.*■ .w

^^^r%,:

THE LIBRARY

UNIVERSITY OF WESTERN ONTARIO

THE J. D. BARNETT TEXT-BOOK COLLECTION

W

University of Western Ontario LIBRARY

LONDON - CANADA

Class .UT\t>.D\

THE

TUTOR'S ASSISTANT;

BEING A

COMPENDIUM OF ARITHMETIC,

AND

COMPLETE aUESTION-BOOK ;

CONTAINING,

I. Arithmetic in whole numbers; being a brief explanation of all its Rules, in a new and more concise method than any hitherto published ; with an Application to each Rule, consisting of a great variety of questions in real Business, with their answers annexed.

II. Vulgar Fractions, which are treated with a great deal of plainness and perspicuity, in. Decimals, with the extraction of the Square, Cube, and Biquadratc Roots, after a

rery plain and familiar manner ; to which are added, Rules for the easy calculation of Interest, Annuities, and Pensions in arrears, &c., either by Simple or Compound Interest.

rv. Duodecimals, or Multiplication of Feet and Inches, with Examples applied to measuring and working by Multiplication, Practice, and Decimals.

V. A Collection of Questions, promiscuously arranged, for the exercise of the scholar ki the foregoing rules.

TO WHICH ARE ADDED,

A new and very short method of extracting the Cube Root, and a General Table for readily calculating the Interest of any sum of money, at any rate per cent. ; Rents, Salaries, &c.

The whole being adapted either as a Question-Book for the use of Schools, or as a Remembrancer and Instructor to such as have some knowledge of Accounts.

This Work having been perused by several eminent Mathemfl cians and Accountants, w recommended as the best Compendium hitlierto published, for the use of Schools, or (or private persons.

aY FRANCIS WALKINGAME,

WRITING-MASTER AND ACCOUNTANT.

TO WHICH IS ADDED,

A COMPENDIUM OF BOOK-KEEPING,

BY ISAAC FISHER.

MONTREAL—ARMOUR & RAMSAY.

KINGSTOK— RAWSAY, ARMOUR & CO,

HAMILTON— RAMSAY & M'KENDRICK.

•1845.

-^"LAt.

PREFACE,

The public, no doubt, will be surprised to find there is another attempt made to publish a book of Arithmetic, when there arc such numbers already extant on the same subject, and several of them that have so lately made their appearance in the world ; but I flatter myself, that the following reasons which induced me to compile it, the method, and the conciseness of tlie rules, which are laid down in so plain and familiar a manner, will have some weight towards its having a favourable reception.

Having some time ago drawn up a set of rules and proper ques- tions, with their answers annexed, for the use of my own school, and divided them into several books, as well for more ease to myself, as the readier improvement of my scholars, I found them, by experience, of infinite use ; for when a master takes upon him that laborious, (though unnecessary,) method of writing out the rules and questions in the children's books, he must either be toiling and slaving himself after the fatigue of the school is over, to get ready the books for the next day, or else must lose jhat time which would be much better spent in instructing and •opening the minds of his pupils. There was, however, still an inconvenience which hindered them from giving me the satis- faction I at first expected ; i. e. where there are several boys in a class, some one or other must wait till the boy who first has the book, finishes the writing out of those rules or questions he wants, which detains the others from making that progress they otherwise might, had they a proper book of rules and examples for each ; to remedy which, I was prompted to compile one in order to have it printed, that might not only be of use to my own school, but to such others as Avould have their scholars make a quick progress. It will also be of great use to such gentle-

a3

It prepack.

men as have acquired some knowledge of numbers at school to make them the more perfect ; likewise to such as have com- pleted themselves therein, it will prove, after an impartial perusal, on account of its great variety and brevity, a most agreeable and entertaining exercise-book. I shall not presume to say any thing more in favour of this work, but beg leave to refer the unprejudiced reader to the remark of a certain author,* con- cerning compositions of this nature. His words are as follows : —

" And now, after all, it is possible that sonve who like best to tread the old beaten path, and to sweat at their business, when they may do it with pleasure, may start an objection, against the use of this well-intended Assistant, because the course of arith- metic is always the same ; and therefore say, that some boys lazily inclined, when they see another at work upon the same question, will be apt to make his operation pass for their own But these little forgeries are soon detected by (he diligence of the tutor: therefore, as different questions to different boys do not in the least promote their improvement, so neither do the questions hinder it. Neither is it in the power of any master (in the course of his business) how full of spirits soever he be, to frame new questions at pleasure in any rule : but the same question will frequently occur in the same rule, notwithstanding his greatest care and skill to the contrary.

" It may also be further objected, that to teach by a printed book is un argument of ignorance and incapacity ; which is no less trifling than the former. He, indeed, (if any such there be,) who is afraid his scholars will improve too fast, will, un- doubtedly, decry this method : but that master's ignorance can never be brought in question, who can begin and end it readily ; and, most certainly, that scholar's non-improvement can be as little questioned, who makes a much greater progress by this, than by the common method."

To enter into a long detail of every rule, would tire the reader, and swell the preface to an unusual length ; I shall, therefore, only give a general idea of the method of proceeding, and leave the rest to speak for itself; which I hope the kind reader will find to answer the title, and the recommendation given it. A»

• Oilworth.

PREFACE. V

lo tlie rules, they follow in the same manner as tae table of contents specifies, and in much the same order as they are gen- erally taught in schools. I have gone through the four funda- mental rules in Integers first, before those of the several de- nominations ; in order that they being well understood, the latter will be performed with much more ease and dispatch, according to the rules shown, then by the customary method of dotting. In multiplication I have shown both the beauty and use of that excellent rule, in resolving most questions that occur in mer- chandising ; and have prefixed before Reduction, several Bills of Parcels, which are applicable to real business. In working Interest by Decimals, I have added tables to the rules, for the readier calculating of Annuities, &c. and have not only shown the use, but the method of making them : as likewise an Interest Table, calculated for the easier finding of the Interest of any sum of money at any rate per cent, by Multiplication and Addition only ; it is also useful in calculating Rates, Incomes, and Serv- ants' Wages, for any number of months, weeks, or days ; and I may venture to say, I have gone through the whole with so much plainness and perspicuity, that there is none better extant. I have nothing further to add, but a return of my sincere thanks to all those gentlemen, schoolmasters, and others, whose kind approbation and encouragement have now established the use of this book in almost every school of eminence throughout the kingdom : but I think my gratitude more especially due to those who have favoured me with their remarks ; though I must still beg of every candid and judicious reader, that if he should, by chance, find a transposition of a letter, or a false figure, to excuse it ; for, notwithstanding there has been great care taken in cor- recting, yet errors of the press will inevitably creep in ; and some may also have slipped my observation ; in either of which cases the admonition of a good-natured reader will be verf ac- ceptable to his much obliged, and most obedient humble servant,

F. WALKINGAME. >9

A

lits...

RI'J

DHMETICAL TABLES.

TT

Nl

JMEF

1

12

123

1.234

lATION.

X. of Thoi C. of Thou Millions . . . X. of MilU

I2 34ii

Tt

sands 123,456 i

1,234,567 1

ons 12.345.678 1

Hundred 1 Thousan

8

Js

'

W

i

MULTIPLICATION

1

3 3

2 ~4"

3 6

4

5

6

7 8|

9

18

10 20

11

22 33 44 55

12

~24 36

48 60

8

10

12

14

16

6

9

12

15

18

21

2A

27 36

30 40

4

8

12

16

20

24

28

32

5

10

15

20

25

30

35

40

45

50

6

12

18

24

30

36

42

48

54

60 ~70

80

66

77

~88

99

72

84

96

108

7

14

21

28

35

42

49

56

63

8

16

24

32

40

48

56

64

72

9

18

27

36

45

54

63

72

81

90

10

20

30

40

50

60

70

80

90

100

110

120

132

144

~156

168

180

192

"204

"216

"

22

33

44

55

66

77

88

99 "i08

110 120

121 132

12

24

36

48

60

72

84

96

13

26

39

52

65

7^

91

104

117

130

143

14

1 28

42

56

70

84

98

112

126

140

154

15 j 30

45

60

75

90

105

120

135

150 ~160

Tto

180

165 176

187 198

16 1 32

48

64

80

96

112

128

144

17 1 34

51

68

85

102

119

136 144

153 162

18 1 36

54

72

90

108

126

19! 38

57

76

95

114

133

152

171

190

209

228

20 40

60

80

1 100

120

140

160

180

200

220 240 11

N^E.—

in 4 are 2,

This Tabic i and 2s in C

nay bo applic( are 3, &c.

1 to Di^ion

by reversing it : as the 2s i

ARITHMETICAL TABLEt

VU

PEXCfi.

TABLES OF MO^^EY

sini.i.

.SCiS.

■'-nil

are Is.

Rd.

80 J. are 5s. Sd.

20s.

are

£1 0:!.

;i2Us

are £6 Os.

n

2

0

84 -79

30

I 10

(130

•■ 6 10

:«)

•• 2

f)

90 -76

40

2 0

!i40

..7 0

36

•• 3

0

96 -80

50

2 10

■150

•. 7 10

■to

3

4

100 -.8 4

60

3 0

!160

• •SO

4ft

•• 4

0

103 ••9 0 110 •• 9 2

70

3 10

170

.. S 10

50

• • 4

•>

SO

4 0

ISO

••9 0

fiO

•• 5

0

120 •• 10 0

PO

4 10

190

• 9 10

70

•• 5

10

130 -10 10

100

5 0

j2tK)

•. 10 0

72

•• 6

0

140 •• 11 8

110

5 10

1210

•• 10 10

OF A POUND.

lOs, Oa. is 1 half

8 1 third

0 1 fourth

0 1 fifth

4 1 sixth

6 1 eighth

0 1 tenth

8 1 twelfth

0 1 twentieth

8 1 thirtieth

6 1 fortieth

PRACTICE TABLES.

I OF A SHILLING.

I 6d. is 1 half

4 1 third

3 1 fourth

2- 1 sixth

If 1 eighth

1 1 twelfth

OF A TON.

10 cwt. 1 half

5 1 f'lirth

4 1 fifth

2i 1 eighth

2 1 tenth

OF A CWT.

qrs. 2 or

Q....

lb.

56 ■ 28- • 16- .14.

is 1 half . • • 1 fourth . • •! seventh •••1 eighth

OF A QCABTER.

I4Iba 1 half

7 1 fourth

4 1 seventh

3J 1 eighth

CUSTOMARY WEIGHT OF GOODS. A Firkin of Butter is 56 lbs.' A Stone of Glass 5 lbs.

A Firkin of Soap 64

A Barrel of Soap 256

A Barrel of Butter 224

A Barrel of Candles 120

A Faii^jot of Steel 120

A Stone of Iron or Shot 14

A Barrel of Anchovies 30

A Barrel of Pot Ashes 200

A Seam of Glass, 24 stone, or 120

TABLES OF AVEIGHTS AND MEASURES.

TBOV WEIGHT.

24 sr. make 1 dwt.

20 dwt 1 ounce

12 oz 1 poand

apothecaries'. 20 gr. make 1 scruple

3 scr 1 dram

8 dr I ounce

12 oz I pound

.WOIRDUPOIS.

16 dr. make 1 oz.

16 OZ 1 lb.

14 lb 1 stone

28 lb 1 quarter

4 qrs 1 cwt.

20 cwt 1 ton

I WOOL WEIGHT.

I 7 lbs. make 1 clove

2 cloves 1 stone

2 siene- • • • 1 tod

I 6i tods 1 wey

2 weys 1 sack

il2 sacks^ • • • 1 last

LAND MEASUKE.

9 feet make 1 yard

30 yds 1 pole

40 poles 1 rood

4 roods 1 acre,

{ CLOTH SIEASUBE.

2} inch make 1 nail

4 nails 1 quar.

3 quar 1 Fl. ell

|4 quar 1 yard

io quar 1 En. ell

6 ouar 1 Fr. ell

! SOLID JIEASVBE.

|1723 in. make 1 sol. ft. 27 feet 1 vard

LONG MEASURE.

3 bar. corn 1 inch 12 inches- •• •! foot

3 feet 1 yard

6 feet 1 fathom

5{ yards-" • 1 pole

40 poles 1 furlong

8 fur 1 mile

3 miles 1 league

69 J miles 1 degree

Viii AKITUMETICAL TABLES. ~1

1

OLD ETAKDABO.

ALE AND BEER.

HEW STANDARD.

Gals. 0- P. Gills.

0 0 0 3-93

0 0 1 3-66

0 3 1 3-46

8 3 0 3-17

17 2 1 2-34

S.'i 1 1 0-69

53 0 0 303

70 3 0 1-33

106 0 1 206

4 gills make 1 pint

2 pints I quart

4 quarts- - • • 1 gal. 9 gallons- - • 1 fir. 2 firkins - - - 1 kild. 2 kUderkins-1 bar. 14 barrel--.- 1 hhd.

2 barrels 1 pun.

3 barrels- "-l butt

Gals. a. p. GUIs. 0 0 1 007 1

0 1 0 0-13

1 0 0 0-54 9 0 1 0-91

13 1 0 1-82 36 2 0 3-64 54 3 1 1-45 73 0 1 3-27 109 3 0 2-91

WINE MEASURE.

0 1 0 1-60

1 0 1 2-41 12 0 0 010 21 2 0 3-38 50 1 1 1-22 75 2 0 3-83

100 3 0 2-44 151 0 1 3-66 302 1 1 3-33

2 pints- --1 quart

4 quarts- 1 gallon 10 gallons- 1 anker 18 gallons- 1 runlet 42 gallons- 1 tierce 63 gallons- 1 hogshead 84 gallons -1 puncheon

2 nogs--- 1 pipe

2 pipes--- 1 tun

0 0 1 2-66

0 3 0 2-65

8 1 0 2-68

14 3 1 3-87

34 3 1 3-70

52 1 1 3-55

69 3 I 3-40

104 3 1 311

209 3 I 2-22

DRY MEASURE.

B. P. G. Q. P. GUIs. 0 0 0 10 0-25 0 0 10 0 1-01 0 10 0 0 202 10 0 10 007 2 0 0 2 0 0-14 4 0 10 0 0-28 8 10 0 0 0-56 33 0 0 0 0 2-24 82 2 0 0 1 1-63

2 pints make 1 quart

4 quarts 1 gallon

2 gallons - - - 1 peck

4 pecks 1 bushel

2 bushels 1 strike

4 bushels-'- -1 sack 8 bushels- ---1 quarter 4 quarters- - • 1 chald. 10 quarters- •• 1 last

B. P. G. Q. P. Gills. 0 0 0 0 1 3-75 0 0 0 3 1 302 0 0 13 1 204

0 3 1 3 0 017

1 3 1 2 0 0-35 3 3 10 0 0-70 7 3 0 0 0 1-40

31 0 0 0 1 1-65 77 2 0 1 1 2-13

COAL MEASURR

3 0 0 3 0 0-21 37 0 1 0 0 2-52

3 bushels--! sack 36 bushels- -1 chaldron

2 3 110 0-52 34 3 1 0 1 2-34

1 1

1

1

CONTENTS.

PART 1.— ARITHMETIC IN WHOLE NUMBERS.

Page

Ihtrodoction 13

Numeration 13

Integers, Addition 15

Subtraction 16

Multiplication 16

Division 19

Tables 21

Addition of several denominations. ^

Subtraction 34

Multiplication 37

Division 42

Bills of Parcels 44

Ficduction 47

Single Rule of Three Direct 53

Inverse 56

Double Rule of Three 58

Practice 60

Tare and Tret 67

Simple Interest 70

Commission 71

Purchasing of Stocks 71

Brokerage 71

Compound Interest 74.

Rebate or Discount 75

Equation of Payments 76

Barter 77

Profit and Loss 79

Fellowship 86

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 %

Progression, Arithmetical 97

Greometrical 100

Permutatiop 104

PART II.— VULGAR FRACTIONS.

Reduction 106

Addition 112

Subtraction 112

Multiplication 113

Division 114»

The Rule of Three Direct 114

Inverse.... 115

The Double Rule of Three 1 16^

CONTENTS.

PART III— DECIMALS.

Numeration 117

Addition 118

Subtraction ll'.i

Multiplication 1 1'J

Contracted Multiplication 120

Division >>..'... A2t

Contracted ; . Z] 122

Reduction 123

Decimal Tables of Coin, Weights,

and Measures 12(5

Tho Rule of Three 129

Extraction of the Square Root 1 30

Vulgar Fractions. . . 131

Mixed Numbers 132

Extract of the Cube Root 134

Vulgar Fractions 13G

Mixed Numbers 136

Biquadrate Root 138

Page

A general Rule for extracting the

Roots of all powers 138

Simple Interest 149

for days 14 1

Annuities and Pensions, &c. in

Arreani 143

Present worth of Annuities 147

Annuities, &c. in Reversion 150

Ilel)ate or Discount 152

E(|uation of Payments 154

Compound Interest 155

A nnujtics, &c. in Arrears 157

Present worth of Annuities 160

Annuities, &c. in Reversion 162

Purchasing Freehold or Real Es- tates 164

in Reversion 165

Rebate or Discount 166

PART IV.— DUODECIMALS.

Multiplication of Feet and Inches, 169 Measuring by the Foot Square. . . 171 Measuring by the Yard Square. . . 171 Measuring by the Square of 100 Feet 173

Measuring by the Rod 173

Multiplying several Figures by several, and the operation in one line only 174

PART V.-aUESTIONS.

A Collection of duestions, sot down promiscuously for the greater trial of the foregoing Rules

176

A general Table for calculating Interests, Rents, Incomes and Servants' Wages 181

A COMPENDIUM OF BOOK-KEEPING 184

EXPLANATIOJT OF THE CHARACTERS.

XI

EXPLANATION OF THE CHARACTERS MADE USE OF IN THIS COMPENDIUM.

=EquaI.

— Minus, or Less. -f- Plus, or More. X Mviltiplied by. H- Divided by. 2357

63

: : So is.

7—2+5=10. 9—2+5=2.

The Sign of Equality ; as, 4 qrs.=l cwt. 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 6.

The Sign of Addition ; as 4+4=8, 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 24.

The Sign of Division ; as, 8-5-2=4, that is, 8 divided by 2 is equal to 4.

Numbers placed like a fraction do likewise denote Division ; the upper number being the dividend, and the lower the divisor.

The Sign of Proportion ; as, 2 : 4 : : 8 ; 16, that is, as 2 is to 4, so is 8 to 16.

Shows that the difference between 2 and 7 added to 5, is equal to 10.

Signifies that the sum of 2 and 5 taken fr«m 9, is equal to 2.

Prefixed to any number, signifies the Square Root of that number is required.

Signifies the Cube, or Third Power.

Denotes the Biquadrate, or Fourth Power,. &c.

id est, that is.

i

THE

TUTOR'S ASSISTANT;

B£INe

A COMPENDIUM OF 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 Numeration, Addition, Subtractiok,. 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.

C. Millions. X. Millions. Millions.

C. Thousands. X. Thousands. Thousands.

Hundreds.

Tens.

Units.

9 8 7,

.654,

.321

9 0 0,

.000,

.000

90

.000,

.000

7

0 0 0

. 0 ao

6 0 0

.000

5 0

.000

4

0 0 0

3 0 0

2 0

1

14 NUMERATION.

Rule. There are three periods ; the first on the right hand. Units ; tlu; second, Thousands ; and the third, Millions ; each consisting of three Figures, or Places. Reckon the first Figure of each from the left hand as so many Hundreds, the next a^ Tens, and the third as so many single ones of what is written over them: thus, the first Period on the left hand is read, Nine Hud- dred and Eighty-seven Millions ; and so on for any of the rest.

THE APPLICATION.

Write down in proper Figures the following Numbers.

(') Twenty-three.

{^) Two hundred and Fifty-four.

(') Three Thousand, Two Hundred and Four.

{*) Twenty-five Thousand, Eight Hundred and Fifty-six.

(^) One Hundred and Thirty-two Thousand, Two Hundred and Forty-five.

(') Four Milions, Nine Hundred and Forty-one Thousand, Four Hundred.

C) Twenty-seven Millions, One Hundred and Fifty-seven Thousand, Eight Hundred and Thirty-two.

(') Seven Hundred and Twenty-two Millions, Two Hundred and Thirty-one Thousand, Five Hundred and Four.

(') Six Hundred and Two Millions, Two Hundred and Ten Thousand, Five Hundred.

Write down in Words at length the following Numbers.

n 3.5

{*) 2017

C)

519007 ('») 5207054

n 59

(») 5201

{')

754058 (•») 2071909

P) 172

(«) 207()6

r)

5900030 (>2) 70054008

(•3)(

65700047 ( ' ♦

) 900061057 ('») 221900790

Notation

by Roman Letters.

I

One.

IX Nine.

H

Two.

X Ten.

HI

Three.

XI Eleven.

IV

Four

XH Twelve.

V

Five.

Xni Thirteen.

VI

Six,

XIV Fourten.

VII

Seven.

XV Fifteen.

VIII

Eight.

k

XVI Sixteen.

ADDITION OP INTEGERS.

15

XVII

Seventeen. CCC

Three Hundred.

XVIII

Eighteen. CCCC

Four Hundred.

XIX

Nineteen. D

Five Hundred.

XX

Twenty. DC

Six Hundred.

XXX

Thirty. DCC

Seven Hundred.

XL

Forty. DCCC

Eight Hundred.

L

Fifty. DCCCC

Nine Hundred.

LX

Sixty. M

One Thousand.

LXX

Seventy. MDCCCXII

One Thousand Eight

LXXX Eighty'.

Hundred and Twelve,

xc

Ninety. MDCCCXXXVIl

'. One Thousand Eight

c

Hundred.

Hundred and Thirty

cc

Two Hundred.

Seven.

INTEGERS.

ADDITION

Teacheth to add two or more Sums together, to make one whole or total Sum.

Rule, There must be due regard had in placing the Figures one under the other, i. e. Units under Units, Tens under Tens, &c. ; then beginning with the first row of Units, add them up to the top ; when done, set down the Units, and carry the Tens to the next, and so on ; continuing to the last row, under which set down the Total amount.

Proof. Begin at the top of the Stun, and reckon the Figures downwards, the same as you add them up, and, if the same as the first, the Sum is supposed to be right.

Qrs.

Montlis. £

Years.

(•)275 110

(')1234 (3)75245 7098 37502

(*) 271048 325^176

473

3314 91474

107584

354

6732 32145

625608

271

2546 47258

754087

352

6709 21476

279736

(5) What is

the sum of 43, 401, 9747, 3464,

2263, 314, 974 ^

Ans. 17206.

(») Add 2'46034, 29S705, 47321, 59653,

*nd 640 together.

B2

64218, 5376. 9821 Ans. 730829.

16 SUBTRACTION OF INTEGERS.

(^) If you give A. £56, B. £104, C. £274, D. £391, and E £703, how much is given in all ? Ans. 1528

(•) How many days are in the twelve Calendar Months?

Ans. 365.

SUBTRACTION

Teacheth to take a less Number from a greater, and shows the remainder or ditlercnce.

Rule. This being the reverse of Addition, you must borrow here (if it require) what you stopped at there, always remember- ing to pay it to the next.

Proof. Add the Remainder and the less Line together, and if the same as the greater, it is right

(•)

n

(•)

n

(•)

(•)

From 271

4754

42087

452705

271509

3750215

Take 154

2725

S'iooe

327616

152471

3150874

Rem. 117

Proof 271

MULTIPLICATION

Teacheth how to increase the greater of two Numbers given as often as there are Units in the less ; and compendiously performs the office of many additions.

To this Rule belong three principal Members, viz.

1. The Multiplicand, or Number to be multiplied.

2. The Multiplier, or number by which you multiply.

3. The Product, or Number produced by multiplying.

Rule. Begin with that Figure which stands in the Unit's place of the Multiplier, and with it multiply the firstFigure in the Unit's place of the Multiplicand. Set down the Units, and carry the Tens in mind, till you have multiplied the next Figure in Uie Multiplicand by the same Figure in the Multiplier ; to the pro- duct of which add the Tens you kept in mind, setting down tne Units, and proceed as before, till the whole line is multiplied.

MULTIPLICATION OV INTEGERS.

17

Proof. By casting out the Nines ; or make the former Mul- tiplicand the Multiplier, and the Multiplier the Multiplicand ; and if the Product of this operation be the same as before, the work is right.

MULTIPLICATION TABLE. 1 2 3 4 5 6 7 8 9 10 11 12

2

4

6

8

10

12

14

16

18

20

22

24

3

6

9

12

15

18

21

24

27

30

33

36

4

8

12

16

20

24

28

32

36

40

44

48

5

10

15

20

.25

30

35

40

45

50

55

60

6

12

IS

24

30

36

42

48

54

60

66

72

7

14

21

27

35

42

49

56

63

70

77

84

8

16

24

32

40

48

56

64

72

80

88

96

9

18

28

36

45

54

63

72

81

90

99

108

10

20

30

40

50

60

70

80

90

100

110

120

11

22

33

44

55

66

77

88

99

110

121

132

12

24

36

48

60

72

8-1

96

108

120

132

144

Multiplicand (») 25104736 (2)52471021 (3)7925437521 Multiplier 2 3 4

Product 50209472

(*) 27104107 5

(*) 231047 6

(•) 7092516 7

(') 3725104

8

C) 4215466 9

(•) 2701057 10

(»") 31040171 11

^Vhen the Multiplier is more than 12, and less than 20, multi- ply by the Unit Figure in the Multiplier, adding to the Product the back Figure to that you multinlied.

B3

16 MULTIPLICATION OF INTEGERS.

( ' ' ) 5710592 (' = ) 510725a ( ' =• ) 7653210 ( » « ) 92057165 13 14 15 16

('«) 6251721 (i«) 9215324 (i') 2571341 (»") 3592101 17 18 19 20

When the Multiplier consists of several Figures, there must he as many products as there are Figures in the Multiplier, ob- serving to put the tirst hgure of every Proifluct under that Figure you multiply by. Add the several Products together, and their Sum will be the total Product.

( '») Multiply 271041071 by 5147.

(2 «) Multiply 62310047 by 1669,

(2 ») Multiply 170925164 by 7419.

(") Multiply 9500985742 by 61879.

(2 3) Multiply 1701495868567 by 4768756.

When Cipliers are placed betvveen the significant Figures ia the Multiplier, they may be omitted ; but great care must be taken that the next Figure must be put one place more to the left hand» t. e. under the Figure you multiply by.

(**) Multiply 571304 By STOOy

5140836 39984-28 1142408

Product 15427G48836

(««) Multiply 7501240325 bv 57002. (2 6) Multiply 562710934 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 the Figures, and setting down on the right hand of the total Product as many Ciphers as were omitted.

DIVISION OF INTEGERS. W

(8 7) Multiply 1379500 3400

55180 41385

4C90300000

(2 8) Multiply 7271000 by 52600. (2 ») Multiply 74837000 by 975000.

When the Multiplier is a composite Number, t. e. if any two Figures being multipUed together, will make that Number, then multiply by one of those Figures, and that Product being multi- plied by the other will give the Answer.

(3 0) Multiply 771039 by 35, or 7 times 5. 7X5=35

5397273 5

26986365

(3 >) Multiply 921563 bv 32. (32) Multiply 715241 by 56. (»») Multiply 7984956 by 144.

DIVISION

Teacheth to find how often one Number is contained in another ; or, to divide any Number into what parts you please.

In this Rule there are three numbers real, and a fourth acci- dental : viz.

1. The Dividend, or Number to be divided:

2. The Divisor, or Number by which you divide :

3. The Quotient, or Number that shows how often the Divisor is contained in the Dividend :

4. Or accidental Number, is what remains when the work is nnished, and is of the same name as the Dividend.

Rule. When the Divisor is less than 12, find how often it i? contained in the first Figure of the Dividend ; set it doAvn 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 same

20

DIVISION OF INTEGERS.

till you have gone through the Line; but when the Divisor ift more than 12, multiply it by the Quotient Figure ; the Product subtract from the Dividend, and to the Remainder bring down the next Figure in the Dividend and proceed as before, till the Figures are all brought down.

Proof. Multiply the Divisor and Quotient together, adding the Remainder, (if any,) and the Product will be the same as the Dividend.

Dividend. Rem. (') Divisor 2)725107(1 («) 3)7210472( «) 4)7210416(

Cluotient 3G2553 Proof 725107

(«) 5)7303287(

<•) 7)2532701(

(') 8)2547325(

(5) 6)5231037(

I) 9)25047306<,

Divisor. Dividend. Cluotient.

9) 29)41723' 2P

77(143875 29

127 116

.112

87

1294875 287750

2 rem.

4172377 ProoC

.253 S33

.217 903

.147 145

(»o) Divide 7210473 by 37.

Ans. 194877*f ( » I) Divide 42749467 by 3^17. (12) Divide 734097143 by 5743. (» 8) Divide 1610478407

by 54716. (n) Divide 497340189 1

by 510834. (IS) Divide 51704567874

by 4765043. (1 «) Divide 17453798946123741 by 31479461.

lUSIL

When there are Ciphers at the ".nd of the Divisor, they may be cut off, and as many places from off the Dividend, but they must be annexed to the Remainder at last.

TABLES OF MONEY.

21

(i») 27i;00)254732!2l(039 (>•) 373iOOO)75-^T3|729(2017

(18) 57^1 100)72534721 16(1267 (» 0) 215|O0O)63251O4|997(

23419

When the Divisor is a composite number, i. e. if any two Fi- fiires, 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. But as it some- times happens, that there is a Remainder to each of the Quotients, and neither of them the true one, it may be found by this

Role. Multiply the first Divisor into the last Remainder, to tliat Product add the first Remainder, which will give the true one.

Dir

3210473 by 27.

(S9)

7210473 by 35.

(S3)

6251013 by 42.

(S4)

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.

i Halfpenny 12 Pence 1 Shilling s.

I Three Farthings 20 Shillings 1 Pound... £

Farthings

4 = 1 Penny 48 = 12 = 1 Shilling 960 = 240 = 20 = 1 Pound.

SBILUKGS.

PENCE TABLE.

•. £

«.

d.

3. d.

d.

«.

d.

20 •

• 1 :

0

20 .

•1:8

90 •

7

6

30 •

• I :

10

24 .

•2:0

96 •

• 8

0

40 •

• 2 :

0

30 •

•2:6

100 •

• 8

4

fiO •

• 2 :

10

36 •

•3:0

108 •

9

0

60 .

. 3 :

0

40 •

•3:4

110 •

• 9

3

70 •

• 3 :

10

48 .

•4:0

120 •

• 10

0

80 .

• 4 •

0

60 •

•4:2

130 •

• 10

10

»0 •

• 4 •

10

60 •

•5:0

132 •

• 11

0

100 •

• 5 •

0

70 •

• 5 : 10

140 .

• 11

8

no .

• 6 •

10

72 .

•6:0

144 •

• 12

0

120 •

• 6 =

0

80 •

•6:8

150 •

• 12

6

130 ■

. 6 :

10

84 •

•7:0

160 •

:3

4

52 TABLES OF WEIGHTS.

TROY WEIGHT. Marled

34 Grains make 1 Pennyweight i ^^'^

20 Pennyweights , 1 Ounce ....oz.

12 Ounces 1 Pound lb.

Grains

34 = I Pennyweight 480 = 20 -^ 1 Ounce 5760 = 240 = 12 = 1 Pound

By this Weight are weighed Gold, Silver, Jewels, Electuaries and all Liquors.

N. B. The Standard for Gold Coin is 22 Carats of fine Gold nnd 2 Carats of Copper, melted together. For Silver, is 11 oz 3 dwts. of fine Silver, and 10 dwts. or Copper.

25 lb. is a quarter of 100 lb. 1 cwt.

30 cwt. 1 Ton of Gold or Silver.

AVOIRDUPOIS WEIGHT. Marked

16 Drams make 1 Ounce >

S oz.

16 Ounces 1 Pound lb.

2S Pounds 1 Quarter qr.

4 Quarters or 112 lb 1 Hundred Weight cwt,

30 Hundred Weight 1 Ton ton.

Drams

16 = 1 Ounce

356 = 16 = 1 Pound

7168 = 448 = 28 = 1 Quarter

38673 = 1793 = 113 = 4 = 1 Hundredweight

573440 = 35840 = 2240 = 80 = 20 = 1 Ton.

There are several other Denominations in this Weight that

Are used ir) some particular Goods, viz.

lb. lb

A Firkin of Butter 56 A Stone of Iron, Shot or

Soap 64 Horseman's wt

A Barrel of Anchovies 30 Butcher's Meat... §

Soap 256 A Gallon of Train Oil 7^

Raisins 112 A Truss of Straw 36

A Puncheon of Prunes 1130 New Hay 60

A Foddorof Lead 19 cwt. Old Hay 56

2 qrs. 36 Trusses a Load.

|„

TABLES OF WEIGHTS. !53

Cheese and Butter. A Clove or Half Stone, S lb.

A Wey in Suffolk, ) lb. A Wey in Essex, ) lb.

si Cloves, or ) 256 32 Cloves, or ) 336 Wool.

lb. A Wey is 6 Tods and ) lb.

A Clove 7 1 Stone, or \ 182

\ Stone 14 A Sack is 2 Weys, or 364

A Tod 28 A Last is 12 Sacks, or 4368

By this Weight is weighed anything of a coarse or drossy na- ture ; as all Grocery and Chandlery Wares ; Bread, and all Me- tals but Silver and Gold.

Note. One Pound Avoirdupois is equal to 14 oz. 11 dwts. 15^ grs. Troy.

APOTHECARIES' WEIGT.

Marked

20 Grains make 1 Scruple 3

3 Scruples 1 Dram 3

8 Drams 1 Ounce 5

12 Ounces 1 Pound lb

Grains

20 = 1 Sruple 60 = 3 = 1 Dram K 480 = 24 = 8 = 1 Ounce

f 5760 = 288 = 96 = 12 = 1 Pound.

Note. The Apothecaries mix their Medicines by this Rule, but buy and sell their commodities by Avoirdupois Weight.

The Apothecaries' Pound and Ounce, and the Pound and Ounce Troy, are the same, only differently divided and subdivi- ded.

CLOTH MEASURE.

Slarked

4 Nails make 1 Quarter of a Yard ^ "j.

3 Quarters 1 Flemish Ell Fl. E.

4 Quarters 1 Yard yd.

J Quarters 1 English Ell E. E.

'i Quarters 1 French Ell Fr. E.

91 TABLES OF MEASURES.

Inches

2i = 1 Nail

9 = 4 = 1 CluartCT 36 = 16 = 4 = 1 Yard 27 = 12 = 3 = 1 Flemish Ell 45 = 20 = 5 = 1 English Ell 54 = 34 = 6 = 1 French Ell.

LONG MEASURE.

Marked

3 Barley Coma make 1 Inch , > ^' **

12 Inchea 1 Foot feet

3 Feet I Yard yd.

6 Feet 1 Fathom fth.

Oi Yards I Rod, Pole or Perch rod, p.

40 Poles 1 Furlong fur.

8 Furlongs 1 Mile mile.

3 Miles 1 League lea.

60 Miles 1 Degree ifg.

Barley Coma

3 = 1 Inch

36 = 12 = 1 Foot 108 = 36 = 3 = 1 Yard 594 = 198 = 161= 5i= 1 Pole 23760 = 7920 = 660 = 220 = 40 = 1 Furlong 190080 = 63360 = 5280 = 1760 = 320 = 8=1 Mile

N. B. A Degree is 69 Miles, 4 Furlongs, nearly, though coir monly reckoned but 60 Miles.

This Measure is used to measure Distance of Places, or any thing else tliat hath length only.

WINE MEASURE.

Marked

2 PinU make 1 auart l^

4 Cluarta 1 Gallon gaL

10 Gallons 1 Anker of Brandy auk.

18 Gallons 1 Runlet run.

31 1 (Jallons Half an Hogshead ihhd.

42 Gallons 1 Tierce....... tier.

63 Gallons 1 Hogshead hhd.

2 Hogsheads 1 Pipe or Butt P. or a

2 Pipes or 4 Hogsheads I fva \-

TABLE? OF MEASURES. ^

Inches*

28j= 1 Pint 57i= 2= 1 auart 231 = 8= 4= I Gallon 9703 = 336= 1G8= 42=1 Tierce 14553 = 504= -252= 63=li = l Hogshead 19404 = 672= 336= 84=2 =li=l Puncheon £9lOG =1008= 504=126=3 =2 =1J=1 Pipe 58212 =2016=1008=252=6 =4 =3 =2=1 Tun

All Brandies, Spirits, Perry, Cider, Mead, Vinegar, Honey, »nd Oil, are measured by this measure ; as also Milk, not by Jaw, but custom only.

ALE AND BEER MEASURE.

Marked

2 Pints make. 1 Clnart > ^'

4 Cluarte 1 Gallon gal.

8 Grallons. 1 Firkin of Ale A. fix.

9 Gallons 1 Fiikin of Beer B.fir.

3 Firkins 1 Kilderkin kiL

4 Firkins, or 2 Kilderkins 1 Barrel l>ar.

1| Barrel, or 5-1 Gallons 1 Hogshead of Beer hhd

3 Barrels 1 Puncheon pun.

3 BarreU,or2 Hogsheads 1 Butt butt

BEER.

Cubic Inches

35i= 1 Pint 704= 2= 1 CLuart 282 = 8= 4= 1 GaUon 2538 = 72= 36= 9= 1 Firkin 5076 =144= 72= 18= 2=1 Kilderkin 10152 =288=144= 36= 4=2=1 Barrel 15228 =433=216= 54= 6=3=li=l Hogshead 20304 =576=288= 72= 8=4=2 =U=1 Puncheon 30456 =864=432=108=12=6=3 =2 =1^=1 Butt

ALE. Cubic Inches

35i= 1 Pint 70i= 2= 1 auart 282 = 8= 4= 1 Gallon 2256 = 64= 32= 8 = 1 Firkin 4512 =128= 64=16=2=1 Kilderkin 9024 =256=128=32=4=2=1 Barrel 13536 =384= 192=48=6=3= H=l Hogshead.

• By a late Act of Pailiament, ihe capacities of the Wine, the Ale and Beer, and tb« Cry Measures, have been leilucol to one Standard. For an accurate comparison of these Measures, witii the old stand <ril M":isurps, tlia Student is referred to tba Table ai iht *' Imperial Meaaures," at llie bogin.iins of the work.

C

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 6 gallons.

DRY MEASURE.

1 auart

2 Pints make

2 duarts 1

2 PottIe« 1

2 Gallons ,....l

4 Pecks 1

2 Bushels 1

4 Bushels 1

2 CooRis, or 8 Bushels 1

4 Quarters 1

5 Quarters 1 Wey

2 Wcys 1 Last

Marked )pU.

„ . )q«*-

Pottle .•••■poL

Gallon :.;....'... gal.

Peck fk.

Bushel tni.

Strike strike.

Coom cooDi.

Quarter qr.

Chaldron ' . . " cnal.

wey.

last.

In London, 36 bushels make a chaldron.

Solid Inches 268t= 1 Gallon 537f= 2= 1 Peck 2150f= 8= 4= 1 Bushd 4300*-= 16= 8= 2= 1 Strike 8601 f= 32= 16= 4= 2= 1 Coom 17203i= 64= 32= 8= 4= 2= I Quarter 8(5016 =320=160=40=20=10= 5=1 Wey 172032 =640=320=80=40=20=10=2=1 Last.

The Bushel in Water Measure is 6 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.

This measure is applied to all dry goods. The standard Bushel is 18^ inches wide, and 8 inches deep

TABLES OF MEASURES. ^

Marked

TIME.

60 Seconds make....l Minute ,_

) m.

W Minutes 1 Hour hour.

i4 Hours I Day day.

7 Days 1 Week week.

4 Weeks 1 Month mo.

13 Months, 1 day, 6 hours . . 1 Julian Year yr.

Seconds

60=

1

Minute

3600=

60=

= Hour

86400=

1440=

= 24= 1

Day

604800=

10080=

= 168= r

=1

Week

2419200=

40320=

= 672= 28= d.

=4=

h.

= 1

Month.

w.

d.

h.

31557600=525960=

=8766=365 ;

:6=

=52

: 1 :

6=1 Julian Year

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.

144 Inches make 1 Foot

9 Feet 1 Yard.

100 Feet 1 Square of flooring

Qn2\ Feet 1 Rod.

40 Rods 1 Rood.

4 Roods, or 160 Rods, or 48M yards 1 Acre of land.

640 Acres *. 1 Square Mile. .

30 Acres...., 1 Yardofland.

100 Acres 1 Hideofland.

C9

2S ADDITION OF MONEY.

Inches

144= 1 Foot 1296= 9 = 1 Yard 39201= 2721= 30i 1 Pole 15^8160=1081)0 =l2i0 = 40=1 Rood 62r2640=43560 =4S10 =160=4=1 Acre.

By this measure are nieasurod all things that have lengtli and breadth ; such as land, painting, plastering, flooring, thatcliiug, plumbing, glazing, &c.

SOLID MEASURE.

1728 Inches. . ., make 1 Solid Foot.

27 Feet 1 Yard, or load of earth.

40 Feet of round tiiiiher, ) . , m r i

Or. DO 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.

Role. Add the fir.st 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.

(•) C) (3) («)

£ f- d. £> s. d. £. s. d, £ : d.

a .. 13 .. 51 27 .. 7 .. 2 35 .. 17 .. 3 75 .. 3 .. 7

7 .. 9 .. \\ 34 .. 14 .. 7J 59 .. 14 .. 7* 54 .. 17 .. 1

5 .. 15 .. 4i 57 .. 19 .. 2* 97 .. 13 .. 5i 91 .. 15 .. 4J 9 .. 17 .. G} 91 .. 16 .. 1 37 .. 16 .. 8* 35 .. 16 .. 5l 7 ..,16 .. 9. 75 .. 18 ,. 7f 97 .. 15 .. 7 2<) .. 19 .. 1\

6 .. t4 .. 71 97 .. 13 .. 5 59 .. 16 .. b\ 91 .. 17 .. 3^

39 .. G .. H\

ADDITION OF WEIGHTS. JJ9

MONEY.

c) n V) • {')

£ g. d. £ *. d. £ a. d. £ : d.

257.. 1 .. 5i 5-25. . 2. .4i 2l..l4..7i 73 •• 2 .. li

734.. 3. .71 179 ..3. .5 75..16..0 25! .. 12 .. 7

5d5 .. b..^^ i!30..4..7j 79.. 2 .. 4i 96 .. 13 .. 5}

I59..14..7J 973. .3. .5* 57..l6..5i 76 .. 17 .. 3j

207.. 5.. 4 2M..5..7 36.. 13 ..81 97 .. 14 .. 14

798.. 16 ..71 379. .4. .51 54.. 2.. 7 54 .. U .. 7i

n

1

C) (")

(.8)

£

f.

d.

£

1. d. £ «.

d.

£ ..

d.

127.

. 4 ..

^\

261

.. 17 .. U 31 .. 1 .,

. H

27.. 13 ..

&i

525.

. 3 ..

5

379

. . 13 . . 5 75 . . 13 . .

. 1

16 .. 12 ..

9i

271 .

. 0 ..

5

257

.. 16 .. 71 39 .. 19 .,

. 7}

9 .. 13 ..

34

524 .

.9 ..

1

184

..13 ..5 97.. 17.,

.3*

15.. 2..

7*

379.

.4 ..

3^

725

.. 2 .. 3i 36 .. 13 ..

5

37.. J9..

1

215 .

. 5 ..

81

359

.. 6 .. 5 24 .. 16 .

.3}

56.. 19..

H

TROY WEIGHT.

r')

n

(')

oz.

dwt.

gr.

lb. oz. dwt.

lb. oz- dwt.

.§

5 .

. 11 .

. 4:

7.. 1 .. 2

5 .. 2 .. 15 .

7.

. 19 .

.21

3.. 2 .. 17

3 .. U .. 17 .

. 14

3 .

. 15 .

. 14

5 . . 1 . . 16

3 .. 7.. 15.

. 19

1 .

. 19 .

. 22

7.. 10 .. 11

9 .. 1 .. 13 .

. 21

9 .

. 18 .

. 15

2 .. 7 .. 13

3 .. 9.. 7.

. 23

8 .

. 13 .

. 12

3 .. 11 .. 16

5 .. 2.. 15 .

. 17

AVOIRDUPOIS WEIGHT

(■')

n

(')

lb.

oz.

dr.

cwt. qrs. lb.

t. cwt qrs. 7.. 17. .2 .

. lb.

152 .

. 15 .

. 15

25 .. 1 .. 17

. 12

272 .

. 14 .

. 10

72 . . 3 . . 26

5 .. 5 .. 3 ,

.. 14

303 .

. 15 .

. 11

54 .. 1 .. 16

2.. 4.. 1

.. 17

255 .

. 10 .

. 4

24 .. 1 .. 16

3.. 18. .2

.. IP

17:5 .

. 6 .

. 2

17 .. 0 .. 19

7.. 9 .. 3

..90

635 .

.. 13

.. 13

55 .. 2 .. 16

8 .. 5 .. 1

..24

C3

30

ADDITION OF MEASURES.

A.POTHECARIES' WEIGHT.

lb. 17

9 27

9 37 49

0)

oz.

10 • 6 .. 2

11 " 1 6 •• 6

10 .. e

0 " 7

dr. KT.

7 •• 1 2 2 1 2 0

(«)

lb. oz. dr. Mcr. gr.

7 .. 2 •• 1 •• 0 .. 12

8 .. I .. 7 .. 1 .. 17

9 •' 10 •• 3 •• 0 •• 14 7 .. 6 •• 7 •• 1 •• le 3 .. 9 •• 6 •• 2 •'• 13 7 .. 1 .. 4 .. 1 .. 18

CLOTH MEASURE.

(>)

(«)

(•)

F1.E qr.

n.

yd.

qr.

n.

E.E.

qr. n.

127 .• 2

• 1

135 •

• 3 •

3

272 •

• 2 •• I

15 .• 1

• 3

70 .

. 2 •

2

162 .

. 1 .. 2

237-.. 0

• 2

95 ■

. 3 •

0

79 •

. 0 .. 1

52 •• 1

• 3

176 •

. 1 .

8

166 •

. 2 .. 0

376 •• 2

• 1

26 •

. 0 •

I

79 .

• 3 •• 1

197 •• 1

• S

279 •

• 2 •

1

154 .

. 2 .. 1

LONG MEASURE.

(')

yd.

feet in

bar.

225 •

. 1 .. 9 .

' I

171 •

. 0 •• 3 •

' 2

62 •

• 2 .. 3 .

• 2

397 .

. 0 •• 10 •

• 1

164 •

. 2 •• 7 •

. 2

137 .

. 1 .. 4 ■

• 1

(2>

lea. m. tar. po.

72 .. 2 •• 1 ■• 19

27 .. 1 •• 7 •• 22

85 .• 3 •• 6 •. 31

79 .. 0 .. 6 •• 12

61 •• 1 •• 6 •• 17

72 •• 0 •• 6 •• 21

(»)

a.

r.

P

726 •

• 1 ••

31

219 •

. 2 ••

17

1455 .

. 3 ••

14

879 .

. 1 ..

21

1195 .

. 2 ••

14

LAND MEASURE.

(«)

a.

r.

P

1232

• 1 •

14

327

• 0 •

19

131

• 2 •

IS

1219

. 1 •

18

459 •

. 2 •

17

ADDITION OF MEASURES. 3^

WINE MEASURE.

hhJa.

gals.

qts.

31 .

. 57.

. 1

97 .

. 18.

. 2

76 .

. 13 .

. 1

55 .

,. 46 .

. 2

87.

.38 .

. 3

55 .

. 17 .

. 1

{•")

t.

hhds. gala.

qte.

U

.. 3 .. 27.

.2

19

.. 2 .. 56.

. 3

17

. . 0 . . 39 .

. 3

79

.. 2 .. 16 .

. 1

54

.. 1 .. 19 .

.. 2

97

.. 3 .. 54.

. 3

ALE AND BEER MEASURE.

(•)

A.B.

fir.

gaL

25 .

. 2 .

. 7

17 .

. 3 .

. 5

96 .

. 2.

. 6

75 .

. 1 .

. 4

96.

. 3.

. 7

75 .

.0 ..

5

(n

eh.

bu.

pks.

75 .

,. 2.

. 1

41 ,

.24.

,. 1

29 .

.. 16.

. 1

70,

..13,

.. 2

54 ,

.. 17,

.. 3

79 .

.. 25

.. 1

(M

w.

d.

h.

71

.. 3 .

. 11

51

.. 2 .

. 9

76

.. 0 .

. 21

95

.. 3 .

. 21

79 ,

.. 1 .

. 15

(^)

(3)

B.B. fir. gal.

hhds.

gals.

qts.

37 . . 2 . . 8

76.,

. 51 .

. 2

54.. 1 .. 7

57..

, 3 .

. 3

97.. 3 .* 8

97.,

. 27.

. 3

78.. 2 .. 5

22.

. 17.

. 2

47 . . 0 . . 7

32 ..

19 .

. 3

35"2.. 5

56 .,

. 38.

. 3

DRY MEASURE.

(=)

la.st

wey. q.

bu.

pks

38 .

. 1 .. 4

.. 5 .

. 3

47 .

. 1 .. 3

.. 6.

. 2

62.

..0.. 2

.. 4 .

. 3

45

.. 1 .. 4

.. 3 .

. 3

78 ,

.. 1 .. 1

.. 2.

. 2

29 ,

.. 1 .. 3

.. 6 .

. 2

TIME.

(2)

w.

d.

h.

m.

*•

57 .

. 2 .

. 15 .

. 42 .

. 41

95 .

. 3 .

. 21 .

. 27.

. 51

76 .

. 0 .

. 15 •

• 37.

. 28

53 .

. 2.

. 21 .

.42.

27

98 .

. 2.

. 18 .

. 47 .

. 38

32 ADDITION.

THE APPLICATION.

1. A man was born in the year 1750, when will he be 47 years of age? Ans. 1797.

2. A, B, C, and D, went partners in the purchase of a quan- tity of goods ; A laid out £7, half-a-guinea, and a crown ; B, 49s. ; C, 54s. 6d. ; and D, 87d. What was laid out in all ?

. Ans. £13 : 6 : 3.

3. A man lent his friend, at different times, these several sums, viz. £63, £25 : 15, £32 : 7, £15 : 14 : 10, and four score and nineteen pounds, half-a-guinea, and a shilling. How much did he lend in all ? Ans. £236 : 8 : 4.

4. What is the estate worth per annum, when the taxes are 21 guineas, the neat income 8 score, £19 : 14?

Ans. £201 : 15.

5. There are three numbers ; the first is 215, the second 519, nnd the third is as much as the other two. What is the sum of them all ? Ans. 1468.

6. Bought a parcel of goods, for which I paid £54 : 17, for packing 13s. 8d., carriage £1 : 5 : 4, and spent about the bargain 14». 3d. What do these goods stand me in ?

Ans. £57 : 10 : 3.

7. There are two numbers, the least whereof is 40, their dif- ference 14. I desire to know what is the greater number, and the sum of both ?

Ans. 54 greater number, 94 sum,

8. A gentleman left his elder daughter £1500 more than the younger, and her fortune was 11 thousand, 11 hundred and £11. What was the elder sister's fortune, and what did the father leave them ? Ans. Elder sister's fortune, £1361 1.

Father left them £25722.

9. A nobleman, before he went out of town, was desirous of I)aying all his tradesmen's bills, and upon inquiry, he found tliat he owed 82 guineas for rent ; to his wine-merchant, £72 : 5 : 0 ; to his confectioner, £12 : 13 : 4 ; to his draper, £47 : 13 ^ 2; to his tailor, £110 : 15 : 6; to his coach-maker, £157: 8:0-, to his tallow-chandler, £8:17:9; to his corn-chandler, £170 : 6 : 8 ; to his brewer, £52 : 17 : 0; to his butcher, £122 : 11 : 5; to hi.«! baker, £37 : 9 : 5; and to his servants, for wages, £53 : 18 : 0. I desire to know what money he had to raise in the whole, Mhen he added to the above sums, £100, which he wished to lake with him? Tins. £1032 . 17 : 3.

10. A fatlior was 24 years of age (allowing 13 months to u year, and 28 days to a montli) M'hen his first child was born ; between the eldest and next born was 1 year, 11 months, 14 days ; between the second and third were 2 years, 1 month, and 15 days ; between the third and fourth were 2 years, 10 months, and 25 days ; when the fourth was 27 years, 9 months, and 12 days old, how old was the father ?

Ans. 58 years, 7 months, 10 days.

11. A banker's clerk having been out with bills, brings home an account, that A paid him £7:5:2, B £15 : 18 : 6|, C £150 : 13 : 2^, D £17 : 6 : 8, E 5 guineas, 2 crown pieces, 4 half-crowns, and 4s. 2d., F paid him only twenty groats, G £76 15 : 9^, and H £121 : 12 : 4. I desire to know how much the whole amounted to, that he had to pay ?

Ans. £396: 7 : Gj.

12. A nobleman had a service of plate, which consisted of twenty dishes, weighing 203 oz. 8 dwts. ; thirty-six plates, weigh- ing 408 oz. 9 dwts. : five dozen of spoons, weighing 112 oz. 8 dwts. ; six salts, and six pepper boxes, weighing 71 oz. 7 dwts. ; 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, ; together with sundry other small articles, weighing 105 oz. 5 dwts. I desire to know the weight of the whole ?

Ans. 102 lb. 2 oz. 13 dwts.

13. A hop-merchant buys 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 lb. • the fifth, 2 cwt. 3 qrs. 15 lb. Besides these, he purchased two pockets, each weighing 84 lb. I desire to know the v/eight of the whole ?

Ans. 15 cwt. 2 qrs.

14. A, of Vienna, owes to B, of Liverpool, for goods received in January, the sum of £103 : 12 : 2 ; for goods received in Fe- bruary, £93 : 3 : 4; for goods received in March, £121 : 17; for goods received in April, £142 : 15 : 4; for goods received in May, £171 : 15 : 10; for goods received in June, £142 : 12 : 6 ; but the latter six months of the year, owing to the falling off in the demands for the articles in which he dealt, the amount was only £205 : 7 : 2. I iesire to know the amount of the whole year's bill ?

Ans. £981 : 3 ^ 4.

34

SUBTRACTION.

SUBTRACTION OF MONEY, WEIGHTS & MEASUREt-

Rule. Subtract as in Integers ; only when any of the lower denominations are greater then the upper, borrow as raany of that as make one of the next superior, adding it to the upper, from which take the lower ; set down the difference, and carry 1 to the next higher denomination from what you borrowed.

Proof. As in Integers.

MONEY.

£ Borrowed 715 . Paid 476 .

«. . 2 . . 3 .

d.

. 8J

£

Lent 316 , Received 218 .

(«)

«.

,. 3 .

. 2 .,

d. • 5t

Remains to pay 238 .. 18 .. 10|

Proof 715

7i

(»)

(«)

(»)

^ (*)

£

». d.

£

s. d.

£

s.

d.

£ «.

d

87.

. 2 .. 10

3 ..

15 .. li

25.

.2..

b\

37.. 3.,

.4*

79 .

.3.. 71

1 ..

14.. 7

17.

.9..

8|

25.. 5.,

l5

(')

(«)

(»)

(10)

£

t. d

£

B. d.

£

«.

d.

£ «.

d.

321

.. 17.. U

59

.. 15.. 3i

71 .

,. 2

.. 4

527 . . 3 . .

5i

257

.. 14 .. 7

36

.. 17.. 2

19 .

. 13

.. 71

139 .. 5..

■ 7»

(II)

£ ». d.

Borrowed 25107 .. 15 .. 7

£

Lent 250 J 56

(IS)

375 .

. 5 .

. 5J

271

Paid 259 .

. 2 .

. 74

Received 359

at 359 .

. 13 .

. 41

at 475

diflercnt 523 .

. 17 .

. 3

several 527

times 274 .

. 15 .

. 7i

payments 272

325 .

. 13 .

. 5

150

d 6

13 .

. 7i

15 .

. 3

13 .

. 9*

15 .

. 'i\

16 .

. 5

0 .

. 0

Paid in all Remains to pay

— " SUBTRACTION.

TROY WEIGHT.

lb oz. dwt gr. (1) Bought 52 .. 1 .. 7 .. 2 Sold 39 , . 0 . . 15 . . 7

lb. OK. dwC gr. (S) 7.. 2.. 2.. 7 5.. 7.. 1 .. 5

Unsold

AVOIRDUPOIS WEIGHT.

lb.

oz.

dr.

cwt. qrs.

lb.

t

cwt qiB. lb.

»). 35 .

. 10 .

. 5

(S) 35.. 1 .

.21

(8) 21

.. 1 ..2..7

29 .

. 12 .

. 7

25 .. 1 .

. 10

9

.. 1 .. 3 .. 5

lb. 2

APOTHECARIES WEIGHT.

OE. dr. 8cr. . 2 .. 1 .. 0 . 5 .. 2 .. 1

lb. oz. dr. Kr. gr.

(«) 9 .. 7 .. 2 .. 1 .. 13

5 .. 7 .. 3 .. 1 .. 18

F1.E. qr. (I) 35 .. 2, 17.. 2

CLOTH MEASURE.

yd. qr. n.

(«) 71-.. 1 .. 2

3.. 2.. 1

E.E. qr. n.

(S) 35 ..2.. 1

14.. 3. .2

LONG MEASURE.

yds.

ft.

in.

bar.

(4) 107 .

. 2.

. 10.

. 1

78 .

. 2.

. 11 .

. 2

lea.

(2) 147

58

mL fiir. po. . 2.. 6.. 29 , 2.. 7.. 33

a. r. p.

(>) 175 .. 1 .. 27

59..0.. 27

LAND MEASURE.

a. r. p. («) 325 .. 2.. 1 279.. 3. .5^

36 8tTB7r. ACTION.

WINL MEASURE.

hhd. gal. qts. pt. tun. hhd. gal. qL

(I) 47 .. 47 .. i}.. 1 (8) 42.. 2.. 37.. 2

28 . . 59 . . 3 . . 0 17 . . 3 . . 49 . . 3

ALE AND BEER MEASURE.

A.B. fir. gal. B.B. fir. gal. hhd. gal. qto

(») 25.. 1.. 2 ('\ :n..2..1 (3) 27..27..1

21 . . 1 . . 5 25 . . 1 . . 7 12 . . 50 . . 2

DRY MEASURE.

qu.

bu.

P-

qu.

bii.

P-

ch.

bu.

P-

(»^

72 .

. 1 .

. 2

i^)

G5 .

. 2 .

. 1

P)

79 .

. 3 .

. 0

35 .

. 2 .

. 3 •

57 .

. 2 .

. 3

54 .

. 7.

. 1

TIME.

yrs. mo. w. da. ho. min.

(») 79 .. 8 .. 2.. 4 («) 24 .. 42 ., 45

23 . . 9 . . 3 . . & 19 . . 53 . . 47

THE APPLICATION.

1 . A man was bom in the year 1 723, what was his age in the year 1 781 1

An*. 58.

2. What is the difference between the age of a man bom in 1710, and an- other bom in 17G6 1

Arts. 56.

' 3. A Merchant had five d«'btor8, A, B. C, D, and E, who together owed hira £1156; B, C, D, and E, owed him £737. What was A.'sdebtl

Ans.£4\9.

4. When an estote of £300 per annum, is reduced, on the paying of Uxes te 12 score and £ 14 : 6 What is the tax 1

Ans. £i:y : 1 4.

COMPOUND MULTIPLICATION. 3^

5. What is the difference between £9154, and the amount of £754 added to

Ans. £3095.

6. A horse in his furniture is worth £37 : 5 ; out of it, 14 guineas ; how much docs the price of the furniture exceed tiiat of the horse 7

' Ans. £7 : 17.

7. A merchant, at his out-sstting 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 ; what is the neat balance at the twelve months' end!

^n*. £12212: 10:6.

%te]

8. A gentleman djing, left £45247 between two daughters, the younger to have 15 thousand, 15 hundred, and twice £15. What was the elder r's fortune 1

Ans. £2&7n.

9. A tradesman happening to fafl in busmess, called all his creditors to-

rther, and found he owed to A, £63: 7:6; to B, £105: l); to C, £34:5: i to D, £-28:16:5; to E, £14: 15 : 8; to F, £11 J : 9; and to G, £143 : 12 : 9. His creditors found the value of his stock to be £21^ : 6, and that he had owing to him, in good book debts, £112 : 8 : 3, In-sides £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 niu;-h ?

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 : 1 1 : 6; linen cloth, £112:14:8; tin, £115:10:5. My correspondent, at the same time, informs me, that he has shipped, acrreeably to my order, wines to the value of £250 : 15 ; fruit to the value of £51 : 12 : 0 ; figs, £19 : 17 : 6 ; oil, £19 : 12 : 4; and Spanish wool, to the value of £il5 : 15 : 6. I desire to know how the account stands between us, and who is the debtor 1

Ans. Dae 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, ud add the quotient to the next superior, after it is multiplied.

If the given quantity is above 12, multiply by any two numbers, which mul- tiplied together will make the same numlier ; but it no two numbers multiplied together will make the exact numl>er, then raultijih" the top line by as many. »» w wanting, adding it to the last product.

1)

38 COMPOUND MULTIPLICATION.

Proof. By Division.

. <')

. (')

{')

(«)

£ s. d.

£ «. d.

£ s. d.

£ «. d.

S5:12:7i

75:13:11

62 : 5 : 4t

57 : 2 - 41

2

3

4

5

71 : 5 : 2J

1. 18 yards of cloth, at 98. 6d. 2. 2C lb. of lea, at £1 : 2 : 6

per yard. 9 per lb. 8

9x2=lb 8X3+ =26

4:5:6 9:0:0

2 3

8:11:0 27:0:%

Top line X 2 = 2 : 5 : 0^

29 : ft : 0

3. 21 ells of Holland, at 7s. 8^(1. per ell.

Facit, £8:1: 1(4-

4. 35 firkins of butter, at 15s. 3^(1. per firkin.

Facit, £26 : 15 : 2i.

5. 15 lb. of nutmegs, at 7s. 23cl. per lb.

Facit, £27 : 2 : 2i.

6. 37 yards of tabby, at 98. 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 78. 5d. per gallon.

Facit, £50 : 1 : 3.

11. 74 ells of diaper, at Is. 4id. per elL

Facit, £5:1:9.

12. 6 dozen pair of gloves, at Is. lOd. per pair.

Facit, £6 : 12.

When the given quantity consists of J, \, or 3-

Rule. Divide the given price (or the price of one) by 4 for \, by 2 for J, and fcr f , first divide by 2 for J, then divide that quotient by 2 for t, add them to the proiuct, and their sum will bo the answer rrt^uircd.

COMPOUND MULTIPLICATION. ']9D

13. 25^ ells of holland, at 3 : 4^d. per ell.

5 5X5=25

16:

10^ 5

4:4 0:1

:4i-- :8i=

=25

4:6

:0|=

=25J

14. 75^ ells of diaper, at Is. 3d. per ell.

Facit,£4:14:4i

15. 19^ ells of damask, at 4s. 3d. per ell.

Facit, £4:2: 10^.

16. 35^ ells of dowlas, at Is. 4d. per ell.

Facit, £2:7:4.

17. 7^ cwt of Malaga raisins, at £1 : 1 : 6 per cwt.

Facit, £7:15: 10^.

18. 6J barrels of herrings, at £3 : 15 : 7 per barrel.

Facit, £24 : 11 : 3^.

19. 35^ cwt double refined sugar, at £4 : 15 : 6 per cwt.

Facit, £169 : 10 : 3.

20. 154^ cwt. of tobacco, at £4 : 17 : 10 per cwt.

* Facit, £755:15:3.

21. 117^ gallons of arrack, at 12s. 6d. per gallon.

Facit, £73 :5:7i.

22. 85| cwt. of cheese, at £1 : 7 :8 per cwt.

Facit, £118:12:5.

23. 29i lb. of fine hyson lea, at £1 : 3 : 6 per lb.

Facit, £34 : 7 : 4^.

24. 17| yards of superfine scarlet drab, at £1 : 3 : 6 per yard

Facit, £20:17: 1^.

25. 37^ yards of rich brocaded silk, at 12s. 4d. per yard.

Facit, £23 : 2 : 6.

26. 66| cwt, of sugar, at £2 : 18 : 7 per cwt.

Facit, £166:4 :7i.

27. 96^ cwt. of currants, at £2 : 15 : 6 per cwt.

Facit, £267 : 15 : 9.

28. 45^ lb. of Belladine silk, at 18s. Gd. per lb.

Facit, £42 : 6 : 4^.

29. 87f bushels of wheat, at 4s. 3d. per bushel.

Facit, £18:12: lU. D2

40 COJfPOUKD MULTIPLICATION.

30. 120i; cwt. of hops, at £4 : 7 : 6 per cwt.

Facit, £528 : 5 ; 7^.

31. 407 yards of cloth, at 3s, Qld. per yard.

Facit, £77 : 3 : 2J.

32. 729 ells of cloth, at 78. 7id. per ell.

Facit, £277 : 3 : 5^.

33. 2068 yards of lace, at Os. 5Jd. per yard.

Facit, £977 : 19 : 10.

THE AI'PLICATION.

1. What sum of money must be divided amongst 18 men, lo that each man may receive £14 : 6 : 6^ ?

Ans. £258 : 0 : 9.

2. A privateer of 250 men took a prize, which amounted to £125 : 15 : 6 to each man ; what was the value of the prize ?

Ans. £31443 : 15 0

3. What is the difference between six dozen dozen, and half a dozen dozen ; and what is their sum and product?

Ans. 793 diff. Sum 936, Product 62208.

4. What difference is there between twice eight and fifty, and twice fifty-eight, and what is their product?

.4ns. 50 diff 76o6 Product.

5. There are two numbers, the greater of them is 37 times 45, and their difference 19 times 4 ; their ^m and product are required ? Ans. 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 ?

^715. 31104 Product, 5184 Square of tlieir 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. 144800.

8. What sum did that gentleman receive in doM'ry with his wife, whose fortune was her wedding suit; her petticoat having two rows of furbelowH, each furbelow 87 quills, and in each quill 21 guineas ? Ans. £3836 : 14 : 0.

9. A mercliant had £19118 to begin trade with ; for 5 years together he cleared £1086 a year ; the next4 years he made good £2715 : 10 : 6 a year ; but the last 3 years he was in trade, he had the misfortune to lose, one year with another, £475: 4 : 6» ■sear ; wliat was his real fortune at 12 years' end?

Ans. £33984 : 8 : 6..

COMPOUNB 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: 6, and spends daily £1 : 12: G. I desire to know what is his an nual income? ^ns. £887 : 15: 0.

12. A. tradesman gave his daughter, as a marriage portion, a acrutoire, 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 fortune ? Ans. £3744.

13. Admitting that I pay eight guineas and half-a-crown for a quarter's rent, and am allowed quarterly 15s. for repairs, what does my apartment cost me annually, and how much in seven years ? Ans. In 1 year, £31 : 2. In 7, £217 : 14.

14. A robbery being committed on the highway, an assessment was made on a neighbouring Hundred for the sum of £386 : 15 : G, 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 public charity he bequeathed £.572 : 10 ; to each of his four ne- phews, £750 : 10 ; to each of his four nieces, £375 : 12 : 6 ; to thirty poor housekeepers, ten guineas each, and 150 guineas to his executor. What sum must he have been possessed of at the time of his death, to answer all these legacies ?

^715. £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. 195446.

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. (') Multiply 107 yards, 3 qrs. 2 nails, by 10. (♦) Multiply 33 ale bar. 2 firk. 3 gal. by'U. (5) Multiply 27 beer bar.. 2 firk. 4 gal. 3 qts. by 13. (*) Multiply 110 miles, 6 fiir. 26 poles, by 12. ' d3

42 DITISION.

DIVISION OF SEVERAL DENOMINATIONSv

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.

(')

(')

(•)

n

£ s. d.

£ s. d.

£ s. d.

£ s. d.

2)25:2: 4(

3)37 : 7 : 7(

4)57 : 5 : 7(.

5)52 : 7 : 0(

12: 11 : 2

(») Divide £1407 : 17 : 7 by 243. C) Divide £700791 : 14 : 4 by 1794. C) Divide £490981 : 3 : 7i by 31715. («) 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 : frj-.

2. The clothing of 35 charity boys came to £57 : 3 : 7, what is the expense of each? Ans. £1 : 12 : 8.

3. If I gave £37 : 6 : 4^ for nine pieces of cloth, what did f give per piece ? Ans. £4:2:11.

4. If 20 cvvt. of tobacco came to £27 : 5 : 4^, at what rate is that per cwt. ? Ans. £1 : 7 : 3.

5. What is the value of one hogshead of beer, when 120 are sold for £154 : 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 : 8J.

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 are 2545 bullocks to be divided amongst 509 men, 1 desire to know h'ow 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. 43

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 ?

Ans. Each contributed 8s., each share £160.

12. A trader cleared £1156, equally, in 17 years, how much did he lay by in a year? Ans. £68.

13. Another cleared £2305 in 7^ years, what was his yearly increase of fortune ?

Atis. £374.

14. What number added to the 43d part of 4429, will raise it to 240? ^ns. 137.

15. Divide 20s. between A, B, and C, in such sort that A may have 2s. less than B, and C 2s more than B ?

Ans. A 4s. 8d., B 6s. 8d., C 8s. 8d.

16. If there are 1000 men to a regiment, and but 50 officers,, how many private men are there to one officer?

Ans. 19.

17. What number is that, which multiplied by 7847, vnW make the product 3013248 ? Ans. 384.

18. The quotient is 1083, the divisor 28604, what was the di- vidend if the remainder came out 1788?

Ans. 30979920.

19. An army, consisting of 20,000 men, took and plunderea a city of £12,000. What was each man's share, the whole being equally divided among them ?

Ans. 12$.

20. My purse and money, said Dick to Harry, are worth 12s. 8d., but the money is worth seven times the purse. What did the purse contain ? Ans. lis. Id.

21. A. merchant bought two lots of tobaccs, 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 : €. I desire to know their respective weights and value.

Ans. Less weight, 5 cwt. 2 qrs. 15 lb. Price^ £53 : 10. Greater weight, 7 cwL 1. qr. Price, £61 : 5 : 6.

22. Divide 1000 crowns in such a manner between A, B, and C, that A mav receive 129 more tlian B, and B 178 less than C.

^Tis. A360, B231,C409.

BILLS OF PARCELS.

EXAMPLES OF WEIGHTS AND MEASURE*.

1. Divide 83 lb. 6 ozv 10 dwts. 17 gr. by 8.

2. Divide 29 tons, 17. cut. 0 qrs. 18 lb. by 9. '.i. Divide 1 14 yards, 3 qrs. 2 nails, by 10.

4. Divide 1017 ntiiles, G fur, 38 poles, by 11.

5. Divide 2019 acres, 3 roods, 29 poles, by 26.

0. Divide 117 years, 7 months, 3 weeks, 6 days, 11 liours» 27 minutes, by 37.

BILLS OF PARCELS.

hosiers'..

(>) Mr. John Thomas,

Bought of Samuel Green. May 1, 18

s. d.

8 Pair of worsted stockings, at.. .4 : 6 per pair £

5 Pair of thread ditti 3 : 2

3 Pair of black silk ditto 14 : 0

(> Pair of milled hose , 4 : 2

4 Pair of cotton ditto 7 : 6

2 Yards of fine flannel 1 ; 8 per yard

£7: 12

MERCERS.

(•) Mr. Isaac Grant,

Bought of John Sims. May 3i 18-

.«?. J.

15 Yards of satin at.. .9 : (i per yard£

18 Yards of flowered silk 17 : 4 .'.

12 Yards of rich brocade 19 : 8

10 Yards of sarsenet 3 : 2

13 Yards of Genoa velvet 27 : 6

23 Yards of lutestring 6 : 3

£62 : 2 : 6

BU.LS OF PARCELS. 43

LINEN drapers'.

(') Mr. Simon Surety,

Douffht of Josiah Short. June 4, 18

s. d.

4 Yards of cambric at.. .12 : 6 per yard £

V2 Yards of muslin 8 : 3

J5 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 : IJ

£17:4:6J

MILLINERS.

(•) Mrs. Bright,

Bought of Lucy Brown. June 14, 18

£ jf. d.

18 Yards of fine lace at...O : 12 : 3 per yard £

5 Pair of fine kid gloves... 0 : 2:2 per pair

12 Fans of French mounts 0 : 3:6 each

2 Fine lace tippets....... 3 : 3:0

4 Dozen Irish lamb.... 0 : 1:3 per pair.

<i Sets of knots... , 0 : 2:6 per set..

£23 : 14 : 4

WOOLLEN DRAPERS\

(*) Mr. Thomas Sage,

Bought of Ellis Smith. June 20, 18

£ s. d.

17 Yards of fine serge at...O : 3:9 per yard £

18 Yards of drugget 0 : 9:0

15 Yards of superfine scarlet 1 : 2:0

10 Yards of black 0 : 18 : 0

25 Yards of shalloon 0 : 1:9

17 Yards of drab .0 : J7 : 6

£59 : 5 : 0

48 BILLS OF PARCELS.

LEATHER-SELLERS*.

(•) Mr. Giles Harvis,

Bought of Abel Smith. July I 18

s. d.

27 Calfskins at....3 : 9 per skin £

75 Sheep ditto 1 : 7

36 Coloured ditto 1 : S

15 Buck ditto 11 : 6

17 Russia Hides 10 : 7

120 Lamb Skins 1 : 2^

£38: 17:5

GROCERS

{'') Mr. Richard Groves,

Bought of Francis Elliot. July 5, 18

s. d.

25 lb. of lump sugar at...O : 6t per lb. £

2 loaves of double refined, ) 0 • Hi

weight 15 lb. \ *

14 lb. of rice 0 : 3

28 lb. of Malaga raisins 0 : 5

15 lb. of currants 0: 5^

7 lb. of black pepper 1 : 10

£3 : 2 : 9i

CHEESEMONGERS'.

(') Mr. Charles Cross,

Bought of Samuel Grant. July 6, 18

s. d.

8 lb. of Cambridge butter at...0 : 6 per lb. £

17 lb. of new cheese 0 : 4

^ Fir. of butter, wt. 28 lb 0 : 5^

5 Cheshire cheeses, 127 lb 0 : 4

2 Warwickshire ditto, 151b 0:3

12 lb. of cream cheese 0 :-6

£3:14:7

REOrCTIO.V.

<7

CORN-CI i^isi>LER3

(') Mr. Abraham Doyley,

Bouffht of Isaac Jones.

d.

July 20, 13

Tares, 19 bushels at...l : 10 per bushel £

Pease, IS bushels 3 : 9|

Malt, 7 quarters 25 : 0 per quarter

Hops, 15 lb 1 : 5 per lb

Oats, 6 qrs 2 : 4 per bushel

Beans, 12 bushels ......4 : 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.

£ s. d.

Guinea 1: 1:0

Half ditto 0: 10:6

Sovereign 1 : 0 :0

Half ditto 0:10:0

Crown 0: 5:0

Half ditto 0: 2:6

ShilUng 0: 1:0

Note. There are several pieces which speak their own ralue ; such as sixpence, fourpence, threepence, twopence, penny, halfpenny, farthing.

1 . In £8, how many shillings and pence ? 20

160 shillings. 12

1920

9. REDUCTION.

U. in £12, how many shillings, pence, and farthings?

Alls. 240s. 28S0d. 11520 far

3. In 311520 farthings, how many pounds ?

Ans. £324 : 10.

4. How many farthings aro there in 21 guineas ?

Ans. 21168.

5. In £17 : 5 : 3i, how many farthings ? Ans. 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. ()84iJ farthings.

10. In 52 crowns, as many half-crowns, shillings, and pencf , how many farthings 1 Ans. 21424.

11. How many pence, shillings, and pounds, are there in 17280 farthings ? Ans. 4320d. 3()0s. £18.

12. How many guineas in 21168 farthings ?

Ans. 21 guineas.

13. In 16573 farthings, how many pounds ?

^715. £17:5: 3i i4. 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. 10. In 21424 farthings, how many crowns, half-crowns, shil- lings, and pence, and of each an equal number ? Ans. 62.

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.

nEDITCTION. 49

23. If 103 guineas and seven shillings are to be divided amongst seven men, how many pounds sterling is that each?

A lis. £15 : 10.

24. A certain person had 25 purses, and in each purse 12 gui- neas, a crown, ami a moidore, how many pounds sterling had he in all ?

■4ns. £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 tt) have 2s. (id. — ^ to poor boys, each to have Is. — J- to poor girls, each to have 9(1. 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. 12960.

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. 225S>3.

29. In 8 ingots of silver, each weighing 7 lb. 4 oz. 17 dwts. 15 gr. how many ounces, petmvweighls, and grains?

Ans. in oz. I422I dwts. 341304 gr.

30. How many ingots, of 7 lb. 4 oz. 17 dwts. 15gr. each, arc there in 341301 grains? Ans. 9 ingots.

31. Bought 7 ingots of silver, each containing 23 lb. 5 oz. 7 dwts. how many grains ? Ans. 945336.

32. A gentleman sent a tankard to his goldsmith, that weighed 50 oz. 8 dwts. and ordered him to make it uito spoons, each to weigh 2 oz. 16 dwts. how many had he ?

Ans. 18.

33. A gentleman delivered to a goldsmith 137 oz. 6 dwts. 9 gr. of silver, and ordered him to make it into tankards of 17 oz. 15 dwts. 10 gr. each ; spoons of 21 oz. 11 dwts. 13 gr. per doz, salts 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, •ad 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.

K

fit

REDWCTION.

AVOIRDUPOIS WEIGHT.

Note. — There arc several sorta of silk which are weighed by a great pound of 34 oz. others by the common pound of l6oz. ; therefore,

To bring great pounds into common, mQlti|Jy by 3, and divide by 2, or add one half.

To bring small pounds into great, multiply by 2, and divide by 3, or subtrac one third.

TTiings bought and sold by the TaU.

12 Pieces or things make 1 Dozen.

12 Dozen 1 Gross.

12 Gross, or 144 doz 1 Great Gross.

24 Sheets 1 auire.

20 auires I Ream.

2 Reams I Bundle.

1 Dozen of Parchment.. 12 Skins. 12 Skins 1 RoU.

34. In 14769 ounces how many cwt, 1

Ans. 8 cwt. 0 qr. 27 lb. 1 oz.

35. Reduce 8 cwt. 0 qrs. 27 lb. I oz. into quarters, pounds, and ounces.

Ans. 32 qrs. 923 lb. 14769 oz.

36. Bought 92 bags of hops, each 2 cwt. 1 qr. 14 lb. and another of 150 lb. how many cwt. in the whole ■?

Ans. 77 cwt. 1 qr. 10 lb,

37. In 34 ton, 17 cwt. 1 qr. 19 lb. how many pounds t

^ An*. 78111 ttx

38. In 547 great pounds, how many common pounds 1

Ans. 820 lb. 8 oz.

39. In 27 cwt. of raisins, how many parcels of 18 lb. each 1

Ans. 168.

40. In 9 cwt. 2 qrs. 14 lb. of indigo, how many pounds 1 , ^„ ..

Ans, 1078 lb.

41. Bought 27 bags of hops, each 2 cwt I qr. 15 lb. and one bag of 137 lb.

how many cwt. in the whole 1 , ^, _ « ,« ,w

Ans. 65 cwt 2 qrs. 10 lb.

42. How many pounds in 27 hogsheads of tobacco, each weighing neat 8| •'*^- ■ Ans. 26460.

43. In 552 common pounds of silk, how many great pounds ^ . ggg

44. How many parcels of sugar of 16 lb. 2 oz. are there in 16 «vt. 1 qr.

Ans. 113 parcels, and 12 lb, 14 oz. o%er.

REDtJCTION. 51

APOTHECARIES' WEIGHT.

46. In 27 lb. 7 oz. 2 dr. 1 scr. how many grains T

Ans. 159022.

46. How many lb. oz. dr. scr. are there in 159022 grains ?

Ans. 27 lb. 7 oz. 2 dr. 1 scr.

CLOTH MEASURE.

47. In 27 yards, how many nails ? Ans. 432.

48. In 75 English ells, how many yards ?

Ans. 93 yards, 3 qrs.

49. In 93| yards, how many English ells ? Ans. 75.

50. In 24 pieces, each containing 32 Flemish ells, how many English ells ? Ans. 460 English ells, 4 qrs.

51. In 17 pieces of cloth, each 27 Flemish ells, how many yards ? Ans. 344 yards, 1 qr.

52. Bought 27 pieces of English stuff, each 27 ells, how many yards? Ans. 911 yards, 1 qr.

53. In 911^ yards, how many English ells ?

Ans. 729.

54. In 12 bales of cloth, each 25 pieces, each 15 English ells, how many yards ? Ans. 5625

LONG MEASURE.

55. In 57 miles, how many furlongs and poles ?

Ans. 456 furlongs, 18240 poles.

56. In 7 miles, how many feet, inches, and barley-corns ?

Ans. 36960 ft. 443520 in. 1330560' b. corns.

57. In 18240 poles, how many furlongs and miles?

Ans. 456 furlongs, 57 miles.

58. In 72 leagues, how many yards ? Ans. 380160.

59. In 380160 yards, how many miles and leagues ?

Ans. 216 miles, 72 leagues

60. If from London to York be accounted 50 leagues, I de- mand how many miles, j^ards, feet, inches, and barley-corns ?

Ans. 150 miles, 264000 yards, 792000 feet, 9504000 inches, 28512000 barley-corns.

61. How often will the wheel of a coach, that is 17 feet in circumference, turn in 100 miles ?

Ans. 31058^ times round. s2

SS REDVCTIOHr.

62. How many barley-corns will reach round the world, the circumference of which is 360 degrees, «iirh <l(grro<)9 miles and a half? A?is. 4755801000 barley-corns.

LAND MEASURE,

63. In 27 acres, how many roods and perches ?

Arts. 108 roods, 4320 perches.

64. In 4320 perches, how many acres? ,4^."?. 27.

65. A person having a piece of ground, containing 37 acres, 1 pole, has a mind to dispose of 15 acres to A. 1 desire to know how many perches he will have left?

Ans. 3521 CO. There are four fields to be divided into shares of ? .5 perches each ; the first field containing 5 acres ; the second, 4 acres, 2 poles ; the third, 7 acres, 3 roods ; and the fourth, 2 acres, I rood. 1 de;»ice tp know ho^jiy^^ many shares are contained therein ?

-47/5. 40 shares, 4J^ perches rem.

WINE MEASURE.

97^ Ooitght 5 tuns of port wine, how many gallons and pints f A71S. 12G0 gallons, 10080 pints.

68. In 10080 pints, how many tuns ? Ans. 5 tuns.

69. In 5896 gallons of Canary, how many pipes and hogs- beads, and of each an equal number?

Ans. 31 cf each, 37 gallons over-

70. A gentleman ordered his builei lo bottle oft' I of a pipe of French wine into quarts, and the rest into pints, I desire to ka(j^)^^|[^Vf,0^ay ilozqn of each he had ?

>ln5. 28 dozeii of eacli.

ALE AND BEER MEASURE.

7)U Hn ,46 barrels of beer, how many pints .

71^5.1.3249* 7«5» IlL ,10. barrels of ale, how many gallons and quarts?

An.<i. 321 gals. 1280 qts. "*., ?n 72^hog3head8 of ale, how many barrels ?

Ans. 108. M* In 108 barrels of ale, how many hogsheads ?

Ans. 72.

ClXCCE RULE OF THREE DIRECT. 8S

DRY MEASURE.

75. In 120 quarters of wheat, how many bushels, pecks, g;al- Ions, and quarts ?

Ans. mo bushels, 3840 pecks, 7080 gallons, 30720 qti.

76. In 30720 quarts of corn, how many quarters ?

Ans. 120

77. In 20 chaldrons of coals, how many pecks ?

^715. 288a

78. In 273 lasts of corn, how many pecks ?

Ans. 8736a

TIME.

79. In 72015 hours, how many weeks ?

Ans. 42S weeks, 4 day«, 15 hours.

80. How many days is it since the birth of our Saviour, to Christmas, 1794? Ans. CKi'^2T^bl.

81. Stowe writes, London was !)uilt 1108 yoars before our Saviour's birth, how many hours is it since to ('hristmas, 1794 ?

Avs. 2."vl38*»32 hours.

82. From November 17, 1738, to September 12, 1739, ho\r many days ? Ans. 299.

83. From July 18, 1749, to December 27 of the same year how many days? Ans. 1(52.

84. From July 18, 1723, to April 18, 17.50, how many year« uid days? A71S. 2() years, 9770A days,

reckoning 365 days 6 hours a year.

THE SINGLE RULE OF THREE DIRECT.

Tcacheth by three numbers {riven to find out a fourth, in sucli proportion to the third, as the second is to the first.

Rule. First stale the question, tliut is, place the numbers in Mich order, that the first and third be of one kind, and the second the same as the number recuired ; then brin«r the first and third numbers into one name, and the .second into the lowest term men- tioned. Multiply the second and third numbers together, and

e3

54 SINGLE RULE OF THREE DIRECT.

divide the product by the first, and the quotient will be the an> SNver to the question in tho same denomination you left the second number in.

EXAMPLES.

1. If 1 lb. of sugar cost 4 J, what cost 54 lb. 1 1 : 4i : : 54 4 18

— iln«. £1:0:S.

18 4)972

12)243

208. 3J.

2. If a gallon of beer cost lOd., what is that per barrel 1

Ans. £1 : 10.

3. If a pair of shoes cost 4s. 6d., what will 12 dozen come to 1

An». £32 : 8.

4. If one yard of cloth cost 15s. 6d., what will 32 yards cost at the same rate! Ans. £2i: 16.

5. If 32 yards of cloth cost £24 : 16, what is the value of a yard 1

Ans. I5s. 6d.

6. If I give £4 : 18 for 1 cwt of sugar, at what rate did I buy it per lb. 1

Atm. lOjd.

7. If I buy 20 pieces of cloth, each 20 ells, for 12s. 6d. per ell, what is the value of the whole a Ans. £250.

8. What will 25 cwt 3 qrs. 14 lb. of tobacco come to, at I5jd. per lb. 1

Ans. £187 : 3 : 3.

9. Bought 27i yards of muslin, at 6s. 9id. per yard, what does it amount to 1 Ans. £9 : 5 : OJ, 3 rem.

10. Bought 17 cwt. 1 qr. 14 lb. of iron, at 3|d. per lb., what does it come lo*? ilrw. £20 : 7 : Oi.

11. If coffee is sold for 5id. per ounce, what must be given for 2 cwt. 1

Ans. £82 : 2 : 8.

12. How many yards of cloth may be bought for £21 : 11 : IJ, when 3) yards cost £2 : 14 : 31 Ans. 27 yards, 3 qrs. 1 nail, 84 rem.

13. If 1 cwt. of Cheshire cheese cost £1 : 14 : 8, what must I give for 3 J Ibl Ans. Is. Id.

14. Bought I cwt. 24 lb. 8 oz. of old lead, at 98. per cwt., what does it come tol Ans. lOa. md. 112 rem.

SINGLE RULE OF THREE DIKECT. Od

'5. (i r 'intleman's income is £500 a year, and he spends 19*. 4d. per day, Lu»v 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 ells can I buy for £283 : 17 : 6 at the same rate ]

Ans. 504 Fl. ells, 2 qrs.

17. If b\W Flemish ells, 2 quarters, cost £283 : 17 : 6, at what rate did 1 pa? for 14 yaids I

■ Ans. 10s. lOd.

18. Gave £1 : 1 : 8 for 3 lb. of coffee, what must be gjven for 29 lb. 4 oz 1

^ns. £10:11:3.

19. If one Ejiglish ell 2 qrs. cost 4s. 7d.- what will 39i yards co«t at the same ratel

Ans. £5 : 3 : 5J, 5 rem.

20. If one ounce of gold is worth £5:4: 2, what is the worth of one grain 1

Ans. 2jd. 20 rem.

21 . If 14 yards of broad cloth cost £9 : 12, what is the purchase of 75 yeards 1

Ans. £51 : 8 : 6^, 6 rem.

22. If 27 yards of Holland cost £5 : 12 : 6, how many ells English can I buy for £1001 il7j«. 384.

23. If 1 cwt. cost £12 : 12 : 6, what must I give for 14 cvrt. 1 qr. 19 lb. 1

Ans. £182:0: llj, 8 rem.

24. Bought 7yards of cloth for 173. 8d. what must be given for 5 pieces, each containing 27j yards 1

Ans. £17 : 7 : 0}, 2 rem.

25. If 7 OK. 11 dwts. of gold be worth £35, what is the value of 14 lb. 9 oz. 12 dwts. 16 gr. at the same rate 1

Ans. £823 : 9 : 3}, 552 rem.

26. A draper bought 420 yards of broad cloth, at the rate of l-4s. lOfd. per cil Enfrlish, 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 £2:8:6 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 parcel 10 pieces, and each piece 26 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. £665»).

30. If 24 lb. of raisins cost 6s. 6d. what will 18 frails cost, each weighing neat 3 qrs. 18 lb. ?

Ans. £24 : 17 : 3.

31. If 1 oz. of silver be worth Ss. what is the price of 14 ingots, each weigh- ina 7 lb. 5 oz. 10 dwts. 1 Ans. £313 : 5.

32. What is the price of a pack of wool, weighing 2 cwt. 1 qr. 19 lb. at 8s. (y\. per stone 1

Ans. £8:4: 6i, 10 rem.

33. Bought 59 cwt. 2 qrs. 2^1 lb. of tobacco, at £2 : 17 : 4 per cwt. ; what doea it come to i

ilns. £171 :3: 7i 80 rem.

80 RULE OF THREE INTERSE.

34. Bought 171 tons of lead, at £14 per ton; paid cnrriage and other iiicidcnl charges, £4 : 10. I require the value of the lead, and wliut it stands me in per ii). ?

An.s. £2:}i)N : lu vahie ; l.^d. 432 rem. per lb.

35. If a pair of stockings cost 10 groats, how many dozen mat I buy for £43 : 6 ?

Ans. 21 dozen, 7J pair. 30. Bought 27 dozen 5 lb. of candles, after the rate of 17d. per 3 lb. what did they cost me ?

Ans. £7 : 15 : 4J, 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 sanio price? Alls. £lo7I : 14: 5i.

3S. If my horse stands me in 0.]d. per day keeping, what will be the charge of 1 1 horses for the year ?

Ans. £158 : 18 : OJ.

39. A factor bought 80 pieces of stud', which cost him £517: 19 : 4, at 4a. lOd. per yard ; I demand how many yards there were, and how many ells English in a piece ?

Ans. 21431 yards, 5() rem. and 19 ells, 4 quarters, 2 nails, (ii rem. in u piece.

40. A gentleman hath an annuity of £M{)f) : 17 per annum. I desire to know how much he may spend daily, that at the year'a end he may lay up 200 guineas, and give to the p()t)r quarterly 40 rnoidores? Ans. £1 : 14 : 8, 170 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, ami 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 than Uie second.

Rui.r.. — Multiply the first and second terms together, and di- vide the prodtict by the third, the quoti«Mit will bear such propor- tion to the second as the first does to the tliird.

RULE OF THREE IXTERSB. S7

EXAMPLES.

1. If 8 men can do a piece of work in 12 days, how many days can IG men perform the same in ? Ans. 0 days.

8 . 12 . . 10 . 6

8

10)0(5(0 days.

2. If 54 men can build a house in 90 days, how many can do (he same in 50 days 1

Ans. 97^ men.

3. If, when a peck of wheat is sold for 2s., the penny loaf weighs 8 oz., how much must it weigli when the peck is worth but is. Cd. ?

Ans. lOf oz.

4. How many pieces of money, of 20s. value, are equal to 240 pieces of 12s. each ? Ans. 144.

5. How many yards, of three quarters wide, arc equal in mea- sure to 30 yards, of 5 quarters wide? Ans. 50.

0. If I lend my friend £200 for 12 months, how long ought he to lend me £150, to requite my kindness ?

Ans. 10 months.

7. If for 24s, I have 1200 lb. carried 30 miles, how many pounds can I have carried 24 miles for the same money ?

Ans. ISOO lb.

8. If 109 workmen finish a piece of work in 12 days, how ■nany are sufBcient 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 0 months I

Ans. 500.

10. If £20 worth of wine is sufficient to serve an ordinary of 100 men, when the tun is sold for £30, how many will £20 worth suffice, when the tun is sold but for £24 ?

Ans. 125.

11. A courier makes a journey in 24 days, when the day ie but 12 hours long, how many days will he be going the samt journey, when the day is 10 oours long?

Ans. 18 day*.

58 DOUBLE nVLli OF TIIKEE.

VZ. How much plush is suHicient for a cloak, which has in it 4 yards, of 7 quarters wide, of siulF, for the lining, the plush being but 3 quarters wide ?

Ans. 9^ yards.

13. If 14 piorteers m|^ke« trench in 18 days, how many days will 34 men take to do Ihe same 1

Ans. 7 days, 4 hours, 56 min. -j^, at 12 hours for a day.

14. Borrowed of my friend £04 for 8 months, and he had oc- casion another time to borrow of me for 12 months, how much must I lend him to requite his former kindness to me ?

^715. £42 : 13 : 4. 15." A regiment of soldiers, consisting of 1000 men, are to have new coats, each coat to contain 2^ yards of cloth, 5 quarters wide, and to be lined with shalloon of 3 quarters wide ; I demand how many yards of shalloon will line them ?

Ans. 4160 yards, 2 qrs. 2 nails. 2 rem

THE DOUBLE RULE OF THREE.

Is so called because it is composed of 5 numbers given to find a 6th, which, if the proportion is direct, must bear such a proportion to the 4lh and 5th, as tlie 3(1 bears to the 1st and 2d. But if in- verse, the 6th number must bear such proportion to the 4th and 5th, as the 1st bears to the 2d and 3d. The three first terms are a supposition; the two last, a demand.

Rule 1. Let the principal cause of loss or gain, interest or decrease, action or passion, be put in the first place.

2. Let that which betokcneth 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 their 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. Hut,

5. If the Wank falls under the first or second term, multiply the third and fourth terms for a divisor, and the other three foi the dividend, and the quotient will be the answer.

Proof. By two single rules of three.

DOUBLE RULE 07 THBKE. 99

EXAMPLES.

1. If H horses eat 56 bushels of oats in 16 days, how many bushels will )» eufiicient for 20 horses for 24 days 1

By two antrle rules. "^ or in one stating, worked thus :

hor. bu. Iior. bu. hor. days. bu.

1. As 14 . 56 . . 20 . 80 V 14 . 16 . 56 56 X 20 X 24

days. bu. days. bu. 20.24.— =120

2 As 16 . 80 .. 24 . 120 j 14X16

2. If 8 men in 14 days can mow 112 acres of grass, how many men must there be to mow 2000 acres in 10 days'?

acres, days, acres, days.

1. As 112. 14.. 2000. 250

days. men. days. men.

2. As 250 . 8 . . 10 . 200

men. days, acres. 8 . 14 . 112.8 X 14X2000

^=200

— . 10 . 2000 112X10

3. If £100 in 12 montlis gain £6 interest, how much will £75 gain in 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 the carriajje of 7 cwt. 3 qrs. 14 lb. for 50 miles 1

^ .Irw. 1 : 16 : 9.

5. If a regiment of soldiers, consisting of 136 men, consume 351 quarters of wheat in 108 days, how many quarters of wheat will 11232 soldiers consume in 56 days 1

Ans. 15031 qrs. 854 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 Ai7S. 480.

7. If 40s. will pay 8 men for 5 days' woA, how much will pay 32 men for 24 days' work 1 Ans. £38 : 8.

8. If £ 190 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. in 56 days 1

Ans. 11268.

10. If a family consisting of 7 persons, drink out 2 kilderkins of beer in 12 lays, how many kilderkins will another family of 14 persons drink out in 8 daysl ilns. 2 kil. 12 gal.

11. If the carriage of 60 cwt 20 miles, cost £14 : 10, what weight can I have carried 30 miles for £5 : 8 : 9, at the same rate of carriage 1

Ana. 15 cwt.

12. If 2 horses eat 8 bushels of oats in 16 days, how many horses will eat np 3000 quarters in 24 days 1

Ans. 4000.

13. If £100 in 12 months gain £7 interest, what is the interest of £571 fijr 6 years'?

Ans. £239:16:41, 20 ren*.

60

PRACTICE.

14. If I pay 10s. for the carriage of 2 tons 0 miles, whatmuBk I pay for the carriage of 12 tons, 17 cwt. 17 miles?

Alls. £9 : 2 . OJ.

PRACTICE,

Is so called from the general use thereof by all persons concern ed in trade and business.

Ail questions in this rule are performed by taking aliquot, oi even parts, by which means many tedious reductious are avoided , the table of which is as follows : —

Of a Pound.

s. d.

10:0...is...^

6 : 8 i

6 : 0 i

4 : 0 i

3 : 4 i

2 : 6 i

2 : 0 .^

1 : 8 -A

Of a shilling. d

6 is ^

4 i

3 i

2 i

n i

1 A

Of a Ton. cwt.

10 is

5

4

2i

2

Of a Hundred.

qr3. lb.

2 or 56 is ^

1 or 2'< }

14 i

Of a Quarter.

14 lb i

7 i

4 *

3i 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 b« the answer.

(■*-) i is i)5704 lb. at i

12)1426 2|0)11|8: 10 Facit,£5: 18 : 10

(')7695at^ Facit, £16 :0:7i

(3) .5470 at ^ Facit, £11:7:11

(*)6547at| Facit, £20 : 9 : 2i

(«)4573at 5 Facit, £14 : 5 : OJ

Rule 2. When the price is less then a shilling, take the ali- quot part or purls that are in a shilling, add them together, and divide by 20, as before

PKACTICC

61

(«)is-iV7547atld,

2|0)(52i8: 11

Facit, £31 : 8 : 1 1

(«)lisi^,375!atl|d

I is i 312 : 7

78: If

210)3910 : 8^

Facit, £19 :10.8a.

(') 543^5 at Hd. Facit, £:339 : 10 : 7^

(*)6254at l}(\. Facit, £45 : 12 : 0^.

(5)2351 at 2d. Facit, £19 : 11 : 10.

(«) 7210at2|:l Facit, £07:11 : 10^.

{^)2710at2^d. Facit, £28 : 4 : 7.

(8)3250 at 2id. Facit, £37 : 4 : 9^.

(9) 2715 at 3d. Facit, £33 : 18 : 9.

("») 70B2at3|d. Facit, £95 : 12 : 7^.

(")2147at3|d. Facit, £31 : G : 2^.

(•'')7000Ht3H Facit, £109 : 7 : G.

('')3257at4d. Facit, £54 : 5 : 9.

('♦)2056at4^d. Facit, £30 : 8 : 2.

(•«)3752at4^d. Facit, £70 : 7 : 0.

('«)2107at4|d. Facit, £41: 14:0^.

('^)32I0at5d. Facit, £00 : 17 : 6.

("')2715at5|d. Facit, £59 : 7 : Of

(")3120at5.'d. Facit, £71 : 10 : 0.

(^">7521 ato^d. Fttcit,£180:3:9|.

{•' ') .3271 at 6(h Facit, £81 : 15:6.

(^•^)79I4atO|d. Facit,£206:l:10^.

(^3) 3250atG^d. Facit, £8:^ : U : 5.

(•^<)2708atG5d. Facit, £7G : 3 : 3.

(«'»)3271 at7d. Facit, £95 : 8 : 1.

(•^«) 32.54 at 7id. Facit. £98 : 5 : 1 1 i

(*^)270l !it 7 d. Facit, £'H1 : 8 : U- F

(5^8) 3714 at 7ad. Facit,£U9:I8:7|.

(«9)27l0at8d. Facit, £90 : 6 : 8.

(3«)3514at84d. Facit,£120:15:10^.

(3')2759at8^d. Facit, £97: 14 :3i.

(3^)98-2at85d. Facit, £359 : 8 : 4.

('3)5272 at 9d Facit, £197:14:0.

(3^)6325 at 94d. Facit, £213 : 15 : 6i.

(•«5)7924at9^d. Facit, £313: 13:3.

(*«)2150at9|d. Facit, £87 : 6 : 10^.

('■) (5.325 at lOd. Facit,£2G3:10:10.

(3 8)5724 at lOjd. Facit, £244 : 9 : 3.

(3 9)G327atl0^d. Facit, £270: 4:3^.

(<»).3254atlO.',d. Facit, £142 : 7 : a

{* ')729l atlO^ Facit, £.326 : 1 1 : 6^.

(<'^)325Gat lid. Facit, £149:4 :a

62

PUACTICE.

(««) 7254 at Hid. Facit, £340 : 0 : TJ.

(♦♦)3754atllid. Facit, £179 : 17 : 7.

(*8)7972atll|d. Facit, 390 ; 5 : 11.

Rule 3. When the price is more than one shilling, and less than two, take the part or parts, Avith so much of the given price as is more than a shilling, which add to the given quantity, and divide by 20, it will give the answer.

(»)i:iL2106atl2id. 43 : lOi

210)21419 : 10^ Facit, £107: 9: 10^.

(5) 3215 at Is. Ud. Facit, £177:9: lOi

(«)2790at Is. lid. Facit, £156 : 18 : 9.

(')7904at Is. l|d. Facit, £452 : 16 : 8.

('')3750atls. 2d.r Facit, £218 : 15 : 0.

(«)3291 atls. 2id. Facit, £195 : 8 : 0^.

('0)9251 at ls.2^d. Facit, £559 : 1 : 11.

('•) 72.50 at l?.2|dT Facit, £445:11:5^.

(' = ) 7591 at Is. 3d. F^cit, £474 : 8 : 9.

('3)6325atls.3:ld. Facit, £101 : 1S:0J.

{'*)5271 atlB.3H: Facit, £340 : 8 t 4A.

{')iA3715atl2id. 154 : 9i

2|0)3S6|9 : 9i Facit, £193 : 9 : 9J.

(i5)3254atl8.3gd. Facit, £213: 10: 10^.

(»«)2915atls. 4d. Facit, £194 : 6 : 8.

('^)3270at ls.4id. Facit, £221 : 8 : 1^.

(» 8) 7059 at ls.4id. Facit, £485 : 6 : U.

(»»)2750atls.43d. Facit, £191: 18:6J.

('^'')3725at Is. 5d. Facit, £263: 17 : 1.

(2')7250atls.5id. Facit, £521 : 1 : 10^.

('^)2597at Is. 5.id. Fncit, £189 : 7 : 3^.

(-3) 72l0atls. 5|d. Facit, £533 : 4 : 9,].

('^«)7524at Is. fid. Facit, £.5(54 : 0 : 0.

(')2712atl2|d. Facit, £144 : 1 : 6.

(*)2107atl8. Id. Facit, £114:2:7.

(2") 7103 at Is. 64d. Facit, £540 : 2 : 5|.

(«'')3254at Is.Oid. Facit, £250: 16:7.

(2^) 7925 at Is. 63d. Facit, £619 ; 2 : 9i.

(-^8) 9271 at Is. 7d. Facit, £733 : 19 : 1.

(2»)7210atls. 7id. Facit, £578: 6: Oi-

(3'')2310atls.7id. Facit, £187 : 13 : 9.

(3')2504at Is. 73d. Facit, £206 : 1 : 2.

(8') 7152 at Is. 8d. Facit, £596 : 0 : 0.

(3 3) 2905 at Is. 84d. Facit, £245 : 2 : 2^.

(«<)7104atlf«.8id. Facit, £006 : 16 : 0.

PEACTICE.

(»») 1004 at Is, 8|d. Facit, £85 : 16 : 1.

('«)2104atls. 9(1. Facit, £184 : 2 : 0.

(3^2571 at Is. 9|d. Facit, £227 : 12 : 9|.

('8) 2104 at Is. 9^d. Facit, £188 : 9 : 8.

(»»)7506atls. 9|d. Facit, £680 : 4 : 7^.

(*») 1071 at Is. lOd.

Facit, £98 : 3 : 6.

(*i)5200atls.lOH Facit, £482 : 1 : 8.

{*2)2117atls.l0^d. Facit, £198 : 9 : 4^.

(*3) 1007 at Is. lOf. Facit, £95 : 9 : 1^.

(**)5000atls. lid. Facit, £479 : 3 : 4.

(") 2105 at Is. Hid, Facit, £203 : 18 : 5i.

(*«) 1006 at Is. 11 id. Facit, £98 : 10 : 1.

(*^)2705atls.ll|d. Facit, 267 : 13 : 7i.

(♦8)5000atls. ll^d. Facit, £489 : 11 : 8.

(*9)4000atl8.1l|d. Facit, £395 : 16 : 8.

Rule 4. When the price consists of any even number of shillings under 20, multiply the given quantity by half the price, doubling the first figure of the product for shillings, and the rest of the product will be pounds.

(')2750at2s. Facit, £275 : 0

0.

(«)3254at4s. Facit, £650 : 16 : 0.

(»)2710at6s. Facit, £813 : 0 : 0.

(*)1572at8s. Facit, £628 : 16 : 0.

(5)2102atlQs. Facit, £1051 : 0 : 0.

('')2101 at 12s. Facit, £1260:12:0.

(^)527l at 14s. Facit, £3689 : 14 : 0.

(8)3123at IGs. Facit, £2498 : 8 : 0.

(») 1075 at 16s. Facit, £860 : 0 : 0.

(»») 1621 at 18s. Facit, £1458 : 18 : 0.

Note. When the price is 10s. take half of the quantity, and if any remains, it is 10s.

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.

(1)2703 at Is. Facit, £135 : 3 : 0.

C)

3270 at 3s. 3

210)98110 Facit, £190 : 10 : 0.

(3) 3271 at 5s. Facit, £817 : 15 . 0.

M

PRACTICE.

(<)2715at78. Facit, £950 : 5

0.

OSSHatOs. Facit, £1440:6:0.

(') 3179 at 13s, Facit, £3D(>tt : 7 : 0.

(8) 2150 at 153. Facit, £1012: 10:0.

('•)2150atl98. Fucit, £2042 : 10 : 0

('»)7157at lOs. Facit, £0799 : 3 : f

(•) 2710 at lis. (») 3142 at 178.

Facit, £1490 : 10 : 0. Facit, £2()70 : 14 : 0.

Note. When the price is 5s., divide the quantity by 4, anJ if any remain, it is 5s.

Rule 0. 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 rest, add them together, and divide by 2D.

('')7514at48. 7d. Facit, £1721 : 19:2.

210

i

(')2710at0s. 8d. Facit, £903 : (i : 8.

(•^)3l50at3s. 4d. Facit, £525 : 0 : 0.

(«)2715at2s. Od. Facit, £339 : 7 : 0.

(*) 71oOat Is. 8d. Facit, £5:)5 : 16 : 8.

(«)3215at ls.4d. Facit, £214 :0:8.

(«)7211 at Is. 3d. Facit, £150 : 13 ; 9.

i

(') 2710 at 3s. 2d. 3

8130 451 :8

85811 :8 Facit £429 : 1 : 8.

(9) 2517 at 58. 3d. Facit, £000 : 14 : 3.

(">)25l7at78. 3id. Facit, £928: 11 : lOJ.

(")3271 at53. 9id. Facit, £943: 10:43.

('«)2l03atl5s.4id. Facit, £1010: 13 : 7J.

{'»)7152at 17s.03d Facit, £()280 : 7 : 0.

('<)25IOat 14:7id. Facit, £1832: 16 :5i.

('»).3715at98. 4id. Facit, £1741 :8: 1^.

(")2572atl3:7id. Facit, £1752:3:6.

('^)725I atlls. 6id. Facit, £5324: 19 01

PnACTICE.

65

|(")32I0atl5s. 7H| iFacit, £-i5ll.3. 1.^. |

('»)2710atl9s. 2Jd. Facit, £2002. 14.7.

Rdle 7. 1st, Wlien the price is pounds and shillings, multiply the quantity by the pvounds, and proceed with the shillings, if they are even, as the fourth rule ; if odd, take the aliquot parts, add them together, the sum will be the answer.

2dly, When pounds, shillings, and pence, and the shillings and pence the aliquot parts of a p;)und, multiply the quantity by the pounds, and take parts for the rest.

3dly, When the price is pounds, shillings, pence, and far- things, and the shillings and pence are not the ali |UOt parts of a pound reduce the pounds and shillings into shillings, multiply the quantity by the shillings, take parts for the rest, add them together, and divide by 20.

Note. When the given quantity consists of no more than three figures, proceed as in Compound Multiplication.

^ (5) 2710 at £2. 3. 7i. 43

•. d 2.6

J.

6

(')7215at£7

7

.4

.0

G

50505 1413

U

£51918

i

C^) 2104 at £5 5

.3

.0

6

10520 203 52.12

£10835.12

( ) 2107 at £2 Facit, £50.50 .

.8 10

0. 0.

(<)7150at£5 Facit, £37920 .

0. IG

0. 0.

1

ra

210

116530 1355 338 . 9

11822|3.9

Facit, £5911 .3.9.

(«)3-215at£l .17.0. Facit, £5947 . 15 0.

{^)2107at£1.13.0. Facit, £3476 .11.0.

(») 3215 at £4. 6. 8, Facit, £13931. 13. 4.

(») 21.54 at £7. 1.3. Facit, £15212. 12.0.

e»

PRACTICE.

(K*) 2701 at £2. 3. 4. Facit, £5852 .3.4.

('»)2715at£lTn^i. Facit, £5051 . 0 . 7^.

(> = )2157at£3.15.2i. Facit, £8108. 19. 5i.

( 13)3210 at £1.1 8.6f. Facit, £6189 . 5 . 7^.

(»«)2157at£2.7.4i. Facit, £5109. 7. 10^.

('») 142at£1.15.a|. Facit, £250 . 2 . 6^.

('«)95at£15.l4.7i. Facit, £1494 . 7 . 4|.

(»^)37 at £1.19.53. Facit, £73 . 0 . 81.

(i8)2175at£2.15.4i. Facit, £6022 . 0 . 7^.

(i'')2150at£17.16.1i. Facit, £38283 .8.9.

Rule 8. When the price and quantity given are of several denominations, multiply the price by the integers, and take parts with the parts of the integers for the rest.

1. At £3.17.6 per cwt, what iathe value of 25 cwt. 3 qra. 14 lb. of tobaccol

2

i

i

£3.17.6

5X5=25

19. 7.6 5

lb. 14

96.17.6 1.18.9 9.8J

99.5.11i

2. At £1.4.9 per cwt., what comes 17 cwt. 1 qr. 17 lb. of cheese to? ^ns. £21 . 10 .8.

3. Sold 65 cwt. 1 qr. 10 lb. of cheese, at £1.7.8 per cwt., what does it come to? Ans. £118 . 1 . 0|.

4. Hops at £4 . 5 . 8 per cwt., what must be given for 72 cwt. Iqr. 181b. ? Tins. £310 . 3 . 2.

5. At £1 . 1 . 4 per cwt, what is the value of 27 cwt. 2 qrs. 15 lb. of Malaga raisins ?

Ans. £29 . 9 . 6i.

6. Bought 78 cwt. 3 qrs. 12 lb. of currants, at £2 . 17 . 9 per cwt,, what did I give for the whole ?

^ns. £227.14.

TARE AND tRET. 67

t. Sold 56 cwt. 1 qr. 17 lb. of sugar, at £2 : 15 : 9 the cwt.. what does it come to ? Ans. £157 : 4 : 4|.

8. Tobacco at £3 : 17 : 10 the cwt., what is the worth of 97 cwt. 15 lb. ? Ans. £378 : 0 : 3.

9. At £4 : 14 : 6 the cwt., what is the value of 37 cwt. 2 qrs. 13 lb. of double refined sugar ?

Ans. £177 : 14 : 8^.

10. Bought sugar at £3 : 14 : 6 the cwt., what 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 53 cwt* 17 1b.? Ans. £190 : 0 : A.

TARE AND TRET.

The allowances usually made in this Weight, are Tare, Treti 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 i^ 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 104 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 (jf the alloAvance is deducted from the gross.

Neat is the pure weight, when all allowances are deducted.

Rule 1. When the tare is at so much per bag, barrel, &.C., multiply the number of bags, barrels, &c, by the tare, and sub- tract the product from the gross, the remainder is neat.

W TARK AND THET.

NoTK. To reduce Pounds into Gallons, multiply by 2, and 4iride by 15.

1. In 7 frails of raisins, cnch weighing 5 cwt. 2qr8. 5 lb. gross, tare at 23 lb. per frail, huw much neat weight?

Aus. 37 cwt, 1 qr. 14 lb.

23 6.2. 5 or, 5.2. »

7 7 23

4 . .

28)101(5 38.3. 7=groes 6.1.10

140 1 . 1 .21 =lare 7

. 1.1 .

21 37.l.l4=ncat 37.1.14

2. What is the neat weight of 25 hogsheads of tobacco, wciglw ing gross 103 cwt. 2 qrs. 15 lb., tare 100 lb. per hogshead ?

Alls. 141 cwt. 1 qr. 7 lb.

3. In 10 bags of pepper, each S5 lb. 4 oz. gross, tare per bag 3 lb. 5 oz. how i.iaiiy 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 i? 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. OS cwt. 2 qrs. 181b. 6. In 75 barrels of figs, each 2 qrs. 27 lb. gross, tare in the whole 597 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 file gross, the remainder is neat.

NoTF. 7 lb. is /j, > lb. is -,-l , 14 lb. is -J-, 10 lb. is -f.

0. What is the neat weight of IS butts of currants, each 8 cwt 2 qr«. 5 lb., tare at 14 lb. per cwt. ?

8.2.5

y X 2=19

7G

. 3 ,

. 17 2

4=1 153 .

l'»

3 . . 0 .

6 25^

13^1

. 2 .

81

TARE AND TRET. 60

7. In 25 barrels of figs, each 2 cwt. 1 qr. gross, tare per cwt IC lb., how much neat weight I

Ans. 48 cwt. 0 qr. 24 lb.

8. What is the neat weight of 0 hogsheads of nutmegs, each weighing gross 8 cwt. 3 qrs. 14 lb., tare 16 lb. per cwt. ?

Ans. 68 cwt. 1 qr. 24 lb.

Rule 4. When tret is allowed with tare, divide the pounds suttle by 26, the quotient is the tret, which subtract from the sub- tle, the remainder is neat.

9. In 1 butt of currants, weighing 12 cwt. 2 qrs. 24 lb. gross, tare 14 lb. per cwt., tret4 lb. per 101 lb., how many pounds neatt

12 . 2 . 24 4

50 28

14= i MM gross. 178 tare.

2C)124(> suttle. 47 tret.

11!)!) neat.

' •

10. In 7 cwt. 3 qrs. 27 lb. gross, tare 3Glb., tret 4 lb. per 104 lb., how mauy pounds neat ?

^ns. 826 lb.

n. In 152cwt. 1 qr. 31b. gross, tare 101b. per cwt., tret 4 lb; per 104 lb., how much neat weight ?

Ans. 133 cwt. 1 qr. 12 lb.

Rule 5. When clofT is allowed, multiply the cwts. suttle by 2, divide the product by 3, the qtiotient will be the pounds clolt which subtract from the suttle, the remainder will be neat.

12. What is the neat weight of 3 hogsheads of tobacco, weigh- ing 15 cwt. 3 qrs. 20 lb. gross, tare 7 lb. per cwt., tret 4 lb. per 104 lb., cloff 2 lb. for, 3 cwL I

Ans, 14 cwt. 1 qr. 3 lb.

TO INTEREST.

7=iV 15 . 3 . 20 grcm. 3 . 27J tare.

26)U . 3 . 20| suttlcu 2 . 8 tret

14 . 1 . I2i suttle. 9i cloff.

14 . 1 . 3

13. In 7 hogsheads of tobacco, each weighing gross 5 cwt. 2 qn. 7 Ih . tare 8 lb. per owt., tret 4 lb. per 104 lb., cloff 2 lb. per 3 cwt., how much net* weight •? iln*. 34 cwt. 2 qrs. 8 lb,

SIMPLE INTEREST,

Is the Profit allowed in lending or forbearance of any stun of money for m determined space of time.

The Principal b the money lent, for which interest is to be received.

The rate per cent, is a certain sum agreed on between the Borrower and tlie Lender, to be paid for every £100 for the use of the principal 12 months.

The Amount is the principaJ and interest added together.

Interest is also applied to Commission, Brokage, Purchasing of Stocks, and Insurance, and are calculated by the same rules.

To find 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 yejus given in the question, and the product will be the answer.

3. If there be parts of a year, as months, weeks, or days, work for the months by the aliquot parts of a year, and for the weeks and days ly' the Ryle of Three Direct.

EXAMPLES.

1, What is the interest of £375 for a year, at 5 per cent pex annoml 5

J8I75 20

15|00 Ans. £18 . 15 . 0.

2, What is the interest of £268 for 1 year, at 4 per cent, per annum 7

Ans. £10 . 14 . 4|.

3, What is the interest of £945 . 10. for a year, at 4 per cent, per annum 1

Ans. yf. 16 . 41.

INTEREST. 71

4. "What is the interest of £547 . 15, at 5 per cent, pet 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- num 1 Ans. £bO . Id . 6.

6. What is the interest of £556 . 13 . 4, at 5 per cent, per annum, for 5 yeare 1 Ans. £139 .3.4.

7. My correspondent writes me word, that he has bought goods to the amount of £754 . 16 on my account, what does his commission come to at 2i per cent. ]

Ans. £18 .11 .M.

8. If I allow my factor 3j per cent, for commission, what may he demand on the laying out £876 . 5 . 10 1 Ans. £32 . 17 . Sj.

9. At 110} per cent., what is the purchase of £2054 . 16. South Sea Stock 1

Ans. £2265 .8.4.

10. At 104| per cent South Sea annuities, what is the purchase of 1797 . 14 1

Ans. £1876.6. 11}.

11. At 96i per cent., what is the purchase of £577 . 19 Bank annuities 1

Ans. £559 . 3 . 3|.

12. At £124| per cent., what is the purchase of £758 . 17 . 10, India Stock 1

Ans. £945 . 15 . 4i.

BROKAGE,

Is an allowance to brokers, for helping merchants or factors to persons, to buy or sell them goods.

Rule. Divide the sum given by 100, and take parts from the quotient with the rate per cent.

13. If I employ a broker to sell goods for me, to the value of £2575 . 17 . 6, what is the brokage at 4s. per cent. 1 25)75 . 17 . 6

20 4s.=|^ 25 . 15 . 2

15|17 Ans. £5. 3 . Oi 12

2110

14. When a broker sells goods to the amount of £7105 . 5 . 10, what may he lemand for brokage, if he is allowed 5s. 6d. per cent. °?

Ans. £19 . 10 . 9J.

15. If a broker is employed to buy a quantity of goods, to the value of £975 .6.4, what is the brokage, at 6s. 6d. per cent. "

,4ns. £3.3. 4i.

16. What is the interest of £547 . 2 . 4, for 5i years, at 4 per cent, per an- num 1 Ans. £120 . 7 . 3J.

17. What is the interest of £257 . 5 . 1, at 4 per cent., for a year and three quarters 1 Ana. £18 . 0 . IJ.

IS. What is the interest of £479 . 5 for 5} years, at 5 per cent, per annum 1

Ans. £125 . 16 . Oi.

in INTEnEST.

19. What is the interest ofl^lQ : 2 : 7 for7i years, at 4 J pet cent, per annum ?

Ans. £187: 19: 1^.

20. What is the interest of £279 : 13 : 8 at 6^ per cent, per annum, for 3^ years ?

Ans. £51 : 7 10.

When the interest is required for any nunber of Weeks.

Rule. As 52 weeks are to the Interest of the given sum for • year, so are the weeks given for the interest required.

21. What is tlic interest of £259 : 13 : 5 for 20 weeks, at 5 per cent, per annum ?

Ans. £4 : 19 10^.

22. What is the amount of £375 : 6 : 1 for 12 weeks, at 4J per cent, per annum ? Ans. £379 : 4 : 0^.

When the Interest is for any number of days.

Rule. As 365 days are to the interest of the given sum for • year, so are the days given to the interest required.

23. At 5^ per cent, per annum, what is the interest of £985 . 2 . 7 for 5 years, 127 days I

Ans. £280 . 15 . 3.

24. What is the interest of £2726 . 1 . 4 at 4J per cent, per •nnum, for three years, 154 days ?

Ans. £419 . 15 . 6i.

When the Amount, Time, and Rate per cent, are given to find the Principal.

Rule. As the amount of £100 at the rate and time given : if to £100 : : so is the amount given : to the princi[>al required.

25. What principal heing put to interest, will amount to £403 10 ia 5 years, at 3 per cent, per ainium ?

3X54-100=£1I5. 100.. 402. 10 20 20

2300 8050

100

2:i|00)8(V50|00(£350 AiUk

INTEREST. 73

26 What principal being pnt to interest for 9 years, will amount to £T3i : >i, aiA per cent, per annum ?

Ans. £540.

27. What principal being put to interest for 7 years, at 5 per cent, per annum, will amount to £334 : 161

Ans. £348.

When the principal, Rate per cent., and Amownt are given^ to find the Time.

Rule. Astheinterestof the principal for 1 year : is tol year : : 60 is the whole interest : to the time required.

28. In what time will £330 amount to £402 . 10, at 3 per cent per annum ?

350 Asl0.10:l::52.10:5

3 20 20

10|50 210 2l,0)105|0(5ye8n. An*. 402.10

20 105 350. 0

10|00 52.10

29. In what time will £540 amount to £734 r 8, at 4 per cent per annum ? Ans. 9 years.

30. In what time will £248 amount to £334 : 16, at 5 per cent, per annum ? Ans. 7 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 : : 80 is £100 : to the interest for the same time. Divide that in- terest by the time, and the quotient will be the rate per cent.

31. At what rate per cent, will £350 amount to £402 : 10 in 5 years' time ?

350 As 350 : 52 . 10 : : 100 : £15

20 i

52.10

1050

100

3510)10500iO(:500s.=£l5-f-5=3 per cent

32. At what rate per cent, will £24S amount to £334 : 16 in 7 years' time ? Ans. 5 per cent.

a

74 INTEREST.

33. At what rate per cent, will £540 amount to £734 : 6 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 first year's interest, which add to the princi- pal ; then find the interest of that sum, which add as before, and BO on for the number of years.

a. Subtract the given sum from the last amount, and it will give the compound interest required.

PiAMPLES.

1. What is the compound interest of £500 forborne 3 years, at 5 per cent. pe( annum

?

600 500 525

5-25 26 . ,

S5|00 525=^l;Hyear. 551 .. 5=2d year. 5 5 551 .. 5

2G125 27I56..5 27.11

20 20

5|00 11|25 500 prin.sub

12

578.16..3=3dyear. ►00 prin. aub

78 . 16 . . 3=intere8tfor3yeau».

3j00

2. What is the amount of £400 forborne 3i years, at 6 per cent, per annum, compound interest ?

Ans. £490 : 13 : Hi-

3. What will £650 amount to in 5 years, at 5 per cent, pei annum, compound interest? Ans. £829 : 11 : 7^.

4. What is the amount of £550 : 10 for 3 years and 6 months, at 6 per cent, per annum, compound interest?

Ans. £675: 6 : 5.

5. What is the compound interest of £764 for 4 years and 9 months, at 6 per cent, per annum ?

Ans. £243 : 18 : 8.

6. What is the compound interest of £57 : 10 : 6 for 6 years, 7 months, and 15 days, at 5 per cent per annum ?

^7is.£18:3:8i.

REBATE OR DISCOUNT. 7&

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 given 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 ?

6m^i6 As 103 : 0 : : 487 : 12

20 20

100 2060 9752

3 .

103 j6s.

Q06|0)2935|6<14.4relntei 487 : 12 principaL 206

14 : 4 rebate.

Ana. £473 : 8 present -wonu

865

416=4&

2. What is tlie present payment of £357 : 10, which was agreed to be paid 9 months he.nce, at 5 per cent, per annum ?

Ans. £344 : 11 : 7.

3. What is the discount of £275 : 10 for 7 months, at 5 per cent, per annum? Ans. £7 : 16 : 14.

G2

70. BaCATION OF PAVMENTS.

4. Bought goods to the value of £109 : 10, to be paiil at nine months, what present money will discharge the same, if I am al- lowed 6 per cent, per annum discount?

Avs.fAOi: 15:8i.

5. What is the present worth of £527 : 9 : 1, payable 7 month* hence, at 4} per cent.? Ans. £514 : 13 : 10^.

6. What is the discount of £85 : 10, due September the 8th, this being July the 4th, rebate at 5 per cent, per annum ?

Ans. 15s. 3id.

7. Sold goods for £875 : 5 . 6, to be paid 5 months hence, wkat is the present worth at 4^ per cent. ?

Ans. £859 : 3 : 4.

8. What is the present worth of £500, payable in 10 months, at 5 per cent, per annum ? Ans. £480.

9. How much ready money can I receive for a note of £76, due 15 mouths hence, at 5 per cent. ?

An.^. £70: 11 : 9|. 10; What will be the present worth of £1.50, payable at 3 four months, i.e. one third at four months, one third at 8 months, and one third at 12 months, at 5 per cent, discount ?

Ans. £145 : 3 : 8^.

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. ? A71S. £10 : 1 1 : 4^.

12. What is the present worth of £500 at 4 per cent., £100 being to be paid down, and the rest at 2 six months ?

Ans. £48S : 7 : 8^

EQUATION OF PAYMENTS,

Is when several sums are due at different times, to find a meaii time for paying the whole debt ; to do which this is the common

fttJLE. Multiply each term by its time, and divide the sum o(rth« products by the whole debt, the quotient is accounted the mcaz) time.

EQUATION OF PAYMENT*; *ti

EXAMPLES.

1. A owes B £'369, whereof £40 is to be paid at 3 months, £60 at o raoiiths, and £100 at 10 months ; at what time may th« whole debt be paid tof^ether, without prejudice to either?

£ m.

40 X 3 = 120

60 X 5 = 300

100 X 10 = 1000

2|00)14|20

7 months -j^.

2. B owes C £S0'), whereof £200 is to be paid at 3 months, £100 at 4 months, £300 at 5 months, and £200 at 6 months ; but they agreeinsr to make but one payment of the whole, I de- mand what time that must be ?

Ans. 4 months, 18 days.

3. I bought of K a quantity of goods, to the value of £360, which w;is t:) have been paid as follows : £120 at 2 months, and £200 at 4 months, and the rest at 5 months ; but they afterwards agreed to have it paid at one mean time ; the time is demanded.

Ans. 3 months, 13 days.

4. A merchant bong' t goods to the value of £500, to pay £100 at the end of 3 months, £150 at the end of 6 months, and £250 at the end of 12 months; but afterwards they agreed to discharge the debt at one payment ; at what time was this payment 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 months, -J- at 3 months, i at4 months, | at 5 months, | at 6 months, and the rest at 7 months ; but they agree that the whole should be paid at one equated time ; what is that time ?

Ans. 4 months, 1 quarter.

6. A is indebted to B £120, whereof \ is to be paid at 3 months, j at 6 months, and the rest at 9 months ; what is the equated time of the whole payment?

Ans. 5 months, 7 days- G3

78

BARTER.

BARTER

Is the exchanging of one commodity for another, and inform* the traders so to proportionate their goods, that neither maji 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 43. per lb. must be delivered in barter for 2 cwt., of tea, at 'Jb. per lb. 1 2 cwt., 112

2241b. 9 price.

4)2016 the value of the tea.

504 lb. of chocolate.

2. A and B barter; A hath 30 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 ; what ought B to rate his hops at in barter, and what quan- tity must be given for the 20 cwt., of prunes 1

112 As 4 : 5 : : 32 20 5

40 2240 4)160 12 5

cwt. qr. lb. 40b.

4810)1120|0(23 . 1 , 9H.Ans. 96

160 144

16=1 qr. 9 lb. |f .

3. How much tea, at 9s. per lb. can I have in barter for 4 cwt., 2 qrs. of chocolate, at 4s. per lb. 1

Ana. 2 cwt.

4. Two merchants barter ; A hath 20 cwt. of cheese, at 2l8. (xl. 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 1

Ans. B must receive of A £8 . 2.

5. A and B barter ; A hath 3i lb. of pepper at I3jd. per lb. ; B hath gin- ger at ISjd. per lb.; how much ginger must he deliver in barter for th» pepper 1

Ans. 3 llv 1 oz. J f .

PROFIT AND LOSS.

70

6. How many dozen of candles, at 58. 2d. per dozen, must be delivered in barter for three cwt. 2 qrs. 16 lb. of tallow, at 37s. 4d. per cwt. 1 „ „ . ,

Ans. 26 doz. 3 lb. f^f-,

7 A hath 608 yards of cloth, worth I4s. per yard, for which B givcth him £1-25 . 12. in ready money, and 85 cwt. 2 qrs. 24 lb. of bees'-wai. The ques- tion is, what did B reckon his bees'- wax at per cwt. 1

Ans. £3 . 10.

8. A and B barter ; A hath 320 dozen of candles, at 4s. 6d. per dozen; for which B giveth hL-n £30 in money, and the rest in cotton, at 8d. per lb. ; I desire to know how much cotton B gave A besides the money 1

Ans. 11 cwt. 1 qr.

9. If B hath cotton, at Is. 2d per lb., how much must he give A for 114 lb. of

tobacco, at 6d. per lb. 7

^ ilrw. 481b.-}-^

10. C hath nutmetrs worth 7s. 6d. per lb. ready money, but in barter will have 8s. per lb. ; an3 D hath leaf lobacco worth 9d. per lb. ready money ; how much must D rate his tobacco at per lb. that his profit may be equivalent withC'sl

Ans. 9id. -f J.

PROFIT AN1> LOSS

Is a Rule that discovers what is got or lost in the buying or selling of goods, and instructs us to raise and lower the price, so as to gain so much per cent, or otherwise.

The questions in this Rule are performed by the Rule of Three.

EXAMPLES.

1. If ayanl of cloth is bought for ^ 2. If 60 ells of Holland cost £)8 lis. and sold for 12s. 6d. what is the I what must 1 cU be sold for to gain 8

^ain per cent. 1

As 11 : 1 : 6 : 12

18

12.6

11. 0 11

:100 20

2000 18

)36'300

1.6

12)3-272^8^. 2I0)27|2 . 8

Ans. £13 . 12 . S-£^.

per cent. 1

As 100 : 18 : : 105 108

1100)19144 30

8)80 12

9160 4

2110

12X5=0«

12)19. 8.9J

5)1 . 12. 4J

0. 6. 5|

Ans Ga. 52d.

80 FELLOWSHIP.

3. If i lb. of tobacco cost ICd. and is sold for 20d. what u the gain per crnt.1

Ans. £25.

4. If a jMirccl of cloth be aold for X5G0, and at 12 per cent gain, what wa« the prime cost 1 Ans. £500.

5. If a yanl of cloth is bought for iSs. 4d. and sold again for iCs. \« hat is th« gain per cent. 1 Ana. £20.

6. if 112 lb. of iron cost 278. Cd., what must 1 cwt. be sold for to gwin 15 per cent. ? Ans. £1 . 11 . 7*.

7. If 375 yards of broad clotli be sold for £490, and 20 p<T cent, profit, what did it cost per yard 1 Ans. £1 . 1 . 9i.

8. Sold 1 cwt. of hope, for £6 . 15, at the rate of 25 per cent, profit, what would have been the gain uct cent, if I had sold them for £8 per cwt. 1

i4n*. £48.2. II4.

9. If 90 ells of cambric cost £60, how much must I sell it per yani to gain 18 per cent. 1 Ann. I2s. 7d.

10. A plumber sold 10 fothcr of lead for £204 . 15, (the fotber Iwing 10\ cwt.) and gained afler the rate of £ 12 . 10 per cent. ; what did it cost him [wr cwt. 1 Ans. 186. 8(1.

11. Bought 43r> yards of cloth, at the rate of 8s. Cd. per yard, and sold it for lOs. 4d. per yard : wiiat was tiie Kiiiii of the whole 7

ylr»*. £39 . 19 . 4.

12. Paid £69 for one ton of steel, which is retailed at 6d. |irr lb. ; what is the profit or loss by the sale of 15 toi s 1 /!?/».£ 182 loss.

13. Bought 124 yards of linen, for £32; how should the same be retailed per yard to gain 15 per cent. 1

Ans. 5s. lld.,*ifr-

14. Bought 249 yards of cloth, at 3s. 4(1. per yaid, rrtaiicd the siimc at 4s. 2d. per yard, what is the profit in the whole, and how much ixvcrnt. 1

Alls. £10 . 1 At profit, and £25 percent

FELLOWSHIP

Is when two or more join tlirir stock and trade together, so to determine each pcrt»on's pnrlictilar sliare of the gain or loss, in proportion to his principal in joint stork.

By this rule a haakrnpt's estate many he divided amongst his creditors ; as also le<racies may be adjusted when there is a defi- ciency of assets or effects.

FELLOWSHIP IS EITHER WITH OR WITHOUT TIME.

FELLOWSHIP WITHOUT TIME.

Rule. As the whole stock : is to the whole jrain or loss : : so is each man's share in stock : to his share of the jrain or loss.

Proof. Add all the shares to>rrther, and the sum will be equal to the given gain or loss- -hut the surest wav is, as the whol»

Sain or loss : is to the whole stock : : so is each man's share ot le gain or loss : tu his share in stock.

PELLOWSUIF. 81

EXAMPLES.

1. Two merchants trade together; A pute into stock £20, and B £40, t^ gained £50 ; what is each person's share thereof 1

20+40=60 1 60 : 50: : 20 As60 : 50 : : 40 33 . 6 . 8, B'sshare.

20 40 16 . 13 . 4, A's.

6|0)100|0 6|0)200|0 60. O.OprooC

£16 13.4 £33.6.8

2. Three merchants trade together, A, B, and C ; A put in £^, B £30 «nd C £40 ; they gained £ 180 : what is each man's part of the gain 1

^n*. A£40; B£GO; C£80.

3. A, B, and C, enter 'nU Trartnership : A puts in £364, B £482, and C £500; and they gained £867; what is each man's share in proportion to hi« stock?

Ans. A £234 9 . 3}— rem 70 ; B £310 . 9 . S— rem. W8; C £322 . 1 . 3i— rem. 1028.

4. Foot merchants, B, C, D, and Emake a stock; B put in £227, C £349, D £1 15, and E £439 ; in trading they gained £428 : 1 demand each merchant's share of thegain?

Ans. B £85. . 19 . 61—690; C £132 . 3 . 9—120; D £43. 11 . U— 250; E £166 . 5 . 6i— 7a

5. Three persons, D, E, and F, join in company ; D's stock was £750, E's £460, and F's £500; and at the end of 12 months they gained £684 : what is each man's particular share of the gain ?

Ans. D £300, E £l84, and F £200.

6. A merchant is indebted to B £275 . 14, to C £304 . 7, to D £152, and to E £104 . 6; but upon his decease, his estate is found to be worth but £675 . 15 : how must it be divideti among his creditors 1

Ans. B's share £'222 . 15 . 2— 65St; C's £245 . 18 . U— 15750; D's £122 . 16 . 2J— 12327; and E's £84 . 5 . 5—15620.

7. Four persons trade together in a tmnt stock, of which A has \, B ^, C -J^ and D i ; and at the end of 6 months they gaia £ 100 : what is each nan's share of the said gain 1

Ans. A £35 . 1 . 9-48; B £26 . 6 . 31—36; C £21 . 1 . OJ —1-20; and D £17 . 10 . lOj— 24.

8. Two persons purchased an estate of £l700 per annum, freehold, fer £27,209, when money was at ('» p»r cvnt. interest, and 49. per pound, !ant!-t<LX ; whereof D piid £15,800, and E the rert; sometime after, the interest of the m<*- ney falling to 5 per cent, and 2s. per pound land-tai, ther sell the said estate bt 21 years' purchase : I desire to know eeucb pmsorv's share f

iln#.D £92,500 ;E £18,30©.

SSi FELLOWSRI •

9. D, E, and F, join their stocks in trade ; the amount of their stocks is £647, and they are in proportion as 4, 6, and 8 are to one another, and the amount of the gain is equal to D's stock : wliat is each man's stock and gain ?

Ans. D's stock £143 . 15 . 6^ gain, 31 . 19 . 0^-

E's 215 . 13 . 4 47 . 18 . 6H.

F's _... 287 . 11 . IW 63 . 18 . OjW-

10. D, E, and F, join stocks in trade ; the amount of their stock was £100 ; D's gain £3, E's £5, and F's £8 : what was each man's stock ?

Ans. D's stock £18 . 15; E's £31. 5; and F's £50.

FELLOWSHIP WITH TIME.

Rule. As the sum of the products of each man's money and time : is to the whole gain or loss : ; so is each man's product : to his share of the gain or loss.

Proof. As in fellowship without time.

EXAMPLES.

1. D and E enter into partnership ; D puts in £40 for three months, and E £75 for four months ; and they gained £70 : what is each man's share of the gain ?

Ans. D £20, E £50.

40x3=120 As 420 : 70 : : 120 As 420 : 70 : : 300

75X4=300 120 300

420 42|0)840|0(20 42|0)2100|0(50

840 2100

2, Three merchants join in company; D puts in stock £195 . 14, for three months, E £169 . 18 . 3, for 5 months, and F £59 . 14 . 10, for 11 months; they gained £364 . 18 : what is each man's part of the gain ?

Ans. D's £102 . 6 . 4— .5008; E's £148 . 1 . U- 482802 ; and F's £114 . 10 . Oj— 14707.

ALLIOATIOX. €3

3. Three merchants join in company for 18 months ; D put in £bOO, and at five months' end takes out £200 ; at ten months' end puts in £300, and at the end of 14 months takes out £130 : E puts in £400, and at the end of 3 months £270 more ; at 9 months he takes out £140, but puts in £100 at the end of 12 months, and withdraws £99 at the end of 15 months : F puts in £900, and at 6 months takes out £200 ; at the end of 11 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 ?

.In*. D £50 : 7 : 6—21720 ; E £62 : 12 : 5^—29859 ; and F £87:0:0^—14167.

4. D, E, and F, hold a piece of ground in common, for which they are to pay £36 : 10 : 6. D puts in 23 oxen 27 days ; E 21 oxen 35 days ; and F 16 oxen 23 days. What is each man to pay of the said rent 1

Ans. D £13 : 3 : lf-624; E £15 : 11 : 5—1688; and F £7 : 16 : 11—1136.

ALLIGATION

ALLIGATION IS EITHER MEDIAL OR ALTERNATE.

ALLIGATION MEDIAL

Is when the price and quantities of several simples are given 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 a,t the mean rate, and if it agrees with the total value of the several quantities af their respective prices, the work is rights-

84 ALLIGATION.

EXAMPLES.

1. A farmer mixed 20 bushels of wheat, at Ss. 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 tliis mixture.

20X5 = 100 As 96:288:: 1:3

36 X 3 = 109 40x2= 80

96 288

Ans. 3s.

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. 68. 2Jd.tf .

3. A grocer mingled 4 cwt. of sugar, at I)()s. 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. A71S. £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. 2^1. W-

5. If I mix 27 bushels of wheat, at 5s. (kl. 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 mixture? Ans. 4s. 3|d.ff.

6. A vintner mixes 20 gallons of port at 5s. 4d. per gallon, with 12 gallons of white wine, at 58. per gallon, 30 gallons ol 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.tf.

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. 9. A tobacconist would mix 50 lb. of tobacco, at 1 Id. per lb. with 30 lb. at I4d. per lb. 25 lb. at 22d. per lb. and 37 lb. at 29 per lb. What will 1 lb. of this mixture be worth ?

Ans. lO^d.Jtf

ALLTOATION.

85

ALLIGATION ALTERNATE

Is 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 tlie given price, observe, 1. Place them one under the otlier, 18 2

and the propounded price or mean rate at the left hand of them, thus.

22,

20_L_

24.

28-

2. Link the several rates together by 2 and 2, always observ- ing to join a greater and a Icrjs 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 iind 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, tliey will be the answer required^

Proof. Bv Alligation Medial.

EXAMPLES.

L A vintner would mix four sorts of wine together, of ISd., 20d., 24d., and 23d. per quart, what quantity of each must he have, to sell the mixture at 22d. per quart ?

Ansioer 18 2

2^

"2411 28J

4 2

14

of 18d. of 20d. of 24d. of 28d.

Proof. = 36d. = 120 = 96 = 56

)308

22.

or thus,

18 _6 of 18d.

12 of 20d.

20

'24_J 29

!2 of 24d. |4 of 28d.

Proof. = 108d. = 40 = 48 = 112

14

)308

23d. 22d.

Note. Questions 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., 6il., and lOd. per lb., so as to sell the compound for 8d. per lb. ; what quantity of each must he take ?

Ans. 2 lb. at 4d., 2 lb. at 6d., and 6 lb. at lOd. II

86 ALtI«ATIOM PARTIAL.

3. I desire to know how much tea, at IGs., 14s., Os., and Sa- per lb., will compose a mixture worth 10s. per lb. ?

Ans. 1 lb. at 16s., 2 lb. 14s., 6 lb. at 9s., and 4 lb. at 8s.

4. A farmer would mix as much barley at 3s. 6d. per busliel, 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 raisins of the sun, at 7d. 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 ?

Ans. 7 lb. at 2s., 4 lb. at Is. 6d., and 4 lb. at Is. 3d.

ALLIGATION PARTIAL

Is when the prices of all the simples, the quantity of but one of them, and the mean rate are given to find the several quanti- ties of the rest in proportion to that given.

Rule. Take the difference between each price and the meat rate as before. Then,

As the difference of that simple whose quantity is given' : tc the rest of the differences severally : : so is the quantity given : tt the several quantities required.

EXAMPLES.

1. A tobacconist being determined to mix 20 lb. of tobacco at 15d. per lb., with others at 16d. per lb., 18d. per lb., and 22d. per lb. ; how many pounds of each sort must he take to make one pound of that mixture worth 17d. ? Answer. Proof.

15 ._.

5 20 lb. at 15d. = 300d. 1 4 lb. at 16d. = 64d.

1 4 lb. at 18d. = 72d.

2 8 lb. at 22d. = 176d.

As 5 : 1 : As 5 : 1 :

As 5 : 2 :

: 1 lb. 17d.

:20 :20 :20

4 4

8

A

ins. 361b. 6l2d.

I ALtlGATION TOTAL. 8?

2. A farmer would mix 20 bushels of wheat at 60d. per bush- el, with rye at 36d., barley at 24d., and oats at 18d. per bushel. How much must he take of each sort, to make the composi- tion worth 32d. per bushel ?

Ans. 20 bushels of wheat, 35 bushels of rye, 70 bushels of barley, and 10 bushels of oats;

3. A distiller would mix 40 gallons of French Brandy, at 12s. per gallon, with English at 7s., and spirits at 4s. per gallon. What quantity of each sort must he take to afford it for 8s. per gallon ?

Ans. 40 gallons French, 32 English, and 32 spirits.

4. A grocer would mix teas at 12s., 10s., and 6s., with 20 lb. at 4s. per lb. How much of each sort must he take to make the composition worth 8s. per lb. ?

Ans. 20 lb. at 4s., 10 lb. at 6s., 10 lb. at 10s., 20 lb. at 12s.

5. A wine merchant is desirous of mixing 18 gallons of Ca- nary, at 6s. 9d. per gdlon with Malaga, at 7s. 6d. per gallon, sherry at 5s. per gallon, and white wine at 4s. 3d. per gallon. How much of each sort must he take that the mixture may be sold for 6s. per gallon ?

Ans. 18 gallons of Canary, 31^ of Malaga, 13-^- of Sherryj • and 27 of white wine.

ALLIGATION TOTAL

Is when the price of each simple, the quantity to be compound- ed, and the mean rafe are given, to find how much of each sort will make that quantity.

Rule. Take the difference between each price, and the mean rate as before. Then,

As the sum of the differences : is to each particular differ- ence : : so is the quantity given : to the quantity required.

EXAMPLES.

1. A grocer has four sorts of sugar, viz., at 12d., lOd., 6(1., and 4d. per lb. ; and would make a composition of 144 lb. worth 8(1. per lb. I desire to know what quantity of each he must take ?

68 POSITION, on THE RULE OF FAL8B.

Answer. Proof,

12 , 4 : 48 at 12d. 57C=As 12 : 4 : : 144 : 48

2 : 24 at lOd. 240=A8 12 : 2 : : 144 : 24 2 : 21 at Gd. 1 14=As 12 : 2 : : 144 : 24 4 : 48 at 4d. 192=As 12 : 4 : : 144 : 48

<:2

12 144 )1152(8d.

2. A. grocer having four sorts of tea, at 5s., 6s., 8s., and 98. per lb., would have a composition of 87 lb., worth Ts. per lb. What quantity must there be of each ?

Ans. Hi Ib.'of 5s., 29 lb. of Gs., 29 lb. of 8s., and 14^ lb. of 9s.

3. A vintner having four sorts of wine, viz., white wine at 49. per gallon ; Flemish at Gs. 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 gallop. 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. 6. A druggist having some drugs of 88., 5s., and 48. per lb., nade 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 h« 'ftke for each parcel ?

Ans. 12 lb. of 8s. 30 lb. of Ss.

8 lb. of 59. 6 lb. of 5s.

8 lb. of 48. 6 lb. of 48.

28 lb. at 6s. per lb. 42 lb. at Ts. per lb.

POSITION, OR THE RULE OF FALSE,

Is a rule that by false or supposed numbers, taken at pleasure discovers the true one required. It is divided into two pait*, SxNOLE and Dovdlk.

l^OSITION, OR THE RULE OF FALSE. 80

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 errors : 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 it agrees with the sum it is right.

EXAMPLES.

1. A schoolmaster being asked how many scholars he had, saia^ If I had as many, half as many, and one quarter as many rnoie^ I should have 88. How many had he? Ans. 32.

Suppose he had... 40 As 110 : 89 ; : 40 32

as many 40 40 32

half as many 20 16

i as many.... 10 1110)352j0(32 8

33 —

110 88 proof.

22 22

2. A person having about him a certain number of Portugal pieces, said, If the third, fourth, and (ith of them were added together, they would make 54. I desire to know how n any 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 : 13 : 4, Chaise £40.

4. A, B, and C, being determined to buy a quantity of goods which woidd cost them £120, ajrrecd among themselves that B should have a third part more than .\, and C a fourth part more than B. I desire to know what each man must pay?

Ans. A £30, B £40, C £50. H.?

JO POSITIOKi OR THE RULE OF FALSB.

6, A person delivered to another a sum of money unknown, to receive interest for the same, at 0 per cent per annum, simple in- terest, and at the end of 10 years received, for principal and in- terest, £300. What was the sum lent ? Ans. £187 : 10.

DOUBLE POSITION

Is hy 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 ihe 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 £0 more than A, and C £9 more than B ; how much must each have ?

Suppose A hail 40 Then suppose A had 50 Then B had 4G then B must have 56 and C 54 and C 64

140 too little by GO. 170 too little by 30.

Bup. errors. 40 wGO

50 ^ 30 GO 60 A

30 G6 B

3000 1200 — 74 C

1200 30 divisor.

3i0)180|0

CO Ans. for A.

200 proof.

2. A man had two silver cups of unequal weight, having one cover to both, of 5 07 , now if the cover is put on the less cup, it will double tlie weight of the greater cup ; and set on th« greater cup, it will be thrice as heavy as the less cup. What is the weight of each cup ?

Am. 3 ounces less, 4 greater

EXCHANGE. 91

3. A gentleman bought a house, with a garden, and a horse in the stable, for £500 ; now he paid 4 times the price of the horse for the garden, and 5 times the price of the garden for the house. What was the value of the house, garden, and horse, separately ?

^715. horse £5S), garden £80, house £400.

4. Three persons discoursed concerning their ages : says If, I am 30 years of age ; says K, I am as old as H and ■}- of L ; and says L, I am as old as you both. What was the age of each person ?

Ans. H 30, K 50, and L 80.

5. D, E, and F, playing at cards, staked 324 crowns ; but dis- puting about the tricks, each man took as many as he could : D got a certain number ; E as many as D, and 15 more ; and F got a fifth part of both their sums added together. How many did- «ach get?

Ans. D 127^, E 142|, and F 54.

6. A gentleman going into a garden, meets with some ladies, and says to them. Good morning to you 10 fair maids. Sir, you mistake, answered one of them, we are not 10; but if we were twice as many more as we are, we should be as many above 10 as we are now under. How many were they?

Ans. 5.

EXCHANGE

Is receiving money in one country for the same value paid ir> another.

The par of exchange is always fixed and certain, it being the intrinsic value of foreign money, compared with sterling ; but the course of exchange rises and falls upon various occasions.

I. FRANCE.

They keep their accounts at Paris, Lyons, and Rouen, in livres, sols, and deniers, and exchange by the crowa=4s. 6d. at par.

Note. 12 deniers make 1 sol.

20 sols 1 livre.

3 livers 1 crown.

9S8 kXCHANOE.

To change French into Sterling. RuLK. As 1 crown : is i(» the given rate : : so is the French sum : to tlie sterling required.

To change Sterling info French. Rule. As the rate of ex hani;e : is to 1 crown : : so ia the sterling suni : to the French re juired.

EXAMPLES. 1. How many crowns must be paid at Paris, to receive in London £l8d exchanged at 48. (>d. per crown?

d. c. £ As 54: 1 :: 180:800. 240

54)43300(800 crown*. 43-2

2. How much sterling muft be paid in London, to receive in Paris 758 crowns, exchanged at 5()d. per crown ?

Ans. £176 : 17 : 4. • 3. A merchant in London remits £176 : 17 : 4, to his corres- pondent at Paris; what is the value in French crowns, at 56d. per crown ? Atis. 758.

4. Change 725 crowns, 17 sds, 7 deniers, at 54?d. per crown, into sterling, what is the snm? Ans. £164 : 14 : O^d.^f.

5. Change £1(54 : I 4 : OA sterling, into French crowns, ex- change at 54^d. per crown ?

Ans. 725 crowns, 17 sols, 7 -jVj deniers.

H. SPAIN.

They keep their accounts at Madrid, Cadiz anrl Seville, in dollars, rials, and maravedies, and exchange by the piece of eight x=4s. 6d. at par.

Note. 34 marnvedics make 1 rial.

8 rinli) 1 |iin!>tre or piece of dghL

10 rials 1 dollar.

Rule. As with France.

EXAMPLES. 0. A merchant at Cadiz r" nils to London 2547 pieces of eight, at 5Gd. per piece, how much sterling is the sum ?

^71.?. £594 : a

- EXeHANGE. 03

I

^ 7. IIow manv pieces of eight, at 56 J. each, will answer a bill of £594 : 0, sterlinir ? Ans. 2547.

8. If I p;iv a bill here of £2500, what Spanish money may I draw my bill for at Madrid, exchange at ST.^d. per piece of eight ? Ans. 10434 pieces of eight, 6 rials, 8 mar. ff.

III. ITALY.

They keep their accounts at Genoa and Leghorn, in Hvres, sols, and deniers, and exchange by the piece of eight, or dollar =43. 6d. at par.

Note. 12 deniers make 1 sol. 20 sols 1 livre.

5 livres 1 piece of eight at Genoa.

6 livres 1 piece of eight at Leghorn.

N. B. The exchange at Florence is by ducatoons; the exchange at Venic* by ducats.

NoT£. 6 solidi make 1 gross. 24 gross . . . , 1 ducat.

Rule. Same as before.

9. How much sterling money may a person receive in London, if he pays in Genoa 976 dollars, at 53d. per dollar 1 Aiis. £215 . 10 . 8.

10. A factor has sold goo.ls at Florence, for 250 ducatoons, at 54d. each; what is the value in pounds sterling 7 Ans. £56 .5.0.

11. If 275 ducats, at 4s. 5d. each, be remitted from Venice to London ; what is the value in pounds sterling ■? Ans. £60 . 14 . 7.

12. A gentleman travelling would exchange £G0 . 14 . 7, sterling, for Venice ducats, at 4s. 5d. each ; how many must he receive 1

Ans. 275.

IV. PORTUGAL.

They keep their accounts at Oporto and Lisbon, in reas, ami exchange by the milrea=6s. S^d. at par.

NoTK. 1000 Teas make 1 milrea. Rule. The same as with France.

EXAMPLES.

13. A gentleman being desirous to remit to his correspondent in London 2750 milreas, exchange at Gs. ud. per milrea ; how much sterling will he be the creditor for in London 1 Ans. £882 . 5 . 10.

14. A merchant at Oporto remits to London 4366 milreas, and 183 reas, at &*■ 5td. exchange per niibrea ; how much sterling must be paid in London for this remittance "? Ans. £ 1 193 . 17 . 6|, 0375.

15. If I pay a bill in London of £1193 . 17 . 6i, 0375, what must I draw for on my correspondent in Lisbon, exchange at 5s. 5|d. per milrea ?

An$. 4336 oiilreas, 183 reas.

^i EXCHANGE.

V. HOLLAND, FLANDERS, AND GERMANY.

They keep iheir accounts at Antwerp, Amsterdam, Brussels, Rotterdam, and Hamburgh, some in pounds, shillings, and pence, as in England ; others in guilders, stivers, and pennings ; and exchange with us in our pound, at 33s. 4d. Flemish, at par.

NoT£. 8 pennings make 1 groat.

a groate, or IG pennings .... 1 stiver. SO stivers 1 guilder or florin.

ALSO,

12 groats, or G stivers make. . 1 schelling. 20 schelliiigs, or G guilders. . . 1 pound.

To change Flemish into Sterling.

Rule. As the given rate : is to one pound : : so is the Flemish sum : to the sterling required.

To change 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 . 0 sterling, how many pounds Flemish is the sum, the exchange at 333. Gd. Flemi8h,_per pound sterling ■{ ilTw. £1263 . 15 . 9, Flemish.

17. A merchant in Rotterdam remits £1263 . 15 . 9 Flemish, to be ptdd in London, how much stcrlin;^ money must he draw for, the exchange being at 338. iJd. Flemish per pound sterhng 1 Ans. £754 . 10.

18. If I pay in London £852 . 12 . 6, sterling, how many guilders must 1 driaw for at Amsterdam, exchange at 34 schei. 4§ groats Flemish per pound fterling 1 Ans. 8792 guild. 13 stiv. 14 i pennings.

19. What must I draw for at London, if I pay in Amsterdam 8792 guild. 13 stiv. 14| pennings, exchange at 34 schel. 4} groats per pound sterling 1

Ans. £852 . 12 . 6.

To convert Bank Money into Current, and the contrary.

Note. The Bank Money is worth more than the Current. The diflference between one and the other is called agio, and is generally from 3 to 6 per cent, in favour of the Bank.

To change Bank into Current Money.

Rule. As 100 guilders Bank : is to 100 with the agio added : : ISO is the Bank given : to the Current required.

J

EXCHANGE. 96

To change Current Money into Bank.

Rule. As 100 with the agio is added : is to 100 Bank : : so is the Current money given : to the Bank required.

20. Change 794 guilders, 15 stivers, Current Money, into Bank florins agio 4|- per cent.

Ans. 7(U guilders, 8 stivers. ll-J-f-f- 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 gentleman remits to Ireland £575 : 15, sterling, what will he receive there, the exchange being at 10 per cent. ?

Ans. £633 : 6 : 6.

23. What must be paid in London for a remittance of £633 ', a : 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. 269^.

2. If 12 yards at London make 8 ells at Paris, how many ells at Paris will make 64 yards at London ?

Ans. 42^.

3. If 30 lb. at London make 29 lb. at Amsterdam, how many lb. at London will be equal to 350 lb. at Amsterdam ?

Ans. 375.

4. If 95 lb, Flemish make 100 lb. English, how many lb. En^ glish are equal to 275 lb. Flemish.

Ans. 289tf .

CONJOINED PROPORTION

Is when the coin, weights, or measures of several countries are compared in the same question ; or, it is linking together a varie- ty of proportions.

•When it is required to 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.

Left.

Right.

20

23

155

180

TO

06 PROPOttTION.

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 divisor.

Proof. By as many single Rules of Three as the question requires.

EXAMPLES.

1. If 20 lb. at London make 2.3 11). at Antwerp, and 155 lb. at Antwerp make 180 lb. at Leghorn, how many lb. at London are equal to 72 lb. at Leghorn ?

20 X 155 X 72 = 223200

23 X 180 = 4i40)223200(53i|f.

2. If 12 lb. at London make 101b. at Amsterdam, and 1001b. 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 1 56 braces at Leghorn, and 7 braces at Leghorn eqiial to 4 ells English, how many bra«* ces at Venice are ^qual to 10 ells English?

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. 112WW-

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 right liand ; then mul» tiply the first row for a divisor, and the second for a dividend.

PROeRESSION. 97-

.„, EXAMPLES.

6. If 12 lb. at London make 10 lb. at Amsterdam, and 1001b. at Amsterdam 1201b. atThoulouse, how many lb. atThoulouse are equal to 40 lb. at London ? Ans. 40 lb.

6. If 40 lb. at London make 36 lb. at Amsterdam, and 90 lb. at Amsterdam 116 lb. at Dantzick, how many lb. at Dantzick are equal to 122 lb. at London ? Ans. HlfffJ.

PROGRESSION

CONSISTS OF TWO PARTS,

ARITHMETICAX. 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, 5, 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, tiie sum of the two extremes will be equal to the two middle numbers, or any two means equally distant from the extcemes : as 2, 4, 6, 8, 10, 12, where 6 -f- 8, the two middle numbers, are=12 + 2, the two extremes, and=10 + 4the two means=l4.

When the number of terms are odd, the double of the middle term will be equal to the two extremes ; or of any two means equally distant from the middle term ; as 1, 2, 3, 4, 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.

2. The last teim, L.

3. The number of terms, .... N.

4. The equal difference, D.

5. The sum of all terms, S.

Any diree of which being given, the other two may be found. The first, second, and third terms given, to find the fifth.

Rule. Multiply the sum of the two extremes by half the number of terms, or multiply half the sum of the two extremes

US PROORESSIOir.

Ly the whole number of terms, the product is the total of all the ter^s : or thus,

I. F L N are given to find 8.

N

F+Lx— =8.

EXAMPLES.

1. Hoir many strokes does the hammer of a cloffk strike ia IS hours t

12-}-l=13, then 13x6^78.

2. A man bought 17 yards of cloth, and gave for the first yard Za, and Xgir the last lOs. what did the 17 yards amount to?

Ans. £3 , 2.

3. If 100 eggs were placed in a right line, exactly a yard as- under from one another, and the first a yard from a basket, what length of ground does that man go who gathers up these 100 eggs singly, and returns with every egg to the basket to put it in ?

Ans. 5 miles, 1300 yards.

The first, second, and third terms givep, to .find the fourth.

RtJLE. From the second subtract the first, the remainder divi- ded by the third less one, gives the fourth : or thus

II. F L N are given to find I^. L— F

-=D.

N— 1

EXAMPLES.

4. X man had eight sons, the youngest was 4 years old, and the eldest 32, they increase in Arithmetical Progression, what was tlie common difl^erence of their ages? Ans. 4.

32 — 4=28, then 28-5-8—1=4 common difference.

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 an equal excess, so that the last day** journey may be 58 miles.

PROGRESSION.

-^

what is the daily increase, and how many miles distant is that place from London ? Ans. 5 daily increase.

Therefore, as three miles is the first day's journey,

3+5=8 the second day. 8-}-5=13 the third day, &c. The whole distance is 366 miles.

The first, second, and fourth terms given, to find the third. Role. From the second subtract the first, the remainder divide by the fourth, and to the quotient add 1, gives the third ; or thus,

III. F L D are given to find N. L— F

+1=N.

D

EXAMPLES.

6. A person travelling into the country, went 3 mile* 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=11+1=12 the number of days.

7. A man being asked how many sons he had, said, that the youngest was 4 years old, and the oldest 32 ; and that he increas- ed one in his family every 4 years, how many had he ?

Ans. 8.

The second, third, and fourth terms given to find the first

Rule. 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 ia the country, every day's journey increasing the former by 4, and the last he went was 46 miles, what the first ?

Ans. 10 miles. 4x10— lc=36, then 46 — 36=10, the first day's journey.

0#100 PROOE88IOX.

9. A man takes out of his pocket at 8 several times, so many dift'erent numbers of sliillings, every one exceeding the formei by 6, the last at 46; what was the first? Ans. 4.

The fourth, third, and fifth given, to find the first.

Rule, Divide the fifth by the third, and iVom 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 wliat it is. What will that person have for his pains ? Ans. £8.

4X12—1

360-5-12=30, then 30 =£8 the first payment.

2

The first, third, and fourth, given to find the second.

Rule. Subtract the fourth from the product of the third, mul- tiplied by the fourth, that remainder added to the first gives the second : or thus,

VI, F N D are given to find L. ND— D-1-F=L

EXAMPLES.

11. What is the last number of an Arithmetical Progression, beffinning at 6, and continuing by the increase of 8 to 20 places?

Ans. 158.

20X8—8=152, then 152+6=158, the last number.

GEOMETRICAL PROGRESSION

Is the increasing or decreasing of any rank of numbers by some common ratio ; that is, by the continual multiplication or division of some equal number : as 2, 4, 8, 16, increase by the multipliei 2, and 16, 8, 4, 2, decrease by the divisor 2.

PROGRESSION. 101

Note. "When any number of terms is continued in Geome- trical Progression, the product of th^two extremes will be equa! 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 obser fed as are in Arithmetical, viz.

1. The first tenn.

2. The last term.

3. The number of terms.

4. The equal difference or ratio.

5. The sum of all the terms.

Note. As the last term in a long series of numbers is very tedious to come at, by continual multiplication ; therefore, for the reader finding it out, there is a series of numbers made use ot in Arithmetical Proportion, called indices, (beginning Avith 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.

. 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, the sum of the indices made choice of must always be one less than the number of terms given in the question ; for 1 in the indices is over the second term, and 2 over the third, &c.

Add any two of the indices together, and that simi 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 24- 4= 6 4x16= 64 13

lO% PR00RES9I0N.

In any Geometrical Progression proceeding from unity, the ratio being known, to find atfy remote term, without producing all the intermediate terms.

Rule. Find what figures of the indices added together would gire the exponent of the term wanted : then multiply the num- bers standing under such exponents into each other, and it will give the term required.

Note. >^hen the exponent 1 stands over the second term, the number of exponents must be one less than the number ot 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 gir* for them ? Ans. £2 . 3 . 8L

16=4

0, 1, 2, 3, 4, Exponents 16=4

1, 2, 4, 8, 16, No. of terms.

256=8 8=3

For 4+4+3=11, No. of terms less 1

4)2048=11 No. offer.

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.

PROOBES8IO\. 103

Role. Proceed as in the last, only observe, that every product imist be divided by the first tei-m.

EXAMPLES.

3. A sum of money is to be divided among eight persons, the first to have £20, the next £60, and so in triple proportion ; what will the last have ? Ans. £43740.

640X540 14580X60

20 20

3+3+1=7, one less than the number of terms.

4. A gentleman dying, left nine sons, to whom and to his exe- cutors he bequeathed his estate in the manner following : To his executor.s £50, his youngest son was to have as much more as the executors, and each son to exceed the next younger by as much more ; what was the eldest son's proportion ?

Ans. £25600.

The first term, ratio, and number of terms given, to find the «nm of all the terms.

Rule. Find the last term as before, then subtract the first from it, and divide the remainder by the ratio, less 1 ; to the quo- tient of which add the greater, gives the sum required.

EXAMPLES.

5. A servant skilled in numbers, agreed with a gentleman to serve him twelve months, provided he would give him a farthing for his first month's service, a penny for the second, and 4d. for the third, &.e., what did his wages amount to ?

Ans. £5825 . 8 . 5}.

256X256=65530, then 65536x64=4191304

0, 1, 2, 3, 4, 4194304—1

1, 4, 16, 64, 256, =1399101, then

4+4+3=11 No. of terms less 1, 4—1

1303101+4194304=5592405 farthings.

6. A man bought a horse, and by agreement was to give a far- thing for the first nail, three for the second, &c., there were four shoes, and in each shoe 8 nails ; what was the worth of the horse ?

Ans. £965114aS1693 . 13 . 4.

104 PERMUTATION.

7. A certain person married his daughter on New-year's day, and gave her husband Is. towards her marriage portion, promise ing to double it on the first day of every month for 1 year ; what was her portion ?

Ans. £204 . 15.

8. A laceraan, 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, «fcc., 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 ?

1X2X3X4X5X6X7X8X9X10X11X 12=479001600 changes, which -i- 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, what his diet would cost for a year, who told him £10, but the scholar not being certain what time he should stay, asked him what he must give him for so long as he should place his family, (consisting of 6 persons besides himself) 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 di'l the scholar stay with the gentleman?

Ans. 5040 days.

106

THE

TUTOR'S ASSISTANT.

PART 11.

VULGAR FRACTIONS.

A rRACTioN is a part or parts of an unit, and written with two Agures, with a line between them, as ^, f , ^, &c.

The figure above the line is called the numerator, and the un- der one the denominator ; which shows how many parts th« anit is divided into : and the numerator shows how many of those parts are meant by the fraction.

There are four sorts of viilgar fractions : proper, improper, compound, and mixed, viz.

1. A PROPER FRACTION is when the numerator is less than the denominator, as f , f , f, -Jf , J^, &,c.

2. An IMPROPER FRACTION is when the numerator is eqoal to, or greater than the denominator, as ^, -J, ;J^, -i^, &c.

3. A COMPOUND FRACTION is the fraction of a fraction, and known by the word of, as -^ of f of f of ^^ of A' *;c.

4. A MIXED NUMBER, OR FRACTION, ie composed of a wholes number and fraction, as 8f, 17^, 8f|, (Sec.

106 REDUCTION OF 7tTtOAR 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, fl and if. 1st num. 2d num. 2x7=14 4x4=16, then 4x7=28 den.=;^ and ^f .

2. Reduce -J-, ^, and f, to a common denominator,

Farit JL2. AA A^

3. Reduce f , f , -j^, and f, to a common denominator.

Pnpit g B* n g g* q g n 1 8 g mm

X ai^ll', 8 360* 8 36 0* 8360' 8160'

4. Reduce -/'g, f , -f, and f , to a common denominator.

'Pa<«if 1 n n » «*n g« o win

r <ll.ll, 18 8 0* 1680' 1680 ' 1698 •

5. Reduce ^ f , f, and -J, to a common denominator,

FariL -"SJ-i -Bgfl -a-flo J-Oj.

X-a^li, 8 4 0» 8 4 0' 8 4 0' STo*

(}. Reduce f , f , f , and f , to a common danominator.

'G'npit ?gO ,1 gqo K 4 D i g«« X-itl,!!., 2 I 6 U' 2160' a 1 6 0» 3 16 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 following, till nothing remain : the last divisor is the common measure ; then divide both parts of the fraction by the common measure, and llie quotient will give the fraction required. Note. If the common measure happens to be one, the fraction is already in its lowest term : and when a fraction hath ciphers at the right hand, it may be abbreviated by cutting them off, as f |f .

EXAMPLES. 7. Reduce f^ to its lowest terms. 24)32(1 24

Com. measure, 8)24(3 Facii.

itkfttroTlOW or VVLOAR FRACTIONS. lOT

8, Rednce -^ to its lowest terms. Facit, ^.

9. Reduce -f-g-J- to its lowest terms. Facit, -^.

10. Reduce fff to its lowest terms. Facit, f .

1 1. Reduce fff to its lowest terms. Facit, -ff .

12. Reduce f^ to its lowest terms. Facit, f .

3. To reduce a mixed number to an improper fraction.

Rule. IVTultiply the whole number by the denominator of the fraction, and to the product add the numerator for a new numerator, which place over the denominator.

Note. To express a whole number fraction-ways setl for the denominator given.

EXAMPLES.

13. Reduce 18f^ to an improper fraction. Facit, -^.

18X7+3=129 new numerator=>f ».

14. Reduce 56^ to an improper fraction, Facit, ^ | ^ '.

15. Reduce 183^ to an improper fraction. Facit, '|+'.

16. Reduce \3f to an improper fraction. Facit, *^.

17. Reduce 27^ to an improper fraction. Facit, 'f *.

18. Reduce 514-j^ to an improper fraction. Facit, 'ff *.

4. To reduce an improper fraction to its proper terras. RvLE. Divide the upper term by the lower.

EXAMPLES.

19. Reduce 'f " to its proper terms. Facit, 18^^.

129+7=18f,

20. Reduce *|f • to its proper terms. Facit, 56ff . 31. Reduce »f|8 to its proper terms. Facit, 183/^.

22. Reduce ^ to its proper terms. Facit, 13|-.

23. Reduce 'f to its proper terms. Facit, 27f.

24. Reduce sff » to its proper terms. Facit, 514^^.-

5. To reduce a compound fraction to a single one.

Rule. Multiply all the numerators for a new numerator, and all the denominators for a new denominator.

Reduce the new fraction to its lowest terms bv Rule 2.-

106 REDUCTION OF VULGAR FRACTIONS.

EXAMPLES.

35. Reduce f of ^ of f to a single fraction.

2X3X5= 30

Facit, — reduced to the lowest term=|.

3X5X8=120

36. Reduce ^ of 4^ of -^ to a single fraction.

Facit, f|i=-iVy

37. Reduce -J^ of ^ of f^ to a single fraction.

■parit AflJLa=ii-a raciv, 4 8 7a— a 88'

38. Reduce J of f of ^^ to a single fraction.

racii, 2 4 0 — !«•

39. Reduce f of f of f to a single fraction.

Facit, in=A-

30. Reduce f of ^ of ^^ to a single fraction.

•T dOll, , g g — I g.

' 6 To reouce 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.

31. Reduce -f- of a penny to the fraction of a pound.

Facit, f of ,J-2 of A=T^-

32. Reduce •}■ of a penny to the fraction of a pound.

Facit, r^.

33. Reduce f of a dwt. to the fraction of a lb. troy.

Facit, -nVo*

34. Reduce f of a lb. avoirdupois to the fraction of a cwt.

Facit, tIt-

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 denominations between it, and that yon would reduce it to, for a new numerator, and place it over the given denominator^

REDUCTION OF VULGAR FRACTIONS. 109

EXAMPLES.

35. Reduce y^To of a pound to the fraction of a penny.

Facit, f . 7 X 20 X 12=1680 -}-|f^ reduced to its lowest term=f .

36. Reduce yj^ of a pound to the fraction of a penny.

Facit, i.

37. Reduce •oVo' of a pound troy, to the fraction of a penny- weight. Facit, f .

^. Reduce yfj 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.

RcL£. As the numerator of the given fraction : is to its deno- minator : : so is the numerator of the intended fraction : to its denominator.

EXAMPLES.

39. Reduce f to a fraction of the same value^ whose numera- tor shall be 12. As 2 : 3 : : 12 : 18. Facit, -Jf.

40. Reduce ^to a fraction of the same value^ whose numera- tor shall be 25. Facit, ff^.

41. Reduce -f- to a fraction of the same value, whose numera- tor shall be 47. 47

Facit,

65f.

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 iti numerator : : so is the denominator of the intended fraction : to its numerator.

EXAMPLES.

42. Reduce -I to a fraction of the samevalue, whose denomi- Bator shall be 18. As 3 : 2 : : 18 : 12. Facit, if.

43. Reduce -f-to a fraction of the same value, whose denomi- iiator shall be 35. Facit, -ff.

44. Reduce ^ to a fraction of the same value, whose denomi- Dator shall be 65f . 47

Facit,

65t.

110 BB»VCTION OF VVLaAR FHACTIONS.

10. To reduce a mixed fraction to a single one.

Rule. When the numerator is the integral part, multiply it' by the denominator of the fractional part, adding in the numerator of the fractional part for a new numerator ; then multiply the de- nominator of the fraction by the denominator of the fractional part for a new denominator.

EXAMPLES.

36f 45. Reduce — to a simple fraction. Facit, 44^="

48

36 X 3 + 2=110 numerator. 48X3 =144 denominator. ' 23f

46^. Reduce — to a simple fraction, Facit, i-ff=-i"^.

38 When the denominator is the integral part, multiply it by the denominator of the fractional part,- adding in the numerator of the fractional part for a new denominator ; then multiply the numerator of the fraction by the denominator of the fractional part for a new numerator.

EXAMPLES. 47

47. Reduce — to a simple fraction. Facit, ^1-?^=^.

* e&f

19

48. Reduce — to a simple fraction. Facit, -ftV=*.

44i

1 1 . To find the proper quantity of a fraction in the known parts of an integer.

Rule. Multiply the numerator by the common parts of the integer, and divide by the denominator.

EXAMPLES.

49. Reduce f of a pound sterling to its proper quantity.

3 X 20=60-^4=1 5s. Facit, 1 Ss.

50. Reduce f of a shilling to its proper quantity.

Facit, 4d. 3| qrs.

61. Reduce 4- of a pound avoirdupois to its proper quantity

Facit, 9 oz. 2* dr. "

62. Reduce ^ of a cwt. to its proper quantity.

Facit, 3 qrs. 3 lb. 1 oz. 12f dr.

REDOtrrioir of tvlgar fractions. Ill

33. Reduce f of a pound troy to its proper quantity.

Facit, 7 oz. 4 dwta.

54. Reduce ^ of an ell English to its proper, quantity.

Facit, 2 qrs. 3^ naila.

55. Reduce f^of a mile to its proper quantity'.

Facit, 6 fur. 16 poles.

56. Reduce f of an acre to its proper quantity.

Facit, 2 roods, 20 poles.

57. Reduce -f^ of a hogshead of wine to its proper quantity.

Facit, 54 gallons.

58. Reduce f of a barrel of beer to its proper quantity.

Facit, 12 gallons,

59. Reduce -^ofa. chaldron of coals to its proper quantity.

Facit, 15 Bushels. CO- 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.

RuLC. Reduce the given quantity to the lowest term men- tioned for a numerator, under which set the integral part reduced to the same term, for a denominator, and it will give the fraction required.

EXAMPLES.

61. Reduce 15s. to the fraction of a pound sterling.

Facit, i^=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. 12f dr. to the fraction of a cwt.

Facit, f .

65. 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 elJ.

Facit, \.

67. Reduce 6 fur. 16 poles to the fraction of a mile.

Facit, f.

68. Reduce 2 roods 20 poles to the fraction of an acre.

Facit, -f. •69. Reduce 54 gallons to the fraction of a hogshead of wine.

Facit, |.

Il2 SUBTRACTION OF VULGAR FRACTIONS.

70 Reduce 12 gallons to the fraction of a barrel of beer.

Facit, f .

71. Reduce fifteen bushels to the fraction of a chaldron of coals.

Facit, |-\.

72. Reduce 2 weeks, 2 days, 19 hours, 12 minutes, to the fraction of a month. Facit, f .

ADDITION OF VULGAR FRACTIONS.

Rule. Reduce the given fractions to a common denominator,, then add all the numerators together, under which place the com- mon denominator.

EXAMPLES.

1. Add f and t together. Facit, ^-l-H-=^?=luV

2. Add ^, f and f together. Facit, Ifff .

3. Add ^, 4| and f together. Facit, 4f J.

4. Add 7f and f together. Facit, ^.

5. Add f and f of ^ together. Facit, •^.

6. Add 5f, 6} and ^ together. Facit, n^. 2. When the fractions are of several denominations, reduce

them to their proper quantity, and add as before.

7. Add ^ of a pound to f of a shilling. Facit, 15s. lOd.

8. Add i of a penny to f of a pound. Facit, 13s. 4^d.

9. Add ^ of a pound troy to -J- of an ounce.

Facit, 9 oz. 3 dwts. 8 grs.

10. Add f of a ton to f of a lb.

Facit, 16 cwt. 0 qrs. 0 lb. 13 oz. 5^- dr.

11. Add f of a chaldron to ^ of a bushel.

Facit, 24 bushels 3 pecks.

12. Add -J- of a yard to f of an inch.

Facit, 6 inch. 2 bar. c.

SUBTRACTION OF VULGAR FRACTIONS.

Rule. Reduce the given fraction to a common denominator, then subtract the less numerator from the greater, and place the remainder over the common denominator.

MULTIPLICATION. 113i

2. When the lower fraction 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 f take i 3X7=21. 5X4=20. 21—20=1 num.

4 X 7=28 den. Facit, ^V-

2. From f take f of f . Facit, f}-.

3. From 5f take -i^. Facit, 4f^.

4. From ^ take f. Facit, ■^.

5. 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 -5- 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 ^ of a lb. troy take -J- of an ounce.

Facit, 8 oz. 16 dwts. 16 grs. 16. From -f- of a ton take -§■ of a lb.

Facit, 15 cwt. 3 qrs. 27 lb. 2 oz. lOf drs.

11. From-|- of a chaldron, take -f- of a bushel.

Facit, 23 bushels, 1 peck

12. From -J- of a yard, take f of an inch.

Facit, 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. 4X5=20 den. — j^.

2. Multiply ^ bv f. Facit, ^.

3. Multiply 48f'by \^. Facit, 67^.

4. Multiply 43O1V bv 18f-. Facit, 7935ff .

5. Multiply i\ by i of f of f. Facit, 2»^=if.

6. Multiply fo bv f of f of f. Facit, f.

K3

t|4 SINGLE RX7LE Op THREE DIRECT.

7. Multiply i of f by | of |. Facit, \.

6. Multiply i of f by f. Facit, ^.

9. Multiply 5f by f- Facit, 4iJ.

10. Multiply 24 by f . Facit, 16.

11. Multiply i of 9 by f. Facit, 5*^.

12. Multiply 9J by f . Facit, 31-

DIVISION OF VULGAR FRACTONB.

RcLE. Prepare the given numbers (if they require it) byjth« rules of Reduction, and invert the divisor, then proceed as io Multiplication.

EXAMPLES.

1. Divide^ by f

Facit, 5X9=45 num. 3X20=60 den. — tf=?-

2. Divide J[^ by f Facit, -J.

3. Divide 672380 by 13|. Facit, 48*.

4. Divide 7935f^ by ISf Facit, 430i.

5. Divide i by f of f of ^ Facit, ■^.

6. Divide f of 16 by ^ of J. Facit, 19^1.

7. Divide ^ of f by f off. Facit, f\=^. a Divide 9^,\ by ^ of 7. Facit, 2^^. 9. Divide -^ by 4^. Facit, j.

10. Divide 16 by 24. Facit, f.

1 1. Divide 5205 jV by f of 91. Facit, 71^.

12. Divide 31 by 9^. Facit, f.

THE SINGLE RULE OF THREE DIRECT, IN VULGAR FRACTIONS.

Role. Reduce the numbers as before directed in Reduction. State the question as in the Rule of Three in whole numbers, and invert the first term in the proportion, then multiply the three terms continually together, and the product will be the answer.

tlNOLB RVLE Or THREE IKVESSB. 1 1&

EXAMPLES.

I. If ^ of a yard cost i^ of £1, what will -^ofn yard come to ftt that rate ? Ans. ii=15s.

yd. £ yd. £

for 4X5 X 9= 180 num. ^•^_i,^^ A^Ai/JJ.r and 3X9X10=240 den. °^«^i»-8» 4i8oUo*"

■ 8. If f of a yard cost f of £1, what will -J^- of a yard cost ?

Ans. 14s. 8d.

3. If ^ of a yard of lawn cost 7s. 3d., what will 10^ yard* cost t Ans. £4:19: 10f|.

4. If f lb. cost ^. how many pounds will f of Is. buy?

Ans. 1 lb.y^=^.

5. If A ell of Holland cost j- of £1, what will 12f ells cost at the same rate ? Ans,£l : 0 : &5" i^-

6. If 12^ yards of cloth cost 15s. 9d., what will 48^ cost at the same rate ? Ans. £3:0:9^ -iW.

7. If "iV of a cwt, cost 284s. what will 7^ cwt. cost at the same rate.? ^ns. £118 : 6 : 8.

8. If 3 yards of broad cloth cost £2|^, what will lOf yards cost? ^n*. £9:12.

9. If i of a yard cost f of £1, what will f of an ell English come to at the same rate? Ans. £2.

10. If 1 lb. of cochineal cost £1 : 5, what will 36 -^^ lb. come to? ^ns. £45: 17:6.

II. If 1 yard of broad cloth cost 15f s., what will 4 pieces cost, «ach containing 27f yards ? Ans. £85 : 14 : 3| -f-f^ or f .

12. Bought 3^ pieces of silk, each containing 24f ells, at 63. d^. per ell. I desire to know what the whole quantity cost ?

Ans. £25 : 17 :2\ -}i.

THE SINGLE RULE OF THREE INVERSE, IN VULGAR FRACTONS.

EXAMPLES.

1 . If 48 men can build a wall in 24^ days, how many men can do the same in 192 days? Ans. &^^. men.

2. If 25f s. will pay for the carriage of 1 cwt. 145^ miles, how Car may 6^ cwt. be carried for the same money?

Ans. 225*^ miles.

116 THB POUBLE RULV OF THREE.

3. If 3J yards of cloth, that is 1^ yard wide, be sufficient to make a cloak, how much must I have of that sort which is f yard wide, to make another of the same bigness ?

Ans. 4f yards.

4. If three men can do a piece of work in 4^ hours, in how many hours will ten men do the same work ?

Ans. 1-^0 hour.

5. If a penny white loaf weighs 7 oz. when a bushel of wheal cost 5$. 6d., what is a bushel worth when a penny white loa. weighs but 2^ oz. ? Ans. 15. 4f d.

6. What quantity of shalloon, that is f yard wide, will line 7$ yards of cloth, that is 1^ yard wide? Ans. 15 yards.

THE DOUBLE RULE OF THREE, IN VULGAR FRACTIONS.

EXAMPLES.

1. If a carrier receives £2-^ for the carriage of 3 cwt. 150 miles, how much ought he to receive for the carriage of 7 cwt. 3^ qrs. 50 miles ? Ans. £1 : 16 : 9.

2. If £100 in 12 months gain £6 interest, what principal will gain £3f in 9 months ? Ans. £75.

3. If 9 students spend £10|^ in 18 days, how much will 2C students spend in 30 days ? Ans. £39 : 18 : 4^\.

4. A man and his wife having laboured one day, earned 4f s. how much rjust they have for 10^ days, when their two sone helped them? Ans. £4 : 17 : 1^.

5. If £50, in 5 months, gain £2^^, what time will £13| re- quire to gain £l-iV ? Ans. 9 months.

0. If the carriage of 60 cwt. 20 miles cost £14^, what weight ean I have carried 30 miles for £51^^ ? Ans. 15 cwt.

** ». "117 ^^^

THE

TUTOR'S ASSISTANT.

PART III.

DECIMAL FRACTIONS.

In Decimal Fractions the integer or whole thing, as one pound, one yard, one gallon, &c. 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 ,5 which stands for fg-, ,25 for yW» '123 for -jVo^.

But the different value of figures appears plainer by the fol- lowing table.

Whole numbers. Decimal parts.

7654321, 2 34567

'sis-?' HffiHxog

5 3 Sao, ,=:

3 ^ o o 5

CO {S p

3 3

? w

From which it plainly appears, that as whole numbers increase b> 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 ADDITICX or DECIMALS.

before 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 ft ; ,05 is 5 parts of 100, or -pj-j^ ; ,005 is 5 parts of 1000, or ToVt ; »0005 is 5 parts of 10000, or Tofoo* B"^ ciphers after decimal parts do not alter their value. For ,5, ,50, ,500, &c. are each but -ft 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 whcrcin one or more figure* are continually repeated, as 2,75222.

And 52,275275275 is called a compound recurriho dbci

MAL.

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 an unit more ; thus, for 26,25999, write 26, 26.

In all circulating numbers, dash the last figure.

ADDITION OF DECIMALS.

Rule. 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 du« regard had to the commas, or separating 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 number*.

EXAMPLES.

1. Add 72,5+32,071 + 2,1574-1-371,4 + 2,75.

Facit, 480,8781

2. Add 30,07 + 2,0071 + 59,432 + 7,1 .

5. Add 3,5 + 47,25 + 927,01 + 2,0073 + 1 ,5. 4. Add 52,75 + 47,21 + 724 + 31,452 +,3075.

6. Add 3275 + 27,514 + 1 ,005 + 725 + 7,32. 0. Add 27,5 + 52 + 3,2675 +,574 1 + 2720.

[VLTIPLICATION 07 DKCIHALS.

11

SUBTRACTION OF DECIMALS.

Rule. Subtraction of decimals differs but little trom whole numbers, only in placing the numbers, which must be carefully obserred, as in addition.

EXAMPLES.

1. Prom ,2754 take ,2371.

2. From ,237 take 1,76.

3. From 271 take 215,7.

4. From 270,2 tak« 75,4075.

5. From 571 take 54,72.

6. From G25 take 76,91.

7. From 23,415 take ,3742.

8. From ,107 take ,0007.

MULTIPLICATION OF DECIMALS.

RvLB. Place the factors, and multiply them, as in whole num- bers, and from the product towards the right hand, cut off a» many places for decimals as there are in both factors together j but if there should not be so many places in the product, sup- ply the defect with ciphers to the left hand.

EXAMPLES.

1. Maltiplj ,2365 by ,2435.

2. Multiply 3071 by 2,27.

3. Multiply 27,15 by 25,3.

4. Multiply 72347 by 23,15.

5. Multiply 17105 by ,3257.

6. Multiply 17105 by ,0237.

Facit, ,06758775.

7. Multiply27,35 by 7,70071.

8. Multiply 57,21 by ,0075.

9. Multiply ,007 by ,007.

10. Multiply 20,15 by ,2705.

11. MulUply ,907 by ,0025.

When any number of decimals is to be multiplied by 10, 100, 1000, &c., it is only removing the separating point in the multi* plicand so many places towards the right hand as there are ciphers m the multiplier: thus, ,578 X 10 = 5,78. ,578x100 = 5,78. ,b79 X 1000 = 578 ; and ,578 X 10000 = 5780.

CONTRACTED MULTIPLICATION OF DECIMALS.

RcLB. Put the unit's place of the multiplier under that place of the multiplicand that is intended to be kept in the product, then invert the order of all the other fiijures. i. e. write thera all the

tX) CONTRACTED MVLTIPLICATIOIC.

contrary way ; and in multiplying, begin at the figure in the mul- tiplicand, wjiich stands over the figure you are then multiplying wth, and set down the first figure of each particular product di- rectly one under the other, and have a due regard to the increase prising 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 35, 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.

Common way. 384,672158

384,672158

5438,63

36,8345

115401647

1923360790

23080329

153HG 88632

3077377

1154016474

115402

3077377

264

15387

23080329

48

1923

115401647

4

14169,2065

14169,2066

038510

Fac

it, 14169,2065

13. MulUply 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 375,13758 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 places of decimals. Facit, 299,0699.

DIVISION OF DECIMALS. 121

DIVISION OF DECIMALS,

This Rule is also worked as in whole numbers ; the only dif- ficulty is in valuing the quotient, which is done by any of the fol- lowing rules :

Rule 1. The first figure in the quotient is always of the same ralue with that figure of the dividend, which answers or stands orer the place of units in the divisor.

2. The quotient must always have so many decimal places, •8 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. 1. Divide 85643,825 by 6,321, Fadt 13549.

2 Ditide 48 by 144

3. Divide 217.75 by 65.

4. Divide 125 by ,1045.

5. Divide 709 by 2,574.

6. Divide 5,714 by 8275.

7. Divide 7382,54 by 6,4252.

8. Divide ,0851648 by 423.

9. Divide 267,15975 by l3,2a

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 tlie left hand, as there are ciphers in the divisor.

Thus, 5784-5- 10=578,4. i 5784-*- 1000=^5,784.

57^+100=57,84. 1 5784-f-10,000=,5784.

122

CONTRACTED DITISXOIT.

CONTRACTED DIVISION OF DECIMALS.

RvLE. 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 dividirvg, emU, one figure of the divisor at each following operation.

Note. That in multiplying every figure left out in the divisor, you must carry I, if it be 5 or upwards, to 15 ; if 15, or Upwards, to 35, carry 2 ; if 25, or upwards, to 35, carry 3, &c.

EXAMPLE.

12. Divide 721,17562 by 2,257432, and let there be only thrw places of decimals in the quotient.

Contracted.

2,257432)721,17562(319,467

67T2296

Common way.

2,257432)721,17562(319,467

6772296

439460. 225743

213717.. 203169..

10548., 9030..

1518.. 1354.,

164 158

439460 225743

213717 903168

10548 9029

1518 1354

120

728

3920 4592

163

158

93280 02024

91256

13. Divide 8,759615 by 5,2714167.

14. Divide 51717591 by 8,7586.

15. Divide 25,1367 by 217,35. 10. Divide 51.47512 bv ,123415. 17. Divide 70,23 by 7,9803.

16. Divide 27,104 by 3,712.

RBDUCTION OF DECIMALS. 12S

REDtrCTtON OF DECIMALS.

To reduce a Vulgar Fraction to a Decimal.

Rule. Add ciphers t6 the numerator-, and divide hy the de- nominator, the quotient is the decimal fraction required.

EXAMPLES.

1. Reduce i .rt...^... to a decimal. 4)1,00(,25 Facit.

2. Reduce ^ to a decimal. Facit, ,5.

3. Reduce % ...: to a decimal. Facit, ,75.

4. Reduce ^ to a decimal. Facit, ,375.

6. Reduce ^g- to a decimal. Facit, ,1923076+.

6. Reduce \\ of ■}-§■. to a decimal. Facit, ,6043950-}-.

Note. If the given parts are of several denominations, they n*y be reduced either by so many distinct operations as there •»'e different parts, or by first reducing them into their lowest denomination, and then divide as before ; or,

2ndly. Bring the lowest into decimals of the next superior de- nomination, and on the right hand of the decimal found, place the parts given of the next superior denomination ; so proceeding till you bring out the decimal parts of the highest integer required, by still dividing the product by the next superior denominator ; or,

3dly. To reduce shillings, pence, and ferthings. 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 number of shillings be odd, the second place of decimals must be increased by 5.

7. Red nee 5s. to the decinwl of a £,. Facit, ,25.

8. Reduce 9s. to the decimal of a £. Facit, ,45.

9. Reduce 16s. to the decimal of a £. Facit, ,8.

L2

»

Ml REDUCTION or PECIMALS.

10. Reduce 8s. 4d. to the decimal of a £.

Facit, ,416<k

11. Reduce IGs. 7|d. to the decimal of a £.

Facit, ,8322916.

fint 168. 7)<L

second. 4)3,00

third. 2)16

7ia.

4

19

199

4

12)7,75 2)0)16,64583

,832

31

960)799(8322916

,8322916

12. Reduce 19s. 5^d. to the decimal of a £.

Facit, 9729ia

13. Reduce 12 grains to the decimal of a lb. troy.

Facit, ,002083.

14. Reduce 12 drams to the decimal of a lb. avoirdupois.

Faci^ ,046876.

15. Reduce 2qrs. 14 lb. to the decimal of a cwt.

Facit, ,628.

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, ,071428+.

19. Reduce 2 gallons, 1 quart of beer, to the decimal of a bar< rel. Facit, ,0625.

20. Reduce 52 days to the decimal of a year.

Facit, 142466+.

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 Uie next inferior denomination, cutting off the decimals from tha product; then multiply the remainder by the next inferior deno- mination ; thus proceeding till you have brought in the least known parts of an integer.

REDUCTION OF DECIMALS. 196

EXAMPLES.

21. What is the value of ,8322916 ot&Yb.f

Ans. 168, 7id.+.

2a

16,645832a 12

7,7499840. 4

2,9999360

22. What is the vahie of ,002084 of a lb. troy ?

Ans. 12,00384 gr.

23. What is the value of ,046875 of a lb. avoirdupois ?

Ans. 12 dr.

24. What is the value of ,625 of a cwt. ?

Ans. 2 qrs. 14 lb. 26. What is the value of ,625 of a gallon?

Ans. 2 quarts 1 pint.

26. What is the value of ,071428 of a hogshead of wine ?

Ans. 4 gallons 1 quart, ,999856.

27. What is the value of ,0625 of a barrel of beer 1

Ans. 2 gallons 1 quart 2a What is the value of ,142465 of a year ?

Ans. 51,999725 dart. U

i*r

126

DECIMAL TABLES OF COIN, WEIGHT, AND MEASURE.

TABLE I. English Coin. £ 1 the Integer.

^h.

Dec.

3h.

19

,95

9

18

.9

8

17

,85

7

16

,8

6

15

,75

&

14

,7

4

13

,65

3

13

,6

2

11

,55-

I

10

,&

Dec. ,45

,4

f

,25

,15

,1 ,05

Pence.

Decimals. ,025 ,020833 ,016666 ,0125 ,008333 ,004166

Farth. 3 2 1

Decimals, ,003125 ,0020833 ,0010416

TABLE II.

EInolish Coin. 1 SL

Long Measure. 1 Foot the Integer.

Penoc & Inches 6 5 4 3

I

Decimals. ,5

,416666 ,333333 ,25

,166666 ,083333

Farth. 3 2 1

Decimals. ,0626 ,041666 ,020833

TABLE HI.

Tbox Weight.

1 lb. the Integer.

Ounces the same as Pence in the last

Table.

Dvfts. 10 9 8 7 6 & 4 3 2 1

Grains. 12

n

10 9 8 7 6 5 4 3 2 1

Dcicimals. ,041666 ,0375 ,033333 ,029166 ,025 ,020833 ,016666 ,0125 ,008333 ,004166

Decimals. ,002083 ,001910 ,001736 ,001562 ,001389 ,001215 ,601042 ,000868 ,000694 ,000521 ,000347 ,000173

Grains 12 11 10 9 8 7 6 S 4 3 2 1

Decimals. ,058 ,022916 ,020833 ,01875 ,016661} ,014583 ,0125 ,010416 ,008333 ,00625 ,004166 ,002083

TABLE rV.

Atoir. Wqoht.

112 lbs. the Integer.

Un. 3

1

Decimals^

f

,36

1 oz. the Integer.

Pennyweights the same as Shillings in the first Table.

Poande.

14

13

12

11

U)

9

8

7

6

5

4

3

2

1

Dechnab. ,125 ,116071 ,107143 ,098214 ,089286 ,080357 ,071428 ,0625 ,053571 ,044643 ,035714 ,026786 ,017857 ,008928

Ounces. 8

Dccinuils. ,004464 ,003906

127

DECIMAL TABLES OF COIK, WEIGHT, AND MEASURE.

,003348 ,002790 ,002232 ,001674 ,001116 ,000558

t Oz. 3 3 1

Decimals. ,000418 ,000279 ,000139

TABLE V.

Atoibddpou weight.

1 lb. the Integer.

Ounces. 1 8 7 6 5 4 3 2 1

Diams. 8 7 6 5 4 3 2 1

Decimals.

,4375

,375

,3135

,25

,1875

,125

,0685

Decimals. ,03125 ,027343 ,023437 ,019531 ,015625 ,011718 ,007812 ,003906

TABLE VL

LmUID tlEASTRE.

I tun the Integer.

Dedmals. ,396825 ,3.37142

80

70

60

50

40

SO

20

10

9

8

7

6

5

4

3

,317460

,27

,238095

,198412

,158730

,119047

,079365

,039682

,035714

,a3l746

,027

,023809

,019841

,01587^

,011904

,007936

,003968

Pints. 4 3

1

Decimals. ,001984 ,001488 ,000992 ,000496

A

Hogshead the

Integer.

Decimal. ,476190 ,317460 ,158730 ,142857 ,126984 ,111111 ,095238 ,079365 ,063492 ,047619 ,031746 ,015673

Pints. 3 2 1

Decimals. ,005952 ,003968 ,001984

TABLE Vn.

Mbisitres.

iiqiikL Dry.

1 G&L 1 Or.

Integer.

Pts. 4 3 2 1

Decimals. ,5

,375 .25 ,125

Bosh. 4 3 2 1

apt.

3 2 1

Decimals. ,09375 ,0625 ,03125

Pck. 3 2 1

Decimals. ,0234375 ^015625 ,0078125

aPks.

3 3 1

Decimals. ,005859 ,008906 • ,001953

Pints. 3 2 1

TABLE VIIL

Long Measttre.

1 Mile the Integer.

DecimalsT ,568182 ,511364 ,454545 ,397727 ,340909

K»

DECIMAL TABLES OF COIN, WEIGHT, AND MEASURE.

500 400 300

aoo

100 90 80 70 CO 50 40 30

ao

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

I ,0000158

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 da; the Integer.

TABLE IX.

Time.

1 year the Integer.

Months the same ns Pence in the second Table.

Decimals.

1,000000 ,821918 ,547945 ,273973 .246675

Hours. 12 11 10

9

8

7

G

5

4

3

2

1

Decimals.

,5

,458333 ,416666 ,37.^> ,333333 ,291666 ,25

,20a'^33 ,166666 125

|083333 ,041666

Minnte&

30

20

10

9

8

7

6

5

4

3

2

1

Decimals. ,020833 ,013888 ,006944 ,00625 ,005555 ,004861 ,004166 ,003472 ,002777 ,002083 ,001389 ,000664

TABLE X.

Cloth measure.

1 Yard the Integer.

duarters the same u Table 4.

Decimals. ,125 ,0625

TABLE XI.

Lead Weioht.

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

Decimals. ,025641 ,012820

Pounds.

14

13

12

11

10

9

8

7

6

5

4

3

2

1

Decimals. ,0064102 ,0059523 ,0054945 ,0050366 ,0045787 ,0041208 ,0036630 ,0032051 ,0027472 ,0022893 ,0018315 ,0013736 ,0009157 ,0004578

«ai RVLB or THREE IN DECIMALS. 129

THE RULE OF THREE IN DECIMALS.

EXAMPLES.

If a6i yards cost £3 : 16 : 3, what will 32^ yards come to?

Ans. £4 : 12 : 9^.

yds. £ yds.

26,5 : 3,8125 : : 32,25 : 32,25

26,5)122,953125(4,63974=£4 : 12 : 9^.

3. What will the pay of 540 men come to, at £1:5:6 per^ man? ^ns. £688 : 10.

3. If 7| yards of cloth cost £2 : 12 : 9, what will 140^ yarda of the same cost ? Ans. £47 : 16 : 3 2,4 qrs.

4. If a chest of sugar, weighing 7 cwt. 2 qrs. 14 lb. cost £36 : 12 : 9, what will 2 cwt. 1 qr. 21 lb. of the same cost ?

Atis. £11 : 14: 2 3,5 qrs.

5. A grocer buys 24 ton 12 cwt. 2 qrs. 14 lb. 12 oz. of tobac- co for £3678 : 6 : 4, what will 1 oz. come to ? Aiis. Id.

6. What will 326^ lb. of tobacco come to, when 1^ lb. is sold for 3s. 6d. ? ^715. £38 : 1 : 3.

7. What is the worth of 19 oz. 3 dwts. 5 grs. of gold, at £2 : 19 per oz. ? Ans. £56 : 10 : 5 2,99 qrs.

8. What is the worth of 827J yards of painting, at lO^d. per yard? ^ns. £36 :4 : 3 1,5 qrs.

9. If I lentmy friend £34 for f of a year, how much ought he lo lend me tV of a year to requite my kindness ? Ans. £51.

10. If I of a yard of cloth, that is 2^ yards broad, make a gar- ment, how much that is f of a yard wide will make the same ?

Ans. 2,109375 yards.

11. If 1 ounce of silver cost 5s. 6d., what is the price of a tan- kard that weighs 1 lb. 10 oz. 10 dwts 4 grs. ?

Ans. £6 : 3 : 9 2,2 qrs,

12. If 1 lb. of tobacco cost 15d. what cost 3 hogsheads, weigh- ing together 15 cwt. 1 qr. 19 Ib.T Ans. £107 : 18 : 9.

13. If 1 cwt. of currants cost £2:9:6, what will 45 cwt. 3 qrs. 14 lb. cost at the same rate ? Ans. £113 : 10 : 9|.

14. Bought 6 chests of sugar, each 6 cwt. 3 qrs. at £2 : 16 per cwt., what do they come to? Ans £113 : 8.

t9P JBXTRACTION OF THE SQUARE ROOT.

15. Bought a tankard for £10 ; 12^ at the rate of Bs. 4d. per ouace, what was the weight f

Ans, 39 oz. 15 dwts.

16. Gave £18") : 3 : 3, for 25 cwt. 3 qrs. 14 lb. of tobacco, ai iiirhat rate did I buy it per lb. ?

Ans, U3|d.

17. Bought 29 lb. 4oz. of coffee, for £10 : 11 : .3, what is the value of 3 lb.? Ans. £1:1:8.

18. If I give Is. Id. for 3| lb. cheese, wha. Avill be the value of 1 cwt. ? Ans. £1 : K : 8.

EXTRACTION OF THE SQUARE ROOT.

Extracting the Square Root is to find out such a number as, being multiplied into itself, the product will be equal to the given num- ber.

Rule. First, Point the given number, beginning at the unit's place, then proceed to the hundreds, and so upon every second figure throughouti

Secondly. Seek the greatest square number in the first point towards the left hand, placing the square number under the first point, and the root thereof in the quotient ; subtract the square number from the first point, and to the remainder bring down the next point and call that the resolvend.

Thirdly. Double the quotient, and place it for a divisor on the left hand of the resolvend ; seek how often the divisor is contain- ed in the resolvend ; (preserving always the unit's place) and pu' the answer in the quotient, and also on the right-hand side of the divisor ; then multiply by the figure last put in the quotient, and subtract the product from the resolvend ; bring down the next point to the remainder if there be any more) and proceed as be- fore.

Roots. 1. 2. 3. 4- 6. 6. 7. 8. 9.

Squares. 4. 9. 16. 26. 36. 49. 64. 81.

feXTRACTIOK OF THE S^VARE ROOT. 131

EXAMPLES. ' 1. What i3 the square root of 119025? Ans. 345.

119025(345 9

64)290 256

685)3425 3425

2. What is the square root of 106929 ? Ans. 327+.

3. What is the square root of 2268741 ? Ans. 1506,23+.

4. What is the square root of 7596796 ? 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 nuraher consists of a whole number and deci- mals together, make the number of decimals even, by adding ci- phers to them ; so that there may be a point fall on the unit's place of the whole number.

7. What is the square root of 3271,4007? Ans. 57,19+.

8. What is the square root of 4795,25731 ? Ans. 69,247+.

9. What is the square root of 4,372594? Ans. 2,091+.

10. What is the square root of 2,2710957? Ans. 1,50701+.

11. What is the square root of ,00032754? Ans. ,01809+.

12. What is the square root of 1,270059 ? ^715. 1,1269+.

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 {i. e.) a number where a root can ne- ver be exactly found, reduce it to a decimal, and extract the root from it.

EXAMPLES.

13. What is the square root of ffff? Ans. f.

14. What is the square root of ffff ? Ans. f.

15. What is the square «•"'•*• "»" 'W\ I Ans. f .

12 s 4 4

132 EXTRACTION OF THE 8RUARE ROOT.

ICTIM.

16. What is the square root of fz^? Ans. ,898024.

17. What is the square root of fff-? Ans. ,86602-f-.

18. What is the square root of fH? Ans. ,933094-.

To extract the Square Root of a mixed number.

KtJLE. Reduce the fractional part of a mixed number to its lowest term, and then the mixed number to an improper fraction.

2. Extract the root of the numerator and denominator for a new numerator and denominator.

If the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the square root therefrom.

EXAMPLES.

19. What is the square root of Slf^-^ -^^S' 'i-

20. What is the square root of 27-i^ ! Ans. b\.

21. What is the square root of 9^ ? Ans. 3f ?

aruRDS.

22. What is the square root of 85H • -^w*- ®»274-

23. What is the square root of 8f ? Ans. 2,95194-.

24. What is the square root of Gf? Ans. 2,58194-.

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 «nd 12 ?

Ans. 3 X 12=36. then V 36=0 the mean proportional.

6. What is the mean proportional between 45276 and 842 ?

Ans. 1897,44-

Thfind the side of a square equal in area to any given superficies.

Rule. The square root of the content of any given superficies is the side of the square equal sought.

SXTRACTION OF THK 8QUARK ROOT. 133

EXAMPLES.

27. 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 equal? An«. 27,38612.

The Area of a circle given to find the Diameter.

RvLS. As 355 : 452, or, as 1 : 1,273239 : : so is the area : to the square of the diameter ; — or, multiply the square root of th« 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^ yards ? Ans. 6,136 perches.

The area of a circle given, to find the periphery, or circumference.

Rule. As 113 : 1420, or, as 1 : 12,56637 : : the area to the square of the 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 •ide.

1. The base and perpendicular given to find the hypothenuse.

Rule. The square root of the sum of the squares of the base «id perpendicular, is the length of the hypothenuse.

M

1^1 EXTRACTION OF YHE saVARE BOOT.

EXAMPLES.

32. The top of a castle from the ground is 45 yards high, and 5Uirounaed with a diich tK) yards broad ; what length must a 1»«4- tler be to reach fronti the outside of the ditch to the top of tb«» castle ? Ans. 75 yards.

li

*

•!«

1-

JS

Oi

.Sf

e

X

Ditch.

Base 60 yards.

33. The wall of a town is 25 feet high, which is smrounded by a moat of 30 feet in breadth : 1 desire to know ih'^ length of a ladder that will reach ^rom the outside of the moat to the top of the wall ? Ans. 39,06 feet.

The hypothenuse and perpendicular given, to find the base.

Rule. The square root of the difference of the squares of the hypothenuse and perpendicular, is the length of the base.

The base and hypothenuse given, to find the perpendicular.

Rule. The square root of the difference of the squares of the hypothenuse and base, is the height of the perpendicular.

N. B. The two last questions may be varied for examples to the two last propositions.

Any number of men being given, to form them into a square battle, or to find the number of rank and file.

Rule. The square root of the number of men given, is the otunbcr of men either in rank or file.

34. An army consisting of 331776 men, I desire to know how many rank and file ? Ans. 576.

35. A certain square pavement contains 4S841 square stones, all of the same size. I demand how many are coRtained in one of the sides? .An*. 221.

XXTRACTtON ©F THE C0BE ROOT. ISTl

EXTRACTION OF THE CUBE ROOT

To extFsct the Cube Root is to find out one number, which be- ing multiplied into itself, and then into that productr produceth llie given fturaber.

RlTLE 1. Point every third figure of the cube given, begtnmng at the unit's place ; seek the greatest cube to the first point, and •ufatract it therefrom ; put the root in the quotient, and bring down the figures in the next point to the remainder, for a Resolvend-

2. Find a Divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient.

3. To find the Subtrahend. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3, except the last, and that product by the square of the last. 3. Multiply the divisor by the Tast 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. 8. 9. CoBES. 1. 8. 27. &4. 125. 216. 343. 512. 729.

EXAMPLES.

1. What is the cube root of 99252847 ?

99252847(463 64 =cube of 4

Divisor

Square of 4X 3=48)35252 resolvend.

216=cube of G. 432 =4 X 3 X by square of 6. 28S =divisor X bv 6.

33336 subtrahend.

Divisor

Square of 46 x 3=6348)1 9! 6S47 resolvend.

27=cube of 3. 1242 =46 X 3 X by square of 3. 19044 =divi8or X by 3.

1916847 subtrahend.

M2

f2.

What

3!

What

4.

What

5-

What

6.

What

7.

What

a

What

9.

What

10.

What

11.

What

12

What

136 BXTRACTION Or TBB CUBB ROOT.

8 the cube root of 389017? Ans. 73.

s the cube root of 6735339? Ans. 179.

s the cube root of 32461769 ? Ans. 319.

s the cube root of 84604519? Ans. 439.

s the cube root of 259694072 ? Ans. 63a

s the cube root of 4822S544 ? Ans. 364.

s the cube root of 27054036008 ? Ans. 3002.

8 the cube root of 220()9S10125? Ans. 2806.

s the cube root of 122615327232? Ans. 4968.

s the cube root of 219365327791 ? Ans. 603h

3 the cube root of 673373097125? 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 b« a point fall on the unit's place of the whole number.

13. What is the cube root of 12,077876 ? Ans. 2,35.

14. What is the cube root of 36155,02756 ? Ans. 33,06+.

15. What is the cube root of ,001906624 ? Ans. ,124.

16. What is the cube root of 33»230979937 ? Ans. 3,216+.

17. What is the cube root of 15926,972504? Ans. 26,16+. la What is the cube root of ,053157376? .4ns. ,376.

To extract the cube root of a vulgar fraction. RuLB. 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 ffj? Ans. |.

20 What is the cube root of -^f^ ? Ans. f.

21. What is the cube root of fHo • -A^«- f-

BUHDS.

22. What is the cube root of f ? Ans. ,829+.

23. What is the cube root of \ ? Ans. ,822+.

24. What is the cube root of f ? Ans. ,873+.

To extract the cube root of a mixed number. Rule. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fraction, extract the cube root of the numerator and denominator for a new numerator and deno-

EXTRACTION OF THE CUBE ROOT. 137

minator ; but if the mixed number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom.

EXAMPLES.

25. What is the cube root of 12^? Atis. ^.

26. What is the cube root of S\-^t Ans. 3|.

27. What is the cube root of 405 f|y ? Ans. 7f .

28. What is the cube root of 7^? Ans. 1,93+.

29. What is the cube root of 9J- ? Ans. 2,092+.

30. What is the cube root of 8f ? Ans. 2,057+.

THE APPLICATION.

1. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it con- tain? Ans. 103823.

2. There is a cellar dug, that is 12 feet every way, in length, breadth, and depth ; how many solid feet of earth were taken out of it ? Ans. 172a

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 ?

Ans. 18 and 54. 6. What 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 solid, as a globe, cylinder, prism, cone, SfC.

Rule. The cube root of the solid content of any solid bodj given, is the side of the cube of equal solidity.

M3

m ElTRACTING ROOTS OF ALL POWIRS.

EXAMPLES.

C. If the solid content of a globe is I0G18, what is the side ot a cube of equal solidity ? Atis, 82.

77ie side of a cube being gtverii ft) Jind the side of a cube that shall be double, treble,, o^c^ in ^gA'antity to the cube given.

RuLEk Cube the side glvfeii» and multiply it by 2, 3, &c., the cube root of the produdl is the side sought.

EXAMPLES.

7. Thete is a cubical vessel, whose side is 12 inches, and it is required to find the side of another vessel, that is to contain three limes as much ? Ans. 17,306.

tJXTRACTING 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- ber.

Rule. First extract the square root of the given number, and then extract the square root of that square root, and it will give the biquadrate root required.

EXAMPLES.

1. What is the biquadrate of 27? Ans. 531441.

2. What is the biquadrate of 76? Ans. 33362176.

3. What is the biquadrate of 275? Ans. 5719140625.

4. What is the biquadrate root of 531441 ? Ans. 27. 6. What is the biquadrate root of 33362176? Ans. 76. 6. What is the biquadrate root of 6719140625? 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 given number.

3. Bring down the first figure in the next point to the remain dw, aivl call it thg dividend.

EXTRACTING ROOTS OF ALL POWERS. 139

4. Involre the root into the next inferiar power to that whicli is given, multiply it by the given power, and call it the divisor.

5. Find a quotient figure by common division, and annex it to the root ; then involve the whole root into the given power, and call that the subtrahend.

6. Subtract that number from as many points of the given power as are brought down, beginning at the lower place, and to the remainder bring down the first figure of the next point for a new dividend.

7. Find a new divisor, and proceed in all respects as before.

EXAMPLES. I. What is the square root of 141376 ?

141376(376 9 3X 2=i6 divisor.

— 37 X 37=1369 subtrahend. 6)51 dividend. 37 X 2=74 divisor. 376 X 376=141376 subtrahend,

1369 subtrahend.

74)447 dividend.

141376 subtrahend.

hat is the cube root of 53157376?

53157376(376

27

27)261 dividend.

50653 subtrahend.

4107)25043 dividend.

53157376 subtrahend.

3 X 3 X 3=27 divisor. 37 X 37 X 37=50653 subtrahend. 37 X 37 X 3=4107 divisor. 376 X 376 X 376=53157376 subtrahend.

140 SIMPLE INTEREST.

3. What is the biquadrate of 19987173370 T

19987173376(376 81

108)1188 dividend. 18741G1 subtrahend.

203612)1245563 dividend.

19987173376 subtrahend.

3X 3X 3X 4=109 divisor. 37 X 37 X 37 X 37=1874161 subtrahend. 37 X 37 X 37 X 4=202612 divisor. 376 X 376 X 376 X 376=19987173376 subtrahend.

SIMPLE INTEREST.

There are five letters to be observed in Simple Interest, vix.

P. the Principal.

T. the Time.

R. the Ratio, or rate per cent.

I. the Interest.

A. the Amount.

A TABLE OP RATIOS.

3

,03

5i

,055

6

.06

3i

,035

6

,06

8i

,066

4

,04

6i

,065

9

,09

^

,045

7

.07

9i

,095

5

,05

7i

,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 : ,036.

SIMPLE INTEREST.

141

When the principal, time, and rate per cent, are given, to find the interest.

RcLE. Multiply the principal, time, and rate together, and it will give the interest required.

Note. The proposition and rule are better expressed thus :-^

I. When P R T are given to find I.

Rule. prt=I.

Note. When two or more letters are put together like a word, they are to be multiplied one into another.

EXAMPLES.

1. What is the interest of £945 : 10, for 3 years, at 5 per cent, per annum. Ans. 945,5 X,05 X 3=141,825, or £141 : 16 : 6.

2. What is the interest of £547 : 14, at 4 per cent, per annum, for 6 years ? Ans. £131 : 8 : 11, 2 qrs. ,08.

3. What is the interest of £796 : 15, at 4^ per cent, per an- num, for 5 years? Ans. £179 : 5 : 4 2 qrs,

4. What is the interest of £397 : 9 : 5, for 2^ years, at 3^ per cent, per annum? Ans. £34 : 15 : 6 3,5499 qrs.

5. What is the interest of £554 : 17 : 6, for 3 years, 8 months, at 4^ per cent, per annum? Ans. £91 : 11 : 1 ,2.

6. What is the interest of £236 : 18 : 8, for three years, 8 months, at 6^ 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 OF £1 FOR ONE DAY.

per cent.

Decimals.

per cent.

Decimals. ^

3

,00008219178

H

,00017S08219

3i

,00009589041

7

,00019178082

4

,00010958904

n

,00020547945

^

,00012328767

8

,00021917808

5

,00013698630

8i

,00023287671

5i

,00015068493

9

,000246575.34

6

,00016438356

9i

,00026027397

Note. The above table is thus found : —

As 365 : ,03 : : 1 : ,00008219178. And as 365 : ,085 : : 1 ,00009589041, &c.

EXAMPLES.

7. What is the interest of £240, for 120 days, at 4 per cent, per annum ? Ans. ,000105)58904 X 240 X ia0=£3 : 3 : U-

a "What is the interest of £364 : 18, for 154 days, at B per cent, per annum? Ans. £7 : 13 : 11^.

9. What is the interest of £725 : 15, for 74 days, 'kt 4 per cent, per annum ? Ans. £5 : 17 : 8^.

10. What is the interest of £100, from the 1st of June, 1775, 4e <he 9th of March following, at 5 per cent, per annum ?

Ans. 3C3 : 16 : 11|.

11. When P R T are given to find A. liluLK. prt + p=A.

EXAMPLES.

11. What will £279 : 12, amount to in 7 year«, at 4^ per cent, per annum ? Ans. £36? : 13 : 5 3,04 qrs.

279,6 X ,045 X 7 + 279,6=367,074.

12. What will £320 : 17, amount to in 5 years, at 3^ per cent. ^er annum ? Ans. £376 : 19 : 11 2,8 qis.

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. ^t annum ? ^tis. £1130 : 9 : 0^ ,92 qrs.

14. What will £273 : 18, amount to in 4 years, 175 days, at 3 per cent, per annum ? Ans. £310 : 14 : 1 3,350800(>4 qrs.

III. When A R T are given to find P.

a

RULB. =P.

rt+1

EXAMPLES.

15. What principal, being put to interest, will amount to £367 : 13 : 5 3,04 qrs. in 7 year?, at 4^ per cent, per annum?

Ans. ,015 X 7+ 1=1,315 then 307,674^-1, 31 5=£279 : 12.

16. What principal, being put to interest, will amount to £370 J 19 : 11 2,8, in 5vear&, at 3^ per cent, per annum?

Ans. £320: 17.

SIMPLE INTEREST. 143

l^. Wliat principal, being put to interest, will amount to £1130 : 9 : 0^ ,92 qrs. in 5^ years, at 4 per cent, per annum ?

Ans. £926 : 12.

18. What principal will amount to £310 : 14 : 1 3,35080064 qrs. in 4 years, 175 days, at 3 per cent, per annum ?

Ans. £273 : 18. IV. When A P T are given to find R. a— p

Role. =R.

pt.

EXAMPLES.

19. At what rate per cent, will £279 : 12, amount to £367 : 13:5 3,04 qrs. in 7 years ?

Ans. 367,674—279,6-88,074, 275,6 x 7=1957,2, then 88,074-f-1957,2=,045 or 4^ per cent. 30. 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 5J years 1 Ans. 4 per cent.

22. At what rate per cent, will £273 : 18, amount to £310-, 14 : 1 3,35080064 qrs. in 4 years, 175 days ? *

Ans. 3 per cent V. When A P R are given to find T. a— p

Rdlb. =T.

pr.

EXAMPLES.

23. In what time will £279 : 12, amount to £367 : 13 : 5 3,04 •qrs. at 4^ per cent. ?

Ans. 367,674—279,6=98,074. 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 2,8 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.

26. In what time will £273 : 18, amount to £310 : 14 : 1 3,35080064 qrs. at 3 per cent. ? Ans. 4 years, 175 days.

ANNUITIES OR PENSIONS, &c. IN ARREARS.

Annuities or pensions, &.c. are said to be in arrears, when they are payable or due, cither yearly, half-yearly, or quarterly, and are unpaid for any number of payments.

144 SIMPLE INTEREST.

Note. U represents the annuity, pension, or yearly rent, T il A as before. I U R T are given to find A. ttu — tu

Rule. X r : + tu=A.

3

EXAMPLES.

27. If a salary of £150 be forborne 5 years at 5 per cent what ^vill it amount to ? Ans. £825. 3000

5 X 6X 150—5 X 150=3000 then X ,05 + 6 X 150=£825.

2

28. If £250 yearly pension be forborne ? years, 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 4^ 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 unpaid for 5 years, what would it amount to in that time at 5 per cent t

^715. £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 II. When A R T are given to find U. 2a

Rule. =U.

ttr— tr42t

au8

SIMPLE INTEREST. 145

33. If a salary amounted to £825 in 5 years, at 5 per cent what was the salary ? Ans. £150.

825 X2=1650 5 X 5 X ,05—6 X ,05-f 5 X 2=1 1 then 1650-*- 11=£150.

.34. If a house is to be let upon a lease for 5^ years, and the amount for that time is £363 : 8 : 3, at 4^ per cent, what is the yearly rent? -^ns. £60.

35. If a pension amounted to £2065, in 7 years, at 6 per cent, what was the pension 1 Ans. £250.

36. Suppose the amount of a pension be £263 : 4 in 8 years, at 5 per cent what was the pension ? Ans. £28.

Note. When the payments are half-yearly, then take 4 a, and half of the ratio, and tmce the number of years ; and if quarterly, then take 8 a, one fourth of the ratio, and four times the number of years, and proceed as before.

37. If the amount of a salary, payable half-yearly, for 5 years, at 5 per cent be £834 : 7 : 6, what was the salary ? Ans. £150.

38. If the amount of an annuity, payable quarterly, be £839 ; I : 3, for 5 years, at 6 per cent, what was the annuity ?

Ans. £150.

III. When U A T are given to find R. 2a— 2ut

RtTLB.^ =R.

utt — ut

EXAMPLES.

39. If a salary of £150 per annum, amount to £825, in 5 years, what is the rate per cent. ? Ans. 5 per cent.

825 + 2—150 +5 + 2=150 then =,06

150X5X5—150X5

40. If a house be let upon a lease for 5^ years, at £60 per an- num, and the amount for that time be £363 : 8 : 3, what is the rate per cent ? Ans. ^ per cent.

41. If a pension of £250 per annum, amounts to £2065 in 7 years, what is the rate per cent. ? Ans. 6 per cent.

42. Suppose the amount of a yearly pension of £28, be £263 : 4, in 8 years, what is the rate per cent ? Ans. 5 per cent

N

146 . SIMPLE INTEREST.

Note. When the payments are half-yearly, take 4 a — 4 at for a dividend, and work with half the annuity, and double the num- ber of years for a divisor ; if quarterly, take 8 a — 8 ut, and work with a fourth of the annuity, and four times the number of years.

43. If a salary of £150 per annum, payable half-yearly, amounts to £834 : 7 : 6, in 5 years, what is the rate per cent ?

Ans. 5 per cent.

44. If an annuity of £150 per annum, payable quarterly, amounts to £839 : 1 : 3, in 5 years, what is the rate per cent ?

Ans. 5 per cent. IV. When U A R are given to find T.

2 2a XX X

Rule. First, 1 =x then : V 1 =T.

r ur 4 2

EXAMPLES.

45. In what time will a salary of £150 per annum, amount to £825, at 5 per cent.? Ans. 5 years.

2 826 X 2 39 X 39

1 =39 =220 =380,25

,06 150 X ,05 4

39

V220+380 ,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 1 Ans. 5^ years.

47. If a pension of £250 per annum, being forborne a certain time, amounts to £2065, at 6 per cent., what was the time of forbearance ? Ans. 7 years.

48. In what time will a yearly pension of £28, amount to £263 : 4, at 5 per cent. ? Ans. 8 years.

Note. If the payments are half-yearly, take half the ratio, and half the annuity ; if quarterly, one fourth of the ratio, and one fourth of the annuity ; and T will be equal to those half-yearly or quarterly payments.

49. If an annuity of £150 per annum, payable half-yeaily, amounts to £834 : 7 : 6, at 5 per cent., what time was the pay- ment forborne? Ans. 6 years.

SIMPLE INTEREST.

1«7

50. If a yearly pension of £150, payable quarterly, amounts to <C839 : 1 : 3, at 5 per cent., what was the time of forbearance ?

Ans. 5 years.

PRESENT WORTH OF ANNUITIES.

Note. P represents the present worth ; U T R as before.

I. When U T R are given to find P.

ttr— tr + 2t Rule.- : X »=P.

2tr + 2

EXAMPLES.

51. What is the present worth of £150 per annum, to continue 5 years at 5 per cent. ? Ans. £660.

5 X 5 X ,06—5 X ,05+5 X 2=11 ,5 X ,05 X 2+2=2,5 then 11-*- 2,5 X 150=£660.

52. What is the yearly rent of a house of £60, to continue 5^ years worth in ready money, at 4J per cent !

Ans, £291 : 6 : 3.

53. What is the present worth of £250 per annum, to continue 7 years, at 6 per cent. ? An^. £1454 : 4 : 6.

54. What is a pension of £28 per annum, worth in ready mo- ney, at 5 per cent., for 8 years ? Ans. £188.

NoTB. The same thing is to be observed as in the first rule of annuities in arrears, concerning half-yearly and quarterly pay- ments.

55. What is th« present worth of £150, payable quarterly, for 5 years, at 5 per cent. ! Ans. £671 : 5.

Note. By comparing the last examples, it will be found that the present worth of half-yearly payments is more advantageous than yearly, and quarterly than half-yearly.

II. When P T R are given to find U.

tr + 1 RrLE. : X 2p=U.

ttr— tr + 2t

N2

SIB SIKFLE INTEREST.

•-" EXAMPLES.

56. If the present worth of a salary be £660, to continue 5 years, at 5 per cent., what is the salary ? Ana. £160.

5X,05+1=1,25 5X5 X,05— 5 X ,054-10=11. 1,25

X660x3=£150.

11

57. There is a house let upon lease for 5 J years to come, I de- sire to know the yearly rent, when the present worth, at 4^ pet cent., is £291 : 6 : 3 ? Ans. £60.

58. What annuity is that which, for 7 years' continuance, at 6 per cent., produces £1454 ; 4 : 6 present worth? Ans. £250.

59. What annuity is that which, for 8 years' continuance, pro- duces £188 for the present worth, at 5 per cent. ? Ans. £28.

Note. When the payments are half-yearly, take half the ratio, twice the number of years, and multiply by 4 p ; and when quar- terly, take one fourth of the ratio, and four times the number of years, and multiply by 8 p.

60. There is an annuity payable half-yearly, for 5 years to come, what is the yearly rent, when the present worth, ^t 5 per cent., is £667 : 10 ? Ans. £160.

61. There is an annuity payable quarterly, for 5 years to come, I desire to know the yearly income, when the present worth, at 6 per cent., is £671 : 5? Ans. £160.

III. When U P T are given to find R.

ut— p X 2

IVLE. =R.

2pt + ut— ttu.

EXAMPLES.

62. At what rate per cent, will an annuity of £150 per annum, to continue 5 years, produce the present worth of £660 ?

Ans. 5 per cent.

150 X 5—660 X 2=180,2 X 660 X 5+5 X 150—5 X 6 X 150=3600 then 180-«-3600=,05=5 per cent.

63. If a yearly rent of £60 per annum, to continue 5^ years, produces £291 : 6 : 3, for the present worth, what is the rate per cent. ? Ans. 4^ per cent.

#

SIMPLE INTEREST. 149

54. If an annuity of £250 per annum, to continue 7 years, produces £1454 : 4 : 6, for the present worth, what is the rate per cent ? Ans, 6 per cent.

65. If a pension of £28 per annum, to continue 8 years, pro- duces £188 for the present worth, what is the rate per cent. ?

Ans. 5 per cent.

Note. When the annuities, or rents, &c. are to be paid half- yearly, or quarterly, then

For half-yearly payments, take half of the annuity, &e. and twice the number of years, the quotient will be the ratio of half the rate per cent. — and

For quarterly payments, take a fourth part of the annuity, &c. and four times the number of years, the quotient will be the ratio of the fourth part of the rate per cent.

66. If an annuity of £150 per annum, payable half-yearly, ha- ving 5 years to come, is sold for £667 : 10, what is the rate per cent. ? Ans. 5 per cent.

67. If an annuity of £150 per annum, payable quarterly, ha- ving 5 years to come, is sold for £671 : 5, what is the rate per cent. ? Ans. 5 per cent.

IV. When U P R are given to find T

2 2p 2p XX X

RwLB. l=x then V — | =T.

r n ur 4 2

EXAMPLES.

68. If an annuity of £150 per annum, produces £660 for the present worth, at 5 per cent., what is the time of its continu- ance ? ^715. 5 years

2 660 X 2 660 X 2

1=30,2 =176

,05 150 150 X, 05

30,2 X 30,2

4

30,2

20,1 =5 years.

2

N3

=228,01 then V228,01 + 176=20,1

150 SIMPLE INTEREST.

69. For what time may a salary of £60 be purchased for £291 : 6 : 3, at 4^ per cent. ? Ans. B J years.

70. For what time may £250 per annum, be purchased for £1454 : 4 : 6, at 6 per cent. ? Ans. 7 years.

71. For what time may a pension of £28 per annum, be pur- chased for £168, at 5 per cent. ? Atis. 8 years.

Note. When the payments are half-yearly, then U will be equal to half the annuity, Alc. R half the ratio, and T the num- ber of payments : and,

When the payments are quarterly, U will be equal to one fourth part of the annuity, &c. R the fourth of the ratio, and T the number of payments.

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, &e. taken in re- version.

Rule. Find the present worth of the ttr — tr4-2t

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 =P.

come before the annuity, &c. commences ; tr fl thus,

EXAMPLES.

74. What is the present worth of an annuity of £150 per an- num, to continue 5 years, but not to commence till the end of 4 years, allowing 5 per cent, to the purchaser ? Ans. £550.

5 X 5 X ,05—5 X ,05+2 X 5=4,4 X 150= 660

. =550.

. 4 X ,05+1

5 X ,05 X 2+2

SUfPLE INTEREST. 151

75. What is the present worth of a lease o€ £50 per annum, to continue 4 years, but which is not tO' commence till the end of 5 years, allowing 4 per cent, to the purchaser ?

Ans. £152 : 5 : 11 3 qrs.

76. X person having the promise of a pension of £20 per an- num, for 8 years, but not to commence till the end of 4 years, is willing to dispose of the same at 5 per cent.,, what will be the present worth? Ans. £111 : 18 : 1 ,14+.

77. A legacy of £40 per annum bein^ left for 6. years, to a person of 15 years of age, but which is not to commence till he is 21 ; he, wanting money, is desirous of selling the same at 4 per cent, what is the present worth ?

• Ans. £171 : 13 : 11 ,07596.

2. To find the yearly income of an annuity, &c. in reversion.

Rule 1. Find the amount of the present worth at the given rate, and for the time ptr + p=A. before the reversion ; thus,

2. Change A into P, and find what an- tr 4-1

nuity being sold, will produce P at the .^n _tt

same rate, and for the time of its continu- .^ . ro.* *^~~ ance; thus, ttr-tr+2t

EXAMPLES.

78. A person having an annuity left him for 5 years, which does not commence till the end of 4 years, disposed of it for £550, adlowing 5 per cent, to the purchaser, what was the yearly in- co e ? ^715. £150.

5X,05 + 1,

550 X 4 X, 05+550=660 5 X 5 X ,05— 5 X ,0&+5 X 2= ,113636 X 660 X 2=£150.

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 sell the same for £152 : 6, present payment, allowing 4 per cent, to the pur- diaser, what is the yearly rent ? Ans. £50.

80. A person having the promise of a pension for 8 years, which does not commence till the end of 4 years, has disposed of the same for £111 : 18 : 1 ,14 present money, allowing 5 per cent to the purchaser, what was the pension ? Ans. £20.

152 REBATE OR DISCOUNT.

81. There is a certain legacy left to a person of 15 years of ajje, which is to be continued for 0 years, but not to eomujenoe till lie 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 f Ans. £40

REBATE OR DISCOUNT

Note. S represents the Sum to be discounted. P the Present worth. T the Time. R the Ratio.

I. When S T R are given to find P. s

Rule. =P.

tr + l

EXAMPLES.

1. What is the present worth of £357 : 10, to be paid 9 months hence, at 5 per cent. ] Ans. £344 : 11 : 6| ,168.

2. What is the present worth of £275 : 10, due 7 month* hence, at 5 per cent. ? Ans. £267 : 13 : lOj^j^.

3. What is the present worth of £875 : 5 ; 6, due at 5 months hence, at 4^ per cent. ? Ans. £859 : 3 : 35 T^F•

4. How much ready money can I receive for a note of £75, due 15 months hence, at 5 per cent. ?

Ans. £70 : 11 : 9 ,1764d. II. When P T R are given to find S. Rule. ptr + p=S.

EXAMPLES.

5. If the present worth of a sum of money, due 9 months hence, allowing 5 per cent., be £344 ; 11 : 6 3,168 qrs., what was the sum first due ? An^. £357 : 10.

344,5783 X ,75 X ,05 + 344,5783=£357 : 10.

6. A person owing a certain sum, payable 7 months hence, agrees with the creditor to pay him down £267 : 13 : lO^ff, a! lowing 5 per cent, for present payment, what is the debt?

Ans. £275 : 10.

7. A person receives £859 : 3 : 3| -[fj for a sum of money

I

REBATE OR DISCOUNT. 153

due 5 months hence, allowing the debtor 4^ per cent for present payment, what was the sum due ? Ans. £875 : 5 : 6.

a A person paid £70 : 11 : 9 ,1764d. for a debt due 15 months hence, he being allowed 5 per cent, for the discount, how much was the debt 1 Ans. £75.

III. When S P T are given to find R. —I

EXAMPLES.

s— p RULB. =R

9. At what rate per cent, will £.357 : 10, payable 7 months haice, produce £344 : 11 : 6 3,168 qrs. for present payment?

3575,-344,5783

=,05=5 per cent

344,5783 X ,75

10. At what rate per cent will £275 : 10, payable 7 months hence, produce £267 : 13 : 10^^ for the present payment ?

Ans. 5 per cent.

1 1. At what rate per cent \n\\ £875 : 5 ; 6, payable 5 months hence, produce the present payment of £859 : 3 : 3| yf^ ?

Atis. 4^ per cent. 12^ At what rate per cent, will £75, payable 15 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.

13. The present worth of £357 : 10, due at a certain time to come, is £344 : 11 : 6 3,168 qrs. at 6 per cent, in what time should the sum have been paid without any rebate ?

Ans. 9 months. 357,5—344,5783

=,75=9 months.

344,5783 X, 05

14. The present worth of £275 : 10, due at a certain time to

164 E(1UATI0N OF PA MENT8.

come, IB £261 : 13 : 10^2^, at 5 per cen , in w time should the sum nare been paid without any rebate ?

Ana. 7 months.

15. A person receives £S59 : 3 : 33 ,0184, for £875 : 5 : 6, due at a certain time to come, allowing 4^ per cent, discount, I desire to know in what time the debt should have been discharg ed without any rebate ? Ans. 5 months.

16. I have received £70 : 11 : 9 ,1764d. for a debt of £75 allowing the person 5 per cent, for prompt payment, I d«sire to know when the debt would have been payable without the rebate !

Ans. 15 months.

EQUATION OF PAYMENTS.

To find the equated time for the payment of a sum of money due at several times.

Rule. Find the present worth of each pay- s

ment for its respective time ; thus, =P.

tr + 1 Add all the present worths together, then, »^p=D.

d and; =E

EXAMPLES.

1. D owes E £^200, whereof £40 is to be paid at three months, £60 at six months, and £100 at nine months ; at what time may the whole debt be paid together, rebate being made at 5 per cent. ?

Ans. 6 months, ^ days. 40 60 100

=39,5061 =58,5365 =96,3865

1,0125 1,025 1,0375

then 20a— 39,5061+58,5365+96,3855=5,6719 6,5719

-=,57315=6 months, 26 days.

194,4281 X ,05.

2. D owes E £800, whereof £200 is to be paid in 3 months, £200 at 4 months, and £400 at 6 months ; but they, agreeing to make but one payment of the whole, at the rate of 5 per cent, rebate, the tnie equated time is demanded ?

Ans. 4 months, 22 days.

COMPOUND INTEREST.

153

3, E owes F £1200, which is to be paid as follows : £200 down, £500 at the end of 10 months, and the rest at the end of 20 months ; but they, agreeing to hare one payment of the whole, rebate at 3 per ceniL, the true equated time is demanded?-

Ans. 1 year, 11 days.

COMPOUND INTEREST.

The letters made use of in Compound Interest, are^

A the Amount.

P the Principal.

T the Time.

R the Amount of £1 for 1 year at aay given rate ;.

which is thus found : ^

As 100 : 105 : : I : 1,05. As 100 : 105,5 : : 1 : 1,066.

A Table of the amount of £1 for one year.

BATES

PEB CENT.

3J

4

4i

5

AMOUNTS OF £1.

1,03

1,035 1,04 1,045 1,05

RATES PER CENT.

54 6

64

7

AMOUNTS OP £1.

1,055

1,06

1,065

1,07

1,075

RATES PER CENT.

8

84 9 94 10

AMOUNTS

of£1.

1,08.

1,085

1,09

1,095

1,1

Table showing the amount of £1 fdr any number ef years under 31, at 5 andQ per cent; per annum.

TEARS.

5 RATES, 6

YEARS.

5 RATES. 6 1

1

1,05000

1,06000

16

2,18287

2,54035

2

1,10250

1,12360

17

2,29201

2,69277

3

1,15762

1,19101

18

2,40662

2,85434

4

1,21550

1,26247

19

2,52695

3,02560

5

1,27628

1,33822

20

2,65329

3,20713

6

1,34009

1,4J852

21

2,78596

3,39956

7

1,40710

1,50368

22

2.92526

3,60353

8

1,47745

1,59385

23

3,07152

3,81975

9

1,55132

1,68948

24

3,23510

4,04893

10

1,62889

1,79084

25

3,38635

4,29187

11

1,71034

1,89829

26

3,55567

4,54938

12

1,79585

3,01219

27

3,73345

4,82234

13

1,88565

2,13292

28

3,92013

5,11168

14

1,97993

2,26090

29

4,11613

5,41838 '

15

2,07892

2,39655

30

4,32194

5,74349

166 COMPOUND INTEREST.

Note. The preceding table is thus made — As 100 : 105 : : 1 : 1,05, for the first year; then. As 100 : 105 : : 1,05 : 1,1026, se- cond year, &,c.

I. When P T R are given to find A. Rule. pXrt=A.

EXAMPLES.

1. What will £225 amount to in 3 years'^me, at 5 per cent per annum?

Ans. 1,05 X 1,05 X 1,05=1,157625, then 1,157625 X 226=

£260 : 9 : 3 3 qrs.

2. What will £200 amount to in 4 years, at 6 per cent, per annum ? Ans. £243 2,025fl,

3. What will £450 amount to in 5 years, at 4 per cent per annum ? Ans. £547 : 9 : 10 2,0538368 qrs.

4. What will £500 amount to in 4 years, tt 5J per cent, pei annum ? Ans. £619 : 8 : 2 3,8323 qrs.

II. When A R T are given to find P.

a

Rule. =P.

rt

* EXAMPLEa

5. What principal, being put to interest, will amonnt to £260i 9:33 qrs. in 3 years, at 5 per cent, per annum ?

260,465625

1,05 X 1,05 X 1,05=1,157625 =£225.

1,157625

6. What principal, being put to interest, will amount to £243 2,025s. in 4 years, at 5 per cent, per annum ? Ans. £200.

7. What principal will amount to £547 : 9 : 10 2,0538368 qrs. in 5 years, at 4 per cent, per annum ? Ans. £450.

a What principal will amount to £619 : 8 : 2 3,8323 qrs. in 4 years, at 5^ per cent, per annum ? Ans. £500.

III. When P A T are given to find R.

a which being extracted by the rule of extra«-

RuLE. — =rt tion, (the time given to the question showing p the power) will give R.

I

COMPOVND INTEREST. 157

EXAMPLES.

9. At what rate per cent will £235 amount to £260 : 9 : 3,3 qrs. in 3 years ? Ans. 5 per cent.

260,466625

=1,157625, the cube root of which

225 (it being the 3d power)=l,05=6 per cent.

10. At what rate per cent, will £200 amount to £243 : 2,025s. 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 1 Ans. 5 J per cent.

IV. \Mien P A R are given to 'find T.

a which being continually dirided by R till no-

RvLK. — =rt thing remains, the number of those divisions p will be equal to T.

EXAMPLES.

13. In what time will £225 amount to £960 : 9 : 3 3 qrs. at

5 per eent. ?

260,465625 1,157625 1,1025 1,05

=1,157625 =1,1025 ^^1,05

225 1,05 1,05 1,05

=1, the number of divisions being three times sought

14. In what time will £200 amount to £243 2,025s. at 5 per eent ? Ans. 4 years.

15. In what time will £450 amount to £547 : 9 : 10 2,0538368 qrs. at 4 per cent ? Ans. 5 years.

16. In what time will £500 amount to £619 : 8 : 2 3,832c qrs. at 5^ per cent ? Ans. 4 years.

ANNUITIES, OR PENSIONS, IN ARREARS.

Note. U represents the annuity, pension, or yearly rent: \ R T as before.

t68

COMPOUND IXTEUEST.

A Table showing the amount of £1 annually^ for any. number of years under 31, at 6 and 6 per cent, per annum.

YEAKH.

5 RATES. 6

VKARS.

5 RATES. 6 1

1

1,00000

1,00000

16

23,65749

35,67252

2

2,05000

2,06000

17

25,84036

28,2)288

3

3,15250

3,18360

18

28,18238

30,90565

4

4,31012

4,37461

19

30,53900

33,75999

5

5,52563

5,63709

20

33,06595

36,78559

6

6,80191

6,97532

21

35,71925

39,99272

7

8,14200

8,39383

22

38,50521

43,39229

8

9,54910

9,89746

23

41,43047

46,99582

9

11,02656

11,49131

24

44,50199

50,81557

10

12,57789

13,18079

25

47,72709

54,86451

U

14,20678

14,97164

26 '

51,11345

59,15638

13

15,pl7l2

16,86994

27

54,66912

63,70576

18

17,71298

18,88213

28

58,40258

68,52811

14

19,59868

21,01506

29

62,32271

73,63979

15

21,57856

23,27597

30

66,43884

79,05818

Note. The above table is made thus : — take the first year's amount, which is £1, multiply it by 1,05-|- l=2,05=8econd year's amount, which also multiply by l,05-fl=2;1525=third year's amount.

I. When U T R are givea to find A. ur* — u

RvLE. =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, &G. 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. !

Ans. 1,05 X 1,05 X 1,05 X 1,05 X 50=60,77531250 60,7753125—60

then- =£215 : 10 : 1 2 qrs. ; or,

1,05—1 by the table thus, 4,31012 X50=£215 : 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. ?

( Ans. £248 : 13 : 0 3,27 qrs.

COMPOUND INTEREST. 199

19. If a salary of £40 per annum, to be pakl yearly^ be for* borne 6 years, at 6 per cent., what is the amount 1

Alts. £279 : 0 : 3,0679e096d.

20. If an annuityof £75 per annum, payable yearly, be omit- ted to be paid for 10 years, at 6 per cent., what is the amount ?

^ns. £988.11 ^2,22d-

II. When A R T are given to find U.

ar — a RULB. =U.

rt— 1

EXAMPLES.

21. What Minmty,f being forborne 4 years, will amount to £215 : 10 : 1 2 qrs. at 5 per cent ?

215,50625 X 1,05—215,50625

Ans. <=£50.

1,05 X 1,05 X 1,05 X 1,05—1

22. What pension, being forborne 5 years, will amount to £248 r 13 : 0 3,27 qrs. at 5 per cent. ? Ans. £45.

23. What salary, being wnitted to bepaid Cyears^will amount to £279 : 0 : 3,0579(K)96d. at 6 per cent. ? Atis^ £40.

24. If the payment of an annuity, being forborne 10- years, amount to £988 : 11 :2,22d. at 6 per cent.^what is the annuity'

III. When U A R are given to find T.

ar+tt — a which being continually divided by R till

Rule. =r* nothing remains, the number of those

u divisions will be equal to T.

EXAMPLES.

25. In what time will £50 per annum amount to £215 : 10 : 1 2 qrs. at 5 per cent, for non-payment 1

Ans. 215,50625 X 1,05+50 215,50625=1,21550625

50

which being continually divided by R, the number of the divi- sions will be=4 years.

26. In what time will £45 per annum amount to £248 : 13 qrs. allo\ving 5 per cent for forbearance of payment ?

Alia. 5 years.

leo

:?;mpouki> interest.

27. In -what time TviU £40 per annum amount to £279 : 0 : 3,05796096tL at C per cent. ? Ans. 6 ycare.

28. In what time will £75 per annum amount to £988 : 11 2,22(1. allowing 6 percent, for forbearance of payment?

.An*. 10 years.

PRESENT WORTH OP ANNUITIES, PENSIONS, &*.

A 7\ible showing the present worth of £1 annuity for any num- ber of years under 31, rebate at 6 and 6 per cent.

TEABS.

5 RATES. 6

TEARS.

5 fUTES. 6 j

1

0,95238

0,94339

16

10,83777

10,10589

3

1,85941

1,83339

17

11,27406

10,47726

3

2,7232-1

2,67301

18

11,68958

10,82760

4

3,54595

3,46510

19

12,08532

11,15811

5

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

12,04158

8

6,46321

6,20979

23

13,48857

12,30338

9

7,10782

6,80169

24

13,79864

12,55036

10

7,72173

7;i6008

26

14,09394

12,78336

11

8,30641

7,88687

26

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

Note. The above table is thus made : — divide £1 by 1,05= ,96238, the present worth of the first year, which-*- 1,05=90753, added to the first year's present worth=l ,65941, 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

r*

RlTL*. =P.

r— 1 or by the table thus :

^J^

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* fwer.

COMPOUND INTEREST. IGl

EXAMPLES.

29. What is the present worth of an annuity of £30 per an- num, to continue 7 years, at 6 per cent ?

Ati3. £167 : d : 5 ,l&4d.

30 10.0483

-=19,9517 30— 19,951 7=-10,0483~

1,50363 1,0(>-1

=167,4716. By the table 5,58238 X 30=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 qrs.

31. What is the present worth of a salary of £35, to continue 7 years, at 6 per cent. ? Ans. £195 : 7 : 7 3,968 qrs.

32. What is the yearly rent of £50, to continue 5 years, worth in ready money, at 5 per cent. ? Ans. £216 : 9 : 5 2,56 qrs.

II. When P T R are given to find U.

pr» X r— pr»

R0J-B.— =U.

r'— 1

EXAMPLES.

33. If an annuity be purchased for £167 : 9 : 5 184d. to be continued 7 years, at 6 per cent, what ia the annuity ?

Ana. 167,4716 X 1,50363 X 1,06—167,4716 X 1,60363

=£30.

1,50363—1

34. If the present payment of £258 : 10 : 6 3,264 qrs. be made for a salary of 8 years to come, at 5 per cent., what is the salary? Ans. £40.

35. If the present payment of £195 : 7 : 7 3,968 qrs. be re- quired for a pension for 7 years to come, at 8 per cent., what is the pension ? Ans. £35.

36. If the present worth of an annuity 5 years to come, be £216 : 9 : 5 2,56 qrs. at 5 per cent., what is the annuity?

Ans. £50. 08

163 COMPOUND INTEREST.

IIL When U P R are given to find T.

u which being continually dirided by R till

RpLB. =:ri nothing remains, the number of those di»

p-f-u — pr visions will be equal to T.

EXAMPLES.

37. How long may a lease of £30 yearly rent be had foi £167 : 9 : 5 ,184d. allowing 6 per cent, to the purchaser?

«^ which being continually

, tiMM divided, the number of

167,4716+30-177,5198" ' those divisions will be=

38. If £25S : 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?

Ans. 8 years,

39. If a house is let upon lease for £35 per annum, and the lessee makes present payment of £195 : 7 : 8, he being allowed 6 per cent., I demand how long the lease is purchased for ?

Ans. 7 years.

40. For what time is a lease of £50 per annum, purchased when present payment is made of £216 : 9 : 5 2,56 qrs. at 5 per cent. * Ans, 5 years.

ANNUITIES, LEASES, &c. TAKEN IN REVERSION.

To find the present worth of annuities, leases, iSfC. taken in reversion.

Rule. Find the present worth of the annui- ty, Ac. at the given rate and for the time of its continuance : thus,

-=P.

r— 1

2. Change P into A, and find what principal being put to interest will amount to P at the il!'

same rate, and for the time to come before the a

annuity commences, which will be the present =P.

worth of the annuity, ice. : thus r*

COMPOUND INTEREST. IfiS

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 «ommence till the end of 2 years, allowing 6 per cent, to the purchaser?

Ans. £175 : 1 : 1 2, 048 qrs.

40 40—28,1984 196,6933

=28,1984 =196,6933

1,41852 1,06—1 1,1236 =175,0563.

42. What is the present worth of a reversion of a lease of £60 «er annum, to continue 7 years, but not to commence till the end of 3 years, allowing 5 per cent, to the purchaser ?

Ans. £299 : 18 : 2,8d.

43. There is a lease of a house at £30 per annum, which is yet in being for 4 years, and the lessee is desirous to take a lease in reversion for 7 years, to begin when the old lease shall be ex- pired, what will be the present worth of the said lease in rever- sion, allowing 5 per cent, to the purchaser 1

Ans. £142: 16 : 3 2,688 qrs.

To find the yearly income of an annuity, ^c. taken in reversion.

Rule. Find the amount of the present worth at the given rate, and for the time be- fore the annuity commences : thus, pr^A.

Change A into P, and find what yearly rent

being sold will produce P at the same rate,

and for the time of its continuance, which will pr* X r — prt

be the yearly sum required : thus, «=U.

ft— 1.

EXAMPLES.

44. What annuity to be entered upon 2 years hence, and then to continue 6 years, may be purchased for £175 : 1 : 1 2,048 qrs. at 6 per cent. 1

Ans. 175,0563 X 1,1236=196,6933 then 196,6933 X 1,41852 X 1,06—279,01337

=£40.

1,41852—1

164 COMPOUND INTXREST.

45. The present worth of a lease of a house is £299 : 18 : 28d. taken in reversion for 7 years, but not to commence till the end of 3 years, allowing 5 per «ent. to the purchaser, what is tlie 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 2,688 qrs. what was the yearly rent of the house, when the les- see was allowed 5 per cent, for present payment ? Arts. £30.

PUaCUASINO FREEHOLD OR REAL ESTATE, IN SUCH AS ARK BOUGHT TO CONTINUE FOR EVER.

I. When U R are given to find W.

Rule. *W.

r— 1

EXAMPLES.

47. What is the worth of a freehold estate of £60 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, wortk 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, wliat is the worth, allowing the buyer 6 per cent. ?

Ans. £1250.

II. When W R are given to find U.

RuLB. 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. 1,05— 1=,05, then 1000 X ,05=£60.

61. If an estate be sold for £3500, and 4 per cent allowed to the buyer, what is the yearly rent X Ans. £140.

COMPOUND INTSREST. 165

52. If a freehold estate is bought for £1250 present money, and an allowance of 6 per cent, made to the buyer for the same, what is the yearly rent ? Ans. £75.

III. When W U are given to find R. w + u

Rin.1 • — =R.

EXAMPLES. 5.3. If an estate of £50 p«- 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 ?

Ans. 4 per cent.

55. If an estate of £15 per annum is sold for £1250, what is the rate per cent, allowed? Ans. 6 per cent.

PXJRCHA.6INO FREEHOLD ESTATES IN REVIRSION.

To find the worth of a Freehold Estate in reversion :

u

Rule. Find the worth of the yearly rent, thus — =W

Change W into A, and 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, thus : — =?

EXAMPLES.

56. If a freehold estate of £50 per annum, to commence 4 years henee, is to be sold, what is it worth, allowing the purchaser 5 per cent, for the present payment ?

50 1000

Ans.-^ =1000, then =£822 : 14 : H,

1,05—1 1,2155

57. What is an estate of £200, to continue for ever, but not to cocnmence till the end of 2 years, worth in ready money, allowing the purchaser 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 the end of 8 veers, allowance being made at 6 percent. ?

Ans. £33,58 : 9 : 10 2,24 qrs.

166

REBATE OR DISCOUNT.

To find the Yearly Rent of an Estate taken in reversion.

Rule. Find the amount of the worth of the estate, at the given ri\te and time before it com- wr'=A oes, thus :

Change A into W, and find what yearly rent wr — w=U, being sold will produce U at the same rate, thus : which will be the yearly rent required.

EXAxMPLES.

59. If a freehold estate, to commence 4 years hence, is sold for £822 : 14 : 1^, allowing the purchaser 5 per cent., what ia the yearly income ? Ans. 822,70625 X 1,2155=1000,

then 1000 X 1,05— 1000=£60.

60. A feehold estate is bought for £4022 : 15 : 7 ,44d, which does not commence till the end of 2 years, the buyer being allow- ed 4 per cent for his money. I desii^ to know the yearly In- come. Ans. £200.

61. There is a freehold estate sold for £3359 : 9 : 10 2,24 qrs., but not to commence till the expiration of 3 years, allowing 6 per cent, for present payment ; what is the yearly income ?

Ans. £240.

REBATE OR DISCOUNT.

A Table showing the present worth of £1 due any number of years hence, under 31, rebate at 5 and 6 per cent.

YIARB.

5 RATES. 6

YEARS.

5 RATES. 6 j

1

,952381

,943396

16

,458111

,393646

4

,907030

,889996

17

,436296

,371364

3

,803838

,839619

18

,415520

,350343

4

,822702

,792093

19

,395734

,330513

5

,783526

,747258

20

,376889

,311804

6

,740215

,70-1960

21

,358942

,294155

7

,710682

,665057

22

,341849

,277505

8

,676839

,627412

23

,325571

,261797

9

,644009

,591898

24

,340068

,246978

!0

,613913

,55H394

25

,295302

,232998

11

,584679

,526787

26

.281^10

,219810

1-2

,556837

,496969

27

.267848

,^7368

13

,530321

,468839

28

,255093

,196630

14

,505068

,442301

29

,'^2946

,184556

15

,481017

,417265

30

,231377

,174110

Note. — The above table is thus made : l-»-l,05=,952391. first year's present worth ; and ,952381^ 1,05=,90703, second ycaT; and ,90703-«- 1,0^^,863838 third year. &c.

REBATE OR DIBCOUKT. 1^

I. When S T R are given to find P

s Role.— =P.

EXAMPLES.

1 . What is the present worth of £315 : 12 : 4 ,2d, payable 4 years hence, at 6 per cent. ?

Ans. 1,06 X 1,06 X 1,06 X 1,06=1,26^7, then by the table. 315,61T5 315,6175 =£250 ,792093

1,26247

249,9994124275

2. If £344 : 14 : 9 1,92 qrs. be payabl« in 7 years' time, what is the present worth, rebate being made at 5 per cent. ?

Ans. £5^45. 3- There is a debt of £441 : 17 : 3 1,92 qrs., which is payable 4 years hence, but it is agreed to be paid in present money ; what som must the creditor receive, rebate being made at 6 per cent ?

Ans. £350.

II. When P T R are given to find S. RriB. p X rt=S.

EXAMPLES. '

4. If a sum of money, due 4 years hence, produce £250 for the present payment, rebate being made at 6 per cent., what was the sum due ?

Ans. £250 X 1,26247=£315 : 12 : 42d.

5. If £245 be received for a debt payable 7 years hence, and tn allowance of 5 per cent, to the debtor for present payment, what was the debt ? Ans. £344 : 14 : 9 1,92 qrs.

6. There is a sum of money due at the expiration of 4 years, but the creditor agrees to take £350 for present payment, allow- ing 6 per cent., what was the debt ?

Ans. £441 : 17 : 3 1,92 qrs.

III. When S P R are given to find T.

s which bemg continually divided by R till nothing

Rule.— '=r' remains, the number of those divisions will be p equal to T

106 REBATB OR DISCOUNT.

EXAMPLES.

7. The present payment of £350 is made for a debt of £315 : 12:4 ,3d., rebate at 6 per cent., in what time was the debt pay- abk?

315,6175 which being continually divided, those

Atis. ■• ==1 ,26247 divisions will be equal to 4=the num- 260 her of years.

8. A person receives £245 now, for a debt of £344 : 14 : 9 1,92 qrs., rebate being made at 5 per cent. I demand in what time the debt was payable? Ans 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 th« present payment of £350, I desire to now in what time the sura should have been paid without any rebate ?

An9. 4 years.

IV. When 8 P T are given to find R.

3 which being extracted by the rules of extraction. RuLB T* (the time given in the question showing the pow 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 «ent. that the re- bate is made at ?

315,6175 4

Ans. =1,26247: V 1,26247=1,06=6 per cent.

250

11. The present worth of £344 : 14 : 9 1,92 qrs., payable 7 years hence, is £245, at what rate per cent, is the rebate made"7

Ans. 5 per cent,

12. There is a debt of £441 : 17 : 3 1,92 qrs., payable in 4 years time, but it is agreed to take £350 present payment. I de- sire to know at what rate per cent, the rebate is made at ?

Ans. 6 per cent

169

THE

TUTOR'S ASSISTANT

PART IV.

DUODECIMALS,

OR, WBAT IS GEIUBALLT CALIXfi

Cross Multiplicatiorh o,nd Squaring of Dimensions by Arti- ficers and Workmen.

RULE FOR MVLtlPLTINO DUODECIHALLT.

1. Under the multiplicand write the corresponding denomina- tions of tiie multiplier.

2. Multiply each term in the multiplicand (beginning at the lowest) by the feet in the multiplier ; write each result under its reapectire term, observing to carry an imit for every 12, from eadi lower denomination to its next superior.

3. In the same manner multiply the multiplicand by the primes in the multiplier, and write the result of each term one place more to the right hand of those in the multiplicand.

4. Work in the same manner with the seconds in the multi- plier, setting the result of each terra two places to the right hand of those in the multiplicand, and so on for thirds, fourths, &,c.

p

DUODECIMALS.

EXAMPT.RS.

f. in.

f. in.

1. Multiply

7 . 91

by 3. 6.

Cross Multiplication.

Practice.

Duodecimals. DedmalA.

7y9 3^6

(

3i7-9

7

. 9 7,75

3.6

2

.6 8,5

21.0.0=7X3

23. 3

23.

3— X3 3876

2.3.0=9X3

3 . 10 . 6

3.

10.6X6 2325

3.6.0=7X6 0.4.6=9X6

27 . 1.6

27.

1 . 6 27,125

tJ7.1.6

f.in.

f. in.

f. in.pt&

H. Multiply

8.5

by 4.7

Facit,

38. 6.11

3. Multiply

9.8

by 7. 6

Facit,

72. 6

4. Multiply

8.1

by 3. 5

Facit,

27. 7. 5

5. Multiply

7.6

by 5. 9

Facit,

43. 1. 6

6. Multiply

4.7

by 3.10

Facit,

n. 6.10,,, 25. 8. 6.2.3

7. Multiply

7.5.9

' by 3. 5.3"

Facit,

8. Multiply

10.4.5

by 7. 8.6

Facit,

79.11. 0.6.6

9. Multiply

75.7

by 9. 8

Facit,

730. 7. 8

10. Multiply

97.8

by 8. 9

Facit,

854. 7

11. Multiply

57.9

by 9. 5

Facit,

543. 9. 9

12. Multiply

75.9

by 17. 7

Facit,

1331.11. 3

13. Multiply

87.5

by 35. 8

Facit,

3117.10. 4

14. Multiply

179.3

by 38.10

Facit,

6960.10. 6

15. Multiply

259.2

by 48.11

Facit,

12677. 6.10

16. Multiply

257.9

by 39.11

Facit,

10288. 6. 3

17. Multiply

311.4.7

by 36. 7.5

Facit,

11402. 2. 4.11.11

18. Multiply

321.7.3

by 9. 3.6

Facit,

2988. 2.10.4.6

THE APPLICATION.

Artificeris' work is computed by different measures, viz :-

1. Glazing, and masons' flat work, by the foot.

2. Painting, plastering, paving, &.c. by the yard.

3. Partitioning, flooring, roofing, tiling, &-c. by the square of 100 feet.

4. Brick work, &.c by the rod of 16J feet, whose square is 272i feet.

duodeci;hals. 171

Pleasuring by the Foot Square, as Glaziers' and Masons^ Flat

Work.

EXAMPLES.

19. There is a house with 3 tier of windows, 3 in a tier — the ''eight of the first tier 7 feet 10 inches, the second 6 feet 8 inches, and the third 5 feet 4 inches, the breadth of each is 3 feet 11 inches ; what will the glazing come to, at I4d. per foot ?

Daodecimals. 7. 10 the 6 . 8 heights 5 . 4 added.

feet. 233

in. pts. . 0 . 6 at 14d. per ft.

2d.=J 233

38, 0.

= Is. , 10 = 2d.

19. 10

3=windows in a tier.

. Oi = 6 parts.

2I0)27|1 . £13 . 11 .

lOi

10 J Ans.

59. 6

3 . 11 in breadth.

178.6 54.6.6

233 . 0 . 6

20. What is the worth of 8 squares of glass, each measuring 4 feet 10 inches long, and 2 feet 11 inches broad, at 4|^. per foot?

Ans. £1 : 18 : 9.

21. There are 8 windows to be glazed, each measures 1 foot

6 inches wide, and 3 feet in height, how much will they come to at 7|d. per foot ? Ans. £1:3: 3.

22. What is the price of a marble slab, whose length is 5 feet

7 inches, and the breadth 1 foot 10 inches, at 6s. per foot ?

Ans. £3:1:5.

Measuring by the Yard Square, as Paviers, Painters, Plas- terers, and Joiners.

NoTB. Divide the square feet by 9, and it will give the num- ber of square yards.

P2

173 DUODECIMALS.

EXAMPLES.

23. A room is to be ceiled, whose length is 74 feet 0 inches, end width 1 1 feet 6 inches ; what will it come to at Ss. lO^d p«r yard? ^n*. £18 : 10 : V.

84. What will the paving of a court-yard come to at 4|d. per yard, the length being 58 feet 6 inches, and breadth 54 feet 9

* inches ?

Ans. £7 : 0 : 10.

25. A r»om was painted 97 feet 8 inches about, and 9 feet 10 inches high, what does it come to ai 2s. 8 jd. per yard ?

Ans. £14 : II : U.

26. What is the content of a piece of «R^ain«coting in yards square, that is 8 feet 3 inches long, and 6 feet 6 inches broad)

^and what will it come to at 6s. 7id. per yard?

Ans. Contents, yards •5.8. 7.'6-; cotnes to 1C1 ^ 19 : 5.

27. What will the paving of a court-yard come to at 3s. 2d- per yard, if the length be 27 feet 10 inches, and the breadth 14 feet 9 inches ?

• Ans. £7:4:5.

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 6 feet 6 inches in breadth ^ the foot way is laid with I'urbeck stone, at 3s. 6d. per yard, %lid the rest with pebbles, at 3s. per yard ; what wUl the whole come to ?

Ans. £49 : 17.

^ 29. What Will t^e plastering of a ceiling, at lOd. per yard, -■■ come to, supposing the length 31 feet 8 inches, and the bread tL

14 feeft \0 inches ? ' Ans. £1:9:9.

* 30. What will the wainscoting of a room come to at 68. pe square yard, supposing the height •of the room (taking in the cor- nice and moulding) is l2 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 half work. Ans. £36 : 12 : 2^.

DUODECIMALS.. l'J3>

Measuring by the Square of 100 feet, as Mooring, Partition- ing, Roofing, Tilingi ^c.

EXAMPLES.

31. In 173 feet 10 inches in length, and 10 feet 7 inches in height of partitioning, how many squares ?

Ans.. 13 squ?ires,,39.feet,.8 inches, 10 p.

32. If a house of three storifes^ besides the ground floor, was to be floored at £6: 10 per square, and the house measured 20 feet 8 inches, by 16 feet 9 inches ; there are 7 fire-places, whose mea- «ures are, two of 6 feet by 4 feet 6 inches each, two of 6 feet by 5 feet 4 inches each, and two of 5 feet 8 inches by 4 feet 8 inches each, and the seventh of 5 feet 2 inches by 4 feet, and the well hole for the stairs is 10 feet 6 inches by 8 feet 9 inches : what will the whole come to ?

Ans. £53 : 13 : 3^.

33. If a house measures within the walls 52 feet 8 inches in length, and 30 feet 6 inches in breadth, and the roof be of a true pitch, what will it come to roofing at 10s. 6d. per square ?

Ans. £12:12.:11|.

NoTC. In tiling, roofing, and slating, it is customary to reckon the flat and half of the building within the wall, to be the measure of the roof of • that building, when the said Toof is of a true pitch, ». e. when the rafters are | of the breadth of the building ; but if the roof is noore or less than the true pitch, they measure from one side to the other with a rod or «tring.

34. What will the tiling of a barn cost, at 25s. 6d. per square ; the length being 43 feet 10 inches, and breadth 27 feet 5 inches <m the flat, the eave boards projecting 16 inches on each side ?

Ans. £24 : 9 : 5|.

Measuring by the Rod.

Note. Bricklayers always value their work at the rate of a brick and a half thick : and if the thickness of the wall is mpre or less, it must be redjucedto that thickness by this

p3

174 DVODECIHALS.

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 thickness two bricks and a half, how many rods doth it contain ?

Ans. 25 rods, 8 feet

36 If 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 many squared rods are there in a wall 62 J feel long, 14 feet 8 inches high, and 2^ bricks thick ?

Ans. 5 rods, 167 feet

38. If the side walls of a house be 28 feet 10 inches in length, and the height of the roof from the ground 55 feet 8 inches, and the gablo (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 i brick thick, and the gable at 1 brick thick ; what will the whole work come to at £5 16s. per rod 7

Ans. £48 : 13 : 6i,

Multiplying several figures by several, and the product to he produced in one line only.

Rule. Multiply the units of the multiplicand by the units of the multiplier, setting down the units of the product, and carry the tens ; next multiply the 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 ca> 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 yQw have multiplied the multiplicand all through, by every figure t the multiplier.

DUODECIMALS.

175

EXAMPLES.

Multiply 35234

by ,. 62424

Common way. 35234 52424

Product, 1847107216

I

140936 70468 140936 70468 176170

1847107216

EXPLANATION.

First, 4 X 4=16, tiiat is 6 and carry one. Secondly, 3 X 4-f 4X2, and I that i& carried, is 21 — set down 1 and carry 2. Thirdly, 2 X 4+3 X 2+4 X 4+2 carried=32, that is 2 and car- ry 3. Fourthly, 5 X 4 + 2x2 + 3 X 4+4 X 2 + 3 carried=47, set down 7 and carry 4. Fifthly, 3X4 + 5X 2 + 2X4+3X2 + 4X5 + 4 carried=60, set down 0 and carry 6. Sixthly, 3X2 + 5X4 + 2x2+3 X 5+6 carried=51, set down land 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, ajid the work is finished.

176 THE

TUTOR'S ASSISTANT.

PART V.

A COLLECTION OF QUESTIONS.

1. What is the value of 14 barrels of soap, at ^d. per lb., each barrel containing 254 lb.? Ans. £66 : 13 : 6.

2. A and B trade together; A puts in £320 for 5 months, B £460 for 3 months, and they gained £100 ; what must each man receive? Ans. A £53 : )3 : 9^, and B £46 : 6 : i^.

3. How many yards of cloth, at Hs. 6d. per yard, can I have for 13 cwt. 2 qrs. of wool, at 14d. per lb. ?

^715. 100 yards, 3^ qrs.

4. If I buy 1000 ella of Flemish linen for £90, at what may I sell it per ell in London, to gain £10 by the whole ?

Ans. 3s. 4d. per ell.

5. A has. 648 yards of cloth, at 14s. per yard, ready money, 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. 5120 gallons.

8. If I buy a yard of cloth for 14s. 6d., and sell it for 16s. 9d., what do I gain per cent.? Ans. £15 : 10 : 4^^.

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 8id. per lb. ? Ans. £76 : 13 : 2J.

A COfiLECTION OF QUESTIONS: 1T7

10. If f of an ounce cost f of a shilling, what will f of a lb. cost ? Ans. 17s. 6d.

11. If f of a gallon cost f of a pound, what will f of a tun cost? Ans. £105.

12. A gentleman spends one day with another, £1:7: 10^, and at the year's end layeth up £340, what is his yearly income ?

^715. £848 : 14 : 4^.

13. A has 13 fother of lead to send abroad, each being 19^ times 112 lb. B has 39 casks of tin, each 388 lb., how many ounces difference is there in the weight of these commodities 1

Ans. 212160 oz.

14. A captain and 160 sailors took a prize worth, £1360, of which the captain had -^ for his share, and the rest wa? equally divided among the sailors, what was each man's part ? ■

Ans. The captain had £272, and each sailor £6 : 16.

15. At what rate per cent, will £956 ampunt jto £1314 : 10, in 7| years, at simple interest? Ans. 5 per cent.

16. A hath 24 cows, worth 72s. eachi and B 7 horses, worth £13 a piece, how much will make good the difference, in case they interchange their said drove of cattle ? Ans. £4 : 12.

17. A man dies and leaves £120 to be given to three persons, viz. A, B, C ; to A a share unknown ; B twice as much as A, and C as much as A and B ; what was the share of each ?

An&. A £20, B £40, and C £60.

18. £1000 is to be divided among three men, in such a man- ner, that if A has £3, B shall haye £5, and C £8 ; how much must each man have ?

Ans. A £187^ 10, B £312 : 10, and C £500.

19. A piece of wainscot is 8 feet 6^ inches long, and 2 feet 9 J inches broad, what is the superficial content ?

Ans. 24 feet 0 : 3" : 4 : 6.

20. If 360 men be in garrison, and have provisions for 6 months, but hearing of no relief at the end of 5 months, how many men must depart that the provisions may last so much the longer?

Ans. 28S men.

21. The less of 2numbers in 187, their difference 34, the square af their product is required ? Ans. 1707920929.

22. A butcher sends his man with £216 to a fair to buy cat- tle ; oxen at £11, cows at 40s., colts at £1 : 5, and hogs at £1 : {5 each, and of each a like number, how many of each sort did he buy? Ans. 13 of each sort, and £8 over.

23. What number added to 1 If will produce 36fff •

Ans. 24|i|.

178 A COLLECTION Of QUESTIONS.

24. What number multiplied by f will produce ll-jV T

Ans. 2Gtf.

25. Wliat is the ralue of 179 hogsheads of tobacco, each weigh ing 13 cwt. at £2 : 7 : 1 per cwt. ? Ans. £5478 : 2 : 11.

26. My factor qends me word he has bought goods to the va- lue of £500 : 13 : 6, upon my account, what will his commission come to at 3i per cent ? Ans. £17 : 10 : 5 2 qrs. -j^.

27. Iff of 6 be three, what will i of 20 be ? Ans. 7^. 2a What is the decimal of 3 qrs. 14 lb. of a cwt. ?

Ans. ,875.

29. How many lb. of sugar, at 4^d. per lb. must be given io barter for 60 gross of inkle, at 8s. 8d. per gross ?

Ans. 1386f lb.

30. If I buy yarn for 9d. the lb. and sell it again for 13^d. per lb., what is the gain per cent. ? Ans. £50.

31. A tobacconist would mix 20 lb. of tobacco at 9d. per lb. with 60 lb. at 12d. per lb., 40 lb. at 18d. per lb., and with 12 lb. at 2s. per lb., what is a pound of this mixture worth ?

Ans. Is. 2id.fr.

32. What is the difference between twice eight and twenty, and twice twenty -eight ; as also, between twice five and fifty, and twice fifty-five 1 Ans. 20 and 50.

33. Whereas a noble and a mark just 15 yards did buy ; how many ells of the same cloth for £^ had I ? Ans. 600 ells.

34. A broker bought for his principal, in the year 1720, £400 capital stock in the South-Sea, at £650 per cent., and sold it again when it was worth but £130 per cent. ; how much was lost in the whole ! Ans. £2080.

35. C hath candles at 6s. per dozen, ready money, but in bar- ter will have 6s. 6d. per dozen ; D hath cotton at 9d. per lb. ready money. I demand what price the cotton must be at in barter ; also, how much cotton must be bartered for 100 doz. of candles 1

Ans. The cotton at 9d. 3 qrs. per lb., and 7 cwt. 0 qrs. 16 lb. of cotton must be given for 100 doz. candles.

36. If a clerk's salary be £73 a year, what is that per day ?

Ans. 4s.

37. B hath an estate of £53 per annum, and payeth 5s. lOd. to the subsidy, what must C pay, whose estate is worth £100 per annum ? Ans. lis. Od. ■^.

r

A COLLECTION OF QUESTIONS. 179

38. If I buy 100 yards of riband at 3 yards for a shilling, and too more at 2 yards for a shilling, and sell it at the rate of 5 yards for 2 shillings, whether do I gain or lose, and how much 1

Ans. Lose 3s. 4d.

39. What number is that, from which if you take f , the re- mainder will be i ? Ans. \^.

40. A farmer is willing to make a mixture of rye at 4s. a bush- el, barley at 3s., and oats at 2s. ; how much must he take of each to sell it at 2s. 6d. the bushel ?

Ans. 6 of rye, 6 of barley, and 24 of oats.

41. If f of a ship be worth £3740, what is the worth of the whole? ^ns. £9973 : 6 : 8.

42. Bought a cask of wine for £62 : 8, how many gallons were in the same, when a gallon was valued at 5s. 4d. ?

^715. 234.

43. A merry young fellow in a short time got the better of j- of his fortune ; by advice of his friends, he gave £2200 for an ex- empt's place in the guards ; his profusion continued till he had no more then 8S0 guineas left, which he found, by computation, was -j^- part of his money after the commission was bought ; pray what was his fortune at first ? Ans. £10,450.

44. Four men have a sum of money to be divided amongst them in such a manner, that the first shall have -^ of it, the second i, the third -J-, and the fourth the remainder, which is £28, what is the sum? Ans. £112.

45. What is the amount of£1000 for 3^ years, at 4^ per cent, simple interest? Ans. £1261 : 5.

46. Sold goods amounting to the value of £700 at two 4 months, what is the present worth, at 5 per cent, simple interest ?

Ans. £682 : 19 : 5i iWr-

47. A room 30 feet long, and 18 feet wide, is to be covered with painted cloth, how many yards of J wide will cover it ?

Ans. 80 yards.

48. Betty told her brother George, that though her fortune, on her marriage, took £19,312 out of her family, it was but f of two years' rent. Heaven be praised ! of his yearly income ; pray what was that ? Ans. £16,093 :6:8a year.

49. A gentleman having 50s. to pay among his labourers for a day's work, would give to every boy 6d., to every woman 8d., and to every man 16d. ; the number of boys, women, and men, was the same. I demand the number of each ?

Ans. 20 of each.

190 A COLLECTION* OF QUESTIONS.

50. A Stone that measures 4 feet 6 inches long, 2 feet 0 inches broad, and 3 feet 4 inches deep, how many solid feet doth it con> tain ? Ans. 41 feet 3 inches.

51. What does the whole pay of a man-of-war's crew, of 040 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 43. 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, 0 put in so much that he might receive £60 of the gain. I de- mand how much C put in ? Ans. £960.

54. Of what principal sum did £20 interest arise in one year, at the rate of 5 per cent, per annum ? Ans. £400.

55. In 672 Spanish 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.

57. If 48 taken from 120 leaves 72, and 72 taken from 91 leaves 19, and 7 taken from thence leaves 12, what number is that, out of which when you have taken 48, 72, 19, and 7, leaves 12? ^ns. 158.

58. A farmer ignorant of numbers, ordered £500 to be divided among his five sons, thus : — Give A, says he, f, B i, C -J^, D f , and E ^ part ; divide this equitably among them, according to their father's intention.

Ans. A £1522^1, B £114iHi C £91fii, D £76iif , E £65^.

59. When first the marriage knot was tied

Between my wife and me, My age did hers as far exceed,

As three times three does three ; But when ten years, and half ten years,

We man and wife had been, Her age came then ns near to mine,

As eight is to sixteen.

Ques. What was each of our ages when we were married Ans. 45 years the man, 15 the woman.

181

A Table for finding the Interest of any sum of Money, for any number of months, weeks, or days, at any rate per cent.

Year.

Caltn. Month.

Week.

Day,

£

£ «.

d.

£ 8. d.

£ ». d.

1

0 1

8

0 0 4i

0 0 OJ

2

0 3

4

0 0 9

0 0 u

3

0 5

0

0 1 1}

0 0 2

4

0 6

8

0 1 6

0 0 21

5

0 8

4

0 1 11

0 0 3i

6

0 10

0

0 2 3J

0 0 4

7

0 11

8

0 2 8i

0 0 4i

8

0 13

4

0 3 1

0 0 5i

9

0 15

0

0 3 5J

0 0 6

10

0 16

8

0 3 lOi

0 0 6J

20

1 13

4

0 7 8i

0 1 1}

30

2 10

0

Oil 6i

0 1 7i

40

3 6

8

0 15 4i

0 2 2i

1 50

4 3

4

0 19 2}

0 2 9

1 60

5 0

0

1 3 1

0 3 3i

70

5 16

8

1 6 11

0 3 10

80

6 13

4

1 10 9J

0 4 4i

90

7 10

0

1 14 7i

0 4 Hi

100

8 6

8

1 18 5J

0 5 5J

200

16 13

4

3 16 11

0 10 Hi

300

25 0

0

515 4i

0 16 5i

400

33 6

8

7 13 10

1 1 11

500

41 13

4

9 12 3i

1 7 41

600

50 0

0

11 10 9

1 13 lOi

700

58 6

8

13 9 2f

1 18 4i

800

66 13

4

15 7 8i

2 3 10

900

75 0

0

17 6 U

2 9 31

1000

83 6

8

19 4 7i

2 14 9J

2000

166 13

4

38 9 2}

5 9 7

3000

250 0

0

57 13 10

8. 4 4J

4000

333 6

8

76 18 5i

10 19 2

5000

416 13

4

96 3 Oi

13 13 llj

6000

500 0

0

115 7 8J

16 8 9

7000

583 6

8

134 12 3i

19 3 6}

8000

066 13

4

153 16 11

21 18 4i

9000

50 0

0

173 1 6i

24 13 IJ

10,000

833 6

8

192 6 If

27 7 llj

20.000

1666 13

4

381 12 3J

54 15 10§

30,000

2500 0

0

576 18 5*

62 3 10

a

168

RiTLE. Multiply the principal by the rate per cent., and the number of months, weeks, or days, which are required, cutofT 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 ; for every 10 cut off in weeks, add a halfpenny •,.and for every 40 in the days, 1 farthing.

EXAMPLES.

1 . What is the interest of £2467 lOs. for 10 months, at 4 per lent per annum ?

2467 : 10 900=75 : 0 : 0

4 80= 6 : 13 : 4

7=0:11:8

9870

10 987=82: 6:0

987100 2. What is the interest of £2467 lOs. for 12 weeks, at 5 per cent. ?

2467 : 10 1000=19 : 4 : 7|

6 400= 7 : 13 : 10

80= 1 : 10 : 9^

12337 : 10 50= 0 : 0 : 2^

12 ^_

1480|50=28: 9: 5

1480150: 0 What is the interest of £2467 lOs., 50 days, at 6 per cent. ?

2467 : 10 7000=19 : 3 : 6^

6 400= 1:1:11 2= 0

14805: 0 50

50

= 0:0: li = 0:0: O}

7402150=20:5: 7

7402150 : 0

To find what en Estate, from one to £60,000 per annum will come to for one day. Rule 1. Collect the annual rent or income from the table for 1 year, against which take the several sums for one day, add i))em together, and it will give the answer.

183

▲n estate of £376 per annum, what is that j«r day ? 300=0:16: 5i

70=0 6=0

10 4

376=1 : 0 : 7i

Tojind the amount of any incomer salary, or servants^ wages, for any number of months, 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 11 months, for 3 weeks, and for 6 days ?

270 11

3970

270 6

1620

For 11 months. 2000=166 : 13 : 4 900= 75 : 0:0 70= 5 : 16 : 8

2970=247 : 10 : 0

B'or 6 days. 1000=2 : 14 : 9J 600=1 : 12 : 10^ 20=0 : 1 : li

1620=4: 8: 9^

For 3 weeks. 270 800=15: 7: 8i 3 10= 0 : 3 : lOJ

810 = 15 : 11 : 6^

For the whole time. 247 : 10 : 0 15 : 11 : 6i 4: 8:9i

267 : 10 : 3J

A Table showing' the number of days from any day in the month to the same day in any other month, through the year.

January . . February . March . . .

April

May

Jane

July

August. . . September October . . November December.

365 334

306 275

31 59

365 28

337 365

306 334

245; 276: 304

214! 245' 273

184^ 215

153< 184

122 153

92 123

61 92

3l! 62

243 212 181 151 la) 90

90

59

31

365

335

304!

'l51

120

92

61

31

365

335

304

2121 a42| 273 1821 2121 2-13 181 212 15ll 182

120

89

61

30

365

33-4

274 304 243 273

151

121

181 212

150, 181

122- 153

9ll 122

61

30

365'

335 365

303, 334 273 304 242: 273 2l2i 243

02 O

243 273 212 242 184' 214 153, 18:^ 123 153 92 122 62: 92

31 ;

365

335 365 304 334i 274' 3041

304334

273 303

245 275

214 244

184214

153183

123153

92:122

611 91

31 61

365 30

335 365

02

m

A COMPENDIUM OF BOOK-KEEPING. BY SINGLE ENTRY,

Book-keeping is the art of recording the transactions of persons in business so as to exhibit a state of their ailairsiu a concise and satisfactory manner.

Books may be kept either by Single or by Double Entry, but Single Entry is the method chiefly used in retail business.

The books found most expedient in Single Entry, are the Day- Book, the Cask-Book, the Ledger, and the Bill-Book.

The Day-Book begins with an account of the trader's property, debts, &c. ; and are entered in the order of their occurrence, the daily transactions of goods bought and sold.

Tke 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, CW A 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 tlve folio of the books from which the respective accounts are extrac- ted, by figures placed in a column against the sums. References are also made in the Day-Book and Cash-Book, to the folios in the Ledger, where the amounts are collected. This process is railed 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 i and, on the contrary, the person to whom you sell goods, or to whom you pay money, is Debtor.

In the Bill-Book are inserted the particulars of bW 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 Bilk Payable, which are those that are drawn upon and accepted b\ (he tradesTian himself.

m

I>A.Y BOOK.

(folio I.)

c tx

1

January 1st, 1837.

£ 500

73

60 133

8. 0

12

6

18

d, 0

7

0

7

I commenced business with a caiHtal of Five Hundred Pounds in Cash

■ 1

2d

Bennett and Sons, London.* Gn By 2 hhds. of sugar

act. or. lb. net. or, lb. 13 1 4 12 0 12 3 16 116

gron wt 26 0 20 tare 2 3 6

neat wt. 23 1 14 at 63fl. p«r cwt.

2 chesU of tea

net. qr. lb, lb. 1 0 15 25 1 0 12 25

2 0 27 1 22

1 3 SatGs.perlb

~F

3d

3 3

7

3 5 0

10

8 17

5

10 12

18

0

0 6

6

0 0 6

6

Hall and Scott, Liverpool, Cr. By soap 1 cwt. at 688. .............

candles, 10 dozen at 7s. 9d. ,

1

6th

Ward, William Pr. To 1 cwt of sugar, at 708

14 lbs of tea, at 88

2

6th

0

6

6

Cooper, William Dr. To 1 sugar ho<^shead

* The student may be directed to fill up this and similar blanks in this book and the Ledcej with the names of places familiar to him. '

a3

nm

DAY BOOK.

(folio 2.)

January 9th, 1837. |

£

0

1 1

4

0

17 _ 17

0 0 0 0 1

6

•.

16 17 15

8

6

0 5

9 8 4 8 10

16

d.

6 0 0 6

0

0 0

0 6 9 3 6

0

Johnson, Richard To 2 dozen ©f candles at Sa. 3d

Dr.

3

a

10th,

Ward, William To sugar, 1 ctuk

groaa wt 5 9 10 cask., tare a 10

Dr.

neat 5 0 0 atCSe

r2th. 1

Smith, John To 14 lb. of euffar

Dr.

12 lb of candles

I lb. of tea.

2 •3

14th.

Ilall and Scott, Liverpool, By 2 cwt soap at 68b.

Cr.

17th.

n.

To 21 lb. of goap, at 74«. per cwt

0 0 1

0 0

0

13 16 10

9

4

IF

10 6

4

0 2

2

19th.

Smith, John To 14 lb. of sugar

Dr.

i lb. of tea

2l8t.

0

0

1

18 8

0 3

Smith, John To 28 lb of sugar

Dr.

'T"

6'

3"

187

DAY BOOK.

(folio 3.)

3

January 23d, 1837.

£ 172

16

d. 0

Yates <^' Lane, Bradford, Cr. By 4 pieces of superfine cloth, each 36 yards,

at 24s. per yard . . .

3

23d.

2

8

0

Edwards, Benj. Manchester, Cr. By 2 paces of calico, each 24 yards, at Is. per yard . .

3 ~2~

23d.

0

9

6

6 0 0 6

3

0 0

Smith, John Dr. To 14 lb. of soap

24th.

0 3 5

9 0

145

2

16

14

5

15

8

16 16

Johnson, Richard Dr. To 3 dozen of candles, at 8s. 3d

1 cwt. of soap, at 74fl

1 i cvrt. of sugar, at 70s

2

24th.

Smith, John Dr. To 1 lb. of tea

3

36th.

Mason, Edward Dr. To 3 pieces of superfine cloth, each 36 yards,

at 278. per yard

& pieces of calico, each 24 yards,

at Is. 2d- per yard..

148

12

0

~3^ 3

27th.

50

172

46

55

2

0

105

8^

16

17

7

19

15

19

0

0

1

0 6 6

1

Parker, Thomas, Dr. To 1 piece of superfine cloth, 36 yards, at 28s

31st.

Bill* Payable, Cr. By Yates & Lane's Bill at 2 months, due April 2

L3^

Inventory, January 31, 1837.

cict. qr. lb.

Raw sugar, 14 3 14 at 63s

Tea, 1 2 I6i at 63. per lb

Soap, 0 3 14 at 68s

Candles, 2 dozen, at 78. 9d

198

CASH BOOK.

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189

INDEX TO THE LElMJER.

A

■Newton, Joim

N

... 2

Bernard & Co 1

0

5 Bennett &. Sons, London 1

Bills payable >, 3

Cooper, William, fi

Parker, Thomas

P

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c

D

Q

Bdwaxds, B. MoDcheeter ^

E

R

F

Stock accoant..

... 1

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... a

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G

T

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H

V

Johnson, Ricbajrd 2

I

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W

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K

X

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Y

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Maaon, Edward 3

M

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