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.
Jof9Tȴac. = 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.
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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
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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,
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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.
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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
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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
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To the several countries are appended notices of their physical geography,
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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
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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
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White's Elementary Atlas of Modern Geography.
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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,
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Edinburgh Institution, etc. 5s.
Keid's Introductory Atlas of Modern Geography.
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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
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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
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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.
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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
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3s. 6d.
History of France; with Questions for Examination, and a
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work has given us still greater pleasure. . . . Dr White is remarkably
happy in combining convenient brevity with sufficiency of information,
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apportioning to each subject its due amount of consideration."
Outlines of Universal History. Edited by Dr
WHITE. 2s.
fea
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History. 13
Dr White's Elements of Universal History, on a New
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Part II. History of the Middle Ages ; Part III., Modern History.
With a Map of the World. 7s. ; or in Parts, 2s. 6d. each.
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Outlines of the History of Rome ; with Questions for
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London Review. — "This abridgment is admirably adapted for the use of
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student."
Sacred History, from the Creation of the World to the
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Edited by Dr WHITE. Is. 6d.
Baptist Magazine.—" An interesting epitome of sacred history, calculated to
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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
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D.D. 2s.
Simpson's History of Scotland ; with an Outline of the
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Simpson's Goldsmith's History of Greece. With
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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
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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.
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The Principles of Gaelic Grammar ; with the Definitions,
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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
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JAMES TROTTER. With 340 Woodcuts. 4s. Gd. KEY, 3s Gd.
Trotter's Manual of Logarithms and Practical Mathe-
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A Complete System of Mensuration ; for Schools, Private
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JAMES TROTTER. 2s.
Ingram and Trotter's Euclid. Is. 6d.
Ingram and Trotter's Elements of Algebra, Theoretical
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Music, Drawing, School Registers. 17
Introductory Book of the Sciences. By JAMES NICOL,
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of Aberdeen. With 106 Woodcuts. Is. 6d.
SCHOOL SONGS WITH MUSIC,
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%* This Work has been prepared with great care, and is the result of long
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CONTENTS. — Music Scales. — Exercises in Time. — Syncopation. — The Chro-
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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
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GEOMETRICAL DRAWING-.
The First Grade Practical Geometry. Intended chiefly
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1 8 French.
CLASS-BOOKS BY CHAS. HENEI SCHNEIDER, F.E.I.S.,
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The Edinburgh High School French Conversation-
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The Edinburgh High School French Manual of
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Letter from PROFESSOR MAX MULLER, University of Oxford, May 1867.
" MY DEAR SIR, — I am very happy to find that my anticipations as. to
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" To Mcns. C. II. Schneider, Edinburgh High School."
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First French Class-book, or a Practical and Easy Method
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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
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An Easy Grammar of the French Language. With
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Modern Languages. Is. 4d. KEY, 8d.
Christison's Recueil de Fables et Contes Choisis,
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French Extracts for Beginners, With a Vocabulary
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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
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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.
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2. Latin Delectus ; with a Vocabulary containing an
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Greek Syntax ; with a Rationale of the Constructions, by
JAS CLYDE, LL.D., one of the Classical Masters of the Edinburgh
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of Greek in the University of Edinburgh. 4th Edition, entirely
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Greek Grammar for the Use of Colleges and Schools. By
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22 Latin and Greek.
DR HUNTER'S CLASSICS.
1. Hunter's Ruddiman's Rudiments. Is. 6d.
2. Hunter's Sallust ; with Footnotes and Translations.
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3. Hunter's Virgil; with Notes and other Illustrations.
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Latin Prose Composition : The Construction of Clauses,
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By JOHN MASSIE, A.M. 3s. 6d.
Dymock's Csesar ; with illustrative Notes, a Historical and
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Dymock's Sallust; with Explanatory Footnotes and a
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Caesar ; with Vocabulary explaining every Word in the Text,
Notes, Map, and Historical Memoir. By WILLIAM M'DowALL,
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M'Dowall's Virgil ; with Memoir, Notes, and Vocabulary
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Neilson's Eutropius et Aurelius Victor ; with Vocabu-
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Lectiones Selectae : or, Select Latin Lessons in Morality,
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Macgowan's Lessons in Latin Reading. In Two PARTS.
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Latin and Greek. 23
Mair's Introduction to Latin Syntax : with Illustrations
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Duncan's Greek Testament. 3s. 6d.
Beza's Latin Testament. Revised by the late ADAM
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Athenceum. — " The text of this admirable little work is that of Dindorf,
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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
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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,
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G. Ferguson's Ciceronis de Officiis. Ex Orellii re-
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The Port-Royal Logic. Translated from the French,
with Introduction, Notes, and Appendix. By THOMAS SPENCER
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ITALIAN.
Theoretical and Practical Italian Grammar ; with
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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
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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;
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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.
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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.
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Evan.
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French. Grace.
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Irish.
Jackson's.
Leipsic.
London, New.
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Manchester.
Martyrdom.
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Morven.
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Newington.
New St Ann's.
Norwood.
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Selma. Soldau.
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