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Arithi and more Rule, cone winexed. II. Vulge III. Deci {'ery plain nterest, Ai IV. Duo( iaeMuring V. ACol in the forej A new ai readily cal< flalarieA, *i The who] Remembra This Woi M recommc for private ] A C( THE W TUTOR'S ASSISTANT; BEING A COMPENDIUM OF ARITHMETIC, AND COMPLETE aUESTION-BOOK j CONTAINING, I. Arithmetic in whole numbers: being a brief exnlanation of all its RhIab in . n.» «inl«d * * '^"** ''"*"' of queatlons in real Businesa, wlS» thei? iSaww! II Vulgar Fractiona, which are treated with a great deal of plainDeaa and perapicuity. *«rini«?n^.!i"v''''M.*^® extracUoa of the Square, Cube, and Biquadrate Roota after a &?rS?An^il^^KR^r,r for SJSJycri^SiuJn of TV n ^"""'*f ' and Pensions m arrears, &c., either by Simple or Compound SitereS. «^^f^^^^^ «PP«ed 10 inThe^^^eJcfnlrTuteJ?''^"''"'' promiacuoualy arranged, for the exercise of the scholar TO WHICH ARB ADnSD, for private persons. BY FRANCIS WALKINGAME, WRITINa-MASTBR AND ACCOUNTAJn*. TO WHICH 18 ADDED, A COMPENDIUM OF BOGK-KEEPINgT BY ISAAC FISHER. MONTREAL-AKMOUll «t JIAMbav KIKOfiTOK-RAMSAT, ARlTiuR '& HAMlLTOlf— RA 1I8AT k M^VmUJCK, 1845. CO. i !. 10/ \i ■3 C MULTIPLICATION. 1 2 3 2 4 6 3 6 9 12 15 18 4 8 12 16 20 24 28 5 10 15 20 6 7 8 16 24 9 18 27 36 10 20 30 40 60 60 70 80 90 100 no 120 130 140 150 160 170 180 190 200 11 22 33 44 55 66 77 88 99 110 121 132 143 154 165 176 187 198 209 220 12 24 36 48 60 72 84 96 108 120 132 144 156 168 180 192 204 216 228 240 12 18 24 14 21 4 8 28 35 42 49 32 5 10 25 30 40 48 45 54 6 12 30 35 36 42 7 8 14 16 21 56 63 72 24 27 32 36 40 44 48 40 45 48 56 64 9 18 54 63 70 77 84 91 72 80 88 96 81 90 99 108 10 20 30 33 50 55 60 60 66 72 11 22 12 24 36 13 26 39 52 56 60 65 70 75 78 104 112 120 117 126 135 144 153 162 171 if .i r r t 14 15 28 30 42 45 84 90 98 105 16 32 48 51 54 64 68 80 85 96 112 128 136 144 152 17 18 34 36 102 108 119 126 72 76 90 19 38 57 60 95 114 120 133 140 201 40 80 100 160 180 Note.— This 1 in 4 are 2, and 2a 'able may be applied to Division by reversing it : as the 2b in G are 3, &c. . s. 12,345 123,450 ,234,567 ,345,678 11 12 22 24 33 36 44 48 55 60 66 72 ~77 84 88 96 99 108 110 120 121 132 132 144 143 156 154 168 165 180 176 192 187 204 198 216 209 228 220 240 as the 28 ARITHMETICAL TABLES. Vll FEN'CE. 20d. are Is. 8d. 24 30 36 40 48 50 60 70 72 Wa. 6 5 4 3 2 2 1 1 2 2 3 3 4 4 5 5 6 6 4 2 lU TABLES OF MONEY. s HII.I.INUK. PO.l. are 6s. 8d. i 20s. are £1 Oa. IWa. uvc. £6 Os. 84 • • 7 1 30 •• 1 10 IHO • . 6 10 90 . • 7 6 40 • • 2 140 •• 7 96 . ■ 9 '50 • • 2 10 1.50 . . 7 10 100 . . 8 4 60 •• 3 IGO •• 8 108 . • 9 70 • . 3 10 170 .• 8 10 110 • 9 2 60 •• 4 IHO .. 9 120 • 10 90 •• 4 10 190 • • 9 10 130 • • 10 10 100 •• 6 200 •• 10 140 . . U 8 110 •• 5 10 210 ■• 10 10 PRACTICE TABLES. or A PODND. 01. is 1 half 8 1 third 1 fourth 1 fifth 4 1 sixth 6 1 eighth 1 tenth 8 1 twelfth 1 twentieth 8 1 thirtieth 6 1 fortieth 6d. 4- or A BHILLIN'O. 19 half third 3 1 fourth 2 1 sixth IJ 1 eighth 1 1 twelfth OF A TON. 10 cwt. 1 half 5 1 fourth 4 1 fifth 2J 1 eighth 2 1 tenth OF A CWT. qrs. 2 or 0.... 0.... lb. r>6 •28- ■ 16- .14. is 1 half • 1 fourth ■ • •! seventh • • • 1 eighth OF A QUARTER. 141bH 1 half 7 1 fourth 4 1 seventh 3^'- 1 eighth CUSTOMARY WEIGHT A Firkin of Butter is 56 lbs. A Firkin of Soap 64 A Barrel of Soap 256 A Barrel of Butter 224 A Barrel of Candles 120 A Fa,?gfot of Steel- ■120 OF GOODS. A Stone of Glass 5 A Stone of Iron or Shot 14 A Bairel of Anchovies 30 A Barrel of Pot Ashes 200 A Seam of Glass, 24 stone, or-. 120 lbs. TABLES OF WEIGHTS AND MEASURES. TROY WEIGHT. 21 -;pM»tm~^ma n iin a lktm -li I viii ARITHMETICAL TABLES. 1 0U> 8TANDABD. Gals. 8 17 36 53 70 106 3 3 2 1 3 P. 1 1 1 1 1 "Gins; 3-93 3*86 3>46 17 34 69 03 38 06 ALE AND BEER. 1 12 2i 50 75 100 151 302 1 2 1 2 3 1 1 1 1 1 1*60 2-41 10 38 22 83 44 66 33 4 2 4 9 2 2 U 2 3 gills make 1 pints 1 quarts* ••• 1 Sallons* • • 1 rkins • • • 1 kilderkins* 1 barrel**" 1 barrels* *'«1 banels*«**l pint quart gal. fir. kild. bar. hhd. gun. utt WINE MEASURE. 2 pints* .-l quart 4 quarts* 1 gallon 10 gallons* 1 anker 18 gallons* 1 runlet 42 gallons* 1 tierce 63 gallons* 1 hogshead B. P. G. a P. Gills. 1 2 4 1 1 1 8 1 33 82 2 1 2 1 0*25 1*01 2*02 0*07 0*14 0*28 0*66 2*24 1*63 84 eallons 2 nogs 2 pipes 1 puncheon 1 pipe 1 tun DRY MEASURE. 3 3 37 1 0*21 2*52 2 pints make 1 quart 4 quarts 1 gallon 2 gallons • * • 1 peck 4 pecks 1 bushel 2 bushels* • * • 1 strike 4 bushels* •*•! sack 8 bushels* • • • 1 quarter 4 quarters* • • 1 chald. 10 quarters* * * 1 last COAL MEASURE. 3 bushels* '1 sack 36 bushels* '1 chaldron NIW STANDARD. Gals. 1 9 18 36 64 73 109 Q. 1 1 2 3 3 P. 1 1 1 I GUIs. 0*07 0*13 0-64 0*91 1-82 3 64 1*45 3-27 2-91 1 3 8 1 14 3 34 3 62 1 69 3 104 3 209 3 266 2*68 3*87 3*70 3*55 3*40 3 11 2*22 B. P. G. Q. P. Gills. 1 3*75 3 1 3*02 13 1 2*04 3 13 0*17 1 3 1 2 0*36 3 3 10 0*70 7 3 1*40 31 1 1*66 77 2 1 1 2*13 2 3 110 0*62 34 3 1 1 2*34 I =1 4in>ARD. GUliT. 0-07 013 0-64 0-91 1-82 3-64 1-45 3-27 2'9I T5B 2-65 2-68 3.87 70 56 40 11 22 P. Gills. 1 3-75 302 204 017 035 0-70 1-40 166 213 1 1 1 1 1 0-52 2.34 CONTENTS PART l.~ARITHMETIC IN WHOLE NUMBERS. Introduction 13 Numeration 13 Integers, Addition '.'. 15 Subtraction 16 Mnltiplicadon 16 Diviaion 19 Tables gi Addition of several denominations. 28 — -Subtraction 34 Multiplication 37 Division 43 Bills of Parcels 44 Reduction 47 Single Rule of Three Direct 53 — Inverse.... 56 Double Rule of Three 58 Practice 60 TareandTret 67 Simple Interest 70 Commission 71 Purchasing of Stocks 71 Brokerage 71 Compound Interest !.*.!!.* 74 Rebate or Discount i 75 Equation of Payments 76 Barter 77 Profit and Loss ."*..*.*.*'..! 79 Fellowship. gQ ^without Time ...!.!.' 80 with Time 82 Alligation Medial 83 Alternate 85 Position, or Rule of False 88 Double , 90 Exchange 91 Comparison of Weights and Mea- sures 95 Conjoined Proportion 96 Progression, Arithmetical 97 -Geometrical 100 Permutation 104 PART II.— VULGAR FRACTIONS. Reduction 106 Addition 112 Subtraction ,,,, = ,= 112 Multiplication 113 Division 114 The Rule of Three Direct. 114 'Inverse .... Ijd The Double Rule of Three 116 C0NTEKT8. PART III.— DECIMALS. Numeration.. 117 Addition 1 18 Subtraction 119 Multiplication 119 Contracted Multiplication 120 Division 121 Contracted 122 Reduction 123 Decimal Tablesof Coin, Weights, and Measures 126 The Rule of Three 129 Extraction of the Square Root 130 Vulgar Fractions. . . 13I Mixed Numbers 132 Extract of the Cube Root ...134 Vulgar Fractions . . . 136 Mixed Numbers. 13G Biquadrate Root 138 A general Rule for extrticti»g the Roots of all powers 138 Simple Intereiit 14C for days 14^ Annuities and Pensions, &c. in Arrears l-j|3f Present worth of Annuities 14* Annuities, &c. in Reversion IS0 Rebate or Discount 153 Equation of Payments 154 Compound Interest l&S A nnuities, &c. in Arrears 157 Present worth of Annuities IGO Annuities, &c. in Reversion 169 Purcha«ag Freehold or Real Es- tates 164 in Reversion 165 Rebate or Discount ..)66 PART IV.-~DUODECIMALS. Multiplication qf Feet and Inches, 163 Measuring tiy the Foot Square. , . 171 Measuring by the Yard Square. . . 171 Measuring t^ the Square of 100 Feet 173 Measuring by the Rod. 171 Multiplying sex-cral Figures by several, and the operation in one line only ,, . . 171. PART V.~aUESTIONa A "Collection of Cluestions, set down promiscuously for the greater trial of the foregoing Rules 17C A general Table for ctdeukCinf' Interests, Rents, Incomes and Servante' Wages I8l^ A COMPENDIUM OP BOOK-KEEI*lNa. tat> EXPLANATIOir OF THE CHARAOTIBt. Pftf« ctiHgthe 138~ \4$ 141^ , &c. in U$ iea 14* Ion 1&9 m 154 155 157 ies IGO on 1G9 164 n 166 16* u m urea by n in on* ralatini let dnr and .181' .I8ti I EXPLANATION OP THE CHARACTERS MADE USB OF IN THIS COMPENDIUM. KsEqual. — Minus, or Less. -+- Plus, or More. X Multiplied by. -t- Divided by. 2357 63 : : So is. 7_2-HS*=10. The Sign of Equality; as* 4 qrs.«=l cwu signifies that 4 qrs. are equal to 1 cwt. The Sign of Subtraction ; as, 8—2-6, that is, 8 lessened by 2 is equal to 6w The Sign of Addition ; as 4-f 4=6>, that is. 4 added to 4 more, is equal to 8. The Sign of Multiplication ; as, 4 X 6=24, that is, 4 multiplied by 6 is equal to 2i. The Sign of Division ; as, 8+2=4, that is, 8 divided by 2 is equal to 4. Numbers placed like a fraction do likeAvise denote Division ; the upper number being the dividend, and the lower the divisor. The Sign of Proportion ; as, 2 : 4 : : 8 : 16, that is, as 2 is to 4, so is 8 to 16. Shows that the difference between 2 and 7 added to 5. is equal to 10. Signifies that the sum of 2 and 5 taken from 0, is equal to 2. Prefixed to any number, signifies th* Square Root of that nu^iber is required. Signifies the Cube, or Thlnl Power. Denotes the Biquadnile» or Fourth Power, &c. i.e. id v^U that is. Ar bers, all it Nt MUL Teac and t THE TUTOR'S ASSISTANT BKf. -i A COMPENDIUM OP ARITHMETIC. PART I. ARITHMETIC IN WHOLE NUMBERS. THE INTRODUCTION. Arithmetic is the Art or Science of computing by Num- bers, and has five principal or fundamental Rules, upon which all its operations depend, viz : — Notation, or Numehation, Addition, Subtraction, Multiplication, and Division. NUMERATION Teacheth the different value of Figures by their different Places, and to read and write any Sum or Number. THE TABLE. m 52 = s c o . 9 8 7 9 -i 7 en m c c CO CO ii 3 s S O o » « C 3 2 1 3 2 1 H KUMERATION. consisting of three PiVurerorpN. % '•'"•''. Million. ; each of each from the lef *h»n^' '^= "''■ "««'""' the firat Piem, •i"e„s.andthethi?dassomfnv.t' T^"^ ""'"'""''• "•« "K •hem: ,h„,, the firs IWdoVthlleft'hr* '''"''''',''""«" °»" ^reda„dEighty.sevenMi,,t",f:„'5':„tt;:V^?r THE APPLICATION. WHt. aov,n in proper Figures tke following Number,.. < ( ) Twenty-three. n\ ^r ^^'!.'''■*•' *"'' Fifty-fonr. • rt'Z '^^""'^l'^- Two Hundred and Pour •nd Forty.five. '>"«y-««'» Thousand, Two Hundred fJ„7 «„":'"""• ^'"' ^»''-' ""O Fortyon. Thou.«,d. ThUaa^S. F?:i''Sl7red'^- "'""•"'• "^^ «-Sred and T,n I ) 800061057 (■•) aaiflooTOO I' One. n Two. ni Three. IV Four V Five. VI Six, VII 8even. Vni Eight Notation by Roman Letters. rX Nine. X Ten. Xr Eleven. XH Twelve. Xm Thirteen, XIV Fourten. XV Fifteen. XVi Si: ..en.. f) ADDITION OF INTEGERS. the right hand, » Millions; each I the first F\gu*"9 5ds, the next as It is written over ead, Nine Hud- my of the rest g" lumbers. d Fifty-six. Two Hundred ne Thousand, d Fifty-seven rivo Hundred ur. dred and Ten Numbers. •) 6207064 ') 2071009 ') 70064000* £1900700 e. en. n. XVII XVIII XIX XX XXX XL L LX LXX LXXX XC C CC Seventeen. Eighteen. Nineteen. Twenty. Thirty. Forty. Fifty. oixty. Seventy. Eighty. Ninety. Hundred. Two Hundred. ccc cccc D DC Dec DCCC DCCCC M MDCCCXII MDCCCXXXVII Three Hundred. Four Hundred. Five Hundred. Six Hundred. Seven Hundred. Eight Hundred. Nine Hundred. One Thousand. One Thousand Ei^ht Hundred and Twelve. One Thousand Eight Hundred and Thirty Seven. INTEGERS. ADDITION JAmal Sum.'^^ '""' '' "*''' ®""' '°^^'^^^» '^ -^ke one wholr &c ; then beginning with the first row of uiifs^add ^1;!; he top; when done, set down the Units, and carry the Teis to. Proof. Bc?in at the top of the Sura, and reckor»-thi» Fl»™-. AeTrnte Su™ •'"'' "' ^""Z"" "«'■" "P. and. ?;*:«rn we nrst, the bum is supposed to be ri.rht. Qrs. (')275 110 473 354 271 352 Months. £ r)1234 75215 ^098 37502 3314 91474 «'?32 32145 2646 47258 { ) What ,s the sum of 43, 401. 9747, 3464, 2203, 314, 974f ('•) Add 216031. 29S?fi5. 47l«»i r^^o o.^./I^aJJ^.. d 640 to Ans. 365. SUBTRACTION 'rS ii ^ ^k. ?^'^ «^^'^ _ ^f^ 3^7616 152471 3150»74 Hem. 117 ~" ~ > "■ Proof 871 MULTIPLICATION To this Rule belong three principal Members, viz. 3 The mS-"""''' "' ^r^"' '» ^ "•uUiplied. £391, and E Ans, 1528 Months ? Ans. 365. MULTIPLICATION OF INTEGERS. ^ id shows the nust borrow i rcmember- •gether, and 3750215 3150974 MULTIPLICATION TABLE. s given, as f perfonns VIZ. ng. nit's place the Unit's carry the ire in the ) the pro- down the itipiied. 2 3 4 ^ "f 8 9 10 11 12 2 4 6 8 3 6 9 12 4 8 12 16 6 10 15 20 6 12 18 24 7 14 21 27 8 16 24 32 9 18 28 36 10 20 30 40 11 22 33 44 12 24 36 48 10 12 16 18 20 24 25 30 30 36 35 42 40 48 45 54 50 60 55 66 60 72 14 16 21 24 28 32 35 40 42 48 49 56 56 64 63 72 70 80 77 88 84 96 18 20 27 30 36 40 45 50 54 60 63 70 72 80 81 90 90 100 99 110 108 120 22 24I 33 36 44 48 55 60 66 72 77 84 88 96 99 108 110 120 121 132 132 144 Multiplicand (>) 25104736 Multiplier 2 Product 50209472 (*) 52471021 (») 7925437521 4 037104107 (.) 931047 (•) ,098616 (,) 3,36104 (") 4215466 9 02701057 (^0)31040171 10 11 ^\ .^'?*[l*.^'f.''liiP"*' \' "i"'?*"- 12. and 168, than 20. m„UI. pz^ ^y wici^nu figure in the Multiplier, adding to the ProdZt the back Figure to that you miiUinlied. ^ ^'^ B3 \8 MULTIPLICATION OF INTEGER*. (^')5710502 (' = )6107252 ('3)7663210 (,»*) 02067165 13 14 16 16 (") 6251721 17 ('«) 9215324 18 (*') 2571341 19 ('«) 3592104 20 When the Multiplier consists of several Figures, there must be as many products as there are Figures in the Multiplier, ob- serving to put the first figure of every Product under that Figure you mu tiply by. Add the several Products together, and their feum will be the total Product. ( ' ') Multiply 271041071 by 5147. (' ° ) Multiply 62310047 by 1668, C ') Multiply 170925164 by 7419. (22) Multiply 9500985742 by 61379. (2 3) Multiply 1701495868567 by 4708756. il!}M^u-^-P^'T ^'"^ P^^,'^^ between the significant Figures ih the Multiplier, they may be omitted ; but great care must be taken hat the next Figure must be put one place more to the left hand, t. e, under the Figure you multiply by. (s *') Multiply 571204 By ST- OOD 5140836 3998428 1142408 Prodact 15427f)4883G (' ') Multiply 7501240325 by 57002. (2«) Multiply 562710931 by 590030. When there are Ciphers at the end of the Multiplicand or Mul- tiplier, they may be omitted, by only multiplying by the rest of tne l)Multiply921563by32. (32) Multiply 715241 by 56. (' ») Multiply 79&4956 by 144. DIVISION Teacheth to find how often one Number is contained in another ; or, to divide any Number into what parts you please. In this Rule there are three numbers real, and a fourth acci- dental : viz. 1. The Dividend, or Number to be divided: 2. The Divisor, or Number by which you divide : 3. The Quotient, or Number that shows how often the Divisor is contained in the Dividend : 4. Or accidental Number, is what remains when the work is finished, and is of the same name as the Dividend. Rule. When the Divisor is less than 12, find how often it is contained in the first Figure of the Dividend ; set it down under the Figure you divided, and carry^the Overplus (if any) to the next in the Dividend, as so many Tens ; then find how often the Divisor is contained therein, set it down, and continue the sani« 90 DIVISION OF INTEGERS. till you have gone through the Line; but when, the Divisor is more than 12, multiply it by the Quotient Figure ; the Product subtract from the Dividend, and to the Remainder brine down the next Figure in the Dividend and proceed a» before, till the Figures are all brought down. . " « Proof. Multiply the Divisor and Quotient together, adding rJ.® . ,*'T''^'^^^» (^^ »"y») a»4716. (!♦) Divide 4973401891 by 510834. (» 6) Divide 51704567874 by 4765043 (» •) Divide 17453798946123741 ' by 31479461. ^ When^there are Ciphers at the -^nd of the Divisor, they may ^e cut On, and as many places from oft the Dividend, but thev must be annexed to the Remainder at last. tAllLSS Ot MONEY. 21 {\y) 271|00V254rJ2l21(939 (i s) 6721 100)72534721 16(13e?7 ( I •) 3731000)75247^1729(2017 (a oj 2l5|(k)0)6326l04|997( 29419 When the Divisor is a composite Pumber, i. e. if any two Fi- ffUres, being multiplied together, will make that Number, then, by dividing the Dividend by one of those Figures, and that Quotient by the other, it will give the Quotient required. Bat as it some- times happens, that there is a Remainder to each of the Quotients, and neither of them the true one, it may be found by this Rule. Multiply the first Divisor into the last Remainder, to that Product add the first Remainder, which will give the true one. (SI) (9 9) Div. 3210473 by 27. 7210473 by 35. (9 8) 6251043 by 42. (9 4) 5761034 by 54. 118906 11 rem. 206013 18 rem. 148834 15 rem. 106685 44 rem. MONEY. Marked i Farthing 4 Farthings make 1 Penny d. t Halfpenny 12 Pence 1 Shilling s. i Three Farthings 20 Shillings 1 Pou»d ?.......£ Farthings 4 = 1 Penny 48 = 12 = 1 Shilling 960 = 240 = 20 = i Pound. 8HILUN08. 8. 20 30 40 60 60 70 80 90 100 IIQ 120 130 £ 1 1 2 2 .3 3 4 4 5' e i I I PENCE TABLE. 8. 10 10 10 10 10 10 d. 8. d. 20 . • 1 I 8 24 • ' 2 : 30 • • 2 ; 6 36 • • 3 ! 40 • • 3 . ! 4 48 . • 4 i 60 • • 4 i 2 60 .. 6 : 70 .. 5 : 10 72 .. o ; u 80 •• 6 : 8 ^4 .. 7 J d. 00 06 100 108 110 120 130 132 140 144 150 160 8. d. 7 ! e 8 : 8 s 4 t s 2 10 ! 10 ! 10 11 i 11 : 8 12 i 12 C2 TABLES OF WElOIfTg. 24 O raing mi THOY WEIGHT. ...I Pennyweight. Marl (^ 20 Pennyweirhli ....1 Ounce oT ^•-^ ^«nces 1 pt,und ".'.'.'.lb' \»rains 9A s: 1 Pennvw^iffht 5?(,0 240 = 12 = 1 Pound By this Weight are wrlghed Gold, Silver, Jewels, Electuaries and all Liquors. N. B. The Standard for Gold Coin is 22 Carats of fine Gold and Z Carats of Copper, melted together. For Silver, is 1 1 oi 4 dwts. of fine Silver, and 10 dwts. or Copper. 25 lb. is a quarter of 100 lb. 1 cwt. 20 cwt. 1 Ton of Gold or Silver. AVOIRDUPOIS WEIGHT. Marked 16 Drams make 1 Ounce I? ^""'^^s 1 Pound lb -^ 1^0"" ^^ I Quarter '"qr 4 Quan rs or 112 lb 1 Hundred Weight ",lwt ed Weight 1 Ton ton ^dr. S oz. 20 Hun 'r .Drams 16 = 256 = 7168 = 28672 = 1 Ounce 16 = 1 Pound 448 -= 28 = 1 Quarter n^Q^^A o32?^ " "^ = ^ = 1 Hundredweight 573440 = 35840 = 2240 = 80 = 20 = 1 Ton. There are several other Denominations in this Weight thai tre used m some pa ticular Goods, viz. lb. . Ij, A Firkin of Butter 56 A Stone of Iron, Shot or } , *A B 1 . '"^P ^"^ Horseman's wt I ^"^ A Barrel of Anchovies 30 Butcher's Meat... 5 ^oap 2.56 A Gallon of Train Oil... . 74 Raisins 112 A Truss of Straw 36 A Puncheon of Prunes. JJ20 New Hay 60 urn liav 56 qrs. A) Tru !?.«es a Load ay. i^-. 4 Nails i^ Marl o4 lb. Electuaries )f fine Gold er, is 1 1 oz Marketf >dr. ) oz. lb. qr- ..,....cwt. ton. TABLES OF WElOHTi. Cheese and Butter. . «' . c n. ..^ ^'"''*' "^ Half Stone, 8 lb A \\«y m Snn.lk, > lb. A Way in Essex, J^Uovea, or J 2&6 3iiCluros,or . ^, ^b. A Wey is ti Tods and ^9'^^^ '3' 1 Stone, or ^ ,^^7^ 14 A Sack is 2 Woys, or ^ *"" 28 A Last is 12 Sacks, or 4369 By this Weicrht is wf^ighed anything of a coarse or drossy na- ture ; as ull Grocery ami (Chandlery Wares; Bread, and all Me- tals but Silver aed (lold. Note. One Pound Avoirdupois is equal to 14 oz. 11 dvvts. 15i grs. Troy ^ $330 nb, S 182 304 APOTHECARIES' WEIGT. Marked 3 It 5 .., lb 20 Grains make 1 Scruple 3 Scruples i Dram ,5 J""^"^' 1 Ounce .*;; 13 Ounces i YowxiA Graias 20 = 1 Sruple r>0 = 3 = 1 Dram 480 = 24 = 8 = 1 Ounce 5760 = 288 = 96 = 13 = 1 Pound. Note. The Apothecaries mix their Medicines bv this Rule, but buy and sell their commodities by Avoirdupois Weight. OnT^^ iP*'*^^^"^ ' P«"»d a»^ Ounce, and the Pound and Ounce Troy, are the same, only differently divided and subdivi- OLOTH MEASURE. 4 Nails make ...l Marked Quarter of a Yard 4 g"*^^^''^ 1 Flepnsh Ell :.F1. E. 4 Quarters i v^„i > Quarters , ^ ,. . — z '> Q.uarterfi.., * • * •««.«« ■ 1 English Ell E. E. 1 French Ell Fr. E 24 TABLES OF MEASURES. Inuhes 2i = 1 Nail 9 = 4 = 1 auarter 36 = 16 = 4 = 1 Yard aI =1? = 3 = 1 FlemishEU 45 = ^ = 5 =, 1 EnglishEU 54 = 24 = 6 = i FrenchEU. LONG MEASURE. < 1 Marked 3 Barley Corns. make l Inch > bar. c. 12 Inches... , „ lin. 3 Feet?!;.- :;:•:;;::•• I ?S -St 6 Feet J ^wd j 5* Yards ;;•• - } p!?'^'-,- n* * u' ^h. 40 Poles • ; Rod P^le or Perch rod, p. SFuriongs...... J S?r'°°g fur "^ 3xMile8.T... • ; f*^^ mile. ^ I^egree j^. Barley Corns ^3 = 1 Inch 36 = 12 = 1 Foot 108= 36= 3= lYard 594 = 198 = 164= i_ 1 p^, iS - S = ..^ = ^ = ^ = ' ^-long lyuObO = 63360 = 5280 = 1760 = 320 = 8=1 Muf WINE MEASURE. i> n. Marked ^ ^"*« n«Ae 1 auart ) pt«. iJ 9SSf; 1 GaUon ""^^ 18 gZI::: j Anker of Bra,id>:::::::::S*. 42 Gallons..... * -Half an Hogshead jhhd. 63 Gallons .' , « "if';'"' ^er. 2 pip^or4Hog8'h;«is'.'.::::;::::} tZ^'.T::.::::::: f-"®- Marked } bar. c. ) in. ..feet ..yd. ..fth. .rod, p. ..fur. ..mUe. ..lea. lOUgh com Bs, or any Marked •Jqts. ...gal. . . .anjt* ...ran. • ..} hhd* > . . tier. . . . una. . ..P. or B. TABLES OP MEASURES. 4jg Inches* 28|= 1 Pint 571= 2= 1 Quart oSJ = oo®= 4= 1 Gallon 9702 = 336= 168= 42=1 Tierce iS = S2i= 2^2= 63=lj=i Hogshead 19404 = 672= 336= 84=2 =l*-i Punchpon 29106 =1008= 504=126=3 =2 = U=i P^'°" 58212 =2016=1008=252=6 =4 =3 =2=l^un ,„:^n-?'^"^''''' ®P'"?' ^^"5^' ^^^«^' Mead, Vinegar, Honev .fat ?u^; :u:to~i;' '' ^'^^ "^^^^-^^ ^ ^^ ^^- ^"^^ -^ ^x ALE AND BEER MEASURE. 2 Pints make. lauart &t:::;::::::;: ip£„fAV-'- Firkins, or 2 Kilderkins.::::::;} bS 4 8 9 2 • • • • • • Barrels, or 2 Hogsheads / : : : : : : i Butt . Marked >pt8. •5qt8. .gal. .A. fir. .B.fir. .kil. .bar. .hhd .pun. .butt. _, BEERa Cubic Inches 35i= iPint 70i= 2= lauart 2538 = 72= 36= 9= 1 Firkin iS =iM= J?= 18= ^Joderkin 10152 =288=144= 36= 4=2-1 BanS 15228 =432=216- 54- 6-tii i w u , 20304 =576=4= jt ld-2*-u?fcl 30456 =864=432=108=12=6=1 =2*=lj=TBr rt ,. ALE. Cubic Inches 35i= 1 Pint JOh= 2= 1 Quart 2^ = 8= 4= 1 Gallon !I?S =ii= ^2= 8=1 Firkin 4513 =128= 64=16=2=^1 Kilderkin 9024 =256=128=32=4=2=1 BaS 13536 =384=192=48=6=3=U=YHo.sh.ad. * By a ip.te Act of ParliamAnf »h« -o^^-.-,: i-.u. ,,.; ^ ~ S«U'm*"""«' '^^^^ been "ri"(inced to 'o7e Standard p!!^ '""^ ^'^ *'"* ^«"' «""» "»• hese Measures, with the old stamlard^Ieasures thp ^f,!^?/ ,"■" «""'•'»'« comparison of the "/n.p«nai Measures," at tlie bSig o^the work " ''^'^"'^ '" *^« ^*'''« < 26 TABLES OF MEASURES. In London they compute but 8 gallons to the firkin of Ale, and 32 to the barrel ; but in all other parts of England, for ale, strong beer and small, 34 gallons to the barrel, and 8^ gallons to the firkin. N.B.— A barrel of salmon, or eels, is 42 gallons, A barrel of herrings 32 gallons. A keg of sturgeon 4 or 5 gallons. A firkin of soap 8 gallons. DRY MEASURE. Marked 2 Pints make 1 Quart } P**' f. ^ ' S qts. 2 Quarts 1 Pottle .pot. 2 Pottles 1 Gallon Jal. 2 Gallons ^,...1 Peck Jk. 4 Pecks 1 Bushel bu. 2 Bushels 1 Strike strike. 4 Bushels 1 Coom cooai. 2 Cooms, or 8 Bushels 1 duarter qr. 4 Quarters 1 Chaldron ....'..• ! ."chal. 5 Quarters 1 Wey wey. 2 Weys I Last .laat In London, 36 bushels make a chaldron. Solid Inches 268t= 1 Gallon 537f = 2= 1 Peck 4= 1 Bushel 215af= 8= 4300f= 16= S= 2= 1 Strike 8601f= 32= 16= 4= 2= 1 Coom 17203i= 64= 32= 8= 4= 2= 1 Quarter 86016 =320=160=40=20=10= 5=1 Wey 172032 =640=320=80=40=20=10=2=1 Last. The Bushel in Water Measure is 5 Pecks. A score of coals is 21 chaldrons. A sack of coals 3 bushels, A chaldron of coals 12 sacks. A load of corn 5 bushels. A cart of ditto 40 bushels. ihis measure is applied to ail dry goods. The standard Bushel is 18^ inches wide, and 8 inches deep ■I TABLES OP MEASURES. 27 TIME. 60 Seconds make....l Minute.... 60 Minutes 1 Hour ^ Hours 1 Day J Jays 1 Week ,o J:®®¥ 1 Month U Months, 1 day, 6 hours . . 1 Julian Year Marked ) m. hour. day. week. .mo. .yr. Seconds 60= 1 Minute 3600= 60= Hour 86400= 1440= 24= 1 Day 604800= 10080= 168= 7=1 Week 2419200= 40320= 672= 28=4= 1 Month. d. h. w. d. h. 31557600=525960=8766=365 : 6=52 : 1 : 6=1 Julian Year o... d. h. m. " 31556937=525948=8765=365 : 5 :48 : 57=1 Solar Year. To know the days in each month, observe, Thirty days hath September, April, June, and November, February hath twenty-eight alone. And all the rest have thirty and one ; Except in Leap- Year, and then's the time February's days are twenty and nine. SQUARE MEASURE. *^ {»<^^» make 1 Foot ,n? £""! •• 1 Yard. ^*F^ :::::::::;;• •-•} |2«'«««ffloom* 40 Rods ...'.V.V.V.V.V.V.'.*"* 1 rS 4 Roods, or 160 Rods, or 4840 vaids-V" ' * ' " i A^^\f i-«; 640 Acres 30 -'sv z^z 2eu£u« t:Z 1 Squaw MUe. ^^ Acres V.V.V.'.Vl Hide C3 1 Yard of land, of land. .88 ADDITION OF MONET. Inches 144= 1 Foot 1296- 9 = 1 Yard 39204= 272|= 30^ 1 Pole 1568160=108<.)0 =1210 = 40=1 Rood 6272^=43560 =4840 =160=4=1 Acre. By this measure are measured all things that have length and breadth ; such as land, painting, plastering, flooring, thatching, plumbing, glazing, &c. SOLID MEASURE. 1728 Inches make 1 Solid Foot. 27 Feet 1 Yard, or load of earth. 40 Feet of round timber, ) . , rr-^^ „, t „„,i Or, 50 Feet of hewn timber, \ '* ^ ^°" °' ^'^• 108 Solid Feet, i. e. 12 feet in length, 3 feet in breadth, and 3 deep, or, commonly, 14 feet long, 3 feet 1 inch broad, and 3 feet 1 inch deep, is a stack of wood. 128 Solid Feet, i. e. 8 feet long, 4 feet broad, and 4 feet deep, is a cord of wood. By this measure are measured all things that have length, breadth, and depth. ADDITION OF MONEY, WEGHTS, AND MEASURES. Rule. Add the first row or denomination together, as in In- tegers, then divide the Sum by as many of the same denomination as make one of the next greater, setting down the Remainder under the row added, and carry the Quotient to the next superior denomination, continuing the same to the last, which add as in simple Addition. MONEY. (») («) («) ^ (*) £ s. d. £ s. d. £ s. d. £ s. d. 2 .. 13 .. 54 27 .. 7 .. 2 35 .. 17 .. 3 75 .. 3 .. 7 7 . . 9 . . 4* 34 . . 14 . . 7i 59 . . 14 . . 7i 54 . . 17 . . 1 5 .. 15 .. 4i 57 .. 19 .. 2i 97 .. 13 .. 5i 91 .. 15 .. 4^ 8 .. 17 .. 6i 91 .. 16 .. 1 37 .. 16 .. 8t 35 .. 16 .. 5| 7 .. 16 .. 3 75 .. 18 ., 71 97 .. 15 .. 7 29 .. 19 ..7* 5 .. 14 ,. 71 97 .. 13 .. 5 59 .. 16 .. 5i 91 .. 17 .. 3i ADDITION OF WEIOHTSi' 2J>' MONEY. length and thatching, £ 257 734 695 159 207 798 n s. 1 .. 3 .. 5 .. 14 .. 5 .. 16 .. d. 6i 7f 3 7i 4 7i £ 525 179 250 975 254 379 n s. .. 2 .. 3 .. 4 ,. 3 ,. 4 d. 4* 5 7i 5i 7 51 £ 21 75 79 57 26 54 <') s. 14 . 16 . 2, 16 . 13 . 2 . d. 7* 4i 5i 8J 7 £ s. 73 .. 2 25 . . 12 96 . . 13 76 97 54 17 14 11 H 5i 3i li 7i idth, and 3 , and 3 feet I feet deep, ve length, ASURES. ', as in In- lomination [lemainder Kt superior 1 add as in £ : 127 .. 4 , (•) 525 271 624 379 215 3 9 4 5 • • ^C • c {') oz. (iwt. 5 .. 11 7 .. 19 3 .. 15 7 .. 19 9 . 18 8- . 13 d. 7i 5 5 1 3i 8i £ 261 379 257 184 725 359 8. .. 17 , ..13 ,. 16 , .. 13 , . 6 . d 1* 5 71 5 3i 5 £ 31 75 39 97 36 24 {'') 1 13 19 d. n 1 7i 17.. 3i 13 .. 5 16 .. 34 TROY WEIGHT. . (') Sf- lb. oz. dwt. 4 7 .. 1 .. 2- 21 3 .. 2.. 17 14 5 .. 1 .. 15 23 7 .. 10 .. 11 15 2 .. 7 .. 13 12 3 .. 11 .. 16 £ 27 16 9 15 37 56 «. . 13 . . 12 . . 13 . . 2 . . 19 . . 19 . d. 5i 9i 3i 7i 1 If lb. oz. (Iwt, gr. 5 .. 2 .. 15 .. 22 3 .. 11 .. 17 .. 14 3 .. 7 .. 15 .. 19 9 .. 1 .. 13 .. 21 3 ..' 9.. 7 .. 23 5 .. 2 .. 15 .. 17 AVOIRDUPOIS WEIGHT. (*) s. d. . 3 .. 7 . 17 .. 1 . 15 .. 4* . 16 .. 5} . 19 .. 7i . 17 .. 3* (') lb. oz. dr. 162 .. 15 .. 15 272 . . 14 . . 10 303 .. 15 .. 11 255 . . 10 . . 4 173 . . 6 . . 2 635 . . 13 . . 13 C) cwt, qrsi lb. t. 25 . . 1 . . 17 7 7i . . 3 . . 26 5 M .. 1 .. 16 2 24 .. 1 .. 16 3 17 .. .. 19 7 55 . . 2 . . 16 8 (») cwt. qrs. II). 17.. 2 . 12 5 .. 3 . . 14 4 .. 1 . . 17 18 . . 2 . 19 9 .. 3 . .20 C3. 30 ADDITION OF MEASURES. APOTHECARIES' WEIGHT. lb. 17 9 27 9 37 49 . (0 oz. dr. scr. 10 •• 7 . • 1 5 •• 2 . • 2 11 .. 1 .. 2 6 •• 6 •• 1 10 .. 6 .. 2 .. 7 .. (^) Fl.E qr. n 127 .. 2 .. 1 15 •• 1 •• 3 237 .. •• 2 52 .. 1 .. 3 376 •• 2 .. 1 197 •. 1 •• 3 (») yd. feet m bar. ■ 225 • • 1 « . 9 •• 1 171 • • •• 3 •• 2 62 •• 2 • • 3 •• 2 397 • • • • 10 •• 1 164 .. 2 .. 7 •• 2 137 • • 1 • • 4 •• 1 (^) lb. 7 .. OZ. dr. scr. «. 2 .. 1 .. ..12 3 •• 1 •• 7 .. 1 .. 17 9 •• 10 .. 2 .. .. 14 7 .. 5 " 7 .. I .. ifi 3 .. 9 •• 5 .. 2 .. 13 7 .. 1 •• 4 .. 1 .. 18 CLOTH MEASURE. yd. 135 70 95 176 26 279 LONG MEASURE. (') qr. n. E.E. 3 •• 3 272 . 2 •• 2 152 . 3 •• 79 . 1 .. 3 166 • .. 1 79 . 2 .. 1 164 . qr. n. 2 •• 1 1 •. i .. I 2 •• 3 •• 1 2 •• 1 lea. m. far. no. 72 .. 2 .. 1 .. ii9 27 .. 1 .. 7 .. 22 35 .. 2 .. 6 .. 31 79 .. .. 6 .. 12 51 •• 1 .. 6 .. 17 72 .. .. 6 .. 21 (») a. r. p. 726 •• I .. 31 219 .. 2 •• 17 1455 •« 3 .. 14 879 .. 1 .. 21 1195 •" 2 .. 14 LAND MEASURE. («) a. r. 1232 1 327 131 2 ■ 1219 1 . 459 2 . p 14 19 15 18 17 ADDITION OF MEASURES. WINE MEASURE. 31 (f) tlr. scr. ^y. 1 •• .. 12 7 .. 1 .. 17 2 •• .. 14 7 •. 1 .. Ifi 6 •• 2 .. 13 4 •• 1 .. 18 2.E. i72 .52 79 56 79 54 in qr. n. • • 2 ♦• 1 • . 1 . . 3 • • •• 1 .« 2 .. .. 3 .. 1 • . 2 •• 1 (») hhds. gals. qts. 31 . . 57. . 1 97 . 18. . 2 76 . . 13. . 1 55 . 46 . . 2 87 . . 38 . . 3 55 . . 17. . I t, hhds. gals. qts. 14.. 3 .. 27.. 2 19. .2 .. 56 . . 3 17. . .39 . . 3 79 . . 2 .. 16 . . 1 54 . 1 .. 19 . . 2 97. . 3 ..54. . 3 ALE AND BEER MEASURE. (») A.B. fir. ftal 25 . . 2 . .7 17.. 3 . . 5 96 . . 2 . . 6 75 . . 1 . . 4 96 . . 3 . . 7 75 . . . . 5 (2) B.B. fir. gal. 37. . 2. . 8 54. . 1 . . 7 97. . 3. 8 78. . 2. 5 47. . 0.. 7 35- • 2 .. 5 (») hhds. gals. qts 76 . . 51 . . 2 57. . 3 . . 3 97. . 27 . . 3 23. . 17. . 2 32. . 19 . . 3 65.. 38. .3 lur. po. •• 1 •• 19 •• 7 .. 22 •• 6 .. 31 .. 6 .. 12 . • 6 .. 17 .. 5 .. 21 («) r. P .. 1 .. 14 .. .. 19 .. 2 .. 15 .. 1 .. 18 .. 2 .. 17 ch. 75 , 41 29 , 70, 54 , 79 , (0 bu. pks . 2 . 1 . 24 . 1 . 16. . 1 . 13 . 2 . 17 . 3 . 25 . 1 (M w. d. h. 71 .. 3 ,. 11 51 . . 2 . . 9 76 .. .. 21 95 . . 3 . . 21 79 . . 1 . . 15 DRY MEASURE. TIME. (») last. wey. q. bu. 38 .. 1 .. 4 .. 5 pks .. 3 47 . . 1 . . 3 . . 6 .. 2 62 . . . . 2 . . 4 .. 3 45 . . 1 . . 4 . . 3 .. 3 78.. 1.. 1 ..2 .. 2 29 .. 1 .. 3 .. 6 .. 2 . C^) w. d. h. m. »» 57 . . 2 . . 15 . . 42 . . 41 95 .. 3 .. 21 .. 27.. 51 76 .. .. 15 •• 37.. 28 53 .. 2.. 21 .. 42.. 27 9H = = 2 . . 18 . , 47 . . as 32 ADDITION. THE APPLICATION. of ag^™*" """ *""■" '" *" y*" '''^' «'••«" "-i" he be 47 year. nineteen Dom,;), LlT? '' *'^ " i* = }"' ■""* <■<»"• ^eor* and he lend in^X ' ■'"'f-'-S'""^''' ""d » «h"ling. How much did 4. What is the estate worth per annu^.l^hen'^^e ia™ 31 guineas, the neat income 8 score, £19 : 14? 5. There are three numbers : the first i, ai'^'^t' ^^^ '}^\^ andthe third is as much as the'clt^'^o:' ^fe^^'ot 6. Bought a parcel of goods, for which I naid 1^4 .\w 7. There are two numbers, the least whefeTfif 4V tWr^dif t^^'o'f bothT'^^ '" """^^ -'^' " "•« ^-'- — J:a1i^- a A gentleman left his ellTda^Vh?ef il5W mL'^.b"'";. younger, and her fortune was 11 thoS.ll h^nleLnd Vn What was the elder sister's fortune, and what did Sfetthe"'] fav, Ans. Elder sister's fortune, £1361 1. Q A 1.1 , , . Father left them £25722. 9. A nobleman, before he went out of town, wa7dSrn,„ ,f paying all his tradesmen's bills, and upon inZirThe found th/, he owed 82 guineas for rent ; to his wine-merSt £7? 5 O 10 his confectioner, £12 : 13 : 4 ; to his draper £47 -iV. 9. , ' his tailor, £110 : 15 : 6 ; to his coach-make? £157 S '• n W° jallow-chandler £8 : 17 : 9 ; .0 his cor^.ch;„d,er,£I70 '• 6 s" his brewer £52 : 17 : 0; to his butcher,£I22 • 11 • 5'. tohl' d.',L^\= ^ •■ "'.^""^ '" '•'^ ^"™"'«' <■" wage , £53 :'l8 ie'"reH'",1"Z '!!!l!"r-?_''^''.»i '" ™- in the who.;. whe"n with him"' ""■"'" ' •^'""' '"^"f "« ^''hed to take • ^»s. £1032. 17: 3. ADDITION. 33 10. A father was 24 years of age (allowing 13 months to u vcar, aild 28 days to a month) when his first chiltl was born ; between the eldest and next born was I year, H months, I i days ; between the second and third were 3 years, 1 month, and 15 days >etwcen the third and fourth were 2 years, 10 months, and 25 (la> ; when the fourth was 27 years, 9 months, and 12 Jays bid, how old was the father ? , '. , , , >lw,9. ,5S years, 7 months, 10 days. 11. A banker s clerk having been out with bills, brino-s hom?. an account, that A paid him £7 : 5 : 2, B £15 : Is": 6i C £150 : 13 : 2^, D £17 : : 8, E 5 guineas, 2 crown pieces, 4 half-crowns, and 4s. 2d., F paid him only twenty groats, G £7t$ 15 : 9i, and H £121 : 12 : 4. I desire to know how much the whole amounted to, that he had to pay ? ,o A ,1 1 . .1715. £396 : 7 : 6|. 12. A nobleman had a service of plate, wliich consisted of twenty dishes, weighing 203 oz. 8 dwts. ; thirty-six plates, weigh- ing 408 oz. 9 dwts. ; five dozen of spoons, weighino- 1 12 oz. 8 dwts. ; six salts, and six pepperboxes, weighing"?! oz. 7dwt8. ; knives and forks, weighing 73 oz. 5 dwts. ; two large cups, a tankard, and a mug, weighing 121 oz. 4 dwts. ; a tea.kettle and lamp, weighing 131 oz. 7 dwts. ; togc'.her with sundry other small articles, weighing 105 oz. 5 dwts. I desire to know the weight of the whole ? ,„ ., ^ , ^715. 102 lb. 2 oz. 13 dwts. 13. A hop-merchant buy? five bags of hops, of which the first weighed 2 cwt. 3 qrs. 13 lb. ; the second, 2 cwt. 2 qrs. 11 lb. • the third, 2 cwt. 3 qrs. 5 lb. ; the fourth, 2 cwt. .3 qrs. 12 ». d. 87.. 2.. 10 79 .. 3.. 74 (*) £ s. d. 3.. 15 .. U 1 .. 14 .. 7 («) £ *. d. 25 . . 2 . . 5^ 17 . . 9 . . 8i £ a. d. 37..3..4J 25.. 5.. 2i £ ». d 321 .. 17.. U 257 .. 14. .7 £ 8. d, 59 . . 15 . . 3i 36 .. 17.. 2 (•) £ a. d. 71 .. 2.. 4 19 .. 13 .. 7* (10) £ a. d, 527.. 3 .. 6| 139 .. 5 .. 7J £ s. d. Borrowed 25107 . . 15 . . 7 375 .. 5 .. 5i Paid 259 .. 2 .. 7* at 359 . . 13 .. 4} different 523 .. 17 .. 3 times 274 .. 15 .. 7* 325 .. 13 .. 5 Paid in all Remains to pay (IS) £ a. d. Lent 250156 .. 1 .. 6 7i 3 9« 3i 5 271 .. 13 Received 359 . . 15 at 475 . . 13 several 527 .. 15 payments 272 .. 16 150 .. \ie lower many of e upper, nd carry wed. d. SUBTRACTION. TROY WEIGHT. (I) B*>"ght52..1. 7..2 0) 7.. a.. 2. .^7 Sold 39 . . . . 15 . . 7 5 . . 7 . . 1 . . 5 Unsold AVOIRDUPOIS WEIGHT. lb oz. dr. cwt. qra. lb. t. cwt. qrs. lb. ^ ^ S-- J2--S <") 35..!.. 21 (8) 21..1..^2..7 29 .. 12 .. 7 25 .. 1 .. 10 9 .. 1 .. 3 .. 5 APOTHECARIES WEIGHT. ,,, 't o^- ' H^D ..5.. Fl.E. qr. n. (») 35. .2. .2 17.. 2.. 1 E.R t qr. n. (») 35.. 2.. 1 14 . 3 . . 2 1 (10) . 3 .. d 1 7» ■ 1 .5.. yds. ft, in. (>) 107. .2. .10. 78 .. 2.. 11 . LONG MEASURE. bar. .2 ' lea. nu. fur. po. 147.. 2.. 6.. 29 58 . . 2 . . 7 . . 33 . 6 1 1 . 3* . 9f i H . a. r. p. (^) 175.. l..§7 59..0..27 LAND MEASURE, a. r. p. («) 325.. 2.. 1 279.. 3,. 5 n * 11 !■ 3o t^vtM nxrrios. WINL .MKASUUE. hlid. gal. qts. pt. («) 47.. 47.. iJ .. 1 28 . . 59 . . 3 . . tun. hlid. gal. qt (8) 42.. 2. .^7. .2 17.. 3. .49. .3 ALE AND REER MEASURE. A.B. fir. gal. (>) 25.. 1 .. 2 21 .. 1 .. 5 BR. nr. gal. ( ' ^ :>7 . . '2 . . 1 25 . . 1 . . 7 hhd. gal. q ' (8) 27 .. 27 .. 1 12 . . 50 . . 2 qu. bu. p. (1) 72 .. I .. 2 CD • . /O . . <> DRY MEASURE. (2) 05 bu. , 2 , 2. P- 1 3 eh. bu. p. (3) 79 .. 3 .. 54.. 7.. 1 yrs. «no, w. <.lg, (>) 79 .. 8 .. 2.. 4 23.. 9 .. 3 5 TIME. ho. min. (8) 24.. 42.. 45 19.. 53.. 47 THE APPLICATION. 1. A man \va« born in the year 1723, what was I^s age in the year 1781 1 Ans. 58. 2. What vi the (iiflfcience between the age of a man born in 1710, and an- other born in iToG 1 Ans. 56. 8. A McrclKsnt hud five debtors, A, B, C, D, and E, who together owed hira £1156; B, C, D, and £, owed him £737. What was A.'s debf? ilns. £419. 4. When an rstatc of £300 [ut annum, is reduced, on the paying of taxcn to 12 score and i- 14 : i> What is the lax ] Ans. £45 : 14. COMPOUND iMULTIPMCATIOX. 31 5. What U the dinbrcncc between £9164, aiui the amount of £754 acKlod lo £305? Ana. £8095. G. A horse in his furniture is worth £37: 5; out of it, 14 guineas ; how much docs the price of the furniture exceed that of the horse 1 Ans. £7 : 17. 7. A merchant, at his out-scttino; in trade, owed £750; he had in cash, commodities, the stocks, and good debts, £12510: 7; he' cleared, the first year, by commerce, £452 : 3 : 6 j what is the neat balance at tae twelve montlis' endl ilnj. £12212: 10:6, 8. A gentleman dying, left £452^17 between two daughters, the younger was to have 15 thousand, 15 hundred, and twice £15. What was the elder sister's fortune '] Ans. £28717. 9. A tradesman happening to fail in business, called all his creditors to- gether, and found he owed to A, £63 : 7 : 6 ; to B, £105 : 10 : to C, £34 : 5 : •2 ; toD, £28 : 16 : 5; to E, £14 : 15 : 8; to F, £112 : 9; and to G, £143 : l-i : 9. His creditors found the value of his stock to be £212 : 6, and that he had owing to him, in good book debts, £112 : 8 : 3, besides £21 : 10: 5 mo- ney in hand. As his creditors took all his effects into their hands, I desire to know whether they were losers or gainers, and how much 1 Ans. The creditors lost £146: 11 : 10. 10. My correspondent at Seville, in Spain, sends me the following account of money received, at different sales, for goods sent him by me, viz : Bees- wax, to the value of £37 : 15 : 4; stockings, £37 : 6 : 7 ; tobacco, £125 : 11 : 6; hneu cloth, £112:14:8; tin, £115:10:5. My correspondent, at the same time, informs me, that he has shipped, agreeably to my order, winps to the value of £250 : 15 ; fruit to the value of £5l : 12 : 6 ; figs, £19 : 17 : 6 •, oil, £19 : 12 : 4 ; and Spanish wool, to the value of £115 : 15 : 6. I desire to know how the account stands between us, and who is the debtor 1 Ans. Due to my Spanish correspondent, £28 : 14 : 4. MULTIPLICATION OF SEVERAL DENOMINATIONS. Rule.— Multiply the first Denomination by the quantity given, divide the product by as many of that as make one of the next, set down the remainder, a.nd add the quotient to the next superior, after it is multiplied. If the given quantity is above 1^ multiply by any two numbers, which raul- jp>iru t.ogcttier whi maKe tue same nutubcr ; but if no two numbers multipiit^.ti together will make the exact number, then multiply the top Une by as many «s m wanting, adding it to the last product. D 38 COMPOUND MULTIPLICATION. Proof. By Division. £ s. d. 35: 12: 7} 2 71: 5:2J per yard. 9x2=18 £ s. d. 75:]3:H 3 (3) (♦) £ s. d. £ «. d, 62:5:4i 57 : 2 : 4f 4 5 1, at 9s. 6d. 9 2. 26 lb. of tea, at £1 : 2 : 6 per lb. 8 ftvS-l- —26 4:5:6 2 ^ 9:0:0 3 8: 11:0 27 : ; Top line X 2 = 2 : 5 : 29 : 5 : 3. 21 ellsofHolland, at7s. 8|d. perell. Facit, £8:1: 10^. 4. 36 firkins of butter, at 15s. 3^(1. per fii^kin. Facit, £26 : 15 : 2|. 5. 15 lb. of nutmegs, at 7s. 2^d. per lb. Facit, £27 : 2 : 21. 6. 37 yards of tabby, nt 9s. 7d. per yard. ^ Facit, £17 : 14 : 7. 7. 97 cwt. of cheese, at £1 : 5 : 3 per cwt. ^,^„ . _ Facit, £122 : 9 : 3. 8. 43 dozen of candles, at 6s. 4d. per doz. „ ,^ . Facit, £13 : 12 : 4. 9. 127 lb. of Bohea tea, at 12s. 3d. per lb. ^^^ _ ^ Facit, £77 : 15 : 9. 10. 135 gallons of rum, at 7s. 5d. per gallon. " Facit, £50 : 1 : 3. 11. 74 ells of diaper, at Is. 4|d. per ell. . ^^ . ^ Facit, £5:1:9. 12. 6 dozen pair of gloves, at Is. lOd. per pair. Facit, £o : 1-5. When the given quantity consists of i, i, or ^. Rule. Dlvitle the given price (or the price of one) by 4 for i, by 2 for h and for i, j»ro.iuo first divide by 2 tor i, then divi<1e that quotient by 2 for J, add them to tho t. and Cicir sum will he thc^ niiswor rctumw COMPOUND MULTIPLICATION. 13. 25i ells of holland, at 3 : 4^d. per ell. 5_ 5X5=25 16 : 10^ 6_ 4:4: 4|=25 0:1: 8^=^ 39 4:6: 0^=25^ 14. 75i ells of diaper, at Is. 3d. per ell. Facit, £4:14:4i 15. 19A ells of damask, at 4s. 3d. per ell. Facit, £4:2: lOJ. 16. 354 ells of dowlas, at Is. 4d. per ell. Facit, £2:7:4. 17. 7i cwt. of Malaga raisins, at £1 : 1 : 6 per cwt. Facit, £7:15:10^. 18. 64 barrels of herrings, at £3 : 15 : 7 per barrel. Facit, £24:11 :3^ 19. 354 cwt. double refined sugar, at £4 : 15 : 6 per cwt. Facit, £169 : 10 : 3. 20. 1544 cwt. of tobacco, at £4 : 17 : 10 per cwt. Facit, £755:15:3. 21. 117i gallons of arrack, at 12s. 6d. per gallon. Facit, £73:5: 7i. 22. 851 cwt. of cheese, at £1 : 7:8 per cwt. Facit, £118:12:5. 23. 291 lb. of fine hyson tea, at £1 : 3 : 6 per lb. Facit, £34 : 7 : 4^. 24. 171 yards of superfine scarlet drab, at £1 : 3 : 6 per yard. Facit, £20:17:1^. 25. 374 yards of rich brocaded silk, at 12s. 4d. per yard. Facit, £23 : 2 : 6. 26. 56f cwt. of sugar, at £2 : 18 : 7 per cwt. Facit, £166:4 :7i. 27. 964 cwt. of currants, at £2 : 15 : 6 per cwt. Facit, £267 : 15 : 9. 2a 451 lb. of Belladine silk, at 18s. 6d. per lb. bushels of wheat, at 4s. 3d. per bushel. 87| per D2 Facit, £18:12: lU 40 COMPOUND MULTIPLICATION. 30. 1205 cwt. of hops, at £4 ; 7 : G per cwt. Facit, £528 : 5 : 7^. 31. 407 yards of cloth, at 3s. 9^d. per yard. _ Facit, £77 : 3 : 2^. 32. 729 ells of cloth, at 7s. 7^d. per ell. Facit, £277 : 3 : 6^. 33. 2069 yards of lace, at 9s. 5^d. per yard. Facit, £977 : 19 : 10. THE APPLICATION. 1. "What sum of money must be divided amongst 18 men, so that each man may receive £14 : 6 : 8^ ? Ans. £258 : : 9. 2.^ A p- ivatcer of 250 men took a prize, which amounted to £125 : 15 : 6 to each man ; what was the value of the prize ? ^«5. £31443: 15 3. What is the difference between six dozen dozen, and half a dozen dozen ; and what is their sum and product ? Ans. 792 diff. Sum 936, Product 62208. 4. What difference is there between twice eight and fifty, and twice fifty-eight, and what is their product ? Ans. 50 diff 7656 Product. 5. There are two numbers, the greater of them is 37 times 45, and their difference 19 times 4 ; their sum and product are required? ^tis. 3254 Sum, 2645685 Product. 6. The sum of two numbers is^ 360, the less of them 144 ; what is their product and the square of their difference ? Ans. 31104 Product, 5184 Square of their difference. 7. In an army consisting of 187 squadrons of horse, each 157 men, and 207 battalions, each 560 men, how many effective sol- diers, supposing that in 7 hospitals there are 473 sick ? Ans. 144806. 8. What sum did that gentleman receive in doAvry with hif< wife, whose fortune was her wedding suit; her petticoat haviuir two rows of furbelows, each furbelow 87 quills, and in cacli quiO 21 guineas ? Ans. £3836 : 14 : 0. 9. A merchant had £19118 to begin trade with ; for 5 years toorether he cleared £1080 a year ; tlje next 4 years he made good £2715 : 10 : a year ; but the last 3 years he was in tradc"^ he had the misfortune to lose, one year with another, £475: 4:6a year ; wliru v/a:! IiIb real fortune at 12 years' end? ' ^n«?. £33984 : 8 : 6. COMPOUND MULTIPLICATION. 41 10. -In some parts of the kingdom, they weigh their coals by a machine in the nature of a steel-yard, waggon and all. Three of these draughts together amount to 137 cwt. 2 qrs. 10 lb., and the tare or weight of the waggon is 13 cwt. 1 qr. ; how many coals had the customer in 12 such draughts ? Ans. 391 cwt. 1 qr. 12 lb. 11. A certain gentleman lays up every year £294: 12: G, and spends daily £1 : 12 : 0. I desire to know what is his an nual income ? Ans. £887 : 15 : 0. 12. A. tradesman gave his daughter, ^ a marriage portion, a scrutoire, in which there were twelve drawers, in each drawer were six divisons, in each division there were £50, four crown pieces, and eight half-crown pieces , how much had she to her ^<''*J"«-, . . ^ws. £3744. 13. Admitting that I pay eight guineas and half-a-crown for a quarter's rent, and am allowed quarterly 15s. for repairs, what does my apartment cost me annually, and how much in seven y^a^s ? Ans. In 1 year, £31 : 2. In 7, £217 : 14. 14. A robbery being committed on the highway, an assessment was made on a neighbouring Hundred for the sum of £386 : 15 : 6, of which four parishes paid each £37 : 14 : 2, four hamlets £31 : 4 : 2 each, and the four townships £18 : 12 : 6 each ; how much was the deficiency ? Ans. £36 : 12 : 2. 15. A gentleman, at his decease, left his widow £4560 ; to a pubhc charity he bequeathed £572 : 10 ; to each of his four ne- phews, £750: 10; to each of his four nieces, £375: 12: 6; to thirty poor housekeepers, ten guineas each, and 150 guineas to his executor. What sum must he have been possessed of at the time of his death, to answer all these legacies ? ,_ . , . Ans. £10109: 10: 0. 16. Admit 20 to be the remainder of a division sum, 423 the quotient, the divisor the sum of both, and 19 more, what was the number of the dividend? Ans. 19544«. EXAMPLES OF WEIGHTS AND MEASURES. (») Multiply 9 lb. 10 oz. 15 dwts. 19 grs. by 9. (') Multiply 23 tons, 9 cwt. 3 qrs. 18 lb. by 7. (3) Multiply 107 yards, 3 qrs. 2 nails, by TO. (*) Multiply 33 ale bar. 2firk, 3 oal. bv 11. ' (5) Multiply 27 beer bar. 2 firk. 4 gal.'3 qts. by 12. (') Multiply no miles, 6 fur. 26 poles, by 12. ii I !| ll 42 DIVISION. DIVISION OF SEVERAL DENOMINATIONS. Rule. Divide the first Denomination on the left hand, and if any remains, multiply it by as many of the next less as make one of that, which add to the next, and divide as before. Proof. By Multiplication. £ s. d. 2)25:2: 4( £ s. d. 3)37 :»7 : 7( £ 5. d. 4)57 : 5 : 7( n £ s. d, 5)52 : 7 : 0( 12: 11 :2 («) Divide £1407 : 17 : 7 by 243. (s) Divide £700791 : 14 : 4 by 1794. (■') Divide £490981 : 3 : 7^ by 31715. (8) Divide £19743052 : 5 : 7^ by 214723. THE APPLICATION. 1. If a man spends £257 : 2 : 5 in twelve months* time, what is that per month ? Ans. £21 : 8 : 6^. 2. The clothing of 35 charity boys came to £57 : 3 : 7, what is the expense of each 1 Ans. £1 : 12 : 8. 3. If I ga-e £37 : 3 : 4^ for nine pieces of cloth, what did I give per piece ? Ans. £4:2:11. 4 If 20 cwt. of tobacco came to £27 : 5 : 4^, at what rate is that per c« t. ? Ans. £1 : 7 : 3. 5. What is the value of one hogshead of beer, when 120 are sold for £164 : 17 : 10 ? . Ans. £1:5: 9^. 6. Bought 72 yards of cloth for £85 : 6 : 0. I desire to know at what rate per yard ? Ans. £1:3: 8^. 7. Gave £275 : 3 : 4 for 36 bales of cloth, what is that for 2 bales '. Ans. £15 : 5 : 8^. 8. A prize of £7257 : 3 : 6 is to be equally divided amongst 500 sailors, what is each man's share ? Ans. £14 : 10 : 3^ 9. There au 2545 bullocks to be divided amongst 509 men, 1 desire to know how many each man had, and the value of each man's share, supposing every bullock worth £9 : 14 : 6. Ans. 5 bullocks each man, £48 : 12 : 6 each share. DIVISION. 13 id, and if as make e. ?. d. :0( ne, wliftt 8:6f. : 7, what 12:8. hat did I 2: 11. hat rate :7:3. I 120 are 5:9f to know hat for 2 3:8J. amongst 3 : 3^ 9 men, 1 3 of each S. share. 10. A gentleman has a garden walled in, containing 9625 yards, the breadth was 35 yards, what was the length? Ans. 275. 11. A club in London, consisting of 25 gentlemen, joined for a lottery ticket of £10 value, which came up a prize of £4000. I desire to know what each man contributed, and what each man's share came to ? lo A J ^^^' ^^^^ contributed 8s., each share £160. 12. A trader cleared £1156, equally, in 17 years, how much did he lay by m a year ? ^^5. £68 ^ 13. Another cleared £2805 in 7^ years, what was his vea'rlv increase of fortune ? J J tA ^TTi. , ^^^* £374. . oVo, ^* number added to the 43d part of 4429, will raise it to ^40? "^ Ans.^37. 15.^Divide 20s. between A, B, and C, in such sort that A may, have 2s. less than B, and C 2s more than B ? -^ _ , ^715. A 4s. 8d., B 6s. 8d., C 8s. 8d. 10. 11 there are 1000 men to a regiment, and but 50 officers^ how many private men are there to one officer ? Ans. 19. 17. What number is that, which multiplied by 7847, will make the product 3013248 ? Jns. 384. 18. The quotient is 1083, the divisor 28604, what was the'di- vidend if the remainder came out 1788? Ans. 30979920. 19. An army, consisting of 20,000 men, took and plunderen a city of £12,000. What was each man's share, the whole being equally divided among them ? Ans. 12s> a }^i ^^i P"^^^' '^"^ ^Tf^oney, said Dick to Harry, are worth 12s. 8d., but the money is worth seven times the purse. What did the purse contain ? ^^^^ Us, ij, 21. A merchant bought two lots of tobacc*, which weighed 12 cwt. 3 qrs. 15 lb., for £114 : 15 : 6. Their difference, in point of weight, was 1 cwt. 2 qrs. 13 lb., and of price, £7 : 15 : o. 1 desire to know their respective weights and value. Ans. Less weight, 5 cwt. 2 qrs. 15 lb. Price, £53 : 10. n- • ?^®^^®^ weight, 7 cwt. 1 qr. Price, £61 : 5 : 6. 22. Diviae 1000 crowns in such a manner between A. B. and C, that A may receive 129 more than B, and B 178 less'than C. ^7?5. A360, B 231, C 409. 44 BILLS OF PARCELS.. EXAMPLES OF WEIGHTS AND MEAflVRE».. 1 . Divide 83 lb. 5 oz. 10 dwts. 17 gr. by 8. 2. Divide 29 tons, 17 cvvt. qrs. 18 lb. by 9. 3. Divide 114 yards, 3 qrs. 2 nails, by 10. 4. Divide 1017 miles, 6 fur, 38 poles, by 11. 5. Divide 2019 acres, 3 roods, 29 poles, by 26. G. Divide 117 years, 7 months, 3 weeks, 5 days, 11 hours, 2T piinutes, by 37. BILLS OF PARCELS. 4 12 15 2 14 35 HOSIERS. (') Mr. John Thomas, Bought of Samuel Green. Mity 1,18^* s, di 8 Pair of worsted stockings, at.. .4 : 6 per pair £ 5 Pair of thread ditto 3 : 2 3 Pair of black silk ditto 14 : 6 Pair of milled hose 4 : 2 4 Pair of cotton ditto 7 : 6.... 2 Yards of fine flannel 1 : 8 per yard £7:12:2: mercers'. (•) Mr. Isaac Grants Bought of John Sims. 5. d. May 3, 18 15 18 12 16 13 23 Yards of satin ...at.. .9 : 6 per yard£ Yards of flowered silk 17 : 4. Yards of rich brocade 19 : 8 Yards of sarsenet 3 : 2 Yards of Genoa velvet 27 : 6 , Yards of lutestring 6 : 3 £62 : 2 : 5 18 5 12 2 4 17 18 15 16 25 17 ■ j'u urs, ^ 1,18" 1 :12:2: BILLS OF PARCELS. LINEN drapers'. 45 (') Mr. Simon Surety, Bought of Josiah Short. s. d. June 4, 18 4 Yards of cambric at.. .12 : 6 per yard £ 12 Yards of muslin 8: 3 15 Yards of printed linen 5 : 4 2 Dozen of napkins 2 : 3 each 14 Ells of diaper 1 : 7 per ell.. 35 Ells of dowlas 1 : l^.. £17:4:6i June 14, 18 MILLINERS*. (*) Mrs. Bright, Bought of Lucy Brown. £ s. d. 18 Yards of fine lace at...O : 12 : 3 per yard £ 5 Pair of fine kid gloves : 2 : 2 per pair 12 Fans of French mounts : 3:6 each 2 Fine lace tippets 3 : 3:0 4 Dozen Irish lamb : 1 : 3 per pair. 6 Sets of knots, : 2 : 6 per set.. £23 : 14 : 4 5, IS 1:2:5 June 20, 18 WOOLLEN drapers'. (») Mr. Thomas Sage, Bought of Ellis Smith. £ s. d. 17 Yards of fine serge at...O : 3 : 9 per yard £ 18 Yards of drugget ,,.., : 9:0 15 Yards of superfine scarlet 1 : 2:0 16 Yards of black : 18 : , 25 Yards of shalloon : 1 : 9 ..,. 17 Yards of drab .0 : 17 : 6 £59 : 5 : 46 bills of parcels, leather-sellers'. (^) Mr. Giles Harris, Bought of Abel Smith. s. d. July I 18 27 Calfskins at...,3 : 9 per skin £ 75 Sheep ditto 1 : 7 36 Coloured ditto 1 : 8 15 Buck ditto 11 : 6 17 Russia Hides 10 : 7 120 Lamb Skins.... 1 : 2^ £3S : 17 : 5 GROCERS {') Mr. Richard Groves, Bought of Francis Elliot. July 5, 18 s. d. 25 lb. of lump sugar at...O : 6i per lb. £ 2 loaves of double refined, ) • \\l weight 15 lb. ) * 14 lb. of rice : 3 28 lb. of Malaga raisins : 5 15 lb. of currants 0: 5^ 7 lb. of black pepper , 1 : 10 £3:2: 9^ CHEESEMONOERS\ (') Mr. Charles Cross, Bought of Samuel Grant. s, d. July 6, 18 8 lb. of Cambridge butter at...0 : 6 per lb. £ 17 lb. of new cheese : 4 ^ Fir. of butter, wt. 28 lb : 5^.. 5 Cheshire cheeses, 127 lb : 4 2 Warwickshire ditto, 15 lb : 3 ., 12 lb. of cream cheese : 6 £3:14:7 REDUCTION. n CORN-CI> ^*ni>LERS\ C) Mr. Abraham Doyley. Bought of Isaac Jones. Tares, 19 bushels at...l Pease, 18 bushels 3 Malt, 7 quarters 25 Hops, 15 lb 1 Oats, 6 qrs 2 Beans, 13 bushels 4 July 20, 18 d. 10 per bushel £ 9^ per quarter 5 per lb 4 per bushel 8 £23 : 7 : 4 REDUCTION Is the bringing or reducing numbers of one denomination into other numbers of another denomination, retaining the same value, and is performed by multiplication and division. First, All great names are brought into small, by multiplying with so many of the less as make one of the greater. Secondly, All small names are brought into great, by dividing with so many of the. less as make one of the greater. A TABLE OF SUCH COINS AS ARE CURRENT IN ENGLAND. £ a. d. Guinea. 1: i:0 Half ditto : 10 : 6 Sovereign 1 : 0:0 Half ditto 0: 10: Crown 0: 5:0 Half ditto 0: 2:6 Shilling 0: 1:0 Note. There are several pieces which speak their own value ; such as sixpence, fourpence, threepence, twopence, penny, halfpenny, farthing. rl. In £8, how many sjiillings and pence ? 20 160 shillings. 1920 ^ REDUCTION. 2. in £12, how many shillings, pence, and farthingi? ? Alls. 240s. 2880d. 11520 far 3. In 311520 farthings, how many pounds ? Ans. £324 : 10. 4. How many farthings are there in 21 guineas ? Ans. 21168. 6. In £17 : 5 : 3^, how many farthings ? Aiis. 16573. 6. In £25 : 14 : 1, how manv shillings and pence ? Ans. 514s. 6169d. 7. In 17940 pence, how many crowns ? Ans. 299. 8. In 15 crowns, how many shillings and sixpences ? Ans. 75s. 150 sixpences 9. In 57 half-crowns, how many pence and farthings ? Ans. 1710d. 6840 farthings. 10. In 52 crowns, as many half-crowns, shillings, and pencftj how many farthings ? Ans. 21424. 11. How many pence, shillings, and pounds, are there in .17280 farthings ? Ans. 4320d. 360s. £18. 12. How many guineas in 21168 farthings ? Ans. 21 guineas. 13. In 16573 farthings, how many pounds ? Ans. £17 : 5 : 3^ 14. In 6169 pence, how many shillings and pounds ? Ans. 514s. £25 : 14 : 1. 15. In 6840 farthings, how many pence and half-crowns ? Ans. 1710d. 57 half-crowns. 16. In 21424 farthings, how many crowns, half-crowns, shil- ■lings, and pence, and of each an equal number ? Ans. 52. 17. How many shillings, crowns, and pounds, in 60 guineas ? Ans. 1260s. 252 crowns, £63. 18. Reduce 76 moidores into shillings and pounds ? Ans. 2052s. £102 : 12. 19. Reduce £102 : 12 into shillings and moidores ? Ans. 2052s. 76 moidores. 20. How many shillings, half-crowns, and crowns, are there in £556, and of each an equal number? Ans. 1308 each, and 2s. over. 21. In 1308 half-crowns, as many crowns and shillings, how many pounds ? Ans. £555 : 18. 22. Seven men brought £15 : 10 each into the mint,^to be ex- changed for guineas, how many must they have in all ? Ans. 103 guineas, 7s. over. REDUCTION. 49 23. If 103 guineas and seven shillings are to be divided amongst seven men, how many pounds sterling is that each ? Ans, £15 : 10. 24. A certain person had 25 purses, and in each purse 12 gui- neas, a crown, and a moiilorc, how many pounds sterling had he in all ? Ans. £355. 25. A gentleman, in his will, left £50 to the poor, and ordered that -^ should be given to ancient men, each to have 5s. — ^ to poor women, each to have 2s. 6d. — ^ to poor boys, each to have Is. — }■ to poor girls, each to have 9d. and the remainder to the person who distributed it. I demand how many of each sort there were, and what the person who distributed the money had for his trouble ? Ans, 66 men, 100 women, 200 boys, 222 girls, £2 : 13 : 6 for the person's trouble. TROY WEIGHT. 26. In 27 ounces of gold, how many grains ? Ans, 12060. 27. In 12960 grains of gold, how mar.y ounces ? Ans, 27. 28. In 3 lb. 10 oz. 7 dwts. 5 gr. how many grains ? Ans. 22253. 29. In 8 ingots of silver, each weighing 7 lb. 4 oz. 17 dwts. 16 gr. how many ounces, pennyweights, and grains ? Ans. 711 oz. 14221 dwts. 341304 gr. 30. How many ingots, of 7 lb. 4 oz. 17 dwts. 15 gr. each, are there in 341304 grains ? Ans. 8 ingots. 31. Bought 7 ingots of silver, each containing 23 lb. 5 oz. 7 dwts. how many grains ? Ans. 945336. 32. A gentleman sent a tankard to his goldsmith, that weighed 60 oz. 8 dwts. and ordered him to make it into spoons, each to weigh 2 oz. 16 dwts. how many had he ? Ans. 18. 33. A gentleman delivered to a goldsmith 137 oz. 6 dwts. 9 gr. of silver, and ordered him to make it into tankards of 17 oz. 16 dwts. 10 gr. each; spoons of 21 oz. 11 dwts. 13 gr. per doz.. •alts of 3 oz. 10 dwts. each; and forks of 21 oz. 11 dwts. 13 gr. perdoz. and for every tankard to have one salt, a dozen of spoons, and a dozen of forks ; what is the number of each he must have T Ans* 2 of each sort, 8 oz. 9 dwts. 9 gr. over. m M M if I ! Iji. If fiO JIEDUCTIOK. AVOIRDUPOIS WEIGHT. NoTB.— There are several sorts of silk which are wcij^hcJ by a (jreal pound of 24 oz. others by the coninion pound of 10 oz. ; thcrctorc, To bring great pounda into common, multiply by 3, and divide by 2, or add one half. To bring small pounds into great, multiply by 2, and divide by 3, or subtrM ope third. lyings bought and sold by the TaU. 12 Pieces or things moke 1 Dozen. 12 Dozen 1 Gross. 12 Gross, or 144 doz 1 Great Gross. 24 Sheets 1 auire. 20 Cluires I Ream. 2 Rt^ams 1 Bundle. I Dozen of Parchment.. 12 Skins. 12 Skins 1 Roll. 34. In 147G9 ounces how many cwt.! o^n o^ n. i «. Ana. 8 cwt. qr. 27 lb. I en. 36. Reduce 8 cwt. qrs. 27 lb. I oz. into qwarters, Pounds and ouncw. Ans. 32 qrs. 923 lb. 147b9oz. if 36 Bought 32 bags of hop, each 2 cwt. 1 qr. 14 lb. and anothey of 150 lU. i»ow many cwt. in the whole t ^^ ^^ ^^^ ^ ^^ ^^ ^^ 37. In 34 ton, 17 cwt. 1 qr. 19 lb. how many pounds 1 ^^ TgUlIb. 88. In 547 great pounds, how many common pounds 1 ^^^ ^^ ^^ ^ ^ 39. In 27 cwt >f raisins, how many parcels of 18 lb. each 1 ^^^ ^^ 40. In 9 cwt. 2 qrs. 14 lb. of indigo, how many pounds 1 ^^^ ^^ ^^ 41. Bought 27 bags of hops, each 2 cwt I qr. 15 lb. and one bag of 187 lb. how many cwt in the whole f ^^^ ^^ ^^ 2 ^^ ^q ^^ 43. How many pounds in 27 hogsheads of tobacco, each weighing neat 8t *wt.1 ilns. S64CO, 43. In 552 common pounds of silk, how many great pound* 1 ^^^g^g . . /• 3st term men- tioned. Multiply the second nnd third numbers together, and e3 54 8INGLE RULE OV THREE I^IRECT. divide the product by the first, and the quotient will be the an- swer to the question in the same denomination you left the second number in. EXAMPLES. 1. If 1 lb. of siigar cost 4j, what cost 54 lb. 1 1:44 ::54 4 18 — Ana. £1:0: Z. 18 4)972 12)243 20s. 3d. 3. If a eallon of beer cost lOd., what is that per barrel 1 ^ Ans. £1 ; 10. . 3. If a pair of shoes cost 4s. 6(1., what will 12 dozen come to % Ans. £32 : a 4. If one yard of cloth cost 158. 6d,, what will 32 yards cost at the sazna rate 1 -Aws- £24 : 16. 5. If 32 yards of cloth cost £24 : 16, what is the value of a yard 1 Ans. 15s. 6d. 6. If I ffive £4 : 18 for 1 cwt. of sugar, at what rate did I buy it per lb. 1 Ans. lOjd. 7. If I buy 2a pieces of cloth, each 20 ells, for 12s. Od. per ell, what is the ▼olue of the whole 1 Ans. £250. 8. What will 25 cwt. 3 qrs. 14 lb. of tobacco come to, at I5jil. per lb. 1 Ans. £187 : 3 : 3. 9. Bou'Tht27j yards of muslin, at 6s. 9 id. per yard, what does it amount Ijq'I " Ans. £i) : 5 : 0}, 3 rem. 10 Bought 17 cwt. 1 or. 14 lb. of iron, at 3id. per lb., wliat docs it come tol " Ans. £20:11:01 11. If coffee is sold for 5id. per ounce, what must be given for 2 cwt. 1 yln«. £82 : 2 : 8. 12. How many yards of cloth may l>e bought for £21 : 11 : li, when 3| yards cost £2 : l4 : '3 ? Ans. 27 yards, 3 qrs. 1 nail, 84 rem. %fi Tf 1 nmf nf nh*^«hire cheese cost £1 : 14 : 8. what must I ffive for 3i Ibl Ans. Is. Id. 14. Bouffht I cwt. 24 lb. 8 oz. of old lead, at 9s. iier cwt., what docs it come tol " Ans. 10s. lUd. 113 rem. SIKOLt; RVLB OF THREE DIRECT. as» 13. \i » g. nneman's income is £500 a year, and ho spcnck 19«. 44- per day, liow much does he lay by at the year's end 1 Ans. £147 : 3 : 4. 16. If I buy 14 yards of cloth for 10 guineas, how many Flemish ella can I buy for £ii83 : 17 : 6 at the same rate 1 Ans. 504 Fl. ells. 2 qr». 17. If bOi Flemish ells, 2 quarters, cost £283 : 17 : 6, at what rate did I y&j for 14 yards ] Ans. 10s. lOd. 18. Gave £ 1 : 1 : 8 for 3 lb. of coffee, what must bo given for 29 lb. 4 oz 1 Ans. £10: 11 : 3, 19. If one English ell 2 qrs. cost 48. 7d. what will 39i yards coet at the same ratcl Ans. £5:3: 5^, 5 rem. '20. If one ounce of gold is worth £5:4:2, what is the worth of one grain 7 ilns. 2jd. 20 rem. '21. If 14 yards of broad cloth cost £9 : 12, what is the purchase of 75 yeards 1 iln*. £51 :8:6|,erom. 22. If 27 yards of Holland cost £5 : 12 : 6, how many ells English can I buy for £1001 ilns.384. 23. If 1 cwt. cost £12 : 12 : C>, what must I give for 14 cwt. 1 qr. 19 lb. 1 Ans. £182:0: llj, 8rem. 34. Bought 7 yards of cloth for 178. 8d. what must be given for 5 pieces, each containing 274 yards 1 iln«. £17 : 7 : 0}, 2 rem. 25. If 7 oz. 11 dwts. of g' ' ■ h^ worth £35, what is the value of 14 lb. 9 oi 12 dwts. 16 gr. at the same ,. I Ans. £833 : & : 3}, 552 rem. 26. A draper bought 420 yards of broad cloth, at the rate of 148. lOjd. per ell English, how much did he pay for the whole 1 Ans. £250 : 5. 27. A gentleman bought a wedge of gold, which weighed 14 lb. 3 oz. 8 dwts. for the sum of £514 : 4, at what rate did he pay for it per oz. 1 Ans. £3. 28. A grocer bought 4 hogsheads of sugar, each weighing neat 6 cwt. 2 qrs. 14 lb. which cost him £:i: 8 : 6 per cwt.; what is the value of the 4 hogsheads 1 Ans. £64 : 5 : 3. 29. A draper bought 8 packs of cloth, each containing 4 parcels, each })arcel 10 pieces, and each piece 2() yards, and gave after the rate of £4 : 16 for 6 yards ; I desire to know what the 8 packs stood him to 1 Ans. £G65t5. 30. If 24 lb. of Kusins cost 68. 6d. what will 18 fraib cost, each weighing ueai 3 qrs. 18 lb. 1 Ans. £SM ; 17 : 3. 31. If I oz. of silver t)e worth 5s. what is the price of 14 ingots, cnch wei|:h- ing 7 lb. 5 oz. 1 ) dwts. 1 Ans. £313 : 5. 32. What is the price of a pack of wool, weighing 2 cwt. 1 qr, 19 lb. at 8fc. 6d. i>er stone 1 Ans. £8:4: G}, 10 rem. QT PrftTtrvKf M /•«»♦ # #iiHj OA IK >n.f fr\\\ari*f\ of CO.' 17 • A TtnY rviri • trKnt line* it coi^c to ^ Ans.£llk.Z:li 80 rem. 56 RULE OF THREE INVERSE. 34. Bought 171 tons of lead, at £14 per ton; paid carnage and other incident charges, £4 : 10. I require the value of the lead, and what it stands me in per lb. ? Ans. £2:398 : 10 value ; l^d. 433 rem. per lb. 35. If a pair of stockings cost 10 groats, how many dozen may I buy for £43 : 6 ? Ans. 21 dozen, 7^ pair. 36. Bought 27 dozen 5 lb. of • andles, after the rate of 17d. per 3 lb. what did they cost me ? Ans, £7:15 : 4^, 1^ rem. 37. If an ounce of fine gold is sold for £3 : 10, what come 7 ingots to, each weighing 3 lb. 7 oz. 14 dwts. 21 gr.^ at the same price ? Ans. £1U71 : 14 : 5^. 39. If my horse stands me in 9^d. per day keeping, what will be tlie charge of 1 1 horses for the year ? Ans. £159 : 18 : 6^. 39. A factor bought 86 pieces of stuff, which cost him £617 : 19 : 4, at 4s. lOd. per yard ; I demand how many yards there were, and how many ells English in a piece ? Ans. 2143^ yards, 56 rem. and 19 ells, 4 quarters, 2 nails, 61 rem. in a piece. 40. A gentleman hath an annuity of £896 : 17 per annum. I desire to know how much he may spend daily, that at the yearns end he may lay up 200 guineas, and give to the poor quarterly 40 moidores? Ans. £1 : 14 : 8, 176 rem. THE RULE OF THREE INVERSE. Inverse Proportion is, when more requires less, and less re- quires more. More requires less, is when the third term is great- er than the first, and requires the fourth term to be less than the second. And less requires more, is when the third term is less^ than the first, and requires the fourth term to be greater thfui: tlie second. Rule. — Multiply the first and second terms together, and di- vide the product by the third, the quotient will bear such propor tion to the second as the first does to the thiid.. RUI.fi or TBREB INTEMi' m EXAMPLES. h l( 8 men can do a piece of work in 12 days, hovr many days can 16 men perform the same in ? 4n9* 6 dtiy«. 8 . 12 . . 16 . e 8 16)96(6 days. 2. If 54 men can build a house in 90 days, how many can do Ihe same in 50 days ? Ans, 97^ men. 3. If, when a peck of wheat is sold for 28., the penny loaf weighs 8 oz., how much must it weigh when the peck is worth but Is. 6d. ? Ans. lOf oz. 4. How many pieces of money, of 20s. value, are equal to 240 pieces of 128. each ? Ans. 144. 5. How many yards, of three quarters wide, are equal in mea- sure to 30 yards, of 5 quarters wide ? Ans. 50. 6. if I lend my friend £200 for 12 months, how long ought he to lend me £150, to requite my kindness ? Ans. 16 months. 7. If for 24s. I have 1200 lb. carried 36 miles, how many pounds can I have carried 24 miles for the same money ? Ans. 1800 lb. 8. If 108 workmen finish a piece of work in 12 days, how nany are sufficient to finish it in 3 days ? Ans. 432. 9. An army besieging a town, in which were 1000 soldiers, with provisions for 3 months, how many soldiers departed, when the provisions lasted them 6 months ? Ans. 500. 10. If £20 worth of wine is sufficient to serve aft ordinary of 100 men, when the tiiu is sold for £30, how many will £20 worth suffice, when the tun is sold but for £24 ? An!^. 125. 11. A courier makes a journey in 24 days, when the day is but 12 hours long, how many days will he be going the same- journey, when the day is 10 aours long? Ans. 18 days... 58 DbVBLG RULE OF THREE. 12. How much plush is sufficient for a cloak, which has in it 4 yards, of Tquaiters wide, of stuff, for the lining, the plush beiag but 3 quarters wide ? Ans. 91- yards. 13. If 14 pioneers make a trench in 18 days, how many dayt will 34 men take to do the same ? Ans. 7 days, 4 hours, 50 min. -,-V, at 12 hours for a day. 14. Borrowed of my friend £()4 for 8 months, and he had oc- casion another time to borrow of me for 12 months, how much must I lend him to requite his former kiridness to me ? Ans. £42 : 13 : 4. 15. A regiment of soldiers, consisting of 1000 men, are to have new coats, each coat to contain 2A yards of cloth, 5 quarters wide, and to be lined with shalloon of 3 quarters wide ; I demand how many yards of shalloon will line them ? Ans. 41(50 yards, 2 qrs. 2 nails. 2 rem THE DOUBLE RULE OF THREE, Is so called because it is composed of 5 numbers given to find a 6th, which, if the proportion is direct, mupt bear such a proportion to the 4th and 5th, as the 3d bears to the 1st and 2d. But if in- verse, the 0th number must bear such pro])ortion to the 4th and 5th, as the 1st bears to the '2d and lid. The three first terms are a supposition; the two Uif^t, a demand. Rule 1. Let the principal cause of loss or gain, interest or decrease, action or passion, be put in tlie first place. 2. Let that wliich betokenctli time, distance of place, and the like, be in the second place, and the remaining one in the third! 3. Place the other two terms under tlieir like in the supposi- tion. 4. If the blank falls under the third term, multiply the first and second terms for a divisor, and the other three for a dividend. But, 5. If the blank falls under the first or second term, multiply ♦}w> fl»ii«/l niul fourth terms for a divisor, ?ind the other three fo? the dividend, and the quotient will be the answer. Proof. By two single rules of tlirec. DOUBLE RULE OF THREE. 09 EXAMPLES. 1 . If 14 horses eat 5G bushels of oats in 16 days, how many busheb will b» sufficient fur 20 horses for 24 days 1 By two single rules. ^ or in one stating, worked thus ; hor. bu. hor. Iiu. | hor. days. bu. bu. 1. As 14 . 50 . . 20 . 80 days. bu. days. bu. 2 As IC . 80 . . 24 . 120 14.16.56 56X20X24 20.24.— =120 14X16 2. If 8 men in 14 days can mow 112 acres of grass, how many mea must there be to mow 2000 acres in 10 daysl acres, days, acres, days. 1. As 112 . 14 . . 2000 . 250 days. men. days. men. 2. As 250 . 8 . . 10 . 200 men. days, acres. 8 . 14 . 113 . 8 X 14 X 2000 =200 - . 10 . 2900 112X10 3. If £100 in 12 months gain £G interest, how much will £75 gain kt 9 months 1 Ans. £3:7: 6. 4. If a carrier receives £2 : 2 for the carriage of 3 cwt. 150 miles, how much ought he to receive for tbis carriage of 7 cwt. 3 qrs. 14 lb. for 50 miles 1 Ans. 1 : 16 : 9. 5. If a regiment of soldiers, consisting of 136 men, consume 351 quarters of wheat in 108 dajs, how many quarters of wheat will 11232 soldiers consume in 56 days ] Ans. 15031 qrs. 864 rem. 6. If 40 acres of grass be mowed by 8 men in 7 days, how many acres can be mowed by 24 men in 28 days 'i Ans. 480. 7. If 40s. will pay 8 men for 5 days' work, how much will pay 32 men fur 84 days' work 1 Ans. £38 : 8. 8. If £100 in 12 months gain £6 interest, what principal will gain £3:7: 6 in 9 months 1 Ans. £75. 9. If a regiment, consisting of 939 soldiers, consume 351 qrs. of wheat in 168 days, how many soldiers will consume 1404 qrs. m 56 days 1 Ans. 11968. 10. If a family consisting of 7 persons, drink out 2 kilJerkins of beer in 12 lays, how many kilderkins will another family of 14 persons drink out in 8 daysl Ans. 2 kil. 12 gal. 11. If the carriage of GO cvrt. 20 miles, cost £14 : 10, what weight can I Lave carried 30 miles for £5 : 8 : 9, at the same rate of carriage 1 Ans. 15 cwt. 12. If 2 horses cat 8 bushels of oats in 16 days, how many horses will eat up 3000 quarters in 24 days 1 Ana. 4000. 13. If £100 in 12 inonlbs gaiii £7 inU^rcsl, wba 'm thi^ iuiiirtfit of £571 fo* 6 years 1 Ans. £239; 16:41,20 rem. 60' fRACTICE. 14. If I pay 10s. for the carriage of 2 tons 6 miles, whatmiui I pay for the carriage of 12 tons, 17 cwt. 17 miles? Ans. £9 i2:0^. PRACTICE, Is so called from the general use thereof by all persons concern ed in trade and business. All questions in this rule are performed by taking aliquot, ov even parts, by which means many tedious reductions are avoided , the table of which is as follows : — Of 10: 6: 5: 4: 3: 2: 2: 1: a Pound. d. \j • • •IS*** 2 a i i 4 i X ••••6 6 i 0. 8. 1 -JL. 13 Of a shilling. d 4 3 i 2 H 1 JL 3 J. 6 JL 8 -J- 13 Of a Ton. cwt. 5. i 4. 3i 2 X 6 X 8 10 Of a Hundred. ors. lb. 2 or 56 is f 1 or 28 i 14 i Of a Quarter. 14 lb i 7 i 4 4 3^ i Rule 1. When the price is less than a penny, divide by the aliquot parts that are in a penny ; then by 12 and 20, it will be the answer. (J.)iisi)57041b. at 4: 12)1426 2|0)ni&:10 Facit,£5:18: 10 (2) 7695 at ^ Facit, £16:0: 7^ (3) 5470 at?. Facit, £11 :7:11 (4) 6547 at I Facit, £20 : 9 : 2i (S)4573at| Facit, £14 : 5 : 9^ Rule 2, When thn price is less then a shillinff, take the ali- quot part or parts that'ure in a shilling, add them together, and divide bv 20, as before PRACTICE. 61 i r. i i JL '7 JL 8 (»)is-iV764Tatld. 2|0)63|8:11 Facit, £31 :8: U («)lis-iV3751atlid fisi 312:7 78: 1| 3|0)3910 : 8^ Facit, £19: 10 ;8J. (^) 54325 at Hd. Facit, £339: 10: 7^. («)6254atHd. Facit, £45: 12:0^; (5)2351 at 2d. Facil, £19: 11 : 10. («)7210at2id Facit, £67:11:10^. (')2710at2^d. Facit, £28:4: 7. (8)3350 at 2^d. Facit, £37 : 4 : 9^. (9) 2715 at 3d. Facit, £33 : 18 : 9. (^»)7062at3id. Facit, £95 : 12 : 1^. (»')2147at3id. Facit, £31 : 6 : 2|^. (»-^)7000at3H Facit, £109 : 7 : 6. (>»)3257at4d. Facit, £54 : 6 : 8. ('♦)2056at4^d. Facit, £36 : 8 : 2. (»°)3752at4id. Facit, £70: 7:0. (»'»)2107at4|d. Facit, £41: 14 :0i. (»')32I0at5d. Facit, £66: 17:6. (»«)2715at5^d. Facit, £59 : 7 : 9^. (»»)3120at5id. Facit, £71 : 10:0. («'»)7521 atSH Facit,£180:3:9i (2 1)3271 at6d. Facit, £81 : 15:6. (2 2) 7914 at 6H Facit, £206: 1:10A. (2 3)3250ut6^d. Facit, £83 : : 5. (2 *) 2708 at Of d. Facit, £76 : 3 : 3. (2'-')3271 at7d. Facit, £95 : 8 : 1. (2«)3254at7id. Facit, £98:5: 11|. (2 7)2701 at7<^d. Facit, £84:8: 1|. F (i8)37l4at7|d. Facit,£119:18:7i. (2»)2710at8d. Facit, £90 : 6 : 8. (3 0)35l4at8id. Facit, £120: 15: 10^. (3')2759at8id. Facit, £97 : 14 : 3J. (»2)9872at8|d. Facit, £359:8:4. ('9)5272 at 9d Facit, £197 : 14 : 0. (^*)G325at9id. Facit, £243: 15 :6i. (»«)7924at9^d. Facit, £313:13:2. (3«)2150at9|d. Facit, £87 : 6 : 10^. (3-)r)32oat lOd. Facit, £263: 10: 10. (3s)572-4atlO:Jd. Facit, £244 : 9 : 3. (3 3)6327atlOid. Facit, £270 : 4 : 3|. (*«)3254atlOJd. Facit, £142 : 7 : 3. (^')729latl0|d. Facit, £326: 11: 6^. (*2) 3256 at lid. Facit, £149: 4:8. 69< PHACTlCtJ. i*')??^^! allied. |(**)3754atnid. \r'^)rm^tUid Facit, £340 : : 7i. | Faeit, £179 : l| : 7. | Facit! S^?5 ! U. Rule 3. When the price is more than one shilling, and less than two, take the part or parts, with so much of^ie Vyel price as 18 more than a shilling, which add to the given L^^^^^^ and divide by 20, it will give the answer ^ quantity, (')iTV210(Jatl2|d. 43 : 10^ 2|0)21 1|9 : 10^ (')i«V37]5atl2^d. 154 : 9i 2|0)3S6i9 : 9^^ (0 2712 at 12|d. Facit, £144 : 1 : 6. Facit, £107 : 9 : 10^. I Facit, £193 ; 9 : 9^. (0 3215 at Is. Hd. Facit, £177:9: lOi (0 2790'itls. I^d7 Facit, £156 : 18 : 9. 7904 at Is. l^d. Facit, £452 : 16 : 8. r03254atls.3|d. Facit, £213: ).0: 10^. CO 2915 at Is. 4d~ Facit, £194:0:8. (0 2107 at Is. Id. Facit, £114:2: 7. (0 3750 at Is. 2d. Facit, £218 : 15 : 0. CO 3270 at Is. 4fd. Facit, £221 : 8 : 1^. C0 7103atls.6|d. Facit, £540 : 2 . 5|. ('^03254atld.6^d. Facit, £250 : 16 : 7. CO 7925 at Is. 6fd. Facit, £619 ; 2 : 9|. (0 3291atls. 2id. Facit, £195:8 : 0:^ ("^)925l"^ir2^d~ Facit, £559: I : H. ("^0^725071172^17 Facit^£4l5;ll;5j. C0 75"977T77k Facit, £474 : 8 : 9. CO 6325 at Is. 3^d! Facit, £401 : I8:0|. (J ' )^7Jun73H CO 7059 at Is. 4^d. Facit, £485 : : 1^. CO 2750 at Is. 4|d. Facit, £191: 18:0^. (^0 37257tll"5dir Facit, £263 : 17 : 1. CO 7250 at U.&Jd. Facit, £521 : 1 : 10|. {' 'T25977rrsT5idT Facit, £189 : 7 : 3.1. (-0 ^271 at Is. 7d. Facit, £733 : 19 : 1. ('^0 72I0atls. 7H Facit, £578:6: 0|. (^OSSlTatlTTid; Facit, £187 : 13 : 9. COssoTTls. 7|d. Facii, £206 : 1:2. ('07l727tTr8d.'' Facit, £596 : : 0. C ' ) 72 10 at 1 s. 5id. j ( 3 2905 at I s. 8^d. I;ac^t^£533 : 4 : 9^ Facit, £245 : 2 : 2^. niACTlOB, 63 [*\^n^ ft W. I(*«) 1071 at l8. lOd. I(*M2105atl8 lUA ('"')3l04atls. 9(1. Facit, £184 : 2 : 0. (*')5300atls.l04d. Facit, £482 : 1 : 8. (•')2571atl3.9id. Facit, £227 : 12 : 9|. (»»)2104atls.9^(L Facit, £188 : 9 ; 8. (•»)7506atl8. 9H Facit, £680 ; 4 : 7f (*'^)21l7atl8.10^d. Facit, £198 : 9 : 4^. r^) 1007 at Is. 10^ Facit, £95 : 9 : Ij. (**)500pat Is. lid. Facit, £479 : 3 : 4. (^MlOOOatls.lUd. Facit, £98 : 10 : K (*02705atl8.11|d. Facit, 267 : 13 : 7if. (^«75000atl8.1Ud, Facit, £489: 11:8. (*'')4000atls.llfd. Facit, £395 : 10 : 8. 2750 at 2s. Facit, £275 : : 0. (»)3254at4s. Facit, £050 : 16 : 0. P2102atl08. 1(^)1075 at IBs. Facit, £1051 ; ; . j Facit, £860 : : 0. (')2101 at 12^. Facit, £1260:12:0. 2710 at 6s. Facit, £813 : : 0. {*) 1572 at 8s. ('")l621atl8s. Fat t, £1458: 18:0. Note. When the (')527I at 14s. Facit, £3689 : 14 : 0. price is lOs. take half of the quantity, and if any remains, it is 10s. («) 3123 at 16s. Facit, £628 : 16 : 0. i Facit, £24{)8 : 8 : 0. Rule 5. When the price consists of odd shillings, multiply the given quantity by the price, and divide by 20, the quotient will be the answer. ( ' ) 2703 at Is. Facit, £135 .3:0. (=) 3270 at 3s. 3 2|0)981|0 Facit, £490 : 10 : P2 (3)3271 at 5s. Facit, £817 : 15 . a 64 PRACTICE. («)2715at78. Facit, £950 : 6 : 0. (»)3214 at 98. Facit, £1446 : 6 : 0. (')3179atl3«. Facit, £3066 : 7 : 0. (8) 2150 at 15s. Facit, £1612:10:0. (•)2710at lis. Facit, £1490:10:0. (»»)2160att98. Facit, £2042: 10 :l» (»')7l57atl98. Faeit, £6799 : 3 : i' (») 3142 at 178. Facit, £2670: 14:0. Note. When the price is 5s., divide the quantity by 4, ani if any remain, it is 5s. ' Rule 6. When the price is shillings and pence, and they the aliquot part of a pound, divide by the aliquot part, and it will give the answer at once ; but if they are not an aliquot part, then multiply the quantity by the shillings, and take parts for the irest, add them iygcther, and divide by 20. (»)7614at48. 7d. Facit, £1721 : 19:2. 6:8 2 2j0 i i (•)2710at6s. 8d. Facit, £903 : 6 : 8. ('*)3150at39. 4d. Facit, £525 : : 0. (»)2715at2». 6d. Facit, £339 : 7 : 0. («)7150atls. 8d. Facit, £595 : 16 : 8. («)3215at ls.4d. Facit, £214:6:8. ('')7211 at Is. 3d. Facit, £450 : 13 : 9. ('r2710"at3s. 2d. 3 8130 451 :8 85Sil :8 Facit £429 1 :8. (») 2517 at 5s. 3d. Facit, £660 : 14 : 3. (,0) Fac {'') 1 1 Fac Fac (13) Fac Fac (16) Fac (la) Fac Fac 2547 at 7s. S^d. t, £928: 11: 10^. 3271 at 5s. 9^d. t, £943: 16:4i 2103 at 15s. 4^d. t, £1616: 13: 7^. 7152 at 17s. 65d t, £6280 : 7 : 0. 2510atl4:7ild. t, £1832: 16: 5^. 3715 at 9s. 4 id. t, £1741 :8:"U,. 2572atl3:7^d. t, £1752 : 3 : 6. 72.51 at Hs.8id. t, £5324: 19 0% l^RACTIOE. ~ cwl., what did I give for the whole ? ^ns. £227.14. uixis, ai, v~.>-.^ . 1 * . ;? pur TARE AND TRET. W it. Sold 56 cwt. 1 qr. 17 lb. of sugar, at £2 : 15 : 9 the cwu. what cioes it come to? Ans. £157 : 4 : 44. S. Tobacco at £3 : 17 : 10 the cwt., what is the wcyrth of 07 cwt. 15 lb. ? Ans. £378 : : 3. 9. At £4 : 14 : 6 the cwt., what is the value of 37 cwt. 2 qrs. 13 lb. of double refined sugar 1 Ans. £177 : 14 : 8^ 10. Bought sugar at £3 : 14 : 6 the cwt., Avhat did I give for 15 cwt. 1 qr. 10 lb. ? Ans. £57 : 2 : 9. 11. At £4 : 15 : 4 the cwt., the value of 172 cwt. 3 qrs. 12 lb. of tobacco is required? Ans. £823 : 19 : 0^. 12. Soap at £3 : 11 : 6 the cwt., what is the value of o3 cwt. 171b.? ^ws. £190 : : 4. TARE AND TRET. The allowances usually made in this Weight, are Tare^ TVcf, and Cloff, Tare is an allowance made to the buyer for the weight of the box, barrel, bag, &c., which contains the goods bought, and ia either At so much per box, bag, barrel, &.c. At so much per cwt., or At so much in the gross weight. Tret is an allowance of 4 lb. in every 101 lb. for waste, dust, &.C., made by the merchant to the buyer. Cloff is an allowance of 2 lb. to the citizens of London, on every draught above 3 cwt. on some sort of goods. Gross weight is the whole weight of any sort of goods, and that which contains it. Suttle is when part of the allowance is deducted from the gross. Neat is the pure weight, when all allowances are deducted. Rule 1. Wh^n the tare is at so much nor hnv. barrel, &C., multiply the number of bags, barrels, &c. l)y the tare, and sub- tract the product from the gross, the remainder is neat. 06 TARE AND TRET. Note. To reduce Pounds into Gallons, multiply by ^ and divide by 15. 1. In 7 ffaiN of raisins, each weighing 6 cwt. 2 qrs. 51b. gross, tare at 23lb. per frail, how much neat weight? Arts. 37 cwt 1 qr. 14 lb. 23 ft. 2. 5- or, 5. 2. 5 7 7 S3 4 58)161(5 38.3. 7=gro6». 6.1.10 140 1.1.2l=tare 7 1.1 21 37.1.14=neal 37.1.14 3. What is the neat weight of 25 hogsheads of tobacco, weigh- ing gross 163 cwt. 2 qrs. 15 lb., tare 100 lb. per hogshead? Ans. 141 cwt. 1 qr. 7 lb. 3. In 16 bags of pepper, each 85 lb. 4 oz. gross, taic per bag 3 lb. 5 oz. how hiany pounds neat ? Ans, 1311. Rule 2. When the tare is at so much in the whole gross weight, subtract the given tare from the gross, the remainder if neat. 4. What is tlie neat weight of 5 hogsheads of tobacco, weigh- ing gross 75 cwt. 1 qr. 14 lb., tare in the whole 752 lb. ? Ans. 68 cwt. 2 qrs. 18 lb. 5. In 75 barrels of figs, each 2 qrs. 27 lb. gross, tare in the whole 697 lb. how much neat weight ? Ans. 50 cwt. 1 qr. Rule 3. When the tare is at so much per cwt., divide the gross weight by the aliquot parts of a cwt., which subtract from the gross, the remainder is neat. Note. 7 lb. is -jV, 8 lb. is -^, 14 lb. is i, 16 lb. is -f. 6. What is the neat weight of 18 butts of currants, each 8 cwt 2 qr;?. 5 lb., tare at 14 lb. per cwt. ? / 8.2.5 9x^13 76 3 . 17 2 4=i 153 .3. 6 19 . . 25} 134 . 2 . 8| tARB AND TRET. 83 t. In 25 barrels ot figs, each 2 cwt. 1 qr. gross, tare per cwt, 16 lb., how much neat weight ? Ans. 48 cwt. qr. 24 lb. 8. What is the neat weight of 9 hogsheads of nutmegs, each treighing gross 8 cwt. 3 qrs. 14 lb., tare 16 lb. per cwt. ? Ans. 68 cwt. 1 qr. 24 lb. Ruti 4. When tret is allowed with tare, divide the poundt suttic by 26, the quotient is the tret, which subtract from the aut- tie, the remainder is neat. 9. In 1 butt of currants, weighing 12 cwt 2 qrs. 24 lb. gross, tare 14 lb. per cwt, Iret4 lb. per 194 lb., how many pounds neat? 13 . 8 . ^ 4 50 28 14=1 1424 grom, 178 tare. 86)1246 sutUe. 47 tret. 1199 neat. iO. In 7 cwt 3 qrs. 27 lb. gross, tare 361b., !tret4 lb. per 104 lb., how many pounds neat ? -Atos. 8261b. 11. In 152 cwt 1 qr. 3 lb. gross, tare 101b. per cwt, tret 4 lb. per 104 lb., how much neat weight ? H Ans, 133 cwt 1 qr. 12 lb. Rdlb 5. When cloff is allowed, multiply the cwts. suttle by 2, divide the product by 3, the quotient will be the pounds clofli which subtract from the suttle, the remainder will be neat 12. What is the neat weight of 3 hogsheads of tobacco, weigh- Sug 2i; ewt. o qrs. ^ Ih. gross, tare 7 ib. per cwt, tret 4 lb per m U)., doff 21b. for 3 cwt 7 Ans. 14 cwt 1 qr. 3 Ib. TO INTBRS8T. 7=|V 15 . 3 . 20 grow. 3 . 27itare. 26)14 . 3 . 20i suttle. y 2.8 tret. 14 . 1 . 12^ suttle. 9i cloff. 14 . 1 . 3 13. In 7 hogsheads of tobacco, each weighing gross 5 ewt. d qn. 7 Vt , tare 8 lb. per cwt., tret 4 lb. per 104 lb., doff 2 lb. per 3 cwt., how much ne»* weight 1 Ana. 34 cwt. 2 qrs. 8 lb. SIMPLE INTEREST, b the Profit allowed in lending or forbearance of any sum of money for a determined space of time. The Principal is the money lent, for which interest is to be received. The rate per cent, is a certain sum agreed on between the Borrower and the Lender, to be paid for every £100 for the use of the principal 12 months. The Amount is the principal and interest added together. Interest is also applied to Commission, Brokage, Purchasing of Stoclu^ and Insurance, and are calculated by the same rules. To jind the Interest of any Sum of Money for a Year. Rule 1. Multiply the Principal by the Rate per cent,, that Product divi- ded by 100, will give the interest required. For several Years, 2. Multiply the interest of one year by the number of years given m the question, and the product will be the answer. 3. If there be parts of a year, as months, weeks, or days, work for th« months by the aliquot parts of a year, and for the weeks and days \3y the Rule of Three Direct. EXAMPLES. 1. What is the interest of £375 for a year, at 5 per cent, per annoEal 5 18175 20 15100 Ans. £18 . 15 . 0. 2. What is the interest of £368 for 1 year, at 4 per cent, per annum 1 Ana. £10 . 14 . 4|. 3. What is the interest of £945 . 10. for a year, at 4 per cent, per annum 1 Ana. 3T. 16 . 4|. INTEREST. 71 4. What is ihc interest of £547 . 15, at 5 per cent, per annum, for 3 years 1 Ans. £82 .3.3. 5. What is the interest of £254 . 17 . 6, for 5 years, at 4 per cent, per an- '^"^ ^ Ans. £50 . 19 . 6. G. What ia the interest of £556 . 13 . 4, at 5 per cent, per annum, for 5 y^'^^^ iln*. £139.3.4. /•''iiS^ *^°O^P°"''ent writes me word, that he has bought goods to the amount of £ /54 . lb on my account, what does his commission come to at 2^ per cent. ] ^n*. £18.17. 4i. 8. If I allow my factor 3| per cent, for commission, what may he demand on the laymg out £876 . 5 . 10 ? Ans. £32 . 17 . 2j. 9. At I10{ per cent., what is the purchase of £2054 . 16. South Sea Stock 1 Ans. £2265 .8.4. 10. At 101| per cent. South Sea annuities, what is the purchase of 1797 . 14 1 ilns. £1876.6.111. 11. At 96} per cent., what is the purchase of £577 . 19 Bank annuities! ^ns. £559 . 3 . 3|. la At £124| per cent, what is the purchase of £758 . 17 . 10, India Stock! Ans. £945 . 15 . 4*. BROKAGE, Is an allowance to brokers, for helping merchants or factors to persons, to buy or •ell them goods. RuLR. Divide the sum given by 100, and take parts from the quotient with tne rate per cent. 13. If I employ a broker to sell goods for me, to the value of £2575 . 17 . G. what IS the brokage at 4s. per cent. 1 25|75 ,, 17 . 6 20 . 4s.=|25.15.2 15|17 12 2110 Ans. £5 . 3 . Oi 14. When a broker sells goods to the amount of £7105 . 5 . 10, what may ho lemand for brokage, if he is allowed 5s. 6d. per cent. ? ^ns.£19. 10.9}. c}^r 'J^^^'ojei'ls employed to buy a quantity of goods, to the value of £>\)lo .0.4, what is the brokage, at Gs. 6d. per cent. 1 Ans. £3.3. 4j. 16. What is the interest of £547 .2.4, for 5j years, at 4 per cent, per an- """^ * Ans. £120 . 7 . 3fc. - ■- -.^=»tt, 1= uic inicresi oi s,;io/ . D . i, ai 4 per cent., ior a year and three quarters Ans, £18 . . \\. 18. Whatia the interest of £479 . 5 for 5* years, at 5 per cent, per annum 1 ilns. £125. 16.0J. 72 INTEREST. 19. What is the interest of £576 : 2 : 7 Corl^ years, at4i P^ cent, per annum ? Ans. £187 : 10 : 1^. 20. What is the interest af £279 : 13 : 8 at 5^ per cent, per annum, for 3^ years ? Ans. £51 : 7 10. When the interest is required for uny nunber of Weeks. Rule. As 52 weeks are to the Interest of the given sun^ for « year, so are the weeks given for .the interest required. 21. What is the interest of £259 : 13 : 5 for 20 we* Its, at 5 per cent, per annum ? Ans. £4 : 19 lOi. 22. What is the amount of £375 ; 6 : 1 for 12 weeks, at % per cent, per annum ? Ans. £379 : 4 : 0|. When the Interest is fur at:-^ mimhsr of days* RvLE. As 365 days are to Ihj? i^tiereat of the given sum for & 7««r, so are the days giv^sn ia the intere&t required. 23. At 5^ per cent, per annum, what is the interest of £965 . 2 . 7 for is> years, 127 davs? Ans. £289 . 15 . 3. 24. What is the interest of £2726 . 1 . 4 at 4^ per cent, per annum, for three years, 154 days ? Ans. £419 . 15 . 6^. When the Amount^ Time, and Rate per cent, are given tofin6 the Principal. Rule. As the amount of £100 at the rate and time given : is io £100 : : 80 is the amount given : to the principal required. 25. What principal being put to interest, will amount to £402 10 in 5 years, at 3 per cent, per annum ? 3 X 5+ 100=£115 . 100 . . 402 . 10 20 20 2300 8050 1AA 2W S3HX^}''050|00(£350 Aiu. INTEREST. 73 at 4^ per 9:U. cent, per :710. ^eeks. sum lor eks, at 5 9 10i. ks, at 4^j 4:0i. um for & 15.3. eent. per 5.6|. 71 to find riven : is [uired. ; to £402 26 What principal being put to interest for 9 years, will amount to £734 : 8, at 4 per cent, per annum 1 Ans. £540. 37. What principal being put to interest for 7 years, at 5 per cent, per annum, will amount to £334 : 16 ? Ans. £248. When the principal, Rate per cent., and Amouat are given, to find the Time, Rule. As the interest of the principal for 1 year : is tol year : : «o is the whole interest : to the time required. 28. In what time will £350 amount to £402 . 10, a4 3 per cent, per annum ? 350 3 Asl0.10:l::52.10:5 20 20 10|50 20 210 2ll0)105|0(5yean. 105 iifw. 402.10 350. 10|00 52.10 29. In what time will £540 amount to £734 r 8, at 4 per cent per annum 1 Ans. 9 years. 30. 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 find the Rate per cent. Rule. As the principal : is to the interest for the whole time : : so is £100 : to the interest for the same time. Divide that in- terest by the time, and the quotient will be the rate per cent. 31. At what rate per cent, will £350 amount to £402 : 10 in 5 years' time ? 350 52.10 Ab350:52.10::100:£15 20 1050 100 32. At what rate per cent 7 years' time ? 35|0)10500|0(3008.=£15-i-5=3 per cent ill £248 amount to £33" 10 in Ans. 5 per cent 74 INTEREST. 33. At what rate per cent, will £640 amount to £734 : 8 in 9 years' time ? Ans. 4 per cent. COMPOUND INTEREST, Is that which arises both from the principal and interest ; that is, when the interest on money becomes due, and not paid, the same interest is allowed on that interest unpaid, as was on the principal before. Rule 1. Find the firstyear's interest, which add to the princi- pal ; then find the interest of that sum, which add as before, and flo on for the number of years. 2. Subtract the given sum from the last amount, and it will give the compound interest required. EXAMPLES. 1. Wl»t is the compound interest ^of £500 forborne 3 years, at 5 per cent, per annum I < 26. .5 500 500 5 25 25100 5^5= l#t year. 26125 5^ 551 . . 5=2d year. 5 551.. 5 27156.. 5 27. 11.. 3 20 11125 12 578.16..3=3dyear. 5Q0 prin. sub 3100 78 . 16 . . 3=intere8tforSyeaw. 3. What is tbe amount of £400 forborne 3^ years, at^-p^r cent, per annum, compound interest ? Ans. £490 : 13 : lU- 3. What will £650 amount to in 5 ye^rs, at 5 per cent, per annum, compound intereiSt? Ans. £839 : 11 : 7^. 4. What is the amount of £550 : 10 for 3 years and 6 months, at (3 per cent, per annum, compound interest ? ^»s. £675 : 6 : 5. 5. What is tlie compound Interest of £764 for 4 years and 9 months, at 6 per cent, per annum ? Ans. £343 : 18 : 8. 6. What ifi the compound interest of £57 : 10 : 6 for 5 years, 7 months, and 15 days, at 5 per cent per annum ? Ans. £18 : 3 : 8^. REBATE OR DISCOUNT. 75 7. What is the compound interest of £259 : 10 for 3 years, 9 months, and 10 days, at 4^ per cent, per annum ? Ans. £46 : 19 : 10^. REBATE OR DISCOUNT, Is the abating of so much money on a debt, to be received be- fore it is due, as that money, if put to interest, would gain in the same time, and at the same rate. As £100 present money would discharge a debt of £105, to be paid a year to come, rebate being made at 5 per cent. Rule. As £100 with the interest for the time given : is to that interest : : so is the sum given : to the rebate required Subtract the rebate from the giveii sum, arid the remainder will be the present worth. EXAMPLES. 1. What is the discount and present worth of £487 : 12 for 6 months, at 3 per cent, per annum ? 3 100 103 487 : 13 prindpaL 14 : 4 rebate. Ail03:0::487:12 aO 20 2060 9752 3 £«. Ans. £473 : 8 present wortn 20610)292516(14.4 rebate. 206 IS 824 2. What is the present payment of £357 : 10, which wag agreed to be paid 9 months hence, at 5 per cent, per annum ? ^ns. £344 : 11 : 7. 3. \Yhat is ihe discount of £275 : 10 for 7 months, at 5 per cent, per annum? Ans, £7 : 16 : If. Q3 w EQUATION OF PAYMENTS. 4. Bought goods to the value of £109 : 10, to be p lid at nine months, what present money will discharge the same, if I am al- lowed 6 per cent, per annum discount? Ans. £10^: 15 : 8^. 6. What is the present worth of £527 : 9 : 1, payable 7 month* hence, at 4^ per cent. ? Ans. £514 : 13 : lOf^. 6. What is the discount of £85 : 1^', ^hm September thr 8th, this being July the 4th, rebate at 5 ^^ei C(*'i , ..er annum? Ans. 15s. 3fd. 7. Sold goods for £875 : 5 ; C, to be paid 5 months hence, what is the present worth at 4^ per cent. ? Ans. £859 : 3 : 4. 8. What is the present worth of £500, payabl^^ i.». ki, i.iciths, at 5 per cent, per annum ? Ans. £480. 9. How much ready money can I receive for a note of £75, due 15 mon'ihs hence, at G per cent. ? Ans. £70 : 11 : 9^. 10. Whiit will be tlse present worth of £150, payable at 3 four months, i.e. one tLird at four months, one third at 8 months, and one third at 12 months, at 5 per cent, discount ? A71S. £145 : 3 : 8^. 11. Sold goods to the value of £575 : 10, to be paid at 2 three months, what must be discounted for present payment, at 5 per cent; ? Ans. £10 : 1 1 : 4f . 12. What ifl the present worth of £500 at 4 per cent., £100 being to be paid down, and the lest at 2 six months ? Ans. £488 : 7 : 8^ EQUATION OF PAYMENTS, Is when several sums are due at different time to fird a mean time for paying the whole debt ; to do which thi ^s t- common RvLB. Multiply eaclii term by its time, and div ide the sum of the products by iMe whole debt, ihe quotient is accounted the inea*- time. ii I \l at nine [ am al- :8i. months lOf. he 8th, 3^(1. hence, ?:4. -ic'iths, C480. of £75, ble at 3 months, i:8i. , 2 three at 5 per t., £100 :8^ a mean ommon the sum nteu the EQUATION OF PAYMENTS. EXAMPLES. t/ 1. A owes B £200, whereof £40 is to be paid at 3 months, £60 at 5 months, and £100 at 10 months ; at what time may the whole debt be paid together, without prejudice to either ? £ m. 40 X 3 = 120 60 X 5 = 300 too X 10 = 2 00) 1000 14 20 7 months 1 6 2. B owes C £800, whereof £200 is to be paid at 3 months, £100 at 4 months, £300 at 5 months, and £200 at 6 months ; but they agreeing to make but one payh^at of the whole, I de- mand what time that must be ? Ans. 4 months, 18 days. 3. I bought of K a quantity of goods, to the value of £360, which was to have been paid as follows : £120 at 2 months, and £200 at 4mo'?iths, and the rest at 5 months ; but they afterwards agreed to hav c it pai'^ at one mean time ; the time is demanded. Ans. 3 months, 13 days. 4. A merchar ought goods to the value of £500, to pay £100 at the end of 3 ionth«' '15( at the end of 6 months, and £250 at the end of 12 months , i alter ards they agreed to discharge the debt at one payment ; u. vhat time was this payment made ? Ans. 8 months, 12 days. 5. H is indebted to L a certain sum, which is to be paid at 6 different payments, that is, [ at 2 moi.< s, f at 3 months, f 1 m( nths, ■}- at 5 months, -i- at 6 months, an.j the rest at 7 months ; bu! they agree that the whole should be paid at one equated time i wiiat is that time ? Ans. 4 lonths, 1 qu-^rter. 6. A is indebted to B £1*30, whertoi | is to be paid at 3 months, 4^ -t « months, a^ 1 the rest st 9 months h equated time of the whole payment ? IS. 5 nonths days. G3 ^ IS iii8 78 : BARTER. BARTER Is the exchanging of one commodity for another, and informi the traders so to proportionate their goods, that neither may sustain loss. Rule 1st. Find the value of that commodity whose quantitj is given ; then find what quantity of the other, at the rate pro posed, you may have for the same money, 2dly. When one has goods at a certain price, ready money, but in bartering, advances it to something more, find what the other ought to rate his. goods at, in proportion to that advance, and then proceed as before. EXAMPLES. 1. What quantity of chocolate, at As. per lb. must be delivered in barter for 2 cwt., of tea, at 98. per lb. 1 2 cwt., 112 "224 lb. 9 price. 4)2016 the value of the tea. 5041b. of chocolate. 2. A and B ])arter ; A hath 20 cwt. of prunes, at 4d. per lb. ready money, but in barter will have .5d, per lb. and B. hath hops worth 328. per cwt., ready money whai ottght B to rate his hops at in barter, and what quan- tity muist, bo given for the 20 cwt., of prunes 1 112 As 4: 5:: 32 20 6 8. 40 12 2240 5 -cwt. qr. lb. 4)160 408. 4810)112010(23 . 1 . 9|f .An*. 96 160 144 16=1 qr. 9 lb. ||. 3. How much tea, at 93. per lb. can I have in barter for 4 cwt., 2 qrs. of chocolate, at 4s, per lb. 1 , « . ' *^ Ans. 2 cwt. 4. Two merchants barter ; A hath 20 cwt. of cheese, at 21s. 6d. per cwt. ; B hath 8 pieces of Irish cloth, at £3 . I4s. per piece : I desire to know who must receive the difference, and how much *? ^ «. « ^o « Ans. B must receive of A x-o . ». r>. A and B barti^r •. A hath 3l lb. of nepper at I3id. per lb. ; B hath gin- ger at I5td. per lb.; how much ginger must he deliver m barter for thr ^^^' - Ans. 3 lb. 1 oz. l\. PROnX AND LOSS. 70 6 How many dozen of candles, at 5b. 2d. per dozen, must bo delivered in barter for three cwt. 2 qrs. 16 lb. of tallow, at 378. 4d. per^cwt.^ ^^ ^ ^^ 7. A hath 608 yards of cloth, worth Us. per yard, for which B riyeth him £125 . 12. in ready money, and 85 cwt. 2 qrs. Jt lb. of bcca'-wax. The ques- tion is. what did B rockon his bees'- wax e.t per cwt. 1 , ^o i n 8. A and B I irter ; A hath 320 dozen of candles, at 43. 6tl. per dozen; for which B giveth him £30 in money, and the rest in cotton, at 8d. per lb. ; 1 dcsiro to know how much cotton B gave A besides the money 1 Ans.ll cwt. 1 qr. 9. If P. hath cotton, at Is. 2d per lb., how much must he give A for 114 lb. of tobacco, at 6d. per lb. 1 . ^o n i •? 10. C hath nutmegs worth 7s. Od. per lb. ready money, but in barter will have 8s. per lb.; and D hath leaf lobacco worth 9d. per lb. ready raciicy; how much must D rate his tobacco at per lb. that hia profit may be equivalent PROFIT AND LOS& Is a Rule that discovers what is got or lost in the buying or selling of goods, and instructs us to raise and lower the price, so as to gain so much per cent, or otherwise. The questions in this Rule are performed by the Rule of Three. EXAMPLES. 1. If ayar.' of cloth is bought for 2. If 60 ells of Holland cost £J8 lis. and sold for I'is, 6d. what is the gain per cent. 1 As 11 : 1 : G : : 100 12 20 18 12.6 11.0 2000 18 11)30000 12)3273^. 2I'0)27I2 . 8 Ans. £13 . 12 . S^^. what must 1 ell be sold for to gain 8 per cent. 1 As 100 : 18 : : 103 108 1 100)19144 20 8|80 12 9160 4 12X5=60 12)19. 8.9i 5)1 . 12.41 0. 6.51 2j40 -Ans Gs. 5 id. so FELLOWSHIP. ii '% 2 If 1 lb. of tobacco cost I6d. and is sold for 20d. what is the gain per cent. 1 Ans. £25. 4. If a parrel of cloth be sold for £560, and at 12 per cent, gain, wha^Wai the prime cost-? , ,, . ^ ./""'A^ii. 5. If a yard of cloth is bought for iSs. 4d. and sold again for 16s. what is the gain percent.'? ^ ,^ ^ 4ns. £^. 6. If 112 lb. of iron cost 27s. 6d., what must 1 cwt. be sold for to gam 15 per (jgjjt^ 1 Ans. £1 , 11 . 7i. 7. If 375 yards of broad cloth be sold for £490, and 20 per cent, profit, what did it cost per yard 1 ^"*- £1 • 1 • 9i- 8. Sold 1 cwt. of hops, for £6 . 15, at the rate of 25 per cent, profit, what wouid have been the gain per cent, if I had sold them for £8 per 'cvit. 7 Ans. £48 . 2 . llj. 9. If 90 ells of cambric cost £60, how much must I sell it per yard to gain 18 per cent 1 -^^- ^^^- '^"• 10. A plumber sold 10 fother of lead for £204 . 15, (the fother being 19J cwt) and gained after the rate of £12 . 10 per cent. ; what did it cost him per cy/fi Ans. 18s. 8d. 11. Bought 436 yards of cloth, at the rate of 8s. 6d. per yard, and sold it for 10s. 4d. per yard j what was the gain of the whole 1 „«« , « ^ Ans. £o9 . 19 . 4. 12. Paid £69 for one ton of steel, which is retailed at 6d. per lb. ; what is the profit or loss by the sale of 15 tons 1 , Ans. £182 loss. 13. Bought 124 yards of Uaen, for £32 ; how should the same be retailed per yard to gain 15 per cent. 1 . - , , i a a ^ ^ ^ ^ Ans. 5s. nd.-ffT' 14. Bought 249 yards of cloth, at 3s. 4d. per yard, retailed the sams at 4s. 2d. per yard, wkat is the profit in the whole, and how much per cent. ^ Ans. £10 . 7 . 6 profit, and £25 per cent FELLOWSHIP Is when two or more join their stock and trade together, so to determine each person's particular share of the gain or loss, in ■ •ioportion to his principal in joint stock. By this rule a bankrupt's estate many be divided amongst his creditors ; as alr.o legacies may be adjusted when there is a defi- ciency of assets or effects. FELLOWSHIP IS EITHER WITH OR WITHOUT TIME. FELLOWSHIP WITHOUT TIMK RuLF. As the whole stock : is to the whole gain or loss : : so is each man's share in stock : to his share of the gain or loss. Proof. Add all the shares together, and the sum will be equal */^ *V.o trUron nrain nr \i)ea lint thp. SlirPSt WftV is. aS the wholo gain or loss : is to the whole stock : : so is each man's share o< the gain or loss : to his share in stock. FELLOWBHrP. 81 EXAMPLES, 1. Two merchants trade together; A puts into stock £20, ood B £40, they gained £50 ; what is each persoi.'s share thereof f A 60: 50:: 20 20 As 60 : 50 . : 40 40 610)200 £33 . 6 . 8 33. 6 . 8, B's share. 16 . 13 . 4, A's. 610)10010 £16 13.4 50. O.Oproofl 2. Three merchants trade together, A, B and C ; A IJ^t in £20 B £30, and C £40 ; they gained £180 : what is -ch^-- PJ^ of ^he^|-n ^^ ^^ q A B and C enter 'ntc partnership; A puts in £364, B £4^, and C £500ranrtfey gained £867; what isWh man's share in proportion to to. """^^ ■ Ans. A £234 . 9 . 3i-rem. 70 ; B £310 . 9 . 5^rem. 248 ; C £322 . 1 . 3J— rem. 1028. A Four merchants B C, D. and Emake a stock; P put in £227, C £349, D £m, LTe £^9 ; in tmding th^ gained £428 : 1 d.maUd each merchant's *'"'''*^'^""%n..B£85..19.6j-690; C £132 3 . 9-120 ; D £43. 11 . i|_250 ; E £166 . 5 . 6i— 70. 5. Three persons, D,E, and F,join in company; D's stock wm £750, E's £460 and F^ £500 ; and at the end of 12 months they gained £684 : what is each man's particular share of the gainf ^^^ ^ ^^^^ ^ ^^^^ ^^ p ^2^ 6. A merchant is indebted to B £275 . 14, to C,£304 .7, to D £152, and to E £104 . 6; but upon his decease, his estate is found to be worth but £675 . 15 : how must it be divided among his creditors 1 Ans. B's share £222 . 15 . 1-6584 ; C's £2^ 18 -^^^^^ > D'c £122 . 16 . 2J— 12227; and E's £84 . 5 . 5-15620. 7 Four persons trade together in a tomt stock, of which A has^, B i", C i and D*T and at the end of 6 months tiey gain £100: what is each nan's share of the said gain 1 ^ ^^^ ^ ^_^^^^ ^ ^^6 . 6 . 3J-36; C £21 . 1 . OJ —120; and D £17 . 10 . lOj— 24. 8. Two persons purchased an estate of £1700 per annum, fr«^;J;"W, for £27,200, when money was at 6 per cent, interest, and 4s. per pound, »^nci-tMC , whereof b paid £15,800, and E the rest ; sometmve after, t^e ^nte'f ?iJ^,Itr^ nev fallinff to 5 per cent, and 28. per pound land-tax, the; sell the said estate for 24-years' purchase : 1 desire to know each V^^'^'^^J^' ^^^ g ^^g gog.. sz F£LL0w3Ht J'^. • .iJ?^ ^' i""'.^ ^^^''^ ^*°^^^ ^^ *ra<^e 5 t^ie amount of their stocks IS £647, and they arc in proportion as 4, 6, and 8 are to one another, and the amount of the gain is equal to D's stock : what is each man's stock and gain ? Ans. D's stock £143 . 15 . 6^1 gain, 31 . 19 . (m. f^ 215.13.4 47.18.6|f. F « 287 . 11 . lA 63 . 18 . (V^. 10. D, E, and F, join stocks in trade; the amount of their s ock was £100; D's gain £3, E's £5, and F's £8 : what was each man's stock ? ^^7^5. D's Stock £18 . 15; E's £31. 5 ; and F'» £50, FELLOWSHIP WITH TIME. ' i'.,^J''''''\^fi!^''' V? ^^*^^ products of each man's money and time . IS to the whole gam or loss : : so is each man's product : to his share of the gain or loss. prouuci Proof. As in fellowship without timtj. EXAMPLES. months, and E £75 for four months ; and they gained £70 • what IS eacli man's share of the gain ? ' Ans. D £20, E £50, 40x3=120 75X4=300 As 420 : 70 : : 120 120 As 420 : 70 : : 300 300 420 42|0)840|0(20 '840 42|0)2100|0(50 2100 14 r T"" '"^'•^^^''\"t^ m^ i« company ; D puts in stock £195 . ;f-o. . 14 . 10, for 1 1 months ; they gained £364 . 18 : what is each Hian's part of the gain? Ans. IVs £102 . .4—5008; E's £148 . 1 . IJ- 4S'iS02 ; and F's £114 . 10 . (5|— 14707." ▲LLIOATION. «3 It of their d 8 are to ►'s stock : : of their t^hat was )ney and Jroduct : 3. Three merchants join in company for 18 months ; D put in £500, and at five months' end takes out £200 ; at ten months* end puts in £300, and at the end of 14 months takes out £130 : E puts in £400, and at the end of 3 months £5:70 more ; at 9 months he takes out £140, but puts in £100 at the end of 12 months, and withdraws £99 at the end of 15 months : F puts in £900, and at 6 months takes out £200; at the end of 11 months puts in £500, but takes out that and £100 more at the end of 13 months. They gained £200 ; I desire to know each man's share of the gain ? Ans. D £50 : 7 : 6—21720 ; E £62 : 12 : ^^-29859 : and F £87:0:0^—14167. 4. D, E, and F, hold a piece of ground in common, for which they are ^o pay £36 : 10 : 6. D puts in 23 oxen 27 days ; E 21 oxen 35 ^' /s ; and F 16 oxen 23 days. What is each man to pay of the said rent ? Ans. D £13 : 3 : 1^-624 ; E £15 : 11 : 5—1688 ; and F £7 ; 15 : 11—1136. 4' 3r three id £70 : I £50. ): :300 )0 010(50 } c £195 . and F what is [ . IJ- 707. ALLIGATION ALLIGATION IS EITHER MEDIAL OR ALTERNATE. ALLIGATION MEDIAL Is when the price and quantities of several simples af« g4ven to be mixed, to find the mean price of that mixture. Rule. As the whole composition : is to its total value : : so is any part of the composition : to its mean price. Proof. Find the value of the whole mixture at the mean rate, and if it '^n'^'^^^s «m<1> tiin ♦a^m »-"1"- -f ^i-- - i .•• their respective prices, tlic work is right. :t si '84 ALLIGATION. EXAMPLES. 1. A farmer mixed 20 bushels of wheat, at 5s. per bushel, and 36 bushels of rye, at 3s. per bushel, with 40 bushels of barley, at 2s. per bushel. I desire to know the worth of a bushel of this mixture. As 96 : 288 : : 1 : 3 Ans. 3s. 20X5 = 100 36x3 = 108 40x2 = 80 96 288 2. A vintner mingles 15 gallons of canary, at 8s. per gallon, with 20 gallons, at 7s. 4d. per gallon, 10 gallons of sherry, at 6s. 8d. per gallon, and 24 gallons of white wine, at 4s. per gallon. What is the worth of a gallon of this mixture? Ans. 6s. 2^d. 6 9' 3. A grocer mingled 4 cwt. of sugar, at 56s. per cwt. with 7 cwt. at 43s. per cwt. and 5 cwt. at 37s. per cwt. I demand the price of 2 cwt. of this mixture. Ans. £4.8.9. 4. A maltster mingles 30 quarters of brown malt, at 28s. per quarter, with 46 quarters of pale, at 30s. per quarter, and 24 quarters of high-dried ditto, at 25s. per quarter. What is the value of 8 bushels of this mixture ? Ans. £1 . 8 . 2^d. ° 5. If I mix 27 bushels of wheat, at 5s. 6d. per bushel,*with the same quantity of rye, at 4s. per bushel, and 14 bushels of barley at 2s. 8d. per bushel, what is the worth of a bushel of this »"ixture? Ans. 4s. S^d.^. 6. A vintner mixes 20 gallons of port at 5s. 4d. per gallon, with 12 gallons of white wine, at 5s. per gallon, 30 gallons of Lisbon, at 6s. per gallon, and 20 gallons of mountain, at 4s. 6d per gallon. What is a gallon of this mixture worth ? Ans. 5s. 3|d.-f4. 7. A refiner having 12 lb. of silver bullion, of 6 oz. fine, would melt it with 8 lb. of 7 oz. fine, and 10 lb. of 8 oz. fine; required the fineness of 1 lb. of that mixture ? Ans. 6 oz. 18 dwts. 16 gr. 8. A tobacconist would mix 50 lb. of tobacco, at lid. per lb. with 30 IK at 14d. per lb. 25 lb. at 22d. per lb. and 37 lb. at 2s per lb. What will 1 lb. of this mixture be worth I Ans. 16|d.||f AT.LIOATION. 86 ALLIGATION ALTERNATE (s when the price of several things are given, to find such quanti* ties of them to make a mixture, that may bear a price pro- pounded. In ordering the rates and the given price, observe, 1. Place them one under the other, 18 M and the propounded price or mean «^20 rate at the left hand of them, thus, 23 24- 28- .6 .4 .2 2. Link the several rates together by 2 and 2, ahva , observ- ing to join a greater and a less than the mean. 3. Against each extreme place the difference of the mean and its yoke fellow. When the prices of the several simples and the mean rate are given without any quantity, to find how much of each simple is required to compose the mixture. Rule. Take the difference between each price and the mean rate, and set them alternately, they will be the answer required. Proof. By Alligation Medial. EXAMPLES. 1. A vintner would mix four sorts of wine together, of 18d., 20d., 24d., aiid 28d. per quart, what quantity r^ each must he have, to sell the mixture at 22d. per quart ? or thus. Proof. 18 6 of 18d. = 108d. 2 of 20d. = 40 2 of 24d. = 48 4 of 28d. = 112 22' Answer. Proof. 18 _2 of 18d. = 36d. 20-. 6 of 20d. = 120 24_ 4 of 24d. = 96 28 2 of 28d. = 56 1 4 )308 22d. 22^—1 24j 28 14 )30S 22d. Note. Queations in this rule admit of a great variety of an- swers, according to the manner of linking them. 2. A grocer would mix sugar at 4d., 6d., and lOd. per lb., »o as to sell the compound Ibv 8d. per lb. ; what quantity of each Ans. 2 lb. at 4d., 2 lb. at 6d., and 6 lb. at lOd, H I r 86 ALLIOATION PARTIAL. 3. I desire to know how much tea, at 16s., 14s., 9s., and 89 per lb., will compose a mixture w^orth 10s. per lb. ? Ans. 1 lb. at IGs., 2 lb. 14s., 6 lb. at 9s., and 4 lb. at 8s. 4. A farmer would mix as much barley at 3s. 6d. per bushel, rye at 4s. per bushel, and oats at 2s. per bushel, as to make a mixture worth 2s. 6d. per bushel. How much is that of each sort? Ans. 6 bushels of barley, 6 of rye, and 30 of oats. 5. A grocer would mix ra?sins of the sun, at7d. per lb., with Malagas at 6d., and Smyrnas at 4d. per lb. ; I desire to know what quantity of each sort he must take to sell them at 5d. per lb. ? Ans. 1 lb. of raisins of the sun, 1 lb. of Malagas, and 3 lb. of Smyrnas. 6. A tobacconist would mix tobacco at 2s., Is. 6d., and Is. 3d. per lb., so as the compound may bear a price of Is. 8d. per lb What quantity of each sort must he take ? i ! An.-» T-» r~\ c per lb. I desire to know what quantity of each lie must take? H ■ f! POSITION, OR THE RULE OF FALSE. Answer. Proof. 12 . 4 : 48 at 12d. 576=As 12 : 4 : : 144 : 48 2 : 24 at lOd. 240=As 12 : 2 : : 144 : 24 2 : 24 at 6d. 144=As 12 : 2 : : 144 : 24 4 : 48 at 4d. 192=As 12 : 4 : : 144 : 48 ^'l3 12 144 )1152(8d. 2. A grocer having four sorts of tea, at 5s., 6s., 8s., and 9s. per lb., would have a composition of 87 lb., worth 7s. per lb. What quantity must there be of each ? Ans. 14| lb. of 5s., 29 lb. of 6s., 29 lb. of 8s., and 14^ lb. of 9s. 3. A vintner having four sorts of wine, viz., white wine at 4s. per gallon ; Flemish at Os. per gallon ; Malaga at 8s. per gal- lon ; and Canary at 10s. per gallon ; and would make a mixture of 60 gallons, to be worth 5s. per gallon. What quantity of each must he take ? Ans^ 45 gallons of white wine, 5 gallons of Flemish, ^ 5 gallons of Malaga, and 5 gallons of Canary. 4. A silversmith had four sorts of gold, viz., of 24 caratf fine, of 22, 20, and 15 carats fine, and would mix as much of each sort together, so as to have 42 oz. of 17 carats fine. How much must he take of each ? Ans. 4 oz. of 24, 4 oz of 22, 4 oz. of 20, and 30 oz. of 15 carats fine. 5. A druggist having some drugs of 8s., 5s., and 4s. per lb., made them into two parcels ; one of 28 lb. at 6s, per lb., the other of 42 lb. at 7s. per lb. How much of each sort did he f^ke for each parcel 1 Ans. 12 lb. of 8s. 8 lb. of 5s. 8 lb. of 48. 30 lb. of 8s. 6 lb. of 5s. 6 lb. of 4s. 28 lb. at 6s. per lb. 42 lb. at 7s. per lb. POSITION, OR THE RULE OF FALSE, ha a rirtc that by false or supposed numbers, taken at pleasure discovers the true < ne required. It is divided into two pall^^ SiKoLE and Double. ifosiriofi, OR the rule op false. SINGLE POSITION Is, by using one supposed number, and working with it as the true one, you find the real number required, by the following Rule. As the total of the em r? : is to the true total : : so is the supposed number : to the true one required. Proof. Add the several parts of the sum together, and if \i agrees with the sum it is right. EXAMPLES. 1. A schoolmaster being asked how many scholars he had, said, If I had as many, half as many, and one quarter as many more, I should have 88. How many had he ? Suppose he had... 40 As 110 : 88 : : 40 as many 40 40 half as many 20 } as many.... 10 110 11|0)352|0(32 33 22 22 Ans. 32. 32 32 16 8 88 proof. 2. A person having about him a certain number of Portugal pieces, said, If the third, fourth, and oth of them were added together, they would make 54. I desire to know how many he had ? Ans. 72. 3. A gentleman bought a chaise, horse, and harness, for £60, the horse came to twice the price of the harness, and the chaise to twice the price of the horse and harness. What did he give for each ? Ans, Horse £13:6: 8, Harness £6:18: 4, Chaise £40. 4. A, B, and C, being determined to buy a quantity of goods which would cost them £120, agreed among themselves that B should have a third part more than A, and C a fourth nart more than B. I desire to know what each man must pay? Ans. A £30, B £40, C £50. H3 00 POSITION, OR THE RULE OF FALSE. 5. A person delivered to another a sum of money unknown, to receive interest for the same, at 6 per cent per annum, simple 'i- terest, and at the end of lO years received, for principal and in- terest, £300. What was ilie sum lent? Ans. £187 : 10. DOUBLE POSITION I : Is by making use of two supposed numbers, and if both prove false, (as it generally happens) they are, with their errors, to be thus ordered : — Rule 1. Place each error against its respective position. 2. Multiply them cross-ways. 3. If the errors are alike, i. e. both greater, or both less than the given number, take their difference for a divisor, and the difference of the products for a dividend. But if unlike, take their sum for a divisor, the sum of their products for a dividend, the quotient will be the answer. EXAMPLES. 1. A,B, and C, would divide £200 between them, so that B may have £6 more than A, and C £8 more than B ; how much must each have ? Suppose A had 40 Tlien B had 4G and C 54 j'hf n suppose A had 50 tiuiii B must have 56 2),.n.{/i« oe. Tiooirxr pQ flip Inss Min Whai is the weight of each cup ? Ans, 3 ounces less, 4 greater BXCHANOE. 01 3. A genOeman bouglit a joiist, witli a garden, and ahorse in the staMe, i^r £500 ; now he paid 4 times the price of
ota. What was the age of each person ? Ans. II 30, K 50 and L 80. 5. D, E, and F, playing at cards, staked 324 crowns : but dis- puting about the tri ^k.!*, each man took as many ' uld : D got a certain numb* '" • E as many as D, and 15 ii and F got a fifth part of both thtir ims added together. many did each get? Ans. D 127^, E 14^^, and F 54.. 6. A gentleman g ? into a garden, meets with some ladies, an'^ says to them, Gv . moniing to you 10 fair maids. Sir, you •'listake, answered o le of them, we ar not 10; but if we weie twi^e as many more as we aro, weshou. 1 be as manv above 10 as we are now under. How many were they? JLtis. 5. EXCHANGE Is receiving money in one country for the same value paid in another. The par of exchange is always fixed and cc'-tain, it being the intrinsic value of foreign money, com ared /ith sterling ; but the course of exchange rises and falls upo' various occasions. I. FRANCE. They keep -heir : ccounts at Paris, Lyons, and Rouen, inlivres, sols, and deniers, and exchange by the crown=4s. 6d. at par. Note. 12 denisid make 1 sol. 20 sols 1 livre. 3 livers 1 crown. ■>%, .ail IMAGE EVALUATION TEST TARGET (MT-3) 1.0 I.I 1.25 | 4S m Hi ■ 40 2.: 22 20 1.8 1.4 III 1.6 V] //. v: %% //^ '/ Hiotographic Sciences Corporation 33 WEST MAIN STREET WEBSTER, NY. MSSO (716) 872-4503 ^ >^ rV ^ :\ ,v \ 4 A^^ r^f ^11051 17 Ai (V^H^ whflt must I draw for on my correspondent in Lisbon, exchange at 58. 5f d. per milrea ^ i iln*. 4366 milreas, 183 reM. 94 ^ EXCHANGE. V. HOLLAND, FLANDERS, AND GERMANY. They keep their accounts at Antwerp, Amsterdam, Brussels, Rotterdam, and Hamburgh, some in pounds, shillings, and pence, as in Fngland ; others in guilders, stivers, and pennings ; and 'exchange with us in our pound, at 33s. 4d. Flemish, at par. JS^OTE. 8 pennings make 1 groat. 2 groats, or 16 pennings .... 1 stive?. 20 stivers 1 guilder or florin. ALSO, 12 groats, or G stivers make. . 1 schelling. 20 schcUiiigs, or 6 guilders. . . 1 pound. To change Flemish into Sterling. Rule. As the given rate : is to one pound : : so is the Flemish «um : 10 the sterling required. To cha7ige Sterling into Flemish. % Rule. As £1 sterling : is to the given rate : : so is the sterling given : to the Flemish sought. EXAMPLES. 16. Remitted from London to Amsterdam, a bill of £754 . 10 . sterling, how many pounds Flemish is the sum, the exchange at 33s. 6d. Flemish, per pound sterling '{ Ans. £1263 . 15 . 9, Flenush. 17. A merchant in Rotterdam remits £1263 . 15 . 9 Flemish, to be paid in London, how much sterling money must he draw for, the exchange being at 338. Cd. Flemish per pound sterlinff 'J Ans. £754 . 10. IS. If I pay in London £853 . 12 . 6, sterling, how many guilders must I draw for at Amsterdam, exchange at 34 schel. 4i groats Flemish per pound iterling '\ Ans. 8792 guild. 13 stiv. 14i pennings. 19. What must I draw for at London, if I pay in Amsterdam 8793 guild. 13 stiv. 14i pennings, exchange at 34 schel. 4| groats per pound sterling *? Ans. £852 . 12 . 6. To convert Bank Money into Current^ and the contrary. Note. The Bank Money is worth more than the Current. The difference between one and the other is called agio, and is generajly from 3 to 6 per cent, in favour of the Bank. To change Bank into Current Money. R «rr «• Acs 1 AA /nii1#lAl*ci ■Rank ? ia to 100 with the awio added ; i so ia the Bank given : to the Current required. EXCHANGE* 05 NY. Brussek, nd pence^ igs ; and t par. 3 Flemish e sterling l^rling, how , per pound Flemish. ) be paid in leing at 338. ;75l.lO. tiers must I per pound pennings. 8793 guild, •ling'? 2 . 12 . 6. irary. Current. iO) and is i \ A AUt^A To change Current Money into Bank. RtJL£. As 100 with the agio is added : is to 100 Bank : : so is the Curl at money given : to the Bank required. 20. Change 794 guilders, 15 stivers, Current Money, into Bank florins agio 4f per cent, t Ans. 761 guilders, 8 stivers. \\\^ pennings. 21. Change 761 guilders, 9 stivers Bank, into Current Money, agio 4f per cent. Ans. 794 guilders, 15 stivers, 4^^ pennings. VI. IRELAND. 22. A gentlenlan remits to Ireland £575 : 15, sterling, what will he receive there, the exchange being at 10 per cent. ? Ans. £633 : 6 : 6. 23. What must be paid in London for a remittance of £633 t 6 : 6, Irish, exchange at 10 per cent. ? Ans. £575 : 15. COMPARISON OF WEIGHTS AND MEASURES. EXAMPLES. 1. If 50 Dutch pence be worth 65 French pence, how many Dutch pence are equal to 350 French pence ? Ans. 269f|. 2. If 12 yards at London make 8 ells ?\ Paris, how many ells at Paris will make 64 yards at London ? Ans. 42^. 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 951b, Flemish make 100 lb. English, how many lb. En- glish are equal to 275 lb. Flemish. Ans. 289f|. CONJOINED PROPORTION Is when the coin, weights, or measures of several countries arc compared in the same question ; or, it is linking together a varie- ty of proportions. When it is required to find how many of the first sort of coin, weight, or measure, mentioned in the question, are equal to a given quantity of the last. 9« PROPORTION. ■I I 'I! I I Left. Right. 20 23 155 180 n Rule. Place the numbers alternately, beginning at the left hand, and let the last number stand on the left hand ; then multi- ply the first row continually for a dividend, and the second for a diviu r. Proof. By as many single Rules of Three as the question requires. EXAMPLES. 1. If 20 lb. at London make 23 lb. at Antwerp, and 156 lb. at Antwerp make 180 lb. at Leghorn, how many lb. at London arc equal to 72 lb. at Leghorn ? 20X155x72=223200 23 X 180 = 4140)223200(53fii. 2. If 12 lb. at London make 101b. at Amsterdam, and 100 lb. at Amsterdam 120 lb. at Thoulouse, how many lb. at London are equal to 40 lb. at Thoulouse ? Ans. 40 lb. 3. If 140 braces at Venice are equal to 156 braces at Leghorn, and 7 braces at Leg^horn equal to 4 ells English, how many bra- ces at Venice are equal to 16 ells English 1 Ans, 25^. 4. If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. at Amsterdam make 116 at Dantzick, how many lb. at London are equal to 130 lb. at Dantzick ? Ans. U^lffA' When it is required to find how many of the last sort of coin, weight, or measure, mentioned in the question, are equal to a quantity of the first. Rule. Place the numbers alternately, beginning at the left hand: and let the last number stand on the rio'ht hand ; then mul» tiply the first row for a divisor, and the second for a dividend. the left m multi- icond for question i 156 lb. London ^• id 100 lb. k London . 40 lb. Leghorn, lany bra- nd 90 lb. t London '*4 1 y « • t of coin, qual to a t the left then mul» ividend. rnoGREgflioN. 07 EXAMPLES. 5. If 12 lb. at London make 10 lb. at Amsterdam, and 100 lb. at Amsterdam 120 lb. at Thoulouse, how many lb. at Thoulouso are equal to 40 lb. at London ? Ans, 40 lb. 6. If 40 lb. at London make 30 lb. at Amsterdam, and 90 lb. at Amsterdam 116 ]b. at Dantzick, how many lb. at Dantzick are equal to 122 lb. at London ? Ans. Hl\^, PROGRESSION CONSISTS OP TWO PARTS, ARITHMETICAL AND GEOMETRICAL. ARITHMETICAL PROGRESSION Is when a rank of numbers increase or decrease regularly by the continual adding or subtracting of equal numbers ; as 1, 2, 3, 4, 3, 6, are in Arithmetical Progression by the continual increasing or adding of one; 11, 9, 7, 5, 3, 1, by the continual decreasing or subtracting of two. Note. When any even number of terms differ by Arithme- tical Progression, the sum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the exteemes : as 2, 4, 6, 8, 10, 12, where 6 -f 8, the two middle numbers, are=12+2, the two extremes, ana=10+4thc two means=l4. When the number of terms arc odd, the double of the middle term will be equal to the two extremes ; or of any two means equally distant from the middle term ; as 1, 2, 3, 4, 5, where the double number of 3=5+ 1=2+4=6. In Arithmetical Progression five things are to be observed, viz. 1. The first term ; better expressed thus, F. 3. The last term, L. 3. The number of terms, ....,N. 4. TheequaldifTerence, D. 5. The sum of all terms, S. Any three of which being given, the other two may be found. The first, second, and third terms given, to find the fifth. Dnvvi iur..i^:.,%i.. 4i.« _i» At.- A___ . 1 t. «* .t - a«u2ica i-AUiiipij mc suui ui uiB \wo exiTBTUGs oy naii ine number of terms, or multiply half the sum of the two extreme! 96 PROOR1.88ION. / by the whole number of terms, the product is the totel of «ll the terms : or thus, F. F LN arc given to find S. -N ■ F-fLX— =S. 2 EXAMPLES. 1. How many strokes does the hammer of a clock strike in 12 hours t 12+1=13, then 13x6=79. 2. A man bought 17 yards of cloth, and gave for the first yard 26. and for the last 10s. what did the 17 yards amount to? Ans, £5 . 2. 3. If 100 eggs were placed in a right line, exactly a yard a«^ under from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggi singly, and returns with every egg to the basket to put it in ? Ans, 6 miles, 1300 yards. The first, second, and third terms given, to find the fourth. Rule. From the second subtract the first, the remainder divi- ded by the third less one, gives the fourth : pr thus II. F L N are given to find P. L— F ■'• >,.^D. ■ . , . ..- N— 1 EXAMPLES. 4. A man had eight sons, the youngest was 4 years old, and the eldest 32, they increase in Arithmetical Progression, what was the common difierence of their ages ? Ans, 4. J ,:Ul then 28T*-&-rl=4 common diflference. 5. A man is to travel from London to a certain place in 12 days, and to go but 3 miles the first day, increasing every day by |ta equal excess, so that the last day*8 journey may be 58 miles, PROORBSSION. 90 1 of all the trike in 12 s first yard tto? £5.2. a yarii as- sket, what e 100 egg'i Qt it in ? ) yards. I fourth. inderdivi- '8 old, and sion, what Ans, 4. le. lace in 12 ery day by e 58 mileg, what is the daily increasef and how many miles distant is that place from London ? Ans. 5 daily incrraie. Therefore, as three miles is the first day's journey, 34-6=8 the second day. 84-5=13 the third day, &c. - Tne whole distance is 366 miles. The first, second, and fourth terms given, to find the third. RuLB. From the second subtract the first, the remainder di /ide by the fourth, -and to the quotient add 1, gives the third ; or thus, III. F L D are given to find N. L— F D ^EXAMPLES. 6. A person travelling into the country, went 3 miles the first day, and increased every day 5 miles, till at last he went 58 miles in one day ; how many days did he travel ? Ans, 12. 58— 3=55-»-5=ll+l=12 the number of days. 7. A man being asked how many sons he had, said, that the youngest was 4 years old, and the oldest 32 ; and that he increas- ed One in his family every 4 years, how many had he? Ans. 8. The second, third, and fourth terms given to find the first Rule. Multiply the fourth by the third made less by one, the product subtracted from the second gives the first : or thus, IV. L N D are given to find F. L— DxN— 1=F. EXAMPLES. 8. A man in 10 days went from London to a certain town in the country, every day's journey increasing the fonner by 4, and the last he went was 46 miles, what the first ? Ans. 10 mile?. 4x10—1=36, then 46—36=10, the first day's journey. r too PROOCSSION. 9. A man takes out of his pocket at 8 several times, so many different numbers of shillings, every one exceeding the former by 6, the last at 46 ; what was the first i Ans. 4. The fourth, third, and fifth given, to find the first. Rule, Divide the fifth by the third, and from the quotient subtract half the product of the fourth multiplied by the third less 1 gives the first : or thus, V N D S are given to find F S DXN— 1 — F. N2 EXAMPLES. 10. A man is to receive £360 at 12 several payments, each to exceed the former by £4, and is willing to bestow the first pay- ment on any one that can tell him what it is. What will that jierson have for his pains ? -Ans, £8. 4 X 12—1 360+12=:30, then 30 =£8 the first payment. The first, third, and fourth, given to find the second. Rule. Subtract the fourth from the product of the third, mul- tiplied by the fourth, that remainder added to the first gives the second : or thus, VI. F N D are given to find L. NP— P-f-F=L EXAMPLES. 11. What is the last number of an Arithmetical Progression, beginning at6» and continuing by the increase of 8 to 20 places? Ans. 158. 30X8—8=^152, then 152+6=158, the last number. GEOMETRICAL PROGRESSION Is the incieasiiig or decreasing of any rank of numbers by some common ratio ; that is, by the continual multiplication or division /. -— 1 —.1.-.— . ^m n A Q ta. iwxMvanaA \\tr t\\tk vnilltlTtllAt Oi Home OqiSSU JlUinucr i oa .«, •*, «j, *v, iiti^«\jf»ow vj »«»« m..vi»».^—— % an4 IG, 8, 4, 2> decrease by the divisor 2. fROORKSSIOlV. 19! so many } formei ns. 4. quotient he third , each to irst pay- will that IS. £8. nent. ird, mul' gives the gressioR, 5 places? s. 158. I by some r division Tiiilf'irtllAt Note. "When any number of terms is continued in Geome- trical Progression, the product of the two extremes will be equal to any two means, equally distant from the extremes : as 2, 4, 8, 16, 32, 64, where 64x2 are=4x32, and 8X16=128. When the number of the terms are odd, the middle term multi- plied into itself will be equal to the two extremes, or any two means equally distant from it, as 2, 4, 8, 16, 32, where 2X32^ 4x16=8x8=64. In Geometrical Progression the same 5 things are to be obeer fed as are in Arithmetical, viz. 1. The first term. 2. The last term. 3. The number of terms. 4. The equal difference or ratia 5. The siun of all the terms. Note. As the last term in a long series of numbers is very tedious to come at, by continual multiplication ; therefore, for the reader finding it out, there is a series of numbers made use oJ in Arithmetical Proportion, called indices, beginning with an unit, whose common difference is one ; whatever number of in- dices you make use of, set as many numbers (in such Geomet- rical Proportion, as is given in the question) under them. ^g 1, 2, 3, 4, 5, 6, Indices. 2, 4, 8, 16, 32, 64, Numbers in Geometrical Proportion. But if the first term in Geometrical Proportion be different from the ratio, the indices must begin with a cipher. ^g 0, 1, 2, 3, 4, 5, 6, Indices. 1, 2, 4, 8, 16, 32, 64, Numbers in Geometrical Proportion. When the Indices begin with a cipher, th^ ^ *m of the indices made choice of must always be one less than me number of terms given in the question ; for 1 in the indices is over the second term, and 2 over the third, «fcc. Add any two of the indices together, and that sum will agree with the product of their respective terms. As in the first table of Indices 2+ 5= 7 Geometrical Proportion 4x32=128 Then the second 13 'Z-\- 4= 6 4x16= 64 i02 FKOOBMSION. In any Geometrical Progression proceeding from unity, the ratio being known, to find any remote term, without producing all the intermediate terms. Rule. Find what figures of the indices added together would give the exponent of the term wanted : then multiply the num* bars standing under such exponents into each other, and it will give the term required. ^ Note. When the exponent 1 stands over the second term, the number of exponents Vnust be one less than the number o( terms. EXAMPLES. 1. A man agrees for 12 peaches, to pay only the price of the last, reckoning a farthing for the first, and a halfpenny for the second, ^c. doubling the price to the last ; what must he give for them? Ans, £2,^,9, 16=4 0, 1, 2, 3, 4, Exponents 16=4 1, 2, 4, 8, 16, No. of terms. 256::a8 8=a For 4+4+3=11, No. of terms less 1- 4)2Q48«llNo.offi^. 12)512 2|0)4|2 . 8 £2.2.8 2. A country gentleman going to a fair to buy some oxen, meets, with a person who had 23 ; he demanded the price of them, and was answered £16 a piece; the gentleman bids £15 a piece and he would buy all ; the other tells him it could not be taken ; but if he would give what the last ox would come to, at a farthing for the first, and doubling it to the last, he should have all. What was the price of the oxen ? Ans. £4360 .1.4. In any Geometrical Progression not proceeding from unity, the ratio being given, to find any remote term, without produ- cing all the intermediate terms. PROORECSION. 103 <# RvLS. Proceed as in the last, onl/ observoi that erery prodact rnual be divided by the first term. EXAMPLE. 3. A sum of money is to be divided among eight persons, the first to have £20, the next £00, and so in triple proportion ; what will the last have ? Ans. £43740. 540X540 14580X60 30 20 3+3+1=7, one less than the number of terms. 4. A gentleman dying, left nine sons, to whom and to his exe- cutors he bequeathed his estate in the manner following : To his executors £50, his youngest son Was to have as much more as the executors, and each son to exceed the next younger by as much more ; what was the eldest son*s proportion? Ans. £25600. IChc first tei'm, ratio, and number of termd given, to find the tsilm of all the terms. Rut.?:. Find the last term as before, then subtract the first from it, and divide the remainder by the ratio, less 1 ; to the quo- tient of which add the greater, gives the eum required. EXAMPLES. B. A servant skilled in numbers, agreed with a gentleman to serve him twelve months, provided he would give him a farthing for bis first month^s service, a penny for the second, and 4d. for the third, &e., what did his wages amount to ? Ans. £6825 . 8 . 5:1. 256X256=65536, then 65536X64=4194304 4194304—1 =1398101, then 0, 1, 2, 3, 4, 1, 4, 16, 64, 250, 4+4+3=11 No. of terms less 1, 4 — 1 1398101+4194304=5592405 farthings. 6. A man bought ahorse, and by agreement was to give a far- thing for the first nail, three for the second, &.c., there were four shoes, and in each shoe 8 nails \ what was the worth of the horse ? 4ns. £9651 14681 693- • 1 3 • i. 104 PERMIITATION. 7. A certaii: person married his daughter on New*year*s day, and gave her husband Is. towards her marriage portion, promis- ing to double it on the first day of every month for 1 year ; what was her portion 1 ^ns. £204.15. 8. A laceman, well versed in numbers, agreed with a gentle man to sell him 22 yards of rich gold brocaded lace, for 2 pins the first yard, 6 pins the second, &.C., in triple proportion ; I desire to know what he sold the lace for, if the pins were valued at 100 for a farthing ; also what the laceman got or lost by the sale thereof, supposing the lace stood him in £7 per yard. Ans. The lace sold for £326886 .0.9. Gain £326732 .0 . 9. PERMUTATION Is the changing or varying of the order of things. Rule. Multiply all the given terms one into another, and the last product will be the number of changes required. EXAMPLES. 1. How many changes may be rung upon 12 bells ; and how long would they be ringing but once over, supposing 10 changes might be rung in 2 minutes, and the year to contain 365 days, 6 hours ? 1X2X3 X4X5X6X7X8X9X 10 XllX 12=479001600 changes, which -f- 10=47900160 minutes ; and, if reduced, is=91 years, 3 weeks, 5 days, 6 hours. 2. A young scholar coming to town for the convenience of a good library, demands of a gentleman with whom he lodged, \yiiat his diet would cost for a year, who told him £10, but the scholar not being certain what time he should stay, asked him what he must give him for so long as he should place his family, (consisting of 6 persons besides himself) in different positions, every day at dinner; the gentleman thinking it would not be long, tells him £5, to which the scholar agrees. What time did the^scholar stay with the gentleman? Ans. 5040 days. 106 ear's day, n, promis- ear ; whal 04. 15. k a gentle for 2 pins >ortion; I ere valued )st by the |rard. k . 9. 1.0,9. THE TUTOR'S ASSISTANT. ;r, and the ; and how 10 changes 65 days, 6 479001600 ced, is=9l lience of a le lodged, 10, but the asked him his family, positions, lid not be at time did 10 days. PART 11. VULGAR FRACTIONS, A PR ACTION is a part or parts of an unit, and written with Iw* Igures, with a line between them, as -}, f , f , Ac. The figure above the line is called the numerator, and the un- Jer one me denominator ; which shows how many parts the «init is divided into : and the numerator shoves how many of ihose parts are meant by the fraction. There are four sorts of vulgar fractions : proper, improper, compound, and mixed, viz. 1. A PROPER FRACTION is wheu the numerator is less than the denominator, as f , f , f , -j^, loj., &c. 2. An IMPROPER FRACTION is when the numerator is equal to, or greater than the denominator, as f , f , ff , J-f^, &c. 3. A COMPOUND FRACTION is the fraction of a fraction, and known by the word of, as ^ of f of ■Z'^jf 5^ of j*^, &c. 4. A MIXED NUMBER, OR FriACTioN, IS composed of a whol» number and fraction, as 8^, 2' "a7» 6lc. 106 REDUCTlOBr OF VVLOAR FRACTIONS. REDUCTION OF VULGAR FRACTIONS. 1. To reduce fractions to a common denominator. Rule. Multiply each numerator into all the denominators, except its own, for a numerator ; and all the denominators, for a common denominator. Or, 2. Multiply the common denominator by the several giren numerators, separately, and divide their product by the several denominators, the quotients will be the new numerators. EXAMPLES. 1. Reduce f and f to a common denominator. Facit, ^ and -Jf . 1st num. 2d num. 2x7=14 4x4=16, then 4x7=28 den.=Jf and f|. 2. Reduce ^, f , and -f, to a common denominator, 3. Reduce f , f, -^, and f , to a common denominator. 4. Reduce -f^t f , 'Pacit. 88*fl 824(1 tnifl aBSft fauil, 386 0* M6ff> 8860* 8969' 7> and f, to a common denominator. Facit, 1 n n a 1«80> 8,4 0. ,a.4.o. 1680)1680' 5. Reduce f , f , f-, and -^i to a common denominator, Facit, fH 1686' 6. » 8To» 8«o» Reduce f , f , f, and f , to a common danominator. ttt* Facit, Tan 8 160» iH 0* K A ft jtaaa 2I«0' 2. To reduce a vulgar fraction to its lowest terms. Rule. Find a common measure by dividing the lower term by the upper, and that divisor by the remainder followingr till nothing remain : the last divisor is the common measure ; then divide both parts of the fraction by the common measure, and the quotient will give the fraction required. Note. If the common measure happens to be one, the fraction is already in its lowest term : and when a fraction hath ciphers at the right hand, it may be abbreviated by cutting them off, as W%. EXAMPLES. 7. Reduce |-f- to its lowest terms. 24)32(1 24 Com. measure, 8)24(3 Faciv. RXVUOTlOIf OF VITLOAR FRACTIONS. 107 8. Rednce -^ ta its lowest termv. Facit, ^5-. 9. Reduce -f^ to its lowest ter ^8. Facit, -jVi-. la Reduce ^ to its lowest i; s. Facit, i. 1 1. Reduce fff- to its lowest terms. Facit, f^. 12. Reduce f^^ to its lowest terms. Facit, f . 3. To reduce a mixed number to an improper fraction. Rule. Multiply the whole number by the denominator of (he fraction, and to the product add the numerator for a new numerator, which place over the dienominator. Note. To express a whole number fraction-ways set 1 for the denominator given. EXAMPLES. 13. Reduce 18f to an improper fraction.. 18X7+3=129 new numerator=»f •. 14. Reduce 56^ to an improper fraction. 15. Reduce 183^ to an improper fractioa 16. Reduce 131- to an improper fraction. 17. Reduce 27^ to an improper fraction. 18. Reduce &14|^ to an improper fraction. 4. To reduce an improper fraction to its proper terms. Rule. Divide the upper term by the lower. Facit, Facit, -Lfft. Facit, »|t». Facit, Y- Facit, «f ». Facit, «ff •. EXAMPLES^ 19. Reduce >f » to its proper terms* 129+7=18f, 20. Reduce »ff-» to its proper terms, 21. Reduce 'ff^ ^q j^g proper terms* 22. Reduce ^ to its proper terms. 23. Reduce «f » to its proper terms. 24. Reduce ^ff » to its proper terms. Facit, 18f. Facit, 56ff . Facit, 183^. Facit, 131^. Facit, 27f . Facit, 614tV. 5. To reduce a compound fraction to a single one. -J'^i^^l Multiply all the numerators for a new numerator, and all the denominators for a new denominator. Reduce the new fraction to its lowest terms by Rule 2. 106 REDUCTION OF VULGAR FRACTIONS. il II EXAMPLES. 35. Reduce ^ of f of | to a sixigle fraction. 2X3X5= 30 Facit, — reduced to the lowest term=sf . 3X5X8=120 26. Reduce ^ of 4^ of -J^ to a single fraction. Facit, fif =iVV- 27. Reduce -{i of fj of |i to a single fraction. Facit, iiff =^. 28. Reduce | of f of ^^ to a single fraction. 29. Reduce f of f of f to a single fraction. Pacit •^-^-^=A 30. Reduce f of f of i\ to a single fraction. Facit. »ft — -g- 6 To reauce fractions of one denomination to the fraction of another, but greater, retaining the same value. Rule. Reduce the given fraction to a compound one, by com- paring it with all the denominations between it and that denomi- nation which you would reduce it to ; then reduce that compound fraction to a a single one. EXAMPLES. 3L Reduce f of a penny to the fraction of a pound. 00 T, A . c r, Facit,fof J5 0f,J-,=T^. . Jx, Reduce t of a penny to the fraction of a pound. Facit, j^. 33. Reduce f of a dwt. to the fraction of a lb. troy. Facit, TsVo"' 34. Reduce f of a lb. avoirdupois to the fraction of a cwt. Facit, -yfj, 7. To reduce fractions of one denomination to the fraction of another, but less, retaining the same value. Rule. Multiply the numerator by the parts contained in the several denorainatioiia between it, and that you would rec je it to, for a new numerator, and place it over the given denominator. REDUCTION OF TULOAR FRACTIONS. EXAMPLES. 10» — 148 fe* Ji—2. — Ifi' 68* miction of by com- denomi- •mpound TTsTT* '» •«0' 1200* Wt. » 7 84» iction of d ia the et. ze it ninator. 35. Reduce j^ of a pound to the fraction of a penny. Faci^ 2. 7X aOX 12=1680 -iiii reduced to its lowest term=f .' 36. Reduce y^. of ^ pound to the fraction of a penny. 37. Reduce -oVo of a pound troy, to the fraction of a pennv- ^«ignt- Facit, *. 38. Reduce ^h of a cwt. to the fraction of a lb. Facit, f . 8. To reduce fractions of one denomination to another of the same value, having a numerator given of the required fraction Rule. As the numerator of the given fraction : is to its deno- mmator : : so is the numerator of the intended fraction : to its uenommator. EXAMPLES. 39. Reduce f to a fraction of the same value, whose numera- VIA U ^ ^^' As 2 : 3 : : 12 : 18. Facit, 44. 40. Reduce f to a fraction of the same value, whose numera- tor shall be 25. p^cit, ^. 41. Reduce f to a fraction of the same value, whose numera- tor shall be 47. ^^ Facit, 65^. 9. To reduce fractions of one denomination to another of the same value, having the denominator given of the fractions re- quired. Rule. As the denominator of the given fraction : is to its numerator : : so is the denominator of the intended fraction : to Its numerator. EXAMPLES. 42. Reduce f to a fraction of the same value, whose denomi- ■ator shall be 18. As 3 : 2 : : 18 : 12. Facit, -ff. 43. Reduce ^ to a fraction of the same value, whose denomi- nator shall be 35. Facit, ^^. 44. Reduce f to a fraction of the same value, whose denomi- nator shall be 65f . 47 Facit, 65f. no REDVCTION OF VULCtAK VRACTIONS* 10, To reduce a mixed firaetion to a single one. RuLK. When the numerator lis the integral part, multiply it by the denominator of the fractional part^ addinginthe numerator Of tfte fractional part for a new numerator ; then multiply the de- nominator of the fraction by the denominator of the fractional part foi anew denominator. EXAMPLESi Facit, m=n- 36f 46. Reduce — to a simple fraction.. 48 36 X 3 -f 2=1 10 numerator. 48 X 3 =144 denominator. 23f 46i Reduce— to a simple fraction* Facit,^=f^. When the denominator is the integral part, multiply it by the denominator of the fractional part^ adding in the numerator of the fractional part for a new denominator ; then multiply the numerator of the fraction by the denominator of the fractional part for a new numerator. EXAMPLES. ^ 47. Reduce^to a simple fraction. Facit. 4M-»^ ; 19° 48. Reduce— to a simple fraction. Facit. -A?-- ^ 11. To find the proper quantity of a fraction in the known parts of an integer. Rule. Multiply the numerator by the common parts of the integer, and divide hy the denominator. EXAMPLES. Q V 2ii5S?"5® t^^ * pound sterling to its proper quantity. J X 20=60-4-4=168. ^ Facit, 168 50. Reduce fofa shilling to its proper quantity. Ki u J ^ ^ , ^*ciC 4d. 34- qrs. &1. Keducef of a pound avoirdupois to its proper quantity Ro D J , i. Facit,9oz.2f dr. 04. KedUCe ■*• of a CXVL tn its nrnner /inantU^r Facit, 3 qrs. 3 lb. 1 oz. ISf dr. i KKOOOTIOK Of TVLOAA VRAOTIOITS. Ill nultiply it mmcrator >ly the de- fractional i?=H. HtW. it by the erator of Itiply the fractional e known t8 of the t, 15s. li qrs. }uantity 2f dr. 2idr. I 153. Reduce f of a pound troy to its proper quantity. Facit, 7 oz. 4 dwts. 54. Reduce ^ of an ell English to its proper quantity. Facit, 2 qrs. 3^ nails. oo. Reduce f of a mile to its proper quantity. _ _ , Facit, 6 fur. 16 poles. 66. Reduce f of an acre to its proper quantity. Facit, 2 roods, ao poles. 57. Reduce f of a hogshead of wine to its proper quantity. Facit, 64 gallons. 88. Reduce ^ of a barrel of beer to its proper quantity. Facit, 12 gallons. 59. Reduce i^ of a chaldron of coals to its proper quantity. Facit, 15 Bushels. 60. Reduce f of a month to its proper time. Facit, 2 weeks, 2 days, 19 hours, 12 minutes. 12. To reduce any given quantity to the fraction of any greater denomination, retaining the same value. Rule. Reduce the given quantity to the lowest term men- tioned for a numeratOi , under which set the integral part reduced to the same term, for a denominator, and it will give the fraction required. EXAMPLES. 61. Reduce t5s. to the fraction of a pound sterling. Facit, ii=i£. 62. Reduce 4. 3^ qrs. to the fraction of a shilling. Facit, f . 63. Reduce 9 oz. 2f dr. to the fraction of a pound avoirdupois. Facit, f . 64. Reduce 3 qrs. 3 lb. 1 oz. 12t dr. to the fraction of a cwt. Facit, ^. 66. Reduce 7 oz. 4 dwts. to the fraction of a pound troy. Facit, f. 66. Reduce 2 qrs. 3^ nails to the fraction of an English ell. Facit, f . 67. Reduce 6 fur. 16 poles to the fraction of a mile. V Facit, f . 68. Reduce 2 roods 20 poles to the fraction of an acre. _ _ Facit, -f. 09. Reduce &4 gallons to the fraction of a hogshead of wine. Facit, f . 112 SUBTRACTION OV TULOAR FRACTIONS. 70 Reduce 12 gallons to the fraction of a barrel of beer. 71 . Reduce fifteen bushels to the fraction of a chaldron of coals. Facit A 72. Reduce 2 weeks, 2 days, 19 hours, 12 minutes,\o the fraction of a month. Facit, f . ADDITION OF VULGAR FRACTIONS. .u ^"^j J i?l^"^® *^® ^^^®" fractions to a common denominator, then add all the numerators together, under which place the com- mon denominator. EXAMPLES. 1. Add f and I together. Facit, J4+if=^f=l^. 2. Add i, f and f together. Facit, Ifjf . 3. Add i-, 4i and f together. Facit, 4f •. 4. Add 7f and f together. Facit, S^. 5. Add f and f of f> together. Facit, if. 6. Add 5f, 6f and 4^ together. Facit, 17^^. 2. When the fractions are of several denominations, reduce them to their proper quantity, and add as before. 7. Add f of a pound to |- of a shilling. Facit, 15s. lOd. a Add ^ of a penny to f of a pound. Facit, 13s. 4id. 9. Add 1^ of a pound troy to i of an ounce. Facit, 9 oz. 3 dwts. 8 grs. 10. Add f of a ton to f of a lb. Facit, 16 cwt. qrs. lb. 13 oz. 5J- dr. 11. Add f of a chaldron to f of a bushel. ,„ . ,^ Facit, 24 bushels 3 pecks. 14. Add J- of a yard to f of an inch. Facit, 6 inch. 2 bar. c. SUBTRACTION OF VULGAR FRACTIONS. the^^subtraJI'thpT ^^' ^''^'" fraction to a common denominator, Mien subtract the less numfirator fmm th- "-Psf J -» -^ remainder over the common"dcnomi„Vto;.* ' """ *'""^' "" HULTIPLI0ATI05. 113 2. When the lower L;ii;tion is greater than the upper, sub- tract the numerator of the lower fraction from the denominator, and to that difference add the upper numerator, carrying one to the unit's place of the lower whole number. EXAMPLES. 1. From ^ take i 3X7=21. 5X4=20. 21— 20=1 num. 4 X 7=28 den. Facit, -»-. 2. From -g- take f of |. Pacit, ||! 3. From &| take W- Facit, d%. 4. From ff- take f Facit, ^^. 6. From ^ take | of f . Facit ^-^ 6. From 64| take f of f . Facit, 63|' 3. When the fractions are of several denominations, reduce them to their proper quantities, and subtract as before. 7. From i of a pound take f of a shilling. Facit, 14s. 3d. 8. From -f of a shilling take ^ of a penny. Facit, 7^. 9. From f of a lb. troy take f of an ounce. Facit, 8 oz. 16 dwts. 16 grs. 16. From f of a ton take f of a lb. Facit, 15 cwt. 3 qrs. 27 lb. 2oz. lOf drs. 11. Fromf of a chaldron, fake ^ of a bushel. Facit, 23 bushels, 1 peck 12. From f of a yard, take f of an inch. F&cit, 5 in. 1 b. c. MULTIPLICATION OF VULGAR FRACTIONS. Rule. Prepare the given numbers (if they require it)by the rules of Reduction ; then multiply all the numerators together for a new numerator, and all the denominators for a new denomin- ator. EXAMPLES. 1. Multiply f by f . Facit, 3X3=9 num. 2. Multiply f by f. 3. Multiply 48^3- by 13f . 4. Multiply 430-1%- by 18|-. 5. Multiply fi by f of f of f. 6. Multiply -^,- by f of f of ^. K3 4X5=20 den.—„a 2 0' 2 7* 6723^. Facit, Facit, Facit, 7935f4. fdCll, 2 4 Facit, 4 9' a. 8* 114 SINOLB RVLB Of THRBB PIRBOT. ?. Multiplyfoff byf off 8. Multiply i off by f. 9. Multiply 5^ by f. 10. Multiply 24 by f . 11. Multiply f of 9 by f. 13. Multiply 9^ by f . Facit, f Fmcit, j^. Facit,4t2. Facit, 10. Facit, 5ff . Facit, hi. DIVISION OF VULGAR PRACT0N8. ^'^'f* Prepare the given numbero (if they require it) by the rules of Reduction, and invert the divwofr then proceed a»lo Multiplication. EXAMPLES. 1. Divide^ by f. Facit, &X9=±4&mdm^ZXSO'=60 2. Divide if by f 3. Divide 672A by ISf. 4. Divide 7935af by 18f 5. Divide f by f of f of f 6. Divide f of 16 by f of f 7. Divideioff byf off 8. Divide9f«, by^ofr 9. Divide ^V by 4f 10. Divide 16 by 24. U. Divide 5206jV by f of 9K 12, Divide 3i by 9f den.-4f=f Facit, f Fa^ ve of that tiurt which is f yard wide, to make another of Vht ajm* bigness? Ans. 4| yards. 4. If three men ctn do a piece of work in 4^ hours, in how many hours will tiill ^«en du the same work ? Ans. r§^o hour. 5. If a penny white lual w< i^hs 7 oz. when a buslu ' of wheat cost 6s. 6d., what is a bushel worth when a penny white loal weighs but 2^ oz. ? . Ans. 16. 4fd. 6. What quantity of shalloon, that is f- yard wide, will lineT^ yards of cloth, that is 1^ yard wide? Ans. 15 yards. THE DOUBLE RULE OF THREE, IN VULGAR FRACTIONS. EXAMPLES. K 1. If a carrier receives £2-^ for the carriage of 3 cwt. 160 miles, h* , mu^^ ought he to receive for the carriage of 7 cwt. 3i qrs. 5U mikt Ans. £1 : 16 : 9. 2. If £100 i 12 months gain £6 interest, what principal will gain £3f in 9 months ? Ans. £76. 3. If 9 students spend £10|-in 18 days, how much will 20 students spend in 30 days ? Ans. £39 : 18 : 4^^^^ 4. A man and his wife having laboured one day, earned 4fs. how much rjust they have for 10^ days, when their two sons helped them ? Ans. £4 : 17 : 1^. 6. If £50, in 5 months, fjain £2iVT» what time will £13f re- quire to gain £1 iV ' ^'"'S' 9 months. 6. If the carriage of 60 cwt. 20 miles cost £14|-, what weight can I have carried 30 miles for £5-jV ? -^ns. 15 cwt. 117 fficient to I is f yard r f ards. 8, in how ^Q hour. ' of wheat vhite loat [5. 4td. all line 7^ > yards. LGAR \ cwt. 1 5a of 7 cwt. : 16 : 9. icipal will IS. £75. h will 20 irned 4fs. two sons 17 :1^ 1 £13i re- months. lat weight 15 cwt THE TUTOR'S ASSISTANT. PART III. DECIMAL FRACTIONS. In Decimal Fractions the integer or whole thing, as one pound, one yard, one gallon, dec. is supposed to be divided into 10 equal parts, and those parts into tenths, and so on without end. So that the denominator of a decimal being always known to consist of an unit, with as many ciphers as the numerator has places, therefore is never set down ; the parts being only distin- guished from the whole members by a comma prefixed : thus ,& which stands for -^5, ,26 for VW, ,123 for -iVW- But the different value of figures appears plainer by the fol- lowing table. Whole numbers. Decimal part*. 7654321, 3 34567 •^ ■-•> "^ IHJ MJ >-H 5 3"^ ■ S Q n 2 6'=' 5 p B a From which it piainly appears, that as whole numbers increaie bt a ten-fold proportion to the left hand, so decimal parts decrease in a ten-fold proportion to the right hand ; so that ciphers placed 118 ADDITION OF DECIMALS. befors decimal parts decrease their value by removing them far ther from the comma, or unit's place ; thus, ,5 is 5 parts of 10, or A ; ,05 is 5 parts of 100, or ^h ; ,005 is 5 parts of 1000, or tAo ; »0005 is 6 parts of 10000, or -foioi- B"t ciphers after decimal parts do not alter their value. For ,5, ,50, ,500, &c. are each but -^ of the unit. A FINITE DECIMAL is that which ends at a certain number of places, but an infinite is that which no where ends. A recurring decimal is that wherein one or more figurct are continually repeated, as 2,75222. And 52,275275275 is called a compound recurring deci NAL. Note. A finite decimal may be considered as infinite, by ma- king ciphers to recur ; for they do not alter the value of the deci- mal. In all operations, if the result consists of several nines, reject them, and make the next superior place aa unit more ; thus, for 26,25999, write 26, 26. In all circulating numbers, dash the last fignre. ADDITION OF DECIMALS. RiTL5. In setting down the proposed numbers to be added, great care must be taken in placing every figure directly under- neath those of the same value, whether they be mixed numbers, or pure decimal parts ; and to perform which there must be a due regard had to the commas, or separatikig points, which ought always to stand in a direct line, one under another, and to the right hand of them carefully place the decimal parts according to their respective values ; then add them as in whole numbers. EXAMPLES. I. Add 72,5+32,071 + 2,1574 + 371,44-2,75. Facit, 480,8784. f. Add 30,07 + 2,0071 + 59,432 + 7,1 . ». Add 3,6 + 47,25 + 927,01 + 2,0073 + 1 ,5. 4. Add52,75 + 47,2l + 724+31,452+,3076. $. Add 3275 + 27,514 + 1,005 -f- 725 + 7,32. «. Add 27,5 + 52 + 3,2675+,5741 + 2720, V them far ts of 10, or »f 1000, or phers after 1 1500, 6cc» number of » ore figurefl UNO DECI itc, by TTia- )f the deci- nes, reject ; thuSffor be added, ;tly under- I numbers, st be a due iich ought and to the according ! numbers. »,8784. MVLTIPtSCATION OF DE0IHAL9, II SUBTRACTION OF DECIMALS, ftuLB. Subtraction of decimals differs but little front whole numbers, only in placing the n«mbers, which mutt be carefully observed, as in addition. EXAMPLES. 1. Prom ,5754 takft ,2371. 3. From ,237 take 1,76. 3. From 271 take 215,7. 4. Prom 270,2 take 75,4075. 5. Prom 571 take 94,72. 6. From 625 take 76,91. 7. From 23,415 take ,3742. 8. From ,107 take ,0007. MULTIPLICATION OF DECIMALS. Rule. Place the factors, and multiply them, as in whole num- bers, and from the product towards the right hand, cut off as many places for decimals as there are in both factors together ; but if there should not be so many places in the product, sup^ ply the defect with ciphers to the left hand. EXAMPLES. Facft, ,05758775. 7. Multiply27,35 by 7,70071. 8. Multiply 57,21 by ,0075. 9. Multiply ,007 by ,007. 10. Multiply 20,15 by ,2705. 11. Multiply ,907 by ,0025. 1. Multiply ,2365 by ,3435. 3. Multiply 2071 by 2,27. 3. MulUpIy 37,15 by 25,3. 4. Multiply 72347 by 23,15. 5. Multiply 17105 by ,3257. 6. Multiply 17105 by ,0237. •iwSf^f " *"^ number of decimals is to be multiplied by 10, 100^ 1000, &c, it is only removing the separating point in the multl- plicand so many places towards the right hand as there are ciphers cJ^.!S2!^'P?fI- tJ^"s, ,578X10 = 5,78. ,578X100 = 5,78. ,578X1000 = 578; and ,678X10000 = 5780. CONTRACTED MULTIPLICATION OP DECIMALS. Rule. Put the unit's place of the multiplier under that nlare of the multiplicand that is intended to be kep't in the product,' then invert the order of all the other fiaures. i. e. write them all the 1^ CONTRACTED MULTIPLICATION. contrary way ; and in multiplying, begin at the figure in the mul- tiplicand, which stands over the figure you are then multiplying with, and set down the first figure of each particular product di- rectly one under the other, and have a due regard to the increase arising from the figures on the right hand of that figure you begin to multiply at in the multiplicand. Note. That in multiplying the figure left out every time next the right hand in the multiplicand, and if the product be 5, or upwards, to 15, carry 1 ; if 15, or upwards, to 25, carry 2; and if 25, or upwards, to 35, carry 3, &c. EXAMPLES. 12. Multiply 384,672158 by 36,8345, and let there be only four places of decimals in the product. Contracted way. 384,672158 5438,63 115401647 33080329 3077377 115402 15387 1923 14169,2065 Common way. 384,672158 36,8345 1923 15386 115401 3077377 23080329 115401647 360790 88632 6474 264 48 4 14169,2066 038510 Facit, 14169,2065. 13. Multiply 3,141592 by 52,7438, and leave only four places of decimals. Facit, 165,6994. 14. Multiply 2,38645 by 8,2175, and leave only four places of decimals. * Facit, 19,6107. 15. Multiply :^75,13758 by 167324, and let there be only one place of decimals. Facit, 6276,9. 16. Multiply 375,13758 by 16,7324, and leave only four placef of decimals. Facit, 6276,9520. 17. Multiply 395,3766 by ,75642, and let there be only foui place« of decimals. Facit, 299,0699. DIVISION OF DECIMALS. 121 he mul- tiplying duct (li- ncrease lu begin me next be 5, or ' 2 ; and be only DIVISION OF DECIMALS This Rule is also worked as in whole numbers ; the only dif- ficulty is in valuing the quotient, which is done by any of the fol- lowing rules : Rule 1. The first figure in the quotient is always of the same value with that figure of the dividend, which answers or stands over the place of units in the divisor. 2. The quotient must always have so many decimal places, as the dividend has more than the divisor. Note 1. If the divisor and dividend have both the same num- ber of decimal parts, the quotient will be a whole number. 2. If the dividend hath not so many places of decimals as are in the divisor, then so many ciphers must be annexed to the divi- dend as will make them equal, and the quotient will then be a whole number. 3. But if, when the division is done, the quotient has not so many figures as it should have places of decimals, then so many ciphers must be prefixed as there are places wanting. EXAMPLES. ,2066. ir places ,6994. r places ,6107. only one .276,9. ar placef l,9520« »nly foul l,0699« 1. Divide 85643,825 by 6,321. 2 Divide 48 by 144. 3. Divide 2lt,75 by 65. 4. Divide 125 by ,1045. 5. Divide 709 by 2,574. 6. Divide 5,714 by 8275. Facit 13549. 7. Divide 7382,54 by 6,4252. 8. Divide ,0851648 by 423. 9. Divide 267,15975 by 13,25. 10. Divide 72,1564 by ,1347. 11. Divide 715 by ,3075. When numbers are to be divided by 10, 100, 1000, 10,000, &c. it is performed by placing the separating point in the dividend so many places towards the left hand, as there are ciphers in the divisor. Thus, 5784- 10=578,4. 5784-^100=57,84. 5784-i- 1000=:55784. 6784-10.66o=,5Ti4. 122 CONtttACtfii) DITISIOH. CONTRACTED DIVISION OF DECIMALS. Rule. By the first rule find what is the value of the first figure in the quotient : then by knowing the first figure's denomination, the decimal ]places may be reduced to any number, by taking as many of the left hand figures of the dividend as will answer them ; and in dividing, omit one figure of the divisor at each following operation. Note. That in multiplying every figure left out in the divisor, you must carry 1, if it be 5 or upwards, to 15 ; if 15, or upwards, to 25, carry 2 ; if 25, or upwards, to 35, carry 3, &€. EXAMPLES. 12. Divide 721,17562 by 2,257432i and let there be only thr«« places of decimals in the quotient. Contracted. 2,257433)721 , 17562(319,467 6772296 Common way- 3,357432)721,17562(319,467 6772296 439460. 225743 . 213717.. 203169.. 10548.. 9030.. 1518. 1354. 164 158 13. Divide 14. Divide 15. Divide 16. Divide 18. Divide 27,104 by 3,712. 8,758615 by 5,2714167. 51717591 by 8,7586. 25,1367 by 217,35. 51.47512 by ,133415. ■yn no i... "^ r^opo 439460 225743 21371700 303168 88 120 738 392S 135414592 10548 9029 1518 163 158 93280 02024 91256 RfiOUCTION OF DSCIMAL8. 183 S. irst figure mination, taking as rer them ; following ;e divisor, upwards, miy thrtse 19,467 3 i 280 REDUCTION OF DECIMALS. To reduce a Vulgar Fraction to a Decimal. Rule. Add ciphers to the numerator, and divide by the de- nommator, the quotient i« the decimal fraction required. EXAMPLES. 1. Reduce \ to a decimal. 4)1,00(,25 Facit. 2. Reduce i ....v to a decimal. Facit, ,5. 3. Reduce I k to a decimal Facit, ,76. 4. Reduce f to a decimal. Facit, ,375. 5. Reduce ^ to a decimal. Facit, ,1923076-f-. 6. Reduce W oi\^, to a decimal. Facit, ,6043956+. Note. If the given parts are of several denominations, they n.^y be reduced either by so many distinct operations as there are different parts, or by first reducing them into their lowest denomination, and then divide as before ; ofj 2ndly. Bring the lowest into decimals of the next superior de- nomination, and on the right hand of the decimal found, place the parts given of the next superior denorhination ; so proceeding till you bring out the decimal parts of the highest integer required, by •till dividing the product by the next superior denominator ; of. 3dly. To reduce shillings, pence, and farthiiigs. If the num- ber of shillings be even, take half for the first place of decimals, and let the second and third places be filled with the farthings contained in the remaining pence and farthings, always remem- bering to add 1, when the number is, or exceeds 25. But if the nuTTiber of shillings be odd, the second place of decimals mu»t be increased by 5. 7. Reduce 5s. to the decimal of a £. Facit, ,25. !^. -vr^!.!.,,^. i,c. i„ liic uv;v,iniai Ul ;t 3t. 1' ECU, ,'*0. 9. Reduce UJs. to tlie decimal of a £. Facit, ,a L2 i j \! \%i REDUCTION OF DECIMALS. 10. Reduce 8s. 4d. to the decimal of a £. Facit, ,4166. 11. Reduce 16s. 7|d. to the decimal of a £. Facit, ,8322916, last. 16s. 7|d. 12 199 4 960)799(8322916 second. third. 4)3,00 2)16 12)7,75 ,832 2)0)16,64583 ,8322916 7|d. 4 ii 12. Reduce 19s. 5^d. to the decimal of a £. Facit, 972916. 13. Reduce 12 grains to the decimal of a lb. troy. Facit, ,002083. 14. Reduce 12 drams to the decimal of a lb. avoirdupois. Facit, ,046875. 15. Reduce 2 qrs. 14 lb. to the decimal of a cwt. Facit, ,625 16. Reduce two furlongs to the decimal of a league. Facit, ,0833. 17. Reduce 2 quarts, 1 pint, to the decimal of a gallon. Facit, ,625. 18. Reduce 4 gallons, 2 quarts of wine, to the decimal of a hogshead. Facit, ,0714284-. 19. Reduce 2 gallons, 1 quart of beer, to the decimal of a bar- rel. Facit, ,0625. 20. Reduce 62 days to the decimal of a year. Facit, 1424654-. To find the value of any Decimal Fraction in the known parts of an Integer. Rule. Multiply the decimal given, by the number of parts of the next inferior denomination, cutting off the decimals from the product; then multiply the remainder by the next inferior deno- mination ; thus proceeding till you have brought in the least ku&'wn parts of an integer. REDUCTION OF DECIMALS. 125 EXAMPLES. 21. What is the value of ,8322916 of a lb.? Ans. 16s. 7id.+. ao 16,6458320 12 7,7499840 4 2,9999360 22. What is the value of ,002084 of a lb. troy? oo «ru . , ^^«- 12,00384 gr. 43. What is the value of ,046875 of a lb. avoirdupois ? 24. What IS the value of ,625 of a cwt. ? oc ^uri . , -^"*- 2 qrs. 14 lb. 25. What IS the value of ,626 of a gallon? tui XT^^. . , -^^** 2 quarts 1 pint 26. What IS the value of ,071428 of a hogshead of wine ? o» wu . . ^ •^^*- * gallons 1 quart, ,999866. »7. What IS the value of ,0625 of a barrel of beer ? oo «ri. . , ^^^* 2 gallons 1 quart 28. What IS the value of ,142465 of a year ? il»5. 51,999726 days. 126 ; f DEOIMAL TABLES OF COIN, WBlOfST, AND MEASURE. TABLE I. English Coin. £ 1 the Integer. ^h. Dec. Sh. 19 ,95 9 18 ,i> 8 17 ,85 7 16 ,8 6 15 ,75 5 14 .7 4 13 65 3 13 ,6 2 11 ,55 1 10 ,5 Dec. ,45 I K ,15 |05 Pence. 6 5 4 3 2 1 Decimals. ,025 ,020833 ,016666 ,0135 ,008333 ,004166 Farth. 3 2 1 Decimals. ,003125 ,(0020833 ,0010416 Farth. 3 2 I Decimals. ,06i?5 ,041666 ,020833 TABLE III. Troy Wbioht. 1 lb. the Integer. Ounces the same as Pence in the last Table. TABLE II. Enolish Coin. 1 Sh. Long Measure. 1 Foot, the Integer. Penoe &i Inches. 6 5 4 3 3 1 u, Decimals. ,5 ,416666 .333333 ,25 ,166666 ,083333 Dwts. 10 9 8 7 6 5 4 3 2 1 Decimals. ,041666 ,0375 ,033333 ,029166 ,025 ,020833 ,016666 ,0125 ,008333 ,004166 Grains. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,002083 ,001910 ,001736 ,001562 ,001389 ,001215 ,001042 ,000868 ,000694 ,000521 ,000347 ,000173 Grains 12 11 10 9 8 7 6 5 4 3 2 1 DecimaU. ,058 ,022916 ,020833 ,01875 ,016()6tf ,•14583 ,0125 ,010416 ,008333 ,00625 ,0ft4166 ,002083 TABLE IV. Ayoir. WncHT. 113 Ibe. tlM Integot. an. 3 3 1 Decimals* .5 ,25 1 oz. the Integer. Pennyweights the same as Shillings in the first Table. ' Pounds. 14 13 12 11 10 9 8 w 4 Q. 5 4 3 3 1 Decimals. ,125 ,116071 ,107143 ,098214 ,089286 ,080857 ,071438 ,0625 ,053571 ,044643 ,035714 ,026786 ,017857 ,008928 Cuiicca. 8 7 TA — : — I- ,004464 ,003906 ! \':' m SURE. DecimaU. ,058 ,022916 ,020833 ,01875 ,016(J6« ,•14583 ,0125 ,010416 ,008333 ,00625 ,0^4166 ,002083 -EIV. WacBT. M Integer. Decimala. f _,25 Decimals. ,125 ,116071 ,107143 ,098214 ,089286 ,080357 ,071428 ,0625 ,053571 ,044643 ,035714 ,026786 ,017857 ,008928 r\ : — 1_ ,004464 ,003906 DBCIMAl, TABLES OF COIN, WEIGHT, AND MEASURE. 6 5 4 8 e I ,003348 002790 K)2232 ,i, '*^74 ,001116 ,000558 \ Oz. 3 3 1 Decimals. ,000418 ,000279 ,000139 TABLE V. Atoudopou weight. 1 lb. the lateger. OunccB. 8 7 6 5 4 3 2 1 Dnms. 8 7 6 5 4 3 2 1 Decimals. ,5 ,4375 ,375 »3185 ,25 ,1875 »125 ,0625 Dewnals. ,03125 ,027343 ,023437 ,019531 ,015625 ,011718 ,007812 ,003906 TABLE VL LlQ,DID MEASURE. 1 tun the lateger. CUllons. 100 90 Decimals. ,396825 ,357142 60 70 60 50 40 30 20 10 9 8 7 6 5 4 3 1 ,317460 ,27 ,238095 ,198412 ,158730 ,119047 ,079365 ,039682 ,035714 ,031746 ,027 ,023809 ,019841 ,015873 ,011904 ,007936 ,003968 Pints. 4 3 fi 1 Decimals. ,001984 ,001488 ,000992 ,000496 A Hogshead the Integer. Gallons. Decimals. 30 ,476190 90 ,3m60 10 ,158730 9 ,142857 8 ,126984 7 ,111111 6 ,095238 5 ,079365 4 ,063493 3 .047619 2 ,031746 1 ,015873 Pints. 3 2 1 Decimals. ,005952 ,003968 ,001984 TABLE Vn. Mbasures. Liquid. Dry. 1 GaL 1 Or. Integer. Tti: 4 3 2 1 Decimals. ,5 ,375 ,25 ,125 Ctpt. 3 3 1 Decimals. ,09375 ,0625 ,03125 Bosh. 4 3 2 1 TSET 3 % 1 Decimals. ,0234375 ,015625 ,0078125 ^Pks. 3 3 1 Decimals. ,005859 ,003906 ,001953 Pints. 3 9 1 TABLE VIIL Long Measure. 1 Mile the Integer. Yards. Decimals. 1000 ,568182 900 ,511364 800 JLfJLfAf^ 700 ,397727 600 ,340909 ■HH 128 DECIMAL TABLES OF COIN, WEIGHT, AND MEASURE. ^ 500 400 300 200 100 90 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 ,284091 ,227272 ,170454 ,113636 ,056818 ,051136 ,045454 ,039773 ,034091 ,028409 ,022727 ,017045 ,011364 ,005682 ,005114 ,004545 ,003977 ,003409 ,002841 ,002273 ,001704 ,001136 ,000568 Feet. 2 1 Decimals. ,0003787 ,0001894 Inches. 6 3 1 Decimals. ,0000947 ,0000474 ,0000158 TABLE IX. Time. 1 year the Integer. Months the same as Pence in the second Table. 80 70 60 50 40 30 20 10 9 8 7 6 5 4 3 2 1 ,219178 ,191781 ,164383 ,136986 ,109589 ,082192 ,054794 ,027397 ,024657 ,021918 ,019178 ,016438 ,013698 ,010959 ,008219 ,005479 ,002739 1 day the Integer. Decimals. 1,000000 ,821918 ,547945 .273973 ,246575 Hours. 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,5 ,458333 ,416666 ,375 ,333333 ,291666 ,25 ,208333 ,166666 ,125 ,083333 ,041666 Minutes. Decimals. 30 ,020833 20 ,013888 10 ,006944 3 ,00625 8 ,005555 7 ,004861 6 ,004166 5 ,003472 4 ,002777 3 ,002083 2 ,001389 1 ,000664 TABLE X. Cloth measure. 1 Yard the Integer. (sluarters the same as Table 4. Nails. 2 1 Decimals. ,125 ,0625 TABLE XI. Lead Weight. A Foth. the Integer. Hund. 10 9 8 7 6 5 4 3 2 1 Decimals. ,512820 ,461538 ,410256 ,358974 ,307692 ,256410 ,205128 ,153846 ,102564 ,051282 ars. 2 1 Pounds. 14 13 12 11 10 9 8 7 6 5 4 3 2 1 Decimals. ,025641 ,012820 I Decimals. I ,0064102 ,0059523 ,0054945 ,0050366 ,0045787 ,0041208 ,0036630 ,0032051 ,0027472 ,0022893 ,0018315 ,0013736 ,0(J05/i37 ,0004578 lilt RULl or THRBl IN DlOIMAtS. l^Q THE RULE OP THREE IN DECIMALS. EXAMPLES. If 26i yards co8t £3 : 16 : 3, what will 32i yards come to? Ans. £4 : 13 : 9|. yds. £ y(}g 26,6 : 3,8125 :: 32,25 ; 32^25 26,6)122,ft53126{4,63974==£4 : 12 : 9f ^2. What will the pay of 540 me« come to. at £1 : 5 : 6 per n a 1- «* ^^*' ^11 : 14 : 2 3,6 ars CO for i^78"fi%'^,.'? '.?,7'- *•>"• " •'»' 18 oz. of ^teo J «fr^'° 6:4, what will 1 oz. come to J An, 1A fo 's^e/'^"'" *^ "*• °' '«'""^° <="»« '»• ''''«» ijlru .* Id r VV^ti, the worth of 19 oz. 3 dwt, 6 glrif'J^li V. Is : a wi ;• .u . -^ns- £56 : 10 : 6 2,99 ors yard ' " "" ""'* °^*"* ''''"'» °f P-^W. aflftj^rper .0 Ld 11''^^'^^ ^" t of « y-.^wluV;urhe 10. If J of a yard of cloth, that a 8i yards broad makpT™, rnent, how much that ia * of. yard wVde wmmaK sam^t" ars 14 Ih Z,» f .r™"'' "°". *a = » • 6. V hat will 45 cwt. 3 qrs. 14 10. cost at the samA rate ? a-~ *»i^" ..« #^, ^»,9 £113: a 19Q SXTRXCTION Of THE, fQUARI ROOT. 15. Bought a tankard for £10 \l% at the rate of St. 4d. per uuac«, what was the weight ? ._ijq • Ana. 39 oz. 16 dwti. 10. Gave £18') : 3 : 3, for 25 cwt. 3 qrs. 14 lb. of tobacco, a: Arhat rate did I buy it per lb. T Ans, U 3id. 17. Bought 20 lb. 4 oz. of coffee, for £10 : 11 : 3, what is the rhlueof31b.? ^n^. £1 : 1 : 8. 18. If I give Is. Id. for 3} lb. cheese, wha* will be the value of 1 cwt. ? Ans. £1 : H : 8. EXTRACTION OF THE SQUARE ROOT. Extracting the Square Root is to find out such a number as, being multiplied into itself, the product will be equal to the given num- ber. Rule. First, Point the dven number, beginning at the unit's place, then proceed to the hundreds, and so upon every second figure throughout Secondly. Seek the greatest square number in the first point towards the left hand, placing the square number under the first point, and the root thereof in the quotient ; subtract the square number from the first point, and to the remainder bring down the next point and call that the resolvend. Thirdly. Double the quotient, and place it for a dirisor on the left hand of the resolvend ; seek how often the divisor is contain- ed in the resolvend j (preserving always the unit's place) and pu» the answer in tlie quotient, and also on the right-hand side of the divisor ; then multiply by the figure last put in the quotient, and subtract the product from the resolvend ; bring down the next point to the remainder if there be any more) and proceed as be- fore, , . ^ G1 hiRoOTS, Squares. 1. 2. 3. 4. 6. 6. 7. 8. 9. 4. 9. 16. 26. 36. 49. 64. 81. . 4d. per dwti. tobaceo. sSid. lat is thtf I :». he value W:8. IS, being 'en num- he unU*s Y second rst point ' the first le square ngdown or on the I contain- i) and pa« ide of the tient, and the next ed as be- 9. n. ■XTR40TI01f or THB SQUARI ROOT. 131 EXAMPLES. 1. What i» the square root of 119035 1 Ans. 346. 119090(846 9 64)390 266 685)3425 3425 2. What is the square root of 106929 ? Ans. 327-I-. 3. What is the square root of 2268741 ? Ans. 1606,2if. 4. What is the square root of 7696796 T Ans. 2756,228+. 5. What is the square root of 36372961 ? Ans. 6031. 6. What is the square root of 22071204 ? Ans. 4698. When the given num1>er consists of a whole number and deci- mals together, make the number of decimals even, by adding ci- phers to them; so that there may be a point fall on the unit's place of the whole number. 7. a 9. 10. 11. 12. What is the square root of 3271,4007? What is the square root of 4795,25731 ? What is the square root of 4,372594? What is the square root of 2,2710967? What is the square root of ,00032754? What is the square root of 1,270059? Ans. 57,19-f-. Ans. 69,247+. Ans. 2,0914-. Ans. 1,60701-1-. Ans, ,018094-. Ans. 1,12694- To extract the Square Root of a Vulgar Fraction. Rule. Reduce the fraction to its lowest terms, then extract the square root of the numerator, for a new numerator, and the square root of the denominator, for a new denominator. If the fraction be a surd («. e.) a number where a root can ne- ver be exactly found, reduce it to a decimal, and extract the root from it. EXAMPLES. 13. WTinf Im iKa onvittvA ^^^^^ r^C 2 3 4 ^ 14. What is the square root of ^flf? 1 5. What is the squaw rcu^t o» -^^auLflj. f Ans. 4. Ans. f. I iii Ml 192 EXTRACTION OF THE SRUARE ROOT. SURDS. 16. What is the square root of ff^? 17. What is the square root of ^^f? IS. What is the square root of fff ? Ans. ,89902^ Ans. ,86602+, Ans. ,93309+. To extract the Square Root of a mixed number* Rule. Reduce the fractional part of a mixed number to its lowest term, and then the mixed number to an improper fraction. 3. Extract the root of the numerator and denominator for a new numerator and denominator. If the mixed number given be u surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom. EXAMPLES. 19. What is the square root of 51|^ ? 30. What is the square root of 21-f^ ? 21. What is the square root of 9ff ? SURDS. 22. What is the square root of 8&^? 23. What is the square root of 8f- ? 24. What is the square oot of 6f ? Ans» 7|. Ans. 5}. Ans. 3|? Ans. 9,27+ Ans. 2,9519+. Ans. 2,5819+. To find a mean proportional between any two given numbers. Rule. The square root of the product of the given number is the mean proportional sought. EXAMPLES. 5. What is the mean proportional between 3 and 12 ? Ans. 3 X 12=36. then V 36=6 the mean proportional. 6. What is the mean proportional between 4276 and 842 ? Ans. 1897,4+. To find the side of a square equal in a'^^'.a to any given superficies. Rule. The 18 the side of the square root of the content of any given superficies square equal sought. ? 1* EXTRACTION OF THE SQUARE ROOT. 13a EXAMPLES. 87. If the content of a given circle be 160, what is the side of the square equal? Ans, 12,64911. 28. If the area of a circle is 750, what is the side of the square ®q"*l* iins. 27,38612. 7%c Area of a circle given to find the Diameter, Rule. As 355 : 452, or, as 1 : 1,273239 : : so is the area : to the square of the diameter ;— or, multiply the square root of the area by 1,12837, and the product will be the diameter. EXAMPLES. 29. What length of cord will be fit to tie to a cow's tail, the other end fixed in the ground, to let her have liberty of eating an acre of grass, and no more, supposing the cow and tail to measure 5 J yards ? ^n*. 6,136 perches. TJui area of a circle given, to find the periphery, or circumference. Rule. As 113 : 1420, or, as 1 : 12,56637 : : the area to the square of the periphery ; — or, multiply the square root of the area by 3,5449, and the product is the circumference. EXAMPLES. 30. When the area is 12, what is the circumference ? Ans. 12,279. 31. When the area is 160, what is the periphery ? Ans, 44,839 Any two sides of a right-angled triangle given, to find the third the figures in the next point to the remainder, for a Resolve ivi>. ^ Find a Divisor by multiplyinff the square of the quotient / ^^* ^^^ ^^^ *' " contained in the resolvend, rej«ctin«f the units and tens, and put the answer in the quotient. ' 3. To find the Subtrahend. ! Cube the last figure In the Quotient. 2. Multiply all the figures in the quotient by 3, except the last, and that product by the square of the last. 3. Multiply the divisor by the last figure. Add these products together, for the subtrahend, which subtract from the resolvend; to the re- mainder bring down the next point, and proceed as before. Roots. 1.2. 3. 4. 5. 6. 7. a 9. Cubes. 1. 8. 27. 64. 125. 216. 348, 512. 729. EXAMPLES. 1. What is the pube root of 99252847 ? Divisor- 99252847(463 64 sscube of 4 Square of 4 X 3=48)35252 resolvend. 216=.cube of 6. 432 «4X3Xby square of 6. 298 «divisor X by 6. Divisor- 33o36 subtrahend. Square of 46 x 3=6348)1916847 resolvend. 27=cube of 3. 1242 =46X3xbysquareof3. 19044 =rHvianr V K,r •) 1916847 subtrahend. M3 sa^^M 136 BXTRACTION OF THE CUBE ROOT. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. What What What What What What What What What What Wh^t 8 the cube root of 389017? s the cube root of 5735339 ? s the cube root of 32461750? s the cube root of 84604519? s the cube root of 259694072 ? 8 the cube root of 48228544 ? s the cube root of 27054036008 ? 8 the cube root of 22069810125? 8 the cube root of 122615327232? s the cube root of 219365327791 ? 8 the cube root of 673373097125? Aii^. 73. Ans. 179. Ans. 319. Ans, 439. Ans, 638. Ans. 364. Ans, 3002. Ans, 2805. Ans. 4968. Ans. 6031. Ans. 8766. When the given number consists of a whole number and deci- mals together, make the number of decimals to consist of 3, 6, 9, &c. places, by adding ciphers thereto, so that there may be a point fall on the unit's place of the whole number. 13. What is the cube root of 12,077875 ? Ans. 2,35. 14. Wh«t is the cube root of 36155,02756 ? Ans. 33,06+. 15. What is the cube root of ,001906624 ? Ana. ,124. 16. What is the cube root of 33,230979937 ? Ans. 3,215+. 17. What is tie cube root of 15926,972504? Ans. 25,16+. 18. What is the cube root of ,053157376 ? Ans. ,376. To extract the cube root of a vulgar fraction. Rule. Reduce the fraction to its lowest terms, then extract the cube root of its numerator and denominator, for a new nu- merator and denominator ; but if the fraction be a surd, reduce it to a decimal, and then extract the root from it. EXAMPLES 19. What is the cube root of ffi ? Ans. |. 20 What is the cube root of ^^Vo" ? Ans. f . 21. What is the cube root of i^^f. Ans. |. SURDS. Ans. ,829+. Ans. ,822+, Ans. ,873+. To extract the cube root of a mixed number. Rule. Reduce the fractional part to its lowest terms, and then tue lui&cu number to an improper fraction, extract the cube rooi of the numerator and denominator for a new numerator and deno- 22. W-at is the cube root of f ? 23. What is the cube root of |- ? 24. What is the cube root of f ? EXTRACTION OF THE CUBE ROOT. 137 73: 179. 319. 439. 638. 384. 3002. 2805. 4968. 6031. 8766. and deci- 3f3,6,9, nay be a 2,35. },06+. ,124. 2154-. >,16+. ,376. [1 extract new nu- t, reduce ns. \. ns. -f-. ns. f. 829+. 822+, 873+. and then ube root nd deno- minator ; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. EXAMPLES. 25. What is the cube root of 12^ ? 26. What is the cube root of 31 gVg- ? 27. What is the cube root of 405 ^hr ? SURDS. 28. What is the cube root of 7^ ? 29. What is the cube root of 95- ? 30. What is the cube root of 8f ? Ans, ^. Ans. 3f. Ans. 7f . Ans. 1,93+. Ans. 2,092+. Ans. 2,057+. THE APPLICATION. 1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 mches deep, how many cubical inches doth it con- Ans. 103823 u ^' jI^®^®,^! ^ °,®"f ^"^' *^^* ^^ *2 ^«e* every way, in length, breadth, and depth ; how many solid feet of earth were taken out *^ Ans. 1728. 3. There is a stone of a cubic form, which contains 389017 olid feet, what is the superficial content of one of its sides? Ans. 5329. Between two numbers given, to find two mean proportionals. Rule. Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme, gives the less mean ; multiply the said cube root by the less mean, and the pro- duct will be the greater mean proportional. EXAMPLES. 4. What are the two mean proportionals between 6 and 162 ? e yur, ^ , Ans. 18 and 54. o. Wiiat are the two mean proportionals between 4 and 108? Ans. 12 and 36. To find the side of a cube that shall be equal in solidity to any given sohd, as a globe, cylinder, prism, cone, 4'C. \ Rule. The cube root of the solid content of any solid bodi given, IS the side of the cube of equal solidity. M3 m EXTRACTING llOOtS Ot ALL POWBRS. EXAMPLES. 6. If the solid content of a globe is 10618» what is the side 9t fa cube of equal solidity ? j^ji^^ 22, *rhe side o/ a cube being eivtn^ to find the side of a cube that shall be double^ treble^ ^t?* in quantity to the cube given. UuLE» Cube the side given, and multiply it by S, 3, &c., Um cube root of the product is the side sought. EXAMPLES, % There is a cubical vessel, whose side is 12 inches, and it i« required to find the side of another vessel, that is to contain three times as much ? Ans. 17,306. EXTRACTING OF THE BIQUADRATE ROOT. To extract the Biquadrate Root, is to find out a number, which being involved four times into itself, will produce the given num* bcr. Rule. First extract the square root of the given number, and then extract the square root of that square root, and it will ^t» the biquadrate root required. EXAMPLES. 1. What is the biquadrate of 27? ^n^. 53144 K 2. What is the biquadrate of 76 ? Ans. 33362176. 3. What is the biquadrate of 275? ^«*. 5719140625. 4. What is the biquadrate root of 531441 ? Ans. 27. 5. What is the biquadrate root of 33362176? Ans. 76. 6. What is the biquadrate root of 5719140625? Ans. 275. A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. Prepare the number given for extraction, by pointing off from the unit's place as the root required directs. 2. Find the first figure in the root, which subtract from the giten number. 3. B>fing down the first figure in the next point to Ihf resosia 4er, and call it the dividend. the Bide St ins. 22, I cube that given* B, and it i« itain three 17,306. OOT. ler, which iven num« mber, and t will giT9 S3144K 368176. 140625. ns» 2t. ns. 76. s, 276. ROOTS tinting off from the efeslsia fiXTRACTING ROOTS OF ALL POWERS* m 4. Involre the root into the next inferior power to that which IS given, multiply it by the dven power, and call it the divisor 6. Find a quotient figure by common division, and annex it to the root ; then involve the whole root into the given power, and call that the subtrahend. * ^ 6. Subtract that number from as many points of the given power as are brought down, beginning at the lower place, and to the remainder bring down the first figure of the next point for a new dividend. * 7. Find a new divisor, and proceed in all respects as before* EXAMPLElS. I. What is the square root of 141376 ? 141376(376 6)51 dividend. 1369 subtrahend. 3X 5^6 divisor. 37 X 37=i=1369 subtrahend. 37 X 2=74 divisor. 376 X 376=141376 subtrahend, 74)447 dividend. 141376 subtrahend, 8. What is the cube root of 63157376? 63157376(376 27 27)261 dividend. 50653 subtrahend. 4107)25043 dividend. 53157376 subtrahend, 3X 3X 3=27 divisor. w« /\ o« A a7=ovn30u suoiranend. 37 X 37 X 3=4107 divisor. 376 X 376 X 376=53157376 attbtrahend. ''*^ SIMPLE INTEREST. 3. What is the biquadrate of 19987173376? 19987173376(376 108)1188 dividend. 1874161 subtrahend. 202612)1245563 dividend. 19987173376 subtrahend. 3x 3X 3X 4=108 divisor. 37 X 37 X 37 X 37=1874161 subtrahend. 37X 37X 37X 4=202612 divisor. 376 X 376 X 376 X 376=19987173376 subtrahend. SIMPLE INTEREST. There are five letters to be observed in Simple Interest, viz, P. the Principal. T. the Time. R. the Ratio, or rate per crnU I. the Interest. A. the Amount. A TABLE OP RATIOS. 3 ,03 H ,055 8 ,08 5^ ,035 6 ,06 8* ,085 4 ,04 H ,065 9 ,09 ^ ,045 7 ,07 9^ ,095 5 ,05 n ,075 10 ,1 Note. The Ratio is the simple interest of £1 for one year, at the rate per cent, proposed, and is found thus : £ £ £ As 200 : 3 : : 1 : ,03 As 100 : 3,5 : : 1 : ,035. SIMPLE INTEREST. HI When the principal, time, and rate per cent, are given, to find the interest. Rule. Multiply the pri icipal, time, and rate together, and it will give the interest required. Note. The proposition and rule are better expressed thus :-v I. When P R T are given to find I. Rule. prt=I. Note. When two or more letters are put together like a word, they are to be multiplied one into another. EXAMPLES. 1. What is the interest of £945 : 10, for 3 years, at 6 per cent, per annum. Ans. 945,6 X,05 X 3=141,825, or £141 : 16 : 6. 2. What IS the interest of £547 : 14, at 4 per cent, per annum, lor 6 years ? Ans. £131 : 8 : 11, 2 qrs. ,08. 3. What 18 the interest of £796 : 15, at 4^ per cent, per an- "T wl^J^^t" ^ . ^^«- £179 : 5 : 4 2 qrs. 4. What IS the interest of £397 : 9 : 5, for 2A years, at 34 oer cent, per annum ? Ans. £34 : 15 : 6 3,5499 qrs. 5. What 18 the interest of £554 : 17 : 6, for 3 years, 8 months, i*li?u *^''"** l^^ annum? Ans. £91 : 11 : 1 ,2. 6. What 18 the interest of £236: 18 : 8, for three years, 9 months, at 5^ per cent, per annum ? Ans. £47 : 15 : 7^,293. When the interest is for any number of days only. Rule. Multiply the interest of £1 for a day, at the given rate by the principal and number of days, it will give the answer. ' INTEREST OP £1 FOR ONE DAY. per cent. 3 3i 4 4^ 5 6 Decimals. ,00008219178 ,00009589041 ,00010958904 ,00012328767 ,00013698630 ,00015068493 ,00016438356 percenl. 6i 7 8 8^ 9 9^ Decimals. ,00017808219 ,00019178082 ,00020547945 ,00021917808 ,00023287671 ,00024657534 ,00026027397 Note. The above table is thus found :— As 3G5 : ,03 : : i : ,00008219178. And as 365 : ,035 : : 1 : ,00009589041, &c. 14ft SIH^LB iKTKRKa'f. EXAMPLES. 7. What is the interest of £240, tot 120 days, at 4 per cent, per annum ? Ans. ,00010968904 X 240 X 120=£3 : 3 : U- 8. What is the interest of £364 : 18, for 164 day», at 6 pet cent, per annum? Ans. £7 : 13 : 11|. 9. What is the interest of £726 : 15, for 74 days, at 4 per cent per annum ? Ans. £5 : 17 : 8^. 10. What is the interest of £100, from the 1st of June, 1776, to the 9th of March following, at 5 per cent, per annum ? Ans. £3 : 16 : llf. 11. When P R T are given to find A. RuLK. prt + ps=A. EXAMPLES. 11. What will £279 : 12, amount to in 7 years^ at 4^ per cent, per annum ? Ans. £367 : 13 : 6 3,04 qri. 279,6 X ,045 X 7 + 279,6=367,074. 12. What will £320 i 17, amount to in 5 yeats, at 3^ per cent, per annum ? Ans» £376 : 19 : 11 2,8 qre. When there is any odd time given with the whole years, reduce the odd time into days, and work with the decimal parts of a year which are equal to those days. 13. What will £926 : 12, amount to in 5^ years, at 4 per cent, per annum ? Ans. £1130 : 9 : 0;^ ,92 qrs. 14. What will £273 : 18, amount to in 4 years, 175 days, at^ per cent, per annum ? Ans. £310 : 14 : 1 3,35080064 qra. III. When A R T are given to find P. a RVLB. >=P. rt+l EXAMPLES. 15. What principal, being put to interest, will amount to £367 : 13 : 5 3,04 qrs. in 7 vears, at 4^ per cent, per annum? Ans. ,045 X 7+ 1:^=1,315 then 367,674+1, 31 5=£379 : 12. 16. What principal, being put to interest, will amount to £376 : 19 : 11 2,8, in 5 years, at 3.] per cent, per annum? Ans. £320 : 17. SIMPLE INTBRB8T. 143 per cent. : 3 : U. at 5 per I: lU. per cent. 17 : 8^. ne, 1776, m? \ : llf. per cent. 04 qrs. per cent. 1,8 qrs. rs, recluce »art8 of a per cent^ [)2qni. lays, at^ 64 qrs. It to £367 ?9:12. itto£31G 50: 17. riiln^n**^ principal, being put to interest, will amount lo £1 130 : 9 : 0^ ,92 qrs. in 6^ years, at 4 per cent, per annum ? A.nt, iC926 * 1*^ 18. What principal will amount to £310 : 14 : 1 3,36080064 qrg. m 4 years, 176 days, at 3 per cent, per annum ? IV. When A P T are giyen to find R. RVLE.- a- =R. pt. EXAMPLES. 19. At what rate per cent, will £27tl : 12, amount to £367 i 13:5 3,04 qrs. in 7 years ? Ans. 367,674—279,6=88,074, 275,6 x 7=1957,2. then 88,074+1957,2=,046 or 4^ per cent 20. At what rate per cent, will £320 : 17, amount to £376 : 19:11 2,8 qrs. in 5 years ? ^ Ans. 3^ per cent. 21. At what rate per cent, will £926 : 12, amount to £1130 : 9 : Oi ,92 qrs. in 5| years ? Ans. 4 per cent 22. At what rate per cent, will £273 : 18, amount to £3ia 14 : 1 3,35080064 qrs. in 4 years, 176 days ? xr ^»r. . ^ ^ ^"^' 3 per cent V. When A P R are given to find T. a— p Rule. =T. pr. EXAMPLES. 23. In what time will £279 : 12, amount to £367 : 13 : 6 3,04 qrs. at 4^ per cent. ? Ans. 367,674—279,6:^88,074. 279,6 X ,045=: 12,5820, then ' 88,074+12,5820=7 years. 24. In what time will £320 : 17, amount to £376 : 19 : 1 1 2,& qrs. at 3^ per cent. ? Ans. 5 years. 25. In what time will £926 : 12, amount to £1130 : 9 : 0^ ,92 qrs. at 4 per cent. ? Ans. 5^ years. 28. In what time will £273 : 18, amount to £310 : 14 : I 3,35080064 qrs. at 3 per cent. ? Ans. 4 years, 175 days. ANNUITIES OR PENSIONS, &c. IN ARREARS. Annuities or pensions, f.K' ~.>Ii ire unpaid for any number of payments. quarterly, antf; ^PflJ^.i^Ki-riini.ii'aL" 144 SIMPLE INTEREST. Note. U repreaents the annuity, pension, or yearly rent, T H A. as before. n I U R T are given to find A. ttu — tu Rule. X r : + tu=A. EXAMPLES. 27. If a salary of £150 be forborne 6 years at 6 per cent, what will it amount to ? Ans. £826. 3000 6 X 5X 150—5 X 150=3000 then X ,05 + 5 X 150=£825. 2 28. If £250 yearly pension be forborne 7 year«, what will it amount to in that time at 6 per cent. ? Ans. £2065. 29. There is a house let upon lease for 5^ years, at £60 per annum, what will be the amount of the whole time at ^ per cent. ? Ans. £363 : 8 : 3. 30. Suppose an annual pension of £28 remain uppaid for 8 years, what would it amount to at 5 per cent. ? Ans. £263 : 4. Note. When the annuities, &c. are to be paid half-yearly or quarterly, then For half-yearly payments, take half of the ratio, half of the annuity, &c., and twice the number of years — and For quarterly payments, take a fourth part of the ratio, a fourth part of the annuity, &c., and four times the number of years, and work as before. EXAMPLES. 31. If a salary of £150, payable every half-year, remains un- paid for 5 years what will it amount to in that time at 5 per cent.? Ans. £834: : 7 : 6. 32. If a salary of £150, payable every quarter, was left unpai(' for 5 years, what would it amount to in that time at 5 per cent, t ' Ans. £839 : 1 : 3. Note. It may be observed by comparing these last examples, the amount of the half-yearly payments are more advantageous than the yearly, and the quarterly more than the half-yearly. 11. When A R T are given to find U. RULE." 2a ttr — tr4-2t =U. ly rent, T StMPLI INTBSB8T« 146 rent, what f. £825. 30=£825. hat will it £2065. t £60 per at 4^ per : 8 : 3. 3aid for 8 263 : 4. yearly or all of the D, a fourth of years, mains un- ! at 5 per : 7 : 6. ;ft unpaid ler cent ? : 1 :3. ixamples, ntageous y^early. 38. If a salary amounted to £825 in 5 years, at 5 per cent, what was the salary ? Ana £lMi 826 X2»1660 6 X 6 X ,06-6 X ,06+ 6 X 2-1 1 th ^1650^ ll=£16a 34. if a house is to be let upon a lease for 5* years, and the vrrUntt***' ''""' is ^£363 : 8 : 3, at 4i percent, what ii the yearly rent t ^^^ ^^^ «w '^ * r""*"" '^T"^'*^ t<> ^«065, in 7 years, at 6 ^r cent, what was the pension ? ^ \^^^ £25i) 38. Suppose the amount of a pension be £263 : 4 in 8 years, at 6 per cent, what was the pensioa ? Am, £28. h^fVni ^®" ^\ payments are half-yearly, then take 4a, and ?K ^/?®o*'**'' *".^ ^'^'''^ ^^^ ""™^'' «f years ; and if quart^rlv: of years, and proceed as before. .♦ V' ^^ *^! wnownt of a salary, payable half-yearly, for 5 veart. at 6per cent, be i^ : 7 : 6, wLt was the salary ? 5«1 l/S ^ I .^ ]y, amounts to £834 : 7 : 6, in 5 years, what is the rate per cent ? Ans, 5 percent. 44. If an annuity of £150 per annum, payable quarterly* amounts to £839 : 1 : 3, in 5 years, what is the rate per cent ? Ans, 6 per cent. IV. When U A R are giren to find T. 2 $^ zx z RutE. First,- l=z then : V— -{ =T, V wr 4. 2 fiXAMPLES. 45. In what time will a salary of £150 per annum, amount id £825, at 5 per cent.! Ans» 5 years. 2 896X2 39X39 1^39 =^aO =.380,26 >06 150X^5 4 39 V220-f380 ,25=24 ,5 =5 years. 2 46. If a house is let upon a lease for a certain time, for £60 per annum, and amounts to £363 : 8 : 3, at 4| per cent, what time was it let for? Ans, ^^ yeart. 47. If a pension of £250 per^nnum, being forborne a oertafai time, amounts to £2066, at 6 per cent., what was the time of forbearance? Am, 7 years. 48. In what iime will a yearly pension of £28, amount to £263 : 4, at ^ per cent t Atis, 8 years. Note. If the payments are half-yearly, take half the ratio, and half the annuity; if quarterly, one fourth of the ratio, and on« fourvh of the annuity ; and T will be «qual to those half-yearly or quarterly payments. 49. If an annuity of £150 per annum, payable haff-yearly, amounts to £834 : 7 : 6, at 5 per cent,, whwt time was the pay- ment forborne I Ans, % years. BIXPtE INTERStT. i the num- , and work r of years. ilf-yearly, er cent? ercent. quarterly, er cent ? er cent. 147 imonnt td 9 years. I, for £6d int, what ^yeart. I a oertaia e time of r years, motuit to ) yeari. ratio, and , and on« Bilf-yearly If-ycarly, I the pay- years. «WW . 1 . J, at b per cent, what was the time of forbearance ? Ans, 5 years. PRESENT WORTH OP ANNUITIES. Note. P represents the present worth ; U T R as before. I. When U T R are given to find P. ttr— tr + 2t RutE -; X U=P. 2tr-f 2 EXAMPLES. . . ^TiSt £660* Ans. £071 : 6. thf^^^enfroXS^^^^^^^ '' -^« »>,« ^-nd that thanVly, and qlterW II. When P T R are given to find U. tr-fl ttr— tr-f2t Miiiii i^ SIIC^IB UfTEREST. EXAMPLES. 56. If the present worth of a salary be £660, to continue 5 years, at B per cent., what is the salary ? Ans. £160. 5X,05H-l=«l^a5 5X5X,05— 5X,064-10=11. 1,25 -— X660x3=£160. 11 67. There is a house let upon lease for 6^ years to come, I de- sire to know the yearly rent, when the present worth, at 4* pei ^^f',}^m:6:31 Ans,mV 68. What annuity is that which, for 7 years' continuance, at 6 per cent., produces £1464 : 4 : 6 present worth? Ans. £260. 69. What annuity is that which, for 8 years' continuance, pro- duces £188 for the presen* worth, at 5 per cent. ? 4ns* £28. NpTB. When the payments are half-yearly, take half the ratio, twice the number of years, and multiply by 4 p ; and when quw- terly, take one fourth of the ratio, and four times the number Of years, and multiply by & p. 60. There is an annuity payable half-yearly, for 6 years to come, what is the yearly rent, when the present worth, at 6 toer cent., is £667 : 10 ? ^ns. £160; ?^*. "^^^'^ ** *n annuity payable quarterly, for 5 years to coxpe, I desire to know the yearly income, when the present worth, at ^ per cent., is £671 : 6^ ^n*. £160. - III. When U P T are given to find R. ut— pX2 =1^. 2pt+ut— ttu. EXAMPLES, 62. At what rate per cent, will an annuity of £160 per annum, to continue 6 years, produce the present worth of £660? Ans. 6 per cent. 150 X 5-660 X 2=180,2 X 660 X 6+6 X 160—6 X 6 X 160=3600 then 180-f-3600=,06-5 per cent. 63. If a yearly rent of £60 per annum, to continue 6^ years, produces £291 : 6 : 3, for the present worth, what is the rate Ans. 4^ per cent. ^ i. « |;7i veil I. : SIMPLB lirtEREST. POT cent? • P""*"' ""'■*• '''«" *» *e "» duces £188 for the present worth, whit i, the rate per c7nt " Ans. 6 per cent yo.^'.'r'J^^^I^/Z"^' --"^ *- - »<"- paid "..f- S^t^pereKnT"* *" ''"""*"' '^ "« *« ~^ "^ ^^f 66. If an annuity of £180 per annum, payable half-rearlv h» nag 5 yeare to come. is eolifor £667 :%! Xflft^Sj:; 61 If an annuity of £180 per annum, pay^btqi?/ h.. mg 6 years to come. i. z^ fyr £6T1 : 1, kat iTCSS Jr WW nri ■^^** ^ per cent. IV. When U P R aire given to find T « 2 2p 2p xj. ^ KVLB. i=x then V— f — r n ur 4 2 =T. EXAMPLES. 6a If an annuity of £160 per annum, produces £tm fn* A- present worth, a. 6 per ce„t..\h.. 1. 'tfe tirnHfircoSaJSi! Ans, 6 years 2 660X2 *05 150 30,2 X 30,2 -1=30,2 660X2 ^ITO 150X,06 =228,01 then V2a8,01 +176=90,1 20-1— 30, ter N3 150 SIMPLE INTEREST. 60. For what time may a salary of £60 be purchased for £201 : 6 : 3, at 4^ per cent. ? Ana. 5^ years. 70. For what time may £250 per annum, be purchased for £1454 : 4 : 6, at6 per cent 1 Ana. 7 years. 71. For what time may a pension of £28 per annum, be pur- chased for £188, at 5 per cent. ? Ans, 8 years. Note. "When the paymerts are half-yearly, then U will be equal to half the annuity, dee. R half the ratio, and T the num- ber of payments : and. When the payments are quarterly, U will be equal to one fourth part of the annul ty^ &c. R the fourth of the ratioy and T the number of payments. 72. If an annuity of £150 per annum, payable half-yearly, is sold for £667 : 10, at 5 per cent., I desire to know the number of payments, and the time to come? Ans. 10 payments, 5 years. 73. An annuity of £150 per annum, payable quarterly, is sold for £671 : 5, at 5 per cent, what is the number of payments, and time to come? Ans. 20 payments, 5 years. ANNUITIES. &c. TAKEN IN REVERSION. 1. To find the present worth of an annuity, dee. taken in re- version. Rule. Find the present worth of the ttr — tr+2t yearly sum at the given rate and for the ' ' : X u«P. time of its continuance ; thus, 2tr+2 2. Change P into A, and find what prin- cipal, being put to interest, will amount to a A at the same rate, and for the time to — =F. come before the annuity, dLC. commences ; tr f-1 thus, ^ EXAMPLES. 74. What is the present worth of an annuity of £160 per an- num^ to continue 5 years, but not to commence till the end of 4 years, allowing 5 per cent, to the purchaser ? Ans. £550. 5 X 5 X ,06— 5 X ,06+2 X 6=4,4 X 150= 660 =660. A N/ ARJL1 SIMPLE iNTSREST^ 151 ased for years, ased for years. , be pur- years. I will be the num- 1 to one Of and^ rearly, is I number years. f, is sold ients,and years. :en in re- 75. What is the present worth of » foase of £50 per atinmn, to continued years, but which is not to commence till th« end of 5 years, allowing 4 per cent ta tliepur chaser ? ^919. £1521 1&^ 11 3qi!«. 76. A person having the promise of a pension o€ £20 per an* num, for 8 years, but not to commence till the end of 4 years, is willing to dispose of thie saim« ait 5 percenls.,. what will be the present worth? Ans. £111 : 18 : 1 ,14+. 77. A legacy of £40 per annum being left for d^yeaxs,. Id a person of 16 years of age, but whiich is not to commence till he u 21 ; he, wanting money, is desirous of selling the same at 4 per eent^ what is the present worth ? ^n«.£l71:ia:. 11,0^^96. %. To find the yearly income of an afimiity, &c. in rerersion. RuLB li Find the amount of the present worth at the givea rate, and for the time ptr+p=A. before the reversion ; thus, 2. Change A into P, and find what an- . . - nulty being sold, will produce F at the ^ same ratc> and for the time^of its continu- Z TTT.' a«io« } tlkus, tti:--tri-2t -:X2p=U. t :Xu»P. per an- 1 end of 4 £550. =550. EXAMPLES. 78. A person hating an annuity left him for 5 years, which docs not commence till the end of 4 years, disposed of it for £560, allowing 6 percent., to the purchaser, what was the yearly in- ^ «^ Ans, £IS0, • 6 X, 06 + 1, 650 X4X ,05+650=660 6 X 6 X ,05— 5 X ,05+6 X 2= ,113636 X 660 X 2«£160. 79. There is a lease of a house taken for 4 years, but not to commence till the end of 5 years, the lessee would seU the same for £162 : 6, present payment, allowing 4 per cent, to the pur- chaser, what is the yearly rent ! An8, £50. 80. A person having the promise of a pension for 8 years, which does not conunence till the end of 4 years, has disposed of c«nW to the purchaser, what was the pension ? Ans, £20. 1 162 111 REBATE OR DISCOUNT* 81. There is a certain legacy left to a person of 15 years of age, which is to be continued for 6 years, but not to eommenoe till he arrives at the age of 21 ; he, wanting a sum of money, sells it for £171 : 14, allowing 4 per cent, to the buyer, what was the an- nuity left him » Ana. £40 REBATE OR DISCOUNT Note. S represents the Smn to be discounted. P the Present worth. T the Time. R the Ratio. I. When S T R are given to find P. Rule. =P. tr+1 EXAMPLES. 1. What is the present worth of £357 : 10, to be paid9montli# hence, at 5 per cent. ? Ans, £344 : 11 : 6i ,16a 2. What is the present worth of £275 : 10, due 7 monUw hence, at 5 per cent, t Ans. £267 : 13 : 10^. 3. What is the present worth of £875 : 5 : 6, due at 5 months hence, at 4^ per cent. ? Ans. £859 : 3 : 3| -rts . 4. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent. ? ^«i?.£70:il:9,1764d. II. When P T R are given to find S. Rule. ptr + p=S. EXAMPLEa 5. If the present worth of a sum of money, due 9 months hence, allowing 5 per cent, be £344 : 11 : 6 3,168 qw., what was the sum first due t Ans, £367 : 10. 344,6783 X ,76 X ,06+ 344,6783=£357 : 10. 6# A person owing a certain sum, payable 7 months hence, agrees with the creditor to pay him down £267 : 13 : lOg^, al lowing 5 per cent, for present payment, what is the debt? Ans. £275: 10. % A person receives £b69 : 3 : S| jfj for a sum of m=:,05=6 per cent 344,5783 X ,75 10. At what rate per cent, will £276 : 10, payable 7 months hcnae, produce £367 : 13 : 10^ for the present payment! A.n8. 6 per cent. 11. At what rate per cent, will £876 : 5 : 6, payable 6 months hence, produce the present payment of £859 : 3 : 3| ^h ^ --. ., - ^7w. 4^ per cent. la. At what rate per cent, will £75, payable 16 months hence, produce the present payment of £70 : 11 : 9 ,1764d. ? Ans, 5 per cent IV. When S P R are given to find T. s — p Rule. -=T. rp EXAMPLES. la The present worth of £357 : 10, dne at a certain time to wme, is £344 : 11 : 6 3,168 qrs. at 5 per cent, in what time should the sum have been paid without any rebate ? Ans. 9 months, 357,5—344,5783 •*' ——=,76=9 montiis. 344,6783 X ,06 14. The present worth of £275 : 10, due at a certain time to. imm»j ! ,T:';\,iH„-„ ...mi-f ~'««59385 23 1,68948 24 1,79084 25 1,89829 26 2,01219 27 2,13292 28 2,26090 29 «,dybOO 30 YEARS, I 5 RATES. 6 2,18287 2,29201 2,40662 2^2695 2,65329 2,78596 2,92526 3,07152 3,22510 3,38635 3,55567 3,73345 3,92013 4,11613 4,32194 2,54035 2,69277 2,85434 3,02560 3,20713 3,39956* 3,60353 3,81975 4,04893 4,29187 4,54938 4,82234 5,11168 5.41838 5174349 H I mmmmm 156 COMPOUND INTEREST. Note. The preceding table is thus made— As 100 : 106 : : 1 : 1,05, for the first year ; then, As 100 : 106 : : 1,06 : 1,1025, •©• oond year, d&c. I. When P T R are given to find A. Rule, p X rt=A. EXAMPLES. 1. "What will £325 amount to in 3 years* time, at 5 per cent per annum? Ans. 1,05 X 1,05 X 1,05=1,157625, then 1,157625 X 226« £260:9:33qr8. 2. What will £200 amount to in 4 years, at 6 per cent, per annum? -Atw. £>i43 2,0258. 3. What will £450 amount to in 5 years, at 4 per cent per anmim? ^ws. £547 : 9 : 10 2,0538368 qrs. 4. What will £500 amount to in 4 years, tt 5^ per cent, per annum? ilns. £619 : 8 : 2 3,8323 qrs. II. When A R T are given to find P. a Rule. =P. rt EXAMPLES. 5. What principal, being put to interest, will amount to £960; 9:33 qrs. in 8 years, at 6 per cent, per annum ? ^ ^ 260,465625 1,05 X 1,05 X 1,05=1,167625— «£225. 6 What principal, being put to interest, will amount to^543 2,025s. in 4 years, at 5 per cent per annum ? ^1"^^' 7. What principal will amount to £547 : 9 : 10 2,05383^rs. in 6 years, at 4 per cent, per annum ? J^ £450. a What principal will amount to £619 : 8 : 2 3,8323 qi^n 4 years, at 6^ per cent, per annum t -47is. £500. III. When P A T are given to find R-, ^ , r *- a which being extracted by the rule of extrae- RuLE.~~=rt tion, (the time given to the question showing p the power) will give K. )5: : 1; 025, ter cent .X22B« Sqw. «nt. per ,0258. ent pef 8 qrs. xnt. per ^qrs. to £260; Uo£243 £200. 8368 qrs. £450. qrs. in 4 £500. f extrae- showing COMfO«KB IVTERE8T. EXAMPLES. 167 9. At wb%t rate per cent will £225 amount to £260 : 9 : 3,3 qrs. in 3 years ? Ans. 5 per cent. 260,465625 =1,157625, the cube root of whicli 225 (it being the 3d power):*=l,05=5 per cent. 10. At what rate per cent, will £200 amount to £243 : 2,025«. in 4 years ? An». 5 per cent. 11. At what rate per cent, will £450 amount to £547 : 9 : 10 2,0538368 qrs. in 5 years ? Ans, 4 per cent. 12. At what rate per cent, will £500 amount to £619 : 8 : 2 3,8323 qrs. in 4 years ? Ans. 5 J per cent IV. When P A R are given to find T. a which being continually dirided by R till no- RuLK. — ^=rt thing remains, the number of those divisions p will be equal to T. . EXAMPLES. 13. In what time will £225 amount to £360 : 9 : 3 3 qrs. at 5 per oent. ? 260,465625 1,157626 1,1025 1,05 =1,157625- =1,1026 =-^1,06- 225 1,05 1,06 1,05 =1, the number of divisions being three times sought 14 In what time will £200 amount to £243 2,0258. at 5 per cent ? Atis, 4 years. 15. In what time will £450 amount to £547 : 9 : 10 2,0538368 qrs. at 4 per cent ? Ans. 6 years. 16. In what time will £500 amount to £619 : 8 : 2 3332S qrs. at SJ per cent t Ans. 4 years. ANNUITIES, OH PENSIONS, IN ARREARS. Note. TJ represents the annuity, pension, or yearly rent t \ li T as before. 168 COMPOUMO INTEREST. A Table showing the amount of£l annually, for any number of years under 31, af 6 and 6 per cent, per annum. Y£AR8. 1 2 3 4 5 6 4 8 9 10 11 13 13 14 15 5 RATES. 1,00000 2,05000 3,16350 4,31012 5,52563 6,80191 8,14200 9,54910 11,02656 12,57789 14,30678 15,91712 17,71298 11>,59868 21,57856 1,(>0000 LVXJOOO 3,18360 4,37461 5,(13709 6,97532 8,39383 9,89746 J 1.49131 13,18079 14,97164 16,86994 18,88213 21,01506 23,27597 YEARS. 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 5 RATES. 6 23,65749 25,84036 28,18238 30,53900 33,06595 35,71925 38,50521 41,43047 44,50199 47,72709 51,11345 54,66912 58,40258 62,32271 66,43884 35,67252 28,21288 30,90565 33,75999 36,78559 39,99272 43,39229 46,99582 50,81557 54,86451 59,15638 63,70576 68,52811 73,63979 79,05818 Note. The above table is made thus :--take the first yeer'e amount, which is £1, multiply it by l,06+l=2,05=8econd year's amount, which also multiply by 1,05+1 =2, 1626=third year's amount. I. When U T R are given to find A. ur* — u Rule. =A, or by the table thus : r— 1 Multiply the amount of £1 for the number of years, and at the rate per cent, given in the question, by the annuity, pension, &c. and It will give the answer. EXAMPLES. 17. What will an annuity of £50 per annum, payable yearly, amount to in 4 years, at 5 per cent. ? ^ ^ j Ans. 1,05 X 1,05 X 1,05 X 1,05 X 50=60,77531250 60,7753125—50 then =£215 : 10 t 1 2 qrs. ; or^ 1,05—1 HI-. by the table thus, 4,31012 X 50=£216 : 10 : 1 1,76 qrs. 18. What will a pension of £45 per annum, payable yearly, amount to in 5 years, at 5 per cent. ? COMPOUND INTEREST. t60 19. If a salary of £40 per annuip, to be paiJ yearly, be '"or* borne 6 years, at 6 per cent., what is the amount? _ _ Aw5. £379 : : 3,0679fl096d. ao. If an annuityof £76 per annum, payable yearly, be omit- ted to be paid for 10 years, at 6 per cent., what is the amount ? II. When A R T are given to find U. ar — a Rule. =U. rt— 1 EXAMPLES. 31. What annuity, being forborne 4 years, will amount to £315 : 10 : 1 3 qrs. at 5 per cent. ? 316,50636X1,05—316,60635 atis. — _ .^c:o, 1,05X1,05X1,06X1,06' -I 33. What pension, being forborne t v ?Ar», v ill amount to £348 : 13 : 3,37 qrs. at 5 per cent. ? Am, £45. 33. What salary, being omitted to be paiu <5 years, will amount to £379 : : 3,06796096d. at 6 per cent, ? Ans, £40. 24. If the payment of an annuity, being forbome 10 years, amount to £986 : 11 : 3,33d. at 6 per cent., what is the annuity? III. When U A R are given to find T. ar-|-u — a which being continually divided by R till Ryuj. -jt nothing remains, the number of those u divisions will be equal to T. EXAMPLES. 35. In what time will £50 per annum amount to £316 : 10 : 1 3 qrs. at 6 per cent, for non-payment ? Ans. 316,50635 X 1,05+50 315,50636 =1,31650635 - ^ which being continually divided by R the number of the divi- sions will be=4 years. 36. In what time will £45 per annum amount to £348 : 13 337 qrs, allowing 5 per cent, for forbearance of payment ? 08 Ana, 5 yCure* 160 :??MPOUND INTERbST. 37. In -what time "flrUl £40 per annum amount to £279:0: 8,05796096(1, at 6 per cent. 1 Ans. 6 years. 28. In what time will £76 per annum amount to £988 : 11 2,22d. allowing 6 percent, for forbearance of payment t Ans. 10 years. PRESENT WORTH OF ANN JITIES, PENSIONS, &«. A Thble ehming the present worth of £1 annuity for any num' her of years under 31, rebate at B and 6 per cent. TEAWI. .*> RATES. 6 YEARS. 5 BATES. G 1 0,95238 0,94339 16 10,83777 10.10589 3 1,85941 1,83339 17 11,C7406 10,47726 8 2,72324 2,67301 18 11,68958 10,82760 4 3,54595 3,46510 19 12,08532 11,15811 6 4,32947 4,21236 90 12,46221 11,46992 6 5,07569 4,91732 21 12,82115 11,76407 7 5,78637 5,58238 22 13,16300 12,04168 8 6,46321 6,20979 33 13,48857 12,30338 ,J 7,10783 6,80169 24 13,79864 12,55036 to 7,72173 7,36008 25 14,09394 12,78336 11 8,30641 7,88687 Q6 14,37518 13,00317 13 8,86325 8,38384 27 14,64303 13,21053 13 9,39357 8,85268 28 14,89812 13,40616 14 9,89864 9,29498 29 15,14107 13,59072 15 10,37965 9,71225 30 15,37245 13,76483 NoTB, The above table is thus made : — divide £1 by 1,05= ,95238, the present worth of the first year, which-*-l, 05=90753, added to the first year's present worth=l ,85941, the second year's present worth ; then, 90703-*-l,05, and the quotient added to 185941=2,72327, third year's present worth. I. When U T R are given to find P. u u- RVLB.- =P. r— 1 or by the table thus : Multiply the present worth of £1 annuity for the time and rate per cent, given by the annuity, pension, &c. it will give the an-. :279:0: yeare. :988 : 11 ? years. &«. it. "6 10589 17736 32760 15811 16992 76407 34158 30338 }5036 78336 X)317 21053 10616 )9072 76483 )y 1,06= •=90753, ) second Qt added and rats i the an-. COMPOVirD INTEREST. EXAMPLES. 29. What is the present worth of an annoity of £30 per an- iram, to continue 7 years, at C per cent T Ans. £167 : 9 : 6 ,184d. 30 1,50363 «167,4716. ■«19,9517 30— 19,9517=-10,0483. 10,0483 1,06—1 By the table 6,68238X30=167,4714. 30. What is the present worth of a pension of £40 per annum, to continue 8 years, at 5 per cent. ? Ans, £258 : 10 : 6 3,264 r s. 31. What is the present worth of a salary of £35, to continue T years, at 6 per cent. ? Ans. £196 : 7 : 7 3,968 qrs. 32. What is the yearly rent of £50, to continue 5 years, worth in ready money, at 6 per cent. ? Ans, £216 : 9 : 5 2,56 qrs. II. When P T R are given to find U. -I <■! IMIIWIB II ll»l,ll W liiMM prt X r— pr» RiftE. =U. ft— 1 EXAMPLES. 33. If an annuity be purchased for £167 : 9 : 5 184d. to be Qontinued 7 years, at 6 per cent what is the annuity ? Ans. 167,4716 X 1,50363 X 1,06—167,4716 X 1,60363 1,50363—1 =£30. 34. If the present payment of £258 J 10 : 6 3,264 qrs. be made for a salary of 8 years to eome, at 6 per cent., what is the Mlaryt Ans, £40. 35. If the present payment of £195 : 7 : 7 3,968 qrs. be re- quired for a pension for 7 years to eome, at 6 per cent, what is the pension ? jins. £35. 36. If the present wofth of an annuity 5 years to come, be £216 : 9,: 5 2,66 qrs. at 6 per cent., what is the annuity ? Ans. £50. 08 im COatPOUND liNTEREST. IIL When U P R are given to find T. n which being continually divided by R till RtTtB.-,-.^ — «ft nothing remains, the number of those di- p-ft' — pt visions will be equal to T. EXAMPLES. 37. How long may a lease of £30 yearly rent be had foi £167 : : 6 ,184d. allowing 6 per cent, to the purchaser? 30 107,471frf 30— 177,6188 which being continually =lf 50363 ^^^^^®^» *^® number of ' those divisions will beas to T=7 years. 3a If £258 : 10 : 6 3,264 qrs. is paid down for a lease of £40 per annum, at 5 per cent., how long is the lease purchased for! Atis. 8 yearf . 89. If a house is let upon lease for £35 per annum, and the l«esee makes present payment of £105 i 7 : 8, he being allowed 6 per cent., I demand how long the lease is purchased for ? Ans. 7 years* 40. For what time is a lease of £50 per annum, purcha^ when present payment is made of £216 : : 5 2,56 qrs. at 6 per cent ' Ans. 5 years. ANNUITIES, LEASES, &c TAKEN IN REVERSION. To find the present worth of annuities, leases^ ^c, taken in reversion. Rule. Find the present worth of the annui- ty, &c. at the given rate and for the time of its u^- o(Hi^inuanc6 : thus, u 8. Change P into A, and find what principal being put to interest will amount to P at the same rate, and for the time to come before the annuity oommenceft, which will be the present MVtf^lirn ^\w 4n<% «k vk •>« avi 4«r Av ^ft Ai««««ia H-'l -=p. a ^=p. t.i' COMPOUND INTEREST. Ifi3 by R till those di-* had foi atinually ^ber of (fill beos »eof£40 ised for? yearf. , and the allowed for? ye&r&. ircha<»ed at 5 per years. ISION. iken in u -l EXAMPLES. 41. What is the present worth of a reversion of a lease of £40 per annum, to continue for six years, but not to commence till the end of 2 years, allowing 6 per cent, to the purchaser? Ans. £175 : 1 : 1 2, 048 qrs. 40 40-28,1984 106,6933 — «28,1984 =196,6933 =176,0663. 42. What is the present worth of ft reversion of a lease of £60 •er annum, to coiftinue 7 years, but not to commence till the end of 3 years, allowing 6 per cent to the purchaser ? .„ _ , Ans. £299 : 18 : 2,8d. 43. There is a lease of a house at £30 per annum, which is yet in being for 4 years, and the lessee is desirous to take a lease in reversion for 7 years, to begin when the old lease shall be ex- pired, what wUl be the present worth of the said lease in rever- sion, allowing 6 per cent, to the purchaser ? Ans, £142 : 16 : 3 2,688 qrs. To find the yearly iticome of an annuity, ^c, taken in rever \ Rule. Find the amount of the present worth at the ^ven rate, and for the time be- fore the annmty commences : thus, pr*=A. Change A into P, and find what yearly rent being sold will produce P at the same rate, and for the time of its continuance, which will pr» X r prt. be the yearly sum required : thus, ^zj]. rt— 1. EXAMPLES. 44, What annuity to be entered upon-2 years hence, and then to continue 6 years, may be purchased for £176 : 1 : 1 2,048 qrs. at 6 per cent. ? >ln«. 176,0563X1,1236=196,6933 .. theh 196,6933 X 1,41862 X 1,66—279,01337 164 COMPOVND INTEREST. 45. The present worth of a lease of a house is £299 : 18 : 2 8d. taken in reversion for 7 years, but not to commence till the end of 3 years, allowing 5 per eent. to the purchaser, what is the yearly rent? Ans. £60. 46. There is a lease of a house in being for 4 years, and the leasee being minded to take a lease in reversion for 7 years, to begin when the old lease shall be expired, paid down £142 : 16 : 3 2>689 qrs. what was the yearly rent of the house, when the les- see was allowed 6 per cent, for present payment ? Ans. £30. PURCHASINa FREEHOLD OR REAL ESTATE, IN SUCH AS ARE BOUGHT TO CONTINUE FOR EVBll. '■.-■" '^ I. When U R are given to find W. u Rule. —W. r— 1 EXAMPLES. 47. What is the worth of a freehold estate of £50 per annum, allowing 5 per cent, to the buyer ? 50 Ans. —=£1000. 1,05—1 48. What is an estate of £140 per annum, to continue for ever, worth in present money, allowing 4 per cent, to the buyer ? Ans, £3500. 49. If a freehold estate of £75 yearly rent was to be sold, what is the worth, allowing the buyer 6 per cent. ? Ans. £1250. II. When W R are given to find U. RuLs. wXr— 1=U. EXAMPLES. 50. If a freehold estate is bought for £1000, and the allowance of 5 per cent, is made to the buyer, what is the yearly rent? Ans. 1,06— 1 =,06, then 1000 X ,05=£50. 51. If an estate be sold for £3500, and 4 per cent, allowed to thA bilVAr. iirhftf ia tha troarlv ve-n* f Amo Ttdfl =— ^ —s., — - — ,.. ........ .^.jj- . , ^a.fi/o» Ai(X^V. bOMPOUND INTBRESti %i tm 18:28d. ill the end bat is th« IS. £60. fl, and the f years, to :i42 : 16 : 5n the les- is, £30. t AS ARB T annum, ! for ever, lyer? £3500. old, what £1250. llowance rent? i^£50. [lowed to 52. If a freehold estate is bought for £1250 present taoney, and an allowance of 6 per cent, made to the buyer for the same, whatis the yearly rent ? ^^s, £75. III. When W U are given to find R. HULB. =R. w EXAMPLES. 53. If an estate of £50 pcJr annum be bought for £1000, what is the rate per cent. ? 1000+50 Ans, =1,05=5 per cent. 1000 54. If a freehold estate of £140 per annum be bought for £3500, what is the rate per cent, allowed T ' _ Ans. 4 per cent 5a. If an estate of £75 per annum is sold for £1250, what is the rate per cent, allovf ed ? Ans. 6 per cent. PURCHASING FREEHOLD ESTATES IN RETERSION. To find the worth of a Freehold Estate in reversion : u W Rule. Find the worth of the yearly rent, thus — Change W into A, ami find what principal, being r— 1 put to interest, will amount to A at. the same rate, and for the time to come, before the estate commences, and a that will be the worth of the estate in reversion, Uius: ^P r'. EXAMPIJES. 56. Ijf a freehold estate df £50 per annum, to commence 4 years heose, is to be sold, whatis it worth, allowing the purchaser 5 per cent, for the present payment? 60 1000 4ns.— -=1000, then =£822 : 14 : U. 1,05—1 1,2156 57. What is an estate of £200, to continue for ever, but not to ootnmctMie till the end of 2 years, worth in ready money, allowing the purchasef 4 per cent. ? Ans. £4622 : 15 : 7 ,44d, 58. What is an estate of £240 per annum worth in ready mo- ney, to continue for ever, but not to commence till tfie end of 3 yoars, aUo.wance being made at 6 per cent. ? Arts, £3358 : 9 : 10 2,24 qrs. ' 1C6 B£13aTE or OtflCOtNT. To jind thR Yearly Rent of an Estate taken in reversion. Rule. Find the amount of the worth of the estate, at tke given rate and time before it coin- wr*=!A oee, thus : Chaoge A into W, and -ind what yearly rent wr — wsU, beinff eold will produce U at the same rate, thiis : which will be the yearly rent required. EXAMPLES. 60. If a freehold estate, to comm<:mce 4 years hence, is «old for £822 : 14 : 1^, allowing the purchaser 5 per cent., whai is the yearly Income T Ane, 823,70626 X 1,2166=1000^ then 1000 X i,05^I0^>$>.^£8a 60. A feehold estate is bought for £4622 : 15 : 7 ,4M. which does not commence till the end of 2y«ars, the buyer being alk v- ed 4 per cent for iiis money. I desire to know ihe yeLly In- come.. Ans. £am. 61. There is a frtiehoW est?.* gold for £3368 : 9 : 10 2,24 qr»., but not to commence till the « Aplrs tiou of 3 years, allowing 6 per cent, (or present payment i 4'lmkk th« yearly income? Ans, £240. ^ REBATE OR DISCOUNT. A Table 'fhowing the present worth of £1 due any number of yeara hence, under 31, rebate at 6 and tper cent. 5 EATSa. ^ TKARS. ,953381 ,^7030 ,8G;JS38 ,7835^26 ,746315 ,710683 ,676839 ,644609 ,613913 ,584679 ,556837 ,530331 ,505068 ,481017 ,943396 ,889996 ,839619 ,793093 ,747358 ,704960 ,665057 ,637413 ,59189^ ,55H394 ,536787 ,496969 ,468839 ,443301 ,417365 NoTX.— -The above table is thus made: l-*-l,0&=,9&238I. first year'fl present worth ; and ,952381*«-1,06=,90703, second year ; and ,9070a-*- i,U6=*tJ63838 thi . year, &c. I i ftEBAIl OR DiaCOUK'T. m oerston, VT — Ws=U, se, is «old t., whai u m, which ling alk V- yeatly ta- ?. £20C. illowing 6 Dme? vumberof unt. 13646 ri364 >0343 (0513 1804 mb6 7505 11797 f6978 12998 9610 17368 '6630 4556 =,952381, ^« second I. When S T R are given to find P RuiB.— «P. EXAMPLES. ?. What is the present worth of £316 : 18 : 4 ,2d, payable 4 jem-B hence, at 6 per cent. ? -» r / ■» 4/u, 1,06 X 1,06 X 1,06 X 1,08=1,26247, then by the table. 316,6175 316,6176 -=£250 ,192093 1,26247 249,9984124276 2. ir £344 : 14 : 9 1,92 qrs. be payabk in 7 years' time, what i8 the prcaent worth, rebate being made at 6 per cent T o rwn- Ans, £245. 3. There is a debt of £441 : 17 : 3 1,92 ys., which is payablt 4 years hence, but it is agreed to be paid in present money ; what smn mu£t the creditor receive, rebate being made at 6 per cent,! n. When P T R are given to find S, ^"*' ^^' RiTLB. p X r*=S. EXA^klPLES. 4. If a snm of money, due 4 years hence, produce £2&0 for the.preeent payment, rebate being laade^t 6 per cent., what was the sum due ? r i 't Ans. £250 X l,2fl247=£&16 : 12 : 42d. 6. If £245 be Tecdved for a debt payable 7 years hence, an4 tn allowance of 5 per cent, te ^e debtor fot present payment. •That was the deht T Ans. £344 : 14 : 9 1,92 qrs. 6. There is a^um dT money due at the expiration of 4 years, but tfie creditor ames totake £350 fo^ present payment, allow- ing 6 per cent., what was the debt t t J ^ Ans. £441 : 17 : 3 1,92 qrs. III. When S P R are given to find T. 8 which bemg continually divided by R till nothing tiVLE.— «4^ remains, the number of those di^sions will be ,p equal to T 168 REBATB OR DISCOUNVi EXAMPLES. 7. The present payment of £950 is made for a debt of £316: 12 : 4 ,3d., rebate at 6 per cent., in what time wa» the debt pay- able ? ^^ 315,6176 which being continually divided, thosa Ajis. ■■ =1,86247 dirisions will be equal to 4=th8 num- 96(^ Vs4. In. ber of years. ,8. A person receives £246 now, for a debt of i344 : 14 : 9 r,92 qrs., rebate being made at 5 per cent. I demand in what time the debt was payable ? Ana 7 years. 9. There is a debt of £441 : 17 : 3 1,92 qrs. due at a certain time to come, but 6 per cent being allowed to the*debtor for the present payment of £350, 1 desire to now in what time the sum should have been paid without any rebate)" , ■ /'■i«'***^ii<«4 :. ■■- .4.710. 4 years. ^ IV. When S P T are given to find R. 8 which being extracted by the rules of extraction. RoLB mv* (the time given in the question showing the pow .ik P er,) will be equal to R. EXAMPLES. 10. A debt of £315 : 12 : 4 ,2d. is due 4 years hence, but it is agreed to take £250 now, what is the rate per tent, that the re- bate is made at ? 315,6175 4 Ans, — ^=1,26247 :Vl,26247=l,06i=6 per cent 360 11. The. present worth of £344 : 14 : 9 1,92 qrs., payable 7 years hence, is ^^M6, at what rate per cent, is the rebate made'7 l^i- ■ ' ■ ■ ■'• . ! . * '•■Am X Ans, 6 per cent. 12. There is a debt of £441 : 17 : 3 1,92 qrs., payable in 4 years time, but it is agreed to take £350 present payments I de- sire to know at what rate per cent, the rebate is made at) Ana, 6 per cent. 160 t of £316 : debt pay- cled, tho88 =the num- 44 : 14 : 9 d in what f years. t a certain tor for the e the sum I years* THE TUTOR'S ASSISTANT PART IV. ztraction. ' the pow DUODECIMALS, (, but it is at the re- rcent. payable 7 tte madel IT cent. able in 4 ijiU Ide- at) T cent. OR, WHAT IS QEWERALLT CALLED Cross Multiplicationt and Squaring of Dimensions by Arti' ficcrsand Workmen* RULE FOR MULTIPLYINO DU0DBCinAJ.LV. 1. Under the multiplicand write the corresponding denomina* tions of the multiplier. 2. Multiply each term in the multiplicand (beginning at the lowest) by the feet in the multiplier ; write each result under its respective term, observing to carry an unit for every 12, from each lower denomination to its next superiov. 3. In the same manner multiply the multiplicand by theprhnes in the multiplier, and write the result of each term one place mwe to the right hand of those in the multiplicand. 4. Work in the same manner with the seconds in the muhl- pliwr, setting the result of each term two places to the right hand of thoiie in the muUipiicand, and so on for thirds, fourths, &c, p mill 1 Miiifiii I 11 ii DUODECIMALS. EXAMPLES. f. 1. Multiply 7 CniM MaltipUcation. 3^6 in. f. in. 9 by 3. 6. Practice. 6i7-» 31.0.0=7X3 2.3.0=9X3 3.6.0=7X6 0.4.6=9X6 3 .6 23. 3. 3 10. 6 Duodecimal. 7.9 2.6 Decimals. 7,75 3,5 n X3 J. 6X6 27 . 1.6 27. 1.6 3875 2325 27,125 27.1.6 2. Multiply 3. Multiply 4. Multiply 5. Multiply 6. Multiply 7. Multiply 8. Multiply 9. Multiply 10. Multiply 11. Multiply 12. Multiply la Multiply 14. Multiply 15. Multiply 16. Multiply 17. Multiply 18. Multiply f.in. 8.5 9.8 8.1 7.6 4.7 7.5.9" 10.4.5 76.7 97.8 57.9 75.9 87.5 179.3 259.2 257.9 311.4.7 321.7.3 f. in. 4. 7 7. 6 3. 5 5. 9 3.10 by 3. 5.3" by 7. 8.6 9. 8 8. 9 9. 5 by 17. 7 by 35. 8 by 38.10 by 48.11 by 39.11 by 38. 7.5 by 9. 3.6 by bv by by by by by by Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, Facit, t. iu.pts. 38. 6,11 72. 6 27. 7. 5 43. 1. 6 n. 6.10_,, 25. 8. 6.2.3 79.11. 0.6.6 730. 7. 8 854. 7 G43. 9. 9 1331.11. 3 3117.10. 4 6960.10. 6 12677. 6.10 10?88. 6. 3 il4lr2. 2. 4.11 11 2988. i.10.4.6 TITE APPLICATION. Artificers' work is computed by diffp; '): , "; measure'^, viz :— 1. Glazing, and masons' flat work, by the foot. 2. Painting, plastering, paving, &c. by the yard. 3. Partitioning, flooTing, roofing, tiling, &,c. 'the iquare of 100 feet. 4. Brick work, &lc by the rdd of 16^ feet, whose square is 272)1^ feet. 59. Mec ■S . Decimals. 7,76 3,5 3875 2325 27,125 11 5 6 6.2.3 0.6.6 8 9 3 4 6 10 3 4.1111 0.4.6 viz: — >quar8 of square is /U0DECIMAL8. 171 masnrinff by the Foot Square, as Glazier, d Masons^ Flat Work. EXAMPLES. u '^; JJ'fr i.^ "" ^''"'® "^^^^ ^ ^^^^ of windows, 3 in a tier-the annM 'fv '. ^D ''Z ^. ^'t' ^^ ^"^^^«' ^h« «-«ond 6feet 8 inches and the third 6 feet 4 inches, the breadth of each is 3 feet 1 1 inches ; what wifl the glazing come to, at 14d. per foot Duodecimals. 7 . 10 the 6 . 8 heights 6 . 4 added. 19. 10 3= windows in a tier. ■feet. in. pt«. 233 . . 6 at 14d. per ft. 2d.=f 233 ~ Is. 38 . 10 = 2d. . Oi =6 parts. 59. 6 3 . 11 in breadth. 178.6 54 . 6 , 6 233.0.6 2|0)27|1 . 1 , £13. H . 10^ Ans. r«?;«vs.^-:.TZT£aS2L~;:;a 8". per Ans. £1 : 18 : 9. 90 ixru^.' .u . Ans. £1:3 :S, ^Z. What IS the price of a marble slaV vhose length i. fi ft, 7 inches, and the breadth 1 foot 10 inches, t 6s. pef Joot f Ans. £3:1:5. ^asnringhy the Yard Square, a. Paviers Painters, Plas- terers, and Joiners. ^ Note. DiTide the square feet by 9. and it will rri.. t,- «.,^ P9 m DUODECIMALS. ■! •ii'ii EXAxVfPLES. 23. A room is to be ceiled, whoso length is 74 feet D inch€«^ and width 11 feet 6 inches ; what will it come to at 3s. lO^d par yard? ^715. £18 : 10 : 1. 34. What will the paving of a court-yard come to at 4fd. per yard, the length being 58 feet 6 inches, and breadth 64 feel 9 inches 'i Ans. £7 : : 10. 25^ A room was painted 97 feet 8 inr " os about, and 9 feet 10 inches high, what does it come to at 28, 83d. per yard 1 Ans, £14 : 11 : 1^ 26. "What is the content of a piece of Wainscoting in yards square, that is 8 feet 3 inches Jong, and 6 feet 6 inches broad, and what will it come to at 68. 7^d. per yard? ^715. Contents, yards 5.8.7.6 ; comes to £1 : 19 : 6. 37. What will the paving of a court-yard come to at Ss. ^ per yard, if the length be 27 feet 10 inches, and the breadth 14 feet 9 inches ? Ans. £7:4: 6. 28. A person has paved a court-yard 42 feet 9 inches in front, and 68 feet 6 inches in depth, and in this he laid a foot-way the depth of the court, of 5 feet 6 inches in breadth ; the ioot way ia Isaid with Purbeck stone, at 3s. 6d. per yard, and the rest with pebbles, at 3s. per yard ; what will the whole come to ? Ans, £49 : 17. 29. What will the plastering of a ccaling, at lOd. per yard, come to, supposing the length 31 feet 8 inches, and the breadtli 14 feet 10 inches? ^ M Ans, £1 : 9 : 9. 30. What will the wainscoting of a room come to at 6s. pe square yard, supposing the height of the room (taking in the cor- nice and moulding) 1^ 12 feet 6 inches, and the compass 83 feet 8 inches, the three window shutters each 7 feet 8 inches by 3 feet 6 inches, and the door 7 feet by 3 feet 6 inches ? The shutters and door being worked on both sides, are reckoned work and /4*.« roa .10.01 li DU0DKCIMAL9. in Uea^uri^by tke Square of 100 feet, as Flooring, Partition^ tng. Roofing, Tiling, J^c. EXAMPLES. 81. In 173 feet 10 inches in lenffth, and 10 feet 7 in/.»,«- i« height of partitioning, how many square" ? ^" *" Ati^, 18 square*, 39 feet, 8 inches, 10 p. 8?l. If a house measure* within the walls^RQ ^P» *fi^4^ 'i?^' • ^Sf Wt ^ilf ft^' *"r ^" ,^-dt^nd^h1ro^^^^^ PUUI, what wiUiMorae tp rooang at 10s. 6d. per square? ^^*' £12: 13: ll|. of Aeroof of thatbuilding XLX aKu 0^^ ^e. when the rafter, are J of the breadthof IbuUd neCt^f tae roof is more or less than tho fm*. «;*-k *i! """"^"8 » out it one side to the other wU a" od „ "trfng?*"' "'*'' """"^ ^""^ 34. What will the tiling of a barn cost at 2fi« m r^ ^?w. £24 ; 9 : 5|. Measuring by the Rod, ^v'^^'^ Bricklayers always value their work at thp ick and a half thir-k . ««/! t/^u^ .u.- ., - r*^ " '"« 5fi. it must h. r.j.;;:7;:"^ "*7.r."v^"^^« .'^^ ^^^ wan is brick ^ , ImrU muM be reduc^.d to that thickness by this ^3 rate of a more or in DUODECIMALS. Rule. Multiply the area of the wall by the number of half bricks in the thickness of the, wall ; the product divided by 3, gives the area. EXAMPLES. 35. If the area of -a wall be 4085 feet, and the thicknej^s two bricks and a hal^ how many rods doth it contain? Ans. 25 rods, 8 feet. 26 I^ a garden wall be 254 feet round, and 12 feet 7 inches high, and 3 bricks thick, how many rods doth it contain ? Ans> 23 rods, 136 feet. 37. How maijy squared rod^ are there in a wall 62| feet long, 14 feet 8 inches high, and 2^ bricks thick ? Ans. 5 rod's, 167 feet. 38. If the side walls of a house be 28 feet 10 inches ia length, arid the height of the roof from the ground 55 feet 8 inches, and the gable (or triangular part at top) to rise 42 course of bricks, reckoning 4 course to a foot. Now, 20 feet high is 2^ bricks thick, 20 feet more at two bricks thick, 15 feet 8 inches more at 1^ brick thick, and the gable at 1 brick thick; what will the whole work come to at £5 16s. pe^ rod ? Ans. £48 : iS : 5^. Multiplying several figures by several, and the product to he produced in one line only. RtLE. Multiply thB units of the multiplicand by the units of the multiplier, setting down the units of the product, and carry the tens ; next muMply the tens in the multiplicand by the units of the multiplier, to which add the product of the units of the multi- plicand multiplied by the tens in the multiplier, and the tens car- ried ; then multiply the hundreds in the multiplicand by the units of the multiplier, adding the product of the tens in the multiplicand multiplied by the tens in the multiplier, and the units of the multi- plicand by the hundreds in the multiplier ; and so proceed till yo^i the multiplier. ify cyp ry iigUrc i DUODECIMALS. 175 EXAMPLES. ' Multiply 35234 by 52424 Product, 1847107216 Common wo 35234 52424 140936 70468 140936 70468 176170 ' 1847107216 EXPLANATION. First, 4 X 4=16, that is 6 and carry one. Secondly, 3X4-1- 4X2, and 1 that is carried, is 21— set down 1 and carry 8. Thirdly 2 X 4+3 X 2+4 X 4+2 carried=32, that is 2 and car- ry3. Fourthly, 5 X 4 + 2x2 + 3X4 +4 X2+3 carried=47. set down^7 and carry 4. Fifthly, 3X4 + 5X2 + 2X4 + 3X2 + 4.x 5 + 4 carried=60, set down and car: t 6. Sixthly, 3X2 + 5X4 + 2x2+3X5+6cai:ried-51, setdown 1 and carry 5. Seventhly, 3X4 + 5X2 + 2X5 + 5 carried=37, that is 7 and carry 3. Eighthly, 3X2+5X5 + 3 carried=34, set down 4 and carry 3. Lastly, 3X5+3 carried=18, which being mul- tiplied by the last figure in the multiplier, set the whole down, and the work is finished. . il O 4 176 THE TUTOR'S ASSISTANT PART V. A COLLECTION OF QUESTIONS. 1. What is the value of 14 barrels of soap, at ^d. per lb., each barrel containing 254 lb. ? Arts. £66 : 13 : 6. 2. A and B trade together ; A puts in £320 for 5 month©, B £460 for 3 months, and they gained £100; what must each roan recbive t Arts. A £53 : 13 : 9ff^, and B £46 : 6 : 2^%^ 3. How many yards of cloth, at 17s. 6d. per yard, can I hare for 13 cwt. 2 qrs. of wool, at I4d. per lb. ? Atis. 100 yards, 3^^ qr». 4. If I buy 1000 ells of Flemish linen for £90, at what may I sell it per 6ll in London, to gain £10 by the whole? . ' ^Tis. 3s. 4d. per ell. ' $.^ A has 648 yards of cloth, at 14s. per yard, ready money, but in barter will have 16s. ; B has wine at £42 per tun, ready money : the question is, how much wine must be given for the cloth, and what is the price of a tun of wine in barter ? Ans. £48 the tun, and 10 tun, 3 hhds. 12f gals, of wine must be given for the cloth. 6. A jeweller sold jewels to the value of £1200, for which he received in part 876 French pistoles, at 16s, 6d. each ; what sum remains unpaid ? Ans, £477 : 6. 7. An oilman bought 417 cwt. 1 qr. 15 lb., gross weight, of train oil, tare 20 lb. per 112 lb., how many neat gallons were there, allowing 7^ lb. to a gallon? Ans. 51^ gallons. 8. If I buy a yard of cloth for 14s. 6d., and sell it for 16s. IW., what do I gain per cent. ? Ans. £16 : 10 : ^^f^. 9. Bought 27 bags of ginger, each weighing gross 84f lb., tare at If lb. per bag, tret 4 lb. per 104 lb., what do they come to at 8|d. per lb. ? Ans. £76 : 13 : ^. ■ n n ' A COLLECTION OP atJESTlONS. I77 colli '^^°^ *" '^"'"''^ *''''' « ^^ * '*""""&» ^**** ^>" t Of a lb. cost ? "^ • ""^ * ^*"^" ^'^^^ 8 of a pound, what witt ^ of a tun 13. A gentleman spends one day with another^^^r- 7^^ni and at the year's end layeth up £34^, what^r^s y;^^^^^^^^^^ "rj A u to ^ 1 ^^s. £848 : 14 : 4*. tim^s m Ih ^nt%o ^'f V^"^ ^^'^^^^ '^^^ being * 9i a^nrP« i?i .^?u^^ ^.^'K' °^ *^"' ^^^^ 388 lb., how mani ' ^^-^-r of boys, womL, and mTn! .»«. „,c aaiuc. 1 uerrr.na t i; iiunibe* of each ? Ans. 20 of each. ido A COLLECTION OF QUESTIONS. 50. A Stone that measures 4 feet 6 inches long, 2 feet 9 inches broad, and 3 feet 4 inches deep, how many solid feet doth it con- tain ? Ans. 41 feet 3 inches. 61. What does the whole pay of a man-of-war's crew, of 640 sailors, amount to for 32 months' service, each man's pay being 22s. 6d. per month ? Ans. £23,040. 52. A traveller would change 500 French crowns, at 4s. 6d. per crown, into sterling money, but he must pay a halfpenny per crown for change ; how much must he receive ? Ans. £111 : 9 : 2. 53. B and C traded together, and gained £100 ; B put in £640, C put in so much that he might receive £60 of the gain. I de- mand how much C put in ? Ans. £960. 54. Of what principal sum did £20 interest arise in one year, at the rate of 6 per cent, per annum ? Ans. £400. 55. In 672 Spanish gilders of 2s. each, how many French pis- toles, at 17s. 6d. per piece ? Ans, 76ff . 56. From 7 cheeses, each weighing 1 cwt. 2 qrs. 5 lb., how many allowances for seamen may be cut, each weighing 5 oz. 7 drams? Ans. 356ff. 67. If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence leaves 12, what number is that, out of which when you hav o taken 48, 72, 19, and 7, leaves 12? -4ns. 158. 68. A farmer ignorant of numbers, ordered £500 to be divided among his five sons, thus: — Give X^ myw^hi»i^ 'Blr^^-J^'^. . and £ f part ; divide this equitably among them, according to their father's intention. Ans. A £152f|i, B £114if|, C £91^, D £76iif , E £65iH. 59. When first the marriage knot was tied Between my wife and me, My age did hers as far exceed, As three times three does three ; But when ten years, and half ten ^ars, We man and wife had been, Her age came then as near to mine, As eight is to sixteen. Ques. What was each of our ages when we were married Ans. 45 years the ma^j, 15 the woman. 181 <*» A Table for finding' the Interest of any sum of Money, for any number of months, weeks, or dar% at any rate per cent. Year. Calm. Month. Week. Day. £ £ 9. d. £ a. d. £ 8. d. 1 1 8 4i 0| 2 3 4 9 li 3 5 1 1} 2 4 6 8 1 6 2} 5 8 4 1 11 3i 6 10 2 31 w w -mr ^ 4 7 11 8 2 8i 4i 8 13 4 3 1 5i 9 15 3 51 6 10 16 8 3 lOi 6} 20 1 13 4 7 8i 1 li 30 2 10 11 6i 1 71 40 3 6 8 15 4i 2 2i 50 4 3 4 19 2f 2 9 60 5 1 3 1 3 3^ 70 5 16 8 1 6 11 3 10 80 6 13 4 1 10 91 4 4i 90 7 10 1 14 7i 4 Hi 100 8 6 8 1 18 5i ft 5{ 200 16 J3 4 3 16 11 10 m 300 25 515 4J Q 16 5i 400 33 6 8 7 13 10 1 1 11 500 41 13 4 9 12 3J 1 7 4| 600 50 J 11 10 9 1 19 10| 700 58 6 b ( 13 9 2| V V II I 18 4i 800 66 13 4 15 7 8i 2 3 10 900 75 17 6 1| 2 9 3} 1000 83 6 8 19 4 11 2 H 9k 2000 166 13 4 38 9 5»| a 9) 7 9000 250 57 13 10 i 9 4 4h 4000 333 6 8 76 18 5i ^' ■• X Y i? 19 2 5000. 416 13 4 96 3 Oj ji IS Hi 6000 TjOO 115 7 8\ 16 8 9 7000 583 6 8 134 12 3i 19 3 6t 8000 6(;6 13 4 153 16 11 m» ** ** * 21 18 4i 9000 50 173 1 6i 24 13 11 10,000 833 6 8 192 6 If 27 7 lU 20,000 1666 13 4 384 12 3i 54 15 lOj 30,000 2500 576 18 5i 62 3 10 t82 Rule. Multiply the principal by the rate per cent., and the number of months, weeks, or days, which are required, cut off two figures on the right hand side of the product, and collect from the table the several sums against the different numbers, which when added, will make the number remaining. Add the several sums together, and it will give the interest required. N.B. For every 10 that is cut off in months, add twopence i for every 10 cut off in weeks, add a. half penny ; and for every 40 in the days, 1 farthing. EXAMPLES. 1. What is the interest of £2467 10s. for 10 months, at 4 per ^ent. per annum ? 2467 : 10 900=75 : : 4 80=6:13:4 7=0:11:8 9870: 10 987|00 987=82: 6:0 2. What is the interest of £2467 10s. for 12 weeks, at 5 per < int. ? 2467 : 10 1000=19 : 4 : 7| 5 400=± 7 : 13 : 10 80= 1 : 10 : 9^ 12337 : 10 50= : : 2^ 12 1480|50=28: 9: 5 1480160: 3, What is the interest of £2467 10s., 50 days, at 6 per cent. ? 2467 : 10 7000=19 : 3 : 6^ 6 400= 1 : 1 : 11 2= : : U 14805 ; 60= : : 0| 50 7402|50=20 : 5 : 7 7402150 : To find what en Est fit-, from one fo £60,000 /7er annum will ' ' ' to for one day. Rule 1. ^ t( t the annual r^ mt or income from the table for .. _.i,:,»i. *,.i.^ iU^ 1 /•_„ J together, and it will g v^e the answer 1 J % * d the Lit off from rhich vera) ince, jvery I per >per int.? will ! for 183 An estate of £376 per annum, what is that per day ? 300=0 : 16 : 5:^ 70=0 : 3 : 10 6=0: 0: 4 376=1 : : 7i To find the amount of any inconf salary ^ or servants' wagts^ for any number of mora'tSf weeks, or days. Rule. Multiply the yearly income or salary by the number )f months, weeks, or days, and collect the product from the table. What will £270 per annum come to for 1 1 months, for 3 weeks, and for 6 days ? 270 11 2970 270 6 1620 For 11 months. 2000=166 : 13 : 4 900= 75 : 0:0 70= 6 : 16 : 8 2970=247 : 10 : For 6 days. 1000=2 : 14 : 9J 600=1 : 12 : 10^ 20=0 : 1 : 1^ 1620=4: 8: 9^ For 3 weeks. 270 800=15: 7: 8^ 3 10= : 3 : 10^ 810 = 15 : 11 : ^ For the whole time. 247 : 10 : 15: 11 :6^ 4: 8:9^ 267 : 10 : 3i A Table showing the number of days from any day in the month to the same day in any other month, through the year. rnoM January February.. , March April May June July August September .. Octfshflr November . , December. . . TO 365 334 306 275 245 214 184 153 U3 31 365 337 30G 276 245 215 184 S 122 153 QOI lOO 61 31 92 62 59 28 365 334 304 273 243 212 181 1 CI 120 90 90 59 31 365 335 304 274 243 212 151 121 ^ 120 89 61 30 365 334 304 273 242 181 151 02 S 151 120 92 61 31 365 335 304 273 212 182 A 181 150 122 91 61 30 365 335 303 242 bo s < 212 181 153 122 92 61 31 365 334 uUi 273 2121 243 2l 243 212 184 153 123 92 62 31 365 u O 273 242 214 18:^ 153 122 92 61 30 ot55| obu 3041 334 274' 304 304 334 273i303 245275 214244 18^1214 1531183 123 92 61 3i 365 153 122 91 Gi 30 335 365 184 A COMPENDIUM OP BOOK-KEEPING. BY SINGLE ENTRY. Book-keeping is the art of recording the transactions of i)crsonS m business s > as to exhibit a stnte of their aflairs iu a concise and satisfactory manner. Books may be kept either by Singh or by Double Entry, but Single Entry is the method chiefly used in retail business. The books found most expedient in Single Entry, are the Day- Book, the Cash-Book, the Ledger, and the Bill-Book. The Day-Book begins with an account of tiie trader's property, debts, &c. ; and are entered in the order of their occurrence, the dafly transactions of goods bought and sold. The Cash-Book is a register of all money transactions. On the left-hand page, Cash is made Debtor to all sums received ; and on the right, Cash is made Creditor by all sums paid. The Ledger collects together the scattered accounts in the Day- Book and Cash-Book, and places the Debtors and Creditors upon opposite pages of the same folio ; and a reference is made to the folio of the books from which the res^ ctive accounts are ext ic- ted, by figures placed in a column &Mmithe sums. References are also made in the Day-Book ancl C'dsli-Book, to the folios in the Ledger, where the amounts ai* Uected. This process is called posting, and the following general rule should be remem- bered by the learner, when engaged in transferring the register of mercantile proceedings from the previous books to the Ledger : The person from whom you purchase goods, or from whom you receive money, is Creditor; and, on the contrary, the person to whom ycu sell goods, or to whom you pay rnoney, is Debtor. In the Bill-Book are inserted the particulars of all Bills of Ex- change ; and it is sometimes found expedient to keep for this pur- pose two books, into one of which are copied Bills Receivable, or such as come into the tradesman's possession, and are drawn upon some other person ; in the other book are entered Bilh Payable, which are those that are drawn upon and accepted b> the tradesman himself. an 185 DAT BOOK, ,3 6 o bc it* 1 January Ist, 1837. I commenced buBinwB with • capital of Five Hundred Pounda m Cash sa_ Bennett and Sona, Lonu. t.* By 2 hhdf of sugar Cr. dot. qr. lb. ctDt. gr. lb. 13 1 4 12 13 3 16 116 gtoM wt. 26 20 tarb 2 3 6 neat wt. 23 1 14 at 636. per cwt. 2 chests of t cwt. qr. lb. 1 15 1 12 lb. 96 S5 2 27 1 22 I 3 5at6«.per 'h. 2 3d^ Hall and Scott, JLdverpool, By soap, 1 cwt. at 688 candles, 10 dozen at 78. 9d. Cr. 6th Ward, WiUiam To 1 cwt. of sugar, 14 lbs. of tea, i cwt. of soap, at 708.. at 8s... at 748. Dr. 6th Cooper, William Tol (folio 1.) £ TjOO 'I. iU 60 8 17 6 10 sugar hogshead, Dr. 10 12 18 6 6 6 .n!l I^J ^y*^^"* ?K Tv,^ directed to fill up this and similar blanks in this bc^k and t^ Ledger with the names of places familiar to }»im, a3 e>. ^> IMAGE EVALUATION TEST TARGET (MT-3) A 1.0 I.I 1.25 l^|28 12.5 |50 "^ !■■ 1^ 1^ 112.2 S: lis. 12.0 1.8 1.4 IIIIII.6 V] v] 7J % %^'^> > -^<^ ^"^^ .V w '/ ftiotographic Sciences Corporation 33 WEST MAIN STREET WEBSTER, N.Y. MSBO (716) 872-4503 d^ \ (V \\ V ^'^^ ^ >. '<^q\ "^ ^%. ^:^ '^^ >* °:^^^ ^^^ i/.. 186 DAY BOOK. (folio 2.) p January 9th, 1837. £ 1 1 4 17 17 1 6 1 8. 16 8 5 5 9 8 4 8 10 2 1 2 1 2 2 Johnton^ Richard Dr. To 2 dozen of candles, at 89, 3d d. 6 i cwt. of Boap, at 14» ' h cwt. of suffar. at 708 6 1 10th. Ward, William Dr. To sugar, 1 cask cwt. qr. lb. gross wt. 5 2 10 cask tare 2 10 neat 5 at68B 6 9 3 6 12th. Smith, John Dr To 14 lb. of sugar „ 12 lb. of candles ' 7 lb. of soap 1 lb. of tea 14th. 16 13 16 10 9 4 10 6 4 2 2 Q 3 Hall and Scott, Liverpool, Cr. By 2 cvyt. soap, at 688 17th. Newton, John Dr. To 21 lb. of soap, at 748. per cwt. ..... . 2 dozen of candles, at 88. 3d 19th. Smith, John . n_ To 14 lb. of sugar ilb. of tea ..'*.*..'*".! 1 13 18 8 6 2 21st. Smith, John Dr To 28 lb. of sugar ^ V ST>a Ot CtUxUlCO, •• ..•.. .... .•.••..... —J c 6 ' 6 9 3_ 6 10 4 A 2 9 187 DAY BOOK. (folio 3.) January 23d, 1837. Yates ^_ To 3 pieces of superfine cloth, each 36 yards, o . , r I.O. at 27s. per yard... 2 pieces of cahcp, each 24 yards, at Is. 2d. per yard. 3 16 14 5 15 6 ~6 145 2 27th. 148 Parker, Thomas, JOff.^ To 1 piece of superfine clpth, 36 yards, at SSsy , . . , 31st. 3 Bills Payable, Cr. _^yYate8 & Lane's Bill at 2 months, due April 2... . Inventory) January 31, 1837. Raw suffar, Tea, Soap, Candles,, 2 dozen,. cwt. gr. lb. 14 3 14 at 638 1 2 16^ at 68. per lb. 3 14 at 688 at 73. 9d. . . . 50 172 46 55 2 105 8 16 16 12 8_ 16 17 7 19 15 19 1 6 6 irJ !88 CASH BOOK. ■ II I ■ ( O O o « <«: oo 00 A eoo 00 ss G«0*i4 f2SS fiN •NW MG» i s B ll "" Bt3 «* S w •-< f^cSeo en 00^6 6 o lAQOOiieeo "^ S C 02 g.S t: « o 13 i CO " s si w ~ v! eg C5 •? I ■-S 5 5 m 00. eS ^«5 SO ^ ^ 3 .o© 00 S i-Hl-4 S o P C Q O o • Pi* © CO Ift k« SJ (?) O Q H 4r Li^nazn, » > •aW 0?i> O «o o tn ^H «e(0 QOO 00 c»o A .^© T>" PQ 1-1 1-> gs IS- 00 f-* 01 C5 •II (O H»-> CO C^ •a OCJCO ©P0«O '^ 1—1 >-< O !-• CO C«s ,5S ^ »>. ^-1 .^ «» JKJ CA ©© CO i- • •LBQCi;ER. • ,00 > :. 1 » " « /». 9 iQ^H CO .2 0^ 9 CO o CO o o s oo oo 1^ rj« o 00 o r^ 0)0*^ i-or- •-•W CQ C3 pq J" I »5 ^j-oo O iiOO 2.§ CMOO 1-1 a: « « ^ i a. ^'Z a PQ pq © o o 00 o to iHt0rHCOO»^O' "H ^D O »0 l^ lO 00 ^ C3^ CO I CO I I -« ^rtGndu coco 1 I •a ;f as I 4J o "^Ot-t oOOO ^ S« »eo CO •ju i».eo -4 THE END. . \ » . >tOQ0 COCO d 8