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The whole I Remembranc This Wwk I ia recommeiK I for prirate pe B A C( D. .\ I TUTOR 5S COMPENDTUM OF JUINO ANC COMPLETE QUESTION-BOOK ; C02JTAININ0, 11; n"'?' ^'"•!''"' "'"■^l" »™ '■»•« witk » treat dwlof pLinn... .nd permiculty. [i.L'f.rSi'oTi rat'*""""'' P""»i">">""y •™SH for tl,. ...«i.. of .h, ,chol., TO WHICH ARE ADDED, BY FRANCIS WALKINGAME, WRITINO-MASTER AND ACCOUNTANT. TO WHICH IS ADDED, A COMPENDIUM OF BOOK-KEEPING, BY ISAAC FISHER. NEW-YORK: PUBLISHED BY D. & J. 8ADLIER & CO., IfrS WILLIAM-STREET. BOSTON :~I28 FEDERAL STREET. AND 179 NOTRK-nA vTin BTncr^m Mn»n«».,A. .„ - 1 « 5 6 1 \{. M M PREFACE. The public, no doubt, will be surprised to find ttere is saiother attempt made to publish a book of Arithmetic, when there are such numbers already extant on the same subject, and several of them that have so lately made their appearance in the world; but I flatter myself, that the following reasons which induced me to compile it, the method, and the conciseness of the rules, which are laid down in so plain and familiar a manner, >vill have some weight tow£.rds 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 bo toiling and slaving himself after fhe 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 which hindered them from giving me the satis- faction I at first expected ; ue. where there are several boys in a class, some one or other must wait till the boy who fii-st 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, tJiat might not only be of use to my own school, but to such others as would have their scholars make a quick progress. It will also be of ffreat use f^ a»oh ,.o«»i.,„.„ «- A3 Jul \h PHEFACK. have acquired some knowledge of numbe,. at school to make tl em tberem, ,t will prove, after an impartial perusal, on account of if« me book I shall not presume to say any thing more in (av„ur of «.« work, but beg leave to refer the unpmudiced re.derT th! ^mark of a certain author,, concerning Jm^^ ' tl nlr' His words are as follows:— ° ^ " "m naiure. thereforer:' diffren^uVt ott^ JlCt wfTnol "':, 'T" = ^fm the same ^^tt^^^^^nr/^g- T^^^^^^ .akes^a much ^eair proJlTy L? t^^rrimt To enter into a long detail of every rule, would tire the reader »d swell the preface to an unusual length; I shall, therefore o^' give a general ,dea of the method of proceeding, and leave til ^t to speak for i^elf ; which I hope the k™d rf der wild 1 «3wer the t,tle, and the re«.mmendatio„ given it. As to tht * Dilworth. PREPACB. ^ rules, they fo:iow in the same manner as the table of contents specifies, and in much the same order as they are generally taught m schools. I have gone through the four fundamental rules in In- tege)-s first, before those of the several denominations; in order that they being well undei-stood, the latter will be performed with much more ^ease and dispatch, according to the rules shown, then by the customary method of dotting. In multiplication I have shown both the beauty and use of that excellent rule, in resolvmg. most questions that occur in merchandising ; and have prefixed' before Reduction, severaKBills of Parcels, which are applicable to real busmess. In working Interest by Decimals, I have added tablfea. to the rules, for the readier calculating of Annuities, &c. and have not only shown the use, but the method of making them : as liko- wise an Interest Table, calculated for the easier finding of the Inte* rest of any sura of money at any rate per cent, by Multiphcatie* and Addition only ; it is also useful in calculating Rates, Incomes, and Servants' Wages, for any number of months, weeks, or daysj and I may venture to say, I have gone through the whole with aa 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 kingdom r but I think my gratitude more especially due to those who have favoured me with their remarks ; tliough I must still beg of every- candid and judicious reader, thafif he should, by chance, find a transposition of a letter, or a fdse figure, to excuse it; for, not^ withstanding there has been great care taken in correcting, yet errore- of the press will inevitably creep in ; and some may also have slip- ped my observation ; in either of which cases the admonition of a good-natured reader will be very acceptable to his nmch obliged, and most obedient humble servant, F. WALKmOAME.. fl Units J 7«"' 12 Hundreds 123 Thousands 1,234 *> 3 5 8 To" 11 ii 15 19 20 6 8 10 12 6 9 *2 15 18 14 16 18 20 22 12 24 13 26 21 24 27 30 33 36 39 28 42 30 45 16 32 48 17 I 34 I 51 18 36 I 54 38 I 57 40 60 NUMERATION. X. of Thousands. . ,, 12,345 C. of Thousands 123,450 JJJll'ons .*. 1,234,567 X. of Millions 12,345,673 MULTIPLICATION. 28 32 36 40 44 8 10 12 15 16 20 20 25 24 30 35 6 12 14 18 24 30 36 21 28 8| 9 35 40 45 50 48 52 56 55 1^ 42 48 54 60 66 72 70 78 60 84 68 72 76 80 75 80 85 90 95 100 84 42 49 56 63 70 ^6; 18 32 27 lOi 11 20j 22 30: 33 36 401 45 481 54 56 64 63 72 72 80 77I 88 84 96 81 12 3( 40, 50 55 60 66 70 77 80 88 _1? 60 72fl 90 99 90 84 _96J 99 lOsI 100. 110 108 91 .90 96 102 108 114 120 98 105 112 119 126 104 112 120 128 136 144 140 152 160 117 126 135 144 153 no; 121 120; 132 130 143 120 132 1441 140; 154 150 165 160, 176 162 171 180 170, 187 180, 198 190, 209 156 168 1801 192 2041 2161 2261 200, 220 240 NoTB.— This Table may be applied to Division by reversing it; as tlie 28 in 4 are 2, and 2s in 6 are 3, &c. I £ S • 12,345 123,450 . 1,234,567 . 12,345,673 lOi 11 1^ 20j 2ii 33 24 30; 36 iO, 44 4fi 30 30 ro iO 55 66 7.7 88 60 84 96 ^0, 99 108 >o. 110 120 o; 121 132 10, !o: 132 143 144 156 Oi 154 165 168 180 0, 176 192 0, 187 204m 0, 198 216 1 0. 209 220 228 240 ing it; as the ARITHMKTICAL TABLES. VH PENCK. 20d. are Is. 8d TABLES OF MONEY. 24 2 30 .. 2 6 3G .. 3 40 .. 3 4 48 .. 4 50 .. 4 2 60 .. 5 70 .. 5 10 72 .. 6 80d, are 6s. 8d, HHIt,L.INC8. 84 90 96 100 108 110 120 130 1 140 7 7 8 8 9 9 10 10 11 6 4 2 10 8 20s, are £i Os. 30 40 50 60 70 80 90 100 110 1 10 2 2 3 3 4 4 5 5 10 10 10 10 1208. are £6 Os. 130 140 150 160 170 190 190 200 210 10 7 7 S 8 9 9 10 10 10 10 10 10 OF A POUND. lOs. 6 5 4 3 2 2 1 1 Od. 8.. isl ..1 1 0....1 4....1 6. .. .1 1 o • • • • 1 8 a • • • 1 8 1 • . • 1 6....1 half third fourth fifth sixth eighth tent I twelfth twentieth thirtieth fortieth PRACTICE TABLES. OF A HHILLJNO. 6d. is 1 half 4 1 third 3 1 fourth 2 1. sixth li..*. 1 eighth 1 1 twelfth OF A TON. 10 cwt. 1 half 5 1 fourth 4 1 fifth 2i 1 eighih , 2 1 tenth ' or A CWT. qrs. lb. 2 or 56 is 1 half 1. .. ..28. ...1 fourth 16.... I seventh 14.... 1 eighth OF A QUARTER. Hlbs ...1 half "7 1 fourth 4 \A seventh 3i««-« 1 eighth CUSTOMARY WEIGHT OF GOODS. A Firkin of Butter is 56 lbs A Firkin of Soap 64 A Barrel of Soap 256 A Barrel of Butter 224 A Barrel of Candles 120 A Faggot of Steel 120 A Stone of Glass 5 lbs, A Stone of Iron or Shot 14 A Barrel of A nchovies 30 A Barrel of Pot Ashes 200 A Seam of Glass, 24 Stone, or-. 120 TROV WEIGHT. 24 gr. make 1 dwt. 20 riwt 1 ounce 12 oz 1 pound TABLES OF WEIGHTS AND MEASURES. AV0THECARIE8* 20 gr. make 1 scruple. 3 scr.. ..... 1 dram. 8 dr ...1 ounce 12 oz 1 pound AVOIRDUPOIS. 16 dr. make 1 oz. 16 oz I lb. 14 lb 1 stone 28 lb 1 quarter 4 qrs 1 cwt. on ^...i. 11. •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 CLOTH MEAHURE. 24 inch make 1 nail 4 nails 1 quar. 3 quar 1 Fl. ell 4 quar I yard 5 quar 1 En. ell quar 1 Fr. ell HOLID MEAhURE. 1728 in. make 1 sol. ft. LAND MEAnVRC. 9 feet make 1 yard 30 yards .... 1 pole 40 poles 1 rood 4 roods .... 1 acre LONG MEASURE. 3 bar. corn 1 inch 12 inches..! •foot 3 feet 1 yard 6 fsef 1 fathom 5i yards ... 1 pole 40 poles 1 furlong 8 fur 1 mile 3 miles ... 1 league 69 i miles. . . 1 degree I iviii M ARITHMKTUAL TABLKS. Ot,D STANDARD ' ALE J AND I3EER. NEW STANDARD. GiUs 3.93 3.SG 3.4(3 3.17 2.34 0.69 3.03 1.38 2.06 4 gills make 1 pint 2 P'nfs 1 quarf 4 quarts.... 1 gal 9 gallons.... 1 fir. 2 firkins 1 kild. 2 kilderkins. 1 bar. U barrel.,. ... 1 hh.l 2 barrels 1 pun. _3 barrels 1 butt 1.60 2.41 0.10 3.2S 1.Q2 3. S3 2.44 3.66 3.33 2 pints.. . 1 i quarts. .1 ) gallons.. 1 J gallons..! ! gallons..! ' gallons..! gallons..! i hhds...! pipes.. .! quart gallon anksr runlet tierce hogshead puncheon pipe tun B. P. G. Q. p. Gills 10 0.25 10 1.01 10 2.02 10 10 0.07 2 2 0.14 ^0100 0.28 „8 1 0.50 33 2 24 S2 2 1 1.63 DRY MEASURE. 2 pints make 1 quart 4 quarts i a;allon 2 gallons. ,. i peck 4 pecks I bu.shel 2 bushels....! strike 4 bushels 1 gack 8 bushels 1 quarter 4 quarters... 1 chald 10 quarters... 1 last 3 3 0.21 £10 10 2.52 JOAL MEASURE. Gills. 0.071 0.13 0.54 0.91 1.S2 3.64 1-45 3.27 2.91 3 bushels..! sack 2 3 36 bushels..! chaldron! 34 3 .Gills. 3.75 3.03 2.04 0.17 0.35 0.70 1 40 1.65 2.13 1 1 0.52 1 1 034 I ii TAIVDARD. ^. P. ihlls. 1 0.07 0.13 0.54 1 0.91 1.S2 a 04 1 1.45 1 3.27 2.91 1 2.36 2.05 2.. 'is 1 3.87 1 3.70 1 3.r>5 1 340 1 3.11 1 2.22 ■ Q P.Gills. 1 1 3.7.") 3 1 3.02 3 1 2.04 3 0.17 i 0.35 ) 0.70 ) 1 40 ) 1 1.G5 1 2.13 0.52 ' 1 2.34 CONTENTS. PART I.— ARITHMETIC IN WHOLE NUMBERS. Page, Introduction..... 11 Numeration 13 Integers, Addition 15 Subtraction 16 Multiplication 16 Division .*.... 19 Tables 21 Addition of several denomindtions28 Subtraction 34 M ultiplication 37 Division 42 Bills of Parcels. 44 Reduction 47 Single Rule of Three Direct. . . 53 Inverse.. 56 Double Rule of Three 58 Practice 60 Tare and Tret 67 Simple Interest 70 Commission 71 Purchasing of Stocks 71 Page. Brokerage 71 Compound Interest 74 Rebate or Discount 75 Equation of Payments 76 Barter 1^ Profit and Loss 19 Fellowship 80 without Time 80 with Time 82 Alligation Medial 83 Alternate 85 Position, or Rule of False SB Double 90 Exchange 91 Comparison of Weights and Mea- sures 95 Conjoined Proportion 96 Progression, Arithmetical 97 Geometrical lOO Permutation 104 PART n.— VULGAR FRACTIONS. Reduction Iu6 Addition 112 Subtraction. 1 12 Multiplication 113 Division The Rule of Three Direct. . . . ~ Inverse.... The Double Rule of Three. . . i' M m 114 115 116 ■ CONTENT*. PART III.-.DECIMALS. |i : Numeration jj^* Addition , ] jjo Subtraction...., [[[ ug Multiplication. . ; '.'.'/.'. 119 Contracted Multiplication...*.* 120 Division ^ ^ 221 -^ Contracted '. 122 Reduction 223 Decimal Tables of Coin.Weights, and Measures 126 The Rule of Three .*.*.*.* 129 Extraction of the Square Root.* 130 Vulgar Fractions 131 r Mixed lumbers. 132 K^xtract of the Cube Root.... 134 —-Vulgar Fractions. . . 136 — Mixed ^Numbers..., 136 -Biquadrate Root. ... 13S PpflTA A general Rule for extracting the Roots of all powers 133 Simple Inteiest .' 140 "; TT" for days 141 Annuities and Pensions, &,c. in Arrears 543 Present worth of Annuit'ies.'.*.* 147 Annuities, &c. in Reversion.. . 150 Rebate or Discount 152 Equation of Payments ..] 154 Compound Interest 155 Annuities. &c. in Arrears..... 157 Present worth of Annuities... 160 Annuities, dec. in Reversion. . . 162 Purchasing Freehold or Real Es- tates..... H34 r~7 in Reversion 165 Rebate or Discount 166 PART IV.~.DUODECIMALS. Multiplication of Feet & Inches, 169 Measuring by the Foot Square, 171 Measuring by the Yard Square, 171 Measuring by the Square of 100 Feet 173 Measuring by the Rod 173 Muluplymg several Figures by several, and the operation in one line only 174 'F'f PART V.-QUESTIONS. A Collection of Questions, set down promiscuously for the greater trial of the foregoing Rules f.... 176 A general Table for calculating Interests, Rents, Incomes and Servants' Wages 181 A COMPENDIUM OF BOOK-KEEPING 184 ; aZPLANATION OF THB CHAXACTBSfl. EXPLANATION =Equal. — ^Minus, or Less, -f Plus, or More. X Multiplied by. -r Divided b/. 2357 OF THE CHARAC THIS COMPEN The Sign of signifies that 4 63 : : So ia. 7—2+5=10. 9—2+5=2. The Sign of Sub! ^ is, 8 lessened 1^,2 is The Sign of Addition; as, 4+4=8, that^, 4 added to 4 more, is equal to 8. The Sign of Multiplication; as, 4X6=24, that is, 4 multiplied by 6 is equal to 24. The Sign of Division : as, 8-7-2=4, that is, 8 divided by 2 is equal to 4. Numbers placed like a fraction do likewise denote Division ; the upper number being the dividend, and the lower the divisor. The Sign of Proportion ; as, 2 : 4 : : 8:16, that is, as 2 is to 4, so is 8 to 16. Shows that the difference between 2 and Y added to 6, is equal to 10. Signifies that the sum of 2 and 5 taken from 9, is equal to 2. Prefixed lo any number, signifies the Square Root of that number is required. Signifies the Cube, or Third Power. Denotes the Biquadrate, or Fourth Power, » ! each of each from the left hand L soT' ^'"'? ""^ «■«» %'"» Tern, and the third aa so mlf ■ >'"'"? Hundreds, the next «. THE APPLICATION. «nd Fort^-five. T'""?"'""' Thousand, Two Hundred Fi Hied'"'™'' ^- S-^O-" and Fort,-one ^o^and. () Seven Hundred anr' T.. "'v^ "^^o- ^K?^dJ^"Sf^ lliousand, Five Hundred. ^^"""^' '^^"^ H"«<^red and Ten Write down in Words 3« 2017 P 5P n 5201 Hm ^ (') 20766 (") 65700047 (H) ■dotation T One. II Two. III Three. IV Four. V Five. VI Six. Vn Seven. Vm Eght. ^^ ^^9th the followmg Numhm P ?L'n'^ (") 5207054 () 754058 ("( oo'7iQn„ 5G00030 M Xml 900061057 (^») ^^900790 *y Roman Letters. IX Nine. X Ten. -X^I Eleven, XH Twelve. XIII Thirteen. XI V fourteen. XV Fifteen. . XVI Sixteen. ADDITION OF IN^EOBBS. Uk *e right Band, Millions; each 'he first Figure df» the next an Js written over 2ad, Nine Hun- of the rest. Vumbera xvn XVIII XIX XX XXX XL L LX LXX LXXX xc c 00 Seventeen. Eighteen.. Nineteen. Twenty. Tliirty. Forty. Fifty. Sixty. Seventy. Eighty. Ninety. Hundred. Two Hundred Three Hundred. ^ ' Four Hundred. Five Hundred. Six Hundred. Seven Hundred. Eight Hundred. Nine Hundred. One Thousand. > One Thousand Eight Hundred and Twelve. MDOOCXXXVn One Thousand Eight Hundred and Thirty Seven. COO COCO D DC DCO DCCC DGCCC • M MDCCCXII. ty-six. wo Hundred '6 Thousand, Fifty-seven TO Hundred ed and Ten INTEGERS. ADDITION • Teacheth to add two or more Sums together, to make one wholo or total Sum. Rule. There must be due regard had in placing the Figures one under the other, i. e. Units under Units, Tens under Tens, &c. ; then beginning with the first row of Units, add them up to the top ; when done, set down the Units, and carry 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 of the Sura, and reckon the Figures downwards, the same as you add them up, and, if the same as the first, the Sum is supposed to be right. Qrs. 275 110 473 354 271 . 852 Months. 1234 7098 3314 6732 2546 6709 £ 75245 37602 91474 32145 47258 21476 Years (*) 271048 325476 107584 625608 754087 279736 /*^ "WKof tq flirk qiivr% cif J.^ 401 Q'7:1'7 ^.4.Ai. OOP,1 Q1 4 O^Mi Am. 17206. (') Add 246034, 298765, 47321, 68653, 64218, 6376, 9821, id 640 together. Am, 730828. , b2 ^ % i m SUBTBACTION OP INTEOE5S. Jo\ hnllT-^-'^^-',?; ^^°*» ^' ^2H D. £391, and E. ( ; How many days are m the twelve Calendar Months ? Ans. 366. SUBTRACTION Rule This being the reverse of Addition, you must borrow Way u r rhl :is! ^''" "^^^ ^ «■-«■ ''-^» '— ^ (') (') (') (*) n /'^ From 271 47S4 42087 452705 271508 ^I^cLk Take^ 2725 ^34096 327616 ?5247i 3150874 Rem. 117 " ' ~" Proof 271 ■ ' ~ MULTIPLICATION d?pf!?*»!''''' ^ tT^""^ *^^ ^'^^^^ ^f *^^ N"°^^rs given as tl^JT.n::^'!^!''' '-'"^ ^"' compendiously performs To this Rule belong three principal Members, viz. 1. ^e Multiplicand, or Number to be multiplied. 2. Ihe Multiplier, or Number by which you multiply. 8. Ihe 1 roduct, or Number produced by multiplying. ^^i"''^ ?-^" "^'^^ *^'^* ^^'S""'^ ^^hich stands in the Unit's place fc r„ J^-n]^ n,P'''^"t ®'* ^^^" *^^« Units, and carry the l^ens m mind, till you have multiplied the next Fimir« in fl.! .«uiu,.iicarm oy the same J^igure in the Multiplier; to the nro"- duct of which add the Tens you kept in mind, setting down ^e Um^ and proceed as before, till the whole line is multfplied XULTIPLICATION OF INTB0ER8. Proof. By casting out the Nines ; or make the former Mul- tiplicand the Multiplier, and the Multiplier the Multiplicand ; and if the. Product of this operation be the same as before, the work is i-ight. MULTIPLICATION TABLE. H q q q q q q q q 4 6 2 4 6 3 6 9 4 8 12 5 10 15 6 12 18 7 14 21 8 16 24 9 18 27 10 20 30 11 22 33 12 24 36 8 10 12 12 15 18 16 20 24 20 25 30 24 30 36 28 35 42 32 40 48 36 45 54 40 50 60 44 55 66 48 60 72 7 8 9 10 11 12 n 14 16 18 20 22 24 i i i i i & I i i A 21 24 27 30 33 36 28 32 36 40 44 48 35 40 45 60 55 60 42 48 54 60 66 72 49 56 63 70 77 84 56 64 72 80 88 96 63 72 81 90 99 108 70 80 90 100 110 120 77 88 99 110 121 132 84 96 108 120 132 144 Multiplicand ( j 25104736 (») 52471021 {*) 79254375?! MtJtiplier 2 3 4 Product 50209472 f 27104107 231047 7092516 3725104 5 6 7 8 4215466 9 2701057 10 (") 31040171 11 When the Multiplier is more than 12, and less than 20, multi. ply by the Unit Figure in the Multiplier, adding to the Product the back Figure to that you multiplied. Bd i") 6110592 13 MirLTIPUCATlOJr OF UtTBQMWL (") 5107252 (") 76?S2in /m «„ w ^^ I ; /053210 (") 92057166 ^^ V 16 I n 6251721 17 9215324 ("\2'i^lqA^ ' n,^ j^ U ^571341 (»•) 3592104 ^® 20 W"ng ,0 „^ the firatfiCTre of^lff'T"'," "'« Multiplier, ob- you multiply by aZ X ^v^ ^>'^"«' ""der that Fi^re Sum will Wf total Prodtt:'''''' ^'"^"'"^ '^'*^''^ »<» S P m"!.'-''!^ f 1''«0'?1 by 5147. U ?f" 'iPy "092«! 4 by 7419 ( JWt,py 9S00985742 by 61870 ( ) Multiply 170149S868S67 by 4768756 that the next Figure mLfb^wi *"',' ^""^ "'"^ ">"«' b« tatea i. e. under the Figure y^u mulSpV bj.''''^ '""'" "^ "'' '*'' '""*'• (") Multiply 671204 % 27009 5140836 3998428 1142408 .r Product 15427648836 P u u-P,'^ ^561240325 b7670027~' ( ) Multiply 662710934 by 590030 L *"• Barrel of Anchovies ... so ^'"^™» ' ,"' f " Soap a-iB 4 r. II ^"'«'ie'-'s Meat. . a fiaisins ?i| t S*""" ?f Train Oil. ... n A Puncheon of Prunes... 1120 "^ ^™^ °f Straw '* A Fodder of T^ha 2 qrs. • • .4.a cwt. 36 Trusses a Load New TTflir Old Ha/..... 66 Marked [ dwl dwt oz. • • • • • .lb. • 8, Electuaries of fine Gold^ er, is H oz. Marked Mr • j oz. ' • • . lb, • • t qr. • ..cwt • • •ton. Weight 1. eigijt tli:*t lb. at or ) f 14 • • • • j sat. . 8 • • • • • • • • » B e * ^\f TABLES OF WEIGHTS. Cheese and Butter. A clove or Half Stone, 8 lb. A Wey in Suffolk, ) lb. A Wey is Essex. 32 Cloves, or J 256 32 Cloves, oi Wool. lb. A Clove 7 A Stone .,...,.,,,,,, , 14 AT(J ,.. 28 !'.' lb. 8? 364 4368 A Wey is 6 Tods and 1 Stone, or A Sack is 2 Weys, or A Last is 12 Sacks, or By this Weight is weighed anything of a coarse or drossy na- ture; as all Grocery and Chandlery Wares; Bread, and all Me- tals but Silver and Gold% Note. One Pound Avoirdupois is equal to 14 oz. 11 dwts. 154 grs. Troy. APOTHECARIES' WEIGHT. ^^arked 20 Grains make 1 Scruple a 3 Scruples i Dram 3 8 I^fams 1 Ounce 5 12 Ounces 1 Pound ..lb. Grains 20 = 1 Scruple 60 = 3=1 Dram 480 = 24 = 8 =± 1 Ounce 6760 = 288 = 96 = 12 = 1 Pound. Note. The Apothecaries mix their Medicines by this Rule, but buy and sell their commodities by Avoirdupois Weight. The Apothecaries' Pound and Ounce, and the Pound and Ounce Troy, are the same, only differently divided and subdivi- CLOTH MEASURE. 4 Nails, i make 1 3 Quarters 1 A r\ i 5 Quarters Marked )n. fqr. 1 EnglishEU. 6 Quarters i French Ell , Quartf.r of a Yard Flemish Ell /.:f1 K .yd. E. E. .Fr.E lard ••••••.*..• TABLES OP MBASUBBS. Inches «i = 1 Nail ftn ™ J *=* ^ Quarter '45 * i^ *= ? = 1 Flemish EU ^ ^ ?? ^ ? =" 1 English Ell 34 « 24 « 6 s= 1 French Ell. - LONG MEASURE. :f";:^^^- -ke ,xnch ^:t1 1S^^ IFoor iin • 6 Feet * ^ Yard ^^f- 5iYardV..';:;;;;; j ^^''''"•••••••••••••'^''.'.""fth *0 Poles ; J god. Pole or Perch **"d „ .^"••ws ::::::::::: Si[> ••^'"* •J Miles '"-t mue 60 Miles... ;: League iii'ta n 1 Degree i®*- deg. Barley Corns 3 ^ 1 Inch 36 = J2 == lOS =r 36 = 1 Foot Mile. 594= 19S -_ ,«.— i.Yard 23760 = 7920 — fifin ^.^^'^ ^ Pole 190080 = dl^ - ,g» ; ,==0 = ^40 = . r„rt„jj^^ Jniy^rJon^t 'o Mil'f !"' "^ ^"'°"«»' "-'y. ^^ough con.. WINE MEASURE. 2 P^nts make 4 Quarts 10 Gallons Marked 1 Quart )pt«. 1 Gallon..... ^^^?' gal. 18 Gallons.*.'.'.*.' * } ^^^^^ of Brandy*.*.'.*.*. S' 31* Gallon* 1 Runlet ""'*• 42 r.!!!'ll!' Half an hAo-V::; ."?'?• 63 Gallons " 'leaos • ••....i. Tierce Hogshead. ' i hhd. a pipeior4H'o*g;h;;d;.';:::;''-i p^pforButt .tier, .hhd. 1 Tun... ^^'•B* TABLES OF MEASURES. 36 2 Pints make 1 Quart.. 4 Quarts 1 Gallon . Inches* 292= 1 Pint 571= 2= 1 Quart 231 = 8= 4= 1 Gallon 9702 = 330= 108= 42=1 Tierce 14553 = 504= 252= 63=1 i=l Hogshead 19404 = 072= 330= 84=2 =U=1 Puncheon 29106 =1008= 504=126=3 =2 =li=l Pipe 58212 =2010=1008=252=6 =4 =3 =2 = 1 Tun All Brandies, Spirits, Perry, Cider, Mead, Vinegar, Honey, and Oil, are measured by this measure ; as also Milk, not by [ law, but cu'5tora only. ALE AND BEER MEASURE. Marked, /pts. ^ qts. gal. 8 Gallons... 1 Firkin of Ale A. fir 9 Gallons 1 Firkin of Beer B. fir. 2 Firkins I Kilderkin kil. 4 Firkins, or 2 Kilderkins 1 Barrel bar. li Barrel, or 54 Gallons 1 Hogshead of Beer hhd. 2 Barrels 1 Puncheon pun. 3 Barrels, or 2 Hogsheads. ...... 1 Butt. ,butL <" BESIL Cubic Inches 354= 1 Pint 7(Ji= 2= 1 Quart 282 = 8= 4= 1 Gallon 2538 = 72= 36= 9= 1 Firkin 5070 =144= 72= 18= 2=1 Kilderkin 10152 =288=114= 30= 4=2=1 Barrel 1.5228 =432=216= 54= 6=3=1 4=1 Hogshead 20304 =570=2G3= 72= 8=4=2 =1^=1 Puncheon 30456 =804=432=108=12=6=3 =2 =li=l Butt ALB. Cubic Inches 35i= 1 Pint 704= 2= 1 Quart 282 = 8= 4= 1 Gallon 2256 = 64= 32= 8=1 Firkin 4512 =128= 04=10=2=1 Kilderkin 9024 =250=128=32=1=2=1 Barrel • OROO no J . 5=t3=:.3=i 3=1 nogsnead. I • By a late Act of Parliameqt.thechpacities of lh« Wine, the Ale and ^er, and the Dry MeH8ure«. have heen reduced to one Standard For an accurate comparison of thaw Measure*, with the old standiird Mcasuroa, the Student ii referred to the Table of th« " Impnial Muuuru," at tke beginning of the work. c mv ill TABLES OP UEASUBES. Btrong beer and small M IV ^^^^,f ^^ ^^ England, for ale^ to thf firkin. ^ ^^ ^^"^"* *^ ^' ^^''^^^ «nd 8^ gallons ^' ^'""a barr!.! ""nV?^™ •"' '''' ""^^' " ^2 gallons. DRY MEASURE. Marked iqts. ...pot . .gal k. strike coom, qr. 2 Pints matn , « ™*'^® 1 Quart 2 Quarts , « , 2Pottles ; Po«le 2 Gallons .V;*. } g^lo" 4 Pecks 1 Peck 2Bu9heis**.*.;:;;:;:;;;;; ; ?"?f^^^ .''.'.'.'.'.'.'.'.tu.' 4 Bushels...., ....V;;:"*i conm «tri JQrt:;s:'.^!""^^^^ -■•' Q-'^leV::::::;:::;::::---- ^^^y' ....1 w.v: r^- . ^ last In London, 36 bushels make a chaldron. Solid Inches 2681= 1 Gallon 637f= 2= 1 Peck 21601= 8= 4= 1 Bushel 4300^= 16= 8= 2= 1 Strike 8601|= 32= 16= 4- o-T , p 172031= 64= 32= t~ IZ o ^^"^ 86016 =320=160=40=20=10^ 5-^ w' 172032 =640=320=80=40=20=10=2=?i Last. The Bushel in Water Measure is 5 Pecks. A score of coals jg oi i i j A sack of coals ^^ chaldrons, A chaldron of coals ....;;; , | ^""f'^' A load of corn ^ ®^^'^^- A r-t "f ditto ^ hmh^h.. This mAfl«Mr?» * *^i*: * ; •/ • •.: V • • t .40 bushels. UivlO — measure is The standard bushel Hpplied to all dry goods. 18 m inches Wde, and 8 inches deep. TABLES OF ]£SASX7SSS. 27 TIME. 60 Seconds make....l Minute. 60 Minutes 1 Hour. . . 24 Hours ..i Day.... 7 Days 1 Week.. 4 Weeks i Month. 13 Months, 1 day, 6 hours. . 1 Julian Year yr. Marked 5 m. .hour. • day. . week, mo. Seconds 60 = 1 Minute 3600 = 60 = 1 Hour 86400 = 1440 = . 24 = 1 604800 = 10080 = 168 = 7 2419200 = 40320 = 672 = 28 d. h. 31557600=525960=8766=365 : 6= d. h. 31656937=525948=8765=365 : 5 ; Day = 1 Week = 4 = 1 Month, w. d. h. =52: 1 :6=lJulianYear. m. " 48 :57=1 Solar Year. To know the days in each month, observe, Tliirty days hath September, April, June, and November, February hath twenty-eight alone, And all the rest have thirty and one ; Except in Leap-Year, and then's the time February's days are twenty and nine. SQUARE MEASURE. 144 Inches make i 9 Feet 2™i fS;;.:::::::::::::::::::;::::::::;;} ^r""'"™""* 40 Kods J Rood Roods, or 160 Rods, or 4840 yards' .' .* .* .' .* ,* 1 Acre of land Acres. Foot. Yard. 4 640 30 100 V'-'*''' 1 Square Mile. ^"^^ 1 Yard of land. ^"«» 1 Hide of land. C2 28 ADDITION OP MONEY. Inches ., 144= 1 Foot 1296= 9 ^ 1 Yard 39204= 272i= 304.- i r> i 1568160=10890 =1210 ^r 40 -^1 P . 6272640=^43560 =48io =160' Zii^^f^^^^ bid'th'^ucra^^ «^-t ^-ve length and I^umbing/glaLg, ia^' ^'"^'"^^' P^^^^""S> flooring, thatching, SOLID MEASURE. 1728 Inches ,«,!,« 27 Feet "^""^ ^ Solid Foot, 40 Feet of round timb'er * ) ^ ^*""''' ^"^ ^"'^^ °^ earth. Or, 50 Feet of hewn timber,' J •••• is .... I Ton or Load. 1 inch deep, is a stal^k of wood * ""'' '""""'^ ""^ « fe«' fe a' coMott!" '■• ' ' '"' ''"'^' ^ f-' ''-'^. -d 4 feet dec,, breS'tK td" "'" "^^"'^ "" '•''"8^ *•'■'' tav. length, t;™7!; *"'''''' ™"^^' ^^° «^^«^'^^«- as make one of the imxt^LTo.l- ^ """" <'ei""nination I «naer the row added. and\£rvK'« O^? f^'.'^" K<'n'»i" •• 14 .. -,{ 87 .. ,,, ,- 5g ■■, -r •• ' aa •• 19 7< ^ 39.. o..,4 ■ ~^— ?ii^ii:^i_£L::Jl.:^4 •! ▲DDITION OP WBI6HTS. MONEY. « <•> £ ». d. £ 257 .. 1 .. H 525 734 .. 3 .. n 179 595 .. 5 .. 3 200 139 .. 14 .. 7i 975 207 .. 5 .. 4 254 798 .. 16 .. 7i 379 (•) s. d. .. 2 .. 44 • • 3 • • t) .. 4 .. 74 • • o • • «>4 • • D • • 7 • • 4 . . 5 j £ «. J. 21 .. 14 .. 74 75 .. 16 .. 79 .. 2 .. 44 57 .. 16 .. 5i 26 .. 13 .. 84 54 .. 2 .. 7 £ a. 73 .. 2 25 .. 13 96 .. 13 76 .. 17 97 .. 14 54 .. il d. " H .. 7 .. 5t .. 34 .. u .. 74 (•) £ s. 127 .. 4 525 .. 3 271 .. 624 .. 9 379 .. 4 215 .. 5 .. f (") d. £ ti. d. £ u 261 .. 17 .. 14 31 5 379 .. 13 ..• 5 75 5 257 .. 16 .. 74 39 1 184 .. 13 .. 5 97 3i 725 .. 2 .. 34 36 84 359 • • 6 . . 5 24 (") s. 1 13 19 17 13 16 d. H 1 74 34 5 34 £ 27 ]6 9 15 37 56 (") fi. 13 12 13 2 19 J9 d. 5* 94 H H 1 a oz. 5 7 3 7 9 8 (0 dwt. . 11 . . 19 . . 15 . . 19 . . 18 . 13 . 4 21 14 22 15 12 TROY WEIGHT. lb. oz. dwt. 7 .. 1 .. 2 3 .. 2 .. 17 5 .. 1 .. 15 7 .. 10 .. 11 2 .. 7 .. 13 3 .. 11 .. 16 O lb. oz. dwt. Rr. 5 .. 2 .. 15 .. 22 3 .. 11 .. 17 .. 14 3 .. 7 .. 15 .. 19 .. 1 .. 13 .. 21 3 .. 9 .. 7 .. 23 5 .. 2 .. 15 .. 17 (*) *. d. .. 3 .. 7 .. 17 .. 1 .. 15 .. 41 .. 10 .. 54 .. 19 ''4 .. 17 .. 3i lb. oz. dr. 152 .. 15 .. 15 272 .. 14 ,. 10 303 .. 15 .. 11 255 .. 10 .. 4 inio 835 13 13 AVOIRDUPOIS WETGHT. (•) cwt. qrs. lb. 25 .. 1 .. 17 72 .. 3 .. 20 54 .. 1 .. 16 24 .. 1 .. 16 17 .. .. i(^ 55 16 (') t. cwt. qrs. lb. 7 .. 17 .. 2 .. 12 5 .. 5 .. 3 .. 14 2 .. 4 .. 1 .. 17 3 . 18 .. 2 .. 19 7 .. y .. 3 .. 20 8 1 24 C3 90 ADDITION OF MEA8URBS. APOTHECARIES' WEIGHT. CLOTH MEASURE. ^35 .. 3 .. 3 70 .. 2 .. 2 ,^5 .. 3 .. 176 .. 1 .. 3 26 .. .. 1 273 .. 2 .. i (•) E.E. qr. n. 272 .. 2 .. 1 152 .. 1 .. 2 79 .. .. 1 156 .. 2 .. 79 .. 3 .. 1 154 ., 2 .. 1 yd. feet 225 .. 1 ., 171 .. .. 52 .. 2 .. S97 .. .. 154 .. 2 .. 137 .. 1 LONG MEASURE. in. bar. 9 .. 1 3 .. 2 3 .. 2 10 .. 1 a. 726 219 1455 879 1195 (•) r. 1 2 3 1 2 LAND MEASURE. p. 31 17 14 21 14 (•) a. r. 1232 .. 1 327 .. 131 .. 2 1219 .. I JKn. 10V , , '4 p- 14 19 15 IS 17 ADDITION OF XBA8USE8. m ir- 8cr. gr. * •• .. 12 7 .. 1 .. 17 2 •• .. 14 !••!.. 15 ? - 2 .. 13 ^ •• 1 .. 18 WINE MEASURE. <:. (•) qr. n. •• 2 .. 1 • 1 .. 2 . .. 1 . 2 .. . 3 .. 1 3 2 .. 1 u.. (*> (•) hhds. gals qts. t. hhds. gals, qts 31 .. 57 .. 1 14 .. 3 .. 27 .. 2 97 .. 18 .. 2 19 .. 2 .. 56 .. 3 76 .. 13 .. 1 17 .. .. 39 .. 3 55 .. 46 .. 2 • 79 .. 2 .. 16 .. 1 o7 • • 38 • • 3 54 .. 1 .. 19 .. 2 55 .. 17 .. 1 97 .. 3 .. 54 .. 3 ALE AND BEER MEASURE. (») (•) (•) A.B. fir. gal. B.B. fir. gal. hhds. gals, qts 25 .. 2 .. 7 37 • • 2 . . 8 76 .. 51 .. 2 17 .. 3 .. 5 54 .. 1 .. 7 57 . . 3 . . 3 96 .. 2 .. 6 y/ •• o •• o 97 .. 27 .. 3 75 .. 1 .. 4 78 .. 2 .. 5 22 .. 17 .. 2 96 .. 3 .. 7 47 .. .. 7 32 .. 19 .. 3 75 .. .. 5 *jO • • <6 • • d 55 .. 38 .. 3 DRY MEASURE. fur. j»o • • 1 . . 19 .. 7 ., S2 • . 5 , . 31 .. 6 . 12 . 6 . 17 . 5 21 (« ch. bu. pks. 75 .. 2 .. 1 41 .. 24 .. I 29 .. 16 .. 1 70 .. 13 .. 2 54 .. 17 .. 3 79 .. 25 .. 1 last. wey. q. bu. r 41 38 .. 1 .. 4 .. 5 .. J 47 .. 1 .. 3 .. 6 .. 2 62 .. .. 2 .. 4 .. 3 45 .. 1 .. 4 .. 3 .. 3 78 .. 1 .. 1 .. 2 .. 2 29 .. 1 .. 3 .. 6 .. 2 • 1 . 14 • . 19 2 . 15 1 . IS 2 .. 17 (») w. d. h. 71 .. 3 .. 11 51 .. 2 .. 9 76 .. .. 21 95 .. 3 .. 21 79 .. 1 .. 15 TIME. w. d. h. m. " 57 .. 2 .. 15 .. 42 .. 41 ^5 .. 3 .. 21 .. 27 .. 51 76 i » Q » . ! 5 . . 9.1 oq 53 .. 2 .. 21 .. 42 .! 27 98 .. 2 .. 18 .. 47 .. 38 32 I jii^; ADDITION. THJB APPLICATJON. n ry ,.^ ^''^'^t JS the Slim of "- •^<= ^^-^"^ ^. There are two numbers fh« i . , ^"*- ^^7 : 10 : 3 Eldest sister's fortune, £138,] »• A nobleman, before he „ *^f """• ''^^ "'em £25722 m'"S f his tradesmeit Ml, td™,' °'' '°"".' "^ "-'''^.3 of I'e owed 82 guineas for rent i„ ht ^'*" '"'J""'-^' ''« ^""'1 that to 1..S eonfoctioner, £12 : j 3 • 4 ^ "'"<: "'^'d^ant, £72 : 5 his t-iilor, £i 10 • 1, . fi . , ,-. * ' to "« draper, £47 • 1 o . o ' . ' ta"o,v-eli;u,dlery£8 /7\ t f T"=''-"«l^^^^ 5778^ t\' '.^ to his brewer, ist : 17. n 'Jt'"f "T""^'"""'!'''-. ^^O ■' 6 • R- lOosiretokno.thamf^'LT?".'-^. ^.^ wage^'l^/^^'!;' ;k"S '^^ "'^ ^^- su- iioo,»vS hets'ri -^w*. iJi 032 : 17 ; 3. Ihebe47yeara ^ns. 1797. »se of a quanti- ' a crown; B It in all ? ^13 : 6 ;'3. . ^ several sums, hur score and fow much did 236 : 8 : 4. tlie taxes are £201 : 15. \ second 519, is tlie sum of ^ns. 14(38. ^54: 17, for t the bargain 7 : 10 : 3. 0, their dif- number, and > 94 sum. J'e than the J and £11. father leave £13611, '22. ^Icsirous of found that 72 ; 5 : ; : 2; to I to his - 6 : 8; to his 1 w]ien -1 to take 17:3. 13 : JO 5; i : oh ADDITION. 33 10. A father was 24 years of a^e (allo^ving 13 months to a (rear, and 28 days to a month) when his first child was born • Detween the eldest 'and next born was 1 year, 11 months 14 lays; between the second and third were 2 years, 1 month 'and 15 days; between the third and fourth were 2 years, 10 months md 25 days; when the fourch was 27 years, 9 months, and 12 lays old, how old 'was the father? • ^ ^ins. 58 years, 7 months, 10 days. 11. A bankers clerk having been out with bills, biiuirs home m account, that A paid him £7 : 5 : 2, B £15 • 18 • 6A C £150 : 13 : 2^, D £17 : 6 : 8, E 5 guineas, 2 crown piec(^, 4 Ihalt-crowns, and 4s. 2d., F paid him only twenty groats, (> £7fJ • 115 : 9|, and H £121 : 12 : 4d. I desire to know how'much the [whole amounted to, that he had to pay ? -, o A 11 ^'**' ^^^^ • '^ ' H- 12. A nobleman had 'a service of plate, which consisted of twenty dishes, weighing 203 oz. 8 dwts. ; thirty-six plates, W(nf from .,,eh take thl lowe^f e To""'th*^tL" *" '"-'A^A MONEY. (') Borrowed 715 .. 2 .. 7i Paid 476 .. 3 .. si Remains to pay 238 . . 18 lOi Prsoof 715 .. 2 7i £ *• <'• 87 .. 2 .. 10 79 .. 3 .. 7i *> *. 3 .. 15 1 .. 14 d. H 7 Lent 316 .. 3 .. 5J Received 218 .. 2 .. ij ^^ " Q •• 8i 25 .. 5 .. 2I £ 321 257 o s. 17 14 7 59 36 (') *. 15 17 Paid in all Remains to pay (•) ^i ^ '- d. £ • ^i 71 .. 2 .. 4 527 • 2 19 .. 13 .. 71 139 Borrowed 25107 .. 15 .. 7 ^ .^ 375 .. 5 .. 5k Paid 259 . . 2 . . 7^ at 359 .. 13 .. 41 different 523 .. 17 .. 3 times 274 . . 15 . . 7J 325 .. 13 .. 5 ». d. • 3 .. 54 • 5 .. 7i Lent 250156 271 6 Received 359 . . 15 . . 3 at 475 .. 13 .. 91 several 527 .. 15 .. 3] payments 272 .. 16 .. 5 150 .. .. MEASURES. Y of the lower] ^ as many off to the upper, nee, and carry arrowed. *. d. • 2 .. i| 1 (•) *. d. ^ " 3 .. 44 • •• 5 .. 2i iuBTKAcnoir. 86 TROY WEIGHT. lb. oz. dwt. gr. lb. oz. dwt gr. 0) Bought 52 .. 1 .. 7 .. 2 (•) 7 .. 2 .. 2 .. 7 Sold 39 . . . . 15 . . 7 5 . . 7 . . 1 . . 8 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 .. 1 .. 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 5..7..3..1..16 Fl.E. qr, n. (>) 35 .. 2 .. 2 17 .. 2 .. 1 CLOTH MEASURE. yd. qr. n. («) 71 .. 1 .. 2 o . < « • • 1 E.E. qr. n. (•) 35 .. 2 .. 1 14 .. 3 .. 2 LONG MEASURE. yds. ft. in. bar. O 107 .. 2 .. 10 ., 1 78 .. 2 .. 11 .. 2v lea. mi. fur. po. (•) 147 .. 2 .. 6 .. 29 58 .. 2 .. 7 .. 33 LAND MEASURE. a. 1 O 175 .. 1 . ^M 1' 27 27 9 r. p (•) 325 .. 2 .. ] 279 tirBTRAeriopK, :'a >ttlailfc ..-t WINE MEASURE. hhd. gal. tg 28..39..3.:i <•) tun- tlid. ptil. ,,^ 4'^ •• 2 .. 37 .. a' 17 .. 3 .. 49 .. 3 A.B. fir. eal. O 25 . . 1 . .*^2 • 21 .. 1 .. ft ALE AND BEER MEASURE. O BB. fir. gal. 37 ,. 2 ..*i 25 .. 1 .. 7 hhd. (•) 27 12 gal. qt.| 27 .. 1 ' 50 .. a q«. bir! n O 72 .. 1 .. I' 35 .. 2 .. 3 DRY MEASURE. q«. bu. p. O 65 .. 2 .. 1 57 .. 2 .. 3 (•) 79 .. 3 .. 54 .. 7 .. 1 TIME. yrs. O 79 ino- w. ds. 8 .. 2 .. 4 23 .. 9 .. 3 .. 5. /# . ^f. inf#. (") 24 .. 42 .. 43 19 .. 53 .. 47 THE APPLICATION. «. A man wa. bor„ i„ the year :,83. what wa, hi, age i„ ,he ,ear ,,,,1 , JhZtrn in tmr""' '"'"="" "•« »«= of a man born inlTiO^Ind S. A Merchant had five debforq a t> r. T^ »»*. . •.- ^no.. B. c. D. and tr:at.''iV3?- ^^b^! :!;° JCa-;-" ^ i»ru •^W.f. i;419 to 12 ,c-o;e and £U r'a'I^^Cir.E^Tiir'''""''' ™ ''^ ^n». £45 ; 14. wmmmnitv n ii i Mt, ), COMPOUND MULTIPLICATION, 37 ihd. gal 27 . . 27 12 . . 50 qt.l 1 3 ch. 79 54 bo. • • 3 , , • • 7 • , • p 1 « mffr. -• 42 .. 45 ' 53 .. 47 le year 1781? •^ns. 58, in 1710, and ''ins. 56. gether owed "s del.t ? na. £i\g, 'ing of taxes £45:14. 15. What is the difference between £9154, and the amount of JC754 added I to £305? Ans. £8095. 6. A horse in his furniture ig worth £37 : 5 ; out of it, 14 guineas ; how much does tlie price of the furniture exceed that of the horse ? • Ans. £7 : 17. 7. A merchant at his out-setting in trade, owed £750; he had in cash, coniniodities, the stocks, and good debts, £12510 : 7 ; he cleared, the first [year, by commerce, £152 : 3 ? 0; what is the neat balarce at the twelve months' end ? Ans. £12212 : 10 : 6. a A gentleman dying, left £ <5247 between two daughters, the younger who was to have 15 tliousand, 15 hundred, and twice £15. What was the elder sister's fortune ? Ans. £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, £J05 : 10; to C, £34 • 5:2; to D, £28 : 16 : 5; to E, £14 : 15 : 8 ; to F, £112 : 9; and to G, £J 13 : VI : 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 ? Ans. 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 : 6 : 7 ; tobacco, £125'- 11:6? linen cloth, £112:14:8; tin, £113 : 10 : 5. My correspondent, at the Siime time, informs me, that he has shipped, agreeably 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. RuLR.— Multiply the first Denomination by the quantity given, divide the product by as many of that as make one of the next, se't down the re- mainder, and add the quotient to the next superior, after it is multiplied. Tf thff given quantity is above 19, multiply by any two numbers, which multiplied together will make the same number; but if no two numbers multiplied together will make the exact number, then multiply the top line by as many as is wanting, adding it to the last product. D "! tOtL ..MHilL 88 COMPOUND MULTIPLICATION. Proof. By Division. (') £ s. d. 35: 12 : 7i 2 71 : 5 : n £ a. d. 75: 13: li 3 1. 18 yards of cloth, at 9s. Gd. per yard. o 0X2=IR , 4:5:0 3 £ 9. d. 62:5 : 4i 4 £ s. d. 67 : 2 s 4| 5 8 : 11 :0 2. 20 lb. of tea, at £1 : 2 : 6 per lb. o 8X3X =20 ! i>:0:0 3 27 : : Top line X2=2 :5:0 29 : 5 : 3. 21 ells of Ilollnnd, at Vs. S^d. per ell. 4. 35 firkins of butter, at I5s. 3^d. per firkin. 6. 76 lb. of nutmegs, at Is. 23d. per lb. 6. 37 yards of tabby, at 9s. 7d. per yard. H ci^ , c ^ o Facit, £17: 144 7. 7. 97 cwt. of cheese, at JCl : 5 : 3 per cvvt. ft ^Q 1 r „ ^"^^^ -^122 : 9 : 3. 8. 43 dozen of candles, at Cs. 4d. per dozen. lo^u ei) i . , Facit, £13: 12:4. 9. 127 lb. of Boliea tea, at 12s. 3d. per lb. 10. 135 gallons of rum, at 78. 6d. per gallon. n H . ,, ^ J. , Facit, £60 : i : 3. 11. 74 ells of diaper, at Is. 4tid. per ell. 12. G dozen pair of gloves, at Is. lOd. per pair. Facit, £0 : 12. When the given quantity consists of i, ^, or |. Rule. Divide the given price (or the price of one) by 4 for i bv 2 for 1 them to the product, and their sum will bo the answer re^iui ed COMPOUND MULTIPLICATION, 39 (•) £ s. d. 57 : 2 ! 4| 5 13. 25i ells of holland, at 3 : 4id. por ell. 5 5X5 = 25 - — ea, at £l :2 : 6 ^ 8 j : iO :0 1 3 1 e 27 : X2=2 : Oi :0 1 :0 1 29: 5: 1 1:1: 10^. t : 16 : 2i. ^ : 2 : 2^. 1: Uil. 22 : 9 : 3. 3 : 12 : 4 7 : 15 : 9. )0 : 1 : 3. 55 : 1 : 9. £0 : 12. 01' 4. by 2 for y '-' Tor i, add red. ^ 10: lOi 5 4:4:4^ = 25 0:l:8i = J 4:G:0^ = 25i 14. 75 J elk of diaper, at Is. 3d. per ell. Facit, £4 : 14 : 4J. 15. 19 J ells of dain.'isk, at 4s. 3d. per ell. Facit, £4 : 2 : lOj. 16. 35 J ells of dowla.s, at Is. 4d. per ell. Facit, £2:7:4. 17. H cwt. of Malaga raisins, at £l : 1 : 6 per cwt. Facit, £7 : 16 : 10 J. 18. 6 J barrels of herrings, at £3 : 15 : 7 per barn^I. Facit, £24: 11 : 3^. 19. 35i cwt. doubled refined sugar, at £4 : 15 : por cwt. Facit, £109: 10:3. 20. 154i cwt. of tobacco, at £4 : 17 : 10 per cwt. Facit, £755 : 15 : 3. 21. Il7i gallons of arrack, at 128. Od. i)or gallon. Facit, £73 : 5 : 7 J. 23. 85^ cwt. of cheese, at £l : 7 : 8 por cwt. Facit, £118: 12:5. 23. 29i lb. of fine hyson tea, at £l : 3 : per lb. Facit, £34 : 7 : 4A. 24. I7f yards of superfine scarlet drab, at £l : 3 : i)cr yard. Facit, £20: 17: 1^. 26. 37i yards of rich brocaded silk, at 12s. 4d. per ynnl. Facit, £23 : 2 : 6. 26. 50 1 cwt. of sugar, at £2 : 18 : 7 per cwt. Facit, £160:4 :7i. 27. 96i cwc. of currants, at £2 : 15 : per cwt. Facit, £207 : 15 : 9. 28. 45| lb. of Beiladino silk, at 188. 6d. per lb. nn H, , , , Facit, £42 : 6 : 4i. 29. 87f bushels of wheat, 5it 4s. 3d. per bushel. Facit, £18:12:11*. d2 40 COMPOOWD MULTIPLICATION. 80. 1201 cwt. of hops, at £4 : 7 : 6 per cwt. 31. 407 ya*„f cloth, at 3s. 9Jcl, per v^f ^''' = ' = '*• 32. 729 ells of cloth, at 7s. 7i■ '"•='' '*" diers, supposing th.at in 7 hitrtC 'a 'rr73"3 J*-^""^''™ ^* : 0: A merchant liad £19118 tn h. ' f'",' ^^^'^^ * ^^ •' ^• t^^ther lie cleared £loll^yL^ ^^J^f^ "^^ ' ^'^ ? ^^- ^52715 : 10 : 6 a year- bn tlut l' H" ,^ '''"'' ^"^ '"''^^^'^ ?^f»J had tlie misfortune tt. I. ■^i""'' ^''' ''^' "^ ^'^''^'^J'^^ ^'-^ year, what wjis his real fortune at 12 yeai-s' end ? '^ " * ' ** ^ ^W5. X33984 : 8 : 6. pieces, year; COMPOUND MULTIPLICATION. "4^ : 3 : 2^. : 3 : 5i. : 19 : 10. 18 men, so 5:0: 0. loiinted tc prize ? : 15 : 0. and Iialf a ; 62208. I fifty, and >oduct. 37 times rod net are Voduct. em 144; Ference. each 157 active sol- 44800. with his ■t having I ach quiil I 14 : 0. ' 5 years ^de good ivadoj h'?. :4:6a 8:6. i 10. In some parts of the kingdom, they weigh their coals by a machine in the nature of a steel-yard, waggon and all. Three of these draughts together amount to 137 cwt. 2 qrs. 10 lb., and tlie tare or weight of the waggon is 13 cwt. 1 qr. ; how many coals had the customer in 12 such draujihts? Ans. 391 cwt. 1 qr. 12 lb. 11. A certain gentleman lays up every year £294 : 12 : 6, and spends daily £1 : 12 : 6. I desire to know what is his an- nual income? Ans. £887 : 15 : 0. 12. A tradesman gave his daughter, as a marriagoj portion, a scrutoiro, in which there were twelve drawers, in each drawer were six divisions, in each division there were £o0, four crown pieces, and eight half-crown pieces; how much had she to her tbrtune? ^W5. £3744. 13. Admitting that I pay eight guineas and half-a-crown for a quarter's r<3nt, and am allowed quarterly 15s. for repairs, what does my apartment cost me annually, and how much in seven years? ^HS. In 1 year, £31 : 2. In 7, £217 : 14. 14. A robbery being committed on the liighway, an assessment was made on a neighbouring Hundred for the sum of £380 : 16 : 6, of which four parishes paid each £37 : 14 : 2, four hamlets £31 : 4 : 2 each, and the four townships £18 : 12 : 6 each; how nnich was the deficiency? Ans. £36 : 12 : 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 ? A71S. £10109 : 10 : 0. 16. Admit 20 to be the remainder of a division sum, 423 the quotient, the divisor the sura of bo'h, and 19 more, what was the nunibei of the dividend? Ans. 195446. EXAMPLES OF WEIGHTS AND MEASURES. Multiply 9 lb. 10 oz. 15 dwts. 19 grs. by 9. n Multiply 23 tons, 9 cwt. 3 qrs. 18'lb. by 7. (®) Mniti|)iy 107 yards, 3 qrs. 2 nails, by 10. Multiply 33 ale bar. 2 firk. 3 gal. by 11. Multiply 27 beer bar. 2 firk. 4 gal. 3 qts. by 12. •) Multiply 110 miles, fur. 26 poles, by 12. 1)3 43 DIVISION. DIVISION OF SEVi^ML DENOMINATIOKS. 5! T^;.,:j_ ii My "emains.'ml.ufdv'irbv '«?"""''''■''" °° ""^ k" hand and if' .one of that, which Jd tl X iT'Vt '-'i"^ "<"" '''^^ ."ak! . 'K. , ^)!^4( 3)3f:r:,'( J\f, /\ what did I * : 2 ; 11. t what rate ^1 : V : 3. len 120 are / 5 : Of. 'ire to l^now : 3 : 8^. that for 2 : 5 : 81 tl amongst 10 : 3^. >f'0 men, I ^e of each 1 share. 10. A gentleman lias a garden walled in, containing 9625 lyards, the breadth was 35 yards, what was the length ? Ans. 275. 11. A club in London, consisting of 25 gentlemen, joined for la lottery ticket of £10 value, which came up a prize of £4000. I desire to know what each man contributed, and what each I 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? Ans. £68. 13. Another cleared £2805 in 7^ years, what was his yeaily Increase q^ tbitunfi ? Ans. £374. 14- What number added to the 43d part of 4429, will raise it to 240? Ans. 137. 15. Divide 20s. between A, B, and C, in such sort that A may have 2s. less than B, and C 2s. more than B ? Ans. A 4s. 8d., B 6s. 8d., C. 8s. 8d. 16. If there are 1000 men to a regiment, and but 50 officers how many private men are there to on^ 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 Hhe remainder came out 1788 ? 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 whole being equally divided among them ? Ans. 128. 20. My purse and money, said Dick to Harry, are worth 12r. 8d., but the money is worth seven times the pui-se. What did tlie purse contain? Ans. lis. Id. 21. A merchant bought two lots of tobacco, which weighed 12 cwt. 3 qrs. 15 lb., for £114 : 15 : 6. Their difference in point of weight, wjis 1 cwt. 2 qrs. 13 lb., and of price, £7 : 16 6. I desire to know their respective weights and value ? Ans. Less weight, 5 cwt. 2 qrs. 15 lb. Price, £53 : 10, Greiiter weight, 7 cwt. 1 qr. Price, £61 : 5 : 6. 22. Dividfl 1000 crowns in such a niannor betwctn A, B, and C, that A may receive 129 more than B, and B 178 less than CL Am. A 360, B 231, C 409. 4A BILLS OP PARCELS. EXAMPLES or WEIGnie AND MEASURES. 1. Dividc3 83 lb. 5 oz. 10 dvvts. 11 p.. by 8 2. Divnc e 29 tons, 17 cwt. m-s. 18 lb L o S. DivKo 114 yards, 3 qrs. 2 l.ails, b^! 1? 4. Dmdol017n,iH6V38Hos,^>yll ^ JutSltsf ^^^^^"' ' --ths, 3Us, ^ days, 11 hou., 27 BILLS OF PARCELS. . HOSIERS*. Mr. John Thomas, BoiigJit of Samuel Green. 8 Pair of worsted atockinrrg ^f !* . f" 5 Pair of thVad ditto. . . f 'y^^'-'tlt ^'' ^^'' ^ 3 Pair of black silk ditto ' i ? ' i " 6 Pair of milled hose.!::: ^^^^ •• 4 Pair of cotton ditto I'.i 2 Yards of fine flannel / ' May 1, 18 8 per yard £1 : 12 : 2 MERCERS*. O Mr. Isaac Grant, Bought of John Sims, 15 18 d. May 3, 18 Vards of satin „* q . ^ Yards of flowered silk ! . . .i ! . ..'.H :* 4.!''^ 16 i ards of sai-senet « . ° 13 Yards • • • • t 3 : 2, xds of Genoa velvet ' 9^ '. a 23 Yards of lutestring ..'/.. q l q. iJC2 : 2 : 5 BILLS OF PARCELS. 46 LINEN drapers'. IC) Mr. Simon Surety, Bought of Josiah Short. June 4, 18 s. d. 4 Yards of cambric at. . . 12 : G per yard £ 12 Yards of muslin 8:3 15 Yards of printed Hnen 5 : 4 • 2 Dozen of naplvins ...,2:3 each ... 14 Ells of diaper 1:7 per cU . . 85 Ells of dowlas 1 : li £17 : 4 : C^ MILLINERS. (*) Mrs. Bright, Bought of Lucy Brown. £ s. 18 Yards of fine lace at. ..0:12 5 Pair of fine kid gloves. . . . . .0 *. 2 12 Fans of French mounts : 3 2 Fine lace tippets 3 : 3 4 Dozen Irish lamb : 1 6 Sets of knots : 2 June 14, 18 d. 3 per yard £ 2 per pair 6 each ... :0 : 3 per pair : 6 per set. . £22 WOOLLEN DRAPERS'. (•) Mr. Thomas Sage, Bought of Ellis Smith. 17 Yards of fine serge. . . .at. . .0 : 18 Yards of drugget : 15 Yards of superfine scarlet .. .1 : Yards of black : Yards of shalloon .0 : 16 25 11 June 20, 18 d. ; 9 per yard £ ; ;0 ; ; 9 : 6 ..•...• £59 : 5 : I'l iMi i 46 BILLS OF PARCELS. ^^'^ Giles HarnX "^^"^^'^—^ l^«"glit of Abel Smith. f Calfskins , ^. ^. "^"'^^'^^ ^^5 SlH-ep ditto ...;.; '•**... 3 : 9 per skin £ ^^' Coloured jitto. . . ^'^ 15 Buck ditto... 1 : 8 17 I^ussia Ifides ^ ^ •" 6 . . , . 120 Lamb Skins.;;;; ^0:7....;;; i-Si £38:17:5 ( ) Mr. Richai-d Groves, Bought of Francis Elliot. 25 lb. of Jump sno-ar s. d. 2 loaves of doubFe r;fineV V * ' "^ '' ^* P^^ lb. £ 14 lb. of rice... ^ •" • H^ 28ib.ofMaiaga;;i;i;;;;;; I- « 15 lb. of currants ^' 5 ^ ^'^•^f wackpe;);;;:::;;;;;;;;;0; 5^ July 5, 18 cheesemongers'. (*) Mr. Cliarles Cross, I^ought of Samuel Grant, ^3:2. 3i 8 lb. of Cambrido-e huttor ^' ^• ^l ">. of new checfse ' ^^- • -0 : 6 per lb. £ 5 I'lr. of butter, wt 28* IK ^ ' ^ • • 5 aeshh-e d^^^^^^^^^ 0:5|.. 2 Warwickshire ditto, 15 lb ^ ' ^ ' • 12 lb. of cream cheese ^ •' ^ • . ^ 0:6 .. July G, 18 £3 : 14 : 5 reduction. 47 corn-chandlers'. (•) Mr. Abraham Doyley, Bought of Isaac Jones. July 20, 18 d. [Tares, 19 bushels at. , 1 : 10 per bushel £> Pease, 18 bushels 3 : 9^ Malt, Y ^^'f— '. ^hill,„gs, ^Id'^rence, »280 tahin"T^ f^""' ^'""'"S^- -/ P--!^. - 'tit™ • h. 12. How m'any g„.ne,. in 21168 fJCj^"'- '"''■ ^''■ 13. :a 16573 farthings, how many pouni ? ^'"^ '' ^"■"^^• 14. In 6169 p„„ce, how many shillings ancf ^^.fiV ' ' '*• 15. Jn 6840 farthmgs, how many pence and h.-df-crowns ? 16 In 2149^ f *T,- 1. ^^^' '^^1^^- 5V half-crowns. lin^; Ja pr,^a^|"e-ht\Tarn= "^'Tr^f " 17. How many shilhngs, crownsjand pounds, in 60 ^.ink',.' 18. Reduco 76 moidores into JZ^Z^ZZ""-' '''■ 19. Reduce £102 : 12 into shillings aifd^moS; f "^ ^ ''■ .a^^-pLr '''■'--' - -;•— trs^iifi:;rhow 22. Seven men brono-ht £^^ - m i, • . "f^""' "^^"'^ ' ^^- changed for guineaJZ'wLf J muft l^ I't taH f'' " ^" ^^^ ^WA'. 103 guineas, 7s. over. 23. 24. 26. 21. 28. 29. 15 gr. 33. REDUCTION. 49 11520 far. iJ324 : 10. IS. 21108. ''S. 10573. s. 0109(1. Ans. 299. sixpences. farthinors. and pence, f. 21424. 8 there in Os. £18. gunieas. ; 5 : 3^. 14 : 1. ^ns? crowns. »wns, shil- im. 52. uineas ? s, £03. 2 : 12. >idores. ire tliei*e 1. over, ig.^, how ) : 18. to bo ex- over. 23. If 103 guineas and seven shillings are to be divided amongst seven men, how many pounds sterling is that each ? Ans. £15 : 10. 24. A certain person had 25 purses, and in each purse 12 gui- neas, a crown, and a raoidore, how many pounds sterling had he in all ? Ans. £355. 25. A gentleman, in his will, left £50 to the poor, and ordered that ^ should be given to ancient men, each to have 5s. — ^ to poor women, each to have 2s. Od. — \ to poor boys, each to have Is, — 1 to poor ^'irls, each to have 9d. and the remainder to the person who distributed it I demand how many of each sort there were, and what the person who distributed the money had for his trouble I Ans. 66 men, 100 women, 200 boys, 222 girls, £2 : 13 : 6 for the person's trouble. TROY WEIGHT. 26. In 27 ounces of gold, how many grains? Ans. 12960. 27. In 12960 grains of gold, how many ounces? Ans. 27. 28. In 3 lb. 10 oz. 7 dwts. 5 gr. how many grains ? Ans. 22253. 29. In 8 ingots of silver, each weighing 7 lb. 4 oz. 17 dwts. 15 gr. how many ounces, pennyweights, and grains ? Ans. 711 oz. 14221 dwts. 341304 gr. 30. How many ingots, of 7 lb. 4 oz. 17 dwts. 15 gr. each, are there in 341304 gi-ains? Ans. 8 ingots. 31. Bought 7 ingots of silver, each containing 23 lb. 5 oz. 7 dwts. how many grains? Ans. 945336. 32. A gentleman sent a tankard to his goldsmith, that weighed 50 oz. 8 dw^ts. and ordered him to make it into spoons, each to weigh 2 oz. 16 dwts. how many had he? Ans. 18. 33. A gentleman delivered to a goldsmith 137 oz. 6 dwts. 9 gr. of silver, and ordered him to make it into tankards of 17 oz. 15 dwts. 10 gr. each; spoons of 21 oz. 11 dwts. 13 gr. per dcz. salts of 3 oz. 10 dwts. each; and forks of 21 oz. 11 dwts. 13 gr. per doz. and for every tankard to have one salt, a dozen of spoons, and a dozen of forks ; what is the number of each he must have ? Ans. 2 of each sort, 8 oz. 9 dwts. 9 ^r. over. E I' 4 ! I .11 fiEDUCTION AVOIRDUPOIS WEIGHT add onJ'hSr"' """"''' '"•" "■""»"- """«il.'^ by 3. and divide b^ 5 „ tral^'niThrd'" P""""' ""° S-»'. n.ul.iply by 2, and divide by 3. or . ^ I2 dS.::."""^^ """%• ?"-» i'?9 lb. lb., how mLy cwtfn ,he Thik ? ' ' '^'^ ' "'• ''' "■■ »"'J »"«' bag of 137 42. How many pounds in 97 l,„ i .. , ■^"''- *" ™'- ' I"- "> 'b. SI cwt. > ^ P"™"' •" 2' hogsheads of tobacco, each weighing neat «. In =52 common pounds of silfc. bow n,a„y great pount'' ''''" ^U. How .any p,rce,s of sugar of t5 lb. 2 „.. are there in tflwt'f ,, ^««. 113 parcels, and 12 lb. 14 OZ. over. heel by a great Jrefore, livide by S or i by 3, or 9 \. REDUCTION. 51 APOTHECARIES' WEIGHT. 46. Ir. 27 lo. 7 oz. 2 dr. 1 scr. how many gra'ns? Ans. 159020. 46. How many lb. oz. dr. scr. are there in 159020 grains? Ans. 27 lb. 7 oz. 2 dr. 1 scr. CLOTH MEASURE. m i ■ lb. 1 oz. and ounce*. 14769 oz. ther of 150 ir. 10 lb. '8111 lb, lb. 8 oz. «.«. 16S. , 1073 lb. * ►ag of 131 . 10 lb. hing neat 26460 s. 368. wt. 1 qr. over. 47. In 27 yards, how many nails ? Ans. 432. 48. In 75 English ells, how many yards ? A71S. 93 yards, 3 qrs. 49. In 93f yards, how many English ells? *:^;>:. 75. 60. In 24 pieces, each containing 32 Flemish ells, ho"? ^lany English ells ? Ans. 460 English ells, 4 qrs. 51. In 17 pieces of cloth, each 27 Flemish ells, how many yards ? Ans. 344 yards, 1 qr. 62. Bought 27 pieces of English stuff, each 27 ells, how many yards? Ans. 911 yards, 1 qr. 53. In 911 1: yards, how many English ells? Ans. 729. 54. In 12 bales of cloth, each 25 pieces, each 15 English ells, how many yards ? Ans. 5625. LONG MEASURE. 65. In 57 miles, how many furlongs and poles ? Ans. 456 furlongs, 18240 poles. 56. In 7 miles, how many feet, inches, and barley-corns ? Ans. 36960 ft. 443520 in. 1330560 b. corns. 67. In 18240 poles, how many furlongs and miles? Ans. 456 furlongs, 57 miles. 58. In 72 leagues, how many yards? Ans. 380160. 59. In 380160 yards, how many miles and leagues? Ans. 216 miles, 72 leagues. 60. If from London to York be accounted 50 leagues, I de- mand how many miles, yards, feet, inches, and barley-corns ? Ans. 150 miles, 264000 yards, 792000 feet, 9504000 inches, 28512000 barley-corns. 61. How often will the wheel of a coach, that is 17 feet in circumference, turn in 100 miles ? Ans. 3l058|4 times round. £2 8S t # ml. r itBDUCTION. £.fcn,fera;L^^^^^^^ the worfa, the' LAND MEASURE. «3. In 27 acres, how ma„^ roods and perches' 64. In 4320 perches, how mJZL"? ""^''' *^^° ^"'^'^■ poMat rrd tijip^s^ri « --^^^ ^^ i l^ow ,„an^ perches he wK 14?"'" '" ^•. ^ "--« «» W -!»«• 40 shares, 42 perches rem. WINE MEASURE '' '""="'"' ^ '""' "^ P-' -> how .an, .anons and pints, h.«, ^",^896 gallons of Camre L ^"'- ^ tuna, heads, and of each an equal nur^^ ^"^ """'? P>« "nd hogs. whow.:;tzen':'a:httadr ^-"^ ''^-^ *^e:uT ^n«. 28 d' «'«' "- >»», gallons and ^Zt^T''' ''■ '" '^ ^"S^''-* of ale, how maijaf- '''° ^** »<• In 108 barrels of ale, how many hogsheads f ^'"^ '"^ Ans, 74J 81. fc I Saviour'i 82. I many dj 83. F how raa 84. F and dayi Teachetl proporti( Rule, such ord the sam numbers tioued. the world, the ' 69 miles and jarley-corns. 8IN0LE KVLE OF THREE DIRECT. DRY MEASURE. 58 20 perches. Ans. 27. gf 37 acres. 1 «ire to know ins. 3521. f 75 perches l> 4 acres, 2 cres, 1 rood, n? ches rem. ^nd pints ? BO pints. • 5 tuns, and hop^- Qs over. f>f a pipe • desire to »f each. 75. In 120 quarters of wheat, how many bushels, pecks, gal- Jons and quarts ? Ans. 960 bushels, 3840 pecks, 7680 gallons, ?0720 qts. 76. In 30720 quarts of corn, how many quarters? Ans. 120. 77. In 20 chaldrons of coals, how many pecks? Ans. 2880. 78. In 273 lasts of corn, how many pecks ? Ans. 87360. TIME. 79. In 72015 hours, how many weels? Ans. 428 weeks, 4 days, 15 hours. 80. How many days is it since the birth of our Saviour, to "Christmas, 1794? Ans. 655258^. 81. Stowe writes, London was built 1108 years before our [Saviour's birth, how many hours is it since to Christmas, 1794 ? Ans. 25438932 hours. ! 82. From November 17, 1738, to September 12, 1739, how many days? Ans. 299. 83. From July 18, 1749, to December 27 of the same year, how many days? Ans. 162. 84. From July 18, 1723, to April 18, 1750, how many yearj and days? Ans. 26 years, 9770^ days, reckoning 365 days 6 hours a year. THE SINGLE RULE OF THREE DIRECT. 13248, ? JOqta. . loa* Teacheth by three numbers given to find out a fourth, in such proportion to the third, as the second is to the first. Rule. Fii-st state the question, that is, place the numbers in such order, that the fii*st and third be of one kind, and the second the same as the number required; then bring the fii-st and third numbers into one name, and the second into the lowest term mcn- tioued. Multiply the second and third numbers together, and e3 I •i ■ Mi llfi i IW M •INOLB BULB OP THREB DIRECT. EXAMPLES. 1. If 1 lb of sugar cost 4hd, what cost 51 lb? 1 : 4i : : 54 4 18 ~ -^w*- £l : : 3. 18 4)972 12)213 208. 3d. 2. If a gallon of beer cost lOd., what is that per barrel ? " -^ns. £1 : 10. 3. If a pair of shoes cost 4s. 6.. what will 12 dozen come to .' >ans. £32:8. , J; '* -«^ y-d of cloth cost 15s. 6d.. what will 32 yards cost at the saino •dns. £-24 : 10. 5. If 32 yards of cloth cost £24 : 16. what is the value of a >ard ? -^Jis. 15s. 6d. 8. If I give £4 : 18 for 1 cwt of ,„gar. at what rata did I bu, it per lb ! •/ins. luid. tJdL\)VlZu> '""°"'' "■=" '" ^"'- '"' »«'• M. per ell, ,vhat i. ^ns. £^50. 8. What will 23 cwt. 3 qr,. 14 lb. of tobacco come to. at I.',J,1. per lb f •^ns. £181 :3 :3 «„ou„Uof' "■' ""' °' ■""'""• "' «- »!''• P^' yard, what d„c. it An. £!) : 5 ! Oi, 2 rem. ^^10. Bough. 1, cwt. 1 qr. 14 lb. of iron, at 31 per lb. what doe, it come ^ns. £20 : 7 : Oi. 11. If coffee is sold for 5id. per ounce, what must be given for 2 cwt ' Jina. £82 : 2 : 8. 12. How many yards of cloth 'may be bought for £oi . , , , , / 3* yards cost £2: 14: 3? i„, 27"vard. Tnf- , m'J*' "^'^^"^ -^ns. 4 1 yards, J qrs. 1 nail, 84 rem 31^. Y ^ "'"*• ^'^ ^^''^''^ «^«««« <^«^t £1 : 14 : 8. what m...st F .fvn 'for •^ns. Is. 1(1. .o"e to/"""' ' ""• "* "•• « - °f "O "•0. V >•" ™ '■• """' <'"e» it ■dm. los. Uid. 112 rem. SINGLE BULE OF THREE DIRECT. 55 15. If a gentleman's income is £500 a year, and he spends lOs. 4d. per day, hi.w much does he lay by at the year's end ? jIhs. jCU? : 3 : 4. 10. It I buy 14 yards of cloth for 10 guineas, how many Flemish elLi can I l)uv for £283 : 17 : 6 at the same rate ? ^ns. 504 Fl. ells, 2 qrs. 17. if 504 Flemish ells, 2 quarters, cost £283 : 17 : 6, at what rate did I pay lor M yards ? j?«». IDs. lOd. 18. Gave £1 : I : S for 3 lb. of coffee, what must be given for 2rth of ot\» grain .' Jlns. 2^(1. 20 rem. 21. If 14 yards of broad cloth cost £9 : 12, what is the purchase of 75 yards ? ' Ans. 51 : 8 : G|, 6 rem. 22. If 27 yards of Holland cost £5 : 13 : 6, how many ells English can I buy for £100 ? Aits. 384. 23. If 1 cwt. cost £12 : 12 : 6, what must I give for 14 cwt 1 qr. 19 lb. Ans. £182 : : Hi, 8 rem. 24. Bought 7 yards of cloth for 17s. 8d. what must be given for 5 piecea, each coiitiiiiiing 27^ yards. A71S. £17 : 7 : 04, 2 rem. 25. If 7 oz. 1 1 dwts. of gold bo worth £35, what is the value of 14 lb. 9 oz. 12 dwt. 10 gr. at the same rate ? Ans. £823 : 9 : 3|, 552 rem. 20. A draper bought 420 yards of broad cloth, at the rate of 148. 10|d per ell English, how much did he pay for the whole ? An^. 250 : 5. 27. A gentleman bought a wedge of gold, which weighed 14 lb. 3 oz 8 dwts. for the sum of £514 : 4, at what rate did he pay for it per oz. ? Ans £3. 28. A grocer bought 4 hogsheads of sugar, each weighing neat 6 cwt. 2 qrs. 14 lb which cost him £2:8:0 per cwt. ; what is the value of the 4 hogsheads ? Ans. £04 : 5 : 3. 29. A draper bought 8 packs of cloth, 'each containing 4 parcels, each parcel V) pieces, and each piece 26 yards, and gave after the rate of £4 : 16 for 6 yards ; 1 desire to know what the 8 packs stood him to ? Ans. £6656. 30. If 24 lb. of raisins cost 68. 6d. what will IS frails cost, each weigh- ing neat 3 qrs. 1 8 lb. ? Ans. £24 : 17 : 3, 31. If 1 oz. of silver be worth Ss. what is the price of 14 ingots, each weighing 7 lb. 5 oz. 10 dwts. Ans. £313 : 5. 32. What is the price of a pack of wool, weighing 2 cwt. I qr= 19 lb. at 88. 6d. per stone ? Ans. £8:4: 64, 10 rem. 33. Bought 59 cwt. 9 qrs. 24 lb. of tobacco, at £2 : 17 : 4 per cwt. ; what does it come to ? ^n». £171 : 3 : 74 80 reia. ill! >'il:( 56 JtVLE OP THREE INVERSE. o f^^^^"2rlit 171 tons of lead nr Via and other incident clmro-es £4 ' in t ^'^ *''"' P^'^ ca"Taffe »ead, and what it stands me in p^r lb ? '^''''' *^' ^"^"^ ^^ ^ ^ 35 If a pair ^l^^^^^^f ^32 rem. per lb. I buy for i;43 : 5 ? ^^ ^^ 10 groats, how many dozen mav V 36. Bought ^7 do/en ^^ m c „^^*' ^1 clozen, 7A pair Pe' 3 lb. wLt did ZyU, '^,f -"^K '"fer the rj^m. . 31 If an ounce of fine gold i, .J"/' 5 ='« = **. 1 «"n. •ngob ^ eaoh ,vei?hi„g 3 ^ t ''/° 1 '"' f^ ^ IC «-lmt come V •""fo^ „ ;''■ !' S---, at the same 38. If my },e,se stands me in M T ■^,'°" = 1* = ^i- be the charge of 11 horses for the year P "^ ^''«' "''^' »•«' 89. A factor boH-ylit rk „• . ^"*- ^J58 : 18 : 6i 19 •• 4, at 4s. lod "^t va,r?/ ''"T' I!'™'' ^«' him i^iV : -ere. and how many 'e„s ^^J ^^^J"- -"7 y- ^i pair, fate of 17d. h 1 rem. lat come 7 It the same , what will I 18 : 6|. im £517 : ards there uarters, r annum, the year's quarterly rem. less re- is greatr han the is less er than ind di- propor- BULB OF THREE XNVEHSB. EXAMPLES. 57 1. If 8 men can do a piece of work in 12 days, how many [days can 16 men perform the same in? Ans. 6 days. 8. 12 . . 16 . 6 8 16)96(6 days. 2. If 64 mep can build a house in 90 days, how many can do the same in 60 days ? Jns. 97} men. 3. If, when a peck of wheat is sold for 2s., the penny loaf weighs 8 oz., how much must it weigh when the peck is worth but Is. 6d. ? Ans. lOf oz. 4. How many pieces of money, of 20s. value, are equal to 240 pieces of 12s. each? Ans. 144. 6. How many yards, of three quarters wide, are equal in mea» sure to 30 yards, of 5 quarters 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. 7. If for 24s. I have 1200 lb. carried 36 miles, how many pounds can I have carried 24 miles for the same money ? 4ns. 1800 lb. 8. If 108 workmen finish a piece of work in 12 days, how many are sufficient to finish it in 3 days ? 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 j£30, how many will £20 worth suffice, when the tun is sold but for £24 ? Ans. 126. 11. A courier makes a journey in 24 days, when the day is but 12 hours long, how many days will he be going the same journey, when the day is 16 hours long? Ans. 18 days. Ki ■ :■• m 6& nil! DOUBLE RULE OB THREE. 12 How much plush is sufficient for a cloak, which has in it Will 34 men take to do the same ? / ' ""' "^ny aays ^^^*- ^ d/ys; 4 hours, 56 min. ^V, at 12 hours for a day 14. Borrowed of raj friend £64 for 8 months inrl i.. l, 7' casion another time to borrow of me for T2 monTl . T must I lend him to requite his f^rm" ktdnelr me) ''" ""' Ans. 4166 yards, 2 qra. 2 nails, 2 rem. 1. If 14 )e sufiiciei 1. 2. By h As 1 da As ] 2. hen If 8r 3 be tc THE DOUBLE RULE OF THREE, Is so called because it is composed of 5 numbers ^iven to find n a supposition ; the two last, aidant '" ^''' ''""' ^'^ J. Place the other t»o terms under .heir"like in the supposi- 5^i^^:.rn-t^j^rr'rftt^i^- tY^^'Jlf"' b^ank foils under the first or second term. mnlnVl. PuooF. By two single rules of three. DOUBLE BULB OF THBEE. 59 EXAMPLES. 1. If 14 horses eat 56 bushels of oats in 16 days, how many bushels will )e sufficient for 20 horses for 24 days By two single rules, hor. bu. hor. bu. 1. As 14 . 56 .. 20 . 80 days bu. days. bu. 2. As 16 . 80 . . 24 . 120 or in one stating, worked thus : hor. days bu. 14 . 16 . 56 56X20X24 20 . 24 . — =120 14X16 2. If 8 men in 14 days can mow 1 J 2 acres of grass, how many men must Ithere be to mow 2000 acres in 10 days ? acres, days, acres, days. 1. As 112 . 14 .. 2000 . 250 days. men. days, men. 2. As 250 . 8 . . 10 . 200 me* ■Ia': — . 10 acres. . 112.8X14X2000 -^s^OO . 2000 112X10 3. If £100 in 12 months gain £6 interest, how much will £75 gain in |9 months. ^««- £3:7:6. 4. If a carrier receives £2 : 2 for the carriage of 3 cwt. 150 miles, how Imuch ought he to receive for the carriage of 7 cwt. 3 qrs. 14 lb. for 50 miles ? ' • ^ns. £1 : 16 : 9. 5. If a regiment of soldiers, consisting of 136 men, consume 351 quar- Iters of wheat in 403 days, how many quarters of wheat will 11232 soldiers consume in 56 days ? Ans. 15031 qrs. 864 rem. 6. If 40 acres of grass be mowed by 8 men in 7 days, how many acres I can be mowed by 24 men in 28 days ? Jins. 480. 7. If 403. will pay 8 men fox 5 days' work, how much will pay 32 men for 24 days' work ? -^ns. £38 : 8. ^ 8. If £100 in 12 months gain £6 interest, what principal will gain £3 : 7 : 6 in 9 months ? ^»J^ £75. 9. If a regiment, consisting of 939 soldiers, consume 351 qrs. of wheat in a 168 days, how many soldiers will consume 1404 qrs. in 56 days .' Ans, 11268. 10. 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 in S days ? -^ns. 2 kil. 12 gal. 11. If the carriage of 60 cwt. 20 miles, cost £14 : 10, what weight can I have carried 30 miles for £5 : 8 : 9, at the same rate of carriage ? Ans. 15 cwt. 12, If 2 horses eat 8 bushels of oats in 16 days, how many horses will eat uu 3000 nuarters in 24 davs ? Ans. 4fX)0. 13, If £100 in 12 months gain £1 interest, what is the interest of £571 for 6 years ? Jlns. £239 : 16 : 45, 20 rem. Pa' ■JjH, i i#: mm 60 I*. a PRACTICE. Ans. £9:2: 0^. PRACTICE All questions in tliis rule are performed bv UVn.c. nV . Of a Pound. s. d. 10 : ..is...^ 6:8 ^ 5 4 3 2 2 1 .... .1 4 f « I _i_ 8 _i Of a shilling. G Of a Ton. cwt. 10.. is.. 4 Of a Hundred j qrs. lb. " ..i^...| M".. IS... i 2 or 56 is i 3 2 5. . 1 ••§■ ••12 u w Of a Quarter. To" I ^4 lb 1 1 II 4 1 3i I -5i^£,t- ^;:vrai?i:3t ? O^isi)57041b.ati 12)142G 210)1118:10 Facit,i;5:18:10 ^695 at ^ Facit, £16:0:7^ O 5470 at ^ Facit, £11:7:11 C) 6547 at I Faci^\£20 : 9 : 2^ C) 4573 at f Facit, £14 : 5 : 9f Tjizx^-i-'^'^^.^-'Ss^ti i« tV ' 2 Facit, i l^is i 2I( 1 Facit, £1 '(») 5432i Facit, £3 (*) 6254 Facit, £^ C) 2351 Facit, £] C) 7210 Facit, £C 2710 Facit, £i (') 3251 Facit, £i {') 2715 Facit, £t {'") 700'. Facit, £ (") 214^ Facit, £1 (") 700< Facit, £ PRACTICE. 61 es, what must! £9:2: Oi. [') is tV '754'7 at Id. sons concern-j g aliquot, or| are avoided; )f a Hundred. I rs. lb. 2 or 56 is ^j 1 or 28. ..i a4...| f a Quarter.! t lb 1: ^ I: •••... .4; 2 •••••••J I vide by the j 5, it will 1:^ I 210)6218 : 11 Facit, £.31 : 8 : 11. (»)lisyV375latUd i is i 312 : Y •78 : li 210)3910 : 8t Facit, £19 : 10 8i (") 325*7 at 4d. Facit, £54 : 5 : 8. ('*) 2056 at 4|d. Facit, £36 : 8 : 2. {") 3/52 at 4^d. Faeit, £70 : 7 : 0. n 2107 at4^d. Facit, £41:14:0^ at 1 JO : 9 : 2i 1 at f 4 : 5 : 9f 1 fe the ali 'ether, and (*) 54325 at l^d. [Facit, £339 : 10 : 7^. (*) 6254 at l^d. Facit, £45 : 12 : 0^ {') 2351 at 2d. Facit, £19 : 11 : 10. C) 7210 at 2id. Facit, £67:11 :10i (') 2710 at 2^d. Facit, £28 : 4 : 7. 3250 at 2td. Facit, £37 : 4 : 9^. (») 2715 at 3d. Facit, £33 : 18 : 9. n 7062 at 3id. Facit, £95 : 12 : 7^ (") 2147 at 3^ Facit, £31:6 : 2^. (") 7000 at 3§d. Faeit, £109 : 7 : 6. (") 3210 at 5d. Facit, £66 : 17 : 6. n 2715 at5H Facit, £59 : 7 : 9f , n 3120 at 5id. Facit, £71 : 10 : 0. {'") 7521 at 5id. Facit, £180 : 3 : 9i (") 3271 at 6d. Facit, £81 : 15 : 6. (") 7914at 6id. Facit, £206 : 1 : 10^ (") 3250 at 6|d. Facit, £88 : : 5. (") 2708 at 6 Id. Facit, £76 : 3 : 3. n 3271 at 7d. Faeit, £95 : 8 : 1. (") 3254 at 7id. Facit, £98:5 : Hi (-') 2701 at 7^d. Facit, £84 : 8 : 1^.' F (") 3714 at 7|d. Facit, £119: 18:7^. (") 2710 at 8d. Facit, £90 : 6 : 8. ('") 3514 at 8id. Facit, £120: 15 :10J. {'') 2759 at 8^d. Fiicit, £97 : 14 : 3^. n 9872 at 8H Facit, £359 : 18 : 4. (") 5272 at 9d. Facit, £197 : 14 : 0. (") 6325 at 9id. Facit, £243 : 15 : 6^. {'') 7924 at 9^d. Facit, £313 : 13 : 2. (="■) 2150 at 9^d. Facit, £87 : 6 : lOf ('') 6325 at lOd. Facit, £263 : 10 : 10. n 5724 at lO^d. Facit, £244 : 9 : 3. n 6327 at lO^d. Facit, £270 : 4 : 3|. (") 3254 at lO^d. Fiicit, £142 : 7 : 3. {*') 7291 at 10|d. Facit, £326 : 11 : 6^. {'') 3250 at lid. Fiicit, £149 : 4 : 8. ft. § ■ 11' if'l 03 Fa PBACTICB. ;■) 72S4 at md. |n3754atlUd. inW2aMTW act, £340 : : 7J. I Facit, £179 : 17* : 7. j hlll'lloo :^fh. li^''Z!',T'Z "'" P™" '' """•' """> o"" «Wlling,mKl less Price ^ ?; ; f " !'"'■.,•;.'• f^"*-"' »i"' «« n.ucl,«f1he ri v^ pnce Oh IS more than a s ii n.r wlii.-li ■i.l.l t„ ii,„ • ° • and divide by 20, it will give & anlwe.' ' ^'''" ^""'^^^^' OilV2106atl2|d, 43 : 10| 210)21419 : 10^ Facit, £107:9: 10^. C) 3215 at Is. l|d Facit, £177:9 :10i (') 279"oi;nri7d7 Fa^ ^^^^ 15 ib. of Mala^ra raisins ? 0. Bought 78 cwt. 3 ai-8. cwt., what did I give for the ^vhat comes IV cwt. 1 qr. 17 lb. of „ ^W5. £21 .10.8 lb.ofcheese,at £1.7.8 per cwt, ^W5. £118.1 0* cwt., what must be given for 72 cwt '^iia. 13 the value of 27 cwt. 2 qrs 19 n ^ ^''*. £2D. 9.0? ' ^^^^^''^' ^?w. £227. 14^ TARE AND TKET. 6Y It I. Sold 56 cwt. 1 qr. lY lb. of sugar, at £2 : 15 : 9 the cwt., ehat does it come to ? ^ns. £157:4; 4^ 8 Tobacco at £3 : 17 : 10 the cwt, what is the worth of 97 Ut"l5lb.? Ans.£Sl8 :0'.S. 9. At £4 : 14 : 6 the cwt., what is the value of 37 cwt. 2 qrs. 13 lb. of double refined sugar? „ ^^ . , Ans. £177 : 14 : 8^. 10. Bouo-ht suirar at £3 : 14 : 6 the cwt., what did I give foi |15 cwt. 1 q". 10 lb.? ^ns. £57 : 2 : 9. II. At £4 : 15 : 4 the cwt., the value of 172 cwt. 3 qrs. 12 lb. |of tobacco is required? ^ns. £823 : 19 : 0^ 12. Soap at £3 : 11 : 6 the cwt., what is the value of 53 cwt \l^\\jj Jns. £190 : : 4. TARE AND TRET. i rr 11 . i The allowances usually made in this Weighty are Tare^ Tret^ and Cloff. .CI Tare is an allowance made to the buyer for the weight of the box, barrel, bag, &c., which contains the goods bought, and is eiAher At so much per box, bag, barrel, 10. At 10 14.' 11. At 96 12. At £ Stock ? INTEREST. 71 4. What is the interest of £547 . 15, at 5 per cent, per annum, for 3 ly^ars • ^ns. £82 .3.3. 5. What is the interest of £254 . 17 . C, for 5 years, at 4 por cent, per I annum? ^«s. £50 . 19 . 6. 6. What is the interest of £556 . 13 . 4, at 5 per cent, per annum, for 1 5 years ? Ans. £139 . 3 . 4. 7. My correspondent writes me word, that he has bin ^' a year, so arc the weeks given for thoSsf^lfJ''^'' "'" pJl^r^^:,^^'^"^' "' '''' ■■''■■' f- 20 weeks, at 22. What is the amount of £375 • 6 • ftAV '^= ^"i' . per cent, per a„n„™. it' ii^TO :t; S^! When the Interest is for any number of days. Rule. As 305 days are to the interest „f (1>„ „: year, so are the da/given to the i"4"^cd^ ™" '"'" "^^ 2 .'7 Vot 5 yii:: m d^; r"™- "'='' '^ *•= ""--' »f ^»8s 24. mat is the interest of £2726 1 1"ft 4?^^ ' '^ V^' i annum, f„r three years, 154 days ? •■'•*•■" ^s per cent, pej ^w. £419 . 15 . 6i When the Amount, Time and Hate per cent, are yiven to find t/ie r'nncipal. to £100 '^\ol r'""' "^/- '' ""^ ^^^ '^'^ ^"^ time .iven • i. to £100 . . so IS tlie amount given : to the principal required ' 25. What principal beinjr put to interest, will amount >^ Pim i 10 m 5 yeai-s, at 3 per cent, per annum ? ^ ^^^^ '^ 3X6+100=£I15. I00..402 10 20 20 2300 8050 100 23!00)8050I0Q(£;?50 Ans. INTEREST. 73 26. What principal being put to interest f^r 9 years, •will imount to £734 : 8, at 4 per cent, per annum? Ans. £540. 27. What principal being put to interest for 7 years, at 5 per cent, per annum, will amount to £334 : 16? Ans. £248. When the principal, Bate per cent, and Amount are given, to find the Time. Rule. As the interest of the principal for 1 year : is to 1 year : : |bo is the whole interest : to the time required. 28. In what time will £360 i aiount to £402 . 10, at 3 per cent, [per annum ? 360 As 10 . 10 ; 1 : : 52 . 10 : 6 3 20 20 10150 210 21i0) 10510(5 years. ^ns. 402 . 10 20 105 350.10 10100 52.10 29. Tn what time will £540 amount to £734 : 8, at 4 per cent, per annum ? Ans. 9 years. 30. In what time will £248 amount to £334 : 16, at 5 per cent, per annum ? Ans. 7 3^ears. When the Principal, Amount, and TimCy are given, to find the Rate per cent. Rule. As the principal : is to the interest for the whole time : : BO is £100 : to the interest for the san^j 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 m 6 years' time ? 350 As 350 : 52 . 10 : : 100 : £15 20 52 . 10 1050 100 36i0)10500iO(300s.=£l5-f-5 = 3 per cent. 32. At what rate* per cent, will £248 amount to £334 : 10 in 7 years' time ? Ayis. 5 per cent. }^. h: »; f ' i 74 INTEREST. 33. At what rate per cent, will £540 amount to £734 : 8 in 9 yeai-s'time? .4/is. 4 per cent. COMPOUND INTEREST, Ts that which arises both from the principal and interest; that is, when the interest on money becomes due, and not paid, tlie 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 k^fore, 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. ^Y[\at is the compound interest of £500 forborne 3 years, at 5 per cent, per annum ? 500 500 • 525 ^25 26 . . 5 25100 525 = 1st jesv. 55 1 .. 5 = 2d year. 5 2G125 20 5l00 2*7156. 20 11125 12 — 551 ..5 5 27. 11. .3 578. 16..3=3dyear. prin. sub 500 3!00 78 . 16 . . 3=inter.for3years. 2. What is the amount of £400 forborne 3^ years, at 6 per cent. \m' 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 yeara and 6 months, at 6 per cent, per annum, compound interest ? Ans. £675 : 6 : d. • K 5. What is the compound interest of £764 for 4 years and 9 months, at 6 per cent, per annum ? Ans. £243 : 18:8. 6. What is the compound interest of £57 : 10 : 6 for 5 yeare, 1 months, and 1 5 days, at 6 per cent, per aiinum ? Ans. £18 : 3 : 8f. REBATE OR DISCOUNT. 75 7. What is the compound interest of £259 : 10 for 3 years, 9 months, and 10 days, at 4^ per cent, per annum? Jns. £46 : 19 : 10^. 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 diseharge a debt of £105, to be paid a year to come, rebate being made at 5 per cent. Rule. As £100 with the interest for the time given : is to that interest : : so is the sura given : to the rebate required. Subtract the rebate from the given sum, and the remainder will be the present worth. EXAMPLES. 1. Wliat is the discount and present worth of £487 : 12 for 6 months, at 3 per cent, per annum ? em=^6 . 3 100 103 As 103:0;:487 : 12 20 20 2060 9752 3 487 : 12 principal. 14 : 4 rebate. £ s. 20610)292516(14 . 4 rebate. 206 Ans. £473 : 8 present worth. 865 824 416=48. 2. What is the present payment of £357 : 10, which was agreed to be paid 9 months lience, at 5 per cent. })er annum ? Ans. £S4 4:11:7. 3. What is the discount of £275 : 10 for 7 months, at 5 per cent. ])er annum ? g2 Ans. £7 : 16 : li ri f-| I J 78 Hiii m EQUATION OF PAYMENTS. 4. Boui^ht goods to the value of £109 : 10, to be naid if ni'nJ months what present money will discharge th'e same,^i I 1 "] ' lowed 6 jer cent, per annum discount ? , x am ai 7 Sold goods for £875 : 5 : 6, to be paid f Inths tte what IS the present worth at 4^ per eont. ? ' 8. Wlfat 18 the present worth of £500, payable in 10 months at 6 per cent, per annum ? ^^, j.^'^^ d„!'i . "" T'^ '"''^^ "'^"'>' '-''" ^ ^^^«i^'^ for a note of £75 due 15 months hence, at 5 per cent. ? ^^ ^'o, in WK . Ml 1 , ^^*- ^'^0 : 11 : 9i. 10. H hat will be the present worth of £l50, payable at 3 four months ^. .. one third at four months, one thi rd^^Ts Inhs and one third at 12 months, at 5 per cent, discount? ' 11 an J X ., , . ^ns. £145 : 3 : 8^. 11. Sold goods to tbe value of £575 : 10, to be paid at 2 three months, what must be discounted for prese'nt payment, at 5 per 12. VVhat is the present M-orth of £500 at 4 per cent £lO0 being to be paid down, and the rest at 2 six montli ? ' ^ns. £488 : 7 : 8^. EQUATION OF PAYMENTS, Is when several sums are duo at different times, to find a mean fame for paying the whole debt; to do which this' is the common" RiTLE. Multiply eacli term by its time, and divide the sum of the product, by the whole debt, the quotient is accounted the tDcali U1I10> KQUATION OF PAYMENTS. n t] EXAMPLES. 1 1. A owes B £200, whereof £40 is to be paid at 3 months, I £00 at 5 months, and £100 at 10 months; at wluit time may the whole debt be paid together, without prejudice to either? I;. £ m. 40 X 3 = 120 60 X 5 = 300 100 X 10 2l( = 1000 30)14120 7 months tV. 2. B owes C £800, whereof £200 is to be paid at 3 months, £100 at 4 months, £300 at 5 months, and £200 at 6 months; but they af^reeing to make but one payment of the whole, I de- mand what time that must be ? Ans. 4 months, 18 days. 3. I bought 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 pay £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 ir ide^ Ans. 8 months, 12 day^-f 5. H is indebted to L a certain sum, which is to be paid at dilforent payments, that is, ^ at 2 months, -^ at 3 months, ^ at 4 months, ^ at 5 months, |^ at 6 months, and the rest at 7 months j but they agree that the whole should be paid at one equated time* what is that time 2 Ans. 4 months, 1: quarter. , 6. A is indebted to B £120, whereof i is to be paid al 3 months, ^ at G months, and th.e rest at 9 months; what is the equated time of the whole payment ? Ans. 5 months, 7 days. g3 ^ f«^ hi 'f^' 78 BARTER. BARTER Is the exchanging of one commodity for another, and informjl the traders so to proportionate their goods, that neither raaj sustam loss. J Rule 1st. Find the vakie of that commodity whose quantitv 18 given ; then find what quantity of the other, at the rate prd posed, you may have for the same money. f 2dly. When one h.i-^ goods at a certain price, ready money, but in bartering, advanct. it to something more, find what the other ought to rate his goo. at, in proportion to that advancJ and then proceed as before. ^' m EXAMPLES. n;:i! !!i| 1. What quantity of chocolate, at 4«. per lb. must be delivered in barter tor 2 cwt., of tea, at Us. per lb. ? 2 cwt, 112 224 lb. 9 price. 4)2016 the value of the tea. 504 lb. of chocolate. 2. A and B barter ; A hath 20 cwt of prunes, at4d. per lb. ready moneyj but in barter will have 5d. per lb. andf B. hath hops worth 32s, per cwt,, ready money ; what ought B. to raU his hops at in barter, and what quan- tity must be given for the 20 cwt., of prunes ? ' 112 As4:5::33| 20 5 s. 40 12 2240 5 -cwt. qr. lb. 4)160 408. 4810)112010(23 . 1 . g'll Ans. 96 *' 160 144 10=1 qr. 9 lb. j^ must receive the difference and hfw^iteh^' ^""^ '^'"'' ">'<"°*"l>'l Jl7is. B. miiflf rerpivp of A Xfo o J'ntl Im ^ner'lh'.' if ^''^1 ^^' "^ "P'^^'' ^^ ^^id. per lb. ; B hati gin- 1 pepper.' ^ ' ""^ """"^^ Sanger must he deliver in barter forV '^ns. 3 lb. 1 ozM ..rfc.l PKOFIT AND LOSS. 7» 6 How many dozen of candles, at 53. 2d. per dozen, must be delivered in barter for three cwt. 2 qrs. 16 lb. of tallow, at 373. 4d. per cwt. .' ' J^ns. 26 dozen 3 Ib.fJ 7 A hath 608 yards of cloth, worth 14c. psr yavi, for which B giveth I him £125 . 12. in ready money, and sr. cwt. " -irs. 24 lb. of bees'-wax. The question is, what did B reckon his be s'-wi : at per cwt. ? 8 A and B barter ; A hath 320 dozen of cauJies, at 43. 6d. per dozen ; for which B giveth him £30 in money, an«' m • rest in cotton, at 8d per lb : I desire to know how much cotton I' ,y> ■ ; .^ besides the money ? ■' Ans. 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. z, ^o lu i a Ans. 48 Ib.yf. 10. C hath nutmegs worth 7s. 6d. per lb. ready money, but in barter will have 8s. per lb. ; and D hath leaf tobacco worth 9d. per lb. ready money ; how much must D rate his tobacco at per lb. that his profit may be equiva- t • v#^^ 11 PROFIT AND LOSS Is 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. M9I EXAMPLES. 1. If a yard of cloth is bought for Us. and sold for 12s. 6d. what is the gain per cent. ? As 11 : 1 : 6 : : 100 12 12.3 U .0 1.6 18 20 2000 18 11)36000 12)3272y\ 210)2712 . 8 Ana. £13 . 12 . SrV 2. If 60 ells of Holland cost £18 what must 1 ell be sold for to gain 8 per cent. ? As 100 : 18 : : 108 108 1100)19144 20 8180 12 0160 12X5=60 12)19 . 8 . 9i 5)1 , 12 . 41 . 6 . 54 2140 -Ans. 6s. 5|d 1:1 ,t!i I V HI Ml 80 FELLOWSHIP. ^J. ^ rf 1 lb. of tobacco COS. m. and i, ,„ld fov 20d. wha. i, ,he gain pj wasV/^-ird,"'. ''°"' "" '"" f" ^5«»- -d a. 12 per cenf ^-In^^i. J ., 5. If a yard of cloth is bou-ht for 13s 4d ,n,l a m • '^.^' '^^'^^- I 18 the gain p^i- cent. ? ^- *"^ ^^^'^ ^gam for l(5s. whati 6. if Hi lb. of iron rn<-t oia ca k . -^>^i- £20. I 15 per cent. ? °'^ ^^' ^'^•' ^^^^ must 1 cwt. be sold for to g^inl 7. If 375 yards of broad cloth be sold for £icn f^"' ^^ ■ ^^ ■ n In! it cost per yard ' ^""^ '^'^^^^ ''"'^ 2U per cent, profit, " " ' Ans. jgl . ] . 9 , 8. Sold 1 cwt. of hona for 4>a ^K * xu '^"*- j£l . 1 . 9i What would have been he ^at pi? cent' .f I td ''!', t '' .P^*- ^^"t" P-""'!^ %am percent, it 1 had sold them for jES^per cut ' . 0, If 90 ells of cambric cost £m \, u *^'**- ^^48 . 2 . li i. ' gain 18 per cent. .' ^ '^^"' ^°^ "^"^^ '""^t I sell it per yard to 10 A plumber sold lOfotherof leid for ^oni i. , ,_ *^'**- ^-«- "''f'- cwt.) and gained after the rate of i?9 in ^'^ ' ^^' ^^^^ ^^^^er being 'M per cwt. .^ ^^^^ "^ ^\2 . 10 per cent. ; what did it cost bin! _ 11. IJought 436 yards of plntK „«^ li. ^ ■^w«. 18s, 8d ■t ro. ,0. u. pe.. jj] wbt:!'i^Ve'"yir„f1b?•wti^/ '-"' -" -" 13. Bought 124 yards of linen for i-J?' I, ^ ,. ^ns. £W2 \o»s. per yard to gain 15 per cent " ' ^ ''^""^^ ^'^^^ ^^^^^ l>^ «-etailed ^/«. £10.7.6 proat, and 25 per cent. ' FELLOWSHIP * proportion to Ws'i.rinciplirroS.tocr "' "" «"'" "' '-' - cJftor'r^rau: i::;.tfI;T r^ *? *"<'^<^ -"-g^t i- ciency of a^sete or cftS. ^ "^J"'"'' "''^" ">«■* » "d* FELLOWSHIP IS Eir„™ WITH OH WITHOUT TIME. FELLOWSHIP WITHOUT TIME Proof A. 11 ij *i i " ^^ "^'^ ^'^''^'"<^ ^f the ffain or loss th« gain or la« ; to l,i« sh^l i, tl' ' "^ " '"'^' "'^"' ^•'«' ' euaic UJ A 60 : 50 21 ' 6IO)100li £16 . 13 . 2. Tfireer I and C £40 ; 3. A, B, a j C £500 ; am 1 to his stock : 4. Four n ■] £349, D£l ! iaerchant's s 5. Three !v3 £4(50, a what is eacl 6. A mer and to E £I but £675 . : At 7. Four \ C ^, and D man's share 8. Two \ £27,200, w tax ; where of the mon the said est M FELLOWSHIP. EXAMPLES 81 1. Two merchants trade together; A puts into stock £20, and B £40, Ithey gained £50 ; what is each person's share thereof? A 60 : 50 ! : 20 20 610)10010 £16 . 13 . 4 20+40=60 As 60 : 50 : : 40 40 33 . 6 . S, B's share. 16 . 13 . 4, A's share. 610)20010 £33 . 6 . S 50 . 0.0 proof. 2. Ttiree merchants trade together, A, B, and C ; A puts in £20, B £30, and C £40 ; they gained "ISO : what is each man's part of the gain ? Ans. A £40 ; B £60 ; C £80 3. A, B, and C, enter into partnership ; A puts in £364, B £482, and C £500 ; and they gained £807 ; what is each man's share in proportion to his stock ? Ans. 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 K £439 ; in trading they gained £428 : I demand each merchant's share of the gain ? Ans. B £85 . 19 . 6^—690 ; C £132 . 3 . 9—120 ; D. £43 . 11 . U— 250 ; E £166 . 5 . 64—70. 5. Three persons, D, E, and F, join in company ; D's stock was £750, E's £4(50, and F's £500 ; and at th'^ end of 12 months they gained £684 : what is each man's particular share of the gain ? A?is. 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 upon his decease, bis 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 £215 . 18 . U— 15750 • D's £122 . 16 . 21—12227 ; and E's £84 . 5 . 5—15620. 7. Four persons trade together in a joint stock, of which A has i, B 4, 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 . lOi— 24. S. Two persons purchased an estate of £1700 per annum, freehold, for £27,200, when money was at 6 per cent, interest, and 4s. per pound, land- tax ; whereof D paid £15,000, and E the rest ; sometime after, the interest of the money falling to 5 per cent, and 2s. per pound land-tax, they sell the said estate for 24 pears' purchase ; I ^csirc to knovv eaen person's shaie ? Ans. D £22,500 ; K £18 "00. . \M i l«i liJi >H 01 « "2 7* "4 2T* FELLOWSHIP. Stocks^; ^fiTv^ ^'i^li; *^''"' "^'^'' '"^ ^'^^^'^ ^^^ ^"^^^""t of theiJ stocks IS £647 and they are m proportion as 4, 6, and 8 arc to one another, and the amount of the gain is equal 'to D^s stock what IS each man's stock and gain? , ' Ans. D's stock £143 . 15 . 6J-f gain, 31 . 19 J-s 215. 13 .4 47. 18 Fs 287.ll.iyL.... 63.18 10. D, E, and F, join stocks in trade; the amount of thpirl :^cLrs IT; ^' ^^^^ "^' ^''^ ^^' -^ ^'^ ^« -^- - ^n.. D's stock £18.15; E's £31.5; and Fs £50. FELLOWSHIP WITH TIME. time . IS to the whole gam or loss ; : so is each man's product • to his share of the gain or loss. proauct . Pboop. As in fellowship without time. 2,', M, EXAMPLES. month^ and e1*?'. 'T r'"^*^^^?; ^ ?"*« in £40 for three montns, and h ^75 for four months ; and they gained £7o • what IS each man's share of the gain ? ^ ^ ' Ans. D £20, E £50. 40X3 = 120 As 420 : 70 : : 120 As 420 • 70 • • '^(\c^ 75X4=300 120 -«^s ^-iu . /u . . 300 420 4210)840)0(20 840 4210)210010(50 2100 2 Three merchants join in company; D puts in stock £195 tl mat's ;tw!:f IL git ' ''''' ''''''' ^''' '''-' -^-^ % Ans. D's £102 . 6 . 4-5008 ; E's £148 . 1 uJ! -o«ou;;; HOu i' a JE;n4 . 10 . 6| — 14707 ALLIGATION. 3. Three merchants join in company for 18 months; D put in £500, and at five months' end takes out £200 ; at ten months' end puts in £300, and at the end of 14 months takes out £130 : E puts in £400, and at the end of 3 months £270 more ; at 9 months he takes out £140, but puts in £100 at the end of 12 months, and withdraws £99 at the end of 15 months: F puts in £900, and at 6 months takes out £200^; at the end of 11 moiiths puts in £500, but takes out that and £100 more at the end of 13 months. They gained £200 : I desire to know each man's share of the gain ? Ans, D £50 : 1 : 6— 21*720 ; E £62 : 12 : 5^—29859 ; and F £87 : : 0^—14167. 4. D, E, and F, hold a piece of ground in common, for which they are to pay £36 : 10 : 6. D puts in 23 oxen 27 days; E 21 oxen 35 days; and F 16 oxen 23 days. What is each man to pay of the said rent ? Ans. D £13 : 3 : U— 624; E £15 : 11 : 5—1688; and F £7 : 15 : 11—1136. ALLIGATION ALLIGATION IS EITHER MEDIAL OR ALTERNATE. ALLIGATION MEDIAL Is when the price and quantititc *»f several simples are given to be mixed, to find the mean p-ice of that mixture. Rule. As the whole .position : is to its total value : : so is any part of the oompo'^iuo: : to its mean price. Proof. Find the value of the whole mixture at the mean rate, and if it agrees with the total value of the several quantitias afc tb<'ir respective prices, the work is right. 84 ALLIGATION. li EXAMPLES. As 96 : 288 : : 1 : 3 20X5 = 100 36X3 = 108 40X2= 80 06 288 Ans. 38. 8cl. per irillon nnrl o^ i ^f ,'. ^^ ^^^^^"^ ^^ «^eny, at 6s. eJ- f ,r-^ --g'^d * "wt. of sugar, at 56^ I'^w'/tlt 7 value of 8 hu^i^l^lAi^J.f ^' <'™^'^>-- ™-' - 'he 5. If I mix 27 bushels of wheat at s'^''fi/,'„;^' ^^'\'''^-- .. 6. A vintner mixes 20 gallons of Dort .i t^'lf' ^^'^^f,' with 12 gallons of white Jfe" 1^"V p^fiVn %^rt-"^""'' li"' \^ gaiiorof^ij^, f "r6d per gallon. What is a gallon of this mixture worth ? -7. A refiner having 12 lb. of silver bullion ^T ^'' !^'^-^^- , , melt it with 8 lb. of 7 oz. fine and J lb nf « ^'' ^"'' ''^"^^ the iiaeness of 1 lb. of that 2ture ? ' ''' ^"' ' ^^^"""^^ .8. A tobacconist would nnx 50 lb^of\ob'4o^f n /' ^"•,. with 30 lb. at I4d per lb 2-. lb -.f ooi ^"^^^9^' "^, ^^^'- pt^'' 'b. per lb What .vn in V *^ ^'^' P^' '^' ^"^^ ^7 lb. at 28. per 10. Wiiat will 1 lb. of this mixture bo worth « -n-iii io; it- i.ii ALLIGATION. 85 J bushel, and s of bailey, I bushel of per gallon, ei-iy, at 6s. per gallon. 2^cl.|f tvt. with 1 em-aiid the .8.9. t*28s. per I", and 24 lat is the 2icl.A. '^hel, with ushels of lel of this 31* gallon, ■allons of t 4s. 6d. f.\ ;.;•'■! ^ '-li ^ ' ; rJtm „ii itl ht.i'ir I ' 'IB'' '.II' 86 ALLIGATION PARTIAL. 8 1 desire io know how much tea, at ICs Us 9s anrl ft« per lb, w,ll compose a ,nixturo worth lOs. per Jb. ? ' ' ^i^^. lib. at IGs, 2 lb. 14s., 6 lb! at 9s., and 4. A t^tr.ner would mi^ T ,S ^^i^ -^ 'V!^!/ ^'- 1 ']' , rye at 4s. per bushel, and oat«l Ss tVbutl t TJ ^2"^' fixture worth 2s, Cd. per bushel. 11^^ .tlrl^^l^ 7tt Jns. G busJiels of barley, 6 of rve and SO nf nnfc 6. A tobacconist would mix tobacco at 2s Is fid «nfl U ^A per lb., so as the compound mav bear a dHop of u «1 ^u What quantity of each sort must he take' ^ ' '^* P'' ^^• Ans. 1 lb. at 2s., 4 lb. at Is. 6d., and 4 lb. at Is. 3d. ALLIGATION PARTIAL, 01 them and the mean rate are given to find the several auanti- ties of the rest m projiortion to that given ^ *>,/' !^''/];^^'T.'?r ""^ ^^'^^ '""P^« ^^^'osP qviantity is given • to he rest o the d.tFerences severally : : so is the quantity^ ven '• to the several quantities required. ^ ^ EXAMPLES. ,♦ 1;/ tobacconist being determined to mix 20 lb. of tobacco at I5d. per lb., with others at IGd. per lb., 18d. per lb and 22d per lb.; how many pounds of each sort must he tak"; to m'ke one pound of that mixture worth 17d.? Answer. ^root 16 5 20 lb. at 15d. = .^ood. Ag 5 ,h1C_ ^M8-J 22 1 4 lb. at IGd. = 64d. As 6 1 4 1b. at 18d. =. 72d. As 5 2 8 lb. at 22d. = 170d. 1 : : 20 : 4 1 : : 20 : 4 2 : : 20 : 8 Ans, 30 lb. ■ .1 ii\i. ALLIGATION TOTAL. 87 2 A farmer would mix 20 bushels of wheat at 60d. per bush- el with rye at 36d., barley at 24a., and oats at 18d. per bushel. How much must he take of each sort, to make the composi- tion worth 32d. per bushel ? , , , ,, H^ V. u 1 Ans. 20 bushels of wheat, 35 bushels of rye, 10 bushela of barley, and 10 bushels of oats. 3 A distiller would mix 40 gallons of French Brandy, at 12s. per * gallon, with English at 7s., and spirits at 4s. per gallon. What quantity of each sort must he take to afford it for 8s. per ^^ ^^ Ans. 40 gallons French, 32 English, and 32 spirits. 4 A grocer would mix teas at 12s., lOs., and 6s., with 20 lb. at 4s. per lb. How much of each sort must he take to make the composition worth 8s. per lb.? Ans. 20 lb. at 4s., 10 lb. at 6s., 10 lb. at lOs., 20 lb. at 12s. 5 A wine merchant is desirous of mixing 18 gallons of Ca- nary, at 6s. 9d. per gallon with Malaga, at 7s. 6d. per ga on, shm'v at 5s. per gallon, and white wine at 4s. 3d. per galloni How much of each sort must he take that the mixture may bo sold for 6s. per gallon 1 , ^ ,r i -. oi ^ ci Ans. 18 gallons of Canary, 31^ of Malaga, 13i of Sherry, and 27 of white wine. ALLIGATION TOTAL UAI Is when the price of each simple, the quantity to be compound- ed, and the mean rate are given, to find how much of each sort will make that quantity. Rule. Take the difference between each price, and the mean rate as before. Then, , .. , j/r As ihe sum of the differences : is to each particular ditter- •nee : : so is the quantity given : to the quantity required. ft EXAMPLES. 1. A grocer has four sorts of sugar, viz., at 12d., lOd., 6d., and 4d. per lb.; and would make a composition of 144 lb. worth 8d. ^-I tv T ;i.vo;-,. ♦/-. hnfxxv vahiit nuaiititv of each he must take? ^ h3 r 11 I- 88 fm POSITION, OB THE HUIB OP FALSE. Proof. 48 at ]2cl. 576=As 12 24 at lOd. 240= As 12 24 at 6d. 144=A8 12 48 at 4d. 192= As 12 4 2 2 4 144 144 144 144 48 24 24 48 52 144 )ll52(8d. Wlmt quantity must there be of each ? ' ''' ^' "^ of 60 gallons to he w Jl, I ' ,V '"'"''^ "^''^ » '"'''•«'« each mu?t helkt?'^ ^°"'' ''• f'' S""""- What quantity of ^'"' t'^nlT fij}'*^ ^'".^' ^ S'tlloos of Flemish, I . 6 gallons of Malaga, and 5 gallons of Canary. *■ A silversmith had four sorts of jrold vW of Oi „ . fine nf 99 9n nr.A i c . /» e*^'^* viz., oi zi carata eacrs:rf,:,^eirso'a,rot™ "42:^7,1^frr"'t"f much must iTe take of each < ' ""'*'' ^"'- H"'' Ans. 4 0.. of 24 4 oz. of 22, 4 oz. of 20, and 30 oz. ot 15 carats fine. take for each parcel? ^ ^""^ """='' "^ ^'^'' «»'' *d he ^«». 12 lb. of 8.,. 80 lb. of 8s. 8 lb. of 5s. « lb. of 5,_ J lb. of 4s. 6 lb. of ^ 28 lb. at 69. per lb. 42 lb. at Ts. per lb. POSITION, OE THE RULE OP FALSE, *s;vr^\*''Ui' t' r:;3™?t "'""^^•.'"^«" «' P'"-"- SiHGM and Double ^ ^' «. divided into two parH «?*t*Si«r?^5| POSITION OR THB RULE OF FALHH. 69 SINGLE POSITION, [s, by using one supposed number, and working witb it. as the true one, you find the real number required, by the following Rule. As the total of the errors : is to the true total : : so is the supposed number : to the true one required. Pjioof. Add the several parts of the sum together, and if it agrees with the sum it is right. EXAMPLES. 1. A schoolmaster being asked how many scholars ho had, said, If I had as many, half as many, and one quarter as many more, I should have 88. How many had he ? Ans. 32. Suppose he had. .40 As 110 : 88 : : 40 32 .as many.... 40 40 32 half as many 20 16 i as many . . 10 Ill0)352l0(32 8 33 110 22 22 88 proof. 2. A person having about him a certain number of Portugal pieces, said, If the third, fourth, and 6th of them were added together, they would make 54. I desire to know how many he had? Ans. 12. 3. A gentleman bought a chaise, horse, and harness, for £60, the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness. What did he give for each ? Am. Horse £13 : 6 : 8, Harness £6 : 13 : 4, Chaise £40. 4 A, B and C, being determined to buy a quantity of goods which would cost them £120, agreed among themselves that B should have a third part more than A, and a fourth part more tJian B. I desire to know what each man must pay ? Ans. A £30, B £40, C £50. h3 P 90 POSITION, OR THE RULE OP FALSE. terest, aiul at the end of 10 vol-FZ ''':\i'^' ''^"'•"'n, simple in. t-est, £300. What ^l J^Cle^U "'V^^Xll^/;!^^" DOUDLE POSITION be tlius ordered :— ^^ / ""^7 are, with their errors, to tho quotient will bo th^ a^^-"" "^ ""■"■ l"'"''""'^ ^o' ■•' dividend, EXAMPLES. n.4^i^;c^olSt^::;'V::;;d^d'^ '^'^v ^^t^ - that b must each have? ' ^^ ^^ ™^'*^ *'^^*» I^; liow much '"C Bind 46 Thens.i,.poseAhad.50 and C ^1 ^^'''' ^^ »""«t have 56 140 too little by 60. sup. errors. 40 60 50 '^ 30 3000 1200 1200 60 30 30 divisor. 310)180(0 60 Ans. for A. 170 too little by 30. 60 A 66 B 74 200 proof. covt tots,!':' fort;, yTc """"!"" ''<"«•"• "-"» 0- it will double the wd^ht V tho CT " P"' "" *''« ^"^ ™P. greater c„p, it will te'?hriee S hSt I T' ,™'' ^''^ <"• '••« is the weight of each cup ? ^ "^ ** '"^^ <="P- What -^»». 3 ounces less, i greater. BXCHANOB. 91 3. A gentleman bought a house, with a garden, and a horse in I tiie stable, for £500 ; now lie paid 4 times the price of the horse for the garden, and 5 times the i)rice of the garden for the hoViSe. What was the value of the house, garden, and hoi-se, separately t Ans. horse £20, garden £80, house £400. 4. Three persons* discoursed concerning their ages: says H, I am 30 years of age ; says K, I am as old as H and i of L ; and says L, I am as old as you both. What was the age of each peraon ? Ans. H 30, K 50, and L 80. 5. D, E, and F, playing at cards, staketl 324 crowns ; but dis- puting about the tricks, each pan took as many as ho could : D got a certivin number; E as many as D, and 15 more; and F got a fifth part of both their sums added together. How many did each get ? Ans. D 127i, E 142^, and F 54. 6. A gentkraan going into a garden, meets with some ladies, and says to tliem, Good morning to you 10 fair maids. Sir, you mistake, answered one of them, we are not 10; but if we were twice as many more as we are, we should be as many above 10 as we are now under. How many were they 3r • * Ans. 5. EXCHAN^GE Is receiving money in one country for the same value paid in another. Tlie par of exchange is always fixed and certain, it being the intrinsic value of foreign money, compared with sterling; but the course of exchange rises and falls upon various occasions. L FRANCE. They keep Lu^ir accounts at Paris, Lyons, and Rouen, in livres, sols, and deniers, and exchange by the crown = 4s. 6d. at par. Note. 12 deniers make 1 sol. 20 sols 1 livre. 3 livres 1 crown. ! 4 'I 4 a^. V^, iMAGE EVALUATION TEST TARGET (MT-3) /. 4i_s ,^ f<'' C^x i/.. sr A^ mp. Q!r i/.l 1.0 I.I 1.25 6" IIIIIM 2.0 11111= 1.4 IIIIII.6 P^ //, 'c^ A ' cp Photographic Sciences Corporation is the French , to receive in , to receive in EXCHANOB. 03 I. How many pieces of eight, at 56d. each, will answer a bill of je594 : 6, sterling ] Ans. 2547. 8. If I pay a bill here of £2500, what Spanish money may I Oraw my bill for at Madrid, exchange at SV^d. per piece of eight? Ans. 10434 pieces of eight, 6 rials, 8 mar. |f III. ITALY. They keep their accounts at Genoa and Leghorn, in livres, sols, and deniers, and exchange by the piece of eight, or dollar =4s. 6d. at par. Note. 12 deniers make 1 sol. 20 sols 1 livre 5 livres 1 piece of eight at Genoa. 6 livres 1 piece of eight at Leghorn. N. B. The exchange at Florence is by ducatoons ; the exchange at Venice by ducats Note. 6 solidi make 1 gross. 24 gross 1 ducat Rule. Same as before. 9. How much sterling money may a person receive in London, if h« pays in Genoa 976 dollars, at 53d, per dollar ? Ans. £215 . 10 . 8. 19. A factor has sold goods at Florence, for 250 ducatoons, at 5 Id. each ; what is the value in pounds sterling ? Ans, £56 .5.0. II. If 275 ducats, at 4s, 5d. each, be remitted from Venice to London ; what is the value in pounds sterling ? Ans. £60 . 14 . 7. 12. A gentleman travelling Would exchange £60 . 14 . 7, sterling, for Venice ducats, at 43. 5d. each ; how many must he receive ? Ans. 275. IV. PORTUGAL. They keep their accounts at Oporto and Lisbon, in reas, and exchange by the milrea=Gs. 8^d. at par. Note. 1000 reas make 1 milrea. Rule. The same as with France. EXAMPLES 13. A gentleman being desirous to remit to his correspondent in London 2750 milreas, exchange at 6s. 5d. per milrea; how much sterling will he be the creditor for in London ? Ans. £882 . 5 . 10. 14. A .merchant at Oporto remits to London 4S66 milreas, an' j83 reas, at 53. Sjd. exchange per milrea ; how much sterling must be paid in Lon- don for this remittance } Ans. £1193 . 17 . 6|, 0375. 15. If I pay a bill in London of £1193 . 17 . 65, 0375, what must 1 dravT for on my correspondent in Lisbon, exchange at 58. 5|d. per milrea ? Ans. 4366 milreas, 183 reas. :ir M 0i i'l: ■d EXCHANGE. V. HOLLAND, FLANDERS, AND GERMANY. aa in England ; ^Ct'gZ: S^ ^''^' ""^ ^"^^^ exchange with us in our poufd, at 33s!'Slemfshf :r;r ' "'"1 Note. 8 pennings make , .,„., 1 guilder or florin. ALSO, 20 schellings, or 6 guilders. . . i pound To change Flemish into Sterling, Bun^: to tlt«i~r' '" »- P-"-! = = - - the Flemish ^0 cAawye Sterling into Flemish given -^ i: He::Sl"sS"ugi.r '"^ ^'"" "'«= = ^ '' ^I^^ ^""S EXAMPLES. how -inrpSTrSl^^^^^^ ^ b'? «f ^^54 .10.0 sterling, per pouPd sterling .' ' ^^"^ '"'"' the exchange, at 33^ oa. Flemish . 17. A merchant in Rotterdam remits jEiIg?" f ^^^q 'J,^ ' ^' ^^^'"^^h. in London, how much sterling monTvmtfhl^* V^' Flemish, to be paid at 33s Od Flemish per pouSd TerlZ^?^"' ^' '^'"^ ^''' '^^ "'^'^i^"'^^ ^''^^S 18. If r pay in London £852 lo %' „*^ri:„„ , '^''•*- ^754 . 10. I draw for at Amsterdam. exchan^P «/ ^j u^\ T ""^"^ guilders must ^n«. 852 . 12 . 6. To convert Sank Money into Current, and the cMrary The Zoic: .^tto„*':n7and Tothr" "T. *« «"™»^ generaIl,f..om3toCperc:nU„tou?„7ttBil'^'"' ""<' " ^0 cAaw^e j?aw^ into Current Money EXCHAN6B. To change Current Money into Bank. 95 Rule. As 100 with the agio is added : is to 100 Bank : : so is Current money given : to the Bank required. 20. Change 794 guilders, 15 stivers, Current Money, into Bank florins agio 4f per cent. Ans. 761 guilders. 8 stivers, ll|f^ pennings. 21. Change 761 gui' 'ers, 9 stivers Bank, into Current Money, agio 4f per cent. Ans. 794 guilders, 15 stivers, 4y='^ pennings. VI. IRELAND. 22. A gentleman remits to Ireland £575 : 15, sterling, what 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. bl5 : 15. COMPARISON OF WEIGHTS AND MEASURE S. EXAMPLES. . If 50 Dutch pence be worth 65 French pence, how many Dutch pence are equal to 350 French pence? Ans. 269^f . 2. If 12 yards at London make 8 ells at Paris, how many ells at Paris will make 64 yards at London ? Ana. 42j8j. 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. 289ff. CONJOINED PROPORTION, Is when the. coin, weights, or measures of several countries are compared in the same question ; or, it is linking together a varie- ty of proportions. When it is required to iind how many of the first sort of coin, weight, or measure, mentioned in the question, are oqual to a given quantity of the last. i'i. 96 PHOPOHTIOW. ply the first row coilZm, fir . ^- m ^ ""^ ''f ''i *<"> """Wl a divisor. «»>u«ually for a dividend, and the second fo,l requfrr ^^ "" """^ ^'"Sle Rules of ITiree as the questioj EXAMPLES. are equal & 72 lb. atleShorn^^'"'™' '""' """^ «'• «' ^^-donl Left. 20 Right. 23 155 180 12 20X155X72 = 223200 23X180=4I40)223200(53J3;|. at AmLerdt 12^ Ib'ari'houl! "• f ^'^''''''"•' ""^ ^"O ">■ are equal to 401b. at xiioluseT '' ^"^ "'"^ '"• "' ^""^on Ans. 40 lb. an^lL'rar4i:i;rtoTell%'^?rr ^' ^^»"''-- ces at Venice are equal to ?6 ells EngHsh F"^^"^' ^"^ "'""y '"■''• Am. 25^^^. at tn?!tetdam make°"m aTn ','•'^ f ^"^'^'d'"-. and 90 lb. are equal to mtJZlif '') '"" "^"^ "'• "' '^''<'- weStJ'JfeSi^ln'lio^ed '^ ZZ " l^' '^^ ^'^ "' -"' quantity of the first. ^ 1"^"°"' "^ «q«al to a han''d!'a'ndfeUhett":±rst:lH™'":lf' f"?'™''"'' »' ">» 'e« tipiy the fi.t row frr?d:;i^or,r the" Z:!t:':t^,^. """■ PROGRESSION. 97 EXAMPLES. 5. If 12 lb. at. London make 10 lb. at Amsterdam, and 100 IL. Lt Amsterdam 120 lb. at Thoulouse, how many lb. at Thoulouse iio equal 'o 40 lb. at London ? ^ws. 40 lb. 6. If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. It Amsteidam 116 lb. at Dantzick, bow many lb. at Dantzick are equal to 122 lb. at London? Ans. Ul^^l^. PROGRESSION CONSISTS OF TWO PARTS ARITHMETICAL AND GEOMETRICAL. ARITHMETICAL PROGRESSION lis when a rank of numbers increase or decrease regularly by the continual adding or subtracting of equal numbers ; as 1, 2, 3, 4, 5, 6, are in Arfthraetical Progression by the continual increasing I or adding of one; 11, 9, 7, 5, 3, 1, by the continual decreasing or subtracting of two. Note. "When any even number of terms differ by 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 numbei-s, 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 = 54-1=2 + 4 = 6. In Arithmetical Progression five things are to bo observed, viz. 1. The first term; better expressed thus, F. 2. The last term, L. 3. The number of terms, N. 4 The equal difference, D , 5. The sum of all terms, S. Any three of which being given, the other two may be found. The first, second, and third tern)s given, to find the fifth. Rule. Multiply the sum of the two extremes by half the number of terms, or multiiJiv half the sum of the two extremes m PHOORESSIOiV. I. F L y a re given to find S. F4-Lx~=a 2 EXAMPLES, hoisf °" many strokes does the ha,„„er of a clock strike in l^l 12+1=13, then 13X6=78. «ngly, and returns .ith eve,y egl^thett "t^X M ^^^ -^»«- 5 miles, 1300 yards. Tbe first, second, and third terms ,iven, to find the fourth FLN are given to. find D. L — F -=0. EXAMPLES. the'-eMer32!'1h?incrr; t m-T.'' >"«- <"''. -" was the common di4re~ their "^T'"' ^'"S'^^'on, '„ha. 32-4=28, then 28^8 = 1+4 common difference. day. and"*:; b'^t^^UTe tttv'° " ""•'"■" J"- '" '' an e,„a. excess, so that the ttXt];re^| -ry^-^^y ., what 1. PROGRESSION. 09 B total of all the! ock strike in 12 >r the first yard unt to ? Ans. £5 . 2. itiy a yard as- a basket, what hese 100 eggs 'Ut it in ? 1300 yards. e fourth, mainder divi- iars old, and 'ession, what Ans. 4. ence. place in 12 'Very day hy >e 58 miles, what is the daily increase, and how many miles distant is that place from London ? Ans. 5 daily increase. Therefore, as three miles is the firet day's journey, 3-|-5 = 8 the second day. 84-5 = 13 the third day, &c. The whole distance is 366 miles. The first, second, and ff>ui-th 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 58 miles m one day ; how many days did he travel ? Ans. 12. 58—3 = 55-7-5 = 11 + 1 = 12 the number of days. Y. A man being asked how many sons he had, said, that the yoangest 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. Multiply the fourth by the thii-d made less by one, the product subtracted from the second gives the first : or thus, IV. L N D are given to find F. L— DxN— 1=F. EXAMPLES. 8. A man in 10 days went from London to a certain town in the country, every day's journey increasing the former by 4, and the last he went was 46 miles, what the first? Ans. 10 miles, 4V10 — 1=:S6. then 46 — 36 = 10. the first day's lournev. 12 mv 100 PROGRESSION. it«. i'Li'"Z^^ Tsml''f'' "' « --'•'" times, so n, J by 6, the last nHe-' «tt'"vf,' thSr """""^ "ifT'l Tlio fourth, third, and fifth given, to find the fet sub's h.5r;t SL'^o'Vo'ii'^' ™," ?T."'-' '■-«- less I gives the fi,i^:or thus, ""''"'''"'' ''^ ""^ ""''''I V. JSr D S are given to find F. SDxN— 1 ~F. EXAMPLES. ex<^!d IhT foriHrr.^ i" :' ,f ' r '^' p"^'"^-"^- -'•■ •« ment on any one tKt o'n „)M *'""«. »<; be^t""- '•'<' first pay pemn have for hi ptl?" ''" '"'° ^'''"'' " ■^- What will W ^ ■ -4ws. £8. 4 X 12—1 300-12=30, then 30 £« fi,. a . ' — *8 the first payment. The first, third, and fourth, given to find the second. ' seLid. or thus' "'"^'^ ""^^'^ '" '^^ ^'^^ ^^^« ti^« FN D are given to find L. ND— D4-F=L. EXAMPLES. be^LSe%!!d\S:i;:?^l^t:^tr'rir^^^^^^^^^ -,. „„ Ans. 158. 20X8-8=152, then 152+6=158, the last number. GEOMETRICAL PROGRESSION Is the increasing or decreasing of any i-ank of ni.mKnr= i 2. and 16,\ 4, 2, d^r;^ by the dwi'r"' '^ ""^ ""'";P"'=' PROGRESSION. 101 Note. When any number of terms is continued in Geome- trical Progression, the product of the two extremes will be equal to any two means, equally dist-mt from the extremes': as 2, 4, 8, 16, 32, 64, where 64X- av unity, the EXAMPLES. ^n*. £2.2.8. 16=4 0> 1, 2, 3, 4, Exponents 16 = 4 1, 2, 4, 8, 16, No. of terms.- For 4+4+3=11, No. of terms less 1 256=8 8=3 4)2048=11 No. of far. 12)512 2(0)412 . 8 £2.2.8 2. A country gentleman goinff to a fair tn K,„. c^ meets, ^v'ith a p.rson who had 23 • he demanrlpd fl7 ■ % a"'"' and was answLd £16 a pilVth gtZt btds^i'jr^'^-'"' and he would buy all ; the^the'r tells'h^mTrouH „ot 'be Xn" PSOORESSIOir. 103 RuLB. Proceed as in the last, only observe, that every product must be divided by the first term. EXAMPLES. 3. A sum of money is to be divided among eight persons, the first to have £20, the next £60, and so in triple proportion ; what will the last have ? Ans. £43740. 540X540 14580X60 20; ei; 180; 540; =^^^80, then- =43740 20 20 3 + 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. £25600. The first term, ratio, and number of terms given, to find the sum of all the terms. Rule. Find the last term 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 with a gentleman to serve him twelve months, provided he would give him a farthing for his first month's service, a penny for the second, and 4d. for the third, &c., what did Ws wages amount to ? Ans. £5825 . 8 . 5^. 266X256=65536, then 65536X64=4194304 0, 1, 2, 3, 4, 4194304—1 1, 4, 16, 64, 250, =1398101, then 4+4+3 = 11 No. of terms less 1, 4 — 1 1398101+4194304=5592405 farthings. 6. A man bought 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 ii each shoe 8 nails ; what was the worth of the horse ? Ans. £965114681693 . 13 . 4. 304 PERMUTATION. Ist" M< Iff was lier })ortion? ^ -^ ^"^" ^^^ ^ 7^*^""; ^vliat ^w*. i;204 . 15. man to "TS 2*2 "vXI '" l""'",',™' '«^'*'' »■'"' « ««'* tJ.e fii-st vir fi , ^ I ^ "'^' »"''' '"■»'"«1<"1 iaco, for 2 ,,i„3 d4f';^ k^::;i; >:, j :„ ;i/r"it,: ;?«'"■'"" "'"""'■''" = ■ sale tiw, s«,>posinVtKr!.'\itrx7^:;'.:ir' '^ "'° ^WA*. Iho laco sola fbr i;;]2G880 .0.0. Gain i;32G732 . . Q. PER¥UTATiON Is tho changing or varying of tlio order of things. ^ EXAMPLES. hom-8 ? innrntts, cind the year to contain 3G5 days, 6 ^.ng, ccl n xi,n, ^u, to wiiich tho scholar airroes WhuV t;."," ." i"-^ the scholar stay with tho gcntleniau ? ^ ^' ^""^ ^'"^ ^n*. 6040 days. 105 ^ow-year's day,; tortion, prouiis- r 1 ^enr ; what . il204 . 15. witli a gentle lice, for 2 pins l)io])oition ; I 18 won; valued or lost by the yard, 880 . . 9. 732 . . 9. tlicr, and Iho Is; and how 10 changes 365 days, 6 470001600 need, is =91 40 THE TUTOE'S ASSISTAIT. PART II. VULGAR FRACTIONS. A KUACTiON is a part or parts of an unit, and written with two figures, with a line between them, jis ^, a, f, &c. The figure above the lino is cjilled the numerator, and the un- der one the denominator; whieh shows how many j)arLs tho unit IS divided into : and the numerator sliows how many of tliose parts are meant by the fraction. There are four sorts of vulgar fractions : proper, improper, compound, and mixed, viz. i i > 1. A PROPER FRACTION is wheu the numerator is less than the denominator, as J, f, I, lo., i« i, Slc. 2. An IMPROPER FRACTION is wheu tho numerator is equal to, or greater than tiie denominaior, jis i, ^, j|, i.^j ^fcc. - " -«-•'•--.-. x.xACi,t,n r, tiKj iraetion of a fraction, and known by the word of, as i of f of f of J^ of ^^, &c. 4. A MIXED NUMRER, or FRACTION, is composed of a whole number and fraction, as 8f, 17^, 8J|, &c. i I; f ^t t^ 106 REDUCTION OP VULGAR FRACTIONS. REDUCTION OF VULGAR FRACTIONS. 1. To reduce fractions to a common denominator. ex^li\^lfA^ """'"'"'"'' '"^^ '^^ '^' denominator,! except Its own, for a numerator; and all the denominators for a common denommator. Or, "'uiudiors, loi 2. Multiply the common denominator by the several eiven nu.nc-rato,-s, separately, and divide their product bv the served dei-onnnators, the quotient, will be the now numeral EXAMPLES. 1. Keduce f and 4 to a common denominator. 1st num. 2d num. ^^''''^ ^« ^"^ ^^• l^vV^ .^X^-l^, then 4X7=.28 den.=|f and 4f . 2. heduce I f, and a, to a common denominator. Facit -Si 11 AS. 3. Reduce h a, j\- and f, to a common denominatoV. "*"*' _ Facit, ^11 A 2 3.10 2.016 a.8 8 4. Reduce J» a 1 nnrl 3 f^ „ 3 3ao» 3 36 0', 3 36 oj taeo- *• ^^^<^"ce Jo-, ^, I, and f , to a common denominator. _ Facit i|l 8 8 40 J'JI.O. 84 6. Reduce « a 3 nnrl i f« „ « ' 1 e a ff» 1 e e ^? 1 e a > Te s ff- o. jvcauce 3, f, ^, and ^, to a common denonnnator. Facit, #?a ^6 5 6 4 105 6 Red imp i « a o«;i 3 ^^ « "'840' 840' 8To» ilf- o. xicuuce ^, ^, I, and f , to a common denominator. Facit, 7'%'W, 4-2.J11 j5_4_o_ inofl ' 2 16 0? 2 1 6 o> 2T0 o» 5i e 0" 2. lo 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 folloW ,^ nothnig remam: the last divisor is the common measure Mho divKle both parte of the fraction by the common measurl an the quotient will give the fraction required. Not. If the comnion measure happens to be one, the fraction s already m its lowest term: and when a fraction hath ciphe,-s at the right hand, it may be abbreviated by cutthig them off, as ^If EXAMPLES. 1. Reduce §f to its lowest terms. 24^32(1 24 Com. measure, 8)24(3 Facit, REDUCTION OF VULGAR FRACTIONg. 107 8. Reduce y'/s to its lowest terms. 9. Reduce f |f to its lowest terms, 10. Reduce ||f to its lowest terms, 11. Reduce m to its lowest terms, 12. Reduce |^f | to its lowest terras. Facit, -^j. Facit, VL3y. Facit, ^. Facit, f f. Facit, f . 3. To reduce a mixed number to au improper fraction. Rule, Multiply the whole number by the denominator of die fraction, and to the product add the numerator for a new numerator, which place over the denominator. Note, To express a whole number fraction-ways set 1 for the denominator ffiven. O EXAMPLES. 13. Reduce 18| to an improper fraction. Facit, If A. Facit, i|i5. Facit, ^\K Facit, y. Facit, a I A. Facit, &f.f 1. 18x7+3 = 129 new numerator=ifa. 14. Reduce 56if to an improper fraction. 15. Reduce 183/y to an improper fraction. 16. Reduce 13 a to an improper fraction. 17. Reduce 2V| to an improper fraction. 18. Reduce 514|^ to an improper fraction. 4. To reduce an improper fraction to its proper terms. Rule. Divide the upper term by the lower. EXAMPLES. 19. Reduce J-f «- to its proper terms. 129-7.-184. 20. Reduce J-||i to its proper terms. 21. Reduce ^{^ to its proper terms. 22. Reduce ^ to its proper terms. 23. Reduce aA£ to its proper terms. 24. Reduce ^\^^ to its proper terms. 6. To reduce a compound fraction to a single one. HuLE. Multiply all the numerators for a new numeraV ^r. Jind all the denominators for a ne^y denominator. Reduce the new fraction to its tSVest terms by Rr'e 2. Facit, 18f Facit, 6Gif. Facit, 183^y. Facit, 13f. Facit, 27a., Facit, 614^. Is' Ml h i 108 SEDUCTION OP VULGAR FB ACTIONS. EXAMPLES. 25. Reduce f of f of f to a single fraction. 2X3X5= 30 Facit, —reduced to the lowest tenn=i- 3X5X8=120 ^' 26. Reduce f of 4 of ii to a single fraction. T?o/ii» 32 55 27. Reduce H of U of U to a single fraction' "'~'^"- 2.8. Reduce J of f of ^ to a single fractio!!""' ' •^*=*-- 29. Reduce 1 of f of J to a single fraction. *''''*' ^''"^" Facit i5i 7 80. Reduce f of ^ of j\ to a single fraction. ' ^« «-^5- Facit, eVo^eV- ^'J"* J''^^''''^'f^^c^''ons of one denomination to the fraction of I another, but greater, retaining the same value. Rule Reduce the given fraction to a compound one, by com- pa rg ,t with all tlie denominations between it «nd tha denol nation which you would reduce it to; then reduce that compos i traction to a sin.ole one. 'i>^um \ EXAMPLES. 31. Reduce | of a penny to the fraction, of a pound. 32. Reduce } of r. penny to the frTcdon of I'^pound"''"^*' 33. Reduce | of a dwt. to the fraction of a lb. troj.""''^' ^'^' F'loit Sl. 34. Reduce 4 of a lb. avoirdupois to the fraction of a'cwl"^" Facit, :yiy. 1. To reduce fractions of one denomination to the fraction of another, b'-t less, retaining the same value. Rule Multiply the numerator by the parts contained in the several denomn,atK>ns between it, and tl.at^-ou woul H dl to, for a new numerator, and place it over the given denonSol REDUCTION OF VULGAR FRACTIONS. 109 !io fraction of EXAMPLES. 35. Ke<.luce j^\^ of a pound to the fraction of a penny. Facit, |. ^ TX 20X 12 = 1680 ifiA reduced to its lowest term=f S(]. Ucduco -^l^ of a pound to the fraction of a penny. ,- u 1 » Facit, |. 3 1. Keduce j^\^ of a pound troy, to the fraction of a nenny- 38. Reduce ^f ^ of a cwt. to the fraction of a lb. ' *' Facit, ^. 8. To reduce fractions of one denomination to anothor of the same value, having a numerator given of the required fraction. ItLLE. As the numerator of the given fraction : is to its deno- linmator : : so is the numerator of the intended fraction : to ita |deiJummator. EXAMPLES. 30. Reduce § to a fraction of the same value, whose numera- |tor sha.l be 12. As 2 : 3 : : 12 : 18. Facit, jf. 40. Reduce 4 to a fraction of the same value, whose numera- |tor shall be 25. p.^^.,-,.^ 24 41. Reduce 4 to a fraction of the same value, whose nmnera- |tor shall be 47. 47 Facit, 65f. 9. To reduce fractions of one denomination to another of the Isaino value, having the denominator given of the fractions re- Iquired. PiULE. As the denominator of the given fraction : is to its numerator : : so is the denominator of the intended fraction • to jits numerator. EXAMPLES. 42. Reduce f to a fraction of the same value, whose denomi- luator sliall be 18. As 3 : 2 : : 18 : 12. Facit, {K 43. Reduce f to a fraction of the same value,, whose i'fiJmi- Itor shall hp. n.'i. p.-. o, 41. Keduce 4 to a fraction of the same value, whose denomi- itor shall be C5f . 47 4' \ Facit, 65f no II* ft 4 REDUCTION OF VULGAR FRACTIONS. 10. To rrduc^ a ,inixorl fraction to a sinr^le one. TluLB When the numerator is the intoirral part, multiplvi by the aenorninator of the fractional i,art, add\na; in the nunierat or tiio fractional part for a new numerator; tiien niultit)!v the d nominator of the fraction by the denominator of the fraction part tor a new denominator. ^ EXAMPLES. 45. Reduce— to a simple fraction. Facit ii4=ii 3 ()X " + 2 = 110 numerator. 48X3 =144 denominator. 234 46. R(^,duce — to a simple fraction. 38 When the denominator is the integral part, multiply it bv tli denonmu.tor of the fractional part, adding in the numerator of the fractional part for a new denominator; then multiply thJ iiumerator of the fraction by the denominator of the frac-tioiiJ part for a new numerator. ■ / EXAMPLES. 47 47. Reduce — to a simple fraction. Facit aiss _s Oof ' '^'■^^ '' 19 48. Reduce — to a simple fraction. 44^ n. To find the proper quantity of a fraction in the knows parts of an integer, I ^ Rui.K. l\ftiltiply the nuniorator by the ccmmon parts of tliJ int-ger, and divide by the denominator. EXAMPLES. 49. Reduce f of a pound sterling to its proper quantity. 3X20 = 60—4=158. p\jj,j|.' igg 60. Reduce f of a shilling to its proper quantity. Facitj 4d. 3i qrs. 61. Reduce 4 ot a pound avoirdupois to its proper quantity 63. R 64. E 55. R 56. R 57. R 68. R( 59. R< > 60. Hi 12. To lenomina RrLE. ionod for the sai qui re d. Facit JL7_ — 3 52. Reduce | of a cwt. to its proper quantity Facit, 9 oz. 2^ dr. Facit, 3 qrs. 3 lb. 1 oz. 12 J dr. 62. lie 03. Re( 64. R<^( 65. Rec 66. Rec n. Rec 68. Red 09. Rec JIEDUCTION OF VULGAR FRACTIONS. Ill 63. Reduce f of a pound troy to its proper quantity. > . „ , . - „ „ Facit, 7 oz. 4 dwts. 54. Ueduce f of an ell English to its proper quantity. _. p , . . ., . I^^'icit, 2 qrs. 3 a nails. 5o. Jteduce f of a mile to its proper quai.tity. Kc Tf 1 . c . ■^'''^^*'^' ^ ^"''- ^^ poles. 60. Keduce f of an aero to its proper quantity. fcH r» 1 « i. , , , •^''*^'^' - ^^^^^y 20 poles. 5/. Ivxluce f of a hogshead of wine to its proper quantity. CO r> 1 n Pixdt, 54 gallons. 58. Keduce f of a ban-el of beer to its proper quantity. rn r. 1 ^ l^'ixdl, 1 2 gallons. 59. Keduce -,% of a chaldron of coals to its proper quantity. ^A i> 1 « . ^'^^^'^^ ^5 bushels. 00. Jveduce | of a month to its proper time. Facit, 2 Aveeks, 2 days, 19 hours, 12 minutes. • 12. To reduce any given quantity to the fraction of any greater penoniination, retaining the same value. RrLR. Reduce the given quantity to the lowest term mon- lioneil tor a numerator, under which set the integral part reduced jo the same term, for a denominator, and it wiirgive the fraction Required. EXAMPLES. 61. Reduce 15s. to the %l4gly 1 by §. 4. Muhi|)Iy 43Uy6j by 18-| 6. Multiply ^^ by 'i of f of f . 6. Multiply ^^ by I of t of f Facit, 672gi'y. Facit, 7935^. Facit, ^»\ = H. Facit, 3. 'y '-,! f'\ 1**1 k3 114 SINGLE RULE OF THREE DIRECT 1. Multiply f of f by f of l 8. Multiply ^ of I by -f. 9. Multiply 5f by |. 10. Multiply 24 by f. 11. Multiply ^ of 9 by |. 12. Multiply 9^ by f. Facit, f Facit, J/y. Facit, 431. Facit, 16. Facit, 5ff. Facit, 31. DIVISION OF VULGAR FRACTIONS. Rule. Prepare the given numbere (if they require it) by tbel 3. If rules of Reduction, and invert the divisor, then proceed as in I cost ? Multiplication. ■ 4 If r m 4 St. I f' EXAMPLES. 1. Divide ^^ by |. Facit, 5X9=45 num. 3X20: 2. Divide 14 by f . 8. Divide 6723V by 13f 4. Divide 193o^ by 18^. 5. Divide f by f of ^ of I 6. Divide f of 16 by 4 of i 1. Divide i of f by f of f . 8. Divide 9/2 by ^- of 7. 9. Divide /g^ by 4^. 10. Divide 16 by 24. 11. Divide 520oy\ by f of 91. 12. Divide 3} by 9^ '^ ■ 60 den.— If =f. Facit, |. Facit, 48|. Facit, 430|. Facit, i^j. Facis, 19 |i. Facit,! A = |. Facit, 2if Facit, ^. -Facit, f. Facit, 111 Facit, }. THE SINGLE RULE OF THREE LIRECT, IN VULGAR FRACTIONS. ^ Rule. Reduce the numbers as before directed in RM"^>Jnn.l ^Z. f 1 ?™ ;" ^^^' proportion, then multiply the tiir(.' terms continually together, and the product will be the answer. BINQI.E fiVLB OF THBEE INVERSE. lift Facit, f Facit, J/y. Facit, 431. Facit, 16. Facit, 5|f. Facit, 31. DNS. [uiro it) by thel t proceed as in I Facit, |. Facit, 48|. Facit, 430|. Facit, VV. ''acis, 19 |i. acit, fA = |. Facit, 2if Facit, ^. -Facit, f. Facit, 7 If Facit, |. :n vulgar! in Reduction: numbers, and | iply the ihw \ le ausw.>r. EXAMPLES. 1. If f of a yard cost | of £1, what will ^^ of a yard come to lat that rate ? Am. ^| = 153. yd. £ yd. £ ^s^ : f :: A : if = 153. for 4 X 5 X 9 = 180 num. 5 v< j,_^,5. 4Um5.£ and 3X8 X 10 =^40 den. "^^»« «» ^;«oVao*- 2. If f of a yard cost f of £1, what will |^ of, a yard cost? Ans. 14s. 8d. 3. If ^ of a yard of lawn cost 7s. 3d., what will 10^ yards cost? Ans. £4 : 19 : 10||. 4. If I lb. cost fs. how many pounds will f of Is. buy ? Ans. 1 lb.2f9 = 3V« 5. If f ell of Holland cost i of £1, what will 12f ells cost at the same rate? Ans. £7:0:8^ ^f. 6. If 12^ yards of cloth cost 15s. 9d., what will 48 J cost at the same rate? Ans. £3 : : 9^ y'/^. 7. If ^^ of a cwt. cost 284s. what will 7^ cwt. cost at the same rate? ^ns. £118 : 6 : 8. 8. If 3 yards of broad cloth cost £2|, what will 10^ yards cost? Ans. £9 : 12. 9. If I: of a yard cost f of £1, what will | of an ell English come to at the same rate ? Ans. £2. 10. If 1 lb. of cochineal cost £l : 5, what will 36y\ lb. come to? Ans. £45 : 17 : 6. 11. If 1 yard of broad cloth cost 15|s., what will 4 pieces cost^ Ans. £85 : 14 34 H or each containing 27^ yards ? 12. Bought 3^ pieces of silk, each containing 24f ells, at 6s, 9|(1. per ell. I desire to know what the whole quantity cost? Ans. £25 : 17 H lit. CUII THE SINGLE RULE OF THREE INVERSE, IN VULGAR FRACTIONS. EXAMPLES. 1. If 48 men can build a wall in 24|- days, how many men the sarne in 192 davs? Ans. Qt-^~ ay; 'Tee 2. If 25 |s. will pay for the carriage of 1 cwt. 145^ miles, hpw. far may 6^ cwt. be carried for the same money 2 Ans. 22^y miles. I \: f % J a.i' I 3 116 m. !■ THE DOUBLE ROIE OF THREB. «ide, to .nak; auother of Zll^^L^^ "" "''"'' *» * rJ weighs but 2| oz ? ''''^' '^^'^'^ » P^n^y white loaf -dn*. 15 yards. THE DOUBLE RULE OF THREE, IN VULGAR FRACTIONS. ' EXAMPLES. !• If a carrier receivos £9 i ^.^ *t. miles, ho^v-much ought helo^iL^vp 5' tr"'^ ^^ ^ ^^^- '^O H q.^. 50 miles? ^ '"^^''^ ^^' *^^« ^•«'-"«i?e of 7 cwt 2. If £100 in 12 months ^ain £^ ' . /? ^* ' ^^ ' ^• gain £3f in 9 months^ ^ "" ^"^ '''^'''^> ^^^^ principal will 3. If 9 students spend £lOi in 1« ^ i ^^'''' ^'^• students spend in 30 Ws. ''^ ''^37171^1' '' helped them? ^ *''' ^^^ ^^>'«' ^^^^n their two sons qu'; to ^n'll^T"^'^' ^^^" ^'^' -^^* «- tii/l/s^;, 6 If the caniage of 60 cwt. 20 miles cost /ui TT'''" . can I have carried 30 miles for £'5 z. a ^^' '"^^'^^ ^^'^'^t T« • -dn*. 15 cwt. parts, 117 be sufficient to which js I y^rdj '««• 4 1 yards, iiours, in how THE TUTOR'S ASSISTANT. mshel of wheat I nny white loaf «*. ]5s. 4|d. K will line 7^ i*. 15 yards. l^ULGAR ^ 3 cwt. 150 t?e of 7 cwt '1 : 16 : 9. principal will Aiis. X75. inch will 20 4JL6 145 J* earned 4|s. *ir two sons : H: 4. ill £13^ re- ^ months, tvhat weight ^ 15 cwt. PART III. DECIMAL FRACTIONS. In Decimal Fractions the integer or one yard, one gallon, &c. is supposed parts, and those parts into tenths, and So that the denominator of a deci consist of an unit, with as many ci places, therefore is never set down ; guished from the whole members by which stands for Vo. .25 for J/^, ,123 But the diflferent value of figures lowing table. whole thing, a.s one pound, to be divided into 10 equal so on without end. mal being always known to phers as the numerator has the parts being only distin- a comma prefixed: thus ,5 for -1-2JL ^"' 100 0* ^ appears plainer by the fol- Whole numbers. Decimal parts. 7654321 ,2 34567 S ss » r o a CO S3 as o o o o o o ^ ^ s a> g- p o g- tr o O S3 g- From whicli it plainly appears, that as whole numbers increase in a ten-fold proportion to th* left hand, so decimal parts decrease in a ten-fold proportion to the right hand ; so that ciphers placed in '% 118 ADDITION OF DECIMALS. li'- before decimal parts dccroa'so flwo'i. ,-oi.,^ u To ? j'-'"^ IS t) l)tUtS OI 100 or —5. • nnr 4„ - \ ,. • ' ^^A„d 52,275275275 is called' a 'compouko n.cuHa.^o „kc- In all circulating numbers, dash the last figure. ADDITION OF DECIMALS. KuLE. In setting down the proposed numbers to \^ icUoA away o hnd t TT'":' v' '"P'"''''"'"? P"""^' ^>"«l' ""S'' to thcr respect, ve values; then add them as in while numbelf^ EXAMPLES. 1. Add '72,5+32,071+2,1574+371,4 + 2,75. 2. Add 30,07 + 2,0071+50,432 + 7 1 ' ^''^'' '''''^''• 3. Add 3,5+47,25 + 027,01+2,0073 + 1,5. r A r S:V^ + '*'''^1 ^^24 + 31,4o2+,3075. p An ;^^+27,514+l,005+725 + 7,32. 6. Add 27,5 + 52+3 2G75+,574I +2720 MULTIPLICATION OF DECIMALS. 119 11 number of JRRINO DECI- SUBTRACTION OF DECIMALS. R.LK. Subtraction of decimals differs but little ^J^^ numbers, only in placing the numbers, vvhich must be caiefully observed, as in addition. EXAMPLES. 1. From ,2Y54 take ,2371. 2. From 2,37 take 1,70. 3. From 271 take 21o,7. 4. From 270,2 take 75,4075. 5. From 571 take 54,72. C. From 625 take 70,91. 7. From 23.415 take ,3742. S. From ,107 take ,0007 MULTIPLICATION OF DECIMALS. Rule. Place the factors, and multiply tbem as i" ^^'^^^Ic nmn- belaud from the product towards the nght hand, cut of as ^rny%es for decimals as there are in both actors tog.the; Tut if tiiere should not be so many places in the product, .ui>- ply the defect with ciphers to the lett hand. EXAMPLES 1. Multiply ,2305 by ,2435. Facit, ,05758775. 2. Multiply 2071 by 2,27. 3. Multiply 27,15 by 25,3. 4. Multiply 72347 by 23,15. 5. Multiply 17105 by ,3257. 6. Multiply 17105 by ,0237. 7. Multiply 27,35 by 7,70071. 8. Multiply 57,21 by ,0075. 9. Multiply ,007 by ,007. 10. Multiply 20,15 by ,2705. 11. Multiply ,907 by ,0025. When any number of decimals is to be multiplied by 10, 100, 1000 &c it is only removirii,^ the separatrng point m the multi- Sn; many plLs toward the ri^t hmul ^ ^l^^^}';:;i in the multiplier : thus, ,578X10-5,78. ,578X 100_o,/8 , ,o.8 X1000=578; and ,578X10000 = 5780. CONTRACTED MULTIPLICATION OF DECIMALS. RuLK. Put the unit's place of the multiplier uu^hv that place of the multiplicand that is intended to be kept in the F'>; ;'^' ; ^''j;" invert the order of all the other figures, i. c. write them all the ■SI 1'^ ♦ I ri 1 CONTRACTED MULTIPLICATION. alZt, iT'"'! •" .7""''''^"° ""' ^g"™ '«ft 0'" CT«>-y time ncM EXAMPLES. foufpla1^'Sof''f^'^? ''y f'^34^' »"'' •«' *ere be only lour places ot aecinials in the product. ^ Contracted way. 384,6721.58 5438.63 Go.Timon way. 384,672158 36,8345 115401647 23080329 3077377 115402 15387 1923 1923 15386 115401 3077377 23080329 115401647 14169,2065 14169,2066 360790 88632 6474 264 48 4 038510 Facit, 14169,2065. of tiS"'^ '''''''' "^ '"''*''• »^ ■-? ^.fy^z '""'■^' ,,,,.., ^ I^acit, 105,6994. 14. Multiply 2,38645 by 8,2175, and .eave only four places '^t"^^ . Faci[, 19,6ll)7 Dli;!') ; -^ ^'"^^^'^^^^^ ^^ ^^^^^'*' '-^"^ ^'^ '^''^'<^ ^^ only one pla^e of decuiials ^^,^ ,^. •! , (>. Multiply 375,13758 by 16,7324, and leave only four plac.s '^''""*'^'- Facit, 6270,9520. onlv four of 17. Multiply 395,3756 by ,75642, and let tliere be places of decimal; Facit, 299,0099. DIVISION OP DECIMALS. 121 DIVISION OF DECIMALS. This Rule is also worked as in whole numbers ; the only dif- ficulty is in valuing the quotient, which is done by any of the fol- lowing rules : Rule I. The first figure in the quotient is always of the same value with that figure of the dividend, whiph answers or stands over the place of units in the divisor. 2. The quotient must always have so many decimal places, as the dividend has more than the divisor. Note 1. If the divisor and dividend have both the same num- ber of decimal parts, the quotient will be a whole number. 2. If the dividend hath not so many places of decimals as are :n the divisor, then so many ciphei-s must be annexed to the divi- dend as will make them equal, and the quotient will then be a wiiole number. 3. But if, when the division is done, the quotient has not so many figures as it should have places of decimals, then so manj Jiphers must be prefixed as there are places wanting. EXAMPLES. 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. C. Divide 5,714 by 8275. I. Facit, 13549. 7. Divide 7382,54 by 6,4252. 8. Divide ,0851648 by 423. 9. Divide 267,15975 by 13,26. 10. Divide 72,1564 by ,1347. 11. Divide 715 by ,3075. VVhen numbers are to be divided by 10, 100, 1000, 10,000, (fee. It is performed by placing the separating point in the dividend *J many places towards the left hand, as there are ciphers in the divisor. 1-^ t- i\ i If i ! * ,1 Tims, 5784-^ 10=578,4. 6784-M 00=57,84. 5784-r- 1000=5,784. 6784-M0,000=,5784. It 1 123 CONTRACTED DIVISION. 11 ' 1^ CONTRACTED DIVISION OF DECIMALS. RifLK. By tlH3 fii*st rulo find what is tlie value of the fii^st fij^ure in the qiioiic t : then by knowing the lii-st figure's denomination, the decimal 'laces may be reduced to any number, by taking as many of tJK icft hand figures of tlie 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 26, carry 2 ; if 25, or upwards, to 35, carry 3, &c. EXAMPLES. 12. Divide 721,17562 by 2,257432, and let there be only three places of decimals in the quotient. Contracted. Common way. 2;357432)721, 17562(319,467 2,257432)721,17562(319,467 6772296 6772296 439460 225743 213717.. 203169.. 10548... 9030... 1518.... 1354.... i64 158 9 6 13. Diude 8,758615 by 5,2714167. 14. Divide 51717591 by 8,7586. 16. Divide 25,1367 by 217,35. 16. Divide 51,47542 by ,123415. IV. Divide 70,23 by 7,9863. 18. Divide 27,104 by 3,712. 439460 225743 2 2 213717 203168 00 88 10548 9029 1518 1354 163 158 120 728 3920 4592 93280 02024 6 91256 ILS. e fii-st fijrure ^nomination, by taking aa [isvvor them ; ch following \ tlie divisor, or np wards. )e only three way. 502(319,407 96 60 43 2 o 17 68 00 88 48 29 120 728 18 54 3920 4592 ^ 63 58 93280 02024 6 91256 BBDUCTION OF DECISIALS. 123 REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to a Decimal. Rule. Add ciphers to the numerator, and divide by the do- 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 % to a decimal. Facit, ,75. 4. Reduce f to a decimal. Facit, ,375. 5. Reduce 2^ to a decimal. Facit, ,1 923070 -h. 6. Reduce \\ of |f . to a decimal. Facit, ,6043950+. Note. If the ji^iven parts are of several denominations, they may be reduced either by so many distinct operations ai. 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 proceeding till you bring out the decimal parts of the highest integer required, by still dividing the product by the next superior denominator ; or, Sdly. To reduce shilling's, pence, and Hirthings. If the num- ber of shillings bo even, take half for the first place of decimals, and let the second and third places be filled with the farthings contained in the remaining pence and fartliings, 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 6. 7. Reduce 5s. to the decimal of a £, 8. Reduce ^s. to the decimal of a £. 9. Reduce 16s. to the decimal of a X. l2 Facit, ,26. Facit, ,45. Facit, ,8. s;.'* ^ I .1.. Ir. i*f :t m ml H! 4 rli 124 REDUCTION OF DECIMALS. 10. Reduce 8s. 4d. to the decimal of a £. 11. Reduce 16s. T^d. to the decimal of a £. first. IGs. VH 12 199 4 960)799(8322916 second. 4)3,00 12)7,75 210)16,64583 ,8322916 third. 2)16 ,832 Facit, ,4166. Facit, ,8322916. 7|d. 4 31 12. Reduce 19s. 5^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. 15. Reduce 2 qi-s. 14 lb. to the decimal of a cwt. Facit, ,625. JO. Reduce two furlongs to the decimal of a league. Facit, ,0833. 17. Reduce 2 quarts, 1 pint, to the decimal of a gallon. Facit, ,625. 18. Rccluco 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 52 days to the decimal of a year. Facit, ,142465 -f. To find the value oj any Decimal Fraction in the known parts of an Integer. Rule. Multiply the decimal given, by the number of parts of tlip. v.oxi ijiff'rior dt^nornination, cutting off the docimaI§ from th« product ; then multiply the remainder by the next inferior deno- mination ; thus proceeding till you have brought in the leasl known parts of an integer. REDUCTION OP DECIMALS. 125 EXAMPLES. 21. What is the value of ,8322916 of a lb. ? Ans. 16s. 7^d.+. 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 ? Ann. 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 qre. 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. 61,999725 days. L3 ■ %l ;l h 126 DECIMAL TAULKS OF COIN, M'EIOHT, AND MEASURE. TABLE I. English Coin. iJ 1 the Integer. Sh. 19 IS 17 16 15 14 13 12 11 10 Dec. ,95 .9 ,S5 ,8 ,75 ,7 ,65 ,6 ,55 ,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 ,0010116 TABLE II. English Coin. 1 Sh. Long" Measure. 1 Foot, the Integer. Pence & Inches. 6 5 4 3 2 1 Decimals. ,OOuoJ J 25 !l 66666 ,083333 Farfh. 3 2 1 Decimals. ,0625 ,011666 ,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 025 ',020833 ,016666 ,0125 ,008333 ,004166 Grains. Decimals. 12 ,052 11 .022916 10 ,020833 9 ,01875 8 ,016666 7 ,014583 6 ,0125 5 ,010416 4 ,008333 3 ,00625 2 ,004 1 (i6 1 ,002(J83 TABLE IV. Avoir. Weight. 112 lbs. the Integer. Grains. 12 11 10 9 8 7 6 5 4 3 9 Decimals. ,002083 ,001910 ,001736 ,001562 ,001380 ,001215 ,001012 ,000868 ,000694 ,000521 ,000347 ,000173 1 oz. the Integer. Pennyweights the same as Shillings in the first Table. Qrs. Decimals 3 ,75 2 ,5 1 ,25 Pounds. 14 13 . 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,125 ,116071 ,107143 ,098214 ,089286 ,080357 ,071428 ,0625 ,053571 ,041643 ,035714 ,026786 ,017857 ,008928 Ounces. 8 7 Decimals ,004464 ,003006 U Avo 1 T)un' Dra L.I 1 Gail K 127 SURE. Decimals. ,022910 ,020833 ,01873 ,01(1000 ,0Nr)&3 ,0125 ,010^110 ,OOS:J33 ,00025 ,004 1 00 ,0020.'-3 .E IV. Weight. le Integer. Decimals. ,75 ,5 _^25 Decimals. ,125 ,110071 ,107M3 ,098211 ,089280 ,080357 ,071428 ,0025 ,053571 ,041043 ,035714 ,020780 ,017857 ,008928 Decimals. ,004404 ,003000 DKCIMAL TABLES OF COIN, WKIOHT, AND MEASURE. ,003348 5 ,002790 4 ,002232 3 ,001074 2 ,001 no I ,000558 4 0z. Decimals. 3 ,000418 2 ,000279 1 ,000139 TABLE V. Avoirdupois weight. 1 lb. the Integer. Ounces. 8 7 (\ 5 4 3 2 1 Decimals. .5 ,4375 ,375 ,3125 ,25 ,1875 ,125 ,0025 Drams. 8 % 6 5 4 3 2 1 Decimals. ,03125 ,027343 ,023437 ,019531 ,015025 ,011718 ,007812 ,003906 TABLE VI. LIQUID MEASURE 1 tun the Integer. Gallons. 100 90 Decimals. ,390825 ,357142 bO 70 GO 50 40 30 20 10 9 8 7 6 .1 4 3 2 1 ,317400 ,27 ,238095 ,198112 ,158730 ,119047 ,079365 ,0-^9082 ,035714 ,031746 ,027 ,023809 ,019841 ,015873 ,011904 ,007936 ,003908 Pint-j. 4 3 2 1 Decimals. ,001984 ,001488 ,000992 ,009490 Hogshead the Integer. Gallons. 30 20 10 9 8 7 6 5 4 3 2 1 Decimals. ,470190 ,317100 ,158730 ,M2857 ,120984 ,111111 ,095238 ,07i)305 ,003492 ,047019 ,031746 ,015873 Tmts. 3 2 1 Decimals. ,005952 ,003968 ,001984 TABLE VII. Measures. Liquid, Dry. 1 Gal. 1 Qr. Integer. Pts. Decimals. Bush A ,5 4 3 ,375 3 2 ,25 2 1 ,125 1 Q. pt 3 2 1 Decimals. ,09375 ,0025 ,03125 Pck. 3 2 1 Decimals. ,0234375 ,015025 ,0078125 Q. Pks. 3 2 1 Decimals. ,005859 ,003900 ,001953 Pints. 3 2 1 TABLE VIL. Long Measure. 1 Mile the Integer. Yards. 1000 900 800 700 600 Decimals. ,568182 ,511364 ,454545 ,397727 ,340909 l^^ 128 DECIMAL TAHLKS OF COIN, WKIOHT, AND MKASUKE. DECIMAL TA 5iJiJ 1 ,2ftl0'.>l 4W ,'227'27-J 3(J0 • ,170151 2(.)(J ,lL'{(i;ji) luo ,050818 UO ,051130 SO ,015151 70 ,03U773 60 ,03-lUUl 50 ,0-iS4l)'J 40 ,02->727 30 ,017015 90 ,011301 10 ,005682 9 ,0(J5114 8 ,001515 7 ,003i<77 6 ,00310f> 5 ,0028-11 4 ,002273 3 ,001704 2 1 ,0011 Co ,000508 Feet. 2 1 Decimals. ,0003787 ,0001891 Indies. 6 3 1 Decimals, ,0000917 ,0000174 ,0000153 TABLE IX. Time. 1 year the Integer. Months the same as Pence in the second Table. Decimals. 1 ,000WO ,821918 ,517915 ,273973 ,240575 70 (>0 50 40 30 20 10 9 8 7 6 5 4 3 2 1 ,219178 ,191781 ,104383 ,13098*) ,10958y ,082192 ,054794 ,027397 ,024057 ,021918 ,019178 ,010438 ,013098 ,010959 ,008219 ,005479 ,002739 1 day the Integer. Hours. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,5 ,458333 ,410066 ,375 ,333333 ,291006 ,25 ,208333 ,160006 ,125 ,083.^33 ,041600 Minutes. 30 20 10 9 8 7 6 5 4 3 2 1 Decimals. ,020833 ,013888 ,000944 ,00625 ,005555 ,004801 ,004166 ,003472 ,002777 ,002083 ,0U1389 ,000004 TABLE X. Cloth meahure. 1 yard the Integer. Quarters the same as Table 4. Nails. o Decimals. ,125 ,0025 TABLE XI. Lead Weight. A Foth. the Integer. Hund. 10 9 3 7 6 5 4 3 2 1 Decimals. ,512820 ,401538 ,410256 ,358974 ,307092 ,250410 ,205123 ,153840 ,102564 ,05!-:2«J2 Qrs. 2 1 Decimals. ,025041 ,012820 Pounds. 14 13 12 11 10 9 8 7 6 IJ 4 3 2 1 Decimals. ,0(J04 1 02 ,0059523 ,0054945 ,0050306 ,0045787 ,0041208 ,0030030 ,0032051 ,0027472 ,0022893 ,0018315 ,001 3736 ,0009157 ,0004578 THB BULB OF THREE IN DECIMALS. 129 THE RULE OF THREE IN DECIMALS. EXAMPLES. If 26 J yards cost £3 : 16 : 3, what will 32^ yards come to? Ans. £4 : 12 : 9^. 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 oian- A71S. £688 : 10. 3. If 7f yards of cloth cost £2:12:9, what will 140i yards of the same cost? Ans. £47 : 16 : 3 2,4 qrs. 4. If a chest of sugar, weighii^-r 7 cwt. 2 qrs. 14 lb. cost £36 : 12 : 9, what will 2 cwt. 1 qr. 2i lb. of the same cost? Ans. £11 : 14 : 2 3,5 qi-s. 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 ©f 827| yards of painting, at 10|d. per ^^^^\^^, Ans.£3Q :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.°5l. 10. If I of a yard of cloth, that is 2^ yards broad, make a gar- ment, how much that is f of a yard wide will make the same ? _, _. ' Ans. 2,109375 yards. 11. If 1 ounce of silver cost 5s. 6d., what is the price of a Um- kard that weighs 1 lb. 10 oz. 10 dwts. 4 grs. ? Ans. £6 : 3 : 9 2,2 qrs. ^ 12. If 1 lb. of tobacco cost I5d. what cost 3 ho'xsheads wei-^h- rng together 15 cwt. 1 qr. 19 lb. ? Ans. £m : 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. £il3 : 10 : 9|. 14. Bought 6 chests of sugar, each 6 cwt. 3 qrs. at £2:16 per L. |, cwt., what do they come to ? Ans. £113 : 8. lao EXTHACTION OP THE SQUARE ROOT. 15. Bought a tankard for £10 : 12, at tho rate of 58. Ad. per ounce, what was tho weight ? Ans. 39 oz. 15 dwt 16. Gave £187 : 3 : 3, for 25 cwt. 3 qrs. 14 lb. of tobacco, at what rate did I buy it per lb. ? Am. Is. 3^d. 17. Bouorht 29 'b. 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 l)c the value of 1 cwt.? Ans. £l : 14 : 8. EXTRACTION OF THE SQUARE ROOT. m 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 tliroughout. 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 the 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. Squares. 1. 2. 3. 4. 5. 6. 7. 8 9. 4. 9. 16. 25. 36. 49. 64. 81. EXTRACTION OF THE SQUARE ROOT. 181 Ans. 345. EXAMPLES. 1. What is the square root of 119025 ? 119025(345 9 64)290 256 685)3425 3425 2. What is the square root of 106929 ? Ans. 327+. 3. What is the square root of 22C8741 ? Ans. 1506,23-j-. 4. What is the square root of 7596796 ? Ans. 2756,228-f- 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. AVhat 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+. ^7is. 69,247+. Ans. 2,091+. Ans. 1,50101+. ^ns. ,01809 + . Ans. 1,1269 + To extract the Square Boot 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 square root of the denominator, for a new denominator. If the fraction be a surd (i. e.) a number where a root can ne- ver be exactly found, reduce it to a decimal, and extract the root from it. EXAMPLES. £301? £184' 13. What is the square root of 14. What is the square root of fi||? 15. What is the square root of xVsVt • Ans. f . Ans. |.. Ans. |. I' * I ^! - . 133 EXTn ACTION OF THE l^QUABE ROOT. SURDS. I? 16. What is the square root of ||f ? 17. What is the square root of m ? 18. What is the square root of f Jf ? Ans. ,898024-. Ans. ,86(302+. Ans. ,933094-. To extract the Square Hoot of a mixed number. Rule. Reduce the fractional part of a mixed number to its lowest term, and then the mixed number to an improper fraction. 2. Extract the root of the numeratoiSvand 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 5l|-»- ? 20. What is the square root of 27/^ ? 21. What is the square root of 9ja 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. 5\. Ans. 3^. Ans. 9,27+. Ans. 2,95 19-t-. Ans. 2,5819-1-. To find a mean i^voportional between any two given numbers. RuLK. The square root of the product of the given number is the mean proportional sought. EXAMPLES. 6. What is the mean ])roportional between 3 and 12 ? Ans. 3 X 12 = 36, then ^36 = 6 the mean proportional. 6. What is the mean proportional between 4276 and 842 ? Ans. 1897,4+. To find the side of a square equal in area to any given otcX/cr/ttcco, RuLK. The square root of the content of any given superficies is the side of the square equal sought. )8024-. 30024-. 33094-. r. ber to its • fraction. ator for a fractional extract the ins. 1}. ins. 5^. [ns. 34. 9,27+. 519+. 819 + . 'imbers. 1 number tional. 342? )V,4+. iven iuperficiea EXTRACTION OF THE SQUARE ROOT. EXAMPLES. 133 27. If the content of a given circle be 160, what is the side of tlie square equal ? ^W5. 12,64911. 28. If the area of a circle is 750, what is the side of the square equal? ^ris. 27,38012. The area of the circle given to jind the Diameter. Rule. As 355 : 452, or, as 1 : 1,273239 : : so is the area : to the square of the diameter; — or, multiply the square root of the area by 1,12837, and the product will be the diameter EXAMPLES. 29. What length of cord will be tit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating ail acre of grass, and no more, supposing the cow and tail to measure 5| yards? Am. 6,136 perches. The area of a circle given, to find the periphery^ or circumference, , Rule. As 113 : 1420, or, as 1 : 12,56637 : : the area to the square of the p i S3 J-f 3 '^ Sc 13 -> a -5 53 a» w Ba »0 yards. S3. The wall of a i wn is 25 feet high, which is surrounded by a moat of 30 feet m breadth: I desire to know the lencrth of a ladder that w)ll reach from the outside of the moat to the top ^^^^^^^^^^"? ^n*. 39,05 feet. ^ The hypothenuse arid perpendicular ffiven, to find the base. KuLE. The square root of the difference of the squares of the bypothenuse 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 hypjHhenuse and base, is the height of the perpendicular. JN. Ji. Ihe two last questions may be varied for examples to tno two last proposiiions. '^ 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 natibcr of men given, is \h number of men cither in rank or file. 34. An army consisting of 331770 men, I desire to know how many rank and hie 2 Ans. 6^. oo. .V conaiu square pavement contains 48841 square stones, all ot the same size. I demand how many are contained in one r. ynwh high, and >th must a lad- tho top of the ns. 75 yards. is surrounded the length of oat to the top . 39,05 feet. d the base. squares of the base. >endicular. squares of the iular. r examples to to a square Jile. given, is the to knr)\v how Ans. 516. square stones, taiiji'd in one Ans. 221. EXTRACTION OF Tl| P i L'ilE uOOT. EXTRACl'ION OF THE CUBE ROOT. 135 To extract the Cube Root is to find out one luin.b.M-, which be inrr multiphcd nito itself, and then into that product, pimhicoth the given number. Rule. 1. I>oint every tliird figure of the cube given, betrinnino- at the units place; seek the greatest cube to thc^ first point, and subtract It therefrom ; put the root in the quotient, and brin-.- down the figures in the next point to the remainder, for a Rksolv^^nd 2 huul a Divisor by multiplying tho ^,quare of the quotient by 3. _ bee how often it is contained in the n."solvcnd, reiectincr the units and tens, and put the answer in the quotient. " 3. To find the Subtrahend. 1. Cube the last firrure in the quotient. 2. Multiply all the figures in the quotient by 3, except the last, and that product by the square of the last. .3. Muitir/lv the divisor by the last figure. Add these products to-n-tluM-, for the subtrahend, which eubtract from the resolvend ; to the re- mainder bring down the next point, and proceed as before. Roots. 1. 2. 3. 4. 5. G. 7. 8. 9. Cubes. 1. 8. 27. 64. 125. 216. 343. 512. 729. EXAMPLES. 1. What is the cube root of 99252847 ? Dlvisor- 99252847(463 64 =cube of 4 S(^uare of 4X3=48)35252 resolvend. Divisor- 216=cubo of 6. 432 = 4 X 3 X by square of 6. 288 = di visor X by 6. 33336 subtrahend. Square of 46X3 = 6348)1916847 resolvend. 27=cube of 3. 1 242 = 40 X 3 X by square of 3. 19044 =divisorXby 3. 1916847 subtrahend. * -f If A % s%. 'f 136 EXTRACTION OP THE CUBE ROOT. I 2. What is the cube root of 380017 ? 3. What is the cube root of 57'.^5',iS0 ? ;U. What is the cube root of ;?24f)l759 ? 5. What is the cube root of 84(504519 ? 6. What is the cube root of 25!:>0y4072 ? V. "v7hat is the cube root of 4822«544 ? 8. What is the cube root of 2705403G008 ? 9. What is the cube root of 22009810125 ? 10. What is the cube root of 122615327232 ? 11. What is the cube root of 219365327791 ? 12. What is the cube root of G73373097125 ? Ans. Ans. Ans. Ans. Ans. Ans. 73. ITD. 319. 439. (i38. 301. Ans. 3002. Ans. 2805. Ans. 4908. Ans. 6031. Ans. 8705. Wlien the given number consists of a whole number ami deci- mals together, make the number of decimals to consist of 3, 6, 9, &€. places, by adding ciphers thereto, so that there may be a point fell on the unit's place of the whole number. 13. What is the cube root of 12,077876 ? Ans. 2,35. 14. What is the cube root of 36155,02756? Ans. 33,06+. 15. What is the cube root of ,001900624 ? Ans. ,124. 16. What is the cube root of 15926,972504 ? Ans. 3,215+! 17. What is the cube root of 15926,972504? Ans. 25,10+. 18. What is the cube root of ,053157376 ? Ans. ,376. To extract tke cube root of a vulvar 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 a surd, reduce it to a decimal, and then extract the root from it ? EXAMPLES. 19. What is the cube root of f f -« ? 20. AVhat is the cube root of y^^V ? 21. What is the cube root of if|§ ? Ans. ^. Ans. ^. Ans. |. SURDS. 22. What is the cube root of | ? 23. What is the cube root of ^- ? 24. What is the cube root of I ? Ans. ,829+. Ans. ,822+". 3' -^««. ,873+. To extract the cube root of a mixed number. Rule. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fractio.5, extract the cube root of the numerator and denominator for a new numerator and done St EXTRACTION OF THE CUBE ROOT. 137 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 121a ? 26. What is the cube root of Sl^^j? 27 What is the cube root of 405 A^ ? SURDS. 28. What is the cube root of 1} ? 29. What is the cube root of 9 J- ? 80. What is the cube root of 8^ ? THE APPLICATION. Ans. 2^ Ans. 3f Ans. 7|. Ans, 1,93+. Ans. 2,092-f-. Ans. 2,05 7-|-. 1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep,, how many cubical inches doth it con- ^""^ ^, , Ans. 103823. 2. There is a cellar dug, that is 12 feet every way, in h.'nglh, breadth, and depth; how many solid feet of earth were takt-n'out ^^^^^^ . Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 Bolid feet, 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 ; multij.ly 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 ? Ans. 18 and 54. 5. What are the two mean proportionals between 4 and 108 ? Ans. 12 and 36. To find the side of a cube that shall he equal in soliditv to an^> given solid, as a globe, cylinder, prism, cone, r. 37x 37 = laf^o w^btrahend. * 37X 2 = 74 /I'^S'^ir. 376X376=141?''«^ wbtrahend • 74)447 dividend. 141376 subtrahend. 2. What is the cube root of 53157376 ? i-- IE ROOTS 53157376(376 27 27)261 dividend. 50653 subtrahend. 4107)25043 dividend. 63157376 subtrahend. 3X 3X 3=27 divisor. 37X 37X 37 = 50653 subtrahend. 37 X 37 X 3 = 4107 divisor. 376X376X376 = 63157376 subtrahend I'i M 3 s*ii fe *. »:, ;.:*»» ' SIMPLE INTEREST. 3. What is the biquadrate of 19087173376 ? 19987173376(376 81 108)1188 dividends 1874161 subtrahend. 202612)1245503 dividend. 19987173076 subtrahend. 3X 3X 3X 4=108 divisor. 37X 37X 37x 37 = 1874161 subtrahend. 3<: 376X376X376 = 19987173376 subtrahend. SIMPLE INTEREST. There are five letters to be observed in Simple Interest, -vift P. the Principal. * T. the Time. R. the Ratio, or rate per cent. I. the Interest. A. the Amount. , A TABLE OF RATIOS 3 ,03 3^ ,035 4 ,04 ^ ,045 5 ,05 8 ,08 sh ,085 9 ,09 H ,095 10 ,1 Note. The Ratio is the simplefnterest of £1 for Ifie rate per cent, proposed, and is found thus one year, at £ £ £ As 200 : 3 : : 1 2. ' for 6 j 3. ' Dum, i 4. cent. ] 6. aiH 0. montl ,03 As 100 : 3,6 : : 1 : ,036. SIMFLE INTEREST. 141 end. btrahend. rnterestj'vias ,08 ,085 ,09 ,095 J ^ one year, at 35. When the p-incipal^ time, and ate per cent, are given^ to find the infer est. RuLK. Multiply the principal, time, and rate together, and it will give the interest required. NoTK. The proposition and rule aro better expressed thus :— I. When P K T are given to find I. Rule. prt=I. Note. When two or more letters are put together like a word, they are to be multiplied one into another. EXAMPLES. 1. What is the interest of £915 : 10, for 3 years, at 5 per cent, per annum. Ans. 945,5 X,OoX 3 = 141,825, or £141 : 16 : 6. 2. What is the interest of £547 : 14, at 4 per cent, per antuim, for C years-? Ans. £131 : 8 : 11, 2 qrs. ,08. 3. What is the interest of £796 : 15, at 4^ per cent, per an- num, for 5 years? Ans. 179 : 5 : 4 2 qrs. 4. What is the interest of £397 : 9 : 5, for 2\ years, at 3^ per cent, per annum? Ans. £34 : 15 : 6 3,5499 qrs. 5. What is the interest of £554 : 17 : 6, for 3 years, 8 months, at 4^ ])er cent, per annum ? Ans. £91 : 11 : 1 ,2 0. What is the interest of £236 : 18 : 8, for three jears, 8 months, at 5^ per cent, per annum? Ans. £47 : 15 : 7^, ,293. When the interest is for any numbe^' of days only. Rule Multiply the interest of £1 for a day, at the given rate, by the principal and number of days, it will give the answer. INTEREST OF £1 FOR ONE DAY. per cent. 3 4 H 5 5^- 6 Decimals. ,00008219178 ,00009589041 ,00010958904 ,00012328767 ,000)3698030 ,00015068493 ,00016438356 per cent. 6.} 7 8 02 9 Decimals. ,00017808219 ,00019178082 ,00020547945 ,00021917808 ,00023287671 ,00024657534 ,00026027397 NoTP^ The above table is thus found : — As 365 : ,03 : : 1 ,0000821 ,00009589041 78. And as 365 : ,035 : : 1 f f' > * n : I (fee. %, W' jl fi M 142 SliMPLE INTEREST. EXAMPLES. V. What is the interest of £240, for ion Hnv« nt a I... an.nun ? Ans. ,OOOI09.580oix 240X 12 l^a' J" ?' 0. Wi.Ht is the interest of £725 • 15 fnrlT) '^^ • /^ ^ U- per r.nnuni ? '< <^i ^/-o . 15, for 74 d.-.ys, at 4 per cent. 10. What is the interest of i!lon f.. n "f' "^^^ 'r^^ = ^^^' to tiio 9tli of AfM,.., f I ' ^'^"^ ^^'^' I'^t of June, 177o JtJi ot ALuch following, at 5 per cent, per annum ? ' ^ns. £3 : 16 : llf. ir. AVhcn P IZ T arc given to find A. Rule, prt + p=A. EXAMPLES. per a„„I';f "■" ^"^ = ''- "•"-" ^ •; ^ y«-, at 4J per cent ^^^5. £370: 19: 11 2,8 qrs. year which are CaUriho^e di;:' "^'' 'f ''''^''' ^^^ <>^ « per an.r;f "" ^'^^ ^ ^^' ^-^"^f ^<' i?,f^ ^-^ at 4 per con. H. What will £273 : 18, amount^ t ^4^;-.', 'I^^J^^s per cent, per annu,n ? Ans. £310:14: r3,3508006rqrf ' III. AVhen A R T are given to find P. a Rule. — ^=P. rt+l. EXAMPLES. • 10 • n o « L K ' ' 'Z^^,'""' ""^ ""'^■i^'»t, will anjount to £376 1.* . 11 2,8 ,n 5 years, at 3^ per cent, per annum? Ans. £320 : 17. SIMPLE INTEHEST. 148 fit 4 per cent. £3:3: If lays, at 6 per : 13: 11|. at 4 per cent. 5:17: 8}^. 'f June, 1775, uin ? : 16: Ul ' 4^ per cent 5 3,04 qrs. 3^ per cent il 2,8 qrs. years, reduce I parts of a t 4 per cent i ,02 qrs. 5 (lays, at 3 0064 qrs. nt to £30 • 579 : 12. It to £376 20: 17. Vr. What principal, bein^ put to interest, will amount to £1130 : 9 : Oi ,02 qi-s. in 5^ years, at 4 per cent, per annum ? Ans. £020 : 12. 18. Wliat principal will amount to £310 : 14 : 1 3,35080064 ars. iu 4 years, 175 days, at 3 per cent, per annum I ^ ^ A?is. £273 : 18. IV. When A P T are giveu 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'^ ,„.k « Ans, 307,074—270,0 = 88,074, 275,0 xV = 19o7,2, then 88,074-r-10o7,2 = ,045 or 4^ per cent. 20. At what rate per cent, will £320 : 17, amount to £376 : 19 : 11 2,8 qrs. in 5 years? -Ans. 3-h F^* cf"^- 21. At what rate per cent, will £926 : 12, amount to £1130 : 9 : Oi ,92 qrs. in 5^ years ? -^w*- 4 per cent. 22. At what rate per cent, will £273 : 18, amount to £310 : 14 : 1 3,35080064 qrs. in 4 yeare, 175 days? Ans. 3 per cent V. When APR are given to find T. a — p Rule. =T. ' ^*^' EXAMPLES. 23. In what time will £279 : 12, amount to £367 : 13 : 5 3,04 qrs. at 4 4 per cent. ? . ^ws. 307,674-279,0 = 88,074. 27^,6 X ,045 = 12,5820, then 88,074-r 12,5820 = 7 years. 24. In what time will £320 : 17, amount to 370 : 19 : 11 2,8 qrs. at 3^ per cent. ? ^If • /^ yeai-s. 25. In what time will £926 : 12, amount to £1130 : 9 : Of ,92 (p-3. at 4 per cent. ? ^'''jX^''''f!' i 2G. In what time will £273 : 18, amount to £310 : 14 : 1 3,35080064 qrs. at 3 per cent. ? Ans. 4 years, 175 days. ANNUITIES OR PENSIONS, &c. IN ARREARS Annuities or pensions, c\rc. are said to be in arrears, when thc' arc payable or due, cither yearly, half-yearly, or quarterly, fti . are u.'ipnid for any number of payments. t! " k •V I ♦ •!.., 144 SIMPLE INTEREST. Note. U represents the annuity, pension, or ycaily rent, T R A as before. •/ j '^ I U U T are given to find A ttu — til Rule. X r : + tu=A. EXAMPLES. 21. If a salary of £150 k^ forborne 6 years at 5 per cent, what will It amount to? Ans, £825. 3000 6X5X1-50— 5X150=3000 then X ,0o + 5X 150=£825 2 28. If £250 yearly pension be forborne 1 years, what ',ill it amount to in that time at 6 per cent. ? Arts. £2065. 29. There is a house let upon lease for 5^ years, at £60 per annum, what will be the amount of the whole time at 4^ per ^^^- • Ans. £363 : 8 : 3. 30. Suppose an annual pension of £28 remain unpaid for 8 years, what would it amount to at 5 per cent. ? XT ^^', ^^*- ^263 : 4. JNoTE. \\heTi the annuities, Ac. are to be paid half-y early or quarterly, then "^ For half-yearly payments, take half of the ratio, half of the annuity, &c., and twice the number of years — and For quarterly payments, take a fourth part of the ratio, a fourth part of the annuity, &c., and four times the number of yeai-s, and work as before. / EXAMPLES. 81. If a salary of £l50, payable every half-year, remains un- paid for 5 years, what will it amount to in that time at 5 per <^ent- ? Ans. £834 : 7 : 6. 32. If a salary of £150, payable every quarter, was left unpaid for 5 years, what would it amount to in that time at 5 per cent. / Ans. £839 : 1 : 3. Note. It may be observed by comparing these la^t oxami-Ies, the amount of the half-yearly payments are more advantageous thaji the yearly, and the quarterly more than the hali-yeaily. IL When ART are given to find U. UuLE.- 2a ttr— tr-f2t =U. 'caily rent, T )er cent, what drts, £825. :150=£825. , what 'vill it IS. £2005. , at £60 per e at 4^ jjcr «3 : 8 : 3. inpaitl for 8 £263 : 4. ulf-)early or half of the itio, a fourth ber of yx'ixrs, remains un- ne at 5 per 14 : 7 : 6. left unpaid per cent. } 9:1:3. it oxanipies, Ivantatreous iiiiily. SIMPLE INTEREST. 145 33. If a salary amounted to JE825 in fivo years, at 5 per cent what was the sahiry? Ar^. iiloO. 825X2=1660 5X5X,05—5X,054.5X2=:11 thJn 1650 -j- 11 =£150. 34. If a house is to be let upon a lease for 5^ years, and the amount for that time is £363 : S : 3, at 4^ per cent, what is the yearly rent? A71S. £60. 35. If a pension amounted to £2065, in 7 years, at 6 per cent, what was the pension ? Am. 250. 36. Suppose the amount of a pension be £263 : 4 in 8 years, at 5 per cent, what wjis the pension ? Ans. £28. Note. When the payments are half-yearly, then take 4 a, and half of the ratio, and twice the number of years ; and if quarterl}'-, then take 8 a, one fourth of the ratio, and four times the number of years, .and proceed as before. 37. If the amount of a salary, payable half-yearly, for 5 years, at 5 per cent, be i2S34 : 7 : 6, what was the salary ? Ans. £ 150. 38. If the amount of an annuity, payable quarterly, be £839 : 1 : 3, for 5 years, at 6 per cent, what wiis the annuity i Ans. jeiSO. III. When U A T are given to find R. 2a— 2ut Rule.— =R. utt — ut EXAMPLES. 39. If a salary of £150 per annum, amount to iE825, in 5 years, what is the rate per cent. ? Ans. 5 per cent. 150 825-1-2—150+5+2=150 then = 05 150X5X5—150X5 40. If a house be let upon a lease for 5^ years, at £60 per an- num, and the amount for that time be £363 : 8 : 3, what is the rate per cent. ? Ans. 4^ per cent. 41. If a pension of £250 per annum, amounts to £2065 in 1 jrears, what is the rate i)er cent, f Ans. 6 per cent. 42. Suppose he amount of a yearly pension of £28, be £263 : 4, in 8 years, what is the rate per cent ? Ans. 5 per cent per N I- ,1 f ii j SIMPLE INTEREST, NoTR. When the payments are half-3'early, take 4 a — 4 iit for a <:Hviroduces £N54 : 4 : 6 present worth? Ans. £250. 59. What annuity is that which, for 8 years' continuance, nro- ducos £188 for the present worth, at 5 per cent. ? Jus. £28. Note. When the payments are half-yearly, take half the ratio twice the number of years, and multiply by 4 p ; and wlien niiar- t(.'rly, take one fourth of tlic ratio, and four times the number of years, and multiply by 8 p. 00. There is an annuity payable half-yearly, for 5 years to come, what is the j'early rent, when the present worth, at 5 per c = 4^4 V 1 50 — 660 6X,05X2-|-2 4X,06-fl =560. 75. W to contimi of 5 years 7G. A luim, for i willing to iii'i'seut w< 77. A person of is 21 ; he per cent., 2. Tof liULB 1 worth at before the 2. Cha nuity boil Baine rate ance; thu 78. A does not < allowing come ? 550 X 4 79. Tl coimnenc for £152 cliiuser, w 80. A which do the same cent, to t ;»urehased for ?. 5^ years. i>iircl»asccl for IS. 7 yoai-s. nuin, be pur- is. 8 ycara. in U will be T the num* squal to one ratio, and T lalf-yearly, is »■ the number s, 5 yeai*s. •terly, is sold fiynionts, and s, 5 years. HON. taken in re- 2t - : X U=:P. p. SIMPLE INTEREST. 151 1 50 per an- ho end of 4 ns. £550. -=660. 75. 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, allowing 4 per cent.' to the purchaser? Anv. £152 : 5 : 11 3 qrs. VO. A person having the promise of a pension of £20 per an- num, 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 liv.'seut worth? Ans. £l 1 1 : 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 ,07506. 2. To find the yearly income of an annuity, &c. in revereion. ptr-f-p=A. liuLK 1. Find the amount of the present worth at tlie given rate, and for die time before the reversion ; thus, 2. Change A into P, and find what an- nuity being sold, will produce P at the Biune rate, and for the time of its continu- ance; thus, EXAMPLES. tr-fl ttr— tr-f2t •:X2p=U. 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 in- come? Ans. £150. 5 X ,05 -I- 1, 550 X 4 X ,05 + 550 = GGO 5 X 5 X ,05—5 X ,05 + 6 X 2= ,113036 X C60 X 2 = £150. 79. There is a lease of a house taken for 4 years, but not to connnence till the end of 5 years, the lessee would sell the same for £152 : 6, present payment, allowing 4 per cent, to the pur- cliiuser, what is the yearly rent? -^ins. x50. 80. A person having the promise of a pension for 8 yearn, which does not commence till the end of 4 years, has disposed of the same for £111 : 18 : 1 ,14 present money, allowing 6 per cent, to the purchaser, what was the pension ? Ans. £20. 152 REBATE OR DISCOUNT. 81. There is ji certain lei^acy left to a person of 15 years of age, whicli is to be continued for 6 years, but not to commence till he arrives at the age of 21 ; he, wanting a sum of money, sells it for £1*71 : 14, allowing- 4 ]>er 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 Katio. * I. When S T R are given to find P. s Rule. = P. tr+1 due 5 m{ payment, 8. A months li much wa III. ^ Rule." 9. At hence, pr EXAMPLES. t. "What is the present worth of £357 : 10, to be paid 9 months hence, at 5 per cent? Ans. £344 : 11 : 6^ ,108. 2. What is the present worth of £275 : 10, due 7 months hence, at 5 per cent. ^ Ans. £267 : 13 : 10^^^. 3. What is the present worth of £875 : 5 : 6, due at 5 months hence, at 4^ per cent. ? Ans. £859 : 3 : 3^ y|^. 4. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent. ? Ans. £70 : 11 : 9 ,l764d, II. When P T R are given to find S. Rule. ptr-{-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 was the sum first due ? Ans. £357 : 10. 344,5783 X ,75 X ,05-f 344,5783 = £357 : 10. 6. A person owing a certain sum, payable 7 months hencr. agrees with the creditor to pay him down £267 : 13 : lO^j'i^, al- lowing 5 per cent, for present jiayment, what is the debt ? A71S. £275 : 10. 1, A person receives £859 : 3 : S^j^-j for a sum of money I 14. 1 10. A hence, pr 11. A hence, pr 12. A produce IV. ^ Rule. years of age, nenee till he ', sells it for was the au- 4w5. £40. id 9 months I 7 ,108. months : IO^Vt- at 5 months : 3^ T63- of £75, 9 ,l764d. i 9 months i qrs., what 1357 : 10. 10. mths hence, •10 3 8 ol. l>t? 1275 : 10. n of money BEBATE OR DISCOUNT, 153 due 5 months hence, allowing the debtor 4]^ per cent, for present payment, what was the sum due? Ans. £875 : 5 : 6. 8. A person paid £70 : 11 : 9 ,1764d. for a debt due 15 months hence, he being allowed 5 per cent, for the discount, how much was the debt ? Ans. £75 III. When S P T are given to find R. s— p Rule. =R. * tp EXAMPLES. 9. At what rate per cent, will £357 : 10, paj-abL months hence, produce £344 : 11 : 6 3,108 qrs. for present payment? 3575,-344,5783 . z=,05=5 per cent. 344,5783 X ,75 10. At whai rate per cent, will £275 : 10, payable 7 months hence, produce £267 : 13 : lOg^^ for the present payment? Ans. 5 per cent. 11. At what rate per cent, will £875 15:0, payable 5 months hence, produce the present payment of £859 : 3 : 3^ jf ^ ? Ans. 4^ per cent. 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 IV. 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 3,108 qrs. at 5 per cent., in what time should the sum have been paid without any rebate ? Ans. 9 months. 357,5—344,5783 :,75=9 mouths. 344,5783 X ,05 14. The present worth of £275 : 10, due at a certain time to 154 EQUATION OF PAYMENTS. come, is £267 : 13 : lOJ^J^, at the sum have been paid witlujut 15. A person receives £859 due at a certain time to come, desire to know in what time the Q(* witliout any rebate ? 10. I liave received £70 : 1 allowinaf the person 5 per cent, know when the debt would have 5 per cent., in what time should any rebate ? ~ Ans. 7 months. : 3 : 3f ,0184, for £875 : 5 : Q, allov.ing 4| per cent, discount, I debt should have been discharer ceut. per 8323 qrs. it to £260 : 3. nt to £243 ns. £200. 538368 qrs. IS. £150. 23 qrs. in 4 IS. £500. of extrac- on showing EXAMPLES. 9. At what rate per cent, \vill £225 amount to £200 : 9 : 3,3 qrs. in 3 years ? ^ ^ns. 5 per cent. 260,465325 =1,157625, 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 : 2,025s. in 4 years ? -4w5. 5 per cent. 11. At what rat« per cent, will £450 amount to £547 : 9 : 10 2,0538368 qrs. in 5 years ? Ans. 4 per cent. 12. At what rate per cent, will £500 amount to £619 : 8 : 2 3,8323 qrs. in 4 years ? Ans. 6| per cent IV. When P A R are given to find T. a which being continually divided by R till no- RuLE. — =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 qrs. ftt 5 per cent. ? 260,465625 1,157625 1,1025 1,05 . = 1,157625 = 1,1025 =1,05 225 1,05 . 1,05 1,06 = 1, the number of divisions being three times sought. 14. In what time will £200 amount to £243 2,025s. at 5 per cent. ? Ans. 4 years. 15.' In what time will £450 amount to £547 : 9 : 10 2,0538368 qi-s. at 4 per cent. ? Ans. 5 years. 16. In what time will £500 amount to £619 : 8 : 2 3,8323 qi-s. at 5^ per cent. ? Ans. 4 years. ANNUITIES, OR PENSIONS, IN ARREARS. Note. U represents the annuity, pension, or yearly rent A R T as before. ,;.A'' ^U.i '4 m 158 COMPOUND INTEREST. A Table showing the amount of £1 annualhj^ for any number of years under oi, at 5 and 6 ^^cr cent. ^;er annum. YEARH. 5 RATK3. 6 Yr.AR!*, 5 RATK3. 6 1 1 1,0')(JUU 1,00000 10 23,65719 35,67252 2 2,U50OU 2,(X)0(X) 17 25,84036 28,21288 3 3,15250 3,18360 18 28,18238 3U,90565 4 - 4,31012 4,37461 19 30,53900 33,75999 5 5,525r,3 5,63709 20 33,06595 36,78559 6 6,80191 6,97532 21 35,71925 39,99272 7 8,1-1200 8,39383 22 33,50521 43,39229 8 9,51ut to intorest will amount to P at the Bame rate, and for the time to come before the annuity commences, which will be the present worth of the annuity, «&c. : thus u u- =p. r— 1 a 44. Wl to continu at 6 per c I i by R till f those di- B had for r? ontinually umber of will be= 36 of £40 !d for ? 3 years. I, and the y allowed . • J yeare. purchased . at 5 per t years. RSION. ken in 11 =P. -1 =P. COMPOUN.i) INTEREST. EXAMPLES. 163 41. What is the present worth of a reversion of a lease of £40 per annum, to continue for six years, but not to commence till Che end of 2 years, allowing 6 per cent, to the purchaser ? Am. £175 : 1 : 1 2 ,048 qrs. 40 40—28,1984 196,6933 =28,1984 =196,6933 1,41852 1,06—1 1,1236 = 175,0563. 42. What is the present worth of a reversion of a lease of £60 p(^r annum, to continue 7 years, but not to commence till the end of 3 years, allowing 5 per cent, to the purchaser ? Ans. £299 : 18 : 2,8d. 43. There is a lease of a house at £30 per annum, which is yet in being for 4 years, and the lessee is desirous to take a lease m reversion for 7 years, to begin when the old lease shall be ex- pired, what will be the present worth of the said lease in rever- sion, allowing 5 per cent, to the purchaser ? Ans, £142 : 16 : 3 2,688 qrs. To find the yearly income of an annuity, <&c. taken in reversion. Rule. Find the amount of the present worth at the given rate, and for the time be- fore the annuity commences : thus, pr*=A. Change A into P, and find what yearly rent being sold will produce P at the same rate, and for the time of its continuance, which will pr*Xr — prt. be the yearly sum required : thus, =U. r»— 1. EXAMPLES. 44. Wliat annuity to bo entered upon 2 years lience, and then to continue 6 years, may be purchased for £175 : 1 : 1 2,048 qrs* at 6 per cent. ? Ans.. 175,0563 v 1.1 23G = 1 96,6933 hen 196,0933 X 1,41852 X 1,00— 279',01337 =£40. 1,41852—1 k'V i; i 164 C03IP0UND INTEREST. 45. The present wortli of a lease of a house is £209 : 18 : 2 8d taken in reversion for 7 years, but not to commence till the enc of 3 years, allowing 5 per cent, to the purchaser, what is the yearly rent? Ans. 60. 46. There is a lease of a house in being for 4 years, and tlio lessee being minded to take a lease in reversion for 7 yeai-s, to begin when the old lease shall be expir-'d, paid down £142 : 16 : 3 2,688 qi-s. what was the yearly rent of the house, when the les- see was allowed 5 per cent, for present payment? Ans. £30. PURCHASING FREEHOLD OR REAL ESTATE, IN SUCH AS ARK BOUGHT TO CONTINUE FOR EVER. I. When U R are given to find W. % u > Rule. =W. r— 1 EXAMPLES. 47. "What is the worth of a freehold estate of £50 per annum, allowing 5 per cent, to the buyer ? 50 jins. =£1000. 1,05—1 48. What is an estate of £140 per annum, to continue for ever, worth in present money, allowing 4 per cent, to the buyer ? Ans. £3500. 49. If a freehold estate of £75 yearly rent was to be sold, what is the worth, allowing he buyer C per cent. ? Ans. £1250. II, When W R are given to find U. Rule. wXr — 1=XJ. EXAMPLES, 50. If a freehold estats is bought for £1000, and the allowanco of 5 per cent, is made to the buyer, what is the veavly rent? Ans. 1 ,05-- 1 = ,05, then 1 000 X ^05 = £5 0. 51. If an estate be sold for £3500, and 4 per cent, allowed to tho buyer, what is the yearly rent 3 per 52. If and an a what is tl III. W RULE.- 53. If is the rat( 54. If £3500, w 55. If the rate p v\ To Rule. Change put to int for the tir that will 1 56. If years hen< 5 per ceni 57. W cominenet the purch 58. W ney, to cc years, allc Ans. £140. COMPOUND INTEREST. 165 18 : 2 8d :ill the qm, li.'it is the Ans. 60. irs, and tlio 7 yeai-s, to ei42 : 16: len the les- [ns. £30. AS ARE per annum, ue for ever, , £3500. ) sold, what . £1250. 52. If a freehold estate is bought for £1250 present money, and an allowance of 6 per cent, made to the buyer for the same, what is the yearly rent ? ■ j4ns. £75. III. When W U are given to find R. w-j-u Rule. =R. w EXAMPLES. 53. If an estate of £50 per annum be bought for £1000, what Is the rate per cent. ? Ans. 1000-1-50 1000 ; 1,05 = 5 per cent. 54. If a freehold estate of £140 per annum be bought for £3500, what is the rate per cent, allowed ? Ans. 4 per cent. 55. If an estate of £75 per annum is sold for £1250, what is the rate per cent, allowed ? Ans. 6 per cent. PURCHASING FREEHOLD ESTATES IN REVERSION. To find the worth of a Freehold Eatate in reversion : u Rule. Find tlie worth of the yearly rent, thus — =W Change W into A, and find what principal, being r — 1 piit to interest, will amount to A at the same rate, and for the time to come, before the estate commences, and a that will be the worth of the estate in reversion, thus : — =P rt. EXAMPLES. 3 allowaiico rent? 5= £50. allowed to '.s. £140. 66. If a freehold estate of £50 per annum, to commence 4 years hence, is to be sold, what is it worth, allowing the purchaser 5 per cent, for the present payment ? 50 1000 Ans. =1000, then =je822 : 14 : 1^. 1,05—1 1,2155 57. What is an estate of £200, to continue for ever, but not to commence till the end of 2 years, worth in ready money, allowing the purchaser 4 per cent.? Ans. £4622 : 15 : 7 ,44d. 58. AVhat is an estate of .£240 per annum worth in ready mo- ney, to continue for ever, but not to commence till the end of 3 years, allowance being made at 6 per cent. ? Ans. £3358 : 9 : 10 2,24 qrs. * ■ * I 166 REBATE OR DISCOUNT. To find the Yearly Rent of an Estate taJcen in reversicm, liuLR. Find the amount of the worth of the wr- -w=U, estate, at tlie gixon rate, and time before it com- wr* iiienccs, thus : Change A into W, and find what yearly rent being sold will produce U at the same rate, thus : which will be the yearly rent required. EXAMPLES. 59. Tf a freehold estate, to commence 4 years hence, is sold for $822 : 14 : 1^, allowing the purchaser 5 per cent., what is the yearly income ? Ans. 822,70025 X 1,2155 = 1000, then 1000X1,05— 1000=£50. 60. A freehold estate is bought for £4622 : 15 : 7 ,44d, which does not commence till the end of 2 years, the buyer being allow- ed 4 per cent, for his liioney. I desire to know the yeai-ly in- come. Ans. £200. 61. There is a freehold estate sold for £3358 : 9 : 10 2,24 qrs., but not to commence till the expiration of 3 years, allowing 6 per cent, for present payment ; what is the yearly income ? Ans 240 REBATE OR DISCOUNT. A Table shotoing the present ivorth of £l due any number of years hence^ under 31, rebate at 5 o,nd 6 per cent. * YEARS. 5 RATES. G YEARS. 5 RATES. 6 1 ,952381 ,943390 10 ,458111 ,393040 2 ,907030 ,889990 17 ,430290 ,371304 3 ,803838 ,839019 IS ,415520 ,350343 .1 ,822702 ,792093 19 ,395734 ,330513 5 ,783520 ,747258 20 ,370889 ,311804 6 ,740215 ,704900 21 ,358942 ,294155 7 ,710082 ,005057 22 ,341849 ,277505 8 ,070839 ,027412 23 ,325571 ,201797 9 ,041009 ,591898 24 ,340008 ,240978 10 ,013913 ,5.jS394 25 ,295302 ,232995 11 ,584079 ,520787 20 ,281240 ,219810 12 ,550837 ,490909 27 ,207848 ,207308 13 ,.'530321 ,408839 28 ,255093 ,190030 14 ,505008 ,442301 29 ,242940 ,184550 15 ,481017 ,417205 30 ,231377 ,174110 Note. — The above table is thus vmhIq : 1 ~- 1 first year's present worth; and ,952381 -i- 1,05= year; and ,90703H-1,05=,8G3838 third year, (fee. ,05 = ,952381, ,90703, second REBATE OR DISCOUNT. 167 werston, =A wr — w=U, I. When S T R are given to find P. nee, is sold nt., what is = 1000, I000=£50. ,44d, which being allow- e yearly in- is. je200. 2,24 qrs., allowing 6 me ? dns. 240. number of ent, i. 6 J9304G ni364 }r)0343 530513 ill804 !94155 ^nr•>o'^ !(3l7i.(7 140978 !3-299S ! 198 10 1073ns 90030 84 ^SG 74110 == ,952381, ^03, second Rule. :P. EXAMPLES. 1. What is the present worth of £315 : 12 : 4 ,2d, payable 4 vears hence, at 6 per cent. ? Ans, 1,06X1,00X1,06X1,06 = 1,26247, then by the table. 315,6175 315,6175 =£250 ,792093 1,26247 249,9984124275 2. If £344 : 14 : 9 1,92 qrs. be payable in 7 years' time, what is the present worth, rebate being made at 5 ^ er cent. ? Am. £245. 3. There is a debt of £441 : 17 : 3 1,92 qrs., which is payable 4 yeai-s hence, but it is agreed to be paid in present money ; wliat sum must the creditor receive, rebate being made at 6 per cent. ? Ans. £350. II. When P T R are given to find S. Rule. pXr*=S. EXAMPLES. 4. If a sum of money, due 4 years hence, produce £250 for the present payment, rebate being made at 6 per cent., what waa the sum due ? Ans. £250Xl,26247=£315 ; 12 : 42d. 5. If £245 be received for a debt payable 7 years hence, and an allowance of 5 per cent, to the debtor for present payment, what was the debt? Ans. £344 : 14 : 9 1,92 qrs. 6. There is a sum of money due at the expiration of 4 years, but the creditor agrees to take £350 for present payment, allow- ing 6 per cent., what was the debt ? Ans. £441 : 17 : 3 1,92 qrs. III. When S P R are jnven to find T. s which being continually divided b}' R till nothing Rule. — =rt remains, the number of those divisions will be p equal to T. • li-^'fi ■»l' 168 BEilATE OR DISCOUNT. EXAMPLES. 1. The present payment of £250 is made for a debt of £315 : 12 : 4 ,2d., rebate at 6 per cent., in what time was the debt pay- able ? ^ ^ 315,6175 which being continually divided, those Ans. =1,26247 divisions will be equal to 4=the num- 250 ber of years. 8. A person receives £245 now, for a debt of £344 : 14 : 9 1,92 qrs., rebate being made at 5 per cent. I demand in what time the debt was payable ? Ans, 1 years. 9. There is a debt of £441 : 17 : 3 1,92 qrs. due at a certain time to come, but per cent, being allowed to the debtor for the present payment of £350, I desire to know in what time the sum should have been paid without any rebate ? Ans. 4 years. IV. When S P T are giv^n to find R. s which being extracted by the rules of extraction, Rule. — =r* (the time given in the question showing the pow- p er,) will be equal to R. TU EXAMPLES. 10. A debt of £315 : 12 : 4,2d. is due 4 years hence, but it is agreed to take £250 now, what is the rato per cent, that the re- bate is made at ? 315,6175 4 Ans. = 1,26247 : ^1)26247=1,06 = 6 per cent. 250 11. The present worth of £344 : 14 : 9 1,92 qr..., payable 7 years hence, is £245, at what rate per cent, is the rebate m'ade ? Ans. 5 per cent. 12. There is a debt of £441 : 17 : 3 1,92 qrs., payable in 4 years time, but it is agreed to take £350 present payment. I de- sire to know at what rate per cent, the rebate is made at ? Ans. 6 per cent. Cross M 1. Und tions of tl 2. Mult lowest) bj retipective each lovv€i 3. In tl in the m more to t] 4. Woi plio.r, setti of those i 169 , of £315 : debt pay- idcd, those =the num- 44 : 14 : 9 id in what 7 years. it a certain >tor for liiG le the sum 4 years. extraction, g the pow- e, but it is lat the re^ ler cent. payable 1 made? er cent. able in 4 nt. I de- er cent. THE TUTOR'S ASSISTANT. PART IV. DUODECIMALS, OR, WHAT 13 GENERALLY CALLED Cross Multiplication, and Squaring of Dimensions hy Arti- ficers and Workmen, RULE FOB MULTIPLYING DUODECIMALLY. 1. Under the multiplicand write the corresponding denomina- tions of the multiplier. 2. Multiply each term in the multiplicand (beginning at the lowest) by the feet in the multiplier ; write each result under its retipective term, observing to carry an unit for every 12, from each lower denomination to its next superior. 3. In the same manner multiply the multiplicand by the primes in the multiplier, and write the result of each term one place more to the rio-ht hand of those in the multiplicand. 4. Work in the same manner with the seconds in the multi- plier, setting the result of each term two pUices to the right hand of those iuTihe multiplicand, and so on for thirds, fourths, &c. * I <*J lib H-' 170 DUODECIMALS EXAMPLES f. in. f. in. 1. Multiply 7 . 9 by 3 . 6. Cross Multiplication Practice. 7 9 6^ 7 . 9 3 '^6 3 . 6 Duodecimals. 7 .9 2 . 6 Decimals. 7,75 3,5 21.0.0=7X3 2.3.0=9X3 3.6.0 = 7X6 0.4.6=9X6 23 . 3 3 . 10 . 6 27 . 1.6 23 . 3— X3 3.10.6X6 27 . 1.6 3875 2325 27,125 27.1.6 2. Multiply 3. Multiply 4. Multiply 5. Multiply 6. Multiply 7. Multiply 8. Multiply 9. Multiply 10. Multiply 11. Multiply 12. Multiply 13. Multiply 14. Multiply 15. Multiply 16. Multiply 17. Multiply 18. Multiply f.in. 8.5 9.8 8.1 7.6 4,7 7.5.9" 10.4.5 75.7 97.8 57.9 75.9 87.5 179.3 259.2 257.9 311.4.7 321.7.3 by by by by by by by by by by by by by by by by by f. in 4. 7 7. 6 3. 5 5. 9 3.10 3. 5.3" 8.6 8 9 5 7 8 7. 9. 8. 9. 17. 35. 38.10 48.11 39.11 36. 7.5 9. 3.6 Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, f. 38. 72. 27. 43. 17. 25. 79.11. 730. 7. 854. 7. 543. 9. 1331.11. in.pts. 6.11 6 7. 5 1. 6 6.10,,,, 8. 6.2.3 0.6.6 8 9 3 4 6 3117.10. 6960.10. 12677. 6.10 102S8. 6. 3 11402. 2. 4.11.1 2988. 2.10.4.6 THE APPLICATIO.NT. Artificers' work is computed by diftereut measures, viz : — 1. Glazing-, and masons' fiat woik, by the foot. 2. Painting, plastering, paving, 37. How many squared rods are there in a wall 62^ feet lon^ li feet 8 inches high, and 2^ bricks thick ? Ans. 5 rods, 166 feet 6 in. 38. If the side walls of a house be 28 feet 10 inches in length, and the height of the roof from the ground 55 feet 8 inches, and the gable (or triangular part at top) to rise 42 course of bricks, reckoning 4 course to a foot. Now, 20 feet high is 2^ bricks thick, 20 feet more at two bricks thick, 15 feet S inches more at H brick thick, and the gable at 1 brick thick; what will the whole work come to at £5 16s. per rod? Ans. £48 : 12 : 7. Multiplying several figures bg several, and the product to be produced in one line only. Rule. Multiply the units of the multiplicand by the units of the multiplier, setting down the units of the product, and carry the tens; next multiply the lens in the multiplicand by the unit,s of the multiplier, to which add the product of the units of the miilti plicand multiplied by the tens in the multij)lier, and the tens car riod ; then multiply the hundreds in the multiplicand by the uni(:> of the multiplier, adding the jjroduct of the tens in the multi})licati(l multii)li"d by the tens in the multijjlier, and the units of the iniilti- plicand by the hundreds in the multiplier; and so proceed till you have multiplied the multiplicand all through, by every figure of the multiplier. First, 4) 4 X 2, and Thirdly, 2; ry 3. Fou set down 7 -[-4X5 + ^ + 5X44 5. Seven il i'ul carry 1 and carr^ '.ijuied by :md the wo iber of half vided by 3, lickness two ds. 3et 7 inches in? 36 feet 1 in. 2 2 feet long, 06 feet 6 in. s in length, inches, and ?e of bricks, s 2^ bricks les more at lat will the 12 : 7. DUODECIMALS. EXAMPLES. 175 Multiply 35234 by . . . . . . 52424 Common way, 35234 62424 Product, 1847107216 140936 70468 140936 70468 176170 1847107216 EXPLANATIONS. First, 4X4=10, that is 6 and carry one. Secondly, 3X4-f 4X2, and 1 that is carried, is 21 — set down 1 and carry 2 Thirdly, 2X44-3X2 + 4X4-|-2 cnrried--=32, that is 2 and car- ry 3. Fourthly, 5 X 4-f 2 X 2 -f 3 X 4+ 4 X 2+3 carried = 47, set down 7 and carry 4. Fifthly, 3X4-1-5X2 + 2 X 4 +3X2 MX 5 + 4 carried =60, set down and carry 6. Sixthly, 3X2 +-5X4 + 2X2+3X5+6 carried = 51, set down 1 and carry 5. Sevenihly, 3X4 + 5X2 + 2X5 + 5 carried=37, that is 7 md carry 3. Eighthly, 3X2+5X5+3 carried=34, set down I and carry 3. Lastly, 3X5 + 3 carried =18, which being mul- iplied by the last figure in the multiplier, set the whole down, and the work is finished. luct to be t ho units of id carry the ho units of f the multi- le tens car 3y the units nultiplicand f the multi- eed till you y figure of 176 THE TUTOR'S ASSISTANT PART V. A COLLKCTION OF QUESTIONS. 1. What, is the vahie of 14 barrels of soap, at 4|d. per lb , each barrel coiitainiiii^ 254 lb. ? Ans. jCOG : 13 : 6. 2. A and B trade too-ether ; A puts in £320 for 5 months, B £400 for 3 months, and they ijained £100; what must each man receive ? Jns. A £o3 : 13 : 9|f a, and B £4G : 6 : 2/^^. 3. How many yards of cloth, at 17s. Gd. }»er yard, can 1 have for 13 cvvt. 2 qrs. of wool, at 14d. per lb. ? Ans. 100 yards, 3} qrs. 4. If I buy 1000 ells of Flemish linen for £90, at what may I sell it per ell in London, to gain £10 l)y the whole ? Am, 3s. 4d. per ell. 5. A lijus 048 yards of cloth, at 14s. per yard, ready money, but in barter will have 16s.; B h^us wine at £42 per tun, ready money : the question is, how much wine must be ^iven for tlio 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 ffiven for tlio clo' 6. A jeweller sold jewels to the value of j£12j j, for which Iie^ received in part 876 French pistoles, at 16s. 6d. each; what sum J remains unpaid ? o-> # Ans. £477 : G. 7. An oilman bought 417 cvvt 1 qr. 15 lb., gross weight, of train oil, tare 20 lb, per 112 lb., how many neat gallons H'ere there, allowing 1^ lb. to a gallon. ? Ans. 5120 gallons. 8. If I buy a yard of cloth for 14s. 6d., and sell it for 10^. 9d| what do I gain per cent. ? Ans. £15 : 10 : 4f^\. 9. Bought 27 baiys of mnijer, each weischinir sjrross 843 11»., tare .*t If lb. i»er bag, tret 4 lb. per 104 lb., what do they come to at SAd. UQt lb.? Ans. £70 : 10 : liV 10. Iff cost? 11, Iff 12. A jE and at the 13. A h times 1 12 I dilference h 14. A cf wliich the divided am 15 At \ in 7^ years 16. A h £13 a pie tliey intercl 17. A m viz. A, B, < and as IT 15. £10 iicr, that il must each i 19. A pi inches broa< 20. If 3 months, bui 'Hi u must c 2\. The of tlitir jiro^ 22. A 1)1 oxen at £1 each, and o buy? 23. Wk A COLLECTION OF QUESTIONS. n7 10. Iff of an ounce cost | of a sliillin"^, what will 4 of a lb. cost? An.^. 17s. 6d. NT per 11) , each I : 1.3 : 6. ) months, B st each man " • '^ 2 i> 8 • , can 1 have ,s, 31 qrs. what may I (1. per ell. ady money, r tun, ready iven for the 12| gals, of! ■or which he] ; what sum] £477 : G. i weight, of! gallons H'erel gallons. for 1 iU. 9d%j : 4t^x' 84 3 11),, tarej u\y come toj . i .-^ . t 3 11. If f of a gallon cost f of a pound, what will | of a tun cost ? Ans. jElOo. 12. A gentleman spends one day with another, £] : 7 : 10|, and at the year's end layeth up je340, what is his yearly income ? Ans. £848 : 14 : 4^. 13. A has 13 fother of lead to send abroad, each being 19^ times 1 12 lb. 13 has 39 ctisks of tin, each 3SS lb., how many ounces diliercnce is there in the weight of these commodities ? Ans. 212160 oz. 14. A captain and 160 -ailors took a prize worth £1360, of which the captain had } for his share, and the rest was equally divided among the sailors, what was each man's part ? Aus. The captain had £272, and each sailor £6 : 16. 15. At what rate per cent, will £956 amount to £1314 : 10, in 7^ years, at simple interest ? Ans. 5 per cent. 16. A hath 24 cows, worth 72s. each, and B 7 horses, worth £13 a piece, how much Avill make good the difference, in case they intercliange their said drove of cattle ? Ans. £4 : 12. 17. A man dies and leaves £120 to be given to three |)ersons, viz. A, B, C ; to A a share unknown ; B twice as much as A, and C as much as A aiid B ; what was the share of each i Ans. A £20, B £40, and C £60. IS. £1000 is to be divided among three men, in such a man- ner, that if A has £3, B shall have £d, and C £S ; how much must each man have ? Am. A£] ~ : 10, B ^2312 : 10, and C £500. 19. A piece of wainscot is S feet 6^ inches long, and 2 feet 9f indies broad, Avhat is the superficial content ? Ans. 24 feet : 3" : 4 : 6. 20. If 360 men be in garrison, and have provisions for C months, but hearing of no relief at the end of 5 rnonths, how many men must depart tliat the provisions may last so yiuv^'k the longer? Ans. 2SS men. 2\. The less of 2 numbers is 187, their difference 34, the square of tiitir j>roduct is required ? jins. 1707920929. 22. A l)utcher send* his man with iE2l6 to v fair to buy cattle; oxen at Jg 11, cows at 40s., colts at £1 : 5, and hogs at £l : 15 each, and of each a like number, how many of eacli sort did he buy? Ans. 13 of each sort, and £8 over. 23. What number added to 11^ will produce 363 3| ? Ans. 24f}-|. 4 :l* i=i 178 A COLLECTION OF QUESTIONS. 24. AVluit number multiplied by -f will produce llyT- 38. If Ans. 2644. J 100 more for 2 sliilli 25. Wliat is the value of 179 hogsheads of tobacco, each weigh- ing 13 cwt, at £2 : 7 : 1 per cwt. ? Ans. £o478 : 2 : 1 1? 20. ^fy factor sends me word he has bought goods to the va- lue of £500 : 13 : 6, upon my account, what will his coniinissjon come to at 3^ per cent. ? Am. £17 : 10 : 5 2 qrs. j%\. 27. If ^ of 6 be three, what will ^ of 20 be ? Ans. 7^ 28. What is the decimal of 3 qrs. 14 lb. of a cwt. ? Ans. ,875 29. ITow many lb. of sugar at 4^d. per lb. must be given in barter for .GO gross of inkle at 8s. 8d. per gross? Ans. 138Gf lb. 30. If I buy yarn for 9d. the lb. and sell it again for 13|(1. per lb., what is the gain per cent. ? Ans. £oO. 31. A tobacconist would mix 20 lb. of tobacco at 9d. per lb. with GO lb. at 12d. per lb., 40 lb. at 18(1. per lb., and with 12 lb. at 2s. per lb., what is a pound of this mixture worth ? Ans. Is. 2^d. j\. 32. What is the difference between twice eight and twenty, and twice twenty-eight ; as also, between twice five and fifty, and twice fifty-five ? Ans. 20 and 50. 33. Whereas a noble and a mark just 15 yards did buy ; how many ells of the same cloth for £oO had I ? Ans. GOo"^ ells. 31. A broker bought for his principal, in the year 1720, ^£400 capital stock in the South-Sea, at £G50 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 Gs. 6d. per dozen ; D hath cotton at 9d. per lb. loadyj money. I demand what price the cotton must be at in barter; also, how much cotton must be bartered for 100 doz. of cariray what was that? Ans, £16,093 : . 8 a yci... 49. A gentleman liaving 50s. to pay among liis labourers for ( day's w<^i"k, would gi\c lo every boy 6d., to every woman 8d and to every man IGd, ; the num>)er of boys, women, and inei was the same. I demand tjio number of each ? 180 A COLLECTION OF CiUESTIONS. ^ 60. A stone that measures 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches deep, how many soHd feet doth it con- tain ? Ans. 41 feet 3 incites. 51. What does the whole pay of a man-of-war's crew, of 640 sailore, amount to for 32 months' service, each man's pay beino 22s. 6d. per month ? Ans. £23,040. ° 62. A traveller would change 500 French crowns, at 4s. 6d. per crown, into sterh'ig 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 in 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 I7s. 6d. per piece? Ans. 76|f. 56. From 7 cheeses, each weighing 1 cwt. 2 qi-s. 5 lb., how many allowances for seamen may be cut, each weighing 5 oz. 7 drams ? ,; Ans. 3563 ^f 57. If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence leaves 12, what number is that, out of which wh^ you have taken 48, 72, 19, and 7, leaves 1^ ? Ans. 158. 68. A farmer ignorant of numbers, ordered £500 to be dividal among his five sons, thus :— Give A, says he, ^, B i, C i, D i, and E | part ; divide this equitably among them, according to their father's intention. ^ Ans. A £l52f||, B £114iii, C £91^^ D £76ii|, E £65if 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. Ques. What was each of our ages when wo were married ? A71S. 45 years the man, 15 the woman* I" 1. On j should be 2. The from the is the gre? 3. Afte 33, 58 ; tl tracted; v 4. Of t third is s three nun: 5. The which is t 6. A si of age ; v older ? 7. If 2' is 21, thei 8. A ir father's a^ 9. Aft( number tl 10. Th 18 be tal5 will be eq 11. Th ence and 12. Tl 1 50 from Is the gre 13. Tl one and greater ? 14. W greatest 1 A COLLECTION OF QUESTIONS. 181 eet 9 inches cloth it con- 3 inclies. rew, of 640 8 pay being £23,040. s, at 4s. 6d, ilfpenny per 1:9:2. mt in £640, \ rmn. I de- ns. £960. n one year,, ns. £400. French pis- ns.76^. 5 lb., howj ing 5 oz. 7 s. 3563if en from 91 i number is I nd 7, leaves j 4«s. 158. ) be dividend I :, Ci,Di,l ccording to. irried ! 3 womanu SUPPLEMENTAL QUESTIONS. 1. On goods that cost 412s. there was 25s. profit; how much should be sold to gain as much more ? Ans. 462s. 2. The less of two numbers is 17, and after having subtracted 23 from the greater, the remainder is eight more than the less ; what is the greater ? Ans. 4S. 3. After having successively subtracted from a number, 17, 29, 33, 58 ; the remainder is 91 more than the total of the sums sub- tracted ; what is that number ? Ans. 365. 4. Of three numbers, the fii-st is 215, the second is 519, and the third is as much as the other two; what is the sum of the three numbere 3 Ans. 1468. 5. The gmater of two numbers is 56 and the difference is 37 ', which is the less? Ans. 19. 6. A sister is 8 years younger than her brother who is 27 years of age ; what will her age be when her brother will be 7 yeara older ? Ans. 26 years. 7. If 27 be added to the sum of two numbers, the less of which is 21, their total will be 147 ; which is the greater ? Ans. 99. 8. A man 47 years of age has a son 9 years old ; what will the father's age be when the son will be the father's present age ? Ans. 85. 9. After having added successively 17, 29, 33, and 54 to a number the total is 214 ; what is that number? Ans. 81. 10. The age of the ffither and son together is GO yeai-s : and if 18 be taken from the father's age and added to the son's their age will be equal ; what is the age of each ? Ans. 48 and 12. 11. The greater of two numbers is four more than their differ- ence and their sum is 27 ; determine the numbers? Ans. 23 and 4. 12. The smaller of two numbers is 160, and after subtracting 150 from one and 48 from the other, the remainder is 244 ; what is the greater ? Ans. 282. 13. The less of two numbers is 37, and after taking 72 from one and adding 34 to the other their total is 145; what is tho greater? Ans. 146. 14. What are the three numbers whose sum is 3291 and th« greatest 1126 exceeds the smallest hy 79 ? Ans. The smallest 1,046, the mean 1,120. P',s 182 A COLLECTION OP QUESTIOIfg. • , A-t'^'' (dividing a certain sum between 26 persons each re. C(Mve(l 20 /s. ; what was the sum ? 'aus. 6,682s 16. hroMi a ceilaui f'^uni 152 persons took 8l7 each, and thert icinainea .^13 ; wh.-it was the sum ? Jns. $'2597 • / ',* ,^^ 1'2^ i' ^^"^ nmnher that beinjr augmented by 56 and di- v.d. d by 55,the quotient will be 2,854 ? A7is. 156 914 18. What IS the number tlint being divided by 27, 'mves i quotient equal to the product of 1,091 by 3 ? Am 88 371 ' 10. By selling 120 yards of cloth for 3,600s. there was 5s, p.oiit per yard ; what was the buying price ? Ans. 3,000s 20 [ boucrht 150 yards of cloth for 3,750s. and sold them for o , Wr'"'^ ' ''^'''^ '^''^ ^ 8^'"" V ^''^ ^'»''?'"n ? Am. 000s o-n^'.ni i.'"f-''''"''.^ ^'"^ obtained, it; after having multiplied 2o0,o40 by 10 this product should be repeated 2,458 limes? .,o A 1 «. "'^'^*- 6,158,273,200. 22 A man has 83000 revenue and spends ^5 per day; what will he hiy up at the end of 1 years ? Am. $1 1,750 23. A class is composed of a certain number of schol'ars • * if there were 8 more the number would be augmented I ; how many scnolars were there? * '^^^^ 4^ ^ 24 'i'he quarter of the 54th part of a number is 5,454*; wjiat IS tktt number? Jn.. 1,178,064. 20 On the sale of 150 yards of cloth for 29s. per yard, there were 600s. protit ; what was the buying price ? Am. 3,750s. 10. What number being divided by 4 gives a quotient such that, atter subtracting 9, the remainder will be 20 ? Am. 116 27. How many revolutions will the second-hand of a clock make in a year, the year being 365d. 5h. 48min. ? oo A , . ^^^*- 525,948 rev. 28. A number is such that in-taking 9 from its fourth part, the remaindor is 91 ; what is the number? Am. 400. 29. What is the number whose 17th part augmented 54,' is equal to 602? Am. 9,316. 30. What number added to the product of 185 by 27, gives 115 times 155 for total? Am. 12. SSO. 31. The h , , f two numbers is 187, and their difference is 34 required the »(, uare of their product ? Am. 1 ,707,920,929. 32. AVhat number must be added to the square of 125 to pro- duce 20,000 for total ? Am. 4 375. 33. The sum of two nurabei-s is 360, and the less is 144 ; re- quired the result of their product by the square of their difference? Ans. 161.243.136. J — J — . . 84. Two their diffen ence, their 35. The double the what is the 36. Det power of i: 37. Wh is 20 and t 38. The greater ; re 39. Wh 156,970 foi 40. If 2 will be 17£ 41. By 42. If y 1,548 ; reo 43. The by the less, the two nu 44. The 154; what 45. Req 46. The 150 from c the two nu 47. A s his age ; w 48. A f< tlieir ages 49. 1'he the son's j father's; w 50. Fin. of the othe ons each re« y. 6,682s. 1, and thert s. $'2597. 56 and di- 156,914. 27, gives a . 88,371. ;re was 5s, ?. 3,000s. Id tliem for n,s\ COOs. multiplied imes ? 273,200. day; what $11,750. cholars ; if how many Ans^ 40. 454 ; what 178,064. yard, there 3,750s. )tient such ns. 116. 3f a clock 948 rev. h part, the ns, 400. ted 54, is f. 9,316. / 27, gives 12.830. ince is 34 '20,929. 25 to pro- . 4,375. s 1 44 ; re- Jiflfercnce ? 43.136. \\ A COLLECTION OF QUESTIONS. 183 84. Two numbers are such, that the greater is 37 times 45, and their difference 19 times 4; required the two numbers, their differ- ence, their sum and their product ? ins. The two numbers are 1,665 and 1,5R9; difi'. 76 ; sum 3,254 ; prod. 2,645,685. 35. The sums of two numbers is 4,517, and by adding 27 to double the square of 25 you will produce one of the numbers ; what is the other ? Ans. 3,240. 36. Determine the difference that exists between the fourth power of 13 and the triple square of 49 ? Ans. 21,358. 37. What is the sum of the cubes of two numbers, whose sura is 20 and the lest number 9 ? Ans. 2,060. 38. There are two numbers, one is 39 and the other is 27 times greater ; required their sum and the square of their difference ? Ans. Sum 1,092, square of their diff. 1,028,196. 39. What is the number that, being multiplied by 55, gives 156,970 for product? Ans. 2,854. 40. If 256 be multipled by an unknown number, the product will be 1792. ^ Ans. 7. 41. By what number must 54 be multiplied to give 9,990? Ans. 185. 42. If you multiply a certain number by 7, you will augment it 1,548; required the number? Ans. 258. 43. The sum of two numbers is 13, and their product, divided . by the less, is equal to the quarter of the same product; required I the two numbers ? Ans. 9 and 4. 44. The sum of two numbere is 2,45S, and their difference is 154; what are the number*? Ans. 1,152 and 1,306. 45. Required two numbers whose difference is 7, and sum 33 ? Ans. 20 and 13. 46. The difference between two numbers is 100, and after taking I 150 from one and 48 from the other, there remains 244 ; required the two numbers? Ans. 271 and 171. 47. A son is 45 years younger than his father who is four times his age ; what is the age of each ? Ans. 60 and 15. 48. A father is six times {is old as his son, and tlie sum of both their ages is 91 ; required their ages? Ans. 78 and 13. 49. 1'he age of tbe father and son together is 80 years, and if the son's ag»« was doubled, it would be 10 years more than his father's ; what is each of tlieir ages ? Ans. 50 and 30. 50. Find two numbers whose sum is 108 ; and one the one-fifth of the other? Ans. 90 and 18. 184 A COLLECTION OF QUESTIONS. 61. 54 years is the aofe of the father and son together; .nnd the fiftther is 22 years older than the son ; what is the ago of each ? Ans. 38 and 16. 62. Two numbers are such that by adding 150 to the less, they are equal, and their sum is 2,400 ; what are the nuinl>ej*s ? A71S. Greater 1,275, less 1,125. 53. The sum of two numbers is 2,588 and to mako them equal add 1 7 8 to the less ; what are the numbers ? Ans. 1,383 and 1,205. 54. The sum and difference of two numbers are 150 and 100; what is their quotient ? Ans. 5. 55. If I had as many more half dollars as I iiave, after spend- ing 18, 1 would still have 194; how many have I? Ayis. 106. 56. The sum of two numbers is 2,587, to make them equal sub- tract 178 from the greater and add 17 to the less; what are the nurnbci-s? Ans. 1,196 and 1,391. 57. The difference between two numbers is 10, and their quo- tient is three; what are the numbers? Ans. 5 and 15. 58. Re(|uired to divide 60 into three parts, so that the first may be 8 more than the second and 16 more than the third ? Ans. 28, 20 and 12. 59. The divisor and dividend together make 180 and their quotient is 11; determine the divisor and dividend ? Ans. 105 and 15. 00. The product of two numbers is 120, and if you add 4 to the 1g.ss, the product will be 168 ; what are the two numbers? Ans. 12 and 10. 61. The quotient of two numbers is 18, and their sum 1,121; find the numbers? Ans. 1,062 and 59. 02. Divide 250 into two such parts, that their quotient will be 31 ? Ans. 248 and 8. 63. The quotient of two numbers is 37 and their difference 684 ; determine the numbers? Ans. 703 and 19. 64. Divide a number into two such parts that their ditferenco be 240 and their quotient 31 ? Ans. 248 and 8. 65. With 1,350 shillings I paid 76 labourers who worked during a week ; how many would I pay with 1,836 shillings. Ans. 102. 66. If I had |350 more ray stock would be tripled ; what do I possess Ans. $175. 67. The sum of two numbers is 4,545, and one of them is 4 times greater than the other ; what are the numbere ? Ans. 3,636 and 900. 68. If y 5,939, the the two nui 69. Divi part of the parts ? 70. Divi first may b( 24 more tli 71. If I more, I wo 72. If tl divided by 73. Wii give 2,731 74. The the produc bers ? 75. I dt it by 12, a 76. Oi their prodi 77. The determine 78. Wh as 456 by 79. A ] scribes for jicome; to ^ ment ? 80. The second is c 81. TIh their sum 82. Th( their differ 83. Div to the qui] 84. A < in taking I A COLLECTION OF QUESTIONS. 165 er ; and the of each ? 3 and 16. le less, thej 83 1,125. them equal id 1,205. D and 100; Ans. 5. iftcr spend- i7is. 106. I equal siib- vhat are the id 1,891. I their quo- i and 15. le first may • and 12. ' and their 1 and 15, u add 4 to mbers 'i 1 and 10. um 1,121; 1 and 59. ient ivill be 8 and 8. ■rence 684 ; i and 19. r ditferenco 8 and 8. •ked during Ins. 102. what do I IS. $175. them Is 4 and 900. 68. Tf you divide by one another two numbers whoso sum is 5,939, the quotient will bo 12 and the remauider 1 1 ; what are the two numbers? Aiis. 456 and 5,483. 69. Divide 100 into two parts in such sort that the iscventh part of the sextuple of one of the parts may equal 24 ; what are the pyrts 8 Ans. 28 and 72. 70. Divide the number 92 into 4 parts, in such sort that the first may be 10 more than the second, 18 more than th.^ third, and 24 more than the fourth? Ans. 36, 26, 18, 12. 71. If [ had three times more money than 1 have, and $245 more, I would have $2,045 ; what have 1? Ans. $450. 72. If the money 1 have was multiplied by 8 and the jtroduct divided by 7, I would have $24. Ans. ^21. 73. What number being added to the ninth part of 2,457 would give 2,731 for totol ? ^n-^- 2,458. 74. The product of two numbers is 144, and ^^^e sixth part of the product is equal to three times the less ; what are the 2 num- ^rs ? ^^- ^8 and 8. 75. I doubled a number and divided it by 4, then I multiplied it bv 12, and the third of the result was 48 ; what is the number? ^ Ans. 24. 76. One of the factors of a number is 37, and 5 times their product is 10,730 ; what is the other? Ans. 58. 77. The sum of two numbei-s is 374, and their quotient is 21 ; determine the numbers ? Ans. 357 and 1 7. 78. What number multiplied by 12 will give the same product as 456 by 15? ^^^•'^- ^'^^• 79. A person having 445 shillings per month to si>end, sub- scribes for 3,150 shillings in effects, that he must pay out of his m icome ; to what must he reduce his expenses to fulfil his engage- n^ei^t? -^^*- ^ shillings per day. 80. The total of three numbers is 131, the third is 89, aiid tha second is quintuple the first; what are the numbers ? Ans. 7 and 35. 81. The less of two numbers is 7 more than their ditfcivnce, and their sum is 41 ; what are the numbers ? Ans. 16 and 25. 82. The less of two nun.Oers is 12, and by tripling their sum, their ditferenco is 51 ; what is the greater? Ans. 29. 83. Divide 20 into two parts in such sort that one part added to the quintuple of the other will make 84 ? Ans. 1 and 4. 84. A certain person wishing to buy some oranges, finds that in takino" 24 he would have 7^(3. over, and in taking 30 he would i ■• H'. ,0-.. IMAGE EVALUATION TEST TARGET (t4il-2) M. *, ..^^. (P., / 1.0 I.I 25 2.0 1.8 1.25 U 1.6 ■* 6" ► <^ /a .m'^^ 5 ^^ Photographic Sciences Corporation 23 WEST MAIN STREET WEBSTER, NY. 14580 (716) 873-4503 •s? V iV \\ ^R)^ o^ ^ 186 A COLLECTION OP QVESTTONB. want lO^d. more; required the price of the oranges and the monei tJie person had ? • ^ Ans. 8d. each orange ; 6s. 1^3. the money the person had 8o. The sum of two numbers is 450 and the loss is equal to their difference; what are they? Jns. 150 and 300. 86. A fiither has six sons, there are 4 years difference between their ages, and the eldest is three times the age of the youno^est what is the age of each ? Ans. 14, 18, 22 and 26 yeare. 87. Two gsmiblers play a game : the first has 54 shillintrs, the aecond 41. After the game, the first has four times as much money as his comrade ; how much did the second loso ? oo ^Ttri-. , . 1 ^^^- 22 shillings. 88. Which IS the greater of two numbers of which the less is three, and the sum added to the product is 39 ? Ans. 9. 89. Which are the two numbers whose difference is 6, and of which 3 times the less and 5 times the greater make 54 ? Ans. 3 and 9. 90. What two numbers give 116 for sum, and for diffeience double the less? Ans. 29 and 81. 91. The sixth part of 9 times the sum that I have, divided by three and sextupled, gives a result such that its fifteenth part is 30 ; what is that sum ? j^n^^ 150. 92. A gambler being asked how many pounds he had, answered : the quotient of 5 times their number, divided by 1, being muldplied by 13, gives a product equal to 65 ; how many had he ? Ans. £1. 93. The seventh part of a number, multiplied by 3, augmented 4, and divided by 13 gives 4 for quotient; what is that number! Ans. 112. 94. If I add $10 to four times the triple of six times the sura have, I will have |658 ; how many had I ? Ans, $9. FRACTIONS. 95. The sum of two fractions is | and their difference is t*,-;! what are the fractions ? Ans, ^^ and f f . 96. What is the number whose difference between its third and) its fourth part is 16 ? ^^^ 192, 97. What number will differ eight from its | and its |} ? Ans. 20. , 98. With 3i more, the | and the | of a number would be equal ; what is it ? ^^, 28. 113. V of the san: 114. T double th 115. 11 ii ore ; ho IIU. s what 1 hi 111 1 same sun A COLLECTION OF QUESTIONS. 187 ,'*i 99. There is 125*j difFerance between the fifth and the ninth • part of a number ; what is it I Ans. 136. 100. The sum of two numbers is 20, and after subtracting |> from one to add it to the other, they are equal ; what are those nurnbei-s? Ans. 8 and 12. 101. Find a number whose | will be equal to ^ of 14? Ans. 5f . 102. There are two towers side'by side, the first is equal to the ^ of the other, which is 156 feet higher ; what is the height of each ] Ans. 273 and 117 feet. 103. The f and |^ of a ship are under water, and there remains 4 feet over water ; what is its depth ? Ans. 48 feet. ^ 104. The ^ and | of a number make 17^ ; what is it? Ans. 30. 105. If you add the ^ of a number to its half, the total will be 1 , what is the number ? , Ans. ^. 106. A number is such that if you adki ^, ^, ^, of the same number, the total will be 12 ; find that number? Ans. llf-j. 107. Find a number whose i,'| and ^ ma^e 4 J ? Ans. mi, 108. Of two numbers one is 17}, and their ^quotient is | ; what is the other? • ^ws. 15^V 109. The quarter of a number multiplied by f is equal to 1|- ; tyhat is it ? -Ans. 8. 110. If the triple of | be added to its third, the -^di^will be 115; what is it? IfH. 75- 111. After selling the f of a piece of cloth there rel^lis \ of the piece plus 6 yards ; how many yards did it contain ? ^ Ans. 18^1 yds. 112. The 4 plus ^ of a number diminished 64 give for result the I of the same number ; what is the number? Ans. 168. 113. j^ plus I of a number augmented 3, give for result half of the same number; what is that number ? Ans. 15. 114. The I and f of a number and twelve more make just double that number ; what is it ? Ans. 32. 115. If I had i, \ and | of what I have, I would have $150 rvore ; how many have I ? -^ws. $360. 11 U. Some body said : if I had the | and | of the double of what I have, I would 'iave $5 more; how many had he ? ins. $6. 117. The I plus ^^ of the sum I have, plus $29, exceed that BAtne sum by $5 ; what is that sum ? sum sum Ans.^lQO, I'' jl'i 1^ \i i:<'i 188 A COLLECTION OF QUESTIONS. M. Ki^^' \'Zi '' ^''^'"^-^^ '*" '"^^ ^^""^ ^^^^ the 1 is white, t Dia I blue and the remaining 12 feet are red; what is the length I black, the rod ? Am. 49tV ifeet of 119. 1 bought a property, and paid by account the §■ of a of i of the pnce, and I owe yet $60,G35 ; what did it cost ? * " ion Fk- -1 c • . ^ws. $109,143. Kv T*i '' n u '"/"^ ^"^^ '"^^ P^*"^ t^^t the quotient of greater f 1,000 more I would acquit myself entirely and ha^-e $200 over how much have I ? How much do I owe ? ' loo Th. fk- 1 c ■ ^^^- ^400 and $1,200. I possess r ''^ ™^ ""^"'^ "'^"'^^^^ t^« T^V by $35 ; what do i.>o a' „ 1 . , , . ^W5. $1,050. pnfl ; .'^""'"^^^!' '« «"«h that in multiplying its fifth by its sev^ enth ,t IS lessened one-fourt'a ; what is it ? " AnI m whaf is if? ^""^ '"^^''^'^ ^ ^'°"' ^ """'^'•'' ^^^^ ^^" ^« the rem. ; whatl.s it? ^ ""^ ^ ""^ ^ """^^^'' augmented ^ of | will make 11 ; Twiif o-i?.tn"^'"? ' ""'"'''I ^^'^' ^ ^ ^ ^^^" fr<>«^ the't If'its f mil one unity for remainder. ^„^ ^^j « ♦k-^f V^' '^'■^''^^'' '"^'"^^ ^^t 4,395 shilling, was sold for two- third ot five tunes what it cost; what was the gain? 19S f^Jf. • , , •^^*- ^^'255 shillings. A v Ji^termme a number such that, if you multiply it by 4 and divide the product by 4^, the quotient will be'l6 ? ^ . dirifnkh!^ *on^ ^f\i ^^'\'T ^ ^^'^^" ^' ^"^""^^ to the same^L k diminished $20; what have I? j^^ 125 crivl^fiV.T''''^ "T^^' """'^ ^^ ^^^^^ to the I and the i of 32 to give 671 for sum? ^«^ , T I. 1 ^''f tain person said : I have spent the a of the | of what I hacl, and 1 have yet $10 ; how many Lad he ? ' H Vl 1a' 1 ', * ^""^'^^ ^ ^^^"'^ payniy debts; $20 less I woujd^pay but the i ; how much money havj I ? How mucirdo i^q' wi.n* X , , ^«5. $46 and $75. Ifi.n Jl m"T """u ^ «"^tracted from the ^ and the i of >o to reduce thaf, nnmKoi. trt I'fo a 3 - ' . _ ^ j4ns. 64. 168 to reduce that number to its a ? ^„^ jj4 «oi!rf:o !h! T'li^'^f Ir ""'"^'''' ^'' ^''^^^' ^"^ their difte^ence ia equal to the third of the greater; what are the two numbers ? Afis. 3,436 ana ^,304* 135. By 136. By and a half ! 137. Of is f , and tl 13vS. To part; what 139. Th their differe 140. Tl] by the gre 141. '^Di equal to I 142. Tl the father 7^ ; what 143. A while the j of the grt leaps must 144. A on what p 145. It 146. Tl meet froir 147. 1 148. I the same together ^49. I in 5 houi would fill 160. 1 empty it be dry t A COLLECTION OF QUESTIONS. 189 te, } black, i length of )p7 feet. ■ of a of I 109,143. of greater and 41. ebts; with ^200 over; $1,200. ; what do $1,050. by its sev' IS. 26^. the rem. ; . 3,437. make 11 ; ins. 12. le § of its for two- lillings. ^ it by f ns. 96. atne sum s. ^^o. of 32 to s. 15f I of what 9. $20. 10 less, I much do i\ $15. the i of IS. 64. erence ia irs? 2,004 135. By what number must you multiply a sum to lessen it |! 1.36. Bv what number must you divide a sum to render it once ^"13^ " Of^three fractions the second is double the first, the third is I and their sum^s ^ ; what are the two first fractions \ ^' ' Ans. jig and j\. 138. To double a number you must multiply its | by its ninth 1 4. • ;t 8 Ans. 27. part ; what is it f , , . /. ^1 j 139 'Wa a- of one number is equal to the f of another, and their aifferenci 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 greater is ^ ; what are they \ Ans. 2i and 1|. 141 ^Divide 60 into two such parts that the 4 of one may be equal to 1 of the other ? ^^ ^ris. 32 and 28. 142 The father and son together are 70 years old, the age of the fadicr multiplied by 3 is equal to the son's age multiplied by . . .1 Arts 9.0 nnfl .'lO. what are the ages ? 75; wnao lire tuo «j^ca ; 7' ^ . v j 143 A greyhound starts after a hare that is 82 4.aps ahead, while the crreyhound makes 9 leaps the hare makes 13, but 3 leaps of the greyhound are equal to 6 leaps of the hare ; how many leans must the greyhound make to catch the hare ? . ^ Ans. 369 leaps. ' 144. A watch marks 12, and both hands are together ; required on what part of the dial the> will next meet ? Ans. 1 o'clock o^j minutes. 145 It is iust six o clock ; when will the hands meet? Ans. 32y"V minutes past 6. 146 It ^s just twelve ; required how many times the hands shall meet from twelve till midnight, and at what o'clock each time? Ans. 147. The I and i of a number make 39 ; what is that number ? * Ans. 60. 148 A man can do a piece of work in ^ day, his wife could do the same in 1 and their son in i day ; what time would tlio three together take to do it ? ^ ^ ^«^' \fj- .^ 149 A sprincr would fill a baain in 3 hours, another would fa I it in 5 hours : if the two run together, required in what time they would fill it? . , . , ^ ^n.9.1jhour. 160. A pump would empty a dit«h in ^t ther can vill it bo r day. UCbVTSCU A COLLECTION OF QUESTIONS. ;9i her three daughters ; the youngest receives the f of the whole, the second 1, and the third 12 for her part; how many were there, and wliat was the part of each 3 . Ans. Total 45 ; 12, 15, 18, resi^ectively. 163. The I and the ^ of what I have in my purse, with $10 more, would make $Q more than I have; what does the pura» contain ? , /"^- ^^O' 164. The triple of a sum added to the i and \ of the same sum, and $5,000 more, would make $22,200 ; what is that sum ? Ans. $4,800. 165. The difference between the f and the | of a number i3 + 12 : what is that number ? Ans. 216. 166. The total of the f and the f of a number, diminished the I of the same number gives 14 ; what is the number ? Ans. 24. 167. Two cocks running together would fill a basin in 2 hours ; one alone would fill it in 5 : in what time would the other fill it if it were to run alone ? -'^««- H 1^« ^ t . . 168. I spent the | of what I had in my purse, and if I add ^44 to what remains, the sum it contained fii-st will be augmented \ ; what did it contain ? •^^^- ^*8. 169. One cock runs 11 gallons in 8 minutes, another runs 7 gallons in 5 minutes ; which runs the most ? Ans. The second, by ^V S^^- P^^ "^^"• 170. A basin receives 45| gallons of water per hour by a cock, and leaks by a hole 37f gallons ; how many gallons does it retain per hour? .1^.9. 7fi gallons. ^ 171. A certain person not recollecting what he paid for an arti- cle, only remembers that there were $14 difference between the f and the | of the price ; what is it ? -Ans. $40. 172. The -f of a number diminished the | of the same number gives 18 fur 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 ? Ans. 5^ hours. 175. Three cocks running together would fill a basin in 4 hom-s; one of them would fill it alone in 10 houre, another would fill it in 12 hours ; what time would the third running alone take to till Uio basin? -'I'**- iSlionrs 176. The quarter of a field is sown with wheat, the ^ with barley f 19? 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 tlie whole field and that of each part? Ans Whole extent 56 acres; U wheat, 24 barley, ^8 oats 177. I have already sold the | of a basket of eggs, and if I add 3y eggs to what remains, the primitive value of the basket will be augmented one-half, how many eggs were there in the basket? 178. A steam-loom weaves 6 yards of cloth in 3 houiCanother 12 yds. m 7 hours ; which has the greater power? Ans. The latter weaves Jj- yd. per , HO A -1 u . ^^^^ ^^^^ *^^»» tbe former. len tl i " ""^ ''''^ '"'^ ^ ^^'^ ""^ • ^^' ^"^^ ' ^'^'^^ ^^" '^ ' AtlS 1 -^ vd 180. A tradesman can do a piece of work in 5| days ; in what time will he do the i of the work ? Ans 4aa days 181. A ship sails at the rate of 16^ miles an hour; how many miles will she sail in 3^ hours ? Am. 63^ miles. ^ 1 82. A weaver weaves 7 yards of linen in 8 hours ; how many yards will he weave in 4f hours ? Ans 4ii yds 183. A man weaves 7 yards of linen in 8 hours ; what time will he take to weave 4| yds. ? Ans. 5X1 houi-s. 184. If o gallons of wine be mixed with 7 gallons of water ; re- quired what quantity of water in f gallon of the mixture ? 1 Q- T^ .1 « ^ . ^^**- * 8 ^^'^^" ^^ ^^'"*^ H of water. , mo. It the f ot the | of a number make 120, what is it ? 186. A person being asked the time of day answered ; it is the i of ^ of f of 24 hours; what o'clock was it? ' Ans. 10 o'clock. ' 187. Divide a succession between three heirs in such sort that the hrst may have the ^ of the whole, and the second the a of the remainder ; what is the part of each ? 1 Qo A ^ "^^^^' ^^^^ ■^' ^*^^^"^ ^'t' ^^^ ^^^^ t^»' only travi than the 213. ^ holds 50 other 4A 214. 1 out ferficies of a rectangular garden, being 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 r f a yf^vd forming a trapezium one of whose sides is 34 yards and the other 56, its height being 25 yards? ^w*. 1,125 yards. 235. What is the area of a rhombus, whose base is 44-i7j and height 38a yards? Ans. l,716^f yards. . 236. V/hat is the superficies of a pillar 17 yards high and 1 yards in circumference? Ans. 119 yards. 237. The circumference of a cone is 12 yards, and the distjince from the summit to the base is 6 yards ; what would the painting of it cost at 3 shillings the square yard ? Ans, 108 shillings. A Tahle^ num 197 A Tahle for finding the Interest of any sum of Money for oji, number of months, weeks, or days, at any rate per cent. Year. Calen. Month. Week. Day. £ £ s. d. £ 8. d. £ !> d. 1 1 8 4i . Oh 2 3 4 9 u 14 * 3 5 1 a 2 '^ 4 6 8 1 6 2i 5 8 4 1 u 34 ^ 6 10 2 31 4 > 7 11 8 2 84 4i 8 13 4 3 1 54 g 15 3 5i 6 10 16 8 3 104 6i 20 1 13 4 7 84 1 U 30 2 10 11 6i 1 74 40 3 6 8 15 4i 2 24 50 4 3 4 19 2| 2 9 60 5 1 3 1 3 3i 70 5 16 8 1 6 11 3 10 80 6 13 4 1 10 94 4 4i 90 7 10 1 14 74 4 114 100 8 6 8 1 18 5i 5 55 200 16 13 4 3 16 11 10 Hi 300 25 5 15 4i 16 54 400 33 6 8 7 13 10 1 1 11 500 41 13 4 9 12 3h 1 7 4| GOO 50 11 10 9 1 12 lOi 700 53 6 8 13 9 21 1 IS 44 800 66 13 4 15 7 8| 17 6 U 2 3 10 900 75 2 9 34 1000 83 6 8 19 4 74 2 14 9i 2000 166 13 4 38 9 2| 5 9 7 300r 250 57 13 10 8 4 4i 4000 333 6 8 76 18 5i 10 19 2 5000 416 13 4 96 3 Oi 13 13 Hi 6000 500 115 7 84 16 18 9 7000 583 6 8 134 12 3i 19 3 6| 8000 606 13 4 153 16 11 21 18 44 9000 50 .173 1 64 24 13 n 10,000 833 6 8 192 6 11 27 7 114 20,000 1666 13 4 384 12 34 54 15 lOi 30,000 2500 576 18 5i 82 3 10 - 1/ i I 198 RuLK. Multiply the principal by the rate per cent,, and the number of mtniths, weeks, or days, which are required, cut off two finrures 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 sums 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-penny; and for every 40 in the days, 1 farthing. EXAMPLES., 1. Whr.t is the interest of £24G7 10s. for 10 months, at 4 pei cent, per annum ? 2467 : 10 900=75 : : '1 80= 6 : 13 : 4 ■ 7= : 11 : 8 9870 : 10 987100 987=82 : 5 : cent. 2. What is the interest of £2467 10s. for 12 weeks, at 5 per It.? 2407 : 10 1000 = 19 : 4 : 7^ 400= 7 : 13 : 10 80= 1 : 10 : 9i 50= : : 2^ 12337 : 10 12 1480150 : is the interest 2407 : 10 6 14805 : 60 1480150=28 : 9 : 5 3. What is the interest of £2467 10s., 50 days, at 6 per cent! 7000=19 : 3 : H 400= 1:1: 11 2= 0:0: u 50= 0:0: 0* 7402)50=20 : 5 : 7 7402150 : To Jind what an Estate, from me to £60,000 per annum ivill come to for one day. Rule 1. Collect the annual rent or income from the table for 1 year, against which take tlie several sums for one day, add them together, and it will give the answer. 199 An estate of £3*76 per annum, what is that per day f 3G0— : 16 : 5i 70=0 : 3 : 10 6=0 : : 4 376=1 : 0: 1i To find the amount of any income, salary, or servants^ wages, for any numhei of months, weeks, or days. RuiF Multiply the yearly income or salary by the number What will £270 per annum come to foi 1 1 montns, lor o vy<. , and for 6 days^ 270 11 2970 270 6 1620 For 11 months, 2000=166 : 13 900= 75 : 70= 5 : 16 4 8 270 3 For 3 weeks. 800=15 : 7 : 8i 10= : 3 : 10^ 2970=247 : 10 : For 6 days. 1000=2 : 14 : 9^ G00 = 1 : 12 : 10^ 20=0 : 1 : li 810 = 15 : il : 6^ For the whole time. 247 : 10 : 15 : 11 : 6i 4 : 8 : 9^ 1620=4 : 8 : 9^ 267 : 10 : 3^ A Tabic showiny the number of days from any day in thi JnTto the saL day in^nyoth^^r^^ FROM January . ■ February . March . . April . . . . May June . . • July August. . September October . November ecember El! 200 A COMPENDIUM OF BOOK-KEEPING. BY SINGLE ENTRY. Book-keeping ,s tlie art of recording the transactions of persons m business so as to exibit a state of their aflairs in 1 n^^ and satisfactory manner. ^" ^ ^^^'^'^^ Books^may be kept either by Single or by DouMp JPnh.. i » Single Entry is the method difly ufed in rLflh^Ll ""' ^"* The books found most expedient in Sino-le Entrv s.^^ ♦!.« n Book, the Cash-Eook, the Z V, and theK^^ '^' ^"^ debts &c. ; and are entered in the order of their occu JrcT he' daily transactions of goods bought and sold. "^'^""ePce, the The Cash-Book is a register of all money transactions. On the left-hand page^ Cash is made Debtoks from wliich the respective accounts are extrat ted, by figures placed in a column against the smns. Keferencet are also made m the Day-Book ancf a-.h-Bcok, to the S in the Ledger, where the amounts are collected. This r^r^Z s Jr^y n' ^''""J' ''^T '"^^^^^^ ^'^ transferring the reSsSr of mercantile proceedings from the previous books to the ui^^ The person from whom you purchase goocfo, or from whom rJCJ" "'^f^' ^\(^-ditor; ind, on thf conC, th^ pet" to whom you sell goods, or to whom yon pay mon,y;L Dehtor In the Bill-Book are inserted the particulars of all Bills of Ej, change; and jt is sometimes found expedient to keep for tht f r- ^t^^r " i^tnthe^^-ett::,^s tWu!;fhirer ^ '''' ''' ^*^^ "^- -' ^^tdt 201 DAY BOOK. (folio 1.) W -a January Ist, 1837. f; 1 commenced business with a capital of Five Hun- dred Pounds in Cash 2d. Bennett and Sons, London,* By 2 hhds. of sugar cwt. qr. II. 13 1 4 12 3 16 Cr. cwt.qr.lb. 12 1 1 6 gross wt. 26 20 tare 2 3 6 neat wt. 23 1 14 at 633. per cwt. 2 chests of tea cwt. qr. lb. 1 15 1 12 2 27 1 22 lb. 25 25 1 3 5 at 6s. per lb. £ 500- s. 73 3d. Ha/l and Scott, Liverpool, By soqp, 1 cwt. at 68s. . . . candles, 10 dozen at 7s. 9d, Cr. 60 133 6th. Ward, William To 1 cwt. of sugar, 14 lbs. of tea, 4 cwt. of soap, J}r. at 703. at 83. . at 749. 6lh. Cooper, William To 1 sugar hogshead. Dr. 10 d. 12 18 8 17 5 10 12 18 6 • The sT'.Jdfint mav be directed to fill on this and similar blanks in thif book and the Ledger with the names of pi;\(;es familiar to him m 202 DAY BOOK. (folio 2.) 2 1 2 1 2 o January 9th . 1837. " £ 1 1 4 17 17 1 6 1 1 s. 16 17 15 8 5 5 9 8 4 8 10 16 13 16 10 9 4 13 IS 8 G 1 d. 6 6 6 9 3 6 10 6 4 2 2 3 n Johnson, Richard To 2 dozen of candles, i cwt. of soap, i cwt. of sugar. at 8s. 3d. . . JJr. at 74s at 70s 10th. Ward, William To sugar, 1 cask cwt. qrs. lb. gross wt. 5 2 10 cask tare 2 10 Dr. neat 5 at 68s 12th. Smith, John To 14 lb. of sugar Dr. 12 lb. of candles 7 lb. of soap * 1 lb. of tea 14th. Hall and Scott, Liverpool, By 2 cwt. soap, at 6Ss .... Cr. 17th. JVewton, John To 21 lb. of soap, 2 dozen of candles, Dr. at 74s. per cwt.... at 8s. 3d 19th. Smith i John To 14 lb. of sugar Dr. i lb. of tea ....'.'.'.'. 21st. Smith, John To 28 lb. of sugar Dr. 12 lb. of candles , .' 2 203 ; DAY BOOK (t oho 3.) 2 3 2 2 2 3 3 3 3 January, 23d., 1837. £ A d. G 6 Vates Sc Lane, Bradford, ^*'' By 4 pieces of superfine cloth, each 36 yards, at 243. per yard. . . 172 2 3 5 9 145 2 148 50 172 4C 5f C I or 16 8 9 16 14 5 23d. Edwards, Benj. Manchester, ^r. By 2 pieces of calico, each 24 yards, at Is. per yard 23d. Smith, John ^' To 14 lb. of soap » 24th. Johnson, Richard ^^- 1 2 dozen 01 canuies, ^.t oa. ou, • • • • • 1 cwt. 01 soap, o\. /'is. • 1 1 o\k\ r\i alicrnr at 70s 15 8 16 16 12 8 16 17 > 7 ! 19 ) 15 > 19 6 3 1 6 6 24th. Smith, John ^' • To 1 lb. of lea 26th. Mason, Edward f^- To 3 pieces of superfine cloth, each 36 yards, at 278. per yard .... 2 pieces of calico, each 24 yards, et Is. 2d. per yard.. 27th. Parker, Thomas £»'• To 1 piece of superfine cloth, 36 yards, at 28s.. . 3 1 St. Bills Payable, , , ^ C-r By Yates & Lane's Bill at 2 months, due April 2 Inventory, January 31, lb37. cwt. qr. lb. Kaw sugar, i'* o i'* ai u.jo. ........•• Tea, 1 2 16i at 6s. per lb o^„.» n ^ 1 4. at fiSs soap, " o it - '»• «'»j m 204 CASH BOOK. o o o a y o *'o «^ o oo o o 00 \fi CO o CO «ooo«oooo«o «o OOQOi-iQOXiOcO 00 I— I I— I »-< tH CO CO CI a CO o CITS V , -^ > — ■"^" 1 = 1 Oca; », o x) « 'rt a ffipqa,W> Bennett & Sons, London ... 1 | XJ Bills payable 3 Cooper, William 2 C Parker, Thomas P . 3 )- D Q Edwards, B. Manchester ... 3 E R F Stock account C^ <5mil-h Tohn .. 1 .. 2 D G T Hall & Scott, Liverpool 1 H V Johnson, Richard 2 I Ward, William W . . A K X L Yates & Lane, Bradford. Y . .. i< I Mason, Edward M 3 z 206 LEDGER. C ^. Wo O O r-i I ,^ 00 o o ! CO I F-l Oi o o 'JO n t^ to I - o CO o eo • 0) • : o - .*; CO a) *j I - I -^et i — S o c!5 J «5 2 "So o So*-* <5 a o B d pq ^ OJ o Tf -I . CM * s . 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