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I D ' THE TUTOR'S ASSISTANT; BEING A COMPENDIUM OP PRACTICAL ARITHMETIC, FOR TH13 USE OF SCHOOLS, OR PRIVATE STUDENTS : CONTAINING, 1. Arithmetic in WfwU Numbers ; | the Rules of which are expressed J in a clear, concise, and intelligi- | ble manner ; and the operations | illustrated by examples worked at ; length, and by numerous explan- atory notes and observations ; with an ample variety of Exam- ples for the exercise of Learners, calculated to initiate them in the Knowledge of Real Business. Also the Neio Commercial Tables, adapted to the present Legisla- tive regulations of Weights and Measures, and the modern pra/:- tice of Trade. tl. Vulgar Fractions ; cxplamed m an easy and familiar manner ; in the practice of which the most elegant and abbreviated modes of operation arc peculiarly incul- cated. III. Decimal Fractions ; elucidated with the utmost perspicuity ; and the calculation of Interest and Annuities, on an extended scale. IV. Duodecimals ; or the Multipli- cation of Feet and Inches ; with numerous examples for practice, adapted to the various business of Artificers. V. A Collection of Qitcsiions, pro- miscuously arranged : intended as recapitulatory exercises in the principal Rules of Arithmetic. VI. A Compendious System of Book-keeping. BY FR^J^CIS WALKI^TGAME, Writing' master and Accountant. The TiniiT£ENTH Edition ; revised, corrected, and enlarged, by the addition of Superficial Mensuration^ and a Compendium of Book-Kr.eph^gj by Single Entry ; By WILLIAM B I R K I N, Master of an Academy in Derby. dTe R B Y : rHINTED AND PUBLISHED DY J. JOHNSON AND SON; AND SOLD .'N LONDON BY THE VRINCirAL BOOKSELLERS. 1837. I The imi even in extensiv of the pr Toad among C incr and ciencies with its to he cs Science to the I Amor may be of Not! Practice tions of Evolutii metical ExampI exercise Rules fi Theorei and Ge( of nsef Keepin Such worth c tion mi left to I V/itl tion of qualify tension; such ar before hopefu Deri Ir R E F A C B. The immense circulation of Walkingame's Tutor's Assistant, even in its original form, is sufficiently evinced by the very extensive and uniformly increasing demand which the Proprietors of the present Edition have for many years experienced. To advance the utility of a work held in such high estimation among Conductors of Schools ; by simplifying the Rules, correct- ing and modernizing the antiquate.^ phraseology, supplying defi- ciencies where there was a paucity of Examples, and incorporating with its original matter such emendations and additions as appear to be called for by the present improved state of Arithmetical Science ; will, it is presumed, be rendering an acceptable service to the Public. Amongst the various improvements introduced in this Edition, may be enumerated, a more intelligible elucidation of the system of Notation; of Direct, Inverse, and Compound Proportion, Practice, Interest, Progression, &c. ; more perspicuous illusira- tlons of the theory and practice of Vulgar and Decimal Fractions, Evolution, Duodecimals, &c. ; the substitution of the new Arith- metical and Commercial Tables ; the insertion of many additional Examples, (particularly in the elementary Rules) adapted to exercise ana improve the judgment of the Learner ; also of Rules for the particular cases in Profit and Loss, of Involution, r " Theorems for the solution of all the possible cases in Arithmetic and Geometrical Progression, Superficial Mensuration, a numb of useful Supplemental Questions, and a Compendium of Boo Keeping. Such are the attempts that have been made to enhance the real worth of this popular Treatise on Arithmetic. How far the inten- tion may have been judiciously or successfully executed, must be left to a candid Public to determine, V/ith a confident reliance, however, on the favorable considera- tion of those whose judgment and experience most essentially qualify them to discriminate between realities and specious pre- tensions to improvement, and duly to appreciate the difficulties of such an undertaking ; the work is respectfully submitted (o atrial before the tribunal of its lc;);itima*'^ j«3 {'30) 584() 10070840 (7) 333 (19) 70M703 (31) 584filOo708400 (3) 127 (20) 311311 , (32) 37G1 35902001 16 (9) 17^ (21) 113113 (33) 5U08000400000 (10) 217 (20 131131131 (34) 601008000180070 (H) 271 (23) 7()8&07780 (35) 37000000000075048 (12) 712 (24) 807078087 * Tk« figuroi in parenthesei refer to the Editor's Key to ihii work. ASSISTANT.] NUMERATION, 7 Express in figures the following numbers. (\) Nine; ninety; niuely-nine ; nine hundred; niae hundred and nine J nine hundred and ninety; nine hundred and ninety-nine. . i j i • u.„ (^) One hundred and ei^>ht ; one hundred and eighty ; e'At hundred and one ; eight hundred and ten ; one hun- dred and sixteen ; one hundred and sixty-one; six hundred ard eleven. (3) One hundred and twenty-three; one hundred and thirly.two; two hundred and thirteen; two hundred and thirty-one; three hundred and twelve; three hundred and twenty-one. (4) Two thousand fiv hundred and seventy-two. (5) Seventy-two thousand five hundred and seventy-two. (6) Five hundred and seventy- two thousand five hundred and seventy-two. (7) Ten thousand nine hundred and t(3n. (8) Nine hundred and ten thousand nine hundred and ten. (9) One hundred and nine thousand nine hundred and one. (10) One hundred and ninety tliousand and ninety-one. (11) Nine hi' -dred and one thousand and nineteen. (12) One hundred and fourteen millions, one hundred and forty-one thousand four hundred and eleven. (13) Four hundred and six millions, six hundred aud four thousand four hundred and sixty. (14) Six hundred and forty millions, forty-six thousand and sixty-four. , , . (15) Seven millions, seventy thousand seven hundred, (IG) Seven hundred millions, seven thousand and seventy. (17) Ten millions, one thousand one hund.ed. (18) One hundred and one millions, eleven thousand one hundred and ten. , , . „ (19) Twelve billions, seventeen thousand and nine mil- lions, i\n<\ eighty. nine. ^ ^ (20) Seven thousand Hve hundred and four trillions, sixty thousand millions, eight hundred thousand. • [{omur> Numerals. I 1 One. 2 Two. Ill IV 3 Three. 4 Four. V VI 5 Five. 6 Six. I i 8 VII VIII IX X XI XII xm XIV XV XVI xvn xvui XIX XX XXX XL L ADDITION OF INTEGERS. [the TUTOR'S 7 Seven. 8 Eight. 9 Nine. Ten, 1 Eleven. 2 Twelve. 3 Thirteen. 4 Fourteen. 5 Fifteen. 6 Sixteen. 7 Seventeen. 8 Eighteen. 9 Nineteen. 20 Twenty. 30 Thirty. 40 Forty. 60 Fifty. LX LXX LXXX XC c cc ccc cccc D DC Dec 60 Sixty. 70 Seventv, 80 Eighty. 90 Ninety. 100 One hundred. 200 Two hundred. 300 Three hundred. 400 Four hundred. 600 Five hundred. 600 Six hundred. 700 Seven hundred. DCCC 800 Eight hundred. DCCCC900 Nine hundred. M 1000 One thoiistind. MDCCCXXX 1830 One thousand eight hun- dred and thirty. All operations in Arithmetic are comprised under four ehmcntarj^ orfundamental Huks : viz. AdLion, Suhtracihl MuUtplication and Division. ^uuiianwuy ADDITION. Teaciibs to find the sum of several numbers. Rule. Place the numbers one under another, so that unit* may stand under units, tens under fens, &c. ; add the unit set down the un.ts in their sum, and cany the Um as so many ones to the next row ; proceed thus to the last row under which set down the whole amount. If FuT""^' .^^r^'" ^* I**® ^''P "^^ ^^^ *^^ ^'g"*-" downwards : right. *""* ""*^ "' ^'^'""'' '* *' presumed to be ASSISTANT.] ADDITION OF INTEGERSt 110 473 354 271 352 (5)590046 73921 400080 4987 19874 2014S6 9883 )1234 1 (3)75245 7093 37502 3314 91474 G732 32145 2546 47258 6709 21476 (6)370416 j (7) 2890 60872 998 47523 9836 j (4)271048 ( 325470 107684 625608 754087 279736 781943 66820 1693748 300486 920437500 78632109 9408175 26627 (8) What is the sum of 43, 401, 9747, 3464, 2263, 314, ^^9) Add 246034, 298765, 47321, 58653, 64218, 5376, '^/oH? ItsTs! B £104. C £m. D £1 390. E £7003 F £1500. and G £998. ; how much is the whole amount of eir money^ many days are in the twelve calendar months ? 12 Add 87929,135594,7964,3«21,27123, 8345, 35921, «374 64223, 42354, 3560, and 152165, together. (13) Add 6228,27305,7856,287, 7664, 100, 1423, 25258, 528, 3135, and 838. . ^ r . - ,u. ^e ri4)llow many days are there in the first six months of the year ; how many in the last six ; and how many in the ^ (°15) In the year 1832, how many days from the Epiphany or Twelth-day (Jan. 6th) to the last day of July } (16) In the common year how many days from each Quar- ter-day to the next ) That is, from Ladyday to Midsumraer- • Say 2 and 1 are 3 nmil^T, an<1 3 are 10, and 5 are 15, act down 5 and carJyY; 1 and 5 arc 6. and 7 are 13. and 5 ^-^^ l«'„^"j ' "^^ ^''a^^^, 1 are •>« and 7 are 33 set dowi. " and carrv 3 ; 3 and 3 nre b, and -i are 1. a"d 3 arc 11 and i are 15, a..d 1 are 16, and 2 are 18, aet down 18 ; "';tr'7ri;.!Sg a few cample., it will be batter 'or 'he learner ,„ add iU llgure. wUl.ou, nanong .hem. ''" ^ '"jf ';;«,»'' | ^^^^ mdumn ot ilio abovo examply, auy 2, J, 7, 10, lo, . set Uuwn a ana ""•riilB meihod will tend both lo qnicUnraa and preciiion. A 2 \Jnffy pr//-/- -vf // -^- 10 SUBTttACTlON OP INTEGERS. [the TOTOR^ fn'iildrSoJlV^'-^r'"'';-^'^' I'''' '^'^'^ *« Christ, rnas day^od-frora Christmas-day to the ensuing Lady-day ? (17) Whenwill the lease of a farm expire, which tas granted iii the year 1799, for ninety-nine year! ) ^2^00 U'!Z ^\'"^^^'', l^ft his widow in possession of ins^ To h 'f ^"",,^"'^^"*^^, property of the value of fnn A u '' *r ^^^h®' '^"5 h« bequeathed a thousand ceededt'h.'^ "'"/^.%' *\hi« ^-ghter ; whose portion ex- ceeded the property left to her mother by ^500. A nephew ^lor.TnVh- f^'^"'" «^^525,eac];tapublfcXit; fmor"'[ ^hl Z 'f "'"*' !^' ^^"'^ «"» to be divided prpefly > ^^ '•'' '''"' "^' "^^^8^*« ^'"*'""' of his ♦1,5 f!n ^^" *^^ ''T' ^""^ '^Snificaiion of the 5«;-« put between hi ,^?"7'"S."""',»^e" •• and find what they are iqual to, as the sign requires ? / i > «» 1724+649+17+54004-12+999. C20) Required the.vMm of forty-nine thousand and sixteen • four thousand e.ght hundred and forty ; eight millions, seven hundred and seven thousand one hundred; nifle hundred ""/ol^^wr"'"''.?. '"^ *^''^" thousand one hunded and ten. rl2 IT will a person bom in 18 1 9, attain the age of 45 ? (22) Henry came of age 13 years before the birth of hij rTJ^r'' ^}'''^ °'^ ^^'^ "'^"'■3' ^« ^hen James is of age ? {^p ilomer^the celebrated Greek poet, is supposed to have lU)ur.shed 007 years previous to the commencement of the u..?j? f" ^'li Admitting this to be fact, how many years ua t from Homer's time to the close of the ISlh centiry ; and how long- to A. D. 18'27 > ^ SUBTRACTION Teaches to take a less number from a greater, to find the remainder or Difercuce. ^ ' The number to be subtracted is the Subtrahend, and the othrr IS called the Minuend. UuLn. Having placed the Subtrahend under (he Minuend (in the same order as in Addition) be^^in at fhe units, and subtract each figure from that aho.e it, selling down the re- rnalnder underueath. iJut uhcn the lower figure is the /• X E tutor's to Chrlst- ady-day ? hich was iession of : value of thousand rtion ex- i nephew c charity e divided nt of his between lal to, as sixteen $ 18, seven hundred and ten. 5eof45? th of his ' of age ? i to have It of the ay years [^ntury ; ind the nd the inuend Hf and the re- is the ASSISTABT.] SDLTRJ|n(« cftMi&A 1 ^^ fract : set down u.e '«T?*^.ri^SSf U - (2) 42087^96 34096187 (3) 45270509 32761684 (5^ 375021699 (8^ 2746981340 278104609 1 1095681539 (6) (9) 666740825 109348172 400087635 9 184267 (10) From 123456789^ibtmct 9878|432. nn From 31147680975 subiract 767380799, fA{ Subtract 641870035 from 1630054154. f {3 RfquTre^^^^^^ D#^«^^^ between 240914 and 24091. 140 How much does twentyfive thousand and four ex- . Jwi fhousand five hundred and sixty pounds. Mr. Lem- rnron S hasan income of seven thousand e.ght h«n- ted^and right^en pounds .er^a^num. How much .s the '°"('?T^rGrr^*^^»^MThis^ccessio^ 1820 wa^^he 58th year of his age. In what year was he S and how long had he reigned on the 29th of January, 1829. tlie anniversa ry of his access iuiW rrom 32S06E47 tiiMrnct 821046S. 3290654-7 Minuend 82104G8 Subtrahend 246y60ZD Difference. 82906547 Proof. Say 8 from 7 1 cannot ; borrow 10, and 7 are 17 8 from 17, 9 remain ; let down 9 and carry 1 Ll and are 7, 7 from 4 I cannot ; borrov^ lb. and 4 n.e lU 7 from 14, 7 ; »ct down 7 and carry 1—1 and 4 are 6, 5 from 5, nothing ; set down (0^ nouuht—O from 6. 6 ; f t «]own G.— 1 (rom 1 cannot ; but 1 from 10, 9 ; let Proceed in like manner to tlie end. ^lown 7 and carry l,&c. 12 MUZ-TIMJCATION OP INTEGERS. [tHE TUTOR's (18) The sum of two numbers is 36570, and one of them IS twenty thousand and twelve : what is the other 1 (19) Thomas has 115 marbles in two bags. In the green bag there are 68 : how many are there in the other 1 i.T^^^^K'^n^ brothers who were sailors in Admiral Lord Nelson s fleet, were born, the elder in 1767, and the younger m 1775. What was the difference of their ages, and how old was each when they fought in the battle of Trafalgar, in 1805t (21) Henry Jenkins died in 1670, at the age of 169. How long prior tg his death was the discovery of the continent of America by Columbus, in 1498 ?— Also, how many years hav© elapsed from his birth to 1827? (22) Borrowed at various times. £644... £957., £90,. £1378., and £1293.; and paid again the different sums of £763., £591., £1161., £1000., and £847.— What remains unpaid? (23) Explain the name and signification of the sign used • and work the two following examples. ' 1G874 — 9999 51170 — 50049 (24) John is seventeen years younger than Thomas : hair old will Thomas be when John is of age ; and how old \5^1 John be when Thomas is 50? \ MULTIPLICATION Teaches to repeat a given number as many times as there are units in another given number. The number to be multiplied^r" called the multiplicand: that by which we multiply is the multiplier ; and the num- ber produced by multiplying is the Product. Rule. When the multiplier is not more than 12, multiply the units' figure of the multiplicand, set down the units o^ the product, reserving the tens ; multiply the next figure, to the product of which carry the tens reserved : proceed thus till the whole is multiplied, and set down the laut product in full.* Example. Multiply 718097 by 4, ' ""' '^ Say 4 times 7 arc 28, set down 8 and carry 2 • 4 t»mei9 are 36 and 2 are 88, set down 8 and carry's- 4 time. (nought) and 8 are ? set down 8 ; 1 times li are 12, set down 2 and carry 1 ; 4 times 1 arc 4 and i are S^set down 5 ; 4 t4i|||^»r»28, let down 28. 718097 4 2852888 - — "t:' A \ ^,»mA«T.] MU.Tm.IC.TION OV tNT.GBB.. MULTIPLICATION TABLE. 13 (1) Multiply a510«36 i,; 2. 1 (7) MuHlply 3725,04 by 8. )2) Multiply 52171021 by 3. ("{Sinly 2701057 by ».• >3 Multiply 79254^7521 by 4. g) '^'Sy 3104.0171 byll. i « 2^S ^y ^6-. P)£U.^y73998063byl2. $ Multiply 7092516 by 7. 1 ^ ^^^ ,o. (12) Multiply 780149326 by 3, 4, 5, 6, 7, 8, 9, and i\r\ Multio'v 123456789 by 4, 5, 6, 7, 8, and ». 1! S?l'y 987654321 by^9 10 ll,and. last figure multiplied .f ,. , ^/j Wom ooifSqa4X 18 iSl as 5 IKS S!S 5 K:p, = .... (17) 7653210 X 15.1 ^ : • ,. , , rn „„nei a cli>l.er to the muUipUcaod, for the EXAMPLV8. t MuUiply 96043 ».y 16. ^^,^n and carry 4: 6 timet Srty 5 times 8 arc 40, sci "";^" ^ . j^„ 2 aud 96048 4 a?e20 «-' 4. arc 24 «n^ 8 a.c 82 .e^ do ^^^ ^^^ ^ 1 5 carry 3 ; 5 t.rnc* ad 3 a . e 3, and ^ ^ ^ ^^^^ ^ _,,^ .- 7; 5 times 6 are SO, *«^'l°';" % ^-t down 4 and carry JT^O nre 45 and 8 are 43, and 6 are o4. .Ct tl_ 6 ; 5 Had 9 are 14, «t down 14. ■%. y u M'JLTIPLICATION OP INTEGERS. [tHE TUTOR's When the multiplier consists of several figures, multiply b}' each ofthem separately, observing to put the first figure of every product under that figure you multiply ,by. Add the several products together, and their sum will be the total product.:}: Proof. Make the former multiplic«nJ the niul(jplier, and the multiplier the multiplicand ; and if the work is right, the products of both operations will correspond. Otherwise, A presumptive or probftble proof (not a positive one) may be -obtained thus : Add to- gether the figures in ««. /i/rt< /or, casting out or rejecling the win^a in the sums as you proceed ; set down the remainders oh each side of a cross, multiply them together, and set down Ihz excess above the nines in their product at the top of the cross Then cast out the nine« from the product and place the excess below the cross. If these two correspond, the work is probably right : if net, it is crrtainly ivrong. (22) 27104.1071 X 5147. (23) 62310047 X 1668. (24) 170925164 X 7419. (25) 9500985742 X 61879. (26) 1701495868567 X 4768756. When ciphers are intermixed with the significant figures in the multiplier, they may be omitted ; but great care must be taken to place the first figure of the next product under the figure you multiply by.* W' Ciphers on the right of the multiplier or multiplicand (if omitted in the work) must be placed in the total product.f t Multiply 7C047 bv 249. 76047 249 684423 Product by 804188 do. by 152094 do. by 9. 40. 200. ProoL 6X6 18935703 Total product, Examples. • Muitiplv 31864 by 7008. 81864 ' 7C03 Proof. 6 2.';i912 4X6 223048 6 S33S02912 r t Multiply 63850 by 5200. 6SS50 5200 12770 Proof. 1 4X7 1 8B2020000 ^\ .^^. ASS STANl (27) (28) 7561 (29) 562 A numi is called a it are calh tiplier is a factors} i total prod' (33) 77 (34) 92 (35) 71! (36) 67 (40) A can five 1 hours in i (41) I many mi a year ? (42) different (43)1 K0TJ£. All Teachi ther: oi Thei which 1 t Mul / •t Jl ASS STANT.] DIVISION OF INTEGERS. 15 (27) 571204 X 27009. (28) 7561240325 X 57002. (29) 56271093^ A numbei; is called a ^ it are call tiplier is a factors ,* total prodi (33) 771 (34.) 92156^ (35) 715241 (36) 679998 X 132. (30) 1379500 X 3400. (31) 7271000X52600. (32.) 74837000 X 975000. tiplyinjg two numbers together, theivvo numbers producing ,ent parts. When the mul- ay multiply by one of the by the other will give iho ^(3J) 7984956 X 144. (38) 8760472 X 999.§ (39) 7039654 X 99999. (40) A boy can point 16000 pins in an hour. How many can five boys ^o in six days, supposing them to work 10 clear hours in a day ? nr i i u (41) If a person walks upon an average 7 miles a day, how many miles will he travel in 42 years, reckoning 365 days to a ■year ^ (42)" Multiply the sum of 365, 9081, and 22048, by tho difference between 9081 and 22048. (43) Required the continued product of 112,45, 17, and 99. KoTK. Multiply all the numberB one uiio auother. DIVISION Teaches to find how often one number is contained in ano- ther : or to divide a number into any equal parts required. The number to be divided is called the Dividend; that by which we divide is the Divis(yr ; and the number obtained by 1 Multiply 6Sl751)j 45. 63175 5x9=^ § For an Hbridged method of niul- tipljing by a series of jiJMes lee the Key. :. 81 £875 9 V 2&42875 • r \n .-* -M 16 DIVISION OP INTEGERS. [tHE TUTOR'g dividing is the Quotient ; which shows how many times the divisor is contained in the dividend. When it is not contain- ed an exact number of times, there ig a. part of the dividend left, which is called the Remainder^ " ^ ' **" -Rule. When the divisor is ijp^more than \% find how often it is contained in the first fi§|d[re*<(or two figures) of the divi- dend j sei down the quotient uniBerii^tfr,and carry the over- plus (if any) to the next in the dividend^ as so many tens ; find how often the divisor is contained therein, set it down, and continue in the same manner to the end. . • ,. *" ■. When the divisor exceeds 12. find the number of times it is contained in a sufficient part of the dividend, which may be called a dividual ; place the quotient figure on the right, multiply the divisor by it, subtra:ct the product fro:*n the di- vidual, and to the remainder bring down the next figure of the dividend, which will form a new dividual : proceed with this as before, and so on, till all the figures are brought down. Proof. Multiply the divisor and quotient together, add- ing the remainder (If any) and the product will be the same as the dividend. (1) Divide (2) Divide (3) Divide (4) Divide (5) Divide (6) Divide (7) Divide (8) Divide 725107 by 2.* 7210472 by 3. 7210416 by 4. 7203287 by 5. 5231037 by 6. 2532701 by 7. 2547325 by 8. 25047306 by 9. (9) Divide 70312645 by 10 (10) Divide 12804763 by !1. (11) Divide 79043260 by 12. (12) Divide 37000421 by 3, 5, 7, and 9. (13) Divide 111111111 by 6, 9, 11, and 12. « ExAMPtB. Divide 7S28105 by 4. Divisor 4)7828105 Dividend. Quotient 1832026—1 Rem. Say the fours in 7» once and 8 over ; tlie fours in 33, 8 timet 4 are 32 and 1 over ; tiio fours iu 12, S times ; the fours in 8, twice ; ths foui-s in 1, o and 1 over j the fours in 10, twice 4 are 8, and 2 over ; the fours in 25, six fours arc 24 and 1 over. 7328105 Proof. ^ >*> mvisioM or integers. 17 (20)1745379894.fil23741 -r 31479461. (21) 25473221- H-27100.t (22) 725347216 -f-572100. (23) 752473729 —373000. (24) 6325104997 -i-215000. AssistASrt.] ri4) 7210473 -r37.* (15)42749467 -r347. (16)734097143 —5743.1 (17) 1610478407 -r547i 6. (18) 4973401891 -^510834. (19)51704567874-r4765043. When the divisor is a composite number, you may divido the dividend by one of the component parts, and that quotient by the ot/ier ; whi£;h will give the quotient requireu. But the true remainder must be found by the following Rule. Multiply the second remainder by the first divisor : to that product add the first remainder, which will give tha true one, (25) 3210473 -r 27.§ I (27) 6251043 -r 42, (26) 7210473 -f- 35. | (28) 5761034 -f- 54. A number may be divided by 10, 100, 1000, &c. by merely cutting off one, two, three, &c. Ggures on the right : the other figures are the quotient, those cut off are the remainder. • Example. Divide 40855 by 29. Dividend. Divisor 29)40855(1408 Quotient. 29 29 118 12672 116 2816 23 Remainder. 255 232 40855 Proof. 23 f When the dtvi»or is large, the quotient figures «r« mott easily found by trials qf the fir»t figure (or tvoo) in the leading figures of tttte dividQnd. .; X Ciphers at the right of the divisor maybe cut off, iihd as many figures from the right of the dividend , but these mutt b« annexed to the remainder at last. i ExAMFLK. Divide 314659 by 21. 21«7X 3)314659 7)104886—1 -^ }.=Sx8+l=«l6rem. 14983—5 J I p -.•.fBdHPif- J ^ 18 DIVISIOIf OF INTEGERS. [tHE TUTOB's Thus 76390-f- 10=7639 ; 2384£7-M 0=23845 and 7 rem. And 4 698663-t-l 000=4598 and 653 rem. (•29) Cr)94I089-r 10 (31) 18043329 -r- 10000. (30) 7208J65-rlOO (32) 7406672—1200. (33) What is the difference between the 12th part of 107724, and the 23rd part of 346610 ? (34) If a ship bound to Jamaica set saP from Liverpool on the 26th of January, 1828, and arrived at that island on the 8th of March, what was the velotjityof her sailing per day and per hour j the distance being 4558 miles ? ><)TB. This ia the direct distance. The circuilouB course of ihe ship wuulil be cnnsiiierably more. (35) The period of Jupiter's revolution in his orbU round lliR sun, which ic tlie year of that planet, is 4330 of our days. How many of our years, reckoning 365 days to the year, are equal to five yrars of Jupiter ? (36) I would plant 2072 elms in 14 rows, the trees in fiuh row 17 feetasutider ; what length will the grove be? (37; If a chest of oranges, 1292 in number, be distri- buted, one rnoiety among \^ boys, that ther among 17 girls : how many will fall to the share of each ? (38) 'J he circumference of the earth's orbit, or annual path round the sun, I'S about 696440000 miles. Supposing ilie year to be exactly ^Qh\ days, or 8766 hours, how many miles in an hour, and how many in a minute) are we carried by (his motion ? (39) Required the sum, the difference, the product, and the quotient, of 3679 and 233 : and also the quotient of the product divided by the sum, (40) The sum of two numbers is 4290 ; the less number is 143 : what is their difference, product, and quotient; and the quotient of the product divided by the difference ? (41) The product of a certafn number multiplied by 694, when 320 are addec), 1» equ,:\l! to 500000: what is that number t (42). Allowing the eu.th fa revolve on its axis in eiactly 24 hours, and the circumference at the equator to be 24864 milts ; at what rate per hour and per minute are the inhabi ^ants of that part carried round by the revolution ? Also, a what rate are the inhabitants of London carried rounds the ciicumference in that latitude being 15480 miltis? 4 12 5 20 Id, 8 10 12 14 16 18 20 22 24 26 23 SO /■-, ASSISTANT.] TABLES OF MONET. ' ^^ ARITHMETICAL AND COMMERCIAL TABLES. STERLING MONEY. 4. farthings (qrs.) make 1 penny, di 12 pence 1 sh'Hing, s. 5 shillings 1 crown, cr. 20 shillings, 1 pound, or sovereign, £, d. denotes a farthing, \d. a halfpenny, and |J. three far- 1 4 things. Qrs 4 =' 1 penny. 48 == 12 = TshiHing. 240 = 60 = 5 = 1 crown. 960 = 240 = 20 = 4 = 1 pound. A moid ore. 27 j«. A noblf, 6s 8ci. A A and there is no nlter- OBSOLETE COINS. A guinea (weights Hwls Hg^^"-) 'a>ue 21». pistole, 17s. A mat k 13s. 4J. An angel, 10s. tester, %i. A groat, 4(/. NoTKs. Gold is considered the standard metal , ation in tlie new coin, either in finenesg or weiglM, from that of former coinages ; 21 sovereigns being equal in weight to 20 guineas. 1869 sover- eigns weigh exactly 40 fts. troy. A sovereign is therefore a little more tlmn 5 dwls. Zi gri. (Sdwts. 3'274 ^rs.) and a half sovereign rather exceeds 2 duls. IS.f grs. (2 diets. 13'637 gr*.-) The new silver com » of the same fineness as that of former coinages ; but 1 lb of silver is now coined into 66s, instead of 62s. as it was formerly, *o that one shilliug row weighs 3 dwls. ISA^rs., and other MUer piecei. in proportion. The mint valu^ of gold is f3..17..10|. per ounce, and of silver 5» 6r/. The standard for gold coin is 22 parts (commonly called carats) of hue, gold, and 2 parts (or carato) of copper, melted together. For silver com 11 oz, Qdiils. of fine silvet alloyed with lb dwta. of copper. MONEY TABLE. /*\vtbinns. qrs. d 4are 1 6 8 10 12 14 16 18 20 22 24 26 28 30 U 2 2i 3 H 4 I H ^\ 6 I ejl 7 I 74 fitrikiniis. qrs. d S^Hi-e 8 34 ... 8| 36 ... 9 38 ... 9j 40 ...10 42 ...10^ 44 ...11 46 ...Hi 48 ...Is. Pence, d. s. 12 are 1 24 ... 2 Hmte. 1 d .. 36 areSi 48. . 4 60. . 5 72. .. 6 84. 96. .. 8 108. .. 9 120. ..10 132. ..II 144. ..12 156. ..13 168. ..14 180. ..1.5 ! 192. ..16 Pence (I. s, 20 are 1 8 6 4 2 10 8 6 4 2 120.. .10 130.. .10 10 140.. .11 8 150. ..12 6 30... 40... 50... GO... 70... 80... 90- •• 100... 110... 2 3 4 5 5 6 •T 4 8 9 Pen'f. d. ,«. d. 1-60 are 13 4 170. ..14 2 180. ..15 190...15 10 200.. .16 8 Shillings s. £. 20 are I 30... 1 10 40... 2 50 60 70... 3 80 «re4 4 S 2 10 3 U) 90 100 no 120 130 140 150 160 170 ISO l!>0 200.. .10 21X) 10 10 6 fi 7 7 S .*. 10 10 10 oi 10 8 10 9 9 10 20 "WEIGHTS AND MEASUHES. [tuE TOTOK's Notb;. When the units' Sgure is cut off from nny number of tbil- lines, iialf the remaininj; ^kuits will be the pounds. Thus, 256«.=r ^'12. 168. because half of 35=12 ^ and the one over prefixed to the 6, Kivcfi IGs. WEIGHTS AND MEASURES. TROY WEIGHT. 24 grains (gT') make 1 pennyweight, dwt» 20 pennyweights . 1 ounce, - oz, J2 ounces . . 1 pound, . *. Grains 24 = 1 pennyvyeight. 4R0 = 20 = I'ounce. 5760 =240 =12 = 1 pound. Gold, silver, and gems, are weighed by tliis weight. apothecaries' weigat. 20 grains (gr.) make 1 scruple, • B 3 scruples . . 1 dram, . . 3 8 drams ... 1 ounce, . . 5 12 ounces ... 1 pound, . . ft. Grains. 20 == 1 scruple. 60 =» 3 = 1 dram. 480 = 24 = 8 == i ounce. 5760 = 288 = 96 = 12 = 1 pound. Tliis is used only in the mixing of medicines. These are the vame grain, ounce, and pound, hs tho*e in Troy Weight. AVOIRDUPOIS WEIGHT. ] 6 drams (dr.) make .... 1 ounce, • . oz, 16 ounces . 1 pound, . IL Impounds » • 1 stone, . st* 28 pounds, or 2 stones .... 1 q arter, , yr. 4 quarters, or 8 st. or 112 It . 1 hundred, cwt, 20 hundreds • • 1 l«n> • • '• Drams. 16 = 1 ounce; 256 "ta 16 =3 1 pound. 3584 == 224 = 14 =^ 1 stone. 7168 « 448=8 28 =■ 2=3 l quart£r. 28672 « 1792 *» 1 12 ==: 8=4=1 cwt. ^'7iA.A.i\ ^^ a5«40 ^: ^240 ^ 160 == 80 == 20 = 1 ton. UTOK*S of tbit- 2565 = a the 6, I ASSISTANT.] WEIGHTS AND MEASURES. It. lV«ight. or. St. gr, cwt, t. I ten. fl Bv this weiiil.t tifnrly all the common nec»-^snri«of life are w*»P*""^- AUussor haU^Sm f.n.i one of .t.a.v=36 ft. A loud '^ 3fi tru..,.. A i.eck loaf vvci«i.8 17 fl>. G oz. 1 . WOOL. When wool is purchased from the grower, the legal atona of 14 ft. and the tod of '28 ft. are used. But in the deaUuga between woolstaplers and manufacturers, 45 pounds are . . I stone. 2 stones, or 30 ft. . 1 tod. 8 tods, or 2-10 tt . . 1 pack or sack. COMPARISON OF WEIGHTS. A grain is the elementary or standard weight. 1 ounce avoirdupois is . . 4<37^ grains. 1 ounce troy .... 4^80 1 pound troy .... 5760 1 pound avoirdupois . . 7000 175 pounds troy=l44 pound*^ avoirdupws. 175 ounces troy=<92 ounces avoirdupois. Wc may, therefore, reduce ll.s. T.ay iuto Avoirdupoii by mulU- y]yU\g them by Ui, aiul dividing by 175, &c. LINEAL, OR LONG MEASURE. foot, . yard, . fathom. yd. fa. 1 pole, rod, or j^erch, j». I land-chain,* ch. fiu'long, mile, , league, fur. I. 11 inches (in.) make . . 3 feet, or 3G inches . . *i yards, or 6 feet • . fli yards, or lt)| feet . . 4. poles, or 2'2 yards . • 40 poles, or 10 ch., or '220 yds 8 furlongs, or l760 yards 3 miles . . • • • Uailcy-cornt. 8 =a 1 inch. 86 » 12 =- 1 foot. lOB — 86 « 3 = 1 y«rd. 694— 198= Ui.!-- Si== 1 pole. *87r.O — 7920 ... 1560 ^ 220 =- 40 ^= 1 furlong. iSoObO '-^ 633GU -^.-. 5280 === 17G0 = 330 «- 8 — 1 mile. - " •~n,V"^U^"Ti^MUir^Tu0^i»i«, e«i:h Tiiik bting — 7W ltt«li«* b2 J^ TABLES OF MEASURES. [tHB TUTOR'J NoTB. It fs commonly supposed that t lie EnglMi inch was originally tal;en liom tliree grains ol" barley, selected Irom the middle ol' the ear, and 'vell dried. A twelfth part of an inch is calkil a line. 4 inches are a hand, un-d in measuring the height of horses. B feet are a pace. A cubit = 1| feet nearly. This measure determines the length of lines. A line has the di- Biensiua of length only, witliuut breadth or thickness. CLOTH MEASURE. 2| inches (in.) make . 1 nail, , . n. 4) nails, or 9 inches . . I quarter, . gr. 4 quarters .... 1 yard, . . yd, 5 quarters .... 1 Englii^h ell, E. e. A Flemish ell is 3 qrs. A French ell 6 qrs. Used for all drapery goods. SUPERFICIAL OR SQUARE MEASURE. IH square inches (sq. in.) make 1 square foot sq.ft. 9 square feet . • 1 square yard, sg. yd. 30| sq. yards, or 272^ sq. foct 1 sq. rod, pole, or perch. Also, in tho measure of land. 40 perches make . . .1 rood, . r. 4 roods or 4840 yards . 1 acre, . a, 10,000 square links . . 1 square chain, sy^ 2 ==^ 1 peck. 2218 = 8=. 4 = 1 bushel 17744 Bsi 64 :=> S3 == 8 = I quarter. OF COALS, S bushels make . . 1 sack. 12 sacks, or 30 bushels 1 chaldron. 21 chaldrons . - 1 Hciirc. All iho measures used for heaped goods arc to be of cylm- 4rtcalform; the diameter being at least double the depth. The height of the raised cono to bo equal to threo-fourOw ol the depth of the measure. The old dry gallon containetl 268^ cubicinches. N(»TB, 'rhti IhibIihI, lor ineasuriiiK liPn|'»'dBtn>il«. must be 17.fil \r\chrn In diomeier, and U.J04 innhos il*-**!) ; or if mnde IH incliPS in diamoter, lii« depUi will ha 0.717 iiicliei. The cone lo b« rnised (i -iO"«1^*»-'" ^'^''^ ^»*^»« «'^*^' "^ ^'"' *"wiNE. Multiply by 6, an^ divide by 5, ..r add I part. OlitCtSVlbC, MuUipiy by 749, and divide bj 624;. TIME- 60 fleconda (sec.) make .... 1 minute, . ^m. 60 minutes j ^"«'-» ^ * f\ 24 hours i/^y' • • ;v 7 days 1 week, . . t./c. 52 weeks, 1 day, 6 hours, or J ^ j jy^ajj year, yr. 365 days, 6. hQurs . . - > , rru c .u- vpar + 365 days 5 hours, 48 min, 51^ seconds The Sulai; >e^K,t 100 years 1 century. Seconds. 60 =. 1 niiiiute. 3600 = 60 = I bour, 86400 = 1440 = 24 = 1 day. 604800 = 10080= 168 = 7 = J. ^^«J- ., , juijan vtar.' 31557600=525960=8766=865 bir yw • 'IM^^Tr is diS im« 13 Ca^^ mo«tlu; January. February, Ma'X Tpiii; aiai'tluue. July, August, Sept.. «Uer, Ocu^b- , N-...b<., i^^lj:";. thirty i.S.pten^er, | Ajf j^^-t^-: -^«^^- III Aniil June, and Mi November ; 1 out every icai'-vcn,' o ■iv.t,';; 4"," iu rJbru.,, .10.... 1 To nuu»r> t>v.„ y...„.e. The Zcop-ror are those which can be exact y >»v.aed by 4 , aM821.,l8k&c. Hence it appears that the ye?^ '» "^ counted 385 days, for t/^ee yec« "f '*^ ' X ,f /L^eVo the fouriA ; the average being 36o| ''"X^" //^^.f ; "L 8^«« four weeks are frequently called a month ; but ni Ihu sens* ti.n« couM.ts or years muutlu, weeks, 4c. allow - wetk* to a IS months to a year. ORMOMETllT. 60 seconds ("j make . 1 nmxutQ, 60 minutes . . . • ' ^^fS';^^ ° 360 degrees . . . . ^ ^_^ ■ «Ad«vU the lime iirwliidrtbelarth revolve, once upon /t. "i«; l.yU and Islon. it i. rec-k.-ned from .mdi.ight ,o «.id».«bt ; but ih, a.tronomical dny bfgius at ni)on. . ^ ^ t The Sol-r, or true year, in tlmt |H..lion of time lU vvi.iui lue fa niakes uue eulise revolution lOUIld the sun. ^ it DEFINITIONS. [the tutor's Many highly important calculations in the mathematical sciences are founded on this division of the circle. In Astronomy, the great circle of the echptic (or of the zo- diacj IS divided into 12 signs, each 30° In Geography, a degree of latitude, or of longitude on the equator, measures nearly 69 J^ British miles. But a minute 01 a degree is. called a geographical mile. articles sold by tale. 12 articles of any kind, arc 1 dozen. 13 dozen 1 gross. in *'^°.". ^ e""**' gross. 20 article* 1 score. 24 sheets of paper 1 quire. 20 quires . . 1 ,eam. 2, reams . . 1 bundle. DEFINTIONS. - 1. A NUMBER id called abstract, when it is considere'd simply, or Without reference to any subject j as seven, a thousand, &c. 2. when a number is applied to denote so many of a par, icular subject, It IS a concrete number ; as seven pounds, a thousand yards, &c. ■' ' 3. A denomination is a name of any particular distinctive part of mouey, weight, or measure ; as penny, pound, yard, &c. 4. I he association of a concrete number with its subject, forms a quantity. *' ' 5. A simple quantity has only one denomination ; as seven pounds. ' ^* 6. A compound quantity consists o^more denominations than on J ; as seven pounds five shillings. £. 8. 8 e 20 d. 6i 168*. < la ft^ 2022 d. 4 8000 qr» Am, EXAMPJ.B. Reduce M..%,,Q\. into farthings. The i'R. being multiplied by 20, nnd tbp 8*. added, h.Hke IOSj. ; tlieso bein^ niultiplied by 12. and the b^/.Hdded. make 202.V ; which beid, multiplied by 4 and the 2 farthings added. iuak« in the\vhol3 80D0/aW/i?W«. assistant] REDUCTION. REDUCTION 57 la the method of changing quantities of one denomination into another denomination, retaining tlie same value. Rule. Consider how many of the less name make one of the greater ; and multiply by that number to reduce the gi-eater name to the less, or divide by it to reduce the less name to the greater. , r l- • (1) In jei2. how many shillings, pence, arid farthmgal ^Tis. 2405. 2880t/. 11520 qrs, (2) In 3 1 1 520 farthings, how many pounds 1 Ans. X324..10. (3) CJhange 21 guineas into farthings. Ans.^UQS qrs, (4) In Jei7..5 .3i. how many farthings? Ans. 16573 qrs» (5) In £25.. 14.. I. how many pence 1 Ans. 6l69(/. (6) Reduce 17940 pence to crowns. Ans, 299 crowns. (7) In 15 crowns, how many shillings and sixpences ? Ans. 75s. 150 sixpences, (S) Change 57 half-crowns into threepences, pence, and farthings. Ans. ^"^0 threepences, MlOd 6S4>0 farthings. (9) How many half-crowns, and how many sixpences, ar«, equivalent to JE25..17..6 1 Ans. 207 half-cr, 1035 sixpences. (10) Convert £17.. 11.. 9. into threepences. Ans. 1407 threep. (11) Change £10.13. .lOi. into halfpence, ^n*. 5133, (12) In 52 crowns, as many half-crowns, shillings, and pence, how many fartliings 1 (13) Convert 17380 farthings into £. (14) In 21424 farthings, how many Bliillings, and pence, of each an equal (15) Reduce 60 guineas to shillings, Ans. 12605, 252 crowns, £63. (16) Reduce 76 moidoresf into pounds. Ans. £j02..12. ^ns. 2 1424 /ar. Ans. £\S,.%.l. crowns, half-crowns, number? Ans. 52. crowns, and poimds. C6nver8e to the preceding Example. In 8090 farthings, how many pounds ? Dividing the farthings by 4, we obtain m)22ii and 2 oter, which arc farthings, because the re- mairder is a part of the dividend. Divide 2023 by 12, and we obtain 168s. and 6d. over; thtse shillings divided by 20, give £B. 8«. t» that tlia Buiwer is j^..8..6.f . 4)8090 qr8 12)2022} d. 20)1681 6 Jd. An$. ^8..8..6i. f 27 shillingi. The looidore is current in Portugal, but not in t^ RSDUCTION t THE tutor's «■ (17) How many shilliiigs, half-crowns, and crowns, an equal number of each, are there in £556. 1 jlns. 1 308 of each, and Is, over. (?8) In 1308 crowns, as many half-crowns, and as many Bhiilings, how many pounds % ^^ns. £555.. 1 8. (19) Seven men brought jei5..lO. each into the mint, to bo exchanged for guineas ; how many would they have ? Arts, 103 guineas arid Is. over. (20) In 5^25 American dollars, at 4.*. 6d. each, how many pounds sterling 1 Jim. £ 1 18..2..6. WEiaHT AND MEASURE. TROV WfilGHT. (21) in 27 ounces of gold, how itiany grains ? ^nL 12960, (22) Reduce 3 lb. 10 02?. 7 dwt. 5 gr. to grains t Ans. 22253. (23) In 8 ingots of silver, each ingot weighing 1 lb. 4> oz. n dwts. 15 gr. how many gt^ains t Ans. 34? 1304 grs. (24<) How many ingots weighing 7 lb, 4 oz. \1 dwts. 15 gr. each are thfere in 34«i304< grains? Jlns. 8 ingots. apothecaries' weight (25) In 27 ft- 7 5 . 2 3 . 1 9. 2 ^T. how many grairisl Jlns. 159022 grains. (26) In a compound of 9 5 . 4 5. 1 9. how many pills of 5 grains each t Jif^s. 916 pills. AVomrtjipois weight (27) In 14769 ounces, how many cwt. t jins. 8 cwt. 6 qr. 27 lb. 1 cz. (28) In 34 tons, 17 cwt. \ qr. 19 lb. how m^ny pounds t ^ns. 78111 lbs. (29) In 9 cwt. 2 qrs. 14 lb. of indigo, how many half stones, and how many pounds 1 Ans. 1 54 half stones, 1 078 lb. (30^ how many stones and ponnd^ are ther6 in 27 hogs* heads of tobacco, each weighing net Si cwi. ? Ans. 1 890 5f ones, 26460 lbs. (31 ) Bougnt 32 bags oi hop , each bag 2 cwt. 1 qr. 14 lb. ana anoihfer of l50 lb. how mai - ^t^. are there in the whole t Ank 77 cwt. 1 qr. 10 M. C$2) In 27 cwi. of rai3ins,how many parcels of 18 lb. each 1 ASSISTANT.] COMPOUND ADDITION. 29 an (32) In 27 cwt. of raisins, lipw many parcels o(lS lb. each ^ Jlns. 168. CLOTH MEASURE. .dns. 4.32. (33) In 27 yards, how many nails ^ (34) In 75 English ells, how many yards 1 Ans. 93 yardSf 3 qrs. (35) In 24 pieces, each containing 32 Flemish ells, how many English ells 1 Ans. 460 English ells, 4 (^rs. (36) in 17 pieces of cloth, each 27 Flemish ells, how many yards 1 •^^** ^^^ yards, 1 qr. (37) In 91 1 ^ yards, how many English ells 1 Ans. 729. (38) In 12 bales of cloth, each containing 25 pieces, of 15 English ells how many yards? •^ns, 5625. LONG MEASURE. (39) In 57| miles, how many furlongs and poles 1 ^■ Ans. 460 furlongs, 1 8.40,0 poles. (40) In 7 miles how many feet and inches 1^ Ans.Se^QO feet, U35^0 inches. (41) In 72 leagues, how many yards 1 Ans. 389160 yards. (42) If the distance from London to Bawtry be accounted 1 bO miles, what is the number of leagues, and also the num- ber of yards, feet, and iches 1 Ans. 50 leagues, 264000 yards, 191000 feet, 9504000 inches (43) How often will the wheel of a coach, that is 17 feet in circumference, turn in 100 miles ? Ans. 3 1058 J^ times round. (44) How many barley-corns will reach round the globe, the circumference being 360 degrees, supposing that each de- gree were 69 miles and a half "? ^ns. 4755801600. See Table of Geometry, page 30. LAND MEASURE. (45) In 27 a. 3 r. 19 p- how;many perches? ^ns. 4459. (46) A person having a piece of ground, containing 37 a- cres, 1 perch, intends to dispose of 1 5 acres : how many per- ches will he have left ? ^ns. 3521 perches. (47) There are 4 fields to be divided into shares of 75 per- ches each : the first field contains 5 acres ; the second 4 acres 30 REDUCTION. [the tutor's 2 perches ; the third 7 acres, 3 roods ; and the fourth 2 acres, 1 rood : how many shares will there be ? Ans. 40 shares, ^Iperches, rem. (48) In a field of 9 acres and a half, how many gardens may be made, each containir.g 500 square yards 1 Ans. 91, and 480 yards rem. IMPERIAL MEASURE. (49) In 10080 pints of port wine, how many tuns ? Ans 5 tuns, (50) In 35 pipes of Madeira, how many gallons and pints % Ans. 4410 gals, 36^80 pints. (51) A gentleman ordered his butler to bottle off f of a pipe of French wine into quarts, and the rest into pints. How many dozen of each had he 1 Ans' 28 dozen of each. (52) In 46 barrels of beer, how many pints ? Ans. 13248. (53) Iq 10 barrels of ale, how many gallons and quarts ? Ans. 390 gals, 1440 gts. (54) In 12480 pints of porter how many kilderkins? Ans. 86 kit. 1 fir. 3 gals. (55) In 108 barrels of ale, how many hogsheads? Ans 72- (56) In 120 quarters of corn, how many bushels, pecks, gal- lons, and quarts ? Ans. 960 bu. 3840 pks. 7680 gal. 30720 gts. (57) How many bushels are there in 970 pints? Ans. \5 bu.l gal. ^pts. (58) In 1 score, 16 chaldrons of coals, how many sacks and bushels ? Ans, 444 sacks, 1 332 bushels, TIME. (59) In 72015 hours, how many weeks ? Ans. 428 weeks, 4 days, 1 5 hours. (60) How many days were- there from the birth of Christ, to Christmas, 1794, estimating 365 ^ days to the year ? Ans. 655268f days. (61) Stowe writes, that London was built 1108 years before our Saviour's birth. Find the number of hours to Christmas, 1794? Ans. 25438932 hours. f^ I -- \ \ i ASSISTANT.] COMPOUND ADDITION. 31 m) From July l8th, 1799, to April l8th, 1826, how many days 1 ^ns. 9110\ days, reckoning 365i days to a year. {63) In a lunar month, containing 3 days, 12 hours, 44 minutes, 2 seconds and eight-tenths, how many tenth parts of seconds] ^«s. 25514428. (64) How many seconds are there in 18 centuries, estima- ting the sole, year at 365 days, 5 hours, 48 minutes, 51f seconds^ ^ns. .6802476700 seconds. COMPOUND ADDITION. Teaches to find the sum of Compound Quantities. Rule. Add the numbers of the least denomination ; divide the sum by as many as make one of the next greater; set down the remainder (if any) and carry the quotient to those .,fthe next greater: proceed thus to the greatest denomination, which add as in Simple Addition. Proof. As in Simple Addition. £. s. 2 13 7 9 5 15 9 17 7 16 5 U (1) d. H '4- EXAMVLB. £. Km (f> 15.. 7.. 4J 7..18..10I 11..19.. 5 6..10..11{ 4.. 0.. 9i 45..17.. 4i MONEY. £. s, d, 27 7 34 14 lOi 57 19 91 16 75 18 97 13 71 5 £. s. d. 35 17 59 14 10| 97 13 lOi 37 16 " 97 15 59 16 7 Of 75 54 91 35 29 91 (4) s. 3 17 15 16 19 17 d. 1 U 3i Say 1, 2, 5, 7 farthings are 1 penny 3 fai- thiDgs; set down i and carry Id. — li 10, 11, 16, 20, 30, 40d. are Zs. Ad. ; set down 4(/. and carry 3s.— 3, 12, 20, 27, 37, 47, 57,a are j£2. 17a. ; set down 17s. and carry £2. Tlie rest as in Sim- ple Addition. In Addition of Money, the reduction of one denomination to the next greater is generally done without the trouble of dividing, by the knowledge previously acquired of the Money Tables. /2 / COMPOUND ADDITION. [the tutor's £, s. d. £. (7) s. d. (9) £. '. rf. je. (11) s. d. 257 1 H 21 14 ".- 127 'i 7| 31 1 4 734^ 3 H 75 16 525 3 10 75 13 1 595 5 3 79 2 4J 271 39 19 7| 159 14. n 57 16 ^ 524 9 1 97 17 31 207 5 4. 26 13 ^ 379 4 01 36 13 5 798 16 •71 54. 2 i' 215 5 n| 24 16 31 x>. s. t/. je. (8) if. d. *. *. a. £. (12) s. d. 525 2 4<^ 73 2 u 261 17 11 27 13 51 179 3 5 25 12 7 379 13 5 i^ 12 101 250 4. 7| 96 13 5f 257 16 7| 9 13 Of 975 3 H 76 17 3f 184 13 5 15 2 10| 254. 5 7 97 14 1^ 725 2 31 37 19 379 4 54. 11 7| 359 6' 5 56 19 H > ASSIST (17) (20) WEIGHTS AND MEASURES. I TROY WEIGHT. (13) ! (14) 03. dwt. gr. lb, oz, dwi. gr. ; 5 7 3 < 9 8 11 19 15 19 18 13 4 21 14 22 15 12 5 3 3 9 5 2 11 7 1 9 2 15 17 15 13 7 15 22 14 19 21 23 i7 APOTHECARIES' WEIGHT. (15) ft 17 10 9 5 27 11 9 5 37 10 5 49 7 7 2 1 6 1 o 2 1 I 2 I (16) 5- 2 1 10 5 9 1 1 7 o 7 5 4 9. iir. 12 1 17 14 1 15 2 13 1 18 jfds. 13r 7( 9J 17( 2( 27i \ \ ASSISTANT. COMPOUND ADDITION. AVOIRDUPOIS WEIGHT S3 (17) lb, oz. dr, 152 15 15 14 15 10 6 13 272 303 255 173 635 10 U 4 2 13 :wt,qrs. lb. (19) t,cwt,qrs.lb. 25 I 17 7 17 2 12 72 3 26 5 5 3 14 54 I 16 2 4 I 17 24 I 16 3 18 2 19 17 19 7 9 3 20 55 2 16 8 5 I 24 LONG MEASURE. (20) "gas. ft- iri. 225 1 9 171 3 52 2 6 397 10 154 2 7 137 1 4 (21) lea. m.fur.po, 72 2 1 19 27 1 7 22 35 2 5 31 79 6 12 51 I 6 17 72 5 21 (22) m.fuwtjds, 39 6 36 14 7 214 3 4 160 45 3 202 17 1 19 32 4 176 CLOTH MEASURE. V2 17 14 Ifi 13 18 (23) j)ds. qrs. n. 135 3 3 70 2 95 3 176 I 26 O 279 2 (24) E.e, qrs. n. 272 2 I 152 I 2 79 O I 156 2 79 3 I 154 2 1 LAND Mr.ASURE. (25) a. r. p. 726 1 31 219 2 17 1455 3 14 879 1 21 438 2 14 757 C26) rt. r. p. 1232 I 14 327 18 131 2 15 1219 1 18 223 2 8 236 9 C2 34 COMPOUND ADDITION. [tHE TUTOr's IMPERIAL MEASURE Vv'INE . lii.!U (-27) hlids-^als,qts. 31 57 I 97 18 2 76 13 1 55 46 2 87 38 3 55 17 1 (28) t. hhd.gals^fjts. 14 3 27 2 ly 2 56 3 17 39 2 75 2 16 I 54 I 19 2 97 3 54 3 ALE AND BEER. (29) bar.Jir. 25 2 7 17 3 5 96 2 6 75 1 8 96 3 7 75 5 (30) hhd. gal. q(s. 76 51 2 57 3 3 97 27 3 22 17 2 32 19 3 55 38 3 DRY. (3 1) qrs. h. 300 2 167 369 7 50 3 74 6 p- 1 I 2 3 6. 16 21 15 3 (32) p, gal. qts. 2 3 I 2 I 1 1 2 3 2 1 TIME (33) w. d. 71 51 76 95 79 h w. 11 157 9' 9 5 21 176 2l|53 1598 (34) d, h. 2 15 3 21 15 2 21 2 18 m. s. 42 41 27 51 37 28 42 27 47 38 I (35) A, B, C, and D, were partners in the purchnse of a quantity of goods: A laid out ^7. half-a-guinea, and a crown; B, 49.9, C, 54«. Gd. and D, H7t/. What was the purchase ? Jtns. «^I3..6..3. C36) A man lent his friend at different times these several sums. W3.^63.— ^25 .15.— 5g'32,.7.— £^I5..14..10. nnd four score nnd nineteen pounds, half-a-guinra, and a shilling. How much was the whole loan ? Ans. j{p236..8..4. (37) Bought goods, for which I paid ^54.. 17 ; for packing ] 3*. 8c/; carriage £l . 5..4 ; and expenses over makine the bar- gain 14«. 3 ' Jns. 58 years, 7 mor'hs, 10 days. (40) A clprk having been out collecting debts, presented an account that A paid him £7..5..2 ;-B £l5..18..6i ;_ CJPl50..l3..2i; — Djgl7..6..8;— K 5 guineas,2 crown pieces, 4. half-crowns and 4*. 2rf ,— F paid him only twenty groats ;— • G je76..15..9i ;— and H JE121..12..4. How much was the whole amount? -^ns. .£396..7..64. (41) A nobleman had a service of plate, which consisted of twenty dishes, weighing 203 oz. 8 dwts j 36 plates, 408 oz, 9 d'ots. ; 5 dozen spoons, 112 oz. S dwts. ; 6 salts, and 6 pepper-boxes, 71 oz. 7 dwts. ; knives and forks, 73 oz. 5 dwts ; two large cups, a tankard, and a mug, 121 oz. 4 dwts. ; a tea- urn and lamp, 131 oz. 7 dwts. ; with sundry other small ar- ticles, weighing 105 oz. 5 dwts. The weight of the whole is required 1 ^ns. 102 lb. 2 oz. 13 dwts. (42) A hop-merchant buys 5 bags of hops, of which the first weighed 2 cwt. 3 qrs. 13 lb. ; the second, 2 cwt.^grs. 11 lb. ; the third, 2 cwt. 3 grs. 5 lb. ; the fourth, 2 cwt. 3 grs. nib.; the fifth, 2 cwt. S (jrs. U) lb. He purchased also two pockets, each pocket weighing 84 lb- I desire to know the weight of the whole. ^ns. 15 cwL 2 qrs. COMPOUND SUBTRACTION Teaches to find the difference of Compound Qunntiiies, Rule. Suhtraci .^- ';>•*' 36 COMPOUND SUBTRACTION. [the tutor's occasion) as many as are equal to one of the 7iext greater de- nomination : observing to carry one to the next for that which was borrowed.* Proof. As in Simple Subtraction. ASSIST: (16) B( P di MONEY. (I) je. s. From 715 2 Take 476 3 * ExAMPLB Subtract i€5'i..l7..9j. from ,^9..12..7J. S.. $. il. Berauie S rHrlhiiiKR cannot be taken rruni 2 say S 89. .12.. 71 from 4, 1, and 2 Hie 3 ; ict down 8 and cany I — I 54.. 17.. 9i and 9 are 10, ]0 from 12, 2, and 7 are 9 : Kct down 9 I and carry I 1 and 17 are 18, 18 from 20, 2. and 12 84.. 11.. 9} are 14 ; set down 14 and carry 1 to the pouudn. ^ .J^'^W^If^*' Pai( Remaii ( lb, oz, 51 1 39 C ^ yds. 107 78 ( ASSISTANT.] (16) £' ••*• '^• Borrowea 350 COMPOUND SUBTRACTION. (17) 37 Puidat f26 5 different J 73 10 6 times i 41 9 8i IdQ 14 9 Paid in all Remains to poy £. s, d Lent 577 10 n • 1 r95 10 O Received ', ^^ ^j q at several < ^^ j^ y times |^,^3 17 41 \. WEIGHTS AND MEASURES. TKOY (18) lb» oz. dxot, gr. 62 1 7 'i 39 15 7 WEIGHT. (19 lb. oz. dwt. gr, 7 2 2 7 5 7 15 APOTHECARIES (20) ft. 5. 5. 9 5 2 10 2 5 2 1 ES WEIGHT. (21) ft. 9 7 2 1 13 5731 18 ^■■^■* AVOIRDUPOIS WEIGHT. /i. OS. f/r. 35 10 5 29 12 7 (23) cu'/. gr. 35 1 25 I lb. 21 27 (24) ^ cu4. qrs. lb 21 1 2 7 9 11 3 16 LONG MEASURi;. C2r)) 1 (26) V(is. ft, in. \lea,mi.fur,po. yds. qrs. n. 107 2 10 H7 2 () 29 71 1 2 CT.OTH MEASURE. (27) ! CJH) E. e. grs. n. 78 2 11 58 2 7 33 3 2 1 3.5 2 1 14 3 2 ^l '>''' \ / 38 COMPOUND SUBTRACTION. [tHE TUTOR's LAND MEASURE. (30) a. r, p. 325 2 \ 279 3 5 V a. r. P- 175 1 27 59 37 1 IMPERIAL MEASURE — WINE. (31) hM.gnl.qts.pts. 47 47 2 1 28 59 3 (32) lurt. hlid. ffiil.qts. 42 2 37 2 17 3 49 3 ALE AND BEER. (33) «/ . fir. gal. 37 2 1 25 1 7 (34) hhd. gal. qts. 27 27 1 12 50 2 CORN AND COAL. (35) qr. b. p, 65 2 1 57 2 3 (36) sc. ch. sa, b, 3 16 1 2 12 2 1 79 8 2 4 23 9 3 5 TIME. (38) h. m. sec. 24 42 45 19 53 47 t (39) yrs. m, d. 10 7 20 5 8 29 (40) When an estate of ^300. per annum is reduced by the payment of taxes, to 12 score and ^14..6. what are the taxes \ Ans. ^45.. 14. (41) A horse with his furniture is worth ^37.. 5. ; without it, 14 guineas ; how much does the price of the furniture exceed that of the horse % Ans. £1..\1. (42) A mercliant commencing trade, owed ^750 ; he had in casii, commodities, the stocks, and good debts, ^125J0..7 ; he cleared the first year by commerce ^452..3..6. "Vi^hat was he then worth ? Ans. ^12212. 10..6. (43) A gentleman left *^45247. to his two daughters, of which the younsfer was to linvp i^^ tUouoaud, lo hundred, and twice .^15. What was the elder sister's fortune ? Ans. 5^28717. (44) A tradesman being insolvent, called all his creditors together, and found he owed to A ^''53..7..6 ;— to B j^l05..i0 : =— to C ^^34.5.2 ;— to D ^^28.. 16. 5 j— to E £U.,\^..S ;— to * In this example allow 4 iveeks to a month, and 13 montha to tiio year. t In tiiia, reckon SO days to a month, and 12 luouths to the year. W— '•» I COMPOUND MULIPLICATION. 39 ASSISTANT.] F ^112..9 ;— and to G ^143..12..9. The value of his stock was ^212..6; and the amount of good book-debts was ^112..8..3 ; besides ^21..10..5. money in hand. How much would his creditors lose by taking the whole of his effects ? Jns. The creditors lost 6^146.. 11.. 10. (45) My agent at Seville, in Spain, renders me the fol- lowing account of money received for the sale of goods sent him on commission viz. for bees' wax ^37..15..4 ; stock- ings ^37..6..7 ; tobacco ^125.. 11. .6; linen cloth ^112..14..8 ; tin ^115. .10.. 5. He informs me at the same time, that he has shipped, agreably to my order, wines, value ^250. .15; fruit ^51..12,.6 ; figs ^19. 17.6 ; oil ^19..12..4 ; and Spanishwool, value ^115.. 15 ,6. How stands the balance of the account between us 1 Ans. Due to the agent ^28..14..4. (46) The great bell at Oxford, the heaviest in England, is stated to weigh 7 tons, 11 cwt. 3 qrs. 4 lbs. that of St. Paul's in Lo\idon, 5 tons, 2 cxst. 1 qr, 22 lbs. and that of Lincoln, called the Great Tom, 4 tons, 16 cwt. 3 qrs. 16 lbs. How much is the aggregate weight of these three bells inferior to that of the great bell at Moscow, which is 1 98 tons ? Jns. 180 tons, 8 cwt. 3 qrs. 14 lbs, COMPOUND MULTIPLICATION Is the method of multiplying Compound Quantities, Rule. Multiply the least denomination ; reduce the product and carry to the next as directed in Compound Addition ; and the same with the rest. When the multiplier is a composite number above 12,multiply (as before direct'^l) by its component parts. For other numbers, multiply by the jactors of the nearest composite ; adding to the last product, *o many times the top line as will supply the deficiency ; or subtraction so many times, if there is an excest. MONEY. • (1) sS. s. d. 35 12 7i o (2) £. s. d. 75 13 1{ 3 (3) s£. s. d, 62 5 41 4 (4) £. s. d. 57 2 4| 5 71 5 S\ . .. . I * la this example, say twice 3 are 6, 6 rnrihliiKi are i|ff. »ft Ouwh ja. and carry 1 ; twice 7 are Hand 1 are 15, 15d. are 1«. 8d. set down 'id. 40 COMPOUND MULTIPLICATION. [tHF TUTOR's !., (6) O) (8) £. (13)0 (1^)1 (10)0 (21) per R). 1 (22) £. 57 81 64 118 s. d. s. 18 9 10 6 d. Hi 5 H X X X X 6. 7. 8. 9. s. 9 2 7 V XlS.f (16) 15 (17) 7 9 6 6 X264 81X21. (9) (10) (11) (12) d, 3^X35. 135 13 d. 79 247 119 16 14. 7 je. (19) 1 (20) 4 111 5| 10. 11. 12. 12. s. d. 5 3x97. 4x43. X X X X (18) 2|X75. 7 X37. I f s the value of 127 lb. of scnichong tea, at 12s. 3df. Ans. ^77.. 15. .9. 135 stones of soap, at 7s. M, per stone? Jlns. ^50..!., 3. (23) 74 ells of diaper, at \s. ^d, per ell ? Aus. 5^5.. 1 ..9. (24) 6 dozen pairs of gloves at Is. lO^/. per pair? Am. £^.A2, Note. When the fraction f, .f , or J is cnimected with the multiplier, take ftrt//the given price (or the price of one) for f , hn{f of that for J, and for f . add them together. § J and carry 1 : twice 12 are 24 and 1 are 25, 2;)S. are .41.. 5. set down Ss. and curry 1 ; twice !) are 10 and 1 are 11, set down 1 and carry 1 ; twice 3 tue (> and 1 are 7, set down 7. s. (i. ^- »• ''• i. ., 6 X 1" 2.. 6 ■ " -o^g— 18 8xS + 2==26 19.. 9 ^Ji 11.. Ans. j5 EXAMFI.B. What is the value of 'llj ttn. oftCH,iitlO,<(.9,/. per ib- ! 9.. 0.. 8 27.. 0.. Multiplicand X 2- 2,. 5.. s. (I. .-£29.. .G.. ^ns. iXlO.. 9 11 jf.';..18.. S =the value of 11. I X 5" 4,i==... do .{. 2.. 8|==... do i. .i'6. G.. 8i 4»iy. *1^ ASSISTANT.] COMPOUND MULTIPLICATION. 41 r25^ What is the value of25i ells of Holland at 3s. 4\d, perein ^ws. Je4..6..0|. (26) 75| ft. of hemp, at Is. 3d. per ft 'i Ans, £4...14..4|. (27 ^ 1 9i yds. of muslin, at 4s. M. per yd. % Ans ;fi4j..2..10|. (28) 35i cwt. of raw sugar, at X4'..15..6. per cwtl (29) 154i cwt. of raisins, at ig4..l7..10. per cwt."? ^ * Ans. jg755..15..3. (30) 1 17i gallons of gin, at l2s. 6c?, per gallon 1 AnSt Jw7u..o.. <2« (31^ 85i cwt. of logwood, at jei..7..8.per cwt. 1 ^ ^ * Ans. £i\S.A2..5. <'32'i 17I yards of superfine scarlet cloth, at jgl..3..6. per yard] Ans £20.A'7..1l,. (33) 37ift.ofhysontea,atl2s.4d.perft.l .^ns. X23..2..6 r34) 56| cwt. of molasses, at jg2..l8..7. per cwt.1 ^ ^ * .^ws. dei66..4..7i. (35) 871 ft. of Turkey coffee, at 4s. 3c?. per ft. 1 (36) I20i cwt. of hops, at £4..7.,6. per cwt. J ^ "^ * ^ns. £628..5..7l. When the multiplier is l^rge, multiply the given quantity (or price) by a series of tens, to find lO, lOO, lOOO times, &c , as far as to the value of the highest place of the multiplier ; mul- tiply the last product by the figure in that place, and each preceding product by the figure of corresponding value ; that is, the product for lOO by the number of hundreds, the produc- for 10 by the number of tens, and the original quantity, by the units' figure, iJfc. The sum of the products thus obtained will be the total product.* ] ,« Example. Multiply ^7..14..9i. by 3645. £, s. d. 7..14.. 9}X5' 10 £. s. d. 38..13..11^ times. The product for 10 The product for 100 77.. 7..U X4- 309..11.. 8 10 7"" .i9.. 3 X6= 10 4643..15.. The rroductfor 1000 77S9..11.. 8 x 8=28218. .15..0 £ 40 600 8000 <»« ilfii. 28210..16..7J - 8645 \ / 42 COMPOUND MULTIPLICATION. [tHE TUTOR's (37) 407 ft of gall-nuts, at Ss. d^d. per ft 1 Ans. £11 .3..2| r38) 729 stones of beef, at 7s l^d. per stone 1 Ans. ^277..3..5i (39) 2068 yards of lace, at 9«. b\d. per yard ? Ans. £977.,19..10. (40) What is the produce of a toll-gate in the course of the year, if the tolls amount, on an average, to lis. l\d, per day % Ans. £212..3..1|. (41) How much money must be equally divided amonn; 18 men, to give each ^14..6..8f 1 Ans. £^d:j,a}..9. C42) A privateer manned with 250 sailors captured a prize, of which each man shared 5^125.. 15..6. What was the value of the prize? Ans. £3] 443. .15. (43) What sum did a gentleman receive as a dower with his wife, whose fortune was a cabinet with two divisions, in each division 87 drawers, and each drawer, containing 21 guineas? Ans. £3S36..\4. (44) A merchant began trade with 5^19118 ; for 5 years together he cle-^red jgl086. a year ; and the next 4 years 5^27 1 5.. 10.. . a year j but the last 3 years he was in trade he had the misfortune to lose upon an average, j^475..4..6. a year. What was his real fortune at the end of the 12 years ? Ans. £339S4..S..6. (45) In many parts df the kingdom coals are weighed in the waggon or cart upon a machine, constructed for the purpose. If three of these draughts amounted together to 1 37 cwt. 2 grs. 10 lb. ; and the tare, or weight of the waggon, was 13 cwt, 1 qr. ; how many coals had the customer in 12 such draughts ? Ans. 391 cwt. 1 yr. 12 lb. (46 J A certain gentleman lavs up every j^ear jg294..12..6. and spends daily -gl..l2..6. What is his annual income ? Ans. £887.. 15. WEIGHTS AND MEASURES. r47)MultipIy 9 lb. lO oz. 15 dwts. 19 gr. by 9, 11, and 12. /(48)Multiply 23 tons, 9 cwt. 3 grs. 18 lb. by 7, 8, and 9. /^ (49)Multiply 107 yards, 3 grs. 2 nails, by 10, 17, and 29. • (50)Multiply 33 bar. 2fr. 3 gal. by 11, and 12. (5l)Multiply 1 10 miles, 6 fur, 26 poles, by 12, 13, and 39. '^f*- i COMPOUND DIVISION. 43 ASSISTANT.] (52)A lunar month contains 29 days, 12 hours, 44 miii. 3 seconds nearly. What time is contained in 13 lunar months ] COMPOUND DIVISION Teaches to find any required part of a Compound quaniity. Rule. Divide the greatest denomination : reduce^ the re- mainder to the next hss, to which add the next ; divide that, and proceed u. 'jefore to the end. When the divisor is above 12, the work must be done at lengvh : unless it is a composite number, for which observe the directions in Simple Di vision .—Proo/ by Multiplication. MONEY. *(0 ^ £. s» a. 2)25 2 4 £' (5) 78 (6) 25 (7) 16 (8) 124 (13) 66 (14) 596 (15) 564 (19) (20) (21) (22) s. 10 19 % 15 6 12 4 Divide Divide Divide Divide (2) £ s . d. 3)37 7 7 d. H -r 6. n -r 7. 8. 2|~- 9. (3) £, s. d. 4)57 5 7 (4) £ s. d. 5)52 7 7i £' s. (9) 87 14 (10) 68 (11) 49 14 (12) 496 8 d. by 10. by 11. 7 by 12. 6 by 12. -i. 25. (16) 248 17 4 by 99. ^36. (17)928 12 8 by 110. 6 -L 63. (18) 608 13 9 by 144. jgl407..l7..7. by 243. Jg70079l..l4..4. by 1794: £490981. .3..7|. by 3l7l5. 5ei9743052..5..7i. by 214723. ♦Example, DividedE27..14..11i. by 5. Say the fives in 27, 6 limes 5 are 25 and 2 over ; S. s. d. £2. are 40s. and 14 are 54, the fives in 54, 10 times 6)37.. 14 Hi 5 are 50 and 4 over ; 4s. are 48d. and 11 are 69, 1_ the fives in 69, 11 fives are 55 and 4 over; M. are 5..10..11i § 16 qr*. and 2 are 18 the fives in 18, 3 times five are ,.._ .._ 15, and 3 over, or J. 44 COMPOUND DIVISION. I I ■:: I [the tutor's (23) Ifa man spend jg257..2..5. in 12 months, what is that per month ? ^/w. £2l..8.3^ f^. (24) The clothing of 35 charity boys came to ^^57..3..7. what was the expence of each boy 1 Ans. ^1 .. l2..8^f . (25) Iflgave^37..6..4|. for nine pieces of cloth, what was that per piece % Arts. ^4..2.. 1 1|. (26) If 20 cwt. of tobacco cost £%l,.b,A\ \ at what rate did I buy it per cwt, 1 Ans. £l..l..^^. C27) What is the value of one hogshead of beer, when 120 hogsheads are sold for £154..17..10 1 Ans. £1..5..9| ^%. (2S) Bought 72 yards of cloth fur £85..6. What was the price per yard ? Arts. ^1..3..8| ff* C29) Gave j^275..3..4. for 18 bales of cloth. What is the price of one bale ? Ans. jgl5..5..8| 4^. C30) A prize of jg7257..3..6. is to be equally divided among 500 sailors. What is each man's share ? Ans. £14..10..3| |^^. rSl) A club of 25 persons joined to purchase a lottery ticket of £10. value, which was drawn a prize of j^4000. What was each man's contribution, and his share of the prize- money ? Ans. each contribution Ss, and share of prize £160. (32) A tradesman cleared ^2805. in 7^ years ; what was his yeariy profit ? ^ns. 374. (33) What was the weekly salary of a clerk who received 5^66..18..1|. for 90 weeks 1 Ans, £2..19..3|. (34) If 100000 quills cost me ^187..17..1. what is the price per thousand ? Ans. ^I..l7..6| j^. WEIGHTS AND MEASURES. i (35) l)ivide 83 lb. 5 02. 10 dwts. 17 gr. by 8, 10, and 12. (36) Divide "19 tons, 17 cwt. qrs. 18 lb. 0, 15, and 19. . (37) Divide 114 yards, 3 grs. 2 ndils, by 10, and 16. (38) Divide 1017 miles, 6 fur, 38 poles, by 1 1, and 49. (39) Divide 2019 acres, 3 rods, ^29 perches, by 26. (40) Divide 117 years, 7 months, 26 days, 11 hours, 27 rninutes, by 37. ASSISTANT.] PROMISCUOUS EXAMPLES. PROMISCUOUS EXAMPLES. 45 (1) Of three numbers, the first is 21 5, the second 519, and the third is equal to the other two. What is the sum of them all? Ans. 14)68. (2) The less of two sums of money is £40, and their dif- ference j614'. What is the greater sum, and the amount of both 1 Ans. £54. the greater, ^694. the sum. (3) Wiiat number added to ten thousand and eighty-nine, will make the sum fifteen thousand and forty ? Ans. 4951. (4) What is the difference between six dozen dozen, and half a dozen dozen ; and what is their sum and product 1 Ans. diff.192, sum 936, product 62208. (5) What difference is there between twice eight and fifty and twice fifty-eight, and what is their product 1 Ans. 50 diff. 1656 product. (6) The greater of two numbers is 37 times 4?5, and their difference is 19 times 4 : required their sum and product 1 Ans. 3254 sum, 264}56S5 pi'oduct. (7) A gentleman left his elder daughter jg 1 500. more than the younger, whose fortune was 11 thousand, 11 hundred, and ^1 1. Find the portion of the elder, and the amount of both. Ans. Elder^s portion £13611. amount £25722. (8) The sum of two numbers is 360, the less is 144. Wnat is their difference and their product ? Ans. 72 difference. 31 104 product. (9) There are 2545 bullocks to be divided among 509 men. Required the number and the value of each man's share, supposing every buU^-' ./orth jg9..14..6 ? Ans. Each man had 5 bullocks, and £48..12..6. for his share. (10) How many cubic feet are contained in a room, the length of which is 24 feet, the breadth 14 feet, and the height 11 feet?* Ans. 3696. (11) A gentleman's garden containing 9625 square yards, is 35 yards broad : what is the length % Ans. 275 yards. (12) What sum added to the 43rd part of £4129. will make the total amount=£240 ? Ans. £137. (13) Divide 20s. among A, B, and C, so that A may have 2*. less than B, and C 2.s. more than B. Ans. A 4s M. B 6s. %d . and C Ss. Sd. ** Multiply the three dimensions continually together. D 2 46 PROMISCOOUS EXAMPLES. [THE tutor's i/S^ ( 14) In an army consisting of 187 squadrons of horse, each 157 men, and 207 battalions of foot, each 560 men, how many effective soldiers are there, supposing that in 7 hospitals there are 473 sick 1 ^ns. 144806. (15) A tradesman gave his daughter, as a marriage portion, a scrutoire, containing 12 drawers ; in each drawer were six divisions, and in each division there were j£?50. four crown pieces, and eight half-crown pieces. How mu"\ had she to her fortune] ^rts. jg3744. (16) There are 1000 men in a regiment, of whom 50 are officers : how many privates are there to one officer] ^dns. 19. (17) What number must 7847 be multiplied by, to produce 3013248 ] Jlns. 384. (18) Suppose I pay eight guineas and half-a-crown for a quarter's rent, but am allowed 1 5s. for repairs; what does my apartment cjst me annually, and how much in seven years'? *dns. In one year, jg3l..2. In seven, £'2 17.* 14. (19) The quotient is 1083; the divisor 28604; and the remainder 1788 : what is the dividend ? Jins. 30979920. (20) An assessment was made on a certain hundred, for the sum of ^386.. 15.. 6. the amount of the damage done by a riot ous assemblage. Four parishes paid £37.. 14.. 2. each; four hamlets £3l..4..2. each; and four townships £18..12..6. each : how much was deficient ^ns. je36..12..2. (2l)Ana;my consisting of 20,000 men, got a booty of jP 12,000 ; what was each man's share, if the whole were equally divided among- them ] ^ns. 12s. (22) A gentleman left by will, to his wife, ^£4560; — to a public charity, £572.. 10 ; — to four nephews, £750.. 10. each ; — to four nieces, £375.. 12.. 6. each ; — to thirty poor house- keepers, 10 guineas each ; — and to his executors 150 guineas. What was the amount of his property ? Jlns. £10109. .10, (23) My purse and money said Dick to Harry, are worth .V2s. Sd. but the m /ney is worth seVen times the value of the purse : what dia . le purse contain? *^ns, lis. id* (24) Supposing 20 to be the remainder of a division, 423 the quotient, and the divisor the sum of both, plus 1 9 ; what is the dividend ? ,/7ns. 195446. (25) A merchant bought two lots of tobacco, which weigh- ed 12 cwt, 3 (p-s, 15 lb. for ^114..I5..6 ; their difference in PROMISCUOUS EXAMPLES. four 47 Re- 423 ASSISTANT.] weight was 1 cwt. 2 grs. 13 lb. and in price ^7.. 1 5. .6. quired their respective weights and value 1* Jlns. Ch'eater weight 7 cwt. 1 qr. value j^61..5..6. Less weight 5 cwt. 2 grs. 15 /A. value ^63.. 10. (26) Divide 1000 crowns in such a manner among A, B, and C, that A may receive 129 crowns more than B, and B 178 less than G. ^ns. A 360 crowns, B 231, C 409. (•27) If 103 guineas and Is. be divided among 7 men, how many pounds sterling is the share of each ? Aji ^\b..\0, (28) A certain person had 25 purses, each pursj contain- ing 12 guineas, a crown, and a moidore, how many pounds sterling had he in all \ Ans. ^355. (29) A gentleman, in his will, left 5^*50. to the poor, and ordered ihat \ should be given ' ild men, each man to have 55. — i to old women, each w< :'■ i to have 2s. 6e/. — f to poor boys, each boy to have \s. — \ poor girls, each girl to have 9c?. and the remainder to the person who distributed it : how many of each sort were there, and what remained for the person who distributed the money 1 Ans. 66 men, 100 women, 200 boys, 222 girls: £2..\S..6.for the distributor. , (30) A gentleman sent a tankard to his goldsmith, thi ^ weighed 50 oz. 8 dwts. to be made into spoons, each weighing 2 02. 16 dwts. how many would he havel Ans, 18. (31) A gentlemen has sent to a silversmith 137 oz. 6 dwts. 9 gi\ of silver, to be made into tankards of 17 oz. 15 dwts. 10 gr. each ; spoons of 21 oz. 11 dwts. ISgr. per dozen ; salts, of 3 oz. 10 dwts. each ; and forks, of 21 02;. 11 dwts. l3 gr. per do'^en ; and for every tankard to have one salt, a dozen spoons, and a dozen forks : what number of each will he have 1 Ans. Two of each sort, 8 oz, 9 dwts. 9 gr. over. (32) How many parcels of su^r of 16 lb. 2 oz. each are there in 16 cwt. 1 gr. 15 lb- 1 Ans. I \3 parcels, and 12 lb. 14 oz. ovei\ (33) In an arc of 7 signs, 14<^ 3' 53", how many seconds 1 Ans. 806633''. (34) How many lbs, of lead would counterpoise a ma>,s of * Add the diflference to the sum, and diviile by 2 for the greater ; sub- tract the difference from the sum, and divide by 2 for the /ea«. is PROPORTKON. fTHE TUTORS bullion weighing 100 lbs. Troy 1* Jlns. 82 lb. 4. oz. 9-^j\ dr. (35) If an apothecary mixes together 1 lb. avoirdupois of white wax, 4* lbs. of spermaceti, and 12 lbs. ot olive oil, how many ounces apothecaries' weight, will the mass of ointment weigh, and how many ma-ses of 3 drams each will it con- tain ? ^ns. the whole 247 oz. 7 jW dr. and 66 1 of 3 dr. each. PROPORTION. Proportion is either direct or inverse. It is commonly called the rule of three ; there being always three num- bers or terms given, two of which are terms o[ supposition ; and the other is the term of demand: because it requires a fourth term to be found, in the same proportion to itself, as that which is between the other two. General rule for stating thk question. Put the term of demand in the third place ; that term of supposition Svhich is of the same kind as the demand, the first ; and the kther, which is of the same kind as the required term, the \cond.** Also the terms being thus arranged, reduce the first and third (if necessary) i to one name, and tlie second into the lowest denominatio.. mentioned. THE RULE OF THREE DIRECT Requires tiic fourth term to be greater than the second, when the third is greater than the first ; or the fo^irth, to be less tiian the second, when the third is less than ihe first. Rule. Multiply i\\p second uiid third together, and divide • Bullion ii tlie term denoting gold or siNer in ihc -vau. I.ead it weiffhtd by Atoirdupois wt-iglit. Se<( tbe Table of CuMPAnisoN ov Wkiohth. ♦• Some modern authorg prefer t>lavii)g (lie teim rf demand the wcawi/, and that Ai'mi/ur to the mytiirrd per day, what is his annual saving ? Ans. jei47..3..4. (16) If 14 j^ards of cloth cost 10 guineas, how many Flem- ih ells can I buy for ^283..17..6. ? Ans. 504 Fl. ells 2 grs. ^ ■'(17) If 504 Flemish ells, 2 quarters, cost je283..17..6- what IS the cost of 14 yards ? ^ns. ^10.. 10. (IS) Attherateof ^l..i..8.for3/i.ofgum acacia what must be given for 29 lb. 4 oz. 1 Ans. ^10.11. .3 (19) If 1 English ell, 2 quarters coat 4s. Id. what will 39* yards cost at the same rate ? Ans. ^5..3..5| 4. (20) If 27 yards of Holland cost «^5..12. 6. how many English ells can I buy for i>100. 1 Ans. 384 ells. ' (21) If7 yards of cloth cost 17s..8c/.. what is the value of 5 pieces, each containing 27| yards ? Ans. J* 17.. 7.. 01 ' (22) A draper bought 420 yards of broad cloth, at the rate of 14*. 10 id. per ell English : what wao he amount of the purchase money ? ^„5. ^^"250.. 6. (23) A grocer bought 4 hogsheads of sugar, each hogshead weighing neat 6 cwt. 2 qrs. 14 lb. at Je2..8..6. per cwt. what 18 the value ? ^„s. £64..5..3. (24) A draper bought S packs of cloth, each pack contain- ing 4 parcels, eacli parcel 10 pieces, and each piece 26 yards ; at the rate of jg'4.. 16. for 6 yards: what was the jnirchase ASSISTANT.] RULE OF THREE INVERSE. H (25) If 24 lb. of raisins cost 6s. 6c?. what will Ifl frails cost, each frail weighing neat 3 qrs. 18 Ib.l Ans. ^24..i7..3. (26) When the price of silver is 5s. per ounce, what is the value of 14 ingots, each ingot weighing 7 lb. 5 oz. 10 dwts. ? Ans. £^3 13.. 5. (27) What is the value of a pack of wool, weighing 2 cwt. I qr. \9lb. at 17s. per tod of 28 lb. .* Ans. £8,.4..6l^% (28) Bought 171 tons of lead, at ^g 1 4. per ton ; paid car- riage and other incidental charges, 5g'4..10. Required the whole cost, and the cost per lb. ? Ans. £2398.,lO, the w^ole cost, and the cost per lb. li\v many must be sent away, that the provisions may last 5 months 1 Jlns. 400. (10) If ^20. worth of wine be sufficient to serve an ordi- nary of lOO men, when the price is ^^30. per tun ; how many will 5^20. worth suffice, when the price is only £24. per tun ? Jlns. 1*2.5 men. (11) A courier makes a journey in 24 days, by travelling 12 hours a day : how many days will he be in going the same journey, travelhng 16 hours a day 1 Jlns, 18 days, (12) How much will line a cloak, which is made of 4 yards of plush, 7 quarters wide, the stuff for the lining being but 3 quarters wide ? Jlns, 9| yards. DIRECT AND INVERSE PROPORTION PROMISCUOUSLY AR- RANGED. (1) If 14 yards of broad cloth cost «^9. 12. what is the purchase of 75 yards ? uins, 5l..8..6|: •^^. (2) If 14 pioneers make a trench in 18 days, in how many days would 34 men make a similar trench ; working in both cases, 12 hours a day ? Jns, 7 daySi 4 ho^irSt 56-,^ minutes. THE DOUBLE RULE OP THREE. 63 oz. with ASSISTANT.] (3) How much must I lend to a friend for 12 months, to requite his kindness in having lent me ;fi64j. for 8 months? ^ Ans, ^4S..13..4. (4.) Bought 59 cwt. 2 qrs. 21 Ih. of tobacco, at £X,\1»A. per cwt. whatdoes it come tol ^ns. ^171.2.. 1. (5) A woollen draper purchased 14?7 yards of broad cloth at I4s. Qd. per yard. Suppose that he sold it in pieces for coats, each 1|: yard, how much must be charge for each, so as to gain ^16..10..9. by the whole ? Ans. j81..9..3|. (6) If ^100. gain £4.1.0. interest in 12 months, what sum will gain the same in 18 months'? Ans, ^66..13..4. (7) A draper having sold 147 yards of cloth, at the rate of ^l..9..3|:. for 1^ yard, found that he had gained ^16.10..9. What did the whole cost him, and how much per yard % Ans. the whole ^10G..11..6. and Us. Qd.per yard. (8) If^lOO in 12 months gain £4..lO. interest, m what time will ^66..13..4 gain the same interest? Ans. 18 months. (9) If a draper bought 147 yards of cloth, at 14s. 6d.^ per yard, and sold it out in pieces for coats, each If yarcl, for ^1..9..3|. ; bow much would he gain per yard, and by the whole ?, Ans 2s. 3d. per yard, ^16..10..9. by the whole. (10) If lcM>f. cost ^I2..i2..6. what must be given for 14 cwt. \qr. \9Jb.? ^ws. £l82..0..11i^. (11) If £100. gain £4.. 10. in 12 months, what mterest will i'375. gain in the same time 1 ^/Ins. «^16..l7..6. ( 12) A regiment of soldiers, consisting of lOOO men, are to have new coats, each to be made of 2i yards of cloth, 5 qviart^rs wide, and to be lined with shalloon of 3 quarters wide. F/ow many yards of shalloon will line them. Ans, 4166 yards, 2 qrs. 2|- nails. THE DOUBLE RULE OF THREE Has^w terms given, three of supposition and two of demand, to find a sixth, in the same proposition with the terms of de- mand, as that of the terms of supposition. It comprises Jwo operations of the SINGLE Rule.— But it may comprise three f^ir ^•.^.nr/.nnornfinnH ofthe Sincle Rule; as there may be 54 DOUBLE RULE OP THREE. [the tutor's seven terms given to find an eighth, or nine to find a tenth, &c. In this respect it is unlimited ; and is therefore more properly called compound proportion. Rule 1. Fut the terms of demand one under another m the third place; the terms of supposition in the same order in the first place ; except that which is of the same nati^re as the required term, which must be in the second place. Examine the statings separately, using the middle term in each, to know if the proportion is direct or inverse. When direct, mark the ^r^f term with an asterisk: when inverse mark the third term. Find the product of the marked terms for a Divisor, and the product of a// the rest for a Dividend: divide, and the quotient will be the answer.* Rule 2. (1) Of the conditional terms, put ilie principal cause of action, gain or loss, &c. in theirs* place. (2) Put that which denotes . time or distance, Ac. in the sc'tont/, and the othei' in the third. (S) Put the terms of demand under the like terms of supposition. (4) If the blank falls in the third place, multiply the first and second terms for a (Utisor, and the other three for a dividend. (5) But if the blank is in ^\it first or second place, divide the product of the rest by the product of the third and fourth terms, for the answer. Note. It will save much labour to write the terms of the Dividend over, and those of the Divisor under a line, like those of a compounrl fraction, and to cancel them accordingly. See Reduction of Vulgar Fractions, Case 6. Proof. By two operations of the Single Rule of Three. (1) If 14< horses eat 56 bushels of oats in 16 days, how many bushels will serve 20 horses for 24 days If (•2) If 8 men in 14 days can mow 112 acres of grass, how many men can mow 2000 acres in ten days 1 Ans, 200 men, (3) If ^100. in 12 months gain jEO. interest, how much will ^75. gain in 9 months? Ans. ^3..7..6. See also Snpplemenial Questions, Nos. 6 and 7. By two single rules. "] or in one stating, worked thus : )u. I " hor, bu. hor. bi hor. days. bu. 1. As U 56 .. 20 . 80 > 14 . 16 . 56 5G X 20 X 24 days. bu. days, bu | 20 . 24 . = 120 2. As 16 80 ..24 . 120 J 14X 16 % ASSISTANT.] PRACTICE. 55 (4) IfjeiOO, in 12 months gain ^6. interest, what prin- cipal will gain £3. .7.. 6. in 9 months? Jns. £lb, (5) If 5^100. gain £Q, interest in 12 months, in what time will £75. gain £3..7..6. interest] Ans. 9 months. (6) If a carrier charges j^2..2. for the carriage oiScwt 150 miles, how much ought he to charge for the carriage of 7 cm;/. 'Sqrs. U lb. 50 miles'? Jns. ^1..16..9. (7) If 40 acres of grass be mown by 8 men in 7 days, how many acres of grass can be mown by 24 men in 28 days'? Jns. 480 (8) If £2, will pay 8 men for 5 day's work, how much will pay 32 men for 24 day's work ? Jns. ^£'38.. 8. (9) If a regiment of soldiers, consisting of 1360 men, con- sume 351 quarters of wheat in lOS days, how much will 11232 soldiers consume in 56 days ? Jns. 1503^^3- 9^^ (10) If 939 horses consume 351 quarters of oats in 168 days, how many horses will consume 1404 quarters in 56 days? ^ns. 11268. (11) If I pay 5g*14..10. for the carriage of 60 cwt. 20 milee, what weight can I have carried 30 miles for £d..S..9. at the same rate ? Jns. 1 5 cwt, (12) U li4 threepenny loaves serve 18 men for 6 days, how mdii\y fourpenny loaves will serve 21 men for 9 days ? Jns. 189. PRACTICE Is so called from its general use among merchants and tradesmen. It is a concise method of computing the value of articles, &.C. by taking aliquot parts. The General Rule is to riippose the price one pound, one shilling or one penny each. Then will the Riven nuniler of article*, consiJ, eiiHJ accordingly hi poumh, or ihWinns, or pence, be the supnomi vulue of the whole; out of which the nlitiuot part or iwla are to be taken for the real price. Note. An aliquot part of a number is such a part a« being taken a certain number of times will produce the number e.rnctly : thus, 4 is aa aliquof. part of 12 ; because 3 fours are 12. 56 PRACTICE. ALIQUOT PARTS. [the tutor's Of a pound, s. d. £. 10 are | 6 8 ... J 5 ... f 4 ... J 5 4 ... § 2 6 ... -5 2 0...,^ 1 8 ... T^ 1 4 ... T^ 1 s ... y, 1 ... 5^ 8 ... gl, 6 ... ,1, 2 qrs, are ^d. 1 gr. is ^d. O/a quarter. lb. qr, 14 are | 7 ... i 4 ... ^ 3,f ... i 2 ... ^, If ... -j^ 1 IS ^g Of an oz. Troy. The same as the parts of a £. changing the names from shillinga to dwts. Of a ton. ctct. ion 10 are | 5 .... f 4 .... ^ 2Sgr.l2/ft.^ 2i .... 1 2 .... -jIj I is ^ Of a lb. oz. lb» 8 are | 4 ... i 2 ... i 1 » Vg Of a dwt, gr. dwt, )2 are i 8 ... } 6 ... f 4 ... I 3 ... 1 2 ... ^ If ... i, I is ^, 0/ rt cvot. qr. lb. civt. 2i or 56 are ^ l,or28 ... i 16 ... 4 14 ... i 8 ... tJj 7 • • ••• IJ O/a ihilling. d. s. 6 .... i 4 .... ' 3 .... 1 2 .... ^ H .... * I .... Vj O/a /6. Troi/. oz. lb. 6 are i 4, &c. as in the parts of a shilling. RuLB 1. When the price is less than a penny, call the given num- J)er pence, an J take the aliqmt parts that are in a penny ; then divide by 12 and 20, to reduce the answer to pounds. (I)iis45704/6.ati 12)1426 2i0)lli8..10. (2) 7695 at ^d. Jns.£5..lS.,W. (3) 5470at 4(/. ^ns. 5^11. .7.. 11. (4^ 6547 at id. Jns,£'i0..9.,\l, (5) 4573 at |c?. Jns. £lit..5..9l. Hur.K 2. When the price is less than a shilling, call the Riven number sfti/Zing'*, take the aliquot part or |>ar/g?is.£45..l2.0|. (^572351 at 2r/. ^n*. ^19.. 11. .10. (G) 7210 Rt-Z^d. Jns,£61.Al.A0^. (7) 27J0at2^J. y//i5. s^'28..4..7. (8)3250at2|t/. ^ns. ^37..4..9|. (y>27l5at3(/. v^ns.e^33..18..9. PRACTICE. (16) 2107 at 4|c?. Ans. ^41..14..0], ("17)3210 at 5c?. " Ans. ^66..17..6. (18)2715 Sitdid. Ans. ^59..7..9|. 67 (3i; 2759 at 8|(/. ^«y. ;g97..14..3^. ~(32) 9872 at 8|X j4ns. £359.A8.A^ (33) 5272Vt 9t/r~ ^«^. £197..14. (19) 3120 at 5^d. Ans. £H.AO. {W) 7521 at 5|t/. ^ns.^l80..3..9|. (21) 3271 at 6d. Ans.£Sl.A5..6 C22) 79l4at6|c?. ^715. ^|^206..1..l0i. (23) 3260 at 6{d. Ans. £8S .0..6. (10)7062at3^(/. Ans. £9o.A'2..7i. (11) 2147 at 3i39i0..8j i£l0..10..8|.^ns. £2 W'^ 5^ PRACTICE. [the tutor's KcleS. When the price is more than one shilling, and less than t,vo, take the part ov parts for the excess above a shilling, add the c^uot.ents to the given quantity, and reduce the whole to pounds as before. Or, when convenient, take the aliquot part of a r^ound *(l)2106at I2|ri. ('i)3715ai l<2\d. (3)27l2atl2lf/. (4) 2107 at Is. Id. (5)32l5at ls:]id. Jns. ^177..9..10| . (0) 2790 at J s. \\d //W5. ^'156 .18.9. (7) 7904 at U.l^d (16) 2915 at Is. 4c^ ^yis. agl94..6..8. (31)2504 at Is. 7|(/. ^ns. ^06..!.. 2. (17) 3270 at Is 4ic?.|(3^^)715^4 at Is. 8rf. ^ws. 5^221..8..1| (18) 7059 at Is. 4id. ^?is.£485..6..1i. (19)2750atls. 4ic/ ^^ws^^l^..l8..e)l (20) 3725 at Is.bd (21)7250 at Is.Sic/ ^«s. jg>596. (33)2905 at Is. 8i8^.,\. .S, (42)21 17at is. lOPf. ^ns. jei98..9. 4f. (43)1007atls.l0irf. ^ns.^95..9..Iirf. (44) 5000 at is. lid. ^ns. .^479..3 .4. 2106i. (175. 6) 48..10i (45)2lo5atls.llic/. ^ns.3e203..1S..5i. This eiample is worked by taking T-i. and then f of that ; because a larlhing is j^ of a shilling; ,»hich IS == Ti of i, or i of ji, herau'o 4 <«cf/res are 48^ ' " ' IE TUTOU'a id less than lling, add tbe pounds us und. ' at Is. 7 id, g^06..K.2, i at Is, Sd. !ns. £596. '} at Is. 8^d. '245..2. .2|. ^at ls.8{d', ratl5.8(/|. >86..16..i. 1 at Is. 9d^ at l5. 9 if/; 'I7..\2. .9i, at 15. 9ic/. at Is. 9icl. g Q..4..7i. ~at 1*. 10 at is. lid. ?479..3 A atls.ll^c/. >3..1S..5i. ! by taking because a ing ; which because 4 > 3 ASSISTANT.] PRACTICE. 59 (46)1006 at ls.llirf.|(47)2705atls.lli. take the tenth, and at 10«. take the hil/oiso many £ (10) 1621 at iSs ^ws. £1458.. 18. Rule 5. When the price is an odd number of shillings, work by Rule 4th. for the greatest even number, and add ^^ of the given quantity fur the odd shilling. — Or, take such pa rts of a pound as will make th« given price. (4) 3214 at 9s. _^7is.^l446..6. (5) 2710 at lis. ^ns.^ 1490.. 10. .(I) 3270 at 3s. w57is.je490..lO. (2) 3271 at 5s. ^ns. £817..15. (3) 2715 at 7s. .dns. Je950..5. (7) 2150 at 15s. »^7is. .£1612..10. (8) 3142 at 17s. _^ns. je2670..14. T9F2r50 at 1 9s. " •^ws. ^2042.. 10. (6) 3179 at 13s. ^ns. £2066 .7- RtJLE 6 When the price consists of shillings and pence, sup{>o!>e the given number to be pounds, an3..\6.Ai. (12)2103 atl5&'.4 id PRACTICE. (I3)7l52atl75.6|l3931..l3.4. (4)7156 at £5..6. ^ns^£31[926 A6. '(5)'27l6at^2.;3..7| Ans, £5911. .3..9. (9)2l54at£7..1..3. ^n5.£l52l2..i2..6. <13)3210ati;i..l8..6|. ^W5.£6l8^9.5.^ (14)2l57at£2.7.4i. Ans.£5\09../..\0i, Cl0,2701at£2..3..4. «^/ig.5g58 52..3..4 . (Il)2715atjei..l7..2i: .^«5. 5051..0..7|. (12)2157at£3:.15.T2|. .^tt5.£8l08..19..5i (lo)l42at£l.i5.'-4. Ans.£^50..2,.6i. (I6;95at£l5.l4.7i. ^»5.£I494 .7..43, (T7)37at£ I. .19.517 *^ns. £73..0..8|. (18)2175at£2..15.T4i. ^n5.£6022p^7|. (b')3^il5at£l..l7. v^?i5.£5947..15. RtJLB. 8 When the given quantity consists of several denomina. lions, multiply the price by the number of the highest and take ali quot parts for the inferior denominations, ^ ' ''^ ^''" (1) AtX.,.17..6. per cwt. what is the value of 250 cw/. a (p's. \4lb. of soap ? t * 4=1 7215 50505 1443 t2gr«i=4|^S..I7..6 J51948 ins. 5X5=25 5 |y6..i7..6 1..18..9 l^ ASSISTANT.] PRACTICE. 61 (i) At £l.,4}.9. per cwt. what is the value of H c^y^ \ qr. lllb.1 Ans. £^\.,IQ.,^^ (3) Sold 85 cwt. 1 qr. 10 /6. of iron, at ^1..7m8. per cu>*. what is the value of the whole ? ^rw. j^I 18..1..0|. (4) If hops are sold at je4..5..8. per cwt. what must be given for 72 cwt. 1 yr.l8 /6. 1 Jins. £310..3..2. (5) What is the value of 27 cwt. 2 qrs. 15 /6. of logwood, at j^l..l..4. per c«>l 1 '^^s. £9„9..9..6i. (6) Bougb^ ^8 cwt. 3 yrs. 12Z.3. of molasses, at £2.J7..9, per c!^/. what must I give for the whole 1 Ans. ^227.. 14. (7) Sold 56 cwt. \qr.\l lb. of sugar, at £2..1 5..9. per cwt. how much is the whole charge % Ans. ^^l 57..'i..4^ (8) What is the value of 97 cwt. 15 lb. of currants, at £3. 17..10. per cwt. 1 Ans. jg'378..0. 3. (9) At ^4..14..6. the cwt, whr.t is the value of 37 cwt, 2 qrs. 13 lb. of raw sugar 1 Ans. £l77..14..8|. (10) Bought sugar at jg3..14..6. the cwt. what did I give for lb cwt. I qr. 10 lb. 1 ^ns. Jg57 .2..9. (11) Required the value of 17 oz. 8 dwts. 18 grs. of gold, at je3..17..10i. per ounce. Ans. ^67-.17..11. (12) At JE37..6..8. per cwt. the value of 1 cvjt. 2 qrs. lOf lb. of cochineal is required. Ans. £b9..\0. (13) Required the value of 13 Mc/f. 42 gaZs. of Champagne wine, at £25..13..6. per hhd. Ans. £350..17..l0. (14) A gentleman purchased at an auction an estate of 149 ff. 3r. 20 p. at ^'54.. 10. per acre. What was the whole purchase money, including the auction d ty of Id. i the £. the attorney's bill for the deeds of conveyance, ^33..6..8. and his surveyor's charge for measuring it, at Is. per acre 1 Ans. £8447..5..0i. RuLB 9. To find the price of 1 lb. at a given number oj shillings Multiply the shillings by 3 and divide the product by 7; the quo- tient will be the price of 1 lb. in farthings.^ (1) What is the price of 1 lb. at 44s. 4 c?. per cwt. ft * Multiplying by 3 reduces the shillings to /(Wirpencf », and 7 four- pences (or2.«. 4(/.) are the value of 1 cwt. at Ifarlhwg per lb. t 44s. 4.'/. 3 7^ 19 fairiiiugs=4i(7. per ia. Ans. 62 TARE AND TllET. [tub tutor's ri) Wliatistlic value of72 7/.A. nt 'ic kj i * . per yard ? ^"** ^L^*' ^c^- anJ fit 14^. 7ul liclli If Tret is allowci?, it it i „f ,i,„ «• "«. ''-'''i^ -Inch .ub,r«ct to find i;^ ^""''^^ ^ -'^ ^« tl.. TARE A^'D TRET. 63 ASSISTANT.] (1) In 7 frails of raisins, each weighing 5 cwt. 2 qrs. 5 lh» gross, tare at 23 lb. per frail, how much neat weight ?* (2) What is the neat weight of 25 hogsheads of tobacco, weighing gross 163 cwU 2 yrs. J 5 lb. tare 100 lb. per hogs- «ead ? ^^,5. 14,1 ^^.^. , y^^ 7 ^^^ (cJ) In 16 bags of pepper, each weighing 85 lb. 4 oz gross, tare per bag, 3 lb. 5 oz. how many pounds neat? /.x Txru - . ^^*- ^311 lb. (4.) What IS the neat weight of 5 hogsheads of tobacco, weighing gross 75 cwt. \ qr. \4 Ih. tare in the whole 752 lb. ? ./Ins. 68 cwt. 2 qrs. \% lb, (D) In 75 barrels of figs, each 2 r/?-*. 27 lb. gross, tare in the whole 597 lb. how much nea weight ? •>?W5. 50 cwt. 1 or. (6] What is the neat weight of 18 butts of currants, each S cwt. 2 yr*. 5 /6. gross, tare at 14. lb. per cwt. ?f (7) In 25 barrels of fi^ ;, each 2 cm;^ 1 yr. gross, tare per cwt, 16 lb. how much neat weight ? . , -/Ins. 48 cwt. gr. U lb. (8) What IS tiie neat weight of 9 hogsheads of sugar, each weighing gross 8 cwt. 3 qrs. u lb. tare 16 /A. per cwt. ,^^ _ •^^5. ()8 cwt. 1 yr. 24. M. (9) In 1 butt of currants, weighing 12 cwt. 2 qrs. 24. lb. cwt. qr. lb. •5. .3.. 5 groH. 2:t (Hie. 6..1..I0 neat of 1 frail. 7 i4w#. 87..1..Uneatof the whole. cwt. qr. lb. t 8. . 2. .5 9X2="18 ;/>. - - ^ 14=.^ 15S..8..6 whole grow. l!)..0..35itare. .In*. 184.278* neat. 64 TARE AND TRET. \l 1 [the tutor's gross, tare 14 lb. per cwt. tret 4 lb. per 104 lb. what is the neat weight 1* (10) In 7 cwt. 3 grs. 27 /6. gross, tare 36 lb. tret according to custom, how many pounds neat 1 .^ns. 826 lb. (11) In 152 cwt, Igr. 3 lb. gross, tare lO lb. per. cwt. tret as usual, how much neat weight? »dns. 133 cwt. 1 qr. 12 /^. (12) What is the neat weight of 3 hogsheads of tobacco, weighing 1 5 cwt. 3 grs. 20 lb. gross, tare 7 lb. per cwt. tret and doff as usual 1\ (13) In 7 hogsheads of tobacco, each weighing gross 5 cwt. 2 g7S. 7 lb. ; tare 8 /i. per cwt. tret and doff as usual, how much neat weight I J^ns. 34 cwf . 2 ^r*. 8 lb. INVOICES, OR BILLS OF PARCELS. (I) Mrs. Bland, London, Sept. 1. 1830. Bought of Jane Harris. 1 5 pairs worsted stockings at 1 doz. thread ditto . at ^ doz. black pilk ditto, at 1 1- doz. milled hose . at 2 doz. cotton ditto . at 17 pairs kid gloves . . at s. 4 3 8 4 7 1 d. 6 per pr, 2 . 3 . 2 . 6 . 8 . £• s. d. s y ^ £21.. 18.. 4 Li i \h, cwt. qr*. lb. • 14-=i 12.. 2..2't RiMii. 1.. 2..10 tare. Ih. 4^^ 11 . 0..14 Butlle. 1..19 tiet. Am. 10.. 2..23 neat. Ih. cwt. qr$, lb. t 7=iij IS.. 3 .20 Rro*. ji..27| tare. 26)14.. f^..20i luttle. 2.. 8 tret. 14. 1..12i 14x2-rS= H cloff. Ana. 11.. 1.. S neat. E tutor's lat is the according 826 lb. r. cwt. tret qr. 12 lb, tobacco, r cwt. tret 088 5 cwt. sual, how assistant.] invoices. (2) Mr. Isaac Pearson, Derby, Bought of John Sims and Son. *. d. 15 yds. satin . . . at 9 6 per yard 18| yds. flowered silk . at 17 4 12 yds. rich brocade . at 19 8 16-J yds. sarcenet . .at 3 2 1 3| yds. Genoa velvet at 27 6 23 yds. lustring . .at 6 3 (3) iMiss Enfield, Nottingham, . Bought of Joseph Thompson. s. d, il yds. cambric . . at 12 6 per yard. 12i yds. muslin . . at 8 3 1 5 yds. printed calico at 5 4 2 doz. napkins . . at 2 3 each . . . 14 ells diaper. . . at 1 7 per ell... 35 ells dowlas . . at 1 1| Received the above. 65 June 3, 1830. £. s. d. 's. 8 lb. 1. 1830. ?• s. d. y ^ /S ^ ^62.11..9| June 4, 1830. C ' ^ i_-^ jei7..14..11 ..18. .4 ) Kroas. ''I tare. i luttle. tret. 4 \ cloff. neat. Joseph Thompson. (4) Mrs. Mary Bright sold to the Right Honorable Lady Anna Maria Lamb, 1 8 yards of French lace at 12s. 3c/. per yd. b pairs of fine kid gloves at 2s 2(/. per pair, 1 dozen I rench fans at 3s. 6(/. each, two superb silk shawls at three gumeas each, 4 dozon Irish lamb at Is. 3r/. per pair, and (> fietaofknot8at2«6(/. perset.—PlcasG to make the Invoice ^"^ k\^ _ Total ammmt £19..U .4. (5) Mr. Thomas Ward sold to James Russell Vernon, Esq. 17| ynrd8 ol fine serge at 3s. M, per yd. 18 yds. of drugget at 9s, per yd. 15| yds. of superfine scarlet at 22s. per yd. 'If ' '( 1 i li ^^ BILLS OP BOOK-DEBTS. [THE TUTOR^S 16| yds. of Yorkshire black at 18*. per yd. 25 yds. of shalloon at Is. 9d. per yd. and 17 yds. of drab at IKi. 6d per yd.— Make an Invoice of these article*. Total amount o£'6o..l0..5^. (6) Mr Samuel Green of Wolverhampton, sent to Messrs. Wright and Johnson, agreeable to order, 27 calf skins at 39. 6(/. each, 75 sheep skins at 1*. Hd. 39 coloured ditto, at 1*. Sd. 15 buck skins at 1 1*. 6d. 17 Russia hides at 10*. Id. and 125 lamb skins at 1*. 2|(/.— Draw up the Invoice. Total amount £39.. 1..8i. (7) Mr Richard Groves sent the following articles to the Rev. Samuel Walsingham ; viz. 2 stones of raw sugar at 6{d. per ft. 2 loaves of sugar, l5i ft. at ll\d per ft- a stone of East India rice at 3\d. per ft. 2 stones Carolina rice at 5d. per ft. 15 oz. nutmegs at ,5ld. per oz. and half a stone of Dutch coffee at I*. lOd. per R).— Make a copy of the Invoice. Total amount £3.,5..5l. BILLS OF BOOK-DEBTS. (8) Mr. Charles Crosa, Chester. To Samuel Grant, and Co., Dr. , 1830. s. d. £, April. 14-. Belfast butter, 1 cwt at 6jperft. ( heese, 7 cwt.Sgrs. 111b. at56 long cwt. May 8. Butter, ^firkin, 2S//>. atO 5| per tb. July 17. SCheshire cheeses, 127M. at Q^ .... Sept. 4.. 1 Stilton ditto. 15 /d. at lOf .... Cream cheese, 13 /*. at 8^ . . . . s. d. i?30..1..6|. .i H Dec. 28. Received the contents, Samuel Grant. ASSISTANT.] SIMPLE INTEREST. 67 (9) Mr. Charles Septimus Tvvigg, Newark. To Isaac Jones, Dr. 1829. s, d. £ s, d. Oct. 22. Tares, 39 bushels at 1 10 per bush. 1830. Pease, 18 bushels at 30 4 per qr. Feb. 18. Malt, 7 qrs. at 63 6 per qr. Hops, 2 cwt. 1 qr. at 1 5 per ft- Feb. 20. Oats, 6 qrs. . at 2 4| per bush. Beans, 17 qrs. at 37 4 per qr. ^S4...9..ll. 1830, July 1. Received the above for Isaac Jones, Thomas West. SIMPLE INTEREST Is the premium allowed for the loan of any sum of money during a given space of time. The principal is the money lent, for which Interest is to be received. The Rate per cent, per annum, is the quantity of Interest (agreed on between the Borrower and the Lender) to be paid for the use of every J^lOO. o( the principal, for one year. The Amount is the principal and Interest added together. 1. To find the Interest of any Sura of J^oney for a Year. Rule. Multiply the Principal bv the Rate per cent, and that Product divided by 100, will give the Interest required. NoTK. When the Rate is an aliquot part or 100, the Interest may be calculated more ezpeditiQUkly by taking such part of the Principal. Thus, for per cent, take ^g ; tor 4 per cent. ^, or 1 of J ; for 2 per cent. ^,; for 2^ per cent. ^; for 8 per cent. ^, }>{»« | ofthat ; &c. This Rule is applied to the calculation of commission, Bro- kerage, Purchasing Stocks, Insurance, Discounting of Bills, &c.* * To discount a Bill of Exchange !• to advance tiie caih for it be- fore it becomei due ; deducting the Interest for the tune it hat to run. Bankeri always charge Discount as the Interat of the mm. 6S I; I! f 't ■ SIMPLE INTEREST. [the tutor's IT. For several years. Multiply the Interest of one year hytiie number or years, and the product will be the answer. For parts of a year, as months and days, kc. the Interest may be found by taking the aliquot parts of a year: or bv ne Rule of Three: and it is customary to allow 12 months to the year, and 30 days to a month f (1) WhatistheiL:erestof£375. for a year, at £5. per cent, per annum ?* j > x, i^^i (2) What is the interest of£945..10. for a year, ntM ^''A?'Ju' '""r. • ^^*- ^37. . 16. . 41 (3) Whatis themterestofje547..l5.ati?5.percentper "'?i!^'XT^'T- ^ns. £82. .3.. -B. (4) What IS the interest of £254. . 17. .6. for 5 years, at i,4. per cent, per annum ? ^ns. £50. .19. .6. Note. Commhsior, and Brokerage Ccommonly called lirohtee^ are allowances of so much per cent, to an agent or broker, JZfngo, selling goods, or transacting business for another. ^ f At the rate of 5 per cent, the interest of ^1. for a year is la : or onep,my for a month., -herefore, the principal X X\it mmber Z mtnths, gives {he interest in pence. ^ numuir q^ :,, ?iV/^' *''^ ^T- '*[,'' ^'''r f«'' »''« »"«»"**. out of « »,fl„y */«7i,»« J» there are /wuwd* in the principal. Thus to find the interest of ^0..10. for 2 months, say 40.i(/. X 2 «)*, 9 At s^l lOj. per cent, what is the purchase of £2054 . . 16. South Sea Stock .' ^ns, £2265. . 8 . . 4-. (9) At £104|. per cent. South Sea annuities, what is the p rchase of £1797. . 14. -? JIns. ^1876..6..l l|. (10) At £96 1 . per cent, what is the purchase of £577.. 19. Bank annuities ? ^ns. £559..3..3|. (1 1 ) -At £124.-5 ..er cent, what is the purchase of £758.. 17..10. India stock 1 Jns, £945..i5..4.i. (12) What sum will purchase £ 1^84. of the 3 per cent. Consol^^ at^59|-. per cent. ; including the broker's charge of I-. or 26. 6t/. per cent, on the amount of stock ? Ans. £770..7..11|. (13) If I employ a broker to buy goods for me, to the amount of £2575..17..6. what is the brokerage at 4*. per cent. ?* (14) What is the broker's charge on a sale amounting to £7105,.5..1O. at 55. 6c?. per cent. ? Ans. ^19..10..9^. ♦4=.^ 2575 .. 17 .. 6 £. SiliJ .. 3 .. 6 20 Oft Ml s. S|OS 1|6S F % = 1 70 SHVIPLE INTEREST. [tHE TUTOH's (15) What is the brokage on goods sold for £9'1!5.,6. 4. at 6s. 6d. per cent. 1 Ans. £^.,Z.A\, (16) What is the interest of £257.. 5.. 1. at j§4. percent, per annum, for a year and three quarters ? Ans. ^^18..0..1|. (17) What is the interest of ^479. 5. for 5| years, at £h. per cent, per annum ? Ans jt^l25,.l6..0|. (18) What is the-amountof ^576..2..7. in 7i years, at jg4|. per cent, per annum ? Arts. £764<..1..8|. (19) What is the interest of i§259..l3..5. for 20 weeks, at i^'S. percent, per annum 1 jlns, £4f..\9..lO^. (20) What is the interest of i?2726..1..4.. at £4{ per cent, per annum, for 3 years, 154 days 1 Ans jg*4l9..l5..6j. (21) Compute the interest of £155, for 49 days, and for 146 days, at £5. per cent, per annum ? Ans.£1..0..9l. and £3..^..0. (22) What will a banker charge for the discount of a bill of ^76.. 10. and another of £^54. negotiated on the 18th of May ; the former becoming due June 30, and the latter July 13 ; discounting at ^5. per cent? /ins. 8s. ild. and Ss. 3d, Wfien the Amounty Time, and Rate per cent are given, to find the Principal, Rule. As the amount of ^lOO. at the rate and for the time given, is to j^lOO., so is the amount given, to the principal required. (23) What principal being put to interest will amount to £a01.,.\0' in 5 years, at 3 per cent per annum If (24) What principal being put to interest for 9 years, will amount to de734..8. at £4. per cent, per annum ? Jlns. £540, (25) What principal being put to interest for 7 years, at ^5. per cent per annum, will amount to ^334. .16. ? An3, .^48. W/ien the Principal, Rate per cent, and Amount are given, to find Vie Time. RuLG. As the interest for 1 year, is to 1 year, so is the whole interest, to the number of yeai^. t,€3x5f,100«iEn5. &. £. £. «. £. At 115 : 100 : ; 402 .. 10 : 850 Am. ASSISTANT.] DISCOUNT. 7l (26) In what time will £350. amount to j^402..10. at ^3. per cent, per annum ?• (<27) In what time will £540. amount to £734..8. at £4, {)er cent, per annum ? Ans, 9 years. (28) In what time will £248. amount to £334.. 16. at £5. per cent, per annum % Jins. 7 years. When the Pri'iu:ipal, Amount^ and Time are given, to find the Rate per cent. Rule. As the principal, is to the whole interest, so is £100. to its interest for the given time. Divide that interest by the number of years, and the quotient will be the rate wer cent. (29) At what rate per cent, will £350. amount to £402.. 10. in .5 years ?t (30) At whatrate per cent, will £248. amount to £334 . 16. in 7 years ? Ans. £5. per cent. (31) At what rate per cent will £540. amount to £734.. 8 in 9 years ? Ans. £4. per cent. V- \ DISCOUNT 7 Is the abatement of so much money, on any sum received before it is due, as the money received, if put to interest, would gain at the rate, and in the time given. Thus £I00. present money would discharge a debt of £105. to be paid a year hence, Discount being made at £5 per cent. RuLB. As £l00. with its interest for the lime given, is to that interest ; so is the sum given, to the Discount required. a, * 850x3 =aj£10..I0. tbe interest for 1 year. ' 100 f 402.. 10 fS50.=,€52..10. the whole intereit. A8JE10..10 : I Mar : : ^52.10 : 5 years. Ant. \ A8;£S50 : i£52.10 : : iClOO : i£15=the interest of jeiOO. for 5 years. Then l6-7-5»=^'S. the rate per ccnu 72 DISCOUNT. Also, As that Jmsunt of ^^lOO. is to ag^lOO. [the tutor's ihe given sum, to the Present worth. But if either the Discount or the Pmew^ worM be found bv the proportion, the other may be found by substracting that koni, aie given sum. ^ (1) What are the discount and present worth of ^^386 5 for 6 months, at £6. per cent, per annum U *"^^»"^- HpS^^''QL"'1"n^'^'^a^ '^'^^'^^ ^^ P''^^^"t payment for a I^'tfr J"""' ^^ ^ "'^"'^'. ^"""^ ' allowing discount at £0. per cut. per annum. Jns. £34i<..ll. 6^ « s (3) What is the discount of ^275..10. for 7 months, at 5ti). percent, per annum? jJns. £7..i6 P ±5 ^1i ^^L'A^^^ P'^'^"* '^^^^ of £527..9..1. payablJin 7 months, at £4| per cent, per annum ? l > "^^ / /K 13 • , , ^w«.i&5I4..l3..lo| ^«J^o . r.iL ToA *^^ P^^e^t worth of £875..5..6.'d^n 5 months, at £4-1 . per cent, per annum 1 months , at .£5. per cent, per annum ? J J. ^4^. ofSr^^i"'! "^"fn '^^l "^°"^J ^"S^»^ I to receive for a note of £75. due m 15 months, at £5. per cent, per annum ? ^ns. dg70..1 1..9|y. * G a).=| iES — d. IOO+3=10S=»amount of ^100. in 6 montlis. £' £. £, 8, As 103 : 8 : : 386.. 5 £ s, 3 386.. &. — ^' 103)1158,.15C "..5 discount. 25 20 375.. present worth. lOS)515c= 5s. COMPOUND INTEREST. 73 ASSISTANT.] (8) What will be the present worth of £l50. payable at 3 instalments of four months ; i. e. one third at 4 months, one third at 8 months, and one third at 12 months, discounting at 5g>5. percent per annum? Jns. £l4<5..S.S\. (9) Of a debt ofJ^blo.AO. one moiety is to be paid in 3 months, an 'le other in 6 months. What discount must be allowed for esent payment, at ^^5. per cent, per annum 1 Am ^i0.11..4|. (!0) What is the present worth of ^500. at £4>. per cent, per annum, ^lOO. being to be paid down, and the rest at two 6 months ? Ans ^488. 7..8I. (11) Bought goods amounting to ^109.. 10- at 6 months' credit, or £3\ per cent, discount for prompt payment. How much ready money will discharge the account ?. Jns. ^105..13..4|. NoTK, The Rule to find the present worth of any sum of monev is precisely identical with that ca?e in Sinople Interest in which the Antount, Time, and Rate per cent, are given to find the Principal. See page 70. COMPOUND INTEREST Is that which arrises from both the Principal and Interest : that is, when the Interest of money, having become due, and not being paid, is added to the Principal, and the subsequent Interest is computed on the Jmount. Rule. Compute the first year's interest, which add to the principal : then find the inter-st of that amount, which add as before, and so on for the number of years. Subtract the given sunt '. om the last Amount and the remainder will be the Compovnd Interest. * The discount in cases of this sort is lo much per cent, on the sum, without regard to lioie. It is, theiefore, computed as a year's intereit. 74 EQUATION OF PAYMENTS. [tHE TUTORS ( 1 ) Wliat is the compound interest of j£'500. forborne 3 years, at £5. per cent, per annum ?t (2) What is the amount of 5^400. in 3f years, at ^5, })ercent. per annum, compound interest? Jlns. £414<..1%S^. (3) What will £650. amount to in 5 years, at ^5. per cent, per annum, compound interest? ^ns. ^S29., 11. .7^, (4) What is the amount of ^550. .10. for 3^ years, ai^6. per cent, per annum, compound interest ? /Ins. ^675. .6.. 5. (6) What is the compound interest of ^754. fo- 4 years and 9 months, at £6. per cent per annum ? Ms, ^2i'3..l8..S. (6) What is the compound interest of £57..10..6. for ft years, 7 months, and 1 5 days, at -^5. per cent, per annum ? ^ns. £18..3..S^. (7) What is the compound interest of ^259.. 10. forS years years, 9 months, and 10 days, at ^'4'f . per cent, per annum ? ^ns. £46 .\ 9.. \0^. EQUATION OF PAYMENTS. Is when c^verai sums are due at different times, to find a mean time for paying the whole debt ; to do which this is the common Rule. Multiply each term by its time, and divide the sum of the products by the whole debt ; the quotient is accounted the mean time. ' t k ^500 25 .^ 525 amount to 1 yr. 26.. 5 551. .5 do. io 2 yrs. 27..11. S 578.. 16.. 3 amount in 8 years. 500 . CO principal subtract. £7B..l6. 3 Ans. ASSISTANT.] BARTER. 75 (I) A owes B £200. whereof £40. is to be paid at 3 months, ^60. a I i; u^ iiths, and £lOO. at 10 r niL:r. at what time may the w' oii^ *3bt be paid together, vvltht..: prej idice to either? 40X 3= 120 60 X 5= 300 100X10=1000 2i00)l4i20 Ans. 7f^ months. (2 r .v/es C £800. whereof £2o0. is to be paid at 3 months, £l00. at 4 months, ^300. at 5 months, and *^200. at 6 months ; but they agree that the whole shall be paid at once ; what is the equated time ? Jus, 4 months^ 18|- days, (3) A debt of £360. was to have been paid as follows : vh. ^120. at 2 months, £200. at 4 months, and the rest at 5 months ; but the parties have agreed to have it paid at one mean time ; what is the time ? Jlns, 2 months, \^ 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 monthe, and £250. at the end of 12 months ; but it was afterwards agreed to discharge the debt at one payment : rer lired the t™e. . jins, 8monthSt ^-idays, (5) H is indebted to L a certain sum, which is to be paid at 6 different payments, that is | at 2 months, -^ at 3 months, ■^ at 4 months, 4: at 5 months, | at 6 months, and the rest at 7 months ; but they mutually agr-^^ that the whole shall be paid at one equated time : what is that time 1 Ans. 4i months. (6) A is indebted to B ^120. whereof^ is to be paid at 3 months, | at 6 months, and the rest at 9 months : what is the equated time of the whole payment 1 Jns, 5^ month^. BARTER Is the exchange of commodities. Rule. Compute, by the most expeditious method, the value of the article whose quantity is given : then find what quantity of tl^e other, at the rate proposed, may be had for the same money. ^,.**^ • 76 BARTER. [the TUTOR's Note. Sometimes one tradesman, in bartering, advances bis goods above the ready money price. In this case, it will be necessary to pro- Ru'KTliree" "'***' ^"'''^^''" ''^ '*'* '^''y '"oney price, by the (1) What quantity of chocolate at 4s. per lb, must be ex- changed for 2 cwt. of tea, at 9s. per/6. ?• (2) A and B barter: A has 20 cwt, of prunes, at4rf per lb. ready money, but in barter will have bd. per lb. and B has hop« worth 32s. per cwt. ready money : what ought B to charge his hops, ard what quantity must he give for the 20 cwt. of prunes ?t (3) How much tea at 9s. per lb, can I have in barter for 4 cwi. 2 qrs. of chocolate, at 4s. per lb. 1 ,^ns. 2 cwt, (4) A exchanges with B 23i cwt, of cheese, worth 52s 6c/. per cwt. ior 8 pieces of cloth containing 248 yards, at 4s. ^d. per yard ; the difference to be paid in money. Who receives the balance, and how much ? Ans. A receives £7.19.1. r ^f? ,?^^ '"'•^^^^ ginger at \bid. per /d. must be exchanged for ^ Ib^ of pepper, at 1 3{d. per lb, ? .^ws. 3 lb. 1 |f oz. (6) How many dozen of candles, at 5s. Id. per dozen, must be bartered for 3 CMJ^ ^ qrs. 16 lb. of tallow, at 37s. 4c/. ^'^^' ^ , ^:e^. 26 dozen, 3|-| /6. (7) A exchanges with B 608 yards of cloth,* worth 14s. per yard, for 85 ewt. 2 qrs. 24. lb, of bcos' wax, and 5g'125..12; In cash. What was the wax charged per cwt. 1 Ans. ^"3 . 10. • 224X9=2016*. the value of the tea. As4«. : lib. : : 2016». : 504 /ft. of chocolate. Ans. t As 4(i. : Stock wa^ £100 ; D's gain was je3 ; E's £5 j and F's £8 ; what was ;Mch man's stock? ^ns. D's stock £18.. 1 5. E's £3 1 ..5. and £F's 60. * The «rtle nf « propei t y for fo manu years^ purchase. Is understood to he, for so much present money m tku annual rent or tuIuc ^ thj^t number of ye.nry. days ASSISTANT,] FELLOWSHIP, 81 the'"' FELLOWSHIP WITH TIME. Rule. As the sum of the products of each person's money and time, is to the whole gain or loss ; so is each individual product, to the corresponding gain or loss. (1) D and E enter into partnership; D puts in j^40. for three months, and E £75. for four months, and they gain £10. What is each man's share of the gain 1\ (2) Three tradesmen joined in company: D put into the joint stock £195..14. for three months; E £l69..18.3. for five months ; and F i^59,.l4..10. for eleven months : they gained ^364. .18. What is each man's share of the gain ? ^ns. D's £102..6..4— 5008. E's i:i48..1.. 11—482802. andF's £114. .10..6|— 14707. (3) Three merchants joined in company for 18 months : D puts in ^5oo. and at 5 months' end takes out jg^OO. at 10 months' end puts in £300. and at the end cf 14 months takes out £1 30 ; 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 theendof 12 months, and withdraws £99. at the end of 1 5 months; F puts in £900. and at six months takes out £200. at the end of 11 months puts in £5C0. but lakes out that and £100. more at the end of 13 months. They gain £200, Required each mans share of the gain 1 Ans. D £50..7..6— 21720. E £62,.12..5J— 29859. andF £87..0..0i— 14167. (4) D, E, and F, hold a piece of ground in couimon, 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£I3.,3,. 1^—624. E £15.. 11. .5— 1688. «»c/F£7..15..11— 1136. t 40x3=120 75X4=800 Ai 42j0 : 7i0 : . 5120 : • |800 : £20=D'i slmre. 60— E'i share. 420 70 Proof. ' ■■ -»«-■ 2 ■~ fi i( :'l 82 I ALLIGATION. ALLIGATION the [the tutor's rule by which we ascertain the mean ^ .^ pound formed by mixing ingredients of various prices ; or the quantities of the various articles which will forma mixture ut* a certain mean or average value. It comprises four distinct cases. Case 1. Alligation Medial. The various quantities and prices being given, to find the mean price of the mixture. Rule. Multiply each quantity by its price, and divide the sum of the products by the sum of the quantities.* (1) A grocer mixed 4 cwt. of sugar, at 565. per cwt. with 7 fwt. at 43*. per cwt. and 5 cwt. at 375. per cwt. What is the value of 1 cwt. of this mixture? Jns. £2..4..4i. (2) A vinter mixes 15 gallons of Canary, at 8^. per gallon, with 20 gallons, at 7*. 4(/. per gallon ; 10 gallons of sherry, at C)s. Sd. per gallon ; and 24 gallons of white wine, at 4s. per gallon. What is the worth of a gallon of this mixture ? Jlns. 6s. ^ j^. (3) A malster mixes 30 quarters of brown malt, at 28*. !)er quarter, with 46 quarters of pale, at. 3r5. per quarter, and '24 quarters of high dried ditto, at 25*, per quarter. What is one quarter of tlie mixture worth ■? Ans.^l . 8 . 2 i ^-d. (4) A vinterrilxes20 quarts of port, at 5^. ^d. per' quart, with 12 quarts of white wine, at 55. per quart, 30 quarts of Lisbon, at 65. per quart, and 20 quarts of mountain, at 45. 6r/. per quart. What is a quart of this mixture worth ? Jlns. 55. 3| |§ri. Example. bushrlt of s. s. 36 buviiels 20x5 = 100 with 40 36x2 = 108 per bushel. 40x2 =: 80 el of this — , ^ 96 i'6)288(S*. yina ASSISTANT.] ALLIGATION. 83 rices : or (5) A refiner melts 12 /d. of silver bullion, of 6 02. fine, with 8 lb. of 7 oz. fine, and 10 lb. of 8 oz, fine ; required the fineness of 1 lb. of that mixture ] ^ns. 6 oz. 18 dwt. 16 gi\ Case 2. Allegation Alternate. The various prices being given, to find the quantities which may be mixed, to bear a certain average price. Rule. Arrange the given prices in one column, with the proposed average price on the lef^. Link each less than the average with one greater. Place against each term the difference between that with which it is linked and the mean ; and the respective differ- ences will be the quantities required. Note. Questions in tiiis rule admit of a great variety of answers, according to the manner of linking them : also by taking ottier numbers proportional to the ansvcers found. ( 1 ) A vinter would mix four sorts of wine together, of 1 8(/. 20c?. 24(/. and 28.-> 18- 20_ 24- 28 or thus, Pro,,/. Cof la/. = 108d. 2 of 20(/. = 40(/. 2 of 3.1*/. = 48(i. 4 of 2bi/. = n2iL 14 14)308 22. lb, 200(1. Ai 6 : 1 : : 20 ; 4 64(/. 72(/. As S : 2 : : 20 : a 176/. 61 2(/. : : 1/6. ; i7(/. EIE tutor's ASSISTANT.] COMPARISON OF WEIGHTS AND ME /VSURES. 85 (1) A grocer has four sorts of sugar at 12c?. IQd. 6d and 4c?, per lb. and would make a composition of 144 lb. wortii 8d. per lb. What quantity of each sort must he take ?* (2) A grocer having 4 sorts of tea at 5s. 6s. 8s. and 9s. per lb. would have a composition of 87 lb. worth 7s. per lb. What quantity must there be of each sort 1 Ans. \^ lb. of 5s. 29 lb. of 6s. 29 lb. ofSs. and l^ lb. of 9s. (3) A vinter haviiig 4 sorts of wine, viz. white wine at l6s. per gallon, Flemish at 24s. per gallon. Malaga at 32s. per gallon, and Canary at 40s. per gallon j .vould make a mixture of GO gallons worth 20s. per gallon. What quantity of each sort must he take ? Ans. 45 gallons of white wine, 5 of FlemUh, 5 of Malaga, and 5 of Canary. (4) A jeweller would melt together four sorts of gold, of 24, 22, 20, and 1 5 carats fine, so as to produce a compound of 42 oz, of 17 carats fine. How much must he take of each sort 1 Ans. 4 oz, of 24, 4 oz. of 22, 4 oz. of 20, and 30 oz. of 15 carats fine. COMPARISON OF WEIGHTS AND MEASURES. This is merely an application of the Rule of Proportion. ( 1 ) If 50 Dutch pence he worth 05 French pence, liow many Dutch pence are equal to 350 French pence rf (2) If 12 yards at London make 8 ells at Paris, how many ells at Paris, will make 64 yards at London 1 Ans. 42|. *12 10 4. Anm-cr. 4 48 at 12 "2 24 at 10 2 24 at 6 4 48 at 4 Pronf. lb. Ih. 576 As 12 : 4 : : 144 : 48 240 As 12 : 2 : : 144 : 24 , 144 192 Sum 12 144 144}1152(8rf. t As 6;i or, at IS 50 10 3500 S50 : — 13 =2(J9|j. Ans. 86 VULGAR FRACTIONS. [tHE TUTOR's (3) If 30 lb. at London make 28 lb. at Amsterdam, how many tb. at London will be equal to 350 lb. at Amsterdam ? Ans, yib. (4) If 95 /6. Flemish make 100 lb. English, how many lb. English are equal to 275 lb lemish ? Ans, 289|^ ii il PERMUTATION la the changing or varying of the order of things. To find the number of chang'is that 7nay be made in the position of any given number of things. Rule. Multiply the numbers 1, 2, 3, 4, &c. continually together, to the given number of terms, and the last product will be the answer (1 ) How many changes may be rung upon 12 bells ; and in what time would they be rung, at the rate of lO changes in a minute, and reckoning the year to contain 365 days, 6 hours ? 1x2x3x4x5X67x8 X 9XlOXIlx 12=479001600 changes, which -^ 10 = 47900160 minutes=9l years, 20 days, 6 hours. (2) A young scholar, coming to town for the convenience of a good library, made a bargain with the person with whom he lodged, to give him £40. for his board and lodging, during 80 long a fane as he could place the family (consisting of 6 persons besides himself ) in different positions, every day at dinner. How long might he stay for his £4.0. 1 Ans. 5040 days. >'. u VULGAR FRACTIONS. DEFINITIONS. 1 . A Fraction is a part or parts of a unit, or of any whole number or quantity ; and is expressed by two numbers, called the terms, with a line between them. VULGAR FRACTIONS. 87 ASSISTANT.] ?. The upper term is called the Numerator, and the lorver term the Denominator, The Denominator shows into how many equa parts unity is divided ; and the Numerator is tlie number of those equal parts sigtiified by the Fraction » • .uV. ^'■^^^'«" ™ay be understood to represent Divi- 7^ di^s^T''^^''' ^^'"^ '^^ '^''''''^''''^' "'"'^ '^^ Denominator, Fractions are distinguished as follows : 4. A Simple Fraction consists of otic numerator ^m\ one denominator : as i, 1^, Sac. c«'m c/»c 5. A Compound Fraction, or fraction of a fraction, consists of two or more fractions connected by the word ./.' "eUe^arfrSiot; ^'" ^"^^"^^ '^"^^^^ ^^^ ^^^-^ ^^ ^;. ^ Proper Fraction, is one which has the numerator /mthai the denominator: as J,|,, i, i*., &c4 "'"^^'^^^^'^ 7. An Improper Fraction' is one which 'has the nume- rator cither egual to, or greater than the denominator : as 4 6. A Mixed Numbrr is composed of a whole number and a fraction, as H, Hi, 8||. &c. ""^^ 9. A Complex Fraction has a fractional numerator or * In thefracUof^Jire-tweirths (^) the Damminator 12 shows that th^ unil oryhole quantity is supposed i» be divided into 12 equal r)«if. • . that .fU be one shiliii,^, each part will be one-twelfth^"? j,lr on;^ penny The A«^,m,for shows that 5 is the number of those twelm! parts intended to be taken ; so f, of a shilling are the san^e as 5 p^t fj of a foot, the same as 5 inches. pcme , t The fraction ^5 signifies not only « of a unit, but 5 units divHp^ into 12 parts or a twelfth part of 5 : 'and it is obvious thatX^/A /jar *of on.sh.ll.n.. (or five pence) is the same as o„e twrnZf^r. ^Y^ualto «„,7,when the tenns are%,al, and ^r.«,"r S T«^' ^^^ the»?tt»jfra/oris the/j-rffl/er. """y wneu Thus |, or ||, or ?;|, is each =1 ; and 5=]|, |=2, ^^l-^g^j. IMAGE EVAIUATION reST TARGET (MT-S) ^ /. {•/ ^ ///// ^^ 4y ^^ w p mi.r t/'. m 1.0 I.I us Ut Km mil 2.0 M Li 1.25 1.4 11.6 1= = < 6" -- ► '^4 /,. '/J M Photographic Sciences Corporation as WIST MAIN STRUT WliSTIR.N.Y. 14510 (7)6) •73-4503 Zt ^. isSs ^^> ^ ^B o \ ^ I 88 REDUCTION. [the tutor's denominator : but this denotes Division of Fractions, Tiius, 8 — , two-thirds divided by five-sixth, — , tight divided by one H and two.thirds. 10. A Common Measure (or Divisor) is a number that will exactly divide both the ter?ns. When it is the greateat number by which they are both divisible, it is called the Greatest Common Measure. NoTF. A. prime number has no factor, except itself and unity. A multiple signifies any product of a number ; and is therefore divisible jy the number of which it is a multiple : thus 14, 2], 28. &c. are multiples (if seven. Also 14 is a multiple of 2 and 7 ; 21, of 3 and 7, &c. hi REDUCTION Is the method of changing the form of fractional numbers or quantities, without altering the value. Case 1. To reduce a fraction to its lowest terms. Rule. Divide both the terms by any common measure that can be discovered by inspection ; which will produce an equivalent fraction in lower terms. Treat the new fraction in a similar manner ; repeating the operation till the lowest terms are obtained.* When the object cannot be accomplished by this process, flivide the greater term hy the less, and that divisor by the remainder, and so on till nothing remains. The last divisor ♦ This first method of nhhmiating fraction* is, when practicBble, always to be preferred : and in ih» application of it, the following ob- (•ervations will be found exceedingly useful. An even number '.- divisible by 2- A number is divisible by 4, when the t number divisible by 3)3, 7» 4, 9, 5 ali the giveu nuiubeii. 1,7,4,3, 5 f 2 X 7 = 147 4X4 = 16 J numerators. Ans. y and i^. 4 X 7 « 28 the denomioator. v~ ASSISTANT.] SUBTRACTION. 95 (3) Reduce g, i, i§, and §, to a common denominator. /J'»Q 7?^ ^H ^o* rrnA 7?" »/ini>' a405 SllJj 8»0j t*im D40. (4) Reduce |, ^, and |, to a common denominator. (5) Reduce ti, 7, and — of 2, to a common denominator. 15 Jns. M, i?f°, awfl? T^2?. (6) Reduce 11,24, and ^ ^^ Uj ^^ a common denomi- nator, yins. U) % and 3g. ADDITION. Rule. Reduce the given fractions to a common denomina- tor, over whicii place the sum of the numerators. (1) Add I and *. together. itf=i^+i|-=||=l-|T- ^ns. (^) Add I, f, and f . (3) Add A, 4Und^.* (4) Add 7| and f togetlier. (5) Addf., and I of I . (6) Add 5|, 6|, and4i. (7)Addl|, 3|, andiofT. (8) Add fa of 6| and 4 of 7f (9) Add i of 9|- and I of 45. Fractional quantities may be reduced to iheirproper values, and the sum found by Compound Addition. ( 10) Add I of a pound to -*- of a shilling. Jns. Ss. 4d. ( 1 1) Add {d. 1$. and £1. .ins. 14s. (12) Add I lb. troy, ^ 02. and |- oz. Ans, 7 oz. 1 9. duis. 20 gr. (13) Add I of a ton to | of a cwi. Jlns. 12 cw/. 1 yr. I^IO. (14) What the sum of |. o{ £17 . .1 . .6d., | of j^ If, and I of a crown 1 ^n*. ^j^-^lS. . 0. . 2^. (15) Add f of 3 rt. 1 r. '20 p.^ f of an acre, and | of 3 roods, 1 r? perchea. .^^/ts, 3 « . 2 r, 33^ />. SUBTRACTION. Rule. Reduce the given fractions to a common denomina- tor, over which place the difference of the numerators. * When there are integers among the given nunibert, first find the $um 0/ the fractions, to which add the integers. Thus iu l-;x. 3, i+g-3; then 3+i=fj+/5=|J ; and H{J=4y. Ans. 96 VULGAR FRACTIONS. [tHE TUTOH's When the numerator of tlie fractional part in the subtrahend >s greater than the other numerator, borrow a fraction egml U) umty, having the common denominator ; then subtract, and r.arry one to the integer of the subtrahend. (I) From I taice |. |--.5^.j<_^o^^.^. jir^s, (•2) From |. take |. (G) From G4.. 19.. \t (4 If I lb. cost I*, how much will ^s. buy 1 Ans. Ih (6) If 48 men can buihl a wall in ^24| days, how many men can do the same in 1 92 days 1 Ans, Cia men. (G) If I of a yard of Holland cost £^i what will 12§ ells cost at the same rate ? Ans. £1..0..H\' (7) If 31 yards of cloth, that is 1 J yard wide, be sufficient to make a cloak, how much that is f ofa yard wide, v/ill make another of the same size .' Ans. 4-J- yards. * A number ti.rcr/fd becomes the reciprocal of that nu tiber j which is file quotient arising from dividing uni/i/ b) the Riven number: thui l-r7»=^, the reciprocal of 7; l-r2=a' ^^ciprocul of 2- I 98 THE DOUBLE RULE OF THREE. [tHE TUTOR's (8) If 12{ yards of cloth cost 15*. 9d. what will 48^ yards cost at the same rate ? Jjis. £3. .0..9^ h. :9) If '25fS. will pay for the carriage of 1 cwt.,U5^ miles, how far mry 6| cwt. be carried for the same money ? j4ns. 'i^^it miles, (10) U% oi cwt. cost £u.As. what isthealue of 71 ""^l U.J. ^ ,• ^ns. £^U8..^..8.' (lO lt + /6. ot cochineal cost £l..5. what will 36,' Ih. come to 1 ^^g^ £61,. 3. .4. (12) How much in length that is 7^a inches broad, will Jns. 204. inches. make a foot square ? _^, (13) What is the value of 4 pieces of broad doth" each '27 j yards at i 5^s. per yard ? Jns. £85.. 1 4..3 1 f. (14) If a penny white loaf weigh 7 g^. when a bushel of wheat costs D5. bU what is the bushel worth when a penny white loaf weighs but '21 02:. ? yins. I5s. 4d. 3l grs. (15) What quantity of shaloon th t is ^ of a yard wide will line 7-,- yards of cloth, that is 1{ yard wide 1 Ans. 15 yardi. (16) Bought ^ pieces of silk, each containing 24-3- q\\^^ ^^ 6s. Old per ell. How must I sell it per yard, to ^ain £b bv the bargain? .^ns. 5*. 9ic/. |||. THE DOUBLE RULE OF THREE. (1) If a carrier receive ^2i^. for the carriage of 3 cwt. 150 miles, how much ought he to receive for the carriage of 7 cwt. '6\ qrs. 50 miles ? Ans. £1..16..9. (2) It £100. in 12 months gain £b\. interest, what princi- pal will gam £3|-. in 9 months ? Ans. £Sb..U. 31 s (3) If 9 students spend ^lOf in IS days, how much will 20 students spend in 30 days ? Ans. ,:;£^39,.18. 412. (4) Two persons earned 4 |s. for one day's labour :' how much would 5 persons earn in 10 1 days, at the same rate ? .,,/•/! Ans.£Q..\.Ai\. (^) If ^50. in 5 months gain £'l\^. what time will £134-. "^^f J|;^t? fin a^l \-^. ? Am. 9 inmths. ^ (b) It the carriage of 60 cwt.^ 20 miles, cost ^141. what weight can I hav carried 3o miles for ^5i^. ? Ans. 1 5 cwt. tutor's 8 1 yards miles. g miles. le of 7| 8.. 6. .8. 36/, Ih. )a(J, will inches. :h, each 31 5 lushel of a penny grs. nde will 5 yards. ells, at ^5. by 3 6q T5T* 3 cu'^ riage of ,6..9. princi- 1 ^1 5 ich will 420 : how ite? HI ai/cs. \. what \5cwt. ASSISTANT.] DECIM.^L FRACTIONS. DECLMAL FRACTIONS. 99 In Decimal Fractions the nnit is suppcied to uC divided into tenths, hundredths, thousandth parts, &c. consequently the denominator is always 10, or 100, or rjOO, &c. In our System of Notation, the figures of a whole member follow each other in a decimal ( or tenfold ) proportion. Hence, the numerator of a decimal Fraction is written as a whole number, only distinguished by a separating pcint pie- jixed to it. Thus -5 for f^, -25 for f A_, -123 for 1-^4^. The denominator is, therefore, not expressed ; being al- ways understood to be 1 , with as many ciphers affixed, as there are places in the numerator. The diiferent values of figures will be evident in the an- nexed Tab'i'^. Integers. 7 fj .5 4 3 2 ^ a^ H H a H d _. C _ X <- -< "3 Decimal parts, 1 . 2 3 4^ 5 G 7, &c. o 3 CO S o O OJ - 3 3 - 3 C- "I Cu ^ — 09 2- = 2 == 3 S ^^ =^- g^ ^ JT- OS -, >:3 00 » .- 2. From this it plainly appears that the figures of the r/m?7w/ fraction decrease successively from left to right in a tenfold proportion, precisely as those of the whole number.* Ciphers on the right of other decimals do not alter their value: for •2=.f^,.20=^§„ .2()0=-^-".O-, are all equal. But one cipher on the left diminishes the value ten times^ two ciphers, one hundred times, &c. for -02=^%^, •002=7^^,&c. • The first, second, thin', fourth, &c. places of decimals are called primes, seconds, thirds rourlhs, &c. respectively; and .iecimals are read thus: 67- £7 fifty-seven, and fire, snun, of a decimal; thut is fiftr- seven, and fifty-seven hundredths, 206-043 two hundred and sit, aiid mughl^four, three t that is, 20G, and foity-three thousandth*. i 'A 100 DECIMALS. [the TUTOR's A vulgar fraction having a denominator compounded of % or 5, or of both, when converted into its equivalent decimal fraction, will be finite : that is, will terminate at some certain number of pi jes. All others are injiniie ; and because they have one or more figures continually repeated without end, they are called Circulating Decimals. Tlie rppeating figures are called repetends. One repeating figm-e is called a single repetend ; as -522, &c. ; generally written thus, -2/ ; But when more than one repeat, the decimal is a compound repetend ; as -36 S6, &c., or -142857 142857, &c. Pure repetends consist of the repeating figures alone ; but mixed repetends have other figures before the circulating decimal begins: as 045', -90^354'. • , ,. Finite decimals may he considered as mfinite, by making riphers to recur, which do not alter the value. Circulating decimals having the same number of repeating /liTwres are called similar repetends, and those which have an 'unequal number are dissimilar. Similar and contermin&Ui repetends begin and terminate at the same places. ADDITION. Rule. Place the numbers so that the decimal points may stand in a perpendicular line: then will units be under units, kc. according to their respective values. Then add as in integers. (1) Add 7'2-54.32071 4.2-1 67^.371'44.9-75. (2) Add 30-074-2-0071 + 69-4.32+7-1. (3) Add 3-5+4-7 2.5+927-014.2 0073+ 1-5. SUBTRACTION. Rui.v;. Plnce tho subtrahend under the minuend with the decimal points as in Addition ; and subtract as in integers. (1) From •27r,nake -2371. (5) From 571 take 54-72. U) From 2-37 take I 70. (6) From G25 take 76-91. (3) Frcm 27l take 215-7. (7) From 23-415 take -3742. (4) From 270-2 take 75-407.5. (6) From -107 take -0007. li E tutor's ASSISTANT.] DECIMALS. MULTIPLICATION. 101 Rule. Place the factors, and multiply them, as in whole numbers ; and in the product point off as many decimal places as there are in both factors together. When there are not so many figures in the product, supply the defect with ciphers on the left. (1) Multiply 2 071 bv 2-27. | (7) (2) Multiply 27- i 5 by 24-3.^^ I (8) (3) Multiply -2365 by -2435. } (9) (4) Multiply 72347 by 23-15. (10) (5) Multiply 17 1 05 by -3257. (11) 27 •35x7-700 11. 57-21 X -0075. -007X.007. 20'15X'2705. •907x-0025. (6) Multiply 17105 by -0327. (12) •3409&03x'00162l8. When the multiplier is 10, 100, 1000, &c. it ii only removing (he separating point in the multiplicand so many places towards the right at there are ciphers in the multiplier : CONTRACTED MULTIPLICATION. RcLB. Write the multiplier under the multiplicand in an inverted order, the unyillbeofthe8ame value as that figure of the dividend which stands ot^gr Jte units in the first product of the divisor: so that the point must be placed accordingly ; ciphers being prehxeu, when necessary. * Note I After proceeding throuj;!, the dividend, to ascertain if tl.e quotient ..correctly pomtcd, ob.eiTe that the dJcimal p aci" i h! dmjor Hnd quotient together, must equal in number those of the dj.i! 2. When there are fewer decimal places in the dividend than in the tion to any degree of exactness desired. '•""niiue uie o|.eia. (1) Divide 217-75 by 65. (2) Divide 709 by 2-574 (7) 7382-54 -r 6-4252. (8) -0851648-7-423. Contracted method. U84G72158 886 2 II 5401 G5 SS08088 807788 11540 1416-747'6 Common method, 3841)72158 8G83 11540 307737 23080X2 1154016474 16474 7264 948 1416-7475 57914 if ASSISTANT.] DECIMALS. 103 (3) Divide 125 by -1045 (4) Divide 48 by 144. (5) Divide 5-7U by 8275. (6) Divide 715 by 3075. To divide by 10, 100, 1000, &c. diiridend so many places towards divisor, and the thing is accomplished * Thus 5784 -r 10 := 578-4, 5784 5-784, 5784 -J- 10000 = -5784. (13) 3719 -r 10. I (15) (14) 3-74 •?• 100. (16) (9) 267^6975 -r- 13-25. (10) 72-1564 -r -1347. (11) 85643-825 -r- 6-321. (12) 1-T- 3-1416. remove the separating point in the the left, R8 there are ciphers in the 100 = 57-84, 5784 -r 1000= 130-7 -f- 34-012 -r 1000. 1 0000. CONTRACTED DIVISION. Ascertain the value of the first quotient tigure : from which it will be known what number of figures in the quotient will serve the purpose icquired. Use that tiumbev of the figures in the divisor, (rejecting tlie others on the ri(>ht) and a sufftcient number cf the dividend, to find the first figure of the quotient; make each remainder a new dividual, and for each succeeding figure reject another from the divisor: but observe to carry to each product from the rejected figures as in Contracted Multiplication. Note. When there are feiver fi){ures in the divisor than the number wHHted in the quotient, proceed by the common rule till those in the divisor are just aa many as remain lO be found in the quotient, and then use the contraction. ( 17) Divide 70*23 by 7-9863, to three places of decimals.* .... * Contracted Method, 7-9863)70-230(8-793 G3890 Common Method. 7-9863)70 •2S00(8798 638904 0840 5590 750 719 6839 60 5690 41 749 718 190 767 81 S4 80!42S0 239589 6:4641 1 m 104 REDUCTION OP DECIMALS. [the TUTOr's (18) Divide 721.17562 by 2-257432, to the extentofonly three places of decimals in the quotient ^cn^ fv'^-t ??*}^f ^y ^^^•^^' ^« *h^ ^«"rth decimal. ^9? ^- /oi'VJi^^V^^^^^^^ to the second decimal. (21) Divide 27.104 by -3712, the integral quotient only, REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to a Decimal, Rule. Add ciphers to the numerator, and divide bv the denominator, the quotient is the decimal fraction required! EXAMPLES. I . Reduce I . . . . to a decimal. 4)1,00(,25 Facit. Z S^^"^® f to a decimal. Facit, ,5. 3. Reduce i to a decimal. Facit, ,73. 4. Reduce | to a decimal. Facit, ',375. 0. Keduce ^5- . . to a decimal. Facit, , 1 923076 + b. Keduce i-| offo . . to a decimal. Facit, ,6043956+. Note. If the giren parts arc of several denominations, li.ev D>av be reduced either by so many distinct operation, as there are difrere„"Zu divKrbe'/^ier^of/'^'" ""^ ''-'' '--' '^--ination, and^n 2udly. Bring the lowest into decimals of the next superior denomj- nation, and on the right band of the decimal found, Jiace the ua^ ^ given of the next superior denomination ; so proceeding till you brioir out the decimal pBrts of the highest integer required, bj still dividnf the product by the next superior denominator ; or, ** 3dly. To reduce shillinss, pence, and farthings. |fth« number of shillings be even, take hah for the first place of decimals, and let the second and third places be filled with the farthings contained in the remaining pence and farthings, always remembering to add 1. when th^ number is, or exceeds 25. But if the number of shillings b^ odd the second place of decimals must be increased by 5. 7. Reduce 5s. to the decimal of a £, Facit 25. S. Reduce 9s. to tlie decimal of a £. Facit,' *45.* 9. Reduce 16«. to the decimal of a £. Facit,* 8* U). Reduce 8s. 4d to the decimal of a £. Facit, ,4166*. 11. Reduce 16s. 7id. to the decimal of a £, Facit, ,8322910. A8SISTANT.J REDUCTION OP DECIMALS. second. third, 4)3,00 2)16 12)7,75 ,832 210)16,64583 first. If^s. 7|d. 199 4f 960)799(8322916 ,8322916 105 7fd. 4f 31 12. Reduce l9ij. 5|d. to the decimal of a £. Facit, ,97"2916. 13. Reduce 12 grains to the decimal of a lb. troy. 4 Facil, ,002083. 14j. Reduce 12 drams to the decimal of a lb. avoirdupois. Facit, ,046875. 15. Reduce 2 qrs. 14- lb. to the decimal of a cwt. Facit, ,625. 16. Reduce two furlongs to the decimal of a league. Facit, ,0835. 17. Reduce 2 quarts, 1 pint, to the decimal of a gallon. Facit, ,626. 18. Reduce 4> gallons, 2 quarts of wine, to the decimal of a hogshead. Facit, ,071428+. 19. Reduce 2 gallons, 1 quart of beer, to the decimal" of a barrel. Facit, ,0625. 20. Reduce 52 days to the decimal of a year. Facit, ,142465 +. To find the value of any Decimal Fraction in the known parts of an Integer^ Rule. Multiply the decimal given, by the number of pirts of the next inferior denomination, cutting off tlie decimals from the product; then multiply the remainder by the next inferior denomination ; thus proceeding till you have brought in the least known parts of an integer. I 2 "-r-jt-^-V; 106 I REDUCTION OP DECIMALS, EXAMPLES. [the tutor's 21. What is the value of ,8322916 of a lb.? Ans, 16s. 74(1. +. 20 16,6458320 12 7,7499840 4 ^ 2,9999360 22. What is the value of ,002084 of a lb. tory ? ^o nxr. . , •^««' 12,00384 gr. 23. What IS the value of ,046875 of a lb. avoirdupois? «^ «ri . . , , '^^' 12 dr. 24. What IS the value of ,625 of a cwt. ? e ,,r. ^^^' 2 qrs. 14 lb. 25. What IS the value of ,625 of a gallon ? 26. What IS the value of ,071428 of a hogshead of wine? c-i ^xn .1 . :^"*' * gallons 1 quart, ,999856. 27. What IS the value of ,0625 of a barrel of beer? OQ wu * • .u , ^ '^^*' ^ gallons 1 quart. ^8. What IS the value of , 1 42465 of a year ? Ans. 51,999725 days. 1 TUTORS ASSISTANT.] DECIMALS. 107 Decitnal Tables of Coin^ Weight, and Measure. TABLE I. SteKLING MONEY. jei. the Integer. s. dec. 19 •95 18 •9 17 •85 16 •8 15 •75 14. •7 13 •65 12 •6 11 •55 10 •5 s. dec, 9 ^45 8^ 7 6 5 4 3 2 I •4 •35 •3 •25 •2 •15 •1 •05 6d. 5 4 3 2 1 3qrs. 2 1 •025 •020833 •016666 •0125 •008333 •004166 •003 125 •0020833 •0010416 TABLE II. Eno. Coin. Is. Long Meas. 1 Foot the Integer. Penee or Inches. Q 5 4 3 2 Decimals. •6 •416666 •333333 •25 •166665 '083333 3grs 2 1 •0625 •041666 •020833 table iii. Troy Weight. 1 lb. the Integer. Ounces the same as Pence in Table ii. dwts* 10 9 8 7 6 5 4 3 2 1 Decimals. •041666 •0375 •033333 •029166 •025 •020833 •016666 •0125 •008333 •004166 12 gr. II 10 9 8 7 6 5 4 3 2 1 •002083 •001910 •001736 •r01562 •001389 •001215 •001042 •000868 •000694 •000521 •000347 '000173 ] oz.tliQ Integer. Penny-weights tlie same as Shillings in the first Table. Grains, 12 11 10 9 7 6 5 4 3 2 I Decimals •025 •022916 •020833 •01875 *016666 •014583 •0125 •010416 •008333 •00625 •004\ 66 •002083 TABLE IV. Avoir. Weight. 1 cwt. the Integer. Qrs. 3 2 1 i4lbs. 13 12 II 10 9 8 7 6 5 4 3 2 I Decimals. •75 •5 •25 •125 •116071 •107143 •098214 •089286 •080357 •071428 '0625 •053571 •044643 •035714 •026786 •017857 •008928 Soz. \ •0044*4 7 i •003906 108 DECIMALS. {the tutor's Decimal Tabl es of Coin.WeighU and Measure. 6oz. •003348 5 •002790 4 •002232 3 •001674 2 •001116 1 •000558 4- '0004-18 i •000279 •000139 TABLE V, Avoir. Weight. 1 lb. the Integer. Ounces 8 7 6 5 4. 3 2 1 Decimals. •5 •4375 •375 •3125 •25 •1875 •125 •0625 Sdr. 7 6 5 4 3 2 1 •03125 •027343 •023437 •019531 •015625 •011718 •007812 •003906 TABLE VI. Liquid Measure. 1 Ton the Integer. Gallons 100 90 Decimals^ •396835 •357142 80^ 70 60 50 40 30 20 10 9 8 7 6 5 4 3 o 1 4p/s. 3 o 1 •317460 •277777 •238095 •198412 •158730 •119047 •079365 •039682 •035714 •031746 •027777 •023809 •019841 •015873 •011904 •007936 •003968 Bpts. 2 1 •005952 •003968 •001984 TABLE VO. Measure. Liquid. Dry. 1 Gal. 1 Qr, the Integer. •001984 •001488 •000992 •000496 Hogshead the Integer. [Decimals, •02343T5 •015625 •0078125 Gallons. Decimals. 30 -476190 20 -317460 10 -158730 9 -142857 8 -126984 •7 -111111 6 i -095238 5 '079365 4 '063492 3 '047619 2 '031746 ~ '""'If"? table viii. Long Measure. 1 Mile the Integer. Yards. 1000 900 800 700 0(Ky Decimals. •568182 •511364 •454545 •397727 I ASSISTANT.] DECIMALS. 109 bOOyd. 400 300 200 100 90 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 Decimal Tables of Coin, Weight, and Measure. •^284091 •227272 •170454 •113636 •056818 •051136 •045454 •039773 •034091 •028409 •022727 •017045 •011364 •005682 •005114 •004545 •003977 •003409 •002841 •002273 •001704 •001136 •000568 2/f. I 6in. 3 2 1 •0003787 •0001894 •0000947 •0000474 •0000315 •0000158 TABLE IX. Time. 1 Year the integer. Months the same as Pence in Table 11. Decimals. •821918 •547945 •273973 •246575 md. 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 I •219178 •191781 •164383 .136986 •109589 •082192 •054794 •027397 •024657 •021918 •019178 •016438 •013698 010959 •008219 •005479 •002739 I Day the Integer. I'ihrs. 11 10 9 8 7 6 5 4 3 2 1 •5 •458333 •416666 •375 •333333 •291666 •25 •208333 •166666 •125 •083333 •041666 30wi. 20 10 9 8 7 6 5 4 3 2 1 •020833 •013888 •006944 •00625 •005555 •004861 •004166 •003472 •002777 •002083 •001388 •000694 TABLE X. Cloth Measure. 1 Yard the Integer. Qrs. the same as Table iv. JVails. 3 2 1 Decimals. •1875 •125 •0625 TABLE XI. Lead Weight. A Foth. the Integer. Hund. 10 9 8 7 6 5 4 3 2 1 Decimals. •512820 •4 61538 •410256 •358974 •307692 •256410 •205128 •153846 •102564 •051282 Sqrs, 2 1 •038461 •025641 •012820 libs. 13 12 11 10 9 8 7 6 5 4 3 2 I •0064102 '0059523 •0054945 •0050366 •0045787 •0041208 •0036630 •0032051 •0027472 •0022893 •0018315 •0013736 •0009157 •0004578 I I . i ' II 7 110 Decimals. [the tuto^i'js THE RULE OF THREE. (1) If 26| yards cosi ^3..16..3. what will 32i yds. cost?"" (2) If 7| yards of cloth cost ^2..12..9. what will 140* yards of the same cost ? ^ns. £4>'7..16..3i (3) If a chest of sugar, weighing 7 cwt. 2 qrs. 14 lb. cost £^6..12..9. what will 2 cwt, 1 qr. 21 lb. of the same cost 1 Ans. ^11. 14..^^. (4) What will 326| lb. of coffee be worth when' If '/L* is sold for 35. 6d. 1 ^ns. £33.A..3, (5) "What is the vnlue of 19 02.3 dwts. 5 grs. of gold, at «^..19. peroz. % £56.A0..5..2-3 qrs. (6) What is the charge for 827| yards of painting, at 10|d. per yard 1 Ans. ,g>36..4..3 .1-5 qrs. (7) It I lent my friend ^34. for I of a year, how much ought he to lend me for |^ of a year 1 Jlns. £b 1 . (8) If I of a yard of cloth, that is 2| yards broad, make a garment, how much of |- of a yard wide will make a similar- one 1 ^ns, 2 yds. 175 nails. (9) If 1 oz. of silver is worth Bs. 6 J. what is the price of a tankard that V;eighs i lb. 10 oz. 10 dwt. 4 grs. 1 ^ns. £6..3..9..<2'2 qrs. (10) Whatis the value oi 15 cwi. 1 qr. 19 lb. of cotton, at 15k\ H'' i|: 1 1 li ' ^ 112 EXCHANGE. [the TUTOn's Table. 12 deniers maie 1 sol. 20 sols ... 1 jivre. 3 livres . . 1 6cu or crown. ( 1 ) How many crowns must be paid at Paris, to receive in London ^180. exchange at 4s. 6d. per crown ?* (2) How much sterling must be paid in London, to receive in Paris 758 crowns, exchange at 4s. 8a. per crown ? Jns. £176..17..4. (3) A merchant in London remits jg* 176.. 17.. 4. to hig cor- respondent at Paris ; what is the value in French ciowns, at 4s. 8c?. per crown ? Jn,. 758 crovms, (4) Change 725 crowns, 17 sols, 7 deniers, at 4s. G^d, per crown, into sterling money. Jns. ^164.A4..01. fi^. (5) Change ^164..14..0i. sterling into French crowns, exchange at 4s, 6ld. per crown. Jns. 725 Clowns^ 1 7 sols 'j V iTM 109 deniers. II. SPAIN. Accounts are kept at Madrid, Cauiz, and Seville^ in dollars, rials, and roaravedies, and exchange is made by the piece of eight=4s. bd. at par. Table. 34 maravedies make 1 rial. S rials . • , 1 piaster or piece of eight. 10 rials ... 1 dollar. (6) A merchant at Cadiz, remits to London 2547 pieces of eight, at 4s. 8c?. per piece, how much sterling is the sum 1 Jns. ^5 94., 6. (7) How many pieces of eight, at 4s. 8c^. each, will ans- wer a bill of ^594..6. sterling? ,/5ws. 2547. *. d. cr, * As 4..6 i . 2 9 sizp. . Id 40 9)7200 sixp. 800 crowns, jins. THE tutor's ASSISTANT.] EXCHANGE. 113 (8) If I pay here a bill of «£2500 , for what Spanish money may I draw my bill at Madrid, exchange at 4fS. 9\d. per piece of eight 1 Jns* IQ^S4> pieces of eighty 6 rials, 8 f §■ mar. III. ITALY. Accounts are kept at Genoa and Leghorn, in livres, sols, anddeniers, and exchange is made by the piece of eight, or dollar=4s. 6c?. at par. Table. 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 ; at Venice by ducats. Table. 6 solidi make 1 gross. 24 gross . , 1 ducat. (9) How much sterling money may a person receive in London, if he pay in Genoa 976 dollars at 4}S, 5d. per dollar ? ^ns. £21o.A0..8, (10) A factor has sold goods at Florence, for 250 duca- toons, at 4S' 6d, each : what is the value in pounds sterling ? ^ns. 5^56.. 5. (11) If 275 ducats, at 4*. 5c?. each, be remitted from Venice to London, what is the value in pounds sterling ? Jns. £60.A4.,7. (12) A traveller would exchai^o;'^ 5^60..14..7. sterling for Venice ducats, at 4iS, 5d, each : how many must he receive 1 ^ns. 275. IV. PORTUGAL. Accounts are kept at Oporto and Lisbon, in reas, and ex- change is made by the milrea==6*. Sid. at par. Table. 1000 reas make 1 milrea. (13) A gentleman being desirous to remit to his correspon* dent in London, 2750 milreas, exchange at 6s. bd, per milrea, for how much sterling will he be creditor in London ? Ans. ^882.,5..10. r'« i lltH HI 114 EXCHANGE. [Tfa[E TUTOR's ( 14) A merchant at Oporto remits to London 4366 milreas, 183 reas, at 5i'. b\d. exchange per milrea : how much ster- ling must be paid in London for this remittance ? Ans. jen93..17..6..3'0375 qrs, (15) Iflpay a bill in London of ^1193..17. 6..3'0375 ^*. what must I draw for on my correspondent in Lisbon, ex- change at bs. b\d, per milrea 1 Jns. 4366 milreas^ 183 reas. V. HOLLAND, FLANDERS, AND GERMANY. At Antwerp, Amsterdam, Brussels, Rotterdam, and Ham- burgh, some accounts are kept in pounds, shillings, and pence, as in England ; others in guilders, stivers and pennings : ex- change with London, at 33^, to 865. or 38^. Flemish per pound sterling. Table. 8 pennings make 1 groat. 2 groats, or 16 pennings. . .. 1 stiver. 20 stivers 1 guilder, or florin. ALSO 12 groats, or six stivers make 1 schelling. 20 schellings, or 6 guilders 1 pound. (16) Remitted from London to Amsterdam, a bill of e£l~d\!. . 10 Sterling : how many pounds Flemish is the sum, the exchange at 335* Q>d. Flemish per pound sterling ? Ans. £1263. 15..9. Flemish. ( 17) A merchant in Rotterdam remits ^^1263 .15 .9. Flem- ish to be paid in London, how much sterling money must he draw for, the exchange being at 33s. 6V. Flemish per pound sterling? Jlns a£-'/54..10. (18) If I pay in London ^^852.. 12.. 6. sterling, how many guiUlers must I draw for at Amsterdam, exchange at 34 schel- lings, 4| groals flemish ])cr pound sterling? Jns. 8792 iruilders, 13 stiv. 1 gr. G] pennings. (19) What must I draw for in Lo..don, if I pay in Am- sterdam 8792 guild. 13 .3liv. 11^ pennings, cxclinnge at 34 «chellings, 4| groats per pound sterling ? ./Ins. ^852..12..C. To convert Bank money into Currency ; and f he contrary, Aa 100: lOO plus the agio: : tlio Bank-money: iho Currency. W HE TUTOR'S 366 milreas, much ster- 0375 qrs. .3-0375^5. Lisbon, ex- w, 183 Teas. T. , and Ham- , and pence, inings : ex- ''lemish per or florin. ling. , a bill of is the sum, in^? Flemish. l5.9.Flem- ley must he li per pound ?7 54..10. liow many at 34 schel- )ennings. ay in Am- innge at 34 ?852..12..C. contrary. loney : the ASSISTANT.] EXTRACTION OF THE SQUARE ROOT. 115 As 100 plus the agio : 100 : : the Currency : the Bank- money. (20) Change 794 guilders, 1 5 stivers, Current money into Bank florins, agio 4|- per cent. Ans. 761 guilders, 8 stivers^ 11||-|" P^nnings. (■21) Change 761 guilders, 9 stivers Bank, into Current money, agio 4|- per cent. Jns. 794 guilders, 15 stivers, 4^-^ pennings. VI. IRELAND. The par of Exchange, long established with Ireland, was «^108..6..8. Irish =£100. English. That is, ^l..l ..8. Irish =5:^1. English ; or 13g?. Irish= Is. English. But the English and Irish currency are now assimilated. (22) A gentleman remitted to Ireland £575. .15. sterling: what would he receive there, the exchange being at £l0. per cent? y/m. £633..8..(). (23) What would be paid in London for a remittance of £633 .Q,.Q, Irish, exchange at £l0. per cent. 1 Ans.£o75..\5. EXTRACTION OF THE SQUARE ROOT. Extracting the Square Root is to And out such a number, as, being multiplied into itself, the product will be equal to the given ri umber. Rule. First, Point tho given number, beginning at fiie unit's place, then proceed to the hundreds, and so upon every second figure throughout. Secondly. Seek the greatest square number in the first point towards the left hand, placing the square number under tho first point, and the root thereof in the quotient; subtiact the square number from the first point, and to the remainder brin down the next point and call that the resolvend. Thirdly. Double tho quotient, and place it for a divisor on tlie loft hand of the resolvend ; seek how often the divisor it* contained in tho resolvend ; (preservir\g always the unit's place) and put tho answer in tho quotient, and also on the right-hand side of the divisor; then multiply by the figure last put in the quotient, and subtract tho product from tho rosol- I« 116 EXTRACTION OF THE SQUARE ROOT. [THE TUTOr's vend ; bring down the next point to the remainder (if there be any more) and proceed as before. Roots. 1. 2. 3. 4. 5. 6. 7. 8. 9. Squares. 1, 4. 9. 16. 25. 36. 49. 64. 81. EXAMPLES. 1. What is the square root of 1 1 9025 ? 119025(345 9 64)290 ^56 Ans* 345. 685)3425 3425 2. What is the square rootdM06929 ? ^^L^«I t* 3. What is the square root of 2<2C874 1 1 ^w«- » ^06,^3 +. 4. What is the square root of 7596796] Jns. 2756,228 +. 5. What is the square root of 363729611 ^ns, 603 1. 6. Whatis the square root of 2207 12041 -^ns. 4698. Wlien tlie given number consists of a whole number and decimals together, make the number of decimals even, by adding ciphers to them ; so that there may be a pomt lall on the unit's place of the whole number. 7. What is the square root of 3271,4007? -^w*- ^^.^.t' H. What is the square root of 4795,2573 1 1 Ans. 69,247 +. 9. What is the square root of 4,372594'? Ans. 2,091 +. 10. Whatisthe8qunrerootof2,27l0957l y/ns. 1,50701 +. 1 1 . What is the square root of ,00032754 1 Ans. ,0 1809+. 12. What i3 the square root of 1,270059 ? Jlns, 1,1269 H- To extrart the Square Root of a Vulgar Fraction. Rule. Reduce the fraction to its lowest terms, then ex- tract the square root of the numerator, for a new numerator, and the square root of tao denominator, for a new denominator. If tho fraction be a surd (t*. e.) a number where a root can never be exactly found, reduce it to a decimal, and extract the root from it. ASSISTANT.] EXTRACTION OP THE SQUARE ROOT. 117 EXAMPLES. 13. What is the square root of ||-f|1 14>. What is the square root of -Htt • 15. What is the square root of^ff^f ? Arts, jlns. J9ns. 2 T- 4 T' 6 y SURDS. 16. What is the square root of ||f ? 17. What is the square root of l^'^l 18. What is the square root of 44 4 74 1 TT7 • .^715. ,89802 +. Jns. ,86602 J.. Jlns, j93309 +. To extract the Square Root 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 numerator and denominator for a new numerator and denominator. If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract tlie 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 9^? Ans, 7|. yins. 5|. jins. 3|. SURDS. 22. What is the square root of 85{|? ulns. 9,27 +. 23. What is the square root of 8f 1 Ans, 2,951 9 . . 24. What is the square root of 6^? Am, 2,5819 4. Tl find a mean proportional between any two given numbers. Rule. The square root of the product of tlie given number is the mean proportional sought. EXAMPLES. 5. What is the mean proportional between 3 and 1 2 1 Ans, 3 X 12=36, then \/ 36=3 the mean proportional, K 2 mf: Hill : I 118 EXTRACTION OP THE SQUARE ROOT. [tHE TUTORS 6. What is tlie mean proportional betvveet 4276 and 842 ? ^ns, 1897,4^. To find the side of a square etjual in area to any given super- ficies. Rule. The square root of the content of any given super- hcies IS the side of the square equal sought. EXAMPLES. 27. Ifthe content of a given circle be 160, what is the side ofthe square equal ■? Jns. 12,64911 *8. h the area of a circle is 750, what is the side of the square equal? ^^. 27,386)2. The Area of a circle given to find the Diameter, Rule. As 355: 452, or, as i: 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 fit to tie to a cow's (ail, the othe end fixed in the ground, to let her have liberty of eating an acre ot grass, and no more, supposing the cow and tail to measure 5^ yards ? ^ns. 6, 1 36 perches. The area of a circle given, to find tfie peripJiery, or drcum- ference. Rule. As 113: 1420, or, as 1: 12,56637: : the area to ttie square of the periphery ;— or, multiply the square root of area by 3,5449, and the product is the circumference. examples. 30. When the area is 12, what is the circumference ? «i Mm .u ^ *^^^' 12,279 31. When the area la 160, what is the periphery ? A . •, ^ . *^'»*- 44,S39. thiMaldr^ ^ right-angled triangle given, to find the if nil ASSISTANT,] EXTRACTION OP THE SQUARE ROOT. 119 1. The base and perpendicular given to find the hypote- nuse. Rule. The square root of the sum of the squares of the baae and perpendicular, is the length of the hypotenuse. EXAMPLES, 32. The top of a castle from the ground is 4 5 yards high, and surrounded with a ditch 60 yards broad ; what length must a ladder be to reach from the outside of the ditch to the top of the castle ? Ans. 75 yards. Base 60 yards. 33. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth : I desire to know the length of a ladder that will reach from the outside of the moat to tlie top of tiie wall ? Jlns 39,05 feet. The hypotenuse and perpendicular given, to find t/ie base. Rule. The square root of the difference of tlie squaree of the hypotenuse and perpendicular, is the length of the base. The ba^c and hypotenuse given, to find the perpendicular » RrjLR. The square root of the difference of the squares of the hypotenuse and base, is the height of the perpendicular. N. B. The two last questions may bo varied for examples to the two last propositions! 120 EXTRACTION OP THE CUBE ROOT. [tHE TutOR's Jlny number of men being given, to form them into a sgucre Oattlej or tojind the number of rank and fie Rule. The square root of the number of men given, is the number of men either in rank or file. > " "»c 34. An army consisting of 331776 men, I desire to know how many rank and file ? ji^^ 57g^ 35. A certain square pavement contains 48841 square 8ton^, all of the same size. I demand how many are con. tamed m one of the sides % J^^^ ^gl. EXTRACTION OF THE CUBE ROOT. To extract the Cube Root is to find out one number, which bemg multiplied into itself, and then into that product, pro- duceth the given number. ^ ^ * RuLD I. Point every third figure of the cube given, begin- ning at the unit's place ; seek the greatest cube to the first point, and subtract it therefrom ; put the root in the quotient, and bring down the figures in the next point to the remainder tor a Resolvend. ' 2. Find a Divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, reiectinc the units and tens, and put the answer in the quotient. 3. To find the Subtrahend. 1. Cube the last ficnire 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! Multiply the divisor by the last figure. Add these products toge her, for the subtrahend, which subtract from the resol- Roots. 1. 2. 3. 4.. 5. G. 7 8 9 Cubes. 1. 8. n. 64. 125. 2lO. 343. 512.' 729,* EXAMPLES. 1. What is the cube root of 99252847? THE Tutor's into a sgucre 'file given, is the sire to know dns, 576. 3841 square ny are con. dns, 221. OT. nber, which 'oduct, pro- iven, begin" to the first [le quotient, ! remainder, ;he quotient d, rejecting int. Lst figure in )tient by 3, le last, 3. 3e products the resol- , and pro- 9. 729. ASSISTANT.] EXTRACTION OF THE CUBE ROOT. 121 • • . 99252847(463 64 =cube of 4 Di visor Square of 4 X 3=48)35252 resolvend. 21<3=cube of 6. 432 =4 X 3 X by square of 6. 288 =divisor X by 6. 33336 subtrahend. Divisor- Square of 46 X 3=6348)1916847 resolvend. 27=cube of 3. 1242 =46 X 3 X by square of 3. 19044 =di visor X by 3. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. What is What is What is What is What is What is What is What is What is What is What is the cube the cube the cube the cube the cube the cube the cube the cube the cube the cube the cube 1916847 subtrahend, rootof 389om root of 57353391 rootof 324617591 rootof 846045191 rootof 2596940721 root of 4S2285441 root of ^^7054036008 1 root of 22069810125 1 rootof 1226153272321 rootof 2193653277911 rootof 6733730971251 Ans. 13, Arts, 179. Ans, 319. Jlns. 439. Ans, 638. Ans, 364. Ans, 3002. Ans, 2805. Ans, 4968. Ans. 6031. Ans, 8765. When the given number consists of a whole number and decimals together, make the number of decimals to consist of 3, 6, 9, &c. places, by adding ciphers thereto, so that there may be a point full on the unit's place of the whole number. 1 3. What is the cube root of 12,077875 1 Ans. 2,35. 14. What is the cube root of 36 1 55,02756 1 Ans. 33,06+, 1 5. What is the cube root of ,001 906624 ? Ans. ,124. 16. What is the oubo root of 33,230979937 Ans. 3,215+, m ! I H 122 EXTRACTION OF THE CUBE ROOT. [tHE TUTOR's 17. What is the cube root of 1 5926,972504 ? Jns. 25,16-1-. 18. What is the cube root of ,053157376 Jns. ,376. To extract the cube root of a vulgar fraction. Rule. Reduce the fraction to its lowest terms, then ex- tract the cube root of its numerator and denominator, for a new numerator 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 1'-° ? *20. What is the cube root of |§l-? 21. What is the cube root of -lAz'o 1 SURDS. 22. What is the cube root of -f ? 23. What is the cube root of | ? 24. What is the cube root of ^ ? Ans.^, Jns. 4-. Jns. ,829 -f. Jns. ,822 + . ^715. ,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 fraction, extract the cube root of the numerator and denominator for a new numer- ator and denominator ; 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 12^2. ? 96. What is the cube root of ^i\^j ? 27. What is the cube root of 405 \^-^ ? Ans, 2^. Ans. 3f . An^. 7|-. SURDS. Am, 1,93+. Ans. 2,092+. Ans. 2,057 + . 28. What is the cubs root of 7| \ S9. What is the cube root of 9f \ 30. What is the cube root of 8^ ? THE APPLICATION. 1. If a cubical piece of timber be 47 inches long, 47 inches liToad, and 47 inches deep, how many cubical inches doth it ^oni^m'\ jins. 103823, 1 tutor's ASSISTANT.] EXTRACTION OF THE CUBE! ROOT. 123 2. There is a cellar dug, that is 12 feet every way, in length, breadth, and depth j how many solid feet of earth were taken out of it ? ^ns. 1728. 3. There is a stone of a cubic form, which contains 389017 solid feet, what is the superficial content of one of its sides ? ^ns. 53^9. Between two numbers given, to find two mean proportionals. Rule. Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme, gives the less mean ; multiply the said cube root by the less mean, and the product will be the greater mean proportional. EXAMPLES, 4. What are the two mean proportionals between 6 and 162 ? Jlns. 18 and 54. 5. What are the two mean proportionals between 4 and 108 1 Ms. 12 and 36. To find the side of a cube that shall he equal in solidity to any given solid, as a globe, cylinder, prism, cone, S^c. Rule. The cube root of the solid content of any solid body given, is tlie sides of the cube of equal solidity. examples. 6. If the solid content of a globe is 10648, what is the side of a cube of equal solidity ? Jlns. 22. The side of a cube being given, to find the side of a cube that sJuiil be double, treble, SiC in quantity to the cube given. Rule. Cube the side given, and multiply it by 2, 3, &c., the cube root of the product is the side sought. EXAMPLES. 7. There is a cubical vessel, whose side is 12 inches, nnd it is required to find the side of another vessel, that is to con- tain three times as much? Jns. 17,306. 124 EXTRACTING ROOTS OP ALL POWERS. [tHE TUTOh's EXTRACTING OF THE BIQUADRATE ROOT. To extract the Biquadrate Root, is to find out a number, which boing involved four times into itself, will produce the given number. Rule. First extract the square root of the given number, and then extract the square root of that square root, and it will give the biquadrate root required. EXAMPLES. 1, What is the biquadrate of 27 ? Ans. 531441. 2. What is the biquadrate of 76 ? Ans. 33362176. 3* What is the biquadrate of 275 % Ans, 5719140625. 4. What is the biquadrate root of 531441 ? Ans, 27. 5. What is the biquadrate root of 33362176 ? Ans. 76. 6. What is the biquadrate root of 5719140625 ? Ans. 275. A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. Prepare the number given for extraction, by pointing oflf from the unit's place as the root required directs. 2. Find the first figure in the root, which subtract from the given number. 3. Bring down the first figure in the next point to the re- mainder, and call it the dividend. 4. Involve the root into the next inferior power to that which is given, multiply it by the given power, and call it the divisor, 5. Find a quotient figure by common division, and annex it to the root ; then involve the whole root into the given power, and call that the subtrahend. 6. Subtract that number from as many points of the given power as are brought down, beginning at the lower place, and to the remainder bring down the first figure of the next point for a new dividend. 1, Find a new divisor, and proceed in all respects as before. HE tutor's : ROOT. t a number, produce the en number, root, and it ns. 531441. 33362176. ri9140625. jlns, 27. •^ns. 76. Ans, ^75* i THE pointing off )tract from to the re- verto that call it the and annex the given ' the given place, and next point as before. ASSISTANT.] EXTRACTING HOOTS OP ALL POWERS. 125 EXAMPLES. 1, What is the square root of l^lS'^d ? 141376(376 9 6)51 dividend, 1369 subtrahend. 3 X 2=«6 divisor. 37 X 37=1369 subtrahend. 37 X 2=74 divisor. 376 X 376=141376 subtrahend. 74)447 dividend. 141376 subtrahend. 2. What is the cube root of 53157376 ? • » 53157376(376 27 27)261 dividend. 50653 subtrahend. 4107)25043 dividend. 53157376 subtrahend. 3 X 3 X 3=27 divisor. 37 X 37 ^ 37=50653 subtrahend. 37 X 37 X 3=4107 divisor. 376 X 376 X 376=33157376 subtrahend. 3. What is the biquadrate of 19987173376 ? I m i_ii I 1216 SIMPLE INTEREST. [tHE TUTOR's 19987173376(376 81 108)1188 diviilend. 1874161 subtrah^. 202612)1245563 dividend. : 19987173376 subtrahend. 3 X 3 X 3 X 4=108 divisor. 37 X 37 X 37 X 37=1874161 subtrahend. 37 X 37 X 37 X 4=202612 divisor. 376 X 376 X 376 x 376=19987173376 subtrahend. SIMPLE INTEREST. There are five letters to be observed in Simple Interest, viz. : P. the Principal. T. the Time. R. the Ratio, or rate per cent. I. the Interest. A. the Amount, TABLE OF RATIOS. 3 ,03 5i ,055 8 ,08 H ,035 ,06 8f ,085 4 ,04 ^ ,065 9 ,09 ^ ,045 7 ,07 H ,095 5 ,05 n ,075 10 ,1 Note. The Ratio is the simple interest of ^1. for one year, at the rate per cent, proposed, and is found thus : As 100 : 3 1 : ,03. As 100 ,035. HE TUTOR'S SIMPLE INTEREST. 127 end. terestj viz. : for one year, 1 : ,035. ASSISTANT.] When the principal, time, and rate percent, are given, to find the interest. fiuLE. Multiply the principal, time, and rate together, and it will give the interest required. Note. The proposition and rule are better expressed thus : — I. When P R 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. interest of £945: 10, for 3 at 5 (1) What is vsent. per annum. ^ns. 945,5 X ,05 x 3=141,825, or £141 : IG : 6. (2) What is the interest of £547 : 14, at 4 per cent, per an- num, for 6 years ] Jns. £13 1:8:11,2 qrs. ,08. (3) What is the interest of ^'796 : 15, at ^'^ | per cent, per an- num, for 5 years t ^ns. £119 : 5 : 4. 2 qrs. (4) What is the interest of je397 : 9 : 5, for 2i years, at 3\ percent per annum 1 Jlns. ^34 : 15 : 6. 3,5499 qrs. (5) What is the interest of £554 : 17: 6, for 3 years, 8 months, at 4f per cent per annum 1 Ans. £91 : 11 : 11 ,2. (6) What is the interest cf£236 : 18 : 8, for 3 years 8 months, at b\ per cent, per annum % Ans. £47 : 15 : 1\ ,293. When the interest is for any number of days only. Rule. Multiply the interest of £ I for a day, at the given rate, by the principal and number of days, it will give answer, INTEREST OF ^1. FOR ONE DAY. Decimals. per cent. Decimals. ,00008219178 61 ,00017808219 ,00009589041 7 ,00019178082 ,00010958904 7i ,00020547945 ,00012328767 8 ,00021917808 ,00013698630 8| ,00023287671 ,00015068493 9 ,00024657534 ,00016438356 n ,00026027397 I '■"-'-'-•''™*— --i-'r 1| !i i 128 SIMPLE INTEREST. [tHE TUTORS Note. The above table is thus found :— As 365 : ,03 : : 1 : ,000082 19178. And as 365 : ,035 : : 1 • ,00009389041, &c. EXAMPLES. (7) What is the interest of jg240, for 120 days, at 4 per cent per annum? ^ Arts, ,00010958904X240X 1 20=£3 : 3 : J i. (8) What is the interest of je36 4 : 18, for 154 days, at 5%er c^nt. per annum ? jins. ^PT : 1 3 : 1 1 1 (9) What is the interest of ^725 : 15, for 74 days, at 4** per cent, per annum ? ^ns. £b : 17 :SK ( 10) What is the interest of X 100, from the Jst June, I775 to tlie 9th March following, at 5 per cent per annum ? ' -^ns. £3: 16: 111. II. When P R T are given to find A. * Rule. prt+p—A. ANiVUITIES OR PENSIONS, &c. IN ARREARS. Annuities or Pensions, itc. are said to be in arrears, when they are payable or due, either yearly, half yearly, or quar- terly, and ar<^ unpaid for any number of payments. Note. U represents the annuity, pension, or yearly rent T II A as before. * I U R T are given to find A. ttu— tu Rule.- S -X r : +tu=A. EXAMPLES. (II) If a salary of ^150 be foreborne 5 years, at 5 per cent, what will it amount to ? £i47is. 825 3000 nx5xl50— 5X 150=3000 then- 2 -X,05+5xl50=£825. (12) Ifer cent, to £2065 3 per cent. )f£28, be 5 per cent. 4 a — 4" ut [id double 8 a— 8 ut, times the ASSISTANT.] (27) If a salary of £150 per annum, payable half-yearly, amounts to £834 : 7 '• 6, in 5 years, what is the rate per cent. 1 Ans, 5 per cent. (28) If an annuity of £150 per annum, payable quarterly, amounts to ^839 : I : 3, in 5 years, whai s the rate per cent. ? Ans. 5 per cent. IV. When U A R are given to find T 2 2a Rule. First, l=x then : V XX X 4- =T. ur EXAMPLES. (29) In what time will a salary of ^150 per annum, a- mount to £825, at 5 per cent. ? jlns. 5 years. 2 826X2 39X39 1=39 —220 =380,25 ,05 150X,05 4 V229+380 ,25= 39 .24 ,5 = 2 :5 years. (30) If a house is let upon a lease ura certain time, for £60 per annum, and amounts to £363 : 8 : 3, at 4^ per cent, what time was it let for ? Jlns. 5\ years. (31) If a pension of £250 per annum, being forborne a certain time, amounts to £2065, at 6 per cent , what was the time of forbearance ? */9ns. 7 years. (32) In what time will a yearly pension of £28, amount to £263 : 4, at 5 per cent. 1 Ans. 8 years. Note. If the payments are half-yearly, take half the ratio, and half the annuity ; if quarterly, one fourth of the ratio, and one fourth of the annuity ; and T will be equal to those half- yearly or quarterly payments. (33) If an annuity of £l50 per annum, payable half- yearly, amountB to £834 : 7 : 6, at 5 per cent., what lime was the payment forborne 1 Ans. 5 years, (34) If a yearly pension of 4^150, payable quarterly, a- mounts to £839 : 1 : 3, at 5 per cent., what was the timo of forbearance 1 Ans. 5 years. r" 132 I SIMPLE IJJTeREST. [thE TTJTOR'i PRESENT WORTH OF ANNUITIES. Note. P represents the present worth ; U T R as before. I. When U T R are given to find P. ttr— tr+^t RULE,- 2tr4-2 : +u=P. EXAMPLES. (35) What is the present worth of jClSO per annum, to continue 5 years, at 5 per c ent. ? Jlns. £QQO. '3l<:yxT05-^^5X~,05+5x2«ll ,5x,05X2-|-2='^^,5 then Il-f-2,5 X 150=^660. (36) What is the yearly rent of a house of ^60, to con- tinue b\ years, worth in ready money, at 4| per cent. ? Jins.£29\ : 6: 3. (37) What is the present worth of ^250 per annum, to continue 7 years, at 6 per cent. ? ^ns ^£1454 : 4 : 6. (38) What is a pension of j^28 per annum, worth in ready money, at 5 per cent., for 8 years? Ans £\^S, Note. Tiie same thing is to be observed as in the first nile of annuities in arrears, concerning half-yearly and quarterly payments. (39) What is the present worth of -g^l 50, payable quar- terly, for 5 years, at 5 por cent ? Ans. £67 1 : 5. Note. By comparing tho last examples, it will 1x3 found that the present worth of iialf-yearly payments is more advan- tageous than yearly, and quarterly than half-yearly. II. When P. T. R. are given to find U. tr-l-l Rule. : x 2p=:U. ttr— tr -f.2t examples. (40) If the present worth of a salary be £6G0, to continue 5 years, at 5 per cent , what is the salary ? Ans, £l 50. HE TTJTOa'l I ASSISTANT.] SIMPLE INTEREST. 13.3 as before. 5 X j05 + 1 = 1,^5 5 X annum, to s. £660. =%5 then 50, to con- ;nt.? 1:6:3. annum, to 1:4:6. ih in ready 5^188. B first nile I quarterly ible (juar- 671 : 5. Ijc found 3re advan- I X ,05- 1,'25 -5 X ,05+10=11. X6'60 X ^2=^150. 11 (41) There is a house let upon lea*e for 5| years to come, I desire to know the yearly rent, when the present worth, at 4.i per cent., is £291 : 6 : 3 ? ./Ins. £60. (42) What annuity is that which, for 7 years' continuance, at 6 per cent., produces £1454 : 4 : 6 present worth ? A.'is. £lbO. (43) What annuity is that which, for 8 years' continuance, produces 5^188 for the present vvortli, at 5 per cent. 1 jlns, J^IS. Note. When the payments are half-yearly, take haifthe ratio, twice the number of years, and multiply by 4p; and when quarterly, take one fourth of the ratio, and four limes the number of years, and multiply by 8 p. (44) There is an annuity, payable half-yearly, for 5 years to come, what is the yearly rent, when the present worth, at 5 per cent. , is ^6667 : 1 1 Ans. £ 1 50. (4.5) There is an annuity payable quarterly, for 5 years to come, I desire to know the yearly income, wlien the present worth, at 5 per cent., is d^(i71 : .5 ? Ans. £150. III. When U P T are given to find R. rt— p X 2 Rule. =R. 2pt ^ ut — ttu. • continue IS. £l50. EXAMPLES. (46) At what rate per cent, will an annuity of £150 per annum, to continue 5 years, produce the present worth of 4^660 ? Ans. 5 per cent. 150 X 5— 6^0X 2=180,^ X 660 X 5 + .5 X 150—5 X 5X 150=3600 then i80-r3600=,O5=5 per cent. I 1345 SIMPLE INTEREST. [the tutor's (47) If a yearly rent of ^60 per annum, to continue 5$ years, produces ^§'291 : 6 : 3, for the present worth, what is the rate per cent 1 Ans. 4| per cent. (48) If an annuity of j§'250 per annum, to continue 7 years, produces ^^1454: 4 : 6, for the present worth, what is the rate per cent. ? Ans. 6 per cent. (49) If a pension of £'28 per annum, to continue 8 years, produ V? £188 for the present worth, what is the rate per cent. . jins. 5 per cent. Note. When the annuities, or rents, &c. are to be paid half-yearly, or quarterly, then For half-yearly payments, take half of the annuity, &c. and twice the number of years, the quotient will be the ratio of half the rate per cent, — and For quarterly payments, take a fourth part of the annuity, &c. and four times the number of years, the quotient will be the ratio of the fourth part of the rate per cent. (50) If annuity of J^ 1 50 per annum, payable half-yearly, having 5 years to come, is sold for £667 : 10, what is the rate per cent. ? Ans, 5 per cent. (51) If an annuity of £150 per annum, payable quarterly, having 5 years to come, is sold for £671 : 5, what is the rate lier cent. 1 Ans, 5 per cent. IV. When U P R are given to find T. 2 2p 2p XX X Rule. l=x then ^ — J- u ur 4 2 ^T, EXAMPLES. (52) If an annuity of ^150 per annum, produces si'GGO for the present worth, at 5 per cent., what is the time of its continuance 1 Ans, 5 years. IE TUTOR'S ASSISTANT.] SIMPLE INTEREST. 135 2 660x2 ,05 150 30,2 X 30,2 —1=30,2 660x2 1 SOX ,05 :176 .=228,01 then v/ 228,01 + 176=20,1 20,1- 30,2 -=5 years. (53) For what time may a salary of 6^60 be purchased for ^'291 : 6 : 3,. at 4i per cent. 1 ^ns. 5f years. (54) For what time may £250 per annum, be purchased for 5^1454: 4:6, at 6 per cent. 1 Arts. 7 years. (55) For what time may a pension of ^28 per annum, be purchased for igl88, at 5 per cent. ? *dns. 8 years. Note. When the payments are half-yearly, then U will be equal to half the annuity, &c., R half the ratio, and T the number of payments : and, When the payments are quarterly, U will be equal to one fourth part of the annuity, &c., Rthe fourth of the ratio, and T the number of payments. (56) If an annuity of £150 per annum, payable half- yearly, is sold for £667 : 10, at 5 per cent., I desire to know the number of payments, and the time to come ] Jns. 10 payments, 5 years. (57) An annuity of £150 per annum, payable quarterly, is sold for £671 : 5, at 5 per cent , what is the number of payments, and time to come 1 Ans» 20 payments, 5 years. ANNUITIES, &c. TAKEN IN REVERSION. I. To find the present wouh of an annuity, &c., taken in reversion. Rule. Find the pr jnt worth of the ttr — tr+2t yearly sum at the given rate and for the X : u=P. time of its continuance ; tlius, 2tr+2 f- ^"" i^^ 136 SIMPLE INTEREST. [the tutor's 2. Change P into A, and find what principal, being put to interest, will amount to A at the same rate, and for the^ time to come before the annuity, &c, commences ; thus, EXAMPLES, tr+ 1 :P. (58) What is the present worth of an annuity of ^^150 per annum, to continue 5 years, but not to co':i: . uce illl the end of 4< years, allowing 5 per cent, to the purchaser 1 Ans. £bbQ, 5x5x,05—5x, 05+ 2x5=4,4X150= 660 4x,05+ 1 -=550. 5X,05x2+^ (59) What is the present worth of a lease of ^50 per an- num, to jntinue 4 years, but which is not to commence till the end of 5 years, allowing 4 per cent to the purchaser ? Ans. £151 : 5 : 11 3 qrs. (60) A person having the prouiise of a pension of ^g'SO per annum, for 8 years, but not to commence till the end of 4 years, is willing to dispose of the same at 5 per cent, what will be the present worth 1 Ans, £\\\ : 18:1 ,144.. (61) A legacy of ^£"40 per annum being left for 6 years, to a person of 1 5 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 1 Ans. £\7V : 13: 11 ,07596. 2. To find the version. yearly income of an annuity, &c., in re. Rule 1 . Find the amount of the present wortli at the given rate, and for the time ptr+psarA. before the reversion ; thus, 2. Change A into P, and find what an- nuity being sold, will produce P at the tr 4. 1 same rate, and for the time of its continu unce ; thus, ttr— tr+'^t :X2p=U. !E tutor's ASSISTANT.] REBATE OR DISCOUINT. 137 EXAMPLES. (62) A person having an annuity loft 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 tlie yearly income '? Jlns. £150. 1 5X,05+1, 550 X 4 X ,05+ 550=660,5 X 5 X jOJi— 5 X ,05+5 X 2 = ,113636 X66Ox2--£150. (63) There is a lease of a house taken for 4 years, but not to commence till the end of 5 years, the lessee would sell the same for ^152 : 6, present payment, allowing 4 per cent, to the purchaser, what is the yearly rent ? Ans. £50. (64) A person having the promise of a pension for 8 years, which does not commence till the end of 4 years, has dis- ix>sed of the same for ^(^l 1 1 : 18: 1 J 4 present money, allow- ing 5 per cent, to the purchaser, what was the pension 1 Ans. £20. (65) There is a certain legacy left to a person of 15 years of age, which is to be continued for 6 years, but not to com- mence till he arrives at the age of 21 ; he, wanting a sum of money, sells it for j^171 : '4, allowing 4 percent, to the buyer, what as the annuity left hira ? »dns, £^^0, REBATE OR DISCOUNT. NoTB. S represents the sum to be discounted. P the Present worth. T the Time. R the Ratio. I. When S T R are given to find P. s Rule. =?. tr+l EXAMPLES. 1. What is the present worth of ^357 : 10, to be paid 9 months hence, at 5 per cent, % Jlns. £d44 : 11 : 6| ,168. M I i i ^^S REBATE OR DISCOUNT. [tHE TUT0R*S 2. What is the present worth of 5^275 : 10, due 7 montlis lience, at 5 per cent. ? Jlns. £'267 : 13 : 10l|y. 3. What is the present worth of ^^875 : 5 : 5, due at 5 months hence, at 4| per cent. ? ^ns. £859 : 3 : 3| j^-^. 4<- How much ready money can I receive for a note ofirl^ due 1 5 months hence, at 5 per cent. 1 Ans. j£70:ll:9,1764d. 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 montlis hence, allowing 5 per cent, be £U^ : II : 6 3,168 qrs., what was the sum first due ? Jns. ^357 : 10. 344,5783x,75x,05+344.,5783=£357 ; 10. 6. A person ov\lng a certain sum, payable 7 months henoe agrees with tlie creditor to pay him down £2Q1 1 13 : ]{)11 ' allowing 5 per cent, for present payment, what is the debt^l ^ , Ans. ^275 : lo! 7. A person receives £^d^ ; 3 : 3| :f|y fora sum of monev, due 5 months hence, allowing the debtor 4| per cent, for pre- sent payment, what was the sum due ? Ans. j^875 : 5 : 6. 8. A person paid £70: 11: 9 ,17641. for a debt due* 15 months hence, he being allowed 5 per cent, for the discount, how much was the debt ? j]fis, jPt'S. III. When S P T are given to find R. s— p RULE.- =R. tp EXAMPLES. 9. At what rate per cent, will ^367 : 10, payable 7 monUis hence, produce £^^^'. il : 6 3,168 qrs. for present payment? 3575,-344,5783 — _-. 05__.5 pg^ pgj^^^ 344,5783 X ,75 HE tutor's e 7 montJis ASSISTANT.] 10. At what months hence, paymenf? 11. At \Ahat months hence, *• "54 ITT* 12. hX what lience, produce REBATE OR DISCOUNT. 139 rate per cent, will £275: 10, payable 7 produce £267: 13: 10^^^ for the present Jns. 5 per cent, rate per cent, will £875: 5: 6, payable 5 produce the present payment of £859: Ans. 4| per cent, rate per cent, will ^75, payable 15 months the present payment of £70 ; 11 : 9 ,l764d.? Jns. 5 per cent. rV. When S P R are given to find T, s— p RULE.- -=T. ip EXAMPLES. 13. The present worth of £357 : It), due at a certain time to come, is £344: 11: 6 3,168 qrs. at 5 per cent., in what time should the sum have been paid without any rebate 1 Jns, 9 months. 357,5—344,5783 344,5783 X ,05 s,75=9 months. 14. The present worth of £275: 10, due at a certain time to come, is £267 : ^'"' lOg^^y, at 5 per cent., in what time should the sum have been paid without any rebate "? Jns, 7 months, 15. A person receives ^859: 3: 3| ,0184, for £875: 5 : 6, due at a certain time to come, allowing 4^ per cent, discount, I desire to know in wliat time the debt should have been discharged without any rebate ? ,dns. 5 months. 16. I have received £70: 11: 9 ,1764d. for a debt of ^75, allowing the person 5 per cent, for prompt payment, I desire to know when the debt would have been payable without the rebate 1 Jlns. 15 months. III #";l lilt 14fO EQUATION OF PAYMFNTS. [thE TUTOR's EQUATION OF PAYMENTS. To find the equated lime for the payment of a sum of money due at several times. Rule. Find the present worth of each pay- ment for its respective time j thus, Add all the present worths together, ther. trXl 8— p=D. d and «=E. pr EXAMPLES. I. D owes E 5^00, whereof ^40 is to be paid at three months, £60 at six months, and ^^lOO at nine months; at what time may the whole debt be paid together, rebate being made at 5 per cent. ] Jtns. 6 raontlis, 26 days. 40 60 100 «=39,5061 =58,5365 1,0125 1,025 1,0375 =96,8855 then 200—39,506 1+58,5365+96,3855=. 5,5719 5,5719 :,57315=6month3, 26 days. I91.,4281xj05. 2. D owes E ^800, whereof £200 is to be paid in 3 months, ^£200 at 4 months, and ^^400 at 6 months ; but they, agreeing to make but one payment of the whole, at the rate of 5 per cent, rebate, tlie true equated time is demanded ? jins» 4 months, 22 days. 3. E owes F jei200, which is to be paid as follows: £200 down, £50^ 1. the end of 10 months, and the rest at the end of 20 months ; but they, agreeing to have one payment of tlje whole, rebate at 3 per cent, the true equated time is demanded^ jlns. 1 year, 11 days. i, IE tutor's ASSISTANT.] DUODECIMALS* 14 1 DUODECIMALS, OR, WHAT IS GRNERALLY CALLED Cross Multiplication^ and Squaring of Dimensions by Artificers and Workmen. RULE FOR MUT.TIPLYING DUODeCIMALLY. 1. Under the multiplicand write the correspunding denomi- nations of the multiplier. 2. Multiply each term in the multiplicand (beginning at the lowest) by the feet in the muliip'.'er ; write each result under its respective term, observing '-> carry an unit for eve.y 12, from each lower denominati. i its next superior. 3. In the same manner *. .tiply the multiplicand by the primes in the multiplier, and writ«= the result of each term one l)lace more to the right 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 places to the right hand of th.se in the multiplicand, and so on for thirds, fourths, &c. EXAMPLES. • f. in. f. in. Multiply 7 , . 9by 3 . 6. Cross Multiplicatiun. Practice. Duodecimali. Decimnls. 7 9 61 7 . 9 7 . 9 7,75 3^6 « 3.6 2.6 3,5 21.0 0=7X3 23 . 3 23 . 3— X3 3875 2.3.0—9X3 3.6.0=7X6 3 . 10 . 6 3 . 10 .6x6 2325 0.4..6=9X6 27 . 1.6 27 . 1.6 27,125 27.1.6 f.in. f.in. f. in. pts. 2. Multiply 8,5 by 4. 7 Facit, 38. 6.1i 3. Multiply 9.8 by 7. 6 Facit. 72. 6 4. Multiply 8.1 by 3. 5 Facit, 27. 7. 5 5. Multiply 7.6 by 5. 9 Facit, 43. 1. 6 142 6. Multiply 7. Multiply 8. Multiply 9. Multiply 10. Multiply 11. Multiply 12. Multiply 13. Multiply U. Multiply 15. Multiply K). Multiply 17. Multiply 18. Multiply DUODECIMALS. 4.7 1.5.911 10.4.5 75.7 97.S 57.9 75.9 87.5 179.3 259.2 257.9 311.4.7 3217.3 by 3.10 by 3. 5.3/^ by 7. 8 6 by 9. 8 by 8, 9 by 9. 5 by 17. 7 by 35. 8 by 38.10 by 48-11 by 39.11 by 36. 7.5 by 9. 3.6 Facitj Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, [the tutor's 17. 6.10/,;;,/ 2.5. 8. 6.2.3 79.11. 0.6.« 730. 7, 8 854. 7 543. 9. 9 1331.11. 3 3117.10. 4 6960.10. 6 l'i677. 6.10 10288. 6. 3 11402.2.4.11.11 2988.2.10.4.6 THE APPLICATION. Artificers's work is computed by diffenjat measuree, viz :^ i. Glazing, and masons' flat work, by the foot. 2. Painting, plastering, paving, &c., by the yard. of'loo^S'''"''''' ^"^""^^ '■^^"«' ''''"^' ^'^ by the square re i« 272tfeet.''''^' ^'•' ^^ '^'' '^ "^ *^'^ ^'^^^ ^'^^^ «^1"« Mmiring hy the Foot Square, as Glazier,^ and Masms^ Flat Work, EXAMPLES. 19. There is a house with 3 tier of windows, 3 in a tier— the height of the first tier 7 feet 10 inches, the second 6 feet 8 inches, and the third 5 feet 4 inches, the breadth of each is 3 ^eet luTcliesj what will the glazing come to, at 14c/. per ASSISTANT.] Duodecimals. 7 . 10 the Q . 8 heights 5 . 4 added. DUODECIMALS. 143 feet in. pts. 233 . 0.6 atl4: 9: ^ Measuring by the Bod, Note. Bricklayers always value their work at the rate of a brick and a half thick ; and if the thickness of the wall is more or less, it must be reduced to that thickness by this Rule. Multiply the area of the wall by the number of half bricks in the thickness of the wall j tho product divw^ed l)y 3, ^ve« tlie area. EXAMPLES. 35. If the area of a wall be 4085 feet, and the thkknee« two bricks and a half, how many rods doth it contain? Jns. 25 rods, 8 feet 36. If a garden wall bo 254 feet round, and 12 feet 7 inchea higli, and 3 bricks ♦hick, how many rods doth it contain j ^ ' Ans. 23 rods, 1 36 feet. 37. How many squared rods arc there in a wall 62 i feet long, 14 feet 8 inches high, and 2\ bricks thick ? ^ Jns, 5 rods, 167 feet. 14S DUODECIMALS. - [the tutor's 3S, If the side walls of a house be 28 feet 10 inches in length, and the height of the roof Oom 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| biicks thick, 20 feet more at two bricks thick, 15 feet 8 inches more at 4 brick thick, and the gable at I brick thick; what will the whole work come to at £5 16b per rod? ^w*. ^48: 13: 5|. Multiplying several figures by several, and the product to b% 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 j next multiply the tens in the multiplicand by the units of the multiplier, to which add the product of the unita of the multiplicand multiplied by the tens in the multiplier, and the tens carried j then multiply the hundreds in the multi. phcand by the units of the multiplier, adding the product of the tens in the multiplicand multiplied by the tens in the multi- plier, and the units of the multiplicand by the hundreds in the multiplier; and so proceed till you have multiplied the multi- plicand all through, by every figure in the multiplier. EXAMPLES. Multiply 35234 by 59424 Common way. 36234 62424 Product, 1847107216 1 40938 70468 140936 70468 176)70 1847107216 IE TU Ton's ASSISTANT.] A COLLECTION OF QUESTIONS. m A COLLECTION OF QUESTIONS. 1. Wliat is tho value of 14:' barrels of soap, at 4id per lb. each barrel containing 254 lb. ? *dns. £36 : 13 : 6. 2. A and B trade together ; A puts in j^320 for 5 months, B £160 for 3 months, and they gain ^100 ; what must each man receive 1 JJns. A £53 : 13 : 9l|^, and B £46 : 6 : £147. 3. How many yards of cloth, at 17s. 6d per yard, can I have for 13 cvvt, 2 qrs. of wool, at 14d. per lb. jJns. 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 JGIO by the whole ? Jns. 3s. 4d. per ell. 5. A has 648 yards of cloth, at 14s. per yard, ready money., but in barter will have 16s. ; B has wine at ^^42 per ton, ready money : the question is, how much wine must be given for the cloth, and what is the price of a ton of wine in barter? jl7L8. £48 tho ton, and 10 tons, 3 hhds. 12* gals, of wine must be given for the cloth. 6. A jeweller sold jewels to the value of £1200, for which \ie received in part 87G French pistoles, at 16s. 6d. each ; what sum remains unpaid ? JIns. £*11 : &. 7. An oilman bought 417 cwt. 1 qr. 15 lb., gross weight, of train oil, tare 20 lb. per 112 lb. how many neat gallons were there, allowing 1\ lb. to a gallon ? Ans. 5120 gallons. 8. If I buy a yard of cloth for 148. 6ci. and soil it for 168. 9d , what do I gain per cent ? Ans. £15 : 10 : 4f|:j-. ■ 148 A COLLECTION OF QUESTIONS. [tHE TUTOa's 4>; £ 9. Bought 27 bags of ginger, each weighing gross 84i Ib^ tare at If lb. per bag, tret 4> lb. per 104 lb , what do they come to at 8id. per lb. ? ^ns. £76 : 13 : 2\. 10. If I of an ounce cost |- of a shilling, what will |- of a lb. cost? ./5ns. 17s. 6d. 11. If ^ of a gallon cost I- of a pound, what will |- of a tun cost? ' Jns. £i05. 12. A gentleman spends one day with another, £1: 7: 10 i and at the year's end layeth up £340, what is his yearly 13. A has 13 fother of lead to send abroad, each being lyf times 112 lb. B has 39 casks of tin, each 388 lb how many ounces difference is there in the weight of these commodities? Jns, 212160 oz. 14. A captain and 160 sailors 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 pan? Jns. The captain had £272, and each sailor £6: 16. 15. At what rate per cent, will £956 amount to £1314 : 10, ill 7i years, at simple interest? ^ns. 5 per cent. 16. A hath 24 cows, worth 72s. each, and B 7 horses, worth £l3 a piece, how much will make good the difference, in case they interchange their said drove of cattle ? ^ Ans, £^'. 12. 17. A man dies and leaves *^120 to be given to threo persons, viz. A, B, C ; to A a share unknown ; B twice as nuch as A, and C as much as A and B ; what was the shaco of each ? ^ns. A £20, B £40, and C £60. 18 £l000 is to be divided among three men, in such a manner, that if A has £3, B shall have £^, and C £8; how much must each man have ? „ ^^^^ ,^ , ,. xjkaa Arts. A ^187 : 10, B £312: 10, and C «^500. 19. A piece of wainscot is 8 feet 6^ inches long, and 2 feet 91 inches broad, what is the superficial content ? * ' Ans. 24 feet : 3'' : 4 : 6. 20 If 360 men be in garrison, and have provisions for 6 months, but hearing of no relief at the end of 5 months, how many men must depart that the proviaions may last so much the long^-r ? ^^'' 288 men. E TUT0E8 ASSISTANT.] A COLLECTION OP QUESTIONS. 149 21. The less of 2 numbers is 187, their difference 34, the square of their product is required 1 Ans, 1707920929. 22. A butcher sends his man with ^216 to a fair to buy cattle ; oxen at £11, cows at 40s , colts at ^1 : 5, and hogs at ^1 : 15 each, and of each a like number, how many of each sort did he buy 1 Jlns. 13 of each sort, and £S over, 23. What number added to 11|- will produce 36if|- 1 Ans. 24f44. 24. What number multiplied by f will produce 11|-^ 1 Ans. 26|f . 25. What is the value of 179 hogsheads of tobacco, each vi?eighing 13 cwt. at ^^2 : 7 : 1 per cwt. ? Ans. ^5478 : 2 : 11. 26. My factor sends me word he has bought goods to the value of j£500 : 13 : 6, upon my account, what will his com- mission come to at 3^ per cent. ? Ans. £n : 10 : 5 2 qrs. |^^. 27. If -^ of 6 be three, what will I of 20 be ? Jns. 7|. 28. What is the decimal of 3 qrs. 14 lb. of a cwt. 1 Ans. ,875. 29. How many lb. of sugar, at4|d. per lb. must be given in barter for 60 gross of inkle, at 8s. 8d. per gross ? Ans. 1386|. lb. 30. If I buy yarn for 9d. the lb. and sell it again for 13^d. pi lb., what is the gain per cent. "? Jns. £50. 31. A tobacconist would mix 20 lb. of tobacco at 9d. per lb. with 60 lb. at 12d. per lb., 40 lb. at 18d. per lb., and with 12 lb. at 2s. per lb., what is a pound of this mixture worth i Ans. Is. 2|d. f^. 32. What is the difference between twice eight and twenty, and twice tvventy.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 ^g'SO had I ? Ans. 600 ells. 34. A broker bought for his principal, in the year 1720, £400 capital stock in the South Sea, at £650 per cent., and Bold il again when it was worth but ^130 per cent. ; how much was lost in the whole ? Ans, «^2080. N il 150 A COLLECTION OP QUESTIONS. [tHE TUTOR'S 35. C hath candles at 68. per dozen, ready money, but in barter will have 6s. 6d. per dozen ; D hath cotton at 9d. per lb. ready money. I demand what price the cotton must be at in barter ; also, how much cotton must be bartered for 100 doz. of candles ? Jns, The cotton at 9d. 3 qrs. per lb.;, and 7 cwt. qrs. 16 lb. of cotton must be given for 100 doz. candles. 36. If a clerk's salary be ^73 a year, what is that per day ] 37. B hath an estate of £^53 per annum, and payeth 5s. lOd. to the subsidy, what must C pay, whose estate is worth ^100 per annum 1 -^ns. lis. Od. -f^. 38. If I buy 100 yards of riband at 3 yards for a shilling, and 100 more at 2 yards for a shilling, and sell it at the rate of 5 yards for 2 shillings, whether do I gain or lose, and how jjjmjh 1 ^ns. Lose 3s. 4>d. 39. What number is that, from which if you take |, the remainder will be 4- ^ '^"** t^* 4-0. A farmer is willing to make a mixture of rye at 4s. a bushel, barley at 3s., and oats at 2s. ; how much must he take of each to sell it at 2s. 6d. the bushel ? Jns. 6 of rye, 6 of barley, -^d 24 of oats. 41. If I- of a ship be worth £3740, what is the worth of the whole 1 ^ns. £9973 : 6 : 8. 42. Bought a cask of wine for £62 : 8, how many gallons were in the same, when a gallon was valued at .5s. 4d. ? ^ns.234. 43. A merry young fellow in a short time got the better of :« of his fortune; by advice of his friends, he gave £2200 for an exempt's place in the guards ; his profusion continued till he had no more than 880 guineas left, which he found by computation, was l^^ part of his money after the commission was bought ; pray what was his fortune at first 1 Jns. £10450. 44. Four men have a sum of money to be divided amongst them, in such a manner, that the first shall have \ of it, the second |, the third |, and the fourth the remainder, which is ^'28, what is the sum ? ^ns. i^ll2. HE tutor's I ASSISTANT.] A COLLECTION OF QUESTIONS. 15! 45. What is the amount of jglOOO for Sf years, at 4>i per cent , simple interest ? '^^s. £1261 : 5. 46. Sold goods amounting to the value of ^700 at two 4 months, wliat is the present worth, at 5 per cent., simple interest ? ^ns. ^682 : 19 : 5^ i||^. 47. A room 30 feet long, and 18 feet wide, is to be cover- ed with painted cloth, how many yards of | wide will cover it ? ^ns. 80 yards. 48. Betty told her brother George, that though her fortune, on her marriage, took ^19312 out of her family, it was but I of two years' rent, Heaven be praised ! of his yearly in- come ; pray what was that ? ^ns 16093 : 6 : 8 a year. 49. A gentleman having 50s. to pay among his labourers for a day's work, would give to every boy 6d., to every woman 8d., and to every man 16d. ; the number of boys, women, and men, was the same. I demand the number of each 1 /Ins. 20 of each. 50. A stone that measures 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches deep, how many solid feet doth it contain ? ^ns, 41 feet 3 inches. 5 1 . What does the whole pay of a man-of-war's crew, of 640 sailors, amount to for 32 months' service, each man's pay being lis. 6d. per month ? ^ns, ^£23040. .52. A traveller would change 500 French crowns, at 4s. 6d. per crown, into sterling money, but he must pay a half- penny per crown for change ; how much must he receive 1 Ans.£\\\ : 9: 2. 53. B and C traded together, and gained £lOO ; B put in 5^0'40, C put in so much that he might receive ^^GO of the gain. I demand how much C put in ? Arts ;£960. 5i. Of what principal sum did £20 interest arise in one year, at the rate of 5 per cent, per annum 1 Jns. ^400. .55. In 6/2 Spanish gilders of 2s. each, how many French pistoles, at 17s.' 6d. per piece ? Ans. 76^^- 56- From 7 cheeses, each weighing I cwt. 2 qrs. .5 lb., how many allowances for seamen may be cut, each weighing 5 oz. 7 drams ? -^w*. 356|f . .57. If 48 taken from 120 leaves 72, and 72 taken from 91 loaves 19, and 7 taken from thence lea ve.^ 12, what number 152 A COLLECTION OF QUESTIONS. [tHE TUTOr's is that, out of which when you have taken 48, 72, 19, and 7» leaves 12 ? Ans, 158. 58. A farmer ignorant of numbers, ordeiod £bOO to be divided among his five sons, thus : — Give A, says he, |, B i, C|, Di, and E y part ; divide this equitably among them, according to their father's intention. Ans. A^152m,B^114HfjC £^1^, 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 we were married . Arts* 45 years the man, 15 the womanl 153 A Table for finding the Interest of any sum of Money, for any number of months, weeks or dm/s, at any rate per cent. were married . 1 5 the woman'? Year. £ 1 2 3 4 5 6 7 S 9 10 20 30 40 50 60 70 80 90 100 200 300 40(i 500 600 700 800 900 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 20000 30000 Calen. Month. £ 1 2 3 4 5 5 o 5. d. 1 3 5 8 4 7 8 16 25 33 41 50 58 66 75 83 166 250 333 416 500 583 666 750 833 1666 2500 6 8 8 4 10 11 8 13 4 15 16 8 13 4 10 6 8 3 4 16 8 13 4 10 6 8 13 4 6 8 13 4 6 8 13 4 6 8 13 4 6 8 13 4 6 8 13 4 6 8 13 4 Week. £ s. d. 41 9 1 It 1 6 1 11 2 3f 2 8i 3 1 3 5i 3 lOi 7 8i 11 6h 15 4^ 19 2i 13 1 1 6 11 1 10 9\ 1 14 1\ 1 18 5i 3 16 11 5 15 4i 7 13 10 9 12 3^ 11 10 9 13 9 2»^ 15 7 8i 17 6 M 19 4 7i 38 9 2^ 57 13 10 76 18 5i 96 3 0^ 115 7 8\ 134 12 3i 153 16 11 173 1 6^ 192 6 1^ 384 12 3 i 576 18 51 N 2 Day. £ s. 1 1 2 2 3 3 4 4 5 10 16 1 1 1 7 1 12 1 18 2 3 2 9 2 14 5 9 8 4 10 19 13 13 16 8 19 3 21 18 24 13 27 7 54 15 82 3 d. Ok n 2 %■ 34 4 4i 5\ 6 6^ U n 9 31 10 4.^ Hi 5i lid 5i 11 M 10^ 4 10 31 9i 7 4.i 2 lU 9 4i in 104 10 '! 154 Rule. Multiply the principal by the rate per cent.,, and the number of months, weeks or days, which are required, cut off two figures on the right hand side of the product, and collect from the table the several sums against the different numbers, which when added, will make the number remaining. Add the several sums together, and it will give the interest re- quired, N. B. For every 10 that is cut off in months, add two- pence ; for every ten cut off in weeks, add a halfpenny ; and for every 40 in the days, 1 farthing. EXAMPLES. I. What is the interest of £2467 : 10, for 10 months, at 4 per cent, per annum 1 2467 : 10 900=75 : : 4 80= 6:13:4 7= 0:11:8 9870: 10 987=82: 5:0 987 100 2. What is the interest of ^"2467 10s. for 12 weeks, at 5 per cent. 1 2467: 10 1000=19: 4: 7^ 5 400= 7:13: 10 80= 1:10: 9| 12337:10 50= 0: 0: 2| 12 1480i50=28: 9: 6 1480|50: 3. What is the interest of £-2467 10s , 50 days, at 6 per cent, 1 2467 : 10 7000=19 : 3 : 6i 6 400= 1:1:11 2= 0:0: 11 14805: 50 7402|50:0 60= 0:0: Oi 7402|50^20 : 5 : 7 155 'To find what an Estate, from one to ^60;000 per annum will come to for one day. Rule 1. Collect the annual rent or income from the table for 1 year, against which take the several sums for one day, add them together, and it will give the answer. An estate of ^376 per annum, what is that per day ? 300=0: i6: 5| 70=0 : 3 : 10 6=0: 0: 4 376=1 : O: 7| To find the amount of any incorMf salary , or servants' wages, for any number of months, weeks, oi' days. Rule. Multiply the yearly income or salary by the number of months, weeks, or days, and collect the product from the table. What will £^0 per annum come to for 1 1 months, for 3 weeks, and for 6 days ? For i I months. 270 2000=166:13:4- 11 900= 76: 0:0 70= 5:16:8 •2970 6 1620 2970=247: 10:0 For 6 days. 270 1000=2:14: 9^ 600=1: 12:104 20=0: 1: 1^ 1620=4: 8: 9| For 3 weeks. 270 800=15: T: S^ 3 10= 0: 3: 10^ 810 = 15:11: 6\ For the whole time. 247: 10:0 15:11:61 4: 8:91 267:10:3^ i ■ II . 156 A COMPENDIUM OF BOOK-KEEPING, BY SINGLE ENTRY. Book-keeping is the art of recording the transactions of persons in business, so as to exhibit a slate of their affairs in a concise and satisfactory manner. Books may be kept either by Single or by Double hntiy, but Single Entry is the method chiefly used in retail business. Tlie books found most expedient in Single Entry, are tlie Day -Book, the Cash-Book, the Ledger, and the Bill- Book. The Day-Book begins with an account of the trader's pro- perty, debts, &c, ; and are entered in the order of their occur- rence, the daily transactions of goods bought and sold. The Cash- Book is a register of all money transactions. On the left-hand page, Cask i made Debtor to all sums received ; and on the right, Cas/i is made Creditor by all sums paid. The Ledger collects together the scattered accounts in the Day-Book ar.d Cash-Book, and places the Debtors and Cre- ditors upon opposite pages of the same folio ; and a reference is made to the folio of the books from which the respective accounts are extracted, by figures placed in a column against the sums. References are also made in the Day-Book and Cash-Book, to the folios in the Ledger, where the amounts are collected. This process is called posting, and the following general rule should be remembered uy the learner, when engaged in transferring the register of mercantile proceedings from the previous books to the Ledger : — The person from whom you purchase goods, or from whom you receive money, is Creditor ; and, on the contrary, the person to whom you sell goods, or to whom you pay money, i^ Debtor. i- u n-n r In the Bill-Book are inserted the particulars ot all Bills of Exchange; and it is sometimes found expedient to keep .^<;<' this purpose two books, into one of which are copied Bills Receivable, or such as come into the tradesman's possession, and are drawn upon some other person ; in the other book are entered Bills Payable, which are those that are drawn upon and accepted by the tradesman himself. m PING, sactions of affairs in a !»/g Entiyy I business. ry, are tlie ll-Book. ader's pro- leir occur- Id. tions. On 3 received ; IS paid, nts in the 3 and Cre- L reference ! respective mn against -Book and mounts are ; following ner, when jroceedinga from whom )ntrary, tlie lay money, all Bills of o keep for opicd Bills possession, ,er book aro Uawn upon 157 DAY-BOOK. (Folio 1.) O bD O OJ fa J 1 I commenced business with a capital of Five Hun- dred Pounds in Cash Bennet and Sons, London,* C'r. By 2 hhds, of sugar, cwt. qr. lb, cwt. qr. lb. ' 13 1 4 12 12 3 16 116 January 1st, 1837. £ 500 • 2d gross wt. 26 20 tare 2 3 6 neatwt. 23 1 14 at 633. per cwt. 2 chests of tea cwt. qr. lb, 1 15 1 12 16. 25 2S s. d. 73 2 27 1 22 1 3 5 at 6s. per lb. 3d [Inlland Scatty Liverpool, Cr. Bysoap, Icwt. at 68s candles, 10 dozen at7a. 9d 60 12 6th Wurdf William, To 1 cwt. of sugar, 14 lb. of tea, i cwt. of soap, at 708. at 83. . at 74s. Vr. 8lh 133 3 3 18 8 17 10 •--Cooper^ William, 2 To 1 sugar hogshead . Dr. 1*^ In 6 6 6 6 • The student may bo directed to fill up this and similar blanks in this Book and the Ledger with the names and places familiar to him. 1 I 158 r? (Folio 2.) DAY-BOOK • 2 I 2 1 2 sssa 2 January 9th, 1837. £ 1 1 8. 16 17 15 6 Jolmsoii, tiichaidf To 2 dozen oi'candles. at 89. 3d. .. Ui. A cwt. of soau. at 74s. . . . ^ 1 i cwt. of sugar, at 70s • • » • • • _4 17 8 5 6 10th WanU William, To sugar, 1 cask, act. qr. Ih. gross vvt. 5 2 10 ca tare 2 10 Dr. 3K • • • • nent 5 at 689 .... 17 5 9 8 4 8 6 9 3 12th lirT Smith, Jo/in, To 14 11). of susrar 12 Ih. of caiidlps. .... .. .. .....••• 7 1b. nf soan...... .......... ...••••• ■• Uj, of tea • 1 6_ 10 J_6_ 13 16 6 10 6 14th Hall and Scott t Livetitool, Ev 2 cwt. soat). at 68s. . • Vr. 17th tSewton, John, To 21 lb. of soap, at 749. per 2 dozen of candles, at Ss. 3d.. Or. cwt... ^_1 10 9 4 13 18 8 4 2 2 S3sa 3 19th Dr. Smith, John, To 14 1b. of 8u?ar •.. ^ lb. of tea •■• 2l8t Smilh, John, Tn 'is \b nf Hilfrnr Dr, 1 2 \h of candles. .....•• 1 6 3 159 DAY-BOOK. (Folio 3.) 8. d. ) 16 6 [ 17 [ 15 1 8 6 ) 5 r r 5 3 9 ) 8 6 3 4 9 ) 8 3 \. 10 6 ~~* 5 J_6_ 13 10 ) 16 6 1 10 4 9 4 2 13 2 -*• == sstsss 18 8 3 1 1 6 3 2 3 2 2 2 3 January 23d, 1837. £ 172 _2 __0 3 r, 16 8 _9 16 5 6 6 Yates 8f Lane, Bradford, l-'*- By 4 pieces of superBne cloth, each ub yards, at 24a. per yard 23d. Edwards, Benj. Manchester, Cr. By 2 pieces of calico, each 24 yards, at Is. per yard. . . . 23d. Smith, John, "^• To 14 lb. of soap • • • 24th. Joknfon, Richard, ^^' 1 cvvt. soap, at 743 ij CWu OJ oUgarj aL*va»« •••••• 9 145 2 15 8 16 16 3 24th. Smith, John, O*"' To 1 lb. of tea 26th. Mason. Kduard, l^r. To 3 pieces of superfine cloth, each 36 yards, at 27s. per yard .... 2 pieces of calico, each 24 yards, at Is. 2d per yard, 148 J2 3 3 3 27th. 50 _172^ 46 55 2 8 _16 1 17 7 19 ir> \ 1 6 1 f'lu-krr, Thomiis, ^^• To 1 piece ef superfine cloth, 36 yards, at Z»a 31. Ht. . Hills liUinhle, •">*; By Yates &, Lane's Bill at 2 months, due April ^ invoxtorv, Jntmnrv 31 •'t. 1837. cut. n CO o CO o o «o O 00 "-H o 00 o to o «o o «o o in ' 00 o o rH T-4 CO ec w !!3 CO • S <5 S § CO K o c eq ca '^ •2 : IS en o o Si CO ^ o o ^ — ' g ^1 » § to u u a o c CO „ S"" o "O n3 j^jj bD H ffl c3 Sh' rt **• to r»-3 fee-, Qj s c ■ o o 5 P3 to C^0 . 00 C to CO O th C< CO CO o to o o o in o 00 CO CO to kn o 00 CO o Oi vn CO o o o o o o o o to to 00 (N o o ,-1 n\ T^ TH CO f^ 161 ;o o 00 t-H M (N >• ■H o »-H -< INDEX TO THE LEDGER. A Newton. John 2 N RjirnJird 8l flo -*- ....••••i o "■r^ Rpnnpffe. Sons. •••1 ■ ■ nilla nn vahlft.. ....«••• •••.3 Farker, Thomas 3 C P D a Edwards, B.^ Manchester,. .. 3 E R F stock account 1 C4 Smith, John 2 W G T Hall Sc ScoU, Liverpool .... 1 H V Johnson, KicUord ....2 1 Ward, William ...1 w K 1 X L Yatot» &, Laiic, Bradford 2 Y ' Mason> Edward •. ..•• Z j ._ _- _. i M o 162 LEDGER. o "o 1. U LEDGER. 163 o • CO • «o ^ o o» CO '^l QOO 00 -H 1 '^O ■<# <3) CO tn M o o O '* Oo o o c o S eo o 00 a o m o t^ ^(M w Q a .03 en «^ pa u 00 -1 00 CO cd ^ pa a> u c 6 cd cd CO CO •^ »~« t^ t- —1 fc** CO • e»5 . 00 . 00 a 00 a 2§ ^ cd Ti cd >-i •-» <-> ^» ss 07 00 • 00 • 00 § 2 § l-» >-i % in .5 '2 O o coo CO . or.' c ,-1 cd 164- LEDGER. o ^ »o 1-1 CO CO Oi 93 s 3 O w s « o PQ «J 5 0> ■^ O »-< 5 h- lO 00 li O O O N) C* M M o CO -^ a -