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Ariihmeticin Whole Numbers : the Rules of which are expressed in a clear, concise, and intt-lligible manner ; and the opt-rations illus- trated -by examples worked at length, and by numerous explana- tory notes and observations ; with an ample variety of examples for the exercise of Learners, calculat- ed'to initiate them in the Know- ledge of R^al Business. Also, the New Cuininercial Tables, adapted to the present Legislative Regulations of Weights end Measures, and the modern prac- tice of Trade. II. Vulgar Fractions; explained in an easy and familiar manner, in the practice of which the mo.'^i ele- gant and abbreviated modes of op- eration are peculiarly inculcated. III. Dcrimnl Frnctiitns ; eliicidnted wiih tile uiino.-t piT-piniiiiy ; with Involuiion, Hvoiulion, Po.-ition, rrocrrcs.-iioii, Jiiiil the calculatioa of Itiicresi and Annuities, on an cxtenik'd pcale. IV. Duodcciindls ; or the ^NIulMpli- caiioa of Fwi and Inches ; with nuiiKToua examples for praclicf, ailapfc'd to the various business of Ariilicers. V. JSleiisuratinn of Superficies; preceded by pltin and concii-e G'ioinr.tiiral Dcjl n i'iuiis. VI. A Collrction of Qiieslinps.prn- niisciioiisly arian<;i-'d ; intended as ri'capitiilatory exercises in tho principal Rules of Ariihmeiic. VII. A compendious System of JDook-Iieepiiig. BY FRANCIS WALKINGAME, Writing' Master and Jicconntant, FROM THE FORTY-THIRD DERBY EDITION. Revised, Corrected, and enlarged by the addition of SUPERFICIAL MENSURATION, AND .K COMPENDIUM OF BOOK-KEEPING BY SINGLE ENTRY, BY WILLIAM BlRKiN, Editor of the^Improved Edition of Jones' Dicrionanj ; Author of th^ Rational English Expositor, ^o. No. MONTREAL. R. ^ A. MILLER, 10, ST. FilANCOIS-XAVlER STRUET 1552. I . -5 r ! •-4 ■ i > ""**" I'liita 'J'l-IU lllllulT 'I'h'ius 'IVns I C. ui MiUio X. of C. ol 4 I n 1 JO .V I 1 I '■•1 -1 3 1 »J 4| S 5| 10 6| 12 7| 11 S| Iti 9 1 18" 10 1 L'O U 1 22 I2|24 P OF) S( s.d. 10 Oa 6 8 5 4 3 4 2 fi 2 1 8 1 4 1 3 1 OF . d. 6 s 4 . 3 . 2 . 1^ 1 . OF Urth ' 2 I . 1 jLfJ^O ■ AHITHMKTICAL TABLES. J.ONU MEAslKE. NIMEUATION. 'i'l-iij • . . . 'rh'>u«k-'li 'IViiS 111 'lililiMliiiS ('. Ill '1 linu.niiils X. ':( MllllI'lH 1 C. ol Millidiis I i I I 2 -! J t 4 I :"■ :> I 1 i 2 a 34 1 ■> . •> ' > 9 f ruuKN r M(im;v. 4 Farlliiri?s iiir.ke - I l'r"inv \1 IVr.r,' • - ■ I . !.';',.< 20 Sliiilmgs, 1 I''.uiiiJ ti| riDViu-i^u r d. 2(» 31 'M) :w, ■I-> >''M M 70 72 M» Kl •.til n; liiO KM K arc . H S . '.» .10 h et ril \;irils II pIlll'S IS liirhing 3 tniirs '.i' miles I loot 1 yard 1 liilh, 1 pole. 1 liir. 1 mile .... llras. 1 dti:- MUJ/ni'LICATION. 1 I -■I 3| ■>! '5 I 61 71 M »l 101 III 12 I 21 ■i\ 10 I 12 1 111 Hi I IS I JO I ii I 24 1 3 I CI 9| 1^1 I", I IXj 2I| ^■11 27 1 30 ;w| 3ti| ■»l 1^1 10 I 21) I 2s| :i2| ;ib| 40 I 14 I 4.s| 5 I (> 10 112 It I \^ 20 I J I 2'i I 30 30 I 3r, 3.i I 42 40 I 4S r. I '.4 :.o I 00 :>■> I hii 60 I 72 I -\ I 14 1 i-;i i 12-1 I .!'- I I 42 ! I lil I I lii I I 70 I I" I |h4| -Ml 3.' I iO I 4-<| ;-.. I 04 1 72 1 K)| SS| M I ' 1-1 4-. I ■.4 I 03 I 72 1 M I 00 I Oil I 10^ I I'.'i 20 I 30 ; 10 I -„.l Oo, 71' I 11 I 22 i 31 I ■111 Y, ; 00 I 30, 00 ~> M I.A.N :) MKA.ilKK. 9 Icit rnako 1 yard ',\"\ vds. 1 pole 40 p()lt!S 1 rood 4 id'xls 1 acre ( l.olll .MKA.^IKK. 2.', iiK :li. iiiako 1 nail 4 iiaiis 1 (|uar 3.|is 1 Fl.ell 4 (jis 1 yard .'i (js 1 hn.ell (i iir.s I Fr. ell .VYf. This trill.- niny Oe .i). pill 1 to Diiisioii ' y u vM>ii:K il : a.s M.f 2's 111 1 ;Hf 2 ; tlit 2's :u .irt' 3, 4tC. >0 ! 8S "'JO r'w 00 lOS 100 1 no |20 no 1 i>\ \2o'\ 132 132 ri44 STablcs of 5[i['*cifll)t.uii,fj. piiAcruK. OF A I'dlNn, OR SOVKUKRIN. s.d. £. 10 0; irc 1 half 6 8 1 tlurd 5 1 fdiirtli 4 () 1 (iftli 3 4 1 si.xth 2 1 fii^iith 2 1 tClltll 1 8 1 Iwcllth 1 4 1 tiflccuth 1 3 .. Isi.XlCL'lltll OF A Ti).\. Cwt. T. 10 are 1 half 5 .. 1 i.oirth 4 .. 1 I, Mil '2\.. 1 .•i-hfh •2 .. 1 Irlilh 1 is 1 tvviMiUclh 10 is 1 lv\ eiiflli OF A IHNDIIK.I). nr.lh. Ctct. '2 are 1 half 1 IS 1 lourlli IC) are 1 .sevoiit! 14 .- 1 cisiitli Al'dTliKCARIFS'. ur. make 1 srniple ."^cr 1 drain dr 1 oiirice o7. 1 [)ouiid WdOI.. It), make 1 rlove rlovcs 1 ^l()lle StOIK'S 1 t')d I loil.s 1 ucy v\ ( vs 1 sack sacks 1 la.st. OF A SHILLING d. 6 arc 4 .... 3 2 .... 1^ is 1 .... 1 half 1 third 1 fduith 1 sixth 1 oi;;lith 1 lUi.lKli TKOV. 24 ST. TiKike 1 ilw t 20d\vt. .. 1 (I/.. 12 oz. .. lib. AI.K AM) HK.KU. 1 2 pints make 1 ijoart 4 quarts . ,',) gallons ... :2 lirkins ... '2 kildcrkiii.s ,\i l.arrol 2 barrel.-; 3 luirrcls ) yalloi) 1 lirkin 1 kiia. 1 iiarref 1 hlol. 1 pilllC. 1 l)iitt TIME. 60 sec. 1 minute 60 inin. 1 lioiir 24 hours 1 day 7 days 1 week 4 weiks — 1 mouth DRY MEA.-.URE. 2 frail make 1 ])eck 4 pocks — 1 bushel 4 liii-hels ■■ 1 .sack 5 I)u>Ih'Is •- 1 quarter 4 (luarlers - 1 ehuld. 10 quarters - 1 last .SOLID MEASURE. 172S inches 1 solid loot 27 feet -• 1 yard COAL MEASURE. 3 bu-;hels -• 1 sack 30 bushels -- 1 chald. A\0IRI)n'01.S. 10 dr. niako 1 oz OF A rK.NNY. fnrth d. 2 are 1 half 1 is 1 fonrlh 10 oz 11 11> 28 1b. .... 4 <\\\ 20 cwt. .., ] lb. wiNp;. 2 pints make 1 quart I quarts 1 f:al. 10 i,'all()us 1 stone 4-1 -all. Ills 1 qi'. iV.i ._rall(iiis 1 cAvt, 1 o hiuis. I I'Oi. I 2 pipes. 1 aiik. Itierrt; 1 hhd. 1 i)!pe 1 lull CIT.STOMARY WEIOHT OF U00D3 lbs. A firkin of butter is 66 A tirkiii of soap — 64 A barrel of pot ash. 200 A barrel of ancho. SO A barrel of soap -- 256 . A barrel of butter 224 ' A fotherof lead, 19 cwt. 2 qis. or •- 2184 A barrel of candles 120 A Rtone of iron or shot - 14 A f^albui of train oil 7i A l.t.got of steel — 100 A stouc of glass — fl A Koam of glass 24 Htone, or 120 A rcil of parchment, h do/ en skins .V barrel of figs from nearly 90 to 360 j THE AUTHOR'S PREFACE. i ! •: Tn« public will, no doubt, be surprised to find there is •►'4.'/ pH tempt made to publish a book of AUITHMETIC, when ther* «#t ^t^ huinbers already extant on the same subject, and several of %^>ti oy8 3o not ill tlio loiirtt proiiiotc llitii- iiii|ii-o\< nwiit. so iicitlicr do tli«« HHinu questions iiMprvlc it. iStiilui- is ii in tin- |)o\vrr ol" iiny iiiiisft'i-, (in tlie course ol' liis business, ) linw lull of spirits stx-vfi- lie itiny !)«•, to Ir.Miio new I'xarnploM nt pk'iisiire in iMiiy IfuNs hut tlio siunc (jui'stiou \vil» IVo- quenlly nccur in tli<' smiui* Kulo, iiipt\villi>t;iu(iiiiir liis f,'rc;itt'st c;ue »md skill It) tlif »'(>uti:iry. " It niJiy also In- iliitlior ohjcdod, tli.il to I'lidi hy a printed btmk is nil iir;.niui«Mit ol" iLruoiMiirc and iiK :i|;;i( ily, whic-li is no less tritlinii than tlio fonin'r. lie, iiidi'fd, (if siicli a «in(' tlioro he.) who is aliaid Iii.s Bcholars ', ill iniproM- too last, will utidoidilt'dly dtcry this lucthod. hut that luastcr's iL'iiorauco can in vur ho liroULlit in (|ncslioii who fan hfiriu and end it ivadily; and. most ( eilainly, lh;it s(li()l;ii"'s inni-iinproveuicut can he as little <|nestinnrd \\ ho makes a miu-li j:realer progress by tliitj than by the eomuinii method." To enter into a lou^' detail of every Ride would tire tlie reader, and swell the prelaee to an unusual length; I shall, therefore, only fiivo h general idea of the method ol' proceeding', and le;ive the rest to speak lor itself, which, 1 hope, the reader will lind to answer the title, aial tho reconimendation iriven it. As to the Hides, they follow in the sanio manner as the table ol contents ^|leciiies, and in much the same ord«;r as they are f:eiieiiilly tauirhl in .«iclii)ols. 1 havi" ^'oiie lhrouj,'h the four fuiidaineiital liules in Inleirers lirst, before thos(> of several denomina- tions, ill order that, they beinir well niuleri-tood. the latter w ill be per- formed with much more ease and despaicli. accoiS,('.. and have not only shown the use, but the iiii'lliod of niakin:: them; likewise a Table calculated for find- ing the Interest of Moiiev I'or aiiy nniid)er of days, ;it any rate per cent., by Miilti[ilicalioii and Addition only, which may also be applied to tlio calculation of Incomes, Salaries, or Wa^-s, for any numbi'r of days; and I may venture to say, T have i!one through the whole with so much plainness and |VM\spi(iiity, that tia-rt" is niau' better extant. I have nothing farther to add but a return of iiiv sinci re thanks to all those gentlemen, schoolmaslrrs, and others, whose k'lid apf)robation and eucouraLM'iiient have now estaili!^.hed tla; \i>c of this book in almost every school of eminence thrtai'.diout the kiiiirdom, but I think my j:^rati- tiide more especially due to thos*- who hav(> l;i\oiired me with their re- marks ; tlioniih I must still be:: of every c;;n>lid and judicious reader, that if he should, by clianc(>. Hud a l!'ans].(t.-itioii of a letter, or a false figure, to excuse it; for, uotw ilh.--tandinLr there has been izreat cave taken in correctinji, yet errors of the pie-s w ill inevitably creep in, and some may also have slipped my observaliou ; in either of which cases the ad monition of a U(jod-iiatiiii'd n^atler will be very acceptable to his Mucli obliged and most obedient humble senant, V. WALKIN.GAME CONTENTS. Ml ^ 1 11 ) KoMERATioN and notation Adilltiou \3(S INTKGEKS. |Sll!)lr!lr8 ^•2,.^^.,,^,„^,^ ^^ g«^""V""" ^ Conjuimd IVoi-orticm 110 Bed..ct.on ':^; Involution.... 112 Compound Addition I^j Kvolnlion 112 ' cmbtractiou >' i>; .,,,1. f«i?v.» na ,, ,. ,. . .,J5i(iMa;r'«»"» Ilsin^'lo r..siunn .... Promiscuous Lxamp es ;^;: D.ml.lo I'osition. .. . Rule of 1 hree Direct 4ft ' ' Inverse .6M 119 120 121 Proizression Arithmetical 123 (icoiiietrical 128 Double Rule oi Three o3 ^.j,^ j,. j.^^^,.^^^ ^,,j Annuities .ASj J''"'^^'^^ '? Discount 133 140 154 Tare and Tret Jli K.iuation of ravineiits Invoices, 7(') Alligation 78 Comparison of Weights and Measures 81 Permutation 82 &c. Viuiliy EfOll . 1S1 Ti'ian-le 165 TraiK'ziuni.&c. 165 an liTegular 16« a Regular Poly • ....-" 1C6 VULGAR FRACTIONS. Definitions, &c ^- Circle U>7 R^ductiou l]\-^^Avouof\\[V.y.'.'.V.['.V.'.m Adniiies 7 miuus \ equal 3: that is, 4 taken from 7 leaves 3. 4X3=12 is read 4 info 3 eqval \2 : that is, 4 multiplied by 3 equiil \2. 13-f-4=3 is read 12 //y 4 fywoZ 3. But Division is more con- voniently expressed in the form of a Fraction: thus, y r=3 ; twelve divided by four equal three. As 2 : 4 : : 8 : 10 ; As 2 are to 4, no are 8 TUTOR'S ASSISTANT, BKINO A COMPEin)IUM OF PRACTICAL ARITIBIETIC. Integers, or Whole Numbers. Arithmetic is the science of numbers ; or the an of numert, cal computation. A whole number is a unitt or a collection o^ units. Nambers are expressed by ten written characters called figures, or digits: viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, which arft significant figures, all declaring their own values by the names ; and the cipher, or naught (0) an insignificant figure, indicating po lalue when it stands alone. NUMERATION AND NOTATION. A figure standing alone, or the first on the right of others, denotes only its simple value, as so many units, or ones ; th0 second is so many tens ; the third, so many hundreds, &c., in: creasing continually towards the left in a tenfold proportiori. Numeration is the art of rending numbers expressed in fii^ nrcs ; and Notation, the art of expressing numbers by figwes THE TABLE. 1) a .2 fa CQ 3 . o 5 7 3 5 -i H 8 a c • •-• H Cm O 09 •73 I 6 9 X ai ^ z s o 2 7 3 'I 2 a - © a 8 4 1 . ' Millions' period. Units' periml. Tliw Table might be inrmitfly jyxteiwied A 2 I i( ?» NUMERATION (tutor's NoTK. To read any Numocr. Divide it into periods of six figiireB each, bepinnin;^ at the right hand : iuid rach period into scmi-periodt with a di/fcrctii mark, for tlie sake of distinction. Tlie Jirftt on tho right hand is the Uuila^ pm-iod, llio second the Millions' \)cv\o(\, &c. Be- pinning at tho left ol)sei've tiiat tho three liiriires of every complete semi' period Tiiust bo reckoned as so ni;!"' hiaulrrds, tmis, and unils; join- ing the word tliousaiuh when yoi; ennie to llie iniddio of the period, and the proper name of the period at the end of it. 2. To express ani/ {riven. Nmnlier in Fiut observe, that though every semi- period but the lii'sl on the left must have its complete number o[ Ihrec fig- ures, that may 1)e incomplete, and consist of only one or two iigures: alsO) where signijicant Jif^i/res are not required in any [lart of a number, no semi-period nmst be omittcil, but the places nuist be hlled up with ciphers. 'Exami)le. Write in ficrnT(>s, povf^ity thnnsand four hundred billions, two hundred and ten thousand millions, and ninety-six. Fh'st, write 70 (seventy) with a comma, these being thousands ; theii 4OO (four hundred) with a siinacoion, denoting the end of the period; jiext, write 210 (two hundred and ten) and. Ijccause they are thousands, put a comma after tliem, and then 000 (thi'ee ciphers, tl)ere being no piore millions) followed by a semicolon, to deiKJte the completion of ^he period; again, put 000 (three more ciphers, denoting the absence of thousands) with a comma after them, and then 096, (ninety-six,) Vbich will comi)leto the number: thus, 70, 100 ; 210,000 ; 000,096. 1 ^ I (2) 3) 4) 5) (6) (7) (8) (9) (10) (11) (12) EXERCISES IN NUMERATION AND NOTATION. Read, or icritc in icords the foUomng numbers 3 30 33 300 303 330 333 127 172 217 271 712 (13) (14) (15) (10) (17) (18) (19) (20) (21) (22) (23) (21) 721 900 4291 94294 294294 3703 703703 311311 113113 131131131 708807780 807078087 (25) .500050005 (20) 1010100 (27) 11 1101 01 (28) 49090494 9 (29) 3584G00987 (30) 584610070840 (31) 5846100708100 (32) 37613590200110 (33) 5008000400000 (34) 601008000180070 (35) 37000000000075048 * The figures hi parentheses refer to the Editor's Key to thij Afi«lSTANT.j NUMERATION. 11 Express in figures the following numbers. (1) Nine; ninety; ninety-nine; nine hundred; nii.e Luv..- dred and nine ; nine hundred and ninety ; nine hundred and ninety-nine. (2) One hundred and eight ; one hundred and eighty ; <^ight hundred and one ; eight liundred and ten ; one hundred and sixteen ; one hundred and sixiy-one ; six hundred and eleven. (3) One hundred and twenty-throe ; one hundred and thirty- two ; two hundred and thirteen ; two liunch-ed and thirty-one* three hundred and twelve ; three hundred and twenty -.one. (4) Two thousand five hundred and seventy-two. (5) Seventy-two thousand five hundred and seventy-two. (6) Five hundred and seventy-two thousand five huiidreil ^nd seventy-two. (7) Ten thousand nine hundred and ten. (8) Nine hundred and ten thousand nine hundred and teau (9) One hundred and nine thousand nine hundred and one. (10) One hundred and ninety thousand and ninety-one. (11) Nine hundrjed and one thousand and nineteen. (12) One hundred and fourteen millions, one hundred and foriy-one thousand four hiuidrcd and eleven. (13) Four hundred and six millions, six hundred and four jthousand four hundred and sixty. (14)' Six hundred and forty millions, forty-six thousand aad sixty-four. -.. (15) Seven millions, seventy thousand seven hundred. (16) Seven hundred millions, seven tliousand and seventy. (17) Ten millions, one thousand one hiin.u-ed. (18) One hundred and one millions, eleven thousand cn6 hundred and ten. (19) Twelve billions, s&venteen thousand and nine millions and eighty-nine. (20) Seven thousand five hundred aiid luur trillions, sixty thousnnd million*, eight hundred thousand. Roman Nii/ticrt . I I ' I I Oue. ♦ I 6 Six. :•; I 1 1 F'leven. IT 2 Two. ni 7 Sevcc- f Xil 12 'I\\eive. i" 3 Throe ' III 8 Kii^ht. XIII 13 Tiiirteeij |V 4 Four. X 9 Nino XIV 14 Frmrf /t J y 5 Five X 10 Ten XV l.= Filti /, 'i i 12 ADDiriON OF IN'TEGERS. [tutor'i SVI 19 Sixteen. XVII 17 Seventeen. XVIII 18 Eighteen. XIX 19 Nineteen. XX 20 Twenty. XXX 30 Thirty. XL 40 Forty. L 50 Fifty. LX 60 Sixty. LXX 70 Seven t>^ LXXX «0 Eighty. XC 90 Ninety. C 100 Ojie hundred. CC 200 CCC 300 CCCC 400 D 500 DC 600 DCC 700 DCCC 800 DCCCC 900 M 1000 MDCCCXXX Two hundred. Three hundred. Four hundred. Five hundred. Six hundred. Seven hundred. Eight hundred. Nine hundred. One thousand. 1830 One thouMiu) ei^ht hundred and thu*ty. Note. A less numencsJ letter standing before a greater, must b© ta- ken from it, as I before V or X, and X before L or C, &c., thus, IV. Four; IX. Nine; XL. Forty; XC. Ninety, &c. And a less numerical Ititter standing after a greater, is to be added to it, thus, VI. Six ; XI. Eleven; LX. Sixty; CX. One Hundred and Ten. All Operations in Arithmetic are comprised under four elemeiitary or fuiidamentcd Rules: viz., Addition^ Suhtraeiion, Multiplication, and Division. ADDITION Teaches to find the sum of several numbers. Rdxe. Place the numbers one under another, so that onitB may ttVBiH tinder units, tens under tons, &c. ; add the units, set down the mnto'Si their sum, and carry the if^w* as so many ones to the XiBXtYtrfr\ pro- ceed thus to the last row, under which set down the Tvhole amount. Proof. Begin ac the top and add the figures downwards : if tht» .luwt is found the same as before, it is presumed to be right. •(I) 275 (2) 1234 (3) 75245 (4) 271048 110 7098 37502 325476 473 3314 91474 107684 954 6732 32115 625608 271 2540 47258 T54087 352 6709 21470 27 ^36 tjay 2 and J are 3, and 4 are 7, and 3 ai-e 10, and 5 are 15, set down 6 fuid cany 1 ; 1 mid ^s are f), and 7 are 13, and 5 are 18, and 7 are 25, and one are 20, and 7 are 3:], set down 3 and carry 3 ; 3 and 3 ai-e 6, and S ;.re 8, and 3 are 11, and 4 are 15, and 1 are 16, and 2 are 18, set down 10: so iho snm i.s ]«:]."). r Ailor practising a finv examples, it will be bettor for the leam*r to i I 1 1' i.! j #1 <) ) ; .)f , 'ii ' it-' ftw-i lAsl rnl I 981 itfilSTAN'T.] ADDITION OF INTEGERS. (5> 590046 (6) 370416 (7) 781943 73921 2890 56820 400080 60872 1693748 4987 998 300486 19874 47523 920437500 201486 9836 78632109 9883 26627 9408175 15 (8) What is the sum of 43, 401, 9747, 3464, 2263, 314, 074? (9) Add 246034, 298765, 47321, 58653, 64218, 6376, 9821, and 640 together. (10) If A has X'56. B i:i04. C £274. D £1390. E X7003. F £1500. and G £998. ; how much is the whole amount of their money ? (11) How many days are in the twelve calendar monllis? (12) Add 87929, 135594, 7964, 3621,27123,8345,35921; 2374, 64223, 42354 j 3560, and 162165 together. (13) Add 6228, 27305, 7856, 287, 7664, 100, 1423, 25258, 028, 3135, and 838. (14) How many days are there in the first six months of ti^ year ; how many in the last six ; and how many in the wtole ? (15) In the year 1832, how many days from the Epiphany W Twelfth-day (Jan. 6th) to the last day of July r ' (16) In the common year how many days from each Quar- ter-day to the next? That is, from Lady-day to Midsummer-day, from thence to Michaelmas-day, from thence "to ' Christmas- day, and from Christmas-day to the ensuing Lady-day ? (17) When ^v•ill the lease of a farm expire, which was granted in the year 1799, for ninety-nine years? ■ (18) A person deceased left his widow in possession of £2500. His eldest son inherited property of the value of £11340. To his two other sons he bequeathed a thousand pounds each more than to his daughter ; whose portion ex- ceeded the property left to her mother by £500. A nephew 8|)d a niece had legacies of £525. each ; a public charity £105. ; and his four servants the same sum to be divided add the figures without naming them. Thus, in adding the first col- umn of the above example, say 2, 3, 7, 10, 15 ; set down 5 and carry 1, &c. r •T^ ^i^ethod will tend both to quickness and precision. . : *' ' '1 J } ' ■ I 14 SUBTRACTION OF INTEGERS [tutor 9 amoiigst tbem. What was the aggregate amount of his pro- perty? (19) Tell the 7iamc antl .signi-fication of the sign put between the following numbers : and rind what they are equal to, as the sign requires. 172 l-f6 19-}- 174-5400-f 12+999. (20) Required the sum of forty-nine thousand and sixteen; four thousand eight hundred a:id forty ; eight millions, seven hundred and seven lliousand one liundred; nine hundred and ninety-nine ; and eleven thousand one hundred and ten. (21) When will a person, born in 1819, attain the age of 43 ? (22) Henry came of age 13 years before the birth of hig cousin James. How old will Henry be when James is of age ? (23) Homer, the celebrated Greek poet, is supposed to have flourished 907 years previous to the commencement of the Christian era. Admitting this to be fact, how many years was it from Homer's time to the close of the 1 8th century ; and howlongto A. D. 1827? (1 SUBTRACTION Teaches to take a less number from a greater, to find the re- niainder or Difference. The number to be subtracted is the Subtrahend, and the other is called the Minuend. Role. Ilaviniz placed the Sublrfiu«»!i(l unclrr the Mlnueud, (in tho same order as in Adilitiou.) lieyiii at the unit-'., and subtract each tifrure from that above it, setting down the leniuinder iiriderneath. But when tho lower figure is the greater, b.irrow trn; wliich add to the upper, and then Bubtract: pet down tlie reuiri'iKder, and cany one to tho nextfigur© ^f the Subtrahend for the fen tknt, was borrouinL Proof. Add the Difference to tlie Subtrahend, and their sum will bo the Minuend. (1) From 27147.04 Take 1542725 (^) 27150S300 72811699 (7) 1000000000 987654321 (2) 4208730G 34090187 (^) 37.0021509 278101609 (8) 2746981340 1095681539 (3) 45270.'509 32761684 («) 40008763.^ 91S42G7 (^) 666740825 109348173 « AaeiSTAN'T.] SUBTRACTION OF INTEGERS. u (10) From 123456789 subtract 98765432. (11) From 31M7680975 subtract 767380799. (12) Subtract 641870035 from 1630054154. (13) Required the difference botweon 240914 and 24091 (14) IIow mucli docs twenty-five thousand and four exceec' eixteen thousand tliree hundred and ninety ? (15) If eiijhly-loiu- thousand and forty-eight be deducted from half a million, what will remain? (16) The amnial inr-ome of Mr. Lommington, senior, is twelve thousand five hundred and sixty pounds. Mr. Lem- inintrton, junior, has an income of seven thousand eight hun- dred and eiLiliteen pcnnids per annum. How much is the son's income less than his fatlsor's ? (17) George tlie Fourth, at his accession to the throne, in 1820, was in the 58th year of his age. la what year was h© born, and how long had he reigned on the 29th '^f January, 1829, the anniversary of his accession? (18) The sum of two numbers is 36570, and one of then^ is twenty thousand and twelve : w hat is the other ? (19) Thomas has 115 marbles in two bags. In the green bag there are 68 : how many are there in the other? (20) Two brothers who wore sailors in Admiral Lor^ Nelson's fleet, were born, the elder in 1767, and the youngr er in 1775. What Avas the dilleronce of their ages, and how old was each when they Ibuglit in the battle of Trafalgar, in 1805? (21) Henry .Jenkins died in 1670, at the age of 169. How long prior to his death was tlie discovery of t!ie continent of America by Columbus, in 1498 ? — Also, how many years have elapsed from his birth to 1S27 ? ExAMPT.E. From 3':f)0G.')47 siiV-tract n-?]Oin;i. 32n0(;r)47 Miunciid. 8210408 Siibtruheu'l 24^()079 DilTci-cncc. 329065 17 ProDf. Siiy 8 tVoiii 7 I ciiiiiiDt; borrow 10. and 7 are 17. V, from 17, D rfMimia; sot, down 9 and oaiTV 1. — 1 \n\'\ (i ;iiv 7. 7 f'-om 4 T cannot; borrow 10. niid 4 are 14. 7 from 14. 7 ; set down 7 and rairv 1. — 1 nnd 4 aro ."), .') I'mm T), nothing; set down {'•)) nanidit. — fi'oin C, (! ; sot (iovvii (i. — 1 fiom Icnnnot; bnt I from 10, 9; set down i) and cany 1. i'locrecl in like niainier to the cud. When the pupil is initialed in the priiftlee by worlwincr an example or two, he may sim])lifv tlie work by oml'tiiiir t() express some of the par- ticulars. Thus, in the preceJinc examj>le, it A\ill be suhiciont mere'" to Bay, 8 from 17, f); .set dor.-a !) a-!.! (Tii-ry ] ; 1 and 6 ai-e 7, 7 '* 7; Bet down 7 and can-y 1. Sec. ' I I I t ' \ \ u MULTIPLICATION OF INTEGERS. iTWTOIl'i (22) Borrowed at various times, JC644., jC957., dC90., iri378., and jC1293.; and paid again the different sums of X763., JC591., JC1161., £1000., and je847.— What remains unpaid ? (23) Explain the name and signification of the sign used ; and work the two following examples. 10874 — 9999 51170 — 50049 (24) John is seventeen years younger than Thomas : how old will Thomas be when John is of age ; and how old wilJ John be when Thomas is 50 ? I A8S181 (1)M (2) M (3) M (4) M (5) M (6) M MULTIPLICATION Teaches to repeat a given number as many times as there are units in another given number. The number to be multiplied is called the Multiplicand ; that by which we multiply is the Mvltiplier ; and the num- ^r produced by multiplying is the Product, Rule. When the multiplier is not more than li miiltiply the units' fignre of the mxiltiplicand, set down the. units oi thiB prodnct, reserving ^ Uns; multiply the next fipnirc, to the product of which carry the Uns reserved : proceed thus till the whole is multiplied, and set down th« last product in full.* MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 & 10 11 12 8 9 10 11 12 4 6 8 10 12 14 16 18 20 22 24 6 9 12 15 18 21 24 27 30 33 36 8 12 16 20 24 28 32 36 40 44 48 10 15 20 25 30 35 40 45 50 55 60 12 18 24 30 36 42 48 54 60 66 72 14 21 28 35 42 49 56 63 70 77 84 16 24 32 40 48 56 64 72 80 88 96 18 27 36 45 54 63 72 81 90 99 108 20 30 40 50 60 70 80 90 100 no 120 22 33 44 55 60 77 88 99 110 121 132 24 36 48 60 72 84 96 108 120 132 144 * Example. Multiply 71.1097 by 4. Say 4 times 7 nre 2S, set down 8 and carry 2 ; 4 times 713097 9 are 36 aiul 2 art> 33, set down 8 and carry 3; 4 times 4 (iiauirlit) and 3 are 3, set down 3; 4 times 3 are 12, S852383 ^^"^ down 2 niid carry 1 ; 4 times 1 are 4 and 1 are 5, set dov n 5; 4 tiiup-; 7 ure 23; Rvt down 23. I ASSISTANT.] MULTIPLICATION OF INTEGERS. 17 (7) Multiply 3725104 by 8. (8) Multiply 4215466 by 9. (9) Multiply 2701057 by 10 .• (10) Multiply 31040171 by 11. (11) Multiply 73998063 by 12. (1) Multiply 251 04736 by 2. (2) Multiply 52471021 by 3. (3) Multiply7925437521by4. (4) Multiply 27104107 bv 5. (5) Multiply 23104759 bv 6. (6) Multiply 7092516 by>. (12) Multiply 780149326 by 3, 4, 5, 6, 7, 8, 9, and 10. (13) Multiply 123450789 by 4, 5, 6, 7, 8, and 9. (14) Multiply 987654321 by 9, 10, 11, and 12. When the niuhlplier is between 12 and 20, multiply by the units' figure in the multiplier, adding to each product tho last figure multiplied. f (15) 5710.592X13. (18) 2057165X16. (19) 6251721X17. (20) 9215324X18. (21) 2571341X19. (16) 5107252X14. (17) 7653210x15. When the multiplier consists of several figures, multiply by each of them 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 pioduct.^ Proof. Make the former Tnultiplicand the multiplier, and the multi- plier the multiplicand ; and if the work is right, the products of both operations will con-espond. Othertcise A presumptive or probable Eroof (not a positive one) may be obtained thus : Add together the gures in each factor, casting out or rejecting the nines in the sums as you proceed ; set down the remainders on each side of a cross, multiply them together, and set down the excess above the nines in their product • To multiply by 10, annex a cipher to the multipUcand, for the pro- duct. To multiply by 100, annex two ciphers, &c. Examples. i Multiply 96048 by 15. Say 5 times 8 are 40, set down and carry 4 ; 5 times 4 are 20 and 4 are 24, and 8 are 32, set down 2 and carry .3 ; 5 times and 3 are 3. and 4 are 7, set down 7 ; 5 times 6 are 30, set down and carry 3 ; 5 times 9 are 45 and 3 are 48, and 6 are 54, set down 4 and carry 5 ; 5 and 9 ar« 14, set down 14. X Multiplv 76047 by 249. 76047 249 Proof. 684423 Product by 9. 304188 do. by 40. flX6 15 2094 do. by 200. 18r/3.')703 Total product 96048 15 1440720 !M ; « ' I« MULTIPLICATION OF INTEGERS. [TUTOR'! At the top of the cross. Then cast out tlie nines from the product, and place the cx^ss bolow the crDrtrf. 11" these two coiTespond, the work la probably right: il' not, it in certain! y wrong. (22)271041071X5147. (23.) 6231 0047 X1GG8. (21) 170925161X7419. (25) 9500985?.i2X 61879. (26) ir01495868567X47687j6. "When ciphers arc intermixt J with the siirnificant figures in ,lhe multiplier, they may be omitted ; but great care must be taken to place the lirst figure of the next product under the figure you multiply by.* Ciphers on the right of the multiplier or multiplicand (if .omitted in the work) must be placed in the total product.f (27) 571204X27009. (28) 7561240325X57002. (29) 562710934X590030. (30) 1379500X3400. (31) 7271000X52600. (32) 74837000X975000. A number produced from multiplying two numbers to- gether is called a composite number; and the two numbers producing it are called the factors, or component parts. When ihe multiplier is a composite number, you may multiply by one ,of the factors ; and that product multiplied by the other wiljl give the total product. | ,(33) 771039X35. .(34) 921563X32. (35) 715241 X. 56. (36) 679998X132. (37) 7984956X144. (38) 8760472X999.^ (39) 7039654X99999. (40) A boy can point 16000 pins in an hour. How many .can five boys do in six days, supposing them to work 10 cleai Jiours in a dav ? Examples. • Multiply 318G4 by 7008. 318G4 700 8 254912 8530 48 S2330i2.912 Proof. 6 4X6 Q t Muhiply 63850 by 5200. 63350 5200 Proof. 1 4X7 I 12770 31925 ^3202 0000 t Multiply 63175 by 45. 63i75 5X9=45 315875 9 2842875 § For jin abridged method of multiplying by a series of ninet, see the Key. 9. ASSISTANT.] DIVISION OF INTEGERS. If (41) If a person walks upon an averajLje 7 miles a day, how many miles will he travel in 42 years, reckoning 365 days to a year ? (42) Multiply the su77i of 365, 9081, and 22048, by the difference between 9081 and 22018. (43) Required the continued product^ 112, 45, 17, and 99. NoTK. Multi|)ly ill! tho miiiibrrs one into another. DIVISION Teaches to find how often one nuiniier is contained in aix- other: or lo divide a nuni!)ei- into ;uiy Cijual parts required. The number to be divided is called the Dividend ; that by which we divide is the Divisor ; and the number obtained by dividing is the Qnotient ; which siiows how many times the divisor is contained in tlie dividend. When it is not contain- ed an exact number of times, tiiere is a part of the dividend left, which is called the Remornder. Rule. When the divisor is iu)t more than 12, find how ofter^ it is contained in the first figm-e (or two figures) of the divi- dend ; set down the quotient underneath, and carry the over- plus (if any) to the next in the dividend, as so many tens; find how often the di\ isor is contained therein, set it down^ and continue in the same manner to the end. Wheii 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; ])lacc the quotient figure on the right, multiply the divisor by it, subtract the product from the di- vidual, and to the remainder bring down the next figure of the dividend, which will f(»rm 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, adding the remainder (if any) and the product will be the same &9 the dividend. (1) Divide 725107 by 2.* | (2) Divide 7210472 by 3. * ExAMiT.K. Divido 7.:!'28105 by 4. Divisor 4)7:?0o10r) Dividend, S:iy the fours in 7, once and 3 overj Quotient T83^-J()— 1 Rem. ^^"^ *'^'"^'« "^ P- ^ ?^^^^'^ "* 'V^ ^- °^^ 4 1 over; the lours in 1-2, 3 innes ; the — r-— -rr ,, c fours in 8, twice; tlie fours hi 1, and ^'^'-'^'"•^ 1 root. J ^^.^,^.. ^jjg j;„j,.^ i„ iQ^ j^^.j^.g 4 jjyg g'^ arvl 2 over; the fours in 25, six fours aie 24 tuid 1 over. 9d DiriSlON 07 IXTEGEnS Ittjtoi $ (3) Divide 7210416 by (4) Divide 7203287 by (5) Divide 5231037 by (6) Divide 2532701 by (7) Divide 2o 17325 by (8) Divide 2501730lf'l)y (9) Divide 70312(515 by 10. (10) Divide 12801763 by 11. (11) Divide 79013260 by 12. (12) Divide 37000121 by 3, 5, 7, and 9 5. 6. 7. 8. 9. (14)7210473 -r37.« (15)42749467 —347. (16)734097143 -r5743.t (17)1610178407 -r-54716. (18)4973401891 —510834. (19)51704567874-4765043. (20) 17453798946123741-r 31479461. (21)25473221 -^27100.t (22)725347216 -f-572100. (23)752173729 H-373000. (24)6325104997 -t-215006. (13) Divide 111111111 bv 6, 9, 11, and 12. When the divisor is a composite numher, you may divide the dividend by one of the component parts, and that quotient by the other; which will fjive the quotient required. But the true remainder must be found by the following Rule. Multiply iho second remainder by the first divisor; to that product add the first remainder, which will give the true one. (25) 3210473-^27.§ (26) 7210473-^-35. (27) 6251043-i-42. (28) 5761034-^54. • Example. Divide 4085.5 by 2D. Dividend. Di\'isor 29)40«55(l'Jn3 Quotient. 2!) 29 rii I2(i72 116 2316. 255 23 Remainder. 232 40855 Proof. "23 t When the di\'isor is large, the quotient figures are most easilj found by trials of the first figure (or tivo) in the leading figuret '*f the dividend. ' X Ciphers at the right of the divisor may be cut off, and as TMJXf figures from the right of the dividend ; but these must be annexed «• the remainder at last. * $ Example. Divide 314659 by 21. 21=7X3)314659 7)104886—1 14983—5 -=5X3+l=rl6 rem. ho at th! ar p? t 16. 34. 043. )0.t 00. )00. )00. .4 ieSISTANT.] DIVISION OF INTEGERS. at A number may be divided by 10, 100, 1000, 40... 3 4 180.. .15 100... 5 10 ...2^ 38... 9k 72... 6 50... 4 2 190... 15 10 110... 510 12 ... 3 40.. .10 84... 7 60... .5 200... 16 8 120... 6 14 ... 3^ 16 ... 4 42... 10^ 44.. .11 96... 8 108... 9 70... .5 10 80... 6 8 130... 6 10 140... 7 Skill ill gs. 18 ...4^ 46.. .lU 120.. .10 90... 7 6 s. £. s. 150... 7 10 20 ... 5 43... Is. 132.. .11 100... 8 4 20 ai-e 1 160... 8 22 ...5^ ' H4...12 110... 9 2 30... 1 10 170... 8 10 24 ... G Pence. I 56... 13 120.. .10 40... 2 180... 9 26 ...6i d. s. 168... 14 130. ..10 10 50... 2 10 190... 9 10 28 ...7 30... 7i 12 are 1 180.. .15 140.. .11 8 60... 3 200.. .10 04 192. ..16 ^^150... 12 6 70... 310 210. ..10 10 1*h€ trui are 8« 4'SftISTANT.] WEIGHTS AND MEASURES. S^ V N0T8. When the units' figure is cut off from any ntimber of shil- lings, half the remaining figures will be the pounds. Tims, 256«.3m £12. 16». becaiise half of 20=12 ; and the one over prefixed to the 6, gives 16 1. T.. > 4 4 10 5 5 10 6 6 10 7 7 10 8 8 10 9 9 10 10 10 10 it- * WEIGHTS AND MEASURES. TROY WEIGHT. 24 grains (gr.) make 1 penny w^eight, dwt. 20 pennyw^eights 1 ounce . oz. 12 ounces . . 1 pound, . fij. Grains. 24 = 1 pennyweight. 480 = 20 = 1 ounce. 5760 «= 240 = 12 = 1 pound. ^ (jold, silver, and gems, are weighed by this weight apothecaries' weight. 20 grains {gr.) make 1 scruple, . . 3 3 scruples ... 1 dram, . . 3 8 drams ... 1 ounce, . . 3 12 ounces ... 1 pound, . . R. Grains. 20 = T scruple. 60 = 3=1 di-am. 480 = 24 = 8=1 ounce. 5760 = 288 = 96 = 12 = 1 pound. This is used only in the mixing of medicines. These ai'e the same grain, ounce, and pound, as those in Troy Weig&l» AVOIRDUPOIS WEIGHT. 16 drams (drr.) make .... 1 ounce, . . oz, 16 ounces 1 pound, . . lb. 14 pounds 1 stone, . . st. 28 pounds, or 2 stones .... 1 quarter, . qr. 4 quarters, or 8 st. or 1121b. . 1 hundred, . cwt, 20 hundreds 1 ton, . . t. Drams. 16 = 1 ounce. 256 = 16 = 1 pound. 3584 = 224 = 14 = 1 stone. 7168 «. 448 = 28 = 2=1 quarter. 28672 =« 1792 =112= 8=4=»:1 cwt. 573440 = 35340 -. 2240 = 160 = 80 = 20 = 1 ton. By this weight nearly all the common necessaries of life are weigh- «d. A truss of hay«=56 lb. and one of 8traw=30 lb. A load is 3$ trusses. A peck loaf weighs 17 »>. 6 oz. 1 dr. In the metropolis, 8 «>• are a stone of meat. A fother of lead is 19^ cwt. In some districts) goods of various descriptions (as cheese, coai, &c, are sold by the lonf^ u WEIGHTS AND MEASURES [tutor'A WOOL. Wheii wool is purchased from the grower, the legal stone of 141b. and the tod of 28 Hj. are used. But in the deal- ings between woolstaplers and manufacturers, 15 pounds are . . 1 stone. 2 stones or 30 ft, 1 tod. 8 tods, 01 240 lb. . 1 pack or sack. COMPARISON OF WEIGHTS. A gra.in is the elementary or standard weight 1 ounce avoirdupois is . . 4371 grains. 1 ounce troy 480 1 pound troy 5760 1 pound avoirdupois . . 7000 175 pounds troy =144 pounds avoirdupois. 175 ounces troy=192 ounces avoirdupois. We may, therefore, reduce lbs. Tnjy into Avoirdupois by multiplying them by i44, and dividing by 175, &c. LINEAL, Oa LONG MEASURE. 12 inches (in.) make . 3 feet, or 36 inches . 2 yards, or 6 feet 5^ yards, or 16^ feet 4 poles, or 22 yards . 40 poles, or 10 ch., or 220 yds. 8 furlongs, or 1760 yards . 3 miles 1 foot, . ... ft 1 yard, . . . yd. 1 fathom, . . . fa. pole, rod,^ or perch, p. 1 land chain,* 1 furlong, 1 mile, 1 league, . ch. fur. tn. Barley-corns. 3 ■» 1 inch. 36 = 12 = t foot. 108 ». 36 . 3 z=e i yard • 594 =. 198 r=. I6fi » 5^ »r 2.3760 =. 7920 = 660 =x= 220 = 1 pole. 40 = 1 furlong. 190080 -= 63360 =« 5280 =1760 = 320 = 8 = 1 mile. Noric. It is commomy supp"«*"rl tnat tno Eus.ish inch was origfinal ly taken from three grains of barley, selected iVom the middle of th« ear and well dried.' A twelfth part of an inch is called a line. , 4 inches are a hand, used in measuring the height of horses. 5 feat are apace. A cubit = 1^ feet neai'ly. This measure determines the length of lines. A line has the dimea ilon of length only, -without breadth or tliickness. i • The chaw consists cf 100 Anks, each link bebg » 7>&Q inchei. •OR'd 1 !1! !1? .•i 1 ,di AMlftTANT.J TABLES OF MEASURES. 26 CLOTH MEASURE. 2\ inches (in.) make . . 1 nail, . . n. 4 nails, or 9 inches . . 1 quarter, . qr. 4 quarters 1 yard, . . yrf. 5 quarters 1 English ell, E. e. A Flemish ell is 3 qrs. A French ell 6 qrs. Used for all drapery goods. SUPtRFICIAL, OR SQUARE MEASURE. 144 square inches {sq. in.) make 1 square foot, sq.ft. 9 square feet 1 square yard, sq. yd. 30| sq. yards, or 272} sq. feet 1 sq. rod, pole, or perch Also, in the measure of land, 40 perches make .... 1 rood, . . r. 4 roods, or 4840 yards . . 1 acre, . . a. 10,000 square links I square chain, sq. c. 10 sq. chains, or 100,000 links 1 acre, . . a. 640 acres 1 square mile, sq. m. Inches. 144= 1 foot. 1296 = 9=1 yard. 39204 = 272i = 30^= 1 pole. 15681^0 = 10890 = 1210 = 40 = 1 rood. 6272640 = 43500 =4840 = 160 = 4 = 1 acre. Roofing, flooring, (&c., are commonly charged by the Square, contain ing 100 square feet. By this measure is expressed the area of any snpei*ficie8, or surface. \ superficies has measul*able length and breadth. CUBIC, OR SOLID MEASURE. 1728 cubic inches {in.) make . 1 cubic foot. 27 cubic feet 1 cubic yard.* 40 feet of round timber, or ) . * , 50 feet of hewn timber J ' 42 feet .1 ton of shippmg, A cord of wood is 4 feet broad, 4 feet deep, and 8 feet long, being 128 cubic feet. A stack of wood is 3 feet broad, 3 feet deep, and 12 feet long, being 108 cubic feet. This determines the solid contents of bodies. A solid has three di- mensions, length, breadth, and thickness. * A eoHd yard of earth is called a load. B iMi , } r 26 TABLES OF MEASURES. Ftutor's IMPERIAL MEASURE. This is the standard now established by Act of Parliament, as a general measure of capacity for liquid and dry articles. 2 pints (pt.) make ... 1 quart, qt. 4 quarts 1 gallon, gal. The imperial or standard gallon must contain 10 lbs. Avoir- dupois weight of pure water, at the temperature of 62'^ of Fahrenheit's thermometer. This quantity measures 2771* cubic inches ; being about one-ffth greater than the old wine measure, one-thirty-second greater than the old dry measure, and one-sixtieth less than the old ale measure. IN DRY MEASURE, 2 gallons (gal.) make , . 1 peck, pk. 4 pecks 1 bushel, b. 8 bushels 1 quarter, qr. Corn to be stricken off the measure with a round stick, or roller. Obsolete. A cooni = 4 bushels ; a chaldron = 4 quarters ; a wey xs 5 quai'ters ; a last =i 2 vveys. Solid inches. 277^ = 1 irallon. r)j4.| =2=1 peck. 221}! = 8= 4=1 bushel. 17744 = 04 = 32 = 8=1 (luarter. OF COALS, 3 bushels make .... 1 sack. 12 sacks, or 36 bushels . 1 chaldron. 21 chaldrons 1 score. All the measures used for heaped goods are to be of cylin- drical form ; the diameter being at least double the depth. The height of the raised cone to be equal to three-fourths of the depth of the measure. The old dry gallon contained 268f cubic inches. Note. The bushel, for moasuruig heaped goods, must be 17*81 inches in diameter, and 8.i)04 inches deep; or if made 18 inches in diameter, the depth v/ill be 8717 uiclies. The cone to be raised 6'6 inches hi height. In WL\E AND SPIRIT MEASURE, the olu gallon contained 231 cubic inches. 63 gallons were .... a hogshead, hhd. 2 hogsheads, or 126 gallons a pipe or butt. 4 hogsheads, or 252 gallons a tun. More accurately, 277 274 cubic iHches. if uToa'a lament, Lcles. . Avoir- 62'^ of i 2771* Id wine leasure, n>ller. a wey ; of cylin- le depth, burths of t be 17-81 i inches ic raised 6*8 ained 231 hhd, t. ASSISTANT.] TABLES OF MEASURES. 27 Some other denominations hiivo baeu long obsolote ; as, an anker (10 gallons;) a runlet (18 gallons;) a tierce (4i2 gallons;) a puncheon (84 gallons.) But casks of m(ist dcscriplious ai'e generally charged accord- ing to the number of gallons contained. Solid inches. 34f4 = 1 pint. 69yV = 2= 1 quart. 2771- = 8 = 4=1 frallon. 17466^ = 504 = 25-2 = 63 = 1 hocrshead. 34933^ = 1008 = 504 = 126 = 2 = 1 pipe. 69867' = 2016 = 1009 = 252 = 4=2=1 tun. In ALE, BEER, or PORTER MEASURE, the old gallon contaia ed 282 cubic inches ; and measures of the following denomi nations have been in use : — A firkin, containing ... 9 gallons. A kilderkin 18 gallons. A barrel 36 gallons. A hogshead 54 gallons. A butt 108 gallons. Cubic inches 34 14 = 1 pint. 69f^ff=: 2= 1 quaTt. 2771- = 8= 4= 1 gallon. 2495} :::. 72 = 36 = 9=1 firkin. 4990i- =144= 72= 18= 2 = 1 kilderfcin-. 998 1 " = 288 = 144= 36 = 4 = 2 = 1 barrel. 1497U ==432 = 216= 54= 6 = 3 = U = 1 hogsh'd'. 29943' = 864 = 432 = 108 = 12 = 6 = 3' = 2 = 1 butt- •rules for changing old MEASURES TO IMPERIAL. Ale Multiply by 60, and divide by 59; or add ^^ part. (True' within Yoxiro P^'^'^* of the whole.) Or, multiply by 179, and divide by 176. (True, within y ooo do o part.^ Dry. Multiply by S2, and divide by 33 ; or deduct ^^ part (Error, less than ^^'^o P^rt.) Wine. Multiply by 5, and divide by 6, or deduct ^ part. (Error,; less than jfji^o part.) Or, multiply by 624, and divide by 749. (Error, less than fl^Q^pp g, part.) *RULES FOR CHANGING IMPERIAL TO OLD MEASURES. Ale. Multiply by 50, and divide by 60, or dedtict ^ part. Or, multiply by 176, and divide liy 179. Dry. Multiply by 33, and divide by 32, or add -jV part— Tha* is, add one peck m every quarter, one quart in every bushel,.er half a pint in every peek. Wine. Multiply by 6, and divide by 5, or add \ part. Otherwise, multiply by 749, and divide by 624. * Examples applying to these Rules will b® found in the Misoollk neous Questions in the latter part of the book. n /' t'.r ..r, t) TIME. •tutor's TIME. 60 seconds (sec.) make .... 1 minute, . min 60 minutes 1 hour, . . hr. 24 hours 1 day,* . . d. 7 days 1 week, . . wk. 52 weeks, 1 day, 6 hours, or > i t i- o^ci Au >...l Julian year, yr. 305 days, 6 hours ... J j ^ y 365 days, 5 hours, 48 min., 51^ seconds The Solar year.f ]fO years 1 century. Secouds. fiO = 1 minute. 3f)00 = f)0 = 1 hour. 86400= 1440— 24rrr 1 tlilV. 604800 = 10080 = l(i8 = 7=1 week. 315.57600 = 525960 = 87ii6 = ;J65 d. Gk. = 52 w. 1 d. Sh. = 1 Julian year. 31556931 = 525948 = 8765 = 3(i5 d. 5 h. 48 m. 51^" =1 Solar year. The year is divided into 12 Calendar months; January, February, March, April, May, June, July, August, September, October, November, December. The days are thirty in Septem1)er, In April, .Tune, ;m(l in November; Twenty-eight in February alone, And in each other thirty-one : But every leap-year we assign To February twenty-nine. The Icap'ycars are those which can be exactly divided by 4 ; as, 1824, 1828, &c. Hence it appears that the year is accounted 365 days, for three years together ; and 366 days in the fourth: the average being 365i days. {The Julian Year.) Four weeks are frequently called a month ; but in this sense it is better to avoid the term. Note. In all questions in this book, where the proposed or required time consists of years, months, weeks, &c., allow 4 weeks to a month| and 13 mouths to a year. GEOMETRY. 60 seconds {'^) make . . 1 minute, ' 60 minutes 1 degree, ° 360 degrees 1 circle. Many highly important calculations in the mathematical sciences are founded on this division of the circle. In Astronomy, the great circle of the ecliptic (or of the zo- diac) is divided into 12 signs, each 30°. * A day is the time in which the eiu-th revolvea once upon its axis, by law and custom it is reckoned from midnight to midnight; but the afitronomical day begins at noon. t The Solar, or true year, is that portion of time in which the earth makes one entire revolution round the sun. 'I Mi 4S8ISTANT.] REDUCTION. 29 In Geography, a degree of latitude, or of longitude on the equator, measures nearly 69j'„ British miles. But a minute of a degree is called a geographical mile. ARTICLES SOLD BY TALE. 12 articles of iny kind, are 1 dozen 12 dozen 1 gross. 12 gross 20 articles 1 score 1 great gross. 24 sheets of paper 1 quire, 20 quu-es ... 1 ream. 2 leams ... 1 bundle. DEFINITIONS. 1. A NUMBER is called abstract, when it is considered simply or without reference to any subject ; as seven, a thousand, &c. 2. When a number is applied to denote so many of a par- ticular 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 money, weight, or measure ; as penny, pound, yard, &c. 4. The 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 of more denominations than one ; as seven pounds five shillings. REDUCTION Is the method of changing quantities of one denomination into another denomination, retaining the same value. Rule. Consider how many of the less name make one of the greater ; and multiply by that number to reduce the great- er name to the less, or divide by it to reduce the less name to the greater. Examples. Reduce ;C8..8..6|. into farthings. The £S. being multiplied by 20, and the 8». add- ed, make 16Ss. ; these being multiplied by 12, and the Gd. added, make 2022y 20, give £8 iis., so that the answer is £S..8..()^. • 27 shillings. The moidore ia current in Portugal, but not in Eng land. 4)8000 qrs. 12)2022^ d. 2|0)l(i|8^. Gjjd Ans. £8..8..C)li. UTOR*i gs? ( qrs. 24. .10. \ qrs. J qrs. md. owns. es ? nccs, ce, and '(i)igs. ces, aro ences. hreep. 5133. lY igs, and 14 far. ..2..1. ■crowns i5. 52. pounds. , je63. )2..12. )wns, an f. ouer. as many 55. .18. lint, to be I? s. over. ow many 8. .2. .6. liiin 2022i. luse the re- )ivide 2022 »ver: these HO that the not in Eng •) i AISISTANT.^ REDUCTION. WEIGHTS AND MEASURES. TROV WEIGHT. f21) In 27 'ounces of gold, how many grains? Ans. 12960. (22) Reduce 3 /6. 10 oz. 7 diet. 5 gr. to grains ? Ans. 22253. (23) In 8 ingots of silver, each ingot weighing 7 lb. 4 oz. 17 dwts. 15 gr. how many grains ? Ans. 341304 gr. (24) How many ingots weighing 7 lb. 4 oz. 17 dmts. 15 gr, «ach are ^here in 341304 grains? Ans. 8 ingots. apothecaries' weight. (25) In 27ib. 7§. 2 5. 1 3. 2gr. how many grains? Ans. 159022 grains. (26) In a compound of 9 §. 4 3. 1 3. how many pills of 5 grains each? Ans. 916 pills. AVOIRDUPOIS weight. (27) In 14769 ounces, how many cwt. ? Ans. 8 cwt. qr. 27 lb. 1 oz. (28) In 34 tonSi 17 cwt. I qr. 19 lb. how many pounds ? Ans. 781 1U&. (29) In 9 cwt. 2 qrs. 14 lb. of indigo, how many half stones^ and how many pounds? Ans. 154 half stones, 1078 lb. (30) How many stones and pounds are there in 27 hogs- heads of tobacco, each weighing neat Sf cwt. ? Ans. 1890 stones, 26460 lb. (31) Bought 32 bags of hops, each bag 2 cwt. 1 qr. 14 /i., and another of 150 lb., how many cwt. are there in the whole ? ^n^. 77 cwt. 1 ^r. 10 /6. (32) In 27 cwt. of raisins, how many parcels of IS lb. each ? u4n5. 168. CLOTH MEASURE. (33) In 27 yards, how many nails ? Ans. 432. (34) In 75 English ells, how many yards ? Ans. 93 yards, 3 qrs (35) In 24 pieces, each containing 32 Flemish ells, how many English ells ? Ans. 460 English ells, 4 qrs. (36) In 17 pieces of cloth, each 27 Flemish ells, how many yards ? Ans. 344 yards, 1 qr. (37) In 91 11 yards, how many English ells ? Ans. 729. (38) In 12 bales of cloth, each containing 25 pieces, of 15 English ells, how many yards ? Ans. 5625. LONG measure. (39) In 57^ miles, how many furlongs and poles ? Ans 460 furlong f, 1 ?400 poles. Iff K 32 REDUCTION fTDTOR'S (40) In 7 miles, how many foet and inches ? Ans. '30960 feet, 443520 inches. S41) In 72 leagues, how many yards ? A/us. 380160 yards. 42) if the distance from London to Bawtry be accounted 150 miles, what is the niiniber of leagues, and also the num- ber of yards, feet, and inches ? Ans. 50 leagues, 20 1000 yards, 792000 feet, 9504000 inches. (43) How often will the wheel of a coach, that is 17 feet in circumference, turn in 100 miles? Ans. 31058|^ times round. (44) How many barley-corns will reach round the globe, the circumference being 360 degrees, supposing that each degree were 09 miles and a half? Ans. 4755801600 See Table of Geometry, page 28. LAND MEASURE. (45) In 27 a. 3 r. \9p. how many perches? Ans. 4459. (46) A person having a piece of ground, containing 37 acres, 1 perch, intends to dispose of 15 acres: how many perches will he have left? Ans. 3521 perches. (47) There are 4 fields to be divided into shares of 75 perches each ; the ](irst field contains 5 acres ; the second 4 acres, 2 perches ; the third 7 acres, 3 roods ; and the fourth 2 acres, 1 rood : how many shares will there be ? A71S. 40 shares, 42 perches rem. (48) In a field of 9 acres and a half, how many gardens piay be made, each containing 500 square yards ? 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. 35280 pints. (51) A gentleman ordered his butler to bottle off | of a pipe of French wine into quarts, and the rest into pints. How many dozen of each had he ? Ans. 28 dozen cfeach. (52) In 46 barrels of beer, how many pints ? Ans. 13248 (53) In 10 barrels of ale, how many gallons and quarts ? Ans. 300 gals. 1440 qts. (54) In 12480 pints of porter, how many kilderkins? Ans. 86 kil. 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. ?>MOpks. imO gal. 30720 qts. ■iw [tutor's inches. 50 yards. ccounted the num- 30 inches. 17 feet in ^^5 round. he globe, hat each 01600 . 4459. lining 37 ow many lerches. res of 75 second 4 ;he fourth ASSISTANT.] COMPOUND ADDITION. 33 1 1 (57) How many bushels arc there in 070 pints? Ans. If) b^i. 1 gal. 2 pts. (58) In 1 score, 16 chaklrons of coals, how many sacks and bushels? Ans. 444 sacks, 1332 bushels. TIME. (59) In 72015 hours, how many weeks? A?is. 428 ircrks, 4 days, 15 hours. (60) IIow many clays were there from tlie l)irth of Christ, to Christmas, 1794, estimating 3G5|- days to tlie year? Ans. 655258^ days. (61) Stowe writes, that JiOndon was built 1108 years be- fore our Saviour's birth. Find the number of hours to Christ- mas, 1794. Ans. 25438932 hours. (62) From July 18th, 1799, to April 18th, 1826, how many days ? Ans. 9770 l days, reckoning 365i days to a year. (63) In a lunar month, containing 29 days, 12 hours, 44 minutes, 2 seconds and eight-tenths, how many tenth parts of seconds? .4^6-. 25514428. (64) How many seconds are there in 18 centuries, estima- ting the solar year at 365 days, 5 hours, 48 minutes, 51^ seconds ? Ans. 56802476700 seconds. es rem. Y gardens ds rem. s? 5 tuns. nd pints ? pints. )ff f of a nto pints. ?n c^ each, ris. 13248 quarts ? 140 qts. tins? 3 gals. ' Ans. 12. )ecks, gal- 30720 qts. m 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 (tf any) and carry the'^uotient to those of the next greater: proceed thus to the greatest dertomination, which add aa in Simple Addition. Proof. As in simple Addition. Example. £. s. d. 15.. 7.. Ah 7. .18. .101 11. .19.. 5 6.. 10.. Hi 4.. 0.. 9J 45.. 17.. 4| Say 1, 2, 5, 7 farthings are 1 penny 3 far- things ; set down I and carry ''d. — 1, 10, 11, 16, 20, 30, 40(7. are 3s. U.; set down Ad. and carry 3s.— 3, 12, 20, 27, 37, A7, 57s. are £2. 17s. ; set down 175. and carry jC2. The rest as in Simple Addition. In Addition of Money, the reduction of one denomination to the next greater is generally done without the ti'ouble of dividing, by the- knowledge previously acquired of the Monej Tables. B2 M COMPOUND ADDITION. [tutor's £ 8. d. MC (4) £. s. d. )i\ (7) £. s. d. £. 8. d. 1 ^7' 2 13 ^)i 75 3 21 14 71 261 17 1^ m 7 J 41 54 17 1 75 16 379 13 5 ■ 5 15 41 91 15 111 79 2 41 257 16 7^ s 9 17 61 35 16 If 57 16 51 184 13 5 1 7 16 3 29 19 111 26 13 ^ 725 2 34 5 14 7| 91 17 31 54 2 7 359 6 5 (2) £. 8. d. (5) £ s. d. (8) £. s. d. (11) £. s. d. (20) 27 7 257 1 51 73 2 11 31 1 1^ 34 14 101 734 3 7f 25 12 7 75 13 1 57 19 21 595 5 3 96 13 51 39 19 71 91 16 159 14 71 76 17 31 97 17 3| 75 18 7| 207 5 4 97 14 11 36 13 5 97 13 5 798 16 71 54 11 71 24 16 3^ (3) £. s. d. (6) £. s. d. £. 5. d. (12) £. 8. d. ( 35 17 525 2 41 127 4 71 27 13 5^ yds. 59 14 101 179 3 5 525 3 10 16 12 10|. 135 97 13 101 250 4 71 271 9 13 0^ 70 37 16 81 975 3 51 524 9 1 15 2 10| 95 97 15 7 254 5 7 379 4 01 37 19 176 59 16 01 379 4 5|- 215 5 1 If 56 19 U 26 - 279 EIGHTS AND MEASURED 5. W TROY W] EIGHT. APOTHECARI] ES' WEIGHT. (13) OX. dwt. gr. ih. (14) oz. dwt. gr. (15) ft. I. 3. 3. (16) 5. 3. 3. ^r. ... ( 5 11 4 5 2 15 22 17 10 7 1 2 1 12 Ihdt.i 7 19 21 3 11 17 14 9 5 2 2 1 7 1 17 31 3 15 14 3 7 15 19 27 11 1 2 10 2 14 97 7 19 22 9 1 13 21 9 5 6 1 5 7 1 15 76 9 18 15 3 9 7 23 37 10 5 2 9 5 2 13 55 8 13 12 5 2 15 17 49 7 1 4 1 18 87 55 ASSISTANT. 1 COMPOUND ADDITION. 35 d. 5 7| 5 J 5 i: ) d. 1 7i 5 3i GHT. L6) 12 1 17 14 1 15 2 13 1 18 1 m^ lb. 152 272 303 255 173 635 oz. 15 14 15 10 6 13 dr. 15 10 11 4 2 13 AVOIRDUPOIS WEIGHT, (18) cwt. qrs. lb. 25 72 54 24 17 55 1 3 1 1 2 17 26 16 16 19 16 (ly^ t. cwt. qra.lh. 7 17 2 12 5 3 14 4 1 17 18 2 19 9 3 20 5 1 24 5 2 3 7 8 (20) yds. ft. in. 225 1 9 171 3 52 2 6 397 10 154 2 7 137 1 4 LONG MEASURE. (21) Zea. m. fur.po. 72 2 1 19 27 1 7 22 35 2 5 31 79 6 12 51 1 6 17 72 5 21 (22) m. fur. yds. 39 6 36 14 7 214 3 4 160 45 3 202 17 1 19 32 4 176 CLOTH (23) yds. qrs. n. 135 3 3 MEASURE. 70 95 176 2 3 1 2 3 26 1 279 2 1 (24) E. e. qrs. n. 272 2 152 1 79 156 2 79 3 154 2 1 2 1 1 1 LAND ME (25) a. 726 219 1455 879 438 r. 1 2 3 1 2 757 r- 31 17 14 21 14 ASURE. (26) a. -. p. 1232 1 14 327 19 131 2 15 1219 1 18 223 2 8 236 9 IMPERIAL MEASURE. (27) khds.gals. 31 57 qta. 1 97 18 2 76 13 1 55 46 2 87 38 3 55 17 1 1 t.) 14 (28) ihd.gal.qts. 3 27 2 1 19 2 56 3 17 39 2 75 2 16 1 54 1 19 2 97 3 54 3 1 ALE AND BEER (29) har. fir gal 25 2 7 17 3 5 96 2 6 75 1 8 96 3 75 7 5 30. hhd. ^ 42 41 . 27 51 ) 37 28 42 27 I 47 38 ise of a crown ; ha$.e ? ..6..3. several ,nd four shilling. ..8..4. packing the bar- 7..10..3 [1 to dis- that he '3..5;— t3..2;— ..8 ; — to r0..6..8; 11..5;— wages, iker, in- 17..3. onths to it child • ths, and I month, months, ths, and days. ■esented 1 pieces. 4 half-crowns and 4^. 2d. ; — F paid him only twenty groats ; — 0:C76..15..9i;— and H jei21..12..4. How much was the >»hole amount ? Ans. £396..7..6i: 4 (41) A nobleman had a service of plate, which consisted of twenty dishes, weighing 203 oz. 8 diots. ; 36 plates, 408 oz. 9 dwts. ; 5 dozen spoons, 112 oz. 8 dwt.<;. ; 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. Ans. 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 Ih. ; the second, 2 cwt. 2 qrs. 1 1 lb. ; the third, 2 cwt. 3 qrs. 5 lb. ; the fourth, 2 cwt. 3 qrs. 12 lb.; the fifth, 2 cwt. 3 qrs. 15 lb. He purchased also two pockets, each pocket weighing 84 lb. I desire to know the weight of the whole. Ans. 15 cwt. 2 qrs. COMPOUND SUBTRACTION Teaches to find the difference of Cornpound Quantities. Rule. Subtract as in integers : but borrow (when there is occasion) as many as are equal to one of the nexi greater denomination: observ* ing to carry one to the next for that which was borrowed.* Proof. As in Simple Subtraction. (1) £. From 715 Take 476 s. 2 3 8 f (2) MONEY. £. s. 316 3 218 2 (3) £. $. d. 87 2 10 79 3 n • Example. Subtract i;54..17..9|. from je89..12..7i. £. s. d. 89..12..7.i M..17..9I 34..14..9I Because 3 farthings cannot be taken from 2, say 3 fi-om 4, 1, and 2 are 3 ; set down 3 and carry 1. — 1 and 9 are 10, 10 from 12, 2, and 7 are 9 ; set down 9 and carry 1.— 1 and 17 are 18, 18 from 20, 2, and 12 aX9 14 ; set down 14 and . y 1 to the paunds. ( I »8 COMPOUND SUBTRACTION. (4) £. 8. d. 3 15 u 1 14 7 (5) £. 8. d. 25 2 5^ 17 9 ^ 1 1 (6) £. 8. d. 37 3 4i 25 5 2:f £. 321 257 (7") 17 14 (8) £. s. 59 15 36 17 £. 71 (9) 2 19 13 d. n H d. 4 7f (16) jG. 5. d. Borrowed 350 Pf'^fi *! aiu at ^^ jQ g different ^^ 9 8l ^'"""^ [66 14 9 Paid in all Remains to pay (10) £. «. 527 3 139 5 7-^ ' 2 (11) £. s. d. 300 15 296 15 10 (12) £ s. d. 08 13 9 44 19 lOl (17) [tutor's (13) X. 8. a* 10 7 6 9 19 7 (14) £. 8, d. 500 499 19 11| (15) £. 8. d, 779 12 689 13 6 £. 8. d. Lent 577 10 95 10 80 Received at several^ kv^ i e t\ ] 74 15 9 times ^ 23 j^ ^, I "i ASSIS' } i f yds. 107 78 yds. 71 3 a. 175 59 WEIGHTS AND MEASURES. apothecaries' weight. TROY WEIGHT. (18) li. oz. dwt. gr. 52 1 7 2 39 15 7 (19) lb. oz. dtot. gr. 7 2 2 7 5 7 15 (22) lb. oz. dr. 35 10 5 29 12 7 (20) ft. 5. 3. 3. 5 2 1 2 5 2 1 (21) ft. i. 3. "S.gr 9 7 2 1 13 5 7 3 1 18 AVOIRDUPOn WEIGHT. (23) cwt. qr. lb. 35 1 21 25 1 27 (24) ^ cwt. qrs. lb. 21 1 2 7 9 11 3 15 [tutor's :i3) d. 7 6 19 7 :i4) d. 19 111 [15) s. d. 12 13 6 i. d. 5 9 7 41 IGHT. 3. 3.^ 1 13 B 1 18 2 7 3 15 ASSISTANT.] COMPOUND SUBTRACTION. 30 LONG MI3ASURE. IMPERIAL MEASURE WINE. (25) yds. fi. in. 107 2 10 (2fi) Z^Y^ vii. j iir.fo. 147 2 G 29 (31) hhil. tal. qfs.pfs. 47 47 2 1 (32) tun hhd. gal. qts. 42 2 37 2 78 2 11 58 2 7 33 28 59 3 17 3 49 3 CLOTH MEASURE. ALE AND BEER. (27) yds. qrs. n. 71 1 2 (28) E. c. qrs. n. 35 2 1 (33) nr. Jir. gnl. 37 2 1 (34) hhd.gal. qts. 27 27 1 3 2 1 1 4 3 2 25 1 7 12 50 2 LAND J MEASURE. CORN AND COAL. (29) fl. r. ;>. 175 1 27 (30) «. r. p. 325 2 1 (35) (jr. b. p. 65 2 1 (36) sc, ch. sa. h. 3 16 1 59 37 279 3 5 57 2 3 2 12 2 1 TIME. •(37)yr». mo. w. d, ' 79 8 2 4 23 9 3 5 (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 jC300. per annum Is reduced by the payment of taxes to 12 score and jG14..6. what are the taxes ? Ans. JG45..14. (41) A horse with his furniture is worth JC37..5; without it, 14 guineas ; how much does the price of the furniture ex- ceed that of the horse ? Ans. £7.. 17. (42) A merchant commencing trade, owed £750 ; he had in cash, commodities, the stocks, and good debts, JC12510..7; he cleared the first year by commerce JC452..3..6. What was he then worth ? Ans. £12212. .10..6. (43) A gentleman left £45247. to his two daughters, of • In this example allow 4 weeks to a month, and 13 months to the year. t In this, reckon 30 days to a month, and 12 months to the year. I I 10 COMPOUND MULTIPLICATION. [tutor's which the younger was to have 15 thousand, 15 hundred, and twice jC15. What was the elder sister's fortune ? Ans. je28717. (44) A tradesman being insolvent, called all his creditors together, and found he owed to A jC53..7..6 ; — to B £105. .10 , —to C JC34..5..2 ;— to D jC28..16..5 ;— to E i:i4..15..8 ;— to F jE:112..9 ;— and to G i:i43..12..9. The value of his stock was JC212..6; and the amount of good book-debts was JG112..8..3 ; besides JC21..10..5. money in hand. How much would his creditors lose by taking the whole of his effects 1 A71S. The creditors lost jC146..11..10. (45) My agent at Seville, in Spain, renders me the follow- ing account of money received for the sale of goods sent him on commission, viz. for bees' v.ax jC37..15..4 ; stockings je37..6..7; tobacco jei25.. 11. .6; linen cloth i:i 12. .14. .8 ; tin JG115..10..5. He informs me at the same time, that he has shipped, agreeably to my order, wines, va]"3 jG250..15 ; fruit JC51..12..6; figsjC19..17..6; oil jC19..i2..4; and Spanish wool, value JC115..15..6. How stands the balance o^ the account between us? A?is. Due to the agent jC28..14..4. (46) The great bell at Oxford, the heaviest in England, is stated to weigh 7 tons, ] 1 cwt. 3 qrs. 4 lbs., that of St. Paul's, in London, 5 tons, 2 cwt. 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 198 tons? Ans. 180 tons, 8 cwt. 3 qrs. 14 lbs. COMPOUND MULTIPLICATION Is the method of multiplying Compound Quantities. Rule. Multiply the least denoinination ; reduce the product and carry to the next, as directed in Compound Addition ; and the same with the rest. When the multiplier is a compdsite number above 12, mul- tiply (as before directed) by its component parts. For other numbers, multiply by the factors of the nearest compdsite ; adding to the last product, so many times the top line as will supply the deficiency ; or subtracting so many times, if there is an excess. [tutor's idred, and JC28717. creditors 105..10, 5.. 8 ; — to his stock ebts was ow much effects ? I.11..10. e follovv- oods sent stockings 114. .8; tin at he has .15 ; fruit nish wool, le account 1.14.A. ngland, is Bt. Paul's, f Lincoln, bs. How inferior to 14 lbs. AMI8TANT.J COMPOUND MULTIPLICATION. e product tion ; and 12, mul- For other ompdsite ; \e as will , if there £. (1) t. d. 35 13 7f 1 5 3| (2) £. 8. d. 75 13 3 MONEY. 62 (3) t. d. 5 41 41 X. A. a. 57 2 4J 5 £. s. d. £. s. d. (5) 57 18 71 X 6. (9) 135 13 6f X 10. (6) 81 9 111 X 7. (10) 79 16 71 X 11 (7) 64 10 5 X 8. (11) 247 14 111 X 12. (8) 118 6 4| X 9. (12) 119 7 53 X 12. £ s. d. s. d. £. s. d. (13) 9 6 X18.r (16) 15 31X35. (19) 1 5 3X97. (14) 1 2 6 X26.t (17) ' r 2|X75. (20) 6 4X43. (15) 7 8^X21. (18) 1 ) 7 X37. (21) What is the value of 127ttJ. of souchong tea, at 12^. 3(f. per ftj. ? Ans. £ll..\b..9. (22) 135stonesof soap, at7^. 5c?. perstone? Ans. £bO..\.M. (23) 74 ells of diaper, at 1^. A\d. per ell? Ans. JC5..1..9. (24) 6doz.pairsof gloves, at Is. lOd per pair? Anc. £6.. 12. Note. When the fraction \, i, or | is connected with the multiplier, take half the given price (or the price of one) for i, half of that for \, and for |, add them both together. $ * In this example, say twice 3 are 6, 6 farthings are li^d. set down J^d. and carry 1 ; twice 7 are 14 and 1 are 15, \bd. are Is. ^d. set down Zd. and carry 1 ; twice 12 are 24 and 1 are 25, 25s. are XL. 5. set down 5». and carry 1 ; twice 5 are 10 and 1 are 11, set down 1 and carry 1 ; twice 3 are 6 and 1 are 7, set down 7. s. d, £• s. d. t 9.. 6 t 1- 2.. 6 2X9=18 8X3+2=2C 19.. 9.. 0.. 6 3 7 3 XS-.H.. Ans. $ Example. What is the value of llflbs. of tea, at 10s. 9rf. per lb. f 27.. 0.. MultiplicandX2=2.. 5.. £29.. 5.. Ans s. d. iXlO.. 9 U JC5..18.. 3 =^theva]ueof 11. iX 5.. 4^= .. do 2.. 8|« .. do £6.. 6.. 3| Ans. r '^ 42 COMPOUND MULTIPI ICATION [tutor's assists (25) What is the value of 251 ells of Holland, at 3*. 4^d >er ell ? Ans. i:4..6..0f . (26) 751 Rj. of hemp, at 1*. 3d. per lb. ? Ans. JC4..14..41. (27) 191 yds. of muslin, at 4^. 3d. per yd. ? Ans. -£:4..2..10i. (28) 351 cwt, of raw sugar, at X'4..15..G. per cwt. ? Ans. JC169..10.3. (29) 1541 cwt. of raisins, at jC4..17..10. per cwt.? Ans. i;755..15..3. (30) 1171 gallons of gin, at 12*. 6c?. per gallon ? A?is. JC73..5..71. (31) 85^ cwt. of logwood, at jC1..7..8. per cwt. ? Ans. JC118..12..5. (32) 17f yards of superfine scarlet cloth, at £l..3..6. per yard? ^n5. jC20..17..1i. (33) 371 lb.ofhysontea,atl25.4^.per ib.? Ans. jC23..2..6 (34) 56f cwt. of molasses, at jC2..18..7. per cwt.? A?is. jei66..4..7i. (35) 873 fij. of Turkey coffee, at 4*. 3d. per fb. ? Ans. i;i8..12..11|. (36) 120f cwt. of hops, at jC4..7..6. per cwt.? Ans. jC528..5..7|. When the multiplier is large, multiply the given quantity (or price) by a series of tens, to find 10, 100, 1000 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 100 by the number of hundreds, the product for 10 by the number of tens, and the original quantity by the units^ figure, ^c. The sum of the products thus obtained will be the total product.* * Example. Multiply £7..1A..9^. by 3645. £. s. d. £. s. d. 7. .14.. 9.JX5= 38..13..11i= 10 The product for 10 77.. 7. .11 X4= 309.. 11.. 8 = 10 The product for 100 773.. 19.. 2 X6= 4G43..15.. = 10 time8> 5 40 600 (37) (38) (39) (40) year i (41) men, t( ; m \ of whi of the (43) his wi '. each ( guinea (44; \ togeth £2111 " had X\ year. (45 wagor If 3 ( 10 lb. 1 qr. ; (46 ands] 1 I* Wi The product for 1000 7739. . 11.. 8 X3=23218..15.. = 3000 Ans. 28210.. 15,. 7^=> 3645 (47 (4e (4[ (5C (51 (5'. 3 se mont [TUTOR*a ASSISTANT.] COMPOUND MULTIPLICATION. 49 il !■ at 3^. A^d 4..6..0f. ..14..41. e4..2..10i. n.l 9..10.3. t.? II5..15..3. '3..5..71. 8..12..5. 1..3..6. per ..17.. If £23..2..6 t.? i6..4..7i. ? .12..111. I8..5..7f m quantity times, &c., plier ; mul- , and each isiiue ; that ihe product Uity by the )tained will times. 5 40 600 3000 3645 (37) 407 lb. of gall-nuts, at 35. 9ld. per ib.? Arts. je77..3..2j. (38) 729 stones of beef, at 7^. 7}f?. per stone ? Ans. jC277..3..5^. (39) 2068 yards of lace, at 9.^ 5U- per yard ? vl/i.v. X977..19. .10. (40) What is the produce ot a toll-jrate in the course of the year if the tolls aniuiint, on ;ui uvcnigc, to II5, 7^!. by 214723. (23) If a man spend JC257..2..5. in 12 months, what is that per month ? Ans. jC21..8..6i j\. (24) The clothing of 35 charity boys came to JC57..3..7. what was the expense of each boy ? Ans. jC1..12..8i|. (25) If I gave £37..6.A^. for nine ^jieces of cloth, what was that per piece ? Ans. jC4..2..11|. • Example. Divide £'27.Ai..U^. by 5. Say the fives in 27, 5 times 5 are 25 and 2 ovi ; £2. are 40s. and 14 are 54, the fives in 54, 10 times 5 are 50 and 4 over; 4s. are 48d. and 11 are 59, the fives in 59, 11 fives are 55 and 4 over; 4d. are 16 qrs. and 2 are 18, the lives in 18, 3 times five are 15, and 3 over, or |. £. s. d. 5 )27.. 14.. 11^ 5..10..11I f I (35) (36) (37) (38) (39) (40) minuter lTWrOR'8 QuantUy reniaindof to oceed aa be length r'ain 3ns iu Siaplo I ; ASSISTANT.] PROMISCUOUS EXAMPLES. 4a I (4) s. d. 7 "bv 10. by 11. by 12. by 12. by 99. by 110 by 144 8..6i 8 . jE:57..3..7. .812.. .12. cloth, what t..2..11f and 2 ovi ; I 54, 10 times id 11 are 59, )ver ; Ad. are 3 times five what is that i ■ (26) If 20 cwt. of tobacco cost jC27..5..41 ; at what rate did I buy it per cwt. ? Ans. £\..l..^^. (27) What is the value of one ho<^shead of beer, when 120 iiogsheads are sold for X'154..17..10 ? Ans. jC1..5..9? ,-Vo. (28) Bought 72 yards of cloth for jC85..6. What was the price per yard? Ans. jC1..3..8i ^^. (29) Gave X275..3..4. for 18 bales of cloth. What is the price of one bale? Ans. jC15.,5..83 ||. (30) A prize of jC7257..3..6. is to be equally divided among 500 sailors. What is each man's share ? Ans. jC14,.10..31 f |f' (31) A club of 25 persons joined to purchase a lottery Jticket of X'lO. value, which was drawn a prize of jC4000, What was each man's contribution, and his share of the prize- money ? Ans. Each contribution 8s. and share of prize £160. (32) A tradesman cleared £2805. in 7^ years. What was his yearly profit ? Ans. jC374. (33) What was the weekly salary of a clerk who received JC266..18..1 L. for 90 weeks ? Ans. jC2..19..3f (34) If ibOOOO quills cost me JC187..17..1. what is the pric" per thousand? Ans. jC1..17.6^ jYo* WEIGHTS AND MEASURES. 1 (35) Divide 83 lb. 5 oz. 10 diets. 17 gr. by 8, 10, and 12. j (36) Divide 29 tons, 17 cwt. qrs. 18 lb. by 9, 15, and la I (37) Divide 114 yards, 3 qrs. 2 nails, by 10, and 16. (38) Divide 1017 miles, fur. 38 poles, by 11, and 49. I (39) Divide 2019 acres, 3 roods, 29 perches, by 26. * (40) Divide 117 years, 7 months,' 26 days, 11 hourSf 27 i minutes, by 37. I PROMISCUOUS EXAMPLES. (1) Of three numbers, the first is 215, the second 519, and the third is equal to the other two. What is the sum of them all ? Ans. 1468. (2) The less of two sums of money is .£40. and their dif- ference X'14. What is the greater sum, and the amount of both ? Ans. £5i. the greater, jC94. the sum. (3) What 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 ? Ans. Diff. 792, sum 936, product 62208. ' V I if; l| 46 PROMISCUOUS EXAMPLES. [TUTOR'S .A8SI8TA^ (5) What (lifTcrcnce is there between twice eight and fifty and twice fifty-eight, and what is their product? Ans. 50 ilijf. 7 G56 product. (6) The greater of two numbers is 37 times 45, and their difference is 19 times 4 : required their sum ; ;ul product. Ans. 3'2C)\ siini, 2iy{i68i> product, (7) A gentleman h-ft his elder daughter X'l; 00. more than the younger, whose fortune was 11 thousand, 11 hundred, and jCll. Find the portion of the ehU'r, and the amount of both. Ans. Elders portion X'13611. amount X*25722. (8) The sum of two nuinlicrs is 300, the less is 144. "What is their difierence and their product 1 Ans. 72 dijf'crcnce, 3ll0'i product. (9) There arc 2545 bullocks to be divided among 509 men. Required the number and the value of each man's share, sup- posing every bullock worth .£9..14.,r). A71S. Each man had 5 bullocks., and jC 18..12..6.ybr his share. (10) How many culiic feet are contained in a room, the length of which is 24 feet, the breadth 14 feet, and the height ]1 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 43d part of X'4429. will make the total amount =jC240 ? Ans. XI 37. (13) Divide 20s. among A, B, and C, so that A may have 2s. less than B, and C 2s. more than B. Ans. A As. Sd. B 6s. 8d. and C 8s. 8d. (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? Ajis. 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 X50. four crown pieces, and eight half-crown pieces. How much had she to her fortune ? Ans. X3744. (16) There are 1000 men in a regiment, of whom 50 are officers : how many privates are there to one officer ? Ans. 19. (17) What number must 7847 be multiplied by, to produce 3013248? Ans. 384, (18) g quarter's apartmer I (19) 1 Iremaindt • (20) A sum of . triotous a ifour ham •;each : h< (21) J^ 1X12,000 "lequally d I (22) A ipublic ch — to foul |keepers, fWhat wn I (23) ^ ll2s. 8d. )urse : v (24) Ihe quoti Is the di' (25) |;d 12 cii height V iiquired th A I (26) I |nd C, t 178 less (27) 1 lany po (28) 1 iig 12 g [terling 1 Multiply the tliree dimensions coutiuually together. 'Add t ibtract tl 1(1 [tutor's t and fifty oroduct. and their oduct. product. more than hundred, amount of ;25722. >s is 144. product. ; 509 men. share, sup- r" his share. room, the the height s. 3696. lare yards, 5 yards. will make . i:i37. may have ' 8*. Sd. orse, each how many )itals there 144806. gQ portion, r were six bur crown lad she to JC3744. om 50 are Ans. 19. to produce ns. 384. ICK. ASSISTANT.] PROMISCUOUS EXAMPLES. 47 (18) Suppose I pay eight guineas and half-a-crown for a ujuarter's rent, but am allowed 15.v. for repairs ; what does my ■ apartment cost me annually, and how much in seven years ? ^«.v. In one year^ jL'31..2. In seven^ X'217..14. (19) The quotient is 1083; the divisor 28604; and the remainder 1788 ; what is the dividend ? Ans. 30979920. (20) An assessment was made on a certain hundred, for the sum of JC386...15..6. the amount of the damage done by a riotous assemblage. Four parishes paid X'37..14..2. each; ifour hamlets JC31..4..2. each; and four townships .£18. .12. .6. •each: how much was deficient? Ans X'36..12..2. (21) An army, consisting of 20,000 men, got a booty of jC12,000 ; what was each man's share, if the whole were equally divided among them ? Ans. I2s. (22) A gentleman left, by will, to his wife, jC4560 ; — to a j)ublic charity, i;572..i0 ; — to four nephews, jC750..10. each ; — to four nieces, -C375. 12..6. each; — to thirty poor house- 'keepers, 10 guineas each ; — and to his executors, 150 guineas. AVhat was tlie amount of his property? Ans. jC10109..10. (23) My purse and money, said Dick to Harry, are worth 125. 8d. but the money is worth seven times the value of the ipurse : what did the purse contain? Ans. lis. Id. (24) Supposing 20 to be the remainder of a division, 423 the quotient, and the divisor the sutn of both, plus 19 ; what Is the dividend? Ans. 195446. (25) A merchant bought two lots of tobacco, which weigh- ed 12 cict. 3 qrs. 15 lb. for jC114..15..6. ; their difference in weight was 1 cwt. 2 qrs. 13 Uj. and in price JC7..15..6. Re- quired their respective weights and value.* Ans. Greater weight 7 cwt. 1 qr. value jC61..5..6. Less weight 5 cwt. 2 cfs. 15 lb. value jC53..10. (26) Divide 1000 crowns in such a manner among A, B. »nd C, that A may receive 129 crowns more than B, and B, 1 78 less than C. Ans. A 360 crowns, 5 231, C 409. I (27) If 103 guineas and 7s. be divided among 7 men, how lany pounds sterling is the share of each? Ans. JC15..10. (28) A certain person had 25 purses, each purse contain- ig 12 guineas, a crown, and a moidoire, how many pounds iterhnff had he in all ? Ans. £355. 1 ■( I I I < I I • Add tlie diffeieiue to the sum, imd divide by 2 for the gi eater f abtract the difl'ereuce from the sum, and divide by 2 for the lesi u., ;^ii 48 PROPORTION. [nitOR'S (29) A gentleman, in his will, left jC50. to the poor, and ordered that i should be given to old men, each man to have 5*. — I to old women, each woman to have 2*. 6d. — i to poor boys, each boy to have Is. — ^ to 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 ? Ans. 66 men, 100 women, 200 boys, 222 girls , £2. .13. .6. for the distributor. (30) A gentleman sent a tankard to his goldsmith, that weighed 50 oz. 8 dwts. to be made into spoons, each weigh- ing 2 oz. 16 dwts. how many would he have ? Ans. 18. (31) A gentleman has sent to a silversmith 137 oz. 6 dwts 9 gr. of silver to be made into tankards of ]7 oz. 15 dwts. 10 gr each; spoons, of 21 oz. 11 dwts. 13 gr. per dozen; salts, of 3 ozi 10 dwts. eacli ; and forks, of 21 oz. 11 dwts. 13 gr. pet dozen ; and for every tankard to have one salt, a dozen spoons and a dozen forks : what number of each will he ha >e ? Ans. Two of each sort, 8 oz. 9 dwts. 9 gr. over. (32) How many parcels of sugar of 16 lb. 2 oz. each are there in 16 cwt. 1 qr. 15 lb. 1 Ans. 113 parcels, and 12 lb. 14 oz. over. (33) In an arc of 7 signs, 14° 3' 53^^, how many seconds? Ans. 806633"^ (34) How many lbs. of lead would counterpoise a mass of bullion weighing 100 lbs. Troy ?* A71S. 82 lb. 4 oz. 9 j^^^j dr. (35) If an apothecary mixes together 1 lb. avoirdupois of white wax, 4 Ihs. of spermaceti, and 12 lbs. of olive oil, ho^ many ounces, apothecaries^ weight, will the mass of ointment weigh, and how many masses of 3 drams each will it contain ? Ans. The whole 247 oz. 7/^2 ^''- ^^^^ 661 o/" 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 of supposition; and the other is the term of demand : because it requires a fourth * Bullion is the tenn denoting gold or silver in the mass. Lead it weighed by Avoirdupoip weight. See the Table of CompariAow 07 Weights. «. [lUTOR'S Assistant.] RULE OF THkEK DIRECT. 4tt poor, and n to have 1 to poor rl to have I it : how d for the ! girls , r. nith, that ;h weigh- ins. 18. 2. 6 dwts vts. 10 gr ; salts, of 13 gr. pel- en spoons lave 1 rr. over. . each are 7z. over. ■ seconds 1 06633'^ a mass of Q-3_5_ dr. rdupois of ^e oil, hoMi f ointment it contain ? ir. each* commonly three num- sition; and es B. fourth «B. Lead is MPARI0ON 07 term to be found, in the same proportion to itself, as that which is between the other two. General rule for stating the question. Put the term of defnand in the third place ; that term of supposition which IS of the same kind as the demand^ the first ; and the other, which is of the same kind as the required term, the second* AlsOi the terms being thus arranged, reduce the first and third (if necessary) into one name, and the second into the lowest denomination mentioned. THE Rule of three otregt RisQUiRES the fourth term to be greater than the second, when the third is greater than \\\q first ; or the fourth to be less than the second, when the third is less than the first. Rule. Multiply the second and third together, lind divide their pro duct by the first: the quotient will be the luiswer, in the same denomi- iiation as the second. t The following methods of contracting the oi)eration8 in the Rule of TriAKK are highly important, and should never be lost sight of: — 1. Let the first and third terms be reduced no lower than is necessary to make them of the same denomination. 2. Let the dividing term and either (but not both) of ike other terms he divided by any number that will divide them exactly ; and use the quotients instead of the original numbers. 3. When it is conveniently practicable, work by Compound Multipli- cation and Division, instead of reducing the terras. * Some modem authors prefer placing the term of demand the ieeond, and that similar to the required term the third. This airangertient will answer the purpose equally well, observing that those of like kind must be reduced (if necessary) to the same name. t The following General Rule comprehends both the cases c€ Direct and Inverse Proportion under one head; which is consider- ed by many scientific men of the present day as a more systematic arrangement. Rule. The question being stated, and the terms pi-epared, consider, from the nature of the case, whether the required term is to be greater or less than the second, or term of similar kind: if greater, multiply that similar to the answer by the greater of the other two, and divide the product by the less; if less, multiply it by the less and divide the product by the greater. In either case tht^ quotient will be the term requirtid, in the same denomination as the similar term. Note. It is evident that the above Rule will answer generally, 'Whether the term of demand is put in rtie sfjcond or third place. c :/■ 1 60 RULE OF THRER DIRECT. [tutor's (1) If one lb. of sugar cost 4iJ. what will 54 lb. cost?* (2) If a gallon of beer cost 10c?. what is that per barrel? Ans. jei..lO. (3) If a pair of shoes cost As. 6d. what is the value of 12 dozen pairs ?t (4) If one yard of cloth cost 15^. 6d. what will 32 yards cost at the same rate? Ans. jC24..16. (5) If 32 yards of cloth cost £24. .16. what is the value of one yard ? Ans. I5s. 6d. (6) If I gave £4.. 18. for 1 cwt. of sugar, at what rate did I buy it per lb. ? Ans. \0\d. (7) Bought 20 pieces of cloth, each piece 20 ells, for 12j 6c?. per ell, what is the value of the whole ? Ans. £250. (8) What will 25 cwt. 3 qrs. i4 lb. of tobacco come to, at \b\J. per lb. ? Ans. jC187..3..3. (9) Bought 21\ yards of muslin, at 6*. ^^d. per yard, what is the amount of the whole ? Ans. £9..5..05 i. (10) Boifght 17 cict. \ qr. 14 lb. of iron, at 3jcZ. per lb what was the price of the whole ? Ans. £26..7..0i. (11) If coffee is sold for b\d. per ounce, what will be the price of 2 cwt. ? Ans. £82. .2. .8. (12) How many yards of cloth may be bought for £21. 11. .11. when 31 yards cost £2. .14. .3. ? Ans. 27 yards, 3 qrs. 1 Jy nail. (13) If 1 cwt. of Cheshire cheese cost £1.. 14. .8. what must I give for 3^ lb. ? Ans. Is. Id. (14) Bouglit 1 cwt. 24 lb. 8 uz. of old lead, at 9*. per cwt. what did the lead cost? Ans. \0s. Hi ^^^d. (15) If a gentleman's income be £500. a year, and he spend \9s. Ad. per day, what is his annual saving ? Ans. £147. .3. .4. (16) If 14 yards of cloth cost 10 guineas, how many Flem- ish ells can I buy for £283..17..6. ? Ans. 504 Fl. ells, 2 qrs. (17) If 504 Flemish ells, 2 quarters, cost £283..17..6. what is the cost of 14 yards? Ans. £10. .10. I f t lb. d. Ih. J}T. s.d. ■^rt. t A« 1 : 4i 4 18 : : 54 18 4)97'i qrt. 12^243 d. ' As 1 : 2^ 4..6 12 14. .0 12 : : 144 in.». 'U. — £1 .0. .\i. Ant. je32 ..8..0. An*. ASSISTANT.] RULE OF THREE DIRECT. 51 (18) At tlie rate of i;i..l,.8. lor 3 Ih. of gum acacia, what must be given for 29 Ih. 4 oz.1 Ans. 10..11..3. (19) If 1 English ell, 2 quarters, cost 4^. Id. what will 39|- yards cost at the same rate ? Ans. £^..l..b\ ^. (20) If 27 yards of Holland cost jC5..12..6. how many Erglisl (lis can I buy for £100.? Ans. 384 ells. (21) U 7 yards of cioth cost lis. 8d. what is the value of 5 pieces, each containing 27-1- yards? Ans. 17. .7. .01 |. (22) A draper bought 420 yards of broadcloth, at the rate of 14^. lO^d. per ell English: what was the amount qf the purchase money ? Ans. JC250..5. (23) A grocer bought 4 hogsheads of sugar, each hogshead weighing neat 6 cict. 2 qrs. 14 lb. at jC2..8..6. per cwt. what is the value ? Ans. i:64..5..3. (24) A draper bought 8 packs of cloth, each pack contain- ing 4 parcels, each parcel 10 pieces, and each piece 26 yards ; at the rate of X'4..1G. for 6 yards: what was the purchase ^noney? Ans. JC6656. 25) If 24 lb. of raisins cost 6s. 6d. what will 1 8 frails cost, •M.h frail weighing neat 3 qrs. 18 Ib.l Ans. jC24..17..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.1 Ans. £313.-5. f27) What is the value of a*pack of wool, weighing 2 cwt, 1 qr. 19 lb. at 175. per tod of 28 lb. ? A71S. JC8..4..61 If. (28) Bought 171 tons of lead, at £14. per ton ; paid car- riage and other incidental charges, £4.. 10, Required the whole cost, and the cost per lb. Ans. £2398.. 10. the loliolc cost, and the cost per lb. l^d. a-ffly. (29) If a pair of stockings cost 10 groats, how many dozen pairs can 1 buy for £43. .5. ? Ans. 21 doz. 7| pairs. (30) Bought 27 doz. 5 lb. of candles, at the rate of 5*. 9d a dozen: what did they cost? Ans. £7..17..7|. (31) A factor bought 80 pieces of stuff, which cost him £5 17.. 17.. 10. at 4.v. lOd. per yard. IIow many yards were there in the whole, and how many English ells in a piece ? Ans. 2143 7/ards ; end 19 rJIs. 4 qrs. 2f^- nails, in a piece. (32) A gentleman has an annuity of £896.. 17. What may he spend daily, that at the year's end he may lay up 200 guin- eas, after giving to the poor quarterly 10 moidores? Ans. JC1..14..8 yVj. 1 I I , '■, i| 5e RULE OF THREE INVERSE. [TUTOR'S THE RULE OF THREE INVERSE Requires the fourth term to be less thau the second^ whei* the third is greater than the first ; or the fourth to be greater than the second, when the third is less than the first. Rule. Multiply the first and second together, and divide their pro* duct by the third : the quotient will be the answer, as before. (1) If 8 men can do a piece of work in 12 days, in how many days can 16 men do the same ?* (2) If 54 men can build a house in 90 days, how many men can do the same in 50 days ? Ans. 97^ men. (3) If, when a peck of wheat is sold for 2s. the penny loaf weighs 8 oz.; how much must it weigh when the peck is worth but \s. 6d. 1 Ans. 10| oz. (4) How many sovereigns, of 20^. each, are equivalent to 240 pieces, value 12*. each? A7is. 144. (5) How many yards of stuff three quarters wide, are equal in measure to 30 yards of 5 quarters wide ? Ans. 50 yds. (6) If I lend a friend jC200. for 12 months, how long ought he to lend me jC150. ? Ans. 16 months. (7) If for 24s. I have 1200 Ih. carried 36 miles, what weight can I have carried 24 miles for the same money ? Ans. 1800 lb. (8) If I have a right to keep 45 sheep on a common 20 weeks, how long may I keep 50 upon it ? Ans. 18 weeks. (9) A besieged town has a garrison of 1000 soldiers, with provisions for only 3 months. How many must be sent away, that the provisions may last 5 months ? Ans. 400. (10) If JC20. worth of wine be sufficient to serve an ordi- nary of 100 men, when the price is jC30. per tun ; how many will jC20. worth suffice, when the price is only jC24. per tun? Ans. 125 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, travelling 16 hours a day ? Ans. 18 days. (12) How much will line a cloak, which is made of 4 yards ol plush, 7 quarters wide, the stuff for the lining being but 3 quarters wide ? Ans. 9^ yards. m. d. m. f^ • As 8 : 12 : : 16 : l_iir=6 days. Ans. 16 -^ TUTOR'S 6?, whei* e greater their pro* in how iw many L men. nnv loaf peck is Of oz. valent to r. 144. are equal yds. ng ought lonths. It weight 300 Ih. nmon 20 weeks. ers, with !nt away, ?. 400. an ordi- ow many per tun ? 5 men. ravelling the same ) days. f 4 yards ing but 3 yards. ASSISTANT.] DOUBLE RULE OF THREE. 93 DIR£CT AND INVERSE PROPORTION PROMISCUOUSLY ARRANGED. (1) If 14 yards of broadcloth cost jC9..12. what is the purchase of 75 yards ? Ans. i:51..8..6f y\. (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 ? Ans. 7 days, 4 hours, 56 j%^- minutes. (3) How much must 1 lend to a friend for 12 months, to requite his kindness in having lent me £64. for 8 months ? Ans. JC42..13..4. (4) Bought 59 cwt. 2 qrs. 21 lb. of tobacco, at JC2..17..4 per cwt. what does it come to? Ans. £171.. 2.. 1. (5) A woollen draper purchased 147 yards of broadcloth, at 14^. 6d. per yard. Suppose that he sold it in pieces for coats, each 1| yard, how much must he charge for each, so as to gain jei6..10..9. by the whole ? Ans. i:i..9..3f (6) if jCIOO. gain £4. .10. interest in 12 months, what sum will gain the same in 18 months? Ans. jC66..13..4. (7) A draper having sold 147 yards of cloth, at the rate of i;i..9..33-. for If yard, found that he had gained X16..10..9. What did the whole cost him, and how much per yard ? Ans. The whole JC106..11..6. and \As. 6d. per yard. (8) If JCIOO. in 12 months gain je4..10. interest, in what time will jC6G.. 13. .4 gain the same interest ? Ans. 18 months (9) If a draper bought 147 yards of cloth, at 14^. 6c?. per yard, and sold it in pieces for coals, each If yard, for jC1..9..3f.; how much would he gain per yard, and by the whole ? Ans. 2s. 3d. per yard, JC16..10..9. by the whole. (10) If 1 ctut. cost £12. .12. .6. what must be given for M civt. 1 qr. 19 lb.? Ans. j£:i82..0..11i y|^. (11) If jClOO. gain £4. .10. in 12 months, what interest will £375. gain in the same time? Ans. £16. .17. .6. (12) A regiment of soldiers, consisting of 1000 men, are to have new coats, each to be made of 2^ yards of cloth, 5 quarters wide, and to be lined with shalloon of 3 quarters wide. How many yards of shalloon will line them? Ans. 4166 yards, 2 qrs. 2| nazb. THE DOUBLE RULE OF THREE Has jl^re terms ffiven, three of supposition and two of demand, to find a .nxth, in the same proportion wiih the terms of de- mand, as that of the terms of supposition. It comprises two 11 t'., J || M DOUBLE RULE OF THREE. [tutor's operations of the sini. ^. rule. — But it may comprise *hree four, or more operations of the Single Rule ; as there may be seven terms given to find an eighth, or nine to find a t€7ith, &;c. In this respect it is unlimited ; and is therefore more properly called COMPOUND PROrORTION. - Rule 1. Put the (rr)}^ of dcmavd one under anotlier in the third place; the terms of suppusifiou in the sanie order in \he first place; except that wliich is of ihe sainc ludi/re us the required term, which must be in the second place. Examine the S'tatiiiirs scjiarately, usiuu the middle term in each, to know if the ]>roj)ortion is direcl or inverse. ^Vhen direct, mark the first term witli an asterisk: wlieii inverse, mark llie third, term. Find the prothict of \\ic marked terms fur a Divisor, and the product of all the rest for a Dividend : divich:-, and tiie quotient will be the answer.* Rule 2. (1) Of the conditional tenns, put the principal cause of action, gain or loss, &c., in \\m first ])hice. (;}) Pnt that which denotes time or distance, Si^c, in the second, and the other in the third. (3 Put the terms (»f demand nnder the lilie terms of suppositicm. (4) If the blank falls in the third place, nndtiply the first and second terms for a divisor, and the other three for a dividend. (5) But if the blank is in the first or second phice, divide the product of the rest by the pro- duct of the third and four.'h terms, for tlie answer. Note. It will save nmch labonr to write the terms of the Dividend over, and those of the Divisor mider a line, like those of a compound 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 5G bushels of oats in 16 days, how many bushels will serve 20 horses 24 days ?t (2) many (3) Willi (4) will g (5) will (6) See also Su])plemental Questions, Nos. G and 7. t By Rule 1. h As*14 d •16 6^ h. 56 By Rule 2. h. d. b. 14 : 16 : 56 rf 20 : 24 : — ^ 10 12 1 ^ 10 12 t '^ 1 1 120 bushels. By two single statings h. h. h. h. h. (1) As 14: 56: : 20: 80 120 d. h. d. h. (2) As 16: 80:: 24: 120 '.' [tutor's se *'hTee may be '»M, (fee properly the tkiri rst place; rm, which (1 each, to mark the cm. le product ill be the cause of h denotes lird. (3 1. (4) If »nd terms the blank f the pro- Dividend rnmpoimd of Vulgar Three. ys, how ASSISTANT.! PRACTICE. 55 (2) If 8 men in 14 days can mow 112 acres of grass, how many men can mow 2000 acres in ten days ? Ans. 200 men (3) If JCIOO. in 12 months gain £Q. interest, how much will £75. gain in 9 months ? .472.9. jC3..7..6. (4) If jCIGO. in 12 months gain jC6. interest, what principal will gain JC3..7..6. in 9 months l Ans. £75. (5) If JCIOO. gain X'6. interest in 12 months, in what time will jC75. gain £3. .7. .6. interest? Ans. 9 months. (6) If a c?\rrier charges £"2. .2. for the carriage of 3 cwt. 150 miles, • / . i ought he to charge for the carriage of 7 act. 3 grs. i4 lb. ; miles? Ans. jC1..16..9. (7) If 40 acres of grass be mown by 8 men in 7 days, how many acres can be mown by 24 men in 28 days ? Ans. 480. (8) If £2. will pay 8 men for 5 days' work, how much will pay 32 men for 24 days' work ? ' Ans. jC38..8. (9) If a regiment of soldiers, consisting of 1360 men, con- spme 351 quarters of wheat in 108 days, how much will 1 1232 soldiers consume in 56 days? Ans. 1503-gi'j. qrs. (10) If 939 horses consume 351 quarters of oats in 168 days, how many horses will consume 1404 quarters in 56 days? Ans. 11268. (11) If I pay £14. .10. for the carriage of 60cit'^ 20 miles, what weight can I have carried 30 miles for £5.. 8. .9. at the same rate? Ans. 15 cwt. (12) If 144 threepenny \oQ,yes serve 18 men for 6 days, how many fourpenny loaves will serve 21 men for 9 days ? Ans. 189. I '\\' I 'k ihels. tatings L h. 0:80 I. b. 4: 120 PRACTICE Is so called from its general use among merchants and tradesmen. It is a concise method of computing the value of articles, (fee, by taking aliquot parts. The General Rule is to suppose the price one pound, one shilling, or one penny each. Then will the given number of articles, consider- ed accordingly as pounds, or shiUings, or pence, be the supposed value of the whole ; out of which the aliquot part or parts are to be taken for the real price Note. An alicfuot part of a number is such a part as being taken a cert^-iiu numl)er of times will produce the number exactly : tlius, 4 is aa aliquot part of 12 ; because 3 fours are 12. \ I «« PRACTKJE. h^TOIi'l ALIQUOT PARTS. Of a pound. s. d. £. 10 are 4 6 8 5 4 3 2 2 1 1 1 1 4 6 8 4 3 2 I } I B j_ 1 2 1 1.? 1 1 " o ,. ¥ 6 ... 1 Ofa shilling. d. 5. 6 are I 7> 4 3 I 1 2 1 8 IS 1 1 2 O/* fl penny. 2 yr*. are 4*^. 1 qr. is frf. /f. Of a ton. ciut. 10 are 5 ton. 1 o 1 4 Of a quarter. lb. 14 are -k 2 3 qr la /i | *.> 2' ..... 1 is 1 1 Iff I ITS Ofa civt. qr. lb. cwt. 2, or 56 are \ ^,or28 ... I 16 ... \ 14 ... I 8 ... yV 7 • / ... j^ 7 4 2 1 IS qr. 1 1 1 1 I TF 1 2l Of a lb. oz 8 4 2 1 are 18 V I ! 1 H 1 IT O/a/i. Troy. oz. lb. 6 are ^ 4, &c., as in the parts of a shilling. Of an oz. Troy. The same as the parts of a £. changing the names from shilltnf^s to dwis. 12 8 6 4 3 2 U 1 dwt. I are ^ 18 I RuLi 1. When the pHce is l^ss than a penny, call the given nuTU ber pence, and take the aliquot- parts that are in a penny ; then divide by 12 and 20, to reduce the answer to pounds. (1 ) lis 1570 IM. all 12)1426 2ionT;s..io Ans. JC5..18..10. «(1) 7547 at 1(/. Ans. X31..8..11. (2) 7695 at iJ. Ans.£l6..b..7l. (4) 6547 at ^d. Ans. i:20..9..2^. (5) 4573 at |u price. — Or, work for the ghillings as in the preceduig Rules, and lake parts ft)r the residue. t(7) 2710 at 3;?. 2 J. Ans. JC429..1..8. (13) 7152 at 17a-. 63(/. Ans. £6280.. 7. (14) 25 1 oTit" uZfirf. 1721..]9..2.!.4;/.v. X*lS32..16..5i. __. _ _ 4^ (8) 7514 at 4.y. 7d. Ans. (9) 2517 at 5s. 3(/.:(15) 3715 ai 95. 4i<;. Ans. i:G60..14..3.! Ans. X:i741 ..8..U. t(l) 2710 at 6s. 8d. Ans. i:90 3..6..8. (2) 3150 at 3s. 4 J. A71S. X525. (3) 2715 atr2s.'6d. An s. X;339..7 ..6. (4) 7150 at Is. 8c?.~|(10) 2547 at 75. 31(^.(16) 2572 at 135. 7i(/. Ans. jg595..16..8.L4;^s\ X928 . . 1 LJ. 5| . I Ans. £1752. .3' 6. (5.) 3215 at l5. 4d. 1(11)3271 aTb^. 9]7L(T7)725rat 145. 8}d. A ns. JC214..6. .8., .4 ;;5. jC943.. 16..4| Ans. £ 5324. .\9..0^. (6) 72U at I5. 3^7 '( 1 2) 2 1 03 at 1 57. 4iJ. (18) 32": at 1 5s. 7^d Ans. i:450..13..9..4«5. i;i616..13..7J-.i 4«5. X'2511..3..1ia R01.B 7. When the price consists of pounds, shillings, and pence multiply the given quantity by the number of pounds, and ake aliquot parts for the residue. — Or, work for the shillinirs as in il: e preceding Rules, &c. — Or, vv^hen the given number of articles is etjT. Jarge, work by Compound Multiplication. « s. s. 5. d. £. s. d. £. • 2=A 3270 t 6..8- 4 2710 t2 .6='-- - * 2710 1=* 327 Ans . i:903...' 8. c^- - 338. .56 163. .10 »~ ' " 9( « Ans. £490. 10 "424 m m 1 1 1 ' 'r*. 1 ' 1 " 1 m PRACTICi;. [TUTOR^t *(l)7215ut jC; 4. An$. X'51018. (2) 2104 at i:5..3r Ans. i;i0835..12, (3) 2107 ut £2..Q. Aug. £ri0bCu.]6. ni 7156 tit £r,..(u Ans. JL:i7[>-2r,.AC,. (5)S710atX2..:}..7i. Am X ftUll.M. A (6)3215!itX*r..ir. Ans. i;oy47..15. (13)32l0ati;i..l8..C|. Ans. JC61 89..5..7J (f4)2157 at je2..7..4i. Ans. £5109..7..10i. (7) 2107 lit XM. .13. Ans. Jf3 47H..ll. (8y3Trral~Jt47.7(il at X"-.»..3..4. ! A lls. £' ) V,b'2.M.A . !(Trf27r.J at jei..l7..2/i.j( 17) 37 aFi:L7igL3|7 i Aus^ £imi .■0..7| .l ^ «* . Je73..0..8| . |(12) 21 07 iitX'3..1.-).. 24". 1(1^72175 at je2..15..4i. ; Alls. XG103..19..5il /Ira.?. j£:6022..0..7i (15) 142atX1..15..2|. A>is. i:250..2..g^. (Hi) 95 at X;ir)..14..7f An. (10) Bought sugar at X*3..14..G. the tirt. what did 1 give for 15 cwt. 1 qr. 10 lb. ? Ans. Jt;57..2..9. (11) Required the value of 17 oz. 8 dwts. IH grs. of gold, at i:3..17..10f per ounce. Ans. jC67..17..11. (12) At JC37..6..8. per cwt. the value of 1 cwt. 2 qrs. 10| lb. of cochineal is required. Ans. £b\)..\0. (13) Required the value of 13 hhds. 42 gals, of Champagne wine, at je2o..l3..6. per hhd. Ans. jC350..17..10. (14) A gentleman purchased at an auction an estate of 149 a. 3 r. 20 p. at jC54..10. per acre. What was the whole purchase money, including the auction duty of 7d. in the £. the attorney's bill for the deeds of conveyance, jr33..6..8. and his surveyor's charge for measuring it, at Is. per acre 1 Ans. t;8i47..5..0^. Rule 9. To find the price of 1 lb. at a given number of shillings per ewt. Multiply the shillini^s by 3 and divide tue prodi. . : by 7 ; ' -e quotient will be the price of 1 lb. in farthings." (1) What is the price of 1 lb. at 44j. 4c?. per ::wt.?j (2) What are the respective prices per / - t 86^. 4d. ; 1 ' s. ; and 116j. 8d. per cwt.? Ans. 9{(i., ^^d., and Is. O^d. Rule 10 It is sometimes expedient to change ♦he price id the quantity for each other. Thus 43 yards at 2s. Od. will be 'equivalent to 33 yards at 4s. ; because 2s. 9d. = 33d. and 4s. = iSd. (1) What is the value of 72 yds. at 3^ 5d, and at 145. 7c?. ;»er yard ? Ans. £12. .6., and i:52..10. (2) 80 yds. at 15^. 3d. and at 16*. 8d. per yard ? Ans. £61., and £66.. ld..4. (3) 42 lbs. at ll^c?. and at Is. 3ld. per lb. ? ^ Ans. JC2..0..3., and £2..\3..4\. TARE AND TRET. Gross weight is the -,v -ight of any goods, together with that of the package which contains them. g» * Multiplying by 3 reduces the shillings to fourpences, and 7 four- pence* (or 2«. id.) are the value of 1 cwt. at I farthing per lb. t 445. U. 3 7^33 19 fi^rtfeings = i%d. per lb. Ans. ■■1 li * j| 13 'lAKi: AND TUET. [TUTOR'S IKnit ttriii/it is thai )!' llio tirtirles alone, or what remains ulu>r lh»* ihMhu'tioii ol' ail allowaiu'os. '/'«/>• is an allowanct' lor the weight of the package. It is oilh<*r so imii'h in the \vlioU\ or at so ninch per hag, hex, bar- rel, i^i'., or at so iniu'h in the owt. 'I'rt't is an allin\aiu-e ot' 1 l'>. in 10 1 //». (or tj'^ part) for waste. (V;;r/"is an allowaiu-e ol* //». ni 3 cicL on i-onie goods : but both these art' nearlv obja the Tc-c is at inui'h {H"r -vc;. rake tho a'ij't.-t y^zrt.i ci" the Cras tor the Tire. Subtfiu-t the Y'.i'-f tVi'ra tr.o (7'.\<.<; tlie rv:iia:L.J.er ia the Siat: aniens tl UM\> is 7' illo\vi-< If 7 rt't IS auoueei traeteU tVvnu ir. the ren\a:.;:de: -.5 :he r.: •^ S\l wnioh b^ir/' s' Tih- if C '.r*r ai3*j u to Ih' iiUowevl. the 3. will Iv tho lb ^ C. ••^\ w niu 'tiDiied Hv -2, aa'l diviaed bv . ^ . .^>. b nn,i :n- V (^n In : fr.ii ^^•,M WU.i: IS bead' ' N p : 'r r-.i: h wei^hiri 5 a/-*. :2 w ::".:.:h ne-i" we['~i" 4/:/, u ■ : -J ! » .( jU ^' ;) I a lo ba^^i ri:a wcLi:n.:"i .:>. 4 ')z. jros3. lart? p«r bv.ii:. 'W ::-2"v rcuibii zci: ' Aij. L3IL/J V 4^ W 'J.: Wc::^: J I'M'sieaaj Of tcoacco. weiiiaiai; i^ro^^s "ne -i^it.'ie . qU i-5 J. 7 J iht* ^H^^ lu :*:> bar' w?A\t" j;' 'W J -J >.7 r"':ss. *^re m An.! ZK rvt. i :u::j :i .-..miita. -jticG 15 pai 1 do2 1 7 do2 H do2 2 do2 17 pai Jx":=:a iM. <^ . 1 "Ir.'ii' •( ■!i< V>i' •'.■■).. :5 -;-3r». [tutor's remains ye. It is box, bar- waste. : but both } has been , the ichole When i: • the Tare. at; unless al**j u to ASSISTANT.] INVOICES. 63 (7) In 25 barrels of figs, cacli 2 cwt. 1 qr gross, tare per cwt. 16 lb. how niurh neat weight ? A?is. 48 cji-^. qr. 24 ZZ>. (8) What is tlie neat weight of 9 liogslieads of sugar, each weighing gross 8 cwt. 3 qrs. 14 lb. tare 16 Ih per cwt. ? ^n.s. 68 cwt. 1 ryr. 24 lb. (9) In 1 butt of currents, weighing 12 cict. 2 qrs. 24 /i. gross, tare 14 lb. per 6-u;/. tret 4 /^. per 104 lb. what is the neat weight ?* (10) In 7 cwt. 3 qr.s\ 27 lb. gross, tare 36 lb. tret according to custom, how many pounds neat? Ans. 826 lb. (11) In 1.52 crct. 1 yr. 3 lb. gross, tare 10 /^. per act. tret as usual, how much neat weight? Ans. 133 cwt. 1 ryr. 12 lb. (12) What is the neat weight of 3 hogsheads of tobacco, weighing 15 cwt. 3 qr.s. 20 lb. gross, tare 7 lb. per. cw^ tret and cloff as usual ?t (13) In 7 hogsheads of tobacco, each weighing gross 5 cwt. 2 qrs. 7 /^. ; tare 8 Z^a per cwt. tret and cioft" as usual, how much neat weight? Ans. 34 cwt. 2 yr-s. 8 lb. ■I I -^'rj*. 5 lb. t. :' '.ocacco. '}z. jT'^sa. L 3 1 L i^ tooacco. I^ 'J). i. :ar:r m r£t. I -/r l.iia. -iach INVOICES, OR lilLLS OF PARCELS. (1) Mrs. Bland, London, Sept. 1, 1830 Bought of Jane Harris. s. (1. £. d. 15 pairs worsted stockings at 4 1 doz. thread ditto . . at 3 i doz. black silk ditto . at t li doz. milled hose . . at 4 2 doz. cotton ditto . . at 7 17 pairs kid gloves . . at 1 6 per pr. 2 3 2 6 8 JC21..18..4 ■ '.• .tfr~^iw ■'•tr. t lb. ciot. qrs. lb. 14 = ] 12.. 2.. 24 gross /J 1..2.. in tare. 4=r TS" 11. 0. 1. 14 suttle. 19 tret. An.) 66 SIMPLE INTEREST. [rOTOK I SIMPLE INTEREST Is the premium allowed for tlie loan of any sum of money during a givon space of time. The PrinJpal is the money lent, for which Intertsi is to be received. The Rate per cent, per annum, is the quantity of Interes, (agreed on between the Borrower and the Lender) to be paid for the use of every X*100. of the Principal, for one year. The Amount is the Principal and Interest added together. I. To find the Interest of any Sum of Money for a Year. Rule. Multii)ly the Principal by the Rate per cent, and that Pro- duct divided by 100 will give the Interest required. Note. When the Rate is an aliquot part of 100, the Interest may be calculated more expeditiously by taking such part of the Principal. Thus, for 5 per cent, take ^^ '■> for 4 per cent. ^Sf, or \ of \\ for 2 per cent. ^^ ; for 2^ per cent. -^^ ; for 3 per cent. ^, vlus ^ of that ; &c. This Rule is applied to the calculation of Commission, Bro- kerag-^. Purchasing Stocks, Insurance, Discounting of Bills, &c.* II. For several Years. Multiply the Interest of one year by the number of years, and the product will be the answer. For parts of a year, as months and days, &c., the Interest may be found by taking the aliquot parts of a year ; or by the Rule of Three : and it is customary to allow 12 months to the year, and 30 days to a month. f * To discount a Bill of Exehiinge is to advance the cash for it befwe it becomes due ; deducting the Interest for the time it has to rur Bank- ers alwfays cliarge Discount as the Interest of the sum. t At the rate of .5 \^eY cent the iiitfM'est of £\. for a yea: A Ij. , or one pen-uy for a montii. Therefore, the principal X the number of mon/lis, gives the interrsf m poire. Or. take the parts of a yeiir for tlie months, out of as many shillingt as there are pnnnds in the prijic/pal. Thus, to find the interest of i."10..10. for 2 months, say 40.^^. X 2 = 8ld. = Gs. 9d.', or, 5 montlis being ^ of a year, 40s. Grf. -f- G = 6s. Sd. Ans. For days, take the ali(|uot jiarts of a month. The interest for days at 5 per cent, may also l)e found ]»y multijdying the j)rincipal by the number of days; and the jtroduet divided by 3fi.') will give the answer in shillings; or divided by 7301) ^—iiCo x ~0) will give the answer in pounds. selling (8)^ Soutli ^ (9)^ purchas (10) Bank a (11) 17..10. (12) H-^i? Consols i },or2^ < 'Ik ;tdtok ■ f money t is to be Interes, be paid year. ogether. 1 Year. I that Pro. e Interest art of the cent. ^, sion. Bro- of Bills, \ one year answer. e Interest or by the nonths to br it befwe •ur Bank- >T is 1*. , or number of ny shillingi Oii. X 2 = G = 65. 9a. >Bt for data ipal by the the answer 3 answer in ASSISTANT.] SIMPLE INTEREST. 67 (1) What is the interest of £375. for a year, at £5. per cent, per annum ?* (2) What is the interest of JC945..10. for a year, at £4. per cent, per annum ? Ans. £37. AG. A^. (3) What is the interest of JC547..15. at £5. per cent, per annum, for 3 years? Ans. jC82..3..3. (4) What is the intercr-^t of i:254..17..G. for 5 years, at £4. per cent, per annum? A7is. £50. .19. .6. (5) What is tlie amount of £556. .13. .4. at £5. per cent per annum, in 5 years ? ^4^/5'. £695.. 16. .8. Note. Cfimminsion and Brohrrnge, (commonly culled Brokage) are allowaiict'S of so much per cent, to an agent or broker, tor buying or selling goods, or transacting business for another. (6) My correspondent informs me that he has bought goods to the amount of £754. .16. on my account, what is his com- mission at £21. per cent.? Ans. £18..17..4J. (7) If I allow my factor £3|. per cent, for commission, what will he require on £876. .5.. 10 ? Ans. £32..17..2i. NoTK. Stock is a general term to designate the Ca])ital8 of oitr Trad- ing Companies; or to denote Proprrtii in the Public Finids ; which means the Money paid by Govennnent for the interest of the National Dclif. The quantity of Slock is a nominal sum, for which the owner receives u certain rate of interest while he holds the same. (8) At £1101. per cent, what is the purchase of £2054. .16. Soutli Sea stock ? Ans. £2265. .8..4. (9) At £104|. per cent. South Sea annuities, what is the purchase of £1797.. 14. ? Ans. £1876..6..1]f (10) At £96f . per cent, what is the purchase of £577.. 19. Bank annuities? Ans. £559..3..3f. (11) At £1241". per cent, what is the purchase of £758.. 17.. 10. India stock ? Ans. £945..15..4i. (12) What sum will purchase £1284. of the 3 per cent. Consols, at £by|. per cent. ; including the broker's charge of \, or 2.9. 6(L per cent, on the amount of stock ? Ans. £770..7..11f. * £. 375 5 £. 18|75 20 s. 15iOO Better thus £. £. ^n.?.£l8..15. Ans. £18.. 15. Cuttiiig the two figures in the above divides the number by 100 : Bee Division, p. 21 \\ 68 SIMPLE INTEREST. fTUTOR«« (13) If I employ a broker to buy goods for me, to the amount of ;e2575..17..6. what is the brokerage at 4^. pei cent. ?* (14) What is the broker's charge on a sale amounting t( £7105. .5.. 10. at 5s. 6d. per cent. ? Ans. i:i9..10..9f (15) What is the brokage on goods sold for jC975..6..4. at 6s. 6d. per cent. ? A?is. jC3..3..4i. (16) What is the interest of jC257.. 5.. 1. at jC4. per cent, per annum, for a year and three quarters ? Ans. jG18..0..1j. (17) What is the interest of jC479..5. for 5} years, at £5. per cent, per annum ? Ans. jC125..16..0|. (18) What is the amount of i:576..2..7. in 7^- years, at jC4i. per cent, per annum ? Ans. £764.. I. .8^. (19) What is the interest of jC259..13..5. for 20 weeks, at £5. per cent, per annum? Ans. jC4..19..10{. (20) What is the interest of £2726.. I. .4. at jC41. per ceni per annum, for 3 years, 154 days ? Ans. jC419..15..61. (21) Compute the interest of £155. for 49 days, and for 146 days, at £5. per cent, per annum. Ans. £1..0..9i. and £3..2..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. ? Ans. 8s. lid. and 8s. 3d. When the Amount, Time, and Rate per cent, are given, to Jind the Principal. EuLE. As the amount of iJlOO. at the rate and for the time given i to jCIOO., so is the amount given to the principal required. (23) What principal being put to interest will amount to ' £402.. 10. in 5 years, at £'^. per cent, per annum ?t [ (24) What principal being put to interest for 9 years will fi amount to £734. .8. ar £4. per cent, per annum ? , Ans. £540. 18 ;;;, i f 8. £. s. d. - hence. * 4=1 2575.. 17.. 6 03 £. 5|15.. 3.. C 12 £ 20 s. 3 03 42 4 ^ • 350 1( Ans. i;5..3..0i. 1 68 i £. tX3X5-i-J00«£ll3. As 113 : £. 100 : £. s. : 403..10 : £ 350 An*. t As years. frUTORte e, to the t 4^. pel mntmg ti 10..9f 5. .6. .4. at ..3..4i. cent, per !..0..1i. ■s, at £5. .16..0f s, at £4^. L.1..81. weeks, at .9..101. per ceni .15..61. 5, and for 3..2..0. t of a bill 18t,h of latter July nd 8s. 3d. en, to Jind me given is t t amount to { j . years will l . £5A0. '' ASSISTANT.! DISCOUNT. 69 (25) What principal, being put to interest for 7 years at £5. per cent, per annum, will amount to JE334..16. ? Ans. je248. When the Principal, Rate per cent, and Amount are given, to Jind the Time. RuLK. As the interest for 1 year is to 1 year, so is the whole in- terest to the number of 'years. (26) In what time will jC350. amount to £402.. 10. at £'A. per cent, per annum ?* (27) In what time will £540. amount to £734..8. at £4. per cent, per annum ? Ans. 9 years. (28) In what time will £248. amount to £334. .16. at £5. per cent, per annum ? Ans. 7 years. When the Principal, Amount, and Time are given, to Jind the Rate per cent. Rule. As the principal is to the whole interest, so is JCIOO. to it* interest for the given time. Divide that interest by the number of years, and the quotient will be the rate per cent. (29) At what rate per cent, will £350. amount to £402. .10. in 5 years ?t (30) At what rate per cent, will £248. amount to £334..ie. 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. DISCOUNT Is the abatement of so much money, on any sum received be- fore it is duGj as the money received, if put to interest, would gain at the rate, and in the! time given. Thus £100. present money would discharge a debt of £105. to be paid a year hence. Discount being made at £5. per cent. ^;ijl: Atu. £. * 350x3 - ■■ = iJlClO. the interest for 1 year. je401>..10. — i:350. = je52..10. the whole interest As JC10..10 : 1 year : : £52. .10 : 5 years. Ans. i As je350 : JE52 .10 ; : XlOO : £15 = the interest of XlOO. for 5 years. Then 15 -f- 5 = £8 the rate per cent. I ■' ^ 70 DISCOUNT. [tutor's Rdlk. As jCIOO. willi its interest for the time given is to that in- terest, so is the sum given to the Discovnt required. Also, As that Amount of iJlOO. is to iJlOO., so is the given sum to the Present worth. But if either the Discount or the Present worth ho found by the pro- portion the other may be found by subtractlrig that from the given sum. (1) What are the discount and present worth of JC386..5. for 6 months, at £6. per cent, per annum ?* (2) How much shall I receive in present payment for a debt of £357. .10. due 9 months hence, allovp^ing discount at £5. per cent, per annum? Ans. jC344..11..6^ \^. (3) What is the discount of £275. .10. for 7 months, at £5. per cent, per annum? Ans. £7. .16. .If -jW- (4) What is the present worth of £527 .9..1. payable in 7 months, at £4^. per cent, per annum ? A71S. £5\4..13..10^ J>^%yL. (5) Required the present worth of £875. .5. ,6. due in 5 months, at £4^. per cent, per annum. Ans. £859. . 3. .3^ joh- (6) What is the present worth of £500. payable in 10 months, at £5. per cent, per annum ? A?is. £480. (7) How much ready money ought I to receive for a note of £75. due in 15 months, at £5. per cent, per annum? Ans. £70..11..9yV. (8) What will be the present worth of £150. 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 £5. per cent, per annum? Ans. £145..3..8-i-. (9) Of a debt of £575.. 10. one moiety is to be paid in 3 months, and the other in 6 months. What discount must be allowed for present payment, at £5. per cent, per annum ? Ans. £10..11..4|. (10) per am C mont credit, much I (1) years, 6 m. = i i;^ £. 100 -f- 3 = 103 = amount of i;iOO. in 6 months £. £. £. s. As 103 : 3 : : 386".. 5 £. s. 3 386..5 103)1158.. 15( 11.. 5 discount. 1133 25 20 37 5.. present worth. 103)515 = 5«. [tutors to that in* sum to the Dy the pro- the given i:386..5. lent for a scount at ,61 ¥3' hs, at £5. I 3 _9_5, ^4 24T' ^able in 7 _6JL3JL 2 4 5 » 5 • due in 5 ■"-'4 407S' ble in 10 . JC480. for a note »um ? .1..9^-V. yable at 3 nths, one- [)unting at 3..8f 3aid in 3 must be nnum ? 11..4|. ASSISTANT.] COMPOUND INTEREST. 71 (10) What is the present worth of jCSOO. at £4. per cenl. per annum, jCIOO. being to be paid down, and the rest at two 6 months ? . Ans. JC488..7..81 (11) Bought goods amounting tc JC109..1G. at 6 months' credit, or X'3}. per cent, discount for prompt payment. How much ready monev will discharge the account ?* Ans. jC105..13..4i. Note. The Rule to find the present worth of any sum of money is precisely identical with that case in Simple Interest in ■'vhich the Amount, Time, and Rate per cent, arc given to find the rrincipal. See page 63. COMPOUND INTEREST Is that which arises 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 Amount. Rule. Compute the first year's interest, which add to the principal: then find the interest of tliat amount, which add as before, and so on for the number of years. Subtract the given sum from the last Amount, and the remainder will be the Compound Interest. (1) What is the compound interest of jC500. forborne 3 years, at JC5. per cent, per annum ?t (2) What is the amount of £'400. in 3l years, at £5. per cenl. per annum, compound interest 1 Ans. jC474..12..6i. (3) What will £050. amount to in 5 years, at £5. per cent, per annum, compound interest? Ans. £829. .11. .71. (4) What is the amount of £550. .10. for 3i years, at £6 per cent, per annum, compound interest ? Ans. £675. .6. .5. (5) What is the compound interest of £764. for 4 years and 9 months, at £6. per cent, per annum ? Ans. £243..18..8. • The discount in cases of this sort is so much per cent, on the sum, without regai'd to time. It is, therefore, computed as a year's interest. t -^ £500 25 oV £551.. 5 27.. 11. 2» 525 amount in 1 yr. 26.. 5 578.. 16.. 3 amount in 3 years. 500.. 0.. principal subtract. 551.. 5 do. in 2 yrs. £78.. 16.. 3 Ans. \ 72 EQUATION OF PAYMENTS. [TUTOR'P (6) What is the compound interest of JC57..10..6. for 5 years, 7 months, and 1 5 days, at X'5. per cent, per annum ? Ans. £IS..3..8\. (7) What is the compound interest of X'259..10. for 3 years, 9 months, and 10 days, at £4J^. per cent, per annum? A?is. jC46..19..10i. 40X 3= 120 60 X 5= 300 100X10=1000 2|00)f4|20 Ans. 7y'^ thonths. EQUATION OF PAYMENTS Is when several sums are due at difierent times, to find » tiiean time for paying the whole debt ; to do which, this is iho common Rule. Multiply each term by its time, and divide the sum of thtr products by the whole debt, the quotient is acconuted the mean limb £. (1) A owes El £200. whereof £A0. is to be paid at 3 months, jCeO. at 5 months, and XMOO. at 10 months : at what time may the whole debt be paid together without pre- judice to either ? (2) B owes C £800. whereof X200. is to be paid at 3 months, XI 00. 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? Ans. 4 months, 18| days. (3) A debt of X'3r)0. was to have been paid as follows: VIZ. : XI 20. at 2 months, X200. at 4 months, and the rest at 5 months ; but the parties have agreed to have it paid at one mean time : what is that time ? Ans. 3 months, 13^ dai/s. (4) A merchant bought goods to the value of X500. to pay XI 00. at the end of 3 months, XI 50. at the end of 6 months, and X250. at the end of 12 months; but it was afteirwards agreed to discharge the debt at one payment : required the time. Ans. 8 months, 12 days. (5) H is indebted to L a certain sum, wuich is to be paid at 6 different payments, that is i at 2 months, | at 3 months, ■i at 4 months, i at 5 months, i at 6 months, and the rest at 7 months ; but they mutually agree that the whole shall be paid at one equated lime : what is that time ? Ans. 4} months. (6) A is indebted to B X120. whereof i 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 ? Ans. 5^ months. ' '"if [TUTOR'P I ioSJSTANT] BARTEA. 73 ..6. for 3 annum ? ^..3..8i. 10. for 3 r annum ? I9..10i. to find 9 this is tho sum of thfr mean tim6 20 00 00 20 J^ nionths. paid at 3 ind £200. le paid al J| daijs. 5 follows: e rest at 5 lid at one i days. 30. to pay 6 months, ifterwards juired the 2 days. to be paid 3 months, he rest at e shall be li months. paid at 3 'hat is the tnonths. DARTER ■-.■.., I ,. . Is the exchanging of commodities. Bor.K. Compute, by the most expeditious metbod, iav value of tbo artfcle wbose quantity is given: tbeii find what quaiitity of llie other at the rate proposed, may be hud for tho same laonry. NoTK. Sometimefl one tradesman, in barUirinir, aclvanreH his goods above the reticly money price. In this ca.se, it will hv necessary to pro. girtionate the other's bui'tering price to liis reiuJy money price, by tho ule of Three. ^ J ■ • ' 7 ■ t (1) What quantity of chocolate at 4.?. per Ih. must be ex- changed for 2 cict. of tea, at 9.y. per Ih. ?* (2) A and B. barter : A has 20 cwt. of prunes, at 4c?. per ^, ready moiiey, but in barter will have 5(/. per //>., and H has hops worth 32.v. per met. ready money : what ougiii V. to charge his hop^, and what quantity nuist he give for tho 20 cwt. of prunes ?} / , , (3) How much tea at ^s. per //;. can. t have in barifcf lor 4 cvol. 2 qrs. of chocolate, at \s. per Ih.J . Ans. 2 act. (4) A exchanges with B 23^ cwt. of cheese, worth 52.v Gr/. per cjr^, for 8 pieces of cloth, containing 248 yards,, at 4^. 4f/. per yard \ the difference to be paid in money. Who receive.'^ the bxilance, and how much ? Ans. A. receives X'7..19..1. (5) How much ginger at loJ^Z. per lb. must be exchanged for 31 Ih. of pepper, at \a\d. per Ih. ? Ans. 3 lh..\^\ oz. (6) How many dozcii of candles, at oa-. ^d. per dozen, must be bartered for 3 cvut. 2 qrs. IG Ih. of lullow, at 37,y. \d. i^tx CMnt.l , ' ., Ans. "i,^ dozen., '^\%lh. , (7) A exchanges with B 608 yards of cloth, worth M^. per ♦ 224 X - 20165. the value of tho tea. As 4.T. \ \lh. '. \ 20165. 504 lb. of chocolate. Ans. •..■■■ • .'■■■■'' > t As 4f/ : hd. : : 325. : 405. the price per cwt. to be charge J for the hops. 2^cxot, =32407*. ^5 I1200 equal the selnnc; pnce. * I minus the loss, ) ^ ° * The selliug price minus the jirime cost equal the gnin. the gain equal the prime cost. The sellmg price plus tlie loss cq7cal the prime cost. Gain or loss per cent, means so mnch on .£100. pnrchaBe money, or vrime cost: therefore, when .£20. per cent, are gained, £120. is the gelling price per cent. ; when £20. per cent, are lost, £80. is the tell tag price. Case 1. Given, the prime cost and the selling price of an integer 01 quantity, to find the gain or loss per cent. As the prime cost given: the gain or loss '. '. £100. : the gain or Ion per cent. Case 2. Given, the prime cogt as before, with a proposed gain or /o« per cent, to find the selling price. As £100 • ^ £100. phis the gai?i ) ; : the prime cost : the selling \ or £100. minus the ^«s y price. Case 3. Given, the selling price of an integer or quantity, and the gain or loss per cevt. to find the prime cost As £100. l>lus tlie nain \ n-mn *r. ;;• • *i, • . -TiAA „ • *i ; I ' £100. : : the selkne pnre : the vnme e&ti, or, £100. mums the /o.s.t J ^' ^ Case 4. Given, the selling price of an integer, and the gain per ceni, to find the gai?i- per cent, at some other proposed price. As the selling price '. £100. phis the gain : : the proposed price '. ths aellivg price per cent, from vvhicli deduct £100. for the gain per cent, reqvircd. Secondly, To find another selling price, at a dijj'ercnt gain per cent. As £100. phis the gain, '. the selling price : : £100. plus l\iG propotei gain : the selline vrtce required. A much greater variety of cases may occur ; but it is presumed that ihe student woo accaius u due luiowledge of these will easily compr* bend tue reiu. AflSlSl ^hat 1 (2) for to , (3) is the (4) cent. \ for ICi (6) for to ^ (7) cent, p (8)! cent, p they hi (9)1 yard to (10) gained him pe] (ii) (12) letailed (13) same b( (14) retailed gain, an As 1 Sixp. S tA8£ |incc. 1 X For [tutor's :i25..12. in \s. JC3..10. 4*. 6c?. pei What was cwt. 1 qr. )e given foi s. 48f lb. AflStstANT.] in the buy. to adjust the jr cent., &c. or Practice Mice. ase money, of £120. is the 50. is the tell an integer or le g-ain or hit d gain or lott : tie selling , and the gain le prime eoti> gain per cent. cd price : tlia ^ain per cent, ain per cent. the proposed )re»nTned that asily compr* fROflt AND LOSS. n (1) If 1 yard of cloth cost lis. and is sold for 12^. Gdi tvhat is the gain per cent. ?* (2) If GO ells of Holland cost jC18. what must 1 ell be solct for to gain £8. per cent. ?t (3) If 1 Ih. of tobacco cost IGJ. and be sold for 20c?. what is the gain per cent. ? Ans. £25. (4) If a parcel of cloth be sold for jCoGO. gainirtg jC12. pet cent, what is the prime cost ? Ans. JC500. (5) If a yard of cloth be bought for I3s. 4c/. and sold agaiit for 16^. what is the gain per cent. ? Ans. jC20. (6) If 112 lb. of iron cost 275. 6d. what must 1 cwt. be sold for to gain =£15. per cent. 1 Ans. £l.Al..7^. (7) If 375 yards of cloth be sold for jC490. at jC20. pef cent, profit, what did it cost per yard ? Ans. jC1..1..9{ i^|. (8) Sold 1 cwt. of hops for X'6..15. at the rate of £25. pejf cent, profit. What would have been the gain per cent, if they had been sold for £8. per cwt. ? Ans. £48. .2. .11^ f (9) If 90 ells of cambric cost £60 how must I sell it pef yard to gain £18. per cent. ? Ans. I2s. 7j^j\d. (10) A plumber sold 10 fothers of lead for £204.. 15. and gained after the rate of £12. .10. per cent. What did it dost' him per cM. f Ans^ I8si 8di (U) What was the profit orl 436 yards of cloth, bought a* 8*?. 6fi. and sold at 10^. 4d. per yard ? A7is. £39. J9..4. (12) Bought 14 tons of steel at £69. per ton^ which wm iGtailed at 6di per Ibi What was the loss sustained ? e, fluinbei ♦ 40; X5; i ^•^troii'i be divicl^cl determine il partneils ires of the on of cofifl* ^SBISTAN^.J FELLOWSHIP. n r, 80 IS each' wtiolo gain £20. ancf B ; B £30 each mans . C £80. nt stock ; B they gainei q Q JJL „o..ir 113' I, .5.. 61 ji^. 's stock was of 1 2 months ticular share 1 F £200. )'C£304..7, decease his wr must it ..1^—15750, 5—15620. t capital ; of the end of 6 >ionths t^ey gain jCIOO. What is each person's share of tho gain? ' A/w. ^ jC;35.4'..y— 12. B £26. .6.. 3^— 9. C £21. .1. .01— 30. and 1) .£17..10..10i— 6. (7) Two persons joined in the purchase ol" an estate yield- ing '-£1706. \>er annum, \'qr £27200. whereof D paid £15000. and E the rest : some time after, they sokl it for 24 years purchase. What was each person's share 1* ^ Ans. D £22500. E £18300. (8) D, E, aiul F, form a joint capital of £647. Their re- spective shares are in proportion to each other as 4, G, and 8 • and the gain is equal to D's stock. Required each person's stock and gain. Ans. D's stock £ri43..15..6^ gain, £:31..19..0/^. E's . . . 215..i3..4 . . . 47..18..6j\. >^'6' . . . 287..11..1jf . . . 63..18..0/^. (9) D, E, and F, joined in partnership ; the amount of their stock was £*100 ; D's gain was jC3 ; E's £5 ; and F's £8; ^hat was each man's stock ? Ans. D's .stock jC18..15. E's .^'31. .5. and Ps £50. FELLOWSHIP WITH TIME. Rur.r,. Aa ll^e siim of the products of each person's money and tijne, h to the whole pain or loss ; so is each individual product to the cor- respondhig gain or loss. (1) D and E enter into partnership ; D puts in £^40. for three months, and II £75. for four months, and they gain £70. What is each man's share of the gain ?t (2) Three tradesmen joined in company ; I) put into the joint stock £195.. 14. for three months ; E £169.. 18. .3. for 5 Tjionths ; and F £'59. .14. .10. for 11 months: they gained £364.. 18. What is each nvWs sl^are of the gain? Ans. D's £102. .6..4— 5008. E's £148. .1..U— 482802. anrZ P* £114. .10..6J— 14707. (3) Three merchants join in company for 18 months: D, puts in £500, and at 5 months' end t ikes out £200. at 10 * The sale of a property for sn many ycarf; fxircliasc, is understopd \o he, for so much present money as the annual rent or value X that niimhor of yeiirs. t 40X3--12q Z5X4— .'{pq 4'^ As 4210 : 7|0 : : 120 : £20=D' share. ;}00 : 50::= E's share. 70 Proof. r8 AhhlGXTlOJft [TUTOE^f months' end puts in jC300. and at the end of 14 it'onths takes out JC130 ; E puts in £:400. and at the end of 3 months i:270 more, at 9 months he takes out jC140. but puts in X'lOO. at the end of 12 months, and withdraws jC99. at the end of 15 months ; F puts in jC900. and at 6 niontlis takes out X'200, ,3,t the end of 1 1 months puts in £500. but takes out that and JCIOO. more at the end of 13 months, They gain jC200. Required each man's share of the gain. Ans. D X'50..7..6— 21720. E X'02..12..,5]— 29859, and /'\£'87..0..0J— 14167. (4) D, E, and F, hokl a piece of ground in common, for which they are to pay £'3G..10..G: D puts in 23 oxen 27 days; E 21 oxen 35 days; and F 16 oxen 23 days. What is each nian to pay of the said rent ? Ans. D X'13..3..1i— G24. E jC15..11..5— 1G88. and 7^X7..15..li— 1130. ALLIGATION Is a rule by which we ascertain the mean price of any corn* pound formed by mixing ingredient? 'f various prices; or the quantities of the various articles which will form a mix-^ ture of a certain ?ncan or average valw;. It comprises four distinct cases. Case 1. Alligation Medial, The various quantities ftnd prices being given, to find the mca7i price of tho mixture. Rule. Multiply each quantity by its price, and divide the sum of the products by the sum of tho quantities.* (1) A grocer mixed 4 cict. of sugar, at 56^. per cwt. with 7 cwt. at 43.S. per cwt. and 5 cwt. at 375. per cwt. What is the value of 1 ctvt. of this mixture ? Ans. £2..i..U (2) A vintner mixes 15 gallons of canary, at 8.s'. per gallon, with 20 gallons, at 7s. 4d. per gallon ; 10 gallons of sherry, Example. * A farmer mixed 20 busljnla of wheat, at fts. per l)usliel, and HH bushels of rye, at '3s. pt-r bushel, with 40 buslu^lf' of ])!irlfy, at Us. per bushel. What is tho worth of bushel of this mixtare ? cox 5 = 100 3(i x.'^ = H)8 40 X 2 ^ fiO i)U U0)21J8(; An* [tutor** I ASSISTANT.] ALLIGATION. 79 )ths takes ihs i:270 X'lOO. at md of 15 ut £200, t that and tin je^OO. 9859. and nmon, for oxen 27 . What 1688. and any com. prices; or rm a mix^ )rises four quantities mixture. the 6UU) o| ' cwt, will] What is 2 4 4J- per gallon, of sherry, t. An* at 6s. 8d. per gallon ; and 24 gallons of white wine, at 4* per gallon. What is the worth of a gallon of this mixture ? Ans. 6s. 21 ##tZ. I U 9' (3) A malster mixes 30 quarters of brown malt, at 28^. per quarter, with 46 quarters of pale, at 30.v. per quarter, and 24 quarters of high dried ditto, at 25*. per quarter. What is one quarter of the mixture worth ? Aus. jC1..8..2J- j%d. (4) A vintner mixes 20 quarts of port, at 5.?. id. por quart, with 12 quarts of white wine, at 5s. per ([uart, 30 quarts of Lisbon, at 6s. per quart, and 20 quarts of mountain, at As. 6d. per quart. What is a quart of this mixture worth I Ans. bs. 3 J f^(?. (5) A refiner melts 12 lb. of silver bullion, of 6 0,3." fine, with 8 Ih. of 7 oz. fine, and 10 lb. of 8 oz. fine; required the fineness of 1 lb. of that mixture. Ans, 6 02-. 18 dvct. 16 gr. « Case 2. Alligation Alternate. The vanovs prices being given, to find the quantities which may be mixed, to bear a certain average price. Rule. AiTange the given prices in one column, with the proposed average price on the left. Link each less than the average with one greater. Place against each tei-m the ditFerence between that with which it Is linked and the mean: and the respective differences will be the quantities required. Note. Questions in this rule admit of a great variety ' f answers, nccording to the manner of Ihiking ti ^ i ; » also by taking other nuni- ber» proportional to the answev.i t, und (1) A vintner would mix four :!orts of wine together, of 18c?. 20c?. 24d!. and 28c?. per quart what quan it^^ of each sort must be take to sell the mixture ol 21d. per quart "* (2) A grocer would mix sugar at 4 J. 6d. and 10c?. per Ih 00 18 — 20 •24 28 — Answer. 2 of Ud. G of 20./. 4 of 24rf. 2 of 2M. Proof. ■ \\Gd. ■■ 120 : 96 ; 56 14 14)303 22(/. 00 18- 24- 28- Or tint a : Proof. (i of ]i]d. = lond. 2 of 20./. = 40 2 of 2 4./. = 4a 4 of 2lic/. = 1 12 i4 14)308 22i I BO ALIJGATION. [TUTOR'S SO as to sell the compound lur 8(/. per lb. Wha| quantity of each kind must he take ? Ans. 2 lb. at 4t/. 2 lb. at Gd. and 6 lb. at 10 J. (3) How much tea at IG.v. 14.?. 9^. and 8s. per lb. will com- pose a mixture worth lOs. per lb.1 Ans. 1 lb. at ]6s. 2 lb. at 14s. 6 lb. at 9s. and 4 lb. at Ss. (4) A runner would mix as much barley, at 3a\ &d. pei bushel, rye at 4^. per bushel, and oats at 25. per bushel, a? will make a mixture worth 2s. ikl. per bushel. How much of each sort ? A?is. G b. of barley, (3 of rye, and 30 of oats. (5) A tobacconist would mix tobacco at 2.?., Is. 6d., and Is. 3d. per lb. so that the compound may be worth Is. 8d. -per lb. What quantity of each sort must he take? Ans. 7 lb. at 2s. 4 lb. at Is. 6d. and 4 lb. at Is. 3d. Case 3. Alligation Partial. This is similar to Case 2 except that one of the quantities is limited. Rule. Link the prices, and place tho differences as before. Then, as the diiVereiico opposite to that whose quantity is given, is tf' each other ditl'erence ; no is the given qucUitity to each required quantity. (1) A tobacconist intends to mix 20 lb. of tobacco at \5d. per lb. with others at 16^/. 18f/. and 22d. per lb. How many pounds of each sort must he take to make one pound of the mixture worth \7d.l* (2) IIo'.v much coffee, at 3^. at 2^. and at 1^. 6d. per lb. \*ith 20 lb. at 5s. will make a mixture worth 2s. Sd. per lb.'>. Ans. 35 lb. at 3s. 70 lb. at 2s. and 10 lb. at Is. &d. (3) A distiller would mix 10 gallons of F'rench brandy, at ASs. per gallon, with British at 285". and spirits at 16^. per gallon. What quantity of each sort must he take to afford it for 32^-. per gallon ? Ans. 8 British, and 8 spirits. ' (4) What quantity of teas at 12^. 10^. and 6^. must ue mixed with 20 ft. at 4.s\ per ft. that the mixture may be wort*; Ss. per ft. ? Ans. 10 lb. at Qs. 10 lb. at lOs. 20 lb. at ]2s * Ti Ansirer. Proof. lb. lb. 5 20 lb. at 15d. = :irM. As 5 : 1 : : 30 : 1 J 16 99 1 4 lb. at 16^. = e,4d. 1 4 lb. at IM. =c 7Qd. As 6 : 2 : : 20 : 8 2 8 //;. at 22cf. ^ 176rf. As 36 '* 612rf. 1 n>. : 17d. [TUTOR'S lantity of at 10 J. will com- . at 8j. r. 6c?. pel )ushel, as V much of of oats. . 6d., and th Is. 8d. Is. 3d. 7 1 to Case 2 •e. is given, is ch required ;co at \5d. iow many und of the 6d. per lb. per lb. ? t Is. 6d. brandy, at at \6s. per to afford it ? spirits, s. must ue ,y be wort'j 6. at I2s lb. 4 4SSISTANT.] COMPARISON OF WEIGHTS AND MEASUREa. M Case 4. Alligation Total. This is also similar to Case 2, except that the whole (piantity of the compound is limited. Rule. Lifik the prices, and place the Jiffcrences as before. Then, As tlie sum f)f the (litllTcnces is to each paiticular difference; BO is the quantity given to each required quantity. (1) A grocer has four sorts of sujrar at I2d. \0d. 6d and Ad. per lb. and would make a composition of 144 lb. worth Sd. per lb. What quantity of each sort must he take ?* (2) A grocer having 4 sorts of tea at bs. Qs. Qs. and 9*. per lb. would have a composition of 87 lb. worth Is. per lb. What quantity must there be of each sort ? Ans. 141 lb. of 5s. 29 lb. of 6s. 29 lb. of 8s. and 141 lb. of 9s. (3) A vintner having 4 sorts of wine, viz. white wine at I6s. per gallon, Flemish at 24^. per gallon, Malaga at 32.y. per gallon, and Canary at 40.?. per gallon ; would make a mixture of 60 gallons worth 20s. per gallon. What quan- tity of each sort must he take ? Ans. 45 gallons of white wine, 5 of Flemish, 5 of Malaga^ and 5 of Canary. (4) A jeweller would melt together four sorts of gold, of 24, 22, 20, and 15 carats line, so as to produce a compound of 42 oz. of 17 carats fine. How much must he take of each sort? Ans. 4 oz. of 24, 4 oz. of 22, 4 oz. of 20, and 30^ oz. of 15 carats fine. m COMPARISON OF WEIGHTS AND MEASURES. This is merely an application of the Rule of Proportion. (1) If 50 Dutch pence be worth 65 French pence, how many Dutch pence arc equal to 350 French pence ?t (2) If 12 yards at London make 8 ells at Paris, how many ells at Paris will make 64 yards at London ? Ans. 42§- 8 J2 1 6 4 Ansiccr. Proof. lb. lb. 4 48 at 12^. = 576 As 12 : 4 : : 144 : 48 2 24 at lOri. = 240 As 12 : 2 : : 144 ; 24 2 24 at Gd. = 144 4 48 at Ad. = 192 Sum 12 144 144)1152(8rf. t As 65 or, as 13 50 10 350 3500 13 = 269yV D2 Ant 82 VULGAR FRACTIONS. [tutor's (3) If 30 lb at London make 28 lb. at Amsterdam, how many lb. at London will be equal to 350 lb. at Amsterdam ? Ans. 375. (4) If 95 lb. Flemish make 100 lb. English, how many lb, English are equal to 275 lb. Flemish 1 Ans. 289/^ I PERMUTATION Is the changing or varying of the order of things. To find the number of changes that may he made in the position of any given number of things. Rule. Multiply the mimbevs 1,2, 3, 4, S^r., cojitinually together, to the given number of terms, and the last product will be the answer. (1) How many changes nay be rung upon 12 bells, and n what time would they be rung, at the rate of 10 changes in a minute, and reckoning the year to contain 365 days, C hours 1 1X2X3X4X5X6X7X8X9X10X11X12 = 479001600 changes, which — 1Q= 47900160 minutes = 9X years^ 26 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 jC40. for his board and lodging during so long a time as he could place the family (consist- ing of 6 persons besides himself) in different positions, every 4ay at dinner. How long might he stay for his jC40. ? Ans. 5040 days. 6 and 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 termSy with a line between them. 2. The upper term is called the Numerator, and the lower tmriy the JMnomtnator. 'I'he Denominator shows into how m. [tutor's lam, how erdam ? s. 375. many lb, 289/^ ASSISTANT. 1 VDLGAR FRACTIONS. 85 ke position toffother, to 3 answer. bells, and 3 changes 365 days, 79001600 = 9X yearSf •nvenience irson with nd lodging y (consist- ons, every 40.? iO days. any whole )ers, called 1 the lower i into how iiany equal parts u?iiti/ is dividec* • and tlie Numerator is the number of those equal parts signified by the Fraction.* 3. Every Fraction may be understood to represent Division; the Numerator being the dividend, and the Denominator thft divisor.\ Fractions arc distinguished as follows : 4. A Simple Fraction consists of one numerator aad one denominator: as |, j-^-, &c. 5. A Compound Fra(;tion, or Iraction of a fraction, con- sists of two or more fractions connected by the word of: as J of -•)- of fj, &c. This properly denotes the product of the several fractions. 6. A Proper Fraction is one which has the numerator less than the denominator : as -}, f, {', \l-, &c."| 7. An Improper Fraction is one which has the numera- tor either equal to, or greater than the denominator : as |-, |, 8. A Mixed Number is composed of a whole number and a fraction, as 1|, 174, 8^4' ^^• 9. A Complex Fraction has a fractional numerator or denominator : but this denotes Division of Fractions. Thus, 2 . -8 J-, two-thirds divided by five-sixths, — , eight divided by one and two-thirds. * In the {ia,ci[on f.rc-t7vclflhs (f\,) the Denominator 12 shows that the- unit or uthole quantity is supposed to be dividpcl into 12 equal parts : so that if it be one shilhnjj, each part will be one-twelfth of Is. or one penny. The Numerator shows that 5 is the number of those twelfth parts intended to be taken ; so ^V of a shilling are the eame as 5 pence ; ,-% of a foot, the same as 5 inches, f The fraction y^^^ signifies not only ,'V of a unit, but 5 units divid- ed into 12 parts, or a twelfth part of five : and it is obvious that^ye twelfth parts of one shilling (or five pence) is the same as one. twelfih port 0^ five shillings. This mode of considering Fractions removes many of the student's difficulties. X A proper fraction is always less than unity : thus J wants one fourth, and j^- ivants one-twelfth of being ecuiu to 1. But an im- proper fraction is ec/ual to unity when ?}« t(n;^f are equal, aaK fj;rentrr than unity when the numerator is '.ta< p'ttui^. Thus \, or It, or 4^, is each —1 ; and Z-.--:! \.jtm^ ^■}r=3^'*j. 84 VULGAR FRACTIONS. [TUTOJl'i 10. A Common Measure (or Divisor) is a number that wiir exactly divide both the tfirms. When it is \\^ greatest npniber by which they are both divisible, it is called t|le Greatest Common Measure. Note. A prime number has no factor, except itself and nnity. A trMltiple signifies any product of a number; and is therefore divisi- Ido by the number of which it is a multiple: thus 14, 21, 28, &C., ar« wivUipIrs of seven. Also 14 is a common multiple of 2 zmd 7; 21, of 3 and 7, &c. ," • ,.. , , , . . • ' REDUCTION is the method of changing the form of fractional number* or "quantities, without altering the value. Case 1 . To reduce a fraction to its lowest terms. Rule. Divide both the terms by any common meatnre that can be discovered by inspection; which will produce an equivalent fractioQ in lower terms. Treat the new fraction in a similar manner; repeating the operation till the lowest terms are obtained.* ' When the objecv cannot be accompl'shed by this proeess, divide th« greater term by tho less, and that divisor by the remainder, and eo on till nothing remains. The last divisor will bo the greateat common -measure ; by vvh.»nh divide both terms of the fraction, and the quotients will be the lowest terms. (1) Reduce j^*^ to its lowest terms. Arts. ^. (2) Reduce f-^f to its lowest terms. Ans. -j^,-. (3) Reduce i^| to the least terms. Ans. \. (4) Reduce Iff to the least terms. Ans. ^{. (5) Abbreviate |^f^ as much as possible. Ans. ^. (6) Reduce |f ||:f to its lowest terms. ' Ans. ^. * This first method of ahlr^-niattng fractions is, when practicably, alvrays to be prefeiTed : and in the application of it, the following OD- nervations will be found exceedingly useful. ' An even nuniber is divisible by 2. A numbfr is divisible by 4, when the tens and units are so; and by 8, when the hundred.'^, tens, and imifs arc di^^sihle by 8. A number is a multiple of 3, or of 9, when the .turn of its digits ie a multiple of 3, or of 9. A ."i or a in the units' place, admits of division by 5 ; one cipher ad rvh^ of divn.Hion by 10, two, by 100, \Ss», laeo (2) terms. ruTOR'i Ik ber that greatest lied t)fie ity. ire divifii- , &c., are 7 ; 21, of mbera; or IS. aat cua be fraction is repeatmg divide the ajid 60 on ai eommon i quotients ns, ^f racticebl^, owing ob- 80 ; and by s digits IB a e cipher sd iSSIBTANT. REDUCTION. ei> (7) Re (8) WI f'uce /iVisVir *o ^^6 lowest terms. What are the lowest terms of as a Of Case 2. 7o reduce an improper fraction to its equivalent number. Rule. Divide the upper term by the lower. This is evident from Definition 3. (1) Reduce f2> Reduce Reduce Reduce Reduce '^' to a mixed number. ^i^* */ to its equivalent number. 8f ' to its equivalent number. 'II* to its equivalent number. 3|4.9 to its equivalent number. (6) Reduce ' |^ ' to its equivalent number. = 184. Ans. Ans. 13^. Ans. 27f . Ans. 56^f Ans. 183|. Ans. 1\\\. Case 3. To redy.ce a mixed number to an improper fraction.* EoLf. Multiply the whole number by the denominator of the (rac- don, and to the product add the numerator (or the numerator required, wnich place oVei" the denominator. hi Note. Any whole number may be expressed in a fractional form by putting 1 for the denominator : thus 1 1 = y . (1) Reduce 18^ to the form of a fraction.! A number is a multiple of 11, when the sum of the 1st, 3d, 5th, &c., digits =: that of the 2d,^ 4th; 6th^ &c., digits, after retrenching the elevent contained in each. ! A multiple of both 2 and 3 is, of course, a multiple of 6 ; and a mnl- tiple of 3 and 4 may be divided by 12. All prime numbers, except 2 and 5, have 1, 3, 7, or 9, in the units' place : all others are composite. Examples. (1) Reduce \^l to ^he hast terms possible. '.'\0 -r9 -r2 (2) Reduce ^f|{§ to the lowest terms. -5 -r3 -rll mn fjOS 836 _ 78 Now, because we cannot easily discover a common measure pro- ceed thus : — 76)133(1 then 19)76 Z? 19)133 57)76(1 57 4> Ant. greatest com. meas. 19)57(3 57 • This is the converse of Case 2. 7 86 VULGAR FRACTIONS, [tutor's (2) Reduce 56i| to an improper fraction. (3) Reduce 183/x- to an improper fraction. (4) Reduce 13f to its equivalent fraction. (5) Reduce 272- to its equival'^nt fraction. (6) Reduce 514/^ to a fraction]] form. Ans. 1^ \' Ans. 3|f' Ans. 3f» A?is. 8|f» Case 4. To reduce a fraction to another of the same value, having a certain proposed numerator or denonunator. Rule. As the pre.sont numerator is to iho denoniiimtor, so is tho proposed imnieiiitor to its deiioiiiiiiatov. Or, as tlie present (leiiomina- tor is to the numerator, so is tlie proposed denominator to its nume- rator. (1) Reduce ^ to a fraction of the same value, whose nu- merator shall be 12. As 2 : 3 : : 12 : 18. Ans. |f. (2) Reduce f to a fraction of the same valuc^ whose nu merator shall be 25. Ans. |f . (3) Reduce ^ to a fraction of the same value, whose nu merator shall be 47. . 47 Ans. 651- • (4) Reduce | to a fraction of the same value, whose de- aiiOminator shall be 18. Ans. ||. (5) Reduce 4 to a fraction, of thu same value, whose de- nominator shall be 35. Ans. |f. (6) Reduce f to a fraction of the same value, whose de- nominator shall be 19. " 16^ Ans. — . Case 5. To reduce complex and compound fractions to a simple fqrm. Rule. For a complex fraction, reduce both toi*ms to simple fractions: then by inverting the lower fraction, they may bp considered as the terms of a compound fraction. And to reduce a compound fraction, ar- range all tlie numerators above a line, and the denominators below, with the signs of multiplication interserted : divide all the upper and lower tenns that are commensurahle,'* cancelling them with a dash, and placing their quotients above and below thein respectively. Do the same w'th the quotients : th(;n tlu; products of the uncancelled numbers will give the single fraction in its lowest ierms.i * That is, having a common divisor. t This rule is of the highest importance as tending to expedite the ts rUTOR'S 2 I • 8319 \e value^ or. 80 is tho leiiomiriii- its iiume- lose nu- is. If. tiose nu IS 2-5.. bose nu 47 65f hose de- to. J g . hose de- ls 2.5. "• 35 • hose de- 16|. • 19" 7 a simple ) fractions : ■ed as the faction, ar- 3rs below, upper and I dash, and r. Do the d numbers lite the 1% 48SI8TANT.] REDUCTION. 3GI (1) Reduce — 5 to a sunplc fraction. 48 ' 234 (2) Reduce — I to a simple fraction, 38 (3) Reduce . — . to a simple fraction. 19 (4) Reduce . — to a simple fraction. \ 87 1 i ■ Ans. !f i Ans. !%• Ans. 4- Ans. ^. (5) Reduce | of ^- of f to a sinj^le fraction. Ans. I. (6) Reduce |^ of \ of }^ to a siiiiplo fraction. Ans. ^g^^. •iaess of compatation, by a. lev^aUng to the utmost all fractional ©x* pressious, as we ju'oceed. Examples. 7- (1) Reduce the complex fraction -4 to a simple form. 2 3 ^J.z=-^— — . — —- = ^Ans. 1 7 H (2) Reduce y'o of f of f of j'^^ to a simple fraction. 3 111 ^ X^ Xll X $ _a- of i of ^ of -5- 1'0 X$ X$ XU 2 14 2 Tfl"* 'txJlS,* (3) Reduce the annexed fractional expression to its proper quan- 16 14 5 2 72 93 T7 667 of iJ-I „r.-— il of ft ■• ^ — _ V — !^- V 1^ V J^ V 11 "* 108 °' e Wj ^^ - 11 X 108 ^ ^8 X 3_^o X-y^ 1 7 11 1 1 ;^ ^x 10 X xm X 1 x;^^ X t$ X o^xiax W *ii X mxUxx^xmxU^xUx U' \ ^ \ ^ u u ^ ^ 1 1 $ 4 1 *4:3..8|5. = £2..8..1f ^tt5. I 77 J^- "32 =2^i '% -•!*! I iMAGr EVALUATION TEST TARGET (MT-3) /> {/ ,V4 :/- 7i V] <^ //, / 1.0 I.I IAS 128 |2.5 £: us i 20 Ui us IL25 1 1.4 Photographic Sciences Corporation 1.6 23 WEST MAIN STREET WEBSTER, N.Y. M580 (716) 872-4503 V iV \\ '^b'"'^^ ^^\ ^q\ V ■^' ? ..V4E>. r/. 7 * W VULGAR FRACTIONS. tTUTOR'f 1 ^4 28 (If) Reduce |1 of — i_of to a simple fraction. ^ 14if 38| ^„,. ^||. (8) Reduce f of ^ of llf to a single fraction. Ans. ^, (9) Reduce yW of 37J of 5 to its equivalent number. , , , Ans. 112^. (10) Reduce -1 of 3f« of Xto its equivalent number. 14 4 Ans. 71 Case 6. To reduce a fractional quantity of a given denomtna tion to an equivalent fraction of another denomination. Rule. Consider what numbers would reduce the greater denomina tion to the lets; then to reduce to a greater name, multiply the denom inator by those numbers, and to reduce to a less name, multiply the uumeratdr: the compound thus produced, when reduced to a sunplo form, will be the fraction required. Reduce ^ of a penny to the fraction of a pound.* Reduce ^d. to the fractip^ of a crpAyn. Ans. ^^^ er. Reduce | dwts. to the fraction of a lb. troy. Ans. -^^j^ lb Reduce 4 lb- avoirdupois to the fraction of a cwt. Ans. Y^j cwt. Reduce j-^^jf of a pound to the fraction of a penny .f Reduce £j^z' *° ^^® fraction of a penny. Ans. |rf. Reduce ^^^ of a pound troy to the fraction of a -weight. Ans. | dwt. Reduce j^j cwt. to the fraction of a lb. Ans. ^ lb. Case 7. To find the proper value of a fractional quantity. Rule. Reduce the numerator to such lower denomination as may be ueceBBory, and divide by the denominator; abbreviating as much as ppesible in valuing the remainders. Note. It is evident, from Definition 3, that this Case is precisely thtf of Compound Division. (1) Reduce | of a pound sterling to its proper value.| 8X12X20 _ 7X20X12 7X12 ^920 3X20 t£J = .1. ?^fi?. Ans, •• 15f . An^. 96 3X5 JH^ '■t ■ ASSISTANT.] HEDUCTION. 89 (3) Reduce |*. to ^ts proper value. Anf. Ad. 3 J qrs. (3) Reduce | of a lb. avoirdupois to its proper va)ue. Ans. 9 oz. 2^ dr (4) Reduce i cwt. to its proper value. Ans. 3 qrs. 3^ lb (5) Reduce | of a /6. troy to its proper value. Ans. 7 oz. 4 c^ti;^* (6) Reduce ^| of an ell English to its proper value. Ans. 2 qrs. 3} naikX. (7) What is the value of S^i^l Ans. I9s. 10^ ^ (8) J[leduce ^f^ of a mile to its proper value. Ans. 6 fur. 105 yds. (9) Reduce |f of an acre to its proper yalu». Ans. I a. 2 r. 3^ per. (10) Find ihp value of if {| J cwt. Ans. I qr. 22 M. y|. Case 8. !fo reduce any given quantity to the fraction of a greater denomination. Bulk. Reduce the given quantity (if compound; to the lowest de- nomination mentioned, tlmt it may assume a simple fomi : then multiply i^lM denominator, as in Case 6. (1) Reduce 15^. to the fraction of a pound sterling. l6*.=je^f = i:f Ans, (2) Reduce Ad. 3^ qrs. to the fraction of a shilling. Ans. f (3) Reduce 9 oz. 2^ dr. to the fraction of a 76. avoirdupois Ans. \ lb. (4) Reduce 3 qrs. 3^ lb. to the fraction of a cwt. Ans. I ewU (3) Reduce 7 oz. 4 dwts. to the fraction of a lb. troy. Ans. I lb. (6) Reducp ;^ qrs. 3^ nails to the fraction of an English jjtll. Ans. ^ ell. (7) Reduce 14;. Q\d. ^ to the fraction of a £. Ans. £^j. (§) Reduce 4d. l{^ qrs. to the fraction of a crown. Ans. ^^ji cr. (Q) What fraction of an acre are 3 roods, 32 perches ? ' 4ns- if a. (10) liyhat part of a shilling are | of 2d. Ans. ^s Case 9. To find the least common multiple of two or moi$ numbers. BoLB. Arrange the given numbers in a line, (omitting any one that is tk factor of one of the others,) and divide any two or mord of them fai^ # ««mmoj» divisor, placing the quotients and usdivided numbers below ; Ml 90 VULGAR FRACTIONS. [TUTOR 8 proceed with these in the same manner, and repeat the procesB till thero remain not any two numbers commensurable : the contmued product of the divisors, quotients, and undivided numbers, will be tlie least common multiple. (1) Required the least common multiple of 2, 3, 4, 5, 6, 7, , 9, and 10.* (2) Find the least number divisible by 3, 4, 5, 6, 7, and 8. Ans. 840. (3) What is the least common multiple of 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12 ? • Ans. 27720. Case 10. To reduce fractions to a common denominator. Rdle 1. Multiply each numerator into all the denominators, except its own, for a numerator; and all the denominators for a cciraon de- nominator. Or, Rule 2. Find the least common mnlliplc of the denominators, which will be the h'ast common denominator. Divide this by each denomina- t»)r, and multiply the several quotients by the respective numerators for the required numerators. (1) Reduce (2) Reduce (3) Reduce (4) Reduce (5) Reduce (6) Reduce f and I to a common denominator.f ^, |, and I, to a common denominator. ^ris. -sa, Ai, and |^ ; or f , |, and f . 8^> e» tV> ^^^ f J ^^ ^ common denominator. JXn&. g4„, 84o» 840» "'*" 8 4 0* f , 1, and |, to a common denominator. 31 Ans. 4f , ^A, and |^ y\, ^, and ■— of 2, to a common denominator. -^"*' TT5S» TTfil' ""** TTsJ* Ij, 2}, and i of 1^, to a common denommator. >lnv J-a. LPJ rtM/7 2.5. f ■i * 2 and 4, being factors of 8, 3 a factor of 9, and 5 a factor of 10, may be omitted. Thus, 2)6, 7, 8, 9, 10 3)3, 7, 4, 9, 5 1, 7, 4, 3, 5 Then 2X3X7X4X3X5 = 42 X 60 = 2520, the least number divisible hj all the given numbers. A^ A If ( numerators. Ans. ^J and ^§ 4 X 7 = 28 the denominator. . hiil ISSItTAKT.] SUBTRACTION. 91 ADDITION. BoLB. Bedace the given fractions to a common denominator avsr which place the $um of the numerators. (1) Add ^ and f together. a+4=|i+|f=|a=i^«_.. Arts. (2) Add f, 3-, and f . (3) Add i, 4J, and §.* (4) Add 7^ and | together. (5) Addf, and a-" of a. (6) Add 5|, Gi, and 4i. (7) Add lA, 31, and i of 7. (8) Add -\ of Gi, and 4 of 7^. (9) Add 1 of 0}, and | of 4f . Fractional (juantitics may be reduced to their proper values, ond the sum found by Compound Addition. (10) Add f of a pound to ^ of a shilling. Ans. 8s. 4 J. (11) Add ^d. Is. and £^. Ans. I4s. (12) Add f lb. troy, J- 02. and f 00. ^;i.y. 7 oz. 19 divts. 20 ^r. (13) Add 4 of a ton to ^ of a civt. Ans. 12 cwt. I qr. l^ lb. (14) What is the sum of f of £l7..7..ed., 1 of £1^. and 2 of a crown Ans. i;i3..0..2i. (15) Add f of 3 a. 1 r. 20 p., f of an acre, and | of 3 roods, 15 perches. Ans. 3 a. 3 r. 331 SUBTRACTION. Rule. Reduce the given fractions to n common denominator, over which i)lace the difference of the numeratore. When the numerator of the fractional part in the gnbtrahend is great- er than the other numerator, bon-ow a fraction equal to unify, lisiving the common denomujator ; then subtract, and cairy one to the integer pf the subtrahend. (1) From ^ tako f ^— 4^|i_|fi=^u, Ans. (2) From | take f (3) From 5| take -f^ of f . (4) From If take f of 1. (5) From |f take | of |. (6) From 641 take a of J. (7) From 15i take i2jV. (8) Subtract ff from If (9) Subtract |i from I of 9. Fractional quantities may be reduced to their proper values, 68 directed in Addition • When there are integers among the given numbers, first find the turn of the fractions, to which add the integers. Thus in Ex. 3, J+f=|; then Hl=iT4-fV-n; and 4+|J==» 4t|. Ans. Is ll 193 yp^akB. FRACTIUNB. (tut^r^ (10) From f of a pound take f of a shilling. Ans. 7s. l\d (11) From l^s. take f of 7id. A?^^. U. dpi. (12) What is the difference between f of £1^^.; 9iic| ^ of £1^^. 1 Ans. 2d. 3^ qrs. ' (13) 3iibtj:actf cwt. from 4 ^^^' ■^^^' 10 c?/;/. 2 qrs. 10^/^. (14) ^fom i of 5 lb. troy subtract ^ of 3^ oz. Ans. 3 lb. 2 oz. 1 dwt. 2f ^r« (^5) Subitr.acjt 7\l\ fjurlongs from l^y mile. Ans. 4 y^r. 9 yds. MULTIPLICATION. Ruj^K. Prepare the given numbers (if they require it) by the rules of Re4uction: then multiply all the numerators together for the numer- ator of the product, and all the denominators for the denominator. (1) Multiply f by |. (2) Multiply I by |. (3) Multiply 48f by 13|. (4) Multiply 430^ by 18f (6) Multiply 1 of i by f (7) Multiply 5f by f . (S) Multiply 24 l^y |. (9) Multiply f of 9 by f (5) Multiply If by f of 4. (10) Multiply je3..15..9i \ by ^j of 5. Ans. £\b.,9..\ll^j, (U) Multiply ^\ fniles by 4 of 4ff ' Ans. 8 m. 2 f. I884 yds. (12) Required the product, in s^quare feet, of 14 jft. 7 t^. by8/^9tn. Ans. 127 ^^ sq. ft. DIVISION. Rdlz. Prepare the given numbers (if they rpquire it) by the rules of Seduction ; then invert the divisor, and proceed as in Multiplication. ■ 1) Divide ^^ by ft 2) Divide l\ by | i (3) Divide 672/^ by 13|. (4) Divide 7935^1 by 18f (5) Divide 16 by 24. («) Divide T^ by 4f (7) Divide Hi by tV of 1^. (8) Divide 9^ by 1 of 7. (9) Divide i by f of ^ of |. (10) Divide a of 16 by 4 of i. • A number inverted becomes the recipi'ocal of that number ; which is the quotient arising from dividing unity by the given num ber: thus 1-7-7: :|, the reciprocal of 7 ; l-*-|=§, the reciprocal of j* 3 1 . U' 4 I Ans, 1 U\ i^ftisfANT.] ii^Li OP TMREfi. «r (11) nivido £^^^ by y of U- -^'*-^' i^3..17..10i |. (12) Divide Is. 4^(/.f ,by | of '/. Ans. Gil. 3^ qrs. (13) Divide 3 qrs. 24|f lb. by y\ of 1^, in the fraction of a kwt.; and yalue the quotient., ,_ Ans.l cwt. \qr. \b\ lb, (14) Wi^at must JC7..14..6. be multiplied by," to profluce i:21..17..9? Ans.2\. THli ItULt OF THlt^E. , Rule. Prepare the torms, previous to stating, so that no snbseqtaent deduction will be necessary: then, having stated the (j^nestion, as pre- viously directed, invert t^e dividing term, and tbe cuntiuucd product of ibe tlu'ee will be the answer. (1) If I of. a yard. cost jCf . what will |%- of a yard cost?* (2) If i ijd. cost £%. what will J i yd. cost ? Ans.. 14*. 8rf (3) If ^ of a yard of lawn cost Is. 2d. what wi|l lO^i yards dost ? . . , Ans. Jt;4..19..10^ f . , (4) If 1 lb. cost ^s. howinuch will f*. buy ? Ans. \r}^ lb. (5) If 48 men can build a wall in 24j days, how many Inen can do the s&ame in 192 days ? Ans. 6Jy men. (6) If f of a yard of Holland cost jCf what.\vill 122- plls' cost at the Same rate f vl»s. JC7..0..8J a. (7) ,If 3^^ yards of cloth, that is 1} yard wide, be sufficient to make a cloak, hovir much that is ^ of a yard wide .will makq another of th6 same size ? . . Ans. 4| y^rds. (3) If 121 yards of .cloth cost 15*. 9f/. what \yill 48^ y^rds cost at the same rate ? Ans. JC3..0..9^ ^. , (9) If 25^5. will pay for the carriage of 1 cwt. 1451 miles; how far may 6i cwt. be carried for the same money ? ■ . Ans. 22^*^ miles. , (10) If ^^ oi a cwt. cost J&14. 4s. what is the value of 7^ cwL/. J . , , , . . ^'»*- £US,.6.S. (11) If ^ lb. of cochineal cost £l..5. what willt 36Vy,^ come to?, ^n*. JC61..3..4.. , (12) How much in length that is 7^ inches broacl will make a foot square ? , ,,, Ans. SOyfj inches.., . (13) What is the value of 4 pieces of broadcloth, each 27| ^ards, at' 15f j. per yard ? Ans. jC85V.l4..3i. j. ' 1 3 1 * . y^' ^ . . yd' A k d ^' ♦ Ab| : f : ; Vy : .?.X|gX-^- = 2 = I5i. Ahi: 2 2/ t... II 94 DfiCIMAL rilACTlONd. ittJtOlffl (14) If a penny white loaf weigh 7 oz. whe» a bushel of wheat coats bs. GiL what is the bushel worth when a penny white loaf weighs but 2\ oz. ? Ans. 15*. 4c?. 3} qrs. (15) What quantity of shalloon that is ^ of a yard wide will line ?! yards of cloth that is 1^ yard wide ? Jin*. 15 yards, (10) Dought yj pieces of silk, each containing 24f ells, at 6.V. Offi. per ell. Ilow must I sell it per yard to gain jC5. by th3 bargain? Ans, 5*. ^\d. ^||. THE DOUBLE RULE OF THREE. (1) If a carrier receive £2^^. for the carriage of 3 ewtt 150 miles, how much ought he to receive for the carriage of 7 cwt, 3^ qrs., 50 miles? Ans. i;i..l6..9. (2) If JCIOO. in 12 months gain £b\. interest, what prin- cipal will gain X'3f. in 9 months? Ans. jC85..14..3i ^, (3) If 9 students spend .£10^. in 18 days, how much will 20 students spend in 30 days ? Ans. jC39..18..4f f (4) Two persons earned 41*. for one day's labour : how much would 5 persons earn in 10| days, at the same rate? >ln*. jC6..1..4f f (5) If £50. in 5 months gain £2^^^. what time will jC13|=* require to gain £1-^-^. ? Ans. 9 months, ^6) If the carriage of 60 mt.^ 20 miles, cost £14i., what Weight can I have carried 30 miles for £^^^> 1 Ans. l&cu^l. n f 1 ! DECIMAL FRACTIONgf. In Decimal Fractions the unit is supposed to be dividetf into tenths, liundredths, thousandth parts, (fee, consequently the denominator is always 10, or 100, or 1000, &c. In our system of Notation, the figures of a mhole number follow each other in a decimal (or tenfold) proportion. Hence, the numerator of a decimal Fraction is written afr a whold number, only distinguished by a separating point prefixed to it* Thus '5 for T^^, '25 for ,Vff, '123 for ^^. The denominator is, therefore, not expressed ; being alwaya understood to be 1, with as many ciphers ajjixed as thcro art places in the numerator. The didercnt values of figures will be ovkkflt in the Mh ii«xe(l Table. :!. ' t1; JtfllSTANT.] DECIMALS 95 Tntppcrn. Decimal parts. 7 6 5 4 3 2 1 . 2 3 4 5 6 7, &0, c 2 • o g S 5 "^ S F^ ^3 g S c ^ g, « * 5 ?■ I 2. ff From this it plainly appears that the figures of the decimal 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 \alue : for •2=-y2^, -20=//^, -SOO^yV/o' ^i"^ all equal. But one cipher on the left diminishes the value ten times, twa ciphers, one hundred times, &c., for ■02==|-f 7, *002=»«f ^5^^, &c. A vulgar fraction having a denominator compounded of 2, or 5, or of both, vehen converted into its equivalent decimal frac' tion, will be finite : that is, will terminate at some certain number of places. All others are infinite ; and because thev have one or more figures continually repeated without end, they are called Circulating Decimals. The repeating figures are called repetends. One repeating figure is called a single repetcnd ; as '222, &:c.; generally written thus, •2''; or thus •%. But when more than one repeat, the decimal is . compound repetend; as '36 36, &;c., or 142857 142857, &c. These may be written -'36^ and -U 42857^ ; or -$0, and •14285;». Pure repetends consist of the repeating figures alone ; but mixed repetends have other figures before the circulating deci- mal begins : as '045^, •96^354'. Finite decimals may be considered as infinite, by making ciphers to recur, which do not alter the value. Circulating decimals having the same number of repeating jigures are called similar repetends, and those which have an unequal number are dissimilar. Similar and conterminous rep- etends begin and terminate at the same places. • The first, second, third, fourth, &c., places of decimals nro called primes, seconds, thirds, fovrtTis, Sec, respectively ; and docimals ore read tb 's : 57-57 fifty-seven, and Jive, seven, of a decimal ; that is, fifty «even) and fifty-seven hundredths. 20fi-043 two hundred and «ix, aiKl«mrf%» fvur, thrc€ ; that is, 206, luid forty-three thousaudtht « 'I ,: i| !i I DECIMALS. ADDITION. llCltil'8 48S ,,^^VL^J , Place- the nu^ibers so tiiut the decimal |)uint« may stiuid b,,, ]>e,n)en,aicnlar line : then will unitn be un^er uiutu, &c., according to ' eu I i) , (5) Add 3275+27-5144- 1005+725-f7-32. {6) Add 27-5+52+3-26754-574 1+2720. ( their respective values. Then add as in integers. [{) Add 72-5+32071^21574+37l-4-f2-75. (2) Add3007+^007l+59-432+71. (3) Add 3-5+47-25+92701+20073+1-5. (4) Add 52-754-47-21+724+31-452+-3075. SUBTRACTION. RutE. Place the subtrahend under the jninuend with the deciii&al points as in Additiun ; and subtract as in integers. 5) From 571 take 54-72. 6) From 625 take 76-91. (7) From 23-415 take -374^-i n.) From •275'4 take -2371. (2) From 237 take 1-76. (3) From 271 take 215-7. (4) From 270-2 take 75-4075. (8) From' -107 take 0007. MULTIPLICATION. t ■• '. ■ . . I I . ■ . ' "* • ■• »•• .^ I Rule. Place the fifctors, and multiply themas in whole nUmberai, ani in the product pouit off as nuuiy decimal f)laces as there are in bot^ factors together. When there are not so many figures in the product, supply the defect with ciphers on the left. Ii •tlUi uikI niw It riitr iiicli N i ^ir^•< ] iirtij , ( four ( 5 plac t ( gers C j)lac lac \i) Multiply 2-071 l)y2-2!7. 2) Multiply 27-15 hj 24-3.* 3) Multiply -2365 by -2435. (4) MMltiply 72347 by 23- 15. h) Multiply 17105 by -3257. (6) Multiply 17105 by 0327. (8) (9) (10) (in' (12} -3409803 X ^7-^5 )<1'l00\\: 61-21 X 0075. •00'7 X -007. 20-15 X •2?705, -907 X -0025. . . 0016218. When the multiplier is 10, 100, 1000, &c., it is only removing th^ ^cpj^^-ating pouit \\i thQ multiplicand. so many places ttnvards the right as there are ciphers in the multiplier: thus, -$78 X l6^s;=5-78, -578 X 100 = 57-8, -578 X 1000 = 578, and -578 X 10000 = 5780 CONTRACTED MULTIPLICATION. , Rule. Write the multiplier under the mnltiplicana U aninveri^il order, the units' figure under that place which is intended to be retain' | ed in the product. „ * The i^ad example may, be moltiplied.in /(co productii, GrBt„by.?f| «^ that product by 8 for 34'. the 3rd, 6th, 7tb, and 12th may be cna tracted in a similar way. ■''* ■ ] ' H.I • I •• • •» licitra [« may stnna Ui,,, iic., accoroiug to 5. \ 48SldfAN'T.J DIVTSTO.V. or i ). ii^aij with the deci: [ take 54-72. ) take 76-91. 415 take •374'2 7 take 0007. whole numbers I , I there are in botlji !s in the produvt, X^tfOOll. X 0075. X 007. X •2^705, X -0025. . . X 0016218. '■ ■■ I nly removing th(^ I :(>wardR the right | 6^ 578, -578 X I rso ad ifl an inverted ided to be retain' iduct», firstly .?> I 12th may be con j In multiplying, brplu wiih thut li«ure ol iho miillljilicnii.l which •lAridH over llio iiiulliplyiiig li',MMp. ivj.tctiiif,' all on tlio ri-lii of tlnif; iij)d set down the first figure- . ol iill llir inuilucts in a )>L-rpoiidiculii/ row. Inci-enso tlio first fi/niiv of ontli {H-nd.irt l.y carri/imrf,, if wh;it would nrisc IVorii niulliplyiiia llm /w,i mxf rcictcir liirnn-s on ili- ri,t:li), Jit tlio l-iitr of ovr from 5 to 11 iii.lu.^ivc, /w/Zfroiii I., i , -1, l]^,■t^, froui Jj to J I iiH-lusivp, \t. Note. If porfert mTurfl''V nn fnr as flir l;i.-.< drciiiKil fi-iin' l>o o eliyiljle to liiiil one ligiire more in tlio piuiliu;t tliau i» iictuiilly w;inl«'(l. (13) iMulliply 381ti7-^i:.8 by 3-ns;{, and let thuro bo on!/ (our places of decinials \\\ the proibu-^.* (11) Mtiltiply 31115serve tliut fll^l dei-iniul places iti the di- Vipor and quotient together inusi equal in nii;iiber llioso of the dividend. '2. When there are /«jrrr decimal pluces in tJ-c dlwdcnd lliaii in thf» ilivisor, equalize them Ity o^rtng ciphfrs; and the cptotient, to tknt extent, will be a ichole number. * Covh'flctrd mrfhod. Common mrfhod :<« 4 672158 .381 ()7'JI.")S 386.3 3fi8.3 I15401fi.5 1 l.'i'JO 10474 2308033 307737 7:'2fii 3077 ::.3 ?30f:033 948 1 1540 1J.)-1(>I(;4 74 1416747«" 1416-7475 Vflli 08 DECIMALS. [tutor's 3. Cl|>h»'ra nviy Im* Fiiljoinotl to tlio ti. Thus .'^.7;M — 10 = r>78-4, 5734 -i- 100 «= 57-84, 5784 -H 1000 = ^784. 5734 -H 10000 —-•r)781. 15! 7382-54 -r 6-425». •0851648-^423. 267- 1 5975 -r- 13-25. 72-1564 -f- -1347. 85643-825 H- 6-321. l-r31416. 11? 3719-^ 10. 3 74-^ 100. (15) 130-7 -r looa (10) 34012 -r 10000. CONTRACTED DIVISION. Ascerts/m the valne of the first quotient figure, from wliich it will be known what nunilxn' of figuies in the quotient will sei-ve the purpose required. Use that number of the ligures in the divisor, (rejecting the others on the riglit,) and a svfficienl number of the dividend, to find the tirst figure of tlie (juolient; make each remainder a new dividual, and for cacli succeedhig figure reject another fnim the divisor: but observe to carry to each product Iroui the rejected figures, us ill Contracted Multij)iicalioii. Note. Wlinn there are /etc«r figures iu the divisor than the number wauteil iu the (piotient, proceed by the common nde till those in the divisor are just (ix many as remuiii to bo found iu the quotient, and then use the coutructicju. (17) Divide 70-23 bv '^•9863, to three places of decimals. • (18) Divide 721 17,562 by 2-257432, to the extent of only three places of decimals in the quotient. (19) Divide 25-1367 by 2 17- 35, to the fourth decimal. .... * Cnntrpcteil Melhod. 7-08G3)70-23C(8-79;j 6LW)C 719 31 34 7 Common Method. 7-98':i)70-2300(8-793 C.18904 C339|60 .'.5y0i41 749il90 718:767 .'i0423U ~(i,4G4l 'il^ ■4^' ASSISTANT.] CmCULATINO DECIMALS. &t^ 1000 » 5'784, (20) Divide 5117512 by 123115, to the second decimal. (21) Divide 27104 by -3712, the integral quotient only* CIRCULATING DECIMALS. To reduce a circulate to a vulgar fraction. kui.E. 1. Kor a pure rcpofriul, mako iho circulalivg figuret ttd numerator^ to as many vines for llu! ih'iwminntor. 2. For u mixed r<'p«!teiul, siihtract tlu; JlnVc part from the whole, ttiA mako tho difference iho mnntratur ; tlio Jcuominalor to which will coiiHint of as many nines as tliert' are rcpctcnds, with at many ciphcri Bulgoincd as thero arc finite iiyures. Examples. (1) Reduce -V, 3', 9^ -^Or, and ^42857^0 their equivalent vulgar fractions. •r=i; -3'=^ = ^; •9^=11 = 1; ''OV^i^\ and -^42857' ¥ddXtt¥ lllll 3T0 3? 3 36 1 ?• (2) Reduce OSMS' and 3oU26' to equivalent vulgat fritc* tions. 345—3 / •03M5' = 3-5^126'- 9900 35126—35 JL4_2_ U U u I To a. 3-5^26'= 3 + 9990 512G~5 = a-5_(L!»j = aaal e u 1110 3JL«_9^. Or thus J 9990 3112.1 = S-ZLO, ^Si99Q ''I 117- In Addition and Subtraction of Circulating Decimahi make them similar and conterminous, and carry to the figures on the right whatever would arise from the repctends being continued. Note In all cases, when tho ropetend is 9, mako it a cipher, oni add 1 to the ticxt figure: for -999, &c. = 1. In Multiplication, carry to the product of the right hand figure what would arise from the product of the repetends con* tinned ; and in finding the sum of the products, observe what is directed in Addition. In Division, it is only necessary to observe that the opera* tioa. fnay be carried on with the repeating figures of the divi* dcncfcirto any extent required. Note. When the Multiplier or tho Divisor is a circnlate, the filOtl cuuveuieiit i^^ethod is, to chuugu it into a common fraction. .«•) aoo DECIMALS. Jtutoe'4 1 ii ■(': S ii ExAMPr.F.g. (3) What is the sum of 25^42857', 10 3'90', 12-035', and 4-02^567^? ^'/'niihir. Similar and ronfcrminova. 25-^42857' = 25- 1428o7' = 25- 14^285714' 10-3^90" = 10-3^909090' == 10-39^090909' 12-035' = 12-0:r555r)55' = r2-03'555555' 402^507' ■■= 402^507007' = 4-02^507567" Sum 5T-59M9"9Tid' (4) What is the diircrcnco between 5G7-'367' and 55 0^9729'? Also, between 57, and 498" 53''? ' 567-^367' = 507-3^673673073673' 57- 55-0^9729' = 550^972997299729' 49-8^53' diiTerence 512-2^700076373943' diff. 7-r46' .. K {5) Midtinly 65-316' by -753. 65-316' •753 195950 3265833' 45721666' product 49-183450 3.v3G^=3n| (6) Multiply 13-M5'bf 3-^36'. = 3-4 37- 13-M5' 37 94^8' 403^63' (7) Divide 1509^045' by 33. 3 )150-9^045' 11)50-3^015' 4-5^728637' quotient. ll)49 7-^Sr product 45-256198, &6. (8) Divide 178054' by 3-6'. 3-6'^ 3^ 9 3| = V-. 53-4163' 17-8054'XJ- = 4-856^03' quotient. H (9) What are the equivalents to 00 1^354' and 65-00063^648'f ^«-^- of f 0' and eS^yVVVo- (10) What is the sum of 57-575 + 3-59463' +210-16' + •<)6^3759'? Ans. 271-397*057674235892'. (11) Required the difference between 36-30M5207 and 47-280^43'. Ans. 10-975^9135982268'. (12) Multiply 4-V128571' by 347; and 170^54' by 6-^M6'; Ans. 1536-^714285'; and 104-85^38720^'. (13) Divide 1536-^714285' by 347; and 10485^387205' \jf 6-448'. Ans 4-M28371'; and 15:^0^5^'. lA 4^§19T4NT.] REDUCTION, m \ REDUCTION. To reduce a Vulgar Fraction tu a Decimal. ftur-K. Add ciplif^rs to tlio mnnerf.tor, and divide by tho denomiua tor; the quotient will Ik; the deeiniul IVaclinn reijuirt'd. (1) Reduce ], \, |, and ^, to decimals. Aius. -25, -5, -75, and -375!. (2) Reduce i, i, J^, and i, to decimals. Ans. -3^ -2, •n42857^ and -V, (3) Reduce -V to a decimal. Ans. -1^923076^ (4) Reduce |} of J-f to a decimal. Ans. -0^043950^ To reduce a given quantity to the Decimal of any denomination required. Rule. Reduce those of the lowest denomination to decimal parts of llie next superior, on the lel't of which place the given quantity of that ^lenominalion ; reduce this to the next, and proceed as before, till it ia if the denomination required. ' (5) Reduce 5^.* 9.y. and \Qs. to the decimals of a pound. Ans. £25, £45, and £:S. (6) Reduce 8s. 4d. to the decimal of a £. Ans. jG-416^ (7) Reduce 16*. 7pl. to the decimal of a £.\ (8) Reduce Ids. 5^d. to the decimal of a £. An.s. jC-972916'. (9) Reduce 12 grains to the decimal of a lb. troy. Ans. lb. -002083^ (10) Reduce 12^^^ drams to the decimal of a lb. ayoirdvpois. Ans. /6. •0476(i8-f . (11) Reduce 2 qrs. \\\ lb. to the decimal of a cwt. Ans. cwt. -62723+. (12) Reduce 2 furlongs, 161^ yards to the decimal of a mile. " Ans. •3417Gr36' mile. (13) Reduce 5|^ pints to the decimal of a gallon. A71S. ■7'i[6\gal. (14) Reduce 4i gallons of wine to the decimal of a hogs- head. Ans. •0^714285'/ » o/ )r)Oo#. £25 Ana. t 4 12 20 300 qrs. 7-75 d. l(J(i4.j;{3' g. £-n;V22i)]6' Ans N, ' I m DECIMALS. [TUT0R»8 i (15) Required the mixed decimal mimber equivalent to (16) Express 7 weeks, 3 days, in the decimal of a year. Ans. yr. -142465-1-. ^ojind the proper value of a Decimal Fraction of any Integer. Rule. Multiply the given decimal by the proper number to reduco it to the next hiforior denomination, pointing off the given number of decimals in the product ; reduce these to the next, and so on to tlie low- est; and the v^liole numbers on ll^e lv"ft (being collected together) will be the value required. A decimal of a £. may be thus valued by inspection. Double the ientht for shilHngs, and cnll the number in the second and third, far- ththgs, abating one above 12, and two above 37. But it" the second is 5, qr upwards, tall the 5 one s/uUiiig, and reckon only the excess above five with the third. By reversing these directions, any given sum in shillings, &c., may be expressed in the decimal of a £. — Thus, half the shillings are tenths. and an odd shilling, 5 hundredths; the rest (in fartliings) add into the ■ecnnd and tliird places, increasing one above 11 farthings, and iw^^ above 36. (17) What is the value of -8322916 of a £.* (18) Reduce jC-740596 to its proper value. Afis. lis. 9(i.2-97216 qrs. (19) What is the value of -082084 of a lb. troy? Ans. 19 dwts. 1 6' 80384 grains. (30) What is the value of -4909375 lb. avoirdupois ? Ans. 7 oz. 13-68 drams. (31) What is the value of .£:• 19895 ? A71S. 3;?. 11J..2-992 qrs. (22) What is the value of -625 of a cn-t. ? Ans. 2 qrs. 14 /*. (23) What is the value of -071428 of a hogshead of winf» ? Ans. 4 gal. 1-999856 qts (24) What is the value of -0625 of a barrel of beer ? Ans. 2 gallons, 1 quart (25) What is the value of 142465 of a year? Ans ■)1 -999725 days vrr ?i C >: ■i;f • £. -8322916 20 s. 16-645832Q 12 d. 7-74!}.984 4 .♦. By inspection. £. a. d. •a = 16..0 j;)32 1= __0. 7% ~{i?,^ =z l(lT.7| qr». 9-99!>9.'U) Ans. 1G..7] very nearly, 11 ■ I !:!*' [TUTOR'8 luivalent to ;3-45471. of a year. 42465-f. 3«y Integer. l)er to reduce ;n immber of )n to tlie low- ogetlu'i) will Double the 1(1 third, far- e seco7i(l is 5, excess above gs, &c., may 'i» aro tenths. a Integer. Decimals. 568182 511364 454545 397727 34090^ Decimol Ta bit's of Coin, ~mli." Weight; ai 219178 'id Measure. " 500 yd. 284091 TABLE X. • 400 227272 70 191781 300 •170454 60 164383 Cloth Measuub. 200 •113636 50 136986 1 Yard the Integer. ' 100 056818 ■10 1095S9 (Jrs. the same as 'r„i.i« ... 90 •051130 ■045154 30 20 082192 054794 I at lie 1 V . ?Q Nuils. Decimals. 70 •039773 10 027397 3 •1875 60 •034091 9 024657 2; •125 40 •0284Ca •022727 3 7 021918 019178 1 •0625 ; TABI.T^ XT. 1 30 •017045 6 016438 20 •011364 5 013698 Lead/ Weight. 10 9 •005682 •005114 4 3 010959 008219 A Foth. 1 he Integer. , Hand. Decimals. 8 •004545 2 005479 10 •512820 7 6 •003977 •003409 1 002739 9 8 •461538 •410256 1 D;iv the Tntocer. 5 •002841 VZhrs. 5 7 •358974 4 •002273 11 458333 •307692 3 •001704 10 416G66 5 ■256410 2 •001136 9 375 4 •205128 1 •000568 8 7 333333 291666 3 2 •153846 •102564 2ft. • 0003787 1 0001894 6 5 25 208333 1 •051282 6 i?i. • 0000947 ^qrs •038461 3 0000474 4 166666 2 •025641 2 1 0000315 0000158 3 3 1 •125 0S3333 041665 1 •012820 Ulhs. 13 12 •0064102 •0059523 •0054945 TABLI : IX. 30 m. 020833 TIM E. 20 013888 11 •0050360 10 006944 10 •0045787 1 Year the Liteger. 9 00625 9 •0041208 Months the 5 same as 8 005555 8 •0036630 Pence in ' Fable ii. 7 •004861 7 •0032051 6 5 •004166 •003472 6 5 •0027472 •00-22893 Days. 1 Decimals. 300 821918 4 •002777 4 •0018315 : 200 5479! 5 3 •002083 3 •0013736 ' 100 273973 2 •001388 2 •0009157 90 .• 240-575 1 •000694 1 •0001578 ^ ^ 1] a06 DECIMALS. [tutor*! THE RULE OF THREE. (1) If 261 yards cost JC3..16..3. what will 32i yds. cost?^ (2) If 7^ yards of cloth cost JC2..12..9. what will 140^ yards of the same cost? Ans. jC47..16..3^. (3) If a chest of sugar, weighing 7 cwt. 2 qrs. 14 lb. cost 4(736. .12. .9. what will 2 cwt. 1 qr. 21 lb. of the same cost? Ans. jeil..l4..2f. (4) What will 826i lb. of coffee be worth when li lb. is $old for 3^. 6d. ? Ans. jGSS .1..3. (5) What is the value of 19 o^ 3 dwls. 5 grs. of gold, at ^2..19. per 0^. .? A?is. £56. .10. .5. .23 qrs. (6) What is the charge for 827f yards of painting, at lO^d. per yard? Ans. JC36..4..3..1-5 qrs. (7) If I lent my friend jC34. for f of a year, how much ought he to lend me for ^5_ of a year ? Ans. £51. (8) If 5 of a yard of cloth, that is 2| yards broad, njake fL garment, how much of | of a yard wide will make a similar pne ? Ans. 2 yds. 1*75 nail. (9) If 1 oz. of silver is worth 5^. 6c?. what is the price of $1 tankard that weighs I lb. 10 os. 10 dwts. 4 grs. ? Ans. ^6..3..9..2-2 qrs. (10) What is the value of 15 ciot. 1 qr. 19 lb. of cotton, at Ibd.^eilb.? Ans. £101.. IS.. 0. (11) If 1 cwt. of currants cost jC2..9..6. what will 45 cwt. 3 qrs. 14 lb. cost at the same rate ? Aiis. JC113..10..93. (12) Bought 6 chests of sugar, each 6 cwt. 3 qrs. at jC2.,16. per cwt. What do they come to? Ans. jC113..8. (L3) Bought a tankard for jC10..12. at the rate of 5s. Ad. per oz. What was the weight? Ans. 39 oz. 15 dwts. (14) Gave i:i87..3..3. for 25 cwt. 3 qrs. 14 lb. of coffee: fit what rate did I buy it per lb. ? Ans. Is. "i^d (15) Bought 29 lb. 4 oz. of snuff for £\0..U..3. What is the value of 3 lb. ? Ans. ^1..1..8. (16) If I give \s. Id. for 3^ lb. of rags, what will be the vtLixie of 1 cwt.? Ans. jei..l4..8. 4 ^If ; I . yds. • As 26-5 £. 3-8125 1/ds. 32-25 £. 4-63974 26-5)122-p53125(4-63974 r-fA 12. .y|. Ana^. [TUTOR'i iSSISTANT.] EXCHANGE 107 11« !l V 'ds. cost ?• will 140* .16..3^. 14 lb. cost le cost? ..14..2f n 11 lb. is 38.1. .3. of gold, at 2-3 qrs. g.atlO^rf. .1*5 qrs. how much IS. £b\. oad, make ) a similar 75 nail. le price of .2-2 qrs. ' cotton, at r..\S..9. ill 45 CM?^ .10..93. atj£:2.,16. :il3..8. of 5^. 4J. .5 dwts. of coffee : \s. 3^(^ EXCHANGE fe the act of bartering the money of one jjlace for llmt of another, by means of a written instrument called a Bill of Exchange. The ojierations in this Rulo consist in finding the qnanlity of one ao^t of money that will be equal to a given sum of tke other, according lo the existing Course of Euxhange. Par of Exchange signifies the equality in the intrinsic value of two sums of money of different countries, and shows how n)i)oh of the one is worth a constant sum (or piece of coin) of the other. Course of Erchangr is the comparative value between the money of two different countries at any particular time, which often fluctuates above or below the Par. Agio is a difference of so much per cent, in the value of the JUank-moneij and the Current-money of some foreign coun- tries, the former being of superior value. To change Foreign Money into British Sterling Money, or Ster- ling into Foreign, according to a given Course of Exchange. Kin.E. As the quantitj^ of Foreign mentioned in the given course erf exchange is to the quantity of Sterllne, so is any other sum of thje For- eign to its con'esponding value in Sterliiij,' money. And by mutually chfmfjhig the words j-oreign and Sterling, the Rule will serve for changing Sterling into Foreign money. I. FRANCE. Accounts are kept at Paris, Lyons, and Rouen, in livres, syls, ftnd deniers, and exchange is made by the ^cu, or crown = \s. 6d. at par. Table:, 12 deniers make 1 sol. 20 sols ... 1 livre. 3 livres . . 1 ecu, or crown. (1) How many crowns must be paid at Paris to receive ir Loiia(»Q Jl?180. exchange at 4^. 6^?. per crown ?* *; d. er, * As ■, 6 : 1 9 oixp. £. : : 130 40 9)7200 sixp. £QQ crov-na 4«« V'lj/ !i( .108 EXCHANGE. [TUI; . 1 • (2) How mucr. tf»er.-)ng must be pari in London, to receive in l^aris 758 crowns, excliange at 4s. 8il. per crown ? Ans. £170.. 17. 4. (3) A merchant in London remits £1,76. .17-. 4. to his cor- respondent at Paris : what is the vahie in French crowns, at 4.y. 8d. per crown? Ans. 758 crowns, (4) Change 725 crowns, 17 sols, 7 deniers, at 4s. 6^d. p«r crown, into sleihng money. Ans. jC164..14,.0i. ^^f. (5) Change £li\4..\4..0^. sterling into French crowns, ex- change at 4s. 6^d. per crown. Ans. 725 crowns, 17 sols, 7-^^ deniers. II. SPAIN. Accounts are kept at Madrid, Cadiz, and Seville, in dollars, Tials, and maravedies, and exchange is made by the piece of eight = 4.s. Off. r.t par. Table. 34 maravedies make 1 rial. 8 rials .... 1 piastre, or piece of eight. 10 rials .... 1 dollar. (6) A merchant at Cadiz remits to London 2547 pieces of ^ight, at 4s. 6d. per piece : .how much sterling is the sum ? Ans. ^594..6. (7^ How many pieces of eight, at 45. 8d. each, will answer a bill of i:594..6. sterling ? Ans. 2547. (8) If 1 pay here a bill of £"2300., for what Spanish money may 1 draw my bill at Madrid, exchange at 4s.§^d. per pkicd of eight? Ans. 10434 jjieccs of eight, 6 rialsj 8|a mar. HI. ITALY. Accounts are kept at Genoa and Leghorn in livres, sols, and deniers, and exchange is made by the piece of eight or dollar=45. 6d. at par. Table. 12 deniers make 1 sol. 20 sols ... 1 livre. 5 livres . . } piece of eight at Genoa. 6 livres . . 1 piece of eight at Leghorn. N. 13. 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 ho r)ay in Genoa 970 dollars at 4.y. 5c?. per dollar? ' " Ans. JC215..10. 8. ASaiSTANT.} E^^CUANOL. lod I I (10) A factor has sold goods at Florence for 250 duca- tooQS, at 4^. 6d. each : what is tho vahic in pounds sterling ? Ans. JC56..5,. (11) If 275 ducats, at 4^. 5d. eacli, be remitted from Venico to London, what is the value in pounds sterling ? Ans. JC60..14..7. (12) A traveller would exchange JC60..14..7. sterling fox Venice ducats, at 4s. 5d. each ; how many must he receive ? Ans. 275,. IV. PORTDGAL. Accounts are kept at Oporto and Lisbon, m reas, and ex- change is made by the milrea = 6s. 8irf. at par. Table. 1000 reas make 1 milrea. (13) A gentleman being desirous to remit to his corres- pondent in London 2750 milreas, exchange at 6s. 5d. per. milrea, for how much sterling will he be creditor in London I Ans. i;882..5..10. (14) A merchant at Oporto remits to London 4366 milreas, 183 reas, at 5*. 5|viU stand on the left. Divide the product of the antecedents by tlie product of the conse* quents for the answer. Proof. By as many single statements as the question requires. (1) If 20 lb. at London make 23 lb. at Antwerp, and 155 lb at Antwerp make 180 lb. at Leghorn, how many lb. at London are equal to 72 lb. at Leghorn \* (2) If 12 lb. at London make 10 lb. at Amsterdam, and 100 lb. at Amsterdanj 120 lb. at Toulouse, how many lb. at London are equal to 40 lb. at Toulouse ? An.f. 40 lb. (3) If 140 braces at Venice be equal to 156 braces at Leg' horn, and 7 bra,ccs at Leghorn equal to 4 ells English, how many braces at Venice are equal to IG ells Enghsh? Ans. 25^^. (4) If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. at Amsterdam make 116 lb. at Dantzic, how many lb. at London are equal to 130 lb. at Dantzic? Ans. \V2-^^. Case 2. When it is required to find how many of the la^t sort mentioned are equal to a given quantity of the jirst. Rule. Place the antecedent and consequent terms as before ; but th? last term, being a conscqvent, will st;ind on the right. Divide the pvQ duct of the consequents by that of the antecedents. (5) If 12 lb. at London make 10 lb. at Amsterdam, and 100 lb. at Amsterdam 120 lb. at Toulouse, how many lb. at Toulouse are equal to 40 lb. at London ? Ans. 40 lb. (6) If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. at Amsterdam 116 lb. at Dantzic, how many lb. at Dantzio are equal to 122 lb. at London? Ans. 14m D ARBI- ir measures comparison r countries, of the jirsti I last If * Antecedents. Consequents. 20 lb. London = 23 lb. Antwei-p. 155 lb. Antwei-p = 180 lb. Leghorn. 72 lb. Leghorn = how many London T 1 8 ^0X155X:^^ 1240 ^_, „ . :53|^ lb. An^, 23 ■* in) I ua BVOLUTIO.V. [TUTOR'S f rst re, and the ed on tha ve powers. ; the small ntinually one ccond, twice her will pro* hus, the set- luse the sum 2174. 726276 420489 thout fiii»l- 757441 015625 [t is there- dch it will ll 4 ASSISTANT.] SQUARE ROOT. 113 be obvious, that the S-> c s / Ditch. Ph 0) '5 i Base ()0 y.iids. (27) The wall of a town is 25 feet high, and is siirrpunde'l by a moat of 30 feeit in broadth : required the length of a lad- dier that will reacli from the outside of the moat to the top of the wall? ^;i5. 3905/ee^ N. B. These two questions may be varied for examples to the second and tliii'd cases. (28) In an army consisting of 331776 men, how many must be in rank and file to form a solid square ? Ans. 576. (29) A certain square pavement contains 48841 equal squaro stones. How many are contained in one of the sides ? ^n-^. 221. CUBE ROOT. Rule 1. Point every third figure of the given number, beginning at the units' place ; find the gretitest cube in the first period, and subtract it therefrom ; put the I'oot in the (juotient, and brhig down the figures VI the next period to the remainder for a Resolvend. 2. Multiply the square of the root found by 300 for a Divisor, and annex to the root the number of times vvhich that is contained lu the Resolvend. 3. Add 30 times the preceding figure (or figures) multiplied by tho last and the square of the last, to the divisor, and multiply the snm by the last for a Subtrahend; subtract it from the Resolvend, and rei)eat the process as fur as necessary.* I * The Bubjoined Theorems (deduced from Problem 91, page 2G6, Enif erson'a Algebra) are very convenient approximations for the Cubo Koot. 4n — 1^ l^ + l A'^h^Mir-^^"] [tutprHi Assistant.] CUBE ROOT. 117 5iirrpund^art. * Questions belong to this Rule which require the addition or sub-, traction of a number, &c., whicri is not any known part of tlie number required. The results are, therefore, not proportional to their suppo- sitions. f The following Rule will, in some cases, bn found more eligible: Multiply the dijference of the supposed numbers by the less error ^ and di\-ide the produot by the difference of tht errors \\ hen they are 01 F M 123 POSITION. truiOR^i (1) A, B, and C, would divide jC200. among them, so that B may have £6. more than A, and C £8. more than B. How much must each have ?* (2) A man had two silver cups of unequal weight, having one cover fo both of 5 ounces. Now if the cover is put on the less cup, it will double the weight of the greater ; and put on the greater cup, it will be thrice as heavy as the less. What is the weight of each ? Ans. 3 ounces the less, and 4 the greater. (3) Three persons conversing about their ages; says K, ** My age is C(]ual to that of H, and i of L's ;" and L says, " I am as old as botli of you together." Required the ages of K and L ; H'.i being 30. Ans. K 50, and L 80. (4) D, E, and F, playing at cards, staked 324 crowns ; but disputing about the tricks, each man seized as many as he could : E got 15 more than D ; and Y got a fifth part of both their sums added together. How many did each person get? Ans. D 127 j, E 142-^, and F 54. (5) A gentleman meeting with some ladies, said to them, " Good morning to you, ten fair maids." " Sir, you mistake," filre kinds, or by their sjim, when Tinlike ; the quotient will be a cov rcclional number; wliich being added to the nenrcsi svpposition when defective, or subtracted lixnn it when excessive, will give the number required. £. £. 40 2nd. Suppose A's share = 70 * 1st. Suppose A's shnre = tlion IVs = and C'lS =» r;4 Sum Tur then B's = 70 and C's =- 84 Sum 230 Therefore the eiTor ?a — GO, or 60 too little. svp. 40 70 X err. 60 30 Here the error ia + 30, or 30 t«o much. £. A GO 12U0 4-200 divisor. l^^O CO-f 3U=«ylO)540iO diviJeua. A's share. B 66 74 SrOO Proof. 70 — 40 X 30 jC'GO Or, by the Eule in the Note. : 10, the oowectloBttl nambet*. 60 -f 30 Theii 70-^ 10 -eo 3£ X_30^ iJO ■so A'k 6l*are, as before. lem, so that an B. How ight, having er is put on greater ; and as the less. le greater. 3S ; says K, and L says, i the ages of and L 80. crowns ; but many as he part of both person get ? and F 54. laid to them, ou mistake," will be a cor- pposttion when ve the number ihare = 70 1 B's = 7fi 1 C's =- 84 iSSISTANT.] PROGRESSION. 123 Sum 230 -f- 30, or 30 tm e. ;o 16 Proof. anmbei*. answered one of them, " we are not te but if we were three' times as many as we are, we should be as many above ten ai we arc now under." How many were they 1 Ans. 5. ARITIIMETICJAL PROGRESSION. Ax Aruhmctical Progression is a series of numbers increasing or decreasing unilbniily by a continued equal difference. Thus, o' r' o' it' i/ f ■> are mcreai'//?^* Arithmetical Series'. 2, 5, 8, 11, 14, &c. S 1 c' 1 o' o' / r! s ^ 3.re decreasing Arithmetical Series. 16, 12, 8, 4, 0, &c. S ^ Observe, that the terms of the first series are formed by adding suc- cessively the coniniou clilleieuce 1, aud the second by the common dit ference 3. The tenris of the third aud the fourth dimmish continually by the subtractiou of 1 aud 4 respectively. In an odd number of terms, the double of the mean (or middle term) is equal to the sum of the extremes^ or of anv two terms equidistant from the mean. Thus, in 1,2, 3, 4, 5',- the double of 3=1+5 = 2+4 = 6. In an even number of terms, the sum of the two means is fequal to the sum of the extremes, or of any two equidistant terms. Thus, in 2, 4, 6, 8, 10, 12, 6 + 8 = 2 + 12 = 4 + 10=14. To give Theorems or Rules for the solution of the varioaaf cases, the terms are represented by symbols, or letters. Thus let a denote the less extreme, or least term^ z the greater extreme, or greatest terrn^ d the conunon difference, n the number of terms, and s the sum of all the term^. Any three being given, the others may be found. Note. The twenty cases in this Rule may be resolved by the follow ing Theorems :— / , N 7 25 s ' a-=z — hi — l)tt= — — z= — 71 n ■ifZ(n— l) = J(f h V (i(f +.*)*— 2 J^w *~a-\-{n—l)d= a n d^ •a -an n nz — s d{n—\)^ V{^d—af I 2ds—idi n — 1 n — 1 fi n- 2 _{z-\-a).{z — a) n 'Zs — a — at i2i PROGRESSION. (TUTOit'i k=- -4-1=. I ,' d • Moreover, when the least term a = nothings the Theoremi become z=d{n — 1,) and s=^nz. base 1. The two extremes and the number of terms being given, to find the sum. Rule. Multiply the sum of the extra res by the number of termfl, tod half the product will be the answer.* (1) riow many strokes does the hammer of ai clock strikeil in 12 hours ?t (2) A man bought 17 yards of cloth, and gave for the first ^ard 2^. and for the last 10.?. What was the price of the 17 yards ? Ans. £5..2. , (3) If 100 eggs be placed in a right line, exactly a yari! irora each other, and th'^ first a yard from a basket, how far inust a person travel to gather them all up singly, and returoi •With fevery egg to put it into the basket ? Ans. 5 mites, 1300 yard^» Case 2. The same three terms givehj to find the common difi ference^ ■ Rule. Divide the difference of the extremCvS by the number of terra§ less 1, and the quotient will be the answer. (4) A man had eight sons, whose ages were in arithmetical progression ; the youngest being 4 years old, and the eldest 32. What was the common difference of their ages 1% (5) A man travelling from London to a certain place; went 3 miles the first day, and increased every day by art equal excess, making the twelfth day's journey 58 miles; I ■ ■ , 1 !-. r 1 ' 1 % • The learner should find each of these cases among the preced. Ikg Theorems. Thus, the present Rule will be found designatod b'^ i^^n(a-f-2r,) &c. f 12 + 1X6 = 13X6 = 78. Ang. \ 3(2 — 4 - 8—1 = 28 -^ 7 = 4 y«orj. Ant, [TUTOit*! A99I8TANT.] PROGRESSION. m — d(n— 1.) ) Theoremi being given, 3er of terms, :;Iock strike? for the first e of the 17 IS. £5..2. , ictly a yard et, how far , and retunoi 00 yardd^ common dij' mber of terra^ arithmetical d the eldest »rtain place; y day by art y 58 miles; ig the preced designatod by ' What was the dailv increase, and how far did he tru -i lu 13 ^ays ^ Ans. 5 miles daily increase^ the nhole distance 366 mtl»$, p^so 3. The two extrcincs and the common difference betng given, to find the number of tcr?ns. Rule. Divide the difference of the extremes )by the common diffefr j^iice, and the quotient increased by uniti/ is the number sought. (6) A person travelling into the country, went 3 miles the first day, and increased every day 5 miles, till at last he went 68 mil^s in one day. How many days did he travel ? Ans. 12. (7) A man being asked how many sons he had, said, that the youngest was 4 years old, and the eldest 32 • and that ills family had increased one iu every 4 years. How many pau up An^- 8. pase 4. The greater extreme, the number of terms, and th$ common difference being given, to fnd the less extreme. ^ULE. Multiply tl>e qomixion difff renc^ by the num)J!?r of terms lesij ^ ; subtract the product from t^ie greater extreme, uud the difference >vill be the less extreme. (8) A man went from London to a certain town in the country in 10 days ; every day's journey exceeding t|>e formey \>y 4 miips, 9,j[>d the last being 46 miles : what was the first ? Ans. 10 miles. (9) A man took out of his pocket, at 8 several times, so piany different numbers of shillings, every one exceeding the former by 6, the last being 46 : what was the first ? Ans. 4. pase 5. The common difference, the number of terms, and thf sum being given, to fnd the less extreme. Rule. Divide the sum by the number of tenns : from the ^ 'otient Bubtract half the ])roduct of the common difference into the number of terms less 1 ; and the remauider wall be the less exti-emei (10) A man is to receive jC360. at 12 severi' payments, each payment to exceed the former by £4. and is willing to hestoyir the first payment on any one that can tell hjm what i }9. What will that person have for his pains ? Ar^s. £8. Pase 6. The less extreme, the common difference, and the num her of terms being given, to find the greater extreme l^tLi Multiply the number of teims less 1 by the common difief 120 PROORESSIOX. (tutor '« V ! I ' !' i fence; to this product wlJ the less extreme, nuil the sum will b,e tho greater extreme. (11) What is the last numl)or of an arithmetical progres- sion, beginning at G, and continuing by the increase of 8 to CO places ? Afis. 158. GEOMETRICAL I'ROGRESSION. A Geometrical Progression is a series of nmnbers increasing pr decreasing uniformly by a cnnnnon ratio; thrit is, by tho continual multiplication or division of some particular mun- ber. Thus, 1, 2, 4, 8, 16, 32, (fee, is an incrcasnur G eoinetrical Series^ in which the terms are formed by multiplying successively by the ratio 2. 81, 27, 9, 3, 1, 7^^, &c., is a decreasing Geometrical Series, m which the terms are formed by dividing successively by the ratio 3. it is evident that either of these may bo continucf;! without end. In an odd number of terms, the square of the ?nean is equal to the product of the extremes, or of any two terms equidistant from the mean. Thus, in 3, 6, 12, 24, 48; 12X12=3X48 ^6X24=144. In an even number of terms, the product of the two means is equal to the product of the extremes, or of any two equidis- tant terms. Thus, in 32, IG, 8, 4, 2, 1 ; 8X4=32X1<^ J6X2=32. To give Theorems, or Rules expressed in symbols, for the fiolution of the various cases, as in Arithmetical Progression^ let a denote the less extreme, X the greater extreme, r the ratio, n the number of terms, and s the sum of all the terms. Any three being given, the others may be found. KoTE. The twenty Theorems following solve all the possible c^sm }Xi. Geometrical Progression. Theor, I. r'*~*= z • a or. Loff. Log. r log.a^^^ ■n. I \'\ '1 m z • In tbis case, if the quotient of — be divided continually by r, til| m>t)iing remains ; tho number of divisions -f 1 vill give n. (tutob'« will bp tho il progres- Lse of 8 to fis. 158. increasing' is, by tho ular niim- cal Scrips^ !ssively by ? Scries, m ely by the continue);! in is equal equidistant 12=3X4^ two means CO cqitidis' =32X1'=- •Is, for the •ogression^ und. Bsible C£^e9 111/ by r, til^ *,S8ISTANT.] PROCllESSION. 127 II rz — a ■'S ; or, s — i*. III. (—)■'■-' =*r; or, Log. ^ — log. a-r'i — l = Log. r ,iy. ^+ z — a VI (^)- s — a Xn-i V. = r. j: — 2 •e) = £.• ^'■o'^i which w may be found, as ip a ' Theorem I. or Loff. ^ — log. a , , ,.,, ,^1 ' Log. [s— a)— log. {s—z) yiU.A^2l=s. IX. (--'>+" ^..- /—I r -a .2X2 X. ^=(l=l)i+l ; or, Log.(>--l)^+a-log^^, Log. r a i»i Tn—l XI. zXs — z\ =^aXs — a\ ; whence z maybe found by Double Position, found in the same manner. ST s-^~a XII. r" — ; whence r may be a a XIII. a— I a. XIV. $l=y=, j-n y.i»— I XV. r^— (r— iy=a. XVL /*" j_ T^ — (r — 1)^ ; or, Log. ^ — log. r;^ — (r — l)s ft ' T " ■ 1-1 =W. •Log. r o-^l XVII. aXs—a\ =^zXs—s in— 1 XVIII. ll-f— />=--^ 5 — z s « From ih<-' two preceding, a and r are to be found by Double Position. XIX. trni^a. XX. ^=C r"— 1 y'^— 1 XS=^z. In a Geometrical series decreasing ad infinitum, a becomes kQ, and n is infinite, or greater than any assignable nvrober. ia3 lien of such a scries. PROGRESSION [tutqf^ Ilenco the three following will exhibit all the various cases - rz z I. s^—=z.{- r — 1 / — 1 II. 5= .(.-1) III. z \ ■ Note. In these cascH, when the ra/io ia a proper fraction, r must bo talcen = the reciprocal of the fraction. Thus, when the ratio is 5» Case 1 . The less extreme, the ratio, and the number of terms being given, to lhul the greater extreme {or any remote term) without producing all the intermediate terms. Rule 1. Wlieu the least terai is equal to the ratio. Write down a few of tlie leading terms of the series, and over them the arithmetical Beries, 1, 2, 3, 4, dec., as indices or exponents. Find which of the in- dices, added together, will give the index of the term sought, and the continual product of the terms standing under those indices will be the tenn sought. 2. When the least term is not equal to the ratio. Write down the leading terms as before, an ^, >• i j i o n i Geom.series, 1, 2, 4, 8,16.1 Then 4 -f 4 + 3 =. 11 =.»-!. Hence 1(» X 16 X 8 = 2048 qrs. = z, and 2048 qr$. = £Q.X.B Ant. I: ■ . .■ . . . I , ■ t (tutor!* ious cases I. r=s — — . )n, r must bo e ratio is ^, T of terms 'mote term) iTrite down a arithmetical ;li of the i»- gkt, and tho 3 will be the te down the , 2, 3, 4, &o. one less than such indices rst term, and he price of Ifpenny for What must oxen, met im jCIG. a ) parties, it, )rice of the d doubling 69..1..4. diviflion. ^.8 Ant. 48R18TANT,3 FROGRESSIOK. 129 I I (3) A sum of money is to be divided among 8 persons, the first to have jC20. the second £60. and so on in triple pror portion. What will the last have ?* Ans. jC43740. ' (4) A gentleman dying left nine sons, to whom and to his executors he bequeathed his estate in the manner following : To his executors jC50., to his youngest son twice as much as tp the executors, to the next, double that sum, and so on tq ^he eldest. What \yas his fortune ? Ans. JC25600. pase 2. The less extreme, the ratio, and the number of terrn^ being given, to find the sum of all the terms. Rule. Find the greater extreme as before, and divide the difference between the extremes by the ratio less 1 ; to the quotient add the great-, tr extreme for the snm- required. This is Theorem II. Or, by Thepr rem VIII; without finding z. (5) A young man conversant with number?, agreed with i^ gentleman to serve him twelve months, provided he would give him a farthing for his first month's service, a penny foy the second, and 4c?. for the third, &c. What did his NVE^geg amount to ?t Ans^ £5825.. 8.. 5^. (6) A man bought a horse, and by agreement was ^o give a farthing for the first nail, three for the second, Asc. Now supposing there were 8 nails in each of his four shoes, what ivas the price of the horse? Ans. je965^1 4681693. .13. .4. (7) A person whqs§ daughter vvas married on new-year*i5 Bay, gave her husband 1^. towards her portion ; promising tp Rouble the sum on the first day of every month during the year. What \yas her portion ? Ans. JG204..15. (8) A laceman, well versed in numbers, agreed with a^ gentleman to sell him 22 yards of rich brocaded gold lace, for 2 pins the first yard, 6 pins the second, &c., in triple pro- portion. What was the price of the lace, valuing the pins * Indices Geom. serie tefsS: GO, m. 540; ] Then 3+3+1=7=,-!. Hence 5i?|<5i5^=540X27X3=437.10=z. otherwise ar^'«=20X3^=43740=z. t Here a=l, r=«4, and z=12. Tlitiefore s — 5592405 qrs. F2 4'»— 1 _^ 167772151 4 — 1~ 3 ^ I I > >90 Simple iNTEREsf. [tutor's at 100 for a farming ? Also, what did the laceman gain sup- posing the laco to have cost him £1. per yard 1 Ans. Thelacesoldfor £32Q886..0..9. Gaifi £3267 32.. 0..Q. Case 3. The first term, and the ratio, being given, to find th^ sum of an infinite decreasing series. BuLE. Divide the square of the first term by the difference boHffeen the first and second.* (9) What is the sum of the circulating decimal -9^, or the series ^^ + j^i^ -\- jTf'o "o» *-^^-5 continued ad infmiliim? Ans. 1. (10) Required the sum of the infinite series ^ + 5-~i"f> &c.; also of the series -J- + ^ "i" 2V' <^^- ^^'^^- 1' ^"^ "I- (11) Suppose a body to be put m motion by a force which gives it a velocity of 10 miles the first minute, (or any given fspace of time,) 9 miles in the second equal space, and so on jp the ratio of ^V ; how many miles would it pass over, if pontinued in motion for ever ? Ans. 100 miles. ■ 1; J t; ;'; 14 '' . ■ > it :i I '•r ft" ■hi '^ SIMPLE INTEREST, BY DECIMALS. To give Theorems for the solution of the different cases ir) pimple Interest, let p denote the principal, r the ratioj t the fime^ (in years,) i the interest, and a the amount. Note. The Ratio i? the interest of £\. fur one year, at the rate per ^nt. proposed, and may be found by Proportion : thus, at X5. per cent, per annum, say, ^ As JCIOO ;e5 : : IjC : £'05, tlic ratio. Therefore the ratio at 3 per cent, is 3| . . . 4 . . . •03 •035 •04 41 per cent, is . . . -045 5 -05 51 -055, &c Cstse 1. When the principal, rate per cent, and time are given, to find the interest. Rule. Multiply the principal, ratio, and time together, and the pro 4uct will be the interest required. That is, prt = i. (1) What is the interest of JC945..10. for 3 years, at £5 per cent, per annum ?t • See the third /orm?//flr, Theorem I. for infinite series, page 128. t |>r r.] SIMPLE INTEREST. 131 (2) What is the interest of jC54*.7..14 , at jC4. per cent, per annum, for 6 years? Ans. jC131..8..11..20S qrs. (3) What is the interest of £796. .15., at jC4J-. per cent, per annum, for 5 years? Ans. £179. .5. .4^. (4) What is the interest of JC397..9..5. for 2^ years, at £31. per cent, per annum? Ans. £34. .15.. 6. .3-55 qrs. (5) What is the interest of £554. .17. .6. for 3 years, 8 months, at £4^. per cent, per annum? A/is. £91..11..1 05cf. (6J What is the interest of £236. .18. .8. for 3 yearg, 8 inontns, at £5^. per cent, per sniiunri ? Ans. £47. .15. .7. .2 293 qrs. When the intaest in for any number of days only. RuiK. Multiply the intereBt of £1. Lr a Jay, at tlie given rat^, by .he principal and the number of days, and the product will be tlio answer. The Interest of £\. for one day. At£2percent.=£-00005479452*at£4:=-00010958904 2^ =- -00006849315 4i==--0001 2328767 3 =., -00008219178 5 =-C0013698G3O 31 =' -00009539041 51== -0001 5058493, , r, and / are give:!, to f-r^d a. Rule. prt-\-p'^ a. (11) What will £279.. 12. amount to ia 7 years, at £4^ per cent, per annum -^ (12) What will £320.. 17 amount to in 5 years, at £3^ per cent, per annum \ Ans. £376. .19. .11. .2-8 qrs. * The table is foi-med thus : — Ab 3G5 days : £00 : : 1 rhy * -00010953904 X 210 X 120 =.f 31.V^lfi 13o~^ --=--^3..n..li •< 279-6 X -OlS X 7 H-5>^3'6 --^ £3i3<-'-6r 1 ^£';6r..]3,.5..:^-a} qr$. £•00005479455, &c. '"^ '^ vi Ana. ] I Is 1 32 SIMPLE IXTERKST. [tutor* (13) What will £926.. 12. amount to in 5| years, at £4. per cent, per annum ? Ans. jEril30..9..0..r92 qrs. (14) What will jC273..18. amount to in 4 years, 175 days, at £3. per cent, per annum? Ans. J^310.. 14. .1.. 3*3512 qrs Case 3. When a, r, and t are given, to find p. Rule. rt-\-l (15) What principal, being put to ir^terest, will amount to JG367..13..5..304 qrs. in 7 years, at jC4|. percent, per annum?* (16) What principal will amount to jG376..19..11..2-8 qrs. in 5 years, at £3^. per cent, per annum ? Ans. £320.. 17; (17) What principal will amount to £1130..9..0..1-92 qrs. m 5i years, at £4. per cent, per annum? Ans. JC926..12 (18) What principal will amount to £310..14..1..3'3512 grs. in 4 years, 175 days, at £3. per cent, per annum ? Ans. £273.. 18. Case 4. When c, p, and t are given, to find r. Rule. ~~^=zr. pt (19) At what rate per cent, per annum will £279. .12, ^mount to £367..13..5..304 qrs. in 7 years ?t (20) At what rate per cent, per annum will £326..!?. amount to £376. .19. .11. .2-8 ^-r*. in 5 years? Ans. £3^. per cent. (21) At what rate per cent, per a-nnum will £926.. 12, amount to £1130. . 9. .0.. 1-92 qrs. in 5i years T A,ns. £4. per cent. (22) At what rate per cent, per annum will £273.. 18 amount to £310. . 14. .1.. 3-3512 qrs. in 4 years, 175 days? Ans. £3. per cent Case 5. When a, p, and r are given, to find t,. _ a — n pr f i II if 1 1 * -045 X 7 -I- 1 = J 31.5 ; then 367-674 -r 1-315 = £279-6 =«je279..12 Anis. ' \ 367-674 — 979-6 88-074 ' 279-G X 7 1957-2 •045, or jC4i. per ce»t. Ans. Jtutor* rs, at £4. "92 qrs. 175 days, •3512 qrs AS0ISrA>'T.] SIMPLE INTEREST. 133 amount to r annum?*' 1..2-8 qrs. 320.. 17: ..1-92 qrs. JC926..12 .1.. 3-3512 urn ? 273..18. r. £279,.l2. ^320..17. per cent. i:926..12 per cent. je273..18 days ? 3. per cent (23) In what time will i:279..12 amount to £367. .13. .5.. 3*04 qrs at £4^ per cent per annum ?* (24) In what timp will £320..17. amount to JC376..19..11.. 2*8 qrs. at £3^. per cent per annum ? Ans. 6 years. (25) In what time will £926.. 12. amount to £1130..9..Q.. 1*92 qrs. at £4. per cent, per annum? Ans. 5^ years. (56) In w^at time 'vjrill £273. .18. amount to £310.. 141.1.. 3*3512 qrs. B.i £3. per centi per annym? Ans. 4 years i 175 diys. ANNUITIES. An annuity is a yearly income or rent. Perpetual Annuitm *re those which are to continue for ever ; Terminable Annuu ties are to cease within a limited time ; and Life Annuities 8r« to continue during the tfirm of lif^ of one or more persons. ' The Amount of Annuities in Arrears. Let u denote the annuity^ /, /, and a, as before. Case 1. Given, u, r, and ty to find a. BciB. (^•-+0 X tu=:a. (27) If a salary of £150. be forborne 5 y^ars, at £5. pn eent. per annum, what will be the amount?! (28) If £250. yearly pension be forborne 7 years, what will it amount to at £4. per cent, per annum ? Ans. £1960. (29) There is a bouse let upon lease for 5^ years, at £60. per annum, what \yill be the accumulated rent, allowing in- terest at £4^. per cent, per annum? Ans. £363..8..3 (30) Suppose an annual pension of .;^28. remain unpaid for $ years, what would it amount to at £5^. per cent, per annum I ■ ' ^ " ' ''''• ' ''^ "■ ^" ' Ans. £263..4. Note. When the annuity is payable half-yearly, or (]^uarterly < will denote a single payment, r, the interest of £1. for that interval oi time, and t, the number of payments. ' ' (31) If a salary of £150., payable every half-year, remain unpaid for 5 years, what will it amoyr^t to in that time, allow- ihg interest at £5. per cent, per annum ? Ans. £834..7..6. =X279..12 Ans. :t • 367-674 — 279-6 ~ 88-074, and 2796 X -045 = 12582 ; then 88-074 -r 12-582 = 7 years, ^ns. t /42. I J- J '« >. I: ;*^^ ^'-^^ 134 SIMPLE INTERF.ST. [TUTOR'S (32) If a salary of £150., payable every quarter, were left unpaid for 5 years, what would it amount to in that time at £5. per cent, per annum ? Ans. jC839. 1..3. Note. It njay be observed by companng the results of the 27th, 3l8t, and 32ntl examples, that half-yearly payments are njore advant* geous than yeai'ly, and quartei'ly more than lialf-yearly. Case 2. When a, r, and t, are given, to find v. Ruts. u. (33) If a salary amounted to X'825. in 5 years, at £5. per cent, per annum, what was the salary '.* (34) If a house has been let upon lease for 5^ years, and the amount in that time is ^363. .8. .3. at £4^. per cent, per annum, what is the yearly rent ? Ans. X60. (35) If a pension amounted to £\960. in 7 years, at £4. per cent, per annum, what was the pension 1 Ans. JC250. (36) Suppose the amount of '^. ^jension was £!263..4. in 8 years, at £5. per cent, per annum, what was the pension ? Ans. j£^28. .. (37) If the amount of a salaiy, payable half-yearly, bo JC834..7..6. in t years, at £i>. per cent, per annum ; what ig the salary per ycar?t -^ns. jC150. (38) If the amount of an annuity, payable quarter!}', was JE?839..1..3. in 5 years, at £5. per cent, per annum ; what was tjie annuity? Ans._£l50. Case 3. "When w, a, and t are given, to find r. {t—l)Xnt (39) If a salary of jC160. per annum amounts to jC825. in 6 years, what is the rate per cent. ?| (40) If a house has been let upon lease for 5^ years, at jE60. per annum, at what rate per cent, would it amount to X363..8,.3 ? Ans. £4^. per cent, 825 X2 1650 1650 11 (•)5X4 + 25)X5 22X5 t Bee note, p. 133. ^ (825 — ISOIT S ) X 2 825 — 750 4 X 150 X'S "^ 2 X 150 X 5 ■ r; therefore the rate is £5. per cfmt. £150. Ana, 75 1 150 X 10 20 OS* [tutor's JkSSISTAKT.] SIMPLE INTEREST. 135 were left It time at )f the 27th, )re advantar at £5 per years, and cent, per IS. X60. irs, at £4. J. ^250. 63. .4. in 8 ension ? ns. £23. ■yearly, be ; what is £1^0. rterlv, was ; what wai s.,£'l50. d r. to jC325. in I years, at amount to per cent. t (41) If a pension of i:250. per annum amoimts to jC1960. in 7 years, what is the rate per cent. ? Ans. £L per cent. (42) Suppose the amount of a yearly pension of jC28. be ;jC26J..4. in 8 years, what is the rate per cent, per annum ? Ans. £5. per cent. (43) If a salary of jC150. per annum, payable half-yearly, amount to .£^834.. 7. .6. in .o years, what is the rate per cent. ?* Ans. £3. per cent. (44) If an annuity of £^150. per annum, payable quarterly, amount to jC839..1..3. in 5 vears, what is the rate ner cent. ? Ans. £j. per cent. Case 4. When 21, a, and r are given, to find ^ (•2-r) RCLS. •v/8r 1-(2 — r)2- It 2r = t. Or, put i = ?« / then \/ 1 \- m^ nr ■ m- (45) In what time will a salary of jGISO. per annum amount to jG825. at £5. per cent. If (46) If a house is let upon lease at £30. per annum till it amount to jC3G3..8..3. at X*-iJ-. per cent, per annum, for what term of years was it let? A71S. 5^ years. (47) If a pension of jC250. per annum, having been for-* borne a certain time, amount to i^l960. at £i. per cent., how long has been the time of forbearance ? Ans. 7 years. (48) In what time will a yearly pension of £2S. amount to JC263..4. at jC5. per cent, per annum ? " Ans. 8 years. (49) If an annuity of jCIdO. per annum, payable half- yearly, amounted to .C834..7..G. at £3. per cent, what time was the payment forborne 1* Ans. 5 years. (50) If a yearly pension of .Gl 50., payable quarterly, amounts to JG839..1..3. at £3. per cent, per annum, what has been the time of forbearance ? Ans. b years. 20 0& • See Note, p. 133 t V/8X OoX 8125 1.jO _|-(2 — -Oo)- — (2~-0r)) 2 X -05 »/ '40 X 5-5 + n-8025 — 1-95 2-45 — 1-95 >= -5 -f- '1 = 5 years. Am* ■ fao SIHPLS INTEREST. ('fUTOIl'# A Table by which the Interest of any sum from JCl. to JC30000. may bt easily computed for any number of days, at any rate per cent. JVd. X. 8. d. qrs. No. 9. d. qrs. No. qrs. '30000 82 3 10 Oil 200 10 u 203 1 2-63 2Q000 54 15 10 2-74 100 5 5 301 0-71 0-9 2-37 10000 27 7 11 1-37 90 4 11 08 210 9000 24 13 1 3-23 80 4 4 1}-41 0-7 1-84 8Q00 21 18 4 110 70 3 10 Oil 0'6 1-58 7000 19 3 6 2-96 60 3 3 i-81 0-5 1-32 6000 16 8 9 0-82 50 2 8 3'51 Q-4 105 5C00 13 13 11 2-68 40 2 2 1-21 0-3 0-79 4000 10 19 2 0-55 30 1 7 2-90 Q-2 6-53 3qoo 8 4 4 2-41 20 1 1 0-60 01 0-26 2000 6 9 7 0-27 10 6 2-30 0-09 0-24 1000 2 14 9 214 9 5 3-67 008 0-21 900 2 9 3 312 8 5 104 0-07 018 800 2 3 10 Oil 7 4 2-41 006 016 700 1 18 4 110 6 3 3-78 005 013 600 1 12 10 208 5 3 115 004 on 500 1 7 4 3 07 4 2 2-52 003 008 400 1 1 11 005 3 1 3-89 002 005 300 16 5 104 2 1 1-26 001 003 ; ,' i^ ^- < ;('■'' ^-^ 1 1: Si; ' c ! id * The above Table is thas constructed; as 365 days : £1. : : 1 day i S^'63 OTs., &c. Hence it appears that the several tabular sams are tho#p yhicn answer to the respective numbers of days, at the rate of £1. per Tear. lo a similar Table in Reea't Cycloptpdia, there are no fewer than Id errors. In Dr. glutton's Table, (Arithmetic, page 84, 12th edition,) ^ers is one error. The aljoye may be depended on as accurate. RvLE. Multiply the principal by the rate, both in pounds, and th^ product by the number of days : divide the last product by IQO, colled^ nt>m the Table the several sums answermg to tne several parts of th^l <}uotient, and the aggregate amount will be the Int^resT: required. Example 1. What is the interest of je370..10. for 220 days at £i-^. per cent, per annum ? £. s. d. qrs. 370-5 Against 3000 stands 8 4 4 2*41 4-5 600 ... 1 12 10 2-08 18525 60 ... 3 3 1-81 14820 7 ... 4 2-41 1667-25 0-9 ... 2-37 220 005 ..000 0-13 3334500 3667-95 10 11 3-21 An^ 333450 True to the last decimal. 3667 9500 • II*' 1 ■ . -~ [TUTOR'f 000. may bt er cent. qr$. 2-63 2-37 210 1-84 1-58 1-32 105 Q79 6-53 0-26 1 0-24 1 0-21 ' 018 • > 016 i 013 I Oil J 008 005 L 003 : : 1 day s ms are tho#p te of Xl. per iwer than 10 3th edition,) ;urate. nds, and th^ Y IpO, collect . parts of th9 squired. >r 220 days 2-41 208 1-81 2-41 2-37 0-13 3-21 An^ decimal. 49SI8TA1IT.] SIMPLE INTEREST- \^i? Example 2. Taking Ex. 8, page 131, we have 364' 9 5 £. s. d. qrs, 1824-5 Against 2000 stands 5 9 7 027 154 800 ........ 2 3 10 on 7OQQ0 9 5 3-67 273675 ^7 ISJ . ■ 003 •• 008 .1 LiV 2809-73 7 13 11 VdlA ns Example 3. What is the interest of i:i7..10. for 117 4ay» ^ £4^. per cent, per annum ^ 17-.5 " 4-75 £. s. d. qrs. 875 4g^^"®' 9^ *^*"^s ^ "^ ^1 0'71 1225 • 7 ........ 4 2'4i 700 0-2 ..... 0-53 8TT25 005 ... OOP oisi 11-^ 97-25 5 3 3-78 iinf, 581875 914375 97|25-6 25 IjTo find the amount cf a yearly income or salary, ^e , Jpr ^ number of days. Multiply the number of pounds per year by the number qf lays ; collect the Tabular sums answering to the product, at ^efore, and their aggregate will be the answer. Example. What will a person receive for 4^ days, t^i tl^« |8te of i:i05. per annum ? 105 £. s. d. qrs. 45 Against 4000 stands 10 19 i 0'56 — 700 - • } 18 4 MP 4i^0 20 1 1 0-60 i^ , _5 3 115 ' ' 4725 12 18 10 3-40 4yij. NoTl. Any of the preceding examples of intereat^for daya, in pa«# I9I> or examples 30 and 21, page 68, may be worked by this metbo4«' i( M )»8 DISCOUNT. {tutors* A Table <*howuig the number of tbiys from any day in the month to the same day in imy other month through the year To a • Apr. ^ S a July bJD c > • Q 'Jan. 3G5 31 59 90 120 151 181212'243 273 304 334 Feb. 334 365 28 59 89 120450 181212 242 273 303 Mar. 306 337 3G5 31 61 92422153184 214'245^275 Apr. 27o'30G 334365 30 6J 91 122:153 183 214 244 May 245276 3041335 365 31 61 92123 153 184 214 s o J me 2 14^245 273304 334 365 30 61 92 122 153 183 2 ^ July 184215 243 274'304 335 365 31 62 92 123 153 Aug. 153 184:212:243 273.304 334 365 31 61 92 ■22 Sept. 122 153;i81212 242 273:303 334 365 30 61 91 ,Oct. 92 123 151 182 212:243 273'304 335 365 31 61 Nov. 61 92 120151 il8l|212'242 273 304 334 365 30 _Dec. 31 62 1 90;i21 |151!182;212 243 274 304 333 365 ii 1 6 <'■■ :« i ?• F H ■ ■ 1 ' I >!;. ui V. -1 S'. DISCOUNT. Let s represent the sum to be discounted, r the ratio, t thf timCf (in years,) and p the present worth. Case 1. Given s, r, and ^, to find p. RUL£. rt-^l (1) What is ihe present worth of je357..1p. to be paid 9 months hence, at jC5. per cent, per annum ?* (2) What is the present worth of jC275..10 due 7 months hence, at jC5. per cent per annum? Ans. jC267..13..10-152(f. (3) What is the present worth of JC875..5. 6. due 5 months hence, at jC4^. per cent, per annum ? A71S. jC859..3..3..3'01824 qrs. (4) How much ready money can I receive for a note ol ^75 due 15 months hence, at £b. per cent, per aniium? Ans. j£:70..11..9-1764(f. -• 357-5 -r -05 X 75+ 1 =344-5783 •= je344.. 11.. 6.. 31G8 qrs. Anf. I ■%,■? (TUTOR^i iioctli to the 2> > o • u Q 73 304 334 12 273 303 14 245 275 83 214 244 53184 214 22 153 183 92 123 153 61 92 .'22 30 61 91 {65 31 61 534 365 30 i04 i335 365 le ratio, t thf to be paid 9 ue 7 montlus .13..10-152rf. lue 5 mouths )1824 qrs. or a rote ol annum ? ,..9-1764rf. G8 qrs. Atif, i DISTANT.] DISCOUNT. 139 Case 2. When p, r, and t are given, to find s. Rur.E prt-\-p = s. (5) If the present worth of a sum of money due 9 months hence, allowing £5. per cent, per annum, be jC344..1i..G. 3" 168 qrs.y what was the sum due ?* (6) A person owing a certain sum, payable 7 months hence agrees with the creditor to pay him down X'2G7..13..10152c?., allowing £5. per cent, per annum for present payment : what is the debt? A ?is. £'27 H. .10. (7) A person receives .€859. .3. .3.. 30182 1 qrs. for a sum of money due 5 months hence, allowing the debtor £^. per cent, per annum for present payment : what was the sum due ? An.'i. JC875..5..6. (8) A person paid ;C70..11..9-1764f?. for a debt due 15 months hence, being allowed X'5. per cent, per annum for the discount. How much was the debt ? Aiis. £lb. Case 3. When s, p, and t are given, to find r. Rule. — = r. pt (9) At what rate per cent, per annum will jC357..10., pay- able 9 months hence, produce jC344..11..6..3-168 qrs. for present payment ?t (10) At what rate per cent, per annum will JC275..10., payable 7 months hence, be worth X2G7..13..10-152(f. for present payment 1 Ans. £5. per cent. (11) At what rate per cent, per annum will jC875..5..6., payable 5 months hence, produce the present payment of JC859..3..3..3-01824 qrs. ? Ans. £'\\. per cent. (12) At what rate per cent, per annum will jC75., pay- able 15 months hence, produce the present payment of JC70..11..91764J. Ans. £5. per cent. Case 4. When s, p, and r are given, to find t. Rule. ~ = t pr * 344-5783 X "05 X 75 + 344-5783 = i:3r)7.. 10. Ana, t 357-5 — 344-5783 344578;^ X -75 • = -05 or jC5. per cent. Ant. ^40 EQUATION OP PATMENTS. (tutor?^ (13) The preseiDt wojrth of JC357..10., due at a certain time to come, is JC344..1 1 ..6. .3- 168 qrs. at jC5. per cent, per annum : \xi what time should the sum have been paid without any (discount?^ (14) The present worth of jC275..10., due at a certain tinfjQ jto come, is je267..13..10152t/. at £!j. per cent, per annum: ^n what time should the sum have been paid without dis- count ? Avs.l months. (15) A person receives JC859..3..3..301824 qrs. for jC875.. 5 .6., due at a certain lime to come, allowing £A^. per cent, jper annum discount : in what time should the debt have beei> jjischarged without any discount ? Ans. 5 months. (16) I have received je70..11..91764 months. ebt of £15., for prompt ible without 5 months. isSISTANT.j COMPOUND INTEREST. Hi (1) D owes E jC200., whereof £A0. is to be paid at threa months, jC60. at 6 months, and -CI 00. at 9 months: at what time may the whole debt be paid together, discount being allowed at £b. per cent, per annum ?* (2) D owes E jCSOO., whereof je200. is to be paid in 3 months, je200. at 4 months, and X'400. at 6 months ; but they agree to have the whole paid at once, allowing discount at the rate of £b. per cent, per annum'. The equated time is required. Ans. 4 months, 22 days. (3) E owes F i^l200., which is to be paid as follows: jt200. down, £500. at the etid of ten months, and the rest a^ the end of 20 months ; but they agree to have only one pay- ment of the whole, discounting at jC3. per cent, per annum. The equated time is required. Aris. 1 year^ 1 1 datjs. offiioney duf M='- equated time. r COMPOUND INTEREST.! The same symbols are adopted in this as in Simple Interes^i' and denote the sa'pie things ; except that the ratio (r,) whicl!^ in Simple Interest denotes the ititerest of JCI., signifies in tliii Rule the amount of £1. for a year. It may be thus found by Proportion. As £100 : i:i05 : : £1 : i:i05, the ratio at £5 per cent, per annum. The ratios are, therefore, at 3 per cent. 3^ " 4 li 1-03 1-035 104 at 4| pfer cent. 5 (( It 1045 I 05 1055, &c; jomething le^s mrpoee of real ion of two pay- \ is founded oil fter it beconet aid before it it I in expression plification than y be regarde(|, UTOR. 'i 39-5061; 6fl( 58-5365 ; 100 end ' 1-025 ' 10375 then 266 — 39-5061 +58-5365 4 96-3855 = 5-5719; 55719 ■ss-57315=>6 months, 26 days. Ans. 96-3855; 194-4281 X -05 ♦ The law of England does not allow the lender to receive Campoun Interest for his money when the receipt of the Interest has been fo borne. But in the granting or purchasing of Annuities, Leases, Ac either immediate or m reversion, it is customary, and indeed necessary to compute them on the principles of Compound Interest, for otherwia Aie celctilatfoii would invoh-e most egregious injustice and abturdily. ui COMPOUND INTEREST. [TUTOi . 1 it k A Table of the A mount of £l.for years. Yrs. 3 fcr Cent. '.}h per Cent. 4 per Cent. A^perCcnt. 5 per Cent. 1 o 3 4 5 1-0300000 l-0()09000 1-0927270 1-1255088 1-1592741 1-0350000 1-071-2250 1-1087179 1-1475230 1-I8768(i3 1-0400000 1-0816000 1-1248640 1-1698586 1-2166529 1-0450000 1-0920250 1-1411661 1-1925186 1-2461819 1-0500000 1-1025000 1-1576250 1-2155062 1-2762816 6 7 8 9 10 1-1910523 l-22!)8739 l-26(i770l 1-3017732 1-3439 in 4 l-22!)2553 1-2722793 1-3168090 l-3(.28974 1-4105988 l-459!)697 1-5110687 1-5639561 1-61869I5 1-67 53 '18 8 1-7339860 1-7946756 l-85748<»2 l-9-.>2r.013 1-9897889 2-0594315 2 1315116 2-2061145 2-2833285 2-3632450 2-4459586 2-5315671 2-6201720 2-7118780 2-8067937 2-9050315 3-0067076 3-1119123 3-2208603 3-3335904 3-45026()l 3-5710254 3-6960113 3-8253717 3-9592597 1-265:) 190 1-3159318 1-3685690 1-4233118 1-4802443 1-3022601 1-3608618 1-4221006 1-4860951 1-5529694 1-3400956 1-4071004 1-4774554 1-5513282 1-6288946 11 12 13 14 15 1 •384-2339 l-4257(i09 l-4()85337 1-5125897 1-5579(J74 1-5394541 1-6010322 1-6650735 1-7316764 1 -8009435 1-6228530 1-6958814 1-7721961 1-8519449 1-9352824 1-7103394 1-7958563 1-8856491 1-9799316 2-0789282 16 17 18 1,9 20 1.60470(;4 1 -052 8 170 1-702 i.rn l-75350fi0 1-80()11T2 1-8729812 1-9479005 2-0258 165 2-1068492 2-1911231 2-2787681 2-3(;99183 2-4647155 2-5633012 2-6658363 2-7721693 2 •8833686 2-fl987033 3-118(;514 3-2433975 2-0223702 2-1133768 2-2084788 2-3078603 2-4117140 2-1828746 2-2920183 2-4066192 2-5269502 2-6532977 21 22 23 24 25 l-860294() 1-9161034 1-9735865 2-0327941 2-0937779 2-5202412 2-6336520 2-7521663 2-8760138 3-0054345 2-7859626 2-9252607 3-0715237 3-2250999 3-3863549 26 27 28 29 30 2-15(;5913 2-2212890 2-2879-377 2-3565655 2-4272625 3-l)067i)0 3-2820096 3-4297000 3-5840365 3-7453181 3-5556727 3-7334563 3-9201291 4-1161356 4-3219424 31 32 33 34 35 2-50()()8()3 2-5750827 2-6523352 2-7319053 2-8138624 3-3731331 3-5080587 3-6483811 3-7P131(i3 3-9460890 3-f) 138574 4-0899810 4-2740302 4-4663615 4' 6673478 4-5380395 4-7649414 5-0031885 5-2533479 5-5160153 36 37 3J S9 40 2-8982783 2-9852266 3-0747834 3-1670269 3 2620378 4-1039325 4-2680898 4-1388134 4-6163660 4-8010206 4-8773785 5-0968605 5-3262192 5-5658991 5-8163645 5-7918161 6-0814069 6-3854773 6-7047511 7-0399887 These tabular juinibcrs arc the successive powers oi"?-; thus, l-05'=s M025, &C.'* * The amouzit ofiJl. in t years is the last term of an iiicreashig gea tneti'ical series, f)t' wliicli the first term = the ratio, and the number ol iftirrns = t, because the first yearns amount is idetitical with the ratio} and as i : r : : r : ^-'^ = the amount in 2 years, as 1 : r : : r* : : ?•' = ther lurount iu 3 years, &(\ The sutcessive amounts r, r^, i^, &c., are fvid^ently in geometrical progree^ioH, and the amount In t years if •ill ASSISTANT.] COMPOUND INTEREST. 143 Case 1. When p, r, and t are given, to find a. Rule. pr^=a. Or, 1oir. rXt-\-\o^. p=log. a. Or by the Table. Miiltiiily tho t;il)u!!ir amount of £1. by the princi* pal, aiid the product will be the amount required. (1) What will £22o. amount to in 3 years, at £b. per cent. per annum ?* { 2) What will jC200. amount to in 4 years, at £5. per cent per annum? Ans. jC243..2..0..1-2 qrs (3) What will jC 150. amount to in 5 years, at jC4. per cent. per annum Ans. .£r)47..9..10..20538368 qrs. (^4) Wliat will jCoOO. amount to in 4 years, at £4i. per cent, per annum ? Ans. jC59G .5..2-232075(i. Case 2. When a, r, and t are given, to find p. Rule. a ■■p. Or, lug. a— iog. r X t = log. p. (5) What principal, being put to interest, will amotinl to £260. .9. .33. in 3 years, at £5. per cent, per annum ?t (6) What principal, being put to interest, will amount to £243. . 2. .0.. 1-2 qrs. hi 4 years, at £3. per cent, per annuny? Ans. £200. (7) What principal will amount to £547..9..10..2-0538368 qrs. in 5 years, at £4. per cent, per annum? Ans. £450. «= /••, because the index always corresponds with the time. By refer* ring to Theorem VII, Gromctricnl Progression, it will tilso be seen that bucli lad term =r X /•*— ^ = 'H, \Ahen a = r. The immense inci'oase of money accunuilaling at Compound Interest for a long period is sutlicient to nstonioh tho human mind, and to stag- ger the credibility of persons \^ll() are not in some degree conversant with the properties of Cieomctrical Progression. The amount of a faiy thing, placed out at Compound Interest at the commencement of the Christian era, and continued to the conclnsion of tire eighteenth centu- ry, would be 14403"^ (]aiutillions of pounds. But of the magnitude of iliis sum, spoken of in the abstract, no just conception can be formed When, however, by a further calcMdati(jn, we have ascertained that t* coin such a quantity of mmiey (were it possil)le) in sovereigns of the [u-eseut weight and fineness, W(j(dd require GO, 308170 solid globes of t;old, each as large as the earth, we are enabled to entertain a more ade« •mate idtsa of tlio sum whoic vastness, without having recourse to this Hdacititious assielaace, placed it almost beyond the reach of our li? ited understandings. The amount at Simple Interest, for the same penod, would be only b" lQ%d. — EiuTon. • 1 059x2a5=157G2r)X2ii5 = 2G0-405G2.'>=.C2C0..9..3i Ant' ^ •?60' 465625 250-465625 ^.^ . rT— =»-rTrrTrr;r-=»^22o. Ant. \ OV 1157C2.0 I I )' t44 COMPOUND INTEREST [tutor's , (8) What principal will amount to jC596..5..2-232075d. in | 4 years, at £4^ per cent, per annum ? Ans. jE^SOO. Case 3. When p, a, and t are given, to find r. the root ot whichi, being extracted, will give r. log. p -f- < = log. r. k VLB. P Or, lo g. a (p) At what rate per cent, per annum will jC225. amount to jE;2feo..9..3f . in 3 years ?• (10) At what rate per cent, per annum will i^200. amount to je243..2..0..r2 qrs. in 4 years ? A71S. £5.. per cent. (11) At what rate per cent, per annum will £450. ?imouti't o i:547..9..i0..2 0538368 qrs. in 5 years? Ans. £4. per cent. (12) At what rate per cent, per annum will jC500. amount to £596-2593003125 in 4 years ? Ans. £4^. per cent. Case 4. When p, a, and r are given, to find i. a P which being continually divided . by r til' rif remains, the number of the divisions will bv: to t. a Or, log. a — log. jj-rlog. r—t.i (13) In what time will jC225, amount to i:260..9..3f at £5. per cent, per annum ?| (14) In what time will £200. amount to £243.. 2-025^, a £5. per cent, per annum? Ans. 4 years. (15) In what time will £450. amount to £547..9..10 2*0538368 qrs. at £4. per cent, per annum? 4^-s- 5 years. ., (16) In what time will ,£500. amount to £5D'6-2593003125 St £4^. per cent, per annum ? Ans. 4 years. THE AMOUNT OF ANNUITIES IN ARREARS. f^prx. u represents the annuity, pension, or yearly rent; a, r, and ( •8 before. • 2;52^!^^'=l-157625,andVll57625=rl05,or j:5.jjfirc«n/. Am t In all cases of this nature t cannot be found without LogarithmSr odesB it be a whole number. .260-465625 , ,^.,^„^ M57625 , ,„„^ M025 , ^^ 1-05 ,. t —— _=I.157626;_^— ==M025;..^:^=105;^«1- the nnmber of diviskins being three, which gives the time sought 9 3 years. Am. I'll !■ iul.'Jl ^f.i. tTUT0R*8 I iiSSlSTANT.J COMPOUND INTEREST. 14i? .2-232075rf. in Ans. i:500. find f*. •d, will give r. 4 jC225. amount JE^200. amount £5. ..per cent. £450. §imouh't s. £4. per cent. jE^500. amount 4^. per cent. ) find t. ...by r til; nr. ;•'-*>•, 7 ionavfiU !>; c. a je260..9..3|. at ^ :243..2025^. a Ans. 4 years. JC547..9..10 Ans. 5 years. 0'6:2593p03125 Ans. 4 yfitffj. SARS. y rent ; a, r, uid \ ^wS. pBT eent. Ant thoat Logarithms, 5 -.1.05. 125-1. — .105,— -.1 ) tima aoagbt ts 3 ••■■=! A Table oj the amount of £\. annuity for years. Vrs. T n 09000 4-1836270 5-3091350 3^perCrr ('rnf.. l-0()0(mO(!i 2-04U0n!l!); 3-121fiG0l): 4-2U;4(J40: 5-4l(;3v>-2»i (r550l52-:; 7 •7.7940751 9-051(ifUiri' ()-4()84099 7-6624022J •8r89233()l| 1:0-1591062| 10-3()8495;j| 11-4633794| 11-7313932 13^17 !)?)20 14-n019G17' KM 13030) 12-8077958 14-1920297 15-617790G 17-08()324>3' 17-(i7(;9f)(;5 lSr59S9140| 19-295(i810 204508814 i '20'-97l 02!)^ 21-7(515878 22-7050158 23-4144334' 24-499n9 14 25-1 168685! 2« -357 i80H 26-87037451 2^;2796819 '28''-764857j 30-2694708 30-.! 3678031 32-3289023 32- 528837 1 34-4604439 34-4!364702! 3r)fUi';.;284' 36-4592643' 38-9493569 38-5530422} 41 3131019 40-7096335: 43-7590605 42-9309225, 46-290(i276, 45-2188502 48-9107996 47:.57.5'll57! 51-6226776 36 37 t'8 39 40 50-0026782|54-'12947i:} 52-5027535! 57.-3345C28 550778412; 60-3412104 57-7301764:63-4531527 60-46208.17| 66-6740130 63-277)9441 1^ (V0i')7 60'3 4 : 66-1742-224 73-4573695^ 69-1594490 77'0288949 72-2342324: 80-724906-3 75-4012593, 84-550277!». 6 (i:?2n75'; 7-89329i5: 9-2J422ii3 10-582795:?: 12-0061071; Kn!]r)35N 15-0;25P,05")' ]|{-29inil2. 20-0235",7(; 2M;24^mi' 23-(i!>75l-^o 25-()45il28 27-6712293' 29-7780735 3ri)(5920i6! 34-24796971 36-6ivii:;n5 39-082(;010^ 41'64590'i2 44-3T{7445i '17-0842143, 4;)-:l675;;2fi' 52-96(i28'r2 5()0349376 '5'>-32« 3:551 62 70l4::3:v 66^2095272: 69'35/9(-33; 73-65222 )(): 77-5983 I :<.; 81-7 022 16 I' 85-97 C335n* 90-409 149:> 95-02551 5':, iS fry Crnf. "1-(V)00()00 2-0450000 3-J::7025!) 4-2r;n9i! 5--i7070'^7 ~7r7T689TTr 8-0191517 9-3;;ooi35 10-8021141 12-2;J82(192 15-4640316 17- 15;):) 130 18 9:V2l09l 20-7!M0540 "22-7i9:53fy^]l 21-741706() 26-i;550S34 29-0635r;22 31-3714225 "'3:5-78^^565 36-303:57 77 38-9:570297 41-6891960 44-5652098 T7~7i70'()i'i3" 50-711:52:53 5:5'9!):5:5:5;]9 57-!23n3:jn .';!-0070t;!)4 T4-752:i;!75' 6fi-6662449 72-7562259 77 -OS 02561 8]-4!)66lr6 lurT(i3li'65-r 91-0 113 139 96-1332044 10.1-46442.,6 107-0 503227 5 jter Cent. 1-0000009 2-050()U00 3-1525000 4-3101250 5-5256312 6-8019128 8- 14200:54 9-5491038 11-0265642 12-5778924 ~l~r20iT78'70 15-9171264 17-712!t3-27 19-5in56318 21'5785()34 "t^W;57¥<>16' 25-8403662 23-1:52:5845 30-5390037 33-065!)539 '"?-rr7l925l6 33-5052142 41-43047 19 44-5019986 47-7270!)35 "51.^1134534 i4-6691261 58-40J5;!24 62-:52271I5 66-4388 171 T0-7(i07"895 75-2988290 80-0637704 85-0fi6!)589 90-:52()3068 "95^5363221 101-6281382 107-7035451 ] 14-()!*50224 i 120-7997735 NoTK. The preceding 'Hible is fonnefl ihu? : tlic tirsf yoarV nmonnf lP:i)l.. ajid 1 X 1-05 -f- 1 --= 2-05, the RrOfMinI \r,(r\-. aiuiMi.i! ; 2-0v> X I'Od -f 1 = 3 1525, th*:" tlunl ^.-orn-".- nnr-tTiiit. vS.c. 146 COMPOUND INTEREST [TUTOR'S Case 1 . When u, t, and r are given to find a. r« — 1 Rule. -Xtfaso. ;• — 1 Or, by the Table. Multiply the tabular amount of £1. annuity by tlie given anunity, and the product will be the amount required. (17) What will an annuity of i^50. per annum amount to in 4 years, at £5. p&Y cent, per annum 1* (18) What Avill a pension of jC45. pdr anntim, payable yearly, amount to in 5 years, at £5. per cent, p^t a'ntiura? A71S. £248.. rSs..3^i7 qrs. (19) i[ nn annual salary of .£40. be forborne & feaiS,- at jC4. per cent, per annum, what is the rfmouW^? Ans. i;2G5..6..4..2-25775616 qri^ (20) If an annui'iy of £75., payable yearly, be Omitted id be paid for 10 years, what is the amount at £5. per cent', pti annum? Ans. i;943..6.. 10-0656 J.-j- Crse 2. When «,• r, and / are given, to find U. Rule. -, — - x o = «. (21) What annuity, being forborne 4 years, will amount id jC215..I0..1i. at £5. per cent, per annum?! (22) Wh;it pension, fcfi'borne 5 years, will amount to' jC248. 13.s\.3-27 qr.s\ at £5. per cent, per annum? Arts. £45. (23) What salary, being omitted to be paid 6 years, wi'ft amount to i;265..6.'.4..2-2577561G (rrs. at £4. per cent. pftJ annum ? Ans. JC40. (24) If the payment of an aimtiity, being forborne 10 yeai*?, amount to jC943..6..10-0656(f. at £5r. ptT tttii. per annuii¥, what is the aunuitv? Ans. £75. t'* ■I i ' Li.l Case 3. \\'hcn //, a, and r are given, to find t. * — -'irp -^ X iO « ( I •V15.')0625 — 1 ) X 1000 « 215-50C?5 »= X^lf . 10.. li- An$. (h, la the Tubh\ thus : 4-3'10'105 X 50' « i5215'50<)y5, M beR*iv. t [TUTOR'i nd a. £1. annuity by equired. im dimount to mm, payable fer alnlnrai ? s..3'2'7 qrs* e 6 yeart,- at 75616 qri. be dmitted id per cent', pfti 0-0656J.+ find it. (vill amount id ount to' JC248. Ajis. jG45. 6 years, w??! per cent, pfel 7ln.9. jC40. lorne 10 years, t. per annuii*, ^«j. je75. find t. iisiSTANT.] (r-l)a COMPOUND INTEREST* 147 Bulk. u which being continually divided by r till ■{-l = r*, nothing remains, the number of divirioni will be equal to t. (25) In what time will jC50. per annum amount to JC215..' 10..!^. at £5. per cent, per annum, for non-payment?* (26) In what time will je45. per annum amount to i&248 . 13j..3*27 qrs. allowing £5. per cent, per annum fdr ftfrbear- aiice of payment ? Aiis: 5 yeiars. (27) In what time will £40. per arihum amount to £265 . 6. ;4.. 2-25775616 qrs. at £4. per cent, per annum? Ans. 6 years. (23) In what time will £75. per annum amotmt to £943.i 6..10-0656(f. allowing £5. per cent, per annum for forbeairahco of payment? Ails. 10 years. Note. The examples relating to the Present iVortJi of Annuities at Simple Interest are now expunged from this work, because, bein^ en- tirely useless, except as d mere drithmelical exercise, it is presumed ihai the judicious teacher will pi-efcr the substitution of other matter of moroi i-enl utility which is introduced to supply tlieir place. The Theoremi are, however, retained in a note, page 148, in order that thfe ingenious" student, who may wish to calculate any example both ways, may have an opportunity of indulging his curiosity, and of compai-ing the true and the false results. That the principle of computing their Value by Sim* flic Interest is erroneous and absurd, will be manifested by the follow- ing observations : — The present worth of an annuity of £150., to. continue only 40 years; calculated at j£5. per cent, per annum, 'jimple Interest, (by Theorem 1,' Note, page 148,) would be £3950. but this sum, put out at the same rate, will produce £197.. 10. annual interest (or £47. .10. a year more than the proposed annuity ) /or ever. If computed on the true principle^ (by the Theorem, Case 1, page 149,) the present value is £2573..17..3, The present value of any perjpctual annuity (great or small) computed 8,1 Simple Interest, is an ti?ifimited or infnite siim; but by using Comi^ jpovnd Interest, we shall obtain a rational result. For instance, an an- nuity of £150., to continue /or ever, will, (by Case 1, Perj)etual Annui- ties, page 152,) at £5. per cent., be worth £3000. purchase; which, \i is evident, is the sum that ioill yield £150. annual interest. — EDitoii. 50625 "=X21-t. j25, M b«R>r». \) j^ jeSO. Alii- 0-5 X 21550625 50 -f- 1 = -001 X 215-50625 + 1 = 1-21550625; whici 4 being continually divided by 1-05, the number of divisions will bi 4; I the years required. Ans. m COMPOUND INTEREST. [TUToit'J •■■J I I if THE PRESENT WORTH OF ANNUITIES* A Tabic of the present worih of XI. annnifi/ for years. Yrs. 26 27 28 29 3d 31 32 33 34 35 3 r 3 10 3 per Cent. 1 -.9 134697 2-JJ286114 3-7170084 4-5797072 5-4171915; 6-230-2,'5;30; 7-019()922 7-7f}{)108!> 8-530-2n2!> 9-252G241 9-95101)10 10-f):M!)55:5 U •2900731 11-9:379350 12-5011019 13-1061183 13-7535129 14-3237989 14-8774747 IhperCent, 0-966 ]!{36 1-8996943 2-8016370 3-6730792 4-5150524 15-4150240 15-9369165 16-4436083 16-9355420 17-4131476 5-32855:30 6-1145110 0-8731)556 7.-6076866 8-316()054 ~]H)015:311 ,9-6(i:3:i.?44 10-302738: 10-920520: 11-5174109 'l2^0'94lT68 12-6513206 13-1896817 13-7098:374 14-2124033 l4-(79797'T2 15-1671248 15-6204104 16-0533675 16-4815145 1 per Cent. 0-9615:385 1-S8()0947 2-7750910 3-6298952 4-4518223 5-2421368 6-0020546 6-7:327448 7-435:3315 8'- 1108956 "876047^)5 9-38507:55 9-985647(i 10-5631227 Wfe.r Cfiit. 0-9569:378 1 -8720677 2-7489043 3-5875256 4-3899766 11-6522954 12-1656086 12-65929()7 13-13:3,9391 13-5903260 "lT029r5!)(i 14-4511150 14-3568413 15-2469628 15-6220796 5-1578723 5-8927.008, 6-5958859 7-2687903 y-9127180 8-5289167 9-1185806 9-68285-22 10-2-228251 10-73;)5455 5 per Cent. 0-9523810 1-8594105 2-7232481 3-5459506 4-:3294768 5 0756922 5-786373,5. 6-4632129, 7-1078218 7-7217351 8-3,064144 8-8632518 9-3935732 9-8986412 10-3796583 11-2340148 10-3377698 11-7071912 11-2740065 J2-1599916|ll-6895872 12-59:32934112-085:3210 13-0079363 12-4622105 I ;- 17-8768423 18-3-270314I 18-7641082 19-I384546j 19-6004414: 20 ' 20 20 21 21 21 22 22 f)0 16-89()3522i 17-28536441 17-66701871 18-0357669| 18-3!)20t53 00042861 18 3887656: 19 7657919! 19 ]3183(i8; 19 4872202, 20 832 2. 526, "^20 1672355; 20 4924017120 8082153121 1147721121 15-!»827()88 16-:3295854 16-66:30029 16-98:37143 17-2920330 13-4047237 13-7844240 14-1477747 14-4954782 14-8282038 12-8211529 13-1630028 13-4885741 13-7986420 14-0939448 •7362757 -0688654 -3!»02081| -700ii842 -0006611! •2T)()-i?)ai3: -57052541 -84108731 -10249981 17-58:M933 17-8735512 18-1476454 18-4111975 18 66 4()i:30 18' 19' 9032817 1425785 15-1466113 15-4513027 15-7428734 16-0218884 16-2888884 16-7888907 17-0228619 17-2467578 17-4610122 14-3751855 14-6430338 14-3981-274 15-1410737 15-3724511 15-5928106 15-8026768 16-0C25493 16-1929041 16-:3741944 17-6()60404 16-5468518 17-8622396J16-7112874 16-8678928 19-3(i786:39j 18-0499900 19-58448151 18-2296555 355072:8! 19-7.9277351 18-4015842 17-0170403 17-1590865 :.i If' ' '! jl * Present Worth nf Annuities at Simple Interest. thocr. I. ^^.=:n±i 2tr-\-2 'Xta-p. II [TUToit'li TIES.* ?r years. 5 per Cent. 8 7 13 iG i6 23 59 03 m) 67 Of) •2--> 51 55 Of)5'23810 1-8594105 27-232481 3-5459506 4-3294768 5 0756922 5-7863735. 6-4632129, 7-1078218 77217351 8-3064144 8-8632518 9-3935732 9-8986412 10-3796583 1148 10-8377698 •2740665 •6895872 12-0853210 12-4622105 12 13 13 13 14 )12 11 )16 11 !)34 363 237 216 747 782 088 ri3 027 734 384 884 mis 907 ;619 57 [; 1122 iT04 1396 821155J9 1630028 •4885741 •7986420 •0939448 U-3751855 14-6430338 14-8981274 15-1410737 15-3724511 15 15 16 16 16 i7j 16 5928106 8026768 0C25493 •1929041 3741944 54685 18 •7112874 900' 16-8678928 17-017040R 17-1590865 i842 tntsrest. + 2).< X2/ = ». .* ASSISTANT.] COMPOUND INTEREST, 149 Note. The above table is thus formod: jCl.-f-l -95 = -9523810, the present worth of the first year; thi8 H- 1-05 = -9070295, which, added to the first year's present worth, gives 1-8594105, the present worth of 2 years; then -9070295-;- 1 05, juid the quotient added to 1-8594105== 2'7232481, the present worth of 3 years, &x,. j(/'ase J . When w, t, and r are given, to findp, the present worth, 11 Rule. (« f)-r(»' — ^)=P-' Or, f> ■ th" given the required. ble. I\Iultjply the tabular present worth for the time ^::n annuity, and th-^* Di-oduct will be the present wortlj (29) What is the present worth of" an annuity of jC30., tc^ continue 7 years, at jC5. per cent, per annum?* (30) What is the present worth of a pension of £40. per annum, to continue 8 years, at £5. per cent, per annum 1 Ans. £258..10.6..3 264: qrs. (31) What is the present worth of an annual salary of jC35 , to continue 7 years, at £i. per cent, per annum ? Aihs. X*210..1..5-04ff. (32) What is the yearly rent of £50., to continue 5 years, worth in ready money, at £0. per cent, per annum ? Ans. £2ie..9..5..2 08 qrs. |lULK Case 2. Whenp, t, and r are given, to find m. pr^ir—l) r*—l ■ u. (33) If an annuity be purchased for jC173..11..1008c?., tp III. IV. Tut (^« — ;>)X2 _ (:Zp-{t-i)it).t -"*• \ tt 2 -^ ^ ru ' ^ For Annuities in Rcverfiion, it is only necessary to observe the Rules lot Reversiouai-y Aiuiuilies at Compound Interest, and to csiiculate (ac- cording to tlie directions therein given) by the Theorems for Simplo interest. • 30 30 1-057 30 ^1- 1-4071 30 — 21-3204 = 8-6796 and 8-6796 ~ -05 == X173-592 = i;i73..11..10-08i. Aiis. pr.hythe T'aJ/f, 5-7863735 X 30=i;i73-591205. Ana. m COMPOUND INTEREST. [TUT03»# I ( ) < be continued 7 yo jrs, at £5. per cent, per annum, what is tho finpuity ?♦ (34) If ^^58. .10. .6. .3-264 qrs. be paid doxyn for a salary 8 years to come, at jC5. per cent, per annum, what is the salary? Ans jG40. (35) If the present payment of jC210..1..504^Z. be required for a pension for 7 years to come, at £'i. per cent, per annum, what is the pe..oion ? Ans. jC35. (36) If the present worth of an annuity, 5 years to come, be JC216..9..5..208 qrs. at jC5. per cent, per annum, what isf ^e annuity ? Ans. £50. Case 3. When u, p, and r are given, to find t. Bulb. u Pi- u- which being continually divided by r til] — — = »-*» nothing renuiins, the 7mmbcr of divisions P^ will be equtU to f. (37) How long may a lease of jC30. yearly rent be had for jC173..11..1008(]?., allowing £5. per cent, per annum to the purchaser ?t (38) If je258..10..6..3-264 qrs. is paid down for a lease of jG40. per an'^ im, at £5. per cent, per annum, how long is the lease purch ;d for ? Ans. 8 years. (39) If a house is let upon lease for jC35. per annum, and the lessor disposes of the leasp for JC210..1..504J., allowing after the rate of £4. per cent, per annum, what term of the lease remains unexpired ? Ans. 7 ye^rs, (40) For what time is a lease of jC50. per annum purchas- ed, when present payment is made C(( jG216..9..5..2'08 qrs,^ at £5. per cent, per annum ? Ans. 5 years. 4'- \']-l ', ANNUITIES, ETC., IN REVERSION. To find the present worth of annuities in reversion. Rttle 1. Find the present worth of the annuity, for the time of itt eontinuatf-c?, £^s if it wesre to commence immediately, by Case 1, page 149. Then find what principal will amount to that sum iu the givea ? 173-592X 1-4071 X -05 -r -4071 = 12-213 -r -4071 =£30. Ana. Or, by the Table, 173-592 -^ 5-7863735 = £30. Ans. t 173-592-1-30 — (173-592 X 1-05} = 203-592— 182-2716 = 21-3204; and 30 — 21-3204 = 1-4071 ; which being continually divided by 1-05, the number of divisions will be 7 : therefore t = 7 years. Ans. Or, b^ ^he Table, 173592 -f- 30 = 5-78G4 ; and referring to the colunm of 5 per ^rt, we find the number 5-7363735 against 7 years. !i ■« ] 'I n, what is the for a gajary what is the Ans je40. rl. be required It. per annum, Ans. JC35. ears to come, num, what isf Ans. £50. find t. ivided by r til] bcr of divisions 3nt be had for annum to the for a lease of )w long is the ns. 8 years, iv annum, and }4d., allowing it term of the IS. 7 yeOfTs, num purchas- '..5..208 qrs.y IS. 5 years. version. • the time of it$ y Case 1, ^age u) iu the giveq *£30. Ans, 716 = 21-3204; ivided by 1-05, . Ans. Or, f>^ :oluniu of 5 f er ASSISTANT.] COMPOUND INTEREST. lot time before the annuity commences, (by Case 2, Compound Interest, page 143,) which will be the present worth. Bulk 2. Find the present wortli of a similar annuity snppoBe«l to eommcnee immcdiatehj, and continue dui-ing the vholc period; and also the present worth of the same till ike thne when the r^evcisic-nary an- nuity actually commences, and the dijfcrenpe of these two will \)e the present value required. NoTB When calculating by the Table, tliis is the most eligibla method . I Rule 3. Find the amount of the annuity at the time of its cessation. (by Case 1, page 146,) and the present worth of that amount (l)eing found by Case 2, Compound Interest, page 143) will be the vidue re- quired. (41) What is the present worth of a reversion of a lease ot jC40. per annum, to continue for 6 years, but not to commence \]\\ the epd of 2 years, avowing £b. per cent, per annum to the purchaser ?* (42) What is the present worth of a reversion of a lease o» jC60. per annum, tp cpntinue 7 years, but not to comnienco till the end of 3 yeara, allowing £b. per cent, per annum to the purchaser ? Ans. £29^..]S.2-\Qd. (43) A house i^ let at jC30. per annum on a lease, of which 4 years arie yet unexpired, and which the lessee is desirous of renewing at the same rental, to continue 7 years beyond the tern) pf the present lease. What will the lessor expect as a bonus for such a renewal of the lease, considering the house to be worth double the present rent^ and allo\ying interest for the money now advanced at £b. per cent, per annum ? Ans. jei42..16..3..M52 qrs. To find the annuity in reversion which a given sum will pur? chase. RcLS. Find the amount of the given sum for the time hefore tlie anr nuity commences, by Case 1, Compound Interest, page 143, which will be the value of the annuity at its commencement. Call thia valiie p, and then find tiio aniuiity as in Case 2, page 149. (44) What annuity, to be entered upon 2 years hence, and 40 40 ■•=40 40 40 — 29-84361= 1O-15130; aixi 105« ~ ' 1-3400956 10-15139 r- -05 «= 2030278; then 2030273 — lOS^ =: £1841022 =- il84..3..0..2ll2 ^rs. Ans. 451? COMPOUND INTERESV. [tutor's ' ll If: ! f * Uli .J! i^J. then to coiitinup G years, may be purchased for JC184..3..0.. ^•112 f/rs. at XT), per cent, per annum?* (45) Tho present worth of a lease, taken in reversion for 7 years, but not to commence till the end of three years, is je299..18..2-10(/., allowing X5. per cent, per annum to the purchaser : what is the yearly rent ? Ans. jC60. (46) There is a lease that has yet 4 years to run, and tho lessee has purchariMl tho reversion of a renewed lease, at thd same rental of £'30. per annum, for the term of 7 years, com- rnencing at the expiration of the present lease, for which he has paid down X'142..10..3..ri52 grs. What increase of rent is reckoned on the proj)erty according to this contract, alloW' ing jC5. [)t^Y cent, per annum for present payment ? Ans. JC30. FERTETUAr. ANNUITIES, OR FREEHOLD ESTATES Case 1, When u and r are given, to find p, the present worth, or purchase money. , • Elui.E. — — -=«.t r — 1 (47) What is the worth of a freehold estate of jC50. yearly rent, allowing £o. per cent, per annum to the buyer ?^ • (48) What is a real estate of jC140. per annum worth in present money, allowing £1. per cent, per annum to the pur- chaser ? Ans. JC3500. (49) What must the purchaser give for a freehold estate of JC437..10. yearly reni, so as to make £3^. per cent, per an- ;num by the investment of liis capital ? Ans. £12500. Case 2. When p and r are given, to find u. Rule, p X r — l = u. (50) If a freehold estate is bought for jClOOO., what mus< * lS4-l.r22 X !• 1025 = 203-0278; then 203-0J78 X 1-3400956 X -0.5 2030278 ~ -3400956 -3400i)56 SQ3-02 78 X 1 -i- (203-0278 X -3400956) ^ 10a!)5G •-}- 203-0278 X -05 = 8000005 X -05 = jC40. Ans. Or, hv the Tatde, 184-1522 -;-(6-4632129— 1-8594105)= 184-1522-i- 4-6038024 = XM(). Ans. t This rule is dedaced from the formula in page 149 ; for, in Annoi ties coutinuiiig for ever, t is infinite, aiid the Bubtractive qnantity % *'-r*=.o; ihfirefore the theorem assumes the above form. ■ t 56— -OS^jCIOOO, Ans. [tutor'* jC184..3..0.i evcrsion for ee years, is nnum to the Ans. jC60. run, and the lease, at thd years, com- or which he rease of rent ntract, allow-^ Ans. i;30! TATES oresent worth, f jC50. yearly lyer ?:^ lum worth in im to the pur- ns. irasoo. hold estate of cent, per an* s. JC12500. id u. [)., what must 2030278 3400956 )=1841522-r ; for, in Annoi avd quantity h n. ^ I ASSISTANT.] COMPOUND INTEREST. J 53 be the yearly rent, to pay the purchaser £5. pey cent, per an- jftum interest for his money ?* (51) If an estate be sold for jC3500., what is the yearly rent allowing to the purchaser £i. per cent, per annum? Ans. jei40. {b'l) If a freehold estate is bought for jC12500., and will yield the purchaser X'3^. per cent, per annum, what is the yearJy^ rent? Ans. jC437..10. {lULE. Case 3. When;? and u are given, to find r. -f- 1 = r. u (53) If an estate of jC50. per annum is bought for jGIOOO., what is the rate per cent, per annum ?t (54) If a freehold estate of jC140. per annum is sold fo^ jp3500., what interest- will it pay to the purchaser? Ans. jC4. per cent. (55) If an estate in perpetuity of JC437..10. per annum i£| 5old for JC12500., what interest will it pay to the purchaser) Ans. £3^. per cent. TREEHOLD ESTATES IN REVERSION. To find the present worth of a freehold estate in reversion. RuLR. Find the value of the estate, supposing it were to come intoj {mmediate possesnion, as in Case 1, i)age 152. Then suppose that valuQ (p) to be a, and find what principal will amount to a in the iirne to come, previous to possession, by Case 2, Compound Interest, page 143. Sucli| principal will be the present value. (56) What must be paid down for the purchase of a freer hold of jC50. per annum, to be entered upon 4 years hence, allowing the purchaser at the rate of £b. per cent, per annuir^ for his purchase money ?| (57) What must be paid down for the reversion of a rea^ estate of £200. per annum, so as to pay the purchaser jC4. per cent, per annum for his capital, supposing 2 years to elapsei before ^he estate comes into possession ? Ans.£^&22..lb..'7..\-lQqrs. • 1000 X -05 = £50. Ans. + 1 = 1'05 = £5. per cent. Ans. _50_ lOOU X 50 -^ -05 = 1000 ; then 1000 — 1:2155 = £8227067 = £822..l4..^., ^•432 qrs. Ane G3 ' 1 I » w OOMPCVND INTERB8V. (tutor a ; I ( r i ' .11 f ■ ■ " ; • ■ (58) A freehold, producing jC280. annual rent, is to be dis- posed of, with a reserve of the next 3 years' rent to the pre- sent proprietor. What is it worth in ready money, allowing £3^. jJer cent, per annum to the purchaser ? Ans. ^7215..10..9..3-36 qrs. To jind the yiearly rent of an estate in reversion, having its present value given. Bulk. Find the amount of the given present value in the time before po$set»ion : thus, pr* := a. Then consider that amount to be the present Value (p) of the perpetual annuity, and find the annuity thus: pX (r — !)=:». (59) What must be the rent of a freehold property, to come into possession 4 years hence, for which jC822..14..1..2-432 ors. is paid down, allowing the purchaser i7d. per cent, per pinum?* (60) A freehold estate is sold for jC4622..15..7..1-76 grs.^ ^e vender reserving to himself the first two years^ rent. Re- quired the annual value to pay the purchaser £4. per cent, per annum for his capital. Ans. jC200. (61) A freehold estate has been purchased for jC7215..10.. 0..3'36 qrs., the possession of which is not to be given up till ffter the expiration of 3 years. What must be the annual lent to pay the purchaser at the xate of Jp'S^. per cent, per annum ? Ans. £280. DISCOUNT, ON THE PRINCIPLES OF COMPOUND INTEREST.f Note. The following Table is constructed by the continual division |)f 1 by the ratio (r) : thus 1 -H 1"05 = -0523810, the first year's present inrorth; then -9523810 -h 1-05 = -.9070295, the second year's present Urorth; and •9Q70295-r 105 = -8638376, the third, &c. ' 822-70625 X 1-2155=1000; then 1000 X -05 = :e50. An$. t This is merely a repetition of the vai-ious cases in Compound In* lerest. For instance, to find the present worth of any debt due some iime hence, is precisely the same operation as findhig what principal wili amount to that sum in the given time ; and this obHervation will fsqually apply to the identity of the other cases. The entire omission, therefore, of Discount, (at Covipound Interest,) arranged under that spe- pific head, would be no detriment to the learner. It is, however, re-, fained here, for the sake of those who may think some repetition jf thai |ifbjject desirable. — Editor. I f: \ 'Li' '%i [tutor • to be dis- to the pre- , allowing •36 qrs. having its time before the present thus : j> X ty, to come U..1.. 2-433 r cent, per ..1-76 grs.f rent. Re- . per cent. s. £200. £:7215..10.. fiven up till the annual )Y cent, per IS. ^280. TEREST.t inual (livisioQ f^ear's present ear's present Ans. ompound In* ;bt due soma hat principal ervation will ire omission, ider that spe- however, re», etitioQ ^f th* ASSISTANT.} COMPOrXD INTEREST. 1^5 A J'flhU of the pretent worth of XI. dve ar/y numhcr of ycart hrnce from 1 to 40. Yrs. 3 per Cent. ^jjerCrnf. 4 per Cent. ■i\pcr Crnf.\^) per Cmt. 1 2 3 4 5 •97087:38 •9425959 •9151417 •8884870 •8G26088 •9(i61836 '9335107 •9019427 •8714422 •8419732 961538.5 •9245562 •88399«3 •8548042 •8219271 9509378 9157299 •8702966 3335613 ■30-J4510 9.523310 9070295 8fi3rj376 8227025 7335202 6 7 9 10 •8374843 •8130915 •7894092 •7664167 •7440939 •813500G •7859910 •7594116 •7337310 •7039183 •7903145 •7599173 •730ri902 •7025367 •6755641 *7ti789.')7 •7343235 •7031351 •6-29044 ■6439277 7402154 ■7100313 C703394 ■6446089 ■6139133 11 12 13 14 15 •7224213 •7013799 •6809513 •6611178 •6418619 •6849457 •6617833 •6394041 •6177818 •5968906 •6495809 •G24;i970 •60057'' : •5774751 •5552645 •61G1987 •5896639 •564'^-16 •53r '729 ■n-'i?j04 •5840793 •5.568374 ■5303214 5050680 4810171 16 17 18 19 20 •6231669 •6050164 •58739^^6 •5702860 •5536758 •5767059 •5572038 •5383611 •5201557 •5025659 •53390^2 •5133 3;: •493ri81 •4746424 •4503869 •4.^44093 ■4731764 •4528004 ■4333018 •4146429 4581115 i3G2967 415.5207 3957340 •37 68895 21 22 23 24 25 ' •5375493 •5218925 •5066918 •4919337 •4776056 •4855709 •4691506 •4532S56 •437.0571 •4231470 •4388336 •4219554 •405r'lf^3 •3901215 •3751168 •39()7874 •3797009 •3633501 •3477035 •332730S ■3589424 ■3413499 ■325,5713 3100679 2953023 26 27 98 Q9 30 31 32 33 34 35 •4636947 •4501891 •437Q768 •42434G4 •4119868 •4088377 •3950122 •3316543 •36P''482 •35''": '• i ■3606892 3468166 3334775 3006'514 3033187 •31C4025 ■3046914 •2915707 2790150 •2670000 2812407 2673483 ■2550936 2429463 2313774 •3.099872 •3883370 •3770263 •3660449 •3553834 •3442304 •332.') 8 97 ■.]2 134-27 •3104761 •2909769 29()4(;03 2350 ;>7 9 2740942 2635.321 2534155 255.-(hM 2J4-!nf)f) 2339712 2238959 2142514 2^03595 209ii')^T2 1 .^^98725 1903543 1812903 36 37 38 39 •3450324 •3349829 •3252262 •3157536 •3065568 •23983'27 •2800316 •2705619 ■2614125 •2525725 24301)87 2342968 22.52854 2166206 2082390 21150^282 I'lninp^ 1877504 I7rt^'l.f.5 r ■f'J;7 1720574 1644356 15(^f)054 1491480 14204.57 a 5a CCMPOUND INTEREST* (tutor'«, I ) > ;■. Case 1 . To Jlnd the present viorth. of any sum due after a eer tain period. RuLK. The same as in Case 2, Compound Interest, conaiderujg a oB the debt whose present value is required. (1) If i'344..14..9..1-92 grs. be payable in 7 years' time, •hat is the present worth, discount being made at £5. pdf •3nt. per annum ?* (2) A debt of .-€409. .9-009925.. payable 4 years hence, U agreed to be paid in present money : what sum must the creditor receive, discounting at £4. per cent, par annum t Ans. JC350. Case 2. To find the debt whose present worth is given, KuLE. See Case 1, Compound. Interest. (3) If i^245, be received for a debt payable 7 years hence, allowing £b. per cent, per annum to the debtor for present payment, what is the debt ?t (4) There is a sum of money due at the expiration of 4 years, but the creditor agrees to take jC35Q. in ready raoney^ allowing £\. per cent, per annum discount. Whav was the debt? -4.nj. je409.. 9-009925. • Case 3. When the rest are given, to find the time. Rule. See Case 4, Compound Interest. (5) A person receives £245. now for a debt of ^^344.. 14. .9.. 192 qrs., discounting at £5. per cent, per annum: in what time was the debt payable ?| ' (6) There is a debt of i;4 09.. 9-009925. due a certain time hence, but £4. per cent, per annum being allowed to the debt- or for the present payment of JS350., it is required to find in what time the sum was to be paid. Ans. 4 years. Case 4. When the rest are given, to find the rate per cent. Rule. As in Case 3, Compound Interest. (7) The present worth of je344..14..9..1-92 qrs.y payable '(r * * 344-7395 — 1 •4071 = . £i24r). Ans. Or, by the Tabic, -7100813 X 344-7395 = £245. Ana. t 245 X 1-4071— ■X'.^44-7395 = X^344..14..9..1-92 qrs. Ans. i 344-7395 4- 12 15 — ] •407 1 ; the cM^nHnual divisions of which by 1'05, will be 7 i= the iiuuibci of yuai-6. Ans. J^39ISTAXT.] DUODECIiHALS. 15T 7 years hence, is jC245., at what rate per cent, per annum is discount allowed ?* (8) There is a debt of i:409.. 9 009925. payable in 4 years, |)ut it is agreed to take dC350. present payment. Required ioe rate of discount. Ans. £i. per cent. EQUATION OF PAYMENTS AT COMPOUND INTEREST. KoLE 1. Find the present worth of each payment respectively! an^ add them together for the whole present worth : then the time in which that present worth will amount to the sum of the debts will be the true exulted time required. 2. Find the amount of each debt from the time of its becoming due till the time of the last payment, and add the respective amounts and the last payment into one sum. Then find the time in which the suni of the debts would amount to that sum of the amounts: subtract ikid |(rotn the time of the last payment, and the difference wiU be the true tquated time. • < (1) Required the true equated time for the payment of a debt of jC400. of which jC320. is now due, and the rest at tjiei end of 5 years ; reckoning compound interest at the rate oif £5. per cent per annum. Ans. •90714 years. (2) If jClOO. will become due one year hence, and jC104. three years hence, what is the true equated time for payment of the whole, allowing compound interest at J^4. per cent, per annum? - 'ns. 2 years. (3) If a person will have to receive JE^200. at the end of 3^ years, and jC80,. more at the end of 5 years, in what time ought he to receive the whole at one payrtient, allowing £5 per cent, per annum, compound interest ? * Ans. 3' 551B years. X DUODECIMAL'S Are so named from the integer of each denomination contati\- ifig twelve of the next inferior. They are in general use, among artificers for computing the quantities of their materials and labour, both in Superficial and Solid Measure.^; • 344-7395 ^245 »= 1-4071 ; and V 1'4071 = 1-05 ; which gives £«. per cent. Ans. t For a clear and intelligible oxplaiiation of the different M«a4ure«i Bee the Tables, page 24^ &c. ■•'•^' lw DUODECIMALS. [tutor'* ' . I ^ ■f 1 ^2 inches (' ) make 1 foot. }2 seconds (" ) 1 inch, or prime. 12 thirds ("') 1 second, &c. To multiply duodeeimally . (tuLE 1 Under the multiplicand write the conesponding terms pf tha multiplier. 2. Multiply by the feet ^n t]|p multiplier, ohgprving tp earfy one for every twelve, from each lower deuomiuation to the next superior. p. In the same manner multiply by the inches in the multiplier, 9pU tjng the result from each term one place farther to the right. 4. Proceed in like manner with the remaining denominations, and the •um of the products will be the total product. Note 1. Length and breadtli multiplied together produce the area of a superficies f and tliis multiplied py the tl^icl^ess, produces the solid content of a body. 3. It is generally more eligible to take allqvot parfi out of the mnlti plicand for the inches, &c., in the multiplier. 1) Mult. 2) Mult, 3) Mult. 4) Mult. [5) Mult. lei) Mult. (7) Mult. (8) Mult. (9) Mi|lt. (10) Muit. (11) Muit. (12) Mult. (13) Mult. (14) Mult. hd) Mult. 259 2 (16) Mult. 257 9 (17) Mult. 311 4 (U) Mqlt. 321 7 7 9 8 5 9 8 8 1 7 6 4 7 7 5 10 4 75 7 97 8 57 9 75 9 87 5 179 3 // by by by by by by by by by by by fi- 3 4 7 3 5 3 3 7 9 8 9 n 6* 7 6 5 9 10 5 8 8 9 5 7 8 fi- Ans, 7 3 by 17 by 35 by 38 :0 by 48 11 by 39 11 by 36 7 5 by 9 3 6 a. a. a. a. a. «• a. a. a. a. a. a. a. a. a. a. 38 72 27 43 17 25 79 11 730 7 854 7 543 9 1331 11 3117 10 6960 10 12677 6 10288 6 U402 2 2988 2 I 6 11 6 7 5 1 6 6 10 8 6 8 9 3 4 6 10 3 4 10 /// fff 2 6 3. 6. 11 4 11. 6 3 6 ll3~3 ^ 10 6" in. Otherwise 3 23 3 3 10 6" Pronf by Decimnlt. 775 3-5 3875 2325 27 1 6 or. 125 sq. feet if 'j ^M- ■!,(■■; 48BI8TANT.] OrODSCIMALS. ^50 g teriM pf tlia eaxvy one for iviperipr • nultiplipr, gpt- ght. itioDt, and the duce tbe area produces the at of the molti f -^ /// in 6 11 6 7 5 1 6 6 10 8 6 2 3. 1 6 6. 7 8 7 9 9 1 3 10 4 10 6 6 10 6 3 2 4 11 11. 2 10 4 6 Glazings Masons' flat work, and some parts of Joiners^ work, are computed at so much per square foot. Painters\ Plasterers', Pavers', and some descriptions of Joiners' work, are estimated by the square yard. Roofs, Floors, Partitions, <^c., by the square of iOO feet. Bricklayers' work by the square rod, containing 272^ feet (19) A certain house has 3 tiers of windows, 3 in a tier ine height of the first tier being 7 feet 10 inches, the second 6 feet 8 inches, and the third 5 feet 4 inches, and the breadth of each w^ndoAV is 3 feet 11 inchas. What will the glazing cost at 14«?. per scjuare fpot?* (20) What is the price of 8 squares of glass, each measiiTr ing 4 feet lO inches long, and 2 feet 11 inches broad, at 4|^. per square fooji ? Arts. jG1..18..9. (21) What is the value of 8 squares, each measviring 3 fee^ by 1 fbot 6 inches, at 7|d. per square foot ? Ans. JC1..3..3. ' (22) What is the price of a marble slab 5 feet 7 inches j^ong and 1 foot 10 inches broad, at Qs. per square foot? Ans. J^3..1..5. (23) What will be the expense of ceiling a room the length of which is 74 feet 9 inches, and the wicllh 1 1 feet 6 inches, Sjit 3j. lOirf. per square yard? Aris. JC18..10..1. (24) What will the paving of a court-yard cost at A^d. per square yard, the length being 58 feet 6 inches, and the breadth 54 feet 9 inches ? . Ans. £7,. O.AO. (25) The circuit of a room is 97 feet 8 inches, and the height 9 feet 1 inches : what is the charge for painting it, at 2j. 8|rf. per square yard ? Ans. jC14..11..2. (26) What is the expense of a piece of wainscot 8 feet 3 inches long and 6 feet 6 inches broad, at ds. 7^4- ppr squar« yard? ' Ans. i:i..l9..5. ft. in. * 7 10 6 8 5 4 in. ft. in. 6— i 19 10 11 19 10 the whole height. 3 11 3 in. 218 2 3=i 9 11 4 11 6" \l 9 the whole breadth 1= » 233 6 at Ud. d. 2 = ^£11 13 the value at 1#. 1 18 10 the value at 2<|, value of 6" —000^ £13 11 lOi Ant. ^eo DUODECIMAL^. [TUTQH^f I .. (27) What will the paving of a court-yard cost at 3^. 2d. per square yard, the length being 27 feet 10 inches, and the preadth 14 feet 9 inches? Arts. £7. A. .5. (28) A certain court-yard is 42 feet 9 inches in front, and 08 feet 6 inches long ; a causeway the whole length, and 5 feet 6 inches broad, is laid with Purbeck stone, at 3.y. 6d. per Square yard, and the rest is paved with pebbles, at 3^. per square yard. What is the expense ? Ans. jC49..17..0^. (29) What will the plastering of a ceiling cost at lOd. per square yard, supposing the length 21 feet 8 inches, and the breadth 14 feet 10 inches? Ans. jC1..9..9. (30) What will the wainscoting of a room cost at 6^. per square yard, supposing the height of the room (including the pornice and moulding) is 12 ifeet 6 inches, and the compass 83 feet 8 inches ; the three window shutters each 7 feet 8 inches h' 3 feet 6 inches, and the door 7 feet by 3 feet 6 inches ? The shutters and door being worked on both sides, are reck- oned work and half-work. ' A7is. £36. .12. .2^. (31) In a piece of partitioning 173 feet 10 inches long, and JO feet 7 inches in hsight, how many squares? • Ans. 18 squares, 39 feetf 8' 10". (32) A house of three stories, besides the ground floor, measuring 20 feet 8 inches by 16 feet 9 inches, is to be floored at JC6..10. per square : there are 7 fire-places, two of which pleasure 6 feet by 4 feet 6 inches each, two others 6 feet by 5 feet 4 inches each, two others 5 feet 8 inches by 4 feet 8 inches each, and the seventh 5 feet 2 inches by 4 feet, and the well-hol^ for the stairs is 10 feet inches by 8 feet 9, inches. What will the whole amount to ? Ans. je53..13..3^. (33) If a house measures within the walls 52 feet 8 inches in length, and 30 feet 6 inches in breadth, the roo* being of a true pitch, what will it cost roofing at 10*. 6c?. per square ? Ans, jei2..12..11^. NoTB. A roof is said to be of a frue pitch when the rafters are | of the breadth of the building. In this case, therefore, the breadth must li>e accounted equal to the breadth and half-breadth of the building. (34) What will the tiling of a barn cost at 25s. 6d. per square, the length being 43 feet 10 inches, and the breadth 27 feet 5 inches on the flat, the eave-boards projecting 16 inches on each side ? Ans. jC24..9..5f . Note. Bricklayers oompute their work at the i3)t« of a brick aad Itutqr^^ St at 3s. 2d. les, and the £7.A..5. in front, and ngth, and 5 i 3s. 6d. per s, at 3*. per 19..17..0^. t at lOd. per les, and the i:i..9..9. St at 6^. per ncluding the compass 83 'eet 8 inches et 6 inches! as, are reck- 36..12..2|. hes long, and ?c^ 8' 10". ground floor, ; to be floored wo of which ers 6 feet by s by 4 feet 8 y 4 feet, and 3 by 8 feet 9, 53..13..3f feet 8 inches )o» being of a er square 1 2..12..11|. rafters are | of e breadth must he building. 25^. 6d. per i the breadth projecting 16 £;24..9..5f. of a brick aad ASSISTANT.] PUODECIMALS. 161 A half thick ; therefore, if the thickness of a wall is more or less, it ilnustb^ reduced to the standard thickness by multiplying the area of the wall by the number of half bricks in the thickness, and dividing the product by 3. ' " "^ *(35) If the area of a wall is 4083 feet, and the thicknesef two bricks and a half, how many rods does it contain of the standard thickness? Ans. 25 rods, 8 feet. (3G) If a gar'.n wall is 254 feet in compass, 13 feet 7 inche^ liigh, arid 3 bricks thick, how many rods does it con- tain ? ' Ans. 23 rods, 136 feet. (37) How many rods are there in a wall 62^ feet long, 1^ feet 8 inches high, and 2^ bricks thick? ' ' Ans. 5 rods, 167 fee^. (38) The end wall of a house is 28 feet 10 inches in length ; the height of the roof from the" ground is 55 feet 8 inches ; and the gable (or triangular part at the top) rises 42 courses of bricks, reckoning 4 courses to a foot. The wall to the height of 20 feet is 2i bricks thick, 20 feet more, 2 bricks thick, and the remaining part, a brick and a half thick'; and the gable is 1 brick thick. VVhat is the charge ipy the whole wall, at £5..l6. per rod ? Ans. je48.'.i3.!5^. To multiply several figures by several^ and obtain the product in one line only. Rule. Multiply the units of the multiplicand by the units of thd multiplier, set down the units of the piroduct, and carry the tens ; 'ne'x| multiply the tens in the multiplicand hy the units of the multiplier, td Which add the product of the units' of the multiplicand multiplied by thd tens 'in the multiplier, and the tens carried ; then multiply the ^imdreds ih the multiplicand by the units of the multiplier, ac^ding th« product of the tens in the multiplicand multiplied by the tiehs in tha multiplier, and the units of the multiplicand by the hundreds in iHe multiplier ; and so proceed till you have multiplied the multiplicand aH through, by every figure in the multiplier. ' Multiply . . 35234 by . . 52 424 Product 1847107216 KXPLANATION. First, 4 X 4 =t 16, that is 6 and carry 1. Secondly, (3 X 4J + (4 X 5) •nd 1 that is carried = 21, set down 1 and carry 2. Thirdly, (2 X 4) •f (3 X 2^) 4- (4 X 4) -|- 2 carried = 32 ; that is, 2 and carry 3. Fourth- ly, (5 X 4) + (2 X 2) +(3 X 4) -f (4 X 2) -{- 3 earned = 47 ; set down 7 nd carry 4. Fifthly, (3 X 4) -(- (5 X 2) + (2 X 4) + (3 X 2) -f (4 X 5) 'r 4 carried = 60 ; set down and carry 6. Sixthly, (3 X 2) -f- (5 X 4) 4- (2 X 2) + (3 X 5) -f 6 carried = 51 ; set down 1 and carry 5. 8ev * In this and the throe following examples the rod is considered — 572 feet. • '':^^, 102 MENSURATION OF SUPERFICIES. [TUTOR'S enthly, (3 X 4) + (5 X 2) + (2 X 5) + 5 carried = 37 ; set down 7 and carry 3. Eighthly, (3 X 2) -{- (5 X 5) + 3 earned = 34 ; set down 4 and cany 3. Lastly, 3 X 5 -f- 3 carried 3= 18 ; set do\yn 18, and the work ii finished. MENSURATION OF SUP^ERFICIES. ge;ometrical definitions. Geometry is the science which investigates the nature and properties of lines, angles, surfaces, and solid bodies. A point has no parts or magnitude. A line has length only, without breadth or thickness. A line drawn wholly in the same direc- tion, or the shortest distance between two points, is a right or straight line. That which continually changes its direction is a curve. Parallel lines preserve the same distance from each oth r throughout ; and therefore would nev«r mept, ihoiigh infinitely pror duced. An angle is the degree of inclination of two lines, or the opening between them wheij they meet in a point ; which is call- ed \\\p qrigular point. When a Jinie mppting -another inclines c /E aot eithjBr way, but inakes equal angles on pach side, those are called right angles; and the lines are perpendicular to each other. Thus the angle ADC «= the angle BDC* X P B An oblique angle is either acute or obtuse. An acute angU 10 le^9 than a right angle, as BDE ; and ap obtuse ai^gle^ greatef than a right angle, as ADE. -■■ — T ^ When more than two lines meet, forming several ang'ies at the »ame po^it^ it is necessary to designate each angle by three letters, placing that which is at the angular po'ut in the middle. Thus, the anglo pDC is that foi-med by the line* BD and CD. K ASSISTANT.] MENSURATION OF SUPERFICIES. /a9 A superficies or surface is a space contained within -inei ^nd has two dimensions, length and breadth. A solid is a space or body bounded by surfaces, and has three dimensions, length, breadth, and thickness. A triangle is a superficies bounded by three lines. A quadrangle, or quadrilateral, is bounded by four lines. A right-angled triangle has one right angle, (Fig. page 116,) an obtuse-angled triangle has one obtuse angle, and an acute-- angled triangle has all its angles acute. An equilateral triangle has the three sides (and consequently the three angles) all equal to each other. An isosceles triangle has two equal sides. A scalene triangle has all the three sides unequal. A parallelogram is a quadrangle having the opposite sic^^a equal and parallel.* When the angles are right ones, it is called a rectangle.^ And a rectangle having all its sides equal \% a squared A rhombus has all its sides equal, but oblique angles.^ A rhomboid has oblique angles, and only its opposite sides pqual. All other quadrilaterals are trapeziums ;* but a trapezi^n^ that has two sides parallel is called a trapezoid. The base^ of a figure is the side on which it is supposed to stand ; and a line drawn from the vertex, or opposite angle, perpendicular to the base, is the altitude^ or perpendicular in^ipht. Right-lined plane figures of more than four sides are called polygons. A J 'lygon of five sides (or angles) is a pentagon^ one of six a heji..gon, &;c. Vide Table , page 166. A circle^ is a plane figure, contained under pne curve ^ine, pajled the circumference, which is in every part equidistant from the centre, or middle point within it. The circle con? |ains more space than any other plane figure of equal comp^^g. A straight line passing through the centre, and meeting the circumference in two opposite points, is called the di- ameter;^ and half the diameter, or the distance from i\\p pentre to the circumference, is the radius.^ An arc of a circle is any portion of t^e circumference.* • Figs. 1, 2, and 3. *> Fig. 2. « Fig. 1. «» Fig. 3. • Fig. 5. 'The line AB, Figa. 3 and 4, ia the base, and CD the altitude. * Fig. 7. »» The line AB Fig. 7. » AC or I3C. ^ aD or ADP, Fig. % m MENSURATION OF SUPERFICIES. > 'I ( ' '.' f i [TUTOft'? A c/iorc? is a right line joining the extremes of an arcf The versed sine is part of the diameter cut off Jtjy tbo jchord.™ A segment is a space contained between an arc and its chord." A semicircle is a segment, the chord of which is the diameter. A sector is bounded by an arc and two radii." When tho ^wo radii are at right angles it is a quadrant, or fourth part of ^le circle. The circumference of every circle is supposed to b^ divi- ded into 360 equal parts, called degrees ; and each degree into {60 equal parts, called minutes, &c. The measure of ar> angle is determined by the number of degrees in the arc of a circle contained between the two lines forming the angle, described round the angulay point as ^ centr^e Thus t)ie angle ACB (Fig. 8.) is an angle of so many degrees as are contaiuQd in the arc AB. Hence, a right angle contains 90 degrees. An ellipse (or regular oval) is a plane figure bounded by ^ curve called the circumference, returning into itself, and de- scribed from two points, called the foci or focuses, in the (ransverse (or longest) diameter. The shortest diameter, which intersects the transverse at right angles, is called the conju-; gate. The diameters are also called axes.^ MENSURATION. Problem 1. To find the area of a Parallelogram,, whether it hf a Squafe, an oblong Rectangle, a Rhombus, or a Rhomboid. Rule. Multiply the )«ngth by the altitude or Perpendicular breadth : the prbcluct will be the F^^' A B (1) The base of the largest Egyptian fjyramid is a square, whose side is 693 eet. Upon how many acres of ground does it stand ? (2) Required the area of a rectangu- lar board, whose length is 12^ feet, and J)readih 9 inches. (3) What quantity of land does a I AB or AD, Fig. 8. "' DE. « AEBDA. " Fig. 9. ^^6- ?!): ifiSlSTANT.j MENSURATION OF SUrER1ia«IG8. ie6 thombus contain, the base of which is 1490, and the perpei^- dicular breadth 1280 links ? Problem 2. To find the area of a Triangle. Fig. 4. Ru^K. Multiply the base l)y the altitude, ahcl talf the product will be the area. (1) Required the number of square yards in a triangle, whose base is 49 feet, and height 25i feet. (2) What is the area of the gable of a house, the base or distance between the eaves being 22 feet 5 inches, and the perpendicular from the ridge to the middle of the base, 9 feet 4 inches 1 Rui-E 2. When the three sides only are given. — From liaJf the suni of the sides subtract each side scveially : nudtinly the half sum and the three remainders conf.inuully together; and the square root of their product will be the area. (3) The three sides of a triangular fish-pond are 140, 336," and 415 yards respectively. What is the vahie of the laind which it occupies, at J&225. per acre ? I*roblem 3. To find the area ofM Trapezium, or a Trapezoid. Rule. Divide the irapezium into two triangles by a diapnnni : midtiply the diagonal by the sum of the two perpendiculars falling upon it ; imd naif the product will be the ai-ea. Fig. 5. That is, DE + BF X AC = the area. For a trapezoid, Midtiply the sum of the two parallel sides bjr the pprpendicular distance between them; and half the product will be the area. (1) How many square yards of paving are in the trapezi- lim, whose diagonal s 65 feet, and the perpendiculars 28 and 33^ feet? (2) Find the area of a trapezium whose south side is 274Q links, ^ast side 3575, west side 4105, and north side 3755 links ; and the diagonal from the south-west to the noftl!i«^t^ angle 4835 links. ieo MENSURATION OF SUPERFICIES. [TUTORS \ • i ! ,!.■: % i I I )e the area of the whole; Fig. 6. B (3) Required the area of a trapezoid whose parallel a\i69 kre 201 feet and 121 feet respectively ; the perpendicular dis- tance being 10| feet? (4) How many square feet sire in a board, whose length is l2^ feet, atid the breadths of the two ends 15 inches and 11 inches respectively ? Problem 4. To find ike area of an Irregular Figure. Uvj.e: Divide it by drawing diagonals into trapeziums and tnaiiglel Find the ai*ea of each, and their sum will bt ' 1. Required the content of the irregular figure abcdefga, in which are given the following diagonals and perpendiculars : namely, ac =5-5 FD= 5 2 Gc = 4*4 6m = 1-3 Bn= 1-8 Go = l-i Ep= 0-8 rq = 2-3 Problem 5; To find the area of a Regular Polygon. , RItle 1. Multiply the perimeter (or sum of the sides) by the perpeiii 4icular drawn from the centre to one of the sides ; and half the prct duct will be the ai*ea. RtJLE 2. Multiply the st^uare of the side by the corresponding tal>- tilar area, or multiplier opposite to the namie in the following table; and the product will be the ai'ea. 1^ tides. 3 4 5 6 7 8 10 11 11^ Names of Polygons. Trigon, 6r Equilateral Triangle Tetragon, or square Pentagon Hexagon Heptagon Octagon Nonagori Decagorl Undecagon Duodecagon ......,,.. Areas, or Multipliers. 0-4330127 1-0000000 1-7204774 2-5980762 3-6339124 4-6284271 6-1818242 7-6942088 9-3656399 111.9^152.4,. [TUTOR^ allel sid^t icular dis* 3 length is ies and 11 tgure. id trianglel e whole; lygon. i the perperii balf the prct ponding tab- >wing table; as, or '.pliers. 30127 00000 04774 80762 39124 84271 18242 42088 56399 ASSISTANT.] MENSURATION OF SUPERFICIES 1§? (1) Required the area of a regular pentagon whose side is 25 feet; (2i) Requirefd tlie area of an octagon whose side is 20 feet. i*roblera 6 To Jind the diameter or circvmference of a Circle, the one from the other* Fig. 7. RiiL«. As 7 : 22 f ) or, as 113 : 355 1 / : : the diameter : the or, as 1 :, 3-1416$ > aU— -C— )D circumRJre'rice ; dud reversing tlie terras will find the diameter. (1) Required the circumference of a circle tvhose diametel is 12;1| * To find the proportion which the circumference bears to thediame'^ ter, and thence to find the area of a circle, is a problem that has engaged the anxious attention of mathematicians, of all ages. It is now general ly conditi^rfe'd Jtrtpossible to dfe'termlne it exactly ; but various approxi- mations have been found, some of which have been carried to so grestt a degree of accuracy, that in a circle as iimnense in magnitude as tlie o.'bit of the planet Saturn, the diameter of wliich is about 158 millions of miles, w^e are enabled to exjxess the circumference (the diameter be- ing given) so nearly approximnfing to the trvlh, rts not to deviate from it so much as the breadth of a single hair. The three approximations in the Rule are those in general use. t This is the ratio assi^ied La- Archimedes, a celebrated philosopher 6f Syracuse. ^S^ho flourished about two centuries before the Christian era. It answers the purpose sufficiently well when particular accuracy is not required. t This was discovered by Metius, a DutchrrfaH, about two centuries since. It is a very good approximation, agreeing with the truth to the sixth figure. $ This ia an abridgment of the celebrated Van Ceulen's proportion, who was nearly coatemporaiy with Rletius. By a patient and most laborious iuvc^tigHtion, hr determined it tnily to 36 places of figures, (3'1'11^98, &c.) But it bus been since extended to considerably more than .00 places. This proportion is extremely convenient, from the circumstance of the first term being unity; which saves the labour of division, in finding the circumference of axvf other circle whose diam«^ tor is given. It ia not quite so accurate as the preceding. ±1^ = -41 = 37-714285 H AS 22 12 wr, aA 113 : 355 •r.. •• 1 : 3 14M) 12 12 355 X 12 = 37-699115 113 3 1416 ?< 12-37-6993 the circuiulb^ ence. I i t 1 1 !.: % I ; i iJl ! ; { f. «> if!: I I: lit. rfi « / ill 11 i i' ' M i6i MENSURATION 01 »?l : r.RHClE*. [TUTOR^i (2) What is the circumference . Lori ilio diameter is 45? (3) What is the diameter of a column whose circumference is 9 feet 6 inches ? (4) If the circumference of a great circle of the earth (as the equator) were exactly 25000 miles, what would be uie diameter ? Problem 7. To find the area of a Circle. Role 1. The area is Qqual to a fourth pnrt of the product of the cii** eumference into the diameter, or the product of half the circumferenco into half the diameter. 1 V *?■ 1 4tfi Therefore, when the diameter is 1, the area = -t = 7354 ; 4 vhence we have Rule 2. Multiply the square of the diameter by -7854, and the pro duct will be the area. , . . . ■ Rule 3. Multiply the scjuaro of the circumference by -07958 for tha area. (1) Required the area of the circle proposed in Example I. Problem 6.f (2) Find the area of the circle proposed in Example 2, Problem.6. . . ,, ^, , (3) ^haj is the area of the end ol a toller whiose diameter is 2 feet ^ inches ? , ,, . .. (4) Required the area when the circumference is 8i feet. Problem 8. To find the side of a square inscribed in a circle. Rule. Multiply ^he radius by 1*4142, (that is, by i/2,) or multiply the diameter by 7071. t (1), Find the side of the square inscribed in the circle whose diameter is 12. , . -- (2) What is the side of th© square inscribed in a circis whose diameter is 6 feet 5 inches ? • .iI3i2iii = 37-6992 X 3= 113-0976) ■ 4 S the area. gr, 12«X 7854 = 12 X 12 X 7854 = 113-0976 > t The following Rules exhibit the principal relations between fhi circle and its equal squai-e, or insoribecl square. 3. The diameter X 707 1068 ) 4. The oircHDifer. X-22507<91 )-=^ the side of the inscribed squnre b. The area. X -6366107 S s [TUTOR*i Bter is 45 1 rcumferenco he earth (as ould be uie I le. . ■ luct of the cUjr circumferenco 1416 = •7354; >4, and the pro y 07956 for tha ri Example 1, I Example 2, [lose diameter ce is 81 feet. •ed in a circle. /2,) or multip'y in the circle ed in a circle sa. ons between tlM |i jual square. uBcribed equni* kt eh an be ne ani 8. in an an an »1! ASSWTANT.] MENSURATION OF SUPERFICIES. 16^ Problem 9. To fnd the length of a Circular Arc. Fig. 8. KuLB 1. From 8 times the chord of half the arc subtract the chord of the whole arc, aiid i of the difference will be the length of the ai'c, nearly. That is AD X 8 — AB ~ 3 = arcADB. |,' ^^ ^^ Note. Half the chord of the whole ;ii-c, tlio clioi-d ol' h:tlf tiie Hxr\ and the versed sine, are sides of a lii-lil, ;.!i;;1l(1 li'j;i;ii;lc ; any two «if which being given, the third may be iound ;i.s diivcted in page 115. Hole 2. Multiply the number of d'j.izrefs in the arc hy ilio radius,, Bi d the pi-oduct by -017 15, for the length of tiie nvc. (1) The chord of the v/hole arc is LIO, and the versed sine 8 . what is the length of the arc ? (2) What is the length of the arc when the chord of tho h-ilf arc is 10 G25, and its versed sine o "? (3) Required the length of an arc of 12^ 10', the radius biiing 10 feet. Problem 10. To Jind the area of a Sector of a circle. Fig. 9, c Role 1. Multiply the length of t\ie arc by the radius, and half tho p. idnct will be the area. Role 2. As 360° : the degrees in the arc : : the area of the circle : Jbe area of the Scctc»r. (1) Required the area of the sector, when the radius is 16, aiid tiie chord of the whole arc 18 feet. (2) What is the ?-rea of a sector whose arc is 147<^ 29', and the radius 25 ? (3) Required the area of a sector, whose radius is 20 feet; end tho versed sine 1 foot 9 inches.* 6. The side of a square X 1-J14-314 = the diameter ) of its ciroum- 7. The side of a square X 4"442SiS3 = the ''ircumi'. j scribing circle. 8. The side of a square X 1"128379 = the diameter 5of au equal cii'- 9. The side of a square X 3*544908 = the circuuif. ) do. • By the properties of the curcle, the veraed sine X the remaining pajft of the diameter •» the square of half the chord of the mx ; whenca til the requisites may be found. H 170 MENSURATION OF SUPERFICIES. {TUTOlf* (4) What is the area of the sector, when the chord of half its arc is 14 feet 2 inches, and the versed sine 6 feet 8 inches 1* Problem 11. To find the area of a Circular Segment. RuT.B 1. Find the area of the sector; and also the area of the tri- angle formed «>y the chord and the two radii of the sector: their differ- ever, when the segment is less tliau a semicircle, or their sum, when it is greater, will be the area of the segment. Rule 2. Divide the height of the segment by the diameter, and find the quotient in llie coin run of heights in the following table. Multiply tlie coiiespoiuliiig area by the square of the diameter, for th^ area of the ac^rneat.t Tabic of th e Areas of Circular Sesments • Area of Srgmenf. •14 Area ' of ! Segment. . to Area of Segment. bo Area of Segment. ■01 •00133 •06683 26 •16226 •39 •28359 •02 •00375 •15 •07387 27 •17109 •40 •29337 •03 •00087 •16 •08111 28 •18002 •41 •30319 •04 •01054 •17 •08854 : 29 ■•8905 •42 •31304 •05 •01468 •18 •09613 1 30 •19817 •43 •32293 •06 •01924 1 •19 •10390 I 31 •20738 •44 •33284 •07 •02117 i •20 •11132 ! 32 •21667 •45 •34278 •OS •029 14 i •21 •11990 ; 33 •22603 •46 •35274 •09 •03501 22 •12811 !i 34 •23547 •47 •36272 •10 •04088 23 •13647 i! 35 •24498 •48 •37270 •11 •04701 •24 •14494 |: 36 •25455 •49 •38270 •12 •05339 •25 •15355 ' 37 •26il8 •50 •39270 •13 •06000 i 38 •27386 (1) What is the area of a segment, when lli? chord of tha whole arc is 60, and the chord of half the arc 37^? * When the half chord (see AE, Fig. 7) of the arc is found by iha pro]>evties of a riglit tingled trinngle, then AE^ = the versed sine (DE) X the remaininii [)art of the diameter; whence the diameter (and cods*- quently the radius) will be known. t When there is a remainder (or fraction) after the second quotieut figure, in dividhig the heiglit by the diameter; having taken out tbe uv'.-'a answering to the two ligures. add to it ^wch fraetional part of the tl{irerencg'het\\'ee-a that and the nitxf vncceedtng area, fr»r tiie «ake Mi' g abater aotnrac-y. •{TUTOlfi chord of half sine 6 feet 8 • Segment. area of the tn- !tor : their differ* their Bum, when iameter, and find table. Multiply r, for the a'©* « tents. •39 •40 •41 •42 •43 •44 •45 •46 •47 •48 •49 •50 •28359 •29337 •30319 •31304 •32293 •33284 •34278 •35274 •36272 •37270 •38270 •39270 ASSISTANT.] MENSURATION OF SUPERFICIES. m Lii? chord of the I 37^? xc is found by iha 2 versed eine (DE) iameter (and codw- le second quotient ving taken out the letional part of the en, e»r tlie «ake «< (2) What is the area of a segment whose height is 18, and the diameter of the circle 48 1 (3) Required the area of a circular segment whose height is 2 and chord 20. (4) What is the area of the segment of a circle whose radius is 24, the chord of the whole arc 20, and the chord of half the arc 10-2? (5) If the radius of a circle is 10 feet, what is the area of the segment whose chord is 12 feet? Problem 12. To find the circumference of an Ellipse, the trans verse and conjugate diameters being given. Rule. Multiply the square root Fig. 10. of half the sum of the squares of the two diameters by 3" 14 16, and the product will be the circumferenco nearly.* (1) What is the circumfer- ence of an ellipse whose trans- verse diameter is 24, and con- jugate 18 ? (2) The two axes of an ellipse are GO and 45 yards re' Bpectively : what is the circumference 1 Problem 1 3. To find the area of an Ellipse. Rule. Multiply the product of the axes by -7854 for the area. (1) Required the area of a.^ ^Uipse whose axes are 35 and 25. (2) What will be the oxpe .o oi' trenching an elliptic gar- den, whose axes are 70 and Ki Teet, at 3^f/. per square yard ? (3) Required the area o'' the elli|)se in Grosvenor Square, London, the transverse '^^'iameter b \vg 840 chains, and the conjugate 6'12 chains. Problem 14. To find the area of a,* Elliptic Segment, the base being parallel to either axis. Rule. Divide the height of the segment by that axe of which it is a part, and find, in the Table of Circular Segments, a versed sine equal to the quotient. • If the half sum of the two diamelers bo multiplit}d by 3*1416, the Iproduct will give the circumferenco svfficiently near for most practkiol pufpoeee. 172 ▲ COLLECTION OF QUESTIOWS. [tutor's Multiply the correspondln.ir tabular area and the two axes continu ally togothev, and the product will be tlic area reqiiu'ed. (1) What is the arcca of an elliptic segment cut off by a line (called a double ordinate) parallel to the conjugate diarf.- eter at the distance of 3(3 yards < ;om ihe centre, the axes be- ing 120 and 40 yards resj)ectivel.'? (2) Required the number of souare yards in the segment of an ellipse, cut off by an ordina j r»ciVa.;el to the transverse diameter; the height being 5 feet, iid .l.e two axes 35 and 25 feet respectively. A COLLECTION OF QUESTIONS. (1) What is the value of 14 barrels of soap, at 4^d. peif lb., each barrel containiuir 254 lb. ? Alts. jCG6.,13..6. (2) A and B joined in partnership; A put into the joint stock X"320. for 5 months, and B X'doO. for 3 months : they gained jClOO. What is each num's share of the gain? ^1/?^. jVs X'53..13..9f^ft., and JTs £i6..(j..2^\\. (3) How many yards of cloth, at 17.y. 6d. per yard, can I have for 13 cwt. 2 qrs. of wool, at 14^/. per lb. 1 Ans. 100 yards, 34- qrs. (4) If I buy 1000 ells of Flemish linen for X90.,'at what price must I sell it per English ell to gain X'lO. by tho whole ? Ans. 'Ss. 4d. per ell. (5) A has C48 yards of cloth at lAs. per yard, ready money, but in barter will have IGs. B has wine at jC42. per tun, ready money : what must he charge it per tun in barter, and what quantity must be given in exchange for the cloth? Ans. £AH. per tun, and the quantltij, 10 tuns, 3 Iihds., 12^ gals. (6) A jeweller sold jewels to the value of jC'1200., for which he has received in part 876 French pistoles, at 165. 6d. each. How much more is due to him ? Ans. JC477..6. (7) An oilman bought 417 cwt. 1 qr. 15 lb. gross weight of train oil, tare 20 lb. per cwt. : how many neat gallons were there, allowing 7\- lb. to a gallon ? Ans. 5120 gallons. (8) If I buy cloth at i\s. Gd. per yard, and sell it at 16^ 9d., what is the gain per cent.? Ans. £15. .10. .4-^. (9) Bought 27 bags of ginger, each weighing gross 84J Ib.^ tare If lb. per bag, tret as usual : what is the value at 8| 7 horses worth jC13. each. IIov; much will make good the difl'erence, in case they interchange tlicir droves of cattle ? .4;^^. X4..12. (17) A man left £\\10. to he oiven to three persons, A, B, and C ; B to have twice as much as A, and C as much as A and B ; what was the sliare of each ? Ans. A £'20. B £iQ. and C £60. (18) £1000. is to be divided among three men, in such a manner, that if A has £3. B shall have .€3, and C £8. How much will each man ha^■e ? Ans. A £187.. 10. B £312. .10. and C £500. (19) A piece of wainscot is 5 feet 6i inches Ions-, and 2 feet 9^ inches broad: what is the superficial cen;ent? An;: 'Zi fret 0' 3" 4"' (j"". (20) A garrison of 3G0 men, v/lio had originally six month.?' provisions, having endured a si>'ife of 5 month:-;, without ob- taining any relief or fresh supniy, wish to know how many men irmst depart, that the provif-iions u^ay sulfice for the resi- due 5 months longer? Anr. 2 SB tiKn. (21) The less of two numbers is 187; tbe difference 34. The square of their pro of the imperial standard! Jtutor'ii ivns, at 4,t. pay a half- money will 11..9..2. ; B put in ;G0. of the ft\ X960. irise in one s. JC400. h, are equal bis. 76f . 5 qrs. 5 lb., h weighing 3563ff taken from what nutn- 72, 19, and In.?. 158. CSOO. to be he,^, Bi mong them ..7..6I III. ..7..2i ^VV ASSISTANT. 1 SIPPLEMENTAL QUESTIONS. 177 (3) How many «iallons of the oUl wine measure are equal in quantity to 63 gallons, imperial tncasiire ? (4) Red ICC 15 quarters, 3 huslirls, ] peck, old measure, to its equivalent in the im|)erial sfaiulard mcNisure. (5) A lady who was asked the'time of the day, said that it was between tJ/rcr nvA Jour: hut l>eino- desired to name the exact time, she replied, '*'i"ho iiiiuiite hand is advanced half an hour precisely before the hour hand.'' Kequired the exact time. (6) If 7 men can huil 1 a wnll !0 yards Ion?, 4 feet high, and 2 feet thick, in 3'3 days ; how many men will build a wall 240 yards long, Icet high, and 3 feet thick, in 8 days?* (7) 7'he weiglit. of a certain bar of iron 2 feel long, 3 inch- es broad, and 1 inch tiiick, is 'jt) ///.s-. \\ hat is the weight of a bar of similar ((iialiiy wiiicii is 7] feet loi.g, 4 J- inches broad, and 3| inches thick? (8) A person who had fne-uinllis of a mine, made hi;^ younger brother a present (^f htdf his share, and sold half the remainder to his cousin John, wlio soon alter purchased i of the younger br(;ther's sliarc ; hut now oilers to disjiosc of half his interest in the ininc for ,£1."30. il.ihnaling at the same rate, what is the value oi' tlic whole mine, and of each brother's share ? (9) A, travelling from London to Manchester, and B ^rom Manchester to London, set out at the same lime. They yneei at the end of six days, A having travelled 3 miles a day narried T he %L'oman. * Qvieslinns of Compmin:! Propnr'inn in v.liicli tlu^ ic-rms are mimer* fms jn;iv hv f;olvrd by Ki-! • 1. fur ll:- \)::u^.-\c JTuli (.f Throe; but tho followiii;: nu'tiiftd !.- ■"-ik/N.' co-vi ni'M.t . Kl'i.e. Arnuiire llio icni:;- tf tl/o /7?-.s/ r.-ivp.i- iu:(\ (J'>cA in one Hue, and the cori'espoiKhiig teiii;.? of lii' .stc, • // (•:vk, h: d i'Jj'trl px:iclly under them ; mn-'plviu^'; ihe }il;ic'.- *a li.*- u ;].i : ' \\,]\\\ w \\\i ;i;i ({<'i ri>,Ji, and con- noctluir llie confrdnj c; :UM'S ;,i: 1 ..t ;'i-. f;. ])V ir(!^,> liu'^--. M;in. iply cotitin- iiiilly die te^^I^^ ^^'l'earh rrrst' iumI ll.o ..//,:■;• pir-ri: iiivi'i;> ;!.-' |irn(lnct wising iVam iht'. _/><7/ uu!rJ)Oi-r;i" ten.". '■ ,\- {\ir t.'iodiic'. ni' ihd.-i.' Willi which llic blank term i& c*iiuioc'a J, ai.tl IIk; (ai'ii,,. hi wVA ho tiiu ausj'Wer. Sahidon of the ahi.i::- rsainj'^o. ird measure a hogshea^l ice between ould corr86» 7X :!1^ X^^li^X tiX 3 * • ^ ''^ 2 10 . G : 3^ • "7 :>i0x:r'h did I give for liim, what did I expend in keeping him and what ditl I gain ])er cent.? (16) It, has been found by expeviment, tl^at sound is con- veyed through the air at !i;o r;ite of 11 12 feet in a second. How far distant is the cloud, wiien 7 J seconds elapse between seeing the flash of liLilitnini and hcarinn; the thunder? (17) What is the height of a tower that projects a sliadow 75'75 yards long, at the same time that a perpendicular staff, 3 feet high, gives a shade of 1-55 feet in len::th ? (18) A baukrujit owes £'2^jS0. and the vahic of liis effects is jCSIG. and the amount oi' recoverable debts X'35S..r2. be- sides wliich he lias an iii!ex]iired lease tha: b.as ]3 years to nm, valued at .CI 2. a year more tb.an th^ stipulated rent. If the lease be disposed of for i^rc^-cnt nionev, allowing Com- pound Interest at XT), per cent, per annum ; and if the working of the commission and other expenses amount to i?17"J; wha will his creditors have in the pound, jirovided they allow him JC160. to recommence buNiness ' (19) A youth aged 12 years, liaving had bequeathed to him [TUTOR'S nee being 31 hours, ich flcu s it number of when they of 15 per much per eh OS in di- e shillings. A'orn down, es it ? n 7f days, what time in return a r woukl he t at jC5. per nrlh of what pense of his lase. How ecping him )und is con- n a second, pso between idor ? ts a Siiadow licular staff, if liis effects 358. .12. be- ] 3 years to eJ rent. If i)win£f Com- tlic working £[72; wha ■V allow him athed to him ASfilSTANT 1 SUPn.EMFAT.Vl, QVESTION'S. 179 an annuity of £CtO. for 12 years, to commence when he cornea of age, the executors think it will be more advantaj/eous to exchange this for nn annuity to commence immcdiatclv, and continue till he is '21, to enable theuj to give him some edu- cation and a trade. What will be such anrmity, i!!100. being reserved at the conclusion to sol him v.-p in buisiness ? (20) There is an iniand 73 uiiloii i.i cirxUmilercnc'e, to travel round which three pedcsliians all start at the same time : A travels 5 miles a day, B travel* 8, and C 10 miles a day. la how inanv davs will they all ccmc tonelhev arjain, and how many circuits w ill each have made I (21) What will a banker charge ibr d!--counting a bill cl JC52..10. on the 7i\i of April, the bUl beinu ddc on the 19th of May ? (22) My agent iii I n ha'. ii.T advised in? rhvu he has purchased goods on ni;, accKunt lo the aiTi-.«i;nt of /-'Tir)6..10. at six montlis' credit, or £7},. per cent. di-.'V'i'!i ■^vil; in 25 minute. ll' :.tl iIk- il in what time v/iii the biuh b filllux to be uniibrm ? (26) A person who had spont tvo-ihirv's of bis ;-poney at one place, and half tire rcrnaindtvr ;a I'lioiiicr, iou.-id thu ho had £32. .12. left. How nuich had he ;.' Ilr l ? (27) The length, breadth, ar:d ii.^ig^t of a voo;.! are I G iiiica, t;.ij)po gici^s to /row uui- tu'O coci^s, IVorn or.e nvid IV •!.! the ether ■ [ :/ ;'. . -..::;■' :,i-n:% * Sep Mole to nxaHiplr' 1 I.^ Di-cr-unt pTg.-- 71 ^. IMAGE EVALUATION TEST TARGET (MT-3) ..iiig ihc cn^iomary interest of jnoiit V lo l-i' at i: ■■ r'l'!' cf /,'■". por c(,nt. per annuin ? Also, stipnoiiui;' i-.);-.:'ifv>t to hv at X'l. per cent, per annum? (■>2) \''. hat will n.! iuo exponso of covoring and guttering a rnol" \vi:h h^'il :;t l^,^'. >>rr c^i f., tito len'M'h ol' tliO roof being 43 I^';'?, '.lid tli'^ irrad'.li or fiuT o\tr it 32 feet; the guttering !)7 tect Icii ' a.ci 2 l;jf t wide: */.e I'ad ftir th.e former being 9't<3i /''.. rtnd luv the bitUr 7-3''3 /V to tiic scpiare foot I \ I [tutor's ht line thai ich a horse ice, so as to rood, sup- r? hts Pucces- i and half a he remain* ht half the educed his t first ? ie, both in ■ excessive plu viameter of the 9th interval of ;he 15th it ' the 29th, vory rainy, les. How spectively, f ojic inch ; surface of ibove men- for which 5 it upon a !^ out upon a liuildiiifr, the termi- ill be a fair )f continu- al present interest of m ? Also, II? I ffuttering roof being ? guttering nicr being foot I (ISl) A COMPKNDIt'M OF * BOO K-K E E P I N G, BY SINGLE ENTRY ; Intended for the purpose of initiating Youth in the LEADIN(9 PRINCIPLES of that important Branch of Science. "pOOK-KEEPING is the art of recording pecuniary or commercial transactions in a regular and systernalic manner. The sfienro of nook-keepiiig adnnils of iiirnimcrnblc vaiiefies of method : but its gonoriU prinripies are invanahle. These bfiiit; wrll unilersiood, the knowledge of any particular system, adapted to the peculiar concerns of any counting-houBu, wi^ hn easily acquired. Sinple Entry, being the most simple and concise, is the method asu- ally adopted in retail business. The General Kule to be observed in eveiy system of Book-keep- ing, is, To make any person Debtor (Dr.) for money or goods which he re- ceives from me, and to make him Creditor (Cr.) for tchatever I recei^ti from him. The books usually kept in Single Entry, are the Day-Book, the Casb- Book, the Ledger, and the Bill-Book. The Day-Book, when a person commences bu.siness, begins with an inventory of the existing state of his atfeiirs : after which are enteredr in the regular order of time, the daily transactions of Goods bought and sold. The Cash-Book contain? the particulars of all Money transactJonfl. It is ruled in a folio form : on the left hand page, Cash is debited to ali sums received, and on the risht, Cash in credited by all sums paid. The Balance (or quantity which the Dr. side exceeds the Cr.) shows tho amount of Cash in hand. This should be ascertained weekly, and in Bonit^ concerns daily, in order to prove if it coiTcsponds with the real Cash in hand. In the Ledger are collected the dispersed accounts of each person from the Day-Book and Cash-Book, and entered in a concit'ir'i'-\i!..rs of all BiU.i of Errhnn^tt, whether Rernvahh or Pfji,nl)!e. Tlio f rnior arp Ihos" wlncii mine into tho 'I'mdoman's pci session, and are drawn upon some ollior poison ; tiie latter are those winch are drawn upon and accepted liy him.— Printed I)ill-lJo-)ks nia_\ I'D had of any Bookseller. Note. In the following tratisartionf;, Hills Reconahle are considnred as Cash: btit many Accountants do not '.Mitrrthi'm a« such, till (ash hss beoii actually reoely 1 1 for thorn V t *S2) MEMORANDUMS • * * i ■ [ r I it or TRE TR.UI8ACT105S STATED IR TRX rOLLOWIRO BOOKf. ^#1. 5. Received from Allen, ^yild, and Co. of Leeds, on credit, 9 ]nece« wf super blue cloth, ouch 36 yds. at 25«. Hd. per yarti.*— and 2 pieces of narrow brown, 84 yds, at 4*. 9 plf.(«.d boforr, articles for wli^ch a jierson w J?'jt^» «d; but some Bookkc^opcra crrJ' H ' .1 1 '' )64 DAY-BOOK, (page 2.) Himmonfis and Cn., Livrrpnnl By Yellow soap, li rwt. at 7()«. 12 fZoz. caiullrs, at Hj«. fxi. 4 . raw do. at 10^. 1\ lb. Congou tea, at 75. Gd. i lb. Hyson, at 125. Dr. -31- John Herdson To Hops. 10 lb. at 1.?. Id. \ ream cap paper, at 7d. per quire Dr. -Feb. 1.- Willinm TinnUnxnn To 2 5^ y 189 CA8H-B00V< > } 1 ' > ; ■I ( t \ ! ii , o> 1 1 o «0 Oi " o o • « 1 1 o 1—1 SI o o f-4 1>H S) 1 1 ** to it c< 1— t rH 1 CI CO o «* 12 00 «o 1 e* a -H oi eo eo s Ois >> ca CO 2 - • i2 CO s K Oi « • • «t1 «J d teC) o o a» 3 O " n a> « C -a =j a rf o c< on ^ ~ 00 30 CO ^^ C 00 I . 1 -9.H fe 3 ^ A ^ o OS »-( r1 P> >-l CO U4 ^o e> n ■ ■ o oo o e 1 1 7t H® eo n I-H 1 I-H f-H 1— t o 1 1 i*> 1500 o> o 1 1 e« ST S| TZ e* 7t t«. e4 LSDOER INDBZ. INDEX TO THE LEDGERS W eo o • 1 5 ^ Allen, Wild, and Co. o N ■R Bills jKiyjible - - - 2 Q Oats' pnrveyiuico - P t . c 1 D Q E R P' Fletcher Samuel 2 § Stock Siininonds and Co. - Sanderson .lames - 3 G TT Herdson Jolin - - Hazard and Jones - 1 3 'p Taylor .lames - - Tondinson Wm.- - 1 2 IJ uv K w 1 L XY \1 Mason Bernard - - 1 Z * The Ledger has an Alphabetical Index, showing, at ono vi«w, In what foUs ija ijBnon's accoiuit may be tounU. o'>i U <0 ■?T EXAMPLE OF A PARTNERSHIP CONCERN. John Herdson and I have been engaged in a joint concern as pnr»'eyor« of oi.i for the army. I have purchased oats for the joint sloclt to the amount of jC2f)..ll. H has paid, for oats purchased by iiim, £449.. 0.. 3. 1 iiave rcce.ved, for oats that lave N>pn disposed of, £507. .1).. 2. ; and mv partner has receiveii. from tiie same source, £55. .2. .8. I have advanced to liim, at diflerent limes, £4:13. .17. ; and have paid, fot v.-arehou.se room and otiier sundry expenses, £I..13..S. Fium tliese geueial heads, collected from tlie particulars recorded in the Granary Boolt, il is required to stat« tnc transactions. It may, perhaps, be interesting to the learner to ho informed that the above wai ^ real occurrence ; for an accurate statement of wliich the writer was some tiia« 8io©f •ypiied to by the parties conceriiwd, •5 16f LKDOEn, (Tollo 1.) C''^ o> o o u in o 1— o o It o ^ \ « li o O lO O r- 1 rH N. N. n '•c c: t^ CO O CO i-( CO T»< O n CO CO 1 r-< ' C.b CJ cttg 1 . '^ , e • ^ ^ (U . c3 ^ •0 «o 1 o 1 1 4.* lo ^ 2 « ■1 e c a it. o O a o K o sundries ash £332. H HU I u s c > O CO eo . CO q r-^ es CI o CO eo o ■ 2 9 « I \ si €<« 6 a .^ -1 1 s a 6 ^ I •«-<« CJ o UO o C> o o "«< CO CO »-t • ■•^ » o CI ed L EDO fell,' (folio 2.) V6{l 3 Ex ziod vr.i>.3ZRs (folio 3.) -an to 00 O ■V o «o o c. 1— 1 ^ ^ <• N. 00 09 00 1 1-H ts> O 00 O) 00 1-H ?-( <-H 1-1 1-1 i H<^^ n 00 j o- J to 1 1 r O 1-1 O 1^ ■«*• c CO r^. >-• eo 4 H $ ©» « CO o O C< fH §§ ™ « ^W eo rH (N 91 C4 f-l a> n m 'V'^ 1^:2^ (U K.^- 9 et-s e Si* Ipqw-^gqcq CO -O ^H .9) (^ •4M< "♦O" •*f ti .^ M^ -H ' 1— 1 r-t 1-t ^^« eo 00 N. I/? (N IC O \n CO IV. CO ;0 . U3 r^ . eo 1 TJ» O 1 »o -4 ;