UC-NRLF

F THE VERS1TY OF<

BY T. M. HUNTER,

RECTOR TO THE ASSOCIATION FOR THE REVIVAL '.OF SACRED MUSIC IN SCOTLAND.

HUNTER'S Elements of VOCAL MUSIC :

An Introduction to the Art of Beading Music at Sight. 6d.

%* This work has been prepared with much care, and is the result of long practical experience in teaching. It is adapted to all ages and classes, and will be found considerably to lighten the labour of both teacher and pupil. The Exercises are printed in the Standard Notation, and the Notes are named as in the Original Sol-fa System. CONTENTS.

Musical Scales.

Exercises in Time.

Syncopation.

The Chromatic Scale.

Transposition of Scale.

The Minor Scale.

Part Singing.

Explanation of Musical Terms.

HUNTER'S SCHOOL SONGS

Por Junior Classes,

With Preface by Rev. JAMES CURRIE, M.A., Principal of the Church of Scotland Training College, Edinburgh, and Author of the u Elements of Musical Analysis," etc.

%* These "SONGS" consist of Verses composed expressly for this work, or carefully selected from approved sources.

In the Songs for Junior Classes the TUNES are- Simple, and princi- pally arranged for two voices ; while in those for Advanced Classes they are more complex in arrangement, of greater length, and mostly written for three voices. The Melodies, which are either original or adaptations from the great Composers, are printed in the STANDARD NOTATION.

The entire Series is intended as a Musical Manual for the Singing Class or the Family Circle.

FIRST SERIES, containing 60 Songs,- price 4d.

Morning Song. The Fairy Queen. The Merry Month.— (Hound for 3 Voices.) Boyhood.

Friendship. — (Bound

for 3 Voices.) The Fairies1 Dance. Herald of Spring. Charming littleValley.

Bright beams the

Morning. What you've to Do. —

(Round for 3 Voices.} The Swing.

Edinburgh: OLIVER AND BOYD. London: SIMPKIN, MARSHALL, AND Co.

Hunter's School Songs with Music.
FIRST SERIES (for Junior Classes)— Continued.
Little Things.	To the Praise of	Call John the Boat-
All Hail! gentle	Truth.— (Round for	man.— (Round for 3
Spring.	3 Voices.)	Voices.)
Fairy Light.	The Kite.	The Brook. [here.
Cuckoo.— (Catch.)	Now the Sun sinks in	The Summer now is
The Bugle Horn.	the West.— (Round	Evening Sun.
Come,0 Come Away !—	for 3 Voices.)	The Bell doth Toll.— •
(Round for 3 Voices.)	Before all Lands.	(Round for 3 Voices.)
The River.	Music.— (Round for 3	Joyous Spring-time.—
Come, Come Here! —	Voices.)	(Round for 3 Voices.)
(Catch for 4 Voices.)	The Seasons.	Winter Song.
The Cricket.	Like a May -day.—	The Snowdrop.
Play is Done.— (Round	(Round for 3 Voices.)	The Little Spring.
for 3 Voices.)	Hark ! How the	Morning Call.
Sunshine.	Lark.— (Round for 3	The School Bell.
Early to Bed.— (Round	Voices.)	Oft by the Deep Blue
for 3 Voice*.)	The Little Busy Bee.	Sea.
Come, May ! thou love-	The Sea.	Winter's Departure.
ly Linge/er.	Ever Blooming.—	England and Her
Softly, gently flow our	(Round for 3 Voices.)	Queen.
Days.	The Daisy.	Boat Song.
Evening.	Birds are Singing. —	Try Again.
Come, Brothers.—	(Round for 4 Voices.)	Holiday Song.
(Round for 3 Voices.)	The Moon.	Evening.— (Canon for
Boat Song.	The Little Lark.	4 Voices.)
SECOND SERIES, containing 63 Songs, price 4d.
W.-lconiP, bright and	Patriotism.	To the Cuckoo.
sunny Spring.	The Village Green.	Dew-drops.
Sunmu-r Mu ruing.	Sister, Wake.— (Round	The Bugle.
Autumn.	for 6 Voices.) [ing.	The Child's May-day
Little Jack Ilorner.—	Children, join in Sing-	Song.
(Rou>>	To the Rainbow.	The Reaper's Song.
	In the Harvest Morn	I am Merry. — (Hound
. MLJ is rising.	so cheering.	for 3 Voices.)
	The Bells.— (Round for	Be kind to each
Billy and M.-.	3 Voices.)	other.
The Valley.	Sleep, my Baby.	He that would live. —
The Monkey.	Daybreak.	(Round for 3 Voices.)
Music Kvprywhpro.	Tin; Woodcutter's	Shall we, oppressed
Hear the Big Clock.—	Night Song.	with Sadness.
(Round for 3 Voices.)	Sing we together. —	Youth's Desires.
Mountain Boy's Song.	(Round for 4 Voices.)	Come from Toil.
[Continued at end of BooTc.

PRACTICAL ARITHMETIC

SENIOR CLASSES.

HENRY G. C. SMITH,

TEACHER OF ARITHMETIC AND MATHEMATICS, GEORGE HERIOT'S HOSPITAL.

SIXTH EDITION.

EDINBURGH : OLIVER AND BOYD, TWEEDDALE COURT.

LONDON : SIMPKIN, MARSHALL, AND CO.

1871.

Trice 2s. bound.— Answers to Ditto 6d.

E2UCATIOH IlfiB,

Now ready^ a New Edition of PRACTICAL ARITHMETIC FOR JUNIOR CLASSES.

BY HENRY G. C. SMITH.

Price 6d.— Answers to Ditto, 6d.

C.

I

EDINBURGH :

TED BY OLIVER AND BOYD, TWEEDDALE COURT.

**~nrr- fv ~

THIS Manual, which is a Sequel to Practical Arithmetic for Junior Classes, is intended for the use of those who have mastered the Fundamental Rules in Simple and Compound Numbers. Considerable space, in accordance with the importance of the subject, has been devoted to the explanation of Fractions ; and the other branches also have been illustrated with a view to practical instruction. The Exercises, which are copious and original, have been constructed to combine interest with utility. They are arranged in distinct Sections, and are accompanied with Illustrative Processes. As the work is essentially prac- tical, the explanatory remarks in elucidation of the various processes are more of an applicate than an abstract character.

The Answers to the Exercises hi this Manual are published in a separate form.

CONTENTS.

Page

Arithmetical Tables 7

Prime and Composite Numbers 12

Vulgar Fractions 19

Decimal Fractions 51

Continued Fractions 76

Practice 77

Allowances on Goods 90

Simple Proportion 92

Compound Proportion . . 103

Statistics 108

Commission and Brokerage Ill

Insurance 114

Interest 115

Discount 124

Equation of Payments 127

Stocks 128

Profit and Loss 131

Distributive Proportion 137

Alligation 141

Barter 144

Chain Kule 145

Exchange 147

Involution 154

Evolution 157

Scales of Notation 168

Duodecimals 171

Series 175

Compound Interest 180

Miscellaneous Exercises 186

ARITHMETICAL TABLES,

Z. MONEY.

MONEY OF ACCOUNT.

4 farthings = 1 penny d,

12 d. =1 shilling s.

20 s. =1 pound £.

d. for denarius: s. for solidus:

£ for libra.

DECIMAL DIVISION OF £1.

10 mils, m. =1 cent c.

10 c. =1 Horin fl.

10 fl. =1 pound £.

COINS IN CIRCULATION.

GOLD. Sovereign, £1 ; Half-sav. 10s. 1869 sovereigns are coined out of 40 lb» troy of sterling gold.

SILVER. Crown, 5s. ; Hf.-cr. 2s. 6d. ; Florin, 2s.; Shilling, IB.; Sixpenc*,6A.', Groat, 4d.; Thrcepencct3d. 66 shillings are coined out of 1 Ib. troy of sterling silver.

BRONZE. Penny, Id.; Halfpenny, fyl.\ Farthing, Jd. In 100 parts of the bronze metal for these coins, there are :— 95, r; 4, tin; and 1, zinc. To 1 Ib. avoir, there are: — Of pennies, 48; of halfpennies, 80 ; of farthings, 160.

OBSOLETE COINS.

Tester, Gd.; Dollar, 4/6; Noble, 6/8; Seven Shilling piece; Angel, 10/; Half- guinea, 10/6 ; Mark, 13/4 ; Pistole, 16/ ; Guinea,21/; Carolus,23/; Jacobus,25/; Moidore, 27/ ; Joannes, 36/.

The denominations of Scots Money are one-twelfth of the value of the cor- responding names in sterling: thus, £1 Scots = 20d. sterling. lg. • = Id. *

Also, 1 merk, * = 13Jd. »

II. WEIGHT.

The Act 5° Geo. IV. cap. 74, which established the IMPERIAL WEIGHTS AND MEASURES, came into operation on 1st Jan. 1826. By the Act 18° and 19° Vic. cap. 72, the Imperial Standard of Weight is the Pound Avoirdupois, deposited in the Exchequer at Westminster, and of which copies are placed in the Mint, the Royal Society of London, Greenwich Observatory, and the Palace at Westminster.

AVOIRDUPOIS WEIGHT.

Avoir. Wt. is the general weight of commerce. 1 Ib. avoir. = 7000 grains.

16 drams, dr. = 1 ounce oz. 16 oz. = 1 pound Ib.

28 Ib. = 1 quarter qr.

4qr.orll21b.=: Ihundredwt. cwt. 20 cwt. = 1 ton T.

Also, 14 Ib. = 1 stone

In London, a stone of butcher-meat = 8 Ib. In Liverpool, &c., 100 Ib. = 1 cental.

TROY WEIGHT.

Troy Wt. is used in weighing the precious metals and in philoso- phical experiments. 24 grains, gr. = 1 penny wt.dwt. 20 dwt. = 1 ounce oz.

12 oz.or5760gr.= 1 pound Ib.

At the Mint, the ounce is divided into 1000th s.

The fineness of gold is estimated in carats. Pure gold is said to be 24 carats fine. Sterling gold, of which every 24 parts contain 2 of alloy, is 22 carats fine.

ARITHMETICAL TABLES.

The fineness of silver is estimated in oz. and dwt. Sterling silver, of which 1 Ib. contains 18 dwt. of alloy, is 11 oz. 2 dwt. fine.

151$ Diamond carats = 1 oz. troy, which is also = 600 Pearl grains.

APOTHECARIES' WEIGHT. 20 grains, gr. = 1 scruple ^

SQ =1 drachm 3

83 =1 ounce ^

In the above, the ^ is the ounce Troy of 480 grains ; but in the NEW SYSTEM of weights adopted by the General Med- ical Council in October 1862, the Q and 3 have been abolished, and the § is the ounce Avoirdupois of 437 J grains.

III. LENGTH.

By the Act 18° and 19° Vic. cap. 72, the Imperial Standard Measure of Length is the Yard, deposited in the Exchequer at Westminster, and of which copies are placed beside those of the Standard of Weight.

LINEAL MEASURE.

12 lines, I. =1 inch in.

12 in. = 1 foot ft.

3 ft. =1 yard yd.

5£ yd. = 1 pole po.

40 po. = 1 furlong fu.

8 fu. or 1760 yd. = 1 mile ml.

Also

Length of 3 barleycorns = 1 in .

Breadth of 4 barleycorns=l digit = f in. ft.

Palm = 3 Hand = 4 Span = 9

Cubit

Step

Pace

Fathom = 6 ft.

-J

GEOGRAPHICAL MEASURE.

6076 ft. nearly = 1 geog. ml. 60 geog. ml. = 1 degree of the Earth's circumf. 21600 geog. ml. = the Earth's circ.

Also 3 geog. ml. = 1 league

SURVEYORS' MEASURE.

n. = 1 link Ik.

100 Ik. or 66 ft. = 1 chain 80 ch. := 1 mile

ch. ml.

4 nl. or 9 in. 4 qr.

CLOTH MEASURE.

= 1 nail nl.

= 1 quarter qr. = 1 yard yd.

Flemish ell =3 qr. I English ell = 5 qr Scotch //=37in. (French » = 6qr.

IV. SURFACE.

SQUARE MEASURE.

This Table is formed by squaring the corresponding denominations in Lineal Measure.

144 sq. in. = 1 sq. foot sq.ft. 9 sq. ft. = 1 sq. yard sq.yd. 30£ sq. yd. = 1 sq. pole sq.po 40 sq. po. = 1 rood ro. 4 ro. or 4840s. yd. =1 acre ac. 640 ac. =1 sq. mile sq. ml.

Also 100 sq. ft. = 1 square of flooring

36 sq. yd. = 1 rood of building

SURVEYORS' MEASURE. 10,000 sq. Ik. = 1 sq. chain

10 sq. ch. =1 acre

V. SOLIDITY.

CUBIC MEASURE.

This Table is formed by cubing the corresponding denominations in Lineal Measure.

1728 cub. in. = 1 cub. ft.

27 cub. ft. = 1 cub. yd.

Also

5 cub. ft. = 1 barrel bulk B. B. 8 B. B. =1 ton measurement

40 cub. ft. of rough timber = 1 load 50 cub. ft. of hewn timber = 1 load 216 cub. ft. = 1 cubic fathom

ARITHMETICAL TABLES.

VI. CAPACITY.

According to the Act 5° Geo. IV. cap. 74, the Imperial Standard Measure of Capacity is the Gallon, which contains 10 Ib. avoir, of dis- tilled water weighed in air at the temperature of 62° Fahrenheit, the Barometer being at 30 in. The Standard Measure is deposited in the Exchequer at Westminster.

MEASURE OF CAPACITY.

pt.

qt gal.

fc

qr.

4 gills gi. — 1 pint 2 pt. =1 quart

4 qt. =1 gallon

2 gal. = 1 peck

4 pk. = 1 bushel

8 bu. =1 quarter

Also

Pottle = 2 qt I Coomb = 4 bu.

Strike = 2 bu. | Load = 5 qr.

Last = 10 qr.

The Imperial Gallon = 277-274 cub. in , is the highest measure for liquids.

The weight of an Imperial Bushel of wheat varies from 56 Ib. to 64 Ib. : by the Tithe Commutation Act of England it is taken at 60 Ib.

The following" were abolished by the Act 5° Geo. IV. cap. M.

Cub. in. Imperial

Wine Gallon =231= '8331109 Gallon Ale Gallon =282 = 1-0170446 Winchester

Bushel

Heaped Measure, usod for coals, &c., was abolished by the Act 5° and 6° Guliel. IV. cap. 63, which enacted that after 1st Jan. 1836, " all Coals, Slack, Culm, and Cannel of every Descrip- tion, shall be sold by Weight and not ." The bushel was = 1 Winchester bushel -|- 1 quart = 2217 02 cub. in., but when heaped in the form of a cone = 2815-486 cub. in. 3 heaped bushels = 1 sack

12 sacks = 1 chaldron

When the terms Hogshead, Pipe, &c. are used, it is merely as the names of

r I =2150-42= -9694472 Bu.

* The Weights and Measures of the United States of America are the same as those used in Great Britain, with the exception of the Measures of Capacity, which continue to be the subdivisions and multiples of the Winchester Bush- el f r dry goods, and of the Wino Gal- lon for liquids.

casks of wine, &c., and not as meas- ures, for the contents must always be expressed in Imperial Gallons. When the names Puncheon, Tierce, are applied to casks of sugar, molasses, &c., their gross and net weight* must be stated.

APOTHECARIES' FLUID MEASURE. 60 minims 11^=1 fluid drachm f 3

8f3 =1 fluid ounce fg

20 f g =1 pint O

8 O =1 Sallon C

O for Octarius ; C for Congius.

1 f § of distilled water weighs 1 oz.

VII. INCLINATION.

ANGULAR MEASURE.

60 seconds " =1 minute

60'

90'

4 L or 360°

30°

= 1 degree = 1 right angle L •=. 1 circle 0 Also = 1 sign of the zodiac

VIII. TIME.

MEASURE OP TIME.

60 seconds, sec. = 1 minute min. 60 min. = 1 hour ho.

24 ho. = 1 day da.

7 da. =1 week wk.

4 wk. = 1 common month co.mo. 365 da. 1

or 52 w. J-zr 1 common year co. ye. Id. J

365 da. 6 ho. = 1 Julian year Ju. yr.

366 da. = 1 leap year

The year is divided into 12 cal- endar months : —

January

February

March

April

May

June

Da.

31 28 31 30 31 30

July

August

September

October

November

December

Da.

31 31 30 31 30 31

In leap year, February has 29 days. * See Allowancet on Good*. A2

10

ARITHMETICAL TABLES.

QUARTERLY TERMS IN ENGLAND.

Lady Day March 25

Midsummer June 5

Michaelmas Day Sep. 29 Christmas Dec. 25

Easter Day, on which the Movable Feasts depend, is the first Sunday after the Paschal Full Moon, which hap- pens on March 21, or next after it. When the Full Moon is on a Sunday, Easter Day is on the next Sunday.

QUARTERLY TERMS IN SCOTLAND.

Candlemas Feb. 2

Whitsunday May 15

Lammas Aug. 1

Martinmas NOT. 11

The Sidereal Day is = 23 ho. 56 min. 4-09 sec. It is the true time of the earth's revolution on its axis, or the interval between two successive me- ridian transits of the same star. A sidereal clock is always kept in an as- tronomical observatory.

The Apparent Solar Day is the inter- val between two successive meridian transits of the sun's centre. This day varies in length. The difference be- tween Apparent Solar Time as shown by a sundial, and Mean Solar Time as indicated by a well-regulated clock, is termed Equation of Time.

The Mean Solar Day of 24 hours is used for the purposes of civil life. Astronomers in using the Mean Solar Day begin at 12 o'clock noon, and reckon the hours onward to 24. The Astronomical agrees with the Civil Reckoning from noon to midnight; but from midnight to noon the former is a day behind, thus : —

CiTil Time. Astron. Time.

Sep. 10 : 7 p.m. = Sep. 10 : 7 ho.

Sep. 11 : 11 a.m. = Sep. 10 : 23 ho, Since the sun apparently describes a circle or 360° in 24 hours, 15° of longitude correspond to 1 hour of mean solar time; thus the time at a place in 45° E. long, is 3 hours before that of Greenwich, while in 60° W. long, it is 4 hours behind it.

The Periodical Month or sidereal rev- olution of the moon is = 27 da. 7 ho. 43 min. 11-5 sec. It is the time of the moon's revolution round the earth, or the interval in which the moon re- turns to the same place in the heavens. The Lunar Month or synodical rev- olution of the moon is = 29 da. 12 ho. 44 min. 2'87 sec. It is the interval be-

ween new moon and new moon, or between two successive conjunctions of the sun and moon.

The Jews use a year of 12 lunar months of 29 or 30 days each ; and to make it somewhat correspondent to the solar year, intercalate a month of 29 days, 7 times in a cycle of 19 years. The Mohammedans use a year of l! lunar months or 354 days, and add a day to the year 11 times in 30 years.

The Sidereal Year is = 365 da. 6 ho. 9 min. 9*6 sec. It is the time of the earth's revolution round the sun.

The Solar or Tropical Year is = 365 da. 5 ho. 48 min. 49'7 sec. = 365*24224 days. It is the interval between two successive passages of the sun through the vernal equinox. The solar year regulates the seasons, and is there- fore the proper standard for regulating the civil year.

Julius Csesar adopted a nominal year of 365 da. 6 ho. In the Julian Calendar, every year whose number is divisible by 4 contains 366 days. The Julian Calendar was introduced 45 B.C. Its error is = 365'25 da. — 365'24224 da. = -00776 da. p yr., or 3-104 da. in 400 years. In the 16th century an error of 12 days had accumulated, but as it was determined to reckon merely from 325 A. D. — the year of the Coun- cil of Nice— Gregory XIII. ordered ten days to be omitted in October 1582. In the Gregorian Calendar, every year whose number is divisible by 4 is a leap year, except when divisible by 100 and not by 400 ; thus, while 1600 is a leap year, 1700, 1800, and 1900, are common years. 400 years in the Gregorian Calendar or New Style (N. S.) are thus 3 days shorter than 400 years in the Julian Calendar or Old Style (O. S.). The error of the Gre- gorian Calendar in 400 years is there- fore 3-104 da. — 3 da. = -104 da., or •00026 da. %> yr. N. S. was introduced into the British Empire in Septem- ber 1752. O. S. is still used by the Greek Church. The difference be- tween O. S. and N. S. is progressive. In the 16th and 17th centuries it was ten days; in the 18th, eleven; in the 19th, twelve.

MEMORANDA.

Sack of Flour or Meal = 280 Ib.

Barrel >/ » = 196 *

Quire of Paper = 24 sheets

Ream // = 20 quires

Bale " = 10 reams

Roll of Parchment = 60 skins

Pack of Wool = 240 Ib.

Long hundred = 120

Gross = 144

ARITHMETICAL TABLES.

11

METRIC SYSTEM OF WEIGHTS AND MEASURES.

The Use of the Metric System was rendered permissive in the United Kingdom by the Act 27° and 28° Vic. cap. 117.

When the metre was first definitively introduced in France in 1799, it was adopted as the ten-millionth part of the Quadrant from the N. Pole to the Equator, but subsequent calculations have, however, shown that it is not pre- cisely so.

MEASURES OF LENGTH.

Metre. Inches.

Millimetre = -uol = -03937079 Centimetre = -01 = -3937079 Decimetre = -1 = 3'937079 METKE =1- = 39-37079

Metre

Yard*.

Dekametre = 10 = 10-93633

Hectometre = 100= 109-3633

Kilometre = 1000= 1093-633

Myriametre = 10000 = 10936*33

MEASURES OF SURFACE. 8q. Metres. Sq. Yards.

Centiare = 1 = 1 19603326 ARE = 100 = 119-603326

Sq. Metret. Acres.

Dekare = 1000 = -2471143 Hectare = 10000 = 2-471143

MEASURES OF CAPACITY. Cub. M«tr-. Pint.

Centilitre = -UUuul = '0176077 D.-cilitre = -0001 = -176O77 LITRE = -001 = T76077

Cub Metre.

Gallons.

Dekalitre = -01 = 2-20027 Hectolitre = -1 = 22-0097 Kilolitre = 1- = 220*097

Milligram = -U01 = "oil

m = -01 = 'I Decigram = -1 = 1-.V, <Ji:.\.M =1- = 15-4323487

WEIGHTS.

Giami. Pounds Avoir.

Dekagram = 10= -022U46212

Hectogram = 100 = -22046212 Kilogram = 1000= 2-2046212 Myriagram= 10000= 22-046212 Quintal = 100000= 220-46212 Millier = 1000000 = 2204-6212

SCOTCH WEIGHTS AND MEASURES.

These were declared obsolete by the Act 5° Geo. IV. cap. 74.

WBIOITT.

16 drops = 1 ounce

16 ounces = 1 pound 16 pounds = 1 stone There were two kinds of weight:— Troyes or Dutch Wright, of which 1 Ib. = 7608-95 Imperial grains, and Tron Wright, of which 1 Ib. = 9022-67 Im- perial grains. The Standard Stone Troyes -I to Lanark.

LINEAL MEASURE.

87 inches = l ell I 4 falls =1 chain

6 ells = 1 fall I 80 chains= 1 mile

The Standard Ell, kept at Edinburgh,

= 37-0598 Imp. in. The chain =

_'i Imp. chain = 74-1196 Imp. ft.

SQUARE MEASURE.

36 sq. ells = 1 sq. fall 40 sq. falls = 1 rood 4 roods = 1 acre = 1-261183 Imp. acre

LIQUID MEASURE.

4 gills = 1 mutchkin I 2 chopins=l pint

2 mutchk.= 1 chopin | 8 pints = 1 gallon

The Standard Pint, kept at Stirling =

104-2034 cub. in. = '375814 Imp. gallon.

DRY MEASURE.

4firlots=lboll 16 bolls =lchalder

41ippies = lpeck 4 pecks =lfirlot

There were two kinds of Dry Meas- ure, the one for wheat and the other for barley, oats, &c. The Standard Firlots were kept at Linlithgow.

Cub. In. Imp. hu.

Wheat Firlot = 2214-3235 = -998256 Barley Firlot = 3230-3072 = 1-456279

There was great diversity in the measures used in the various counties.

The Standard Scotch boll of meal is usually reckoned at 140 Ib. avoir.

PEIME AND COMPOSITE NUMBEES,

PRIME NUMBERS.

A NUMBER which cannot be divided by any other without leav- ing a remainder is termed a Prime Number or Prime; thus, 1, 2, 3, 5, 7, 11, 13, are primes.

A number composed of two or more primes multiplied to- gether is termed a Composite Number ; thus, 6, 15, 35, are com- posite numbers.

(1) Find the primes in the following series : —

1, 2, 3, 4, 5, -fr, 7,-6-,-e; 46; 11, 4* 13, *fc

By eliding every second number after 2, we cancel all numbers «-H * 2. By eliding every third after 3, we cancel those «-J^ 3. The numbers no£ eZzWecZ are prime.

This process, commonly known as ERATOSTHENEs'f SIEVE, may be abridged as in the following examples : —

(2) Find the primes to 50.

1, 2, 3, 5, 7, -9-, 11, 13, 4% 17, 19, •», 23, 45; &r, 29, 31, -S3; -35; 37, -39; 41, 43, -4% 47, -4fc

Since 2 is the only even prime, we omit all the other even num- bers. In eliding the composite numbers containing any prime, we need not test any below the square of the pi~ime ; for since all the lower composite numbers containing the prime contain a lower prime also, they must have been previously elided ; thus we be- gin to elide those <-h 3 from 9, and those <-h 5 from 25.

For the same reason we divide by no prime whose square is > (greater than) the highest number in the series ; thus we finish by eliding 49, which is <-{-> 7.

(3) Find the primes from 100 to 150.

101, 103,4^ 107, 109,434-, 113,4*5-,4i?-, 119, 121, 423-, 495; 127,1297 131, 433; 435; 137, 139, 444, -H3; 44^44?-, 149.

* The sign <-h», for " divisible %," was introduced by Mr Barlow of Woolwich Academy in 1811. f Eratosthenes, curator of the Alexandrian Library, died B. c. 194.

PRIME AND COMPOSITE NUMBERS.

13

Elide every third number after 105, the first number «-H 3 .. fifth n ,f 105, , „ 5.

* » seventh » n 105, // >, „ „ 7.

» » eleventh » » 121, » n „ „ n. All the composites are now elided, for 149 is < (less than) the

square of the next prime, 13.

Find the primes below 1000, giving those in each hundred as a separate exercise.

2, PRIME FACTORS.

THE primes that make up a composite number are termed its Prime Factors; thus, 2, 2, 2, 3, 7, are the prime factors of 168.

Resolve the following into prime factors.

2 2	KM _ (2X2X2X3X7 i||- tor2'X3X7	2 2	091— 5 2X2X3X7X11 |±|- tor 2*X3X7X11
2	T2	3	231
3	21	7	77
	~		11
1. 6	6. 42	11. 98	16. 143	21. 245	26. 624
2. 12	7. 55	12. 100	17. 154	22. 264	27. 1188
3. 15	8. 66	13. 105	18. 165	23. 275	28. 1331
4. 21	9. 70	14. 110	19. 192	24. 343	29. 1452
5. 30	10. 75	15. 125	20. 242	25. 539	30. 1584

Resolve the following into prime factors, and combine them into sets of three factors each, not greater than 12. 8 1 0^^2 X 3 * X 5

:3'X3*X'-'X5 Itis easier to obtain the 91810=9x9X10 =9X9X10 three factors thus:- -go=9xlo

31. 225	35. 495	39. 405	43. 704	47. 240	51. 960
32. 315	36. 616	40. 448	44. 756	48. 486	52. 1152
33. 392	37. 968	41. 504	45. 792	49. 729	53. 1296
34. 441	38. 1089	42. G72	46. 1056	50. 768	54. 1728

3. GREATEST COMMON MEASURE.

A NUMBER which divides another without leaving a remainder is termed a Measure or Factor of that number ; thus, 8 is a measure of 24. A number which divides two or more num- bers without leaving a remainder is termed a Common Measure of those numbers ; thus, 6 is a common measure of 24 and 36.

14

PRIME AND COMPOSITE NUMBERS.

3» The greatest number which divides two or more numbers without leaving a remainder is termed their Greatest Common Measure (G. C. M.) ; thus, 12 is the G. c. M. of 24 and 36 ; 6 the G. c. M. of 24, 36, and 54.

Numbers whose G. c. M. is 1 are prime to each other. Com- posite numbers may be prime to each other ; thus, 25 is prime to 36.

(1) Find the G. c. M. of 78 and 300.

I. By Prime Factors. 78 = 2 X 3 X 13 ; 300 = 2* X 3 X 5*

G. c. M. = 2 X 3 = 6

Since 2 and 3 are the only factors common to 78 and 300, 2X3 or 6 is the o. C. M. of 78 and 300.

II. By Division. 78)300(3 1

234

66)78(1 2

66 12)66(5 6 G. c. M.

60

G. c. M.~6)12(2 12

6 is a common measure of 78 and 300. For, 6 measures 2X6 or 12; 5X12 or 60; 60 + 6 or 66; 66 + 12 or 78 ; 3X78 or 234; and 234 -j- 66 or 300.

No number ^> 6 is a common measure of 78 and 300. Since every measure of 78 measures 3 X 78 or 234, every common meas- ure of 78 and 300 measures 234 and 300. Now if a number is con- tained a certain number of times in 234 and another number of times in 300, the difference between the quotients is an integer, which is the number of times the number is contained exactly in 300 — 234. Every common measure of 78 and 300 therefore meas- ures 300 — 234 or 66. Since it measures 78 and 66, it measures also 78 — 66 or 12 ; hence also 5 X 12 or 60 ; and finally 66 — 60 or 6. No common measure of 78 and 300 can therefore be > 6 ; but 6 is a common measure of 78 and 300 ; .•. (hence) 6 is their

G. C. M.

Find G. C. M. of

78 66	300 234
12 12	66 60
6

1.48,78 2. 56, 98 3.121,143 4. 342, 665 5. 448, 784 6. 203, 261 7. 375, 525

8.841, 899 9.961, 1178 10. 1243, 1469 11.1001,1287 12.1131,2639 13:9889,986 14.1792,1832

15. 1850, 1517 16. 1792, 1847 17. 3927, 5049 18. 1287, 1551 19. 1537, 1802 20. 3056, 3629 21.1261,22116

22.3243,37976 23.31484,109268 24.82739,57693 25.10759,20405 26.714285,857142 27.49593,43902 28.17641,22243

3. PRIME AND COMPOSITE NUMBERS. 15

(2) Find G. c. M. of 42, 56, and 49.

42)56(1 14)49(3

42 42

14)42(3 G. c. M. 7)14(2

42 14

Every c. M. of 42 and 56 is a measure of their G. c. M., 14 ; .*. every c. M. of 14 and 49 is a c. M. of 42, 56, and 49 ; but 7 is the o. c. M. of 14 and 49 ; .-. 7 is G. c. M. of 42, 56, and 49.

(3) Find G. c. M. of 192, 56, 44, 128, 94.

Take any two numbers, as 44 and 94 ; 2 is their G. c. M. The o. c. M. required cannot therefore be > 2. Now 2 measures the numbers 192, 56, 128; .'. 2 is the G. c. M. required.

Suppose we had selected 56 and 128, their G. c. M. is not a measure of all the rest. G. c. M. of 8 and 44 is 4. G. c. M. of 4 and 94 is 2, which measures 192, and 2 is the o. c. M. required.

To abridge the process, it is expedient to select at first two num- bers whose G. c. M. is among the least of the mutual G. c. measures.

(4) Find G. C. M. of 27, 216, 48, 105, and 405.

3 is G. c. M. of 27 and 48 ; 3 is a M. of 216, 105, and 405. .-. 3 is the G. C. M. of 27, 48, 216, 105, and 405.

Find G. C. M. of

29. 45, 27, 54

30. 90, 84, 81

31. 56, 84, 63

32. 24, 36, 48, 216

33. 32, 40, 64, 108

34. 72, 84, 66, 176

35. 198, 495, 209, 660

36. 146,730,365,219

37. 924, 378, 612, 246

38. Find the Greatest Common Divisor of 12460 and 10769.

39. Find the greatest number cancelling 1859 and 3003.

40. Find the length of the greatest line exactly measuring the sides of an enclosure 216 yd. long and 111 broad.

41. Find the greatest measure of capacity contained exactly in two measures containing respectively 6 gal. 7 pt. and 8 gal. 6 pt.

42. What is the greatest sum of money contained exactly in £2 " 9 » 1 and £2 " 3 " 11?

43. Find the greatest sum of money contained exactly in £34 »7»7 and £70 "12 »2.

44. George, James, and John, wish to spend 2/6, 1/10£, and 3/5^, on the same kind of squibs. Find the price of the dearest squib they can purclin

45. Two apprentices carry 1147 and 961 ivory balls respectively from the workshop to the showroom. The balls are carried in baskets of equal contents, which are filled and emptied several times. How many balls are in a basketful ?

16 PRIME AND COMPOSITE NUMBERS.

46. Two frigates having the same number of guns fire a number of rounds. The one has fired 608, and the other 1102 shots. How many guns has each ?

47. The Nemesis and Mceander frigates having the same number of guns greater than 36, fire a number of rounds. The one has fired 352 and the other 484 shots. How many guns has each ? And why is the limitation " greater than 36" necessary?

48. Two opposition coaches, which have run full during the season for the same number of days, have had 4807 and 3971 pas- sengers respectively. How many days has the season lasted, and how many passengers does the one contain more than the other ?

LEAST COMMON MULTIPLE.

A NUMBER which contains another an exact number of times is termed a Multiple of that number ; thus 48 is a Multiple of 8.

Measure and Multiple are correlative terms : —

7 is a measure of 14, 14 is a multiple of 7.

A number containing two or more numbers an exact number of times is termed a Common Multiple of those numbers ; thus 48 is a Common Multiple of 4, 6, and 8.

The least number containing two or more numbers an exact number of times is termed their Least Common Multiple (L. c. M.) ; thus 24 is L. c. M. of 4, 6, and 8.

When two or more numbers are prime to each other, their L. c. M. is their product ; thus L. c. M. of 3, 5, 7, and 11, is 3X5X7XH.

(1) Find L. c. M. of 15 and 21.

15 = 3 X 5; 21 = 3 X 7. L. c. M. 105 = 3 X 7 X 5.

Every common multiple of 15 and 21 must contain 3, 5, and 7. But 3 X 5 X 7 is the least number containing 3, 5, and 7; .-. 3 X 5 X 7 is L. c. M. of 15 and 21.

L. c. M. of two numbers = Product -— G. c. M. Thus, of 15 and 21; G. c. M.=3; Product = (3 X 5) X (3X7).

^ In finding the L. c. M. of 2 numbers it is thus easier to di- vide one of the numbers by their G. c. M., and multiply the quotient by the other number.

PRIME AND COMPOSITE NUMBERS.

17

(2) Find L. c. M. of 224 and 256.

G. c. M. = 32. L. c. M. = v*4 X 256 = 7 X 256 = 1792.

(3) Find L. c. M. of 384 and 564.

G. C. M. = 12.

L. c. M. = 32 X 564 = 18048.

Find L. c. M. of

1. 27, 36

2. 42, 56

3. 35, 49

4. 72, 48

5. 52, 78

6. 34, 51

7. 144, 180

8. 216, 225

10. 200, 250

11. 224,343

9. 196, 343 12. 324, 360

13. 420, 798

14. 225, 375

15. 234, 390

(4) Find L. c. M. of 16, 18, 21, 24, 30, 32, 36. i.

2 2 2 3	-tft •*% 21,	24,	30,	32,	36
21,	12,	15,	16,	18
21,	6,	15,	8,	9
21,	-B-,	15,	4,	9

7,

5, 4,

We elide 16 and 18, which are respectively measures of 32 and 36.

We divide by the prime 2 so Ions as it is contained in more than one number. Since 2 is not contained in 21, we continue to write 21 until we divide by the next prime. In the 4th line we elide 3, a measure of 9.

The factors contained in the numbers in addition to 2, 2, 2, 3, are 7, 5, 4, 3 ; and as these are prime to each other, L. o. M. = 2X2X2X3X7X5X4X3 = 10080.

IF.

12 | jfr 4fr, 21, 24, 80, 32, 36 7, «j 5, 8, 3

Since the factors of 12 are contained in one or other of the num- bers, we may divide by 12, and find the other factors contained in the numbers.

12 is not a measure of 21, but on dividing 21 by 3, the o. c. M. of 12 and 21, we obtain 7. Similarly we divide 30 and 32 respec-

tively by 6 and 4. The facl

factors contained in the numbers besides those of 12, are 7, 5, 8, and 3 ; and as these are prime to each other, L. c. M. = 12X7X5X8X3= 10080.

In the First Method we divide by a prime so long as it is contained in two or more numbers. In the Second, we divide by a composite number whose factors are contained in one or other of the numbers ; and when any number is not a multiple of the divisor, we divide it by their G. c. M.

18

PRIME AND COMPOSITE NUMBERS.

(5) Find L. c. M. of 21, 24, 25, 27, 28.

2 21, 24, 25, 27, 28

21, 12, 25, 27, 14

3 21, 6, 25, 27, 7, 2, 25, 9

L. c. M.

25 X 9 = 37800.

ii.

12 | 21, 24, 25, 27, 28 7, 2, 25, 9, 7

L. c. M. = 12X7X 2 X 25 X 9

= 37800.

16. 4, 6, 10, 12

17.8,12,15,18

18.12,16,18,20

19.12,16,18,27

20. 10, 6, 15, 12

21.12,15,20,40

22.12,28,35,21

23. 32, 36, 49, 56, 42

24. 20, 24, 25, 27, 45 25.28,30,32,36,42 26.35,40,42,49,28 27.8,14,18,21,32,28

28. 24,27,28,32;36,56

29. 15,21,24,27,28,35

30. 25,32,63,40,35,56,80 31.30,36,32,48,40,54,63 32. 30,33,36,42,48,63,55 33.27,36,45,54,63,72,81 34.35,45,56,63,40,72,28 35.15,21,33,24,35,40,77 36.56,40,24,88,55,21,33

37. Find the least number containing the 9 digits.

38. Find the shortest distance that three rods of 8 ft. 3 yd. and 4 yd. will exactly measure.

39. Find the content of the smallest vessel that may be exactly filled by using a gallon, a 10 pint, or a 12 pint measure.

40. A rides at 10 miles an hour, B drives at 6 miles an hour, and C walks at 3 miles an hour. Find the shortest distance they may all traverse in an exact number of hours.

41. Tom, Dick, and Jack, agree to spend the same sura in pur- chasing fire-wheels at the rate of l£d., 4d., and 2£d. respectively. What is the smallest sum they can expend?

42. What are the prime factors of L. c. M. of 12, 35, 28, and of 21, 15,20?

43. Mention the prime factors of L. c. M. of 12, 28, 35, 21, 55, and of 15, 33, 20, 77, 44.

(6) Find the least number which, when separately di- vided by 2, 3, 4, 5, 6, always leaves the remainder 1.

Least Number <-h 2, 3, 4, 5, 6, is L. c. M. of 2, 3, 4, 5, 6=60; 60+1 or 61 is the number required.

44. Find the least number which, when separately divided by 2, 3 ...... 7, leaves the remainder 1.

(7) Find the least number which, when separately di- vided by 2, 3 ...... 8, leaves the remainders 1, 2 ...... 7

respectively.

L. c. M. = 840 ; 840 — lor 839 is the number required.

45. Find the least number which, when separately divided by 2, 3 ...... 9, leaves the remainders 1. 2 ...... 8 respectively.

19

VULGAR FRACTIONS.

IK a unit is divided into ' - ! - '• - '

three equal parts, and two i 1

of them are taken, the

parts thus taken form ' - * - ! - ! - '

tico-thirds of one unit. 2

If two units as a whole are divided into three equal parts, and one taken, the part thus taken is one-third of two units.

That which we have obtained by either method is written, f . It is termed a VULGAR FRACTION, of which 2 is the Nu- mmifnr and 3 the Denominator.

The Denominator of a Vulgar Fraction indicates the number of equal parts into which a unit is divided ; and the Numera- tor, the number taken. Or, the Numerator indicates the number of units, and the Denominator the number of equal parts into which these units, considered as a whole, are di- vided, and of which one is to be taken.

If 2 units are eacli di-

vided into 5 equal parts, 1

we obtain 10 fifths. The integer 2 is thus reduced to the

fractional form, 13°.

(1) Reduce 3 to an equivalent fraction with denominator 7.

s=y

1. 1!^ luce 9 to equivalent fractions with denominators 4, 8, 7.

2. ,. 33 .. " " " 3, 5, 8.

3. .. 29 " 11,13,20.

4. * 37 * » >• 12, 14, 15.

5. A baker divides 12 rolls into 4 equal parts each. How many fourths has he ?

6. Into how many eighths of a yard can a draper cut 17 yards of cloth ?

Suppose we

have two units 1 2 s

and three-fifths of a unit, by dividing each of the units into fifths, and adding in the three-fifths, we obtain thirteen-fifths.

23=_= 13

~5 5 " ~S '

Every number which, like 2|, is thus made up of an integer and a fraction, is termed a MIXED NUMBER.

20 VULGAR FRACTIONS.

Reduce the following mixed numbers to a fractional form :—

13. 928^3

14.

15.

		T3" ~~~		13
1.	7J	4.	8T6T	7.	90	I7	10.
2.	11|	5.	8T\	8.	79	?&	11.
3.	13*	6.	15jf	9.	23	1 1	12.

16. How many eighths of a yard are in 7§ yd. ?

17. How many twelfths of a penny are in 8T55d. ?

1 8. How many sixteenths of a yard has a draper sold who has disposed of 9T35 yd. ?

If we take the ' ! ! ! : ! ! '• — ^ ! ! —

fraction y, we

find we can make up two units, with three-fourths over. V = 2f.

Every fraction which, like y , has its numerator > its de- nominator, is > 1, and is termed an IMPROPER FRACTION.

Every fraction which, like £, has its numerator = its de- nominator is = 1, and is termed an IMPROPER FRACTION.

Every fraction which, like f , has its numerator < its de- nominator, is < 1, and is termed a PROPER FRACTION.

Reduce the following improper fractions to whole or mixed numbers : —

(1) VV = 14. (2) >A° = 7TV (3) i| = 1.

1. 86

2. H

3. *TV

4.

5.

6.

8. 4300

9. *ff*

10. ij?a

1L 2500

12. 2^f2

13.

15. 3T°°°

16. A grocer, who has sold 89 quarter-pounds of tea, wishes to know how many Ib. he has sold.

17. A draper, who has sold 117 sixteenths of a yard, is asked how many yards he has sold.

18. The average length of a year, according to the Gregorian Calendar, is ' 44fi0°09 - days. Express this as a mixed number.

If we take any fraction, as f, all fractions with the denomina- tor 7, having the numerator < 5, as ^, |, &c., are < 4; and all with the numerator 5, having the denominator > 7, as f, |, &c., are also < f . By diminishing

f

VULGAR FRACTIONS. 21

8. the numerator, or increasing the denominator, we thus dimin- ish the value of a fraction.

Again, all fractions with the ' - « - « - •. - 1 - : _ «

denominator 7, having the nume- ?

rator > 5, as f, ^ &c., are > f ; .....

and all with the numerator 5, but ' « the denominator < 7, as J, £, &c.,

are also > 4- By increasing the ' - ! - ! - ! - • - *•

numerator, or diminishing the de- I nominator, we thus increase the value of a fraction.

(1) Mention 4 fractions with denominator 12, next > T

A i A» Ai i8a' A"

(2) Mention 3 fractions with numerator 5, next < T5T.

A » i**> A» TV

1. Mention 4 fractions with numerator 9, next < T'T.

2- - 3 ...... 6, » > TV

3. » 6 » " denominator 9, •• > $.

4. " 5 ...... 10, , < T',.

5. « 3 " " numerator 13, •> < jf.

6. » 3 •» » denominator 17, •• < if.

We may multiply -^ by 2, by • - ' - ' - '• - •• - « - « — • doubling the number of the parts ; f I I

thus, $ X 2 = $.

We may multiply -| by 2, by — ^ -- :

doubling the magnitude of the s

parts; thus, f X 2 = f .

By multiplying the numerator or dividing the denominator, we thus multiply the value of a fraction.

Multiply the following fractions by integers : —

12 X 2

.{), 12, a common factor of 24 and 60, is contained 2 times in 4. To tor by 2,

a), J.Z, a COI11H1UI1 lildUi Ui if* aim uv, ia WJJIKMUWU «

24. To multijjly ^ by 24 we may therefore multiply the numera- 2, and divide the denominator by 12.

1.	i	X	5	5.	A	X	3	9.	A	X	12
2.	w	X	4	6.	if	X	4	10.	/*	X	14
3.	TT	X	4	7.	U	X	7	11.	H	X	27
4.	w	X	8 / 8.	H	X	16	12.	«	X	15

9.

22 VULGAR FRACTIONS.

13. Seven purchasers each buy g peck of meal. How many pecks have been bought?

14. 1 Ib. troy = -} f 1 Ib. avoir. How many Ib. avoir. =

15. Find the number of degrees = 25 grades, of which each = i9o deg.

1O» We may divide f by 4, by taking one-fourth of the number of parts ; thus, | -f- 4 = f .

We may divide f by 2, by taking as many parts of half the magnitude; thus, | -=- 2 = T3o-

T3o

(MfTiWUUfC ) UlUO, ^ -7- AI TO*

By dividing the numerator or multiplying the denominator, we thus divide the value of a fraction.

Divide the following fractions by integers : —

3X6

In (3), 3, a common factor of 18 and 15, is contained 6 times in 18. To divide £* by 18, we may therefore divide the numerator by 3, and multiply the denominator by 6.

9. 44 — 20

2. 3.

4.

-T- 2

*H2

*- 6

4 — 9

5. 6.

7.

— 5

— 3

— 8

— 6

10. 11.

12. |f — 30

— 12

— 21

11.

13. What part of a mile does a stream flow tp minute which flows 2*5 mile in 7 min. ?

14. 12 oz. troy = ^1 Ib. avoir. What part of 1 Ib. avoir, is 1 oz. troy?

15. 64 squares of a draught-board occupy £f sq. ft. What does one square occupy ?

Having given any fraction, as f , I

by taking one-third of the num- • — \ — i — • — • — • — L_ ber of parts each three times as I

large, we have the same fraction expressed, as f . By dividing the numerator by 3, we divide the fraction by 3 ; and by di- viding the denominator by 3, we multiply the fraction by 3, and thus the fraction is unaltered in value.

By dividing the numerator and the denominator of a fraction by the same number, the value of the fraction remains unaltered.

The fraction f , when expressed as |, is said to be in its LOWEST TERMS. A fraction is in its lowest terms when its numerator and denominator are prime to each other.

1

VULGAR FRACTIONS. 23

11* Reduce the following fractions to their lowest terms : —

In I., we divide the numerator and denominator successively by the common factors 4, 7, 4, selected by inspection. In II., we at once divide the numerator and denominator by their

O. C. M., 112.

The product of the factors 4, 7, 4, used in I., is = G. c, M., 112. (2) J«. By inspection, |»| = iif = > f

(3) Ufi.

I- IS

2. «

4. ff

5. ^

6- Ill 7

8- im

9- Uff

!0- iiil

»• H«

12. ««

O. C.M.

74

13. Hit

14. 4f*f

15- Mft

16- /T% 17. UH

18- i«M

19. HIJ

20. *jJSS 21.

22.

5 1 1

TTJUUT

10212

TTJT^

23. ttitf 24- IBH5

= H-

Of^ 3927

***• 3U4"^?y 26. !iJ«

27. 28. 29. 30. ii|44

31. 32.

33. 33u%]

34. jffi

35. i«$

36. AVA

37. AVA

38.

39.

40. «JH

4.9 285714 ^^" W^-5V5

43. 44.

45t 857142

46. imii

47 616384

48. HHH

12* Having given any fraction, as •J, by taking twice as many parts of half the magnitude, we have the same fraction expressed, as

T8o

By multiplying the nu- merator by 2, we multiply the fraction by 2; and by multiplying the denominator by 2, we divide the fraction by 2 ; and thus by multiplying the numerator and the denominator of a frac- tion by the same number, the value of the fraction remains unaltered.

Take any two fractions, f, {. Of the Common Multiples of 4 and 6, let us take 24, | = Jf , and f = ££. The fractions have thus been reduced to equivalent fractions with a Common Denominator.

12 is the L. c. M. of 4 and 6 ; j = T\ and f = JJ. The fractions have thus been reduced to equivalent fractions with the Least Common Denominator.

24

VULGAR FRACTIONS.

|0

1. 2. 4 7 8 9 11 12	Reduce the following to equivalent fractions ha least common denominator (L. C. D.) : — ;i) i, i, TV, it- L. c. D. = L. c. M. of 3, 6, 12, 16 == 48. Since the denominator of § is contained 16 F- M- times in 48, we multiply its numerator by 1 6. 16 Similarly, we divide the L. c. D. by the deno- minator of each fraction, and multiply the 8 corresponding numerator by the quotient. ^ The number, showing how often the denominator 4 of a fraction is contained in the L. c. D., is that by which the numerator of that fraction is g multiplied, and may be termed the FRACTIONAL MULTIPLIER (F. M.) 4, f, 1 13- 4> ?, t, «, f 4, 1	ving the 2 _ 32 3 "~ 48 5 .__ 40 .6 :"" 48 7 _ 28 12 48 15 _ 45 16 ~~ 48 3 -f
*, J, 1 14- TT, A> A) A» 33, -e* f, {, 1, T\ I5' t) T\, 4, H, -1, iJ j, *, 4 H 17- M» f A? If iV f7 7 11 1Q 11 11 11 11 11 11 , "5", TT, T3 1O* T2", ^4", T¥, 4"¥, "615, Tl" t, T\, «, i8 19. $, {, ii, 44, ft, 11 7 17 19 11 90 4 24 124 624 3124 ¥J ^4~J "32, ^"ff ^U< T) 2"S? T2"5"J "62"5"J "3"T2 5" 23572 91 3711973 T) TTJ TT, T, TT ^i' ^^J Uo, *2> 2O? 24"> 2"ff 359 36 99 555 555 • TJ t> TT> 12> "33 £*" t? IT) Tf7, TfT, 2TJ -5? f3 3 3 3 9Q 11 15 17 35 31 23 > 4~5 T¥) fi?J ^36" £l°' T2^J 1"6, T¥) "3~6^ "3"2"J ^T
(2) *, T9T, TV L. c. D. =7 X 11 X 13 = 1001. When the denominators are prime to each n x ^ other, the F. M. of a fraction is the product of the denominators of the other fractions : thus, the F. M. of f is 11 X 13, which is = 7 X 13 7 X 11 X 13	6 _858 7 ~1001 iT~~Iooi 4 308
7 7XH (3) *, A, T\, 41 - L. c. D. = 7 X 13 X 19 X 23 = 39767. Although the denominators are prime to ™* each other, yet it is often convenient, when there are four or more fractions, to obtain the F.JM. by dividing the L. c. D. by each 3959 denominator in succession : thus, the F. M. ofM-^fiRI 7X13X19X23 ,. . 2093	13 "~ 1001 3 _ 17043 7 "~ 39767 J__ 3059 13 "~ 39767 7 _ 14651 19 39767 12 _ 20748 23 ~ 39767
ui 7 is ou5i — ? which is = 13 X 19 X 23. 1729

VULGAR FRACTIONS.

25

12.

25.	1,	1.	£	31.	i i	i, 1, T'T
26.	1,	f,	3 T	32.	1 1!	?', "
27.	I,	*.	4, A	33.	1, T,	ii, T9F
28.	i,	1,	T? T\	34.	f *,	1, i?
29.	i	|,	T\, T4T	35.	f, *!	f, TV
30.	i,	i	T^J TT	36.	fi i,	f, T8T, 1
con	ipar	in<r	the maernitudes ' —	• i	i i •

i 1

of a number of fractions, as f , J,

<,7_. we may take a line, or, for

the sake of distinctness, as many

equal lines as there are fractions,

and lay off parts corresponding — — ! — ! — ! — • — ' — ! — '• — * — ' — L_I

to them. By an appeal to the T7s 1

eye, or by the aid of compasses, we may then compare the

magnitude of the fractions.

In comparing the fractions arithmetically, we may proceed as in the following examples.

Arrange the following fractions in order of magnitude : —

(1) ii £> if, respectively = «J, «}, «J.

Order of Magnitude, f , j|, J.

By reducing the fractions to a common denominator, we at once discover the order of magnitude.

(2) i I, I, iV *•

Complements, J, i, i, T\y, f Order of Magnitude, r%, |, J, f , f .

That which, when added to a proper fraction, makes up unity, is termed its COMPLEMENT. Of a number of fractious, that which has the least complement is the greatest fraction.

(3) TV, A) A* respectively = ^, ~, ~.

Order of Magnitude, TV, A> «rr-

(4) f y », H, «, respectively = |, 7-&, f , f .

Order of Magnitude, ft, |}, f, |f .

Of the series of fractions T23, 535, /f, 54g, arranged in order of magnitude, let any two, as 5aff and /4, be taken. Reducing tlK-in to the common denominator 20 X 34, we have the nu- merators respectively == 3 X 34 and 20 X 5, of which the for-

mer is the greater.

When fractions are arranged in order of B

26

VULGAR FRACTIONS.

Iflt magnitude, the product of the numerator of any fraction by

the denominator of the next less is > the. product of the de- nominator of the former fraction by the numerator of the lat- ter; thus, in the series { |, ||, |i, §; 15 X 26 > 16 X 23, 23 X 36 > 26 X 31, 31 X 6 > 36 X 5.

1. i,f,S 3. I,' 1 1' I

5312 • 1?2") T) TU

A 2 5 1

O. VV. -^Vi ^

1 1 7 33' 6

8211 7 5 • -3-) 7-5-5 T2-) ¥)

10. *, it, H, i*, I

11. if, if, i}, is

19 9 15 19 16 12 1^' TO) 1F> ¥0) TT) TS

13. 14.

, T, A» A

16. A, A. A» A, **»

n3 4 5 216 ' T7> **» 2^> TT> "5T>

18. TV,

) A» 3T>

Let us ADD the fractions ^, J, and |-. ' By taking any line as the unit, we place the lines representing £, |, f , in a line, and thus obtain their sum. To express the value of the sum, or to add the fractions arithmetically, we reduce them to equi- valent fractions .^ having a common •<<<<••• . . . . i . denominator, and 1 add the numerators of the equivalents, nators, the L. c. D. is generally taken.

I 1

Of common denomi-

(1) i + I + I-

I— A

1 = A i 4. i

*=ii

f f = 2 A

(2) i + i + « + * + A =

Otherwise

4- l

5.

9-4+A-f A-hA

13. »+« + « + TW + H + 5%

14. T'T + A + A + A + T'A + 3*

15. H + » + A + AV + T¥T + ,

VULGAR FRACTIONS.

27

(3) If +11 =2-

In the diagram, having rep- resented 1£ and If by lines, we place the integers together, and tlion the fractions in the same line with them, and thus obtain the sum whose value is 2 + £ == I

1

1 1

I +1

1 2 3 i

In adding Mixed Numbers we need not reduce them to improper fractions.

(4)

tf + 7f

= 19

234 + 8964+ *16 + 429

17. 6i+7|+8J+9$

19. 20. 21.9J+10|+11f+5U

The work may be abridged by combining, in the process of adding, those fractions whose denominators are either the same or have a common measure, as in the following examples :

(5) & + A + TV + i + f +

1 = 1

11

(6)

4 T A — ""24"

2 I 1 1 _ 6 + U

TT "T "5T — 51

• 7

2

3"

81-4+*+*

W-A+I+A+A

28 VULGAR FRACTIONS.

14* (7) A student spends I of the day in teaching, TV in at- tending classes, ^ in study, /¥ in recreation and meals, and £-; in miscellaneous reading. What part of the day is he thus occupied ?

- -

- 4¥ - T6-

37. i of a pole is in sand, and T45 of it in water. What part of the pole is thus below the level of the water ?

38. In an Allied Camp, £ of the soldiers are natives of England, T25 of Scotland, Tyff of Ireland, and 550 °f Wales. What part of the camp is under British colours ?

39. Of the chairs in the University of Edinburgh, 395 of the num- ber was founded in the nineteenth century, £ in the eighteenth, and ^ in the seventeenth. What part of the whole was founded in these centuries ?

40. In 1685, the regular infantry, and the regular cavalry of England, were respectively T|5 and Tt£o of the militia. What part were they together of the militia ?

41 . Of the prismatic spectrum, red occupies J, orange 53o, and yellow T2S. What part of the whole do these three colours occupy ?

42. What part of a piece of cloth has a draper sold, who has cut °ffi3G> S52, &, and ftofit?

43. A treasurer has expended fcfc, ?7g, i§, §75, and ,*, of a given sum. What part of the whole has he laid out ?

44. In 1853, of the number of freshmen belonging to Cambridge iff belonged to Trinity College, 3523g to St John's College, & to Gonvilleand Caius College, and 5|B to Queens' College. What part of the whole did these form ? "

45. Of the water of the Dead Sea, T||B is muriate of lime, ;V? muriate of magnesia, &*>** muriate of soda, 3^ sulphate of lime. What part of the whole are these ingredients ?

15* ^ Let us SUBTRACT a frac-

tion, as -|, from an integer, » - - - : _ ' .......

as 2. We dimmish one of 1 | 2

the units by |; thus, | — | = |. This, with the other unit, makes the whole remainder If. In subtracting a proper frac- tion from an integer, we find the complement of the fraction, and diminish the integer by 1.

(1) 18-« =

1. 18 —

2. 10 —

3. 9 - T

4. 11 —

5. 8-

6. 23 -

VULGAR FRACTIONS.

29

15. Let us subtract f from f . By -T83

reducing the fractions to a com- «'.......

mon denominator, we find f=T\, ?$ T92 1

£ = T9j 5 an(l by taking T85 from T9^ we obtain the remainder T'T.

W f — f - 21	•*• IT*
7- J — J	15.	2 9 36" ~~~	ft	23.	a 15
8. 1-i	16.	1 1 TT ~	T\	24.	•B — i\V
9. § — §	17.	l-l		25.	fl ~" I,
10. | — |	18.	if	T§	26.	TT T3
11. | — |	19.	T9T-	f	27.	if -if
12. it - T7*	20.	«~	T97	28.	19 11 ^l T^
13. T*T — T	21.	1 3 T*	1 9	29.	tt-*A
u- ii — ir	22.	H-	1 7	30.	i! — H
In subtracting a mixed		-2
numuLr, as ^5-, iruin dii- other, as 3£, we find the	1 2 3
difference first between	-T5o
the fractions, and then '-1	i t i t i i	J
between the whole num-	f35 1	s*
bers. Thus, J or -fa from	3 Ol* T8TJ	leaves T35, and	3 — 2=1,
So, 3|-2i=3 — 2 + -	~5— 1 Io~— l

r = 2 + ^^ = 2^

31. 3i — 2J	35. 18| — 10J	39.	17i| — 13	1 2
32. 7^ — 5^	36. 17f — 10T\	40.	18if — 17	sW
33. 17f — 13T%	37. 23|i — 19JJ-	41.	29T\V — 9-	rvv
34. 6| — 3 ,\-	38. 16iJ — 14JJ	42.	m — 2Tv	5
Let us take 1| from 3i. Since we	-i
cannot subtract £ or £ from £ or f , we reduce one of the 3 units to sixths.	i	2
1 \ or | diminished by £ is thus == £.	~6
2 — 1 — 1. So, 3i — 1£ — 2 l-l~	61 2
1 — 1 «	6 X 0

In subtracting the sixths, instead of taking 4 from 9, we may subtract 4 from »'), ;ind add in 3. The prac- tical advantage of this method is illustrated in A, in which we take

= 11 If

B

Units. Sixths 3 » 3 1 » 4 1 • 5

30 VULGAR FRACTIONS.

15 the lower numerator from the common denominator, and add * in the upper numerator. In B, we may consider the mzto as units of a lower name, of which six make up a higher unit. The solution is then obtained as in Compound Subtraction.

44. 16i — 13|

45. 14f — 9|i

46. 16* — 10f

47. 14* — 13

48. 15H — 4

49. 13J — 11$

51. 13i -

52. 6*-

53. 181*—

50. 8| - 4if 54. 23{.| — 22 jf

(5) | of a pole is below the level of a pond, ^ of it is in the water. How much of it is in the ground ?

55. g of a pole is above the bottom of a pool, and T\ is in the pool. What part of it is above the level of the water ?

56. A retail draper who has bought f of a piece of cloth, sells £i of the piece. What part of it has he over ?

57. f of a common is laid out as bleaching- ground. What part of it is over ?

58. A sailor has spent T93 of Ms life at sea. What part was spent before he went to sea ?

59. A person succeeding to a legacy left by an ancestor or de- scendant in the direct line, pays T£o °f the value as duty. What part is over ?

60. Of the prismatic spectrum, the blue, indigo, and violet rays together occupy £, and the blue and indigo together T5g. What part does the violet occupy ?

61. The number of pear and apple trees in an orchard is f of that of the whole, and that of the pear trees is 575. What part of the whole is the number of apple trees ?

62. Of a consignment of guano from Saldanha Bay, f g£ consisted of carbonate of lime and phosphates of lime and magnesia, and ±§ of the phosphates. What part of it was carbonate of lime ?

63. In 1857, the number of parliamentary electors in Scotland was ?y? of the whole number in Great Britain. What part of the whole number was the number in England and Wales ?

2- i-

-i-

4- 1 + I - A + i

5- i + 8+T95-l* 6. f ~

VULGAR FRACTIONS. 31

(2) A traveller has gone £ of a journey on foot, T\ on horseback, £ by rail, and the rest by coach. What part has he gone by coach ?

* _i_ i\ — i /20 + 24 + 45\ is T-*; — J- -^ 180 — )

13. * of a pole is blue, 5 red, and the rest white. What part of it is white ?

14. A student has in three weeks read respectively 55T, f , and £ of the First Book of the JEneid. What part of it has he yet to read ?

1">. A soldier while in the army had spent | of his life in the United Kiii'_rdom, 557 in Canada, T'a in Gibraltar, £ in India, and 5'7 in the Crimea. What part of his life had he spent before rafisti

16. having used T80, 57^, and ££ of an ingot of gold, wishes tci know what part still remains.

17. Of the whole time spent by Professor Piazzi Smyth in the LOmical Ivxpedition to Teneriffe in 1856, T2g was spent in the

lowlands of TencrifFe, TY7 at Guajara, and T2T6, at Alta Vista. What part was spent in the voyage?

18. Of the component elements of albumen, i£ is carbon, T£5 hydrogen, and 5«, nitrogen. What part of it does the remainder, consisting of oxygen, phosphorus, &c., constitute ?

19. Of the whole number of Jehoshaphat's " men of valour " in Judali and Benjamin, the three divisions of Judah were respec- tively Jg, 57S, and 2V What part belonged to Benjamin?

20. Of the black and mulatto population of Cuba in 1850, the free mulottoes were 53Ty§i the free blacks sViV» an(^ *ne mulatto slaves 5? |5. What part was the number of black slaves ?

21. Of the annual salaries of the principal, depute, and assist- ant clerks of the Court of Session, 5 deputes receive T|n each, and 9 assistants T 1 3 each. What part does each of the 4 principals receive ?

In Mri/m-LYiNG a fraction by another, as £by f, we consider that since th<> nultiplier § is £ of 2, the product will be £ of 2 *. Nn-.v '2 X $ = 5, and the required product is = | -^ :$ = ,"-, which is thus = £ X f .

In multiplying fractions together, the product of the numer-

32

VULGAR FRACTIONS.

ators becomes the numerator of the product, and the product of the denominators the denominator of the product.

In multiplying by an integer we repeat the multiplicand as many times as there are units in the multiplier ; in multiplying by a fraction we take that part of the multiplicand which is denoted by the multiplier.

| X | may be expressed as f of •£, or -J of §-, which being the fraction of a fraction is termed a COMPOUND FRACTION, in contradistinction to a SIMPLE FRACTION, as f. A Com- pound Fraction is reduced to the form of a simple one by mul- tiplying the numerators arid the denominators, as in Multipli- cation of Fractions ; thus, f of | = A.

We may consider •§- of -f |

either as i of 2 X 4» or, as • •••••<

in the diagram, we may di- i4s is

vide | into three equal parts, and take two of them. Si- milarly, we may take f of £ ,

A T s

either as -| of 4 X f, or, as in the diagram, we may divide £ intone equal parts and take/owr of them.

4

/IN 8 V 1 7 V _ 6 ft 1 2 3

\ / "S r\ *Tw — 9 ^ 10 — 4"3> — ?!>' 5

Since 2 is a common factor of 8 and 10, we CANCEL these numbers, and write the number of times the factor is con- tained in each. By thus cancelling any numerator with any denominator with which it has a common factor, we obtain the product in its lowest terms.

(f)\ e y 7 — "^"s/"7" — i W TTT * T8 —.35- X ^.— T-S.

5 3

The numerator of the product is = 1 X 1. Unity takes the place of a numerator or a denominator cancelled with any of its multiples.

X

_ 29- v 31 _

1. I X *	9. I X 5|	17 Qi V Q 3 -U. UT A 0-6?
2- * X A	10. * X 7f	1 Q 72 V 2 O O TIT ^ 2~TT
3- % X A	11. f X 161-	19. 19i X 16|
4- | X i$	12. I X 18i	20. 231 X 3i|
5. 4^ X ff	13. 1T\ X «	91 173 v is'* ^*« -1 « ^ X T^ f
6- 4 X «	14. 2A X 4	22. «|J X H-
7. A x if	15. | X 64	23. 4A X 17,1
8- it X if	16. A X 7|	24 * a 3 X 3 l~-

17.

VULGAR FRACTIONS.

(4) * X 5| X 4T«T =| X ^ X £ = Vi

33

25. i X 2| X -»-

26. J X 3i X 1

27. 3| X « X «

28. 2J X 4 X 3f

29. 3T'T X TV X 5J

30. 3f X & X 61

31. | X i

32. 6| X

33. -J X 4

34. T«T X

1 = 10*. £X 8i fr X 6* X 6| r X 41

35. 7J X * X i4

36. 81 X If X *

Reduce the following Compound to Simple Fractions : —

-a- (5) i of * of A of » =i_X f X * X §= TV

2 5

(6) 1-ofi

37. 4 of £ of J

38. f off of if

40. f of f of J4

41. 4 of J of 8}

42. f of 44 of 4 of 4 j

43. f of 2J of a of 64

43. 44

45.

| of 2J of f c

Of T7g- Of ±S.

1 1 nf i 3

'* 01 T¥

3 4

X f X ^ X J| =

11

46. 4$ of, % of 4 J of

47. 4 of | of | of 4 of T% of 7

48. J of 64 of A of if of 5

49. ff of 16f of A°f7°TT

51. 52. 53.

54.

of T4T of

- Of i Of ^

-«Tof8iof8^ - ofT4?of6561

(7) If a train runs £ of a mile in a minute ; how many miles will it run in f of 431 min. ?

ml. 29

-4- 3 87- 29

"5" s^ ~5 /^ T ^"~ Q^ ^N n_ f\ "2" ~~~ "4" ^^ 4 ^'^^*

-3- 2

55. A soldier was in hospital 5\ of the time he served in India, which was 5e, of his life. What part of his life was he in hos- pital?

56. A sailor's share of prize-money is 575 of a midshipman's, whose share is 2-\ of a lieutenant's. What part of a lieutenant's share does a sailor get?

57. Jack, who gets •} of a plum-pudding, gives « of his share to Tom, who gives 1; of his to Harry. What part of the plum-pudding

i I arry get ?

B 2

34 VULGAR FRACTIONS.

17* 58. A schoolboy prepares his lessons at home in £ of the time he plays, which amounts to T% of | of a day. During what part of a day does he prepare his lessons ?

59. On the Geelong and Melbourne Railway, the fare per mile by the third class is ^ of that by the second, which is | of that by the first, which is 3|d. Find the fare per mile by the first.

60. Find the receipts of a railway for a week which amount to $fi of £6384.

61. The* number of registrars employed in the Census of 1851 was Tyy7 of that of the enumerators, of whom there were 38740. Find the number of registrars.

62. If a train runs a mile in f of 3f min., in what time will it run -fo of 23£ miles ?

63. 24 flagstaffs are placed on a road at the distance of § of 73 J yards between each. How many yards are between the first and the last.

The number of spaces between a number of objects placed in a line is one less than the number of objects.

18. In DIVIDING a fraction by another, as f by f , we consider that since the divisor £ is £ of 2, the quotient obtained by dividing by f is 3 times as large as that obtained by dividing by 2. Now | — 2 = |, and the required quotient is = f X 3 = f • f -r" I thus produces the same result as f X f •

In dividing a fraction by another, we invert the divisor, and proceed as in Multiplication of Fractions. A fraction inverted is the RECIPROCAL of the original fraction ; thus, f is the re- ciprocal of f . The product of a fraction by its reciprocal is = unity.

To divide f by £, we may, • — *. — \ — : \ i • • L ."

as ill the first diagram, ac- I | 1 |

cording to the previous explanation, take one-half of f , which is |, and by taking three parts each = |, we obtain |.

Expressing I and | in the same 1 f

denominator as T9F and T\ re- ' .....i...

spectively, we see in the second 13 T9s 1

diagram that if we take 8 twelfths as the unit. 9 twelfths con- tain 9 of those parts of which the unit contains 8. T\ is thus f of T\> or I is the quotient obtained by dividing T\ b*y -^ or is — | -T- f.

(i) i ~ T'T = * x v = « = i&.

(2) ^-7{f=»X=

VULGAR FRACTIONS.

35

18.

1.	1 —	- 1	9.	4 _	- A	17.	19| -	- f £
2.	1 -	- 1	10.	«-	- AV	18.	17A	• , 3 3
3.	* ~	- 1	11.	t? -	- if	19.	41-7-
4.	* -	•«	12.	If ~	- ti	20.		11*
5.	IT "	- if	13.	5| -	- .{..I.	21.	2 '7 _
6.	«-	- **	14.	7* -	- i J	22.	3ii -	- 4
7.	ii-	_ 2£	15.	3f -	- 1	23.	14| -
8.	TT -	- 3T	16.	6* -	- if	24.		2i"
	(3) J	-5- T7r of	3| ='	"*" v ]	i v -8- ^- X 10	= W-

(4) Aof4j-r-f =

25. |} - I of 10*

||i -r- J of 25$

X

26.

27.

28. 29. 30.

0^ 2| -1- 7

f of 12| 33. f of IA -T- Jl of |

1 J — * of 3| 34. | of « H- A of 4

• of 7i 35. A of iii -T- f of 1

f of || -M of A We may write the quotient | -f- f in the following form :

17 v"6" —

•5. X 5 —

31. ^ of j

32.

33.

34.

35.

36.

H-

wliich the dividend becomes the numerator, and the divisor the denominator of a COMPLEX FRACTION.

A Complex Fraction has a fraction in either its numerator or

3 2 5- B3 ninator, or in both of them: — thus, -i, — , -^, 13, are

T ' 4" *^ 'I"

coni]»lcx fractions. The reduction of Complex to Simple Frac- tions is similar to the Division of Fractions.

Reduce the following Complex to Simple Fractions : —

(5) f = A-

We have multiplied the numerator and the denominator of the fraction by 4, the denominator of the numerator. So, when either the numerator or the denominator is an integer, we multiply the numerator and the denominator by the de- nominator of the fractional term.

(c) -i- = * -;-

-T \y/ 13

— o X TT —

[9 = *l

36

VULGAR FRACTIONS.

quotient may evidently be obtained by multiplying es of the complex fraction for the numerator, and

18. The

extremes

means for the denominator.

the the

(7)

_ 12*

—5 35-

±. 21 = 14. •W- 11 J

—3 •£•

We may cancel either of the extremes with either of the means. As the numerator and the denominator will likely he expressed in the lowest terms, we thus cancel the first with the third, as 35 with 77, and the second with the fourth, as 4 with 6.

37.	6 8*	42.	3 T	47.	$
38.	7	43.	7	48.	S
	• "9		Ttf		15"
39.	9 12|	44.	if «	49.	19*
28T7-g
	11				O s
40.	134	45.	«	50.	T?
41.	7	46.	it	51.	24|

(8)

tof_3j | of 34

, 13

x-25-5

13X7 _ 91 6X20 — 120-

(9)

52.

53.

5 5 5 «*- 4^
3 3
T ff
T8T
|of3i	54 tVofiai 3 x. 5 8°f 6}	56. 1. 1
*of9*
		5
*of!3*	^^ 1 °f T 00. -J 7_	57. JL
T T	TJ ii	|
		i

VULGAR FRACTIONS. 37

18* O1) How many pieces, each 30 £ yards, are contained in 114} yards?

14

114{ + 30| =^ X -4= || = 3jJ pieces.

3

58. If a piece of cloth is 29f yards in length, and a remnant 1 {j £ yard ; how many times is the former as long as the latter?

59. How many squares, each $ sq. inch, are contained in 132£ sq. inches ?

60. How many postage- stamps, containing $f sq. in., are in a sheet of 172* sq. in.?

61. How many times can a measure of | pint be filled out of a yessel containing 63 1 pints ?

62. How many times will a coin 2£ inches in circumference turn round in traversing 30 inches ?

63. Mercury is 13f times as heavy as water, and gold is 19 1 times. How many times is gold as heavy as mercury ?

64. The pellicle from which goldbeaters' skin is made is 3^0 inch thick, while gold leaf is 335^05 m« thick. How many times is the former as thick as the latter?

65. The largest scale of the Ordnance Survey Maps is lineally T?*72 °f that of nature, and the smallest is g3joo- How many times is the farmer as large as the latter?

66. The mass of the Earth is 3 55'j 5T, and that of Jupiter is y^ of that of the Sun. How many times is the mass of Jupiter as great as that of the Earth ?

67. A book of 240 leaves without boards is ji inch thick, and another of 180 leaves without boards is T75 inch thick. How many times is the paper of the form eras thick as that of the latter ?

(12) How many men are in a regiment of which T% = 255 men ?

255 -r fi; = 255 X V° = 85° men- The regiment is evidently = '3° of T*5 of the regiment, but TJff of the regiment = 255; 'hence the number in the regiment = '3° of 255 = 850.

68. Find the length of a pole of which f = 18 ft.

69. The Pylades war steamer, having 2 1 guns, has TS3 the num- ber which the Princess Royal war steamer has. Find the number of guns in the latter.

Find the distance from London to Kurrachee, that from the head of the Red Sea to Kurrachee, which is 1700 miles, being Ty3 of it.

38 VULGAR FRACTIONS.

18* (13) Of a pole, T^ is painted white, ^ green, JJ red, and the remainder which is 5 ft. is painted black. Find the length of the pole.

T\r T KG i ii = GO == "e o 2 1 — i = J; 5 ft. -S- J = 10 ft.

71. Of the area of the five great lakes, Lakes Erie and Ontario together contain £, Michigan and Huron together £f , while Lake Superior contains 32000 sq. miles. How many square miles do they in all contain ?

72. Of an army $ is English, 57? Scotch, T$ff Welsh, and the remainder numbers 4796 Irish. How many are there in all ?

73. Of the distance from Edinburgh to London by rail, via Car- lisle, that from Edinburgh to Carlisle is £, from Carlisle to Preston <9<j, while that from Preston to London is 210 miles. Find the dis- tance from Edinburgh to London.

(14) A labourer can do a piece of work in 18£ days. What part of it can he do in a day ?

74. A labourer can perform a piece of work in 12J days. What part of it can he do in a day ?

75. A workman can floor a room in 5^§ days. What part of the room can he floor in a day ?

A can do a work in 8 days, B in 12 da., and C in 16 ,. In what time will they do it working together ?

A can do -| of the work in 1 day. B // TL. // //

C // T^ // //

A, B, and C can do £ + TV + T\, = 6+448+3 = j | Of the work in 1 da. A, B, and C, will thus do the whole, work in as many days as are = 1 -=- £| = ff = 3T9T da.

(16) A can do a work in 10 J da., B in 12 J da., and C in 8| da. In what time will they together do it ?

A lOi = «,

B 12i == V C 8| = V

= T

A = TM da.

VULGAR FRACTIONS. 39

18* 7$- D can do a work in 6 da., E in 9 da., and F in 10 da. In what time will they do it by working together?

77. A cistern can be filled by three pipes in 10, 12, and 18 min. respectively. In what time will it be filled when they are all open ?

78. X can do a work in 3 hours, Y in 4£ ho., and Z in 6| ho. In what time will they together do it ?

79. A can do a work in 10£ da., B in 11£, and C in 12 £. In what time will they do it together ?

80. A can do a work in 3 da., B in 4 da., and C can do as much as A and B together. In what time will they do it working together ?

Of C can do £ -f- J of the work in a day.

81. A can do a work in 7 hours, B in 5 £ hours, and C can work twice as fast as A. In what time will they do it together ?

82. A, B, C, can do a work together in 20 days ; A alone can do it in 40 da., B alone in 60 da. In what time can C alone do it?

<gr C can do ^ — (jv -f- ^,7) of the work in a day.

83. D, E, F, can do a work together in 5 days, D in 16§, and E in 13£ da. In what time can F alone do it ?

84. A, B, C, can do a work together in 7 days, which A and B can do together in 10 da. In what time will C do it?

43T C can do } — T'5 of the work in a day.

85. F, G, H, can perform a work together in 1 day, which G and II can do together in l£ day. In what time can F do it?

86. X and Y can accomplish a work together in 8 days, Y and Z together in 9 da., and Y in 14 da. In what time can X and Z do it separately and together?

&S° X can do | — T^ of the work in a day.

87. A and B can do a work together in 3£ da., B and C together in 4 da., and B in 5£ da. In what time can A and C do it sepa- rately and together ?

-Hi

" 18f*

19. (i) A

v x Tj¥ = a = m

10T 287

1. loft + foff + f oflf

2. fof3J + jofJ + AofS 3- Jof4i + Jof^ + iJof

6| 9*

23|

6 '

llf T 40i

40

VULGAR FRACTIONS.

19. (2) I •

3 of fr = i X = ! X IS = T% =

9625 — 2964 _ 10640 "~

6661 10640*

We place " <-*•» " between two quantities whose difference we wish to find, when the less is written first, or when we are uncertain of their relative magnitude.

7. | of 3| <-> ii of 3J

of 11

H of*

of

10.

£?i -> A Of 24

(3) » — (« + *- *)•

155 — 99 — 40 -f 120 114 34

180 - 180 45'

From f i we are required to subtract |£- -f- f diminished by f . By subtracting A£ -f- 1 we obtain a remainder too little by f . By adding f to this remainder we therefore obtain the re- quired result.

When " — " is placed before a parenthesis, we change the " + " and " — " signs of the enclosed quantities respective- ly to " — " and " +," and add or subtract as indicated by the changed signs ; thus : —

12. H + A- (H 4-

13.

(4)

t "r.f

1 8

^- T! =

15. 16. (

18. *_=.* •

l+S

VULGAR FRACTIONS.

41

19» The following show the difference in value produced by changing the place of the parenthesis : —

19. 20. 21. -T

of

x

X !?--

2O • In REDUCING the fraction of a quantity to a lower name than that in which it is given, we multiply the fraction by the num- ber of times the former is contained in the latter ; thus, in reducing /T foot to the fraction of an inch, we multiply the numerator by 12, and obtain £f inch, which is = £f of ^ foot.

(1) Reduce ^V oz. troy to the fraction of a grain.

2. 3. 4.

&

7. 9.

	1 X 20 OZ. rry 9 ,1.
a £
	, s.
_•»_ cr. .	, s.
	, hfd.
VTcr	sixd.
T	cwt.

Ib. av oz. av.

• 9 ^'

10. .T^lb. tr oz. tr.

11. TVV ac po.

12. ^,da ho.

13. ,jscwt Ib. ^

15. T^£tf ml yd.

16. TyTbu gal.

17. Tfcfu yd.

18. -gVo ho min.

In reducing the fraction of a quantity to a higher name than that in which it is given, we divide the fraction by the num- ber of times the former contains the latter ; thus, in reducing T\ f/rain to the fraction of a Ib. avoir., we multiply the denom- inator by 7000, and obtain 1^33 Ib. avoir., which is = TT£UTy of 7000 gr.

(2) Reduce 4d. to the fraction of a crown.

7 X

10.

20. 21. i

£.

s.

22. $s gu.

23. jib T.

24. |f in yd.

42

VULGAR FRACTIONS.

31. -f min da.

32. £; da co. yr.

33. if qt qr.

34. |£ cub. in cub. yd.

35. |f po ac.

36. ijf gr lb. av.

2O« 25- If sec ............ no-

26. -I gal ............. bu.

27. 4| yd ............. ml.

28. |f oz. tr ......... lb. tr.

29. if pt ............. gal.

30. fl pk ............ qr.

In reducing the fraction of a quantity to a name which is neither a measure nor a multiple of the name in which the fraction is given, we both multiply and divide as in the fol- lowing example : —

lb. av. to the fraction of a lb. troy.

(3) Reduce

720

In multiplying by 7000, we reduce the fraction of a lb. av. to that of a grain, which, when divided by 5760, becomes that of a lb. troy.

42. !!.§ oz. tr oz. av.

43. fflk ft.

37. |fl cr.

38. -Hgu £.

39. Jnl ft.

40. •§ f E. E yd. 45. ^§ co. mo co. yr.

41. T4ihjlb. av lb.tr. 46. if°geog. ml Imp.ml.

In reducing a compound quantity to the fraction of a simple or a compound quantity, we proceed as follows : —

(4) Reduce £1 * 2 * 7 to the fraction of £1 * 13 // 5. £1*2*7 = 271d. £1*13*5 = 401d. £1*2*7 = ift of £1*13*5.

Having reduced the quantities to the same name, we find that since £1 » 2 » 7 contains 271 pence, of which £1 ,, 13 // 5 contains 401, the former is J£J of the latter.

47. 11/6 £1

48. 2/2* £1

49. 2ft. 8 in 1 yd.

50. 3ro. 15 po 1 ac.

51. 6fu. 15 po 1 ml.

52. 6oz. 3dwt. ...1 lb. tr.

53. 4/4 13/8

54. 7/8J 13/3J

55. £l//15//3 £3//13//9

56. 3 oz. 4 dwt

57. 3 fu. 44 yd

58. 2qr. 3nl

59. 2ro. 14 po

60. 7bu.3pk

61. 7 ho. 12mm.

62. 4 da. 17 ho

63. 22°30X

64. 66° 32X 23/x..

2 lb. 6 oz.

3ml.

3yd.lqr.

3ac. 1 ro.

Iqr.Sbu. ..3da.4ho.

lwk.3da.

360° ..90°

VULGAR FRACTIONS.

43

2O« (5) Reduce § s. to the fraction of ±% £., or find what part f s. isof4?£.

£.

3

5 X 20

_ a _ 3 X 27

— TStf — 100 X 10

f , Q ,, OI *•

4. (2).

65. T\£ 6G. f s 67. T3<y ac 68. 69. 70.

yd ...........

£.

fu

Jpo. | ml.

71. 72.

73. 3|s ............... £2f

74. Sjgal ........ ....144 qr-

75. 6|ho ............ | da.

76. ¥fy oz. av ....... ^ oz.tr.

In finding the value of a fraction of a quantity, we may

either in reducing a fraction to a lower name; or,

king as many units of the name in which the fraction

<m as are indicated by the numerator, we may divide

by the denominator as in Compound Division.

(6) Find the value of f } £.

5 s.	8. fl. (1. £	s. d.
£25 X -20- 125	_Kfl 5X43-__2; 25_ — °¥T> 34- — ^ai 7T5 -	= 5//2i.
24	2	£
Otherwise: £j-	1 = 7v of £25 96{g2	25//0//0
77. 44 s.	84	T3A ^' av<	91. *y	y ml.
78. J4 s.	85	-|^ CWt.	92. 4f	ac.
79. J4J £.	86	19 T1	93. *y-	y oz. tr.
80. 44 £.	87	. TyT lb. tr.	94. Tv-	5- bu.
81. Jf cr.	88	• It Jd-	95. ||	pk.
82. 4f4 gu.	89	. 4J sq. yd.	96. TV	Ik.
83. §£fl.	90	. 44 cub. yd.	97. Aq	f Ju. yr.
(7) Find the value of J of 9T\ acres
	ac. ac. ac. ro.	10 po.
Jof9T»¥ac. = JX W	= 3,Y =: 7,7¥, j^ =^_	10 = 23 j;
ao. ro.	j)0. 12 3
7 A ac. = 7 // 0 //	234.

44 VULGAR FRACTIONS.

OQ ac. ro. 10 po. ac. ac. ro. po.

Otherwise: ±x'*'=i*4* = W9 9 A =

9)64 »

7 // 0 //

98. f of 5} cr.

^ofSfhf.cr.

102. T\of3fu. 12 po.

100. | of £3 // 7*6 103. | of 2 ac. 3 ro.

104. 1 of 3| s.

105. $of2ho.34min.

106. | of 3 wk. 6 da.

107. Express a Russian Archine, which is \ of a yard, as the fraction of a mile.

108. Express the height of Ben Macdhui, which is \\\ of a mile, in feet.

109. Schiehallion, where Maskelyne made a series of observations on the Density of the Earth, is nearly | of a mile high. Express its height in feet.

110. Harton Coalpit, where Airy conducted a series of observa- tions on the Density of the Earth, is f \ of a mile deep. Express its depth in fathoms.

111. The velocity of sound is 575 of a mile ^ sec. Express it in ft.

112. Express 5 dwt. 9 gr., the weight of a guinea, as the frac- tion of 1 Ib. troy.

113. In an estate of 3173 acres 20 poles, the roads occupy 66 ac. 1 ro. 8 po. What part of the estate is occupied by roads ?

114. The distance traversed by an express train in T55 hour is run by a goods' train in | of 1 J hour. What fraction is the for- mer time of the latter ?

115. The National Subscription, promoted by Cromwell in aid of the Waldenses, amounted to £38097 » 7 » 3, of which Cromwell gave £2000. Express the latter as the fraction of the former.

116. In November 1855, the Patriotic Fund amounted to £1,296,282 » 4 » 7, of which Glasgow subscribed £44,943 » 1 » 10. What part was the Glasgow subscription of the whole ?

117. Express 58|? yards, the depth of an Artesian well, as the fraction of another which is /^ of a mile deep.

118. What fraction is an oz. avoir, of an oz. troy?

119. Reduce a grain to the fraction of a dram avoir.

120. Express a Ib. troy in avoir, weight.

121. In Scotland, during June 1856, the mean weight of vapour in a cubic foot of air was 3 T7o grains. Express this as the fraction of 1 Ib. avoir.

122. In Scotland, during April 1856, the mean weight of vapour in a cubic foot of air was To§55s Ib. avoir. Express this in grains.

123. Mont Blanc is 15780 feet above the level of the sea, and

VULGAR FRACTIONS.

45

2O»Dhawalagiri is 5g7g'5 miles. Express the height of the former as the fraction of that of the latter.

124. A degree of longitude on the parallel of Greenwich is nearly = £ of a degree of the Equator, which is = 60 X 6076 ft. Find the number of Imperial miles in the former.

Find the sum of f ac., } of 3f ro., and } of 16} po. I. II.

ac. ro. po. ac.

| ac. =0*2 // 16

J of 3| ro. = 0 // 2 // 36} } of 16} po. = 0 // 0 » 4}

1*1* 17} f }j

c.=lac. Iro. 17}po.

In adding fractions expressed in different names, we may, as in I., find the value of the fractions, and then proceed as in Compound Addition; or, as in II., we may reduce the frac- tions to the same name, and having added them, find the value of their sum.

7. £ T. + | cwt. + \ qr.

lo£3fro.=|fro.= |f=

2. A £. + f fl. + } s.

3. I ac. + 2 j} ro. + 3£ po. 9. f f.+7f cwt.+lj|qr.+20flb.

4. f ml. + T3T fu. + T*T po. 10. ,

5. f Ib. + 1} oz. + 2} dwt. 11. T'¥ ft. + | yd. •

G- T6u£-+is-+!iofV<r£- 12- Jof3jpo.+^uml.+^of2|fu.

13. Find the total weight of seven half-chests of tea, containing re- i vdy 1 $ qr., T7g cwt., 534 T., 1 5 qr., y cwt, £ of 55S T., and f cwt.

14. llo\v many acres are in a parish in which cultivated land occupies 2.^ sq. miles; pasture, f of 13T9g sq. miles; and planta- tion, 234f acres?

15. Find the weight, by the old system, of a pill-mass, consisting of 1 5 rhubarb, £ 3 acetate of potash, and j £ § of conserve of roses.

16. The highest part of the woody region of Mount Etna is || of 1 i § ! mile above the level of the sea ; the foot of the cone is 1160^ yd. higher; and the summit is j^ of 1316T6T ft. above the latter. Find the height of the summit above the level of the sea ?

22. (1) From -I of 6} fur. subtract T\ mile ; or find the value of 4 of 6} fu. — T3j ml.

I II.

fu. po. yd.

* of 6} fu. = 3 // 32 // 0 T\ ml. = 1 // 28 tf 3} ' 2 a 3 // 2J

4 of G A ml. 2 fu. 3 ro.

fu. = 3$ =

2T8, yd. = 2,a¥ fu.

46 VULGAR FRACTIONS.

, In finding the difference between fractions expressed in dif- ferent names, we may, as in I., find the value of the fractions, and then proceed as in Compound Subtraction; or, as in II., we may reduce the fractions to the same name, and find the value of their difference. In I. we have the number of yards =o^ — 3£ =2 + •

= 2 .

• t

2. ii cwt. — if qr.

3. T\ cr. ^ | £.

4. cwt. — A T.

5. f oz. — TT3 dwt.

6. k.^fbu.

(2) Find the value of f £. — (J s. + -& cr. — f fl.)

s. d.

|£. = 16 //O

£fl. = 1 // 2J;

17 // 21 1 17 * 21 f — 2 » 4£ = 14 // 9| |.

?.}£. + ft s. — (ft cr. — | fl. + | go.)

8. & ac. — (I ro. + if po. — T35 ac.)

9. By how much does 1 ~y jacobus exceed f Joannes ?

10. A vessel containing f gal. is filled, and 1 of 3£ pt. is then poured out. How much is left in the vessel ?

11. An apothecary prepares ^ § of medicine, which contains 1 5) 4 gr. of conserve of roses. What is the weight of the othei ingredients, by the old system ?

12. The rope of a bucket, while ascending the shaft of a coal pit £ of 212f fathoms deep, snaps while the bucket is £ of 200 * ft. from the top. Through what depth is the bucket precipitated ?

13. The top of St Peter's, Eome, is Tf ^ 'mile above the ground, while that of St Paul's, London, is 5lft mile. Express their dif- ference in feet.

14. A retail grocer having bought f of 58 £ Ib. of tea, sold during six days, $ qr. Tf 5 cwt., i qr. ft cwt., f qr., and £ of ft of 18f Ib. How many Ib. has he still on hand ?

15. A draper having a piece of cloth containing 27| yd., sells I of 7ft yd., $ of 3| yd., and i of 3 qr. What has he over?

16. The astronomical stations chosen by Professor Piazzi Smyth in Teneriffe, in 1856, were respectively j§| of 2£ miles and ^ of 3T49429o miles above the level of the sea. By how many yards did the height of the latter exceed that of the former ?

VULGAR FRACTIONS. 4?

23. Multiply $ £. by 30|.

30| X ? £. = i}* X * £. = £13 A = £13*3*6} J. We multiply the fraction of a quantity as in abstract num- bers, and then find the value of the product.

I- *£ X 17 2. I s. X 29

3. T3r ac. X 18

4. If pk. X 3«

ho. X 3 6. ,«, ml. X ~

N

7. An incumbent has received 40 stipends at an average of £148{£3 each. Find the total amount.

8. If a train runs a mile in a35 hour, in what time will it traverse § of 150 miles?

9. Find the price of 6f pieces, each 29 1 yd., @ ^s. ^ yd.

10. A farmer having found 263 sheep trespassing on his fields, claims by an old statute, as compensation from their owner, £ of I of £T'5 for each sheep. Find the total claim.

1 1. A train runs J mile in a minute; what distance will it run in t of 3 1 hours?

12. The area of Paris is 657/j times as large as that of Frank- fort-on Maim-, which isz:2 2's sq. miles. Express the former in acres.

13. The area of one of the parishes in the smallest county in P.ritain is ; | ? of 4563 acres, while that of the county is 6f £f

times as large. Express the area of the latter in sq. miles.

14. The ?- of a Prussian thaler is pure silver. The weight of a thaler is 52f of a Cologne mark, which is = 7|i oz. troy. How much pure silver is in a thaler?

24. (1) Divide 1T\ acre by 28}.

1 V, ac. +- 28$ = |f ac. X *fo= AVfc ac- = 8i* P°- AVe divide the fraction of a quantity as in abstract numbers, and then find the value of the quotient.

(2) How often is 4 s. contained in T4T £. ?

T4r £• -T- A £• = IOTT times.

In dividing one quantity by another, we reduce them both to the same name, and by finding the quotient, we see how many times the one is contained in the other.

This operation is equivalent to finding the fraction, proper or improper, which the dividend is of the divisor; thus, as in § 20. (5), we find that T\ £. is = Vr* of f s., or that T4T £. is = 10TaT times £ s.

2. J cr. + IJf

3. 1 ft ml. -7-

'-5

4. 8| da. -r- 1,V

5. 9|i ac. -r- i| ac.

6. VTsq. yd. -7- « sq. yd.

48 VULGAR FRACTIONS.

f24« 7- 623 sovereigns are coined out of 1 f of 19 ^ Ib. troy of sterling gold. Find the weight of a sovereign.

8. 155 Napoleon pieces weigh 32 /^ oz. troy. Find the weight of a Napoleon piece.

9. If a cubic foot of air contains 2-^ grains of vapour; what volume of air will contain 1 Ib. avoir, of vapour ?

10. How many crofts, each || of 3| roods, can be portioned out of 121 acres?

11. How many pieces, each *i of 48 yards, are contained in 595 of 683i E. E.?

12. How many Ib. troy, each jf | Ib. av., are = § of §7, cwt. ?

13. In Mid -Lothian, the total area under a rotation of crops was, in 1856, 1 04077 £ acres, and in 1857, 160^| square miles. What part of the former is the latter?

14. An American dollar weighs f.| oz. troy, and a British crown 55j Ib. troy. Express the former as the fraction of the latter.

MISCELLANEOUS EXERCISES IN VULGAR FRACTIONS.

1 . How many hundredths of an inch are in a link ?

2. A student has read 55r of the Sixth Book of the ^Eneid, which contains 903 lines. How many lines has he yet to read ?

3. Find the weight of 200 guineas, each 5| dwt.

4. The sheriff and justices of peace of a county enrolled 54 spe- cial constables in one day, on the next day f of that number, and on the third day f of the number enrolled on the second. How many have been enrolled in all ?

5. A boy who has 36 marbles gains £ of that number, and then loses T35 of what he has. How many marbles has he gained ?

6. In 1855, the population of Texas, amounting to 400,000, in- cluded 35,000 Germans. What part of the entire population was the rest of the inhabitants ?

7. In 1856, 106000 acres in Ireland were occupied in the growth of flax, of which 150 square miles were in Ulster. What part is the latter of the whole ?

8. Of a vessel, worth £5600, A, who has |J, sells | of his share to B, who sells * of his to C. Find the value of C's share.

9. Of a number of sheep on a hill-farm, the Cheviot ewes were £, the black-faced ewes |, the Cheviot hogs T5f, the half-bred hogs 35g, and the remainder consisted of 100 black-faced hogs. Find the total number.

10. In Scotland, in 1855, the number of deaths in February, the month of greatest mortality in that year, was 7227 ; and in Sep- tember, the month of least mortality in 1855, the number of deaths

VULGAR FRACTIONS. 49

25* was 32 more than g of that in February. Find the number in September.

11. If, in small forms in Asia Minor, £ of the produce is given to the landlord who furnishes the seed, and Jg of the remainder to the government as land-tax, what part remains to the tenant ?

12. A gentleman leaves property worth £556 to his cousin, who pays a duty amounting to 5'5 of its value; and £470 to his second cousin, who pays -*$ of it in duty. Find the total duty on both.

13. A bankrupt's effects amount to f of | of his debts. How much can he pay per £. ?

14. A bankrupt pays 11/3 ^ £. What part of his debts are his effects ?

15. In the examination for admission to tlje Royal Military Academy at Woolwich, the number of marks for English amounts to 1250, and is T5^ of the number of marks for Mathematics. Find the number of the latter.

16. Divide £57 f into 4| shares.

17. Divide £819 among 6 men and 5 youths, giving a youth £ of a man's share.

18. Share a bonus of £20 $£ among 1 foreman, 16 journeymen, and 4 apprentices, giving a journeyman | of the foreman's share, and an apprentice -j*0 of a journeyman's.

19. Sir George Cathcart, who fell at Inkerman in 1854, was 16 when he received his commission. He spent §g of his life in the military profession. In what year was he born ?

20. In the end of 1855, the number of widows relieved by the Patriotic Fund, amounting to 2544, was §^ of that of children relieved. Find the number of the children.

21. The copper sheathing of the hull of a vessel which had been seven years in the Pacific was found to contain ^^^ °f ^s weight in silver. What fraction of a Ib. troy of silver would 1 cwt. of the sheathing contain ?

22. In the division in the House of Commons on March 3, 1857, on the Canton disturbances, among those who voted against the Ministry there were 198 Conservatives, and the numbers of Peelites and Liberals were respectively £ and j55 of this number ; while of those who voted with the Ministry the number of Liberals was 5T15 times that of Liberals on the other side, and the number of Con- rives 5% of that of the opposite Conservatives. Find the

majority against the Ministry.

23. In 1856, the number of births in the eight principal towns of Scotland was 31885. Find the number of deaths, which was 527 less than f of that of births.

50 VULGAR FRACTIONS.

25. 24. Montaigne the Essayist's copy of Caesar's Commentaries was bought at a bookstall for T95 franc, and subsequently sold by auc- tion for 1550 francs. How many times does the latter contain the former ?

25. From Montreal to Toronto by the Grand Trunk Railway is 332 miles. Of this, £ mile more than f was opened in November 1855, and the remainder in November 1856. Find the latter distance.

26. The 36 Israelites who fell in the first assault on Ai were 2|5 of the force sent by Joshua. How many were there in all ?

27. Of 909 men of the 23d Foot or Royal Welsh Fusiliers, 32 men more than \ were killed and wounded in the Crimea. How many were killed and wounded ?

28. In the Line, the price of a lieutenant-colonel's commission is £4500, a major's is \\ of a lieut. -colonel's, a captain's T9g of a major's, a lieutenant's T75 of a captain's, and an ensign's T9< of a lieutenant's. Find the price of an ensign's commission.

29. Of 98600 non-commissioned officers and privates in the British service who sailed for the Crimea, 25500 embarked under Lord Raglan. What fraction was the remainder of the whole ?

30. An angler for fishing salmon smolts was fined £IJ$. The expenses of court were 2 ? f times the fine. Find the whole amount.

31. Of the number in the British Army killed and wounded in the Crimea until the fall of Sebastopol, in siege-duties there were 54 men more than J|, in assaults 115 fewer than £|, and in battles 408 more than ig. Find the total number.

32. Of the number of shares in the Atlantic Telegraph Company, 4 shares more than 5'5 are held in America, 1 more than f in London, 16 more than 4 in Liverpool, 2 more than T^ in Glasgow, 225 in Manchester, and g1- in other places in Great Britain. Find the total number of shares.

33. A alone can do a work in 6^ days, and with B's assistance in 3T95 days. In what time will B do it by himself?

34. What number multiplied by8fis = 3£ + £ + £l-J-|§?

35. Multiply the sum of f , $, and f by the difference between | and *, and divide the product by the sum of f and |.

36. Multiply the sum of ^ and | by their difference.

37. Find that number, to which, if we add T95 of 6|, the result will be i of 13i.

38. What number when multiplied by § of 5* gives the product 164?

29

39. Multiply the product of l/^ and j| by the quotient of the former by the latter.

VULGAR FRACTIONS. 51

25t 40. There were 154 fewer wrecks on the coasts of the United Kingdom in 1855 than in 1854, and this difference was ffa of the number in 1854. Find the number of wrecks in 1855.

41. Find the content of a plank 233 ft. long and 5' in. broad.

42. I low many square feet are in a wall 5| yd. long and 6£ ft. high ?

43. What is the circumference of a room whose opposite walls are equal, the length being 30 £ ft. and the breadth 22575 ft.

•1 L How many square yards are in the walls of a room 26 i ft. , 18| ft. broad, and 14T8T ft. high?

45. I lew many cubic ft. are in a box 5§ ft. long, 2| ft. broad, and 23 i in. deep?

46. A can do a work in $ of the time which B can, and C can do it in {f of A's time. They take 10£ days, working together. In what time can each do it?

47. A cistern can be filled by a pipe in 14£ minutes, and emptied by another in 18 minutes. In what time will it be filled when both the pipes are open ?

I n a map drawn on the lineal scale of 3^3 of that of nature, how many inches represent a mile?

49. The height of Kinchin-junga in the Himalayas, above the • >f the sea, is r>jss94 miles, and that of Aconcagua in the Andes

1 ) feet greater than 4£ miles. Reduce the latter to the fraction of the fori;

50. The attraction of gravity at the Equator is less than that at the Poles by 5£5 on account of centrifugal force, and ^^ on ac- count of the earth's oblateness. Find the sum of these fractions, and give a fraction with the numerator 1, to which the sum is nearly equal.

DECIMAL FEACTIONS.

IN Integers we employ the decimal notation, by which the places ascending from right to left have respectively ^ the local value of units, tens, hundreds, thousands, &c. Fractions in which the decimal notation is employed are termed DEC- IMAL FRACTIONS. In Decimal Fractions, the places de- scending from left to right have respectively the local value of t< i> if ix, hundredths, thousandths, &c. Thus, in 4'235, the point is placed to the right of the units' place, and the inures to the right of the point represent 2 tenths, 3 hun- ths, 5 thousandths; '235 denotes T% + T^ + isW == 0+^ + 5 = ||; and 4-235 = 4Tm- Similarly, -0379

I _ 300 + 70 + 9 379

denotes ^ -f- TBVV T TSUSTJ — "'ioooo — 10000'

52

DECIMAL FRACTIONS.

<26» A Decimal Fraction may be expressed in the form of a vul- gar fraction, having the figures of the decimal as the numerator, arid 10, or a power of 10, as 100, 1000, &c., as the denom- inator. The number of figures in the decimal is = the number of ciphers annexed to " 1 " in the denominator of the vulgar fraction.

Ciphers annexed to a decimal do not alter its value ; thus, •36 = -360 = -3600, for T%% = ^ = T3_e_o_o_.

Express the following decimals in the form of vulgar frac- tions : —

(1) -1341 =TV&V (2) '00739 = .

1. -3

2. -27

3. -167

4. -231

(3) -005 = •

13. -8

14. -125

15. -3125

16. -15625

5. 6.

7.

•4153 •8827 •32471 •98347

9. 10. 11. 12.

-009 -0007 -000093 -000107

(4) '0848 = T jjfr =

17. -032

18. -004

19. -0625

20. -7168

21. -0425

22. -46875

23. -00256

24. -000375

Write the following fractions in the form of decimals : — (5) TVo = *71. (6) T^-o- = '003.

25. 26.

27. 28.

29. 30. 31.

71ER50

33. 34. 35.

307

ToooooS"

27* By Amoving the decimal point of a number one place towards the right, we increase the value of the number tenfold ; thus, •34 X 10 = 3-4; -07 X 10 = -7. By moving the decimal point of a number one place towards the left, we diminish the value of the number tenfold: thus, 7'13 — 10 = -713: •79 -i- 10 = -079.

In multiplying a decimal by a power of 10, we move the point as many places towards the right as there are ciphers in the multiplier; and in dividing by a power of 10, we move it as many places towards the left as there are ciphers in the divisor.

(1) Multiply and Divide -00347 by 1000. •00347 X 1000 = 3-47 •00347 -f- 1000 = -00000347.

27.

DECIMAL FRACTIONS.

(2) Multiply 3-219 by 10000.

3-219 X 10000 = 32190.

(3) Divide 7830 by 100000.

7830 -r- 100000 = -0783.

53

1. 2. 3. 4. 5. 6.

•0369 X 1000 ••J17<3 X 100 •42839 X 10000 3-216 X 1000 7-23 X 10000 15-9 X 10000

7.

8.

9. 10. 11. 12.

•273 — 100 •5236 — 1000 •367 — 10000 72-3 — 100 98-475 -f- 1000 8-375 -H 10000

f^° reduce a vulgar fraction, as |, to a decimal, we must multiply the numerator and the denominator by such a num- ber as will produce a power of 10 in the denominator.

Since 1000 is the lowest power of 10 which contains 8, we multiply the numerator and the denominator of £ by -<V>-°, which is = 125. 1 = 1^=^^55: -375, Now, 3X125

= 3 X 1Jir-0 = 3-°H°° 5 |ience tne figures of the decimal are obtained by annexing ciphers to the numerator of the vulgar fraction and dividing by the denominator. The number of places in the decimal is = the number of annexed ciphers.

AYlien we can readily find how often the lowest power of

10, which is a multiple of the denominator, contains it, we

multiply the numerator by the quotient; thus,

Since the prime factors of 10 are 2 and 5, no number con- taining any other prime factor will exactly divide a power of 10. Hence, those Vulgar Fractions only whose denominators in the lowest terms of the fraction have no other prime factor than 2 or 5, produce TERMINATE DECIMALS.

Express the following vulgar fractions as decimals : — •

(1) * = "75.

1. i

2. i

4- t

5- I G. I

!) TJT = -056.
125)±|™ or TJ?	= TSU3 =	•056.
1) ,fe = Tfo Of ;	\ = -0075.
7. A	13.	9	19.	1 3	25.	AV
8. it	14.	*V	20.	Wj	26.	183 •6^^
9. TV	15.	r2ir	21.	iVff	27.	329 ¥on
10. A	16.	7 Yl¥	22.	7	28.	233 23S
11- *i	17.	TVlT	23.	T^TT	29.	1 KIT
12. A	18.	1 1 T5T5	24.	if	30.	*15

54 DECIMAL FRACTIONS.

29« In the ADDITION of Decimals, we place tenths under tenths, hundredths under hundredths, &c., and thus add figures hav- ing the same local value.

(1) 67-37 + -1883 + -0965 + 6-314 + 77-4006.

67-37

We carry as in integers; thus, for 14 ten thou- -1883

sands, we write 4 in the ten thousandths' place, -0965

and carry 1 to the thousandths' column. Simi- 014 larly with the thousandths and the hundredths.

For 13 tenths, we write 3 in the tenths' place, and 77'4(

carry 1 to the units' column. 151-3694

1. -30103 + -47712 + -60206 + -69897

2. -096 + -0096 + 96-0096 + -96

3. 7-0096 + -314 + -326 + 81*093 + 325-73

4. -7146 4. -003 + 94-216 + -314 + 95-279

5. 93-423 + -875 + -329 + 4-326 + 57-916

6. 373-912 + 37-3912 + 3739-12 + 3-73912

7. 247-35 + 9-168 + -709 + 82-361 + 18-017

8. -73 + -0073 + -073 + -00073 + -000073

9. .716 + -00716 + 716-0716 + -0000716

(2) Add J, J, and T5g by Vulgar and Decimal Fractions.

H — jL = -3125

= m = 1-9375

10. i + I + A +

11. i + l + l +

12. '

3O» In the SUBTRACTION of Decimals, we find the difference between figures of the same local value.

(1) .59 _ -043.

By taking 3 thousandths from 10 thousandths, we '59

obtain 7, which we write in the thousandths' place. .Q43

We proceed as in integers, taking 5 from 9, or 4 from

8, £c. '547

1. -5475 — -4212

2. -875 — -525

3. -275 — - -198

4. 5-25 — 3-875

5. 3-125 — 1-9375

6. 8-425 — 5-3875

7. 1-25 — -175

8. 2-834 — 2-786

9. 3-245 — 1-2375

10. 1-1 — -0009

11. 8-75 — 7-00009

12. 9-03 — -90003

DECIMAL FRACTIONS.

55

3O. (2) Subtract -

13. J - |

14. J} - £

V from J£ by Vulgar and Decimal Fractions. ^ = U = "44

JM = TV = -4375 ,fc, = -0025

16- I - |

17. if -

18. is —

In the MULTIPLICATION of Decimals we proceed as in in- tegers, and point off as many decimal places in the product as there are together in the multiplicand and the multiplier.

(1) Multiply -347 by 2-3.

•347 X 2-3 = jfo x *s

— 7 o B i — '7QQ1

1041 "7981

In working by vulgar fractions, we see that the number of ciphers

in tin- denominator of the product is = the sum of the numbers of

hers in the denominators of the factors ; so, the number of

il places in the product is = the sum of the numbers of

the factors.

*53

(2) Multiply -53 by -0047. -0047

371 212 •002491

•74213

TOO

519-491

5-09 67000

3563 3054 _ 341030

-5.3 X -0017 = TVu X TsVro = '002491.

(3) Multiply -74213 by 700.

Since one factor contains five decimal places, and the other ends in two ciphers, we point off three places in the product.

(4) Multiply 5-09 by 67000.

Since one factor contains two decimal places, and the other rnds in three ciphers, we annex one cipher to the product.

1. 5-27X4-83 2. -430 x 2-1!) 89X-76 4. 2-38x3-47 5. 5-G2X-213 6. -278X-547

7. 5-27 X -00483 8. -0436X '00219 !). 18-9X-000076 10. -238X-0347 11. -0562X-0000213 12. -00278 X '000547

13. 52-7X48300

14. 4-36X219000

15. -189X7600

16. -00238X347000

17. -00562X21300

18. 27800X '000547

31.

56 DECIMAL FRACTIONS.

19. 98-7654 X '983427 22. -007639 X 763900

20. -123456 X '654321 23. 87'6591 X 684000

21. 5-78934 X '000763 24. -000009 X '000983

25. 100 X '01 X '001 X -0001 X 1000

26. 300 X '003 X '0003 X 3000 X '00003

27. 5000 X 500 X '0007 X '035 X '00005

28. -003 X '03 X '3 X '0003 X 30000

Find the following products by Vulgar and Decimal Fractions : 29. | X A X 2j 32. 4 X A X

30. 31.

X

i X

X

X

33. 34.

| X 2f X

X

X

32* In the DIVISION of Decimals we divide as in integers, and point the quotient so that it may^ contain as many decimal places as are in the dividend, diminished by the number in the divisor. (1) Divide 228-75 by 30-5; and 6-4 by 25-6. 30-5)228-75(7-5 25'6)6-400(-25

2135 512

1525 1280

1525 1280

= TV X

X

= 4 = <25<

H»A X

In dividing 228-75 by 30-5, since there are two places in the di- vidend and one in the divisor, we point off one in the quotient. In dividing 6'4 by 25'6, since we use three places in the dividend and one in the divisor, we point off two in the quotient.

The following examples illustrate various modifications of the general rule : —

(2) Divide 48-97 by -59 ; and 292-3 by 3-95.

•59)48-97(83 3-95)292-30(74 472 2765

177 1580

177 1580

(3) Divide~768625 by 91500; and 32-1 by 128400. 91500)-68625(-0000075 128400)32-100(-00025

6405 2568

4575 15420

4575 6420

Since in dividing -68625 by 915 we would have -00075, by in- creasing the divisor 100 times we diminish the quotient as many times, and thus obtain -0000075. Similarly, in dividing 32-1 by 128400, the number of decimal places in the quotient is = the sum of the number of decimal places used in the dividend, and of the number of annexed ciphers in the divisor.

DECIMAL FRACTIONS.

57

32.

Divide 2230-1 by -769 ; and 1400 by -00224.

•769)2230-1(2900 1538 6921 6921

•00224)1400-00(625000 1344 560 448 TT20 1120

In dividing 2230*1 by 769 we would have the quotient 2*9. By diminishing the divisor 1000 times we increase the quotient as many times, and thus obtain 2900. Similarly, in dividing 1400 by •00224, we annex as many ciphers to the quotient as there are dec- imal places in the divisor, diminished by the number of decimal places used in the dividend.

We may often find it of advantage to reduce the divisor to an integer, and move the decimal point in the dividend as many places towards the right as we do in the divisor.

According to this method, the examples in (2) and (4) would be expressed in the following manner : —

1897( 395)29230(	769)2230100( 224)140000000(
1. 1-7503-7-7-61 2. 40-3858 -r- 6'34 3. 39-538 -T- -53 4. 392-37-7-31-9 5. 110-i»'Jf> — 1-53 6. 5-2441 -7- 22-9	7. 1750-3 -f- -0761 8. 4038-58 -T- -0634 9. 3953-8 -r- -053 10. 39237 -7- -319 11. 1109-25 -7- -0153 12. 524-41 -7- -0229	13. 175-03 H- 76100 14. -403858 -r- 63400 15. -39538-7-5300 16. -39237-7-3190 17. -110925-7-153000 18. -52441-7-22900
19. -0156366 -7- -0042 20. -03486 -r- 4-98 21. -378816 -7- 5-919	22. 20973-6 -7- -8739 23. 9110-64-7-2900 24. 7-127577 -7- 1-0053

Find the following quotients by Vulgar and Decimal Fractions :

25. f -Mi 27. 1)4-7- 10J I 29. 44 -7- A

26. 7i -f- TS 28. 44-4-2$ I 30. 3j -7- 12f

In an INTEKMINATE DECIMAL, one figure or a series of figures * continuously recurs. The figures which recur form a Period. AVln-n the decimal contains the recurring period only, it is termed a Pure Interminate, as -333, &c., written -3 ; '036036, &c., written '036, AVI i en the decimal contains a terminate as well as an interminate part, it is termed a Mixetf Interminate, as -1666, &c., written -16 ; -159090, &c., written -1590. When the period contains one figure, the decimal is called a Repeater ; but when more than one, it is called a Circulator.

c 2

58

DECIMAL FRACTIONS.

33* rURE INTERMINATE. MIXED INTERMINATE.

Pure Repeater as... -3 Mixed Repeater as... -16

Pure Circulator // ...-036 Mixed Circulator....// ...-1590

A vulgar fraction whose denominator in the lowest terms of the fraction contains neither of the prime factors 2 or 5, pro- duces a pure interminate ; thus, £ = *3 ; y = '428571.

A vulgar fraction whose denominator in the lowest terms of the fraction contains 2 or 5, and one or more of the other primes, produces a mixed interminate ; thus, £ = -16 ; 14^ = -5236.

Express the following vulgar fractions as decimals :

(1) f = -857142.

By annexing ciphers to 6 and dividing by 7, we find that the quo- tient consists of a period of six figures.

(2) ,'T = -3i8.

The interminate part of the decimal begins at the second place, and consists of a period of two figures.

(3) TV = -05882352941 17647.

When the numerator is unity, and the T'T = -05882T67 denominator such a prime as will produce e_ — _ -35294 * a considerable number of figures in the pe- riod, we may work as follows : By taking out "J the decimal, say to 5 places, we obtain T'7 i . = -05882 T67, which, multiplied by 6, gives the decimal for T67. Proceeding similarly with the other final vulgar fractions, as in the subjoined process, we have T'7 = -05882352941 17 6470588 Ty By ^examming where the figures begin to recur, we obtain a period of sixteen figures as above.

1- *

2.

3. 4.

6.

7.

8.

9.

10.

34.

11. 12. 13. 14.

15.

16. 17. 18. 19. 20.

if

li

Express the following interminate decimals as vulgar fractions • (1) '185.

1000 X '185 = 185-185 1 X '185 = -185 Therefore, 999 X '185 = 185

And, 485 = iff = /T

t In reducing a pure interminate to the form of a vulgar frac- tion, we take the period as the numerator, and write " 9 " •as often in the denominator as there are figures in the period.

34.

i.

2. 3.

4.

(2)

(3) •4 •64

•07 •962

(4)

DECIMAL FRACTIONS. 48i = f|i = Jf
076923 = 7	769? r?S53	3 — 5 —	= TT'
	5. •	135			9.	•296
	6. •	288			10.	•023i
	7. •	259			11.	•00369
	8. •	48 1			12.	•02439
38i
	1000	X	•68i	:=:	681-81
	10	X	•681	=	6-8i
re,	990	X	•68i	=	675

13. 14. 15. 16.

59

•428571 •153846 •000407 •047619

And, -681 = f}$ = if.

In reducing a mixed interminate to the form of a vulgar fraction, we take for the numerator the difference between the ml numbers, which respectively contain the figures of the decimal and those of its terminate part ; and for the denomina- tor we write " 9 " as often as there are figures in the period, and annex as many ciphers as there are figures in the terminate part.

,_. .1->'-M 1234 — 12 naa - . en

The following method may also be employed in reducing a mixed interminate to the form of a vulgar fraction : —

•681 = -6 + -081 =

of -81

35.

.!..-»•. ..•••.*. 594 + 81
		T5 1 f •• ' I!	To 1 555	990
		— "so = 55'
17.	•16	21. -7045	25. -0054	29.	•00962
18.	•116	22. -0045	26. -0916	30.	•000216
19.	•0138	23. -0054	27. -0916	31.	•5142857
20.	•416	24. -0054	28. -0916	32.	•1076923
	(10 •	3 -f -8i + -037 + -375.
Since the terminate decimal '375 occupies three places, the interminate part of the sum	*333	333333
MS at the fourth place. The periods, con- ic of 1, 2, and 3 figures respectively, are	•037	181818 037037
extended G places beyond the terminate dec-	•375
imal. Mini as they then recur in the same rela- tive order, the period in the sum thus consists	1-563552188
	of (J places,	which is the L. c.	M. of 1, 2, 3.

In extending periods to as many places as are denoted by the L. c. M. of the number of places in each, we are said to make the periods siinilar.

60

DECIMAL FRACTIONS.

35» I*1 the Addition of Interminate Decimals, having extended the Interminates to the longest terminate part, we make the periods similar and then find the sum.

(2) -3 + 4 + -7 = 1-5.

3 _|_ 4 _|_ 7 — 14. Since, by extending the decimals a place to the right, we would obtain the same sum, we add in 1, and thus obtain the sum = 1-5.

(3) 4-962 + -416 + 5-076923. 4-96 296296 As the periods have been made similar, we '41 666666

first add the columns at the beginning of the 5«07 692307

similar periods to find the number to be car-

ried to the last column. 10*45 655270

(4) -3 + 4 + -5 + -6 = 2.

6 = 18, so with the carrying figure the sum is

When we obtain 9 as a repeater, we write 0 and carry 1.

1. -5 + -1 +'6 + -3

2. -2 + -8 + -7 + '4 + -6

3. -09 + 45 + -27 + -54

4. -36 + -18 + '63+-8i

5. -962 + -26i+-i62+-i85

6. •370-1--259 + -636+-407

7. -509 + -037 + -75

8. -216+-216+-2i6+-2l6

9. -037 + '503 + -142857

12.

r 4- WW 4~

4~ '

36. In Subtracting an Interminate Decimal from another, we make the periods similar, and then find the difference.

•91142857 •3J962962

•5 179894

•275 •j.96|296

•078 703 •030 —- -0300

Having found that we carry 1 from the be- ginning of the period, we take 3 from 7, &c.

(2) -275 — 1962.

In subtracting an Interminate from a Ter- minate, instead of carrying frorn the beginning of the period, we may subtract each of the figures in the inter mina'te from 9 ; thus, having obtained 703 by taking 296 from 999, we carry 1 to 6 in the subtrahend.

1. -16 — -07 2. -216 — -1583 3. -243 — -074 4. -076923 — -0375

5. -234— -1672 6. -285714— '-0093 7. -306 — -009 8. -003 — -0003

9. 10. 11. 12.

DECIMAL FRACTIONS.

61

In Multiplying an Interminate Decimal by a Terminate, we proceed in the following manner : —

•7623

(1) -7623 X 27-5.

In multiplying by 5, we carry 3 from the be- ginning of the period ; similarly, in multiplying by 7, we carry 5 ; and in multiplying by 2, we carry 1. We then extend the periods, and find the sum.

27-5

38118

533663

1524752

•1083 X 4 •216 X 7 •32 X 9 •142857 X 5

5. -962 X 11

6. -753 X 64

7. 8-46 X 846

8. 7-27 X 72

20-96534 9. 3-09 X 37 10. -037 X 23 0033 X 606 09756 X 250

11. 12.

In Dividing an Tnterminate Decimal by a Terminate, we extend the dividend until the quotient recurs.

(2) -148 -f- 12.

12)048148148 •012345679

13. -857142 -r- 6 14. -03523 ---26

15. -0231-7-308 16.57-18-7-37

17. 24-106 -7- 32

18. 33-3 -f- 271

In Multiplying or Dividing by an Interminate Decimal, we reduce it to a vulgar fraction.

(3) -076923 X '285714. •285714=3- -076923

7) -153846 •021978

(4) -536 -7- -5.

•5 = J -536 9

5)4-8272 •9654

We may sometimes reduce both the multiplier and the multipli- cand, or both the dividend and the divisor, to vulgar fractions.

19. -27 X '3

20. -037 X '027

21. -02439 X '00369

22. 2-25037 X '4i8

23. 10 — -3

24. 23 -7- 2-09

38. If wo wish to have 3-14159265358979, &c. correct to 8 dec- imal places, we take 3-14159265; but if we desire to carry it

25. -00369 -7- -00271

26. -02439 -T- -17073

27. ^ X '03

28. A- X A

29. -00813-7-^

30. & -7- If

62 DECIMAL FRACTIONS.

33»t° 4 places merely, it will be more accurate to write 3'1418 than 3-1415, for the fifth decimal place being above 5, the for- mer is nearer to the true decimal than the latter, and is thus a nearer APPROXIMATION.

(1) Give approximations to -8450980400+ from 9 places to 1 place successively.

•845098040+ ; -84509804+ ; -8450980+ ; -845098+ •84510—; -8451—; '845+5 '85—; -8+.

By affixing " + " we mean that the true value of the decimal is > the approximation ; and by affixing " — ", that the former is < the latter.

If -8450980400 had been terminate, we would have written •84509804 merely. But if we had written only eight places in the approximate decimal, it would seem as if we knew not the next two.

Give approximations to the following from 9 places to 1 place successively.

1. -0413926852—

2. -3010299957—

3. -4771212547+

4. -6020599913+

5. -6989700043+

6. -7781512504—

(2) Find the sum -428571 + '39024, to 6 decimal places. "VVe extend the decimals to 7 places, and '1285711 I

finding the sum of the 7th column, we add in •3Q024.3Q-I- the carriage to the 6th, and thus obtain the OJU OJ~T"

sum correct to 6 places. '818815+

(3) Find the sum 1-05 + -571428 + -83 + -39024, to 4 decimal places.

To obtain the last figure as the nearest ap- 1 '055556 —

proximation, it is often necessary to extend the • 57 1 429

decimals two places beyond the number re- *8333

quired. The sum of the 5th column, increased

by the carriage from the 6th, being nearer 20

than 10, we carry 2 to the 4th column. 2 '8506 —

7. 7-27 + 9-2916 + 8-36 to 3 pi.

8. -036 + -036 + -036 to 4 pi.

9. -02439 + -003 + 3-1416- to 4 pi.

10. -91908- + -72428- + -72607+ to 5 pi.

!2- A + Y+ A^to 4 pi.

(4) From -12195 subtract -OG93 to 5 places.

•05264+

(5) From -142857 subtract -00813 to 5 places.

DECIMAL FRACTIONS.

63

38*

TIie extra fi£ure in the Gth place of the re- mainder being > 5, we increase the figure in the 5th place by 1.

•012987013— •002710027+ •01027699—

20. 1-041393 -698970+ to 5 pi.

21. 1 -41497+ — 1-32222- to 4 pi.

22. 7r^_ _|Tto5pl.

r to 5 pi.

•142857+ •008130+ •13473—

(6) From -012987 subtract -00271 to 8 places.

Since the extra figure is>5, we cancel the carriage.

15. -7 — -72916 to 4 pi. 1G. '259 — -0027ito5pl.

17. -962 — -90 to 4 pi.

18. -0625 — -0416 to 3 pi. 23.^ — 7 10. -2 — -0083 to 3 pi.

39. . *n CONTRACTED MULTIPLICATION we obtain a product which is correct to a certain number of places.

If we wish to find the product of the terminate decimals 5*2467 and 4-2635 tv four decimal places merely, it is evident that the fig- ures to the right of the line in A are unnec<-s>;iry.

In B, we commence the first line by multiplying the figure in the fourth place having the local value of 7 ten thousandths by 4 imits ; the second line by multiplying 6 thousandths in the third place by 2 tenths, adding in the carriage of 1 from 2X7; the third line by multiplying 4 hundredths in the second place by 6 hundredths, adding in the carriage of 4 from 6X6, &c. Since the first column on the right has the local value often thousandths, there are thus four decimal places in the product, as required.

To insure accuracy in the last decimal place of the approxi- mate product, we work for one place more than what is re- quired. To accommodate the eye, we invert the multiplier, and put its units' place under the place in the multiplicand whose local value is the same as that of the last decimal place for which we are working.

(1) Multiply 5-2467 by 4-2635 to 4 places.

Working for five places, we invert the multiplier, and put the figure in the units' I'ljK'o of the multiplier under the fifth place of the multiplicand.

In ridding, AVC carry 1 from the last

coin

A 5-2467 4-2635	B 5-2467 4-2635
26 157 3148 10493 209868 22-3693	2335 401 02 4	209868 10493 3148 157 26
0545	. 22-3692

5-24670 Inv. (5362-4)

2098680 104934 31480 1574

262

22-3693

64 DECIMAL FRACTIONS.

. . .. 39. (2) Multiply -02439 by '037 to ^ (730730-)

8 places. 731707 Working for nine places, we place the 170732 inverted multiplier so that its units' place -~~ may be under the ninth of the multiplicand. • ^ We carry 1 from the last column, and as 170 there are five significant figures in the pro- duct, we prefix three ciphers. "00090334

1. 4-5625 X 3-375 to 5 pi.

2. 5-7563 X 3-996 to 3 pi.

3. 69-235 X 2-525 to 3 pi.

4. 14-36738 X 30-61725 to 5 pi.

5. -0842367X52-6739 to 6 pi.

6. -74216 X -8237 to 5 pi.

7. 4-02439 X '5027 to 5 pi.

8. 5-857142 X 8'09 to 5 pi.

Let us find the product of the approximate factors 324*1674+ and 2-12967 +. The former may stand for any number be- tween 324-16735 and 324-16745; and the latter for any be- tween 2-129665 and 2-129675. Since the product of the least values 324-16735 and 2-129665 = 690-367859, and that of the greatest values 324-16745 and 2-129675 = 690-371314, the product of the approximate factors can therefore be guaranteed to two decimal places only, as 690*37+.

As the factors in the ac- 324-1674*

companying process are ap- 2-12967 *

proximate, we see that of the nine decimal places in o

the product seven are inde- terminate. The number of

1 Q

291

determinate places is=9 — 7 ; soon

9 = 5 + 4, the sum of the -? 5 « ~ numbers of decimal places in

*******

2691718 450044* 75066* 0088*

* * *

*******

the factors; 7, corresponding '"

to the number of figures in the factor 324-1 674+, is = 3 + 4, the sum of the numbers of integral and decimal places in that factor. By cancelling the number of decimal places in the fac- tor having the greater number of figures, we have 9 — 7 = 5 — 3. The number of determinate places is = the number of decimal places in the factor having the fewer figures diminished by the number of integral places in the other.

(3) Find the product of 31-7436± by 31-7436

•76321+ to as many places as can be Inv- (12367*) depended on. 222205

Since the number of decimals in the factor 19046

having the fewer figures is = 5, and the num- 952

her of integral places in the other is = 2, the 63

number of determinate places is = 5 — 2 = 3.

We therefore work for 4 places. £

24-227

DECIMAL FRACTIONS. 65

39. Find the following products to as many places as can be depended on : —

9. 2-183+ X -00704±.

flgr -00704, whose significant figures extend over three places, has in all Jive decimal places. The other factor contains one in- tegral place. The number of determinate places = 5 — 1.

10. -000732 ± X 2-8+.

^T In -732+ X 2-8 +, the number of reliable places would = 1. Since we have -000732+ as a factor, we remove the point three places to the left, and thus increase the number of reliable places.

11. -23± X 7142-3±.

43T The number of decimal places in the factor having the fewer figures being less by 2 than the number of integral places in the other, we cannot depend on the last 2 integral places of the pro- duct, and thus can give it in hundreds only.

12. 1-375 X -2304±.

When one of the factors is terminate, the number of determinate places is = the number of decimal places in the approximate fac- tor, diminished by the number of integral places in the terminate.

13. 17-69235+ X 2-00976± 17. -7854- X '0036712±

18. -052+ X 12345-

19. -275 X 3-2463±

20. 2-005 X -00017±

14. 1G-;UG7± X 8-3146±

15. 3-247 ± X -00603 ±

16. 3-1416- X -007009±

4O» In CONTRACTED DIVISION, we obtain a quotient which is cor- rect to a certain number of places.

(1) Let us divide 74-0625 by -3147, of which both are

terminate, so as to obtain three decimal places in the quotient.

By inspection, we find '3,1,4,7,0,0) 74'0625 ( 235'343 +

by dividing 7-1 by -3, that 6294

there will be three integral 11122

places in the quotient. We q, * *

thus require six figures in ^

the quotient. Annexing as 16815

many ciphers to the divisor 15735

as make it contain six fig- ~~i' Turn ures, we find the first figure in the quotient, and then

elide a figure from the divi- ~T36

it each successive step. 1 9/»

need not write the ci- L*°

in the first two partial 10

products. 9

66

DECIMAL FRACTIONS.

'K \ .A-IOKAA / .c\f\c\AC\c.Q I ,5 ) '012500 ( «0<

210 1 o^

26 25

4O« (2) Divide '0125 by 30*725, to obtain seven decimal place? in the quotient.

We find that as there will be three prefixed ciphers in the dividend, the number of figures necessary to make up the required number in the quotient will be 7 — 3, or four. We commence to . divide by 3072, and elide a figure at each successive step.

To obtain a certain number of figures in the quotient of two terminate decimals, we begin the division by having as many figures in the divisor as are = the number of required decimal places increased by the number of integral places in the quo- tient, or diminished by the number of prefixed ciphers in it. We then continue to elide a figure from the divisor at each successive step until it is exhausted.

Find the quotients of the following numbers having ter- minate decimals : —

1. 6-75 -f- 3-25 to 4 pi.

2. 10- -f- 4-75 to 3 pi.

3. 20-6 — 3-3125 to 3 pi.

4. 6-23475 -r -04875 to 3 pi.

5. 4-12189 -^ -04763 to 2 pi.

6. -004365 -H -71215 to 5 pi.

7. -0007 -h 3-125 to 6 pi.

8. -00034625-^631 -247 to 10 pi.

When either the divisor or the dividend, or both, are ap- proximate, we can depend on only a certain number of places in the quotient, as may be seen in the following examples : —

(3) 2-5 -f- -0773±. •0773)2-500(32-3+ 2319 181

155

~26

23

(4) -0031416--- -674. •67)-00314|16(-0047— 268 46 47

(5) 6-143±-^-007354±. •00735|4) 6-143 (835-+ 5883 260 221 39 37

(6) -007316+ -H 7-4. 7-4)-007316 (-000989 666 656 592 64 67

DECIMAL FRACTIONS. 67

4O* IR ftN cases, we first find the initial figure in the quotient and point it.

When the divisor is approximate, and the dividend has more determinate places than are in the divisor, as in (3) and (4), we begin to elide the figures in the divisor after the first par- tial product. When, as in (5), the dividend is approximate, and the divisor can produce more determinate places than are in the dividend, as many figures only of the divisor must be D as will make the first partial product contain no more than are in the dividend. But when, as in (6), the divisor is terminate, and has its significant figures extending over fewer places than the number of the determinate in the dividend, we carry on the division in the ordinary way till the dividend is exhausted, and then commence the contraction.

- — "525

10. -00313± — 7-4

11. 1-0367 ± -7- -94364± 1-J. 12-3± — -8738±

13. 2-575 -;- '234±

14. 10- -7- -5236—

15. 5-2673 + — -06731 ±

16. 2-0167+ -7- -733±

17. 1-0035 — -0417 +

18. 10- — 21-63 ±

19. 1- — 2-302585093—

20. 4- -+ 2-167 +

21. -1 — -000767 +

22. 72-1 — -00312 ±

23. 10- -h -000763 ±

24. -007635 -f- 7-142 +

25. -073167 ± -h 2-25

26. 1- — 12-56637 +

27. 42-75 ~ -00077 ±

28. 630- -f- -0739 ±

29. -0125 -i- 71-23±

30. 10- ~ 2-718281828 +

• In i:i;i»r< IN*; a simple quantity to the decimal of another in a higher name, we annex ciphers to the number of units in the r, and divide by the number which shows how often a unit of the lower name is contained in one of the higher.

(1) Reduce 9d. to the decimal of I/. d.

12)9-00 We thus change T9a to a decimal. - — — -

* i OS.

In reducing a compound quantity to the decimal of a simple quantity, we reduce the number in the lowest name to the dec- imal of"the next higher, to which we prefix the integer in the latter, and so proceed till we obtain the decimal of the required

(2) Reduce 4 Ib. 7 oz. 15 dwt. to the decimal of 40 Ib. The accompanying process is equiv- t,~ ^;

alent to the following:—

ISdwt. = *!<*. = '75 o* z.

7-75 oz. = Lu> Ib. = -64583 Ib. 40 ! 4'64583 Ib.

1 2

3 11). — 4^u_8_3=-] 1614583 of 40 Ib. '11614583

68 DECIMAL FRACTIONS.

41. When the quantities are expressed in mixed numbers con- taining vulgar fractions or decimals, we proceed as follows :

(3) Reduce 4| min. to the decimal of 15*2 hours. 60 ) 4-75 min.

15-2) -07916 ho. (-0052083. We may sometimes cancel thus : — 4-75 _ -25 25

60 X 15'2 — 60 X '8 — 48 ~

In reducing a compound quantity to the decimal of another, we find the vulgar fraction, which shows what part the former is of the latter, and reduce it to a decimal.

(4) Reduce £2 //I I// 8 to the decimal of £5//7*7i- By the method of § 20., No. (4);£2//ll//8 = if If of £5//7//7i

4 f|f= -48006194+

Otherwise: By reducing 11/8 to the decimal of £1, and pre- fixing 2, we obtain £2'583, and are thus said to have re- duced £2//ll//8 to the decimal of £1. Similarly, £5*7*74 reduced to the decimal of £1 =£5-38125.

£2-583 -^ £5-38125 = -48006194+

1. 8d Is.

2. 15cwt 1 T.

3. 30 in 1 yd.

4. 7/6 £1.

5.13/44 £1.

6. 5/6£ £1.

7. 8 oz. 3dwt lib. troy.

8. 3fu. 10 po 1ml.

9. 2ro. 30 po 1 ac.

10. 3qr. 15|lb 10 cwt.

11. 3bu. 3ipk 5qr.

12. 6 ho. 9^ min 3 da.

13. 2/8J 5/3J

14.7/81 15/3

15. 6/7£ 18/9

16. 3oz. 5dwt lib. 3 oz.

17. 2bu. 3pk 5bu 1 pk.

18. 2ft. Sin 3yd. 2ft.

19. 5fu. 8 po 7fu. 20ipo.

20. 5min.l6isec..3ho.l5min.

21. 23°27/37// 90°.

22. 5 cwt. 3qr 2T. 10 cwt.

23. 3 da. 101 ho... .3 wk. 4 da.

24. 6f min 7ho.30min.

25. From Delhi to Bombay the direct distance is 720 miles ; and from Delhi to Madras, 1080 miles. Reduce the former to the decimal of the latter.

26. Westminster Hall is 270 feet long and 75 feet broad. Re- duce the latter to the decimal of the former.

27. Reduce a sidereal day, which is = 23 ho. 56 min. 4'09 sec., to the decimal of a solar day of 24 hours.

28. Reduce the sidereal day of Jupiter, which is = 9 ho. 55 min.

DECIMAL FRACTIONS.

69

41. 50 sec., to the decimal of the Earth's sidereal day, which is 23 ho. 56 min. 4-09 sec.

29. Reduce a solar year, which is =. 365 da. 5 ho. 48 min. 49-7 sec., to the decimal of a sidereal year, which is = 365 da. 6 ho. 9 min. 9*6 sec.

30. Express the height of the Peak of Teneriffe, which is = )2 feet, as the decimal of a mile.

31. Express £3 " 17 " 10 £, the value of 1 Ib. troy of sterling gold, in the decimal of £1 .

32. The Danube is 1630 miles long, and from the source of the >uri to the mouth of the Mississippi the distance is 4000 miles.

uce the former to the decimal of the latter.

33. Reduce the weight of a Cologne mark, which is =, 3608 grains, to the decimal of 1 Ib. troy and of 1 Ib. avoir.

42* In finding the value of a decimal of a unit, we multiply the decimal by the number of times the given unit contains the next lower unit, and so on as far as may be required.

(1) Find the value of £'7895. £•7895 20

£•7895 =

s. 15-7900

12

d.9-48

4

f. 1-92

(2) Find the value of '583 oz. ' troy.

oz. -583 20

dwt. 11-6 24 gr. 16-

By multplying the intermmate decimals, we obtain

•583 oz. = 11 dwt. 16 gr.

The following examples afford additional illustration of finding the values of decimals : —

(3) Find the value of 2'75 of 5'45 acres. 5-45 2-75	(4) Find 5 cwt 5 cwt. 3 28 4	the value of 2'425 of 3 qr. 16 Ib. qr. 16 Ib. = 660 Ib. 2-425 660 14550^ 14550
2725 3815 1090
ac. 14-9875 4 ro. 3-9500 40
1600-5 57 qr. 4 Ib.
14 cwt. 1 qr. 4'51b.
po. 38-00

70 DECIMAL FRACTIONS.

42. 1-

2. £-975

3. 2-875 s.

4. -4375 gu.

5. £1-05416

6. £-7302083

7. -275 Ib. av.

8. -16 oz. tr.

9. 3-142857 cwt.

10. -583 hour - 22. 5-24 of 3«

11. 7-0625 ac.

12. 2-0945 cub. ft

13. -55 of 4-204 ac.

14. 2-75 of -04yd.

15. -003 of 3-6 ml.

16. 4-125 of 243 ac.

17. -075 of 3 bu. 2 pk.

18. 3-0916ofllb.4oz.lOdwt.

19. -325 of 7 ho. 24 min.

20. -432 of 5 cwt. 2 qr. 24 Ib.

21. -037 of 15-201 yd.

23. -725 of 7-76 bu.

24. 3-425 of 4-003 cwt.

(5) Find the value of -0025 ac. -f 3-45 ro. + -0076 ac. + -009 po.

ro. po. ac.

•0025 ac. = 0 // 0-4 = '0025

3-45 ro. = 3 //18- = '8625

•0076 ac. = 0 // 1-216 = '0076

•009 po. = 0 // 0-009

3 //19-625 = -87265625

25. 2-003 ml. + -275 ml. + 1050 yd. + -025 ml.

26. £3-3 — -5 s. + -075 cr. — 285714 guin.

27. -425 ho. + -003 min. — -275 ho. + -925 min.

28. Express the hectolitre, = -343901 qr., in bu. and pk.

29. Express the Linlithgow wheat boll, = -499128 qr., in bu. and pk.

30. Express in grains troy, a weight -00024 Ib. avoir, heavier than a kilogram, which is 2*20462 Ib. avoir.

31. From Paris to Berlin by railway is a distance of 1308 kilo- metres, of which each is = 1093*63 yards. Express the distance in miles and yards.

32. Mercury revolves round the Sun in 87-9692580 days. Ex- press the period of revolution in days, hours, and minutes.

33. Express in avoir, wt. the weight of a Prussian pound, which is -46771 of 2-20486 Ib. avoir.

34. Find the length in inches of the Greek foot, = * *• of the Roman foot, which was — '97075 foot.

35. Find the weight of 3| cubic feet of water at 62-455 Ib. avoir, per cub. ft.

36. The radius of a circle is = -1591549 of its circumference, which contains 360°. Find the angle whose arc is = the radius.

DECIMAL FRACTION'S.

71

43. ^'e cal1 the tenth of a Pound Sterling, a florin. In extend- ing the decimal division of the Pound, it was proposed to call the hundredth a " cent," and the thousandth a " mil." 1 florin = £-1 = 2s. 1 cent = -01 = 2|d. 1 mil = -001 = |ff.

1 shilling = ^ florin; 1 farthing = -JJ or 1^ mil.

To express a sum of money as the decimal of £1, we may work as in § 41. ; but to do it mentally, let us consider the following analysis : —

s. f. fl. m.

14/10i = 14 + 41 = 7 + 42iJ

14/10} = £-7427083.

For the first place, we take half the number of the shillings. For the second and third places, we express the pence and far- things as farthings, and increase the number by 1 if it is > 24. For t\\& fourth and///'/// places, we multiply the number in the second and third, or when the number is > 25, its excess above 25, by 4, and add 1 for every 24. For the sixth and seventh multiply tho number in the fourth and fifth, or when the number is > -Jf>, 50, or 75, its respective excess above 25, 50, or 75, by 4, and add 1 for every 24.

When the number of shillings is odd, we work for the next

r even number of shillings, and add 5 to the second place ;

thus, 15/10J = 14/10J + 1/ = £'7427083 + £'05 = £-7927083.

Jd. =: £-0010416, any sum of money expressed in the deci- mal of £1 contains no more than six terminate places. When there are more than six places the seventh is interminate, being either

:• I).

Reduce the following sums of money to the decimal of £1. (1) 17/5 1 = £-8739583.

	8.		d.	8. d.	8.	d.	8.	d.
1.	12	//	6 7.	14 * 5J	13.	18	"10f	19.	7	// Ox
2.	18	//	8.	16 // 34	14.	12	// 6i	20.	13	» 6f
3.	4	//		9.	12 * 4£	15.	8	" ?f	21.	14	,2J
4.	2	//	7	10.	6 // 4j	16.	3		22.	19	' 8*
5.	12	//	10	11.	8 // 3±	17.	9	// 8i	23.	3
6.	14	//	8	12.	18 //1U	18.	15	x/ 7i	24.	19	//11J

To • -urn of money approximately to three decimal

j;i, or in florins, cents, and mils, we adopt the prin- ciple of approximate decimals (see § 38.), by increasing the

72 DECIMAL FRACTIONS.

43,number of farthings by 1 when it is > 12, or more than half- way up to 24, and by 2 when it is > 36, or nearer to 48 than to 24; thus, 16/4J = £'8197916 = £-819jf, being nearer to £•820 than to £'819, is approximately = £'820.

(2) Reduce 16/7± to three places of the decimal of £1.

16/71 = £-831.

Reduce the sums of money, Nos. 1 to 24, approximately to three places.

Being familiar in § 42. with the common method of finding the value of the decimal of £1, we may now consider the fol- lowing plan : —

Let us find the value of £'9238. £-9,238

By pointing off the first place, we 952

obtain the number of florins, £'9238=18/5i|°f. Now, since 96 farthings = 1

florin, we must multiply by 96. But as 96 = 100 — 4, we put 4 times the multiplicand two places to the right, and then subtract. The number made up of the first two places on the left is the number of farthings.

(3) Find the value of £-7145. -7,145

£•7145 = 14/3HI-

25. £-125

26. -225

27. -375

28. -975

29. £-3125

30. -7625

31. -9875

32. -5375

33. £-4236

34. -5168

35. -8274

36. -4537

13,92

37. £-7219

38. -8437

39. -2914

40. -3853

To obtain the value of the decimal of £1 to the nearest far- thing without a fraction, we proceed as follows : —

Let us find the value of £'7287. We consider it approxi- mately = £-729, which is = 14 s. + 29 mils.

Since 25 mils = 24 f., we subtract 1 from 29, and obtain 29 mils = 28 f. nearly, and £-729 approximately = 14/7.

To obtain the number of shillings, we divide the number of cents in the first two places by 5, the number of mils being = the remainder with the figure in the third place annexed ; thus, £•883 = 17s. + 33 mils ; '824 = 16s. + 24 mils. When the second figure is < 5, we may obtain the number of shillings by doubling the figure in the first place.

In reducing the number of mils to farthings, we adopt the principle of approximate decimals, and subtract 1 when the number is > 12, or more than half-way up to 25, and 2 when > 37 or nearer to 50 than to 25.

DECIMAL FRACTIONS. 73

43. W Express £-768 to the nearest farthing.

£-768 = 15s. + 18m. = 15/4£. Valuate the decimals, Nos. 33 to 40, to the nearest farthing.

••fiT The pupil may now construct a table, showing the correct and the approximate decimals of £1 from £d. to I/, so that by men- tally inserting the decimal for the number of shillings, the decimal of any sum may be obtained.

MISCELLANEOUS EXERCISES IN DECIMAL FRACTIONS.

1. Find the price of 30 Parian statuettes @ £1-775 each.

2. In January 1856, the number of days during which rain fell in Scotland was 13, and the amount which fell was 2*38 inches. Find the daily average for each of the 13 days.

3. How many ac. ro. and po. are in a park containing -08 of 155-1875 acres?

4. If 31-75 poles are feued for £2-38125, how much is it per pole ?

5. Find the sum of £-3125, -4375s., and -75d.

6. In March 1856, in Edinburgh, the thermometer at the highest was 51°' 1, and at the lowest 29°*4. Find the difference or range.

7. Find the value of -00375 Ib. troy of sterling gold @ £3»17»10£

8. Of 100 parts of matter in locust beans, sugar and gum form 61*10, other vegetable matter forms 31*55, and moisture 5. Of how many parts does the remainder, which is mineral matter, consist ?

9. The distance from Paris to Leipsic by railway is 1225 kilo- metres, each 1093*63 yards. Express it in miles.

10. Of the manure of dissolved bones "1571 of its weight is organic matter. Find the weight of organic matter in 80 tons of manure.

1 1 . Express the sum, T»g of 4| + J + j i of 57S + 535, as a decimal.

12. In February 1856, at Sandwick, Orkney, the barometer at the highest was 30-543 inches, and at the lowest 28*843 inches. Find the difference or range.

13. The following rents are drawn from a property : — mansion, £150-15; farm, £470*475 ; parks, £80*875 ; feus, £7 '625. Express the total in £, s., d.

14. Find the price of 14 cwt. 3 qr. 14 Ib. rice @ £'625 jg> cwt.

15. The time of Jupiter's rotation on his axis is 9 ho. 55 min. 50 sec., and the period of his revolution round the sun is 4332-5848 days. Reduce the former to the decimal of the latter.

16. A line in a diagram in a book published in the sixteenth century, which now measures 6*83 inches, has shrunk to *£ of its original length ; find what it had been.

D

74 DECIMAL FRACTIONS.

, 17. A cubic inch of pure water weighs 252-458 grains, find the weight of a cylindrical inch which is '7854 of a cubic inch.

18. A gallon of pure water weighs 10 Ib. avoir. ; and a cubic inch, 252-458 grains. From these data, find the content of a gallon.

19. The period of the revolution of the Earth round the Sun, measured sidereally, is 365-2563612 days, and that of Mars is 686*97964580 days. Reduce the latter to the decimal of the former.

20. The height of the Peak of Mulhacen in Spain, formerly es- timated at 3555 metres, has been found to be -156 kilometre less. Find its height in feet at 39*37079 inches ^ metre.

21. A gallon of pure water weighs 10 Ib. avoir., find the weight in oz. of a pint of whey of which the Specific Gravity is 1-019.

^gT When \ve mention the Specific Gravity (s. G.) of a substance, we show how many times it is as heavy as pure water ; thus, the s. G. of lead being 11'35, any volume of lead is 11-35 times the weight of the same volume of water whose s. G. is 1.

22. Find the weight of 12 gallons of olive oil, of which the s. G. is -915.

23. Find the content of a block of granite 5*5 ft. long, 3-2 ft. broad, and 1*6 ft. deep.

24. A metre is = 39*37079 inches. Reduce an inch to the deci- mal of a metre.

25. What decimal of the whole time necessary to burn a ton of coals continuously at the same rate is that required to burn 2-20486 Ib. ?

26. Divide £31-4 among 6 men and 11 youths, giving a youth •525 of a man's share.

27. The weight of a cubic foot of pure water is 999-278 oz. avoir., find the weight in Ib. avoir, of the air in a room 12 '5 ft. high, 16-25 ft. long, and 10*4 ft. broad, air being 815 times as light as water.

28. In March 1856, the weight of vapour in a cubic foot of air in Edinburgh was 2-24 grains. Find in the decimal of a Ib. avoir, the weight of vapour in the atmosphere of a room 12 ft. in height, length, and breadth, supposing that there was no fire and that the window was open.

29. Reduce \ of ^ of ^ to a decimal.

30. Express the sum, f of If + f of f| + '2, as a decimal.

31. Reduce f guinea to the decimal of £1.

32. Express the sum of SJ7 and 5JT as a decimal.

33. The Polar and Equatorial Diameters of the Earth are re- spectively 41,707,620, and 41,847,426 feet. Express each decimally in miles.

34. Find the number of miles in the Meridional Circumference of the Earth, supposing that it contains 40,000,000 metres, each 39-37079 inches.

DECIMAL FRACTIONS. 75

Gravity of Hydrogen, that of air being 1, is •069, while that of air as compared with water is -0012. Express the relative weight of Hydrogen as compared with water.

ggT Water is the standard for solids and liquids, and air for gases.

36. The s. a. of carbonic acid gas, that of air being 1, is 1-524.

L6 relative weight of carbonic acid gas as compared with

•T.

37. Reduce an oz. avoir, to the decimal of an oz. troy.

38. Keduce a Ib. troy to the decimal of a Ib. avoir.

39. A Winchester bushel is = -9694472 Imperial bushel. Ex- press an Imperial bushel as the decimal of the former.

40. A zinc bar, which at 32° Fahrenheit measures 1 inch, at 212° measures 1 -003 inch ; find the length of a bar of the same metal at

. which at 32° measures 2-25 inches.

41. What decimal multiplied by i of T95 produces fi?

42. Divide £1-3125 equally among a number of almsmen, giving each -375 ilorin. What is the number ?

43. What quantity of sugar @ £-025 ip Ib., will cost 19'575

Hod

•14. Divide the sum of -075 and -0075 by the difference of 7-5 and

45. The yard measure made by Bird in 1758 was 36-00023 inches long. How many times would this measure be contained in a mile.

46. In 1825, the Stirling jug or pint measure was measured in Edinburgh, and found to contain 104-2034 cubic inches. Reduce this to the decimal of an Imperial gallon.

47. On the floor of a room 10 ft. 8£ in. long and 8'25 ft. broad, dust has accumulated to the depth of -075 inch. Express the volume of dust in the room as the decimal of a cubic foot.

48. The maximum delivery of a reservoir is 567-07 cubic ft. of r ^ minute, and its minimum delivery 516*66 cubic ft. Find

the number of gallons, each 277-274 cub. in., delivered on an equal average in 24 hours.

49. The mean diameter of the Earth is 7912-409 miles. Find the surface of a sphere of the same diameter, found by multiplying the square of the diameter by 3-1416.

50. Find the content of a sphere of the same diameter as the earth, found by multiplying the cube of the diameter by -5236.

76 2 ^

/

CONTINUED FKACTIONS.

0lF we take a vulgar fraction, as Jf ±, and divide the numerator and the denominator by the numerator, we obtain ^^ = ^-^ Similarly, T«ft = ^, and « = ^ . We have thus J$J L

i __ 1_ __ 1_ In the last form, we observe that

sffi 3 * 3 L_ every numerator is unity. A com-

2 L_ plex fraction, in which every numer- 3^' ator is unity, and every denominator includes the succeeding parts of the fraction, is termed a CON- TINUED FRACTION.

In the foregoing process, we have obtained the continued

fractions : l - 1 . 1 . 1 These fractious are re-

3 ' 31' 3 i_ 3 i_ spectively = ^ 4, T\,

2 2I 2L- and «i- We find that

3^' * we have reproduced the

original fraction J|i. As the other fractions continually ap- proach to it in value, they are termed Convex-gents. The con- vergents are alternately greater and less than the original fraction.

(1) Find the convergents to £|J.

The practical method of finding the con- vergents is to proceed as in finding the G. c. M. of 121 and 415 (see § 3.).

We may write the quotients 3, 2, 3, 17, i 1 __j ;.L- .I/L - .C_-_.L dace

the

in a column, and opposite the first we p unity in the Numerators' column, and

Quot. 3 2 3 17

Num. 1

2

7 121

Den. 3

7

24 415

first quotient 3 in the Denominators'.

In the second line the numerator is = 2 X 1, and the denomina- tor is = 2 X 3 + 1.

In the third, the numerator is = 3 X 2 -}- 1, and the denomina- tor is = 3 X 7 + 3.

In the fourth, the numerator and the denominator of the original fraction are reproduced.

The convergents are, % f, 27?, |||.

(2) Find the first three convergents to 3-14159. By proceeding as in finding the G. c. M. of 14159 and 100000, we obtain the first three quotients, 7, 15, 1.

The convergents are,

Quot. 7 15 1

Num. 1

15 16

j, and 3 Ty,; or, y, f«|, f *|. Find the convergents to the following fractions : — !• Jft- I 2. fltf¥. | 3. TV*. | 4. f«f

Den.

7

106 113

CONTINUED FRACTIONS.

77

4*5* 5. Find the convergents to yVsV

^" We first reduce the fraction to its lowest terms. But whether we do so or not, the fraction is reproduced in its lowest terms.

6. Find the fifth convergent to -7854.

7. Find the third convergent to '5236.

8. The Specific Gravity of oxygen is Ty5'T of that of carbonic acid gas. Give the fourth convergent to this fraction.

<ET Whenever a remainder is a comparatively small fraction of the corresponding divisor, the convergent obtained may be taken as a good approximation.

9. The Specific Gravity of gold is 19*35, and that of platinum is 21 -47. Find the second convergent to £f f «.

10. Venus revolves round the sun in 224*701 days, and the Earth in 365*256 days. Give the fifth and sixth convergents, which approximately show what part the former period is of the latter.

11. Mercury revolves round the sun in 87*969 days, and the Earth in 365'2f>6 days. Give six convergents.

12. The solar year is = 365*24224 days. Give the fourth con- vergent.

13. A metre is = 39*37079 inches. Find the fourth convergent to the fraction which a yard is of a metre.

14. A Scotch acre is = 1-261183 Imperial acre. Find five con- vergents to the fraction which an Imperial is of a Scotch acre.

PKACTICE. >

PRACTICE is the method of computing by means of Aliquot Parts.

A number contained an exact number of times in another is an aliquot part of it : thus, 7 is an aliquot part of 21 ; 10/ of £1; 6/8 of £1; 14 Ib. of 1 cwt.

ALIQUOT PARTS.

10 //O 6//8 5//0

4//0 3 // 4

s. d.

2//6 2//0

0*8 0//6

46.

(l^Find the price of 794 yards of silk @ 2/6 V yd.

«£794 = price of 794 yd. @ £1 £ 99 // 5 = @ 2/(

d.

6 4 3 2

H

78

PRACTICE.

s. d.

1.	8462	@ 10	a. *0	7.	8472 @	s. 1*	a. 8	13.	7342	@3	u. *4
2.	7926	.. 6	*8	8.	7904 ..	5*	0	14.	9836	.. 5	*0
3!	8248	.. 2	*6	9.	8463 ..	4*	0	15.	9246	.. 0	*8
4.	7923	.. 4	'/O	10.	9527 ..	3*	4	16.	9372	.. 0	*6
5.	7686	.. 3	*4	11.	3513 ..	2*	6	17.	7236	.. 0	*8
6.	7968	.. 2	*0	12.	6723 ..	1*	8	18.	8943	.. 6	*8

(2) What cost 7689 oranges @ lid. and @ fd. each?

		7689s. . .		t VOJl. 3
					4	zoOo7
H	i	I/	961*11 .	@lid.	12 2,0
5766f
£48*l*li
48,0*6
		d.		i.
19.	7268 @	6	23.	8464 @ li	27. 6847 @
20.	8379 ..	4	24.	7932 .. 1	28. 8467 ..
21.	3848 ..	3	25.	7233 .. li	29. 6593 ..
22.	5766 ..	2	26.	7923 .. i	30. 7892 ..

9 (1) Give two aliquot parts which make up 7£d. and 5^d.

d. d.

711 ( 6 = iof I/ r, i ( 4 = £of I/

74^. - ^ +1,= ? of 6d. 5id.=|+1_ I Qf a|

1. Find two aliquot parts which compose the following rates :

3|d. ; 7d. ; 7Jd. ; 4jd. ; 6fd. ; 6jd.

(2) Give two aliquot parts which make up 8/4 and 12/6.

OH _ f 5/ = i of £1 -2 fi _ ( 10/ = J of £1 - t +3/4= j of £1 -| +2/6 = \ of 10/

2. Find two aliquot parts which compose the following rates :

7/6; 3/9; 5/10; 6/3; 2/11; 4/8.

(3) Find the aliquot parts which, when respectively sub-

tracted from I/ and £1, leave 9d and 16/8.

9d

_f I/ ' — \ —3d. = i

of I/

16/8- ~

—3/4 = i of £1.

3. Find aliquot parts which, when respectively subtracted from I/ or £1, leave the following rates :

lOid.; 9d.; Hid.; 17/6; 13/4; 15/. (4) Find three aliquot parts which make up 8^d. and 15/7*.

PRACTICE.

79

47.

f 6 = i of I/

Jd. = 4+2 = £of6d

(+> = ^of2d

( 10/=iof£l

15/7^ = -^+ 5/=i.oflO/ ( +7id. = 4 of 5/

4. Find three aliquot parts which compose the following rates : 9jil.; ?id.; 8|d. ; 7/8J; 11/10J; 13/9.

48. (1) Find the price of 4671 loaves @ 4|d. and @ 9d.

J 1	I	I/ 4,1. Jd. 2,0	4671s. . @ I/	d. 3	'	114671s. . . @ 1//0 I/ 1| 1167* & . .. 0//3
. . @4d. 194* 71 .. 1 97* 3f .. i
2,0)350,3 // 3 . @ 0 // 9 £175//3//3
)184,8//lli @ 4fd.

49.

£92*8*111

(2) Find the price of 846 yards of cotton @ SJd. and ^@T

d.			846s. . . @ I/			846d. . . @ Id.
4	i		•)u->	. @ 4d.
	I	1 r I/	105//9	... 11	i.	d.	6768 . . . @ 8d. O1 1 1 Jtrl
2,0)38,7*9 '7 "9	. @ 5id. J	t i	12	Zll^ . . .. 5U 5B556}7T7@ 7f d-
	2,0)54,6//4
						£27//6//41
33 @	d.	9. 6723 @	d. 2i	17. 6874 @ 2|
3. 4673 .. 4. 8423 .. 8. 87	74 41 5 31 81 94	10. 7247 .. 11. 3475 .. 12. 4672 .. 13. 2435 .. 14. 6724 .. 15. 7233 .. 16. 9894 ..	61 3J li 61 6i 84 11		18. 8674 .. 52 19. 7683 .. Jfc 20. 8267 ..mt 21. 8956 4p)f 22. 5732 Jr9* 23. 746MFHi 24. 8722 .. 7
	(i)	Find the ?> 1/I'-'J	price of 423 yards	of clotli*® 1/10, and
	*i	> *•!	£423	.@£ls d			423s. . . @ 1^0
2/		£1	42//	6..@2//0 J	5 i	•ji	211// 6 . .. 0//6
		2/	3//10//6 .. O// 2 :	7 2 t ?	6d	26// 5i .. 0//OS
			£38// 5^/6 @ 1" 10		2 0)66 O// Hi @ lx/t>l

£33^111

80

PRACTICE.

49. (2) Find the price of 846 cwt. of rice @ 7/9}, and @ 11/7$.

2/6 3|d.	i I	£1 2/6	£846 . . @ £1 s> d.	d. 6 1J	I
211*10 . @5*0 105*15 . ..2*6 13* 4*4J ..0*3|
£330* 9*4J @ 7*9}

846s. 11

9306 . 423 . 105//9

_ s. d.

@11*0 . 0*6

2,0)983,4*9 @ll*7a £491*14*9

(3) Find the amount of 793 railway fares @ 16/8, and @6/9.

5O»

		£793 . . . @ £1* 0*0	1793s. ..©I/
3/4	i £1	132* 3*4 .. 0* 3*4	6 g J
		£660*16*8 @ £0*16*8 d.	4758 . . @ 6*0
			9 i	594*9 . .. 0*9
			2,0)5352*9 @ 6*9
			£267*12*9
		s. d.		s. d.	s. d.
1. 4567 (	® 1* U	17. 798 (	S 10* 6	33. 893 @ 16* 8
2. 3283 .. 1* 7i	18. 742	.. 4* 8	34. 979 .. 17* 6
3. 5687	.. 1* 6|	19. 467	.. 5* 3	35. 894 .. 18* 4
4. 8672	.. 1*10J	20. 923	.. 1*10	36. 897 .. 18* 9
5.	937	. 15* 0	21. 916	.. 4* 6	37. 374 .. 8* 9
6.	423	. 13* 4	22. 743	.. 3* li	38. 968 .. 4* 4i
n t .	341	3// Q o* u	23. 123	.. 2* 4	39. 763 .. 9* 6
8.	876	. 12* 6	24. 732	.. 2* 9|	40. 423 .. 13* 54
9.	827	. 11* 8	25. 428	.. 3* 8	41. 346 .. 9* 9|
10.	729	Q A	26. 293	.. 5* 6	42. 729 .. 18* 74
11.	873	. 4* 2	27. 468	.. 2* 3	43. 777 .. 19* 24
12.	798	. 5*10	28. 736	.. 10* 8	44. 947 .. 4* 8
13.	149	. 10*10	29. 716	.. 1* 54	45. 589 .. 5* 74
14.	824	. 11* 3	30. 637	.. 2* 8	46. 346 .. 5* 5
15.	899	6* 3	31. 468	.. 17* 4	47. 777 .. 9* 9
16.	243	. 2*11	32. 823	.. 7* 84	48. 732 .. 7*10i
(1)	Find the price of 783 qrs. of wheat @ j f^^g ^ qr'
		£783.. . @£l	783 ... @ £1
		3	5
		2349 . . . @£3» 0»0	113915 @£5" O'.-O
ton	£1	391-10 .. 0"10»0	12/6 i £5|| 489» 7-6 .. 0-12.6
l/3 i	r 10/i	48»18»9 .. 0" 1»3	£3425»12"6 @£4" 7-6

£2789" 8»9@£3»11"3

PRACTICE.

81

5O« (2) ^d the price of 379 quarters of barley @ £2//3//5£ ^ qr.

379s.

51.

43	\& j./
1137 >	8. d.
d. 1516 f	@ 43//0
4 i	I/ II 126//4 ... .	, 0//4
a H *	I/ II 47//4' , . ,	» \j> i „ O//H
2,0)1 647,0*8 J . . .	„ 43//5i
£823//10//8£
£ 8. d.	£ s. d.		£ s. d.
1. 916 @ 4//16//0	11. 985 @ 7//11//8	21. 896 @	7//19// 0
2. 169 .. 3//15//0	12. 946 .. 7// 8//9	22. 846 ..	6// 8// 4
3. 843 .. 2//13//4	13. 853 .. 10//16//8	23. 859 ..	2//12// 6
2 .. D'/ 3//4	14. 976 .. 10//12//6	24. 987 ..	4// 7// 6
5. 847 .. 3//12//0	15. 793 .. 12//13//4	25. 739 ..	4//11// 8
6. 974 .. 3// 7//6	16. 847 .. 11//13//9	26. 463 ..	4// 1//10£
7. 874 .. 5//16//8	17. 569 .. 6//13//4	27. 568 ..	Iff I// 6f
8. 734 .. 5" 8//4	18. 279 .. 4//15//0	28. 984 ..	9// 2// Si
9. 986 .. 6//15//0	1(.>. 947 .. 5//18//4	29. 719 ..	25// 9// 8J
10. 793 .. 7//17//6	20. 539 .. 4//18//8	30. 346 ..	27//15// 6f
It is often convenient to employ the FLORIN as	the unit of
computation.
(1) Find the price of 489 tons of coal@ 14/, and @ £1*3 ^ ton.
489 fl. . . @ 2/ 489	fl. . . . @	2/
111
342,3 fl. . . @ 14/ 244	//Is.
£342^6 5379

562,3 fl. Is. . @ 23/

£562*7

The most convenient method of reducing a sum expressed in £ and a. to fl., is to annex half the number of s. to the number of £ ; thus, £3 " 4 = 32 fl.

(2) Find the price of 878 cwt. of sugar @ j £2//15//4 ™

878 fl. . @ 2/

28

7024 1756

d I I II 24584 . . @£2"16«0 8 |}| Fl. || 292*1 »4 .. 0" 0"8 2429,1"0"8 @£2*15»4 £2429"2-8

D 2

JF1.	878 fl. . . 17	@2/ .. 0*	14^0 0-8
6146 1 1 878 J | 292«1"4
1521,8"l,/4	@£1»	14/-8
£1521-17"4

82

PRACTICE.

51. 1. 794

2. 798

3. 823

4. 697

5. 796

6. 267

7. 937

8. 469

9. 835

10. 974

11. 826

12. 563

52. (*) Find the Price of 749TT cwt- @ n/8 V cwt

2 ! 13. 943	@£1//16	25. 763	@£1//14// 8
6 1 14. 879	.. 1//18	26. 269	.. 0//17// 4
8 15. 937	.. 2//12	27. 263	.. 1//13// 6
14 16. 893	3// 4	28. 798	.. 0//16// 6
16 17. 828	'.'. 7// 8	29. 839	.. 0//12// 4
12 18. 726	.. 5//14	30. 346	.. 2//15// 8
18 19. 699	.. 3// 5	31. 876	.. 0//16// 3
7 20. 893	6// 7	32. 732	.. 0//17// 9
11 21. 467	;; 7// 9	33. 356	A "1 A . O
19 22. 796	.. 9*11	34. 797	!.' 1//15//10
13 23. 876	.. 8//17	35. 798	.. 2//18// 2
17 24. 539	.. 11//13	36. 529	.. 3//11//10

io/

1/8

£749

374//10//0 62// 8//4

T6T of 11/8 = Q// 6//4£T\ £437// 4//8JT6T

s. d.

11//8 _6

ll)70//6 6//4J

(2) Find the price of 292T\ Ib. @ ll/5£ & Ib.

IO/ 1/3

£1 io/ 1/3	£292// 8//	9
146// 4// 18// 5// 3// 0//1	6*4

Since ^ of £1 = 8/9, the price of 292 TTg Ib. @ £1 is £292»8"9.

£167//10//1

In the method of (1), we first find the price of the integral num- ber 749 cwt., by taking the parts which make up the rate 11/8 ; and then add in the price of the fractional number, T\ cwt. In the method of (2), we first find the price of the mixed number 292T78 Ib., at the unit of computation £1, and then take the parts whicli make up the given rate. The first method is of more general application than the second, which is only conveniently applied when the denom- inator divides the unit of computation without producing a fraction.

1.216i@13-/ 4 2. 547|.. 16// 8 3.899^.. 9// 6 4. 447 f .. 5// 9 5.967^.. 6//10 6. 793| .. 17// 1 7. 468| .. 16// 6 8. 794f .. 19// 6^

£ s. d. 9. 235| @0//18// 4 10. 324-^ .. 2// 3// 8 11. 829| .. I// 6// 3 12.247TV. 1"13" 4 13. 794T%..2// 5//10 14. 823TV.. I// 9// 4 15. 299|°..0// 8*114 16. 834T\ .. 0//17// 9

£ s. d. 17.273T\@1 //3 //9 18.347f ..0//17// 5J 19. 423£ ..0//17//1H 20. 342^..0// 5// 34 21.827| ..0//19// 2^ 22.286* ..0//12// 9^ 23. 999T\ .. 1//13// 54 24.889^ ..2//14// 71

PRACTICE.

53.

83

ALIQUOT PARTS.

qr.

i = 14 lb.

i = 71b. f = 41b.

i = 3i lb.

These Aliquot parts are given as examples. The pupil having a thorough knowledge of the Arithmetical Tables can easily find ali- '

f the various denominations in WEIGHTS AND MEASURES. (1) Find the price of 7 cwt, 2 qr. 7f lb. i

2qr. 1 qr. IGlb. 14 lb.

I =

i

To —

2ro. Iro. 32 po. 16 po.

£8 * 6*8,

! £8*6*8 & cwt. price of 1 cwt.

7ft. ill., ilb.	f 1 T*	1 cwt. 2qr. 71b.			7	cwt. qr. price of 7 * 0 / ... 0//2/ ... 0*0/ ... 0*0/ ... 0*0/	lb. /O '0 ^7 'OJ '04
58* 4* 0* 0* O//	•0 CO 0 0 0	/ 8^, ^. 4 J 7 / 9O 1 3 ' Z3> T*

£63 * 1 * 4£, T9T price of 7 * 2 * 7-f (2) Find the rent of 353 ac. 2 ro. 10 po. @ £2*7*6 & ac. £353 . . . rent of 353 ac. @ £1 2

			£706		)	;ac. ro. po.
	j	£1	88*	5	( i	. . 353 // 0 * 0 (c	§£2*7*6
2/6	£	5/	44*	2*	6J
2ro.	i	1 ac.	1*	3*	q	.... 0 * 2 * 0 .
lOpo.	i	2ro.	0*			J ... 0*0*10 .	.

£839 // 14 // 2i \ . . 353 // 2 // 10 @ £2 // 7 // 6

the number in the name in which the price of a unit is is small, as 7 cwt. in (1), we find its price by multiplication, ;md then take parts for the numbers in the lower names. But when the number is large, as 353 acres in (2), we may find its price liy taking p.-irts for the rate, and then finding the price of the num- bers in the lower names as in (1).

cwt 1. 1:5	qr. I). " 2// 14	^vcwt. @ £1*17* 4	9.	yd.	£	nl. ' 2	^•yd. @ £0*15* 4
	'/ 3*14	.. 1*	19*	8	10.	19*	3/	' 3	.. 1* 3* 4
	* 2*16	.. 1*	1*	10	11.	227*	2/	1	9/y 7 fi
•1. 122	/ 3* 9	J .. 2*	I//	71	12.	313*	3*	3|	q i •« o
	clwt trr.		gv oz
'15*12 6. 17//11//15	@ -£0* .. 1*		0 8	13.	ac. 17 /	ro. '2*	po. 20	@ £6*10* 0
	fhvt. gr.		^vlr		14.	43 /	'3//	35	.. 4*16* 8
7. !>	'15*23	@£46//		6	15.	365 /	'!//	19	.. 8*13* 4
K \\'	'14^/22;	.. .T7'/	2*	6	16.	49 »	3//	37i	.. 3* 7*11

84

PRACTICE.

53.

qr. bit. pk. ^ qr.

17. 7 // 4* 2 @ £2 // 8 *0

18. 11*7* 2 .. 2//16// 8

19. 0*7* 3 .. 3// 5// 4

20. 301*5* 1}.. 3// 3// 8

gal. pt. gi. ^ gal.

21. 3//5// 2 @ £0* 8* 0

22. 0//7// 3 .. 0*16* 0

23. 125 * 4// 1 .. 1*10* 3

24. 73 //5// 2f.. 1//12'/ 6

(3) Find the price of 195 cwt. 2 qr. 11 Ib. @ £4// 13//4 ^ cwt.

£1 V cwt. = 5/ V qr. = 1/3^7 Ib. = 2|d. V Ib.

Since 195 cwt. 2 qr. 11 Ib. £195*11*1 U

= 195 cwt. -f- 2 qr. -f 7 Ib. 5

-f- 4 Ib., the price at £1 aa* cwt. is = £195 + 2 X 51 + 1/3 +

4 X 2jd. = £195*11 »/llf. 6/8 £ £1 |

Having thus found the price £Q19//1'v/1

of 195 cwt. 2qr. lllb. @£l

^ cwt., we proceed to find it at the required rate.

(4) Find the price of 14 yd. 1 qr. 2 nl. @ 6/4 :

I/ ^ yd. = 3d. ^ qr. = f d. ^ nl.

The price of 14 yd. 1 qr. 2 nl. @ I/ yd. is = 14/ + 3d. + 2 X Jd. =

£977'/19// 65// 3//l

yd.

£1 ^ T. = I/ ^ cwt. = 3d. ^ qr. £1 ^ ac. =. 5/ ^ ro. r=

£1 ^oz. tr.ml/ ^ dwt.m 5/ ^ oz. tr.=3d. ^ dwt.=

T. cwt. qr. #• T.

25. 6 // 13 // 3 @ £5 // 7 // 6

26. 73//19// 1 .. 0//13//4

27. 17 // 3// 2 .. 4* 2*6

cwt. qr. Ib. $> cwt.

28.23* 3//14@£l // 8*4

29. 13 // I// 21 .. 2//10//0

30. 19 // 2//11 .. 0//14//6

Ib. oz. dwt. |v Ib.

31. 3* 7//11 @£5*11*8

32. 43 * 5//17 .. 10//13//4

33. 37 // 9 // 7 .. 3 * 17 // 4

oz. dwt. gr. fv oz.

34. 7* 13*17® £1*16*3

35. 6//17// 9 .. 0//17//3

36. 3// 5//13 .. 0*7*6

l/^ gal. =

ac. ro. po. ^ ac.

37. 13*2*30@£2* 3* 6

38. 14 '/I// 27 .. 3// 3* 4

39. 37//3//11 .. 5//10* 8

yd. qr. nl. ^ yd.

40. 7*2* 3@£0*17* 4

41. 8*3* 1 .. 2* 2* 6

42. 23//2// 2 .. 5// 6 // 8

qr. bu. pk. ty qr.

43. 7//3// 2@£1* 3// 1^

44. 6//5// 3 .. 4//10* 0

45. 15//3// 1 .. 3// 5* 10

gal. pt. gi. & gal.

46. 5*3*1@£0*16* 0

47. 17 //I// 2 .. 0//17// 4

48. 163*0*3.. 2* 2* 0

PRACTICE.

85

The following special methods are useful in A\oirdupois Weight,

£1»1 ^ cwt. = 5/3 sp» qr. = 1/35 & 1 Ib. = 2id. ^lb. Is. ^ cwt. = 3d. ^ qr. = fd. ^ 7 Ib. = f farthing ^ Ib.

(5) Find the price of 4 cwt. 2 (6) Find the price of 16 cwt. qr. 5 Ib. @ £6"6s. ^ cwt. 3 qr. 15 Ib. @ 4/6 per cwt.

4 X £1"1"0 = £4" 4« 0 16/ + 3 X 3d. = £0"16" 9 2 X 0"5"3 = 0"10" 6 2X|d.-j-ff.= 0" 0"

5 X 0"0"2i = 0" 0»llj

£4 "1

5} 6

£28"12"

cwt. qr. Ib.

49. 3 « 3 " 12

50. 27 « 1 w 18

£3» 7"

cwt. qr. Ib.

! £7"7 <$> cwt. 51. 6 " 2 " 14 @ 8/6 ip cwt. 5"5 ... 52. 7 " 1 » 20 .. 16/3 ...

We may now obtain a method for finding the price of 1 Ib. when that of 1 cwt. is expressed in £ and s.

(7) Find the price qp> Ib. @ £7 "5s. per cwt.

£7 " 5s. = £7 " 7s. — 2s. 7 x 2id. — 2 X ?f- = l/3£ } W Ib.

Having given the price of 1 cwt. in £. and s., to find that of 1 Ib., we multiply 2 id. by the number of £., and ff. by the differ- ence between the number of £. and s. ; and increase or diminish the former product by the latter, according as the number of s. is > or < than that of £.

Find the price ^ Ib. at the following rates <$> cwt. : — 53. £8"lls. | 54. £6"10s. | 55. £9"2s. | 56. £ll»ls. Id. ^ Ib. = 2/4 $* qr. = 9/4 ^ cwt. "b. = 7d. {p- qr. = 2/4 ^ cwt.

(9) What cost 13 cwt,

(8) Find the price of 3 cwt,

Iqr. 13lb. s. d. 3 X 9 " 4 1 X 2 » 4 13X0^1	@ 5$d. ^ Ib. £ s. d. = 1" 8" 0 = 0" 2" 4 Q/; J,, J	per Ib. ? s. d. 2 X 9 " 4 3X2-4
£1"11» 5 5} 7»10J 7»17» 1 £8" 4"lli

£

: 0"

: ^

£1*

£16^

@2fd.

s. d. L8" 8 7" 0

5" 8 13

cwt. qr. Ib.

57. 3 » 2 " 5 @ 2d. . 68. 17 . 3 " 19 .. 9;d.

59. 9 cwt. @ 7 id.

60. 26 cwt. .. 4|d.

86

PRACTICE.

Since the numbers 12 and 20 are employed in the Money of Account, we may easily obtain methods for finding the prices of 1 2 and 20 articles with some of their multiples, when that of a unit is given, which may be convenient in MENTAL COMPUTATION.

In finding the price of One Dozen, every penny in the price of the unit becomes a shilling. When the price of the unit is below 1/8, that of the dozen is below £1.

(1) 12 @ 2d. =

(2) 12 .. 3*0. ==

2/ 3/3

(3) 12 @

(4) 12 ..

= £4^12,6

Find the price of one dozen at the following rates ^ unit : —

1. 50. 3. 10iO. 5. 1/3 7. 2/8

2. 7d. 4. 9^0. 6. 1/7 f 8. 3/5$

In finding the price of Two Dozens, every penny in the price of the unit becomes & florin. When the price of the unit is below 10d., that of the dozen is below £1.

(5) 24 @ 4d. == 8/

(6) 24 .. 5|d. = 11/6

(7) 24 @ 1/5 = £l»14s.

(8) 24 .. 2/3$ = £2»14»6

Find the price of two dozens at the following rates ^ unit : —

9. 10.

3d. 90.

11.

12. 7fO.

13 14.

1/6

15. 16.

3/7 5/8|

In finding the price of Four Dozens, every farthing in the price of the unit becomes a shilling. When the price of the unit is below 5d., that of the four dozens is below £1.

(9) 48 @ 3d. = 12/ (10) 48 @ 1/5 J = £3»9s.

Find the price of four dozens at the following rates ^ unit : —

17. 20. 19. 1^0. 21. 7£0. I 23. l/6i

18. 40. 20. 3*0. 22. lOjd. | 24. 1/lOf

In finding the price of Eight Dozens, every farthing in the price of the unit becomes a florin. When the price of the unit is below 2£d., that of the eight dozens is below £1.

(11) 96 @ 20. = 16/. (12) 96 @ 7±d. = £2«18s.

Find the price of eight dozens at the following rates ^ unit : — 25. 2iO.; 26. l£0.; 27. 5f 0. ; 28. l/6f.

In finding the price of Any Number of Dozens, every penny in the price of the unit becomes as many shillings as there are dozens.

(13) 84 @ 40. = £l»8s. (14) 144 @ 7|0. = £4-13s.

Find the price of the following : — 29. 72 @ 3d. | 30. 108 @ 5|d. | 31. 144 @ 6|d. | 32. 60 @ 8*0.

In finding the price of One Score, every skill itif/ in the price of the unit becomes one pound, and every penny becomes 1/8.

PRACTICE.

87

54-. (15) 20 @ 3/6 = £3"10s. (16) 20 @ 7/5^ = £7»9*2. Find the price of one score at the following rates ^ unit : — 33. 7/; 34. 5/3; 35. 7*d.; 36. 2/4$.

In finding the price of Two Hundred and Forty units, every penny in the price of the unit becomes a pound.

(17) 240 @ 5d. = £5. (18) 240 @ 1/2 1 = £14" 15s.

Find the price of 240 units at the following rates yp unit : —

37. 8d.; 38. 7sd.; 39. 1/11 1; 40. 5/7£.

In finding the price of One Hundred units, every penny in the price of the unit becomes 8/4, and every shilling becomes £5.

(19) 100 @ 4d. = £1"13"4. (20) 100 @ 5£d. = £2»5»10. Find the price of 100 units at the following rates ^ unit: — 41. 7d.; 42. 9{d. ; 43.2/3; 44.19/11.

55. MISCELLANEOUS EXERCISES IN PRACTICE.

(1) A bankrupt whose debts are £3075 offers a composition of 1 1/3 ^ £. How much does he pay ? £3075 10/

1537»10 192» 3 <

£1 10/

(2) Find the weight of 124| bushels of wheat @ 63 Ib. sp bushel, cwt. qr. Ib.

1 cwt. 56 Ib.	124	> 1 » 14
62 7	/ 0 " 21 / 3 " 2

I 1 cwt. ^ bu., we

69 » 3 » 23|

1 Living found the weight of 124| bushels I take aliquot parts for 63 Ib., or 2 qr. 7 Ib. 1 . Find the price of 288 dressing-glasses @ 7/9 each. : nd the value of 840 stones of hay @ Tjd. each.

3. I'ind the price of 6 T. 15 cwt. oat manure @ £8»5^9 fj.1.

4. Wh.it does a chemical manufacturer receive for 5 1. Ib cwt. 2 qr. of sulphate of ammonia @ £19«10s. #• T. ?

5. Fin.l the price of 17 cwt. 3 qr. 14 Ib. of marine salt © 2/6

.V bankrupt whose debts are £2016 offers a composition of 14/;;3 -a, £. Find his effects.

7. How mudi is got for a silver epergne, weighing 7 Ib. 3 oz. 10 dv.1 i mi-hand @ -V ^ oz. ?

88 PRACTICE.

8. What does an ensign receive in 365 days @ 5/3 ^ day ?

9. A French sub-lieutenant receives 1350 francs ^ annum. To how much sterling is this equal, reckoning a franc at £5ls?

10. Express in sterling the annual salary of a field-marshal of France, which is = 30,000 francs.

11. Find the value of 300 Austrian florins @ 2/0£ each.

12. Find the value of 325 Rhenish florins @ 1/8 each.

13. Find the value of 360 Prussian dollars @ 2/1 Of each.

14. What is the value of a lac of 100,000 rupees @ 1/1 Oj each ?

15. What is the value in sterling of 5000 rubles @ 3/l£ each ?

16. To what sum in sterling are 1600 West Indian pistoles, each 16/, equivalent?

17. On Oct. 16, 1854, the stock of tea in London amounted to 47,522,000 Ib. Find the duty @ 2/1 ^ Ib.

18. A newspaper sold at 3^d. has a circulation of 3500. How much is received for each issue ?

19. Find the weight of 331 qr. 3 bu. of wheat @ 62 Ib. & bushel.

20. Find the weight of 692 qr. 5 bu. of oats @ 42 Ib. ^ bu.

21. Find the weight of 242 qr. 7 bu. of barley @ 54 Ib. ^ bu.

22. What is the weight of 1248£ bu. of wheat @ 2 qr. 4 Ib. #• bu. ?

23. Find the weight of 720| bu. of barley @ 1 qr. 26 Ib. y bu.

24. What is the weight of 200 bu. of oats @ 1 qr. 16 Ib. y bu. ?

25. Find the import duty on 14 cwt. 2 qr. 14 Ib. prunes @ 7/ ^cwt.

26. Find the import duty on 16 cwt. 3 qr. 21 Ib. Berlin wool @ 6d. y Ib.

27. Find the amount of excise duty charged in England in 1855 on 83,221,004 Ib. of hops @ 2d. y Ib.

28. Find the amount of excise duty charged in the United King- dom in 1855 on 166,776,234 Ib. of paper @ l£d. ^-Ib.

29. Find the whole pay of 34- majors of Dragoon Guards and Dragoons in the British Army for 31 days, @ 19/3 each ^ day.

30. What did a writer's clerk whose income was £110^- annum, pay for income tax in 1855, at the rate of ll£d. ^ £?

31. What did a minister, whose stipend in 1856 was £326"10"5£, pay for income tax @ 1/4 ^ £ ?

32. A bankrupt whose debts are £30,000 pays 8/3 Tr2 y £. How much does he pay?

33. In 1852, 590,767 oz. of gold coin were exported from the United Kingdom. Find the value @ £3"17/'10| y oz.

34. Reckoning the ducat at 4/2J, find the value refused by Shyloclc, when he says : —

" If every ducat in six thousand ducats

Were in six parts, and every part a ducat,

I would not draw them, I would have my bond."

PRACTICE.

89

55. (3) Find the gross rental of the following 5 farms : —

nc. ro. po. £ s.

I. 263 "0 38 @ £1"11 » 6 i 414

II. 457 " 0 39 .. 1« 5 » 0 571

III. 49 " 3 5 .. 2» 5 » 0 112

IV. 156 " 2 32 .. 1"15 » 0 274 V. 146 » 1 39 .. 1-13 »4 244

£1616

6 1*

35. Find the amount of a minister's stipend:— 30 qr. 7 bu. 0575 pk. barley @ 39/6 y qr. ; 12 qr. 2 bu. 3575 pk. oats @ 24/10J f qr. ; 40 bolls oatmeal @ 18/10 ; and £48«6»10|.

36. In the Edinburgh grain market, on 52 Wednesdays ending Oct. 22, 1856: — 42,915 qr. wheat were sold at an average price of 73/7 ; 42,206 qr. barley @ 42/1 ; 44,558 qr. oats @ 31/5. Find the amount.

37. Find the value of the average annual agricultural produce of a parish :— 1386 qr. wheat @ 65/1 ; 1350 qr. barley @ 40/6 ; 2314 qr. oats @ 29/9; 82,500 stones hay @ 7$d.; 204£ acres turnips @ £12'-2»6; 204$ acres potatoes @ £13"! "8.

38. Find the rental of an estate containing 4 farms :— 375 ac. 2 ro. 30 po. @ £3"12"6 y ac.; 432 ac. 1 ro. 20 po. @ £3"2"6 v ac. ; 280 ac. 3 ro. 25 po. @ £2«12«6 y ac. ; 413 ac. 0 ro. 15 po. @ £2»17 y ac.

(4) Edinburgh,

Mrs Jones, Sept. 14, 1857.

Bot of Adam Coburg, General Draper, s. d. £ s. d,

5 pieces, each 46 yd. merino @ 4." 3 8 .. .. 80 yd. cotton .. 0 " 4

3^ .. .. 54yd. linen .. 2«9

f yd.

48 12

£86

Mr James White, Grocer, Perth,

To Price & Co., Wholesale Merchants, Glasgow.

£ s. d.

1857. June 26.

Sept. 11.

In writing out the following Accounts, supply Names and Dak*. 39. 286 loaves @ 7*d. ; 140 loaves @ 6Jd. ; 89 fancy ^loaves @ 8d. ; 147 doz. biscuit ® 3d. per doz.; 176 Ib. flour @ 2Jd-

5 chests congou, each 2 qr.l lib. @ 3"8 fib. 3 hhd. sugar, each 13 cwt. 2 qr. .. 39»4 .. cwt. 3 cwt. 1 qr. 14 Ib. coffee .. 49"6 .. cwt. 14 cwt. 2 qr. 3 Ib. cheese .. 0"5£.. ID.

61 79 8 37 ;i86

8 13 7 5 14

4 0 Of 8!

90 PRACTICE.

55. 40. 648 silk mantles @ 14/10|; 420 richly trimmed mantles @ 45/; 600 yd. satin @ 8/10 J; 252 silk velvet mantles @ 71/9 ; 140 Paisley shawls @ 47/6 ; 246 foreign shawls @ 66/8.

41. 900 yd. moleskin @ 1/2J; 500 yd. plaiding @ 1/4; 250 yd. flannel @ 1/5; 600 yd. gingham @ 4f d. ; 1800 yd. unbleached cotton @ 3£d. ; 200 yd. twilled linen @ 1/5 ; 80 yd. pilot cloth @ 6/5£; 200 yd. pack sheeting @ 5£d.

42. 348 squares of Windsor soap @ 5£d. q> square; 440 doz. squares of honey soap @ 10/6 ^ doz. ; 200 bottles marrow oil @ 1 If d. ; 288£ pints castor oil@ 1/2 ; 350 pots polishing paste @ 5|d. ;

1 cwt. 2 qr. 7 Ib. starch @ 6£d. f Ib.

43. 740| Ib. coffee, No. I., @ I/; 370£ Ib. coffee, No. II., @ 1/2 ; 561J Ib. coffee, No. III., @ 1/4; 311 Ib. coffee, No. IV., @ 1/8.

44. 4965 qr. wheat @ 41/4; 236£ qr. barley @ 39/2; 483| qr. oats @ 26/1 ; 146^ qr. beans @ 39/5.

45. 14 pieces, each 37£ yd. @ 10/5 ^ yd.; 11 pieces, each 53£ yd. @ 12/4 f yd. ; 19 pieces, each 44| yd. @ 13/8£ ^ yd. ; 23 pieces, each 59£ yd., @ 16/7J ? yd.

46. 124 qr. 7 bu. wheat @ 55/5 f qr. ; 88 qr. 4 bu. barley @ 45/3 w qr. ; 138 qr. 3 bu. oats @ 23/8 y qr. ; 181 qr. 5 bu. beans @ 40/8 ^ qr.

47. 6 chests congou, each 2 qr. 17 Ib. @ 3/9 f Ib. ; 13 hhd. brown sugar, each 13 cwt. 1 qr. 18 Ib. @ 36/4 ^ cwt. ; 3 casks molasses, each 7 cwt. 2 qr. 14 Ib. @ 13/9 ^ cwt.

48. 14 cwt. 2 qr. 14 Ib. Cheshire cheese @ 50/ y cwt. ; 17 cwt. 3 qr. 14 Ib. Wiltshire @ 40/ $* cwt. ; 23 cwt. 1 qr. 18 Ib. Gouda @ 28/^cwt. ; 15 cwt. 2 qr. 13 Ib. American @ 35/ ^ cwt.; 27 cwt. 3 qr. 16 Ib. Carlow butter @ 77/ f cwt.; 39 cwt. 1 qr. 14 Ib. Waterford @ 72/ ; 47 cwt. 2 qr. 20 Ib. Dutch @ 84/ #• cwt. ; 23 cwt.

2 qr. 7 Ib. Limerick @ 66/8 ^ cwt.

56. ALLOWANCES ON GOODS.

IN selling goods by weight, an ALLOWANCE is made for the box or package containing them.

The weight of any commodity, with that of the box or package containing it, is termed Gross Weight ; the weight of the box, Tare; and the weight of the commodity, Net Weight.

If a chest of tea weighs 80 Ib., and the empty chest 16 Ib., the Gross Weight is 80 Ib., the Tare 16 Ib., and the Net Weight 64 Ib.

Draft is an allowance given to a retailer to enable him to tarn the scale in selling a commodity in small quantities.

Gross Weight.	Tare.
cwt.	qr. lb.	qr. lb.
0 //	3 //10	0 // 17
0 //	3 // 7	0 // 16
0 //	3 // 5	0 // 14
0 //	3 // 4	0 // 15
3 //	0 //26	2 // 6
0 //	2 // 6

ALLOWANCES ON GOODS. 91

56. A wholesale merchant in selling a chest of tea may deduct 1 lb.

The Commercial Allowances Tret and Cloff are now obsolete Cloff was similar to Draft. Tret was an allowance given on goods liable to waste.

(1) Find the net weight of 4 chests of tea, of which the gross weight and tare are respectively as follow : —

I.

II.

III.

IV.

2 // 2 // 20 Net Weight.

1. How much honey is sold, when in placing a jug weighing 7f oz. in one scale, weights amounting to 3 lb. 3f oz. are placed in the other ?

2. A railway truck weighing IT. 16 cwt. 3 qr., when loaded witli wool, weighs 8 T. 11 cwt. 1 qr. What is the weight of the wool?

3. A two-horse cart, weighing 13 cwt. 2 qr. 21 lb., when loaded with compost, weighs on the machine of a toll-bar 2 T. 1 cwt. 1 qr. 7 lb. What is the weight of the compost?

4. Find the net weight of a barrel of flour; gross weight, 1 cwt. 3 qr. 10 lb.; tare, 12 lb.

5. Find the net weight of 12 drums of Turkey figs ; gross weight, 24 lb. 8 oz. ; tare, f lb. each.

Find the net weight of 3 tierces of coffee, of which the gross • •wt. 2 qr. 9 lb. each, and the tare 2 qr. 17 lb. each.

7. Find the weight of coal brought up by a train of 20 trucks depot, the average of each truck being 10 T. 17 cwt. 2 qr.

gross, and 3 T. 0 cwt. 3 qr. tare.

8. Find the net weight of 3 hogsheads of sugar, of which the gross weight and the tare are as follow:

I. 13 cwt. 2 qr. 14 lb. gross; tare, 1 cwt. 1 qr. II. 12 cwt. 1 qr. 13 lb. gross f tare, 1 cwt. 20 lb. III. 13 cwt. 1 qr. 20 lb. gross; tare, 1 cwt. 1 qr. 7 lb. Find the net weight of 3 tierces of coffee, of which the gross ht is respectively 5 cwt. 2 qr. 13 lb. ; 4 cwt. 1 qr. 12 lb.; 6 cwt. 0 qr. 17 lb. ; and the average tare 2 qr. 7 lb. $>• tierce.

92 ALLOWANCES ON GOODS.

(2) Find the net weight of 9 bales of wool, each 3 cwt. 3 qr. 14 Ib. gross ; draft, 2 Ib. ^ bale ; tare, 16 Ib. V cwt.

cwt. qr. Ib.

3 // 3 // 14 Gross wt. of 1 bale 0 // 0 // 2 Draft "

12 Draft Suttle 9

16 Ib. 4 1 cwt.

34 // 2 // 24 * » 9 bales 4 // 3 // 23} Tare "

29 // 3 // 0± Net weight

10. Find the net weight of 231 cwt. 2 qr. 3 Ib. gross; tare, 14 Ib. w cwt.

11. Find the net weight of 200 cwt. 1 qr. 4 Ib. gross; tare, 20 Ib. f cwt.

12. Find the net weight of 8 chests, each 1 cwt. 2 qr. 14 Ib. gross ; tare, 1 6 Ib. ^ cwt.

13. Find the net weight of 29 chests, each 1 cwt. 1 qr. 7 Ib. gross ; tare, 12 Ib. ^ cwt.

14. Find the net weight of five half- chests of tea, tare being 20 Ib. y cwt., and gross weight respectively, I qr. 19 Ib. ; 1 qr. 18 Ib. ; 1 qr. 20 Ib. ; 1 qr. 21 Ib. ; 1 qr. 16 Ib.

15. Find the net weight of 4 chests tea, weighing respectively 75 Ib., 84 Ib., 63 Ib., 83 Ib. ; draft, 1 Ib. ^ chest; tare, respectively 13 Ib., 17 Ib., 14 Ib., 15 Ib.

16. Find the net weight of 20 casks madder, average gross weight of each cask being 15 cwt. 2 qr. 14 Ib. ; draft, 5 Ib. ^ cask ; tare, 17£ Ib. ^ cwt.

SIMPLE PEOPOBTION.

IN comparing two numbers, by finding how many times the one is as large as the other, the quotient obtained expresses the relation or RATIO of the dividend to the divisor ; thus, the ratio of 16 to 8 is y5 ; of 14 to 5, V 5 of 2 to 9, f .

In expressing the ratio of two numbers, as of 16 and 8, we write it thus, 16:8. The first term, 16, is called the Ante- cedent, and the second, 8, the Consequent.

Four numbers are said to be Proportional when the ratio of the first to the second is equal to the ratio of the third to the fourth. On examining the four numbers, 14, 8, 35, 20, we find V = f & or 14 : 8 = 35 : 20, and say 14 is to 8 as 35 is to 20, which we write as follows— 14 : 8 : : 35 : 20.

SIMPLE PROPORTION. 93

Since V = it, -H-*? = |ff, and U X 20 = 8 X 35. When four numbers are proportional, the Product of the Means is = the Product of the Extremes.

According to the arithmetical interpretation of Definition of Proportionality in Euclid (Book V. Def. 5), four numbers are pro- portional when the first, or a multiple of the first, contains the second as often as the third, or a like multiple of the third, contains the fourth.

Let us take 8, 2, 28, 7 ; 8 contains 2 four times, and 28 contains 7 four times ; hence 8 : 2 : : 28 : 7.

Again, take 27, 48, 63, 112 ; sixteen times 27 = nine times 48, and sixteen times 63 = nine times 112 ; hence 27 : 48 : : 63 : 112.

In SIMPLE PROPORTION, we are required to find a number to which a given number may have a given ratio.

t Find a number to which 56 may have the ratio of 24 to 63.

Let x be the required number, then 24 : 63 : : 56 : x ; and

since the product of the means is = the product of the ex-

tremes, 24 times the required number, or 24# = 63 X 56,

therefore the required number, x = — ^ — = 147.

The fourth term in a proportion is termed the Fourth Pro- portional to the other three. We have seen it is obtained by multiplying the second term by the third, and dividing the product by the first.

(1) Find the fourth proportional to 21, 30, and 28. 21 : 30 : : 28 : x

10 4

, =^=40.

-*"

We may cancel the common factors of the first term with

those of the second or the third.

Find fourth proportionals to the following numbers :

1. 6, 14, 12

2. 8, 24, 5

3. 7, 18, 21

4. 3-6, 4-2, 6-6

5. -27, 11-7, 2-1

6. 15-3, 2-89, -171

(2) Find the fourth proportional to 5j, 9$, and 5J : 9| : : i : x V : V : : * : *

2 ± — 14

&• — T*

94 SIMPLE PROPORTION.

57. Find the fourth proportionals to the following numbers :

7. 3i, 5J, 8,V 9. i, |, A

8. 3f, 6f, 1'Ji 10. i£, 7|, «

When there are three numbers, of which the first is to the second as the second is to the third, the third is termed the Third Proportional to the first and the second, and the second is the Mean Proportional between the first and the third.

(3) Find the third proportional to 9 and ll£. 9 : 11J : : llj : x

5

rf — 45v 45 * Q — 45 X-4&- — 2 2 5 — 1 4 , - * X , — * - 4X4X9- — ^ - 14T«

14-jjg. is the third proportional to 9 and llj, and 11 J is a mean proportional between 9 and 14T]7.

Find the third proportionals to

11. 9, 15 I 13. 5,

12. 49, 56 ! 14. 2-7

58. (!) K 27 cwt. sugar cost £51, what cost 63 cwt. ?

cwt.	£
27 ...	. . 51
63 ...	. . x
cwt. cwt.	£ £
27 : 63 :	: 51 : x
7	17
-33- >	C~^~ 4?11Q

-3-

The greater the quantity of sugar, the greater will be the price. Since we multiply the third term by the second and divide by the first, in order to obtain the fourth term greater than the third, the second must be greater than the first. Having stated £51 in the third term, we place 63 cwt. in the second and 27 cwt. in the first.

Having stated the number which is of the same kind, or is homogeneous to that which is required, we place the greater, or the less of the other two homogeneous numbers, in the second term, according as the fourth term should be greater or less than the third.

The following method may sometimes be adopted : Since 27 cwt. cost £51 1 cwt. costs £§}

and 63 cwt. cost je^LXJL1 = £119.

SIMPLE PROPORTION. 95

58. (2) If 39 men can do a work in 168 days, in how many days can 72 men do it?

men. days.

39 168

72 x

men. men. days. days.

7'2 : 3<J : : 168 : x

7 13

# = t—^— =91 days.

-3-

The greater the number of men, the less will be the time. Having stated 168 days in the third term, we place 39, the less number of men, in the second term, and 72, the greater number, in the first.

now give the following method:

Since 39 men can do a work in 168 days

1 man in 39 X 168 days

and 72 men in 89 *2168 = 91 days.

In (1), since the quantity increases as the price increases, the one ! to vary Directly as the other. In (2), since the number of workmen increases as the number of days decreases, the one is said to vary Inversely as the other. The former is an example of Direct Proportion, the latter of Inverse Proportion. In Direct Proportion, the term connected with the fourth is always placed in the second term. In Inverse Proportion, the term connected with the fourth is always placed in t\\Q first term.

Every question in Proportion admits of four variations.

(1)

I. If 27 cwt. cost £51, what cost 63 cwt. ? II. If G3 cwt. cost £119, what cost 27 cwt.? III. If 27 cwt. cost £51, how many cwt. may be had for £119 ? I V. If 63 cwt. cost £119, how many cwt. may be had for £51 ?

(2) I. If 39 men can do a work in 168 days, in how many days can 72

men do it ?

If. If 72 men can do a work in 91 days, in how many days can 39 m.-ii doit? . .

III. If 72 men can do a work in 91 days, how many men can do it in

168 days?

IV. If 39 men can do a work in 168 days, how many men can do it

in 91 days ?

1 . If 20 cwt. of rice cost £12, what cost 35 cwt. ?

2. If 12 tons of linseed cake may be had for £99, how many may iind for £231?

96 SIMPLE PROPORTION.

58* 3. A labourer earns £35 in 40 weeks, in what time will he earn £14?

4. An express train runs 40 miles in 64 minutes ; how far will it run in 24 minutes ?

5. If 110 acres of a West Indian plantation can produce 200 hogsheads of sugar, find the produce of 176 acres.

6. If 48 reapers cut 20 acres in a week, how many acres will 156 reapers cut in the same time ?

7. If 20 reapers can cut a field in 6 days, in what time will 30 reapers do it ?

8. If 42 men can do a work in 165 days, how many men will do it in 45 days ?

9. How many loaves at 8d. are equal in value to 240 loaves at 7d.?

10. A lends B £420 for 30 days ; how long must B lend A £360 to return the obligation ?

11. D lends E £525 for 64 days; what sum must E lend D for 48 days to return the favour ?

12. If 63 oxen can be grazed in a field for 16 days, how long may 84 oxen be grazed as well in it ?

13. The number of copies in the first edition of the Lady of the Lake, which was 2050, was to that in the second as 41 to 69. Find the number of copies in the second edition.

14. The length of the steamer track from Liverpool to Quebec, which is 2502 miles, is to that from Liverpool to Boston as 139 to 155. Find the length of the latter track.

(3) If 27 Ib. of coffee cost £l//12//3, what cost 38 Ib. ?

27 Ib £l//12//3

38 Ib x

Ib. Ib. £

27 : 38 : : 1//12//3 : x

We reduce the third term -77 . 0 ~~ £b,l2,,3to pence. Wecancel

the first and the third terms 114 32s.

by 9, and obtain the fourth 152 12 term in the same name as

that to which the third was 3)1634 387d.

reduced, viz. =^2 pence 12)544§d. ~43

= 544|d. = £2»5//4§. 2,6)4,6s._4d.

£2/75Mjd.

(4) If 25 yards of cloth cost £1//14//4J, what cost 35 yards ?

25yd £l//14//4£

35yd x

SIMPLE PROPORTION.

58. We are sometimes Jd- 3rd-

able to obtain the fourth zr : 55

term easily without re- 5 7 ducing the third term.

- —^ ,^ay be had for £8//l .. much sugar may be obtained for £3//15//6£?

4cwt.2qr.241b £8,/n//8

x .... £3//15//6i

£ £

cwt. qr. lb.

8,11,8 : 3*15*6J : : 4V2* 24 _20 20 j

We reduce the first 171s. 75s. 18 or

and the second terms to 19 10 OQ

farthings, and the third ^777: , — - _

term to lb. We cancel 2°60 d. 906 d. 148

the first and the second 4 4 38

terms by 5, and the first QOJA f QAOK f KOCMU

and the third by 16, — _*' £625 f. 528 lb.

and obtain the fourth 1648 725 33

,. "103 33

2175 2175

103)23925(232T%lb.

= 2 cwt. 0 qr. I

15. If a labourer in 37 weeks saves £5"10«2£, how much may he save in 50 weeks ?

16. Nine dozen loaves of refined sugar cost £48"7"6; what cost 73 loaves ?

17. If 41 lb. of raisins may be had for £l//17//7, how many lb. may be had for £5" 11 »4£ ?

18. If a steamer from Liverpool to Portland can make the pas- sage of 2750 miles in 11 da. 6 ho., in what time would the passage of 2980 miles from Liverpool to New York have likely been made ?

19. In April 1857, the duty on 3 qr. 5 lb. of tea was £6"6"1. Find the duty on 2 cwt. 1 qr. 20 lb. at the same time.

20. If a commercial traveller can drive between two towns 13 miles distant in 1 ho. 25 min., in what time can he drive 9 miles ?

21. In what time will an express train, which runs at the rate of 40 miles an hour, traverse a distance which a parliamentary train, at 24 miles an hour, runs in 3 ho. 15 min. ?

22. If 1 cwt. 1 qr. 25 lb. of Mocha coffee may be had for £7"4'-4£, for what may 2 cwt. 3 qr. 11 lb. be obtained ?

98 SIMPLE PROPORTION.

58» 23- If 4 cwt 3 V- 13 lb< of rice cost ^"^"lO, how much may be bought for £3"10"8?

24. The annual feu-duty of a site containing 10,588 square yards is £207 "11 1/2 J. How much is it y acre ?

25. If the penny loaf weighs 8 oz. avoir, when wheat is at 41/3 ^ quarter, what should it weigh when wheat is at 49/6 ?

26. If a sum is sufficient to pay the wages of 112 workmen who get 17/6 each, how many whose wages are 24/6 may be paid with the same sum?

27. A tierce of crushed sugar, containing 8 cwt. 3 qr. 14 lb., costs £27"10"3. What cost 7 tierces, each 7 cwt. 3 qr. 21 lb. ?

28. A box of pale soap, containing 2 cwt. 2 qr. 7 lb., costs £5"7"10£. Find the price of 7 boxes, each 2 cwt. 15 lb. ?

29. If 7 chests of tea, each 3 qr. 5 lb., cost £54»10"3, what cost 13 chests, each 3 qr. 13 lb.?

30. If 30 yards of iron-rail weigh 17 cwt. 1 qr. 18 lb., how far will 1600 tons reach?

(6) A bankrupt's debts are £535//10/>5, and his assets £321*6*3. How much can he pay ^ £1?

Debts. Assets.

£635*10*5 £321*6*3

£1 x

£535 // 10 // 5 : £1 : : £321 // 6 // 3 : x

Here we say as £535//10»5 of debt is to £1 of debt, so is £321»6»3 of assets to x of assets. We may however state and work as fol- lows : —

£535*10*5 : £321*6*3 : : £1 : x _20 20 20S. 10710 s. 6426 s. _JL2 12 128525 d. 77115 d. 25705 15423

20

25705)308460(12 s. 308460

When all the terms are homogeneous, we can state the proportion in two ways.

31. A tenant whose rent is £53"6"8 pays a tax of £l"13/>4. Find the tax on a rent of £36.

32. Find the rent of a tenant who pays 13/9 of poor's-rates, at the rate of 5Jd. f £.

SIMPLE PROPORTION. 99

33. A bankrupt's debts are £525" 10^6, and his assets £375*7, 6. How much can he pay f £ ?

bankrupt whose assets are £3420 pays a composition of 9/6 ^ £. Find the amount of his debts.

;>."). The tax paid on an income for the year ending April 5, 1856, was £19"4, at the rate of 1/4 ^ £. Find the income.

36. A clerk, after paying £2/<2"l of income-tax for the year end- ing April 5, 1854, found that he had £98«17//11 over. What was the rate #• £ ?

37. If the shadow of a staff 3 ft. 7 in. high measures 4 ft. 9 in., find the height of a steeple whose shadow is 158 ft. 4 in.

38. A farmer inadvertently used stone weights of 26 Ib. 8 oz. each for 28 Ib. What would 2 T. 13 cwt. of grain appear by these

rhts to be ?

39. A merchant used weights of 27 Ib. 12 oz. instead of 28 Ib. Find the true weight, which would appear 1 cwt. more by the false weights.

(7) If a person can walk 8£ miles in 2| hours, how far can he walk in 3^ hours ?

84 miles ..... 2i hours x miles ..... 3^ hours ho. ho. ml. ml.

2i : 3J : : 84 : x

.

40. If 26{| yards of cloth cost £8±f, what cost 111| yards?

41. If 39S38 cwt. of rice cost £18^, how much rice may be had for£3i|?

42. If 93| yards of damask cost J of £45f $, what cost 113T',

yards ?

43. If 54 men can do a work in 29| days, how many men will

. do it in 35iJ days?

44. For every 5J miles that A walks, B goes 4J miles. ^Ho long will B take to traverse a distance walked by A in 6

hours?

V train at the rate of 25| miles an hour traverses a distance in 3i hours. In what time will one at the rate of 24£ miles an

. A mile in * of 2| hours. In whattime

can he walk J of 1 { $ mile at the same rate ?

47. If ft of a vessel is worth £1393, what is the value of | of f. • of the vessel ?

100 SIMPLE PROPORTION.

58* (**) If 2'45 cwt. cost £22'75, how many cwt. may be had for £11 -7?

cwt. cwt.

£22-75 : £11-7 : : 2-45 : x 1-75 -9 49

"35 J_ 1

"5 5)6-3

1-26 cwt.

We have cancelled the first and the second terms by 13, the first and the third terms by 5. By multiplying the first and the third terms by 100, we clear the decimal points, and obtain two numbers which when cancelled by 7 are 5 and 7.

48. If 4-06 cwt. of rice cost £3*480, how much rice may be bought for £7-625?

49. A wall whose height is 9*1875 ft., casts a shadow of 10*5 ft. Find the length of the shadow of a steeple 93 -8 ft. high.

50. A bar of cast-iron, whose Specific Gravity is 7*207, weighs 80 Ib. Find the weight of a bar of cast-brass of the same size, whose s. G. is 8-100.

51. A jar of honey, whose s. o. is 1-450, weighs 4| Ib. Find the weight of olive-oil, whose s. a. is -908, contained in the same jar.

52. A block of Parian marble, whose s. G. is 2*560, weighs 2J tons. Find the weight of a block of Carrara marble of the same size, whose s. G. is 2*716.

We now give some MISCELLANEOUS EXERCISES, which include several important Applications of Proportion.

53. After paying 7d. qp* £ as income-tax for the year ending April 5, 1854, a gentleman had £971"16»1 over, on what had the tax been charged ?

£T £1« 0 n 0

0» 0*7

£0/49 // 5 : £971»16//1 : : £1 : x

54. A person paid 1 1 ^d. ^ £ as income-tax for the year ending April 5, 1856, and had £104"14»7 of net proceeds. Find his income.

55. The ratio of the diameter to the circumference of a circle was given by Peter Metius as 113 : 355. Find the circumference of a fly-wheel 10 ft. in diameter.

56. A cistern can be filled by a pipe running 3| gallons y minute in 54 minutes ; in what time can it be filled by another running 4£ gallons ^ minute ?

57. If 300 labourers can make an embankment in 48 days, in how many more days will 60 fewer do it?

SIMPLE PROPORTION. 101

58. 77 tailors can execute an order of regimental clothing in 30 days ; how many more must be engaged to fulfil the order 8 days sooner ?

59. If 33 masons can build a wall in 47 days ; and if, after work- ing 11 days, 15 leave ; in how many days after the 15 leave will it be finished ?

<6T 33 masons csmjinisJi the wall in 47 — 11 or 36 days. Since 15 masons have left, 18 remain.

masons. days.

Hence, 18 : 33 : : 36 : x = the number of days after the 15 have left.

60. If 17 men can do a work in 89 days ; and if, after working 33 days, 3 men leave ; in how many days in all will the work be done ?

61. If 64 men can perform a work in 57 days ; and if, after work- ing for 12 days, notice is sent to finish the work 9 days before the stipulated time ; how many additional men must be engaged ?

62. If 3 men can do as much as 4 youths; and if 13 men can do a work in 9 days ; in what time can 12 men and 8 youths do it?

youths. men.

£ZT 4 : 8 : : 3 : x = 6

6 + 12 = 18 men. men. days.

18 : 13 : : 9 : x

63. If 4 men can do as much as 7 youths ; and if 15 men can do a work in 16 days; in what time can 16 men and 14 youths doit?

64. Find the Horse Power of an engine which can raise 5 tons of coals per hour from a pit whose depth is 66 fathoms.

%3T The labour necessary to raise 1 lb. through 1 foot is termed the Unit of Work (U. W.) Watt found that a horse could do 33,000 units of work ^ minute. 1 H. P. = 33,000 U. W.

5 tons = 11200 lb. 66 fathoms = 396 ft. 11200 X 396 = 4435200 U. W. ^ ho. 6,0) 443520,0

73920 U. W. <p min. Qr ^^ = no. of H. P.

u. w. u. w. n. P. H. P. bu * ojuuu

33000 : 73920 : : I : x

65. Find the II. P. of an engine which can pump 4500 gallons of water ^ hour from a mine whose depth is 77 fathoms.

66. A watch, set on Saturday at 8»30 p. m., loses 1$ minute in 30 hours. What time does it show, next Thursday, at 4 p. m. .

From Saturday, 8»30 p. m., to next Thursday, 4 p. m., is hours.

ho ho. min. min. . .

30 : 11. "5$ : : }\ ' % = number of mm. before 4.

102 SIMPLE PROPORTION.

58* 67. A watch, set on Friday at 9 p. m., gains 45 seconds in 12 hours. What time does it show next Monday at 3 p. m. ?

68. A clock, set on Wednesday at 6 p. m., loses 2£ minutes daily ; what is the correct time when the clock strikes 6 next Satur- day morning ?

$3T 24 hours of correct time = 23 ho. 57| min. of dock's time, ho. min. ho. min. min.

23 " 57J : 60 : : 2| : x — number of min. after 6 by the correct time.

69. A sets out in a gig at the rate of 7 miles an hour. In £ hour, B follows at the rate of 10 miles an hour. In what time will B overtake A ?

flST f X 7 — 5£ miles, the distance between A and B when B starts.

10 — 7 = 3 miles, gained by B on A every hour, ml. ml. ho. ho.

3 : 5J : : 1 : x = the time in which A will be overtaken.

70. C starts from a hotel at 6 a. m., driving at the rate of 6J miles an hour. At 7 "45 a. m., D follows at the rate of 9| miles an hour. When will D overtake C ?

71. A luggage train starts at 5 » 45 a. m., at the rate of 20 miles an hour. A parliamentary train starts from the same station at 6"20 a. m., at 25 miles an hour. At 8"20 a. m., the luggage train shifts rails, and waits till the parliamentary train passes. When does the latter pass ?

72. When do the hour and the minute hands of a watch coincide between 8 and 9 o'clock ?

^ST The hour-hand moves through 5 minute-spaces while the minute-hand traverses 60. Since the minute-hand moves 12 times as fast as the hour-hand, the former in moving through 12 spaces traverses 11 spaces more than the hour-hand.

When the hour-hand is at VIII, the minute-hand being at XII is 40 minute-spaces behind it. Now if the minute-hand to gain 11 spaces must move through 12, how far must it move to gam 40 spaces ?

spaces. min.

1 1 : 12 : : 40 : x =. number of min. after 8.

*^T The pupil may construct a table showing all the times when the hour and the minute hands coincide.

73. Two trains start simultaneously from the opposite termini of a railway 100 miles long: one goes at the rate of 20 miles an hour, and the other at 25 miles an hour. When and where will they meet ?

<jgr The trains approach each other at the rate of 20 -J- 25 or 45 miles an hour.

SIMPLE PROPORTION. JQ3

ml- ml- h°- ho.

45 : 100 : : 1 : x = number of hours in which the trains meet.

ml. ml. ml. ml.

45 : 100 : : 25 : x = number of miles from one of the termini.

100 — x = number of miles from the other terminus. . The distance from Edinburgh to Berwick by the North British Railway is 58 miles. A train starts from Edinburgh at the same time as from Berwick ; the former at the rate of 24, and the latter at 30 miles $>• hour. When and where do they meet ?

7."). From Carlisle to Preston is 90 miles. A train leaves Car- lisle at 12 "15 a. m., at 40 miles f hour, and Preston at 2 a. m., at 36 miles f hour? When and where do they meet?

®T Find where the Carlisle train is when the Preston train starts, and then proceed as in the other examples.

COMPOUND PBOPOBTION.

59. "WE have seen that the ratio of one number to another may be expressed by a fraction, of which the antecedent is the numer- ator and the consequent the denominator. Thus, the ratio of 4 to 5 is = £, and the ratio of 6 to 7 is f . Since the com- pound fraction | of -f is = £$, we say that the ratio of 24 to 35 is compounded of the ratios of 4 to 5 and of 6 to 7. Hence, if one number is to another in the ratio of 24 to 35, it is in the ratio compounded of the ratios of 4 to 5 and of 6 to 7. Thus, since 24 : 35 : : 48 : 70, the ratio of 48 to 70 is compounded of the ratios of 4 to 5 and of 6 to 7. We write these numbers in the following form : —

* if}:: 48: 70.

45 = 4 of f = i£|.

48 X 5 X 7 __ 4 X 6 X 70 70 X 5 X 7 ~~ 5 X 7 X 70'

and 48 X 5 X 7 = 4 X 6 X 70. ; he Product of the Means is = the Product of the Extremes.

In COMPOUND PROPORTION we find a number to which a given number may have a ratio compounded of two or more ratios.

Find a number to which 72 may have a ratio compounded of the ratios 4 : 5 and 6 : 7.

104 COMPOUND PROPORTION.

59* Let x be the number,

• yO . /•£

Then since the product of the extremes is = the product of the means,

4X6X^ = 72X5X7

The required consequent is = its antecedent X the other con- sequents -=- the other antecedents.

(1) If 4 horses plough 45 acres in 10 days, in what time will 6 horses plough 81 acres?

Before stating, we may write the terms in two rows. This me- thod is particularly useful in writing down a question to dictation.

4 horses, 45 acres, 10 days 6 horses, 81 acres, x days.

Horses 6 : 4\ Acres 45 : 81 j :

27,0)324,0 (12

We follow the same method as in Simple Proportion ; thus 6 horses will take a less number of days than 4 horses ; hence 6 : 4. Again, 81 acres will require a greater number of days than 45 acres ; hence 45 : 81. We thus consider each pair of terms separately in refer- ence to the required number.

\Ve may work every question by resolving it into questions in Simple Proportion.

The foregoing question may be resolved as follows : —

I. If 4 horses plough 45 acres in 10 days, in what time will 6 horses plough the same number of acres ?

Horses. Days.

6 : 4 : : 10 : «

II. Now if 6 horses can plough 45 acres in ^y- days, in what time will the same number of horses plough 81 acres? Acres. Days.

45 : 81 : : «LXJ : x - "x*81 _ 12 dilys.

COMPOUND PROPORTION. 105

59. (2) If 21 reapers cut 3 ac. 3 ro. of corn in 4| days, in what time will 24 reapers of the same strength cut 16 ac. 1 ro. ?

21 reapers, 15 roods, \3 days 24 reapers, 65 roods, x days.

Reapers 24 : 21 Roods 15 : 65

13

) } : :

= VW = 17jft days.

5

In (1) the number of days is in the inverse ratio of the number of -, and in the direct ratio of the number of acres. In (2) the number of days is in the inverse ratio of the number of reapers, and in the direct ratio of the number of roods.

'nay illustrate (1) as follows: — 4 horses plough 45 acres in 10 days 1 horse ploughs do. in 10 X 4 days 6 horses plough do. in ^~ days do. plough 1 acre in ^|— days

do. plough 81 acres in ~£~^ days.

Similarly we may illustrate (2) or any other exercise in Com- iiid Proportion.

1 . If 3 families of 6 persons each consume 28 loaves in a week, how many will 9 families of 5 persons each consume in the same time?

^ This Question, and others similar ^ to it, may easily be worked by one statement in Simple Proportion. 2. A housekeeper having used 6 pots of jelly with 14 loaves each 12 slices, wishes to know how many will be used with 8

h 7 slices?

:;. If 13 bushels of oats serve 3 horses for 11 days, how many bushels will serve 7 horses for 12 days?

1 f G boys are boarded for 10 months for £270, for what ought i:; Ix-vs to he boarded for 7 months?

5. If 8 labourers earn £14«8 in 12 days, what will 17 labourer • •••mi in 5 days? , 0- r t

If 22,500 types are used in setting up 12 pages each 25 lines, how many types will be required in setting up 17 pages typr- nnd breadth each 31 lines?

106 COMPOUND PROPORTION.

59. 7. A family may live for 3 months in the country for £24" 10, what will be required to maintain them in town for 9 months, sup- posing £3 in the country to be equivalent to £4 in town ?

8. If a traveller walks 140 miles in 8 days walking 7 hours a- day, how many miles may he accomplish in 12 days walking 8 hours a-day ?

9. If 3 tailors make 5 vests in 1 1 hours, in what time will 1 1 tailors make 15 vests ?

10. If 64 yards of carpet, 3 qr. wide, cover the floor of 4 equal 'rooms ; how many yards of carpet, 1 yd. wide, will cover 3 of them ?

11. If the 4 Ib. loaf costs 8d. when wheat is @ G4/ ^ qr., find the weight of the penny loaf when wheat is @ 56/.

12. A bootmaker who employs 15 men fulfils an order of 25 dozen pairs of Wellington boots in 4 weeks, in what time may he accomplish an order of 45 pairs by employing 3 additional men ?

13. If 24 cakes can be made out of 3/ worth of oatmeal when meal is @ 18d. ^ pk., how many cakes can be made out of 10/3^ worth when meal is @ 1 3d. ?

14. Captain Basil Hall, in computing the time in which Sir Walter Scott might execute the MS. of Kenilwortli, introduces the following: — if 120 pages of 777 letters each may be written in 10 days, in what time would 3 volumes of 320 pages of 864 letters each be written ?

15. A railway company charges 18/ for the carnage of 9 cwt. 40 miles,

(1) What should be charged for carrying 10 cwt. 54 miles?

(2) What weight should be carried 27 miles for 54/?

(3) How far should 3 cwt. be carried for 15/?

16. If 7 compositors set up 15 sheets in 6 days,

(1) In how many days will 21 compositors set up 30 sheets?

(2) How many sheets will 27 compositors set up in 14 days ?

(3) How many compositors will set up 25 sheets in 7 days ?

17. If 36 labourers clear 513 yards for a railway in 6 days,

(1) How many will clear 3800 yd. in 10 days?

(2) How many yd. will be cleared by 156 labourers in 18 days ?

(3) In how many days will 16 labourers clear 190 yd. ?

18. If 4 masons build 27 yards of wall in 5 days working 9 hours a-day, in how many days will 32 masons build 81 yards of a similar wall working 10 hours a-day?

19. If 12 boys are boarded 10 months for £498, find the board of 18 boys for 9 months, supposing that the cost of boarding 4 of the former = that of 3 of the latter.

20. If £5 is sufficient to maintain 8 labourers for a fortnight

COMPOUND PROPORTION. 107

59* when corn is at 28/ qp> qr., how much will be required to maintain 6 labourers 29 days when corn is at 32/ ^ qr. ?

21. If 20 men, of whom the average strength is f of an ordinary man's strength, can load 81 trucks in 8 hours; in what time will 32 men, of the average strength of ^° of an ordinary man's strength, load 63 trucks ?

22. If 7 labourers mow 50 acres in 9 days of 8 hours each,

(1) IIow many acres will 14 labourers mow in 3 days of 6 ho. ?

(2) How many labourers will mow 25 acres in 18 days of 7 ho. ?

(3) In how many days of 9 hours each will 14 labourers mow

icres?

(4) By working how many hours a-day will 20 labourers mow

500 acres in 2 1 days ?

23. 8 men can dig a trench 200 yards long, 2 ft. broad, and 6 ft. deep, in 12 days,

(1) How many will dig another 160 yd. long, 3 ft. broad, 5 ft.

deep, in 6 days ?

(2) What length of trench will 7 men dig in 11 days, suppos-

ing it 4 ft. broad and 7 ft. deep ?

(3) What breadth of trench will 6 men dig in 8 days, sup-

posing it 50 yd. long and 6 ft. deep ?

(4) What depth of trench will- 12 men dig in 15 days, sup-

posing it 50 yd. long and 4 ft. broad ?

24 If 17 men cat 33/ worth of bread in a week when the 4 Ib. h at Bd., what value of bread will 9 men eat in 2 weeks when the 2 UK loafisat4Jd.?

2 -» 1 f a family by using 2 gas-burners ?i hours a-day pay £1-5 - when gas is @ 10/ F 1000 cub. ft., what will a family Mirners 4 hours a-day pay p quarter when gas is @ 7/6 v

. Tnfcldles, of which 8 weigh 1 Ib., serve 4 winter even- ings from 5 to 11 P. M. ; how many candles, of which 6 weigh 1 Ib., will serve 3 spring evenings from 7 to 11 P. M.?

27. If 330 slices, A inch thick, are obtained from 12 rounds of

beef, how many similar rounds will supply 495 slices, 1 inch thick ,

28 If the part representing land cut out of a map of a countiy

/l()sn miles in extent weigh 384 grains; find the extent ot

drawn on the scale of 56'

on that of 100 sq. ml. to the sq.

hound run while the hare runs 420 yards ?

108 COMPOUND PROPORTION.

59* 30. If the horse Flying Dutchman takes 10 strides while the horse Nonsuch takes 9, but if 6 strides of the former are equal to 5 of the latter, what distance will the latter run while the former runs 1200 yards?

31. If 6 bars of metal, 2 ft. long, 6 in. broad, and 3J in. thick, weigh 126 Ib. ; find the weight of 7 bars 3 ft. long, 41 in. broad, and 3 in. thick.

32. The weight of 35 cubic inches of gold, of which the Specific Gravity is 19'258, is 355-270 oz. troy; find the weight of 49 cubic inches of silver, of which the Specific Gravity is 10-474.

33. A slab of granite containing 3,ss cub. ft. weighs 541 Ib., find the weight of a piece of pumice stone containing If cub. ft., the s. o. of the former being to that of the latter as 175 to 61.

34. A contractor having engaged to lay ten miles of railway in 150 days, finds that 90 men have finished 3 miles in 80 days ; how many additional men must be engaged to finish it within the time ?

35. A stabler lays in 80 bushels of oats to feed 15 horses for 16 days ; at the end of 4 days he receives other 5 horses ; how many additional bushels will be required for the given time ?

36. The diameter of the Sun is 882,000 miles. His apparent diameter as seen from the Earth is 32' 1-8". Find the apparent diameter of a globe of fire as large as the Solar System, 5,700,000,000 miles in diameter, viewed at the distance of the nearest fixed star 206,265 times as distant as the Sun.

COMPUTATIONS made at a certain rate per hundred (per cen- tum) are termed PER-CENTAGES.

Per-centages are used in Commercial Arithmetic in finding Commission, Interest, &c. They are often employed in ques- tions of Statistics.

STATISTICS.

©©.STATISTICS treats of the numerical data of any subject.

Thus, if we examine the number of persons who pay Income Tax, the amount annually paid, &c., we are said to inquire into the Statistics of the Income Tax. Again, if a Table gives the amount of Tea annually imported and consumed in Great Britain, with the amount of duty paid, &c., it is said to furnish the Statistics of the Tea Trade.

The Statistics of a country treats of its population, rev- enue, and general resources.

STATISTICS. 109

6Ot (1) Of 93,498 births registered in Scotland in 1855, 47,872 were males. Find the per-centage.

93498 : 47872 : : 100 : x = 51-201 per cent.

1. Find the per-centage of alloy in sterling gold, of which 1 Ib. troy contains 1 oz. alloy.

2. In 1851, of 335,966 emigrants from the United Kingdom, 257,372 were Irish. How much per cent, was the latter number of the whole ?

3. In 1855, the produce of silver in the United Kingdom amounted to 561,300 oz., of which 4947 were from Scotland. Find the per- centage that the latter number was of the whole.

(2) The number of poor relieved in Scotland for the year " ending 14th May 1848 was 100,961 ; for 1849, 106,434 ;

and for 1850, 101,454. Find the increase per cent, from 1848 to 1849, and the decrease percent, from 1849 to 1850. 106434 100961

100961 : 5473 : : 100 : x = 54-2091 per cent, of

increase. 106434 101454

106434 : 4980 : : 100 : x = 46*7896 per cent, of

decrease.

4. The number of letters delivered in the United Kingdom in the year preceding Dec. 5, 1839, when penny postage was gener- ally introduced, was 82,470,596; and in 1840, 168,768,344. Find the increase per cent.

In 1854, the number of letters delivered in the United King- dom was 443,649,301 ; and in 1855, 456,216,176. Find the increase per cent. .

6. The total number of railway tickets issued in the United Kingdom in 1850 was 66,840,175; and in 1851, the year of the Great Exhibition, 78,969,623. Find the increase per cent.

7. The population of Ireland in 1841 was 8,175,124; andin!851, '.52,385. Find the decrease per cent.

(3) A sample of bone manure was found to contain 15-83 per cent, of sulphate of lime. Find the weight of the sulphate in 12 tons of manure.

Tons.

100 : 15-83 : : 12 : x

T.

x = '1583 X 12 = 1'9

110 STATISTICS.

6 Ot When the rate per cent, contains an approximate decimal, the result

can be obtained to a certain number of decimals only (see § 39). In some cases the required result is necessarily a whole number.

8. The number of representatives in the House of Commons is 658. Of this number, or even of 654, which was for some years the number of representatives, the per-centage for Scotland is 8'1. Find the number of the Scottish representatives.

9. A sample of turnip manure was found to contain 20'5 per cent, of sulphate of lime. Find the weight of the sulphate in 20 tons of manure.

10. The Queen's Remembrancer in Scotland has a salary of £1250 ^ annum. Find the salary of his chief clerk, which is 44 per cent, of his own.

(4) The Estimate for the Science and Art Department in Scotland, for the year ending 31st March 1856, was £1763. Find the estimate for the succeeding year, which gave an increase of 3*165 per cent.

100 : 103-165 : : £1763 : x = £1818*16 x — 1-03165 X 1763 = £1818-8

11. In 1855, the number of marriages registered in Scotland was 19,639. In 1856, the increase was at the rate of 4-318 y cent. Find the number of marriages in the latter year.

12. The population of Scotland in 1841 was 2,620,184. Find the population in 1851, which had increased at the rate of 10*2496 per cent.

13. The population of England and Wales in 1841 was 15,914,148. Find the population in 1851, which had increased at the rate of 12-65202 per cent.

(5) In 1855, the per-centage of deaths, amounting to 62,154, was 2-06884 of the estimated population. Find the estimated population.

2-06884 : 100 : : 62154 : x = 3004300.

3,004,300 is the reliable number obtained from the given number of decimal places. Had we taken the rate per cent, as 2-07, we would have obtained 3,000,000 merely. To obtain 3,004,290, the correctly estimated population, we require 6 decimals in the rate per cent.

In statistical computations we can reproduce all the places of whole numbers only when a sufficient number of decimals in the per-centage is given.

14. In 1856, when the number of acres in Scotland on which wheat was cultivated was 70,522 more than in 1855, the increase

STATISTICS. 1 1 1

6O.WAS at the rate of 36'8646 per cent. Find the number on which wheat was cultivated in 1855.

15. In Scotland, during the year ending May 14, 1855, the de- crease in the number of registered poor was 3217 from the former

t As the decrease was at the rate of 3-09992 per cent., find the number relieved during the year ending May 14, 1854.

(6) ^December 1856, the number of deaths in London was 14, GIG. This was an increase of 2*482 per cent, over the number of deaths in December 1855, in which the num- ber showed a decrease of 17-408 per cent, from that in December 1854. Find the number of deaths in December 1855 and in December 1854.

100 + 2-482 = 102-482 : 100 : : 14616 : x = 14262 100 — 17-408 = 82-592 : 100 : : 14262 : x = 17268

16. In 1851, the population of the United Kingdom, which was 27,674,352, had increased from 1841 at the rate of 73-361818 per cent. Find the population in 1841.

17. In 1812, the census of China in the seventeenth year of Kiaking amounted to 362 millions. This gave an increase of 8'7 per cent, since 1792, when a statement was made to Lord Macart- ney in the fifty-seventh year of Kienlung. Find the census in

18. In 1856, the number of deaths in England and Wales was 391,369; the decrease per cent, was 8-18103 from the previous year ; find the number in 1855.

COMMISSION AND BBOKEBAGE.

61* COMMISSION is a per-centage allowed to an agent for buying • or selling goods.

/ jt BROKERAGE is a per-centage allowed to a broker lor trans- ferring the right of property, or for assisting in the sale or purchase of goods. ,

A merchant often allows a per-centage to a customer when

he pays goods in Ready Money. This allowance, termed

JCOUNT, must be distinguished from Bank Discount (see

§ 64), in whose calculation the element of time is introduced.

1 Express the following per-centages as allowances

40; 33i; 25; 20; 12$; 5, percent.

112 COMMISSION AND BROKERAGE.

61* 2. Express the following per-centages as allowances ^' s. : 25 per cent. = ^ = J. J of Is. = 3d.

50; 33£; 16f ; 12 J per cent.

3. Express the following allowances as per-centages : 7/6 sp- £ = § = }. | of 100 = 37^ per cent. 10/; 5/; 2/6; I/; 8d. ; 6d. ^ £. 6d.; 4d. ; 3d.; 2d. ; IJd.; Id. ^ 5.

(1) Find the commission on £578*10'/6} @ 2} per cent.(°/0) £100 : £578*10*6} : : £2} : x

We therefore multiply the sum by the rate per cent., and divide by 100.

£578*10* 6} £578-526

2* 2}

8)1735*11* 6} 8)1735578

216*18*ll}f 216947

1157* 1* 0£ 1157052

£13,73*19*11}} £13-7401 = £13*14*91 20

14,79 £13*14*91 ,%

12

9,59

4

(2) Find the brokerage on £347*12//6 @ ^ °/0, and @ 8/4 c/0.

8) £347*12*6

,43* 9~^0| 5/ i £

20 3/4 i £

8,G9 8/8 i

£347*12* 6 86*18* 1£ 57*18* 9

£1,44*16*10-

12 20

8,28 p6

A __!?.

M5 =,*& = & 1T62

4

a

COMMISSION AND BROKERAGE. 113

61» Find the commission on :

4. £1260 . . . @ 5 %

5. 1274*17*8 .... 4 °/0

6. 375* 7*6 .... 2» 7o

7. 840*11*6 . . 5£ °/0

8. £375*15 .... @ 31 c

9. 509*10*6 4|c

10. 846*17*3 44 e

11. 723*11*6 4/6 c

12. £8467*10*6

13. 3176*13*4

Find the brokerage on :

14. £5260*12*6 . . @2/8°/0

15. 324* 3*4 . . .. 7/3 °/0

16. A commission agent sells goods to the amount of £536»10.. Find his commission @ 2i %.

17. A broker sells 50 shares of the Bank of Scotland, each £196. Find his brokerage @ i °/0.

18. A traveller for a sugar-house transacted business in a pro- vincial town to the following amount : — Raw sugar, £620 ; crushed sugar, £547"10; refined sugar, £320/45; molasses, £200»12»6. Find his commission @ 3 °/0.

19. An agent is allowed 5£ °/0 f°r selling goods and guarantee- ing the debts to his employer. His sales in a year amount to £15,375" 10»6, and his losses to £375"4»2. Find his income.

20. An agent is allowed 5£ % for selling goods and guaranteeing the debts. His sales amount to £13,756»10»8; his bad debts to £200" 15 ; and his doubtful debts, amounting to £500" 16, are valued @ 12/6 tf- £. Find his probable income.

21. An agent is allowed 5f °/0 for sales and risk of debts. Sales amount to £15,246"10; debts, amounting to £609"15, are valued @ 10/6 q? £. Find his probable income.

22. An invoice, containing an account of goods purchased, is sent by an agent to his employer. The price of goods is £409" 12"6; charges for packing, &c.,£7"12"9; commission on the whole, @ 2* °/0. Find the amount of the invoice.

23. An agent sent to his employer in St Vincent's an account of ' the sales of 56 tierces of sugar, each 8 cwt. 3 qr. 16 Ib. average net

weight, @ 62/^ cwt.; deducting commission @ 2$°/0j duty, 15/ V cwt. ; freight, &c., £180"12"9. Find the net proceeds.

24. An agent, who is offered a commission of 5J °/0 on amount of sales with risk of bad debts, or a commission of 3f °/0 on amount of sales without any risk, accepts the former. The sales amount to £8500, and the bad debts to £147 "15. How much has he gained or lost by his choice ?

114

INSUEANCE.

62«^NSURANCE *s a contract by which a company engages to in- demnify the value of property against loss.

The owner, whose property is insured, pays to the Insur- ance Company a certain per-centage or Premium on the sum insured, on which a Government Duty also is chargeable. The deed of contract between the Insurance Company and the owner of the property is termed the Policy of Insurance.

(1) Find the expense of insuring a cargo valued at £525* 12//6; premium, 2 guineas °/o j duty, 3/ °/0 5 commission to agent for effecting the insurance, | c/0-

•fcl

£525*12*6 4-	£525*12*6 £600 @ 3/ 70 = 18/
2	2,62*16*3 90 When the sum insured is not a multiple of £100, 12,56 the duty is charged on
£1051* 5*0 52*11*3
£11,03*16*3	12	the next greater multiple.
20	6,75
,76	4
12	§T
9,15
Premium, .		£11* 0*9
Commission		2*12 * 6|
Duty, . .		0*18 * 0

£14*11*3]

1. Find the premium on insuring an hospital for £3400 @ 3/6 °/0.

2. Find the premium on insuring farm stock for £530 @ 2/6 %.

3. Find the expense of insuring household property to the amount of £469" 10 @ 1/6 °/0; duty, 3/ %.

4. What was paid for insuring a house for £750 @ 2/6 % ; duty,

5. What was paid for insuring a cargo for £1250 @ £1"17"6 %; duty, 2/°/o?

6. An agent insures a cargo for £1370 @ 3 guineas % ; duty, 4/ °/0 ; commission oil the sum insured @ £ °/0. What is the total expense ?

7. A house factor insures four houses for £560, £940, £420, and £780 respectively, @ 1/6 °/0; duty, 3/ %. Find the expense.

8. Insured £3250 on a ship @ 3£ % ; duty, 4/ °/0 ; commission, \ °/o- Find the expense.

INSURANCE. U5

62. 9. An agent insures £4530 on a cargo® 4£ guineas %• duty 4/ % ; commission, £ °/0. Find the expense.

10. A ship, worth £5500, had a cargo worth £2670. All the expenses connected with insuring the ship and the cargo to their full value amounted to £4"1»8 °/0. How much was paid?

(2) Find what sum must be insured on property worth 3846, so that, in case of total loss, the whole, including the expense of insurance, may be recovered. The ex- pense is— premium, 3 gum. °/0 ; comm?, 1%; duty,4/0/0. £3// 3 10 4

£100 — £3*17 = £96//3 : £100 : : £3846 : £4000

The expense of insuring £4000 @ £3»17 8|0 = £154. The net sum thus recovered = £4000 — £154 = £3846. By insuring £4000 over all the expenses.

11. What sum must be insured to cover £1530, the expense of insuring being £4"7"6 "/.?

12. How much must be insured to cover £3890; premium, 2 guin. •/. ; commn ,£"/,; duty, 3/ °/o ?

1.;. I I«>\v much must be insured to cover £5005; premium, £3"! "/.; commn, J °|0; duty, 4/°/«?

14. What sum must be insured to cover £429, all the expenses connected with the insurance being £2 "10 °/0?

15. A cargo is worth £2442, and the expense of insuring it amounts to £2»17"6°/0. What must be insured to cover the value?

INTEBEST.

63JNTEREST is a per-centage charged for the loan of money.

The money lent is termed the Principal, and the sum of the Principal and the Interest is termed the Amount.

(1) Find the interest on £280//13//6 for 1 year @ 3J % & annum.

£100 : £280//13//6 : : £3* : x

£280*13*6 X 3£ ~~

The Interest on a sum for 1 year = Principal X Rate % .1- 100. For conciseness, we may use the Initials in a formula.

116 INTEREST.

63. i = P.XJ*

100

£280*13*6 £280-675 3£ -035

140* 6//9 1403375

842* 0//6 842025

£9,82// 7//3 £9-823625

20

16,47

12 £9*16*5J JJ = £9*16*5! nearly.

5,67 _4

2,68

Find the interest for 1 year on

1. £320 @ 3°/0

2. 647//15//6 .. 4 %

3. £802*11*6 @ 3i °/0 p* ann.

4. 772*16*9 .. 4} % ..

(2) Find the Int. on £567//5//6 for 7 yr. @ 4J. % V ann.

Prin. £100 : £567//5//6 ) ~. 1Int Yr. 1 : 7 } : : £4^ : *

£567//5//6 X 7 X 4J 100

Int. on a sum for a number of years = Principal X NO of Years X Rate % -f- 100.

y P X Y X R

100

£567// 5//6 £567-275

7 -315

3970//18//6 1701825

4J 8509125

1985* 9//3 178-691625

15883*14*0 178,69* 3//3

20 13,83 £178//13//9| { | = £178*13*10 nearly.

12

9,99

4

INTEREST.

Find the Int. on

8. £564,13H **,o .. o .. .. 3 v/0 9 361,14 fi

2"17,6.. 4 .. .. 2J% 10.' 874*18*8 .'.' 8 '.'. '.'. 2j«/. (3) Find the Int. on £321,15,4' for 2 yr. 5 mo. @ 3J °l°

5. £750 for 7 \

6. 216, 4,6 .. 5 }.:* :i°i':

£321,15,4^ 2

4 mo. £ 1 yr. 1 mo. I 4mo.

643,10,9 107, 5,U

£321-769 2.

12)1608845_ 134070~ 643538

-7//

2332824 100) £2527-226 w o^ii^ 25-27226

= £25,5,5^ Find the Int. on 11. £374,17,3 for5mo.@3r/0 14.£876,14,6^..2y.3m.@2»78

12. 769,13,3 .. 8

13. 467, 2//4J.. 5

.. 47

...8

15. 723//16,3|..3y.llm...3i%

16. 846,12,6 ..2y.7m...57

(4) Find the Int. on£220,4//7 from April 1 to Sept. 11,

(Qf * /o« Da.

, £4

Prin. £100 : £220,4,7 ) Da. 365 : 163 f

J

___ £220//4*7X 163X4, 36500 >

or, with a more convenient divisor,

__ £220//4//7 X

.

31

Of)

163

Int. on a sum for a number of days = Principal X N<? of Days X Double the Rate %-:- 73000.

* 5 °/0 = I/ tp £. 2} °/0 = 6d. ^ £.

In finding Interest at the following rates, we may first take it @ 5 or 2£ °/0, and then increase or diminish it as follows : —

6 e|0 = 5 e/. -f one fifth I 5A % = 5 °/0 -f- one tenth I 3 °/. = 2 £ °/. + one fifth 4 •/. = 5 °/, — one fifth | 4| °/8 = 5 % — one tenth | 2 °/. = 2j °/, — one fifth

118

INTEREST.

63.

T P X D X 2R»

73000

By treating 73,000 in the adjoining manner, we obtain 100,000.

A number, increased in the same man- ner, and divided by 100,000, produces the same quotient as when divided by 73,000.

We may work (4) by this method, known as the Third, Tenth, and Tenth rule.

In order to obtain the result within a farthing, we do not require the decimal in the product. The correction to be made at the end is to subtract 1 for every 10,001, or as 10,000 is sufficiently correct, we point off four figures, and subtract those to the left. This correction is, however, unnecessary, as in the example, when it does not affect the approximate value of the number of mils in the result obtained by dividing by 100,000. (See Decimal Coinage, § 43).

Find the Int. on

73000

i = 24333| of i = 2433£

100010

—10

100,000

£220-229

163

660687 1321374 220229 35897-327

287178-616 95726 9572 957

39,3433

—39 £3-93394 £3//18//8j

17. £420 for 73days@3% 18. 674 ..219 .. .. 3i°/0	20. £294*18 for231da.@3% 21. 360//17 .. 120 .. ..2J°/0
19. 547//10.. 88 .. ..	41%	22. 301//12//6.. 79 .. .. 4°/0
23. £720 from	May 29 to July 3 @ 4°/0
24. 330	June 8 .. Sep. 11 .. 3°/0
25. 690	March 10 .. May 29 .. 2i°/0
26. 2160	April 1 .. Sep. 11 .. 5%
27. 467//17//4 ..	April 16 .. June 8 .. 4l°/o
28. 164// 8//5.1 ..	Jan.	7 .. Mar. 29 .. 3f%
29. 876//14//6' ..	April 2, 1856, to Mar. 8, 1858,® 3i°/o
30. 561// 8//31 ..	July 26, 1855, .. Feb.27,1860, .. 3|%

* The following may easily be verified : —

Int. for 73 days =

Int. for 219 days =

P X 6R

1000 P X 8R

1000

INTEREST.

119

63.

(5) Borrowed £302*17*6 on April 1 ; Paid, £100 on April 29 ; £50//10 on June 8 ; and the Balance on September 11. Find the Interest due @ 3£ °/0.

Dates.	Dr.	Cr.	Balances.	Da.	Products.
April 1 April 29 June 8 Sept. 11	£302-875	£100-000 50-500 152-375	Dr. Dr. Dr.	£302-875 202-875 152-375	28 40 95	8480-500 8115-000 14475-625
31071-125 7

73000 ) 217497-875 £2"19"7 *44-9g

Sums borrowed are placed in the Debtor (Dr.] Column, and sums paid in the Creditor (Or.) Column.

31. Borrowed £600 on June 1 ; Paid, £200, July 1 ; £300, Aug. 1. Find Int. @ 5 °/0 due on Oct. 1.

32. Lent £950 on May 28; Received, £200, June 12; £300, July 4 ; Balance, Aug. 2. Find Int. @ 2 J °/0.

Sums lent are placed in Cr. column ; sums received in Dr. column.

33. Lent £500 on Candlemas (Feb. 2) ; Received, £300 on Whit- sunday (May 15) ; £100 on Lammas (Aug. 1). Find Int. @ 4J °/0 due on Martinmas (Nov. 11).

34. Borrowed £525 on Lady Day (March 25); Paid, £200 on Midsummer (June 24) ; £150 on Michaelmas (Sep. 29). Find Int. © 2 2 °/0 due on Christmas (Dec. 25).

A barrister having borrowed £500 at the beginning of Hilary Term on Jan. 11, paid £200 at the end of Easter Term on May 8; £125 at the end of Trinity Term on June 12 I ^d the Balance at the end of Michaelmas Term on Nov. 25. Find Int.

(5) 3i °/ .

36. Borrowed £300,15 on Jan. 1; ^°^t\°\U.^' ^ " one-fifth on the 1st of every second month (May, &c.) ti

Pt'. A^pulfadlced.aOOOonJan. ,1856, and.ceiv.d £500 on the 1st day of every quarter till the whole was pud. I

In38@B4oI°;wed£506»l2»6on June 12, 1858. Paid £|00,,19 on 4 15 ; £190»7»6 on Dec. 14; and £30-10 on Jan. 5, 1859. I Int. @ 3 7» due on April 5, 1859.

120

INTEREST.

63. (6-) Borrowed £3000 on Jan. 1, 1856; £500 on Feb. 1; £1200 on March 10; £300 on July 4. Paid the whole on Aug. 2. Find Int. @ 4°/0.

Dates.	Dr.	Cr.	Sums.	Da.	Products.
Jan. 1 Feb. 1 Mar. 10 July 4 Aug. 2	£3000 500 1200 300	£5000	Dr. Dr. Dr. Dr.	£3000 3500 4700 5000	31 38 116 29	93000 133000 545200 145000
916200 8

73000 ) 7329600

£100"8"1$.

II.

Dates.	Dr.	Da	Products.
Jan. 1 Feb. 1 Mar. 10 July 4 Aug. 2	£3000 500 1200 300	214 183 145 29	642000 91500 174000 8700
916200 8

73000)7329600

In the second me- thod, the days are reck- oned to the final date : — thus from Jan. 1 to Aug. 2 = 214 da.

39. A freshman at Cambridge borrows 30 guineas at the begin- ning of Michaelmas Term, Oct. 10, 1856 ; 25 guineas at the be- ginning of Lent Term, Jan. 13, 1857 ; £30 at the beginning of Easter Term, April 22, 1857. Find Int. @ 4 °/0 due at the end of Easter Term, July 10, 1857.

40. The inventor of a patent machine borrows £200 on Jan. 13, £100 on Apr. 3 ; £50 on May 6 ; £75 on JulylS. Find Int. @ 4 °/. due on Dec. 31.

41. An Oxonian receives 50 guineas in loan on the first day of Lent, Easter, Trinity, and Michaelmas Terms, viz. Jan. 14, Apr. 22, June 3, and Oct. 10, 1857, respectively. Find Int. @ 5°/0 due on Dec. 17, 1857.

42. Lent £509»12"6 on April 1, 1858; £392«15»6 on June 8; £96"8"6 on June 26; and £341»17»6 on Sep. 11. Find Int. @ 4£ °/0 due on Dec. 31, 1858.

INTEREST.

121

63. (7) Flnd the Interest to June 30, 1856, on the following Account- Current, allowing the Clydesdale Banking Com- pany 6 °/0, and Mr David Deans 3i °/0.

Dr. Clydesdale Banking Co. in Ace* with Mr David Deans Cr
1856.		£	s.	a.	1856.		£	8.	d.
Jan. 10 Apr. 1	To Cash . .	310 100	0 0	0 0	Feb. 14 May 12	By Cash . .	275 300	12 10	6 0
« 29	" "	50	If)	0	June 3	n it	50	13	6
May 17	" "	61	0	0	» 30	" Balance	96	3	8
June 24	" "	J200	0	0
" 30	» Interest	1 1	4	8		^^^
	~T22	19	8		722	19	8

The following shows the form of working Interest on the fore- going Account in the Deposit Ledger of the Bank.

Dates.	Dr.	Cr.	Balances.	Da.	Dr.Products	Cr.Products.
Jan. 10		310-000	Cr.	310-000	35		10850-000
Feb. 14	275-625		Cr.	34-375	47		1615-625
Apr. 1		100-000	Cr.	134-375	28		3762-500
" 29		50-750	Cr.	185-125	13		2406-625
May 12	300-500		Dr.	115-375	5	576-875
" 17		61-000	Dr.	54-375	17	924-375
June 3	50-675		Dr.	105-050	21	2206-050
» 24		200-000	Cr.	94-950	6		569-700
» 30						3707-300	19204-450
						12	• 7

44487-600134431-150

44487-600

73000)89943-550

Interest due by the Bank, . . . £1»4»7£ |J£f

The sums paid into the Bank are entered on the Dr. side of the pass-book, and in the Cr. column of the Bank Ledger ; thus, when Mr Deans pays £310 into the Bank, the statement in the pass-book Jlank Dr. to Mr Deans for £310, becomes in the Bank Ledger Mr Deans Cr. by Bank for £310. Similarly, sums drawn from the Bank are entered on the Cr. side of the pass-book, and in the Dr. column of the Bank Ledger.

The Interest on the Dr. sums in the Bank Ledger is calculated at the rate charged by the Bank, and that on the Cr. sums at the rate given by the Bank. In banks when the Dr. and Cr. Products are found, the Interest is obtained by tables ; here, however, we multiply the sum of the Dr. Products by double the rate charged, and that of the Cr. Products by double the rate given, and then divide the difference of the products by 73,000. The Interest being on the Cr. side of the Bank Ledger is entered on the Dr. side of the pass-book. When the account is balanced on June 30, we find that Mr Deans has £96*3-8 in the Clydesdale Bank.

122

INTEREST.

g3 43- Find tbe Int- to Dec- 31» '• 46- Find tte Int- to MaF 15»
1855, @ 3 °/0 on the following	1857, on the following account
account of the Savings' Bank	of the National Bank of Scotland
with Mr Colin Careful.	with Mr Purdie, allowing the
Dr. 1855.	1855. Cr.	Bank 6C/0, and Mr Purdie 3£ °/0.
June 8 £15 July 6 10	Aug. 7 £12 Oct. 23 8	Dr. 1857 Jan. 6. ..£700	1857. Cr. Feb. 10. ...£350
Sep. 5 20	Dec.10 10	Mar. 3. .. 120	Mar. 31. ... 850
Nov.13 10		May 1. .. 200	May 5. ... 315
t * T7"I_,1 ±1~~. T~.l. 4-n. T\~f* O1	n 11. .. 420 A1 T? ,-.A 4-V.rt	T«4- 4-n Tia,n O1

44. Find the Int. to Dec. 31, 1855, @ 2 °/0 on the following account of the Union Bank of Scotland with Mr John Jar-

Dr. 1855. Mar. 10 ...£200 May 29... 100 Oct. 30 ... 300

1855.

Cr.

Apr. 29 ... £50 Aug. 5 ... 200

45. Find the Int. to Dec. 31, 1856, @ 3£ °/0 on the following account of the Commercial Bank of Scotland with Mr James Worthy.

Dr. 1856. Julyl.£155"12»6 Aug.29. 74»15»0 Oct. 11.100»10"0 Nov.25. 31»17»6 Dec. 6. 42</12"6

1856.

Cr.

Aug.l.£63«12»0 Oct. 1. 24- 2»6 Nov.11.26* 5"0

47. Fnd the Int. to Dec. 31,

1855, on the following account of the British Linen Company with Mr D.iwson, allowing the Bank 5£°/0, and Mr Dawson 3°/0. Dr. 1855. 1855. Cr. Feb. 6. ...£800 Mar. 5. ...£300 Apr. 2. ... 600 May 31. ... 700 July 4. ... 250 Aug. 13. ... 850 Oct. 9. ... 700 Nov.30. ... 600 Dec. 4. ... 500

48. Find the Int. to June 30,

1856, on the following account of the Bank of Scotland with Mr Henderson, allowing the Bank 6 %, and Mr Henderson 3£ °/0.

Dr. 1856.

Jan. l....£1250

Feb.lL... 125

» 18.. Mar.29.. May 10..

78 231 366

1856. Feb. l.. Mar. 1..

n 15..

Apr. 15..

Cr.

.£875 . 565

. 200

. 310

(8) Deposited £200 in the Royal Bank of Scotland on April 10, 1855, when Interest was 3 °/0. On May 15, Int. fell to 2i °/0 ; on June 30, to 2 °/0 ; and on Oct. 8 it rose to 3 %. Find the Int. due on Nov. 7.

This is an example of finding the Interest on an Interest Receipt for a period during which the rate varies.

Dates.	Da.	Double Kate.	Products.
April 10 May 15 June 30 Oct. 8 Nov. 7	35 46 100 30	6 5 4 6	210 230 400 180
1020 200

73000)204000 £2»15»10}fi

INTEREST. 123

63. * iud the lnterest on the following Interest Receipts :—

49. £300 from Sep. 24 to Sep. 30, 1853, @ 2 °/0; and to Oct 15 @ 2 J °/0.

50. £500 from Aug. 1 to Oct. 7, 1856, @ 2£ °/0 ; to May 15 1857 @ 3J °/0 ; and to July 10 @ 4°/0.

51. £400 from April 1 to May 15, 1856, @ 3 %; to June 30 <§> 2J %, and to July 16 @ 2 °/0.

On examining the process in (8), we see that£200X1020=£200X102x2x5

73000 73000

= Interest on £200 for 102 da. @ 5 °/..

As rates of interest may be reduced to 5 °/0, we May 1 1 may consider the following plan on which Interest 12 .

Tables used in some banks have been constructed. 13 f

Let a sum be deposited on May 11, when Int. is 54 ' at 3 °/0. By writing J or '6 opposite May 12, add- ing -6 continuously till the rate changes, say on May

15, to 2£ °/0, and then adding jr- or '5 continuously, 17 .

we can at once see how many days @five °/0 will pro- 18 . duce the required interest.

Int. on £200 from May 11 to May 15 @ 3°/0, and to May 18 @ 2J0/, £200 X (4 X 6 -f- 3 X 5) _ £200 X 39 __ £200 X 3-9 X 10

73000 ~" 73000 ~" 73000

Int. on £200 @ 5 °/0 for 3'9 days as given in the table.

(9) What Principal will produce £210 of Interest in 5 years @4'/0?

Prin.

. x _ ^>£4L° = £1050.

1-2 1-8 2-4 2-9 3-4 3-9

For Years : P = ™g For Days: P =

52 . What principal will produce £384 of Interest in 6 years @ 4°/0 ? 53. What principal will produce £153 of Interest in 4J years @

4A °/0 ?

54°. Find the principal of which the Interest for 50 days @ 4 °/0

is £14" 12.

(10) What Principal will amount to £1260 in 5 years

Int. on £100 for 5 yr. @ 4°/0 Amount of £100

£12~1260 : : £100", =

124 INTEREST.

63* 55. Find the principal which in 4 A yrs. @ 4£ °/0 will amount to £962.

56. What principal will amount to £1017" 15 in 6£ yrs. @ 3 °/0?

57. What principal lent from March 10 to May 22 @ 5 % will amount to £712 "9"5?

(11) At what rate must £730 be lent for 95 days to amount to£739//10?

Prin. £730 : £100 ) Jjnt;ft, SGSOOXO* «. Da. 95 : 365 } ' ' *'9"10 : x = 736x95 = £5'

For r«,: R =

58. At what rate must £424 be lent for 2£ yrs. to produce £26" 10 of Interest ?

59. At what rate must £255" 10 be lent from April 1 to June 20, to produce £2 "16 of Interest?

(12) Lent £1825 @ 3°/OJ when will £10// 13 of Int. be due?

Prin. £1825: £100 \ J^' 36500 x 213 71 ~ Int. £3 :£lO//13f : : 365 : x== i825xeo =71

For Years: Y = For Days: D =

60. How long must £670 be lent to produce £134 @ 5 %.

61. How long must £91 "5 be lent to produce £2 of Int. @ 5 %?

62. Lent £511 on Jan. 1, 1856, @ 4J°/0, when will it amount to £517*13?

64' DISCOUNT.

DISCOUNT is a per-centage charged for the payment of money before it is due.

£200. London, March 15, 1858.

Three months after date, I promise to pay to Mr William Jones, or order, Two hundred pounds for value re- ceived. James Brown.

II.

£200. I London, March 15, 1858.

Three months after ^ date pay to me or order. Two hundred pounds for value | received.

To Mr James Brown. ^ William Jones.

DISCOUNT. 125

64* No' I' is the form of a Promissory Note, in which Mr James Brown

promises to pay £200 in 3 months after the given date.

No. II. is the form of an Inland Sill, drawn by Mr William Jones and sent to Mr James Brown, who on accepting it writes his name across the bill, and becomes bound to pay £200 in 3 months after the given date.

If Mr Jones who holds the note or the bill cashes it at the bank before it is due, as on April 19, the bank charges discount for ad- vancing the money.

The value of a bill when it is discounted is termed its Present Value. The value of a bill when it becomes due is termed its Future Value.

We may compare the Present Value and the Future Value to Ready Money and Credit Price. Goods which may be had on credit for a certain sum may be bought for less ready money. The Credit Price is the Present Value of the goods increased by Interest ; the Keady Money is the Future Value diminished by Discount. The Bank or Common Discount is the Interest on the Future Value of the bill.

The True Discount is the Interest on the Present Value of the bill.

The Present Value lent out when the bill is discounted amounts to the Future Value when the bill becomes due. The True Discount is the difference between the Future Value and the Present Value.

Common Discount (C. D.) = Int. on Future Value (F. V.)

e Discount (T. D.) = Int. on Present Value (P Hence, C. D. - T. D. = Int. on (F. V. - P. V.)

But, F. V. — P. V. = T. D. Hence, C. D. — T. D. = Int. on T. D.

The difference between the Common and the True Discount on a bill is = the Interest on the True Discount.

In Great Britain and Ireland, Three Days of Grace are given on all bills except those drawn " at sight," which are payable on presentation. When a bill, running for a number of months, and dated on the 31st of a month, becomes due in a month having fewer than 31 days, it is nominally due on the last day of the month, and legally due on the third of next month. Find the Common and the True Discount on a bill for £200 drawn March 15, 1858, at 3 months; discounted April 19, @ 4 °/0.

Nominally due, June 15

From April 19 to June 18 = 60 days.

Legally due, June 18

Amount or Future Value .

Common Discount, or Int. on £200 I . . I// 6//3£

for 60 days @4°/, .... Not Proceeds

126 DISCOUNT.

64.

Int. on £100 for 60 days

Future Value. Present Value.

£1004« : £200 : : £100 : x = £198*13*10j|fJ|

If we wish the answer correct within a farthing, we may express the fraction decimally, and use contracted division.

£100-6575 : £200 : : £100 : x = £198'693 = £198*13»10J.

Amount or Future Value £200

True Net Proceeds or Present Value .

True Discount £ I// 6//

Proof.

True Discount £l<r6

Int. on the True Disc, for 60 da. @ 4 70 0"0 Common Discount £l//6/'3£f£

Find the Common and the True Discount on the following bills :

Drawn. Discounted.

1. £300 . . . Mar. 25 for 3 months. . . April 16 @ 4 %

2. 600 ... June 23 // 3 * ... July 15 * 4

3. 275 ... Aug. 4 » 2 * ... Aug. 31 // 5

4. 360 ... Mar. 19 // 2 * ... April 10 // 3

5. 275 ... Mar. 11 // 3 // ... April 1 // 5

6. 720 . . . Oct. 19 // 2 // ... Nov. 10 // 3

7. 137//10. . Mar. 7 // 2 // ... April 3 » 5

8. 315//10. . July 10 // 4 * ... Sept. 11 // 3±

9. 480//12//6 Jan. 1 // 6 » ... Mar. 31 * 4

10. 157*15. . Nov. 30 // 3 » ... Dec. 30 * 3£

11. 68*15. . Oct. 31 * 4 // ... Jan. 25 // 5

12. 240//6//3 . Oct. 31 // 4 * ... Nov. 28 // 4

13. What sum will at the rate of 5 °/0 amount in a year to £75 ?

14. Find the present worth of £89 due in a year @ 5 %•

15. The price of goods, allowing 6 months' credit @ 5 °/0, is £4"8"10. Find the ready-money price.

16. What ready money is equivalent to 30/6 with 4 months' credit at 5 °/0 ?

17. The credit price of a newspaper per annum is £2 "4. Find the ready money payable in advance, taking true discount @ 10°/0.

18. What sum due in one day will produce Id. of true discount at5°/o?

19. What sum due in one day will produce I/ of common dis- count at 5 °/o ?

20. Find the common discount on a sum for 1 yr. @ 5 °/0, of which the true discount for the same time and rate is 5/5.

127

65. EQUATION OF PAYMENTS,

EQUATION OF PAYMENTS shows when a number of debts pay- e at different times may be adequately paid at once.

(1) Find the equated time for paying £90 due in 80 days £30 in 92 days, and £120 in 105 days.

£90 X 80 = 7200

30 X 92 = 2760

120 X 105 = 12600

240 )22560(94 days.

216 96 96

Suppose 94 days to be the equated time for the payment of the sums mentioned in (1), at the equated time interest will be charge- able on £90 for 14 days, and on £30 for 2 days. But if £120 which is paid 11 days before due be lent out at the same rate, the interest produced by £120 in 11 days would balance the interest chargeable on £90 and £30. 11 days must, hoAvever, elapse before this interest can be had, so that the True Discount and not the Interest on £120 should be = the interest chargeable on £90 and £30 at the equated time.

This approximate method, which is, however, sufficiently accurate for practical purposes, furnishes the correct answer to the following : —

Lent £90 for 80 days, £30 for 92 days, £120 for 105 days. In what time will their sum produce the same interest ? Int. on £90 for 80 days = Int. on £1 for 7200 days. // 30 // 92 // = // " 2760 // // 120 //105 // = " " 12600 // Total interest . . . = " " 22560 // = Int. on £240 for 94 days.

(2) £80 is payable to-day, £80 in 30 days, £90 in 40 days, £50 in 60 days. Find the equated time.

80 X 0 = 0

80 X 30 = 2400

90 X 40 = 3600 _50 X 60 = 3000 300 )9000(30 days. 9000

128 EQUATION OF PAYMENTS.

65, Exercises like (1) may also be performed somewhat simi- larly, thus : —

90 X 0 = 0 30 X 12 = 360

120 X 25 = 3000 Dayg. Da79.

240 )3360 (14 + 80 = 94

3360

Find the equated time approximately for paying the following sums due in the following number of days : —

1. £40 in 54 days, £80 in 36 days.

2. £30 in 58 days, £90 in 26 days.

3. £19 in 12 days, £22 in 24 days, £31 in 36 days.

4. £360 in 15 days, £140 in 20 days, £400 in 17 days.

5. i of a debt in 6 mo., TS5 in 7 mo., | in 8 mo., and the remain- der in 9 mo.

6. £ of a debt in 3 mo., f in 4 mo., and the remainder in 4J mo.

7. £190 payable to-day, £220 in 12 days, £310 in 24 days.

. 8. £95 payable 3 days ago, £110 in 9 days, £155 in 21 days. 9. | of a debt payable to-day, f in 48 days, and the remainder in 64 days.

Find the date on which the sum of the following debts can bo

adequately paid : —

. 10. £115 due on Mar. 2; £300 on Mar. 20; £600 on Mar. 21 ; £500 on Mar. 29.

11. £30 due on Apr. 1 ; £50 on Apr. 16; £30 on Apr. 26; £25 on May 1 ; and £15 on May 21.

12. £64 due on Apr. 1; £60 on Apr. 13; £50 on Apr. 18; £30 on Apr. 20 ; £28 on Apr. 24.

66. STOCKS,

STOCK is the money or capital belonging to any company. Government Stocks consist of the various loans granted to gov- ernment which form the National Debt. The different kinds of government stock are designated according to the annual rates of interest they yield ; thus the Three per cents yield £3 on every £100 of stock. The price of stock is estimated & £100; thus when the 3£ ^ cents are at 95, the value of £100 stock is £95 sterling.

(1) Find the annual income derived from £450 of stock in the 3J per cents.

STOCKS. 129

Stock. Income.

£100 : £450 : : £3£ : x = £14//12//6.

1 . Find the annual revenue derived from £56525 stock in the 3 per cents.

2. Find the annual income obtained from £10,871 "10 stock in the 3 per cents.

(2) Find the value of £1350 in the 3 per cents @ 82.

Stock. Sterling.

£100 : £1350 : : £82 : x = £1107.

A person on buying or selling stock per a stockbroker pays •J 3/0 of brokerage on the amount of stock.

(3) Find the buying price of £650 stock @ 80}..

Stock. Sterling.

£100 : £650 : : £(80} + i) ' x = • £525*13*9.

(4) Find the selling price of £825 stock @ 9l£.

Stock. Sterling.

£100 : £825 : : £(91J — i) : x = £750//15.

3. Find the value of £800 stock @ 95&.

4. Find the value of £450 stock @ 88.

Find the buying price of £375 stock in the- 3 per cents @ 70g, allowing brokerage @ & °/o.

6. What was paid for £650 stock in the 3£ per cents @ 91*, al- lowing brokerage @ J °/o?

7. Find the selling price of £330 stock in the 3 per cents . paying brokerage @ £ e/o.

8. How much was obtained for £570 stock in the 3} per cents @ 94 J, allowing brokerage @ & °/o ?

(5) Find the quantity of stock @ 92 equivalent to £828.

Sterling Stock.

£92 : £828 : : £100 : x = £900.

(6) How much stock may be bought for £361 @ 90|?

Sterling. Stock.

£(90£ + i) : £361 : : £10° : x ~

(7) How much stock of the 3 per cents @ 93| has realized £1235//17?

e^.i:«» Stock.

F2

130 STOCKS.

66* 9- Find tne quantity of stock @ 81 £ worth £655.

10. Find the quantity of stock @ 83 f worth £502 "10.

11. How much stock @ 93 1 may be bought for £750, allowing brokerage @ £ % ?

12. How much stock @ 81 i may be bought for £434, allowing brokerage @ £ °/0 ?

13. Find the quantity of stock @ 96 £ which will realize £576, allowing brokerage @ £ % ?

14. Find the quantity of stock @ 92 £ which will realize £739, allowing brokerage @ i %•

(8) Find the rate of interest obtained from capital invested in the 3 per cents @ 92*.

£92f : £100 : : £3 : x = 3}JJ °/0.

15. Find the rate of interest obtained when the 3£ per cents are @95£-

16. What rate of interest is obtained when the 3* per cents are

(9) How do the 3J per cents stand when they yield 40/0?

£4 : £3} : : £100 : x = £81J.

17. How do the 3 per cents stand when they yield 4 °/<, ?

18. How do the 3£ per cents stand when they yield 3£ °/o?

(10) Find the annual income derived from a capital of £617//10 invested in the 3 per cents @ 95.

£95 : £617*10 : : £3 : x = £19*10.

19. What income is derived from a capital of £6 11 "5 invested in the 3£ per cents @ 81 £ ?

20. Find the income derived from £308 invested in the 3£ per cents @ 82.

(11) What sum must be invested in the 3£ per cents @ 85 to produce £24//10 of annual income?

£3£ : £24i : : £85 : x = £595.

21. What sum must be invested in the 3| per cents @ 84 £ to produce an income of £50 ?

22. How much must be irvested in the 3£ per cents @ 92 £ to produce an income of £504 ?

23. A legacy of £2000, reduced by a duty of 3 '/«, has been invested in the 3£ per cents @ 97 1. Find the amount of the an- nually derived income.

24. Bought £300 stock @ 90£, and sold it @ 95i ; what was

STOCKS. 131

^gained, allowing | °/8 for brokerage on both the buying and the selling price ?

25. When the 3 per cents are @ 89, at what rate must the 3$ per cents stand to produce the same rate of interest?

26. Find the difference in the rate of interest between the 3 per cents @ 90 and the 3£ per cents @ 98.

27. A person buys £800 stock @ 91, and sells out @ 93 J. What does he gain, allowing £ */0 for brokerage on the buying and the gelling price ?

28. Invested £1380 in stock @ 91 1, and sold out @ 90|. How much was lost, reckoning the usual brokerage on the buying and the selling price ?

67. PKOFIT AND LOSS.

IN PROFIT AND Loss we consider the difference between the Buying and the Selling prices of commodities.

The Bwiing Price or Prime Cost (p. c.) is the sum at which goods are bought ; the Selling Price (s. P.) is that at which thev are sold. ...

the difference between the buying and selling prices is termed Gain or Loss, according as the Selling Price is greater or less than the Prime Cost.

(1) How much is gained by selling 234 yards of cloth @ G/5J, bought @4/3itf- yd.?

6 , 5 ' S P & yd. 234 yd. @ 2/2 yjp.C., " £ £25,7 Total Gain.

2//2~~ G. " "

(2) How much is lost by selling 12 cwt 3 qr 16 Ib. sugar @ 4td. V Ib., bought @ £2,4,4 V cwt. ?

4,d.rlb.=£2:2 Sp.Pctcwt. li^^SS

-Si Loss, * Total Loss.

. What is gained by selling 367 yards of cloth @ 7/9, bought @ ^w Lch is gained

fisswa

bought @£2"4-4^ cwt?

Lch is gained by selling 3 cwt 1 V. of cheese @ 6,d.

132 PROFIT AND LOSS.

C 4. Find the loss on 364 qr. of wheat, bought @ 65/6 f qr., and sold @ 7/11 3 ^ bushel.

5. What is gained by selling 10 dozen of pears at two for IJd., bought at the rate of 5 a-penny ?

6. What did a publisher gain by buying the remainder of an edition consisting of 420 copies for £57»10»6, and selling 300 copies @ 3/6, and the remaining number @ 3/?

7. Bought 3 cwt. 1 qr. 9 Ib. of soap @ £2"! 1"4 ^ cwt., and sold it @ 6d. y Ib., but found that the soap had inlaked 27 Ib. What was gained or lost by the transaction ?

8. Bought 2 cwt. 27 Ib. sugar @ 58/4 y cwt., and sold 1 cwt. 3 qr. @ 7£d. f Ib., but by a fall of the market was obliged to sell the remainder @ 5d. ^ Ib. What was gained or lost by the trans- action ?

(3) Find the selling price of 14 cwt. 3 qr. 21 Ib. of coffee bought @ £6"10//8 ^ cwt., and sold with a profit of •'5d. Vlb

£6'/10//8 P. C. W cwt. 14 cwt. 3 qr. 21 Ib. @ £8* . = 2// G//8 G. // // 17//4 ^ cwt. = £132//

8//17//4 S. P. // // 8//11 Total S. P.

(4) What must a corn merchant pay for 500 stones of hay, so as to sell it @ 8£d. with a gain of l£d. ^ stone?

8jd. S. P. ^ stone. 500 stones @ 7d.

lid. G. // // = £14//11 // 8 Total P.C.

7d. P. C. // //

9. How must 288 yd. of cloth, bought @ 4/5 £ ^ yd., be sold #• yd. to gain 12 guineas by the transaction?

10. How must 3 pieces of cloth, each 89 yd., bought for £73 » 8 »6, be sold fr yd., to gain £2 "4 "(5 ^ piece?

.11. Find the prime cost of 6 chests of tea, each containing 2 qr. 27 Ib., sold @ 4/8 ^ Ib. with a total gain of £15»2»6.

12. At what rate ^ cwt. must a merchant purchase a lot of Cumberland hams, so as to retail them @ 9d. ^ Ib. with a gain of 1-^d.^lb.?

13. What was paid for 4 cwt. 3 qr. 16 Ib. of Cheshire cheese, sold at 6*d. ^ Ib. with a gain of 4/8 ^ cwt. ?

14. At what rate must soap be retailed ^ Ib. so as to gain l£d. y Ib. on 3 cwt. 2 qr. 14 Ib., purchased in all for £8"9"2 ?

15. What is the prime cost ^ cwt. of 6 cwt. 3 qr. 17 Ib. of coffee, sold @ 1/8 ^ Ib. with a total gain of £12"! »6| ?

16. How much does a retailer receive for 3 cwt. 2 qr. of raisins, bought at 42/ ^ cwt., and sold with a profit of 2|d. ^ Ib. ?

PROFIT AND LOSS. 133

t Pin °/0 b^ sellinS Dutch butter @ 10 id ., bought at the rate of 84/ ^ cwt.

lOid. S. P. ^ lb. cwt. = 9d. P. C. // //

ltd. Gain. 9d. : ltd. : : 100 : x = 16f °/0.

(6) Find the loss % by selling 50 copies of a work @ 7/6 50 copies @ 4/, and the remainder of the edition for £12 the c6st of publication being £72//10. 50 copies @ 7/6 = £18//15 80 // ©41 = 16

Remainder = 12 _ £72//10 : £25//15 : : 100: S. P ..... £46Vl5 * = 35ti LOSS °/0.

P. C ..... 72//10 Loss .... £25715

17. What was gained % by purchasing goods for £16»12*6, and selling them for £17»10"1£?

18. Find the gain °/0 by selling butter @ 7£d. ^ lb., bought @" £2«6'-8 ^ cwt.

10. Bought 2 cwt. 1 qr. 7 lb. soap for £4"17«1 }, and sold it @ 5£d. %» lb. What was gained % ?

20. What was lost °/0 on tea, bought @ 2/7 ^ lb., duty 2/1 y lb., and sold @ 4/4 ^ lb. ?

21. Bought 37 yd. of cloth @ 13/6 f yd., sold 34J yd\ @ 16/, and the remnant @ 2/6 below prime cost. What was gained %?

22. Bought 3 cwt. 3 qr. of coffee for £23"12»6, but on account of damage was obliged to sell one-half @ 1/1 y lb., and the other half @ I/ #• lb. What was lost % ?

23. How much does a photographer gain % by buying frames @ 29/6 ^ doz., and selling them @ 4/6 each ?

24. Bought a sloop for £180, paid £40 for new mast and anchor, sold her for £275. What was gained %, allowing \ % on the sell- ing price for commission agency ?

25. Bought 26 cwt. 2 qr. 14 lb. of cheese @ 52/ f cwt. ; sold 20 cwt. wholesale @ £3" 10 f cwt., and retailed the remainder @ 9d. y lb. What was gained °/0 ?

26. A picture-seller who paid £250 for engraving a picture, sold 12 India proofs @ 3 guineas each, and 240 prints @ £1"11"6 each. What was gained °/0 by the transaction?

(7) At what rate must cheese, bought @ 50/ ^ cwt., be sold ^ lb. so as to gain 12 °/0 ?

134 PROFIT AND LOSS.

67. 100 : 112 : : 50/ : x = 56/ S. P. ^ cwt.

= 6d. S. P. & Ib.

(8) Find the buying price of cloth, sold @ 9/6 V yd. with a loss of 24%?

100 _24 76 : 100 : : 9/6 : x = 12/6 P. C. & yd.

27. At what rate must starch bought @ 42/ ^ cwt. be sold qp> Ib. so as to gain 33£ °/a ?

28. Find the prime cost of coffee $• cwt. sold @ 1/10 ^ Ib. with a profit of 10 °/e.

29. What was the prime cost of goods sold for £26"5 with a loss of 12£°/0?

30. Bought 7 cwt. 3 qr. Java rice for £4"10»5. How must it be sold y Ib. to gain 20 °|0?

31. Find the prime cost of a work of 10 vols, sold @ 10/6 ^ vol. with a profit of 16§ °,0.

32. A contractor gains 16£ °/0 by performing a piece of work for £233"! 9"5. What is his outlay for workmanship and materials?

33. A paper merchant bought 100 reams of foolscap, and sold 50 reams @ £1"5, with a gain of ll£ °/0; 25 reams @ £l>-8 ; and the remainder, being damaged, @ 17/8. Find the total prime cost, and the gain or loss °/0.

34. Find the weekly outlay of the proprietor of an omnibus who receives on an average £3»15"3 every lawful day, and thus clears 75 %.

35. At what price must cloth bought @ 5/6 sp- yd. be rated so as to allow 4 °/<, discount for ready money and gain 9T'T e/o by the money received ?

36. Suppose a bootmaker pays on an average 6/4 for the leather and furnishings of a pair of boots, and 6/4 for the workmanship ; what must he charge his customer so as to allow him a discount of 5 e!o, and gain 50 °/« by the money received ?

(9) Sold goods for £225//10 with a gain of 12} %• What would have been gained or lost 70 by selling them for £187*10?

100 £225//10 : £187*10 : : 112} : x = 93}

~6j Loss %

(10) Sold a bale of leather for £14*14, and gained 17| c/0. How should it have been sold to have gained 18 °/0 ?

117| : 118 : : £14/14 : x r= £14*15 S. P.

PROFIT AND LOSS. 135

67. 37 • AjDookseller having bought two copies of the seventh edition of the Encyclopaedia Britannica at the same price, sold one @ £25 with a profit of 9j'T "/.. How much did he gain °|. by selling the other @£27»10?

38. A merchant of Lyons by selling silk @ 10 francs #• metre gained 20 "/.. What did he lose % by selling silk of the same prime cost @ 8 francs ^ metre?

39. Lost 36 °/« by selling cloves @ 8d. ^ Ib. What would have been gained or lost °/0 by selling them @ 1 £d. ^ oz. ?

40. Gained 13 °/« by selling paper @ 9/5 y ream. What was lost °/0 by selling paper. of the same value @ 8/3 y ream?

41. Sold a bale of leather for £15, and lost 25 e/.. How should it have been sold to have gained 33 °/0 ?

42. Sold pencils at the rate of 3 for 2d., and gained 33£ °/0. What would have been gained or lost °/0 by selling them @ 5£d. q? doz. ?

43. A bootmaker by selling boots @ 24/ ^ pair gains 50 °/0. What must he have charged to have given a discount of 5 °/0, and to have gained 78& °/0?

(11) Find the prime cost and selling price of goods sold with a gain of 32 °/e, and of £16/'17/'4 in all.

32 : 100 : : £16*17*4 : x = £52*14*2 P. C.

16*17*4 Gain.

£69*11*6 S. P.

44. Sold goods with a loss of 20 °/0, and lost £57"6"8 by the transaction. What was the prime cost?

45. Find the selling price of goods by which there was a loss of 2 °/0 or of £54" 10 by the whole transaction.

46. What does a draper receive for 39 yd. of cloth which he sells with a gain of 2/ $x yd. and of 26§ °/0 ?

47. Sold cheese with a gain of 2^d. $>• Ib. or of 62£ %. At what was it bought and sold ^ cwt. ?

48. Sold 39 casks of cod-liver oil, each containing 52£ gallons, with a loss of 1§ °/0, and of £8»10«7i on the transaction. What was the prime cost $>- gallon ?

49. Find the original outlay of a publisher who sold 2000 copies of a guide-book, with a gain of 6d. ^ copy and of 25 %.

50. Find the outlay of a publisher who sells 500 prints of an en- graving with a gain of 5/6 f print and of 35 ^f °I0.

(12) How much sugar, bought @ £2*13*8 V> cwt. was sold @ 5d. W Ib., with a total loss of £3*18*9 <

136 PROFIT AND LOSS.

67. £2*13*8 V cwt. = 5fd. P. C. & Ib. £3*18*9

5_d. S. P. * J78S<

id. Loss. * 945^

3)3780 3780f.

"I2601b. = llcwt. Iqr.

(13) How many prints of an engraving must a picture - dealer sell @ £l//ll//6, so that he may gain 5l£ °/0 on an outlay of £250 ?

100 : 151J : : £250 : x = £378 S. P. £1*11*6 £378

31 s. 7560 s.

63 sixd. )15120 sixd. (240 prints.

51. Bought a cargo of oranges @ 12/6 ^ chest, and sold it with a gain of 30 °/0, and of £18" 15 in all. How many chests were in the cargo ?

52. How many yd. of cloth bought @ 13/2J ^yd. must a draper sell @ 16/6 to gain £3"19"6?

53. What quantity of butter bought @ £2»13»8 y cwt. must be sold @ ?id. ^ Ib. to clear £4»18 ?

54. Bought haddocks @ 3/4 y long hundred (120). How many must be sold at 7d. ^ dozen to gain 12/6?

55. How much sugar bought @ 42/ ^ cwt. must be sold @ 6d. ^lb. to gain £20 in all?

56. Bought 10 cwt. of sugar @ 44/ ^ cwt., and sold it at 4^d. ^ Ib. . How much tea bought @ 3/1 y Ib. must be sold @. 4/4 ^ Ib. to cover the loss on the sugar ?

57. Sold iron @ £5 "6 y ton, with a profit of 6 °/0, and of £21 «10«6 in all. What quantity was sold ?

58. A drysalter purchases goods @ 58/4 f cwt., and by retailing them gains £2 "17 "6£, being at the rate of 4 °/0. What quantity was sold ?

59. A grocer buys sugar @ 37/4 $>• cwt., and by selling it @ 62 ^ % profit gains £5"5"5. What quantity does he sell ?

60. Bought a cargo of oranges @ 15/ sp- chest, and sold one-half of them @ 19/6 ^ chest, and the other with a loss of 10 %, but gained £27"7»6 on the whole. How many chests were bought?

(14) Bought goods for £53, and sold them for £75, with one year's credit. What was gained °/0?

Let us first find the Present Value of £75, reckoning the rate of interest here and in all the following examples at Five per cent.

PROFIT AND LOSS. 137

£105 = Future Value of £100 in 1 yr. @ 5 °/ .

105 : 100 : : £75 : x = £71* P. V. of S.°P. The question is now reduced to the following : — Bought goods for £53, and sold them for £7 If °/0 ; what was gained °/0 ?

£53 : £71f : : 100 : x = 134f «« S. P.

Gain °/0 = 34f f « .

These two statements may be united as follows : 105 : 1001 . . 1on . £53 : £75 f ' ' 10° ' x

(15) How must cloth, bought @ 6/9 ^yd., with 3 months' credit, be sold so as to gain 5 °/0, and allow 9 months' credit.

F. V. of £100 @ 5 °/0 for 3 mo. and 9 mo. = £101J and £103J. 10H : 103| ) s. a. s. a.

100 : 105 | : : 6*9 : x = 7*3T»V S. P.

61. Bought goods for £59, and sold them for £89 with one years credit ; what was gained °/0 ?

G2. What was lost by selling 288 yards of cloth for £182 »8, bought 6 months ago @ 12/6 ^ yd. ?

63. Bought goods for £70, and sold them for 70 guineas with twelve months' credit ; what was gained or lost % ?

G4. I low must goods be sold to gain 5 %, and give 9 months' credit, bought the same day for £81 with 3 months' credit?

65. What is gained or lost % by selling goods @ £47*13 "4 y cwt. bought 6 months ago @ 8/ ^ lb. ?

66. What is lost % by selling goods with 6 months' credit, bought 6 months ago for the same money ?

68. DISTRIBUTIVE PROPORTION.

IN DISTRIBUTIVE PROPORTION we divide or distribute a given number into parts which have a given ratio to each other. (I) Divide £376//5 of gain among three partners in an ad- venture whose risks are respectively £225, £150, and

£250.

£225
150
250
625	225		£376//5	: x =	£135* 9
625 625	150 250		37G//5 376//5	: x = : x =	90 v 6 150//10
					£376* 6

138 DISTRIBUTIVE PROPORTION.

68 ^ie sum °^ ^e risks — £625. As the whole risk is to each risk,

50 is the sum to be divided to the share of each. The sum is thus divided into parts proportional to 225, 150, and 250, which may be cancelled by their common factor 25.

The following method is often convenient : —

225 150 250

* =

9 X £15*1 = 135* 9 6 6 X 15*1 = 90* 6 10 10 X 15"! = 150*10

25)£376"5(£15*1 £376* 5

(2) A sum of £1000 was bequeathed to four relations, and by an inadvertency in the will, it was stated that they were to receive J, £, J, and £ of the sum respectively. How much should each receive according to the spirit of the will?

6 6 X £66*13*4 = £400

4 4X 66*13*4= 266*13*4

3 3 X 66*13*4 = 200

2 2 X 66*13*4 = 133* 6*8

I5)£1000(£66*13*4 £1000

We divide £1000 in the mutual ratios of £, J, £, £. The sum of these fractions = { | is greater than unity. T'5 is therefore one- fifteenth of the sum. Dividing £1000 by 15, we multiply by 6, 4, 3, 2, successively to obtain the respective shares.

1. Divide 84 into parts having the mutual ratios of 2, 3, 7.

2. Divide 1200 into parts having the mutual ratios of 11, 12, 13, 14.

3. Divide a line 4 feet long into parts having the ratios of the first four odd members.

4. Divide 100 into parts having the ratios of the cubes of the first three numbers.

5. Divide 390 into parts having the ratios of £, £, J.

6. Divide 1331 into parts having the ratios of the reciprocals of the first three even numbers.

7. Apportion a house tax of £6»18"8 among 3 joint proprietors, who pay in the proportion of the annual values of their properties, which are £30, £40, and £60 respectively.

8. A vessel is divided into 64 equal shares, of which A, B, C, D, have 6 shares each; E, 12; F, 16; Gr, 4 ; and H the remainder. Find their respective shares in sustaining a joint loss of £158" 10" 1.

9. Divide a profit of £689 among 3 partners, of whom the first owns T23 of the joint stock and the second T53.

10. A, B, C, D, invest £450, £230, £190, and £110 respectively

DISTRIBUTIVE PROPORTION. 139

«*n a speculation. Find their respective liabilities in a joint loss of £313»12.

11. Three partners respectively claim ^, {|, and ,»s of the gain of an adventure amounting to £1260. Give to each a proportionate share.

12. Divide 5 guineas among George, James, and Henry, who respectively claim §, *, and £, so that they may have proportionate shares.

13. An analysis of the manure of dissolved bones gives the following results for every 100 parts: — Water, 13*97; Organic Matter, 15-71 ; Soluble Phosphates, 21'63 ; Insoluble Phosphates, 11-43; Sulphate of Lime, 15'83; Sulphuric Acid, 15'63; AJj^jline Salts, 1-10; Silica, &c., the remainder. Find the weight of each in a ton of dissolved bones.

14. Oil of vitriol (HO, S03) contains by weight, 1 of Hydrogen, 32 of Oxygen, and 32 of Sulphur. Find the weight of each in a gallon of oil of vitriol which weighs 18| Ib.

(3) D, E, and F, gain £564 : D's capital of £300 has been in trade for 6 months ; E's, which is £400, for 3 mo. ; Fs, which is £500, for 2 mo. Find the share of each.

D, £300X6=1800,9 9x£28//4=£253//16

E, 400X3=12006 6x£28//4= 169// 4

F, 500X2=100015 £ 5x£28//4= 141

20)564_ £564

£28//4

The use of £300 in trade for 6 mo. is equivalent to that of 6 times £300 for 1 mo. Similarly, £400 for 3 mo. is equivalent to 3 times £400 for 1 mo. ; and £500 for 2 mo. to 2 times £500 for 1 mo. Taking the time of 1 month alike for D, E, F, we see that the shares are proportional to 1800, 1200, and 1000.

(4) A commences trade with £3000 : in 3 months B joins him with £4000 ; at the end of the next 2 months A takes out £1000 ; in 1 mo. after C joins them with £2000, and B adds £1500; in 2 mo. after C takes out £500: at the end of 12 months they divide £2760 of gam. What is the share of each ?

A has £3000 in trade for 5 mo., and £2000 for 7 mo. B // £4000 " » 3 // and £5500 // 6 » C // £2000 » " 2 // and £1500 // 4 *

. ( £3000 X 5 = 1 5000

A{ 2000 X 7 = 14000

R (£4000 X 3 = 12000 145000

B ! 5500 X 6 = 33000 f4 '

140 DISTRIBUTIVE PROPORTION.

68. c | £2000 x 2 = 4000) 10000

0 \ 1500 X 4 = 6000 f10'00

29 X £32'/17//lH =£ 952//17//l^ 45 X 32*17*1 J$ = 1478*11*5? J 10 X 32//17//l£f = 328*11*58$ 84)£2760 £2760* 0//0

£32*17*1$ *

15. In a copartnery, A's capital of £400 has continued for 9 mo. ; B's of £350 for 8 mo. ; C's of £600 for 2 mo. Divide £570 of gain among them.

16. Three cattle-dealers rent a field of 9 acres @ £5 ^ acre: A puts in 6 cows for 2 months ; B, 9 cows for 1 mo. ; C, 12 cows for 3 mo. How much does each pay ?

, , 17. At the end of 12 months, D, E, F, having a joint capital of £6000, find that they have lost £625. D's capital of £2500 has been in trade for 12 mo., E's of £1500 for 8 mo., and F's for 4 mo. What is the loss of each?

18. A and B enter into partnership, the former with £1800, the latter with £900 : in 8 months B adds £300 to his capital. Divide a profit of £840 between them at the end of 12 months.

19. A has £300 in trade for 7 months, when B joins him with £400. At the end of the next 3 months C joins them with £300. Divide £549 of gain among them after 18 months' trade.

20. A, B, and C, enter into partnership on Jan. 1, 1856, with a capital of £1000 each. On April 30, B withdraws £400, and C makes up the sum. On Aug. 28, A withdraws £200, and C makes up the sum. On balancing their books for the year they find they have a gain of £365. What is the share of each ?

21. Three graziers rent a field from May 11 to October 19, 1857, for £43. A agrees to pay £13 for grazing 12 oxen; B, £18 for 18 oxen; and C the remainder for 20 oxen. To how many days is each grazier entitled ; and if the oxen go into the field in the order A, B, C, on what days do B's and C's severally enter?

<£gr The times are proportional to the sums paid for 1 ox. A pays m ; B, £}f ; C, ££$ for 1 ox.

22. 3 men and 4 boys are loading carts with sand. A man takes 7 shovelfuls for a boy's 6, and 4 shovelfuls of a man's = 5 of a boy's. Divide £3 "1 proportionally among them.

141

69. ALLIGATION.

ALLIGATION treats of the prices and quantities of a compound and its ingredients.

In Alligation Medial, the prices of the ingredients are given and the price of the compound is obtained by finding the average price.

(1) A merchant mixes 45 gallons of spirits @ 7/4, 20 @ 6/6, -84 @ 6/8, and 21 gallons of water. What is the price of the compound ^ gal. ?

45 @ 7/4 = 330s.

20 .. 6/6 = 130

84 ..6/8 = 560 21

170 ) 1020 (6/V gal.

The average is thus found by multiplying each price by the corresponding quantity, and finding the sum of the products by the sum of the quantities.

1. Find the average price of 4 gal. @ 5/, 5 @ 4/, 8 @ 2/6, and 7 @3/.

€5" We may thus often find the average price merely, without considering that the whole has been compounded.

2. Find the average price of 100 Ib. rice @ Id. y lb., 300 Ib. @ 2d., 400 @ lid., and 100 @ 4d.

3. Find the price y gal. of a mixture of spirits of 50 gal. @ 4/6, 40 @ 4/2, 45 @ 4/4.

4. Find the average price of 23 qr. wheat @ 40/, 32 @ 48/, 12 @ 69/, 24 <§> 38/, and 17 @ 50/.

5. On Feb. 6, 1856, the following quantities ol wheat were sold at the six highest prices in the Edinburgh Grain Market: — 8 quarters @ 96/; 4 @ 84/; 21 @ 78/; 13 @ 76/; 1 @ 75/; 2 @ 74/. Find the average price ^ qr. as deduced from these prices and quantities.

In Alligation Alternate, we find the proportional quantities of ingredients of given prices which will produce a compound of a given price.

(2) Mix spirits @ 8/3, 7/9, 6/6, and 8/4 ^ gal. so that the compound may be worth 8/ ^ gal.

142 ALLIGATION.

I.

d. gal.

f 78—, 3 X 78 = 234

qr\ 93 -, 4 X 93 = 372

99_l is x 99 = 1782

1 100 — ! _3 X 100 = 300

28 28) 2688

96 n.

d. gal.

f 78 -i 4 X 78 = 312

nJ 93-] 3 X 93 = 279

J01 99 J 3 X 99 = 297

[ 100 J 18 X 100 = 1800

28 28)2688

96

We express the prices in the same name.

To obtain a compound at 96d. we must mix two ingredients, of which the one is dearer and the other cheaper than the com- pound.

We may, as in Method L, connect 78d. with 99d., and 93d. with lOOd.

The act of thus connecting or binding the prices together is the reason why the rule is termed Alligation.

If spirits worth 99d. qp gal. are sold @ 96d. there will be a loss of 3d., and if spirits worth 78d. are sold @ 96d. there will be a gain of 18d. Since 18 X 3d. = 3 X 18d., the loss on 18 gallons worth 99d. will balance the gain on 3 gallons worth 78d. We therefore write the difference between 96 and 78 or 18 opposite its alternate number 99 ; and the difference between 99 and 96 or 3 opposite its alternate number 78. We proceed similarly with 93d. and lOOd.

In Method II. we may connect 99d. with 93d. and 78d. with lOOd.

When the differences between the price of the compound and that of a dearer and of a cheaper ingredient connected together are equal) we may take any equal quantity of each of the latter ; thus, instead of 3, 3, 4, 18, we may take a:, a, 4, 18, where x may be any quantity.

6. Find the proportional quantities of sugar @ 5d. and 8d. that must be sold to make the average price 7d. ^ Ib.

7. What proportional quantities of potatoes @ 2/, 3/, and 3/6 #• bushel must be sold to make the average price 2/9 y bushel ?

8. Mix tea @ 4/6, 4/2, 3/4, and 3/9 y Ib., so that the compound may be worth 3/1 1 y Ib.

9. What proportional quantities of wine @ 15/, 12/, 18/, 19/, and 21 1 y gal. must be sold to make the average price 16/ ^ gal. ?

(3) What quantities of tea @ 5/3, 4/5, and 2/9, must be mixed with 21 Ib. @ 6/1, to make the whole worth 5/

mr\, TU Q

69. f33-j 3X3 = Ib9 ® 33d.

53 1

ALLIGATION. 143

3X3 = Ib9 ® 33d. 13 X 3 = 39 .. 53d.

60 -, X = 39 .. 53d.

OU1 63-J 27 X 3 = 81 .. 63d.

173 J 7 . . . . . 21 .. 73d.

[33-, -, 13 + 3= 16 X fi = 9|f @ 33d. W|S H « =13Xfi= 8 A .. 53d.

-J 27 = 27Xfi = 16}| .. 63d.

[73- J 27 +7 = 34 . . . . 21 .. 73d. Having found the proportional quantities as formerly, we multi- ply them by the ratio of the given quantity to its corresponding proportional quantity.

Similarly, when the quantity of the compound is given, we mul- tiply the proportional quantities by the ratio of the given quantity to the sum of the proportional quantities.

10. How much wheat @ 42/ and 56/ must be sold with 13 qr. of wheat @ GO/ to make the average price 50/ y qr. ?

11. How much sugar @ lOd. and! Id. must be mixed with 9 Ib. of 7d. sugar to make the whole worth 8£d. ?

1 2. How many gallons of water must be mixed with 63 gallons of spirits @ 8/ so that the prime cost may be 7/ f gal. ?

^* We alligate Sj with 0. Or we may solve this by proportion, s. s. gal. pal. 7 : 8 : : 63 : 72. .-. Number of gal. of water = 72 — 63.

13. How many gallons of water must be mixed with 47j gallons of spirits @ 6/3 to make the prime cost 5/ ^ gal. ?

14. How many gallons of each kind of wine @ 15/3, 16/4, 17/2, and 18/1, must be sold to make the average price of 154 gallons 17/fgal.?

15. The Specific Gravity of an alloy of gold and copper is 16-65, while that of gold is 19-2, and that of copper 9. Find the weight of gold and copper in 144 oz. of the alloy.

16. A crown made of gold and silver weighs 150 oz. and displaces 13-824 cub. in. of water. Had it been gold it would have displaced 12-96 cub. in. of water, and had it been silver it would have displaced 23-04 cub. in. Find the weight of gold and silver in the crown.

&F This question is founded on the story of Archimedes and Hiero. Hiero had given a goldsmith a certain quantity of gold to make a crown. In course of time, the artificer presented a crown of the same weight as that of the quantity of gold ; but as Hiero suspected a fraud, he requested Archimedes to discover if any baser metal had been alloyed with the gold. Archimedes considered that if the crown contained any metal lighter than gold, it would be larger than a pure gold crown of the same weight. Having o tained a mass of pure gold and of the other metal, each of the same weight as the crown, he found the quantity of water which each ol the three displaced, and from these data discovered the proportion of each metal in the crown.

144

7O. BAETEE.

IN BARTER, two parties mutually give goods of equal value in exchange.

(1) Exchanged 164 Ib. of tea @ 4/8 W Ib. for coffee @ 1/7 ^ Ib. How many Ib. of coffee were received?

x Ib. of coffee @ 1/7 = 164 Ib. of tea @ 4/8.

(2) In return for 146 qr. wheat @ 70/ ^ qr., an agent re- ceived Wilts cheese @ 88/ ^ cwt., and Dunlop cheese @ GO/ ^ cwt., obtaining 6 cwt. of Wilts for every 5 of Dunlop. How many cwt. of each were received ?

6 X 8*8 = 528 5 X GO = 300

828s. = the price of 1 parcel of both kinds of cheese in the given proportional quantities.

x parcels @ 828s. = 146 qr. @ 70/ = 10220s.

o« - 146 X 70 - 10220 — 1971 T^orpplc

x — ~828 — ~T*¥ -- lzaST parcels. Each parcel contains 6 cwt. of Wilts and 5 cwt. of Dunlop.

6 X 12^0-V = 74A cwt. = 74//0// 6|J Wilts. 5 X 12^V = 6HSf cwt- = 61//2//245W Dunlop.

Proof 4 74A cwt' @ 88/ = roof ^ 61^« cwt @ 6Q/ = 37026 .s>

10220s.

1. How many yd. of cloth @ 2/3 are worth 54 Ib. of tea @ 4/1?

2. What is the price ^ yd. of cloth, of which 200 yd. are worth 2 cwt. 2 qr. 25 Ib. @ 93/4 y cwt. ?

3. How many gallons of brandy @ 24/6 ^ gal. are worth 35 doz. loaves of refined sugar, each 16 Ib. @ 70/ ^ cwt. ?

4. Exchanged a tierce of sugar weighing 8 cwt. 3 qr. 14 Ib. for 31 cwt. 0 qr. 7 Ib. rice @ 18/ y cwt. Find the price of the sugar ^lb.

5. How many yd. of linen cambric @ 5/6 must be given in ex- change for 15 dozen pairs of boots @ 18/ y pair, and 13 dozen pairs of shoes @ 8/ y pair ?

BARTER. 145

7O* 6. ^ baker, who has run an account with a grocer for 12 £ Ib. tea @ 4/2, 60 Ib. sugar @ 6£d., 3£ Ib. coffee @ 1/8, and 13 drums of sultana raisins, each 20 Ib., @ lid. f Ib., has a contra-account of 23 dozen loaves @ 7£d. ^ loaf. How many loaves @ 8£d. will settle the account ?

7. A dairyman, who has supplied a baker with 90 pints of milk @ 2£d., 13£ pints of cream @ 10d., and 80 Ib. of butter @ 10d., agrees to take an equal number of loaves @ 7d. and 7£d. How many of each does he get ?

8. Exchanged 28 Ib. of tea @ 4/2 for coffee, and got 5 Ib. of coffee for 2 Ib. of tea. How many Ib. of coffee were got, and what was its price ^ Ib. ?

9. In return for 80 qr. barley @ 56/ ^ qr., £ of the value was re- ceived in bone-dust @ £8»8 & ton, and the rest in money. How much money and how many tons of bone-dust were received ?

10. In return for 165 cwt. flour @ 15/ f cwt., an agent received 3 chests of tea, each 81 Ib., @ 4/4 ^ Ib., and 8 doz. loaves of re- fined sugar, each 19| Ib. What was sugar ^ Ib. ?

1 1. In return for 14 cwt. 2 qr. 20 Ib. Glo'ster cheese @ 77/f cwt. ; beef @ 8d. y Ib., and mutton @ 7d. f Ib., were received in the ratio of 7 Ib. of beef for every 3 Ib. of mutton. How much of each was received ?

12. Exchanged 6 cwt. 2 qr. 3 Ib. salmon @ 1/6 y Ib., 20 tur- bots @ 4/2, 16 dozen haddocks @ 4/6 & doz., and 15 pints of shrimps @ 6d., for 2 cows @ £9"13 each, 160 Ib. beef @ 7£d., 240 Ib. pork @ 5d., and 80 pairs of fowls @ 3/9 ^ pair. How many Ib. of mutton @ 7d. must be given for the balance ?

71. CHAIN EULE.

(1) IF 5 pheasants are worth 4 grouse; 5 grouse, 8 par- tridges ; 2 partridges, 5 snipes ; how many snipes may be had for 10 pheasants ?

x snipes = 10 pheasants 5 pheasants = 4 grouse 5 grouse = 8 partrid; 2 partridges = 5 snipes

Having arranged the pairs of equal values or equations, so that numbers of the same name are on different sides, wo examine the equations as follows . —

146 CHAIN RULE.

Snipes. Snipes.

1 partridge = f \ S[ouse I- ?_XJJ<1

or 5 pheasants ) 2X5

or85 Pg±edgeS }= ^ 1 Peasant = j£

1 grouse = |£f 10 pheasants = ^

We see then that the number of snipes = 10 pheasants is obtained by dividing the product of the numbers on one side by the product of those on the other.

N<? of snipes = 5* x* Vs™ which by cancelling = 32.

This method is known as the CHAIN RULE. Each equation is a link in the chain ; each link begins with the name with which the preceding link ended, and the chain is complete when the last ends with the name in the first link, whose number is wanted.

(2) How many francs are = a lac of 100,000 rupees, each 1/10J ; 25-22 francs being = £1.

x francs = 100,000 rupees 1 rupee = 89 f. 960 f. = 25-22 francs

1. 9 old ale gallons = 11 old wine gallons of which 9 = 20 Scotch pints, and 8 Scotch pints = 3 Imperial gallons. How many Imperial gallons = 54 ale gallons?

2. How many^j Linlithgow barley firlots = 3 Winchester bushels of whicfy 33 =>' 32 Imperial bushels or Linlithgow wheat firlots, and 16 LHnlith£ow wheat firlots = 11 Linlithgow barley firlots?

3. How many Scotch acrek= 100 Irish acres, 121 Irish acres = 196 Imperial "Stores, and 126\Imperial acres = 100 Scotch acres ?

4. 8 Scotch miles = 9 Imperial miles; 14 Imperial miles =11 Irish miles. How many Irish miles = 112 Scotch miles?

rfgT The mutual ratiosTirtue preceding examples are convenient approximations.

5. 2 quarts of plums are worth 3 of pears ; 6 of pears = 5 of apples ; 8 of apples cost 2/4. Find the price of 3 quarts of plums.

6. In 1855, the mutual ratios of the weights of bales of cotton imported at Liverpool from the following places were as follow : — 2 from Bombay = 3 from Egypt ; 9 Brazil = 4 United States ; 7 Brazil = 5 Egypt ; 7 Calcutta = 5 Madras ; 14 United States = 15 Madras. How many from Calcutta were = 50 from Bombay?

CHAIN RULE. 147

7. By examining the average weight of the bales of cotton im- ported at Liverpool in 1843, the following were obtained :— 55 from Egypt = 69 from W. Indies ; 35 from Alabama = 43 from the Upland U. States, from which 207 = 350 from Egypt; 91 from Alabama = 215 from Brazil, from which 27 = 13 from E. Indies. Ilbw many from W. Indies = 165 from E. Indies ?

8. From t^c Imperial averages for the week ending 30th April IS^jt^aj^eared that the price of 39 quarters of barley = that of 73 of oats ; 68 of barley = 73 of beans ; 27 of beans = 28 of pease ; 39 of wheat = 58 of rye, of which 153 = 143 of pease. How many quarters of wheat = 638 of oats ?

9. 4 talents were =#75 Ib. avoir., and each talent contained 3000 shekels. Find thd weight of a shekel in oz. avoir.

10. 273 quarters of irheat = 638 of oats, of which 73 = 39 of barley, sold @ 42/7 ^ quarter. Find the price of 1 quarter of wheat.

11. By a comparison of the apothecaries' grains of different countries, it was founfx that 17 German = 20 British; 85 German = 86 NeapolitaTTpS7 Spanish = 45 Austrian; and 185 Spanish = 172 Neapolitan. How many British = 90 Austrian?

12. A mile = 8|0 chains = 63360 inches; a chain = 100 links. How many inched are in a link ?

13. 176 Ib. troy = 144 Ib. avoir., each 7000 grains, of which 3608 = 1 Cologne mark. How many Cologne marks = 451 Ib. troy?

14. If a *aeire = 39-37079 in. be taken as 40)0o0>000of the earth's circumference, how many miles are in the earth's circumference?

15. 4 nautical miles = a German mile; the earth's circumfer- ence contains 5400 German miles = 40,000,000 metres. How many feet are in a nautical mile ?

72. EXCHANGE.

EXCHANGE is the method of changing the money of one country into that of another.

The Par of Exchange is the real comparative value of the money of two countries, estimated by the weight and fineness of the coins. , f .,

The Course of Exchange is the comparative value ot t money of two countries, which fluctuates according to tne circumstances of commerce.

In Exchange, £1 is generally adopted as the unit ;of '^PJ"££ Thus the r»ar of exchange with France is 25 francs 22£ centii ! When £1 is tSe unit, the equivalent i* foreign money vane,

148 EXCHANGE.

in the course of exchange ; thus, £1 may be exchanged at one time for 24 fr. 30 c., and at another for 25 fr. 50 c. When a foreign coin is taken as the unit, the equivalent in sterling varies in the course of exchange ; thus, while the par with Naples is 39f d. ^ ducat, the exchange may at one time be 38d., and at another 40d. ^ ducat.

CANADA. — Accounts are kept in £, s. D. Currency, of which £1, being taken as = 4 dollars of the Nominal value of 4/6 each, is = 18/ sterling. Hence the nominal par is £100 cur- rency = £90 sterling. But as the real average value of the dollar is 4/2, £1 currency = 16/8 sterling, and the real par is £108 currency = £90 sterling. The Nominal Par is taken as the standard, and a Premium is added to show the course of exchange. At a premium of 8 70) £108 currency=£90 sterling.

WEST INDIES. — The old currencies are now superseded by sterling. Of the foreign coins in circulation, the principal are the dollar = 4/2, and the doubloon = £3//4.

(1) How much sterling is = £327 currency, at a premium of9c/o?

B>j the Chain Rule.

£l09r-e£327 • • £9eO * or- Ster' * = £327Curr. £109 . td27 . . 190 . x or

x = ^^ = £300 Ster. iuy

(2) How much currency is = £8460 sterling, at a premium of91°/o?

*onterlcnQ8icn £"™?cy' f Curr. z=£8460Ster. £90 : £8460 : : £1091 : * or j Ster.£90=£109JCurr.

= £10293

1. How much currency will an emigrant to Canada receive for £135"7«6 sterling, at a prem. of 8£ °/0?

2. An emigrant on arriving at Toronto changes 6 crowns, 7 hf.-crowns, 37 shillings, and 5 sixpences sterling, to currency, @ the rate of 15d. for I/ sterling. How much currency does he receive ?

3. How much sterling is = £324"2 »3 currency, remitted from Montreal, at a premium of 8 % ?

4. An agent at Quebec wishes to remit to his employer in Lon- don £489"12"l£d. To how much sterling will this be equal at a premium of 8 J °/0 ?

5. How many dollars @ 4/2, or how many doubloons @ £3 "4, must a Jamaica merchant receive from his correspondent at Cuba, who Is due £320 ?

EXCHANGE. 149

6. An agent changes 3000 dollars to sterling @ 4/2 ^ dollar, at Kingston, Jamaica, on embarking for Halifax, Nova Scotia, and on arriving changes the sterling to currency @ 8 °/0 premium. How much currency does he receive ?

UNITED STATES. — Accounts are kept in dollars and cents. 10 cents* = 1 dime ; 10 dimes= I dollar* ($) ; 10 dollars = 1 eagle. The par of exchange, deduced from the gold coins, is $1 = 4/l£ nearly; from the silver coins, $1 = 4/2 J nearly. In custom-house valuations, $1 = 4/2. The nominal par of exchange is $1 = 4/6 ; hence, $40 = £9, or $100 = £22//10. We take the nominal par as the standard, and add a premium to $100 ; thus, at a premium of 9} °/0, $109£ = £22*10.

(3) How much sterling is in $1 11-55, at a premium of 9-| °/o?

ft ft

109-125 : 111-55 : : £22*10 : x = £23.

(4) How many $ are in £2560, at a premium of 9 % ?

£ £ ft

22*10 : 2560 : : 109 : x = $12401-77}.

7. How much sterling is in i$3390, remitted from New York, at a premium of 8 °/0 ?

8. How much sterling is in ft994'25, remitted from Philadelphia, at a premium of 9& °/0 ?

9. How many ft are received at Boston for £738, at a premium

of 9 - - °/0 ?

10.° How many ft are in £7659, received at New Orleans, at a

premium of 10 °/0 ?

The following Illustrative Processes may suffice for the rest of the Exercises : —

(5) Change £999//12 to florins at Vienna, at the rate of 'lO florins 50 kreuzers ^ £1.

fl kr <fl.z=£999-6

£l:£999//12::10//50:z=10829n. or-] £1 ==f£kr-

( kr.bU=l n.

(6) A merchant remits 717 thalers 12 groschen from Berlin, at the rate of 6 thalers 24 groschen V £1. To how much sterling is this equivalent ?

th.gr. th.gr. £ ( £a=717-4th.

6*24 : 717//12 : : 1 : a= -

The names of the coins quoted in Exchanges are put in Italics.

150 EXCHANGE.

72* (?) How many dollars may be had at Malaga for £809//15//10, at 4/2 y dollar?

j,o ^QAQ IK m diU' ooo7 rtj doll.z= 194350 d. 4/2 : £809"15'/10 : : 1 : x = 3887 or-j ^ 59 == i <j0ii

(8) What is the value in sterling of 733 oncie, remitted from Palermo, @ 10/5 ^ oncia?

oncie. ( £* = 733 one.

1 : 733 : : 10/5 : : £381//15//5 or < onc.l = 125 d.

( d.240 = 1 £

FRANCE ; BELGIUM. — Accounts are kept in francs and cen- times. 100 centimes = 1 franc = 9^d. nearly. The par of exchange, deduced from the gold coins, is 25 fr. 22£ c. per £1 ; and from the silver coins, 25 fr. 57 c. per £1. The franc weighs 5 grammes, and is coined of silver T% fine.

11. How many francs are = £525»10"G, remitted to Marseilles, @ 25 fr. 22 c. $> £1 ?

12. How many francs must be remitted from Brussels to pay a bill of £987«14»6, @ 25 fr. 10 c. $* £1 ?

13. How much sterling must be remitted to Paris to settle an account of 9900 francs, @ 24 fr. 75 c. f £1 ?

14. Plow much sterling must be sent to Antwerp to be equivalent to 25663 fr. 75 c., @ 24 fr. 50 c. ^ £1 ?

HOLLAND. — 5 cents = 1 stiver ; 20 stivers or 100 cents = 1 florin or guilder = 1/8. Par of exchange, 12 florins = £1.

15. How many florins must be paid at Amsterdam in order to liquidate a debt of £1500"8 ; exch. 12 fl. 6 c. f £1 ?

16. A merchant at Gouda consigns cheese to the amount of 8993 florins to an agent in Scotland. How much sterling must the latter remit @ 1 1 fl. 50 c. ^ £1 ?

SWITZERLAND. — 10 rappen = 1 batz ; 10 batzen = 1 franc = 1/2 nearly. The French coinage is also used.

17. A London jeweller remits £701"12«6 to a watchmaker in Geneva @ 25 fr. 30 c. f £1. How many francs does the latter receive ?

18. A merchant of Geneva, on coming to Berne, changes 518 French to Swiss francs @ 148 French for 100 Swiss francs. How many Swiss francs does he receive ?

AUSTRIA. — 4 pfennings = 1 Icreuzer; 20 kreuzers = 1 zwanziger; 3 zwanzigers or 60 kreuzers = 1 florin = 2/0£ nearly. Par of exchange, 9 fl. 50 kr. = £1.

EXCHANGE. 151

72* 19- How many florins will be received at Vienna for £786 "14"6, exch. 10 fl. 30 kr. ^ £1?

20. The sum of 19868 fl. is remitted from Augsburg in Bavaria, where the Austrian coinage is used. Find the value in sterling @ 10 fl. 45 kr. ^ £1.

SOUTHERN GERMANY. — 4 pfennings=l Jcreuzer ; 60 kreuzers = 1 florin = 1/8 nearly. Par of exchange, 120£ fl. = £10.

21. How many florins will be received at Frankfort-on-the- Maine for £767, exch. 119J?

22. An agent at Munich remitted 2241 florins. Find the value in sterling, exch. 124|.

PRUSSIA; HANOVER, &c. — 12 pfennings = 1 groschen; 30 groschen = 1 thaler = 2/10f nearly. Par of exchange, 6 thai. 27 gr. = £1.

23. How many thalers may be had at Dantzic for £726»15"6, exch. 6 th. 20 gr. ?

24. How much sterling is = 36473 th. 20 gr. remitted from Hanover, exch. 6 th. 10 gr.?

Bremen.— 6 schwaren = 1 grote ; 72 grotes = 1 rixdollar = 3/3i nearly. Par of exchange, 609 r. d. = £100.

25. How many r. d. may be had at Bremen for £575, exch. 608 ?

26. How much sterling is = 1517 r. d. 36 gr. remitted from Oldenburg, where the Bremen coinage is used, exch. 607 ?

Hamburg; Lubec.— l2 pfennings = 1 schilling ; 16 schil- lin __ i mar/Cf Money is distinguished into Banco and Currency. Banco is used in Hamburg in exchanges in whole- sale transactions and in Bank business. Currency consists of coins in circulation; the marks current of Hamburg, and Lubec are, from the latter, termed marks Lub. The agio or difference between banco and currency varies from 20 to Zbl» Par of exchange, 13 ink. 10J sch, banco = £1 , 1 mark banco = 1/51 nearly; 1 mark current = 1/2* nearly.

27. How many mk. banco are = £876;8, exch 13 mk * scl. ?

28. How much sterling will be received m London for 27783 mk. banco, remitted through the Bank of Hamburg, exch. 1,

1229!CA merchant pays 6461 mk. cur, into the Bank of Hamburg How much banco is entered on the books, agio being 24i fci

equivalent, agio 21 § °/0?

152 EXCHANGE.

DENMARK. — 16 shillings = 1 mark ; 6 marks = 1 Rigsbank dollar = 2/2 £ nearly. Par of exchange, 9 R. d. 10 sk. = £1.

NORWAY. — 24 shillings = 1 mark ; 5 marks = 1 species dollar = 4/5 nearly. Par of exchange, 4 sp. d. 53 sk. = £1.

SWEDEN. — 48 shillings = 1 rixdollar banco = 1/8 nearly. Exchanges are generally effected through Hamburg.

31. How many Rigsbank dollars are in £432, remitted to Copen- hagen; exch. 9 R. d. 10 sk. ^ £1 ?

32. How many species dollars are in £1050"! 0, remitted to the Bank of Norway at Trondheim ; exch. 4 sp. doll. 53 sk. ^ £1 ?

33. How much sterling is = 5300 species dollars, remitted through a branch of the Bank of Norway at Bergen ; exch. 4 sp. doll. 50 sk. ? £1 ?

34. How much sterling is = 740 rixdollars banco, sent from Stockholm; exch. 12 r. d. 16 sk. banco y £1 ?

RUSSIA.— 100 copecs = 1 ruble = 3/1 £.

35. A British merchant sends £867" 14^6 to an agent at St Petersburg; what does the latter receive @ 3/l£ ^ ruble?

36. How much sterling must be remitted to Riga to discharge a bill of 1200 R<? 50 c.

PORTUGAL. — 1000 reas = 1 milrea ($) = 4/9 £ nearly.

37. How much sterling is =. a conto or 1000 $, remitted from Oporto @56d. tf-lftl?

38. How many $ are = £2270, sent to Lisbon @ 56|d. ^ $1 ?

SPAIN. — 34 maravedis = 1 real vellon ; 20 reals vellon = 1 hard dollar = 4/2 nearly.

Gibraltar. — 16 quartos = 1 real current; 12 reals current = 1 hard dollar.

39. A soldier on landing at Gibraltar changed 23 hf. sov. to dollars @ 50d. ^ dollar. How many did he receive ?

40. How much sterling must be remitted to Madrid to discharge an account of 1230 reals @ 50^d. ^ dollar?

AUSTRIAN ITALY. — 100 centesimi = 1 lira = 8|d. nearly. Par of exchange, 29 1. 52 c. = £1 or 48|d. = 6 Austrian lire.

SARDINIA. — 100 centesimi = 1 lira nuova = 9£d. nearly = French franc. Par of exchange, 25 1. 22 c. = £1.

TUSCANY. — 100 centesimi = 1 lira = 7|d. nearly. Par of exchange, 301. 68 c. = £1.

41. How many Austrian lire are = £375'- 10, remitted to Milan ; exch. 29 1. 52 c. f> £1 ?

EXCHANGE. 153

72. 42. How many Austrian lire are = £341»5, remitted to Venice ; exch. 48| d. for 6 Austrian lire ?

43. How much sterling is in 2300 lire nuove, remitted from Genoa ; exch. 25 1. 30 c. y £1 ?

44. How much sterling is in 4590 Tuscan lire, remitted from Florence ; exch. 30 1. 60 c. $* £1 ?

ROMAN STATES. — 10 bajocchi = Ipaolo ; 10 paoli = 1 scudo or crown = 4/2 nearly. Par of exchange, 48 paoli or pauls

45. On visiting Home, an Englishman changes £37 "12 "6. How many pauls does he receive at the rate of 47 £ pauls ^ £1 ?

46. How much sterling is = 3697 scudi, 68 baj., remitted from Ancona, at the rate of 48 pauls y £1 ?

NAPLES.— 100 grani = 1 ducat = 3/3J nearly. SICILY. — 600 grani = 1 oncia = 10/3£ nearly.

47. A merchant in Naples receives a bill from London to the amount of £861. To how many ducats is this equal; exch. 41d. y ducat?

48. How much sterling is = 846 oncie; exch. 123d. y oncia?

TURKEY. — 40paras=l^'«^re=2d. ; about 120piast.=£l. EGYPT. — 40 paras = 1 piastre=2%d. ; // 100 // =£1.

49. A traveller pays an interpreter at Constantinople the sura of 500 piastres. What is the value in sterling at 120 piastres f £1 ?

50. Change £125"! 0 to piastres at Alexandria @ 97 \ piastres f £1 . GREECE. — 100 lepta = 1 drachma = 8|d. nearly. Par of

exchange, 28 dr. 15 Ip. = £1.

51. Find the difference in Sterling and in Greek money between £44»16 and 1317 dr. 42 Ip., exch. at par.

EAST INDIES. — 12 pice = 1 anna ; 16 annas = 1 rupee = 1/10*.

52. A Calcutta merchant makes a payment of a lac or 100,00( rupees. Find the amount in sterling @ 1/10J.

53. Sent to Bombay goods worth £299"16»3. To how many rupees is this equivalent @ 1/1 0£ each?

CHINA.— 1000 le or cash =. 1 leang or tael, reckoned by the East India Company @ 6/8. 720 taels = 1000 dollars of 4/9 i nearly.

54. How much sterling is = 5400 taels, paid at Canton, reck- oning them @ 6/6 each ?

55. A merchant of Hongkong sells goods to the amount of £846 *13»4. How many taels does he receive @ 6/8 each?

G 2

154 EXCHANGE.

» INDIRECT EXCHANGES between two countries are effected through the medium of another. It is seldom that the medium is effected through more than one intermediate place.

(9) How much sterling must be paid in London to pay 749 Rigsbank dollars in Copenhagen through the me- dium of Hamburg; exch. 13 mk. 6 sch. banco = £1; 200 R. d. = 300 mk. banco.

£ x = 749 R. d.

R. d. 200 = 300 mk. b.

mk. b. 1 = 16 sch.

sch. 214 = £1

x == 749 X 300 X 16 __. £g^

56. How many francs = £250, sent to Paris through Hamburg ; exch. 13 mk. 14 sch. banco = £1 ; 185 fr. = 100 mk. ?

57. Find the number of mk. curr. = £180, remitted through Hamburg ; exch. 13 mk. 12 sch. banco = £1 ; agio 20 °/0.

58. How much sterling must be remitted to Berne through Paris to be equivalent to 6325 Swiss francs ; exch. 25 fr. 30 c. *p» £1 ; and 148 French = 100 Swiss francs?

59. How much sterling is = 60,180 paras; exch. between Con- stantinople and Vienna, 210 paras =r 1 florin ; between Vienna and London, 9 fl. 50 kr. = £1 ?

60. How much sterling is = 530 th. 7£ gr. ; exchange between Berlin and Paris, 3 fr. 60 c. ^ 1 th. ; between Paris and London, 25 fr. 20 c. y £1?

73. INVOLUTION.

INVOLUTION is the continued multiplication of a number by itself.

The continued product thus obtained is termed a Power of the given number ; and the number of times the number is used as a factor denotes the Index of the power. Thus 2X2X2X2X2X2 = 64 = sixth power of 2 = 26.

(1) Find the seventh power of 27. 27 X 27 X 27 X 27 X 27 X 27 X 27 = 10,460,353,203

Instead of multiplying by the number successively, we may use those powers of which the sum of the indices is equal to the index of the required power ; thus,

INVOLUTION.

73.

155

27X27X27= .... 19,683 .... - 27'

19683 X 19683 = 387,420 489 = 27»X2?i - 27« 387420489X27 = 10,460,353,203 = 27«X27 = 27' Find the following powers : —

1. 173

2. 32*

3. 36«

4. 5. 6.

98* 99 5

101 •

7. 8. 9.

II7 158 149

10. 13'°

11. 3095

12. 1002*

(2) Find the sixth power of T3r.

3- = 729 (T3T)e = 11° = 1771561

13. 14.

(I)3 (A)4

15. 16.

(*)•

(U)a

17. 18.

(f)8 (A)5

19. 20.

(*)' (fl)4

(3) Find the 5th power of 1-025 true to 6 decimal places (see § 39.)

1-025 XI '025 = 1-050625 l-050625xl'025 = 1-076891 1-076891 X 1-050625 = 1-131408

21. 1-04* to 4 pi,

22. 1-05* .. 6 ..

23. 1-03' .. 4 ..

24. 1-0256 to 6 pi.

25. 1-045' .. 7 ..

26. 1-035'°.. 7 ..

27. 2-625' to 6 pi.

28. 3-165" .. 4 ..

29. 9-9994 .. 4 ..

30. Find the area of a floor 19| ft. square.*

31. Find the cubic content of a die whose side is }| inch.

32. How many sq. ft. are contained in the aroura, 50 Greek ft. (each 1-01146 ft.) square?

33. How many sq. yd. are in the are, 10 metres (each 39'37079 in.) square?

34. Find how many cub. ft. are in the stere or cubic metre.

35. How many flagstones 14 in. square will be required to floor a kitchen 2 1 ft. square ?

36. Find how many cubes |£ inch in the side can be cut out of 7 cub. ft. 74 cub. in., allowing 3 cub. in. for waste.

Circles are proportional to the SQUARES of their diameters.

37. How many times is a circle 27 ft. in diameter as large as another 15 in. in diameter?

38. The paving of a circular floor 25*6 ft. in diameter cost £9" l.°,»4 ; what cost the paving of a similar floor 38-4 ft. in diameter?

* When we say a surface is 19| ft. square, we mean it contains 19| X 19£ square ft. A surface 10 ft. square is 10 times as large as a sur- face containing 10 square ft.

156 INVOLUTION.

73* Spheres are proportional to the CUBES of their diameters.

39. TJhe weight of a metallic ball £ inch in diameter is -398 oz. Find the weight of another of the same metal f inch in diameter.

40. A ball £ inch in diameter displaces '128 oz. of water; how many oz. will another 2£ in. in diameter displace ?

41. How many times is the Earth, whose mean. diameter is 7912 miles, as large as the Moon, whose diameter is 2140 miles?

A body, in falling, traverses 16'1 ft. during the first second, 4 X 16* 1/2. in two seconds, and so on, the SPACES traversed being proportional to the SQUARES OF THE TIMES.

42. Through what space will a body fall in 2J seconds?

43. To what height must an aeronaut ascend so that a ball let fall from his balloon may reach the ground in the quarter of a minute ?

I. The square of the sum of two numbers 9+7

is = the sum of their squares increased by 9+7 _

twice their product.* 92 _i_ 9x7

Thus, (9 + 7)2 = 92+2X9X 7 + 72 9X7 +7«

or 162 =81 + 126 + 49 = 256. 92+2(9X7)+72

Similarly, (40 + 3)2 = 402 + 2 X 40 X 3 + 3* or 432 = 1600 + 240 + 9 = 1849.

II. The square of the difference of two 9 — 7 numbers is = the sum of their squares dimin- 9 — 7 ished by twice their product.* 9^ 3X7

Thus, (9 — 7)-'=92 — 2X9X7 + 7' — 9X7 +72

or 22= 81 — 126+49 = 4. 92~

Similarly, (50 — 4)- = 50- — 2 X 50 X 4 + 42 or ' 462 = 2500 — 400 + 16 = 2116.

III. The product of the sum and the differ- 9 + 7 ence of two numbers is = the difference of 9 — 7 their squares.* 9219 O~~T"

Thus, (9 + 7) X (9 — 7) = 92— 7s — 9 X 7— 72

or 16X2=81 — 49 = 32. cp HjT

Similarly, (50 + 7) X (50 — 7) = 502 — 7 2 or 57 X 43 = 2500 — 49 = 2451.

* These propositions are more conveniently remembered in their algebraic form.

II. (a — &)* = «« — III. (a + &) (a — b] =a» —

INVOLUTION.

157

From III., we obtain a convenient method for obtaining the square of a number mentally*

By III. (77 + 3) X (77 — 3) = 77*— 3'

Hence (77 + 3) X (77 — 3) + 3* = 772 or 80X74 + 33 = 772

Find the difference between the given number and a num- ber near it ending in 0. Take a third number, so that the dif- ference between it and the given number may be = the former difference. The square of the given number is = the product of the other two numbers increased by the square of the common difference.

Thus, 93 2 = 90 X 96 + 9 = 8649.

From I., we obtain a method applicable when the number to be squared ends in 5 or £.

By I. 752 = 702+2 X70 X5 + 53 = 70X70 + 10X70+59 = 80X70 + 25

When the last figure is 5, the square may be found by multi- plying the number of tens by the next greater number, and then affixing 25. Similarly, (9fc)» = 9 X 10 + £ = 90*. Square the following numbers mentally : —

1. 21 2. 61 3. 33 4. 47

5. 56 6. 89 7. 74 8. 68

9. 72 10. 97 11. 82 12. 64

13. 85 14. 75 15. 35 16. 65

17. 195 18. 895 19. 395 20. 495

21. 19j 22. 22J 23. 17i 24. 25J

EVOLUTION.

EVOLUTION is that process by which we find a number which when multiplied a certain number of times by itself, repro- duces a given number. The number found is termed a root of the given number.

SQUARE ROOT.

. The SQUARE ROOT of a number, when multiplied by itself, reproduces the original number ; thus, 3 is the square root of

9, 3 = ^9 = 9* 5 8 = V64 = 64*'

Take any number, as 43, we know that 43 • = (40 + 3)» = 40* + 2 X 40 X 3 + 3*.

* This is sa Darvel, Ayrsh powers.

tid to be the method ^^^^^^A^ lire, who has acquired some celebrity for her arithmetical

158 EVOLUTION.

74. Let us now in re- 40

producing the number determine the method 2x40+3 of finding the Square Root.

40 '+2X40X3+3 '(40+3 40 2

2X40X3+3* 2X40X3+3'

Subtracting 402, we leave 2 X 40 X 3 + 32. Further to obtain the quotient 3, the divisor must be 2 X 40+3.

No number containing 1 figure can have more than 2 fig- ures in its square. No number containing 2 figures can have more than 4 figures in its square. Since1 1 place in a number corresponds to a period of 2 places in it§ square, ^efore ex- tracting the square root, we point off in periods oftwb places, commencing at units' place.

(1) Find ^1849.

The greatest square root in 18 ^s '4. Sub- 4 ) 18,49 (43 tracting 42, we have 2, which with the next * 1 period annexed is 249. Doubling* 4, we see "

that 8 in 24 is 3 times. Anne^ng;3 to 8^ we subtract 3X 83, and having no remainder, find 43 = ^1849. I ,

We have first subtracted 402 = 1600; we have then sub- tracted 2 X 40 X 3 + 3* or (2 X 40 + 3) X 3 = 249, to make up 43*. '

(2) Find ^12744$.

The greatest sq. root in 12 is 3. Sub- 3 )12,74,49(357 tracting 9, and annexing the next period, 9

we have 374. Doubling 3, we see that (55 \ "374 as we have a figure to annex to 6, the 325

next figure in the quotient will be 5. 7^7 "loiq Subtracting 5 X 65, and taking down the TO Tq

next period, we have 4949. Adding 5 • , to the divisor, we obtain 70, the double of 35. The next fig- ure being 7, we subtract 7 X 707 or 4949 ; and thus find that 357 = ^127449.

We have first subtracted 300« or 90,000. ^ Having then subtracted 2 X 300 X 50 + 50' or (2 X 300 + 50) X 50, we have now subtracted in all (300 + 50) 2 or 350*. We then subtract 2x350X7 + 7* or (2x350 + 7) X 7, fad thus complete the square of 357.

In extracting the square root, no remainder can be greater than twice the root obtained.

Thus, in finding the greatest square root in a number to be 8, it is evident the number is less than (8 -f- 1)* or 8* + 2 X 8 + 1. When 8* is subtracted, the remainder is therefore less than 2X8 -f- 1. or not greater than twice 8.

EVOLUTION.

74.

159

Find the square root of

1. 1024 2. 4225 3. 3136 4. 137641 5. 50625 6. 401956 7. 5499025 8. 9897316

9. 10. 11. 12. 13. 14. 15. 16.

7365796 27415696 20820969 14235529 16232841 70207641 31843449 79263409

17. 80568576 18. 62473216 19. 88887184 20. 22992025 21. 56987401 22. 58415449 23. 236144689 24. 998876025

I, Find ^672-35675 to 5 decimals. )6,72-35,67,5(25-92984 2 )6,72'35,67,5(25'92984

45

4_ 272 225 4735 4581 15467 10364 51849 5103

45

509 5182

509 5182

4

272

225 4735 4581

518588 5185964

. -.50 4666 41

437

0900

5,1,8,4

4148704 22 219600 20743856

1 475744

15467

10364

5103

4666

437

415

22

21

1

In extracting the square root of a number, we need only extract as many figures as the number next greater than half the number of the required figures. In the example before us, we require 5 decimals, and as there are 2 integral places in the root, there will thus be 7 figures in all. We need only extract 4 figures, and then finish as in Contracted Division (see § 40.)

Let us now examine the closeness of the approximation. In comparing the first part of the root which is extracted, with the second part which is required, we must attend to local value, by adding as many ciphers to the former as will give it 7 figures, the required number in the root.

When the square of the first is subtracted, the remainder is = twice the product of the first and second with the square of the second. We now merely divide this by twice the first, so that the quotient = the second with the square of the second divided by twice the first. Now the second contains 3 places, hence its square contains no more than 6 places ; and as twice the first cannot con-

160 EVOLUTION.

74* tain less than 7, the square of the second, divided by twice the first, is a proper fraction, and hence less than 1, so that the quotient is a convenient approximation to the second part of the root.*

(4) Find </-009 to 6 places. (6) Find ^T\ to 5 places.

TT = '63 7 ) '63 (-79772

49

149 1463 1341

1587 12263 11109

15,9,4 1154" 1116 38 32

"6 ^289 = 17; ^3136 = 56

9 ) -0090(-0948G8 81
184	900 736
1888 18,9,6	16400 15104
1296 1138
	158 152
	6

When the root cannot be expressed exactly, carry the decimal to 6 places.

25. 15-7609	35. 11-	45. -042849
26. -180625	36. 45-	46. -081
27. 2889-0625	37. 16-675	47. T%
28. -001296	38. 28-75	48. fH
29. 152-399025	39. 43-384675	49. TV
30. -00494209	40. 3-16227766	50. |
31. 7-	41. 7-0030025	51. 4
32. 2200-	42. -0000016	52. ^ of 114	|
33. -025	43. -00784	53. T^T of 48|
34. -0729	44. -000784	54. 2 |S of5|
The side of a square is found by extracting the square root	»/
its area.
55. Find the side of a square whose area is 1000 sq. yd.

* For conciseness, let a = first part with ciphers having 2n or 2n -f- 1 figures, b = second part with n — 1 or n figures respectively ; then the remainder = 2ab + 62, which divided by the divisor 2a = 6-|~r~- Now 62 cannot contain more than 2 (n — 1) or 2n fig. respectively; and 2a not fewer than 2n or 2n + 1 respectively ; hence ^ is a proper frac- tion, &c. (See Kelland's Algebra, p. 57.)

EVOLUTION. 161

74. 56. Find the length of the side of a square field containing an acre.

57. The area of Great Britain and Ireland is 122,091 square miles ; find the side of a square tract of land of equal extent.

58. How many yd. are in the side of a square, equal in area to a rectangle 972 yd. long and 1296 ft. broad?

59. A rectangle is 240 yd. long and 450 ft. broad; find the side of a square 10 times as large.

60. Find the side of a square of equal extent to 3 fields re- spectively 15 ac. 3 ro. 17 po. ; 11 ac. 3 ro. 36 po. ; 5 ac. 1 ro. 36 po.

Diameters of circles are proportional to the square roots of their areas.

61. Find the diameter of a circle twice as large as another whose diameter is 120 ft.

62. Find the diameter of a circle £ of the area of another whose diameter is 30 ft.

C

In a right-angled triangle, the square of the hypotenuse is = the sum of the squares of the base and the perpendicular.

Thus, AC2 = AB3 + BCa

(Euclid I. 47).

A. E

When the hypotenuse is wanted, we square the base and the perpen- dicular, and extract the square root of their sum. When the base or the perpendicular is wanted, we square the hypotenuse and the per- pendicular or the base, and extract the square root of their difference*

63. Base = 39, Perpendicular = 52 ; find Hypotenuse.

64. Base = 180, Perpendicular =19; find Hypotenuse.

65. Base = 35, Hypotenuse = 91 ; find Perpendicular.

66. Base = 13, Hypotenuse = 85 ; find Perpendicular.

67. Perpendicular = 18, Hypotenuse = 82 ; find Base.

68. Perpendicular = 72, Hypotenuse = 75; find Base.

To obtain integral numbers to represent the sides of a right- angled triangle, take any odd number as the base or the perpendic- ular ; from its square, subtract 1, and divide by 2, for the per- pendicular or the base ; the latter number increased by 1 will be

the hypotenuse. Thus, base = 7 ; perpendicular = — ^- = 24 ; hypotenuse = 25. Any multiple of these numbers will also suffice.*

* Let n — base or perpendicular ; ? ~ = perpendicular or base ; -n-^t^ = hypotenuse. (See Notes in Leslie's " Elements of Geometry " on Euclid I., 47.)

162 EVOLUTION.

» 69. Find the diagonal of a rectangular field whose sides are 20 yd. and 14 yd.

70. Find the diagonal of a wall 28 ft. long and 15 ft. high.

71. Two vessels sail from the same point, the one due north 51 miles, the other due east 68 miles; how many miles are they distant from each other ?

72. How many feet from the base of a house must a ladder 27 ft. long be placed to reach a window 2 1 ft. high ?

73. Find the length of a" cord stretching from the vane of a steeple 95 ft.- high to a point 40 ft. from its base.

74. A cord 287 ft. long is stretched from the top of a column 63 ft. high ; find the distance of its point of contact with the ground from the base of the column.

75. A room is 28 ft. long, 21 ft. broad, and 12 ft. high; find the length of the diagonal of the floor or the roof, of the side walls, and of the end walls.

76. In the same room, find the length of the diagonal from a corner of the roof to the opposite corner of the floor.

^JT The square of the diagonal of a room = the sum of the squares of the length, the breadth, and the height ; for the sum of the squares of the length and the breadth = square of the diagonal of the floor, which increased by the square of the height = the square of the diagonal of the room.

77. Find the diagonal of a hall, 50 ft. long, 30 ft. broad, and 15 ft. high.

78. Find the breadth of a street from a point in which a ladder 50 ft. long reaches a window 40 ft. high on one side, and another 48 ft. high on the other.

When the same number occupies the 2d and 3d terms of a proportion, it is a Mean Proportional between the 1st and 4th. Its square is therefore = the product of the extremes ; and the M. P. of two numbers is hence = the square root of their product ; thus, 24 is M. P. of 18 and 32.

79. Find M. p. of 16 and 49.

80. Find M. p. of 84 and 140.

81. Find M. p. of 5T5 and 3'g of \\.

*J3T The true weight of a body successively weighed in the scales of a false balance is the M. p. between the apparent weights.*

* Let the lengths of the arms of the balance be a and b respectively, x the true weight, m, w, the apparent weights.

x : m : . a . 1 n : x : : a : o \

EVOLUTION. 163

74* 82- A body successively weighed in the scales of a false balance appears to be 12 J Ib. and 12 J Ib. respectively ; find its true weight.

83. A body appears to weigh 5-^ Ib. in one scale and 5| Ib. in the other scale of a false balance ; find its true weight.

^" The times in which bodies fall are proportional to the square roots of the spaces traversed. Since 16-1 ft. is traversed during the first second ; to find the time, we divide the space by 16-1 and extract the sq. root.

84. In what time will a stone fall to the bottom of a coal pit 70 fathoms deep ?

85. In what time would a body fall from the N. or the S. Pole to the centre of the earth, taking the Polar Radius as 20,853,810 ft. ?

75. CUBE ROOT.

When the CUBE ROOT of a number is raised to the third power, the number itself is reproduced ; thus 8 = cube root of 512 = $/512; 83 = 512.

Take any number, as 50'+2(50x9)+9a 59, we know that 59 2 = 50+9

503+2(50*X9)+ (50X9*) 9*. Multiplying by (50*X9)+2(50x9«)+93

" -

50x93+93.

In reproducing 59 or 50 + 9, let us determine the method of rinding the Cube Root.

)503+3x502X9+3x50x9*+93(50+9 503

3x50*+3X50x9+9* ) 3x50*x9+3x50x92+93 3x50l2x9+3x50x9*+93

Subtracting 503, we leave 3 X 50» X 9+3 X 50 X 9* + 93. Further, to obtain the quotient 9, the divisor must be 3X50 * +3X50X9+92, or 300x52+30x5x9+92.

(1) Find y 205379. METHOD i.

A number of one figure has 205,379(59

no more than three figures in 125^

its cube ; a number of two fig- 80379

ures has no more than six. 300 X 52=7500

Since one place in a number 30x5x9=1350

corresponds to a period of 92 = 81

three places in its cube, before 8931 80379

extracting the cube root, we . ~

point off in periods of three places, commencing at units place.

164

EVOLUTION.

75* The greatest cube root in 205 is 5. Subtracting 5s, we leave 80, which with the next period annexed is 80379. As we have to add other numbers to 300 X 52 = 7500, we may require to make repeated trials to obtain the second figure. 7 + " some number to be added" may go 9 times in 80. We then take 30x5x9 = 1350, and 92 =81, and adding them to 7500, subtract 9 X 8931. As there is no remainder, we find that 59 = y 205379. Having thus in the first part subtracted 503, we have next subtracted as much more as makes up 59 3.

We may vary the/orm of working as in the following methods : —

METHOD II.

METHOD in.

205379(59

125

75 80379 159 1431

8931 80379

205379(59 125

300 X 5* =7500 80379 30X5=150 __9 9X159=1431

8931 80379

In Method II., 9(30 X 5 + 9) =30 X 5 X 9+9'.

In Method III., we abridge the process, by omitting the equivalents, and, instead of writing ciphers, we merely attend to the relative local value of the figures.

Find the Cube Root of the following numbers : —

3. 4.

357911

148877

1. 9261

2. 29791

(2) Find ^45499293.

METHOD II.

45,499,293(357

27

300 X 3* = 2700 18499 30 X 3 = 90 _5

5X95=475

3175 15875

300X35'= 367500 "2624293 30X35=1050 _7 7X1057=7399

374899 2624293

5. 6.

103823 474552

95

METHOD III.

45,499,293(357 27

27 18499 475)

3175 1 15875 25) 2624293

3675

1057 7399 374899

2624293

EVOLUTION.

165

75. In METHOD II., having found the first two figures of the root as be- fore, we take 300 X 35*, and finding the third figure to be 7, we make up the divisor as we did for the second figure. In METHOD III., having found the first figure 3, we write 3 X 3 or 9 in one column, and 3X9 or 27 in another. Finding the next figure to be 5, we annex 5 to 9, and by putting 5 X 95 or 475 two places to the right of 27 obtain 3175. By subtracting 5 X 3175 from 18499, we find the remainder 2624. We ob- tain 3 X 35«, by adding 5* or 25 to 3175 and 475. We now triple the last figure of 95, and obtaining 15, write 5 and carry 1 to 9, and thus have 105 = 3 X 35. By annexing 7 to 105, we add 7 to 30 X 35. We now multiply 1057 by 7, and by writing the product two places to the right of 3675, we add 7399 or 7(30 X 35 -f 7) to 367500 or 300 X 35«. We now subtract 7 X 374899, and find 357 = ^45499293.

In the accompany- 2700 = 3X302 ing process we show ( 475 = 3X30X5+5*

why 3 X 35a is ob- -{ 3175 = 3X302-j-3X30X5-}-52 j tained by adding 5a ( 25 = 5»)

to 3175 and 475 :— 3675 = 3X30«-f 6X30X5+3X5'

= 3(30*+2X30X5+5«)=3X35*.

7. 53157376 13. 184608795384 19. 570547876184

8. 62099136 14. 103690516392 20. 455289041557

9. 41421736 15. 102700479987 21. 1881365963625

10. 12812904 16. 305501115375 22. 160288833718161

11. 113379904 17. 597585982967 23. 184676889190123

12. 1458274104 18. 327510203957 24. 497640375631125 We may often shorten the operation by Contracted Division.

(3) Find V 12396-8834.

METHOD III.

)12,396-8834(23-14395 _8 4396

63

691

4167 229883

691)

159391 I 159391 1 ) 70492

5,9.3 _277

160083 __277 16036,0 _j28

16064 2

64144 6348 4820 1528 1446 82

166

EVOLUTION.

75.

(4) ;/27 =

^1331 =

(5) Find

= -95647—

25. 250-047 26. 175-616 27. 87528-384 28. -000068921 29. -000405224

30. -000970299 31. 2126-781656 32. 24212-815957 33. -00027 34. -00008

35. TVV 36. «* 37 - — 38. f'of Jof I,8* 39. T8Tof T\of8j

The side of a cube is found by extracting the cube root of its content or volume.

40. A cube contains 5832 cub. in. ; find the length of its side.

41. The Imperial gallon contains 277-2738 cub. in ; find the side of a cube containing a gallon.

42. The litre, the French standard of capacity, contains 61 '027 cub. in. ; find the side of a cube containing a litre.

Diameters of spheres are proportional to the cube roots of their contents.

43. Find the diameter of a sphere nine times as large as another whose diameter is 150 ft.

44. The Equatorial Diameter of the Earth is 7926 miles; find that of Venus, whose volume is -953 of that of the Earth.

Kepler1 s Third Law:— the SQUARES OF THE TIMES in which the planets revolve round the sun are proportional to the CUBES OF THEIR MEAN DISTANCES from the sun.

45. The periodic time of the Earth is 365*256 da., and of Venus 224-701 da., if the Earth's distance = 1, find that of Venus.

(S65-256)2 ; (224-701)2 : : 1 : a; dist. of Venus = t/x.

46. The periodic time of Jupiter is 4332-585 da., if the Earth's distance = 1, find that of Jupiter.

HORNER'S METHOD.

William G. Homer's Method of Finding Roots is applicable to the solution of ANY HOOT.

(1) Find ^45499293. Having found the greatest cube root in 45 to be 3, we write 3 in one column, 3s or 9 in another, and subtract 3s or 27 from 45. We return to the first column, and by adding in 3 obtain 6. We now add 3 X 6 or 18 to 9 in the second column and obtain 27. Again, we add 3 to 6 in the first column. Making allowance for what	3 3 6 3	9 18 27 475	45,499,293(357 27 18499 15875
95 5	3175 500	2624293 2624293
100 5	3675 7399 374899
1057

EVOLUTION.

107

76 may ^e car"e<^' we find that 27 when increased may go 5 times in 184. * In the first column, we place 5 one place to the right of 9 and obtain 95. In the second column, we write 5 X 95 or 475 two places to the right, and by adding obtain 3175. 5 X 3175 or 15875 being put three places to the right of 18, or under 18499, we obtain the remainder 2624. Re- turning to the first column, we add 5 to 95. In the second column, we add in 5 X 100 and obtain 3675. In the first column, we again add in 5. Finding the next figure in the root to be 7, we annex 7 to 105 in the first column. We place 7 X 1057 two places to the right in the second column, and obtaining 374899, place 7 times this sum in the third. We have thus found the CUBE root of 45499293 to be 357. To facilitate comparison, the figures in this process, which are common to the divisors in Methods II. and III. (see page 164), are printed in a bolder type.

(2) Find

We place 1, the integral part of the fourth root of 12, in the first column ; 1» or 1 in the second; 1» or 1 in the third; and I4 or 1 under 12 in the fourth. In the first column, by adding in 1 to 1 we obtain 2 ; in the second, by adding in 1 X 2 we obtain 3 ; and in the third, we add in 1X3 and obtain 4. Returning to the first, we add 1 to 2 and obtain 3 ; and in the second 1X3 added to 3 produces 6. We again add in 1 to the first column and obtain 4.

Finding the next figure in the root to be 8, we put 8 one place to the right in the first column and obtain 48. We then put 8 X 48 two places to the right in the second, and by adding obtain 984. We write 8 X 984 = 7872 three olaces to the right in the third column, and by adding ob- tain 11872. We then put 8 X 11872 or 94976 four places to the right of 11 in the fourth column and subtract it from 110000.

The work is carried on so that while each figure in the FOURTH root is added four times in the first column, three products are added in the second, two are added in the third, and one is subtracted in the tourtn. A f ter finding the root to be 1 -86 we finish the work by Contracted Division.

1 0

1

1

48 8

732

__6

738

6

7,4,4

2 3 3

384

984 448

12 (1-86120972—

JL 110000

94976

150240000

U7123216 3116784

203148

4428

207576

74

20765^0

7 2^7,7,2

2578°.?<> _ £5

257851,1

168 EVOLUTION.

76* The fourth root of a number may be found by taking the square root * of its square root ; the sixth root, by taking the square root of its cube root, &c.

Find the following roots by Horner's Method :

1. #2 4. #228886641 ' | 7. #21224-09008801

2. #20 5. #35806100625 8. #81-108054012001

3. #200 6. #20730-71593 9. #148035889

10. #17 11. #^iT 12. #|| of A of li.

77. SCALES OF NOTATION,

IN the common notation, the local value of the figures ascends in the SCALE of TEN. We may, however, adopt other scales : In the scale of 6, " 1 " in the second place being six times the value of " 1 " in the first, " 10" represents 6, the lose of the scale. Again, " 1 " in the third place being six times the value of " 1 " in the second, " 100" represents 36, the second power of the base. " 2534 " in the scale of 6, or (2534) e, is = 4 + (3 X 6) + (5 X 6*) + (2 X 63) ; (65284). = 4+ (8 X 9) +

The number of characters used in any scale is denoted by its base. In the scales of 11 and 12, we may represent 10 by D for Decem; and in the scale of 12, 11 by U for Undecim. (1) Express 451 in the scale of 6.

/» j r -t

In dividing 451 successively by 6, we

75//1 find that 451 = (2X6* )-f (OX6«)-f(3X6)

I i

451 = (2031),.

To reduce a number in the decimal scale to its equivalent in another scale, we divide the number successively by the base of the latter, and to the final quotient annex the succes- sive remainders.

1. Red. 666 to scale of 6

2. » 315 // // 4

3. // 225 // // 7

4. Red. 313 to scale of 8

5. // 222 // // 2

6. // 1859 n // 12

(2) Express (1234) a in the decimal scale. 1234 5 7 (1234), = (1X53) + (2X5') + (3x5) + 4

= 5{(1 X 5*) + (2 X 5) + 3} + 4

194

SCALES OF NOTATION. 169

To reduce a number in any scale to its equivalent in the decimal scale, we multiply the left-hand figure by the base of the former, and add in the next figure to the right, and pro- ceed similarly till all the figures are taken in.

Reduce the following to the decimal scale :

7. (423) 5

9. (3567)8 11. (2D98),,

. 5

8. (1243) 6 10. (12345) 9 12. (DU10)la

(3) Express (2143) 8 in the scale of 7.

To reduce a number from one _(2143)6 7298 scale to another, of which neither n 7 42//4

is the decimal, we first reduce to gg gx/Q

the decimal, and then to the re- -— ^ .

quired scale. 298 = (604)*

' 13. Reduce (1001001), to the scale of 3.

14. Reduce (2D43)j , to the scale of 7.

15. Reduce (4U57) l , to the scale of 2.

The pupil will now see that the "higher the base of the scale, the fewer figures are necessary to represent any number; but lit same number of figures is required in two scales, then the left-hand figure in the Mgher is less than that in the lower scale.

(4) Reduce 23, 34, 41, to the scale of 3, add them and prove the work.

23 = (212) 3 (10122).

34 = (1021). *

41 = (1112) . gg

98 = (10122). 98

Tnd in the 4th, 3 = (10),.

16 Reduce 64, 127, 95, to the scale of 2, and find the sum.

17 Reduce 2^ 14B, 79, to the scale of 12, and find the sum.

(5) Reduce 2002 and 1271 to the scale of 4, and find their difference. (23123) 4

2002 = (133102) 4 Tl

1271 = J103313)^ "731 = "723123^ lg

3 from 2 we cannot, 3 from 4 leaves 1, 1 and 2 are 3. 1 and 1 make 2, 2 from 4 leaves 2, &c. H

170 SCALES OF NOTATION.

•7*7 18. Reduce 625 and 367 to the scale of 5, and find their difference.

19. Reduce 237 and 74 to the scale of 9, and find their difference.

The Arithmetical Complement (A. c.) of a number in any scale is obtained by subtracting the number from the base, or the next greater power of the base. The Arithmetical Com- plement of a number is so called because its figures and those of the number together fill up the scale.

In the decimal scale, A. c. of 7 = 10 — 7 = 3 ; A. c. of 213 =r 1000 — 213 = 787.

In the scale of 6, A. c. of (3). = (10) 6 — (3)e = (6)6 ; A. c. of (342) e = (1000) .- (342) . = (214) ..

The best method of finding A. c. is to commence at the left hand, and subtract each figure from the base diminished by one, except the right-hand figure, which we take from the base. In the scale of 8, to find A. c. of (263) 8, we take 2 from 7, 6 from 7, and 3 from 8, and thus obtain (515)8.

20. In the decimal scale, find the A. c. of 43, 726, and 2817.

21. In the scale of 6, find the A. c. of (24) 6, (253)6, and (1243)e.

22. In the scale of 12, find the A. c. of (24) 12, (346) 12, and (28DU)12.

(6) Reduce 1691 and 127 to the scale of 12, and multiply them.

1691 = (U8U)la (I>4345)ia

127 = _(D7)la -124

11837 6D25

20292 9952

__

214757 = (D4345)ia 7 times U = 77 = (65) 12. Write 5 and carry 6, &c.

23. Reduce 2341 and 725 to the scale of 7, and multiply them.

24. Reduce 741 and 1286 to the scale of 6, and multiply them.

25. Reduce 198 and 241 to the scale of 12, and multiply them.

(7) Reduce 753 and 29 to the scale of 7, and find tho quotient.

153 "264 224

(40),

26. Reduce 864 and 72 to the scale of 3, and find the quotient.

27. Reduce 78467 and 317 to the scale of 12, and find the quotient.

171

78.

DUODECIMALS.

IN Duodecimal Multiplication, we descend in the scale of twelve pom the/«tf, which is adopted as the unit of computation. . ^towdfoot is divided into 12 inches or primes (') ; an inch into 12 hues, parts, or seconds (") ; a line into 12 thirds ("'), &c. Descending from the square foot in the duodecimal scale, the names are as follow : twelfth of sq.ft. (') ; 8q. inch (") ; twelfth ofsq. in. ("') ; sq. line ("" or Iv), &c.

Let AB be a lineal foot, di- vided into 12 in. each = BH. The square BD is a .<«?. /oo£, containing 144sq. inches, each = BG. BF is the twelfth of a sq.ft., containing 12 sq. in. The twelfth of a sq. ft., which is often erroneously called "an inch,"* is a surface, whose length BA is a foot, and breadth BE an inch.

			1	1		1
			1	1		1
			1	1		1
			1	1
			1	1
			1	1
			1	|L		1
			1			1
			1			1
			1			1
			1			I c
			1			CM

Find the area of the rectangle AMLK, seven inches long and jive inches broad. The rectangle contains 5 rows of sq. inches, and in each row there are 7 sq. inches. The number of sq. inches in the rectangle = the product of the number of lineal inches in each di- mension = 35 sq. in. We thus see that the product of the number of lineal units in the length of a surface by the number of lineal units in the breadth is = the number of square units contained in the surface.

(1) Find the area of a surface, 4 ft. 3 in. by 3 ft. 2 in.

«. ' 4//3 o c\		4 ft. 3 in. 3 ft. 2 in.	= 51 in. = 38 in. 408
12//9			153
8 n	6		12 1938 sq. in.
13//5'//	6"		12 161' // 6"
ft. 3ft.X3m. =3 3 ft X4 ft — •	ft. sq.ft. X TST = T\ .... 12	Area = 13	i O K/ £•" Id // 0 ft o sq. ft. ; 5 twelfths
2in.x3in. = T\ 2in.x4ft. = T\	XT3T = T$* X 4 = A	of sq. ft.	; 6 sq. in.

* Errors of this kind perpetuated among artificers by their continued ;:eadyReckoners,byconfusingnames,generatefalse ideas, oiw&icn

172 DUODECIMALS.

78. (2) Find the area of a surface 3 ft. 4' * 7" long, and 2 ft. 9'* 10" broad.

ft. ' " ft. • "

3*4* 7 3*4* 7 = 487

2*9 *10 2*9*10 = 406

6*9* 2 2922

2*6 * 5 *3 1948

197722 sq. lines 9*6'* 5"*0'"*10" -16476*10"

Area = 9sq. ft., T\. sq. ft., 5

sq. in., T°5 sq. in., 10 sq. lines.

9(*/ * u

. (3) Find the area of a surface 75 ft. 9'*9" by 16 ft. 4'*7".

When the number of feet is 6*3* 9'* 9"

greater than 12, we may either 1 * 4 * 4*7

keep the number as it is, or -^ — = — p- s

extend the duodecimal scale. 6*d*9* I

The dimensions = (63) . , » 9' 2 * 1 * 3 * 3*0

» 9", and (14) , ,» 4'»7". The 2 * 1 * 3*3 * 0

product = (875) l , sq. ft., &c. 3*8*2*8 * 3

= 1241 sq. ft, &c. /x Q///— oiv

Find the area of surfaces of the following dimensions : —

ft. ' ft. ' ft. ' " ft. ' " ft. ' " ft.

1. 3* 2X 2* 3

2. 5* 3X 6* 7

3. 7*10X 8*11 4.13* 6X 9* 8 5.18* 7X 7// 8 6.18* 9X12*10

7. 7*1*6X 2* 4*3

8. 4//4//6X 5* 6*7

9. 8*9*7 X 9* 6*5 10.19*3*6X 7* 4*9 11.19*8*6X11*10*9 12. 32*3*7 X 9*11*9

13.28*9*11X11*11*11

14.34*5* 6X15* 4* 7 15.43*9*10X28*11*11 16.73*6*11X18* 3* 6 17.64*5*10X16* 9* 9 18.76*9* 5X21*11* 3

19. Find the content of a board 6 ft. 3 in. long, and 4 ft. 7 in. broad.

20. Find the area of a floor 16 ft. 4 in. long, and 14 ft. 8 in. broad.

21. Find the area of a square court whose side is 17 ft. 11 in.

22. What is the content of the ceiling of a square room whose walls are 12 ft. 5£ in. broad ?

the following is an illustration : A master carpenter once stated that he had often been puzzled by the seeming discrepancy between the extent of a surface as measured and as computed. Laying down a surface 15 in. by 13 in. he marked off a square foot, and observed that the true content of the remainder seemed to be different from that given in the computed result His difficulty, however, vanished when he found that the answer was NOT 1 sq. ft. 4 in. 3 pts., but 1 sq. ft. 4 tivelfths of sq. ft., 3 sq. in. = 1 sq. ft. 51 sq. in. The twelfth of a cubic foot is also erro- neously termed an inch.

DUODECIMALS. 173

78. 23- How much sheet-iron will be required to line the lower half of 12 window shutters, each 8 ft. 2 in. high, and 1 ft. 4 in. broad?

24. How much veneering will be required to cover the surface of 6 counters, of which 2 are each 12 ft. 3 in. by 3 ft. 4 in. ; 3 each, 10 ft. 6 in. by 3 ft. 4 in. ; and the other 6 ft. 8 in. bv 2 ft 10 in.?

25. How many sq. yards are in the walls of a room, 18 ft. 3 in. in height, and 96 ft. 8 in. in circuit ?

26. How many sq. ft. of paper are in a book containing 288 pages, each 7 in. by 4| in. ?

27. How much glass will be required for the front windows of a house of 3 flats : the ground floor containing 6 windows each 7 ft. 4 in. by 3 ft. 4 in., and a fanlight 1 ft. 10 in. by 3 ft. 4 in., and each of the upper flats 7 windows, each 7 ft. by 3 ft. 4 in. ?

28. A square court, whose side is 19£ ft., contains a grass-plot 13 ft. 6 in. by 12 ft. 8 in. How much is left to be macadamized?

(4) Find the price of painting a wall 25 ft. 6 in. long, and 14 ft. 4 in. high, @ 1/1} W sq. yd.

25//6/

14//4' Ijd. j I/

40 5

0//8J

357//0 ii of 1/1}

8//6//0 45//8J

365//6'//0" £2//5//8i

9)365} sq. ft. ~~40-}-| sq. yd.

29. Find the price of 12 panes of glass, each 1 ft. 5 in. by 11 in @ 2/3 ty sq. ft.

30. How much must be paid for lining the bottom of a reservoir 32 ft. 3 in. long, and 14 ft. 8 in. wide, with asphalt @ 2/3 «p« sq. yd.

31. Find the expense of whitewashing the ceiling of a square room, the breadth of the wall being 10 ft 6 in. @ 3d. & sq. yd.

32. What should be paid for causewaying a street 62 yards long and 12 ft. 6 in. broad, @ 1/6 tp sq. yd. ?

33. Find the cost of paving a court 58 ft. 9 in. long and 21f ft. broad @ 2/3 ^ sq. yd. ?

34. What must be paid for painting a stair of lo steps, e£ ft. 7 in. broad, 7 in. high, and 10 in. wide, @ 1/6 f sq. yd.?

(5) Find the superficial content of the walls and ceilmg of a room 15 ft. 6 in. long, 12 ft. 4 in. broad, 10 ft. 7 in. high.

DUODECIMALS. ft. ' 55 * 8 10*7	ft. ' 15*6 12*4
556 * 8 32 * 5 *8	186*0 • 5*2 *0

174

78. 15 * 6

+12* 4

27 // 10'

2

55 n 8' Circuit. 589 // r*8" Walls. 191 * 2'*0" Ceiling.

35. How many sq. ft. are in the walls of a room 15 ft. 6 in. long, 13 ft. 4 in. broad, and 11 ft. 2 in. high?

36. How many sq. ft. will be required to line a cistern, without lid, 4 ft. 6 in. long, 3 ft. 8 in. broad, and 4 ft. 5 in. deep?

37. Find the cost of painting the walls of a room 13 ft. 6 in. long, 12 ft. broad, and 9 ft. high, @ 1/6 ^ sq. yard.

38. Find the cost of painting the outside of a box, except the bottom, length and breadth each 3 ft. 4 in., and depth 2 ft. 8 in., @ 1/3 ^ sq. yd.

39. How many sq. yds. of plastering are in the walls and ceiling of a room in the form of a cube 12 ft. each way, deducting for window 6 ft. 3 in. by 3 ft. 2 in., door 7 ft. 6 in. by 3 ft. 6 in., and fireplace 4 ft. 3 in. by 3 ft. 4 in. ?

40. How many copies of a pictorial newspaper of 4 pages, each 28 in. by 20 in., will be required to cover the walls of a country barber's shop, 18 ft. 8 in. long, 14 ft. 4 in. broad, and 8 ft. 10 in. high, allowing for 2 windows each 5 ft. 6 in. by 3 ft. 2 in. ; 2 doors each 7 ft. by 3£ ft. ; and fireplace 3 ft. by 2£ ft. ?

Descending from the cubic foot in the duodecimal scale, the names are : TV of cub. ft. ('), T^T of cub. ft. ("), cub. in. (/x/), TV of cub. in. (lv), T^ of cub. in. (v), cub. line (VI).

(6) Find the cubic content of a solid, 11 ft. 4| in. long, 3 ft. 3T\ in. broad, and 2 ft. 4J in. thick.

11 * 4'*4// 37// 3'*9"*9'" //SIV

3* 3//5 2// 4 *3

34 // 1 *0 74* 7 *7 * 7 *4"

2 *10 //I //O 12// 5 *3 // 3 //2 *8

4//8 *9 *8 9 //3 //ll *5 *5 *0

37 * 3'*9"*9'"*8lir 87/'10'*2"* I0x//*0iv//lv//0vl~

Cubic content = (87 + i§ -f TJT) cub. ft. + (10 + i\ _|- _^) cub. in. -}- 0 cub. lin.

Find the cubic content of solids of the following dimensions . —

ft. ' ft. ' ft. ' ft. ' " ft. ' " ft.

41. 8//11 X 7*8 X 6//7

42. 9// 6 X 6'/6 X 4//3

43. 9// 7 X 6*8 X 5//4

44. 11//3//4 X 6*9*10 X 5*4*6

45. 12*4*6 X 8*6* 8 X 4*6*6

46. 9*6*7 X 3*4* 5 X 5*4*3

DUODECIMALS.

175

78, 47. Find the solidity of a block of granite 8 ft. 4 in. long, 6 ft. 6 in. broad, 5 ft. 7 in. thick.

48. Find the cubic content of a slab of marble 5 ft. 6 in. long, 4 ft. 3 in. broad, 1 ft. 10 in. thick.

49. How many cubic ft. of air are in a room 35 ft. 6 in. long, 20 ft. 8 in. broad, and 12 ft. 4 in. high ?

50. Find the weight of sea- water in a cistern 1 1 ft. 3 in. long, 6 ft. 7 in. broad, and 5 ft. 6 in. deep, the weight of a cubic foot of sea- water being 1025 oz.

51. Find the weight of a log of oak 10 ft. 5 in. long, and 2 ft. 3 in. square throughout, the weight of a cubic foot of oak being 925 oz.

52. Find the cost of a block of lead 1 ft. 3 in. long, 9 in. broad, 8£ in. thick, taking the weight of a cubic foot of lead at 709 Ib. and the price @ £23 "10 & ton.

SEKTES.

A SERIES is a succession of numbers which mutually depend on one another, according to a certain law.

^ ARITHMETICAL PROGRESSION.

An ARITHMETICAL PROGRESSION (A. P.) is a series of num- bers uniformly ascending or descending by a constant deference and is therefore appropriately termed an EQUIDIFFERE

--

7 10

is an ascending

which 1 is the

common difference.

The latter series is as follows :

= 50 =50

= 50 — (1X4) =46 = 50 -(2X4) =42 = 50 - (3X4) = 38 = 50 — (4X4) = 34 &c. &c-

.ION (H. P.) is a series of ™m^s/j££ Reciprocals of the terms of an Arithmetical Pro- gress^; <nus, o, „ "^are in A. P, and |. 4, * *, m H.P. |, i i fc are in A. P., and | 2, 3, 6, in H. P.

The former series is as follows :	The la Term
Term 1st 2d	=	1 1	+ (1X3)	= 1 = 4	1st 2d
3d 4th 5th	"==- '•	1 1 1	+ (2X3) + (3X3) + (4X3)	= 7 = 10 = 13	3d 4th 5th o _
&c.			&c.	&c. <^c

176 SERIES.

79 • To obtain any term in an A. p., we multiply the common difference by the number less by one than the number showing the rank of the term in the series, and add the product to the first term, or subtract it from it, according as the series is ascending or descending.

In the first series, the 100th term is = 1 + (99 X 3) = 298. In the second, the 10th term is = 50 — (9 X 4) = 14.

(1) Find the 36th term in the A. p. 5, 5}, &c. Difference = i ; 36th term = 5 + (35 X J) = 22J.

(2) Find the 20th term in the A. p. 7, 6j, &c. Diff. = J; 20th term = 7 — (19 X |) = 4|.

Find the

1. 10th term in 1, 3, 5, &c.

2. 100th // 2, 4, 6, &c.

3. 25th » 7, 11, 15, &c.

4. 73d " 18, 22, 26, &c.

5. 36th » 1J, 2J, 3, &c.

Find the

6. 13th term in 3|, 4J, 4j, &c.

7. 100th " -015,-02,-025,&c.

8. 50th » 100, 99^,99, &c.

9. 30th » 50, 48}, 4?i, &c. 10. 19th » 12,ll-75,ll-5,&c.

11. A number of nuts is divided among 30 boys. The first gets 120, and each boy gets 3 fewer than the one preceding. How many does the thirtieth get ?

12. A clerk is engaged for £70 the first year, with an increase of 7 guineas for every successive year. Find his salary for the seventh year.

13. A body falls 16'1 feet during the first second; thrice as far during the second ; five times during the third ; and so on. How far would a body fall during the sixth second ?

14. Of seven frigates, the first has 66 guns, the second has 4 fewer, and so on with the same difference. How many has the seventh ?

15. Thirteen trucks are laden with coal ; the first contains 5*65 tons, and each truck has 2-5 cwt. more than the one preceding. How much coal is on the last truck ?

Take the A. p. 8, 11, 14, 17, 20, 23.

We find that 8 + 23 = 31 11 + 20 = 31 14+17 = 31 Sum of the A. p. = 3(8+23) = 3 X 31.

Take the A. p. 70, 63, 56, 49 42.

We find that 70 + 42 = 112

63 + 40 = 112

56 = J of 112

Sum of the A. p. = 1(70+421 — 2J X 112.

SERIES. 177

79. The sum of an A. p is == the product of the sum of the first

the last term by half the number of terms.* Any term in an A. p. is the Arithmetical Mean between two terms equidistant from it; thus, 14 is the A. M. between 11

™d ll; J17' the A' M' between 11 and 23 ; 56, the A. M. between 10 and 42.

(3) Find the sum of the series 2, 5, to 51 terms.

51st term = 2 + (50 X 3) = 152. S = V (2 + 152) = ^*J!! = 3927.

16. FindS. of 4, 10, 16, to 50 terms J7. " " i £, 1, " 30 „ 18. " " I, li,l, " 40 »

19. FindS.of-01,-03,-05,to29terms

20. » »/ 2,1-9, 1-8, "15 n

21. >' n 80,77^,75, "30 »

22. In Venice the clocks strike to 24. How many strokes are made in a day ?

23. A boy gains 10 marbles on Monday, 3 more on Tuesday, and 3 more on each successive day. How many has he gained in six days ?

24. A merchant gained £90 during the first year in business, and £35 more in each successive year than the one preceding. How much has he gained in 20 years ?

25. A labourer saved Id. the first week of the year, and |d. more on each successive week. How much has he at the end of the year?

26. A body falls 16'1 ft. during the first second, thrice as far dur- ing the second, and so on. How far would a body fall in six seconds ?

27. If 20 sentinels are placed in a line at the successive distance of 40 yards; how far will a person travel who goes from the 1st to the 2d and back; from the 1st to the 3d and back; and so on till he goes from the 1st to the 20th and back : and how long will he take at the average rate of 3£ miles f hour?

SO. GEOMETRICAL PROGRESSION.

A GEOMETRICAL PROGRESSION (G. P.) is a series of num- bers uniformly ascending or descending by a common ratio ; and is therefore appropriately termed an EQUIRATIONAL SERIES.

2, 6, 18, 54, &c., is an ascending G. P., in which the common ratio is f or 3. 1, J, J, ft, &c., is a descending G. P., in which the common ratio is \.

* Let a — the first term, d = the difference, S = the sum, and I = the nth term, or the to of n terms; then I orthenthterm=o±(n— IJo,

S=

H 2

JJO«The Dormer series is as follows: Term	Thel? Term
1st	_- 2	= 2	1st
2d	= 2X3	= 6	2d
3d	= 2 X 32	= 18	3d
4th	= 2 X 33	= 54	4th
&c.	&c.	&c.	&c.

178 SERIES.

The latter series is as follows : = 1 =1

= 1 X (I)* = I = 1 X U)3 = i

&C. cNJC.

To obtain any term in a G. P., we raise the common ratio to the power whose index is less by one than the number showing the rank of the term in the series, and then multiply the power by the first term.

In the 1st series, the llth term is = 2 X 310= 118098.

In the 2d series, the 20th term is = 1 X Q) 1 9 = ^ ATT-

(1) Find the 9th term in the G. P. 7, 21, &c. Ratio = y = 3 ; 9th term = 7 X 38 = 45927.

(2) Find the 6th term in the G. P. 2 J, 1 J, &c. Ratio = H -r- 2J = J; Gth term == 2J X (f)5

= ! X AV = A-

Find the Find the

4. 10th term in 81, 27, 9, &c.

5. 7th ., g, 3, i, &c. 6.5th « /«, i§

1. 6th term in 4, 8, 16, &c. 2.5th » 7, 28, 112, &c. 3. 9th " i, 1, 5, &c.

7. Of seven purses, the first contains 1/4; the second, 2/; the third, 3/; and so on in the same ratio. How much does the last contain ?

8. A person who found a potato imitated the example of Samuel Budgett and planted it. At the end of the first season he obtained 25 potatoes; and during each successive season the whole crop of the preceding one was planted and increased in the same ratio. Find the crop at the end of the fifth season.

9. Out of a vessel containing 10 gallons of brandy, T'5 was ex- tracted and replaced with water, T!5 of the content was again extracted and replaced with water, and so on for seven times. How much brandy is finally in the vessel ?

45T The first term is 10, the ratio ^j, and the number of terms 8. Let us find the sum of the G. P. 2, 6, 18, 54. Ratio = | = 3.

3 X Sum = (2X3)+(2X3«)+(2X3*) + (2X 3*)

(3 — 1) Sum = (2X34)— 2 = 2 (34 — 1) Sum = 2 X ^ T = 80.

SERIES. 179

8O« Let us find the sum of the G. p. 9, If, ft, ^

Ratio = If -f- 9 = i.

|XSum= (9X|)+{9xa)2}+{9xg)3} (l-i) Sum = 9-{9X(|)'}=9$l-(i)«} Sum=9x^4=10-§.

To find the sum of a G. P. we raise the ratio to the power denoted by the number of terms, divide the difference between this power and unity by the difference between the ratio and unity, and multiply the quotient by the first term.*

Any term in a G. P. is a Mean Proportional, or a Geome- trical Mean between two equidistant terms; thus, in the G. P. 1C, 24, 36, 54, 81; 36=^16 X 81=s/24x54(see§57&§ 74.)

(3) Find the sum of 7, 14, 28, to 10 terms.

Ratio = V4 = 2. Sum = 7 X ^f = 7161-

(4) Find the sum of |, fa T{T, to 8 terms.

Ratio = ,V -r £ = i-

_ I \s * (?) _ 1 \s 65535 V 4

U1H = f X ~fZ.i" — T X e"f f 36" •* 3^

terms.

10. Find the sum of 2, 4, 8, to 12 j 13. Find the sum of 3, f , T35, to 7

1 1 . ,/ »/ 5, 15, 45, * 8

12.

14. » 15.

16. Of seven boys, the first has 64 nuts, the second 96, and so on in the same ratio. How much have they in all ?

17. Of five brothers, the eldest has £759'-7»6, the second two- tliirds of this sum, and so on in the same ratio. How much have they in all ?

18. A gentleman on taking a house for twelve months ignorantly agreed to pay 1 mil as rent for the first month, 1 cent for the se- cond, 1 florin for the third, and so on in the same ratio. To what would the rent amount ?

The number of terms in a descending G. P. may sometimes be infinite; thus every Interminate Decimal is an infinite descending G. P.

* Let a = the first term, r = common ratio, S = the sum, I = the last of n terms ; then I or the nth term = ar --1,

180 SERIES.

0 In ;7, which is = T^ + Tfo + T^u + &c. ad infin. (co), the ratio is TV

Now a fraction when raised to a power becomes less as the index

of the power becomes greater ; when therefore the index is infinite,

the fraction becomes 0.

Hence, Sum which is = -^ X 1 7" ^V* is=TVXJ-

1 - To * 15

- T?g - _IP_ - 7

" 1 - I* " I** '

The sum of an Infinite descending G. P. is = the first term divided by the difference between the ratio and unity.*

(5) Find the sum of Ty& + TU*«*TO + &c-°°

Ratio = TTfcs ; Sum = Ty& ~ (1 — TUW) = «f • See § 34, No. 1.

19. Find the value of -45, or the sum of TV5 + ToVW + &c-°°

20. Find the value of '037, or the sum of I§

81. COMPOUND INTEREST.

WHEN a sum is lent for a number of periods or terms at COM- POUND INTEREST, the Interest is added to the Principal at the end of each term, and the Amount obtained becomes the Prin- cipal for the next term.

On £600 lent for 5 years @ 5 °/0, the Simple Interest would be £150; and the Amount, £750. But at Compound Interest the Amount would be as follows : — Principal for the first year .... £600 Interest // // // ..... 30 Principal for the second year .... 630 Interest » // // .... 31'5

Principal for the third year ..... 661-5 Interest // // // ..... 33-075

Principal for the fourth year .... 694*575

Interest // // // .... 34'72875

Principal for the fifth year ..... 729-30375

Interest // // '/ ..... 36-4651875

Amount for 5 years ..... 7657689375 Original Principal ...... 600

Compound Interest ..... £165-7689375 * When n is infinite, and r < 1 . rn = 0, S = j (^_ .

COMPOUND INTEREST. JgJ

81. Exercises in Compound Interest may be performed by .s method, but a more concise plan may be obtained by considering the following : —

Interest on £1 for 1 year © 5 °/0 = -05 Amount on £1 // // // // // = 1-Q5

Since the Amount for any year becomes the Principal for the next, we obtain the following proportions :

Principal. Amount.

£> & £ £

1 : 1-05 : : 1-05 : 1-052 = Am*, for 2 years.

1 : 1-05* : : 1-05 : 1-053 = // // 3 //

1 : 1-053 : : 1*05 : 1-05* = * // 4 //

1 : 1-05* : : 1-05 : 1'055 = // // 5 //

Therefore 1 : 600 : : 1-05* : 600X1 -05 *= £765-7689375

To find the AMOUNT of a given sum for a number of terms at Compound Interest, we raise the Amount of £1 for one term to the power denoted by the number of terms, and mul- tiply by the given sum.

(1) Find the Amount of £450 and the Compound Interest on it for 3 years @ 4 %.

Am* of £1 for 1 yr. @ 4 % = £1*04 Am1, of £450 for 3 yr. @ 4 % = 450'X 1 '04s = 450 X 1-124864 = £506-189 = £506*3*9 J Compd Int. = £506//3//9^ — £450 = £56*3*9 J.

In involving the Amount of £1, we take as many places in the powers as will produce the result correct to three decimal places* (see § 39, § 73, & § 43.)

(2) Find the Amount of £547*625 for 4 years @ 5 °/0, payable half-yearly.

Am1, of £1 for 4 yr. © 5°/0 V ann.=£l'025

Am*, of £547-625 for 8 half years © 5 % = 547-625

X 1-0258 = £667-228

* Calculations in Compound Interest are often effected by having the amounts of £1 at the important rates tabulated for a series of years. Exercises in Compound Interest afford good illustration of the advan- tages of Logarithms. The Questions prescribed above are, however, given for such periods as enable them to be easily solved by Invoh tion.

182 COMPOUND INTEREST.

Gl* Find the Amount of the following sums : —

1. £600 for 2 years @3%

2. 300 * 3 // // 5°/0

3. 800 // 4 // // 3°/0

4. 400 // 4 // f 47o

5. 700 » 4 '/ // 2i°/°

6. 834 // 5 // // 3£%

7. £G97'/15//Ofor6yrs.@2i°/a

8. 468//10//6 // 4 // // 4°/0

9. 232// 7//6 * 8 // // 3%

10. 35// 3//9 // 3 // // 3|°/0

11. 666'/13//4 » 5 // // 2J°/0

12. 267//19//2 // 7 // // 4i°/0

13. Find the Amount of £670 for 3 years @ 6 °/0, supposing the interest to become due half-yearly.

14. Find the Amount of £684 for 3 years @ 4 °/0, supposing the interest to be due quarterly.

15. What is the Compound Interest on £764-42 "6 for 4 years @ 5 °/0, due half-yearly ?

16. Find the Compound Interest on £29" 15 for 3£ years @ 3£ °/0, due quarterly.

17. Find the difference between the Simple and the Compound Interest on £750 for 3 years @ 4£ %.

18. A sum of £300 is lent for one year @ 4 °/0 ; find the difference between the Simple and the Compound Interest, due quarterly.

19. To what will a legacy of £500 left to a boy 11 years of age have accumulated at Compound Interest, on his attaining majority at 21 years of age, allowing Interest @ 5 °/0?

20. A legacy of £2500 was left to a young lady in 1852 on con- dition that it should be improved at Compound Interest for a mar- riage-portion. To what will it have accumulated at her marriage in 1860, reckoning Interest @ 5 °/0?

We may require to find the Principal which, improved at Compound Interest, may at a future date amount to a given sum; thus, let us find a sum which in 6 years @ 3J°/0 will amount to £700. .

Am4, of £1 for the given time = £1'0356 =£1-229255

Amount. Principal. ^.^

£1-229255 : £700 : : £1 : x =

We work by Contracted Division (see § 40.), and obtain the result £569-450, the Present Value of £700.

To find the PRESENT VALUE of a given sum due in a given time at a given rate, we divide the given sum by the amount of £1 for the given time.*

* Let P = Principal, A = Amount, R = Kate, n = number of years, A - P (\ 4- AV • P ~ A

P+ ' -

COMPOUND INTEREST. jg3

alUG °f £500//12"6> due in 7 years 1-037 = 1-229874

= £407-054 = £407//1//1.

21.£900duein2yrs.@47o 22. 700 * ,4 , 7 5°/I

23.1200 // // 4 // // 3°/0

Find the Present Value of

24.£1405//ll//6duein4yr.(5)40/a

25. 105//11//3 // // 3 * 7SA.

26. 333// S//4 // // 5 /•/ // 2-»-°/I

27. What sum will in 3 years @ 4 °/0 amount to £100, supposing the interest to be paid quarterly ?

3. Find the sum which, with half-yearly payments of hit- will at 6 °/0 amount in 4 years to £253-354.

29. A merchant who has increased each year's capital by a tenth t finds that at the end of twelve years he has £3985" 16" 1£. Find his original capital.

30. A sloop was bought by A, who sold it to B, by whom it was sold to C, who finally disposed of it. Each gained 30 % on his prime cost. C sold it for £659»2 ; what did A pay for it?

One of the most important applications of Compound Interest is in the calculation of ANNUITIES. An Annuity, as its name imports, is a sum payable yearly for a certain number of years ; an Annuity may, however, be payable at equal intervals of any duration, as half-yearly, quarterly, \£;c.

Suppose a person, entitled to an annuity of £30 ^ annum for 5 years, payable yearly, draws none of it till the end of the time ; to what will it have amounted, reckoning interest at 4°/0?

£1 of the annuity might be lent at the first payment for 4 years, and become at Compound Interest £1'044; £1 at the second payment might be lent for 3 years, and become £1*043 ; £1 at the third payment might be lent for 2 years, and become £1-042; £1 at the fourth payment would in 1 year become £1-04; and to these we would add the fifth payment of £1.

The Amount of an Annuity of £1 for 5 years @ 4 °/0 is thus = £1-04*+ 1-04 3 + 1-042 + 1-04 + 1. The sum of this

/ c of\\ - />1'04»-1 <.1'04»-1.

Geometrical Progression (see § 80.) is = *1.04_1 = * — r^— Having found the Amount of an Annnitv of £1, that of £30

J.Q^S 1

for the same time and rate = £30 X — r^j— •

To find the AMOUNT of an Annuity, we diminish the amount of £1 for the given time and rate by £1, divide the differ-

184 COMPOUND INTEREST.

81 • ence by the interest of £1 for one term, and multiply the quotient by the given Annuity.

(4) Find the Amount of an Annuity of £25 payable half- yearly in 4 years @ 5 °/0.

Int. of £100 for £yr.=£2'5; Int. of £1 for 1 hf.yr.=£'025 Am1. of£lfor8hf.yr.=£l-0258; Annuity for ±yr.=£12'5

Amount of Annuity = £12-5 X -^^ss £12-5 X ^- = £12-5 X 8-736116 = £109-20145.

31. Find the amount of an annual rent of £25 for 8 years @ 5 °/0.

32. Find the amount of an annuity of £60 payable yearly for 6 years @3£°/0.

33. The Lord Justice Clerk of Scotland has an annual salary of £4500. To what would it amount in seven years @ 4 °/0 ?

34. Find the amount of an annuity of £36 payable quarterly for 2| years @2£°/0.

35. A gentleman of fortune, entitled to an annual pension of £200, payable half-yearly, allows it to accumulate for 10 years. Find the amount @ 5 °/0.

36. A salary of £180, payable quarterly, is not drawn for 1^ years. Find the amount @ 5 °/0.

37. Find the amount of 4 half-yearly dividends of £2000 stock in the three per cents, reckoning interest @ 4 °/0.

^gr The half-yearly annuity is one-half of 3 °/0 on £2000.

Suppose a person, desirous of obtaining an annuity of £70 W annum for 10 years, wishes to know how much he must pay for it @ 3 %.

1.A91O 1

The amount of this would be £70 X 03 . The sum to be paid for the annuity would evidently be that which in 10 years would produce this amount. We would therefore re- quire to find the Present Value of the Amount by dividing it by the amount of £1 for the given time (see p. 183).

£70 X 1'03I<1""1 -T- l'03l° = £70 X I~T^R

I03

To find the PRESENT VALUE of an Annuity, we diminish £1 by the Present Value of £1 for the given time and rate, divide the difference by the interest of £1 for one term, and multiply the quotient by the given Annuity.

(5) Find the Present Value of an Annuity of £30//17"6 payable quarterly in 2J years @ 3i °/0.

COMPOUND INTEREST. jgr

81. I^.of£100for:lyr.=:£.875;Int.of£lforlquar.=£.00875 Present Value of £1 for 11 quarters = £ Annuity for 1 qr. = £7'71875 = j Of

Present Value of Annuity = £7-71875 X ^FoogTgn

*00875 —

= £7-71875 X ^»=£7-71875X10-4436 = £80-6115.

UnHmited' the Annuit^ is termed a

A person wishing to obtain a perpetuity of £200 y annum is desirous of knowing the sum to be paid for it @ 5°/0

The amount of any sum, as £1, for an unlimited time being QO (infinite), its reciprocal, or the present value of £1, due in an unlimited time, is hence = 0. Present Value of £1 = £^ = 0. Present Value of Perpe- tuity = £200 X ±=£ = £^ = £4000.

The sum of £4000 lent out @ 5 °/0 will produce £200 in perpetuity.

(6) Find the Present Value of a Perpetuity of £99//2//6 ^ annum @ 3£ %.

Present Value = £ = 3050.

38. Find the present value of an annuity of £40 payable annually for 10 years @ 4£ °/0.

39. Find the present value of an annuity of £62»10 for 3£ years, payable half-yearly, @ 5°/0.

40. Find the present value of a perpetuity of £2 1 0" 17 •• 6 ^ annum

41. The Lord Justice General of Scotland has a salary of £4800 tp annum. Find the present value of this for 10 years @ 3°/0.

42. A tenant, on taking a lease of a house for 7 years @ £19 ^ annum, pays the present value. Find the sum, reckoning interest @ 4 °/0.

* Let a = Annuity, R=Eate°/0, r=^ = Int. on£l, n=N° of years,

Amount of an Annuity = a X -- - - .

i __ L_ Present Value of an Annuity = a X (H-r)».

r

a 100#

Present Value of a Perpetuity = — or -^— .

186 COMPOUND INTEREST.

43. A colonel of the Royal Marines on half-pay has £264" 12 » 6 ^ annum. Find the present value of this annual salary for 6 years @ 4%.

44. What ought to be paid for a property giving an annual rent of £187»8"6, reckoning @ 4| °/0?

45. What sum paid in January 1858 will produce an annuity of £50, payable half-yearly until July 1861, @ 4| %?

82. MISCELLANEOUS EXERCISES.

1. FIND the L. c. M. of all the multiples of 3 from 6 to 27 inclusive.

2. Find the G. c. M. of 25 X 45 and 5 X 3s.

3. What is the G. c. M. of the square of 48 and the cube of 18 ?

4. Find the L. c. M. of the first ten even numbers.

5. Reduce y^l^B to its lowest terms.

6. Arrange §, ^§, ££, £f0 and ji, in order of magnitude.

7. Subtract the sum of $ + jj + u + 13 + it from 5-

8. Find that number of which ({j -f ; — |) is = 51 .

9. Multiply * of 2 § by 2 iiL,

10. Multiply |-f | by J — §, and increase the product by T55 of 1 1.

11. Find that number whose fifth diminished by its seventh is = 3?.

12. From the square root of '000169 subtract the square root of •00016.

13. Find the decimal which when added to the difference of 5|5 and -002775 produces the square of '215.

14. Subtract the cube of 1-6 from 130 times '0325.

15. Find the interest on £-219 for 47 days @ 3'6 %.

16. A grocer by selling sugar @ 6£d. ^ Ib. loses £d. ; find his loss °/0.

17. From Edinburgh to Glasgow by railway is 47 ^ miles. In what time will a train traverse the line at the rate of 990 yards y min., allowing 5S? hour for stoppages?

18. The number of copies in the first edition of the Lay of the Last Minstrel, which was 750, was to that in the seventh as 15 to 7 1 . Find the number in the latter.

- 19. From 1847 to 1857, the Revenue of the City of Edinburgh was £70,629, and the Expenditure £57,684. What per-centage was the difference or surplus of the former ?

20. A, at the rate of 4^ miles an hour, walks a distance in 3T'5 hours ; in what time will B walk the same distance at the rate of 5 of 51 miles an hour ?

MISCELLANEOUS EXERCISES. 187

82* 21- Find tne square root of 10 5 — 316 -.

22. Find the cube root of -296.

23. Find the H. p. of an engine which can raise 4£ tons of coala per hour from a pit 77 fathoms deep.

24. The centre arch in Westminster Bridge, which is 76 feet wide, is the seventh from the side, and each arch is 4 ft. narrower than the adjoining one nearer the centre. Find the width of the first arch.

m 25. Divide £5 among A, B, C, D, in the mutual ratios of £, £, £, and £.

26. A sum of £1343"14»6 collected for a family of orphans was laid out at 6 °/0 per annum. Find the value of a half-yearly payment.

•J 7 . Reduce 1 dwt. to the decimal of 1 Ib. avoir.

Divide 25832 in the ratios of the squares of the reciprocals of the first four odd numbers.

29. The Admirals, the Vice Admirals, and the Rear Admirals of the British Navy are each divided into 3 classes of Red, White, and I Hue, and the classes of each rank contain the same number. The number of Admirals is 57g of that of the whole, which is 99, and that of Men-Admirals is -fr. Find the number in each class of Admirals.

30 The walls of Rome erected by Aurelian have been calculated to contain 1396£ hectares, each 2-47114 acres. Express the area in

^SL Thfl weight of an American dollar is 412 \ grains, of which » is pure silver. Find the weight of pure silver in 100 dollars

32. If on every guinea of selling price half-a-crown is gamed ; find the gain on £1000 of buying price.

33. Victoria Bridge on the St Lawrence is within 50 yards 2 miles in length ; in what time will a train traverse it at the r

°fJ/F^

for 4 years @ 5 •/., and for 5 years @ 4 •/., payable half-yeaily n

; Horticultural Society, founded by Sir Joseph Banks £

,,was remodelled in 1856; been incorporated by Royal Charter, m what year did i

"

lny days elapsed between the Annular Eclipse of 15th

^^^

S! Kri the cube of 11 in the scale of 3. . rom York to London is a distance of 192 miles ,

188 MISCELLANEOUS EXERCISES.

at ^1C same time from each terminus, the one from York at the rate of 40 and the other at 32 miles an hour. How far from London will they meet ?

40. F starts at 12h at 6| miles an hour, and B at 12h. 30™. At what rate must B travel to overtake F at 21? ?

41. 3 Russian versts are = 3500 yards. Reduce a verst to the decimal of a mile.

42. Assuming the length of a glacier, described by Principal Forbes, to be 20 miles, and its annual progression 500 ft. ; how long would a block of stone take to traverse its length ?

43. Of 150 encumbered estates in Ireland, the numbers in the four provinces were respectively as 1, 2, 3, and 4. Find the num- ber in each province.

44. How many metres are in a Scotch mile, taking 1 Scotch mile = 1-123024 Imperial mile; and 1 metre = 39*37079 inches.

45. How many metres are in a Scotch mile, taking the following approximations, 8 Scotch miles = 9 Imperial miles, and 32 metres = 35 yards ?

46. If, in victualling a crew, 80 days are allowed for an outward and homeward voyage to Oporto ; 1g5 of this time for one to Deme- rara ; to Boston, f of that to Demerara ; to Valparaiso, Y of that to Boston. Find the time allotted for an outward and homeward voyage to Valparaiso.

47. In 1851, the population of Glasgow was 3'58866 per cent, of the population of Edinburgh more than double the latter, which was 161,648. Find the former.

48. Of an estate, the uncultivated part is 535, the cultivated part, f , and the remainder under wood contains 65 acres. How many acres are in the whole ?

49. In 1 855, in the naval armament of France, the number of line of battle ships was f of 100; that of frigates, which was 12 less than f of the number of the line of battle ships, was f of that of the smaller vessels, and the number of steam vessels was ^ f of double the number of frigates. HJOW many were there of each?

50. What length of rails requires 873 T. 1 cwt. 1 qr. for their construction @ | cwt. ^ yard ?

51. Find the greatest depth of Lake Erie, which is | of that of Lake Huron, whose greatest depth is f of that of Lake Ontario, which is § of that of Lake Michigan, whose greatest depth is {$ of that of Lake Superior, which is 990 ft.

52. In the Walcheren Expedition, out of an average force ot 40,589, there were 4212 deaths. Find the per-centage.

53. The circulation of a periodical was 38,500 ; of the whole, the

MISCELLANEOUS EXERCISES. 189

of stamped copies was T 1 4. How many copies were un- stamped ?

54. Find the surface of a floor 28 ft. 7£ in. long, and 15 ft. 6| in. broad.

55. Find the sum of V|| -f- S/ffs + if HI-

56. Find that number whose square root is = $ of 5J -f- 7 °f * !•

57. Find the true discount on £22»17/>3f for 3 months at 5£ °/0.

58. In 1850, the states of Ohio and Tennessee, nearly of equal extent, produced 59,078,695 and 52,276,223 bushels of wheat respectively. Find the difference of their weight, reckoning the bushel at & | cwt.

59. If A pays ll|d. ^ £ for income-tax, what is his income when the net proceeds are £116"3"1 ?

60. If a courier traversed a distance of 400 miles in 36 hours, in what time did he traverse | of | of f of 6| miles?

61. No. 1585 of the Athenceum appeared on 13th March 1858 ; on the hypothesis that it has been regularly published once a-week, find the date of No. 1.

62. Find the price of 3 cwt. 2 qr. 13 Ib. carrots @ 16/ ^ 240 Ib.

63. Reduce a talent of 3000 shekels, each | oz. avoir., to the decimal of 1 cwt.

64. A train contains 13 trucks laden with coals; the average weight of a loaded truck is 10 T. 8 cwt. 1 qr., and that of an empty truck 3 T. 16 cwt. 2 qr. Find the weight of coals conveyed by the train. .

65. If, in the Russian tariff, the duty on Scotch herrings is 40 copecs y pood ; how much sterling is this f cwt., a ruble of 1 copecs being = 3/1 i, and a pood being = 36 Ib. avoir. ?

66. A miser collected £370 in packets of pound notes, crowns, half-crowns, florins, shillings, and sixpences. The values of five of the packets were the following fractions of the whole -.-packet of notes, W,; of crowns, ^5 of hf.-crowns, ***; ^ norms, „, ; of shillings, ,V Find the number of notes, crowns, hf.-crowns, florins, shillings, and sixpences. ^

67. Find the value of | cr. + f s. — § n. + £s-

68. A can do a work in 7* days, B in 6f days, and C m 5| days. In what time can they do it by working together ?

69. A field contains 18 ac. 2 ro. 18 po., and another 7 ac. 3 ro. 7 po. Find the side of a square field of equal area to ^«

70. Of the two members of parliament wtm^™^^ in 1857, the number of votes polled *"•"£*** ^ nYthe was to.its excess above the other number as 902 tc

™h citation of a newspaper in the first quarter of a yea,

190 MISCELLANEOUS EXERCISES.

82»was 3200, and in the second quarter 3600. What would the cir- culation in the third quarter require to be to show the same ratio of increase ?

72. An angler, by using a single hook and a tackle of four hooks alternately for equal times during a day, caught 9i Ib. with the former, and 11 Ib. with the latter. On another day he caught 25 Ib. with the former ; what might he have taken with the latter?

73. Texas contains 274,362 square miles. Into how many lots, each 4536 acres, might it be divided?

74. The managers of a congregation buy a site of f rood for £500. How much will they pay for 595 acre ?

75. A person who has paid £6»4»2 of income-tax has £142 "15 "10 over. How much has he paid ^ £ ?

76. 70 masons can build a mansion in 61 days; after working for 10 days, 15 more are engaged. How many days fewer will be occupied than would otherwise have been ?

77. If 7 men can do as much as 11 youths, and if 21 youths can do a work in 13 days; in what time can 14 men and 4 youths doit?

78. A starts on a journey at the rate of 3£ miles an hour, B fol- lows in | hour at the rate of 4 miles an hour. How far on will B overtake A?

79. In the household book of a ducal family we have the follow- ing entry by the steward: — " Given your lordship on New Year's Day to give your grandchildren and the servants and several others, £32"6"6." Taking this as Scots money, which is one-twelfth of sterling, express the sum as the fraction of £100 sterling.

80. Find the 10th term of the series 1, 1|, 2, &c.

81. Find the 10th term of the series 1, 1£, 2i, &c.

82. The population of a country in 1854 was 4,500,000, and if it has increased each year at the rate of 10 per cent, on the preceding year; find the population in 1859.

83. In an estate in Sweden, the arable land contains 200 ttmn- lands ; meadowland, 2 per cent, less than the arable ; and wood- land, 1 £ per cent, less than 7 times the arable. Find the area in acres of the estate, a tunnland being = 1-2312 acre.

84. From Dresden to Prague by rail is 150 kilometres, each 1093-63 yards : a train leaves Dresden at the rate of 48 kilometres q? hour, and in a quarter of an hour afterwards another leaves Prague at the rate of 40 kilometres qp hour. How many miles from Prague will they meet ?

85. Give eight convergents to the fraction which a kilometre is of a mile.

MISCELLANEOUS EXERCISES. 1'Ji

82, 8^- Wlmt sum invested in the 3| per cents at 93 «[ will produce £61 "5?

87. Deposited £500 in the National Bank on 1st September 1856, when interest was @ 2£ %; "on 8th Oct. it rose to 3| °/0, and on 15th May to 4 °/0. Find the interest due on 8th June 1857.

88. At what rate must £273 be lent from 1st January to 27th May to produce £4-914 of interest?

89. Find the price of 19 cwt. 2 qr. 7 Ib. @ £1"8«6 y cwt. by decimals.

90. Find the value of 17-375 cwt. @ £-5625 by decimals and by practice.

91. In the reign of Henry VIII., among the monasteries and religious houses whose revenues were confiscated, there were 186 belonging to the Benedictines with a revenue of £65,87 9" 14, and 173 to the Augustines with a revenue of £18,691"12»6. Reduce the average of one of the latter to the decimal of that of one of the former.

92. Divide the square of 390,404,646 by the square of 123,456,789, and let the quotient be carried out till it contains 3 significant figures in the decimal.

93. A capitalist who had invested £3120 sterling in stock @ 97*, gold £2500 stock @ 98, and the remainder of the stock @ 96. Find his gain or loss.

94. A labourer's wages for 30 days are £3"18"9. Find the wages for the working days in January and February 1860, new year's day being on a Sunday.

95. If 1 Ib. troy of sterling gold is worth £46f £, find the weight of 3465 sovereigns and 1792 hf.-sovereigns.

96. The Brisbane Prize Fund of the Royal Scottish Society of Arts amounts to £175 in the 3 per cent. Government Consolidated An- nuities. Find the value of the fund at 90; and the value of the pria

97 Find the weight of an oaken block 2-25 ft. long, 16 inches broad, and ft of 1 '625 ft. thick; a cubic foot of water weighing 999-278 oz. avoir., and the specific gravity of oak being '925.

98 The deflection of the Earth's curvature is 8 inches

mile, 32 inches for 2 miles, and so on, «* a****^S . -j-ii.~ ^:^4-n-nno, "FTi-nd tlifl heicrnt or a Iignt

portional to the square of the distance. J"^

above the level of the sea which is visible for 14 nautical

each 6076 ft.

c . „ . , » n_ -.wiinnrv

99. 42 men, whose average strength is f of that of •****»% an, can do a piece of work in 4t days which other 5 = men can do

... 4 days. What is the average strength of one of the latte. as compared with that of an ordinary man?

100. The period of the Earth's revolution IB 365 256 days,

man in

192 MISCELLANEOUS EXERCISES.

0that of Mercury 87 '969 days. Express by Kepler's Law the deci- mal that the distance of Mercury from the Sun is of tha.t of the Earth.

101. In a heavy gale, a flagstaff 60 ft. high snaps 28'8 ft. from the bottom, and not being wholly broken off, the top touches the ground. How far is its point of contact from the bottom ?

102. Seventeen trees are standing in a line 20 yards apart from each other ; a person walks from the first to the second and back, thence to the third and back, and so on to the end. How far will he have walked ?

103. If the value of 1 oz. troy of sterling gold { £ fine is £3'89375 ; find the value of 1 Ib. avoir of pure gold.

104. A lunation = 29*53 days, is the period in which the moon passes once through her phases. After a cycle of 223 lunations, known to the Chaldeans as Saros, eclipses recur in the same order and magnitude. Find the date of the eclipse in the next cycle corresponding to the solar eclipse of 28th July 1851.

105. When a body floats in a liquid the weight of the liquid dis- placed is = the weight of the floating body. The effective length of a vessel is 96 ft., the effective breadth 22 £ ft., and the draught of water 9 ft. Find the weight of the vessel, taking the weight of a cubic foot of water roughly at 62 £ Ib.

106. Find the mean discharge per second of the River Tay, sup- posing the area of its basin to be 2400 square miles, the annual fall of rain to be 30 inches, of which £ is lost in evaporation.

THE END.

PRINTED BY OLIVER AND BOYP, EDINBURGH.

EDUCATIONAL WORKS

PUBLISHED BY

OLIVER AND BOYD, EDINBURGH;

SOLD ALSO BY

SIMPKIN, MARSHALL, AND CO., LONDON.

*** A Specimen Copy of any work will le sent to Principals of Schools, post free, on receipt of one half the retail price, in postage stamps.

English Reading, Grammar, etc. Armstrong's Eng. Composition.... P. 6

Eng. Etymology 6

Cennon's English Grammar 4

First Spelling-Book 4

Dalgleish's English Grammars 5

Gram. Analysis. 5

Eng. Composition 5

Demaus's Paradise Lost 7

Analysis of Sentences 7

Douglas's English Grammars 6

Progressive Eng. Reader. 6

Selections for Recitation . 6

Spelling and Dictation.... 6

Ewing's Elocution 7

Fisher's Assembly's Catechism 8

Lennie's English Grammar 6

M'Culloch's Reading-Books 3, 4

English Grammar 4

M'Dowall's Rhetorical Readings 7

Millen's English Grammar 7

Morell's Poetical Reading-Book 4

Reid's English Grammar 5

English Composition 5

n^lish Dictionary 5

Sess. School Etymological Guide 8

Old & New Test. Biographies... 8

Siiakspeare's Richard II 6

Spaldintfs English Literature 7

English Gramniar 7

Wordsworth's Excursion o

Object-Lessons. Object-Lessons on the Vegetable

Kingdom - 7

How to Train Young Eyes and Ears 7

Geography and Astronomy.

Clyde's School Geography P. 8

Elementary Geography 9

Douglas's Introductory Geography. 9

Progressive Geography... 9

Text-Book of Geography. 9

Edin. Acad. Modern Geography — 9

Ancient Geography 9

Swing's Geography 10

Atlas 11

Lawson's Geog. of British Empire. . . 9

Murphy's Bible Atlas 10

Reid's First Book of Geography 10

Modern Geography 10

Sacred Geography 10

Introductory and School At- lases 11

Reid's (Hugo) Elements of Astro- nomy 10

Phys. Geography 10

Stewart's Modern Geography 9

White's Abstract of Geography i»

System of Geography 9

Atlases 11

School Songs. Hunter's Books on Vocal Music School Psalmody

Household Economy. Brewster's Household Economy

History. Corkran's History of England

Simpson's Scotland

Goldsmith's England...

t"f,,, Greece

Rome

4 English Reading, Grammar, etc.

Lessons from Dr M'Cnlloeh's First Reading-Book,

printed with LARGE TYPE, in a Series of Ten Sheets, for Hanging on the Wall. Price Is. ; or mounted on Roller, Is. 8d.

Dr M'Culloch's Manual of English Grammar, Philo- sophical and Practical ; with Exercises ; adapted to the Analytical mode of Tuition. Is. 6d.

English Prefixes and Affixes. 2d.

In all the books of Dr M'Culloch's series, the important object of exer- cising the juvenile mind by means of lessons on useful aud interesting subjects is steadily kept in view. Directions are given relative to the mode of teaching, as well as tables and lists calculated to assist in the process of instruction. On this point the Spectator newspaper, when re- viewing the series, remarked : — "In recommending these books, it must not be conceived that we recommend them as likely to save trouble to the teacher, or to operate by witchcraft on the pupil. At their first introduction they will require some care on the part of the master, as well as the exercise of some patience, to enable the pupil to profit by the lessons. But this once done, their foundation is sound, and their progress sure. And let both parents and teachers bear in mind that these are the only means to acquire real knowledge."

Poetical Reading-Book, with Aids for Grammatical Analysis, Paraphrase, and Criticism ; and an Appendix on English Versification. By J. D. MORELL, A.M., LL.D., Author of Gram- mar of the English Language, etc., and W. IHNE, Ph. D. 2s. 6d.

Containing,

THE DESERTED VILLAGE.

THE TASK (Book I.)

PARADISE LOST (Books I. and V.)

THE MERCHANT OF VENICE. MISCELLANEOUS SELECTIONS. THE PRISONER OF CHILLON.

THE FIELD OF WATERLOO.

Dr Morell, in the preface to his "Grammar of the English Language," says—" As great care was taken to adapt this book [the Poetical Reading- Book] to the requirements of teachers using the Grammar, and special marks invented for indicating the correct analysis of the poetical extracts con- tained in it, I take the present opportunity of recommending it to the attention of the higher classes of schools in the country."

English Grammar, founded on the Philosophy of Language

and the Practice of the best Authors. With Copious Exercises, Constructive and Analytical. By C. W. CONNON, LL.D. 2s. Gd.

Spectator. — " It exhibits great ability, combining practical skill with philo- sophical views."

Connon's First Spelling Book, 6d.

English Reading, Grammar, etc. 5 Outlines of English Grammar and Analysis, f..r

ELEMENTARY SCHOOLS, with EXERCISES. By WALTER SCOTT DALGLEISH, M.A. Edin., one of the Masters in the London Inter- national College. 8d. KEY, Is.

Preface. — "Aims at providing a COMMON-SCHOOL GRAMMAR which shall be fully abreast of the latest developments of the science, and at the same time thoroughly practical and simple in its mode of treating the subject"

Dalgleish's Progressive English Grammar, with EXER- CISES. 2s. KEY, 2s. 6d.

From Dr JOSEPH BOSWORTH, Professor of Anglo-Saxon in the University of

Oxford; Author of the Anglo-Saxon Dictionary, etc,, etc.

" Quite a practical work, and contains a vast quantity of important informa-

tion, well arranged, and brought up to the present improved state of philo-

logy. I have never seen so much matter brought together in so short a sp^ce."

Dalgleish's Grammatical Analysis, with PROGRESSIVE

EXERCISES. 9d. KEY, 2s.

Dalgleish's Introductory Text-Book of English

COMPOSITION, based on GRAMMATICAL SYNTHESIS; containing Sentences, Paragraphs, and Short Essays. Is.

Dalgleish's Advanced Text-Book of English Com-

POSITION, treating of Style, Prose Themes, and Versification. 2s. Both Books bound together, 2s. 6d. KEY, 2s. 6d.

A Dictionary of the English Language, containing

the Pronunciation, Etymology, and Explanation of all Words authc ized by Eminent Writers. By ALEXANDER REID, LL.D., la Head Master of the Edinburgh Institution. Reduced to 5s.

Dr Reid's Rudiments of English Grammar. 6d. Dr Reid's Rudiments of English Composition. 2s.

ne«s in spelling, punctual, on, »« ^^Y^rspfcmty in style, and niont of sentences; Part II. to co -ectness m v F the tice of the

6 English Reading, Grammar, etc.

Lennie's Principles of English Grammar. Comprising

the Substance of all the most approved English Grammars, briefly defined, and neatly arranged; with Copious Exercises in Parsing and Syntax, Revised Edition ; with the author's latest improve- ments, and an Appendix in which Analysis of Sentences is fully treated. Is. 6d.

The Author's Key, containing, besides Additional Exercises in Parsing and Syntax, many useful Critical Remarks, Hints, and Observations, and explicit and detailed instructions as to the lest method of teaching Grammar. 3s. 6d.

Analysis of Sentences ; being the Appendix to Lennie's

Grammar adapted for General Use. Price 3d. — KEY, 6d.

The Principles of English Grammar ; with a Series of

Progressive Exercises. By Dr JAMES DOUGLAS, lately Teacher of English, Great King Street, Edinburgh. Is. 6d.

Douglas's Initiatory Grammar, intended as an Intro- duction to the above. 6d.

Douglas's Progressive English Reader. A New Series

of English Reading Books. The Earlier Books are illustrated with numerous Engravings.

FIRST BOOK. 2d. I THIRD BOOK. la. I FIFTH BOOK. 2s.

SECOND BOOK. 4d. | FOURTH BOOK. Is. 6d. | SIXTH BOOK. 2s. 6d.

Douglas's Selections for Recitation, with Introductory

and Explanatory Notes ; for Schools. Is. 6d.

Douglas's Spelling and Dictation Exercises. 144 pages,

price Is.

Athenceum.—" A good practical book, from which correct spelling and pro- nunciation may be acquired."

Shakspeare's King Richard II. With Historical and

Critical Introductions ; Grammatical, Philological, and other Notes, etc. Adapted for Training Colleges. By Rev. Canon ROBINSON, M.A., late Principal of the Diocesan Training College, York. 2s.

Wordsworth's Excursion. The Wanderer. With Notes

to aid in Analysis and Paraphrasing. By Canon ROBINSON. 8d.

English Composition for the Use of Schools. By

ROBERT ARMSTRONG, Madras College, St Andrews; and THOMAS ARMSTRONG, Heriot Foundation School, Edinburgh. Part 1., Is. 6d. Part II., 2s. Both Parts bound together, 3s. KEY, 2s.

Armstrong's English Etymology. 2s. Armstrong's Etymology for Junior Classes. 4d.

English Eeading, Object-Lessons, etc. 7 History of English Literature ; with an OUTLINE of the

ORIGIN and GROWTH of the ENGLISH LANGUAGE. Illustrated by EXTRACTS. For Schools and Private Students. By WILLIAM SJ-ALDING, A.M., Professor of Logic, Rhetoric, and Metepfc in the University of St Andrews. Continued to 1870. 3s. 6d. Spectator. — "A compilation and text-book of a very superior kind. . . . .Mr Spalding has brought to his survey not only a knowledge of our history and literature, but original reflection, a comprehensive mind, and an elevation of tone, which impart interest to his account, as well as soundness to his decision. The volume is the best introduction to the subject we have met with."

Athenceum. — " The numerous extracts scattered throughout the volume are well chosen for the purpose of throwing light on the authors from whom they are taken, and at the same time exhibiting the gradual advance of our literature from its earliest to its present state. Mr Spalding's critical remarks are discriminating, impartial, judicious, and always well put."

Selections from Paradise Lost ; with NOTES adapted for

Elementary Schools, hy Rev. ROBERT DEMAUS, M.A., F.E.I. S.,

late of the West End Academy, Aberdeen. Is. 6d.

The Selections have been chosen so as to furnish a continuous narrative.

Demaus's Analysis of Sentences. 3d.

System of English. Grammar, and the Principles of Com- position. With numerous Exercises, progressively arranged. 13y JOHN WHITE, F.E.I.S. Is. 6d.

Millen's Initiatory English Grammar. Is.

Ewing's Principles of Elocution, improved by F. B.

CALVERT, A.M. 3s. 6d.

Consists of numerous rules, observations, and exercises on pronuncial pauses, inflections, accent, and emphasis, accompanied with copious extracts in prose and poetry.

Rhetorical Readings for Schools. By WM. M'Dow.u.i,

late Inspector of the Heriot Schools, Edinburgh. 2s. 6d.

Object-lesson Cards on the Vegetable Kingdom. Set

of Twenty in a Box. £1, Is. Sd of great value in the education of the young.

Edinburgh. Is. 6d.

8 Household Economy, Geography, etc.

Household Economy: a MANUAL intended for Female Training Colleges, and the Senior Classes of Girls' Schools. By MARGARET MARIA GORDON (Miss Brewster), Author of " Work, or Plenty to do and how to do it," etc. 2s.

Athenceum. — " Written in a plain, genial, attractive manner, and constituting, in the best sense of the word, a practical domestic manual."

SESSIONAL SCHOOL BOOKS. Etymological Guide. 2s. 6d.

This is a collection, alphabetically arranged, of the principal roots, affixes, and prefixes, with their derivatives and compounds.

Old Testament Biography, containing notices of the chief

persons in Holy Scripture, in the form of Questions, with references to Scripture for the Answers. 6d.

New Testament Biography, on the same Plan. 6d.

Fisher's Assembly's Shorter Catechism Explained. 2s.

PART I. Of what Man is to believe concerning God, II. Of what duty God requires of Man.

GEOGEAPHY AND ASTRONOMY.

IN compiling the works on these subjects the utmost possible care has been taken to ensure clearness and accuracy of statement. Each edition is scru- pulously revised as it passes through the press, so that the works may be confidently relied on as containing the latest information accessible at the time of publication.

School Geography. By JAMES CLYDE, LL.D., one of the

Classical Masters of the Edinburgh Academy. With special Chapters on Mathematical and Physical Geography, and Technological Ap- pendix. Corrected throughout. 4s.

In composing the present work, the author's object has been, not to dissect the several countries of the world, and then label their dead limbs, but to depict each country, as made by God and modified by man, so that the rela- tions between the country and its inhabitants — in other words, the present geographical life of the country— may appear.

Athenceum.— " We have been struck with the ability and value of this work, which is a great advance upon previous Geographic Manuals. . . . Almost for the firsi time, we have here met with a School Geography that is quite a readable book, — one that, being intended for advanced pupils, is well adapted to make them study the subject with a degree of interest they have never yet felt in it. ... Students preparing for the recently instituted University and Civil Service examinations will find this their best guide ."

Geography and Astronomy.

Dr Clyde's Elementary Geography. Corrected

throughout. Is. 6d.

In the Elementary Geography (intended for less advanced pupils), it has been endeavoured to reproduce that life-like grouping of facts— geographical portraiture, as it may be called — which has been remarked with ap'i : in the School Geography.

A Compendium of Modern Geography, POLITICAL,

PHYSICAL, and MATHEMATICAL : With a Chapter on the Ancient Geography of Palestine, Outlines of Astronomy and of Geology, a Glossary of Geographical Names, Descriptive and Proim; Tables, Questions for Examination, etc. By the Rev. ALEX. STEWART, LL.D. Carefully Revised. With 11 Maps. 3s. Gd.

Geography of the British Empire. By WILLIAM

LAWSON, St Mark's College, Chelsea. Carefully Revised. 3s.

PART I. Outlines of Mathematical and Physical Geography. II. Physical, Political, and Commercial Geography of the British Islands. III. Phy- sical, Political, and Commercial Geography of the British Colonies.

Edinburgh Academy Modern Geography. Carefully

Revised. 2s. 6d.

Edinburgh Academy Ancient Geography. 3s.

An Abstract of General Geography, comprehending a

more minute description of the British Empire, and of Palestine or the Holy Land, etc. With numerous Exercises. For Junior Classes. By JOHN WHITE, F.E.I.S., late Teacher, Edinburgh. Carefully Revised. Is. ; or with Four Maps, Is. 3d.

White's System of Modern Geography; with Outlines of

ASTRONOMY and PHYSICAL GEOGRAPHY; comprehending an Aecoiir

Edinburgh. Carefull

Dr Douglas's Progressive Geography. U4 pages, is

In the Press.

Globe. Systematically arranged. Maps, 3s. Carefully Revised.

10 Geography and Astronomy.

First Book of Geography; being an Abridgment of

Dr Eeid's Rudiments of Modern Geography ; with an Outline of the Geography of Palestine. Carefully Revised. 6d.

This work has been prepared for the use of young pupils. It is a suitable and useful companion to Dr fteid's Introductory Atlas.

Endiments of Modern Geography. By ALEX. REID,

LL.D., late Head Master of the Edinburgh Institution. With Plates, Map of the World. Carefully Revised. Is. ; or with Five Maps, Is. 3d. Now printed from a larger type.

The names of places are accented, and they are accompanied with short descriptions, and occasionally with the mention of some remarkable event. To the several countries are appended notices of their physical geography, productions, government, and religion. The Appendix contains an outline of ancient geography, an outline of sacred geography, problems on the use of the globes, and directions for the construction of maps.

Dr Reid's Outline of Sacred Geography. 6d.

This little work is a manual of Scripture Geography for young persons. It is designed to communicate such a knowledge of the places mentioned in holy writ as will enable children more clearly to understand the sacred nar- rative. It contains references to the passages of Scripture in which the most remarkable places are mentioned, notes chiefly historical and descrip- tive, and a Map of the Holy Land in provinces and tribes.

Murphy's Bible Atlas of 24 MAPS, with Historical

Descriptions. Is. 6d. coloured.

Witness. — " We recommend this Atlas to teachers, parents, and individual Christians, as a comprehensive and cheap auxiliary to the intelligent reading of the Scriptures.

Ewing's System of Geography. Carefully Revised. 4s. 6d. ; with 14 Maps, 6s.

Besides a complete treatise on the science of geography, this work contains the elements of astronomy and of physical geography, and a variety of prob- lems to be solved by the terrestrial and celestial globes. At the end is a pronouncing Vocabulary, in the form of a gazetteer, containing the names of all the places in the work.

Elements of Astronomy : adapted for Private Instruction and Use of Schools. By HUGO REID, Memher of the College of Preceptors. With 65 Wood Engravings. 3s.

Reid's Elements of Physical Geography; with Outlines

of GEOLOGY, MATHEMATICAL GEOGRAPHY, and ASTRONOMY, and Questions for Examination. With numerous Illustrations, and a large coloured Physical Chart of the Globe. Is.

Geography and Astronomy, History. 11

REVISED EDITIONS OF SCHOOL ATLASES.

A General Atlas of Modern Geography; 29 Maps,

Coloured. By THOMAS EWING. 7s. 6d.

School Atlas of Modern Geography. Maps 4to, folded

8vo, Coloured. By John WHITE, F.E.I.S., Author of " Abstract of General Geography," etc. 6s.

White's Elementary Atlas of Modern Geography.

4to, 10 Maps, Coloured. 2s. 6d.

CONTENTS.— 1. The World; 2. Europe; 3. Asia; 4. Africa; 5. North America; 6. South America; 7. England; 8. Scotland; 9. Ireland; 10. Palestine.

A School Atlas of Modern Geography. 4to, 16 Maps,

Coloured. By ALEXANDER REID, LL.D., late Head Master of the Edinburgh Institution, etc. 5s.

Keid's Introductory Atlas of Modern Geography.

4to, 10 Maps, Coloured. 2s. 6d.

CONTENTS.— 1. The World; 2. Europe; 3. Asia; 4. Africa; 5. North America; C. South America; 7. England; 8. Scotland; 9. Ireland; 10. Palestine.

H I S T 0 E I.

THE works in this department have heen prepared with the greatest care. They will he found to include Class-hooks for Junior and Senior Classes in all the branches of History generally taught in the hest schools. ^ hik- utmost attention has been paid to accuracy, the narratives have in every case been rendered as instructive and pleasing as possible, so as to relieve tl study from the tediousness of a mere dry detail of facts.

A Concise History of England in Epochs. By J. F.

CORKRAN. With Maps and Genealogical and Chronological Tables, and comprehensive Questions to each Chapter. 2s. 6d. * * Intended chiefly for the Senior Classes of Schools, and for the Junior Student,

of Training Colleges.

in this History of *tg"3^W^ wT^buTt and full impression of its great Lpocns, na t ° ™ * f L aud af the subordination to the ^^J^^^r^^l^tMi^l but where Constitution. He has sum^\\f^Tt into relief, or where the story of illustrious characters were to be bim '8" ""? * \ occupied more space some great achievement merited a Ml ^"^^ Ufor it is his b-Hef that

£$ &,£%££&* &&& <audabie ambitioa tban ^

n^el^^^n^ated with more than usual fulness.

12 History.

History of England for Junior Classes ; with Questions

for Examination. Edited by HENRY WHITE, B. A., Trinity College, Cambridge, M.A. and Ph. Dr. Heidelberg, is. 6d.

Athenceum. — " A cheap and excellent history of England, admirably adapted for the use of junior classes. Within the compass of about a hundred and eighty duodecimo pages, the editor has managed to give all the leading facts of our history, dwelling with due emphasis on those turning points which mark our progress both at home and abroad. The various changes that have taken place in our constitution are briefly but clearly described. It is surprising how successfully the editor has not merely avoided the obscurity which generally accompanies brevity, but invested his narrative with an interest too often wanting in larger historical works. The information conveyed is thoroughly sound; and the utility of the book is much increased by the addi- tion of examination questions at the end of each chapter. Whether regarded as an interesting reading-book or as an instructive class-book, this history deserves to rank high. When we add, that it appears in the form of a neat little volume at the moderate price of eighteeupence, no further recommendation will be necessary."

History of Great Britain and Ireland ; with an Account

of the Present State and Resources of the United Kingdom and its Colonies. With Questions for Examination, and a Map. By Dr WHITE.

Athenceum. — "A carefully compiled history for the use of schools. Th

History of Scotland for Junior Classes ; with Questions

for Examination. Edited by Dr WHITE. Is. 6d.

History of Scotland, from the Earliest Period to the Present Time. With Questions for Examination. Edited by Dr WHITE. 3s. 6d.

History of France; with Questions for Examination, and a Map. Edited by Dr WHITE. 3s. 6d.

Athenaum. — "We have already had occasion to speak favourably of Dr White's ' History of Great Britain and Ireland.' The perusal of the present work has given us still greater pleasure. . . . Dr White is remarkably happy in combining convenient brevity with sufficiency of information, clearness of exposition, and interest of detail. He shows great judgment in apportioning to each subject its due amount of consideration."

Outlines of Universal History. Edited by Dr

WHITE. 2s.

fea

Spectator. — "Distinct in its arrangement, skilful in its selection of leading atures, close and clear in its narrative."

History. 13

Dr White's Elements of Universal History, on a New

and Systematic Plan. In THREE PARTS. PART I. Ancient History;

Part II. History of the Middle Ages ; Part III., Modern History.

With a Map of the World. 7s. ; or in Parts, 2s. 6d. each.

This work contains numerous synoptical and other tables, to guide the researches of the student, with sketches of literature, antiquities, and manners during each of the great chronological epochs.

Outlines of the History of Rome ; with Questions for

Examination. Edited by Dr WHITE. Is. 6d.

London Review. — "This abridgment is admirably adapted for the use of schools, — the best book that a teacher could place in the hand of a youthful student."

Sacred History, from the Creation of the World to the Destruction of Jerusalem. With Questions for Examination. Edited by Dr WHITE. Is. 6d. Baptist Magazine.—" An interesting epitome of sacred history, calculated to

inspire the young with a love of the divine records, as well as to store the

mind with knowledge."

Elements of General History, Ancient and Modern. To

which are added, a Comparative View of Ancient and Modern Geography, and a Table of Chronology. By ALEXANDER ERASER TYTLER, Lord Woodhouselee, formerly Professor of History in the University of Edinburgh. New Edition, with the History continued. With two large Maps, etc. 3s. 6d.

Watts' Catechism of Scripture History, and of the

Condition of the Jews from the Close of the Old Testament to the Time of Christ. With INTRODUCTION by W. K. TWEEDIE, D.D. 2s.

Simpson's History of Scotland ; with an Outline of the

British Constitution, and Questions for Examination at the end of each Section. 3s. 6d.

Simpson's Goldsmith's History of England ; with the

Narrative brought down to the Middle of the Nineteenth Century. To which is added an Outline of the British Constitution. With Questions 'for Examination at the end of each Section. 3s. 6d.

Simpson's Goldsmith's History of Greece. With

Questions for Examination at the end of each Section. 3s. 6d.

Simpson's Goldsmith's History of Rome. With Questions

for Examination at the end of each Section. 3s. 6d.

14 Writing, Arithmetic, and Book-keeping. WRITING, ARITHMETIC, AND BOOK-KEEPING.

THIS section will be found to contain works in extensive use in many of the best schools in the United Kingdom. The successive editions have been carefully revised and amended.

Practical Arithmetic for Junior Classes. By HENRY

Gr. C. SMITH, Teacher of Arithmetic and Mathematics in George Heriot's Hospital. 64 pages, 6d. stiff wrapper. Answers, 6d.

From the Rev. PHILIP KELLAND, A.M., F.R.SS. L. & E., late Fellow of Queens' College, Cambridge, Professor of Mathematics in the University of Edinburgh.

"I am glad to learn that Mr Smith's Manual for Junior Classes, the MS. of which I have examined, is nearly ready for publication. Trusting that the Illustrative Processes which he has exhibited may prove as efficient in other hands as they have proved in his own, I have great pleasure in recommending the work, being satisfied that a better Arithmetician and a more judicious Teacher than Mr Smith is not to be found."

Practical Arithmetic for Senior Classes ; being a Con- tinuation of the above. By HENRY GL C. SMITH. 2s. Answers, 6d. KEY, 2s. 6d.

%* The Exercises in both works, which are copious and original, have been constructed so as to combine interest with utility. They are accompanied by illustrative processes.

English Journal of Education.—1' There are, it must be confessed, few good books on arithmetic, but this certainly appears to us to be one of them. It is evidently the production of a practical man, who desires to give his pupils a thorough knowledge of his subject. The, Rules are laid down with much precision and simplicity, and the ilk iitelligible to boys of ordinary capacity."

Lessons in Arithmetic for Junior Classes, By JAMES

TROTTEH. 66 pages, 6d. stiff wrapper; or 8d. cloth. Answers, 6d.

This book was carefully revised, and enlarged by the introduction of Simple Examples of the various rules, worked out at length and fully explained, and of Practical Exercises, by the Author's son, Mr Alexander Trotter, Teacher of Mathematics, etc., Edinburgh ; and to the present edition Exercises on the proposed Decimal Coinage have been added.

Lessons in Arithmetic for Advanced Classes; being

a Continuation of the Lessons in Arithmetic for Junior Classes. Containing Vulgar and Decimal Fractions ; Simple and Compound Proportion, with their Applications ; Simple and Compound Interest; Involution and Evolution, etc. By ALEXANDER TROTTER. New Edition, with Exercises on the proposed Decimal Coinage. 76 pages, 6d. in stiff wrapper ; or 8d. cloth. Answers, 6d.

Each subject is also accompanied by an example fully worked out and minutely explained. The Exercises are numerous and practical.

precision and simplicity, and the illustrations cannot fail to make them int< ""

Writing, Arithmetic, and Book-keeping. 15

A Complete System of Arithmetic, Theoretical and

Practical ; containing the Fundamental Rules, and their Application to Mercantile Computations ; Vulgar and Decimal Fractions ; Invo- lution and Evolution ; Series ; Annuities, Certain and Contingent. By Mr TROTTER. 3s. KEY, 4s. 6d.

*t* All the 3400 Exercises in this work are new. They are applicable to the business of real life, and are framed in such a way as to lead the pupil to reason on the matter. There are upwards of 200 Examples wrought out at length and minutely explained.

Ingram's Principles of Arithmetic, and their Application

to Business explained in a Popular Manner, and clearly Illustrated by Simple Rules and Numerous Examples. Remodelled and greatly Enlarged, with Exercises on the proposed Decimal Coinage. By ALEXANDER TROTTER, Teacher of Mathematics, etc., Edinburgh. Is. KEY, 2s.

Each rule is followed by an example wrought out at length, and is illustrated by a great variety of practical questions applicable to business.

Melrose's Concise System of Practical Arithmetic;

containing the Fundamental Rules and their Application to Mercan- tile Calculations; Vulgar and Decimal Fractions; Exchanges; Involution and Evolution; Progressions; Annuities, Certain and Contingent, etc. Re-arranged, Improved, and Enlarged, with Exer- cises on the proposed Decimal Coinage. By ALEXANDER TROTTER, Teacher of Mathematics, etc., in Edinburgh. Is. 6d. KEY, 2s. 6d. Each Rule is followed by an example worked out at length, and minutely explained, and by numerous practical Exercises.

Button's Arithmetic and Book-keeping. 2s. 6d. Button's Book-keeping, by TROTTER. 2s.

S^ts of Ruled Writing Books,— Single Entry, per set, Is. 6d.; Double Entry, per set, Is. 6d.

Stewart's First Lessons in Arithmetic, for Junior Classes;

containing Exercises in Simple and Compound Quantities arranged so as to enable the Pupil to perform the Operations with the greatest facility and correctness. With Exercises on the Proposed Decimal Coinage. 6d. stiff wrapper. Answers, 6d.

Stewart's Practical Treatise on Arithmetic, Arranged

for Pupils in Classes. With Exercises on the proposed Decimal Coinage. Is. 6d. This work includes the Answers ; with Questions for Examination. KEY, 2s.

Gray's Introduction to Arithmetic; with Exercises on

the proposed Decimal Coinage. lOd. bound in leather. KEY, 2s.

16 Copy-Books^ Mathematics, etc.

Lessons in Arithmetic for Junior Classes. By JAMES

MACLAREN, Master of the Classical and Mercantile Academy, Hamilton Place, Edinburgh. 6d. stiff wrapper.

The Answers are annexed to the several Exercises.

Maclaren's Improved System of Practical Book-

KEEPING, arranged according to Single Entry, and adapted to General Business. Exemplified in one set of Books. Is. Gd. A Set of Ruled Writing Books, expressly adapted for this work, Is. Gd.

Scott's First Lessons in Arithmetic. 6d. stiff wrapper.

Answers, 6d.

Scott's Mental Calculation Text-book. Pupil's Copy, Gd.

Teacher's Copy, Gd.

Copy Books, in a Progressive Series. By R. SCOTT, late Writing-Master, Edinburgh. Each containing 24 pages. Price : Medium paper, 3d ; Post paper, 4d.

Scott's Copy Lines, in a Progressive Series, 4d. each.

The Principles of Gaelic Grammar ; with the Definitions,

Rules, and Examples, clearly expressed in English and Gaelic; containing copious Exercises for Reading the Language, and for Parsing and Correction. By the Rev. JOHN FORBES, late Minister of Sleat. 3s. Gd.

MATHEMATICS, NATURAL PHILOSOPHY, ETC. Ingram's Concise System of Mathematics, Theoretical

and Practical, for Schools and Private Students. Improved by JAMES TROTTER. With 340 Woodcuts. 4s. Gd. KEY, 3s Gd.

Trotter's Manual of Logarithms and Practical Mathe-

MATICS, for Students, Engineers, Navigators, and Surveyors. 3s.

A Complete System of Mensuration ; for Schools, Private

Students, and Practical Men. By ALEX. INGRAM. Improved by JAMES TROTTER. 2s.

Ingram and Trotter's Euclid. Is. 6d.

Ingram and Trotter's Elements of Algebra, Theoretical

and Practical, for Schools and Private Students. 3s.

Music, Drawing, School Registers. 17 Introductory Book of the Sciences. By JAMES NICOL,

F.R.S.E., F.G.S., Professor of Natural History in the University of Aberdeen. With 106 Woodcuts. Is. 6d.

SCHOOL SONGS WITH MUSIC,

By T. M. HUNTER, Director to the Association for the Revival of Sacred Music in Scotland.

Elements of Vocal Music : An Introduction to the Art of Reading Music at Sight. Price 6d.

%* This Work has been prepared with great care, and is the result of long practical experience in teaching. It is adapted to all ages and classes, and win be found considerably to lighten the labour of both teacher and pupil. The exercises are printed in the standard notation, and the notes are named as in the original Sol-fa System.

CONTENTS. — Music Scales. — Exercises in Time. — Syncopation. — The Chro- matic Scale. — Transposition of Scale. — The Minor Scale. — Part Singing. — Kxplanation of Musical Terms.

Hunter's School Songs. With Preface by Rev. JAMES CURRIE, Training College, Edinburgh.

FOR JUNIOR CLASSES : 60 Songs, principally set for two

voices. 4d. — Second Series : 63 Songs. 4d. FOR ADVANCED CLASSES : 44 Songs, principally set for three

voices. 6d. — Second Series : 46 Songs. 6d.

School Psalmody ; containing 58 Pieces arranged for three voices. 4d.

GEOMETRICAL DRAWING-.

The First Grade Practical Geometry. Intended chiefly

for the use of Drawing Classes in Elementary Schools taught in connexion with the Department of Science and Art. By JOHN KENNEDY, Head Master of Dundee School of Art. 6d.

School Register. PUPIL'S DAILY REGISTER OF MARKS.

Improved Edition. Containing Spaces for 48 Weeks ; to which are added, Spaces for a Summary and Order of Merit for each Month, {'or each Quarter, and for the Year. For Schools in general, and constructed to furnish information required by Government. 2d.

School Register of Attendance, Absence, and Fees :

adapted to the Provisions of the Revised Code, by MORRIS F. MYKON. Each folio will serve 50 pupils for a Quarter. Is.

1 8 French.

CLASS-BOOKS BY CHAS. HENEI SCHNEIDER, F.E.I.S.,

M.C.P.,

Senior French Master in the Edinburgh High School, the Merchant Com- pany's Educational Institution for Young Ladies, the School of Arts and "Watt Institution, etc. ; French Examiner to the Educational Institute of Scotland, etc.

Schneider's First Year's French Course. Is. 6d.

%* This work forms a Complete Course of French for Beginners, and comprehends Grammatical Exercises, with Rules; Reading Lessons, with Notes; Dictation; Exercises in Conversation; and a Vocabulary of all the Words in the Book.

The Edinburgh High School French Conversation-

GRAMMAR, arranged on an entirely New Plan, with Questions and Answers. Dedicated, by permission, to Professor Max Mullcr. 3s. 6d. KEY, 2s. 6d.

The Edinburgh High School New Practical French

READER: Being a Collection of Pieces from the best French Authors. With Questions and Notes, enabling both Master and Pupil to converse in French. 3s. 6d.

The Edinburgh High School French Manual of

CONVERSATION and COMMERCIAL CORRESPONDENCE.

2s. 6d.

In this work, Phrases and Idiomatic Expressions which are used most frequently in the intercourse of every-day life have been carefully collectt-d. Care has been taken to avoid what is trivial and obsolete, and to introduce all the modern terms relative to railways, steamboats, and travelling in general.

Ecrin Litteraire : Being a Collection of LIVELY ANEC- DOTES, JEUX DE MOTS, ENIGMAS, CHARADES, POETRY, etc., to serve as Readings, Dictation, and Recitation. 3s. 6d.

Letter from PROFESSOR MAX MULLER, University of Oxford, May 1867. " MY DEAR SIR, — I am very happy to find that my anticipations as. to the success of your Grammar have been fully realized. Your book does not require any longer a godfather; but if you wish me to act as such, I shall be most happy to have my name connected with your prosperous child. — Yours very truly, MAX MU'LLER.

" To Mcns. C. II. Schneider, Edinburgh High School."

The French New Testament. The most approved

PROTESTANT VERSION, and the one in general use in the FRENCH REFORMED CHURCHES. Pocket Edition, roan, gilt edges, Is. 6d.

Chambaud's Fables Choisies. With a Vocabulary

containing the meaning of all the Words. By SCOT and WELLS. 2s. Le Petit Fablier. With Vocabulary. For Junior Ola By G. M. GIBSOX, late Rector of the Bathgate Academy. Is. Gd.

French. 19

Standard Pronouncing Dictionary of the French and

ENGLISH LANGUAGES. In Two PARTS. Part I. French and English.— Part II. English and French. By GABRIEL SURENNE, late Professor in the Scottish Naval and Military Academy, etc. The First Part comprehends Words in Common Use, Terms con- nected with Science and the Fine Arts, Historical, Geographical, and Biographical Names, with the Pronunciation according to the French Academy and the most eminent Lexicographers and Gram- marians. The Second Part is an ample Dictionary of English words, with the Pronunciation according to the best Authorities. The whole is preceded by a Practical and Comprehensive System of French Pronunciation. 7s. 6d., strongly bound.

The Pronunciation is shown by a different spelling of the Words.

Surenne's French -English and English -French

DICTIONARY, without the Pronunciation. 3s. 6d. strongly bound.

Surenne's Fenelon's Telemaque. 2 vols, Is. each, stiff

wrapper ; or bound together, 2s. 6d.

Surenne's Voltaire's Histoire de Charles XII.

Is. stiff wrapper ; or Is. 6d. bound.

Surenne's Voltaire's Histoire de Eussie sous Pierre

LE GRAND. 2 vols, Is. each, stiff wrapper ; or bound together, 2s. 6d.

Surenne's Voltaire's la Henriade, Is. stiff wrapper;

or Is. 6d. bound.

Surenne's New French Dialogues; With an Introduc- tion to French Pronunciation, a Copious Vocabulary, and Models of Epistolary Correspondence. Pronunciation marked throughout. 2s.

Surenne's New French Manual and Traveller's

COMPANION. Containing an Introduction to French Pronuncia- tion ; a Copious Vocabulary ; a very complete Series of Dialogues on Topics of Everyday Life; Dialogues on the Principal Conti- nental Tours, and on the Objects of Interest in Paris; with Models of Epistolary Correspondence. Intended as a Class-book for the Student and a Guide to the Tourist. Map. Pronunciation marked throughout. 3s. 6d.

Surenne's Pronouncing French Primer. Containing

the Principles of French Pronunciation, a Vocabulary of easy and familiar Words, and a selection of Phrases. Is. 6d. stiff wrapper.

Surenne's Moliere's 1'Avare : Comedie. Is. stiff wrap- per ; or Is. 6d. bound.

Surenne's, Moliere's le Bourgeois Gentilhomme :

Comedie. Is. stiff wrapper ; or Is. 6d. bound.

20 French.

Surenne's Moliere's Le Misanthrope: Come'clie. Le

MARIAGE FORCE : Comedie. Is. stiff wrapper; or Is. 6d. bound.

Surenne's French Reading Instructor, Reduced to 2s. 6d. Hallard's French Grammar. 3s. 6d. KEY, 3s. Gd. Grammar of the French Language. BY AUGUSTE

BELJAME, B.A., LL.B., Vice-Principal of the Paris International College. 2s.

Beljame's Four Hundred Practical Exercises. Being

a Sequel to Beljame's French Grammar. 2s.

*„* Both Books bound together, 3s. 6d.

The whole -work has been composed with a. view to conversation, a great number of the Exercises being in the form of questions and ansv

First French Class-book, or a Practical and Easy Method of learning the FRENCH LANGUAGE, consisting of a series of FRE\< n and ENGLISH EXERCISES, progressively and grammatically arranged. By JULES CARON, F.E.I.S., French Teacher, Edin. Is. KEY, Is.

This work follows the natural mode in which a child learns to speak its own language, by repeating the same words and phrases in a great variety of forms until the pupil becomes familiar with their use.

Caron's First French Beading-book: Being Easy and

Interesting Lessons, progressively arranged. With a copious Vocab- ulary of the Words and Idioms in the text. Is.

Caron's Principles of French Grammar. With numerous

Exercises. 2s. KEY, 2s.

Spectator. — " May be recommended for clearness of exposition, gradual pro- gression, and a distinct exhibition to the mind through the eye by means of typo- graphical display : the last an important point where the subject admits of it."

An Easy Grammar of the French Language. With

EXERCISES AND DIALOGUES. By JOHN CHRISTISON, Teacher of Modern Languages. Is. 4d. KEY, 8d.

Christison's Recueil de Fables et Contes Choisis,

a 1'Usage de la Jeunesse. Is. 4d.

Christison's Flenry's Histoire de France, Racontee

a la Jeunesse. With Translations of the difficult Passages. 2s. 6d.

French Extracts for Beginners, With a Vocabulary

and an Introduction. By F. A. WOLSKI, Master of the Foreign Language Department in the High School of Glasgow. 2s. 6d.

Wolski's New French Grammar. With Exercises. 3s. dl.

Latin and Greek. 21

EDINBURGH ACADEMY CLASS-BOOKS.

THE acknowledged merit of these school-books, and the high reputation of the seminary from which they emanate, almost supersede the necessity of any recommendation. The " Latin " and " Greek Rudiments " form an intro- duction to these languages at once simple, perspicuous, and comprehensive. The "Latin Rudiments" contain an Appendix, which renders the use of a separate work on Grammar quite unnecessary ; and the list of anomalous verbs in the " Greek Rudiments " is believed to be more extensive and complete than any that has yet appeared in School Grammars of the language. In the '' Latin Delectus " and " Greek Extracts " the sentences have be«n arranged strictly on the progressive principle, increasing in difficulty with the Advance- ment of the Pupil's knowledge ; while the Vocabularies contain an explanation not only of every word, but also of every difficult expression which is found in the works, — thus rendering the acquisition of the Latin and Greek languages both easy and agreeable. The Selections from Cicero embrace the portions of his works which are best adapted for Scholastic tuition.

1. Rudiments of the Latin Language. 2s.

%* This work forms an introduction to the language, at once simple, perspicuous, and comprehensive.

2. Latin Delectus ; with a Vocabulary containing an Explanation of every Word and Difficult Expression which occurs in the Text. 3s. 6d.

3. Rudiments of the Greek Language. 3s. 6d.

4. Greek Extracts ; with a Vocabulary containing an Explanation of every Word and of the more Difficult Passages in the Text. 3s. 6d.

5. Selections from Cicero. 3s.

6. Selecta e Poetis Latinis ; including Extracts from

Plautus, Terence, Lucretius, Catullus, Persius, Lucan, Martial, Juvenal, etc. 3s.

Greek Syntax ; with a Rationale of the Constructions, by JAS CLYDE, LL.D., one of the Classical Masters of the Edinburgh Academy. With Prefatory Notice by JOHN S. BLACKIH, Professor of Greek in the University of Edinburgh. 4th Edition, entirely re-written, and enlarged by a Summary for the use of Learners and a chapter on Accents. 4s. 6d.

Greek Grammar for the Use of Colleges and Schools. By

Professor GEDDES, University of Aberdeen. 4s. The author has endeavoured to combine the clearness and conciseness of the older Greek Grammars with the accuracy and fulness of more recent ones.

22 Latin and Greek.

DR HUNTER'S CLASSICS.

1. Hunter's Ruddiman's Rudiments. Is. 6d.

2. Hunter's Sallust ; with Footnotes and Translations.

Is. 6d.

3. Hunter's Virgil; with Notes and other Illustrations.

2s. 6d.

4. Hunter's Horace. 2s.

5. Hunter's Livy. Books XXI. to XXV. With Critical

and Explanatory Notes. Reduced to 3s.

Latin Prose Composition : The Construction of Clauses,

with Illustrations from Cicero and Caesar; a Vocabulary containing an Explanation of every Word in the Text ; and an Index Verborum. By JOHN MASSIE, A.M. 3s. 6d.

Dymock's Csesar ; with illustrative Notes, a Historical and Geographical Index, and a Map of Ancient Gaul. 4s.

Dymock's Sallust; with Explanatory Footnotes and a Historical and Geographical Index. 2s.

Caesar ; with Vocabulary explaining every Word in the Text, Notes, Map, and Historical Memoir. By WILLIAM M'DowALL, late Inspector of the Heriot Foundation Schools, Edinburgh. 3s.

M'Dowall's Virgil ; with Memoir, Notes, and Vocabulary explaining every Word hi the Text. 3s.

Neilson's Eutropius et Aurelius Victor ; with Vocabu- lary containing the meaning of every Word that occurs in the Text. Revised by WM. M'DOWALL. 2s.

Lectiones Selectae : or, Select Latin Lessons in Morality, History, and Biography : for the use of Beginners. With a Vocab- ulary explaining every Word in the Text. By C. MELVILLE, late of the Grammar School, Kirkcaldy. Is. 6d.

Macgowan's Lessons in Latin Reading. In Two PARTS.

Part I., Improved by H. FRASER HALLE, LL.D. 2s. 17th Edition. Part II. 2s. 6d. The Two Courses furnish a complete Latin Library of Reading, Grammar, and Composition for Beginners, consisting of Lessons which advance in difficulty by easy gradations, accompanied by Exercises in English to be turned into Latin. Each volume contains a complete Dictionary adapted to itself.

Latin and Greek. 23

Mair's Introduction to Latin Syntax : with Illustrations

by Rev. ALEX. STEWART, LL.D. ; an English and Latin Vocabulary, for the assistance of the Pupil in translating into Latin the English Exercises on each Rule; and an Explanatory Vocabulary of Proper Names. 3s.

Stewart's Cornelius Nepos; with Notes, Chronological

Tables, and a Vocabulary explaining every Word in the Text. 3s.

Ainsworth's Latin Dictionary. Edited by WM. DUNCAN,

E.G. P. 1070 pages. 9s. strongly bound.

This edition contains a copious index of proper names, a complete list of Latin abbreviations, and other important and useful tables.

Duncan's Greek Testament. 3s. 6d.

Beza's Latin Testament. Revised by the late ADAM

DICKINSON, A.M. 3s. 6d.

Xenophon's Anabasis, Books I. and II. ; with Vocabulary

giving an Explanation of every Word in the Text, and a Trans- lation of the more difficult Phrases. By JAMES FERGUSSON, M.D., late Rector of the West End Academy, Aberdeen. 2s. 6d.

Athenceum. — " The text of this admirable little work is that of Dindorf, and the punctuation generally that of Poppo. Its principal excellence as an introduction to the study of Greek consists in the copious, correct, and well arranged Vocabulary at the end. This contains good translations of difficult passages, with exact information upon points of antiquities derived from the best and most modern authorities."

Grammatical Exercises on the Moods, Tenses, and

SYNTAX OF ATTIC GREEK. With a Vocabulary containing the meaning of every Word in the Text. On the plan of Professor Ferguson's Latin " Grammatical Exercises." By Dr FERGUSSON. 3s. 6d. KEY, 3s. 6d.

%* This work is intended to follow the Greek Rudiments.

Homer's Iliad— Greek, from Bekker's Text. Edited

by the Rev. W. VEITCH, Author of " Greek Verbs, Irregular and Defective," etc. 3s. 6d.

Homer's Iliad, Books I., VI., XX., and XXIV.; with

Vocabulary giving an Explanation of every Word in the Text, and a Translation of the more difficult Passages. By Dr FERGUSSON. 3s. 6d-

24 Latin and Greek.

LATIN ELEMENTARY WORKS AND CLASSICS.

Edited by GEORGE FERGUSON, LL.D., lately Professor of Humanity in King's

College and University of Aberdeen, and formerly one of the

Masters of the Edinburgh Academy.

1. Ferguson's Grammatical Exercises. With Notes,

and a Vocabulary explaining every Word in the Text. 2s. KEY, 2s.

2. Ferguson's Introductory Latin Delectus ; Intended

to follow the Latin Rudiments ; with a Vocabulary containing an Explanation of every Word and of every Difficult Expression. 2s.

3. Ferguson's Ovid's Metamorphoses. With Explanatory

Notes and an Index, containing Mythological, Geographical, and Historical Illustrations. 2s. 6d.

4. Ferguson's Ciceronis Orationes Selectae. Containing

pro Lege Manilia, IV. in Catilinam, pro A. L. Archia, pro T. A. Milone. Ex Orellii recensione. Is. 6d.

5. Ferguson's Ciceronis Cato Major sive de Senectute,

Laelius sive de Amicitia, Somnium Scipionis, et Epistolae Selectae. Ex Orellii recensione. Is. 6d.

G. Ferguson's Ciceronis de Officiis. Ex Orellii re- censione. Is. 6d.

The Port-Royal Logic. Translated from the French, with Introduction, Notes, and Appendix. By THOMAS SPENCER BAYNES, B.A., Professor of Logic, Rhetoric, and Metaphysics, United College of St Salvator and St Leonard, St Andrews. 4s.

ITALIAN. Theoretical and Practical Italian Grammar ; with

Numerous Exercises and Examples, illustrative of every Rule, and a Selection of Phrases and Dialogues. By E. LEMMI, LL.D., Italian Tutor to H. R. H. the Prince of Wales. 5s.— KEY, 5s.

From COUNT SAFFI, Professor of the Italian Language at Oxford. — "I have adopted your Grammar for the elementary instruction of students of Italian in the Taylor Institution, and find it admirably adapted to the purpose, as well for the order and clearness of the rules, as for the practical excellence and ability of the exercises with which you have enriched it."

PUBLISHED BY OLIVER AND BOYD, EDINBURGH;

SOLD ALSO BY SIMPKIN, MARSHALL, AND CO., LONDON, AND ALL BOOKSELLERS.

Hunter's School Songs with Music,
SECOND SERIES (for Junior Classes)— Continued.
The Golden Rule.—	Our Bonny Boat.—	Would you be loved?—
(Round for 3 Voices.)	(Round for 4 Voices.)	(Round for 3 Voices.)
December Night.	Welcome ! Welcome !	Song of the Bees.
The Lambkin.	little Stranger.	Slaves to the World.—
Haste thee, Winter,	Hark ! the little Birds	(Round for 3 Voices.)
haste away.	are singing. — (Round	Queen of the Fairies'
Evening Hymn.	for 4 Voices.)	Song.
Song of the Brook.	Wild Wood Flowers.	Moonrise.
Cradle Song.	Never say Fail.	Work and Play.
The Change of the	The Meadow Spring.	The Year's last Hour
Seasons.	The Mariner's Song.	is sounding. — (Round
Now we're met. —	Echo.— (Round for 3	for 4 Voices.)
(Round for 4 Voices.)	Voices.)	The Garment of Truth.
The Rivulet.	Man the Life-Boat!	The Farewell.
HUNTER'S SCHOOL SONGS
For Advanced Classes.
FIRST SERIES, containing 44 Songs, price 6d.
The Lark.	Ever-flowing, mighty	The Mountain Shep-
Lo! the Heavens are	Ocean.	herd Boy.
breaking.	Home, Sweet Home.	TheTraveller'sReturn.
Gather your Rosebuds.	A Man's a Man for a'	The Wayside Stream.
Freedom's Land.	that.	Bright are the Glories.
Humble is my little	Rule Britannia.	The cloud-cap t Towers.
Cottage. — (Pound	ColdtheBlastmayblow.	While Gladness hails
for 3 Voices.)	Sun, Moon, and Stars.	the parting Year.
Bright are Young Life's	Oh, see the lovely	Mariners, spread the
golden Treasures.	golden Sun !	Sail.
Good Night!— (Hound	A Southerly Wind and	LettheSmilesofYouth.
for 4 Voices.)	a Cloudy Sky.—	The Cuckoo.— (Round
Morning Star.	(Sound for 3 Voices.)	for 3 Voices.)
Hark! the Bonny.—	Around the Winter	Good Night.
(Round for 3 Voices.)	Fire so bright.	Now Sing the girds.—
God Save the Queen.	Come tothe Hills away!	(Mound for 3 Voices.)
When the rosy Morn	Fair Morn is up.	Come, honest Friends.
appearing.	Hark! 'tis the Wild	(Rouncifbr 8 Voices.)
How great is the Plea-	Birds singing. —	Christmas.
sure.— (Round for 3	(Round for 3 Voices.)	Come, conie, my Play-
Voices.)	The Spring breathes	mate's.- \R*G Voices.)
The Eagle.	around us.	Now Autumn rich. —
Sweet Spring is return-	Work while you may.—"	(Ottnon for 4 Voices.)
ing.	(Round for 4 Voices.)	The Shipwreck.

YA

043?0

Hunter's School Songs with Music.

SECOND SERIES, containing 46 Songs, price 6d. Slumber, gentle In- Now we are met.— The Pleasures of the

(Round for 3 Voices.) \ Wood. Now the Sun, his jour- ! Beautiful Primrose, ney ending. JO Toil, from thee

fant.

By and By. The Quail's Call. Life is Onward.

! The Psalm of Life.

I

I comes every Joy. How Sweet tobe Roam- Cursed be the Wretch. ' The Skylark.

ing.— (Round for 3 , (Round for 3 Voices.) I Night March.

Voices.) [ Let us all be up and The Sea-King's Song.

The Fisherman's Cot- doing.

tage. How lovely are

The Lorelei. days of Spring.

My Heart's in the See the Conq'ring

the

Highlands. Ode to Nature. 'Tis Hum - Drum. —

Hero comes. I love to Wander. Murmur, gentle Lyre.

(Sound for 3 Voices.) . The Open Window. The Chapel. | The Wayside Well.

Sweet the Pleasures. — Ye high-born Spanish

(Round for 3 Voices.)

Lordly Gallants.

L' Amour de la Patrie.

The Sun is career- ing.

Noblemen.

Come again.

Come, Follow me mer- rily.— (Round for 3 Voices.)

Patriotic Song.

Home.

Farewell to the Forest.

A Fairy Song.

Autumn Winds.

Silent Night.

Those Ev'ning Bells.

Wind, gentle Ever- j Brightly the Sun is green. — (Round for 3 beaming. — (Round Voices.) j for 6 Voices.)

Forest Song. ! In this little Island.

SCHOOL, PSALMODY;

Containing 58 Pieces, arranged for Three Voices. Price 4d. CONTENTS.

Abbey.

Bedford.

Belgrave.

Bethlehem.

Bishopthorpe.

Bur ford.

Culross.

Dismission.

Doversdale.

Doxology.

Duke Street.

Dundee.

Dunfe rin line.

Dusseldori.

Erfurt.

Evan.

Franconia.

French. Grace.

Harrington.

Huddersfield.

Irish.

Jackson's.

Leipsic.

London, New.

Macclesfield.

Manchester.

Martyrdom.

Mehcombe.

Morven.

Narenza.

Newington.

New St Ann's.

Norwood.

OldC.

Old CXXIV.

Sanctus.

Selma. Soldau.

Southwell.

Stroudwater.

St Ann's.

St Bride.

St David's.

St John's. St Magnus. St Mary's. St Matthias. St Michael. St Paul's. St Stephen's. St Thomas. Tallis. Wareham. Weimar. Winchester. Wittemberg. York.

*,* A Specimen Copy of any of these Books will le sent to Principals of Schools ly OLIVEB AND BOYD, on receipt of one-half the retail price in stamps. Edinburgh : OLIVER AND BOYD. London : SIMPKIN, MARSHALL, AND Co.