 three months, what must be discounted for present payment, at 5 per ;ent.? ^n«. £10 : 11 : 4|. 12. What is the present worth of £500 at 4 per cent., £100 being to be paid down, and the rest at 2 six months ? Ans. £488 : 7 : 8^. EQUATION OF PAYMENTS, *' Is when several sums are due at different times, to find a mean time for paying the whole debt ; to do which this is the common, Rule. Multiply each term by its time, and divide the sum of the products by the whole debt, the quotient is accounted the mean time. ^P^i WM — BQUATIOPJ OF PAYMENTS. W EXAMPLES. 1 A owes B £200, whereof £40 is to be paid at 3 months, £60 at 6 months, and £100 at 10 months; at what time may the whole debt be paid together, without prejudice to either? £ 40 X 60 X 100 X m. 3 5 10 120 300 1000 2100)14120 7 months yV* 2 B owes C £800, whereof £200 is to be paid at 3 monOis, £100 at 4 months, £300 at 5 months, and £200 at 6 months ; but they agreeing to make but one payment of the whole, 1 de- mand what time that must be ? , , « j Ans. 4 months, 18 days. 3. I bouffht of K a quantity of goods, to the value of £360 which was to have been paid as follows: £120 at 2 months, and £200 at 4 months, and the rest at 5 months; but they afterwards agreed to have it paid at one mean time ; the time is demanded. ^ ^ Ans. 3 months, 13 days. 4. A merchant bought goods to the value of £500, to W £100 at the end of 3 months, £150 at the end of 6 months, and £250 at the end of 12 months; but afterwards they agreed to discharge the debt at one payment ; at what time was this payment made i ^ Ans. 8 months, 12 days. 6. H is indebted to L a certain sum, which is to be paid at 6 different payments, that is, i at 2 months i at 3 months, ^ at 4 months, i at 5 months, i at 6 months, and the rest at 7 months, but they agree that the whole should be paid at one equated time; what is that time ? ^ . .i i >+„.. Ans. 4 months, 1 quarter. 6. A is indebted to B £120, whereof i is to be paid at 3 "• -^ ., 1 ii. - i. «* n -M^/^ntKc. • what. IS the months, ^ at 6 moutns, ana luu ict^u »^ o- x^v..vx,., equated time of the whole payment? ^^ ^ ^^^^^ ^ ^^^ e3 mi ll iiii > i 78 BABTER. BARTER is the exchanging of one cc- , for another, and informs the traders so to proporti'^iK\te their goods, that neither may sustain loss. Rule 1st. Find the vahie of that commodity whose quantity is given ; then find what quantity of the other, at the rate pro- posed, you may have for the same money. 2dly. When one tn^s goods at a certain price, ready mone), but in bartering, advancu it to something more, find what the other ought to rate his goov. at, in proportion to that advance, and then proceed as before. EXAMPLES. 1. What quantity of chocolate, at 4s. per lb. must be delivered in barter for 2 cwt., of tea, at 93. per lb. i 2 cwt, 112 2241b. 9 price. 4)2016 the value of the tea. 504 lb. of chocolate. ■'^1 t' 2. A and B barter ; A hath 20 cwt of prunes, at4d. per lb. ready money, but in barter will have 5d. per lb. and B. hath hops worth 328. per cwt., ready money ; what ought B. to rat^ his hops at in barter, and what quan- tity must bfi given for the 20 cwt, of prunes ? 112 As 4: 5:: 32 20 5 8. 40 12 2240 5 cwt qr. lb. 4810)112010(23 . 1 4)160 408. 96 160 144 16=1 qr. 9 lb. j| 9Jf . J?n#. 3. How much tea, at 9s. per lb. can I have in barter for 4 cwt, 2qrg. of chocolate, at 4s. per lb. ? ^ns. 2 cwt 4. Two merchants barter ; A hath 20 cwt. of cheese, at 21s. 6d. per cwt; B hath 8 pieces of Irish cloth, at £3 . 148. per piece : 1 desire to know who must receive the difference, and how much ? ^n«. B. must receive of A JBS . 2. 5. A and B barter ; A hath 3h lb. of pepper at 13 id. per lb. ; B hath gin- ger at 154d. per lb.; how much ginger must he deliver in barter for the pepper .' ^ns. 3 lb. 1 oz.|f «. He in barte 7 Al him £1 The qut 8. A for whic lb. ; I d 9. If lb. of to 10. C have 8s how mu lent wit Is a Ri of gOO( gain so The i 1. If lis. and gain pel 12 11 ^'^^siinMt"- PBOFIT AND LOSS. 79 6. TTow many dozen of candles, at 59. 2d. per dozen, must be delivered Id barter for three cwt. 2 qrs. 16 lb. of tallow, at 379 4d. per cwt. ? Ana. 26 dozen 3 lb.| j 7 A hath 609 yards of cloth, worth Ha. per vard, for which B giveth him £123 . 12. in ready money, and S5 cwt. 2 qrs. 24 lb. of bees'-wax. The question is, what did B reckon his bees'-wax at per cwt. ? Ans. £3 . 10. 8. A and B barter ; A hath 3-^0 dozen of candles, at 4s. 6d. per dozen ; for w^hich B giveth him £30 in money, and the rest in cotton, at 8d. per lb. ; I desire to know how much cotton B gave A besides the money ? Jlns. 11 cwt. 1 qr. 9. If B hath cotton, at Is. 2d. per lb., how much must he give A for 114 lb. of tobacco, at 6d. per lb. Ana. 48 lb. jf. 10. C hath nutmegs worth 7s. nd. per lb. ready money, but in barter will have 8s. per lb. ; and D hath leaf tobacco worth 9d. per lb. ready nioney ; how much must D rate his tobacco at per lb. that his profit may be equiva- lent with C's i ' Ana. 9id. U- 4)160 408. 3S ■IT PROFIT AND LOSS la a Rule that discovers what is got or lost in the buying or selling of goods, and instructs us to raise and lower the price, so as to gain so much per cent, or otherwise. The questions in this Rule are performed by the Rule of Three. EXAMPLES. Ji 1. If a yard of cloth is bought for Us. and sold for 12s. 6d. what is the gain per cent. ? As 11 : 1 : P : : 100 12 20 18 12.6 11.0 1.6 2000 18 11)36000 12)3272y\ 210)2712 . 8 Ans. £13 . 12 . 8j\ 2. If 60 ells of Holland cost £18 what must 1 ell be sold for to gain 8 per cent. ? As 100 : IS : : 108 108 1100)19144 20 8180 12 9160 12X5=60 12)19 . 8 . 9i 5)1 . 12 . 41 0. 6.51 2140 -Ans. 6s. 5|d ■?1 80 FELLOWSHIP. what 3. If 1 lb. of tobacco co'st 16d. and is sold for 20d. what is the gain per lie a parcel of cloth be sold for £500. and at 12 per cent, gain, what ""''lVa'"a?d'of cloth is bought for 13s. 4d. and sold again ^o^^\^^^:^^-^ '" 'e' If U2Tb'of iL cost 27s. Gd.. what must 1 cwt. be sold^fi^ to^in I'inercent > •^"■'- ■^l • 1^ • ^* 7 If 375 yards of broad cloth be sold for £490, and 20 per cent, profit, -^"^ rctr;/H.s, for £3 . .5, at .he rate of ^S^e. ce;„ p,.fit, ,at would have been the gain per cent, if 1 had sold them for £8 pel cv% t. . ^ns. dL4S . A • Ala- 9 If 90 ells of cambric cost £G0, how much must 1 sell it per yard to ,Q i J Jlns. r2s. 7(1. ^*10. A^pl'umber sold 10 fother of lead for £204 . 15, (the fother being lOi cwt.) and gained after the rate of £12 . 10 percent. ; what did » cost him ' J ^ Ans. 18s. 8d. ^^^Ir'^Bought 43G yards of cloth, at the rate of Ss. 6d. per yard, and sold it for lOs. 4d. per yard ; what was the gain of the whole ? 12 Paid £09 for one ton of steel, which is retailed at Gd. per lb. ; what is the profit or loss by the sale of 15 tons. -^ns. £182 loi^s. 13. Bought 124 yards of linen, for £32 ; how should the same he retailed per yard to gain 15 per cent. ? -^^s- os. lld.y^j. 14. Bought 249 yards of cloth, at 3s. 4d, per yard, retailed the same at 48. 2d. per yard, what is the profit in the whole, and how much per cent. .' *^ "^ Am. £10 .7.6 proUt, and 25 per cent. FELLOWSHIP & when two or more join their stock and trade together, so to determine each person's particiilar share of the gain or loss, in proportion to his principal in joint stock. m\ By this rule a bankrupt's estate may be divided amongst his creditors ; as also legacies may be adjusted when there is a de6- ciency of assets or eflfects. FELLOWSHIP IS EITHER WITH OR WITHOUT TIME. FELLOWSHIP WITHOUT TIME. Rule. As the whole stock : is to the whole gain or loss : : m is each man's share in stock : to his share of the gain or loss. Proof. Add all the shares together, and the sum v\ i!i be equJi to the given gain or loss— but the surest way is, as tlif wholf gain or loss : is to the whole stock : : so is each man's share of the gain or loss : to his share in stock. C FELL0W8HIF. 81 HI e gain per *. £-25. ;ain, what ( £500. 108. what t*. mo. for to gain 11 . 7i jnt. profit, . 1 . Oi »nt. profit, I per cwt. .' 2. Hi )er yard to 123. 7d. r being 19i it cost him 183. 8d. d, and sold . 19 . 4. r lb. ; what 182 loss. ! be retailed lid 2_8 the same at 1 per cent. ? per cent. :,her, so to or loss, in I nongst his e is a defi- lE. r loiis : : fO W loSifl. . .. « « lii iKi «q51J.i the wholf 's share of EXAMPLES 1. Two merchants trade together; A puts into stock £2C, and B JC40| they gained £30; what is each person's share thereof? A 60 : 50 : : 20 20 610)10010 £1G . 13 . 4 20+40=00 As 60 : 00 I : 40 40 « 610)20010 33 . 6.8, B'» share. 16 . 13 . 4, A's share. 50 . 0.0 proof. £33 . 6 . S 2. Three merchants trade together. A, B, and C A puts in £20, B £30, and C £40 ; they gained £180 : what is each man's part of the gain ? Ans. A £40 ; B £00 ; C £80 * 3. A, B, and C, enter into partnership ; A puts in £364, B £482, and C £500 ; and they gained £867 ; what is each man's share in proportion to his stock ? Arts. A £234 . 9 . 34— rem. 70 ; B £310 . 9 . 5— rem. 248 ; C £322 . 1 . 3i— rem. 1028. . 4. Four merchants, B, C, D, and E make a stock ; B puts in £227, C £349, D £115, and E £439 ; in trading they gained £42S : I demand each merchant's share of the gain ? Ans. B £85 . 19 . 6|— 690 ; C £132 . 3 . 9—120 ; D. £43 . 11 . 11—250 ; E £166 . 5 . 64—70. 5. Three persons, D, E, and F, join in company ; D's stock was £750, E's £460, and F's £500 ; and at the end of 12 months they gained £684: Trhat is each man's particular share of the gain ? Ans. D £300, E £184, and F. £200. " . 6. A merchant is indebted to B £275 . 14, to C £304 . 7, to D £152, ^and to E £104 . 6 ; but iipon his decease, his estate is found to be worth but £075 . 15 : how must it be divided among his creditors ? Ans. B's share £222 . 15 . 2—6584 ; C's £245 . 18 . U— 15750 • D's £122 . 16 . 21—12227 ; and E's £84 . 5 . 5—15020. 7. Four persons trade together in a joint stock, of which A has i, B |, C |, and D | ; and at the end of 6 months they gain £100 : what is each man's share of the said gain ? Ans. A £35 . 1 . 9—48 ; B £26 . 6 . 3^-36 ; C £21 . 1 . Oi • ; . —120 ; and D £17 . 10 . 10^—24. 8. Two persons purchased an estate of £1700 per annum, freehold, for £27.200, when money was at 6 per cent. interest.''ARd^«t. per pound, land- tpc ; whereof D paid £15,000, and E the rest ; 80metr«7e after, the interest «f the money falling to 5 per cent, and 28. per poiind land-tax, they sell the said estate for 24 pears' purchase : I desire to kno^ each person's shai e ? Ans. D £22,500 ; E £18,300. i*iif ...Ilia m FELLOWSHIP. 9 D E and F, join their stocks in trade ; the amount of their stocks is £647, and they are in proportion as 4, 6 and 8 ^e to one another, and the amount of the gain is equal U> Ds stock, what is each man's stock and gain ? Ans. D's stock £143 . 15 . 6|| gain, 31 . 19 . Oif E'8....o- 215 . 13 . 4 .... 47 . 18. 6|f Ps 287 . 11 . lA 63 . 18 . OJ/7. 10 D E, and F, join stocks in trade; the amount of their stock' wal £100 ; D's gain £3, E's £5, and F's £8 : what was each man's stock ? ^ , t^, r>^r, Ans. D's stock £18 . 15 ; E's £31 . 5 ; and Fs £50. FELLOWSHIP WITH TIME. in- Rule. As the sum of the products of each man's money and time : is to the whole gain or loss : : so is each mans product : to his share of the gain or loss. P]»ooF. As in fellowship without time. EXAMPLES. I I I 04 BXCHANOB. V. HOLLAND, FLANDERS, AND GERMANY. They keep their accounts at Antwerp Amsterdam Bru^^^^^^^ RoSam, and Hamburgh, some in pounds, f ""-f^'^^^r^d as in England; others in guilders, ^^ivers and pennings , ana Txchange lith us in our pound, at 33s. 4d. Flemish, at par. • Note. 8 pennings make J gjoat- 2 groats, or 16 pennings. ... 1 stiver. 2oftivev3 : 1 guilder or florin. ALSO, 12 groats, or 6 stivers make. .1 scheUing. 20 schelUngs, or 6 guilders. . . 1 pound. To change Flemish into Sterling. Rule. As the given rate : is to one pound : : so is the Flemish turn : to the sterling required. To change Sterling into Flemish. Rule. As £1 sterling : is to the given rate : : so is the sterUng given : to the Flemish sought. ' EXAMPLES. at 33s. Gd. Flemish per PO""d ^terljn^. many'guildevs must 18. If I pay in London £8.)2 . 12 . 6. spelling, n J ^ j^,, per Idraw for at Amsterdam. -^^-^;/9^,,^^g^,t!'l3 st?^^^ pound sterling ? j^^^aJ if l nav in Amsterdam 8792 guild. ^ 19. What must I draw for at London, 'f "^^^ >^J^^'l^^'^,,,,^ .terling ? 13 stiv. 14i pennings, exchange at 34 schel. 4i groats pe^r^po ^^ ^^ ^ B To convert Bank Money into Current, and the contrary. Ki^^r. ThA Bank Money is worth more than the Current. ThSer^: ^"eeri^one'and the other is called agio, .nd . generally from 3 to 6 per cent, in favour of the Bank. To change Bank into Current Money. Rule. As 100 guilders Banlc : is to 100 with the agio «> is the Bank given : to the Current requii--^ ed. EXCHAN6S. 99 To chanffe Currents Money into Bank. Rule. As 100 with the agio is added : is to 100 Bank : : so is the Current money given : to the Bank required. 20. Change 794 guilders, 15 stivers, Current Money, into Bank florins agio 4f per cent. Ans. 161 guilders, 8 stivers, llif-f pennings. 21. Change 761 guildei-s, 9 stivers Bank, into Current Money, agio 4f per cent. Ans. 794 guilders, 15 stivers, ij\ pennings. VI. IRELAND. 22. A gentleman remits to Ireland £575 : 15, sterling, wluit will he receive there, the exchange being at 10 per cent? Ans. £633 : 6 : 6. 23. What must be paid in London for a remittance of £633 : 6 : 6, Irish, exchange at 10 per cent. ? Ans. 575 : 15. COMPARISON OF WEIGHTS AND MEASURES. EXAMPLES. . If 50 Dutch pence be worth 65 French pence, how many Dutch pence are equal to 350 French pence ? Ans. 269Af . 2. If 12 yards at London make 8 ells at Paris, how many elb at Paris will make 64 yards at London ? Ans. 42y«y. 3. If 30 lb. at London make 28 lb. at Amsterdam, how many lb. at London will be equal to 350 lb. at Amsterdam ? « Ans. 375. 4. If 95 lb. Flemish make 100 lb. English, how many lb. En- glish are equal to 275 lb. Flemish. Ans. 289f f. CONJOINED PROPORTION, Is v4ien the coin, weights, or measures of stjveral countries are compared in the same question ; or, it is linking together a varie- ty of proBprtions. Whenlfc is required to fii weig)^ or measure, mentioned given quantity of the last. how many of the first sort of coin, in the question, are equal to 96 PROPORTION. Rule. Place the nundfers alternately, beginning at the left hand, and let &e last number stand on the left hand ; then multi- ply the first row continually for a dividend, and the second for a divisor. Proof. By as many single Rules of Three as the question requires. EXAMPLES. 1. If 20 lb. at London make 23 lb. at Antwerp, and 155 lb. at Antwerp make 180 lb. at Leghorn, how many lb. at London are equal to '72 lb. at Leghorn ? 20X155X'72=223200 23X180=4140)223200(53|H- 2. If 12 lb. at London make 10 lb. at Amsterdam, and 100 lb. :^t Amsterdam 120 lb. at Thoulouse, how many lb. at London are equal to 40lb. at Thoulouse ? * ^ Ans. 40 lb. Left. RigEt. 20 23 155 180 72 " K'U. 3. If 140 braces at Venice are equal to 156 braces at Leghorn, and 7 braces at Leghorn equal to 4 ells English, how many bra- ces at Venice are equal to 16 ells English? Ans. 26^. ^ 4. If 40 lb. at London make 36 lb. at Amsterdam, and 90 tb. at Amsterdam make 116 at Dantzick, how many lb. at London are equal to 130 lb. at Dantzick ? Ans. 112,VtV- When it is required to find how many of the last sort of ^n, sight, or measure, mentioned in the question, are equal to a quantity of the first. » Rule. Place the numbers alternately, beginning H the left hand, and let the last number stand on the right hand ; tliefe mul- tiply the first row for a divisor, and the second for a dividend. weig PR06HESSI0N. #t EXAMPLES. f 6. If 12 lb. at London make 10 lb. at Amsterdam, and 100 lb. at Amsterdam 120 lb. at Thoulouse, how many lb. at Thoulouse are equal to 40 lb. at London ? Ans. 40 lb. 6. If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. at Amsterdam 116 lb. at Dantzick, how many lb. at Dantzick are equal to 122 lb. at London? Am. 14mj|. PROGRESSION CONSISTS OF TWO PARTS ARITHMETICAL AND GEOMETRICAL. ARITHMETICAL PROGRESSION Is when a rank of numbers increase or decrease regularly by the continual adding or subtracting of equal numbers ; as 1, 2, 3, 4, 6, 6, are in Arithmetical Progression by the co;>Unual increasing or adding of one; 11, 9, 7, 5, 3, 1, by the continual decreasing or subtracting of two. Note. When any even nurnVar of terms differ i .y Arithme- tical Progression, the sum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the extremes; as 2, 4, 6, 8, 10, 12, where 6+8, the two middle numbers, are=12+2, the two extremes, and=10-|-4 the two means =14. When the number of terms are odd, the double of the middle term will be equal to the two extremes; or of any two means equally distant from the middle term ; as 1, 2, 3, 4, 6, where the double number of 3=5+1=2+4=6. In Arithmetical Progression five things are to be observed, viz. 1. The first term; better expressed thus, F. 2. The last term, L. •i 3. The number of terms, N. J 1., 4 The equal difference, . . .' D 5. The sum of all terms, S. Any three^f which being given, the other two may be found. The nrsi, «i©ond, and third terms given, to find the' fifth. Rule. Multiply the sum of the two extremes by half the number of %rms, or multiply half the sum of the two extremes 06 PBOORE88ION. by the whole number of terms, the product is the total of fdl the terms : or thus, I. F L N are given to find S. N F+Lx— =S. 2 EXAMPLES. i k^. 1. How many strokes does the hammer of a clock strike in 12 hours ? 12 + 1 = 13, then 13X6='78. 2. A man bought 17 yards of cloth, and gave for the firat yard 2s. and for the last 10s. what did the 17 yards amount to? Ans. £5 . 2. 3. If 100 eggs were placed in a right line, exactly a yard as- under from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggs smgiy, and returns with every egg to the basket to put it in ? Ans. 5 miles, 1300 yards. ,%€( first, second, and third terms given, to find the fourth. Rule. From the second subtract the first, the remainder divi- ded by the third less one, gives the fourth : or thus F L N are given to find D. L — F =D. N— 1 EXAMPLES. 4. A man had eight sons, the youngest was 4 years old, and the eldest 32, they increase in Arithmetical Progress!^ what was the common difference of their ages ? ^ ^JW. 4. 32—4=28, then 28-r 8 = 1 +4 common dii||enc^. * 6. A man is to travel from London i days, and to go but 3 miles the fii-st day. mcreasin an equal excess, so that the last day's journey of all the •ike in 12 first yard )9 1 • £5.2. yard as- iket, what 100 eggs in? > yards. irth. ader divi- 3 old, and iitm, what Aim, 4. ce. laee in 12 »jrda3|% 58 miles, PROGRESSION. what is the daily increase, and how many miles distant is that place from London ? Am. 5 daily increase. Therefore, as three miles is the first day's journey, 3-f-6=8 the second day. • 8+5 = 13 the third day, &c. The whole distance is 366 miles. The first, second, and fourth terms given, to find the third. Rule. From the second subtract the first, the remainder divide by the fourth, and to the quotient add 1, gives the third ; or thus, III. F L D are given to find N. . L^F — +1=N. D EXAMPLES. 6. A person travelling into the country, went 3 miles the first day, and increased every day 5 miles, till at last he went 68 miles in one day ; how many days did" he travel ? An%, 12. 68—3=55-7-5 = 11 + 1 = 12 the number of days. 7. A man being asked how many sons he had, said, that the youngest was 4 years old, and the oldest 32 ; and that he increas- ed one in his family every 4 years, how many had he ? Ans. 8. The second, third, and fourth terms given to find the first. Rule. Multif)ly the fourth by the third made less by one, the product subtracted from the second gives the first : or thus, IV. L N D are given to find F. ■ ■^W*4, L— DxN— 1=F. EXAMPLES. 8. A man in 10 days went from London to a certain town in the countryi every day's journey increasing the former by 4, and i Ans. 10 miles. 4X10—1=36, then 46—36=10, the firat day's journey. 12 100 PROGRESSION. 9. A man taues out of his pocket at 8 several times, so many diflferent numbers of shillings, every one exceeding the former by 6, the last at 40 ; what was the first ? ^ns. 4. The fourth, third, and fifth given, to find the first. EuLE. Divide the fifth by the third, and from the quotient subtract half the product of the fourth multiplied by the third less 1 gives the first : or thus, ^ V. N D S are given to find F. SDXN— 1 — F. N2 EXAMPLES. 10 A man is to receive £360 at 12 several payments, each to exceed the former by £4, and is wilHng to bestow the first pay- ment on any one that can tell him what it is. What will that person ha.ve for his pains ? ^^*- *S- 4X12—1 360-12=30, then 30 =£8 the first payment. 2 The first, third, and fourth, given to find the second. Rule. Subtract the fourth from the product of the third, mul- tiplied by the fourth, that remainder added to the first gives the second : or thus, F N D are given to find L. \ ND— D+F=L. EXAMPLES. 11 What is the last number of an Arithmetical Progression, beginning at 6, and continuing by the increase of 8 to 20 places? ° ® Ans. 168. 20x8—8=152, then 152+6 = 158, the last number^ GEOMETRICAL PROGRESSION « .-. . _? J- ^^:«/. y^f n«ir ran\z nf TtllTTlbt^Ra bv SOlQft is tne increasing ui ucvicarsmg ui «t.j -^^ ,.""': " v • '"' common ratio; that is, by the continual multiphcation or division of some equal number: as 2, 4, 8, 10, increase by the mum^lwr 2, and 16, 8, 4, 2, decrease by the divisor 2. \^ PROORB88ION. 101 Note. When any number of terms is continued in Geome- trical Progression, the product of the two extremes will bo equal to any two means, equally distant from the extremes: as 2, 4, 8 16, 32, 64, where 64X2 are=4X32, and 8X16=128. When tlie number pi the terras are odd, the middle term mulU- phed mto itself will be equal to the two extremes, or any two means equally distant from it, as 2, 4, 8, 16, 32, where 2X32 = 4X16 = 8X8=64. . In Geometrical Progression the same 5 things are to be obser- ved as are in Arithmetical, viz. 1. The first term. 2. The last term. * 3. The number of terms. 4. The equal difference or ratio. 6. The sum of all the terms. Note. As the last term in a long series of numbers is very tedious to come at, by continual multiplication ; therefore, for the reader finding it out, there is a series of numbers made* use of in Anthmetical Proportion, called indices, beginning with an unit, whose common difference is one ; whatever number of in- dices you make use of, set as many numbei-s (in such Geomet- rical I roportion, as is given in the question) under them. ^ 1, 2, 3, 4, 5, 6, Indices. 2, 4, 8, 16, 32, 64, Numbers in Geometrical Proportion. But if the fii-st term in Geometrical Proportion be different ti-om the ratio, the indices must begin with a cipher. As ^' ^' 2, 3, 4, 5, 6, Indices. 1, 2, 4, 8, 16, 32, 64, Numbers in Geometrical Proportion. When the Indices begin with a cipher, the sum of the indices made choice of must always be one less than the number of terms given m the question; for 1 in the indices is over the second term, and 2 over the third, &c. Add any two of the indices together, and that sum will affree with the product of their respective terms. Aft in tha firaf taVKl/^ nf l^Ji: ct I ^ t, — — — -..., ,,,,.v ittvit: vri xiiuiuca ;; -J"" 0= 7 Geometri ' '^ Proportion • • • • • 4X32=128 Then the second 2+4= 6 4X16= 64 13 f I 102 PROOHESSION. In any Geometrical Progression proceeding from unity, the ratio being known, to find any remote term, without producmg all the intermediate terms. Rule. Find what figures of the indices added together would eive the exponent of the term wanted: then multiply the num- lerB standing under such exponents into each other, and it will give the t«rm required. Note. When the exponent 1 stands over the second tem^ the number of exponents must be one less than the number of terms. * EXAMPLES. 1. A man agrees for 12 peaches, to pay only the price of the last, reckoning a farthing for the fii-st, and a halfpenny for the second, &c. doubling the price to the last; what must he give for them? ^n.. £2 . 2 . 8. k: 0, 1, 2, 3, 4, Exponents 1, 2, 4, 8, 16, No. of terms. 16=4 16 = 4 256 = 8 8=3 For 4+4+3=11, No. of terms less 1 4)2048=11 No. of far." 12)512 210)412 . 8 £2.2.8 2. A comitry gentleman going to a fair to buy some oxen, meek with a person who had 23 ; he demanded the pnce of them, ^d wJanswered £16 a piece; the gentleman bids £15 a p^e «d Twould buy all ; the'other tells him it could not be^l^ken but if he would iive what the last ox would^come to, at a farming for the first, and doublinc. it to the last, he should hav^all. ^ What was the price of the oxen % ^'''' ^-"^ ''% In any Geometrical Progression not Proceeding fro«a unity, the ratio being given, to find any remote term, without produ- cing all the intermediate terms. PBoaRBisioir. 198 Rule. Proceed as in the last, only observe, that every product must be divided by the fii-st term. EXAMPLES. 3. A sum of money is to be divided amon^r eight persons, the first to have £20, the next £60, and so in triple proportion ; what will the hist have ? Ans. £43U0. 640X640 14580X60 0, 1, 2, 3, 20, 60, 180, 540, k=14580, then =43740 20 20 34-3+1=7, one less than the number of terms. 4. A gentleman dying, left nine sons, to whom and to his exe- cutors he bequeathed his estate in the manner following : To his executors £50, his youngest son was to have as much more as the executors, and each son to exceed the next younger by as much more ; what was the eldest son's proportion ? Ans. £2560Q. The firet term, ratio, and number of terms given, to find the sum of all the terms. Rule. Find the last terra as before, then subtract the first from It, and divide the remainder by the ratio, less 1 ; to the quo- tient of which add the P^reater, gives the sum required. EXAMPLES. 5. A servant skilled in numbers, agreed wit^ a gentleman to serve him twelve months, provided he would give him a farthinjr :f .^' T^^> "month's service, a penny for the second, and 4d. for tne third, &c., what did his wages amount to ? Ans. £5825 . 8 . 6^. 256X256=65536, then 65536X64=4194304 0» h 2, 3, 4, 4194304—1 l'A^.^^;?^^» ^-r =1398101, then 4+4+3=11 No. of terms less 1, 4—1 1398101+4194304=5592405 farthings. I tu'^' ^ ^^*" uoti^t a horse, and by agreement was to give a far- thing for the first nail, three for the second, &c., there were four shoes, and in each shoe 8 nails ; what was the worth of the horse ? ^, Ans. £965114681693 . 13 . 4. / /i tot PERMUTATION. 1 A certain pereon married his daughter on New-years da.y, and'eave her husband Is. towards her marriage portion, proniis- ing to double it on the first day of every month for 1 year ; what waa her portion? ^n5. £204 . 15. 8 A laceman, well versed in numbers, agreed with a gentle- man to sell him 22 yards of rich gold brocaded lace, for 2 pms the fii-st yard, 6 pins the second, &c., in triple proportion ; 1 desire to know what he sold the lace for, if the pins were valued at 100 for a farthing; also what the laceman got or lost by the «ae thereof, supposing '^jj,- ^^^''™J",f ,^^6^^^^^^ . 9. Gain £326732 .0.9. PERMUTATION Is the changing or varying of the order of things. Rule. Multiply all the given terms one into another, and the last product will be the number of changes required. EXAMPLES. 1 How many changes may be rung upon 12 bells; and how long would they be ringing but once over, supposing JO changes might be rung in 2 minutes, and the year to contam 365 days, 6 hours? '"^ ,;^-.,, 1X2X3X4X5>C6X^ X 8 X 9 X 10 X H X 12 = 479001600 changes, which -^ 10=47900160 minutes ; and, if reduced, is=91 years, 3 weeks, 6 days, 6 hours. 2 A young scholar coming to town for the convenience of a good library, demands of a gentleman with whom he lodged, what his diet would cost for a year, who told him £10 but the scholar not being certain what time he should stay, asked him what he must give him for so long as he should place his family, (consisting of 6 persons besides himself) m different J?osition^ ^ .!„„ „* ,K««««. *\^a n-AntlAman thinkinfif it woul* not be every uay «»< m'"'^-^^ j ^"^- & - =? „ti j. !• j;,i long, tells him £5, to which the scholar agrees. What time did the scholar stay with the gentleman? ^ Ans, 6040 days. 105 fdi^ THE t.^ TUTOR'S ASSISTANT. PART II. VULGAR FRACTIONS. A FRACTION is a part or parts of an unit, and written with two figures, with a Une between them, as ^, f , f , &c. The figure above the Une is called the numerator, and the un- der one the denominator ; which shows how many parts the unit is divided into : and the numerator shows how many of those parts are meant by the fraction. There are four sorts of vulgar fractions: proper, improper compound, and mixed, viz. ' 11. A PROPER FRACTION is whcu the numerator is less than the d^ominator, as |, |, J, j-f, |ii &c. "i 2. An IMPROPER FRACTION is when the numerator is ^^qual to, or greater than the denominator, as f , f , ||, iii, &c. ^ 3. 4^ COMPOUND FRACTION IS the fraction of a fraction, and tuowuwy the word of, as ^ of f of f of j\ of y'j, &c. 4. A MIXED NUMBER, or FRACTION, ia composed of a whbla aumber and fraction, as 8|. 17^, 8J|, &c. 106 REDUCTION OF VULGAR FRACTIONS. REDUCTION OF VULGAR FRACTIONS. 1. To reduce fractions to a common denominator. Rule. Multiply each numerator into all the denominators, except its own, for a numerator; and all the denominators, for a common denominator. Or, 2. Multiply the common denominator by the several given numerators, separately, and divide their product by the several denominators, the quotients will be the new numerators. EXAMPLES. 1. Reduce | and 4 to a common denominator. Facit, If and |f Ist num. 2d num. 2X1 -U 4X4 = 16, then 4X'7=28 den.=if and ff. 2. Reduce i. L and 4, to a common denominator. Facit, II, tf , il 3. Reduce |, f , y%, and 4, to a common denominator. Pn(^\f 2-9 4 aaAii a.jOLi6 am. rdUt, 3 3flo^j 3360» 3360) 3360* 4. Reduce y^i h h ^^^ h^^ common denominator. Fa<5it, \nh iWo, M^) tVA- 5. Reduce i, |, 4, and i, to a common denominator. 6. Reduce f , |, |, and f , to a common denominator. Fnpit -TiUL iiAli -i'-M^ J^^l^4 racii, 2 If o» at e 0) 2ioir» 2 if IT* 2. To reduce a vulgar fraction to its lowest terms. Rule. Find a common measure by dividing the lower term by the upper, and that divisor by the remainder following, till nothing remain: the last divisor is the common measure; then divide both parts of the fraction by the common measure, and the quotient will give the fraction required. Note. If the common measure happens to be one, the fraction is already in its lowest term : and when a fraction hath ciphers at the right hand, it may be abbreviated by cutting them off, as f 1^. EXAMPLES. 7. Reduce 44 to its lowest terms. ^ 24^32(1 24 Com. measure, 8)24(3 Facit. )NS. lenominators, minators, for leveral given ' the several rs. ^f and If and if. aa 11 Ai. B4) 04' 64' tor. jt — reduced to the lowest term=i. ' 3X5X8 = 120 26. Reduce f of 4 of \^ to a single fraction. 27. Reduce U of If of |i to a single fraction. X" ttVyl L, 4 8 12 2 a 2 28. Reduce | of | of -i?^ to a single fraction. Fipil 135. = JL 29. Reduce | of f of | to a single fraction. 30. Reduce f of M^ to *<> » »^»g^^ fraction. Fqpit -8JL — J*- .6. To reduce fractions of one denomination, to the fracti^Qn of nnother, but greater, retaining the same value. Rule. Reduce the given fraction to a compound one, by cora- parino- it with all the denominations between it and that denomi- • nation which you would reduce it to; then reduce that compound fraction to a single one. EXAMPLES. # 31. Reduce i of a penny to the fraction of a pound. Facit, i of y'a of 2\'-^T9TtV' 32. Reduce i of a penny to the fraction of a pound. Facit, ^^y. 33. Reduce | of a dwt. to the fraction of a lb. troy. Facit, ysU' 34. Reduce ^ of a lb. avoirdupois to the fraction of a cwt. Facit, Tf\'f. 1. To reduce fractions of one denomination to the fraction oF another, but less, retaining the same value. Rule. Multiply the numerator by the parts contained in the several denominations between it, and that you would reduce it to, for a new numerator, and place it over the given denominator. BEDVCTION OP VULGAR PHACTI0N8. 109 EXAMPLES. 35. Reduce r^\j^ of a pound to the fraction of a penny. Facit,f 7X20X12=1680 ^U reduced to its lowest term=f 36. Reduce ^^^ of a pound to the fraction of a penny. oT r> 1 A !• Facit, ^. d7 Keduce ^a^iro of a pound troy, to the fraction of a penny- weight. 1? •! 4 "^ oo r» J M i> , Facit, 4.* 38. Reduce .^f y of a cwt. to the fraction of a lb. Facit, f . 8. To reduce fractions of one denomination to anothor of the same value, havmg a numerator given of the required fraction. Rule. As the numerator of the given fraction : is to its deno- mmator : : so is the numerator of the intended fraction : to its denommator. EXAMPLES. * 39. Reduce f to a fraction of the same value, whose numera- tor shall be 12. As 2 : 3 : : 12 : 18. Facit, j-a 40. Reduce 4 to a fraction of the same value, whose num'era- tor shall be 25. p^^^jj^ 4. 41 Reduce 4 to a fra<5tion of the same value, whose numera- tor shall be 47. 4Y Facit, 65|. 9. To reduce fractions of one denbmination to another of the same value, having the denominator given of thf. fractions re- quired. Rule. As the denominator of the given fraction : is to its numerator : : so is the denominator of the intended fraction : to Its numerator. EXAMPLES. #. Reduce f to a fraction of the same value, whose denomi- nator shall be 18. As 3 : 2 : : 18 : 12. Facit, |f . 43. Reduce 4 to a fraction of the same value, whose d'enomi- tor shall be 35. Facit M, 44. Reduce 4 to a fraction of the same value, whose denomi- nator shall be 65|. 47 Facit, 65f 110 WEBDUCTION aF VULGAR FRACTIONS. 10. To reduce a mixed fraction to a single one. Rule. When the numerator is the integral part, multiply it by the denominator of the fractional part, adding m the numerator of the fractional part for a new numerator ; then multiply the de- nominator of the fraction by the denominator of the fractional part for a new denominator. EXAMPLES. . 45. Reduce — to a simple fraction. Facit, |J^f — 4f. 48 3 6 X 3 + 2 = 1 1 numerator. 48X3 =144 denominator. ^ 234 46. Reduce— to a simple fraction. Facit, iff =1^3 • 38 When the denominator is the integral part, multiply it by the denominator of the fractional part, adding in the numerator of the fractional part for a new denominator; then multiply^ the numerator of the fraction by the denominator of the fractional pai-t for a new numerator. EXAMPLES. 47.. Reduce— to a simple fraction. Facit, Iff =4- 65f 19 . _ 48. Reduce — to a simple fraction. Facit, 1V3— ^• 11. To find the proper quantity of a fraction in the known parts of an integer. Rule. Multiply the numerator by the common parts of the integer, and divide by the denominator. EXAMPLES. 49. Reduce | of a pound sterling to ita proper quantity. 43X20=60-^4 = 153. Facit, 16s. 50 Reduce ^ of a shilling to its proper quantity. Facit, 4d. 3i qrs. 61 Reduce 4 of a pound avoirdupois to its proper quantity, Facit, 9 oz. 2^ dr. 52. Reduce 1 of a cwt. to its proper quantity. Facit, 3 qi-s. 3 lb. 1 oz. 12^ dr. Itiply it merator the de- •actional = -8JL. 133* t by the rator of iply the ractional 9 known s of the y- it, 15s. 3 1 qrs. mtity, 2f dr. 12^ dr. liBtftCTlON OP VUtGA^fHACTIONS. iir 63. Reduce f of a pound troy to its proper quantity. ^ . ^ . , ^ „ „ Facit, 1 oz. 4 dwts. 64. Keduce f of*an dl English to its proper quantity. •rir T> J * ^ ^ ^»c'*» 2 qrs. 3^ nails. 65. Keduce f of a miJe to its proper quantity. ^- ^ , . Facit, 6 fur. 16 poles. 66. Keduce f of an acre to its proper quantity. Facit, 2 roods, 20 poles. 5/. Keduce f of a hogshead of wine to its proper quantity. Ko T> J 1. Facit, 64 gallons. 68. Keduce | of a barrel of beer to its proper quantity. ^ Facit, 12 gallons. 69. Keduce /^ of a chaldron of coals to its proper quantity. - . _, - ^ Facit, 15 bushels. 60. Keduce | of a month to its proper time. Facit, 2 weeks, 2 days, 19 hours, 12 minutes: 12. To reduce any given quantity to the fraction of any greater denomination, retaining the same value. Rule. Reduce the given quantity to the lowest term men- tioned for a numerator, under which set the integral part reduced to the same term, for a denominator, and it will give the fraction required. EXAMPLES. 61. Reduce ISs. to the fraction of a pound sterling. ^ , Facit, if =|£. 62. Reduce 4. 3} qrs. to the fraction of a shilling. 63. Reduce 9 oz. 2f dr. to the fraction of a pound avoirdupois. . Facit, 4. 64. Reduce 3 qrs. 3 lb. 1 oz. 12f dr. to the fraction of a cwt. Facit, ^. 65. Reduce 1 oz. 4 dwts. to the fraction of a pound troy. Facit, |. 66. Reduce 2 qi-s, 3^ nails to the fraction of an English ell. Facit, f 67. Reduce 6 -fur. 16 poles to the fraction of a mile. Facit^. 1= 68. Reduce 2 roods 20 poles to the fraction of an acre. * Facit, f.' 69^ Reduce 54 gallons to the fraction of a hogshead of wine. Facit, f . It 112 bubtbactionIof vulgar fractions. 70. Reduce 12 gallons to the fraction of a barrel of beer. Facit, ^. 71 Reduce fifteen bushels to the fraction of a chaldron of coals. ^ * # Facit, T*i. 72. Reduce 2 weeks, 2 days, 19 hours, 12 minut^, to the fraction of a month. ^^^^ «* ADDITION OF VULGAR FRACTIONS. Rule. Reduce the given fractions to a common denominator, then add all the numerators together, under which place the com- mon denominator. EXAMPLES. Facit, it +H=H=l7V Facit, HH- Facit, 4^|. c Facit, 8tV- Facit, li. . 1. Add a and ^ together. 2. Add I, if and | together. 3. Add 1, 4^ and f together. 4. Add 7| and | together. 5. Add ^ and | of f together. 6. Add 6i 6^ and 4i together^, Facit. 17^>^. 2. When the fractions are offeeveral denominations, reduce them to their proper quantity, and add as before. 7. Add I of a pound to I of a shilling. Facit, ISs. lOd. 8. Add i of a penny to | of a pound. Facit, 13s. ^a. 9. Add i of a pound troy to ^ of an ounce. Facit 9 oz. 3 dwts. 8 grs. 10. Add I of a ton to I of a lb. Facit, 16 cwt. qrs. lb. 13 oz. 5 J dr. 11. Add I of a chaldron to f of a bushel. Facit, 24 bushels 3 pecks. 12. Add i of a yard to | of an indi. . ^ „ ^ ° Facit, C inch. 2 bar. corns. SUBTRACTION OF VULGAR FRACTIONS. Rule. Reduce the given fraction to a common deuomiuator, then subtract the less numerator from the greater, a%d place the remainder over the common denominator. MULTIPLICATION. 1*13 2. When the lower fraction is greater than the upper, sub- tract the numerator of the lower fraction from the denominator, and to that ^fFerence add the upper numerator, carrying one to the unit's place of the lowei- whole number. EXAMPLES. 1. Fromftakef 3XV = 21. 5X4=20. 21— 20=1 num. 4X7 = 28 den. Facit J- 2. From f take f of f . Facitj 11 3. From 5^ take -j?^. Pacit, 4|5. 4. From II take f Facit, ^y of a yard come to at that rate? ^^^^ la^ig^, yd. £ yd. £ Asf : |::^y : |i = l5s. for 4X 5X9 = 180 num. . „ . ,x , „ and 3 X 8 X 10 = 240 den. ^'* ^X A=U fHf (M^- 2. If f of a yard cost f of £l, vfha,t will ji of a yard cost? Ans. 148. 8d. 3. If I of a yard of lawn cost Ts. 3d., what will 10^ yards cost? Ans. £4 : 19 : 10||. 4. If ^ lb. cost |s. how many pounds will | of Is. buy ? Ans. 1 lb.^f^=^\. 6. If I ell of Holland cost J of £1, what will 12^ ells cost at the same rate ? Ans. £7:0:8^ U. 6. If 12^ yards of cloth cost 15s. 9d., what will 48^ cost at the same rate ? Ans. £3 : : 9^ j\\. I. If ^^ of a cwt. cost 2848. what will 7^ cwt. cost at the same ^^^ ^W5. £118 : 6 : 8. 8. If 3 yards of broad cloth cost £2 A, what will 10ft yards cost? Ans. £9 : 12. 9. If i of a yard cost § of £l, what will f of an ell English come to at the same rate ? Ans. £2. 10. If 1 lb. of cochineal cost £l : 5, what will S6j\ lb. come ^? Jws. £45 : 17 ; 6. II. If 1 yard of broad cloth cost l5fs., what will 4 pieces cost, each containing 27f yards? Ans. £85 : 14 : 3^ |f or 4. 12. Bought 3^ pieces of silk, each containing 24f ells, at 6s. 9|d. per ell. I desire to know what the whole quantity cost ? Ans. £25 : 17 : 2^ ||. THE SINGLE RULE OF THREE INVERSE, IN VULGAR FRACTIONS. EXAMPLES. 1. If 48 men can build a wall in 24i days, how many men can do the sAme in 192 days ? 'Ans. 6-^^ men. 2. If 25if8. will pay for the carriage f^^ 1 cwt. 146i miles, how for may ^^ cwt. be carried for the same -s>ney Ans. 22^ miles. 116 THB DOUBLE RULE OF THRBB. 3. If 3\ yards of cloth, that is 1} yard wide, be sufficient to make a cloak, how much must I have of that sort which is f yard Avide, to make another of the same bigness ! Ans. 4 1 yards. 4. If three men can do a piece of work in 4^ hours, in how many hours will ten men do the same work ? Ans. ly\ hour. 5. If a penny white loaf weighs 1 oz. when a bushel of wheat cost 5s. 6d., what is a bushel worth when a penny white loaf weighs but 2^ ozJ • ^ws. 16s. 4|d. 6. Whaf. quantity of shalloon, that is f yard wide, will line H yards of cloth, that is 1^ yard wide ? Ans. 15 yards. THE DOUBLE RULE OF THREE, IN VULGAR FRACTIONS. * EXAMPLES. 1. If a carrier receives £2^^ for the carriage of 3 cwt. 150 miles, how much ought he to receive for the carriage of 1 cwt 3^ qrs. 60 miles? -^wa- :6l : 16 : 9. 2. If £100 in 12 months gain £6 interest, what principal will gain £3f in 9 months ? -^w*. £76. 3. If 9 students spend £10| in 18 days, how much will 20 students spend in 30 days ? Ans. £39 : 18 : 4t\%\. 4. A man and Lis wife having laboured one day, earned 4f s. how much must they have for lOi days, when their two sons helped them? Ans. 4 : 11 : l^. 5. If £50, in 6 months, gain £2yV4. "^^^^ ^^^ ^^^^ ^^^^ '^^ quire to gain £1tV? ^m. 9 months, C. If the carriage of 60 cwt. 20 miles cost £14^, what weight can I have carried 30 miles for £5t\ ? Ans. 15 cwt. 117 THE TUTOR'S ASSISTANT. PART III. DECIMAL FRACTIONS. • tn Dedmal Fractions the inteo'er or wIioIa fJ.in„ ,. ' , one yard, one gallon, &6. fa ,„^led it dWdfdlnr.rT""'^; Whole numbers. Decimal parts. 1 6 o s CO S-g- t3^ CO CO Orf !» on B " 4321 ,2 34667 g- sr g- s- ST S- i- 2> 2. S o o o e S3 o _ . , Cii e- g a 2 tr' CO CO B CO g § " i| sSiif^^j^iHrss tjg ADDITION OP DECIMALS. before decimri parts decrease their value by removing them far- S from the comma, or unit's place ; thus, ,5 is 6 parte of 10, or X 06 is 6 Paftsof 100. "' tJi! .005 is 6 par 8 of 1000, or „!;• ,0005 is 6 parte of 10000, or „H?- »"Vn'".To tc didmal parts do not alter their value. For ,6, ,60, ,500, &c. " a1'« DlcfMl^i""t which ends at a certain number of nlaces but an infinite is that which no where ends. ^ A kECunaiNO bkcimal is that wherein one or more figures '^A:d'Si'527^2?6''is".SwT-ooMPOUNi> KBCuaaiNa .koi- MAL. Note. A finite deciraal may be considered as ;nfi"^tej)y ma- king ciphers to recur; for they do not alter the value of the deci- ""In all operations, if the result consiste of several nines reject them, and make the next superior place an umt more; thus, tor 26,25999, write 26, 26. , , , , ^ ^ In all circulating numbers, dash the last tigure. ADDITION OF DECIMALS. Rni.B In settin" down the proposed numbers to be added, „^r^re muTbe" taken in plilcin^g every figure directly under- ShXw the same value, whether they be mixed numbe.^, rSre decimal parte; and to perform which there in-t be a due ZLi had to the ««mas, or separating pomts, which ought Sp tostand injilrect line, one under another, and t/^ 4e le G^i iii^^Tii emcare SThandTth r^^rSyliaee the decimal parte acco..,ng KeS respective values; then add them as m whole numb .:. EXAMPLES. 1. AddV||fr3.^Zi.+2,15U+8t. '+2.V|^.^_ ,3o_3,3,. 2, Add 30,0'?4 '-',0O7H-59,432+7,l. _ 8. Add 3,6+i7,25+9'27,Ol+2,0073-hi,&. 4 Add 62,76+47,21 + 724 + 31,452+8075. 6. Add 3275+27,514+1,005+725+7 32. 6. Add27,5+52+3,2676+,5741+2720. MULTIPLIOATIOIf OF DECIMALS. M SUBTRACTION OF DECIMALS. RuLK. Subtraction of decimlfs diflfers but littlo from whole numbers, only in placing the numbers, which must be carefully observed, as in addition. EXAMPLES. 1. From ,2764 take ,2371. 2. Froiii 2,37 take 1,76. ?. From 271 take 215,7. 4. From 270,2 take 75,4075. 6. From 671 take 64,72. 6. From 625 take 76,91. 7. From 23.415 take ,3742. 8. From ,107 take ,0007 MULTIPLICATION OF DECIMALS. .)•>: Rule. Place the factors, and multiply them, as in whole num- bers, and from the product towards the right hand, cut off as many places for decimals as there are in both factors together; but if there should not be so many places in the product, sup- ply th« defect with ciphers to the left hand. EXAMPLES 1. Multiply 2. Multiply 3. Multiply 4. Multiply 5. Multiply 6. Multiply ,2365 by ,2435. Facit, ,05758775. 2071 by 2,27. 27,15 by 25,3. 72347 by 23,15. 17105 by ,3257. 17105 by ,0237. 7. Multiply 27,35 by 7,70071.- 8. Multiply 67,21 by ,0075. 9. Multiply ,007 by ,007. 10. Multiply 20,15 by ,2705. 11. Multiply ,907 by ,0025. When any nui^er of decimals is to be multiplied by 10, 100, 1000, (fee, it is 01% remo\dng the separating point in the raulti- pUcand so many places towards the right hand as there are ciphers in the multiplier: thus, ,578X10—5,78. ,578X100=5,78 ;, 578 X 1000=578; and ,578X10000=5780. CONTRACTED MULTIPLICATION OF DECIMALS. Rule. Put the unit's place of the multiplier under that place of the multiplicand that is intended to be kept in the product, then invert the order of all the other figures, i. e. write them all the 120 CONTRACTED MULTIPLICATION. contrary way ; and in multiplying, begin at tlie figure m the mul- tiplicand, which stands over the figure you are then multiplying with, and set down the first figure of each particular product di- rectly one under the other, and ^ve^ due regard to the increase arising from the figures on the right hand of that figure you begin to multiply at in the multiplicand. Note. That in multiplying the figure left out every time next the right hand in the multiplicand, and if the product be 6, or upwards, to 15, carry 1 ; if 15, or upwards, to 25, carry 2; and if 25, or upwards, to 35, carry 3, «fcc. EXAMPLES. 12. Multiply 384,672158 by 36,8346, and \ei there be only four places of decimals in the product. Contracted way. 384,672158 5438.63 115401647 23080330 307737t 115402 15387 1923 14169,2065 Common way. 384,672158 36,8345 1923 15386 115401 3077377 23080329 115401647 360790 88632 6474 264 48 4 14169,2066 038510 Facit, 14169,2066. 13. Multiply 3,141592 by 52,7438, and leave only four places of decimals. f^^^^^^ 165,6994. . 14. Multiply 2,38645 by 8,2175, and leave only four places of decimals. F^^^*' l^'^^^^' 15. Multiply 375,13758 by 167324, and let there be only one Place of decimals. Facit, 6276,9. 16. Multiply 375,13758 by 16,7324, and H*^«.<^"]y Jf^^ P^^^ of decimals. Facit, 6276,9520. 17. Multiply 395,3756 by ,76642, and let there ^^ o£y^*<>^'f places of decimals. Facit, 299,0699. le mul- :iplying luct di- mcrease u begin ne next >e 6, or 2; and be only DIVISION OP DECIMALS. 121 DIVISION OF DECIMALS. finJf'^- ^""^^ J8. also worked as in whole numbers; the only dif- lowinyrX : "'"^ '^'''^^''''' ""^^^ '' ^"°" ^^ ^"^ ^^ *^^ ^'^l- RuLE 1. The first figure in the quotient is always of the same ot'thTi^'f'n"•'^^/^^^^^^^ ^^^^^ --- -'^^ over the place of units m the divisor. a. ?^ The quotient must always have so many decimal places as the dividend has more than the divisor. ^ * hJ^^A ^' Y *^! ^'r^*' ^"'^ ^^^^^°^ ^^^« lt>«tl^ the same num- ber of decimal parts, the quotient will be a whole number. 3. But if, when the division is done, the quotient has not so ZhlJ^Zt:^ '' f'f^ ^r P^«^^^ ^f ^''^^^ then so man; -^iphers must be prefixed as there are places wanting. ^, EXAMPLES. ) ,2065. ir places ,6994. ir places ,6107. only one 276,9. ur places ,9520. )nly four ,0699. \r 1. Divide 85643,825 by 6,321. 2. Divide 48 by 144 3. ^Divide 217,75 by 65. 4. Divide 125 by ,1045. 5. Divide 709 by 2,574. 6. Divide 5,714 by 8275. I, m r. Divide 7382,54 by 6,4252. 8. Divide ,0851648 by 423. 9. Divide 267,15975 by 13,25. 10. Divide 72,1564 by ,1347. 11. Divide 715 by ,3076. Vyiien numbers are to be divided by 10, 100, 1000 10000 so many places towards tliA l^ft har,A o« *u^t --.i . ., divisor." " '"^'* '*° """""^ "*" wpiiers m tne Thus, 5784-^ 10=578,4. 5784-M00=57,84. 5784~- 1000=5,784. 6784-M0,000=,5784. 3«3 CONTRACTED DIYISIOW. CONTRACTED DIVISION OF DECIMALS. Rule. By the first rule find what is 'the value of the first figure in the quotient: then by knowing the first figure's denomination, the decimal places may be reduced to any number, by taking as many of the left hand figures of the dividend as will answer them ; and in dividing, omit one figure of the divisor at each following operation. Note. That in multiplying every figure left out in the divisor, you must carry 1, if it be 5 or upwards, to 15 ; if 15, or upwards, to 25, carry 2 ; if 25, or upwards, to 35, carry 3, &c. EXAMPLES. 12. Divide '721,1'7562 by 2,257432, and let there be only three places of decimals in the quotient. ! Contracted. Common way. 2,257432)721,17562(319,467 2,257432)721,17562(319,467 6772296 6772296 439460 225743 213717.. 203169.. 10548... 9030... 1518.... 1354. ... 164 168 6 ■ t Hi H rTit Oli i. iOVL 13. Divide 8,758615 by 5,2714167. 14. Divide 15. Divide 16. Divide 17. Divide 18. Divide 439460 2 225743 2 213717 00 203168 88 10548 120 9029 728 1518 3920 1354 4592 163 93280 158 02024 • 5 91256 25,1367 by 217,35. 51,47542 by ,123415. 70,23 by 7,9863. 27,104 by 3,712. UEDUCflON OP DECIMALS. 128 5t figure lination, iking as r them ; allowing ) divisor, ipwards, aly three [319,467 2 2 00 88 120 728 3920 4592 93280 02024 91256 REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to a Decimal. % Rule. Add ciphers to the numerator, and divide by the de- nominator, the quotient is the decimal fraction required. EXAMPLES 1. Reduce ^ to a decimal. 4)1,00(25 Facit. 2. Reduce ^ to a decimal. Facit 5 3. Reduce f to a decimal. Facit '75* 4. Reduce f to a decimal. Facit, ',375.' 5. Keduce ^\ to a decimal. Facit, ,1923076+. 6. Reduce J-i of |f . to a decimal. Facit, ,6043956+. Note. If the given parts are of several denominations, they may be reduced either by so many distinct operations as there are different parts, or by first reducing them into their lowest denomination, and then divide as before ; or, 2ndly. Bring the lowest into decimals of the next superior de- nomination, and on the right hand of the decimal found, place the parts given of the next superior denomination ; so proceedino- till you bring out the decimal parts of the highest integer required, by still dividing the product by the next superior denominator ; or, 3dly. To reduce shillings, pence, and farthings. If the num. ber of shillings be even, take half for the first place of decimals, and let the second and third places be filled with the ferthirif^s contained in the remaining pence and farthings, always remem- bering to add 1, when the number is, or exceeds 25. But if the number of shillings be odd, the second place of decimals must be increased by 5. '7. Reduce 5s. to the decimal of a £. Facit, ,26. 8. Reduce 9s. to the decimal of a £. Facit', [45! 9. Reduce 16s. to the decimal of a £. Facit! ,8! l3 124 SEDUCTION OF DECIMALS. 10. Reduce 8s. 4cl. to the decimal of a £. 11 ^Reduce 168. Tfd. to the decimal of a £. first. 168. I^d. 12 , 199 4 960)799(8322916 second. 4)3,00 12)7,76 third. 2)16 ,832 Facit, ,4166. Facit, ,8322916. 7|d. 4 2iO)16,64583 ,8322916 31 r l 12. Reduce 19s. S^d. to the decimal of a £. Facit, 972916. 13. Reduce 12 grains to the decimal of a lb. troy. Facit, ,002083. 14. Reduce 12 drams to the decimal of a lb. avoirdupois. Facit, ,046875. 16. Reduce 2 qrs. 14 lb. to the decimal of a cwt. Facit, ,625. 16. Reduce two furlongs to the decimal of a league. Facit, ,0833. 17. Reduce 2 quarts, 1 pint, to the decimal of a gallon. Facit, ,626. 18. Reduce 4 gallons, 2 quarts of wine, to the decimal of a hogshead. Facit, ,071428+. 19. Reduce 2 gallons, 1 quart of beer, to the decimal of a bar- rel. Facit, ,0625. 20. Reduce 62 days to the decimal of a year. Facit, ,142465+. To find the valite of any Decimal Fraction in the known parts of an Integer. Rule. Multiply the decimal given, by the number of parts of the next interior denomination, cutting off the decimals from the product ; then multiply the remainder by the next inferior deno- mination; thus pioceeding till you have brought in the least known parts of an integer. ■'>«iiiffiftBir.ii'aifri-tii,ii II l l«.lwil .ll I ■^^' BEDUCTION OP DECIMALS. 125 EXAMPLES. 21. What is the vahie of ,8322910 of a lb. ? Ans. 16s. 7id.-f . 20 16,6458320 12 7,7499840 4 2,9999360 22. What is the value of ,002084 of a lb. troy ? Ans. 12,00384 gr. 23. What is the value of ,046875 of a lb. avoirdupois ? Ans. 12 dr. 24. What is the value of ,625 of a cwt. ? Ans. 2 qrs. 14 lb. 25. What is the value of ,625 of a gallon ? Ans. 2 qrs. 1 pint 26. What is the value of ,071428 of a hogshead of wine ? Ans. 4 gallons 1 quart, ,999856. 27. What is the value of ,0625 of a barrel of beer ? Ans. 2 gallons 1 quart 28. What is the value of ,142465 of a year ? Ans. 51,999725 day^. 126 il h: ¥1 m DECIMAL TABLES OP COIN, WEIGHT, AND MEASURE. TABLE I. English Coijv. £ 1 the Integer. Sh. 19 18 17 16 15 14 13 12 11 10 Dec. ,93 ,9 ,S3 ,8 ,73 ,7 ,63 ,6 ,53 ,5 Sh. 9 8 7 6 5 4 3 2 1 Dec. ,45 ,4 ,35 ,3 ,25 ,2- ,15 .1 ,05 Pence. 6 5 -■' 4 3 2 1 Decimals. ,025 ,020833 ,016666 ,0125 ,008333 ,004166 Farth. 3 2 1 Decimals. ,003125 ,0020833 ,0010416 TABLE n. English Coin. 1 Sh. Long Measure. 1 Foot, the Integer. Pence & Inches. 6 IS O 4 3 2 1 Decimals. ,5 ,333333 ,23 ,166666 ,083333 Faith. 3 2 1 Decimals. ,0625 ,041666 ,020833 TABLE III. Troy Weight. 1 lb. the Integer.. Ounces the same as Pence in the last Table. Dwts. 10 9 8 7 6 5 4 3 2 1 Decimals. ,041666 ,0375 ,033333 ,029166 ,023 ,020833 ,016666 ,01,25 ,008333 ,004166 Grains. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,002083 ,001910 ,001736 ,001562 ,001389 ,001215 ,001042 ,000868 ,000694 ,000521 ,000347 ,000173 1 oz. the Integer. Pennyweights the same as Shillings in the first Table. Grains. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,052 ,022916 ,020833 ,01875 ,016666 ,014583 ',0125 ,010416 ,008333 ,00625 ,004166 ,002083 TABLE IV. Avoir. Weight. 112 lbs. the Integer. Qrs. Decimals. 3 ,75 2 ,5 1 ,25 Pounds. 14 13 12 11 10 9 8 7 6 5 4 3 o a 1 Decimals. ,125 ,116071 ,107143 ,098214 ,089286 ,080357 ,071428 ,0625 ,053571 ,044643 ,035714 ,026786 ,008928 Ounces. • 8 7 Decimals. ,004464 ,003906 127 3ASURE. Decimals. ,052 ,022916 ,020833 ,01S75 ,016066 ,014583 ',0125 ,010416 ,008333 ,00625 ' ,004166 ,002083 BLE IV. . Weight. the Integer. Decimals. ,75 ,5 ,25 Decimals. ,125 ,116071 ,107143 ,098214 ,089286 ,080357 ,071428 ,0625 ,053571 ,044643 ,035714 ,026786 ,ui ioo s ,008928 Decimals. ,004464 ,003906 DECIMAL TABLES OF COIN, WEIGHT, AND MEASURE. 4 3 o 1 ,003348 ,002790 ,002232 ,001674 ,001116 ,000558 k Oz. 3 2 1 Decimals. ,000418 ,000279 ,000139 TABLE V. AvOIRDli^'OIS WEIGHT. 1 lb. the Integer. Ounces. 8 7 H 5 4 3 2 1 Decimals. ,5 ,4375 ,375 ,3125 ,25 ,1875 ,125 ,0025 Drams. 8 7 6 5 4 3 2 1 Decimals. ,03125 ,027343 ,023437 ,019531 ,015625 ,011718 ,007812 ,003906 TABLE VL lilQiriD MEASURE 1 tun the Integer. Gallons. 100 90 Decimals. ,390825 ,357142 80 70 00 50 40 30 20 10 9 8 7 6 5 4 3 2 1 ,317400 ,27 ,238095 ,198412 ,158730 ,119047 ,079365 ,039082 ,035714 ,031740 ,027 ,023809 ,019841 ,015873 ,011904 ,007930 ,00396s Pints. 4 3 2 1 A Hogshead the Integer. Gallons. 30 20 10 9 8 7 6 5 4 3 2 1 Tints. 3 2 1 Decimals. ,005952 ,003968 ,001984 Decimals. ,001984 ,001483 ,000992 ,009496 TABLE yil. Measures. Liquid. Dry. 1 Gal. 1 Qr. Integer. Pts. 4 3 2 1 Decimals. ,5 ,375 ,25 ,125 Decimals. ,476190 ,317460 ,158730 ,142857 ,126984 ,111111 .095238 ,079365 ,063492 ,047619 ,031746 ,015873 Q.pt. 3 2 1 Decimals. ,09375 ,0625 ,03125 Decimals. ,0234375 ,015625 ,0078125 Decimals. ,005859 ,003906 ,001953 TABLE VIL. Long Measure. 1 Mile the Integer. Yards. 1000 900 800 700 600 Decimals. ,^568 182* ,511364 ,454545 ,397727 ,340909 128 DECIMAL TABLES OF COIN, WEIGHT, AND MEASURE. 5UU 400 300 200 100 90 80 70 CO 50 40 30 20 10 9 8 7 5 4 3 2 1 ,284091 ,227272 ,170454 ,113630 ,056818 ,051136 ,015451 ,039773 ,034091 ,028409 ,022727 ,017045 ,011364 ,005682 ,005114 ,004545 ,003977 ,003409 ,002841 ,002273 ,001704 ,001138 ,000568 Feet. 2 1 Decimals. ,0003787 ,0001894 Inches. 6 3 1 Decimals. ,0000947 ,0000474 ,0000158 TABLE IX. Time. 1 year the Integer. Months the same as Pence in the second Table. Davs 365 300 200 100 90 Decimals. i,6bo6oo ,821918 ,547945 ,273973 ,246575 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 ,219178 ,191781 ,164383 ,136986 ,109559 ,082192 ,054794 ,027397 ,024657 ,021918 ,019178 ,016438 ,013698 ,010959 ,008219 ,005479 ,002739 1 day the Integer. Hours. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. .5 ,458333 ,416666 ,375 ,333333 ,291666 ,25 ,208333 ,166666 ,125 ,083333 ,041666 Minutes. 30 20 10 9 8 7 6 5 4 3 2 1 Decimals. ,020833 ,013888 ,006944 ,00.625 ,005555 ,004861 ^004166 ,003472 ,002777 ,002083 ,001389 ,000664 TABLE X. Cloth measure. 1 yard the Integer. Quarters the same as Table 4. ails. Decimals 2 ,125 1 ,0025 TABLE XI. Lead Weight. A Foth. the Integer. Hund. 10 9 d 7 6 5 4 3 2 1 Decimals. ,512820 ,461538 ,410256 ,358974 ,307692 ,256410 ,205128 ,153846 ,102564 ,051282 Qrs. 2 1 Decimals. ,025641 ,012820 Pounds. 14 13 12 11 10 9 8 4 3 Q Decimals. ,0064102 ,0059523 ,0054945 ,00503r>6 ,0045787 ,0041208 ,0036630 ,0032051 r\r\nri Ann 16022898 ,0018315 ,00137^6 ,00091^7 ,0004578 THE RULE OF THBEE IN DECIMALS. 129 ■fjfj' THE RULE OF THREE IN DECIMALS. EXAMPLES. If 26^ yards cost £3 : 16 : 3, what will 32^ yards come to? Ans. £4 : 12 : 9i. yds. £ yds. 26,5 : 3,8125 : : 32,25 : 32,25 26,5)122,953125(4,63974=£4 : 12 : 9^. 2. What will the pay of 540 men come to, at £l : 5 : 6 per i"»n? ^W5. £688 : 10. 3. If Vf yards of cloth cost £2:12:9, what will 140| yards of the same cost ? ^W5. £47 : 16 : 3 2,4 qrs. 4. If a chest of sugar, weighing 1 cwt. 2 qrs. 14 lb. cost £36 : 12 : 9, what will 2 cwt. 1 qr. 21 lb. of the same cost? Ans. £11 : 14 : 2 3,5 qrs. 6. A grocer buys 24 ton 12 cwt. 2 qrs. 14 lb. 12 oz. of tobac- co for £3678 : 6 : 4, what will 1 oz. come to? Ans. Id. 6. What will 326^ lb. of tobacco come to, when 1| lb. is sold for 3s. 6d. ? Ans. £38:1: 3. 7. What is the worth of 19 oz. 3 dwts. 5 grs. of gold, at £2 : 19 per oz.? Ans. £56 : 10 : 5 2,99 qrs. 8. What is the worth of 827f yards of painting, at lO^d. per yard? ^W5. £36 : 4 : 3 1,5 qrs. 9. If I lent my friend £34 for f of a year, how much ought he to lend me y\ of a year to requite my kindness ? Ans. 61. 10. If f of a yard of cloth, that is 2^ yards broad, make a gar- ment, how much that is | of a yard wide will make the same ? Ans. 2,1 093 '75 yards. 11. If 1 ounce of silver cost 5s. 6d., wkat is the price of a tan- kard that weighs 1 lb. 10 oz. 10 dwts. 4 grs. ? i' Ans. £6 : 3 : 9 2,2 qrs. 12. If 1 lb. of tobacco cost 15d. what cost 3 hogsheads, weigh- ing together 15 cwt. 1 qr. 19 lb. ? Ans. £107 : 18 : 9. 13. if. 1 cwt. of currants cost £2:9:6, what will 45 cwt. 3 qrs. 14 lb. cost at the same rate? Ans. £113 : 10 : 9#. 14. Bouofht 6 chests cwt., what do they come to ? sugar, each 6 cwt. 3 qrs. at £2 :.16 per Ans, £113 : 8. M\ 1 1 II il :{ sir 130 BXTBAGTION OF THB SaVARB I109T. 15. Bought a tankard for £10 : 12, at the rate of 68. 4d. per ounce, ^\ hat was the weight ? Jns. 39 oz. 15 dwts. 16. Gave £187 : 3 : 3, for 25 cwt. 3 qi-s. 14 lb. of tobacco, at what rate did I buy it per lb. ? Ans. Is. 3id. 17. Bouorht 29 lb. 4 oz. of coffee, for £10 : 11 : 3, what is the value of 3 lb. ? Ans. £1:1:8. 18. If I give Is. Id. for 3^ lb. cheese, what will be the value of 1 cwt.? ^ws. £1 : 14 : 8. EXTRACTION OF THE SQUARE ROOT. Extracting the Square Root is to find out such a number as, being multiplied into itself, the product will be equal to the given num- ber. Rule. First, Point the given number, beginning at the unit's place, then proceed to the hundreds, and so upon every second figure throughout. Secondly. Seek the greatest square number in the first point towards the left hand, placing the square number under the first point, and the root thereof in the quotient; subtract the square number from the first point, and to the remainder bring down the next point and call that the resolvend. Thirdly. Double the quotient, and place it for a divisor on the left hand of the resolvend ; seek how often the divisor is contain- ed in the resolvend; (preserving always the unit's place) and put the answer in the quotient, and also on the right-hand side of the divisor ; then multiply by thg figure last put in the quotient, and subtract the product from the resolvend; bring down the next point to the remainder if there be any more) and proceed as be- fore. * Roots. o,- ^ ilTiS. 1. 2. 3. 4. 5. 6. 7. 8 9. 4. "a. 25. 3fi. ^% 6? P-\. EXTSACTION OF THE SaUARB BOOT. I 131 Ans. 346. EXAMPLES. 1. What is the square root of 119025 ? 119025(345 64)290 256 685)3425 3425 2. What is the square root of 106929 ? Ans. 327+. 3. What is the square root of 2268741 ? Am. 1506,23 -f. 4. What is the square root of 759679^ ? \Ans. 2756,228+ 5. What is the square root of 36372961 ?**» Ans. 6031. 6. What is the square root of 22071204 ? Ans. 4698. When the given number consists of a whole number and deci- mals together, make the number of decimals even, by adding ci- phers to them ; so that there may be a point fall on the unit's place of the whole number. .7. What is the 8. What is the 9. What is the 10. What is the 11. What is the 12. What is the square square square square square square root of root of root of root of root of root of 3271,4007? 4795,25731? 4,372594? 2,2710957? ,00032754? 1,270059? Ans. 57,19+. ^w*. 69,247+. Ans. 2,091 -f. Ans. 1,50101+. Ans., 01809 -i-^ Ans. 1,1269 + To extract the Square Moot of a Vulgar Fraction. Rule. Reduce the fraction to its lowest terms, then extract the square root of the numerator, for a new numerator, and the squiiai root of the denominator, for a new denominator. If m^ fraction be a surd (i. e) a number where a root can ne- ver l^e exactly found, reduce it to a decimal, and extract the root from it. EXAMPLES. 13. What is the square root of f f |f ? 14. What is the square root of f f f a ? 15. Wh.'it is the square root of tVt-'A? » 9 fi ■» 4 Ans. f . Ans. |. Ans. 4- at I ii ;1 liB •i' Ml EXTRACTION OF THS SaUARB ROOT. BURDB. 16. What is the square root of |ff ? 17. What is the square root of |4^ ? 18. What is the square root of }Jf ? Arts. ,898024-. Ana. ,86602+. Ans. ,933094-. To extract the Square Root of a mixed number. Rule. Reduce the fractional part of a mixed nuraber to ita lowest term, and then the mixed number to an improper fraction. 2. Extract the root of the numerator and denominator for a new numerator and denominator. • If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom. EXAMPLES. 19. What is the square root of 5lf-} ? 20. What is the square root of 27/^ ? 21. What is the square root of 9a| SURDS. 22. What is the square root of 85|f ? 23. What is the square root of 8^ ? 24. What is the square root of 6| ? Ans. 7}. Ans. b\. Ans. Z\. Ans. 9,27-|-. Ans. 2,95194-. Ans. 2,6819 + . To find a mean proportional between any two given numbers. Rule. The square root of the product of the given number . is the mean proportional sought. EXAMPLES. 5. What is the mean proportional between 3 and 121 Ans. 3 X 12 = 36, then fj 36 = 6 the mean proportional. 6. What is the mean proportional between 4276 and 842 ? Ms. 1897,4 + To find the side of a square equal in area to any given superficies. Rule. The square root of the content of any given superficies is the side of the square equal sought. inator for a ■XTRAOTION OP THE SQUABB HOOT. EXAMPLES. 133 27. If the content of a given circle be 160, what is the side of the square equal? ^n^. 12,64911. eouaH ^^ ^^^ ^'"^ ""^ * ""'""^^ ^ ^^^' ""^^^ ^ *^® «*^« «^ t^« square ^ ^w«. 27,38612. The area of the circle given to find the Diameter, Rule. As 355 : 452, or, as 1 : 1,273239 : : so is the area : to Leabrri2l37^ntr''"'r'; "s^^a^^^ «^"^- ^^^'-^'^^ area Dy 1,12837, ana the product will be the diameter vV EXAMPLES. . oth!r"»nT/' ^r^Y ""^ T" ^ "' to tie to a cow's tail, the an ITS"* '" '^ ^™""'^' '^ '^' ■>«' have liberty of e;tinK t/j jeuus » ^^^ g j3g perches. TAe area o/ a circle given, to find the periphery, or circumference, sau^r^f tL"^ •• i^^^' ^'' "^ ^ • ^2,56637 : : the area to the Xhv s^iioP'"i'^r^''~r' ™"^'^P^^ *^^ «q«^'-^ root of the area by 3,5449, and the product is the circumference. EXAMPLES. 30. When the area is 12, what is the circumference ? Ans. 12,279. 31. When the area is 160, what is the periphery? Ans. 44,839. sid^"^ two sides of a right-angled triangle given, to find the third fa^^ „n(i j^ci^cuuiuuxar given lo nna the hypothenuse. an?n«f ^^^ square root of the sum of the, squares of the base and perpendicular, is the length of the hypothenuse. I .' 134 BX'ERACTIOTr OF THE SaVJ^E ROOT. /w * EXAMPLES. 32. The top of a castle from the ground is 45 yards high, and surrounded with a ditch 60 yards broad ; what length must a lad- der be to reach from the outside of the ditch to the top of the castle ? -4ws. 75 yards. S3 I a Ditch. a> ^^ ■J§ o , CO 1 «4-l t^ o «3 ""t feD a> w Base 60 yards. 33. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth : I desire to know the length of a ladder that will reach from the outside of the moat to the top of the wall ? -Ans. 39,06 feet. The hypothenuse and perpendicular given, to find the base. Rule. The square root of the difference of the squares of the hypothenuse and perpendicular, is the length of the base. The base and hypothenuse given, to find the perpendicular. Rule. The square root of the difference of the squares of the hypothenuse and base, is the height of the perpendicular. N. B. The two last questions may be varied for examples to the two last propositions. Any number of men being given, to form them into a square battle, or to find the number of rank and file. Rule. The square root of the number of men given, is the number of men either in rank or file. 34. An army consisting of 331776 men, I desire to know, how ok and tile ? -Ans. 57G. L certain square pavement contains 48841 squaji-e stones, e same size. I demand how many are contained in one many r ar,. all of t c" the sides ? Ans. 221, BXTBACTION OF THE CUFE ROOT. 1^ EXTRACTION OF THE CUBE ROOT. To extract the Cube Root is to find out one number, which be* ing multiplied into itself, and then into that product, produceth the given number. Rule. 1. Point every third figure of the cube given, beginning at the unit's place ; seek the greatest cube to the first point, and subtract it therefrom ; put the root in the quotient, and bring down the figures in the next point to the remainder, for a Resolvend. 2. Find a Divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient. 3. To find the Subtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in Ihe quotient by 3, except the last, and that product by the square of the last. 3. Multiply the divisor by the last figure. Add these products together, for the subtrahend, which subti^t from the resolvend ; to the re- mainder bring down the next point, and proceed as before. Roots. Cubes. r. 2. 3, 4._, 6. 6. 1. 8. 9. li^^ 1. 8* 2'rr'61E. J25. 216. 343. 512. 729. EXAMPLES. 1. What is the cube root of 99252847 ? 99252847(463 64 =cube of 4 Divisor- Square of 4X3=48)35252 resolvend. 216=cube of 6. 432 =4X3Xby square of 6. 288 %divisorXby 6. Divisor- 33336 subtrahend. Square of 46X3=6348)1916847 resolvend. , 27=cube of 3. 1242=46X3Xby square of 3. 19044 =divisorXby 3. 1916847 subtrahend. M2 / A / 186 EXTRACTION OP THE CUBE ROOT. 4 2. What is 8. What is 4. What is 6. What is 6. What is 1. What is 8. What is 9. What is 10. What is 11. What is 12. What is the cube root of 389017 ? the cube root of 5735339? the cube root of 32461759 ? the cube root of 84604619 ? the cube root of 259694072 ? the cube root of 48228544 ? the cube root of 27054036008 ? the cube root of 22069810125 ? the cube root of 122615327232 ? the cube root of 219365327791 ? the cube root of 673373097125 ? Ans. 73. Ans. 179. Ans. 319. Am. 439. Ans. 638. Ans. 364. Ans. 3002. Ans. 2805. Ans. 4968. Ans. 6031. Ans. 8765. When the given number consists of a whole number and deci- mals together, make the number of decimals to consist of 3, 6, 9, &c. places, by adding ciphers thereto, so that there may be a point fall on the unit's place of the whole number. 13. What is the cube root of 12,f77875 ? Ans. 2,35. 14. What is the cube root of 36155,02756 ? Am. 33,06+. 15. What is the cube root of ,001906m.? . Ans. ,124. 16. What is the cube root of l^^9fm^4 ? Am. 3,2164-. 17. What is the cube root of 15926,972504 ? Am. 25,16-|-. 18. What is the cube root of ,053157376 ? Ans. ,376. To extract the cube root of a vulgar fraction. Rule. Reduce the fraction to its lowest terms, then extract the cube root of its numerator and denominator, for a new nu- ^ merator and denominator ; but if the fraction be st surd, reduce it to a decimal, and then extract the root from it ? EXAMPLES. 19. What is the cube root of f ff ? 20. What is the cube root of jU^*^ ? 21. What is the cube root of -Raa ? Am. 4- Ans. |. Ans. |. \ SURDS. .m^ 22. What is the cube root of 4 ? Am. ,829+. 23. What is the cube root of ^ ? Am. ,822+. 24. What is the cube root of | ? Am. ,873+. To extract the cube root of a mixed number. Rule. Reduce th^ fractional part to its lowest terms, and then the mixed number to an improper fraction, extract the cube root >f the numerator and denominator for a new numerator and deno- BXTRACTIOrr OP THE CT7BB ROOT. f87 minator; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. EXAMPLES. 25. What is the cube root of 12Af ? 26. What is the cube root of 31 JJL ? 27 What is the cube root of 405 A?- ? 13 5 Ans. 2|. Ans. ^. Ans. 7|. SURDS. 28. What is the cube root of 1\ ? 29. What is the cube root of 9| ? 30. What is the cube root of 8f ? THE APPLICATION. Am. 1,93+. Ans. 2,092-f . Ans. 2,0574". 1- If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 mches deep, how many cubical inches doth it con- ^^ • Ans. 103823 2. There is a cellar dug, that is 12 feet every way, in length, breadth, and depth; how many sohd feet of earth were taken out . Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 solid teet, what is the superficial content of one of its sides ? Ans. 5329. Between two numbers given, to find two mean proportionals. Rule. Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme, gives the less mean ; multiply the said cube root by the less mean, and the pro- duct will be the greater mean proportional. EXAMPLES. 4. What are the two mean proportionals between 6 and 162 ? e TI71 . 1 ■^***- 18 and 64. o. VVhat are the two mean proportiQnals between 4 and 108 ? Ans, 12 and 36. ^ :■ jvrc^z- wco ciae vj t* o-«t/c mai snau oe equal m solidity to any given solid, as a globe, cylinder, prism, cone, ■*• at 4i per cent, per annum" ' ' "/"'' L^'f^'^ « """'h^- 0. What is L in.«"st of £236 • 18 S^'Vfr V " = ' '' months, at 5i per cent, per annumf Z £«'ri5 YTi '293' ' ^<^ Ike interest is for any number 0/ days miv.' INTEREST OF £1 FOR ONE DAY. Decimals. ,00008219178 ,00009589041 ,00010958904 ,00012328767 ,00013698630 ,00015068493 per cent. 6i 7 8 9 r\ 1 Decimals. ,00017808219 ,00019178082 ,00020547945 ,00021917808 ,00023287671 ,00024657534 ,00026027397 Note. The above table is thus found :— As 365 : ,03 : : 1 : ,00008219178. And as 365 : ,086 • • 1 ,00009589041, (fee. ^ o..i 142 SIMPLE INTEREST. EXAMPLES. I. What is the interest of £240, for 120 days, at 4 per cent, per annum \ Ans. ,00010958904 X 240 Xl20=£3 : 3t 1^. 8. What is the interest of £364 : 18, for 154 days, at 5 per cent, per annum? Ans. £7 : 13 : 11 J. 9. What is the interest of £725 : 16, for 74 days, at 4 per cent, per annum? Ans. £5 : 17 : 8^. 10. What is the interest of £100, from the 1st of June, 1775, to the 9th of March following, at 5 per cent, per annum ? Ans. £3 : 16 : llf. II. When P R T are ffiven to find A. Rule, prt + p=A. EXAMPLES. 11. What will £279 ; 12, amount to in 7 years, at 4^ per cent, per annum ? Ans. £367 : 13 : 5 3,04 qrs. 279,6 X ,045 X 7 + 279,6=367,674. 12. What will £320 : 17, amount to in 5 years, at 3^ per cent, per annum? Ans. £376 : 19 : 11 2,8 qrs. When there is any odd time given with the whole years, reduce the odd time into days, and work with the decimal parts of a year which are equal to those days. 13. What will £926 : 12, amount to in 5^ years, at 4 per cent, per annum? Ans. £1130 : 9 : 0^ ,92 qrs. 14. What will £273 : 18, amount to in 4 years, 175 days, at 3 per cent, per annum? Ans. £310 : 14 : 1 3,35080064 qrs. in. When A R T are given to find P. a Rule. =P. rt,+ 1. EXAMPLES. ^ 15. What principal, being put to interest, will amount to £36 13 : 5 3.04 ora. in 7 vears. at 4* ner cent. Der annum ? Ans. ,045X''7+1=1,315 then'367,674-rl,315=£279 : 12. 16. What principal, being gut to interest, will amount to £376 19 : 11 2,8 in 5 years, at S^'per cent, per annum? Ans. £320 : 17. SIMPLE INTEREST. 143 i 11. What principal, being put to interest, will amount to £1130 : 9 : 0^ ,92 qrs. in 5| years, at 4 per cent, per annum ? Ans. £026 : 12. 18. 5Vhat principal will amount to £310 : 14 : 1 3,35080064 qrs. in 4 years, 175 days, at 3 per cent, per annum? Ans. £273 : 18. IV. When A P T are given to find R. ♦ a — p Rule. ==R. pt EXAMPLES. 19. At what rate per cent, will £279 : 12, amount to £367 : 13 : 5 3,04 qrs. in 7 years? Ans. 367,674—279,6 = 88,074, 275,6x7=1957,2, then 88,074-M957,2=,045 or 4^ j^er cent. 20. At what rate per cent, will £320 : 17, amount to £376 19 ; 11 2,8 qrs. in 5 years? ^ Ans. 3^ per cent. 21. At what rate per cent, will £926 :*12, amount to £1130 9 : 0^ ,92 qrs. in 5^ years ? Ans. 4 per cent. 22. At what rate per cent, will £273 : 18, amount to £310 14 : 1 3,35080064 qi-s. in 4 years, 175 days? Ans. 3 per cent. V. When A P R are given to find T. a — p Rule. =T. pr. EXAMPLES. 23. In what time will £279 : 12, amount to £367 : 13 : 5 3,04 qrs. at 4^ per cent. ? Ans. 367,674—279,6=88,074. 279,6X ,045=12,5820, then 88,074-f- 12,5820=7 years. 4 24. In what time will £320 : 17, amount to 370 : 19 : 11 2,8 qra. at 3^ per cent. ? Ans. 5 years. 25. In what time will £926 : 12, amount to £1130 : 9 : 0|- ,92 qrs. at 4 per cent. ? Ans. 5| years. 26. In what time will £273 : 18, amount to £310 : 14 : 1 3,35080064 qi-s. at 3 per cent. ? Ans. 4 years, 175 days. ANNTTTTTFS OR Pror of years for a divisor ; if quarterly, take 8 a— 8 ut, and work with a fourth of the annuity, and four times the number of years. 43. If a salary of £150 per annum, payable half-yearly, amounts to £834 : 7 : 6, in 5 years, what is the rate per cent. ? Ans. 5 per cent. 44. If an annuity of £150 per annum, payable quarterly, amounts to £839 : 1 : 3, in 6 years, what is the rate per cent. ? Ans. 5 per cent. IV. When U A R are given to find T. =T. 2 2a XX Rule. First, 1 =x then : ^ [ r ur 4 2 EXAMPLES. 45. In what time will a salary of £150 per annum, amount to £825, at 5 per cent. ? Ans. 5 years. 2 826X2 39X39 1=39 = 220 =380,26 ,05 160 X, 05 4 39 v/220-f 380 ,25=24 ,5 =5 years. 2 40. If a house is let upon a lease* for a certain time, for £60 per annum, and amounts to £363 : 8 : 3, at 4J per cent., what time was it let for ? Ans. 5^ years. 47. If a pension of £250 per annum, being forborne a certain time, amounts to £2065, at 6 per cent., what was the time of forbearance ? " Ans. 1 years. 48. In what time will a yearly pension of £28, amount to £263 : 4, at 5 per cent. ? ' Ans. 8 years. Note. If the payments are half-yearly, take half the ratio, and half the annuity; if quarterly, one fourth of the ratio, and one fourth of the annuity ; and T will be equal to those half-yearly or niM\vfnr\\T •nQXTmnnl-o 49. If an annuity of £150 per annum, payable half-yearly, amounts to £834 : 7 : 6, at 5 per cent., what time was the pay- ment forborae ? A71S. 5 years. " SIMPLE INTEREST. 147 J2; ^1 ^ yearly pension of £150, payable quarterly, amounts to AB3y . 1 : 3, at 6 per cent., what was the time of forbearance? Ans. 6 years. PRESENT WORTH OF ANNUITIES. Note. P represents the present worth ; U T R as before. I. When U T R are given to find P. ttr— tr + 2t Rule. : x u=P. 2tr + 2 EXAMPLES. .61. What is the present worth of £150 per annum, to continue 5 years at 5 per cent, 'i J^^^ ^gg^^ 5X5x;05l-5x, 05+5X2^11 ,5 X ,05X2+2=2,6 then 11-^ 2,5X150=£660. 52. What is the yearly rent of a house of £60, to continue 5* years worth m ready money, at 4 J per cent. ? Ko wu . • .1. , •^^*- ^291 : 6 : 3. 53. What IS the present worth of £250 per annum, to continue IT'^^t ^ P'' '^'"^' • ^^'' ^1454 : 4 : 6. .54. What IS a pension of £28 per annum, worth in ready mo- ney, at 5 per cent, for 8 years ? Ans. £l 88. Note. The same thing is to be observed as in the first rule of annuities m arrears, concerning half-yearly and quarterly pay- 65. What is the present worth of £150, payable quarterly, for 5 years, at 5 per cent. ? Ans. £611 : 5i Note. By comparing the last examples, it will be found that the present worth of half-yearly payments is more advantageous than yearly, and quarterly than half-yearly. II. When P T R are given to find U. Rule.— tr-f 1 ttr— tr + 2t X 2p=U. ^9 148 SIMPLE INTERB8T. EXAMPLES. 56. If the present worth of a salary he £660, to continue 5 years, at 5 cent., what is the salary? Ans. £150. 6X ,05+1 = 1,25 5 X 5 X ,05—5 X ,05 + 10 = 11. 1,25 X 660 X 2 = £150. 11 67. There is a house let upon lease for 5^ years to come, I de- sire to know the yearly rent, when the present worth, at 4^ per cent, is £291 : 6 : 3 ? Ans. £60. 58. What annuity is that which, for 1 years' continuance, at 6 per cent., produces £1454 : 4 : 6 present worth? Ans. £250. 59. What annuity is that which, for 8 years' continuance, pro- duces £188 for the present worth, at 5 per cent. ? Ans. £28. Note. When the payments are half-yearly, take half the ratio, twice the number of years, and multiply by 4 p ; and when quar- terly, take one fourth of the ratio, and four times the number of years, and multiply by 8 p. 60. There is an annuity payable half-yearly, for 5 years to come, what is the yearly rent, when the present worth, at 5 per cent., is £667 : 10 ? Ans. £150. 61. There is an annuity payable quarterly, for 5 years to come, I desire to know the yearly income, when the present worth, at 5 per cent., is £671 : 5 ? Ans. £150. III. When U P T are given to find R. R^tE.- ut— p X 2 =R. 2pt + ut — ^ttu EXAMPLES. 62. At what rate per cent, will an annuity of £150 per annum, to continue 5 years, produce the present worth of £660 ? Ans. 5 per cent. 150X5-660X2 = 180,2X660X5 + 5X150-5X5X150=3600 then i80-^3600:=:,05=5 per eont. 63. If a yearly rent of £60 per annum, to continue Bl yoars, produces £291 : 6 : 3, for the present worth, what i^ the rale per cent. ? Ans. 4^ per cent. SIMPLE INTBRBBT. JNv 64. If an annuity of £250 per annum, to continue 1 yeai-s, produces £1454 : 4 : 6, for the present worth, what is the rate per cent. ? ji^s. 6 per cent. 65. If a pension of £28 per annum, to continue 8 years, pro duces £188 for the present worth, what is the rate per cent. ? Ans, 5 per cent. Note. When the annuities, or rents, &c. are to be paid half- yearly, or quarterly, then For half-yearly payments, take half of the annuity, &c. and twice the number of years, the (quotient will be the ratio of half the rate per cent. — and For quarterly payments, take a fourth part of the annuity, &c. and four times the number of years, the quotient will be the ratio of the fourth part of the rate per cent. ^ 66. if an annuity of £150 per annum, payable half-yearly, ha- ving 5 years to come, is sold for £667 : 10, what is the rate per ^^^^- ^ Ans. 5 per cent. ^ 67. If an annuity of £150 per annum, payable quarterly, ha- ving 5 years to come, is sold for £671 : 5, what is the rate per ^^^^' • Ans. 5 per cent IV. When U P R are given to find T. 2 2p * • 2p xlc X Rule. 1 =x then ^ — | =T. t n ur 4 2 r annum, EXAMPLES. 68. If an annuity of £150 per annum, produces £660 for the present worth, at 5 per cent., what is the time of its continu- ance? ^ws. 6 years. 660X2 ,05 ■ 150 30,2X30,2 —1=30,2 660X2 =176 150 X, 05 =228,01 then ^/228j01-f-176— 20,1 30,2 20,1- =5 years. y/i 150 SIMPLE INTEREST. 69. For what time may a salary of £60 be purchased for £291 : 6 : 3, at 4^ per cent. ? Ans. 5^ yeare. 70. For what time may £250 per annum, be purchased for £1454 : 4 : 6, at 6 per cent.? Ans. 1 years. 71. For what time may a pension of £28 per annum, be pur- chased for £188, at 5 per cent. ? Ans. 8 years. Note. When the payments are half-yearly, then U will be equal to half the annuity, &e. R half the ratio, and T the num- ber of payments : and, When the payments are quarterly, U will be equal to one fourth part of the annuity, &c. R the fourth of the ratio, and T the number of payments. 12. If an annuity of £150 per annum, payable half-yearly, is sold for £607 : 10, at 5 per cent., I desire to know the number of payments, and the time to come ? Ans. 10 payments, 5 years. 73. An annuity of £150 per annum, payable quarterly, is sold for £671 : 5, at 6 per cent., what is the number of payments, and time to come ? -y Ans. 20 payments, 5 years. ANNUITIES, &c. TAKEN IN REVERSION. 1. To find the present worth of an annuity, &c. taken in re- version. Rule. Find the present worth of t^e ttr — tr+2t yearly sum at the given rate and for the time of its continuance ; -thus, 2. Change P into A, and find what prin- cipal, being put to interest, will amount to A at the same rate, and for the time to come before the annuity &c. commences ; thus, EXAMPLES. 2tr4-2 : X u=P. a =P. tr-fl 74. What is the present worth of an annuity of £150 per an- num, to continue 5 years, but not to commence till the end of 4 years, allowing 5 per cent, to the purcliaser ? Ans. iE550. =650. 6 X, 05X2+2 4X,05-fl SIMPLE INTEREST. 151 15. What is the present worth of a lease of £50 per annum, to continue 4 years, but which is not to commence till the end of 5 years, alloAving 4 per cent, to the purchaser? An6'. £152 : 5 : 11 3 qrs. 16. A person having the promise of a pension of £20 per an- nunri, for 8 years, but not to commence till the end of 4 years, is willing to dispose of the same at 5 per cent., what will be the present worth? Ans. £111 : 18 : 1 ,14 + . 77. A legacy of £40 per annum being left for 6 years, to a person of 15 years of age, but which is not to commence till he is 21 ; he, wanting money, is desirous of selling the same at 4 per cent., what is the present worth ? Ans. £171 : 13 : 11 ,07596. 2. To find the yearly income of an annuity, &c. in revei*sion. ptr-}-p=A. Rule 1. Find the amount of the present worth at the given rate, and for the time before the reversion ; thus, 2. Change A into P, and find what an- nuity being sold, will produce P at the same rate, and for the time of its continu- ance; thus, tr+1 ttr— tr+2t :X2p=U. EXAMPLES. 78. A person having an annuity left him. for 5 years, which does not commence till the end of 4 years, disposed of it for £550, allowing 5 per cent, to the purchaser, what was the yearly m- come? . ^W5. £150. 5 X ,05 + 1, 550 X 4 X ,05 + 550 = 660 5 X 5 X ,05—5 X ,05 + 5 X 2= ,113636 X 660 X 2 = £150. 79. There is a lease of a house taken for 4 years, but not to commehce till the end of 5 years, the lessee would sell the same for £152 : 6, present payment, allowing 4 per cent, to the pur- chaser, what is the yearly rent ? Ans. £50. 80. A person having the promise of a pension for 8 years, which does not commence till the end of 4 years, has disposed of the same for £111 : 18 : 1 ,14 present money, allowing 5 per cent, to the purchaser, what was the pension ? Ans, £20. 152 BEBATB OB DISCOUNT. 81. There is a certain legacy left to a person of 15 years of age, which is to be contiriued for 6 years, but not to commence till he arrives at the age of 21 ; he, wanting a sura of money, sells it for XI 71 : 14, allowing 4 per cent, to the buyer, what was the an- nuity left him ? Ans. £40. REBATE OR DISCOUNT. Note. S represents the Sum to be discounted. P the Present worth. T the Time. R the Ratio. I. When S T R are given to find P. RULE.- tr+1 ■=P. EXAMPLES. 1. What is the present worth of £357 : 10, to be paid 9 months hence, at 5 per cent.? Ans. £344 : 11 : 6^ ,168. 2. What is the present worth of £275 : 10, due 7 months hence, at 5 per cent. I ' Ans. £267 : 13 : lO-i^ij. 3. What is the present worth of £875 : 5 : 6, due at 5 months hence, at 4| per cent. ? Ans. £859 : 3 : 3f ylg. 4. How much ready money can I receive for a note of £75, due 16 months hence, at 5 per cent. ? Ans. £70 : 11 : 9 ,l764d. II. When P T R are given to find S. Rule. ptr-t-p=S. EXAMPLES. 5. If the present worth of a sum of money, due 9 months hence, allowing 5 per cent., be £344 : 11 : 6 3,168 qrs., what Avas the sum first due? Ans. £B57 : 10. 344,5783 X ,75 X ,05-f 344,5783=£357 : 10. 6. A person owing a certain sura, payable 7 months henoe, agiees with the eieditor to pay him down £267 : 13 : lO/y^, al- lowing 6 per cent, for present payment, what is the debt ? Ans. £275 : 10. 7. A person receives £869 : 3 : 3|y|j for a sum of money BBBATE OR DIBGOUNTi 153 due 8 months hence, allowing the debtor 4i per cent, for present payment, what was the sum due? Ans £875 • 5 • 6 8. A person paid £70 : II : 9 ,1764d. for' a debt due *15 months hence, he bemg allowed 6 per cent, for the discount, how much was the debt ? ^^ £^g III. When S P T are given to find R. s — p Rule.- =R. tp examples: 9. At what rate per cent, will £367 : 10, payable 7 months hence, prpduce £344 : 11 : 6 3,168 qi«s. for present payment? 3575,-344,5783 ■ — -=,05=5 per cent. 344,5783 X ,75 10. At what rate per cent, will £275 : 10, payable 7 months hence, produce £267 : 13 : lO^jV for the present payment? 11 Ax I- ^ . ^'^*" ^ P^^ cent. 11. At what rate per cent, will £875 : 5 : 6, payable 5 months lience, produce the present payment of £859 : 3 : 3f -a_ ? io A 4. 1 X ■^^*- "^i percent. 12. At what rate per cent, will £75, payable 15 months hence, produce the present payment of £70 : 11 : 9 ,l764d. ? Ans. 5 per cent. rV. When S P R are given to find T. s— p Rule.- =T. rp EXAMPLES. 13. The present worth of ^£357 : 10, due at a certain time to come, IS £344 : 11 : 6 3,168 qrs. at 6 per cent., in what time should the sum have been paid without any rebate ? Ans. 9 months, 357,5—344,6783 =,75=9 months. 344,5783 X ,06 14. The present worth of £275 : 10, due at a certain time to 154 EaUATION OF PAYMENTS. I come, is £267 : 13 : 10^^' a^ the sum have been paid without 15. A person receives £859 Uu' a certain time to come, de^.'-e to knew in what time the ec' without any rebate? • 16. I have received £70 : 1 allowing the person 5 per cent, know when the debt would have 5 per cent., in what time should any rebate ? Ans. 1 months. : 3 : 3| ,0184, for £875 : 5 : 6, allowing 4| per cent, discount, I debt should have been discharg- ers. 5 months. 1 : 9,l764d. for a debt of £75, for prompt payment, I desire to been payable without the rebate ? Ans. 15 months. EQUATION OF PAYMENTS. To find the equated time for the payment of a sum of money due at several times. Rule. Find the present worth of each pay- ment for its respective time ; thus, Add all the present worths together, then, s tr+1 s — p=D. d and =E pr EXAMPLES. 1. D owes E £200, whereof £40 is to be paid at three months, £60 at six months, and £100 at nine months; at what time may the whole debt be paid together, rebate being made at 5 per cent. ? Ans. 6 months, 26 days. 40 60 100 =39,5061 =58,5365 =96,3855 1,025 1,0375 1,0125 then 200— 39,5061 + 58,5365 + 96,3855=5,5719 6,6719 — =,57315=6 months, 26 days. 194,4281 X ,05. 2. D owes E £800,' whereof £200 is to bo paid in 3 months, £200 at 4 months, and £400 at 6 months ; but they, agieeiug W make but one payment of the whole, at the rate of 5 per cent. rebate, the true equated time is demanded ? Ans. 4 months, 22 days. COMPOUND INTEREST. 155 3. E owes F £1200, which is to be paid as follows : £200 down, £500 at the end of 10 months, and the rest at the end of 20 months ; but they, agreeing to have one payment of the whole, rebate at 3 per cent., the true equated time is demanded ? Ans. 1 year, 11 days. COMPOUND INTEREST. The letters made use of in Compound Interest, are, A the Amount. P the Principal. T the Time. R the Amount of £l for 1 year at any given rate ; which is thus found : As 100 : 105 : : 1 : 1,05. As 100 : 105,5 : : 1 : 1,055. A Table of the amount of £l for one year. RATES AMOUNTS RATES AMOUNTS RATES AMOUNTS PKtt CENT. OF £1. PKR CENT. OF £1. PER CENT. OF £1 3 1,03 • 5i 1,035 8 1,08 H 1,035 6 1,00 8i 1,085 4 1,04 6^ 1,065 9 1,09 4i 1,045 7 1,07 9i 1,095 5 1,05 7-i 1,075 10 1,1 Table showing the amount of £l for any. number of years under 31, at 5 and 6 per cent, per annum. YEARS. 5 RATES. 6 YEARS. 5 RATES. 6 1 1,05000 1,06000 16 2,18287 2,54035 2 1,10250 1,12360 17 2,29201 2,69277 3 1,15762 1,19101 18 2,40662 2,85434 4 1,21550 1,26247 19 2,52695 3,02560 5 1,27628 1,33822 20 2,65329 3,20713 6 1,34009 1,41852 21 2,78596 3,39956 7 1,40710 1,50363 22 2,92526 3,60353 8 1,47745 1,59385 23 3,07152 3,81975 9 1,55132 1,68948 24 3,22510 4,04893 10 1,62889 1,79084 25 3,38635 4,29187 11 1-71034 1,89829 26 12 1,79585 2,01219 27 3,73345 4,82234 13 1,885Q5 2,13292 28 3,92013 5,1 n 68 14 1,97993 2,26090 29 4,11613 5,41838 15 2,07892 2,39655 30 4,32194 5,74349 156 COMPOUND INTERS8T. Note. The preceding table is thus made—As 100 : 106 : : 1 : 1,05, for the first year; then, As 100 : 105 : : 1,05 ; 1,1026, »^ cond year, &c. I. When P T R are given to find A. Rule. pXrt=A. EXAMPLES. 1. What will £225 amount to in 3 years' time, at 5 per cent per annum ? r • Ans. 1,05X1,05X1,05=1,157625, then 1,157625X225= £260 : 9 : 3 3 qrs. 2. What will £200 atoount to in 4 years, at 6 per cent, per «»"""» ? Ans. £243 2,026s 3. What will £450 amount to in 5 years, at 4 per cent, per ^n^"°i ? Ans. £547 : 9 : 10 2,0538368 qrs. 4. What will £500 amount to in 4 years, at 5^ per cent, per ^"^^"^ • Ans. £619:8:2 3,8323 qref n. When A R T are given to find P. a Rule. =P rt EXAMPLES. 6. What principal, being put to interest, will amount to £260 : 9 : 3 3 qrs. in 3 years, at 6 per cent, per annum ? 260,465626 1,05X1,05X1,05=1,157625 =£225. 1,157625 6. What principal, being put to interest, will amount to £243 2,025s. in 4 j^ears, at 5 per cent, per annnm ? Ans. £200. '7. What principa- will amount to £547 : 9 : 10 2,0538368 qrs. in 5 years, at 4 per cent, per annum ? Ans. £450? 8 What principal will amount to £619 : 8 : 2 3,8323 qrs. in 4 years, at 5^ per cent, per annum ? Ans. £50Q. . TT ix^Jlx A XX A aiu giVuu lo una xi. a which being extracted by the rule of extrac- RuLE.— =rt tion, (the time given to the question showing p the power) will give R. COMPOUND IlfTBSfiST. 157 EXAMPLES. 9. At what rate per cent, will £226 amount to £2G0 : 9 : 3 3 qrs. in 3 years ? ^W5. 5 per cent. ' 260,465625 '- ^=1,167625, the cube root of which 225 (it being the 3d power) =1,05=5 per cent. 10. At what rate per cent, will £200 amount to £243 • 2 026' '^VT^. . ^n.. 5 pe; cent. * 11. At what rate per cent, will £450 amount to £547 • 9 • 10 2,0538368 qrs. in 5 years ? Ans. 4 per cent. 1^. At what rate per cent, will £500 amount to £619 : 8 • 2 3,8323 qrs. in 4 years ? Ans. 5j per cent* IV. When P A R are given to find T. a which being continually divided by R till no- KuLE.--=rt thing remains, the number of those divisions p will be equal to T. EXAMPLES. 13. In what time will £225 amount to £260 : 9 : 3 3 ars at 6 per cent. ? ^ 260,465625 1,157625 1,1025 1,05 • = 1,157625 = 1,1025 =1,05— _i *f . . ^'^^ 1.05 1,05 -1, the number of divisions being three times sought. 14 In what time will £200 amount to £243 2,025s. at 5 ner ""Vt wr -u. Ans. 4 ye J 10. in what time will £450 amount to £547 : 9 : 10 2 0538368 qrs at 4 per cent. ? ^n.. 5 years, lb. In what time will £500 amount to £619 : 8 : 2 3 8323 qrs. at 5^ per cent. ? , .^.4 y^^^^ ANNUITIES, OR PENSIONS, IN ARREARS. J^OTE. U represents the annuity, pension, or yearly rent: A U i a£ before. ^ ^ > 158 COMPOUND INTEREST. A Table showing the amount of £1 annually, for any number of years under 31, at 5 and 6 per cent, per annum. YEARS. 5 RATES. 6 YEARS. 5 RATE3. 6 1 1,00000 1,00000 16 23,65749 35,67252 2 2,05000 2,06000 17 25,84036 28,21288 3 3,15250 3,18360 18 28,18238 30,90565 4 4,31012 4,37461 19 30,53900 33,75999 5 5,52563 5,63709 20 33,06595 36,78559 6 6,80191 6,97532 21 35,71925 39,99272 7 8,14200 8,39383 22 33,50521 43,33229 8 9,54910 9,89746 23 41,43047 46,99582 9 11,02656 11,49131 24 44,50199 50,81557 10 12,57789 13,18079 25 47,72709 54,86451 11 14,20678 14,97164 26 51,11345 59,15638 12 15,91712 16,86994 27 54,66912 63,70576 13 17,71298 18,88213 28 58,40258 68,52811 14 19,59868 21,01506 29 62,32271 73,63979 15 21,57856 23,27597 30 66,43884 79,05818 Note. The above table is made thus :— take the first year's amount, which is £1, multiply it by 1,05 + 1 =2,05 = second year's amount, which also multiply by l,05+l = 2,1525=third year's amount. T. When U T R are given to find A. ur" — u RULE.- -=A, or by the table thus : Multiply the amount of £1 for the number of years, and at the rate per cent, given in the question, by the annuity, pension, &c. and it will give the answer. EXAMPLES. 1*7. What will an annuity of £50 per annum, payable yearly, amount to in 4 years, at 5 per cent. ? Ans. 1,05 X 1,05 X 1,05 X 1,05 X 50=60,77531250 60,7753125—50 Ijien =£215 : 10 : 1 2 qrs.; or, 1,05—1 t>y the table thus, 4,31012X50=£215 : 10 : 1 1,76 qrs. 18. What will a pension of £45 per annum, payable yearly, unount to in 5 years, at 5 per cent. ? Ans. £248 : 13 : 3,27 qis. • COMPOUND INTEREST. 150 19. If a salary of £40 per annum, to bo paid yearly, be for- borne 6 years, at 6 per cent., what is the amount? Ans. £279 : : 3,05796008d. 20. If an annuity of £15 per annum, payable yearly, be omit- ted to be paid for 10 years, at 6 per cent., what is the amount? Ans. £988 : 11 : 2,222d. II. When A R T are given to find U. ar — a Rule. =U. rt— 1 EXAMPLES. 21. What annuity, being forborne 4 years, will amount to ^£215 : 10 : 1 2 qrs. at 5 per cent. ? • 215,50625X1,05—215,50625 Ans. : — =^£50. . 1,05X1,05X1,05X1,05—1 22. Wliat pension, being forborne 5 years, will amount to .£248 : 13 : 3,27 qrs. at 5 per cent. ? Ans. 45. 23. What salary, being omitted to be paid 6 years, will amount to £279 : : 3,05796096d. at 6 per cent.? Ans. £40. 24. If the payment of an annuity, being forborne 10 years, amount to £988 : 11 : 2,22d. at 6 per cent., what is the annuity? Ans £i*l^ ni. When U A R are given to find T. ar-j-u — a which being continually divided by R till Rule. =rt nothing remains, the number of those u , divisions will be equal to T. EXAMPLES. 25. In what time will £50 per annum amount to £216 : 10 : 1 2 qrs. at 5 per cent, for non-payment ? Ans. 215,50625X1,05+50 215,50625 =1,21550625. 50 which being continually divided by R, the number of the divi- 01/\Wlo ^ir» I I 1-V/-V .1.1 ■ A ■«▼/>#>«<>• ojivixp TYiix uxi — -x ycaio. 26. In what time will £45 per annum amount to £248 : 13 327 qrs. allowing 5 per cent, for forbearance of payment ? Ans. 5 years, 02 ^ 160 COMPOUND irtTBRBST. 37. In what time will £40 per annum amount to £279 : : 3,05796096(1. at 6 per cent. ? ^ns. 6 years. 28. In what time will £75 per annum amount to £988 : 11 : 2,22d. allowing 6 i)er cent, lor forbearance of payment ? Ans. 10 years. . PRESENT WORTH OF ANNUITIES, PENSIONS, &,c. A Table showing the present worth of £l annuity/, for any num- ber of years under 31, rebate at 5 and 6 per cent. YEARS. 5 RATES, YEARS. 5 RATES. 1 0,93238 0,94339 10 10,83777 10,10589 2 1,85941 1,83339 17 11,^^7400 10,47726 3 2,72324 2,07301 18 11,08958 10,82700 4 3,54595 3,40510 19 12,03532 11,15811 5 4,32947 4,21230 20 12,40221 11,40992 6 5,07309 4,91732 21 12,82115 11,70407 7 5,78037 5,58238 22 13,10300 12,04158 8 0,40321 0,20979 23 13,48857 12,30338 9 , 7,10782 0,80109 24 13,79804 12,55030 10 7,72173 7,30008 25 14,09394 12,78330 11 8,30041 7,88087 26 14,37518 13,00317 12 8,80325 8,33384 27 14,04303 13,210.53 13 9,39357 8,85908 28 14,89812 13,40010 14 9,89804 9,29498 29 15,14107 13, .59072 15 10,37965 9,71225 30 15,37245 13,70483 Note. The above table is thus made : — divide £1 by 1,05=5 ,95238, the present worth of the first year, which — 1,05=90753, addexi to the first year's present worth= 1,85941, the second year's present worth ; then, 90703—1,05, and the quotient added to 185941 = 2,72327, third year's present worth. I. When U T R are given to find P. u u- RULE.- :P. r— 1 Multiply the present worth of £1 annuity for the time and rate per cent, given by the annuity, pension, &c., it will give the m- BWQr. COMPOUND INTUBBBT. 161 79 : : r^eavft. 8 : 11 : ^ears.' J c. ly num- 589 726 7G0 811 992 407 ir)8 1338 i03G 1336 1317 053 (GIC) 1072 i4S3 1,05 = = 90753, I second it added and rate ! the m- EXAMPLES. 29. What is the present worth of an annuity of £30 per an- num, to continue 7 years, at 6 per cent. ? Ans. £167 : 9 : 5 ,184d. 80 -=19,9517 1,60363 = 167,4716. 30—19,9517 = 10,0483 10,0483 By the table 5,58238X30=167,4714. Ii06— 1 30. What is the present worth of a pension of £40 per annum to continue 8 years, at 5 per cent. ? ' . Ans. £258 : 10 : 3,264 qrs. 31. What IS the present worth of a salary of 35, to continue I 7 years at 6 per cent. ? Ans. £195 : 7 : 7 3,968 qra. j 32. What IS the yearly rent of £50, to continue 5 years, worth m ready money, at 5 per cent. ? Ans. £216 : 9 : 5 2,56 qrs. ir. When P T R are given to find U. prtXr— pr* Rule.— — = U. r^— 1 EXAMPLES. 33. If an annuity be purchased for £l67 : 9 : 5 184d. to b« }ntinued 7 years, at 6 per cent, what is the annuity? ^^*' 107,4716X1,50363X1,06—167,4716X1,50363 1,50363—1 =£30. 34. If the present payment of £258 : 10 : 6 3,264 qrs. be nade for a salary of 8 yeai-s to come, at 5 per cent., what is the '^^^•■y- Ans £40 35 If the present payment of £195 : 7 : 7 3,968 qrs. be 're- luired tor a pension for 7 years to come, at 6 per cent., what is Po^ip ? ^\^ P^^^^"* ^^^'^^ ^^ ^^ annuity 5 years to come, 'be t^iO : 9 : 6 2,56 qrs. at 5 per cent., what is the annuity ? Ans. £50. 03 162 COMPOUND INTEREST. in. When U P R are given to find T. u which being continually divided by R till fjuLE. =r* nothing remains, the number of those di- p-f-u — ^pr visions will bo equal to T. EXAMPLES. 37. How long may a lease of £30 yearly rent be had for £167 : 9 : 5 ,184d. allowing 6 per cent, to the purchaser? which being continually 1 f^ncjf Q divided, the number of l,DUdt)d those divisions will be= 167,4716+30-177,5198 ^ ^^^ ^^^^^^ 30 38. If £258 : 10 : 6 3,264 qrs. i^ ^.aid down for a lease of £40 per annum, at 5 per cent, how long is the lease purchased for ? Ans. 8 years. 39. If a house is l6t upon lease for £35 per annum, and the lessee makes present payment of £195 : 7 : 8, he being allowed 6 per cent., I demand how long the lease is purchased for? Ans. 7 years. 40. For what time is a lease of £50 per annum, purchased when present payment is made of £216 : 9 : 5 2,56 qrs. at 5 per cent. ? ■ ■^^«- ^ y^^^"^^- ANNUITIES, LEASES, &c. TAKEN IN REVERSION. To find the present worth of annuities^ leases, <&c. taken in reversion. ^ Rule. Find the present worth of the annni- u ty, &c. at the given rate and for the time of its u • continuance : thus, ^ -=?. r— 1 2. Change P into A, and find what principal being put to inteiest will amount to V at the same rate, and for the time to come before the annuity commences, which will be the present worth of the annuity, ifct ^^ fl».-v %^«i1fi. plicand by the hundreds in the multipher ; and so proceed till you the multiphcand all through, by every figure of the multiplier. * DUODECIMALS. 176 EXAMPLES. Multiply 35234 by #62424 Common way. 35234 62424 Product, 1847107216 140936 70468 140936 70468 176170 1847107216 EXPLANATIONS. Firet, 4X4=16 that is 6 and carry one. Secondly, 3x4+ TV ll''oli ^^f '' '""^^^' ^^ ^^-''^ ^1«^" 1 and carry I Ihirdly 2X4+3X2 + 4x4+2 carried = 32, that is 2 and^cat ry 3 Fourthly, 5 X 4+2 X 2 +3 X 4+ 4x 2+3 carried = 47 '^f7,Vr'^ Z'^ '• ™'^y^ 3 X 4 + 5 X 2 + 2 X 4+3X2 TsX 4^+rx 2+ s'i'V!?"" 'r^ ""^^ '' SixthbT3X2 r Q Ti o + ^ ^ ^+^ carried = 51, set down 1 and carrv Ll r"'^^' r'^i'it 5X2 + 2X5 + 5'carried=37, that's 7 and carry 3 Eighthly, 3X2+5x5+3 carried=34, et down tinlH T'V' , ^fi'^' ' ^ ' + ^ carried=18, which being mul tiphed by the last figure in the multiplier, set the whole down and the work is finished. ' 176 THE TUTOR'S ASSISTANT PART V. A COLLECTION OF QUESTIONS. 1. What is the value of 14 barrels of soap, at 4^d. per lb., each barrel containhig 254 lb. ? Ans. £66 : 13 : 6. 2. A and B trade togetlier ; A puts in £320 for 5 months, B • £460 for 3 months, and they gained £100 ; what must each man i-eceive ? Ans. A £53 : 13 : 9|if , and B £46 : 6 : 2^\\. . 3. How many yards of cloth, at 17s. 6d. per yard, can I have for 13 cwt. 2 qrs. of wool, at 14d. per lb. ? , - ^ , Ans. 100 yards, 3i qrs. 4. 1 I buy 1000 ells of Flemish linen for £90, at what may I sell It per ell in London, to gain £10 by the whole ? Ans. 3s. 4d. per ell. 5. A has 648 yards of cloth, at 14s. per yard, ready money, butm barter will have 16s.; B has wine at £42 per tun, ready money : the question is, how much wine must be given for the cloth, and what is the price of a tun of wine in barter ? Ans. £48 the tun, and 10 tun, 3 hhds. 12f gals, of wine must be given for the cloth. 6. A jeweller sold jewels to the value of £1200, for which he received in part 876 Fiench pistoles, at 16s. 6d. each; what sum remains unpaid ? Ans. k^ll : 6. y. An oilman bought 417 cwt. 1 qr. 15 lb., gross weight, of tram oil, tare 20 ^b. per 112 lb., how many neat gallons were there, allowing n\ ib. to a gallon. ? Ans. 5120 gallons. 8. If I buy a yard of cloth for 14s. 6d., and sell it for 16s. 9d., uu i gam per cent. Ans. 9. ?.ought 27 bags of ginger, each weighing ^ , .^t 1^ lb per bag, tret 4 lb. per 104 lb., what°do they at b^d. per lb. ? \ £15 : 10 : 4yVV Gfross 84-^ lb., tare come to Ans. £76 : 13 : 1-i^ rff* A COLLECTION OP QUESTIONS. 177 cost?' ^^ * ""^ ^"^ ''"''"^ "^'^ ^ ^^ ^ '^""^"°, what will I of H lb. 11 Ti?s ^/? ,1 ^ „ Ans. 17s. 6d. 11, If I of a gallon cost f of a pound, what will f of a tun cost ? 19 A „«„+! 1 ^^*- -£105. 1^. A gentleman spends one day with another, £] • 7 • IQk and at the year's end layeth up ^340, what is his yearly income J tirne^'n.ir If/^'t '^^^'^ ^ ^^"^ ^Cj'e^ch Iting^l'g. times H2 lb B has 39 cii^ks of tin, each 3S8 lb, how many ounces difference is there in the weight of these commodities ? ^ »? TA K + • 1 , ^*^- 212160 oz. 14 A captain and 160 sailors took a prize worth £1360 of which the captam had i for his share, and the rest was equally chvided among the sailors, what was each man's part ? ^ ^ 1 r; A . ^r: ^^^ ^^P^^^"" ^'^^ •^^^^' ^^^^d each sailor £6:16 15. At What rate per cent, will £956 amount to £1314 • 10 '" 16^ r h' fh o7^' ^^''''''' ; ^^^- '5 per cent. ' £13 Aw \ ''""i' """.f ^?'- ^''^^^^' ^''^ ^ 7 J^^^rses, worth £13 a piece, hovv much will make good the difference, n case they interchange their said drove of cattle ? Ans £4 • lo J\ \T- fT^ ^r'' ^}^^ ^^ ^' ^^^^^^ *^ three pers^ons, InA n ' V ^ "" ".^'^'^ unknown ; B twice as much as A and C as much as A and B ; what was the share of each ? IS £innn- f ^ ^--i /'^^- ^ ^^0, B £40, and 6 £60. IS. £1000 is to be divided among three men, in such a imn ner, that if A has £3, B shall hav^ ^5, and C ^8 itw mub rau3t each man have ? ' ia A . .^''^^^ •^^^'^ • ^^' ^ ^312 : 10, and C i^SOO, , 19. A piece of wainscot is 8 feet 6^ inches long, and 2 feet 95 inches broad, what is the superficial content ? ,„ .u u . ? ^^^ ¥ "' garrison, and have provisions for G months, but hearing of no relief at the end of 5 months, how many men must depart that the provisions may last so much the longer'^ 01 Ti 1 /• o , . ^^^^^* 2^S men. 21. The less of 2 numbers is 187, their difference 34, the square no ^Pf ^"^t ^^ ^■'^^""'':^ ^ ^W5. 1707920929. * oxen at f 11, cows at 40s., colts at £1 : 5, and hoo-s fit £l • if. number, how buy? lany of each sort did he Ans. 13 of each sort, and ^£8 ^vJ. What number added to 11| will produce 36^ over. 31 3 I A7is.2i^\l J^ 178 A COLLECTION OF QUESTIONS. 24. What number multiplied by ^ will produce IIW' Ans. 26f f 25. What is the value of 1*79 liogsheads of tobacco, each weiffh- is the :£2: _6_8_ 10 0' Ans. 1^. ing 13 cwt. at £'I: 7 : I per cvvt. ? Ans. £5478 : 2 : 11. 20. My factor sends me word he has bought goods to the va- lue of £'500 : 13 : 6, upon my account, what will his commission come to at 3| per cent. ? Ans. £17:10:52 qi-s. ' " 27. If ^ of G bo three, what will I of 20 be ? as. What is the decimal of 3 qi-s. 14 lb. of a cwt. ? ^ Ans. ,875 29. How many lb. of sugar at 4|d. per lb. must be given in barter for 60 gross of inkle at 8s. 8d. per gross ? Ans. 1386f lb. 30. If I buy yarn for 9d, the lb. and sell it again for 13|d. per lb., what is the gain per cent. ? Mns. £50. 3?. A t^^baci'onist would mix 20 lb. of tobacco at 9d. per lb. with 60 lb. at 12d. per lb., 40 lb. at 18d. per lb., and with 12 lb. at 2s. pel !b., what is a pound of this mixture worth i Ans. Is. 2id. /y. 32. What .s 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 noble and a mark just 15 yards did buy ; how many ells of the same cloth for £qO had I ? Ans. 600 ells. 34. A broker bough': for his principal, in the year 1720, ^2400 capital stock in the South-Sea, at £650 per cent., and sold it again when it was worth but £130 per cent.; how much was lost in the whole ? Ans. ^£2080. 35. C hath candles at 6s. per dozen, ready money, but in bar- ter will have 6s. 6d. per dozen ; D hath cotton at 9d. per lb. ready money. I demand wh; '. price the cotton must be at in barter ; also, how much cotton must be bartered for 100 doz. of candles? Ans, The cotton at 9cl. 3 qrs. per lb., and 7 cwt. qrs. 16 lb. of cotton must be giv^-' for 100 doz. candles.* 36. If a clerk's salary be £12 ay. what is that per day ? '-Aon a Aj» 37. B. hath an estate of £53 per annum, and payetb 5s. lOd. to the subsidy, what must pay whose estate ia^vorth £100 per annum? ^we. lis. Od. yV \ A COLLECTION OF QUESTIONS. 179 38. If I buy 100 yards of riband at 3 yards for a shilling, and 100 more at 2 yards for a shilling, and sell it at the rate of 5 yards for 2 shiUings, whethgA I gain or lose, and how much ? ^^ Ans. Lose 3s. 4d. 39. What number is that, from which if you take f , the re- mainder Avill be g- ? jifig 3£^ 40. A farmer is willing to make a mixture of rye at 4s. a bushel, barley at 3s., and oats at 2s. ; how much must he take of each to' sell it at 2s. 6d. the bushel ? Ans. 6 of rye, 6 of barley, and 24 of oats. , 41. If f of a ship be worth £3740, what is the worth of the whole ? ^1^5. £9973 ; 6 : 8. 42. Bought a cask of wine for £62 : 8, how many gallons were in the same, when a gallon was valued at 5s. 4d, ? Ans. 234. ^ 43. A m#ry young fellow in a shor^ time got the better of i of his fortune ; by advice of his friends he gave £2200 for an ex- empt's place in the guards ; his profusion continued till he had no jnore then 880 guineas left, which he found, by computation, was 5% P'^vi't of his money after the commission was bought ; pray what Avas his fortune at first ? J ws. £10,450. 44. Four men have a sum of money to be divided anion o-st them in such a manner, that the first shall have ^ of it, the second I, the third ^, and the fourth the r^nainder, which is £28, what is the sum? Ans. £112. 45. Wi)at is the amount of £1000 for 5| years, at 4f per cent, simple interest? Ans. £1261 : 5. 46.^ Sold goods amounting to the value of £700 at two 4 months, what is the present worth, at 5 per cent, simple interest ? yi?is. £082 : 19 : 5i yVVr- 47. A room 30 feet long, and 18 feet wide, is to be covered with painted cloth, how many yards of ^ wide will cover it ? Ans. 80 yards, 48. Betty told her brother George, that though her fortune, on her marriage, took £19,312 out of her ftmiily, it was but -f of two ye&m* rent, Heaver be praised ! of his yearly income ; pray what was that? Ans.'xWfif"^, :6:8a year. 49. A gentleman having 50s. to pay ^imr his labourers for a day's work, would give to every boy 6d., to every woman 8d., and to every man 16d, ; the number i)f boys, wonien, and men, was the same. I demand the number of each ? Ans. 20 of each. 180 A COLLECTION OF QUESTIONS. [\ 60. A stone that measures 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches deep, how many solid feet doth it con- *'*";^i* ^^Tu .1 . , , •^^'- 41 ^^^^ 3 inches. oi. What does the whole pay of a man-of-war's crew, of 640 sailors, amount to for 32 months' service, each man's pay being 22s. 6d. per month ? Ans. £23,040. 52. A traveller would change 500 French crowns, at 4s. 6d. per crown, into sterling money, but he must pay a halfpenny per crown for change ; how much must he receive ? Ans. £111 : 9 : 2. 53. B and C traded together, and gained £100 ; B put in £640 C put m so much that he might receive £60 of the gain. I de- mand how much C put in ? • Ans. £960. 54. Of what principal sum did £20 interest arise in one year at the rate of 5 per cent, per annum ? Ans, £400. ' 55. In 672 Spanish guilders of 2s. each, how many French pis- toles, at 17s. 6d. per piece? * Ans. 762.1. 56. From 7 chv^cses, each weighing 1 cwt. 2 qre. 5 lb.,'' how many allowances for seamen may be cut, each weighing 5 oz 7 '^''fi^ V. • ^^*- 3563ii o7. If 48 taken from 120 leaves 72, and 72 taken from 91 loaves 19, and 7 taken from thence leaves 12, what number is that, out of which when you have taken 48, 72, 19, and 7, leaves ' Ans. 158 58. A farmer ignorant of nlfmbere, ordered £500 to be divided among his five sons, thus :r-Give A, says he, i, B ^, C i, D i, and E | part ; divide this equitably among them, according to their father's intention. Ans. A £l52ff |., B £ll4iif, C £9Uf }, I> £76111 E £65i|f . 59 When first the marriage knot was tied Between my wife and me, My age did hers as far exceed, As three times three does three ; But when ten years, and half ten years, We man and wife had been, Her age came then as near to mine, As eight is to sixteen, ues, ¥v flat wa» eaca oi our ages when we were married ? Ans. 45 years the man, 15 the woman. A COLtBCTIOy OP QUESTIONS. 181 SUPPLEMENTAL QUESTIONS. 2. The less of two numbers is 17 nnri oA i, • ^'**- 462s. tracted; what is that number" "^ *<* ™'"' ^"1'- 4. Of three numbers, the fint is 2Ti <• <. . 7 • ;^'"'- ^^^• third is as much as the otW 1 ' u '"'°''"'' '' ^1"' ^d the three numbera! " """' "•>"' '^ «><= s"m of the whL'sufr 5 °' ''^^ ""'""^■^ '^ ^« »<• '"e diff-n'ie^s' 37 ; :^^t^;^,fhra^"^:,--X^^^^^^^^ 9- After having added successivelv 11 20 qo i"!"/^- , .8 b; SLTo^'ttf^ t;"^ r.'^r/i' ''."» ^-- -^' 'f ! "ill be equal ; what is the I^el^e'h t " "'" ""''' "'^'^ "S' is the greater t * "'^''' ""= ■'emainder is 244 ; what Iffreater ? ^ "^ *^^ ""^^^"^ ^^'^'^ total is 145; what is iK- Ans. The smallest 1,046, the mean 1,120. ■4i ■mn ' ^82 A COLLECTION OF UUESTKTHg. h •■^^j ;f/L^'' dividing a certain sum between 26 persons each re. ceived 257s. ; what was the sura ? Ans. 6 682s 16. From a certain sum 152 persons took $17 each* and there remamed $13; what was tlie sura ? Jns. $2597 17. What is the niimber that being augn- anted by 56 mid 'di- vided by 55,the quotient will be 2,854 ? Ans. 156 914 18. What is the number that being divided by 27 gives a quotient equal to the product of 1,091 by 3 ? Ans 88 371 19 By selling 120 yards of cloth for 3,600s. tliere was fls profit per yard ; what was the buying price ? Ans. 3,000s. 20 I bought 150 yards of cloth for 3,7508. and sold them for oi Uu '"^ ' "^^'^^ "^'^ ^ S^"' ^3^ ^he bargain ? Am. 60Qs. o^nl'.nu^*,'"^.'^^''^? ^^ obtained, if, after having multipUed Jo0,540 by 10 this product should be repeated 2,458 times? on A 1 .^^ e>. IMAGE EVALUATION TEST TARGET (MT-3) W^/ J-/ & 4s 10 I.I 2.5 2.0 1.8 1.25 ■ 1.4 1.6 — 6" <^ /A A "% '/ Photographic Sciences Corporation d. M 4 •>S L :\ \ C^ '" \. ^^^' i' 136. By what number must you divide a sum to render it once and a half greater ? j^^g a ^ 137. Of three fractions the second is double the first, the' third IS f , and their sum is ^ ; what are the two first fractions ? /ins 1_ a.Tifl _i_ 138. To double a number you must multiply its V by its ninth prfrt ; what is it ? ^^^ 27. , ]^^:J^^ 3 .of one number is equal to the a of another, and their difference is 6 ; determine those two numbers ? Ans. 18 and 12 140. The sum of two numbers is 4, and the quotient of the less by the grc^ater is 4 ; what are they ? Ans. 2i and la. 141. Divide 60 into two such parts that the I of one may be equal to } of the other ? Ans. 32 and 28. 142. The father and son together are 70 years old, the age of the father multiplied by 3 is equal to the son's age multiplied by 1i ; what are the ages ? Ans. 20 and 50 143. A greyhound starts after a hare that is 82 leaps ahead- while the greyhound makes 9 leaps the hare makes 13, but 3 leaps of the greyhound are equal to 5 leaps of the hare ; how many leaps must the greyhound make to catch the hare ? 144. A watch marks 12, and both hands are together ; required on what part of the dial they will next meet ? , _ _ . . -^^5. 1 o'clock 5/p minutes. 145. It is just six clock ; when will the hands meet ? . ^W5. 3 2y»y minutes past 6. 146. It IS just twelve ; required how many tidies the hands shall mffet from twelve till midnight, and at what o'clock each time? Ans. 147. The f and J of a number make 39 ; what is that number f ■ ._ . • ^ns. 60. 148. A man can do a piece of work in ^ day, his wife could do the same in 1, and their son in ij day ; what time would the three together take to do it ? Ans. ■*- day. 149. A spring would fill a basin in 3 hours, another would fill it in fi V,^««, . if f^e t^Q j.^jj together, required in what time thov would fill it? Ans. 11 hour. 150. A pump would empty a ditch in 8^ days, another would empty it in 7f ; if both work together, in what time will the ditch 1 be dry! Ans. 3|f f dsjm. w> A COLLECTION OF QUESTIONS. 161. A set of workmen can build a wall 45 yards long in days by working 9 hourtj per day ; another set would build it in 8 days by working 7 hours per day : if both work together and work 8 hours a day, in how many days will the wall be built ? Am. 21 hours +tV7> ^^ ^ ^^J^ ^ hours + //,. 152 Two bands of reapers can reap a field : the first in 4 days, and the second in 6 days ; if J the first and I of the second be employed, in what time will the field be reaped ? Ans. 5^^ days. 153. A cock gives 8 gallons of water in 1 minutes, another 6 gallons in 6 minutes ; how many gallons do both give in one minute ? Ans. l^i gallons. 164. A person questioned about his age answei-ed : the f and the ]• of my age plus 7 years, just make my age 3 years hence ; what is his age ? Ans. 30 years. 155. The double of a sum augmented i, } and } of the same jwim and $5 more make $76 ; what is that sum ? Ans. #24. 156. A water-spout would fill a basin in } hour; another would do the same in | hour ; and a third in | hour ; in what tima would the three running together fill it ? Ans. ^V Jio^i* OJ* 3 minutes. 157. I can do a piece of work in 4 days ; my brother can do the same in 5 days ; if we both work together, in v/hat time will it be done ? ^ns. 2| days. 158. A set of workmen can sink a well in 9 days, another can do it in 10 days, and a third in 12 days ; now if I employ \ of the first band, | of the second, and J of the third, in what time will the well be dug out ? Ans. 9|^ days. 159. A basin has three cocks : two destined to fill it, and a third to empty it The'first cock would fill the basin alone in 4 hottts, the second in three hours, and the third would empty it in 6 hours ; now if the three be opened together, in what time will it be filled? Ans. 2 1 hours. 160. A basin has three cocks : two to fill, and one to empty it. The first would fill it alone in 4 hours, the second in 5 hours, and the thii-d would empty it in 2 hours. The basin being already full, the three cocks are opened together ; in what time will it bo ompty ? ^w«. 20 hours. 161. A workman can do a piece of work in | day, another can do it in f day ; if the two work together, in what time will it bo done ? ^ws. \{ day. lar. A mother divides a certain number of sugar-plums between A COLLECTION OP QUESTIONS. 191 her three daughters ; the youngest receives the f of the whole, the second l, and the third 12 for her part; how many were there, and what was the part of each ? - _„ -,, ^ns. Total 45 ; 12, 15, 18, respectively. 163. Ihe f and the i- of what I have in my purse, with $10 more, would make $0 more than I have ; what does the purse contam ? ^^^ ^§q^ 164. The triple of a sum added to the i and \ of the same su'ra, and $6,000 more, would make $22,200 ; what is that sum ? IflR rp, ,..r ^ws. 14,800. 169. liie difference between the | and the f of a number is 12 ; what is that number ? ^^^^ 216. 166. The total of the f and the f of a number, diminished the f of the same number gives 14 ; what is the number ? , -H m , . ^**^- 24. 157. Iwo cocks running togethijr would fill a basin in 2 hours • one alone would fiM It in 5 : in what time would the other SIl it if it were to run alone / Ans. 31 hours. 168. I spent the | of what I had in my purse, and if I add $44 to what remains, the sum it contained first will be auo-mented i • what did it contain ? ^^^.^ $48 169. One cock runs 11 gallons in 8 minutes, another runs 7 gallons m 5 minutes ; which runs the most ? Ans. The second, by ^V gal. per min. 170. A basm receives 45f gallons of water per hour by a cock and leaks by a hole 37f gallons ; how many gallons does it retain P^^,;^^"^'; . A71S. 7fi gallons. 171. A certain person not recollecting what he paid for an arti- cle only remembers that there were $14 difference between the J and the | of the price ; what is it ? ' Ans. $40. _ 172. The f of a number diminished the | of the same number gives 18 for rem. ; what is that number i Ans. 70. 173. The sum of the f and the j\ of a number less one-half of the same number, gives 24 ; what is that number ? Ans. 40. 174. Two workmen can do a piece of work in 3 hours, one alone can do it in 7 hours ; in what time will the other do it alone ? . ^«*. 5:J: hours. 175. Ihree cocks running together would fill a basin in 4 hours; one of them would fill it alone in 10 hours, another would fill it in 12 hours ; what time would the third running alone take to fill .ifVr ^"^' 15 hours. 176. Ihe quarter of a field is sown with wheat, the ^ with barley 1 192 A COLLECTION OP QUESTIONS. and the remainder with oats. The portion sown with barley con- tains 10 acres more than that sown with wheat; required the ex- tent of the whole field and that of each part 1 Ans. Whole extent 56 acres; 14 wheat, 24 barley, 18 oats. 1*77. I have already sold the | of a basket of eggs, and if I add 39 eggs to what remains, the primitive value of the basket will be augmented one-half, how many eggs were there in the basket ? Ans. 30. 178. A steam-loom weaves 5 yards of cloth in 3 houre, another 1 2 yds. in 7 hours ; which has the greater power ? ^ Ans. The latter weaves ^ yd. per hour more than the former. 179. A ribbon was cut into 5 parts of f yd. each ; what was its length? Ans. 1} yd. 180. A tradesman can do a piece of work in 5| days ; in what time will he do the | of the work ? Ans. 4|f days. 181. A ship sails at the rate of 16^ miles an hour; how many miles will she sail in 3-^ hours ? Ans. 63y\ miles. 182. A weaver weaves 7 yards of linen in 8 hours; how many yards will he weave in 4| hours ? Ans. 4i|- yds. 183. A man weaves 7 yards of linen in 8 hours; what time will he take to weave 4f yds. ? Ans. 5if hours. 184. If 5 gallons of wine be mixed with 7 gallons of water ; re- quired what quantity of water in | gallon of the mixture 'i Ans. if gallon of wine f |- of water. 185. If the f of the f of a number make 120, what is it ? Ans. 162. 186. A person being asked the time of day answered ; it is the f off of I of 24 hours ; what o'clock was it ? Ans. 10 o'clock. 187. Divide a succession between three heirs in such sort that the first may have the ^ of the whole, and the second the ^ of the remainder ; what is the part of each ? Ans. First ^, second /y, and the third ■^. 188. A sum of money was employed in four successive pur- chases. For the first purchase the | of the sum was laid out ; for the second, ^ of the remainder ; foi; the third, the f of the second remainder ; and finally, for the fourth the last remainder, which was $5 ; required the total sum, and the amount of each pur- chase ? Ans. Total $50 ; first f , second, ^y, third -^^y, fourth yV* 189. A certain person leaves to his nephew a fortune of $80,000, and orders the ^ of f of the succession to be given to a servant, A COLLECTION OF aUESTIONS. 193 and to his nurse ]■ of | of i of the same succession ; what is the portion of each ? ' .if ,^^%^.,f * ^^ t^e l^^gth of a garden is 48 yards ; what is the length of It? ^n.. 90 yards. .u ; .1* ^'T '''^^^'' "l'"^^*^^ ^^^^^" themselves a sum of money that they had stolen ; the first takes the f of it, and the two othei^ take each half of what remamed ; what part fell to each robber ? Ans. First f of the sum, and the two others j\ each. 192. A stage performs a journey in four days. The first day it tj-avels the i of the whole route ; the second day it travels the i of the remainder; the third day it travels the ^ of the second re- mainder ; and lastly, the fourth day it completes the journey and goes 144 miles; required the length of the journey, and each day^s travelling ? . •' j Ans. Length 540 miles ; first day 108 miles, and each of the other days 144 miles. 193. A tradesman can finish a piece of work in | of a day an- other can do the. same in a day : 1st. What time will they take to do It together ? 2d. What part of the work will be done by each ? 8d. What will be the gain of each, if the whole be worth 4s 7d ? Ans. 1st., r\ day ; 2d., the 1st. ^V of the work, the 2d.' the Jy ; 3d., the 1st. will have 2s. 6d., the 2d. 2s. Id. 194. A little boy playing marbles augmented his number x the hrst day ; on the next day he augmented his last number i ; finally he plays a third day, and augments his last number f , and finds lumselt master of 63 marbles ; how many had he when he beffan ^ ^^^y • Ans. 21, 195. What number multiplied by 3| will give 1 for product! Affis -A- 196. In 8 hours 5f yards are woven, in what time will Vyard be woven ? j^^^ j/ 197. A ship sails at the rate of 29f miles in 3f hours' whit is that per hour? An.. 8^^ miles per hour. lUb. vyiule a locomotive runs the whole route, a stage runs but the y3j-of it ; how many times does the locomotive go quicker than the stage ?_ _ An*. Slimes quicker. 199. A ship IS victualled for 12 days only; and must be kept on sea during IS days; to what must the daily rations of each man be reduced ? Jlw*. f of one. ■. 200. Four labourers work together and are paid equally. Now the first who worked the whole day received 4s. 2d. while the 194 A OOIXBCnON OP auESTioira. second received bnt 38. 4d., the third 2s. 6d., and the fourth Is. 8d. Required what part of the day the three last labourers worked ? Alls. The second f day, 3rd |, 4th | day. 2Q1. A spinning-wheel takes in 1|^ yard of thread every ti^rn it makes ; how many turns should it make to wind up 45f yards ? Ans. 40^J^ turns. 202. An omnibus takes ^ hour to reach its destination, it sta- tions ^ hour, and takes ^ hour to return to its starting place. Admitting that a trip is composed of going to and from the station; how many such trips will the omnibus perform from half-past seven in the morning till 10 o'clock at night? Ans. 14^ trips. 203. A traveller having missed the stage, it is already 29 miles a pront would have been just | of the uiss cost ; what did it cost ? Ans. $60. 21 i. The rail oars start from New York at noon and arrive at Philadelphia at 4 o'clock P. M. A stage started with the cars, and A COLLECTION OP UUBSTI0N8. m went but the I of the route : at what o'clock will the stage arrive ot ^^&^ • t' ^' -^^^- ^^^^ ^^y at 2 o'clock A. M. 212. While a horseman goes the f of a journey a footman can only travel the j^ ; how many times does the horseman go quicker I, n .no ^,*'™^ .^I^'" ^"^"^ water-spouts take to fill a basin that holds 508 gallons, if one runs 5^ gallons per minute, and the other 4A gallons. ^^^,. 48 minutes. J 14. A garrison has provisions for 9 days only, and must hold out tor 12 days : to what fraction must the daily rations of each m^ be reduced? ^r... to | of usual. -*lt). A watch is now regular, gets out of order and advances 5* mmutes in a day ; m how many days will it again mark the exact Ti« TK •. u , ^w«. 130if days. 216. Ihree writers equally clever can write 40 pages each per day ; now if the first write but 30 pages, the second 25, and tiio tnird 20 : required during what portion of the day each worked? oi'r T4. u , . . . , •^'*^- ^^^' ^' ^'^^' ^' 3rd i day. ^17. It a bucket takes 1^ minute to reach the bottom of a well and remains i minute below, then If minute ascending, how many buckets of water may be drawii in 250 minutes. o,o rp. . -^^- "^2 buckets. -il8. iwo couriers start at the same time at a distance of 92* miles apart, to meet emjh other and travel; the first 7 miles an hour the second 13^ miles an hour: in what time will they meet, and what will be the distance travelled by each ? Ans. In 4^ hours. The first travelled 31^ miles, the second 60^ miles. 219. A fox that makes 2^ leaps in a second is already 30f leaps a-head, when a dog. that makes 4^ leaps in a second starts after him. , In what time will the dog overtake the fox ? rton rr • ^^^* ^^A seconds. 220. Iwo couriers start at the same time from the same place for a neighbouring town distant 92^ miles, the fii-st travels 13^ miles an hour, the second 1 idem : how many hours will the first arrive before the second ? Jns. 6^* hours. 221. A courier goes 24 miles in 2 hours. Three "hours aft«r his departure another starts and goes 12 miles in 5 hours, in how ' *«it>«ijr iiuuio \\fiit illQ laiwr uveriake ine lonuer? ftor. fxT. , -^^*' 15 hours. 1^22. With 13t yards of old silk | wide I can line a vestment; now many yards | wide will do the same ? Ans. 12i| yard«. 106 A COLLECTION OP QUESTIONS. 323. There is a levy of $800 to be taken of three villages in proportion to their inhabitants, in the first there are 240, second 510, third 450 inhabitants: what share of the impost will each have to pay ? Ans. Ist. $160, 2nd. $340, 4th. $300. 224. An uncle on his death-bed bequeathes to his three nephews a fortune of $67,500 in proportion to their age. The first is 30, the second 25, the third 20 years of age : required what will fall to each? Ans. Ist. $27,000, 2nd. $22,600, 3rd. $18,000. 225. Divide the number. 1,028 into three parts so that they bo between themselves as the three fractions, f, |, | ? Ans. 320, 420, 288. 226. Divide $450 between three persons so that the second may have the f of the first, and the third the ^ of what the two first have together? Ans. $200, $150, $100. 227. '^Divide 100 shilHngs between two persons, and give the second the f of the first? Ans. 60, 40 shillings. 228. Divide $180 between two persons, and give the second i of the first's part more than the first ? Ans. $80, $100. 229. Divide | into two parts so that they be between themselves as 4 and 7 ? . ^»»*- H' ^' 230. The power of one machine is to that of another as 6 is to 7, while one makes 48 yards of work : how many will the other - j^j^ iQ ? ^^** ^^ yards. 231. Distribute $582 between 3 persons so that the part of the first be to the second as ^ is to |, and that the part of the second be to that of the third as f is to ^J ? Ans. $168, $252, $162. 232. What is the superficies of a rectangular garden, bemg 40 yards long by 30 yards in breadth ? Ans. 1,200 yards. 233. What is the area of a meadow in the form a triangle of 60 yards of base and 48 in height? . Ans. 1,440 yards. 234. What is the area of a yard forming a trapezium one of whose sides is 34 yards and the other 56, its height being 2.5 yaj.(jg^ u4w*. 1,125 yards. 235. What is the area of a rhombus, whose base is 44yV and height 38| yards? Ans. I,7l6|f yards. 236. What is the superficies of a pillar 17 yards high and 7 yards in circumference? •^'**' l^^ yards. 237.. The circumference of a cone iV 12 yards, and the distance irom ine summji lo mv wjwc w w jaiw3_, n^^v tt--v,.-., 2- s of it cost at 3 shillings the square yard ? Ans, 108 shillings. 197 A Table for finding the Interest of any sum of Moneu for «u- numher of months, .eeks, or lays, It any LuT^c^l "*^ JQ *. d. Oi u 2 2i 3i 4 4i 54 6 6i 1 14 1 7| 2 24 2 9 3 3i 3 10 4 4i 4 114 5 5| 10 Hi 16 54 1 1 11 1 7 4| 1 12 lOi 1 IB 44 2 3 10 2 9 3i 2 14 9i 7 4i 2 •5 9 8 4 10 19 13 13 Hi 16 18 9 19 3 6| 21 18 44 24 13 \l 27 7 114 54 15 lOi 82 3 10 y • 198 Rule. Multiply the principal by the rate per cent., and the number of months, weeks, or days, which are required, cut off two figures on the right hand side of the product, and collect from the table the several sums against the different numbers, which when added, will make the number remaining. Add the several Bums together, and it will give the interest required. N. B. For every 10 that is cut off in months, add twopence ; for every 10 cut off in weeks, add a half-penuy ; and for every 40 in the days, 1 farthing. EXAMPLES. 1. What is the interest of £2467 lOs. for 10 months, at 4 per cent, per annum ? ^ ^ ^ 2467 : 10 900=75 : : 4 80= 6 : 13 : 4 , 7= : 11 : 8 9870 : 10 987100 987=82 : 5:0 2 What is the interest of £2467 10s. for 12 weeks, at 5 per cent.! 2467:10 • 1000=19: 4: 7\ 5 400= 7:13:10 „ 80= 1 : 10 : H 12337 : 10 50= : : 2^ ^ 12 1480150=28 : 9 : 5 1480150 : 3. What is the interest of £2467 lOe., 50 days, at 6 per cent.! 2467 : 10 7000=19 : 3 : 6^ 6 400= 1 : 1 : 11 • —_— . • 2= : : U 14805 : 50= : : Oi 60 7402160=20 : 5 : 7 7402150 : To fiiid what an Estate, fnm me to £60,000 per a^nmm will camB to for one day. Rule 1. Collect the annual rent or income from the table for 1 year, against which take the several sums for one day, add them together, and it will give the answer. 109 An estate of £376 per annum, what m that per day t 300=0 : 16 : 5^ 70=0 : 3 : 10 6=0 : : 4 370=1 : : ,7^ To find the amount of any income, salary, or servants' wages^ for any number of months, weeks, or days. Rule. Multiply the yearly income or salary by the number of months, weeks, or days, and collect the product from the table. What will £270 per annum come to for 11 months, for 3 weeks, and for 6 days? 270 11 2970 270 6 1620 For 11 months, 2000=166 : 13 : 4 900= 75 : 0:0 70= 5 16 : 8 2970=247 : 10 : For 6 days. 1000=2 : 14 : 9^ 600=1 : 12 : loj 20=0 : 1 : l| 1620=4 : 8 : 9i For 3 weeks. 270 300=15: 7: 8^ 3 JK)= : 3 : loj 810 = 15 : 11 : 6i For the whole time. 247 : 10 : 15: 11 : 6i 4 : 8 : 9i 267 : 10 : 3| A Table showing the number of days from any day in the mxmth to the same day in any other month, through the year. FROM January February March April . May . . June . July . . August Septembei October . November December TO 334 306 275 245 214 184 153 12-2 92 61 31 4 i 1—5 < ^ S 01 a s 1— 1 9 Aug. 31 59 90 120 151 ISl 212 365 28 59 89 120 150 181 337 365 31 61 92 122 153 306 334 365 30 61 91 122 276 304 335 365 31 61 92 245 273 304 334 365 30 61 215 243 274 304 335 365 31 184 212 243 273 304 335 3fin 153 181 • 212 242 273 303 334 123 151 182 212 243 273 304 92 120 151 181 212 242 273 62 90 121 151 182 212 243 243 212 184 153 123 92 62 31 365 335 304 242 u O 273 242 214 183 153 122 92 61 30 365 334 304 > o 25 304 273 245 214 184 153 123 92 61 31 365 334 303 275 244 214 183 153 122 91 01 30 835| 365 200 A COMPENDIUM OF BOOK-KEEPING. BY SINGLE ENTRY. Book-keeping is the art of recording the transactions of persons in business so as to exibit a state of their affairs in a concise and satisfactory manner. Books may be kept either by Single or by Double Entry ^ but Single Entry is the method chiefly used in retail business. The books found most expedient in Single Entry, are the Day- Bookj the Cash-Book, the Ledger, and the Bill-Book. ** The Day-Book begins with an account of the trader's property, debts, (fee. ; and are entered in the order of their occurrence, the daily transactions of goods bought and sold. The Cash-Book is a register of all money transactions. On the left-hand page, Cash is made Debtor to all sums received; and on the right, Cash is made Creditor by all sums paid. The Ledger collects together the scattered accounts in the Day- Book and Cash-Book, and places the Debtors and Creditors upon opposite pages of the same folio ; and a reference is made to the folio of the books from which . the respective accounts are extrac- ted, by figures placed in a column against the sums. References are also made in the Day-Book and Cash-Book, to the folios in the Ledger, where the amounts are collected. This process is called posting, and the following general rule should be remem- bered by the learner, when engaged in transferring the register of mercantile proceedings from the previous books to the Ledger : The person from whom you purchase goods, or from whom you receive money, is Creditor ; and, on the contrary, the person to whom you sell goods, or to whom you pay money, is Debtor, In the Bill-Book are inserted the particulars of all Bills of Ex- change ; and it is sometimes found expedient to keep for this pur- pose two books, into one of which are copied Bills Receivable^ or such as come into the tradesman's possession, and are 'drawn upon some other person; in the other book are entered Bills Payable, which are those that are drawn upon and accepted by the tradesman himself. 201 • aMM^M DAY BOOK. (foUo 1.) I o »- O V 1 1 1 1 i 2 January 1st, 1837. • £ s. d. 500 C ■ ' 1 commenced business with a capital of Five Hun dred Pounds in Cash " ■ 'IB 2d. 73 60 133 12 6 7 fl Bennett and Sons, London,* Cr By 2 hhds. of sugar cwt. qr. lb. cwt.qr.lb. 13 1 4 12 12 3 16 116 1 gross wt. 26 C 20 tare 2 3 6 1 neat wt. 23 1 14 at 633. per cwt. ■ 2 chests of tea cwt. qr. lb. lb. 1 15 25 1 12 25 1 2 27 1 22 ^^^1 1 3 5 at 6s. per lb H 18 7 6 6 6 6 * ^1 3d. 3 3 7 3 5 10 8 17 5 10 12 18 6 ■ ^ ■ Hall and Scott, Liverpool, Cr By soap, 1 cwt. at 68s .*. . 1 candles, 10 dozen at 7s. 9d. . . . ■ ■ 6th. ■ TVard, William j)r, ~ To 1 cwt. of sugar, at 70s I 14 lbs. of tea, at 8s ' '■ i cwt. of soap, at 74s ■ ■ 6th. ■ Uooper, William j),. ~ To 1 sugar hogshead ,\ .' , . ■ m k«*i '''^^ student may be directed to fill up this and similar 'blanks in th» oook and the Ledger with the names of places familiar to him y 202 DAY BOOK. (foUo 2.) 2 1 2 1 2 2 2 Janxiary 9th, 1837. £ 1 1 4 .0 17 17 1 s. 16 17 15 8 5 5 9 8 4 8 10 d. b 6 6 9 3 6 10 6 4 2 2 3 3 Johnson, Richard To 2 dozen of candles at 8a 3d.. . Dr. i cwt of soar). at 74s« ••••••••••• i cwt" of suff'kr. at 70s. ••••••••••• lOth. Ward, William To sugar, 1 cask cuit. qrs, lb. gross wt. 5 2 ID cask tare 2 10 Br. neat 5 at 68s .... 12th. Smith, John To 14 lb. of auerar Dr. 12 lb of candles 7 lb. of soao ••••••••• 1 lb. of tea 14th. 6 16 13 16 10 9 4 13 19 8 6 Hall and Scott, Liverpool, Bv 2 cwt. scan. at 6Ss. • • Cr. 17th. 1 1 J>tewton, John To 21 lb. of soap, at 74s. per « 2 dozen of candles. at Ss. 3d. • • Dr. :wt.... i9th. Smith, John To 14 lb. of sugar Dr. i lb. of tea • 21 St. Smith-, John To 28 lb. of sugar Dr. 12 lb. of candles : ; , 203 DAY BOOK (folio 3.) ■ 2 3 2 2 2 3 3 3 1 3 January, 23d., 1837. £ 8 d. 6 6 H Yates 8r Lane, Bradford, Cr. By 'i pieces of superfine cloth, each 36 yards, at 249. per yard... « 17i2 ! 16 8 9 ■ 23d. 2 m Edwards, Benj. Manchester, Cr. By 2 pieces of calico, each 24 yards, at Is. per yard '^H 23d. ^^1 s Smith, John Br. To 14 lb. of soap H 24th. 3 5 9 16 14 5 a Johnson, Richard Dr. To 2 dozen of candles. at Ss. 3d 1 1 cwt. of soap, at 74s ■ 1 k cwt. of sugar, at 70s ■ l^M 15 6 1 6 6 J H 24th. 145 2 148 50 8 16 16 12 8 16 17 7 19 15 19 H Smith, Jithn Dr, To 1 lb. of tea * . . ■ 26th. Mason, Edward J)r. To 3 pieces of superfine cloth, each 36 yards, at 27s. per yard.... 2 pieces of calico, each 24 yards, at Is. 2d. per yard.. 27th. H Parker, Thomas Br. To 1 piece of superfine cloth, 36 yards, at 28s.. . 1 31st. 172 ■ Bills Payable, Cr. By Yates & Lane's Bill at 2 months, due April 2. 1 Inventory, January 31, 1837. 46 55 2 105 ■ cwt. qr. lb. Raw sugar, 14 3 14 at 633 ■ Tea, 1 2 16^ at 6s. per lb Soap, 3 14 at 68s M Ca . Id ic'S, 2 dozen, at 7s. 9d '^1 ■ ' m 201 CASH BOOK. O o O O PQ a CQ < O o 'e O OO CO O O «0 O O O CO CO • O OCvrt OOOOthOO'OOcO 00 T-t r-l ?H 1-H «.^ o «o t- « "^ Oco OeOOcOOOO O CO «9 oco oooonoivooo co 00 ^- * ,H iH T-l «^ OO CO i> rjt CO o o o d (N O CO CJ O rH l> ^— \ « 1 i-H r-t -H —I (N .- < tH CJ (N (N CO CO r~t • -hi a • , o months. . aibated 6d) id 5d)... t weeks . . account, hs brough foli( .(J : 00 oo OS CO CQ C< OJ CO CO (N ^ o o o C'\ o> -H 1-1 T-l CI IS CO '^ 'a si (O c» ?9 P3 CO s§ 4 .^J 208 LBDOBR. CO o .o CO .2 CO .5 ^ CO apq o o o o o O 00 1-t Tjt on o ) ^ CO to col (N O i-H f O »> ^-t iH CO o to o CO ^ a pq >> pq i ^% PQ pq CO _• O CO ►-9 CO ^ ao 00 o o 0» 00 o o o o 00 o ^ CO 11 Q S fi^ o CO _ o en «o o r~ (N r~ (N CO CO • CO 00 C 00 r- CM on C <5 PQ m a >H >» PQ O PQ ,- (N rH u 8 Q eq &■ • < : a a o p-^ 9 PQ V tl -S PQ m 1 J? o to CO e a lTPQ to n to Oi Tl< O to O «0 t* »0 00 o> o ■^ OJ o o o Tt d 1> Tjl r~{ CO o to I tH 1-H tl d (N CO CO < o 09 O 00 ^■» <=> o to 1-4 "^ ••H iwH epq^ pm ^'GLWrn'^SrST ?5 ™F REPOEMATION IN EN- «T bJ ^irp'^"" "^ V"''"' "'0 Apostle a,S tZT list. By Sig. PASTOHiNr. Neat English muslin, Kc. * *L«to1fa''S£;f CONFESSIONS. Translated from tl» i.atin, by a Catholic Clergyman. Mnslin, 50 cents. '"jslmr*'''*""^'''' DISCUSSION. 12mo.«ne paper. ™! X'i i'^''.,fP'!";'?°y?FY; V the Key. Dr. Ma.,«. iavu", nn.1 I?, p' 'r ^"'''"'1 '» 'he Lord Bishop ofSt SADLIERS' FIRESIDE LIBRARY. ^?^1* '^ A^ ™^V^ lUustrftted with steel engravini? and ilu- ?.!^iK'^^- A ^!^" "l^*'^' ^"^ backs, 50c. ^ ^ ^'^ i? ull g It, and gilt edges, 75c. Imitation gilt, |1 OO. ^'75'cente.' ""'^ '"'"""■"""l 'Mo. Ml «at sides «nd edges, Imitation morocco, $1 00. 8 1 CATALOGUE OP BOOKS THE CASTLE OF ROUSSILLON ; ob, Qu«bot w tm Six- TKKNTH Century. A talo, trunalatod from the French of Madame Eugknik uk Ul Kociikbk, by Mrs. J. Saduxb. Qoth, gilt back-, 60o. Steel phito, and illuminated title, full gilt aides and edges, 75 cuntD. Imitation morocco, |1 00. DUTY OF A CHRISTIAN TOWARDS GOD. Beinff an im- proved version of the orlj?inal Trent! ^o, written by the Ven- erable J. B. Sall«, Founder of the Brothers of the Christian School. Translated from the French by Mrs. J. Saduki, ■with the l*rayer3 at Moss and the Rules of Christian Polite- ness. 12mo. 400 pages. The same, half bound, 87*0. Muslin, neat, 50c. " gilt edges, 75c. LIFE OF THE RT. REV. DR. DOYLE, Bishop o» Kiujabb AND Leiohun. With a Portrait. Th« lift of thU Pntriot Bishop should .b« in the houaa of every Ctolbolio ; more «fpe- eklly tho« of Irish origiu. It give* • tummarr of bis examination W«for8 tho HouM of Lords, on the Catholic question. It is a work which every Iri»h father should place in the hands of Ms children, as Lis nnme should be engraved indelibly upon the hoarU of Irlshmeo and their oftbpring. BENJAMIN : or, Pupil or thk Christian Brothers. Trans- lated from the Frorich, by Mrs. J. Sadlier. 24rao. muslm, 26o. Full gUt, S7i. THE LIFE OF THE BLESSED VIRGIN— to which is added a Novena in honor of Her Immaculate Conception, with an Historical account of the Origin and Effects of the Miraou- ' Ions Medal. 2 plates, large type, 82mo, revised by the Very Rev. Felix Vakella— full muslin, gilt back, 18|o. Full gilt, and gilt edges, 87ic. A SHORT HISTORY OF THE FIRST BEGINNING AND PROGRESS OF THE PROTESTANT RELIGION, by way of Question and Answer. By Bishop Cualloneb. A moet : excellent and instructive work. ISmo., clear type. Full muslin, 18c; DOUAY TFi^TAMENT, with am*, -ii ions, and chronoiogical -■•' '•'■'^nm 'Mr^i- '' . MV John and analytical tables, as approve^? o^ I y ' ''O Most ' ..PV iSr'iO. full muslin, em- Hughes, Archbishop of New bossed, gilt back, 87io. ART MAGUIRE; or, The Broken Pledoe. By Wiluam r Carleton. Author of " Traits and Stories of the Irish Peas- antry.'" Dedicated to the Very Rev. Theobald Mathew. m _PT7BLISHED BY D. ^ j. s.dlIER 4 CO. THil CATHOLIC CHOIR BOOTT- n^ , EVEMNO 8KRVICR Or THE CW.f. p"' ^""^ ^««^NO AND choice collection of G^-c-'orhm aS ^^'""xV ^'"»«r'»i"K » l*8nlin8, Sacred Ifvinn« Anl\ *',^''^** Mumsos, LitmifeB selected «n,l new/a "ra^t" t V/'"^'^'^^' '^"'^ ^o i?',' first masters. Comnilfi.rf,!. i " ^''? ^compositions of the THE CATHOLIC HARP o . • ^^^ ^^"' ^^- «^ 00. ing Service oYt["a,?i^^o'li?fr'^^ the Morning «nd Even- thenis, VerHlcIes, and x\lSs ..i T' ^"f'?'^ %mn9, An- Bitions of the aU maftetfl 'em„^n '^*t'^ ^^""^ '^^'^ «o\npo- Half bound, 874c, "'^^^'*'' compiled by Philip A. KmK. CATHOLIC PRAYER BOOKS. for b«.„ty of°flS,rr"bilS''oA'I,M" '"'"''"' "" ""• P-P*'. «- <"*»'. bold type «d und., ,he «pprob«Uo„ of the Moet Rev ^ h h ^V'" '"'•"^"^ '*^'»"<« ««» i««««d t..n uvea .„ .he pW„U„, and hMif;:, :^;.f S;:, /o^ir * ^'"-'" "'"" »-- THE GOLDEN M A NTTAT . 1 • tion, Public and Privafe'CornL'd^"'^'' ^° ^'^^h°"« ^evo- 18mo., 1041 pages. ^ofnp'lcd from approved sources. Neatly bound iu aheep, 1 nlate H T-i Joan, plftin P' C^e, f 75. Koan, luarble edges, a {o- " ,^iit, ^.^ « Jj- An.riatnWo,gm Morocco, extra illiimin„f^j « ' 2 00. 2 50. 8 25. Morocco, extra illuminated, 8 clasps, jj 5^5 ^ th;^^'«Ii.h lanKuago. The ^a/er^^'ilrr^T' r"^""' *^"P"'''»''«'' Ihe UUn oriKi,uiU, wherever .uch were k .ow?; ■ . ' ^^^ ^'^ '=°"*t«d with P-lin. here pven ha. bee« co„,truc7ed bv^rol """' '^'"' ^"«""' ^^"'o" of the (to which, in .uutance. it adhereS wUh h! Lve'^Ut;; "' '"".'"""--I »o"ay text t.n.. have been .onctionea for the pT 11 tf dZ °""'.'Lr™0''* ^^ch f^om time to been literally tra„„ated from the R^tr^uver. ' r' I'"' """'^•"""' P^^- '"'ve »«t edition of .he Cosleate Palmet,.r Se^X. ""' °" "«'"'&<"><='"•. and th, t> e., Ac, to which indulgence, are at^^.hed haCt '?""''*'* *'"» ">« Confratemi- -.rce.. and published with the appri. o^oTb^E^ ""^"""''''* ^™™ ""thori^d prewnt edition ha. been enlarirerf wUh Eminence, Cardinal Wi,eraan Th« -nn, and .election. ftT^:^^^:^^^^^ '""•^"'"- '^'" *"* ^'-""^i lit office, of the IHeued Virgin Lrthrlolr°:."''^' together with the complet. pi-ation. of all the f»ii.L7i';z'^i,T.::^2 TtT ''•' '""'' -^ « •fylea of binding :_ ' ""• ^'"''^ «*"» oK b« had in the followinr CATALOGUE OF BOOKS y Fine Paper, Illuminated TiiHe, 12 PUUm. , Morocco, bevelled, $3 00. Morocco, bevelled, clasps, $3 7o. Morocco, antique, $4 50. Morocco, antique, clasps, f5 60. Rich velvet, clasps, $6 00. Biclj velvet, clasps and corners, PaPff cases, |9 00. Kich velvet, medallion on the side, |10 00. Also, various styles, $10 00 to $20 00. THE GARDEN OF THE SOUL. A Manual of fervent Prav-erl pious Eeflections, and solid Instructions, calculated fo^nswe? ?he use of the members of all ranks and con^- ifonsoT the Roman Catholic Church; to which is prefixed, aniistorical Explanation of the Vestments Ceremonies, efc aDPertainin^o the Holy Sacrifice of the Mass. By the r£\i?Kv Dr. Enoiwvnu, late Bishop of Char eston; with a Slat m^y other valuabfe additions under the care f an SeS divine of New York; with ten steel engravings, Se hi the best style, expressly for this work. 18mo. Clasps, antique, $r, 00. ^ellishments, gilt edge^, %f 00. Extra morocco, clasps, gilt edges, $2 To. do. ^'^' ^ " Imitation clasp, do. 150. do. do. do. 1 20. (Jo. plain edges, i o. Strong sheep, do. 50. THE KEY OF HEAVEN. Greatly enlarged and jmproved, lishments, $4 00. Velvet, clasps, gilt edges, $3 00. Super extra morocco, clasps, gilt edging, with beautiful antique embellishments, $3 00. Extra morocco, clasps, gilt edges, $2 00. do. without clasps, gilt edges, f 1 50. Imitation morocco, gilt edges, clasps, |1 25. do. do. do., |1 00. THE KEY OF HEAVEN, 24mo. imitation, morocco, gilt edges, 76c. Imitation morocco, plain edges, 50o. Sheep or muslin, plain edges, dufi. PUBLISHED BY D. & J. SADLIER & CO. THE PATH TO PAEADISE, 32mo, (large edition,) with 12 flue steel enjjravmgs. Rich velvet, clasps, gUt eJies with beautiful aatique embellishments, U Op. ^ ' Velvet, daspa, gilt edges, |3 GO. Super extra morocco, clasps, gilt edges, with beautifU antique embellishments, $3 00. £xtra morocco, clasps, gilt edges, $2 00. do. without clasps, gilt edges, $1 25. Imitation morocco, six steel engravings, gilt edges, 50o. Imitation morocco, four steel engravings, plain edges, 87io. Sheep, one steel engraving, plain edges, mui^lin, 25c THE PATH TO PARADISE, (beautiful miniature edition,) \2 fine steel engravings rich velvet, clasps, gilt edges, with beautiful antique embellishments, $4 00. Velvet, clasps, gilt edges, $8 00. Super extra morocco, clasps, gilt edges, with beautiful an- tique embellishments, $3 00. Extra morocco, clasps, gilt edges, tl 50. do. without clasps, gilt edges, $1 00. Imitation six steel engravings, gilt edges, 50c. ^^ four do. plain edges, 87ic. Sheep, one steel engraving, plain edges, 25c. PAROISSIEN D^ PETITS ENFANTS PIEUX. A beautiful French Prayer Book ; published with the approbation of the Kight Rev. Dr. Burget, Lord Bishop of Montreal. 48mo. Turkey, gilt, clasped, or in tuck, 50c Imitation turkey, do. do. 82c Plain, do. i8c Sheep, 18c JOJRNEE DU CHRETIEN. A very fine French Prayer Book, containing 630 pages, and twelve fine steel enm*- vings. 24mo. Rich velvet, clasps, gilt edges, with beauti- ful antique embellishments, $4 00. Velvet, clasps, gilt edges, do. |3 00. Extra morocco, clasps, gilt edges, $2 00. do. without clasps, gilt edges, $1 60. Imitation morocco, raised bands, six steel emrravinire. gilt edges, |1 00, -© — « » 7 CATALOGUE OF BOOKS. I > h Imitation moroooo, six stoel engruvingfs, gilt edges, 76o. Imitation morocoo, four steel engravings, gilt edges, 60c. NEW POCKET MANUAL, (a very neat Pocket Prayer Book,) with four steel engravings. 48mo. turkey moroooo, gilt edges, 50o. (In tuck,) gilt edges, 50c. Imitation turkey, gilt edges, 81o. Embossed, gilt, 25o. do. plain, 18c. Sheep, 12ic. HISTORY OF lEELAND,— ancient and modem, taken from the most authentic records, and dedicated to the Irish Brigade, by the Abb6 MacGeoghegan. Translated from the French, by Patrick O'Kelley, Esq., with 4 fine steel engravings. " Ha is graphic, easy, and Irish. Ho ii not a bigot, but appareatly a gennine CathoUe. His ioformation as to th« number of troops and other facts of our Irish battles, is superior to any other general historian, and they who know it well need not blush, as most Iriah- men must now, at their ignorance of Irish history."— TAomo* 2>a«w, En»if en IHihHU- tory, THE EISE AND FALL OF THE IRISH NATION, by Sir Josiah Barriugtou ; late Member of tlie Irish Parliament for the Cities of Tuam and Clogher. It contains twenty-nine portraits of celebrated men who figured in the Irish Parlia- ment. The above work iit admitted by competent Jidges to be om of the nost beaatifal pro. ductions of the day. It was sold in Dublin, when first published, at $10. Soch is iti intrinsic merits, that it iias been admitted by the Snperintendent of PubUo Schools iato the District School libraries of the various States. No Irishman, who !• desirooa of be- commg acquainted with that epoch of his countrj-'s history — the rise of the volunteers— the declaration of rights— and the debates conneeted with that tynumoua act, the Legis- lative Union, should iail to possess himself of a copy of this admirable work. VALENTINE MoCLUTCHY, THE IRISH AGENT; by Carleton. 8 vols, of the Dublin edition, in one. Muslin, 76 cents ; half bound 50 cents. In this work Mr. Carleton has depicted the wrongs and suflbrings of his eoontrymen— their patience and forbearance under their sufferings ; and in the character of Valentine McClutchy he shows the villany practised, by the agents of absentee landlords, apon those tenants who are ao nofbrtimate as to hold lands tram them ; and a* a speeimea of a religious attorney, we would challenge creation to produce a more correct picture than that drawn by Carietiiini «f Solomon McShine. We venture to say that there It net •■ Irishman, who waal^iM ten pagee of the work, that would be without it for ten tiniM the cost. ■uMi-.