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A PRACTICAL ARITHMETIC
FOR
ELEMENTARY SCHOOLS
BY JAMES CURRIE, A.M.
PRINCIPAL OF THE CHURCH OK SCOTLAND TRAINING-COLLEGE, EDINBURGH J
AUTHOR OF " EARLY AND INFANT SCHOOL-EDUCATION,"
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I.
FIRST STEPS IN ARITHMETIC.
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PREFACE.
THIS treatise of Arithmetic is designed to comprise all that is
needed by the pupils of common schools, and by those of higher
schools till they have completed their elementary education.
It is not one of theory, since the instruction of pupils of their
standing must be, in the main, practical ; nor, on the other hand, is
it a mere collection of examples, since the only practical instruction
worthy of the name is that which sets the processes before them in a
rational way. It aims throughout at that just combination of theory
with practice which is the greatest merit of an elementary text-book.
The explanations are given concisely, and in the form in which they
are likely to be soonest apprehended by the pupil ; whilst the exer-
cises for practice will be found to be very numerous and carefully
graduated.
In particular, Notation and the four elementary operations, on a
satisfactory knowledge of which the pupil's subsequent progress
depends, are treated with great fulness. An introductory text-book
of Arithmetic should not be a mere condensation of a higher one ; it
should devote the space which it gains from the omission of certain of
the more advanced rules to the ampler treatment of those which are
fundamental. Where the arithmetic of a school is weak at all, it is in
these rules that the weakness almost invariably lies ; and it is in these
rules, according to the testimony of all competent authorities, that
the most material improvement in the teaching of the subject is to be
looked for.
In the arrangement of the treatise the author has kept in view the
requirements of the Privy- Council for Elementary Schools and Pupil-
Teachers, although he has not limited himself by them.
The Miscellaneous Exercises at the end have been taken chiefly
from the papers of the Privy-Council and Dick Bequest Examinations.
For the convenience of junior classes the early chapters, treating of
the elementary operations with simple numbers and with money, and
forming pp. 1-64 of the present work, are published separately under
the title of " First Steps in Arithmetic."
CONTENTS.
PAOB
TABLES OF VALUE, 5
NOTATION, 9
ADDITION, 17
SUBTRACTION, • . . 25
MULTIPLICATION, 32
DIVISION, 39
MISCELLANEOUS EXERCISES, 48
COMPOUND ADDITION — MONEY, 51
COMPOUND SUBTRACTION— MONEY, ...... 54
COMPOUND MULTIPLICATION— MONEY, ..... 55
COMPOUND DIVISION— MONEY, 58
REDUCTION— MONEY, 62
MISCELLANEOUS EXERCISES, 66
COMPOUND RULES— WEIGHTS AND MEASURES, . . 70
MISCELLANEOUS EXERCISES, 79
BILLS OP PARCELS, 83
PRACTICE, .84
RULE OP THREE, 92
COMPOUND RULE OF THREE, 101
MEASURES AND MULTIPLES, 103
VULGAR FRACTIONS, 106
DECIMAL FRACTIONS, 114
SIMPLE INTEREST, 121
COMPOUND INTEREST, 124
DISCOUNT, ; 125
STOCKS, 127
BROKERAGE, 128
INSURANCE, . 129
PROFIT AND Loss, 130
SQUARE ROOT, 131
MENSURATION, 132
MISCELLANEOUS EXERCISES, 137
*
v * y & .
TABLES OF MONEY, WEIGHT, AND MEASURES.
MONEY.
I. Money of Account.
4 farthings, /. = 1 penny, d.
12 pence = 1 shilling, 5.
20 shillings = 1 pound, £
II. Coins In Circulation.
BRONZE.
2 farthings = 1 halfpenny, \d.
2 halfpence = 1 penny.
SILVER.
4 threepenny pieces = 1 shilling.
3 groats = 1 shilling.
2 sixpences = 1 shilling.
2 shillings = 1 florin,./!
2 shillings and sixp. = 1 half-crown.
5 shillings = 1 crown, cr.
GOLD.
10 shillings -\
5 So""" [=1 half-sovereign
2 crowns J
20 shillings "k
8 half-crowns f , „„ „„„,•„„
10 florins f = l s°vereign.
4 crowns J
Paper money is also in use. One
pound-note represents the value of
20s. , or one sovereign ; and there are
also five-pound notes, ten-pound notes,
twenty-pound notes, fifty-pound notes,
and one-hundred-pound notes.
The guinea, formerly a gold coin =
£1, Is., is still recognised as a standard
value, though the coin itself is not in
use : so the half-guinea, or 10s. 6d.
WEIGHT.
m. Avoirdupois Weight
is used for all common goods.
16 drams, dr. = 1 ounce, oz.
16 oz. = 1 pound, Ib.
28 Ib =1 quarter, qr.
4 qrs. or 112 Ib =1 hundred wt. cwt.
20 cwt. =1 ton, T.
Also,
14 Ib =1 stone, st.
IV. Troy Weight
is used for weighing the precious
inetals and jewellery.
24 grains, gr. — 1 pennyweight, dwt.
20 dwt. = 1 ounce, oz.
12 oz. =1 pound, ft
Note.— The Ib Troy = 5760 gr.
The Ib Avoir = 7000 gr.
LENGTH.
V. Lineal Measure
Is used for measuring length, and is
iience often called long measure.
1? inches, in. = 1 foot, ft.
3 reet = 1 yard, yd.
5£ yards = 1 pole, po.
40 poles = 1 furlong, fur.
8 furlongs = 1 mile, ml.
Tradesmen use what is called & foot-
rule of three feet long for measuring
with, on which the feet are divided
into inches, and the inches into ei^ith
parts, tenths, or sixteenths. For
longer measurements, a tape or line
of 22 yards, similarly divided, is com-
monly used.
Obsolete measures, but still used for
special purposes, are the following : —
1 line = Ath inch.
1 palm = 3 inches.
1 span = 9 inches.
1 cubit = 18 inches.
1 hand (for mea-)
suring height of >• = 4 inches.
horses) )
1 fathom (for mea- ) _ fi f .
suring depth) } ~ 6 feet'
1 geographical mile = 1 mile 266 yds.
[nearly.
1 league = 3 geog. miles.
1 degree = 60 geog. miles.
VI. Cloth Measure
is used for measuring cloth.
2J inches = 1 nail, nl
4 nails = quarter, qr.
4 quarters = 1 yard, yd.
Also,
5 quarters = 1 ell, E.
The draper's rod, one yard long, is
divided according to this measure;
though in practice, fractions (six-
teenths) of a yard are more commonly
used.
VTI. Land Measure
is used for measuring land. Sur-
veyors use a chain for this purpose,
called Gunter's chain, 22 yards (or 4
poles) long, and divided into 100 parts
or links.
22 yards = 1 chain of 100 Iks.
10 chains = 1 furlong.
Note.— The link = 75? inches.
TABLES OF MONEY, WEIGHT, AND MEASURES.
SURFACE.
VIII. Square Measure,
sometimes called superficial measure,
is used for measuring surface or area.
144 sq. in. = 1 sq. ft.
9 sq. ft. = 1 sq yd.
30$ sq. yd. = 1 sq. po. (or perch, per.)
40 sq. po. = 1 rood, ro.
4 roods = 1 acre, ac.
640 acres = 1 sq. ml.
Still used for special purposes are
the following measures : —
100 sq. feet •= 1 square of flooring.
*?•*»* °T}=1 rod of brickwork.
36 sq. yd. = 1 rood of building.
Land-surveyors, as stated above, use
the chain of 100 links, though they
express the result of their measure-
ments in this table : —
10,000 square links = 1 square chain.
10 square chains = 1 acre.
SOLIDITY.
IX. Cubic Measure
Is used for measuring the contents of
solid bodies, e.g., masses of stones or
earth (hence often called solid mea-
sure), or of bodies which have tli*i
shape of solids, e.g., rooms, cis-
terns, etc.
1728 cubic in. =1 cubic ft.
27 cubic ft =1 cubic yd.
Shipping is measured by tonnage,
timber by loads, and general goods
sometimes by barrel-bulk, thus :—
42 cub. ft. = 1 ton shipping, T. sh.
40 cub. ft. rough )
timber >• = 1 load, lo.
60 do. hewn )
5 cub. ft =1 barrel-bulk, B.B.
CAPACITY.
X. Measure of Capacity
is used for the measurement of liquids,
and also of dry goods, like grain, etc.
4 gills, gi. = 1 pint, pt.
2 pints = 1 quart, gt.
4 quarts = 1 gallon, gal
2 gallons = 1 peck, pk.
4 pecks = 1 bushel, bu.
8 bushels = 1 quarter, qr.
The peck, bushel, and quarter are
nsed for dry goods only.
For wine and beer, casks of various
sizes are used, of which the Eort
common are —
FOR WINE.
The puncheon = 84 gal.
The pipe = 126 gal.
The tun = 252 gaL
FOR BEER.
The kilderkin = 18 gal.
The barrel = 36 gaL
The hogshead, Jihd. = 54 gal.
But these casks are not standard
measxires, and vary in their capacity.
The imperial gallon contains 277 '274
cubic inches.
TIME.
XI. Measure of Time.
60 seconds, sec. = 1 minute, min.
60 minutes = 1 hour, ho.
24 hours = 1 day, da.
7 days = 1 week, wk.
52 wks. 1 day, or ^ _ r
365 days f ~ Y ' V
366 days = 1 leap year.
100 years = 1 century.
The year is divided into 12 calendar
months : —
January 31 days
February 28
March 31
April 30
May 31
June 30
July 31 days
August 31
September 30
October 31
November 30
December 31
Every year (with very rare excep-
tions) whose number is divisible by 4,
is a leap year ; in which February has
29 days.
Thirty days have September,
April, June, and November :
All the rest have thirty-one,
Excepting February alone,
Which has but twenty-eight days clear,
And twenty-nine in each leap year.
The lunar month = 29 da. 12 ho. 44 min.
The solar year = 365 da. 5 ho. 48 min.
48 sec., i.e., nearly 365 days 6 hours
(the Julian year).
QUARTERLY TERMS.
In England.
Lady-Day, . March 25.
Midsummer, . June 24.
Michaelmas, . Sept. 29.
Christmas, . Dec. 25.
In Scotland.
Candlemas, . Feb. 2.
"Whitsunday, . May 15.
Lammas, . Aug. 1.
Martinmas, . Nov. 11.
TABLES OF MONEY, WEIGHT, AND MEASURES.
The centimes are reckoned, among
Christian nations, in numerical order
from the birth of our Lord (called the
Christian era) : thus the years 1 to 99
are the first century, 100 to 199 the
second, and so on. This is the nine-
teenth century. Any particular year,
e.gr., 1864, is denoted 1864 A.D., i.e.,
Anno Domini, in the year of our Lord.
The years before the birth of our Lord
are reckoned back in order from that
event: thus 1460 A.C., means Ante
Christum, or before Christ.
INCLINATION.
XII. Angular Measure
is used for measuring the angle or
inclination of one line to another.
60 seconds, " =1 minute,
60' = 1 degree,
90° = 1 right angle, L
860" = 1 circle, ©
The following Tables are subjoined
for reference : —
Paper Measure.
24 sheets = 1 quire, qu.
20 quires = 1 ream, re.
21^ quires = 1 perfect ream.
Cloth Measure.
5 quarters = 1 English ell.
3 quarters =. 1 Flemish ell, FI. E.
6 quarters = 1 French ell, Fr. E.
37 inches = 1 Scotch ell, S. E.
Apothecaries' Weight.
OLD MEASURE.
20 grains, gr. = 1 scruple, j^
3 scruples = 1 drachm, 3
8 drachms = 1 ounce Troy, 5
12 ounces =12) Troy.
NEW MEASURE (1862).
437J grains = 1 ounce Avoir.
Apothecaries' Fluid Measure.
60 minims, TT^ = 1 fluid drachm, /. 3
3 fl. drachms = 1 fluid ounce, /. ^
16 ounces = 1 tb
20 ounces = 1 pint, O
8 pints = 1 gallon, C
FOREIGN MONEY.
United States.
10 cents = 1 dime.
10 dimes = 1 dollar, $
1 dollar = 4s. 2d.
France.
100 centimes = 1 franc.
1 franc = 9£d. nearly.
Canada.
Accounts are kept in £ s. d. currency
of which £1 = 16s. 8d. sterling.
East Indies.
16 annas =. 1 rupee.
1 rupee = Is. lO^d,
OLD SCOTCH MONEY AND
MEASURES
still recognised in Scotland for certain
purposes.
Money.
1 shilling Scots = Id. sterling.
£1 Scots = Is. 8d. do.
being one-twelfth of the same
names sterling.
1 merk = Is. l£d.
Lineal Measure.
37 inches = 1 ell.
6 ells == 1 fall.
4 falls = 1 chain.
1 chain = 1J Imp. chain nearly.
Square Measure.
36 sq. ells = 1 square falL
40 sq. falls = 1 rood.
4 roods = 1 acre.
1 acre = 1J Imp. acre nearly.
Liquid Measure.
4 gills = 1 mutchkin.
2 mutchkins = 1 chopin.
2 chopins = 1 pint.
8 pints = 1 gallon.
1 gallon = 3 Imp. gallons nearly.
Dry Measure.
4 pecks = 1 ftrlot.
4 flrlots = 1 boll.
10 bolls = 1 chalder.
The Wheat Firlot was nearly equal
to an Imp. bushel (= '998 bush.); the
Barley Firlot nearly equal to 1J bush.
(= 1-456 bush.) The Boll weighs 140 tt>
Avoir.
PROPOSED DECIMAL COINAGE.
1 mil = one thousandth part of £1,
or = Jd. less j^d.
10 mils =1 cent, one-hundredth of £L
10 cents = 1 florin, one-tenth of £1.
10 florins = £1.
1.
2.
NUMERATION AND NOTATION.
Numbers of One Place.
One finger and one finger make two fingers.
Two fingers and one finger make three fingers.
Three fingers and one finger make four fingers.
Four fingers and one finger make five fingers.
Five fingers and one finger make six fingers.
Six fingers and one finger make seven fingers.
Seven fingers and one finger make eight fingers.
Eight fingers and one finger make nine fingers. Bf.1
One, two, three, four, five, six, seven, eight, nine, are the names
of numbers used in counting.
The naming of numbers is called Numeration.
One, three, five, seven, nine, are called odd numbers.
Two, four, six, eight, are called even numbers.
These nine numbers mean so many ones, or units as they are
called ; thus two means two ones or two units, three means three
ones or three units, and so on.
EXERCISE I. Bf.
1. Repeat the table of units, as given above.
2. Repeat it, using balls, marbles, boys, etc., instead of fingers.
3. Repeat it with the numbers alone, thus, " one and one are two."
4. Count from one up to nine, and from nine back to one.
6. Count the odd numbers from one to nine ; from nine to one.
6. Count the even numbers from two to eight ; from eight to two.
7. Name the two numbers next above five, eight, three, etc.*
8. Name the two numbers next below six, nine, four, etc.
9. Hold up three fingers, five, seven, etc.
10. How many wheels has a cart ? How many halfpence in a penny?
How many pence in a threepenny-piece ? How many letters in the
word " dog" ? How many legs has a cow ? etc.
11. If I have four pence and give one away, how many do I keep ?
If I have six marbles, and get one from James, how many have I ? etc.
The nine numbers are denoted by signs or figures, thus : —
one, two, three, four, five, six, seven, eight, nine,
12 3456789
The figuring of numbers is called their Notation.
I Bf. means that the ball-frame may be used for illustration.
> Etc. means that various other questions of the same kind may be given.
10 NUMERATION AND NOTATION.
EXERCISE II.
1. Write down the figures — (1.) even along ; (2.) up and down.
2. Name the numbers in Ex. iv. sect. 16.
3. Write down the figures for the same numbers.*
3. Numbers of Two Places.
If I count nine on my fingers, I find one finger over.
Nine fingers and one finger make ten fingers ; which is the
whole number of them.
If I wish to count beyond ten, I must begin again and go
round a second time ; that will give me two-times ten or two
tens. Three times round will give three-times ten or three tens ;
and so on, up to nine-times round, which will give nine-times
ten or nine lens.
One ten is called Ten, denoted by 10.
Two tens are „ Twenty, 20.
Three tens „ Thirty, 30.
Four tens „ Forty, 40.
Five tens „ Fifty, 50.
Six tens „ Sixty, CO.
Seven tens „ Seventy, 70.
Eight tens „ Eighty, 80.
Nino tens „ Ninety, 90.
The tens are numbers of two places. They are denoted by
the figures for the units with a cipher on the right.
The value of a figure is increased ten times by its being
written in the second place from the right : thus 3 denotes three
units, but 30 denotes three tens. Hence the notation we use
is called the decimal 2 notation.
The cipher is used to fill up the first or right-hand place,
when that place contains no units or nothing ; hence it is
commonly called nought or nothing. It is never used alone.
EXERCISE III.
1. Repeat the table of tens ; backwards ; by odds ; by evens.
2. Count the tens.
3. Name the tens next above forty, sixty, etc. ; next below thirty,
eighty, etc.
4. How many fingers have six boys ? eight boys ? etc. JBf.
5. How many boys together have thirty lingers ? seventy ? etc. Bf.
6. How many units in eight tens ? six tens ? etc.
7. How many tens in thirty units ? in seventy units ? etc.
1 Either from the copy or to dictation. /The teacher may vary the exercise
by having the figures pointed out on the board from columns written by him-
self. 2 From the Latin word decem, ten.
NUMERATION AND NOTATION.
11
One ten and two units
twelve,
One ten and three units
thirteen,
One ten and four units
fourteen,
One ten and five units
fifteen,
One ten and six units
sixteen,
One ten and seven units
seventeen, ,
One ten and eight units
, eighteen, ,
One ten and nine units
The tens-units are also num
, nineteen, ,
bers of two places
' 8. If I have ninety marbles and give away ten, how many do I keep ?
It 1 have seventy, and get ten more, and other ten, how many have
1 / etc.
9. Write down the figures for the tens below each other.
10. Name the numbers, Ex. vi. sect. 17, Nos. 1, 2.
11. Write down the figures for these numbers.
• One ten and one unit are called eleven, denoted by 11
12
13
14
15
16
17
18
19
the first being
the units' place, the second the tens' place.
The names of the numbers from 13 to 19 are formed by put-
ting the number of the units before that of the tens ; thus
thirteen is three and ten, fourteen is four and ten, etc. The
names of all the other numbers of two places are formed by
putting the number of the tens before that of the units j thus
Two tens and one are called twenty-one, denoted by 21
Two tens and two „ twenty-tivo, „ . 22
Etc. etc. etc.
Three tens and one „ thirty-one, „ . 31
Three tens and two „ thirty-two, „ . 32
Etc. etc. etc.
When numbers of two places are written below each other,
units are written below units, and tens below tens.
EXERCISE IV.
1. Repeat the table of tens-units fron ten to twenty, from twenty
to thirty, etc.
2. Count the tens-units from ten to 'twenty, from twenty to
thirty, etc.
3. If one boy holds up the fingers of his right hand, and other three
boys all their fingers, how many fingers are up ? how many if another
boy holds up his ? if another? if one boy removes his ? etc., Ef.
4. If I hold up seven fingers, how many girls must hold up all their
fingers to make twenty-seven ? to make thirty-seven ? etc., Bf.
5. Count by tens from thirty-one, from forty-two, etc.
Count by tens back from ninety-eight, eighty-seven, etc.
6. How many are 1 ten and 4 ? 2 tens and 6 ? 4 tens and 7 \ etc.
7. What tens and units make up 18, 27, 33, 47 ? etc.
8. Figure from ten to twenty, twenty to thirty, etc.
12 NUMERATION AND NOTATION.
9. Figure 2 tens below 2 units, 3 tens below 3 units, etc. ; 9 ur.Hs
below 9 tens, 8 units below 8 tens, etc,
10. Name the numbers in Ex. vi. sect. 17, No. 3-25.
11. Write down, or tell in order, the figures for these numbers.
5. Numbers of Three Places.
Nine tens and one ten make ten tens.
As we put ten units together, and call them one-ten, so we
put the ten-tens together and call them one hundred. JBf.
One hundred is denoted by . 100
Two hundreds „ . . 200
Three hundreds „ . . 300, and so on.
The hundreds are numbers of three places. They are denoted
by the figures for the units with two ciphers on the right.
The value of a figure is increased a hundred times by its being
written in the third place ; thus 3 denotes three units, tut 300
denotes three hundreds.
The two ciphers are used to fill up the first and second places,
when these places contain no units and no tens.
EXERCISE V.
1. Count the hundreds, backwards, by odds, by evens.
2. Name the numbers in Ex. ix. sect. 19, Nos. 1, 2.
3. Tell in order the figures in these numbers.
4. How many tens in 100, 500, 800 ? etc.
5. How many hundreds in 10 tens, 70 tens ? etc.
6. Figure the hundreds in an up-and-down line.
7. Figure 1 hund. below 1 ten, 2 hund. below 2 tens, etc.
9 tens below 9 hund., 8 tens below 8 hund., etc.
8. Figure 1 h. below 1 1. below 1 u.— 2 h. below 2 t. below 2 u. etc.
9 u. below 9 t. below 9 h.— 8 u. below 8 t. below 8 h, etc.
9. Write down the figures for the numbers Quest. 2.
^- Numbers consisting of hundreds, tens, and units are also
numbers of three places ; the first being the units' place, the
second the tens' place, and the third the hundreds' place.
Their names are formed by combining, in their order, the
number of the hundreds, the number of the tens, and the
number of the units. Thus —
146 denotes 1 h. 4 t. 6 u., and is called one hundred and forty-six.
270 „ 2 h. 7 t. 0 u., „ two hundred and seventy.
804 „ 8 h. 0 t 4 u., „ eight hundred and four.
Where there are no units, or no tens, these are omitted in
the names, as in the last two numbers.
When numbers are written in column, the same places must
be kept below each other.
NUMERATION AND NOTATION. 13
EXERCISE VI.
1. Count from one hundred to nine hundred and ninety by tens,
and from nine hundred and ninety to one hundred by tens.
2. Count from two hundred and forty to two hundred and fifty.
„ five hundred and sixty to five hundred and seventy, etc.
3. Name the numbers in Ex. ix. sect. 19, No. 8-25.
4. Tell in their order the figures in these numbers.
5. Figure below each other two hundred and twenty-two, two
hundred and two, two hundred and twenty, two hundred, twenty,
two : — etc. Repeat the same, beginning with the units.
6. Figure the numbers in Quest. 3.
7. Numbers of One Period.
All numbers of one, two, or three places — that is, all num-
bers from 1 to 999 — are numbers of one period.
Numbers of one place may be written with their period
completed by putting two ciphers to the left hand. Thus,
since 6 units is the same as 0 hundreds 0 tens 6 units, the
number 6 may be written 006, and read no hundred and
six.
Numbers of two places may be written with their period
completed by putting one cipher to the left hand. Thus,
since 6 tens 5 units is the same as 0 hundreds 6 tens 5 units,
the number 65 may be written 065, and read no hundred and
sixty-five.
A cipher placed to the left hand of any figure does not alter
its place, nor, consequently, its value.
EXERCISE VII.
1. What are the numbers whose figures in order are three, two,
one ; four, nothing, six ; six, four ; seven, two, nothing, ? etc.
2. What figures in order denote two hundred, two hundred and six,
five hundred and thirty-two ? etc.
3. What are these numbers made up of ?— Ex. ix. sect. 19.
4. Figure their several parts in order below each other ?
5. Point out the tens' place in them ? units' place ? hundreds' ?
6. What numbers are made up of these parts, 3 h. 2 1. 6 u. ? 4 h.
Ot. 7n.l 7h.4t.0u.? 8h. 4u.? etc.
7. Read these numbers, 7, 17, 20, 34, etc. (1.) as they stand ; (2.)
with their periods filled up ?
8. Read these numbers, 008—080—800—088—880—80, etc.
9. Take any number, as 5. What does it denote with one nought
before it ? with two ? with one after it ? with two ? with one before
and one after it I Which nought increases its value ten times ?
which leaves it unaltered ? What two noughts increase its value one
hundred times ? what two leave it unaltered ? What two increase its
value ten times ? etc.
14 NUMERATION AND NOTATION.
10. Write the numbers, eight, ten, twenty-five, etc. (1.) as incom-
plete periods ; (2.) as completed periods.1
11. Write in figures : fifty-three, thirty-seven, ninety-four, one
hundred and seventy, four hundred and sixty-nine, eight hundred
and eight, seven hundred and fourteen, seventy-eight, two hundred
and eighteen, five hundred and five, six hundred and sixty, three
hundred and thirty-three, nine hundred and forty one, five hundred
and sixteen, etc.
%.* When the pupil has obtained perfect facility in reading and writing num-
bers of one period, he may proceed with their addition, subtraction, and multi-
plication, returning afterwards to the notation of larger numbers.
8, Numbers of Two Periods.
Nine hundreds and one hundred make ten hundreds. As we
put ten tens together and call them one hundred, so we put the
ten hundreds together and call them one thousand.
One thousand is denoted by . . . 1,000
Two thousands, .... 2,000
Ten thousands, .... 10,000
Eleven thousands, .... 11,000
One hundred thousand, . . . 100,000
Three hundred and forty-seven thousand, . 347,000
Any number of thousands is written as if it were units, with
three ciphers on tlie right.
If the number contain also hundreds-tens-units, these are
written in place of the cyphers. Thus —
One thousand five hundred is denoted by . 1,500
Two thousand six hundred and thirty, . 2,630
Ten thousand four hundred and twenty-five, 10,425
Eleven thousand seven hundred and eight, . 11,708
One hundred thousand one hundred end thirty, 100,130
Three hundred and forty-seven thousand three
hundred and forty-seven, . . . 347,347
Every number of thousands has from four to six places,
forming two periods. The first period containing the hundreds
-tens- units, if there are any ; the second the thousands.
%* The two periods are often separated by a comma, as above, to prevent
mistakes in reading numbers ; but by practice the pupil vill soon be able to
do without it
EXERCISE VIII.
1. Read the numbers, Ex. x. sect. 20.
2. Write to dictation the numbers in same Exercise.
3. In 501274 (or any of the numbers in same Exercise), how many
thousands ? hundreds ? tens of thousands ? units ? hundred thousands ?
tens?
i Counters may be used to aid the pupil in writing numbers of one period ;
see Note, section 9
NUMERATION AND NOTATION. 15
4. In 347029 (or any of the numbers in same Exercise), what does
the 3 denote ? the 9 ? 0 ? 7 ? 4 ? 2 ?
5. What figures in order denote six thousand three hundred? or
any of the numbers in the same Exercise ?
6. What numbers are denoted by the following sets of figures in
order, 4, 2, 4, 8 ? 8, 0, 7, 9, 2 ? 3, 6, 5, 2, 0, 1 ? etc.
m Numbers of Three Periods.
Nine hundred thousands and one hundred thousands make a
thousand thousands, which we call one Million.
One million is denoted by ... 1,000,000
Two millions are 2,000,000
Ten millions, 10,000,000
Eleven miUions, 11,000,000
One hundred millions, .... 100,000,000
Three hundred and forty-seven millions, . 347,000,000
Any number of millions is written as if it were units, with
six ciphers to the right.
If the number contain also thousands, hundreds, tens, and
units, these are written in place of the ciphers, thus : —
One million five hundred thousand is denoted by 1,500,000
Two millions six hundred and thirty thousand, 2,630,000
Ten millions four hundred and twenty-five thousand, 10,425,000
Eleven millions seven hundred and eight thousand
five hundred and ten, .... 11,708,510
One hundred millions one hundred thousand and
one hundred, 100,100,100
Three hund. and forty-seven mills, three hund. and
forty-seven thousand, three hun. and forty-seven, 347,347,347
Every number of millions has from seven to nine places,
forming three periods; the first called the units1 period, the
second the thousands', and the third the millions'.
EXERCISE IX.
1. Read the numbers, Ex. xii. sect. 21.
2. In 243,076,549 (or any of the above numbers), how many hun-
dreds ? tens of thousands ? tens of millions ? units ? hundreds of thou-
sands ? etc.
3. In 804395276 (or any of the above numbers), what does the 5 de-
note? 4? 8? 0? 6? 7? etc.
4. What figures in order denote seven millions and thirty thousand,
or any of the above numbers ?
5. What is denoted by the 1st place, 2d period ? 2d place, 1st period ?
1st place, 3d period ? 2d period ? 1st period ? 3d place, 1st period ? etc.1
6. Write to dictation the numbers, Ex. xii. «ect. 21.
1 This questioning may be continued with the help cf three periods of
counters; thus 69G 999 999
These may be also advantageously used in the following exercises in dicta-
16 NUMERATION AND NOTATION.
10. Numbers of more than Three Periods.
Nine hundred millions and one hundred millions make a
thousand millions.
One thousand millions are denoted by 1,000,000,000
Ten thousand millions, . . 10,000,000,000
A hundred thousand millions, . . 100,000,000,000
Thousands of millions are written as if they were thousands,
and six ciphers are added.
If there are also millions, thousands, and units, these are
written in place of the ciphers, thus : —
One thousand two hundred and thirty millions is 1,230,000,000
Ten thousand five hundred and sixteen millions,
five hundred and sixteen thousand, . 10,516,516,000
One hund. and thirty-seven thous.,one hund. and
thirty-seven mills., one hund. and thirty-seven
thousand one hundred and thirty-seven, 137,137,137,137
Every number of thousands of millions contains from ten to
twelve places, forming four periods ; which may be separated
by commas, as above.
Still larger numbers may be expressed by a fifth period, com-
mencing at a million of millions, or, as it is called, a Billion ;
or even a sixth period for thousands of billions, thus : —
B. M. U.
137,137,137,137,137,137
But numbers of more than three periods rarely occur.
J]_> Appendix on the Roman Notation.
Numbers are sometimes denoted by another set of characters, called
Roman.*
These are seven in number, thus : —
1 is denoted by the letter I, 5 by V, 10 by X, 50 by L, 100 by 0,
500 by D, and 1000 by M.
EXERCISE X.
1. Name the letters, with the numbers they denote.
2. Write down the letters, with the numbers they denote.
tlon. Tims the pnpil may be asked to read 28 14 7, or to write numbers
In that way in the first instance, and then to supply the necessary ciphers.
i So called from having been used In the ancient Roman notation. The
ordinary characters are often spoken of as the Arabic, from having come to us
through the Arabs.
ADDITION.
17
12.
To denote other numbers, these seven characters are combined in
two ways — First, a character following another of greater or equal
value adds thereto its own value ; thus VI denotes 5 + 1, or 6. Second,
a character preceding another of greater value subtracts therefrom its
own value ; thus IV denotes 5 — 1, or 4.
The only numbers which are denoted by subtraction are the units
next under V and X, and the tens next under L and C ; thus 4 is de-
noted by IV, 9 by IX, 40 by XL, and 90 by XC. All the rest are
denoted by addition.
I 1
II 2
III 3
IV 4
V 5
VI 6
VII 7
VIII 8
IX 9
X 10
XX 20
XXX 30
XL 40
L 50
LX 60
LXX 70
LXXX 80
XC 90
XI 11
XII 12
XIII 13
XIV 14
XV 15
XLI 41
XLII 42
XLIII 43
etc.
C 100
CO 200
CCC 300
CCCC 400
D 500
DC 600
DCC 700
DCCC 800
DCCCC 900
CX 110
CXX 120
CXXIV 124
CXLIX 149
CCXXX 230
CCCLXI 361
DXC 590
DCCIII 703
etc.
M 1000
MC 1100
MCC 1200
MD 1500
MDLXIV 1564
MDCX 1610
MDCXCII1692
MDCCC 1800
MM 2000
The Roman characters are now used only to denote numbers, e.g.,
the chapters of a book, the hours on the clock, the houses in a street,
and the years ; never to calculate with.
EXERCISE XL
1. What numbers are denoted by V, X, IV, XX, XXII, XL, etc. ?
2. Name, or write down, letters for the numbers, Ex. iv. sect. 16.
3. Name, or write down, letters for the numbers, Ex. vi. sect. 17.
4. Name, or write down, letters for the numbers, Ex. ix. sect. 19.
5. Do. do. 1250, 1365, 1473, 1582, 1624, 1738, 1806, 1835, 1864.
13.
ADDITION.
Ex. — Of four flocks of sheep, one contained 35, the second 29,
the third 50, and the fourth 47. They were put into one field ;
how many sheep were there in all ?
Here we have to find one number as large as four given
numbers together.
The number to be found is called the sum.
The sum is got by adding the four given numbers together.
The process of adding is called addition; and — when the
things to be added are of one kind, as here — simple addition.
The sign of addition is + (plus) : thus 1 + 1 are 2.
We cannot find the sum of the above four numbers at once ;
they are too large. We must therefore add them in parts ;
for which purpose we must leajn the addition of the first nine
numbers.
18
ADDITION.
14.
Addition Table.
%* This Table should be learnt first in lines even along ; thus, 1 and 1 are 2 ;
2 and 1 are 3, etc. ; afterwards in lines up and down. Lf.
1 and |2 and |3and |4and
5 and
5 and
7 and |8 and
9 and
1 are 2,1 are 3.1 are 41 are 5
lare 6
lare 7
1 are 8 1 are 9
lare 10
2... 3,2... 42... 52... 62... 7
2... 8,2... 92... 10
2 ... 11
3... 43... 5;3... 63... 73... 8
3 ... 93 ... 10
3 ... 11
3 ... 12
4... 5
4... 6.4... 714... 84... 9
4 ... 10
4... 114 ... 12
4 ... 13
5... 6
5... 75... 85... 95... 10
5 ... 11
5 ... 125 ... 13
5 ... 14
6... 7
6 ... 8:6 ... 96 ... 106 ... 11
6 ...126 ... 136 ... 14
6 ... 15
7... 8
7 ... 9.7 ... 107 ... 11-7 ... 12
7 ...137 ... 14
7 ... 15
7 ... 16
8... 9
8 ... 108 ... 118 ... 128 ... 13
8 ... 148 ... 15
8 ... 16
8 ... 17
9... 109... 11J9... 129... 139... 14
9... 159... 16
9 ... 17
9 ... 18
EXERCISE I. Bf.
1. Repeat the several lines of the table even along ; backwards ; by
odds and evens.
2. Repeat the several lines up and down in the same orders.
3. 5 and 6 are — ? 8 and 3 are — ? 4 and 9 are — ? etc.
4. 2 + 3 + 5 are — ? 6 -f 3 + 8 are — ? etc.1
5. 2 + 4 + 3 + 7 are— ? 5 + 2 + 2 + 6 are — ? etc.*
6. 2 books and 3 books are — ? I have 5d. and John 4d., how
much have we both ? John had 3 marbles ; if he bought 6 and
gained 7, how many has he now ? etc.
7. Write down the columns of the table in order.
. If one of the numbers to be added contains tens and units,
add the units as if they were alone, and prefix the number of
tens. Thus —
11 and 1 are 12 ; 12 and 1 are 13 ; 13 and 1 are 14.
11 and 2 are 13 ; 12 and 2 are 14 ; 13 and 2 are 15.
Etc. etc. etc.
EXERCISE II.
1. Repeat the several lines of this table from 11 to 19, (1.) even
along, (2.) up and down.
2. Repeat a similar table for 21-29, 31-39, etc.
3. 11 and 4 are — ? 17 and 8 are — ? etc.
4. 5 + 19 + 4 are — ? 17 + 6 + 5 are — ? etc.
5. 16 + 7 + 2 + 4 are — ? 13 i + 4 + 9 + 6 are — ? etc.
6. Write down any line of this Table in order.
EXERCISE III.
Count forward from 1, 2, 3, 4, 5, 6, 7, 8, 9 by twos, then by threes,
fours, etc., up to nines.
i In Ques. 4, the sum of the first two numbers, and in Ques. 5, the sum of
the first three, should not exceed nine.
ADDITION.
19
16. Addition of Numbers of One Place.
Ex. — John had 8 marbles, James had 4, William had 7,
and Henry 5 ; how many had they amongst them ] 8
We can find the sum of these small numbers 4
without writing ; but if we wish to write down 7
the process, we set the numbers below each other, 5
and add step by step, thus —
(5 and 7 are) 12 ; (and 4 are) 16 • (and 8 are) 24— 24
which is the sum required.
%* The words within parentheses may be used for some time by the pupil,
but should be omitted at the earliest moment he can do without them.
The addition may be proved to be correct by adding the
column downwards from the top. The sum of any series of
numbers is the same in whatever order they are added.
EXERCISE IV.
(l)
(2)
(3)
(4)
(5)
(C)
(7)
(8) (9)
(10)
(11) (12) (13) (14) (15) (16) (17) (IS)
(1.
8
9
2
6
8
3
5
6
7
2
1
4
5
6
7
8
9
2
(2.) 7
4
5
1
6
2
9
7
2
0
1
3
6
0
5
9
0
4
(3.)
5
7
7
5
4
1
8
7
5
6
9
2
0
5
4
1
3
2
(4.
6
6
8
4
8
6
4
9
4
3
2
1
0
8
6
9
3
1
(5.
4
0
9
3
5
3
8
7
2
0
1
5
6
8
5
6
9
8
(6-,
3
8
1
2
4
0
0
8
3
6
9
4
5
9
1
7
3
4
CM
2
5
0
7
2
9
5
0
8
6
5
3
1
2
3
4
0
1
(8.) 9
4
6
0
3
7
6
2
0
4
9
3
4
1
5
7
3
0
(9.
0
1
5
9
5
4
3
2
1
0
1
2
3
4
5
6
7
8
(10.
5
1
4
4
4
5
6
7
8
9
8
7
6
5
4
3
2
1
(11.
4
5
3
5
0
1
2
3
4
5
6
7
8
9
0
4
3
8
(12.) 8
9
2
2
6
1
0
5
3
4
7
4
3
8
5
2
2
9
(13.) 7
6
5
1
7
0
5
6
4
3
9
2
1
8
4
6
3
8
(14.
3
2
7
2
6
4
5
4
0
9
1
2
3
5
8
2
0
9
(15.
5
0
6
0
4
1
0
7
3
6
5
4
2
9
2
1
1
7
(16.
8
7
7
9
1
4
7
8
2
1
5
3
9
4
3
6
6
4
(17.
2
6
8
3
0
6
7
5
4
8
6
2
0
1
4
7
2
5
(18.
) 9
5
3
4
8
2
4
3
2
6
0
9
5
4
3
2
0
1
(19.
1 5
1
5
6
1
2
4
6
8
0
2
4
6
8
0
2
4
6
(20.
) 7
8
4
7
1
3
5
7
9
1
3
5
7
9
1
3
5
7
(21.
) 4
2
2
9
4
2
3
6
2
4
7
3
5
8
4
6
9
5
(22.) 0
4
3
0
5
4
3
4
7
8
9
5
6
8
2
0
0
1
(23.
6
6
1
1
3
4
9
6
8
2
0
1
4
3
7
7
6
4
(24.) 8
9
7
2
5
6
2
1
4
9
3
2
0
8
6
4
3
2
(25.) 1
8
8
3
4
5
8
9
1
4
7
6
8
5
1
2
3
0
%* These numbers may be added in parts of columns, or in whole
columns, up — down — from left to right — from right to left. And the pupil
sbould work at them a little every day till he attains expertness in adding.
20
ADDITION.
17.
Addition of Numbers of Two Places.
The Table given, sect. 14, serves also for the addition of
tens, thus : —
If 1 and 1 are 2, 1 ten and 1 ten are 2 tens, or 10 and 10 are 20.
2 and 1 are 3, 2 tens and 1 ten are 3 tens, or 20 and 10 are 30.
Etc. etc. etc.
EXERCISE V.
Perform Ex. i. Quests. 1-5, with tens.
Ex. — Of four flocks of sheep one contained 35, the second 20,
the third 50, and the fourth 47. They were put into one field :
how many sheep were there in all ?
Set the numbers below each other in their places. 35
Then in the units' column : (7 and 9 are) 16, (and 5 29
are) 21 (units ; set down) 1 (in the units' place), and 50
carry 2 (tens to the tens' column). Next, in the tens 47
column : (2 and 4 are) 6, (and 5 are) 11, (and 2 are)
13, (and 3 are) 16 (tens. Set down the) 6 (in the) 161
tens' (column), and (the ten tens as) 1 hundred (in
the hundreds' column).
EXERCISE VI.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10. .
11.
12.
13.
14.
15,
20
70
46
23
14
22
34
54
72
29
13
27
41
64
39
30
40
50
13
43
19
96
34
49
64
70
91
27
36
44
40
7
64
14
50
47
94
18
81
49
17
9
4
51
63
50
20
36
29
69
90
25
60
70
80
90
40
56
4
20
60
60
45
56
24
47
18
26
43
31
83
7
91
54
7
40
9
69
73
33
53
37
43
62
15
24
19
9
48
17
70
80
92
20
41
64
74
51
64
82
39
24
47
64
8
80
10
87
34
76
92
82
27
39
51
63
75
87
99
9
90
5
71
47
92
10
45
14
17
20
23
6
9
2
49
30
50
25
56
85
86
37
35
38
41
44
47
50
53
80
40
30
34
81
24
48
29
94
91
87
84
62
59
72
27
50
40
23
73
37
35
15
62
59
18
60
53
27
9
93
16.
17. 22 + 80+6 + 12 + 15 + 93 + 27 + 36 + 48 + 51 + 70+10 + 29 + 8.
18. 37 + 45 + 15 + 7 + 1 + 27 + 39 + 82+99 + 4+54 + 37 + 10 + 29.
19. 28 + 57 + 3 f 30 + 17 + 37 + 90+25 + 41 + 8 + 59 + 32+87 + 40.
20. 29 + 5 + 16 + 34 + 64 + 72 + 19+7 + 38 + 64+28 + 11 + 58 + 38.
21. 18 + 90+21 + 7 + 9 + 8+15 + 27+47 + 50 + 62+71 + 89 + 69.
22. 30+54+4+23 + 93 + 47 + 50+41 + 39+8 + 17 + 28 + 60.
23. 16 + 84+17 + 30 + 85 + 74 + 32+91 + 11 + 22 + 50+5 + 15 + 66.
24. 93 + 9 + 8 + 17+29 + 40+57 + 85 + 36 + 21 + 73 + 17 + 76 + 82.
25. 87 + 53 + 20 + 6 + 9 + 14 + 65 + 89 + 53 + 28 + 70 + 38 + 67 + 2.
ADDITION. 21
EXERCISE VII.
1. 10 + 11 are — ? 10 + 12 are — ? 10 + 13 are — ? 10 + 21 are — ? etc.
2. 20 + llare — ? 20 + 12are — ? 20 + 13are — ? 20 + 21 are — ? etc.
3. 30 + llare — ? 30 + 12are — ? 30 + 13are — ? 30 + 21 are — ? etc.
4. 40 + llare — ? 40 + 12 are — ? 40 + 13 are — I 40 + 21 are — ? etc.
5. Add the remaining tens in a similar way.
6.50 + 25are— ? 20 + 18are — ? 40 + 29 are — ? etc.
7.22 + 15are — ? 34 + 18are — ? 75+24 are — ? etc.
V* In this last question, it is easier to add the tens first; thus : 34+18
are 4 tens and 12, that is 52.
Addition of Numbers of One or more Periods.
The table given, section 14, serves also for the addition of
hundreds, thousands, etc. ; thus,
If 1 and 1 are 2, 1 h. and 1 h. are 2 hs., or 100 and 100 are 200.
2 and 1 are 3, 2 h. and 1 h. are 3 hs., or 200 and 100 are 300.
Etc. etc. etc.
EXERCISE VIII.
Perform Ex. i. Questions 1-5, with hundreds.
Ex. — Four heaps of bricks were lying in a field. The first con-
tained 208 bricks, the second 349, the third 160, and the fourth
87 ; how many bricks were there in all ?
Set the numbers below each other in their places.
In the units* column — (7 and 9 are) 16, (and 8
are) 24 (units ; set down) 4 (in the units' 208
place), and carry 2 (tens). 349
In the tens' column — (2 and 8 are) 10, (and 6 160
are) 16, (and 4 are) 20 ; (set down) 0 (in the 87
tens' place) and carry 2 (hundreds).
In the hundreds' column — (2 and 1 are) 3, (and 804
3 are) 6, (and 2 are) 8, (set down 8 in the
hundreds' place).
Sum, 804.
%* After some practice in adding, the words within parentheses should
be omitted.
Rule. — Set the numbers below each other in their places ;
and add the columns in their order from the units, carrying the
tens.
19.
4
1.
100
300
500
700
900
400
600
800
2.
200
500
900
100
300
800
600
700
3.
418
296
306
851
628
435
200
753
4.
524
615
500
924
705
396
527
713
ADDITION.
EXERCISE IX.
5. 6. 7. 8.
638 793 814 701
800 215 427 593
524 300 324 414
357 618 650 710
184 509 379 327
225 493 800 967
604 215 930 413
593 336 247 258
9.
649
524
700
810
81
47
913
27
10.
547
64
147
291
17
364
84
913
11.
890
47
562
50
900
73
654
209
12.
736
624
93
14
257
39
572
809
13. 365+210 + 93 + 27+110+345+563 + 207+824+85+127.
14. 241 + 56 + 37+26^+357+842 + 506+37+81 + 190+429.
15. 306 + 194+516 + 70+7 + 829+593+601 + 72+720+18.
16. 501+600+60+372+144+11 + 111+29 + 360+306+71.
17. 76+706 + 760+370 + 307 + 37+377+84 + 804+840+9.
18. 275 + 360+910 + 989+724+57+507+37+7+190+273.
19. 188+560+108+506+56 + 15 + 7+180+18+56 + 566.
20. 673 + 840+737+928+517+349 + 210+500+618+819.
21. 307 + 509+910+117+250+638 + 356+951+117+89
22. 15 + 27+119 + 94+101 + 709 + 364 + 87+2 + 370+241.
23. 293 + 18+573+194+346 + 504+673+936 + 19 + 207.
24. 64 + 604 + 406 + 600 + 640 + 460 + 46 + 83 + 803 + 830.
25. 199 + 96 + 737+307+516+93+7+16 + 738+259+59.
EXERCISE X.
1.
2.
3.
4.
5.
6. 7.
8.
9.
1,000
1000
5000
7000
1896 4567 8456
2408
9406
2,000
1100
500
700
1304 8432 7349
5493
1250
4,000
1200
4000
70
1940 9064 9118
9621
6430
6,000
1300
40
7
1284 2345 2565
8504
8094
8,000
1400
800
600
1700 7298 3894
7632
5432
9,000
1500
9000
4000
1676 5934 5248
4562
8006
7,000
1600
5000
900
1864 6309 7348
3901
9210
5,000
1700
600
6000
1547 7124 9176
2008
5090
10.
11.
12.
13.
14.
15.
16.
17.
3476
2930
8046
10,000
30,000
70000 80000
27,300
593
456
810
30,000
40,000
30000
500
34,000
24
3948
9
50,000
70,000
6000
60
26,900
896
27
9421
90,000
80,000
200 50000
84,200
7208
639
39
80,000
10,000
8000
9000
53,700
5009
7204
840
40,000
30,000
90000
40
85,600
648
408
7240
20,000
60,000
600 30000
28,400
8
3072
384
50,000
50,000
50000
700
61,060
ADDITION. 2;
18.
19.
20.
21.
22.
23.
24.
43,214
73059
83426
29070
45623
82472
19465
28,970
84320
34924
50846
72020
846
3947
36,429
92000
85241
63147
93647
9701
64
82,456
84372
12345
94621
804
35624
94702
93,484
50028
66666
80403
9562
256
876
21,086
90200
93002
70002
93
7
5724
73,481
89301
47020
70020
84756
9470
12730
18,498
56238
13076
70200
7250
85064
9400
25.
26.
27.
28.
29.
30.
100,000
300,000
400000
648,724
910,317
542300
300,000
200,000
8000
720,720
843,256
272484
700,000
700,000
90
843,843
123,000
364862
800,000
60,000
900
920,000
456,700
127859
400,000
50,000
9000
647,000
506,840
730640
900,000
500
80000
564,300
920,100
827938
500,000
800,000
800000
734,310
800,701
910400
600,000
500,000
60000
173,094
308,452
478915
81. 843 + 2465 + 724+17+10934+59470 + 107+20094 + 800.
32. 927 + 250 + 3070 + 601 + 38 + 731 + 1 456 + 1 001 + 27 + 374.
33. 493 + 913 + 67 + 500+610+1100 + 1420 + 3706 + 3076+3760.
34. 39 + 280 + 563 + 730+525 + 3482 + 79 + 2496 + 7314+326 + 89.
35. 470+1493 + 293 + 674+825 + 300 + 93 + 1910+2564+836 + 932
36. 9246 + 29805 + 367934 + 39 + 493 + 9 + 90 + 49321 + 7007.
37. 8439 + 7246 + 297 + 800 + 2094 + 73825 + 493 + 12345 + 936.
38. 4731 + 8472+938+76+3938 + 425 + 18 + 967+2005+6790.
39.4901 + 829 + 736 + 90 + 894 + 3247 + 9694+8482+386.
40. 7000 + 770 + 9382 + 54 + 504 + 5004 + 5040 + 5400 + 7054.
41.348 + 7 + 77+777 + 7777 + 77777+9 + 49+17248 + 34.
42. 2693 + 301 + 4 + 404 + 39456 + 327 + 999 + 45602 + 18.
43. 24962 + 376 42 + 4936 + 2754 + 930 + 18500 + 2590 + 196.
44. 93642 + 80010 + 930 + 18275 + 60600 + 66000 + 60060.
45. 7285 + 93271 + 893 + 7249 + 90000 + 18506 + 375 + 9640.
46. 8546 + 2764 + 94681 + 27600 + 9300 + 71486 + 8206 + 9.
47. 45894 + 318 + 7462 + 80001 + 90309 + 7402 + 70906.
48. 437 + 938 + 94 + 7300 + 1805 + 72468 + 79005 + 9406 + 50.
49. 6293 + 946 + 8001 + 92465 + 716 + 24070 + 807 + 5005 + 397.
50. 5484 + 29367 + 937056 + 720000 + 804906 + 100000 + 9040.
51 . 249356 + 730854 + 272494 + 800800 + 549304 + 20400 + 701.
52. 42836 + 90045 + 89362 + 5279 + 7264 + 7649 + 1200 + 937.
53. 5000 + 50000 + 50 + 505 + 5050 + 5 + 555 + 55555 + 550.
EXERCISE XI.
Below the sum of the following numbers, write the uppermost, and
add again ; below that sum write the second from the top, and add
again ; continue the addition in this way till all the numbers are
taken in, and find the sum.
24
ADDITION.
1. 235 + 196 + 450 + 600 + 801.
2. 342 + 94 + 502 + 86 + 300.
3. 279 + 50 + 116 + 270 + 207.
4. 100 + 50 + 322 + 901+626.
5. 736 + 941 + 257 + 509 + 316.
6. 241+80 + 173 + 428 + 299.
7. 864 + 731 + 279 + 333 + 67.
8. 420 + 204 + 176 + 815 + 700.
9. 304 + 430 + 82 + 73 + 371.
10. 536 + 801 + 78 + 306 + 420.
11. 216 + 39 + 500 + 493 + 811.
12. 340 + 610 + 93 + 217 + 536.
13. 117 + 711+270 + 207 + 453.
14. 820 + 304 + 916 + 732 + 564.
15. 936 + 576 + 429 + 827 + 517.
16. 320 + 600 + 66 + 308 + 201.
17. 524 + 47 + 39 + 809 + 468.
18. 279 + 320 + 809+543 + 397.
EXERCISE XII.
1.
2.
3.
4.
5.
238946
900500
1,000,000
8000000
3,564,236
72400
2736
3,000,000
800000
2,564,304
930
93
8,000,000
80000
2,197,629
645046
84293
4,000,000
8000
8,46D,038
8434
701 856
6,000,000
800
7,382,0<J3
67
73900
7,000,000
80
2,946,904
93248
2784
9,000,000
90000
3,842,460
100484
932043
2,000,000
7000000
8,080,803
6.
7.
8.
9.
10.
3456729
9203564
37,240,000
72,483,624
193,700,070
3040506
964383
93,280,000
8,734,724
270,937,000
3004005
728
87,200,400
9,328
384,256,070
3000400
92100
93,400,860
904,374
930,184,293
2790364
8056720
85,085,023
87,208,936
127,249,130
8710800
5296
62,473,903
97,318
147,234,876
56231*33
931724
24,084,573
9,433,729
310,249,364
7703804
8403203
16,946,004
47,082,970
172,849,564
11. 1234567 + 7238049 + 3947246 + 8420800 + 9220000.
12. 8004930 + 12340 + 7248436 + 9436 + 87 + 72456 + 9384567.
13. 72483624 + 8734724 + 9328 + 904374 + 87208936.
1 4. 27007<~»70 + 2700707 + 94302 + 734 + 85693 + 9438729.
15. 37248734 + 946432 + 87324 + 9256491 + 80724300.
16. 1 25000890 + 700700700 + 193299870 + 240019000.
1 7. 738456938 + 248724807 + 301234563 + 384965724.
18. 2000000 + 7304524 + 5428946 + 7289476 + 1S0050 + 72004.
19. 47849562 + 93859627 + 2507923 + 804974 + 2904 + 93006.
20. 192196924 + 534920815 + 8256293 + 79000600 + 180000018.
EXERCISE XIII.
. 1. John has 38 marbles ; he buys 20 more, wins 17, and gets 11 from
a friend. How many has he now ?
2. In a school, the first class has 15 scholars, the second 24, the
third 27, the fourth 30, and the fifth 31. How many scholars are in
the school ?
3. If I pay 8 shillings for bread, 14 shillings for tea, 7 shillings for
sugar, and 11 shillings for butter and cheese ; how many shillings do
I pay?
SUBTRACTION. 25
4. In a wood there are 41 oak-trees, 18 firs, 63 beeches, and 9 elms.
How many trees in all ?
5. A traveller went 110 miles by train, 62 miles by steamer, 17 miles
by coach, and then he had to walk 2 miles. What was the length of
his journey?
6. England has 52 counties, Scotland 33, and Ireland 32. How many
counties in the whole ?
7. A class of 26 pupils receives 14 new ones. How many pupils has
it now ?
.8. Three apple-trees in a garden were shaken for fruit : if one gave
516 apples, and the other two 620 each, how many apples did they
give in all ?
9. Three omnibuses started on a pleasure-trip : one carried 23 per-
sons, the second 32, and the third 26. If 4 were taken up by the way,
how many persons were there in the party ?
10. A grocer pays £140 for shop rent, £37 for taxes, £11 for rent of
cellars, and he spends £75 on repairs. What is the whole expense ?
11. In a railway train there were 79 first-class passengers, 101
second-class, and 249 third-class. How many passengers in all ?
12. When will a boy born in 1855 be 69 years old ?
13. From Glasgow to Stirling is 30 miles, from Stirling to Perth 31,
from Perth to Aberdeen 90. How far from Glasgow to Aberdeen ?
14. A merchant owes to one creditor £4275, to a second £531, to a
third £300, and to a fourth £3005. How much does he owe ?
15. A basket of eggs contains 232, another contains 35 more than
the first, and a third 101 more than the second. How many eggs
in all?
%* Only a few problems of the very simplest kind are presented at this
stage : the pupil will be able to continue them to more advantage when he has
learnt the four elementary rules. See Ex. § 55.
23. SUBTRACTION.
Ex. — Of 689 trees in a park, 327 were cut down. How many
remained standing ?
Here we have to find the difference between two given num-
bers, or what remains when the less is taken from the greater.
The greater of the two numbers is called the Minuend, which
means the number to be diminished ; the less is called the Sub-
trahend, which means the number to be taken away.
The number which remains is called the Difference or Re-
mainder.
The process of finding it is Subtraction ; called, when the
things are of one kind, as here, Simple Subtraction.
The sign of Subtraction is — (minus) ; thus 2 — 1 is 1.
We cannot find the difference between 689 and 327 at once ;
the numbers are too large. We must, therefore, subtract them
in parts ; for which purpose we must learn the subtraction of
the first nine numbers.
26
SUBTRACTION.
Subtraction Table.
1 from
2 from
3 from
4 from
5 from
6 from
7 from 8 from
9 from
2 is 1
3 is 1
4 is Ij 5 is 1
6 is 1
7 is 1
8 is 1 9 is 1
10 is 1
3... 2
4... 2
5 ... 2
6 ... 2
7 ... 2
8... 2
9 ... 210 ... 2
11 ... 2
4... 3
5 ... 3
6... 3
7 ... 3
8... 3
9 ... 310... Sill ... 3
12... 3
5... 4
6 ... 4
7... 4
8... 4
9 ... 4
10 ... 4
11 ... 412 .. 4
13... 4
6... 5
7 ... 5
8 ... 5
9 ... 5
10 ... 5
11 ... 512 ... 5J13 ... 5
14 ... 5
7 ... 6
8 ... 6
9 ... 6 10 ... 6
11 ... 6
12... 613 ... 614... 615 ... 6
8... 7
9 ... 710... 7|11 ... 712... 7
13... 714... 715... 716 ... 7
9... 810... 8|11 ... 8I12... 813... 8
14 ... 8 15 ... 816 ... 8 17 ... 8
10 ... 911 ... 9jl2 ... 9,13 ... 914 ... 915 ... 9(16 ... 9J17 ... 9J18 ... 9
EXERCISE I.
1. Repeat the several columns— backwards — by odds — by evens.
2. Subtract the units in each column from its highest number.
3. 3 from 8 leaves — ? 4 from 13 leaves — ? etc.
4. 9 less 2 less 3 is — ? 17-8-4 is— ? etc.
5. To 7 add 3 and take away 4? 9 + 8-2-2 is— ?
6. From 5 books take 2, and how many remain ? John had 6
marbles ; if he lost 3 and then 1, how many had he ? Jane has 7
pence ; if she gets 6 pence more and gives away fourpence, what has
she now ? etc.
7. Write down the columns of the Table in order.
24. Subtraction of Numbers of Two Places.
The Table given above serves also for the subtraction of
tens ; thus : —
If 1 from 2 is 1, 1 ten from 2 tens is 1 ten, or 10 from 20 is 10.
If 1 from 3 is 2, 1 ten from 3 tens, is 2 tens, or 10 from 30 is 20.
Etc. etc. etc.
If 2 from 3 is 1, 2 tens from 3 tens is 1 ten, or 20 from 30 is 10.
If 2 from 4 is 2, 2 tens from 4 tens is 2 tens, or 20 from 40 is 20.
Etc. etc. etc.
EXERCISE II.
Perform Ex. i. with tens instead of units.
Ex. — A woman had 76 eggs in a basket ; if she sold 34, how
many had she remaining ?
Set down the subtrahend below the minuend in 76
its place ; then, subtract the places in their order. 34
4 from 6 is 2 units ; set down the 2 in its place.
3 from 7 is 4 tens ; set down the 4 in its place. 42
Total difference, 42.
SUBTRACTION. 27
To prove the result, add together the subtrahend and the
difference ; the sum should be the minuend, since what is taken
away from a number and what is left of it make up between
them the whole number.
. EXEKCISE III.
(1.) (2.) (3.) (4.) (5.) (6.) (7.) (8.) (9.) (10.) (11.) (12.)
84 56 76 48 59 37 29 70 86 91 64 73
32 24 36 25 32 21 19 30 20 31 20 52
13.47-24" 16.39-19 19.81-41 22.85-42
14. 78-51 17. 40-20 20. 56-36 23. 71-31
15. 63-30 18. 93-63 21. 78-47 24. 99-57
25. Though the minuend must always be greater than the sub-
trahend, any place of the minuend except the highest may be
less than the place below it of the subtrahend.
Ex. — A teacher has 45 steel pens ; if he distributes 29 to his
class, how many are over ?
9 from 5 cannot be taken ; change one of the tens
into units, making 15 units in all ; 9 from 15 is 6 45
units, set down the 6 in its place. 29
2 from 3 (the 3 tens remaining) is 1 ten ; set down —
the 1 in its place. 16
Total difference, 16.
Rule. — Write the less number under the greater in its place ;
subtract the columns in their order beginning with the units' ;
change one of the next highest name when necessary.
Or thus,1
9 from 5 cannot be taken ; add 10 units to the 5, 45
making 15 in all ; 9 from 15 is 6 units. 29
Add 1 ten to the 2 tens ; 3 from 4 is 1 ten. —
Total difference, as before, 16. 16
In adding 10 units to the minuend and 1 ten to the subtra-
hend, we have added the same number to both. This does not
alter their difference ; but makes it easier to find, by keeping
each place of the minuend greater than the place below it of
the subtrahend.
Rule. — Write the less number under the greater in its place ;
subtract the columns in their order beginning with the units' ;
add ten to any place of the minuend which is less than the
place below it of the subtrahend, and one to the next place of
the subtrahend.
i Both methods of subtraction are given ; the teacher may choose either.
28 SUBTRACTION.
EXERCISE IV.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
a. 35 47 53 64 71 60 82 91 47 24 63 30 44 28 34 41
17 39 27 35 49 29 35 53 19 17 45 21 27 9 16 27
17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.
6. 44 21 43 94 42 76 48 32 51 36 22 74 52 81 34 45
18 12 24 47 25 39 29 17 37 17 13 49 26 39 27 19
33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48.
c. 53 70 42 80 43 30 52 63 74 85 96 97 50 32 43 77
18 43 19 37 19 12 24 34 45 67 29 38 26 27 17 58
49. 66-17
57. 21-13
65. 74-16
73. 44-27
50. 47-23
58. 38-19
66. 81-25
74. 58-39
51. 23-14
59. 72-43
67. 62-37
75. 86-48
52. 55-27
60. 83-54
68. 50-23
76. 90-54
53. 70-34
61. 51-26
69. 27-9
77. 93-65
54. 84-27
62. 66-37
70. 34-15
78. 45-29
55. 95-46
63. 80-43
71. 53-27
79. 74-36
5ti. 60-24
64. 91-54
72. 67-39
80. 82-43
EXERCISE V.
Perform tlie above exercise mentally.
*#* In doing so, it is more convenient to subtract the tens first, and then
the units ; thus in 35-17, 10 from 35 leaves 25, and 7 from 25 leaves 18.
26. Subtraction of Numbers of One or more Periods.
The Table given, sect. 23, serves also for the subtraction of
hundreds, thousands, &c. ; thus :
If 1 from 2 is 1, 1 liund. from 2 hund. is 1 himd., or 100 from 200 is 100.
If 1 from 3 is 2, 1 Lund, from 3 hund. is 2 hund., or 100 from 300 is 200.
Etc. etc. etc.
If 2 from 3 is 1, 2 hund. from 3 hund. is 1 hund., or 200 from 300 is 100.
If 2 from 4 is 2, 2 hund. from 4 hund. is 2 hund., or 200 from 400 is 200.
Etc. etc. etc.
EXERCISE VI.
Perform Ex. i. with hundreds instead of units.
Ex. 1. Of 689 trees in a park, 327 were cut down : how many
remained standing?
7 from 9 is 2 units ; set down the 2 in its place.
2 from 8 is 6 tens ; set down the 6 in its place. 327
3 from 6 is 3 hund. ; set down the 3 in its place.
Total difference, 362. 362
SUBTRACTION. 29
Ex. 2. How much greater is 6073 than 484 ?
In this example, there is a cipher in the minuend, and the
highest place of the minuend has no place below it in the
subtrahend.
4 from 13 is 9 for the units' place. 6073
8 from 16 (changing one of the next highest name, 484
which is thousands) is 8 for the tens' place.
4 from 9 (the 9 hundreds remaining when the one 5589
thousand was changed) is 5 for the hundreds' place.
0 from 5 is 5 for the thousands' place.
Or thus :
4 from 13 is 9 for the units' place.
9 from 17 is 8 for the tens' place.
5 from 10 is 5 for the hundreds' place.
1 from 6 is 5 for the thousands' place.
EXERCISE VII.
1. 2. 3. 4. 5. 6. 7. 8. 9
796 805 909 483 857 564 769 960 637
454 403 100 150 724 203 456 500 415
10. 758-342 13. 576-420 16. 7345-5135
11. 975-600 14. 874-574 17. 8500-7000
12. 856-326 15. 716-516 18. 2021-1020
EXERCISE VIII.
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.
547 635 248 429 511 924 700 801 540 707 800 600
219 427 154 274 364 519 451 605 229 593 209 405
13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24.
713 391 420 706 300 401 535 297 316 «02 732 194
256 98 301 279 107 208 328 198 49 541 342 94
25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
3429 5642 7396 4524 8527 5418 4000 5040 6080 9004
2763 3804 5409 2790 6050 2980 2534 3956 2500 5084
»5. 36. 37. 33. 39. 40. 41. 42. 43. 44.
7320 8074 2094 7000 5484 9302 7549 1368- 2008 7309
2496 1943 859 3456 2390 1903 5840^,66^.1123 978
Ip if iv ii7
V /> , oar
30
SUBTRACTION.
45. 407-298
46. 630-450
47. 275-87
48. 116-58
49. 730-5G3
50. 805-496
51. 357-192
52. 207-84
53. 476-189
54. 520-218
55. 600-315
56. 2809-939
57. 7340-2093
58. 9008-572
59. 1009-450
60. 7084-3921
61. 8000-1090
62. 5009-3094
63. 9101-9011
64. 7308-5904
65. 8234-4731
66. 2890-1936
EXERCISE IX.
\* In the following, find the first remainder less than the subtrahend.
28.
1. 402-86
2. 530-105
3. 736-209
4. 900-121
5. 437-99
6. 215-67
7. 600-143
8. 816-197
9. 701-156
10. 2760-672
11. 8207-1938
12. 6094-856
13. 9400-2763
14. 8405-1504
15. 3091-750
16. 7463-1976
17. 5000-987
18. 5185-1978
19. 7320-2094
20. 9017-1853
1.
45060
29360
2.
38905
19450
EXERCISE X.
3. 4. 5.
27936 84571 73021
10007 25038 49950
6. 7.
45239 84901
29308 56402
378923
194033
14.
2567283
730946
934856
256094
15.
45070134
29098040
10.
734085
508506
11.
400000
40401
12.
501020
392406
23900140
4015002
17.
50000014
6010305
13.
276408
120394
18.
100200300
100199025
19. 25678-19341
20. 38056-9456
21. 45804-993
22. 50600-5600
23. 89476-4890
24.793246-45600
25.840300-524080
26. 60S409-
27. 900000-
28. 257931-
29. 456890-
30. 8409302-
81. 10000000-
32.57340506-
93560 33.
90909 34.
80002 35.
193456 36.
908567 37.
1001001 38.
8530205
73894219-25934764
170170170-7107100
59340947-20560724
123456789-98764532
10000000-100000
500500500-650650
%* In the following, find the first remainder less than the subtrahend.
39. 56030-9807 43. 60930-9493 47. 730294-165085
40. 10101-3427 44. 127936-29647 48. 100901-10192 ,
41. 27092-5083 45. 982401-109472 49. 605090-92071
42. 47138-7509 46. 273408-84279 50. 400000-101010
29. EXERCISE XI.
1. Count back by twos from 100, from 101.
2. Count back by threes from 102, from 101, from 100.
3. Count back by fours from 100, from 101, from 102, from 103.
4. Count back by fives from 100, 101, 102, 103, 104.
5. Count back by sixes from 102, 101, 100, 103, 104, 105.
6. Count back by sevens from 105, 104, 103, 102, 101, 100, 106.
7. Count back by eights from 100, 101, 102, 103, 104, 105, 106, 107.
8. Count back by nines from 108, 107, 106, 105, 104, 103, 102, 10L
*»* This and the following Ex. should be practised along with the foregoing.
SUBTRACTION. 31
EXERCISE XII.
1. 8+2+9-5-4+1-3 + 8+2-7-1 + 3 + 6-4-
2. 7 + 4-3+5+7-5 + 9 + 6-7-3+9 + 4-9-1 + 8 + 5-7-3.
3. 15-8 + 9-4+9+5-3-7 + 4-5 + 10 + 20-11-7 + 4 + 8-5-10.
4. 22 + 8-11-4+8 + 4-7-1 + 9 + 4-3 + 2-7+9-4-3 + 6 + 7-5.
5. 40+3-7-4+20-7-9-4 + 8-7 + 6-8 + 10-5-7-2 + 8 + 10.
6. 14+3-9+10-6-2 + 20-5 + 9-7-10 + 11 + 11-8 + 13-4-12.
7. 36 + 9-4-8 + 2+5 + 9-12-10 + 7 + 9-4-8 + 5 + 4-6-8 + 20.
8. 19 + 9-5-8 + 7 + 10-5-11 + 20-11 + 9+4-5+5-4 + 9-7-4.
9. 28 + 10+7-12-10 + 6-7-4+3 + 1-9-8 + 11 + 5-7-9 + 8 + 4.
10. 50 + 10-20+30+16-10-10 + 20+7-10 + 20 + 50-30-7-10.
11. 49-5 + 12-9 + 16-10 + 8 + 13-5-8-7 + 11 + 7 + 15-30-10 + 9.
12.- 53 + 8-11+5 + 9-14-15 + 9 + 30-4-9 + 12 +20-5 + 16-13- 9.
Etc. etc. etc.
EXERCISE XIII.
How many are 37 - 29 + 48 - 33 + 79 - 15 ?
Here, instead of subtracting 29 from 37, then adding 48, and so on,
it is shorter to add together the numbers which are +, then add to-
gether the numbers which are — , and find the difference of the two
Bums, thus : —
37 -29 For it is the same thing whether, in find-
+ 48 - 33 ing 9 - 2 - 2, we say 2 from 9 is 7, 2 from
+ 79 - 15 7 is 5 ; or 2 and 2 are 4, 4 from 9 is 5.
164 -77 is 87
j.. 125+37-84-10+76 + 53-101+56+279-184-45+293.
2. 74-40 + 51-9 + 29 + 16-19-5 + 36-27 + 40 + 11.
3. 18+15-10 + 40 + 36-19-14 + 23-39 + 20 + 16-19.
4. 56 + 20-43-27 + 39 + 24-31 + 64-45 + 21 + 10-34.
5. 90 + 45 + 16-49-51 + 6-15 + 39-60 + 49 + 53-19.
6. 36-19+53-29 + 36-24-11-B 4 64 + 17-24-9 + 14.
7. 49+36-29-14+20+36-18-y +25 + 84-59-27+40.
8. 74 + 52-63-10 + 29 + 37-45-37+22-51 + 69-19 + 26.
9. 192-56-14+58 + 213-191 + 64-49+346-154-48 + 90.
10. 724-593 + 824-48+93 + 702-500+293-59-73 + 256-100.
11. 50004 - 8456 - 401 + 4592 + 9400 - 10100 + 734 - 809.
12. 29340 - 4560 - 9390 + 7248 - 15600 + 93402 - 56840.
30. EXERCISE XIV.
1. A woman went to market with a basket of eggs containing 342 :
if she sold 192, how many did she bring back ?
2. John has 95 nuts, but gives 37 to William. How many does he
keep?
3. A teacher gives out pens to a class of 60 scholars, but the box
lias only 37. How many does he want ?
4. A cheese weighs 78 pounds. How much heavier is it than an-
other which weighs only 47 pounds ?
5. A tradesman owes £260, but he has only £137. How much does
he require to pay his debts '/
32 SUBTRACTION.
6. A cask of sugar contains 539 pounds' weight. How much must
be sold to leave 257 pounds ?
7. James has 24 marbles, and his brother gives him 37. How many
must he buy to make up 100 ?
8. If a school has 374 scholars, of whom 27 are in the first class, and
32 in the second ; how many are in the other classes together?
9. A green-grocer received a basket of apples and pears, 264 in all :
157 were apples ; how many were pears ?
10. A house is worth £520, but it will cost £84 to repair it. How
much should it be sold for?
11. Edinburgh to Dunbar is 29 miles, and Edinburgh to Berwick is
57 miles. How far from P-unbar to Berwick ?
12. A tradesman earns 16s. a week, and spends 13s. How much
does he save in four weeks ?
13. A farmer had in his yard 31 fowls, 17 geese, 24 turkeys, and
his ducks made up the entire number of his poultry to 87. How many
ducks had he ?
14. How much of 385 yards remains if 93 yards be cut away from
the piece ? How often may 93 yards be cut away, and what will
remain ?
15. A train started with 374 passengers. At the first station 16
went out and 9 came in ; at the second, 11 went out and 25 came in ;
at the third, 3 went out. How many passengers left the train at the
terminus ?— See Ex. § 55.
31. MULTIPLICATION.
Ex. — Five boxes of oranges contained 125 each, how many
oranges were there in all ?
Here we have to find a number equal to 125 repeated 5
times.
We could find that by adding 125 to itself 5 times ; but a
shorter way is to multiply 125 by 5.
The number to be repeated is called the multiplicand.
The number of times it is to be repeated, multiplier.
Both are sometimes called the . . factors.
The result is called the ... product.
The process is called multiplication ; and, when the multipli-
cand is of one kind as here, simple multiplication.
The sign of multiplication is X (multiplied by) ; thus 2X2
are 4.
We cannot find how much 5 times 125 is "by one step ; the
multiplicand is too large. We must therefore do it in parts ;
for which purpose we must learn the multiplication of the first
nine numbers.
32.
MULTIPLICATION.
Multiplication Table.
33
2 times
1 are 2
2 ... 4
3 ... 6
4 ... 8
3 times
1 are 3
2 ... 6
3 ... 9
4 12
4 times
1 are 4
2 ... 8
3 ... 12
4 ... 16
5 times
1 are 5
2 ... 10
3 ... 15
4 ... 20
6 times
1 are 6
2 ... 12
3 ... 18
4 ... 24
7 times
1 are 7
2 ... 14
3 ... 21
4 ... 28
5 ... 10
6 ... 12
7 ... 14
5 ... 15! 5 ... 20
6 ... 18! 6 ... 24
7 ... 21 7 ... 28
5 ... 25
6 ... 30
7 ... 35
5 ... 30
6 ... 36
7 ... 42
5 ... 35
6 ... 42
7 ... 49
8 ... 16
9 ... 18
10 ... 20
11 ... 22
12 ... 24
8 ... 24 8 ... 32' 8 ... 40
9 ... 27 9 ... 36 9 ... 45
10 ... 3010 ... 4010 ... 50
11 ... 3311 ... 4411 ... 55
12 ... 3612 ... 4812 ... 60
8 ... 48
9 ... 54
10 ... 60
11 ... 66
12 ... 72
8 ... 56
9 ... 63
10 ... 70
11 ... 77
12 ... 84
8 times
1 are 8
2 ... 16
3 ... 24
4 ... 32
9 times
1 are 9
2 ... 18
3 ... 27
4 ... 36
10 times
1 are 10
2 ... 20
3 ... 30
4 ... 40
11 times
1 are 11
2 ... 22
3 ... 33
4 ... 44
12 times
1 are 12
2 ... 24
3 ... 36
4 ... 48
5 ... 40
6 ... 48
7 ... 56
8 ... 64
9 ... 72
10 ... 80
11 ... 88
12 ... 96
5 ...
6 ...
7 ...
8 ...
9 ...
10 ...
11 ...
12 ...
45
54
63
72
81
90
99
108
5 .
6 .
7 .
8 .
9 .
10 .
11 .
12 .
.. 50
.. 60
.. 70
.. 80
.. 90
.. 100
.. 110
.. 120
5
6
7
8
9
10
11
12
... 55
... 66
... 77
... 88
... 99
... 110
... 121
... 132
5 ... 60
6 ... 72
7 ... 84
8 ... 96
9 ... 108
10 ... 120
11 ... 132
12 ... 144
*** This Table should be learnt first in lines even along, then in lines up
and down. The pupil should practise it daily till he has it thoroughly at
command.
EXERCISE. I. Bf.
1. Repeat the several lines even along ; backwards j by odds ; by
evens.
2. Repeat the lines up and down ; backwards ; by odds ; by evens.
3. 4 times 5 are — ? 6 times 9 are — ? 8 times 7 are — ? etc.
5 times 4 are — ? 9 times 6 are — ? 7 times 8 are — ? etc.
4. How many fingers have 8 boys? How many wheels have 9
carts ? How many days have seven weeks ? How many farthings
have four pence ? How many units in 5 tens ? How many marbles
have 9 boys with 11 each? What cost 6 oranges at 2 pence each?
7 fowls at 3 shillings each ? etc.
5. Name two factors of 18, 24, 96, etc.
6. How many times 7 is 63 ? 21 ? 70 ? etc.
7. 36 is 9 times — ? 72 is 6 times — ? etc.
8. 8 times 6 + 2 are — ? 5 times 8 with 9 added are — ? ete.
4 times 12 less 9 are — ? 7 times 5 - 6 are — ? etc.
9. 2 times 4 and 3 times that are — ? etc.
6 multiplied twice by 2 are — ? etc.
10. Write down the several columns of the Tab! 3.
0
34 MULTIPLICATION.
OO. The Table given above serves also for the multiplication of
tens, hundreds, etc. Thus —
If 2 times 1 are 2, 2 times 1 ten are 2 tens, or 2 times 10 are 20.
If 2 times 2 are 4, 2 times 2 tens are 4 tens, or 2 times 20 are 40.
Etc. etc. etc.
If 3 times 3 are 9, 3 times 3 tens are 9 tens, or 3 times 30 are 90.
Etc. etc. etc.
EXERCISE II.
Perform Ex. i. with tens in the multiplicand.
If 2 times 1 are 2, 2 times 1 h. are 2 h., or 2 times 100 are 200.
If 2 times 2 are 4, 2 times 2 h. are 4 h., or 2 times 200 are 400.
Etc. etc. etc.
EXERCISE III.
Perform Ex. i with hundreds in the multiplicand.
Multiplication by Units.
Ex. — Five boxes of oranges contained 125 each, how many
oranges were there in all ?
Set the multiplier below the multiplicand in its place ; then,
multiplying each place in its order,
5 times 5 are 25 units j set down 5 units and
carry 2 tens. 125
5 times 2 are 10, and 2 are 12 tens ; set down 5
2 tens and carry 1 hundred. -
5 times 1 are 5, and 1 are 6 hundreds. 625
Product, 625.
Hule. — To multiply by units, multiply each place of the
multiplicand in order, carrying tens.
The answer may be proved by adding the multiplicand to
itself 5 times ; the sum should be the same as the product. Or
we may multiply by 4, the number next below the multiplier,
and add the multiplicand to the product.
EXERCISE IV.
1. Multiply the following numbers by 2, 3, etc., to 12, in order : —
13 21 31 41 51 61 71 81 91
14
22
32
42
52
62
72
82
92
15
23
33
43
53
63
73
83
93
16
24
34
44
54
64
74
84
94
17
25
35
45
55
65
75
85
95
18
26
36
46
56
66
76
86
96
19
27
37
47
57
67
77
87
97
20
28
38
48
58
68
78
88
98
29
39
49
59
69
79
89
99
80 40 50 60 70 80 90 100
MULTIPLICATION. 3 5
8. Multiply the several columns mentally.
8. 2 times 27 are — ? 3 times 32 are — ? 4 times 48 are — ? etc.
4. Multiply the following numbers by 2, 3, etc., to 12, in order : —
1. Ill
11. 893
21. 2461
31. 24682
41. 34194
2. 222
12. 248
22. 5382
32. 74394
42. 21384
3. 333
13. 604
23. 2081
33. 31208
43. 75689
4. 444
14. 573
24. 4095
34. 24295
44. 38472
5. 555
15. 421
25. 2496
35. 19064
45. 29319
6. 666
16. 298
26. 5162
36. 70538
46. 82964
7. 777
17. 157
27. 7349
37. 25819
47. 70109
8. 888
18. 820
28. 8210
38. 39147
48. 10840
9. 999
19. 659
29. 9347
39. 16731
49. 30028
10. 427
20. 416
30. 1924
40. 42858
50. 90084
%* This exercise Is designed to be performed orally from the book as well
as on slate.
Multiplication by Factors.
.Ek— Multiply 248 by 24.
Since 24 is 6 times 4, we multiply by 24, if 248
we multiply first by 6, and then that product by g
4 ; thus :— r—
The result may be proved by multiplying by /?
3 and 8, or by 2 and 12 ; which are also factors _
of 24, and which should therefore give the same 5952
product.
A number like 24 which is made up of factors (other than 1)
is called a composite number.
A number like 7, 11, or 23, which is not made up of factors,
is called a prime number.
Multiplication by two factors may be used in the case of all
composite multipliers between 12 and 144.
Practice in multiplying will show the pupil that three factors
may often be used for a multiplier with advantage ; thus,
252=4X7X9.
EXERCISE V.
Multiply, using factors : —
1. 536x14, 15, 21, 22. 6. 4732x77, 81. 84.
2. 270x25, 27, 28, 32. 7. 2096x88, 96, 99.
3. 905x33, 42, 44, 45. 8. 8405x108, 121, 132,
4. 827 x 54, 55, 56. 9. 7289 x 144, 160, 270.
5. 638x63, 66, 72. 10. 8175x420, 840.
11. 3497 x 16, 18, 48, 72, in two ways.
12. 7302 x 24, 36, in three ways. "
36 MULTIPLICATION.
36. Multiplication by more than One Place.
A cipher annexed to the right of a figure increases its value
10 times, that is, multiplies it by 10. Therefore, to multiply
by 2 tens or 20, multiply by 2, and annex the cipher ; to mul-
tiply by 30, multiply by 3, and annex the cipher ; and so on.
Similarly to multiply by 200, multiply by 2, and annex two
ciphers ; to multiply by 300, multiply by 3, and annex two
ciphers ; and so on.
Rule. — To multiply by tens, hundreds, etc., multiply by the
left-hand figure, and annex the ciphers.
EXERCISE VI.
1. Multiply the columns in Ex. iv. by 20, 40, 50, 90.
2. Multiply tlie same columns by 300, 600, 700, 800.
O 7 • Ex* — A book contains 356 pages, and each page 237 words :
how many words are in the book ?
Set the multiplier below the multiplicand in its
place ; then multiplying by the 7 units, 237
we have ..... 2492
Multiplying by the 3 tens, we have . . 10680
Multiplying by the 2 hundreds, we have . 71200
Product by whole mutiplier is . . 84372
The result may be proved by interchanging . the multiplier
and multiplicand, that is, multiplying 237 by 356 ; which will
give the same product.
Rule. — To multiply by a number of several places, multiply
by each place in order from the units, and add the several
products.
V* The pupil may by and by omit the ciphers, denoting the tens and
hundreds in the second and third lines of multiplication ; being careful to
place the right-hand figure of each line exactly under that place of the
multiplier which gives it.
Should there be a cipher in the tens or some higher place of the multi-
plier, it is simply passed over in multiplying.
EXERCISE VII.
I 2364 x 29, 37, 43. 5. 8256 x 17, 93, 49. 9. 40001 x 81, 28, 34.
i 4328 x 39, 51, 86. 6. 6439 x 38, 57, 61. 10. 73000 x 47, 59, 92.
3. 5936 x 28, 46, 59. 7. 20480 x 71, 43, 53. 11. 90000 x 27, 64, 79.
4. 9320 x 19, 73, 31. 8. 30093 x 98, 83, 78. 12. 70091 x 75, 88, 99
MULTIPLICATION.
37
EXERCISE VIII.
1. 85627x183, 297,403.
2. 47231x245,318,721.
3. 93086x240, 825, 649.
4. 23456x409,207,308.
5. 73610x930, 470,290.
6. 85093x418,738,562.
7. 72170x936,259, 816.
8. 37293x904, 506,801.
9. 80050x629,350,680.
10. 90000x456,789,910.
11. 70700x843, 529,365.
12. 90280x706, 504,209.
13. 456789x297,399, 536.
14. 724936x840,908,273.
15. 459630x364, 814, 518.
16. 536298x230, 563, 720.
17. 210830x821, 913, 713.
18. 914567x439, 546,208.,
EXERCISE IX.
1. 500606x5423, 6106.
2. 730000x2936, 8492.
3. 700000x4028, 5003.
4. 830830x6300, 7240.
5. 308070x8740, 5007.
6. 934764x23418, 93125.
7. 621930x19728, 73465.
8. 493628x27368, 93480.
9. 840300x19030, 80807.
10. 621934x70029, 54309.
11. 493002x56721, 12765.
12. 2389745x4567, 7394, 6270.
13. 6348576x7321,8492,1029.
14. 2930840x6080, 5090, 7200.
15. 7394900x8936, 2009, 5900.
16. 8002006x7290, 5718, 3290.
17. 7802058x35467,29631.
18. 4932096x84932, 94629.
19. 7007007x93021,80709.
20. 3489493x29100,28101.
21. 9000000x73500,82090.
22. 4290000x80972,50608.
EXERCISE X.
1. 25473809x258956,817456.
2. 73890496x483921,293185.
3. 90900900x259671,798491.
4. 25608709x408506,930850.
5. 70409360x273093,129608.
6. 49328914x506090,709080.
7. 82483949x210000,930039.
8. 72340090x724801,520936.
9. 53042485x493094,891172.
10. 73249000x938950,249056.
11. 490562001x362987, 450893.
12. 293904510x450813,920854.
13. 710842930x293050, 493096.
14. 256849361x259928,936190.
15. 209209209x123456,789012.
16. 600040068x900405,908550.
17. 394620100x736493,856190.
18. 824904561 x 437285, 737292.
19. 296382173x555555,505050.
20. 493084095x828561,400800.
38.
Squares and Cubes.
A figure like this, which has 4 rows of counters, ....
each containing 4, is called a square. The ....
number of counters we see by counting to be
16 ; that is, the number even along (4) multi-
plied by the number up and down (4). Bf.
Similarly 7 rows of trees with 7 trees in each would be a
square of 49 ; 10 lines of soldiers with 10 soldiers in each line
would be a square of 100.
When any number is multiplied by itself, the product is
called the square or second power of that number. The square
of 4 is denoted 4a.
39.
40.
38 MULTIPLICATION.
EXERCISE XI.
1. Repeat the squares of 1, 2, 3, 4, etc., up to 12.
2. Find the squares of 13, 14, 15, 16, 17, 18, 19, 20.
8. Find the squares of these numbers : —
1. 784 5. 3456 9. 23456 13. 75423 17. 50005
2. 937 6. 2930 10. 90307 14. 20056 18. 728946
3. 508 7. 4500 11. 58126 15. 90030 19. 809407
4. 610 8. 7000 12. 37000 16. 80705 20. 916738
"When a number is multiplied twice by itself, the product is
called the cube or third power of that number ; thus 4X4X4—
64. The cube of 4 is denoted 43.
>
%* This may be illustrated by a small cube of wood, or, better still, by a
box of such cubes.
EXERCISE XII.
1. What are the cubes of 1, 2, 3, etc., up to 10?
2. Find the cubes of these numbers : —
1. 789 4. 4506 7. 12000 10. 67809
2. 405 5. 5730 8. 37100 11. 40506
3. 623 6. 9825 9. 24089 12. 12345
EXERCISE XIII.
1. How many eggs in 16 boxes, each having 96 ?
2. How many pupils in a school which has 7 classes of 23 each ?
3. How many hours in 36 days ?
4. How many pence in 47 half-crowns ?
5. How many oranges, at 15 for a shilling, will 25s. buy ?
6. How long a journey shall I make in 27 days, at 18 miles a day ?
7. How many yards of linen in 387 pieces, each 35 yards ?
8. How many bottles in 45 dozen and 5 ?
9. How many pages in a yearly volume, of which a monthly part
has 96 ?
10. What cost a railway 49 miles long, at £4500 a mile ?
11. A postman delivers 29 letters each morning and evening for a
week ; how many did he deliver in all ?
12. A pipe pours into a cistern daily 13410 gallons water ; how
many gallons will it pour in during November ?
13. A house of five storeys has seven windows in each, and twelve
panes of glass in each window ; how many panes of glass are there in
all?
14. Three men, in business together, receive £672 each of the profits
at the end of the first year ; what were the whole profits ?
15. If a baker reckons 13 to a dozen, how many biscuits does he
count to 136 dozen ?
16. A merchant's office occupies 43 clerks at £2 a week each, and
24 at £3 ; what sum is required in a year for their wages ?
17. There are 29 trees in the side of a square plantation ; how
many trees has the plantation I
41.
DIVISION.
DIVISION.
39
Ex. — A box of eggs, containing 852, is to be divided amongst
a number of families, each getting 6 ; how many families will be
served ?
Here we have to find how often 6 is contained in 852.
We could find that by subtracting 6 from 852 successively
till nothing remains, and then counting the number of 6's we
have got, but a shorter way is to divide, 852 by 6.
The number to be divided is called the dividend.
The dividing number is called the divisor.
The number of times the divisor is contained in the dividend
is called the quotient.
The process of dividing is called division ; and, where the
dividend is of one kind as here, simple division.
The sign of division is -j- (divided by) ; thus, 4 -7- 2 is 2.
We cannot find how often 9 is contained in 243 by one step ;
the dividend is too large for that. We must therefore do it
in parts, for which purpose we must learn the division of the
first nine numbers.
42. Division Table.
2 i
n
3 i
a
4 ]
n
5 ii
i
6
i:
a
7 i
n
2 i
s
1
3
3 1
L 4 ]
LS 1
5 is
\ 1
6
i
I
1
7 i
s 1
4
|%
2
6
<
I 8
2
10
2
12
2
14
2
6
3
9
\
\ 12
3
15
3
18
3
21
3
8
>e
4
1
2
i
[ 16
4
20
4
24
4
28
4
10
5
1
5
\
5 20
5
25
5
30
5
35
5
12
6
1
8
(
5 24
6
30
6
36
6
42
6
14
.
7
2
1
\
r 28
7
35
7
42
7
49
7
16
8
2
4
\
J 32
8
40
8
48
8
56
8
18
9
2
7
<
) 36
9
45
9
54
9
63
9
20
10
3
0
1(
) 40
10
50
10
60
]
LO
70
10
22
11
3
3
1]
L 44
11
55
11
66
1
11
77
11
24
-•
12
3
6
IS
I 48
12
60
12
72
]
12
84
12
8
in
9
ID
10 i
n
11
ii
a
12 ii
i
8
is
1
9
is
1
10 i
3 1
11
i
1
1
12 is
1
16
"
2
18
2
20
2
22
2
24
2
24
3
27
3
30
3
33
3
36
3
32
4
36
4
40
4
44
4
48
4
40
£
45
5
50
5
55
5
60
5
48
e
54
6
60
6
66
6
72
6
56
\\
7
63
7
70
7
77
7
84
7
64
8
72
8
80
8
88
8
96
8
72
mt
£
81
9
90
9
99
9
1
08
9
80
M
1C
90
10
100
10
110
1
0
1
20
. 10
88
11
99
11
110
11
121
1
1
1
32 .
. 11
96
IS
108
12
120
12
132
1
2
1
44 .
. 12
40 DIVISION.
EXERCISE I. Bf.
1. Repeat the lines of this Table up-and-down ; backwards ; by
odds ; by evens.
2. Repeat the lines even along in the same way.
3. 2 in 8 is — ? 5 in 35 is — ? 9 in 72 is - ? etc.
4 in 8 is — ? 7 in 35 is — ? 8 in 72 is — ? etc.
4. How many pence in 8 farthings ? Divide 15 shillings among 5
persons. Divide 40 marbles among 8 boys. How many oranges at
2d. each can I buy with 16 pence ? etc.
5. Write down the several columns of the Table.
4:0. This Table serves also for the division of tens, hundreds, etc.
Thus—
If 2 in 2 is 1, 2 in 2 tens is 1 ten, or 2 in 20 is 10.
If 2 in 4 is 2, 2 in 4 tens is 2 tens, or 2 in 40 is 20.
Etc. etc. etc.
If 3 in 3 is 1, 3 in 3 tens is 1 ten, or 3 in 30 is 10.
Etc. etc. etc.
EXERCISE II.
Perform Ex. i., Nos. 1, 2, 3, with tens in the dividend.
If 2 in 2 is 1, 2 in 2 hunds. is 1 hund., or 2 in 200 is 100.
If 2 in 4 is 2, 2 in 4 hunds. is 2 hund., or 2 in 400 is 200.
Etc. etc. etc.
EXERCISE III.
Perform Ex. i., Nos. 1, 2, 3, with hundreds in the dividend.
44. Division by Numbers of One Place.
Ex. — How often is 3 contained in 963 ?
Place the divisor to the left of the dividend.
3 in 9 hundreds is 3 hundreds. 3 ]_963
3 in 6 tens is 2 tens. 3~2i
3 in 3 units is 1 unit.
Quotient, 321.
EXERCISE IV.
Divide—
1. By 2 : 86, 128, 420, 642, 864, 4806, 6428.
2. By 3 : 63, 96, 123, 249, 630, 963, 6093.
3. By 4 : 84, 168, 244, 488, 804, 884, 4084.
4. By 5 : 105, 155, 250, 355, 505, 4550, 5035.
6. By 6 : 126, 246, 306, 426, 5460, 6048, 12660.
6. By 7 : 147, 217, 357, 714, 6377, 7063.
7. By 8 : 168, 248, 320, 880, 1608, 5680.
8. By 9 : 189, 279, 540, 3609, 4599, 8190.
DIVISION. 41
. The places of the dividend do not often contain the divisor
evenly ; there is generally a remainder.
2 in 3 is 1 and 1 over
3 in 4 is 1 and 1 over
4 in 5 is 1 and 1 over
5 in 6 is 1 and 1 over
EXERCISE V. •
in 5 is— ? in 7 is— ? etc.
in 5 is — ? in 7 is — ? etc.
in 6 is — ? in 7 is — ? etc.
in 7 is — ? in 8 is — ? etc.
*** The exercise should be continued up to 12 as divisor.
4:6. Ex. 2. — A box of eggs, containing 852, is to be divided
amongst a number of families, each getting 6 ; how many
families will be served ?
Set the divisor to the left of the dividend. Then 6)852
6 in 8 hundreds is 1 hundred and 2 hundreds over ; T42
set down the 1 in its place, and change the 2 hun-
dreds into tens, making 25 in all.
6 in 25 tens is 4 tens and 1 ten over ; set down the 4 in its
place, and change the 1 ten into units, making 12 in all.
6 in 12 units is 2 units.
Quotient, 142.
Rule.— To divide by a number of one place, divide the
places of the dividend in order from the highest, carrying the
tens.
The result may be proved by multiplying the quotient by
the divisor ; the product should be the dividend.
EXERCISE VI.
Divide
1. By 2 : 98, 258, 374, 454, 526, 598, 638, 694, 738, 876, 938, 972.
2. By 3 : 87, 378, 465, 471, 513, 582, 648, 657, 726, 735, 879, 978.
3. By 4 : 96, 492, 536, 548, 620, 676, 768, 792, 860, 892, 948, 956.
4. By 5 : 565, 590, 675, 680, 745, 775, 865, 880, 930, 975, 7345.
5. By 6 : 150, 672, 726, 744, 804, 852, 918, 990, 6834, 8526, 8730.
6. By 7 : 161, 798, 805, 875, 910, 987, 7847, 7952, 8596, 8764, 9233.
7. By 8 : 256, 896, 960, 992, 8976, 9544, 1896, 1944, 2888, 3976.
8. By 9 : 144, 252, 423, 603, 828, 1026, 2160, 3267, 5040, 6543, 7038.
9. By 10 : 730, 840, 9320, 4500, 7310, 2030.
10. By 11 : 748, 396, 594, 286, 7942, 8503, 25894, 92477 56089.
11. By 12 : 348, 564, 936, 3888, 57372, 20928, 3708, 94020, 67308.
47. Ex.— How often is 6 contained in 24295 ?
Dividing as before, there is a remainder of one 6)24295
after dividing the units. This is annexed to the 4049J
quotient with the divisor below in the form £, which
denotes one-sixtht or the sixth part of one.
42 DIVISION.
In multiplying the quotient in this case by the divisor to
prove the result, the remainder must be added to the product ;
thus, 4049 X 6+ 1= 24295.
EXERCISE VII.
Divide —
1.
2.
3.
4.
5.
6.
1.
By 2,
345
467
931
857
1129
2525
2.
By 3,
472
305
721
922
2684
7055
3.
By 4,
105
653
437
829
5634
8631
4.
By 5,
732
482
911
573
8421
7018
5.
By 6,
515
833
791
273
5927
6381
6.
By 7,
452
635
134
608
3210
7962
7.
By 8,
123
537
817
909
4561
8347
8.
By 9,
258
316
501
823
7082
1293
9.
By 10,
137
259
533
471
2563
9327
10.
By 11,
564
800
601
942
3874
6088
11.
By 12,
373
529
705
637
1949
2009
48. Division by Factors.
In dividing by any composite number up to 144, we may get
the quotient by dividing by its two factors successively. E.g.,
in dividing an apple into 4 parts, we first divide it into 2 parts,
then each of these again into 2 parts.
Ex. — Divide 3568 marbles into parcels of 24.
The factors of 24 are 6 and 4. 6 13568
Dividing first by 6, we have for quotient 4 594 4 \
594 (parcels of 6), and 4 (marbles) over. -j.48 2 1 ^
Dividing next by 4, we have for quotient
148 (parcels of 4 sixes or 24's) and 2 (parcels of 6) over.
Adding now the second remainder (2 parcels of 6, or 12
maibles) to the first (4 marbles), we have for total re-
mainder 16 marbles : 6X2-f4 = 16.
Hence, to get the real remainder, multiply the first divisor
by the second remainder, and add the first remainder to the
product. If there be no second remainder, the first is the
real one.
EXERCISE VIII.
1. 234564-14, 15, 21, 22 6. 905036-7-84, 88, 96
2. 37095-25, 27, 28, 32 7. 249076^99, 108
3. 90851—33, 42, 44, 45 8. 593250-7-120, 132, 144
4. 84379—54, 55, 56, 63 9. 731105-M6, 18, 48, 72, in two ways.
5. 65927—66, 77, 81 10. 847644-r-24, 36, in three ways.
49. Division by more than one Place.
As a cipher annexed to the right of a figure multiplies it by
10, so a cipher removed from the right of a figure divides the
number by 10 : thus, 20 -j- 10 = 2.
DIVISION. 43
If the dividend do not end in a cipher, then the figure in the
nnits' place is removed for a remainder : thus, 21-7-10 = 2-^.
If the divisor contain more tens than one, as 30, divide nrst
by 10 as one factor, and then by the other factor, 3 ; that is,
remove the units' place of the dividend for the remainder, and
divide by the second factor, carrying what is over in this divi-
sion to the remainder. Thus, 63 -*- 20 = 3^y ; 73 +• 20 = 3 J§.
To divide by a number of hundreds, remove the two last
ciphers of the dividend, or the two last figures of it, for re-
mainder, in a similar way. Thus, 200 -i- 100 = 2 ; 564 -i- 200
^"JTJO*
EXERCISE IX.
Divide by 10, 30, 50, 70, 90—
1. 370 7. 1200 13. 2474 19. 32814
2. 290 8. 6600 14. 3935 20. 56732
3. 835 9. 8800 15. 5066 21. 83940
4. 672 10. 7000 16. 7317 22. 50761
6. 425 11. 4800 17. 8058 23. 69005
6. 901 12. 6300 18. 9720 24. 85436
EXERCISE X.
Divide by 200, 400, 600, 800, examples 7-24 in last Exercise.
, Ex.— How often is 234 contained in 849726 ?
234 in 8 or in 84 cannot be taken, but 234) 849726(363 !£,&
in 849 (thousands) is 3 (thousands), 702
and 147 (thousands) over. Set down
the 3 in the thousands' place of the
quotient, and carry the 147 to the
hundreds' place, making the next
part of the dividend 1477 (hun-
dreds) in all.
234 in 1477 (hunds.) is 6 (hundred),
and 73 (hunds.) over. Set down 306
the 6 (hunds.) in its place in the 234
quotient, and carry the 73 (hunds.)
to the tens' place, making the next 72
part of the dividend 732 (tens) in all.
234 in 732 (tens) is 3 (tens), and 30 (tens) over. Set down
the 3 (tens) in its place in the quotient, and carry the 30
(tens) to the units' place, making the next part of the divi-
dend 306 (units) in all.
234 in 306 (units) is 1 (unit), and 72 (units) over. Set the
1 (unit) in its place in the quotient. The 72 units are
remainder.
44
DIVISION.
This form of division, which is required when the divisor
contains more than one place, is known as Long Division.
EXERCISE XL
1. 370374-25, 37, 43.
2. 298354-34, 49, 51.
3. 73632-1-47, 93, 39.
4. 802944-19, 26, 41.
6. 900004-73, 61, 17.
6. 50032-29, 53, 98.
7. 17918-13, 34, 82.
8. 47320-38, 91, 47.
9. 20971-67, 82, 93.
10. 54280—23, 46, 85.
EXERCISE XII.
1. 456824-251, 183, 342.
2. 409364-301, 457, 631.
3. 238434-113, 911, 564.
4. 890404-824, 159, 296.
5. 900004-457, 734, 825.
6. 123844-391, 516, 364.
7. 730274-801, 709, 208.
8. 290414-257, 314, 846.
9. 928814-934, 652, 293.
10. 799484-418, 506, 853.
11. 5608024-293, 791, 846.
12. 2935444-151, 258, 174.
13. 8587414-325, 291, 397.
14. 4853614-851, 702, 813.
15. 9341104-561, 582, 738.
16. 5006364-921, 309, 257.
17. 7000004-416, 526, 736.
18. 205428^-901, 754, 815.
19. 9340654-297, 358, 492.
20. 7144084-824, 964, 708.
EXERCISE XIII.
1. 74893184-37, 74, 89.
2. 2934821-7-41, 73, 97.
3. 73486404-594, 416, 607.
4. 2684816-5-208, 541, 732.
5. 60845164-2342, 5684.
6. 54031444-9348, 2571.
7. 72561544-3040, 8009.
8. 9144608—9401, 5008.
9. 82717504-3075, 4908.
10. 91939324-5671, 2943.
11. 573380644-5473, 3024, 9902.
12. 630927064-2931, 4708, 5004.
13. 72491840-7-3040, 8009, 5231.
14. 20018414^-7298, 6804, 77?4.
15. 921006254-5136, 1984, 2875.
16. 800000004-8345, 6205, 7095.
17. 538054484-4001, 8936, 9027.
18. 7300692-7506, 9324.
19. 9000000—8931, 7295.
20. 8203570—4583, 9308.
21. 2568903684-28, 79, 35.
22. 9314562044-17, 47, 82.
23. 2490860224-457, 329, 704.
24. 3036067964-293, 718, 274.
25. 7240880434-8561, 2793.
26. 3659057804-5006, 2918.
27. 8543724004-9300, 8540.
28. 2936001704-2005, 7009.
29. 8759127804-3054, 7090.
30. 2934000004-7200, 5090.
, To find an Average.
Ex. — A boy gets 23 marks on Monday, 17 on Tuesday, 28
on Wednesday, 31 on Thursday, 25 on Friday, and 14 on
Saturday : what is his average number of marks daily for the
week?
Here the sum of his marks for the whole week is 138. There
is a certain number of marks, which had he got every day of
the week, the sum of his marks at the end of the week would
DIVISION. 45
have been the same as it is now. That is the number we wish
to find.
The average of a series of numbers is that number which, if
repeated as often as there are numbers, will amount to their
sum. It is found by dividing the sum of the numbers by their
number ; thus 138-J-6 = 23.
EXERCISE XIV.
Find the average of the following numbers : —
1. 27, 37, 42, 50, 22, 24. 6. 2738, 3624, 3001,
2. 13, 49, 35, 64, 53, 42. 7. 937, 1001, 1100, 1010, 1110.
3. 93, 87, 59, 67, 73. 8. 856, 1533, 930, 1399.
4. 29, 30, 37, 32, 33. 9. 8973, 10704, 9320, 14976, 9999.
5. 125, 250, 315, 193. 10. 27345, 73421, 85648, 79286.
- Fractional Multipliers and Divisors.
Ex. — A train runs 27 miles an hour for 14| hours ; what
distance will it go in the time ? 27
The distance is 27 miles repeated 14 times -a
and | a time ; which is got by multiplying 27 by
14}.
To multiply by j, multiply by 3 and divide 20£
the product by 4. Then in multiplying by 14,
the right-hand figure of the first line, being units, 27 _
is set in the units' place. 398^
The number f , which is less than 1 is called a fraction.
If one is divided into 2 equal parts, each is called a half ; if
into 3, each is called a third ; if into 4, a fourth ; and so on.
A fraction is denoted by two numbers, the one written below
the other ; thus one-half is written J, one-third £, one-fourth | ;
if more than one part be taken, the upper figure denotes how
many, thus three-fourths is written j. The number 14f , which
consists of a whole number and a fraction, is called a mixed
number.
EXERCISE XV.
1. Find one-half of 38, 57, 108, 265, 798, G357.
2. One-third of 51, 252, 254, 768, 784, 8472.
3. One-fourth of 56, 92, 94, 397, 3828, 8927.
4. Multiply by f : 85, 101, 357, 456, 2456, 7530.
5. Multiply by | : 84, 356, 537, 933, 1272, 7000.
6. 8456x41, 6}, 15J, 27$, 139|, 308£.
7. 93582 xlOJ, 200*, 750§, 30|, 5$.
46 DIVISION.
Ex. — How often is 29 J contained in 9384 ?
The numbers cannot conveniently be 29J 9384
used for divisor and dividend as they 4 4
stand. 117 )37536(320i*A
Multiply both by 4, the fraction in the 351
divisor being fourtlis. This will give a ~243
new divisor and dividend four times greater «„ .
than those given ; but which will be free
from fractions, and will give the same
quotient.
EXERCISE XVI.
1. 3482-^-3$, 6§, S±. 6. 900536-^-12*, 74$, 256$.
2. 8506-Hi|, 5£ 94. 7. 852079-f-5A, 301, 3651.
3. 72584-7-27$, 54f, 79£. 8. 205930-^-152, 85$ , 365J.
4. 59321 H-19J, 68^, 128$. 9. 730526-~29i, 217$, 8342^.
5. 80999-^-15$, 265, 94g. 10. 45067824-i-14|, 58$, lOOfc.
» Multiplication and Division Combined.
Ex. — What number results from multiplying 57 by 16, and
dividing by 24 ?
To multiply by 16 is the same as to multiply by 2 and then
by 8 ; and to divide by 24 is the same as to divide by 3 and
and then by 8. We may strike out the 8 from both terms ;
since to multiply a number by 8 and then to divide it by 8
leaves it unaltered. So that —
57 x 16
24
The striking out of a factor common to a
multiplier and a divisor is called cancelling.
Cancelling may sometimes be performed
more than once in the same exercise ;
thus—
EXERCISE XVII.
Perform the following operations, cancelling where possible,
1. 9x7 8x15 24x12 16x6 33x14 48x24.
2.
8.
3
45x36
5
84x48
18
105x21
8 35
117x48
57x25
72
81
89x32
84
157x81
49
238x63
108
181x36
40
66x45
44
124 x If
108
119
54
99
00
DIVISION. 47
*. 85x9x12 45x16x18 24x15x21 30x14x24 42x16x32
4x18 36x45 40x35 20x28 48x35
5. 59x10x33 63x8x25 18x14x28 50x34x21 9x8x6
11x60 35x32 9x36 14x25 3x4
6. 147x24x18 240x65x8 306x28x63 564x84x33
72x45 16x30 35x102 88x144
Any number is divisible exactly —
1. By 2, when its last place is divisible by 2.
2. By 4, when its last two places are divisible by 4.
3. By 8, when its last three places are divisible by 8.
5* By 9^ } waen tne sum °f its places is divisible by 3 or 9.
6. By 5, when its last place is 5 or 0.
7. By 10, when its last place is 0.
EXERCISE XVIII.
1. 243x316 79x104 348x252 219x573 391x215 893x4128
228 432 384 693 300 376
2. 256x216 750x375 358 x 516 250x700 295x415 312x462
8. 584x2928 73x321 92x840 300x200 843x356x296
3024 412 342 6000 296x560
54. EXERCISE XIX.
1. How many scores in 340 ?
2. How many one-dozen baskets may be filled out of 468 bottles ?
3. How many pieces, each 25 yards, may be got from 6425 yards.
4. How many forms, of 15 each, will hold 675 scholars ?
5. Into how many parcels of 16 may 432 marbles be divided ?
6. How often can I subtract 64 from 2304 ?
7. What must 73 be multiplied by to give 22995 ?
8. How many regiments, each 829, are in an army of 38963 men ?
9. If 2664 be dividend, and 36 be quotient, find the divisor.
10. How many boxes will hold 7000 oranges, if each hold 125 ?
11. If a man divides £728 equally among his 4 children, what is
the share of each ?
12. How many years' rent of a house at £6 is £792 ?
13. If the journey from London to Edinburgh, which is 385 miles,
be made in 11 hours, what rate is that per hour ?
14. What multiplier of 346 gives 81964 as product ?
15. If a tradesman saves 5 shillings a week, in how many weeks will
he save 850 shillings.
16. What is the nearest number to 850 which can be divided evenly
by 27 ? and the next nearest ?
17. The year 1864 began on a Friday, how many Fridays had it I
and how many Sundays ?
48 DIVISION.
18. In a certain city there died in the month of April 23790 persons,
what was the daily number of deaths on an average ?
19. A banker has a box with 7460 shillings, 24 five-shilling pieces,
and 50 florins, how often can he change a pound ?
20. Five trains left London Bridge for the Crystal Palace, the first
with 379 passengers, the second with 250, the third with 483, the
fourth with 579, and the fifth with 294 : what was the average number
in each train ?
21. A regiment of 1170 men had one man killed or wounded in
battle for every 18 men in it : how many remained fit for service ?
22. A cargo of tea, 435 chests, each 180 pounds' weight, is to be
packed in boxes, each containing 54 pounds : how many of these must
be ordered ?
23. What must I add to the square of 154 to contain exactly the
square of 27 ?
. MISCELLANEOUS EXERCISE ON THE FOUR RULES.— I.
1. Printing was invented 1440 A.D., and the first book was printed
in England 34 years thereafter : what was its date ?
2. If a farmer sells 35 oxen for £12 each, 253 sheep for £2 each,
and 159 lambs at £1 each, what does he receive for all ?
3. The circumference of the earth is 24900 miles, in how many days
could a ship sail round it at 9 k miles an hour?
4. How much higher is Mont Blanc, the highest mountain in
Europe, which is 15,680 feet high, than Ben Nevis, the highest in
Britain, which is 4368 feet high ?
5. To half the sum of 85 and 57 add half their difference.
6. A clerk, engaged for five years, receives £80 salary the first year,
and an advance of £15 each year : what is his average yearly salary ?
7. The six largest cities in England are London with 2,362,236 in-
habitants, Liverpool with 375,955, Manchester with 316,213, Bir-
mingham with 232,841, Leeds with 172,000, and Bristol with 137,000 :
what is the population of these cities together ?
8. Sir Isaac Newton was born in 1642 and died in 1729 : how old
was he at his death?
9. Three apples were given to each of 178 pupils of a school, but
672 apples were provided in all : how many more pupils could have
been served ?
10. I met 7 flocks of sheep, of one score each, .on their way to
market, 5 of twoscore and nine each, 6 of threescore and ten each,
and then one of 19 : how many sheep did I pass ?
11. From London to Peterborough is 76 miles, from Peterborough
to York 115 miles, from York to Newcastle 72 miles, from Newcastle
to Berwick 65 miles, from Berwick to Edinburgh 57 miles : what is the
distance from London to Edinburgh ?
12. What number added to 7803 will make up the third part of
87003? -
13. To 7 times the sum of 909 and 98, add 7 times their difference.
14. A train contains 1097 passengers ; of these, 286 are first-class,
and half as many more second-class : how many third-class are there?
15. What divisor of 44934 gives 348 as quotient, and 42 over?
MISCELLANEOUS EXERCISES 49
16. Find the number of days in a leap year.
17. A teacher buys 100 boxes steel-pens, containing one gross each.
He has 563 pupils in school : after serving them with pens 7 times,
how many remain ?
18. The ship " Graceful," from Charente to Leith, discharged 2552
one-dozen cases brandy, 122 two-dozen cases, and 16 three-dozen
cases : how many gross of bottles were in her cargo ? If 6 bottles go
to a gallon, how many gallons of brandy ?
19. A shelf in a library contained — History of England, 10 volumes ;
British Poets, 75 volumes ; Goldsmith's Works, 4 volumes ; Waver-
ley Novels, 25 volumes ; British Essayists, 45 volumes ; and the shelf
below contained exactly the same number : how many volumes were
on both ?
20. What must be added to the third part of 1395 to bring it up to
the fifth part of 3790?
21. Find the product of three numbers, of which the first, 374, ex-
ceeds the second by 93, and the third by twice as much.
22. In what time will 3 pipes empty a tank of 429165 gallons, if
they run off respectively 450, 500, and 535 gallons per hour ?
23. If a stage-coach travel 5£ miles an hour, how far will it go in
two days of 9 hours each ?
24. An army of 69776 men was drawn up in squares of 28 in a side ;
how many squares were there ?
25. Find the difference between the square of 9009, and the cube
of 909.
MISCELLANEOUS EXERCISES-^mftVwed. II.
1. Julius Caesar invaded Britain 55 B.O. : how long was that before
the union of England and Scotland in 1700 ?
2. How often does a clock strike in a year ?
3. A boy, working 8 hours a day, can point in a year 33979280 pins :
how many can he point in an hour ?
4. A travels 3 miles an hour, B 4£ : when B has gone 45 miles, how
far has A gone ?
5. Great Britain and Ireland contain 121385 square miles ; the
British possessions in Europe, 145 ; in Asia, 928610 ; in North
America, 768577 ; in South America, 89000 ; in Africa, 201403 ; in the
West Indies, 73384 ; in Australasia, 560000. What is the whole area
of the British Empire ?
6. Michaelmas is 86 clear days before Christmas : what is the date
of it?
7. January 4, paid into savings' -bank, 14 shillings ; February 1,
paid in 13 shillings ; February 28, drew out 11 shillings ; March 14,
paid in 19 shillings ; March 31, drew out 25 shillings ; April 24, paid
in 17 shillings ; May 3, paid in 9 shillings ; May 25, drew out 15 shil-
lings ; June 1, paid in 16 shillings. My account was then balanced :
how much had I at my credit ?
8. Adam lived 930 years ; Seth, his son, was born when he was 130
years old, and lived 912 years : how long did they live together ?
9. A bag of nuts, containing 3000, was divided among a school ;
the pupils above 9 years got 35 each, and those below 9 (who were
exactly the same number) got 25 each : how many pupils were in the
school?
10. A railway guard makes two journeys every lawful day from
D
50 MISCELLANEOUS EXERCISES.
Edinburgh to Glasgow and back ; if these towns are 47 miles apart,
what distance has he travelled, after being in his situation five years ?
11. Three regiments form squares, the side of the tirst being 33 men,
of the second 29, and of the third 27 : how much stronger is the first
regiment than the second, and the second than the third ?
12. How often will a cart-wheel, 16£ feet round, revolve in going
a mile, which has 5280 feet ?
13. A railway 273 miles long has a station every 10£ miles on the
average : how many stations has it ? And what is the length of a
railway which has 18 stations, distant on the average 7£ miles from
each other ?
14. George I. of England began to reign 1714 A.D., and reigned 13
years ; George n. reigned 33 years, George in. 60, George iv. 10, and
William iv. 7 years. Queen Victoria succeeded William ; in what
year did she begin to reign ?
15. The sea route from London to Hamburgh is 482 miles. When
the London steamer is 130 miles on its way, and the Hamburgh
steamer 210 miles on its, how far are they apart ?
16. If Scotland produced in 1864, 23000 tons pig-iron weekly, what
was the produce for the year ? and at £3 a ton, how much did it add
to the wealth of the country during the year ?
17. A farm has 5 fields, the first containing 89 acres, the second
101, the third 174, the fourth 92, and the fifth the average of the other
four. It is to be divided into as many fields of equal size : how many
acres will each contain ?
18. (a) A legacy of £1595 is left to two charities, of which the one
receives half as much again as the other : what was the share of each ?
(6) Out of a legacy of £8578, £730 were devoted to charitable
purposes ; the rest was to be divided into 9 shares, of which the
eldest son was to get four, the second three, and the youngest two :
how much did each get ?
19. If a candidate at an election is returned by a majority of 291
votes out of 3579, how many voted for the unsuccessful candidate ?
20. If I bought 79 shares in the Great Western Railway at £64
each, and sold out at £69, what did I pay for them, and what did I
gain?
21. The exports from Liverpool to the United States in 1861 were
£8223587 ; in 1862, £11986233 ; and in 1863, £13765217. How much
did the increase in 1862 exceed that in 1863 ?
22. In a journey of 37 hours, I travelled one-third of the time at
24 miles an hour, and two-thirds at 27 miles : what was the length of
the journey ?
23. Divide 318 apples among 18 boys and 8 girls, giving each boy
twice and a half as much as a girl.
24. Handel, the great musical composer, died in 1759, aged 75 ; and
Haydn was born when Handel was in his forty-seventh year; what
year was that ?
25. A tank of water contained 75000 gallons. A supply was drawn
off by 3 pipes, which ran for 10 hours at the rate of 255 gallons each
Eer hour ; but during that time two pipes ran into the tank 335 gal-
>ns each per hour : how much water was left ?
26. Find the difference between the square of the sum of 28 and 39,
and the sum of their squares.
MONEY. 51
27. A sum of money was divided between A, B, c, and D, so that
A got £260, B £375, c the excess of B'S share above A'S, whilst D was
to receive £25 from A, £48 from B, and £17 from c, as his share.
What were the shares of all four ?
28. The sum of 2 numbers is 428, and their difference 194 ; find the
numbers.
%* Half the smn-fhalf the difference gives the greater.
Half the sum -half the difference gives the less.
29. A tradesman, out of his weekly savings for a year, bought a
table that cost 22s., 6 chairs that cost 7s. each, a carpet of 20 square
yards in size at 3s. a yard ; he had besides 32s. over : how much had
he saved every week ?
30. A farmer paid £780 for cows and sheep. Of t]iis sum he paid
£350 for 25 cows ; if a cow cost 7 times as much as a sheep, how many
sheep did he buy with the rest of his money ?
56.
MONEY.
MONEY OF ACCOUNT TABLE I.
Accounts are kept in pounds, shillings, pence, and farthings
sterling.
Pounds are denoted by the letter £, thus £40.
Shillings by the letter s., or by a line ; thus, 3s. or 3/.
Pence by the letter d.9 thus 9d.*
Farthings, which mean fourths of a penny, are denoted by
fractional numbers ; thus, one farthing by £ ; two farthings, or
one halfpenny, by Jd. ; and three farthings by f d.
Pounds, shillings, and pence, when written in columns, are
denoted by £ s. d. placed over the column.
EXERCISE.
Read off the sums in Ex. i. sect. 58.
t COMPOUND ADDITION.
Ex. — If I have paid into the bank in January, ,£27, lls. 3jd. ;
in February, £23, 14s. 8^d. ; in March, .£13, 19s. 9|d. ; in April,
£7, Os. 2jd. ; in May, £2, 7s. lid. ; and in June, 17s. 3jd. : how
much have I put in during the six months ?
We have here to find the sum of the six payments ; which
we do by addition.
Write the numbers below each other, pounds below pounds,
shillings below shillings, and pence below pence.
i £ is the first letter of libra, a Roman weight ; *. and d. the first letter* of
iolidus and dtnariu-s, Roman coins.
52
MONEY.
Then, adding the farthings' column, we have 2-3-6-8-9 farth-
ings, which is 2|d. ; set down the Jd. and carry the 2d.
Adding the pence column, we have, by
simple addition, 29d., that is 2/5 ; set
down the 5d. and carry the 2/.
Adding the shillings' column, we have,
by simple addition, 70/, that is £3, 10s. ;
set down the 10s., and carry the £3.
Adding the pounds' column, we have,
by simple addition, £15.
Sum, £75, 10s. 5|d.
Rule. — Write the numbers below each other so that each
column may be of the same name ; add each column in its
order, carrying as many of the next highest name as are con-
tained in its sum.
The result may be proved, as in simple addition, by adding
the columns from the top downward.
The addition of quantities of different names, as here, is
called compound addition.
£75 10 5
58.
EXERCISE I.
1.
2.
3.
4.
5.
£ s. d
£ s. d.
£ s. d.
£ s. d. 2 s. d.
a. 3 6 10
I 2 10 9^
t 11 9 10s
[ 7 9 4* 5 10 9;
b. 4 16 8.
384.
543
11 9 2;
7 9 6.
c. 930
\ 9 5 6j
729;
507;
1 14 10;
d. 5 14 10;
7 7 103
800
3 17 2,
3 9 e;
e. 10 0 9
4 15 11'
7 2 11-
11 9 5;
0 17 10
/. 6 18 6,
729
0 8 2} 8 2 4j
2 17 93
a. 0 19 3;
12 0 5*
9 0 11
12 0 Oi
8 5 101
*. 11 5 8J
6 16 10;
7 10 9£ 976^
11 11 4a
». Ill;
8 2 3^
11 15 0
0 18 91
f 879^
j. 2 3 9}
756;
4 10 11
10 9 4,
3 11 5J
A. 12 2 10
10 11 11.
12 12 9}
500'
276
& 9 6 5J
2 19 6
1 17 1*
7 0 1<H
L 994
m. 8 10 11;
3 15 4
248;
549i
12 4 10,
n. 5 14 3^
4 17 2J
5 15 71
6 10 10;
1 19 11
o. 0 10 2
5 10 H
10 0 0
0 18 2^
696
p. 1 17 9J
11 9 4
909
11 9 2;
11 4 4
EXERCISE II.
1. Count from Id., 2d., 3d., etc., by lid., 2£d., 2fd., 4|d., etc.
2. Count from 1 sh., 2 sh., 3 sh., etc., by 1/3, 2/4, 1/3 J, etc.
8. Count from £3, £4, £5, etc., by 7/8, 13/4, 12/6£, etc.
V Ex- J- and 11 tor oral practice, whilst the pupil is working the following
Exercise. The same remark applies to the subsequent rulefc,
COMPOUND ADDITION.
53
59.
EXERCISE
III.
1.
2.
3.
4.
5.
£ 8.
d.
£
s.
d.
£
s. d.
£
s.
d.
£
5. d
73 10
91
14
10
8*
47
17 10
63
14
5f
72
8 41
47 5
ol
92
0
4
39
14 8i
28
10
10
17
8 9j
3 7
91
37
16
4
7
19 10;
37
15
71
93
15 2j
13 17
4
29
18
llf
72
12 6l
9
9
9}
82
10 4f
28 9
3|
15
14
oj
84
0 11]
8
5
29
79
1 11
80 5
11
34
8
5
59
10 0;
92
10
10j
9
11 u
6.
7.
8.
9.
10.
£ 9.
d.
£
s.
d
£
s. d.
£
s.
d
£
*. d.
42 8
28 10
59 0
OOOO
147
82
7
2
5
17
CO i—l O
293
118
500
14 10;
10 0;
8 0
\ 673
: 200
[- 74
10
18
0
***
COOrH
l-H
534
1
10 9
0 6
5 Of
72 0
10J
973
0
rj
94
3 6]
* 9
10
4f
904
15 2
5 18
38 14
1
11
459
226
19
4
ill
ol
7
192
18 2
17 5;
28
t 990
14
19
51
o]
673
49
17 0*
15 8J
0 19
3J
305
2
11
201
0 7
, 309
17
6
200
18 2
17 8
el
38
18
6J
802
14 0
25
8
0
55
16 0
11. £934, 18, 6 + £84, 0, 9 + £702, 15, 21 + £39. 4. 0 + £740, 0, 0 +
£85, 16, 2£ + £156, 18, 6| + £529, 5, 1J.
12. £617, 10, 11| + £290, 0, 10j + £38, 5, 6 +£93, 0, OJ + ^549, 7,2|
+ £29, 10, 0 + £709, 18, 4f +£8l£, 16, 1.
13. £127, 14, 81 + £293> 11, 5l + £3±Q, 10, 101 + ^458, 10, 9 +
£500, 17, 7f + £110, 19, 2£ + £301, 1, 11$ + £824, 0, 0^ + £629, 5, 5.
14. £543, 10, 0£ + £H 17 +£7, 10, 6f + £829, 7 +£471, 10,
£28, 15, 0£ + £728, 16, 10| + £840, 0. 11.
15. £293, 18 + £72, 19, l£ + £9, 10, 5? + £820, 15 + £94, 18, 65 +
£571, 15, 4| + £629, 18, 4 +£930, 15, 10 J.
16. £2005, 7, 6 + £943, 18, lJ + £564, 9 + £7248, 0, 9^ + £1508, 10, 8|
+ £592, 8, 0^ + £9408, 2, 10 + £93, 0, 11|.
17. £329, 14, 4^ + £73, 18, 5«i + £493, 9, 4^ + £701, 1, 7i + £592, 10, 11
+ £17, 4, 8 + £9, 7, 6^ + £341,19, 8| + £700, 1, 11.
18. £112, 9, 4£ + £257, 3, 0^ + ^62, 11, 7| + £79, 19, 91 + .£790, 8, 2
+ 173, 13, llf + £459, 12, 10 + £614, 14, 11^ + 998, 19, 5|.
19. £72, 7, 10 + £394, 6, 4| + £593, 0, 8i + £360, 0, llf + £94, 15, 81
+ £250, 11 + £37, 18, 0| + £84, 15, 6f + £4^20, 18, 6 + £13, 2, 1|.
20. £640, 10, 11 + £93, 4, 7^ + ^870, 19 + £250, 0, 9f + £550, 9, 1 +
£709, 13, 6f + £1, 2, 3^ + ^85, 16, 6 + £924, 15, l£ + £9, 2, 8f.
21. £279, 18, 6 + £90, 17, 3| + £250, 4, 10 + £79, 18, If + £100, 15 +
£25, 0, 6^ + £365, 19, 1 + £209, 14, 7| + £99, 18, 4 +£805, 7, 6J.
22. £8408, 14, 10 + £2930, 10, 44 + £6009, ig? Of +£509, 7, 111 +
£93, 10, 6^ + ^793, 10, 0^ + ^209, 18, 1 + £3085, 2, Of + £94, 18, 2J.
23. £2563, 14, l + £846, 10, 01 + £2564, 0, lOi + £865, 17, llf -f
£590, 0, 6 + £859, 2, l£+£9337, 19, 0^ + £820, 7, 6 + £94, 17, 6f.
EXERCISE IV.
Work the auestions Ex. iii. as directed Ex. xi. p. 23.
54 MONEY.
60.
COMPOUND SUBTRACTION.
Ex.— It I pay a debt of £-28, 18s. 5^d. out of a sum of £63,
13s. 4jd., how much have I over?
We have here to find the difference of these two sums of
money ; which we do by subtraction.
Write the subtrahend below the minuend in its place.
2 f. from 1 f. cannot be taken ; change
one of the pence, making 5 f. in all ; 2 f.
from 5 f. leaves 3 f.
5d. from 3d. cannot be taken ; change one
of the shillings, making 15d. in all ; 5d. £34 14 lOf
from 15d. leaves lOd.
18s from 12s. cannot be taken ; change one of the pounds,
making 32s. in all ; 18s. from 32s. leaves 14s.
28 pounds from 62 pounds leaves 34.
Rule. — Write the subtrahend below the minuend so that
each column shall be of the same name ; subtract each column
in its order, changing one of the next highest name when
necessary.
The result may be proved, as in simple subtraction, by add-
ing together the subtrahend and the difference.
The subtraction of quantities of different names, as here, is
called compound subtraction.
Or thus i1
Then, beginning with the lowest name, 2
from 1 cannot be taken ; add Id. or 4 farthings,
making 5f. in the minuend ; 2f. from 5f. is 3f.
Then 6d. from 4d. cannot be taken ; add Is.
or 12d., making 16d. in the minuend ; 6d. from £34 14 10|
16d. is lOd.
Then 19s. from 13s. cannot be taken ; add £1 or 20s., making
33s. in the minuend ; 19s. from 33s. is 14s.
Then £9 from £3 cannot be taken, but 9 from 13 is 4 ; and
3 from 6 is 3 for the tens' place ; making £34.
Rule. — Write the subtrahend below the minuend so that
each column shall be of the same name ; subtract each column
in its order, beginning with that of lowest name, and carrying
as in compound addition ; if any name in the minuend is less
than the same name in the subtrahend, add to it one of the
next highest name changed to its own, and add one to the
next name in the subtrahend.
1 Both methods of subtraction are given as in simple subtraction, sect. 26 ;
the teacher may choose either.
COMPOUND SUBTRACTION.
EXERCISE I.
1. 7J-2A, 5A-31, 8f-5f, 10^-7, 9A-1A, etc.
2. 5j-3|, 71-6?, 71-5$, 9£-7£, 111-8J, etc.
3. 6/5-3/2, 8/11-5/6, 7/9-1/9, 3/6£-2/4A, 14/10^-7/4, etc,
4. 4/3-2/6, 7/2-3/8, 8/4-4/7, 8/41 - 6/5, 13/2J - 8/8, etc.
EXERCISE II.
1. Count back from I/, 2/, etc., by 2£d., 3|d., 43d., etc.
2. Count back from 20/, 19/, etc., by 1/3, 1/4, 2/2, l/7£, etc.
3. Count back from £5, etc., by 10/6, 12/8, 13/4$.
EXERCISE III.
55
1. 2.
£37 8 4A £93 10 3|
19 5 10J 39 6 9
3.
£84 7 101
53 17 I}
£47
29
4. 5.
17 8} £205 2 9
8 10$ 126 12 8
1
6.
7.
8.
9. 10.
£730 2
6.1 £704 14
91
£294 0 9* £360
10
6^ £545 12 0£
428 17
8j
396 0
a
89 10 9
^ 219
19
0}
293 18 OJ
11. £848
0 Oj
?- £274
10
1 19.
£8000
0
1 -
£1793
10
0*
12. 763
10 11
[- 294
18
2|
20.
3030
13
0 -
2594
0
71
13. 540
14. 643
0 0
15 8
J- 290
i- 19°
0
15
9
21.
22.
2000
903
0
0
01-
61-
17
50
0
0
9
7
15. 1938
17 6
I- 209
19
8£
23.
1000
0
0 -
295
0
T$
16. 2467
14 8
- 938
15
6
24.
3724
6
104-
1936
2
11
17. 3091
10 11
- 1857
16
111
25.
5704
13
8 -
2945
2
105
18. 4000
0 0
- 993
1
u
26.
8407
0
7J-
899
19
91
EXERCISE IV.
Find the first remainder less than the subtrahend in-
1. £2761
13
4i —
£564
17
63 10.
£8473
16
4
- £1005
5 10A
2. 4095
14
of-
709
19
1
11.
10000
0
0
- 2946
0 5j
3. 8740
0
?2 ~
1096
10
91
12.
7338
2
11;
- 943
4 9:
4. 5436
10
81-
854
12
0?
13.
7009
9
- 856
0 4^
5. 9425
16
2 -
1906
17
2}
14.
1946
10
10
- 405
16 11.
6. 7464
13
11 -
948
17
6|
15.
6429
14
6.J
- 842
15 10;
7. 4763
0
01-
742
11
4
16.
8754
12
3^
- 947
13 6
8. 6000
0
0 -
823
0
Of
17.
5431
18
l'
- 739
11 41
9. 2346
2
10 -
473
0
0J
18.
9402
14
;:- 1246
16 8f
COMPOUND 1MULTIPLICATIOK
Ex.— What cost 9 chests of tea at £24, 14s. 7Jd. per chest ?
We have here to find 9 times the price of one chest ; which
we do by multiplication.
56 MONEY.
Write the multiplier under the pence column £ s. d.
of the multiplicand. 24 14 7J
Then, beginning with the lowest name, 9 __ 9_
times If. are 9f., which is 2£d ; set down If., £222 11 5£
and carry 2d.
9 times 7d. are 63, and 2 are 65d., which is 5/5 ; set down
Bd., and carry 5/.
9 times 14s. are 126s., and 5 are 131s., which are £6, lls. ;
set down lls., and carry £6.
9 times 24 are 216, and 6 are £222.
Total product, £222, lls. 5^d.
The result may be proved by dividing the product by the
multiplier (see sect. 65), which will give the multiplicand.
Rule. — When the multiplier is not above 12, multiply each
name in the multiplicand by it in order, beginning with the
lowest, and carry as in compound addition. When the multi-
plier is not greater than 144, and has two factors, neither above
12, multiply by each factor in succession.
The multiplication of quantities of different names, as here,
is called Compound Multiplication.
EXERCISE I.
upy e oowng y , , , ec., up o successvey :— .,
3id., 3|d., 4Ad., 51d., 5|d., 6 Ad., 63d.. 71d., 7id., 81d., 8jd., 9Jd.,
9jd., lO
Multiply the following by 2, 3, 4, etc., up to 12 successively :— 2M.,
i
9j
EXERCISE II.
Multiply by 2, 3, 4, etc., to 12 successively : —
1. 6d., 8d., 10d., 1/1, 1/4, 1/8, 2/1, 2/7, 3/4, 3/6, 4/2, 4/10, 5/6, etc.
2. 10A, 10/3, 10/9, ll/, 12/2, 12/8, 13/3, 13/10, 14/4, 15/1, 15/11, etc.
EXERCISE III.
x 2.
x 3.
x 4.
x 5.
x 6.
x 7.
x 8.
x 9.
xll.
x!2.
EXERCISE IV.
1. £7, 8, 4i x 2, 4, 7, 8, 9. 4. £21, 4. 8} x 3, 8, 2, 7, 5.
2. £10, 9, 4x3, 6, 8, 10, 11. 5. £34, 17, Iljx7, 11, 9, 12, 3.
8. £16, 0 5| x 4, 5, 7, 9, 12. 6. £43, 10, 10* x 5, 8, 4, 10, 7.
COMPOUND MULTIPLICATION.
57
7. £87, 9, Of x!4, 15, 21, 22.
8. £92, If, 4* x25, 27, 28, 32.
9. £127, fc, 6| x35, 42, 44.
iO. £209,15,7|x45,48,54. ,
11. £543, 18,21 x56, 60, 63.
12. £708, 13, l| x 64, 72, 77.
13. £900, 0, 91 x 84, 99, 108.
14. £1256, 10, Of x 121, 132, 144.
\* Multiply by three factors.
15. £18, 9, 41 xl!2, 125.
16. £37, 0, 9| x 105, 126.
17. £85, 17,2^x192,216.
18. £90, 14, 83 x 128, 135.
19. £74, 8, 111 x 147, 162.
20. £105, 15,0^x189, 210
197 16 10 price of 8 chests.
64 chests.
4 chests.
68 chests.
63. Multipliers of Two Places.
Ex. — Find the price of 68 chests tea at £24, 14s. 7Jd. per
chest.
The number 68 cannot be re-
solved into two factors under 12. £ «. d.
Take the next less which can, that 24 14 7^X4
is, 64. Find the price of 64 chests 8
(8X8), and add the price of 4
chests ; for 68 = 8 X 8+4.
The price of 64 chests is found -J582 14 g
as above : the price of 4, by mul- 93 18 5
tiplying the price of one (first line) ,„„, — r
by 4 ; the price of 68 by adding
the price of 64 and the price of 4
together.
Other factors which might be used are 9X7 + 5 and
10 X 6 + 8, either of which pairs may be taken to prove the
result,
Rule. — When the multiplier is not above 144, and cannot
be resolved into two factors under 12, multiply by the factors
of the next less number which has them, and add the product
of the multiplicand by the difference between that number
and the multiplier.
It is advisable to take factors for the number next above the
multiplier, when that number exceeds it only by 1, and then
subtract the excess; thus, 39 = 10X4 — 1. In the present
case we might have taken 68 = 10x7 — 2.
EXERCISE V.
1. £2, 14, 2S
2. £7, 10, 9]
x 13, 17, 19, 24, 29, 31.
x 34, 38, 43, 51, 58, 61.
3. £13, 8, 5
4. £34, 3, 2^
6. £60, 0, 9;
fe x 62, 69, 74, 78, 82, 87.
; x 91, 94, 101, 106, 117, 123.
x 129, 135, 142, 145.
U06.
58 MOXEY.
. Multipliers of Three Places.
Ex.— Find the price of 457 chests at .£24, 14s. 7|d. pel
chest.
£24 14 7£ X 7 = X173 2 2| price of. 7 chests.
_ 10
£247 6 Oi X 5 = 1236 10 2| „ 50 „
10
£2473 0 5 X4= 9892 18 ,,400
Total product, £11301 14 1J „ 4^7 „
Rule.— Multiply by factors for 100 (10 X 10). Then multi-
ply the multiplicand by the number of units in the multiplier,
ten times the multiplicand by the number of tens in it, and a
hundred times the multiplier by the number of hundreds in it .
add these three products for the total product.
EXERCISE VI.
1. £9, 13, U x 257, 381, 473. 7. £59, 7, 31 x 915, 638, 187.
2. £13, 10, g^x 319, 459, 542. 8. £73, 8, 10±x562, 784, 268.
3. £19, 8, 5J x417, 534, 629. 9. £83, 15, 71 x 400, 701, 511.
4. £23, 10, Of x 566, 671, 713. 10. £89, 0, 5± x208, 962, 609.
6. £31, 19, 41x647, 738, 825. 11. £93, 14, 2^x354, 849, 276.
6. £43, 1, 11 J x 724, 850, 993. 12. £109, 7, 9£ x 417, 651, 767.
Multipliers of Four Places.
The same method is used for multiplying by thousands.
Rule.— Multiply by factors for 1000 (10X10x10). Then
multiply the multiplicand and the successive products by the
several places of the multiplier in order, beginning with the
units' place ; add these products for the total product.
EXERCISE VII.
1. £13, 18, 5^x1924, 2438. 4. £57, 10, 7| x6234, 7941.
2. £19, 5, 104 x 2741, 3925. 5. £69, 5, 8^ x 8301, 9042.
8. £27, 3, 4£ X4837, 5529. 6. £124, 15/6 jx 4520, 6009.
V* These products are obtained more easily by practice.
B COMPOUND DIVISION.
Ex. 1. — Divide .£93, 15s. 9|d. equally among 7 persons : what
is the share of each ?
Write the divisor and dividend as in simple division.
COMPOUND DIVISION. 59
Then 7 in £93 is X13 and £2
over ; set down the £13, and carry 7 ) 93 15 9|
the £2 to the shillings, making "13 7 ll£ f
55s. in all.
7 in 55 is 7s. and 6s. over ; set
down the 7s. and carry the 6s. to the pence, making 81d. in all.
7 in 81 is lid. and 4d. over ; set down the lid. and carry
the 4d. to tJie farthings, making 19 farthings in all.
7 in 19 is 2 farth. and 5 farth. over ; set down the 2 farth.
and, as the division is now finished, there is a remainder of
5 farthings, divided thus, f .
Quotient, £13, 7s. lljf.
The result may be proved by multiplying the quotient by
the divisor, and adding the remainder, which will give the
dividend.
Ex. 2. — Divide the same sum ^ ~
equally among 28 persons.
Resolve the divisor into its two
13 7 11H-5 (
19f.
factors (7X4), and divide by each 3 611|-f2)
in succession.
Quotient, £3, 6s. llj Jf.
The result may be proved by reversing the order of factors
in dividing, or by multiplying the product by the divisor.
Rule.— When the divisor is not above 12, divide each
name by it in order, beginning at the highest, and carry the
remainder to the next lower. When the divisor is not above
144, and has two factors neither above 12, divide in the same
way by each factor in succession.
The division of a quantity of several names, as here, is
called compound division.
66. EXERCISE I.
1. 2d. 3d. 5d. 6d. 7d. etc. 4-2, 4. 10. 1/3,1/6, 1/9, 2/, 2/3, etc. 4-6/12.
2. lid. 3d. 4£d. 6d. 7£d. etc. 4-3, 6. 11. 1/OJ, 1/2, l/3£, 1/5 A, etc. 4-7.
3. lid. 2Ad. 3M. 5d. 6|d. etc. 4-5. 12. 1/1*, 1/4$, l/7i, 1/10, etc. 4-11.
4. ifd. 3|d. 5jd. 7d. etc. 4-7. 13. £1, £1, 4, £1, 8, etc. 4-2, 4, 8.
5. 2d. 4d. 6d. 8d. etc. 4-8. 14. £l,2/6,£l,5/6,£l,8/6,etc. 4-3,9.
6. 2}d. 4$d. 6|d. 9d. etc.4-9d. 15. £1, £1, 5, £1, 10, etc. 4-5, 10.
7. l/,l/2, 1/4, 1/8, 1/10, 2/, etc. 4-2,4. 16. £1,4, £1,10, £1,16, etc. 4-6, 12.
8. lli,l/l$,l/3|,l/6,l/8|,etc. 4-3,9. 17. £1, 1, £1, 4/6, £1, 8, etc. 4-7.
9. 1/0$, 1/3, 1/5$, 1/8, 1/104, etc. 4-5. 18. £1, 2, £1, 7/6, £1,13, etc. 4-11.
60
MONEY.
EXERCISE
II.
1.
£8
19
72
-2, 3, 4, 5.
13.
£89
14
102-14,
15,
21.
2.
7
0
5i
—3. 4, 5, 6.
14.
91
2
81-
-24,
27,
22.
3.
19
10
3;
-4, 5, 6, 7.
15.
156
17
3j-
-25;
28
100.
4.
27
15
6;
-5, 6, 7, 8.
16.
193
0
5 -
-30,
32,
108.
5.
79
1
111
-6. 7, 8, 9.
17.
279
6
104-
-84,
96,
99.
6.
54
0
0
-7, 8, 9, 10.
18.
309
1
4|-
-80,
81,
35.
7.
60
5
7l
-8, 9, 10, 11.
19.
600
10
101-
-77,
72,
121.
8.
86
14
9'
-9, 10, 11, 12.
20.
793
15
6i-
-70;
64,
18.
9.
43
6
11;
-10, 11, 12, 7.
21.
72
5
6}-
-56,
63,
16.
10.
37
18
1;
-11, 12, 5, 9.
22.
68
7
84-
-48,
50,
144.
11.
5
17
5
-12, 6, 7, 10.
23.
81
19
02-
-42,
44,
132.
12.
3
12
9*
-7, 9, 4, 5.
24.
69
2
7g-
0«
-36,
40;
33.
67.
Divisors of Two or more Places.
-Ex.— Divide ,£93, 15s. 9|d.
among 43 persons.
Rule.— Divide each name
m order by the divisor, be-
ginning at the highest ; and
carry each remainder to the
next lower name.
43)93 15
_86
y
20
)155 s.
129
9|(2 3
*»* The 40 farthings over are written in the quotient with the divisor below
tham, as &
EXERCISE III.
^29, 37, 53, 71, 83.
^19, 41, 67, 86, 91.
f-52, 23, 47, 95, 13.
f-124, 213, 352, 793, 61.
^225, 538, 401, 191, 17.
M15, 116, 237, 73, 85.
-372, 416, 509, 1000, 1937.
f-562, 57, 829, 900, 2340.
^1256, 4073, 236, 800, 158.
^721, 1356, 2943, 673, 78.
7-2905, 7238, 825, 34, 304.
T-59, 97, 652, 8905, 4005.
1.
£567
10
3i
2.
734
18
5
3.
392
15
4J
4.
78
2
11-
6.
27
18
o;
6.
115
0
10J
7.
1897
14
3!
8.
2700
18
0^
9.
8035
17
5;
10.
5682
11
3,
11.
73582
14
7
12.
290732
9
li
COMPOUND DIVISION.
(J3. Fractional Multipliers.
Ex. — What cost; 8} packages if 1
package cost £5, 17s. 9|d. ?
Multiply first by the fraction (}), then
by the whole number (8). Add the
products.
^
V%££0£1
£5 17 91
_ _
4)17 13 3|
3|
2
4
47
EXERCISE IV.
£51 10
1. £7, 10, 3£ x71,9J, 11J.
2. £14, 15, 7|x4f, 6|, 8£.
3. £24, 19, 3 x 15i, 27|, 36?
4. £71, 5, Il£x49f, 84 j, 10(
5. £91, 15, 6£ x 73J, 591, 91 j
6. £256, 14, 10x291 13 J, 681
7. £509, 8, 3} x 231 J, 4501, 6713 .
8. £891, 11, 1^x3071,5931, 713J.
Fractional Divisors.
Ex.— It 17} yards cost £9, 18s. lOjd., what is that per
yard?
We have to divide the 17} £9 18 10 J
whole price by the num- 4 4
ber of yards to get the 71
price of one yard.
Multiply both divisor
and dividend by 4 to re-
move the fraction from
the divisor.
)39 15
20
795
71
85
71
14
12
)174
142
32
4
)128
71
57f.
EXERCISE V.
1. £7, 10, 11J^5.J, 6
2. £11, 14, 5JH-81, 1
3. £29, 5, Oi-r-18i, 2
4. £36, 7, 2|-^2l|, 87J, 52J. 8. £643, 0, 5^83^ 173J, 824J.
Money Divisors.
" Ex.— How Dften is £5, 13s. 6jd. contained in £39, 14s. 7|d. ?
62 MONEY.
Rule. — Reduce divisor and dividend to the same name, and
proceed as in simple division —
£39, 14, 7|-~£5, 13, 6|=38143f.-j-5449=7.
EXERCISE VI.
%* To be performed after reduction has been learnt.
1. £27, 17, 3^-£6, 3, 10. 6. £63, 8, Of -f-£21, 2, 8|.
2. £137, 8, 9-i-£8, 19, 4j. 7. £671, 10, 1— £47, 19, 3.i
3. £361, 2, 9f-7-£72, 4, 6f . 8. £268, 10, 3-4-£100, 9, lOj.
4. £2090, 0, 7j-r-£81, 0, 91. 9. £675, 19, 3£-f-£75, 2, U
5. £459, 18, 2^-£24, 17, 8J. 10. £870, 0, 5£-r-£39, 18, 5J.
70. REDUCTION.
MONEY OF ACCOUNT TABLE I.
From a Higher to a Lower Name.
Ex. — In £7, 13s. 3|d., how many farthings ?
We cannot change this sum to farthings by one step, as it is
too large ; we must therefore do it in parts, changing first the
pounds to shillings, then the shillings to pence, then the pence
to farthings.
Thus, to change the pounds to
shillings, since there are 20/ for £ s. d.
every pound, there will be 20 7 13 3j
times as many shillings as 20
pounds ; multiply 7 by 20, and "153" 3| 8h. in the sum.
add the 13/ already in the sum, 12
making 153 sh jggg- j penee ln the aum.
To change the shillings to 4
pence, since there are 12d in >.„,,„
every shilling, there will be 12 7359
times as many pence as shil-
lings ; multiply 153 by 12, and add the 3d. already in the
sum, making 1839d.
To change the pence to farthings, since there are 4 farth. in
every penny, there will be 4 times as many farthings as pence ;
multiply 1839 by 4, and add the 3 farth. already in the sum,
making 7359f. in all.
Rule. — Multiply each name, in order from the highest, by
the number of the next lower which it contains, adding to each
product the number of the lower in the given sum.
REDUCTION. 63
The process of changing from one name to another is called
Reduction.
The result may be proved by changing back the farthings to
pounds ; dividing by the same numbers by which we have multi-
plied. If ,£7, 13s. 3jd., when changed to farthings gives
7359f., 7359 farthings, when changed to pounds, must give
£7, 13s. 3|d. (See sect. 71.)
EXERCISE I.
1. How many farthings in l|d., l*d., Ifd., 2d., 2}d., etc., to 12d. ?
2. How many pence in 1/1, 1/2, etc., 2/1, 2/2, etc., to 20 ?
3. How many shillings in £1, Is. ; £1, 5s., etc. ; £2 ; £2, 7s ; £10 ?
EXERCISE II.
(1.) To pence— £75; £352; £1001; £2450; £23, 10s; £179, 17s. ;
£305, 19s. ; £5024, 15s. ; £734, 17s. 4d. ; £809, 10s. 8d. ; £2702, Os. lid. ;
£6304, Is. 7d.
(2.) To halfpence— 5/, 7/, 13/, 8/2, 18/3, 14/7^, 53/8^, £15, £23, 17s.,
£27, 9s. 10d., £150, Os. 7£d., £207, 19s. 0*d.
(3.) To farthings-4/, 9/, 24/, 37/, 3/4*, 11/9J, 19AJ, 15/Ofc 29/10J,
72/8£, 13/9|, 194/0*.
(4.) To farthings—
1. £93. 5. £39, 17. 9. £4, 17, 10. 13. £922, 10, OA.
2. £201. 6. £125, 8. 10. £172, 0, QL 14. £507, 19, 11£.
3. £485. 7. £709, 10. 11. £250, 0, Oj. 15. £1854, 0, 3.
4. £7392. 8. £4890, 19. 12. £793, 15, 11J. 16. £3000, 10, 10J.
71. From a Lower to a Higher Name.
Ex. — To what sum of money are 37227 farthings equivalent ?
Here we have to change the farthings to the highest name.
We cannot do this at one step, as the number is too large ;
we must therefore do it by several steps, first changing thj
farthings to pence, then the pence to shillings, then the shil-
lings to pounds, thus : —
To change for the far- 4 | 37227
9306 j = pence in the sum.
77(5 6 j = shillings, etc. in sum.
things to pence, since it
takes 4 farthings to make
1 penny, there be only — — „
one-fourth as many pence ^38 15 6 i=P°unds, etc. in sum.
as farthings ; which is got
by dividing the number of pence by 4, giving 9306|d.
To change the pence to shillings, since it takes 12 pence to
make 1 shilling, there will be only one-twelfth as many shil-
lings as pence ; which is got by dividing by 12, giving 775s.
64 MONEY.
To change the shillings to pounds, since it takes 20 shillings
to make 1 pound, there will be only one-twentieth as many
shillings as pounds ; which is got by dividing by 20, giving
^38, 15s. 6|d.
Rule. — To change a sum of money from a lower to a higher
name : — Divide by the number of the lower contained in the
next higher, and so on till the required name be reached.
The result may be proved by changing back the pounds,
shillings, and pence to farthings. If 37227f., when changed,
give £38, 15s. 6|d., so must ,£38, 15s. 6|d., when changed back
again to farthings, give 37227f.
EXERCISE III.
1. How many pence in 4 f. 5, 6, 7, etc., to 48 f.
2. How many shillings in 12d., 13d., etc., 24d., 25d., etc., to 240d.
8. How many £ in 20/, 40/, etc., 21/, 22/, etc., 30/, 31/, etc., to 200/.
EXERCISE IV.
1. To shillings from farthings— 912, 1344, 1680, 2352, 737, 501,
1079, 1893, 600, 903, 1807, 2356.
2. To shillings from halfpence— 360, 432, 552, 768, 247, 301,
423, 593, 827, 1327, 1613, 2597.
3. To pounds from pence— 6480, 2376, 4800, 11040, 35721,
60089, 23459, 45930, 49087, 780923, 56421, 93000.
4. To pounds from farthings— 23496, 39408, 45082, 69857, 289508,
543306, 60085, 932092, 1000000, 2456793, 4560000, 5369480.
COINS IN CIKCULATION TABLE II.
From a Higher to a Lower Name.
Ex. 1. — In £7, how many half-crowns ?
Since there are 8 half-crowns in £l, £7 will have 8 times
as many half-crowns, that is 7x8 or 56 half-crowns.
Ex. 2. — In 8 florins, how many groats ?
Since there are 6 groats in 1 florin, 8 florins will contain six
times as many groats, that is, 8X6, or 48 groats.
EXERCISE V.
1 £ to cr. and fl.— £75 ; £37, 15 ; £114, 10 ; £204, 5 ; £493, 10 ; £500.
2 £ to halfcr. and sixp.— £83, 7/6 ; £52, 2/6 ; £94, 15 ; £173, 12/6 ; £79.
8 £ to gro. & threep.— £13 ; £28, 10 ; £47, 18/6 ; £52, 10 ; £73, 8, 8.
i. Half-sovereigns to halfcr. and shillings— 59, 107, 293, 408, 96, 315.
73.
REDUCTION. 65
EXERCISE VI.
1. Crowns to half-crowns and sixp.— 345, 201, 793, 1248, 930, 300.
2. Crowns to shillings and groats— 410, 293, 548, 702, 1564, 2738.
3. Half-crowns to shill. and threep.— 450, 379, 901, 763, 1001, 2100.
4. Half-crowns to sixpences and pence— 93, 58, 176, 290, 315, 728.
EXERCISE VII.
1. Florins to shillings and groats— 345, 290, 1000, 1293, 5681, 1807.
2. Shillings to sixpences and threep.— 195/6, 37/6, 87/6, 27/6, 45/, 105/.
3. Shillings to groats— 19/4, 25/8, 37/4, 56/4, 93/8, 82/4.
4. Shillings to threepences -63/9, 70/6, 82/3, 29/9, 71/6, 42/3.
EXERCISE VIII.
1. Sixpences to pence and halfpence— 378, 290, 573, 900, 1856, 2073.
2. Groats to halfp. and farthings— 250, 316, 843, 569, 1789, 3476.
3. Threepences to pence and farthings— 73, 101, 1236, 578, 1936, 2001.
From a Lower to a Higher Name.
Ex. 1. — In 375 florins, how many pounds ?
Divide the number of florins by 10, since there will be only
one-tenth as many florins as pounds ; giving ,£37, 10s.
Ex. 2. — In 720 pence how many crowns ?
Divide the number of pence by 12 to bring it to shillings,
and the shillings by 5, since there will be only one-fifth as
many crowns as shillings ; giving 12 crowns.
EXERCISE IX.
1. Halfpence to groats and shill. :— 496, 728, 916, 236, 1020, 2000.
2. Pence to threep. and sixp. :— 240, 324, 825, 113, 1562, 8249.
3. Farthings to groats :— 960, 376, 420, 810, 1256, 9000.
4. Threepences to shillings :— 240, 813, 190, 1000, 2483, 9267.
EXERCISE X.
1. Shillings to florins and crowns— 324, 290, 732, 1000, 2736, 5028.
2. Groats to shill. and half-crowns— 298, 728, 1000, 2564, 4916, 952.
3. Pence to florins and crowns— 934, 960, 2562, 8426, 3560, 5240.
4. Farthings to sixp. and florins-2456, 8400, 3000, 5249, 738, 7004.
EXERCISE XI.
1. Florins to crowns and half-sovs.— 1248, 4000, 1214, 793, 501, 910.
2. Sixp. to half-crowns and sovs.— 317, 819, 1584, 4008, 704, 3048.
3. Halfpence to florins and sovs.— 726, 8400, 906, 834, 2894, 5000.
4. Threepences to half-guineas- 493, 724, 840, 1000, 4934, 1960.
6. Farth. to groats and guineas- 8400, 9346, 7245, 2309, 6451, 8243.
74.
75.
66 MONEY.
Coins not contained in each other."
Ex. 1. — In 36 crowns how many florins ?
Change the crowns to shillings, and then the shillings to
florins, giving 36*5> or 90 florins.
2i
Ex. 2. — In 9 groats, how many threepenny-pieces ?
Change the groats to pence, and the pence to threepenny-
9x4
pieces, giving — '- — or 12 threepenny-pieces.
3
Rule. — Change the given name first to some lower name,
which is contained evenly in the name required.
EXERCISE XII.
1. Groats to threepenny-pieces— 192, 252, 972, 396, 468. 2076.
2. Threepences to groats— 708, 324, 96, 4782, 725, 589.
3. Crowns to florins— 200, 370, 630, 1000, 484, 1297.
4. Florins to crowns— 450, 995, 857, 500, 21170, 5000.
5. Half-crowns to florins— 120, 840, 1000, 380, 296, 2483.
6. Florins to half-crowns— 660, 3000, 1750, 475, 793, 215.
7. Pounds to guineas- 621, 793, 800, 1750, 2000, 576.
8. Guineas to pounds— 347, 725, 240, 2000, 152, 937.
EXERCISE XIII.— MISCELLANEOUS.
1. A owes me £72, 19s. 3d. ; B £192, 16s. 9fd. ; c £258, 10s. Oid.
I have goods worth £174, 16s, 4d., and in the bank, £62, 18s. 7d. :
what am I worth in all ?
2. My butcher's bill is £7, 19s. 6d. ; my baker's, £9, lls. 5£d. ;
my grocer's, £11, 15s. Ofd. ; my green grocer's, £1, 17s. 6d. ; my
shoemaker's, £1, 5s. 6£d. ; and my tailor's, £2, Os. ll^d. : what sum
will pay the whole ?
3. A house has three storeys, of which the rent of the first is
£60, 10s. ; of the second, £42, 7s. 6d. ; and of the third, £25, 5s. :
what is the entire rent ?
4. What was lost on a cargo which cost £1749, 14s. 6d., and sold
for £1393, 2s. lOf d. ?
5. A workman's weekly wage is 33/, what must he spend to save
4/9 a week ?
6. If I send my servant to pay an account of 17/6 for bread, 9/8 for
butter, 2/4 for eggs, and 3/9| for vegetables, and give him a five-pound
note, what should he bring back ?
7. What cost 27 yards silk at £1, 13s. 5£d. per yard ?
8. What cost 36 tb tea at 3/9^ per Ib ?
9. What cost 11 sheep at £1, 18s. 5d. each ?
10. What cost 16 stones sugar at 7/10% per stone ?
11. What cost 21 tons iron at £3, 7s. 3|d. per ton ?
12. What cost 15 oz. silver at 5/3 1 per ounce ?
13. If 39 yards cloth cost £1, 17s. 4*>d., what is that per yard ?
14. If 26 Ib tea cost £4, 16s 5d.« what is that per ft }
MISCELLANEOUS EXERCISES. 67
15. If 1 cwt. of sugar (112 Ib) cost £2, 13s. 8<L, what is that per tb?
16. If a quarter of wheat (32 pecks) cost 52/, what is that per
peck?
17. If a cask of wine (140 gallons) cost £116, 13s. 4d., what is that
per gallon ?
18. If 7 doz. sherry cost £11, 4s., what is that per bottle ?
19. A gentleman gave 6d. each to a number of poor persons : how
many would he relieve with £100 ?
20. If I hold in my hand one of all the coins in use, and add a
guinea to them, how much have I ?
21. What is the annual income of an art-union which has 963 guinea
subscribers ?
22. A ploughman's wages are 5 guineas a quarter ; he receives also
a yearly allowance of £6, Os. 3d. for oatmeal, and 40/ for potatoes :
what are his yearly wages ?
23. A family finds its monthly account with the grocer as follows :
— 1 R> tea at 4/ ; 6 lb sugar at 5£d. per lb ; 3^ lb soap at 4d. per ib ;
1 lb soda at Id. ; 4 lb butter at Is. 3d. per lb ; 4 lb cheese at lid. per
ft) ; 1 lb currants at 6d. ; 1 Ib raisins at 8d. ; £ lb almonds at 4s.
per lb : what is it in all ?
24. A draper has in bank £39, 14s. 6d. ; goods to the value of
£136, 15s. O^d. ; and credits for £53, Os. ll|d. ; but he owes £47,
16s. 2d., and his bad debts amount to £11, 4s. 3$d. : what is he
worth?
25. A merchant borrowed £700 ; he has paid three instalments of
£150 : what does he still owe, allowing £33, 5s. for interest ?
26. What cost 25 hogsheads beer at £4, 2s. 8d. each ?
27. What cost 33 bales cotton at £7, 15s. 9£d. each ?
28. What cost 58 yards cloth at 18/9 £ per yard ?
29. If a year's wage is £114, what is that per week and per day?
30. If 2 chests tea, each 40 lb weight, cost £16, 3s. 4d., what is that
per lb ?
31. If 19000 cubic feet gas cost £5, Is. 4d., what is that per 1000
cubic feet ?
32. How much has a tradesman drawn during the day, who finds in
his drawer at night 1 sovereign, 3 half-sovereigns, 7 crowns, 9 half-
crowns, 3 florins, 38 shillings, 14 sixpences, 9 groats, 6 threepennies,
with 66 id. in copper ?
33. A Frenchman, about to travel in England, changes 7000 francs
into English sovereigns : how many does he receive at 25 francs for
one sovereign ?
34. The money raised in a penny subscription, which had 12936
names, was divided into three equal shares : what was the amount of
each?
35. A tradesman draws on Monday, £2, 13s. 5d. ; on Tuesday,
£1, 19s. 7|d. ; on Wednesday, £2, Os. 9^d. ; on Thursday, £1, 15s. 3£d.;
on Friday as much as on Wednesday and Thursday together ; and on
Saturday twice as much as on Wednesday : what were his week's
drawings ?
36. A merchant bought tea for £259, 19s. 3d., sugar for £192,
Os. 5|d., and coffee for £207, Us. 6d. : what must he sell the whole
for to gain one-fourth of what he paid for them ?
37. I have 65 guineas, and my friend £60 and 60 crowns : what ii
the difference between us ?
68 MONEY.
38. A sum_of £2765, 10s. is to be divided between A, B, and c ; A
gets £459, 15s., B twice as much : what remains forc.'s share ?
39. I exchange 82 cwts. sugar at £5, 2s. llgd. per cwt. for 19 chests
tea at £23, 15s. 6^d. : how much should I pay besides ?
40. If 1 have put £9 into the post-office savings'-bank during a half-
year, what have I saved per week ?
41. A house rented at £62, 10s. sold for £1125 : how many years'
purchase was that ?
42. A farm of 73| acres is rented at £138, 5s. 7 Ad. : what is that
per acre ?
43. The old pound Scots was 1/8 : how many in £250 ?
44. I paid an account with 25 half-guineas, 25 half-sovereigns, 25
half-crowns, and 25 sixpences : what was its amount?
45. The visitors to a menagerie were 153 at 2/6, 439 at I/ with 52
at half-price, 736 at 3d., and 237 scholars at 2d. each : how much money
did it draw in all ?
76. EXERCISE XIV.
1. The receipts of a railway for the first week of February were
£2075, 16s. 2d. ; for the second, £2192, 19s. 8id. ; for the third.
£1989, Os. P|d. ; and for the fourth, £2530, 17s. : what were its ave-
rage weekly receipts for the month ?
2. A bankrupt paid 5/3.^ per pound on a debt of £425, what was his
estate ? and how much does a creditor lose on a claim of £37 ?
3. Bought 45 railway shares at £23, 10s., and sold out at £25,
16s. 6d : what did I gain?
4. At a collection s* a church door there were in the plate 375d.
749 halfpence, 45 groats, 28 threepences, 7 sixpences, and 3 shillings :
how much in all ?
5. I bought a book in 3 volumes at half-a-guinea a volume ; dis-
counting £ of the price, what did I pay ?
6. A dinner-bill for 23 persons came to £8, 12s. 6d. ; if five were
guests, what had each of the others to pay ?
7. A draper bought 37 yards cloth at 7/9^ per yard ; if he gained
30/, what did he sell it at per yard?
8. Bought oxen and lambs for £193, 17s. 6d. ; if the oxen cost
double of the lambs, what cost each?
9. A factory consumes 11 tons coal per week at 9/7^ per ton, what
is its annual outlay for coal ?
10. In 93 American dollars ($1 = 4/2), how much sterling money?
11. A public work employs 25 labourers at 13/6 a week, and 15 at
15/9 : what sum is expended annually in wages ?
12. If an apprentice's wages are 4/6 a week the first year, and are
advanced 1/6 a week each of his five years' service : how much does
he receive in all ?
13. What is my clear income, if I am assessed 3/4A per pound on
£375?
14. If a tradesman's wages are £95 per annum, what should be his
daily expenditure to save £10 a year ?
15. A farmer's profits for 1860 were £407, 11s. 6d. ; for 1861,
£493, 2s. 8d. ; and for 1862, £430, 3s. lOd. : how much does the increase
for 1861 exceed that for 1862 ?
MISCELLANEOUS EXERCISES. 69
' 16. Bought 38f yards at 17/6J per yard ; retaining 5J yards, I sold
the rest so as neither to gain nor lose ; how did I sell it ?
17. The stipend of 153 clergymen is £150 each, but there is a fund
of £4082, 15s. available for equal distribution among them : to what
does that bring up the stipends ?
18. If the amount of deposits in a savings'-bank is £15645, 14s. 3d.,
and the number of open accounts 935 : find the average amount of each.
19. Divide a legacy of £3000 among 3 sons and 4 daughters, so that
each son shall receive twice as much as each daughter.
20. If 2500 persons cross Waterloo Bridge daily, paying a toll of
id. : how much is raised yearly ?
21. My bank-book for April shows these entries— April 3, paid in
13/6 ; April 9, paid in 7/10 ; April 16, received 7/3 ; April 23, paid in
5/6 ; April 30, received 14/9 : find the increase to my credit forthe month.
22. What is the annual saving to the owners of a factory employ-
ing 550 hands, if wages fall three-halfpence a day ?
,
23. My gas account for the last quarter was 7500 cub. feet at 4/7J
per 1000 : what had I to pay ? and what will I save next year, if I
burn 1500 cub. feet less each quarter?
.
24. A grocer mixes 12 gallons whisky at 18/6 with 18 at 16/6,
and 15 at 14/6 : find the value of the mixture per gallon.
25. I hold a sum of money, consisting of 5 five-pound notes, 25
sovereigns, 25 half-sovereigns, and 16 half-crowns, of which three-
fifths belong to a friend ; how much is mine ?
26. A merchant who began business with a clear capital of £2396,
15s. 6d., increased it by one-third for three successive years : what
was it at the end of that time ?
27. A, B, and c subscribe to a venture of £7260 in 10 shares. A has
2, B 3, c 5 shares : what did each subscribe ?
28. A merchant bought 7 chests tea, each 48 Ibs., for £73, 8s. 3d.
Three of the chests he sells at 4/6 per Ib : what must he sell the rest
at to gain £4, 18s. on the whole?
29. If my income is 500 gs. a year, what income-tax do I pay at 7d.
per pound, and how much more would I pay at 7§d. per pound ?
30. An hospital contains 125 boys. At the beginning of the year
16 leave, and 11 new boys are admitted : what will be its expenditure
for the year, if each boy costs £11, 17s. 4d.
31. Farmer A drove to market 8 oxen, which he sold at £15, 10s.
a head, and two score of sheep which he sold at £1, 18s. a head ; but
he bought a horse for 27 gs., and a gig with harness for 11 gs. : what
money did he bring home ?
32. Fanner B bought 23 oxen at £9, 10s. each. One of them died ;
but he fattened the rest at an expense of 25/ each, and then sold the
lot for 300 guineas : what did he gain on each ?
33. A merchant paid a bill of £257, 10s. Of that £94, 15s. was for
sugar at 5d. per lb, and the rest for tea at 5/6 per Ib : how much of
each did he buy ?
34. In a school of 350 children, the quarterly fee is 5/6 for the
first division, which contains 108, 4/9 for the second, which contains
236, and 3/ for the third, which contains the rest : how much is drawn
in fees for the quarter, and what would be gained if they were raised
6d., 9d., and I/ for the first, second, and third divisions respectively ?
35. A cheque for £63, 15s. is paid in an equal number of crowns,
half-crowns, and shillings : how many of each ?
70 WEIGHTS AND MEASURES.
WEIGHTS AND MEASURES.
• • • General Rule. — For adding, subtracting, multiplying,
dividing, and reducing weights and measures, the rules are the
same as for performing these operations with money.
Avoirdupois Weight — Table III.
EXERCISE I.
1.
2
i.
3.
4.
5.
6.
T. cwt.
qr.
cwt,
qr. Ib
qr. Ib oz.
T. cwt. qr.
cwt. qr. Tb
Ib
oz.
dr.
27 15
2
13
1
18
2 25 14
84 13
2
14 2 15
26
13
12
45 17
1
17
3
15
1 20 11
93 17
1
16 3 12
20
11
10
83 9
3
19
2
16
3 9 10
60 10
3
908
15
9
1
56 0
3
8
3
7
2 18 6
74 19
0
5 1 13
17
15
9
70 8
2
11
0
11
0 13 13
26 8
3
17 2 11
8
10
4
92 19
1
14
1
27
3 21 15
45 15
2
19 1 9
14
7
13
7.
8.
9.
10.
11.
12.
T. cwt.
qr.
cwt.
qr.
Tb
qr. Tb oz.
T. cwt.
qr.
cwt. qr. Ib qr.
Ib
oz. dr.
73 19
1
14
2
19
3 17 11
115 10
3
10 3 12 2
20
7
9
29 7
2
9
3
17
1 23 12
79 15
2
7 3 15 1
23
13
14
13. 36 tons 14 cwt. 3 qr. +17, 18, l, + 94, 10, 3 + 7, 2, 1.
14. 15 cwt. 1 qr. 27 lb + 16, 0, 20 + 8, 2, 19 + 19, 3, 23.
15. 1 qr. 20 Ib 12 oz. + 3, 14, 14 + 2, 19, 7 + 3, 24, 13.
16. 275 tons 1 cwt. 2 qr. + 193, 14, 2 + 400, 15, 2 + 640, 18, 1.
17. 13 cwt. 2 qr. 19 lb + 8, 3, 14 + 19, 2, 11 + 11, 1, 15.
18. 2 qr. 24 Ib 7 oz. 8 dr. + l, 20, 14, 12 + 3, 10, 11, 4 + 1, 8, 7, 11.
19. 157 tons 10 cwt. 2 qr.- 59, 15, 3. 22. 81 tons 11 cwt. Iqr.- 37, 14, 3.
20. 13 cwt 2 qr. 20 Ib - 9, 3, 21. 23. 7 cwt. 3 qr. 14 Ib - 4, 3, 19.
21. 3 qr. 15 Ib 12 oz. - 1, 17, 9. 24. 113 Ib 14 oz. 7 dr. - 75, 15, 10.
Note on Subtraction — Instead of saying 15 dr. from 9 dr. Tb oz. dr.
cannot be taken, change one of the oz., making 25 dr. in all ; 18 11 9
15 from 25 leaves 10 dr., and so on ; it is simpler, in this and 11 14 15
some of the following tables, to say, 15 dr. from 16 leaves 1, ~~Q 12 10
1 and 9 are 10 ; 14 oz. from 16 oz. leaves 2 oz. which with 10
oz. makes 12 oz., etc. That is to say, instead of adding one of the higher name
to the lower befo^t subtracting, we may subtract at once from one of the higher
name, and add the difference to the number of the lower in the minuend.
EXERCISE II.
1. 13 tons 14 cwt. 2qr. x 6, 15, 32i, 63.
2. 5 cwt. 3 qr. 16 Ib x 5, 21, 36|, 79.
3. 3 qr. 9 Ib 14 oz. x 11, 72, 87, 108.
4. 193 tons 19 cwt. 3 qr. 91b x 144, 172, 360.
6. 15 cwt. 1 qr. 24 Ib 7 oz. x 96, 473, 840.
6. 1 qr. 17 Ib 11 02. 15 dr. x 120, 285, 793.
WEIGHTS AND MEASURES. 71
7. 843 tons 15 cwt.
8. 25 tons 1 cwt. 1 or. 20 Ib —8, 36, 32j, 193.
9. 61 cwt. 3 qr. 14 ft
10. 173 tons 5 cwt.
11. 83 tons 15 cwt. 3 qr. 3 Ib 8 oz.
12. 10 cwt. 3 qr. 19 Ib 4 oz. 7 dr.
4, 15, 62i, 279.
•12, 99, 18$, 370.
•10, 21, 132, 562.
•84, 390, 821.
-49, 913, 770.
78
EXERCISE III.
1. 79 tons to Ib 10. 27645780 oz. to cwt.
2. 25 tons 7 cwt. 18 Ib to oz. 11. 72480 oz. to dr.
3. 19 cwt 1 qr. 21 Ib 9 oz. to oz. 12. 17250 Ib to T.
4. 2 qr. 15 Ib to dr. 13. 9456 dr. to oz.
5. 27 ft) 15 oz. 12 dr. to dr. 14. 694721 oz. to T.
6. 10 cwt. to stones. 15. 932450 dr. to cwt.
|7. 127 tons 15 cwt. 1 qr. to ft 16. 57289 oz. to qr.
8. 57 Ib to dr. 17. 123456 oz. to cwt.
9. 15 cwt. 7 Ib to oz. 18. 93000 Ib to T.
Troy Weight — Table IV.
EXERCISE IV.
1.
2.
3.
4.
5.
Ib oz.
dwt.
oz.
dwt.
gr
ft
oz. dwt. gr.
Ib oz. dwt.
oz.
dwt. gr.
125 9
15
27
15
21
29
8
14
22
193 8 13
85
14 19
27 11
19
36
11
20
36
4
12
15
59 11 15
19
15 20
98 5
10
9
14
11
8
0
2
10
6.
7.
193 10
230 0
13
11
50
30
3
10
8
0
50
79
7
11
0
7
18
8
Ib oz. dwt. gr.
193 7 13 15
ft.
96
oz. dwt. gr.
10 13 20
79 7
0
11
0
14
5
10
13
23
85 11 9 19
37
11 7 21
8. 731blOoz. 14 dwt. +94, 9, 9 + 150, 2, 11 + 8i 11, 15.
9. 11 oz. 14 dwt. 20 gr. + 19, 12, 16 + 24, 11, 22 + 30, 7, 18.
10. 453 ib 11 oz. 17 dwt. -87 Ib 11 oz. 19 dwt.
11. 285 oz. 11 dwt. 17 gr. -97 oz. 12 dwt. 23 gr.
12. 64 Ib 7 oz. 13 dwt. 17 gr. -28 Ib 10 oz. 15 dwt. 20 gr.
EXERCISE V.
1. 23 ft> 5 oz. 11 dwt. x 7, 56, 130, 257.
2. 54. oz. 13 dwt. 20 gr. x 9, 24, 560, 365.
3. 18 Ib 10 oz. 14 dwt. 22 gr. x 11, 150, 4/9.
4. 136 Ib 10 oz. 14 dwt. -f- 8, 42, 70, 192.
5. 172 oz. 10 dwt. 15 gr. — 3, 81, 92, 268.
6. 93 Ib 2 oz. 17 gr. -f- 108, 236, 807,
EXERCISE VI.
1. 10 oz. 15 dwt. to gr. 7. 5932 gr. to oz.
2. 7 Ib 8 oz. to dwt. 8. 2400 dwt. to Ib
3. 21 Ib 9 oz. 18 dwt. to gr. 9. 29324 gr. to Ib
4. 14 Ib to gr. 10. 7256 gr. to dwt.
5. 8 oz. 15 dwt. to dwt. 11. 10000 dwt. to Ib
6. 5 Ib 11 oz. to gr. 12. 9000 gr. to oz. ,
72
79.
1.
yds. ft in.
118 2 7
72 1 8
92 0 6
240 1 10
74 2 8
154 1 11
6.
237 1 8
194 2 9
WEIGHTS AND MEASURES.
Lineal Measure — Table V.
EXERCISE VII.
2.
3.
4.
ml.
93
fur.
5
po.
22
po. yds. ft.
87 4 2
fur.
82
po. yds.
24 2
118
2
15
36
2
1
25
20
4
70
0
10
59
5
2
90
1
4
120
7
11
89
0
1
61
17
3
81
6
4
94
3
2
35
10
1
59
4
17
16
4
1
70
6
2
7.
8.
9.
56
5
20
37
2
1
85
21
5
28
6
18
18
3
2
25
22
5
;o. yds. ft. in.
7 3 2 11
27 4 2 10
8518
36 2 0 5
60 1 2 9
1427
10.
17 3 2 10
8 3 2 11
11. 27 yds. 1 ft. 3 in. + 58, 2, 8 + 37, 2, 5 + 84, 1, 11.
12. 84 ml. 3 fur. 20 po. + 17, 7, 8 + 29, 5, 15 + 47, 6, 17.
13. 28 po. 3 yds. 1 ft. +54, 2,.2 + 30, 5, 2 + 25, 1, 2.
14. 19 fur. 20 po. 2 yds. +17, 10, 1+24, 15, 3 + 49, 10, 2.
15. 70 yds. 2 ft. 3 in. + 39, 1, 10 + 25, 2, 8 + 40, 1, 7.
16. 418 yds. 2 ft. 7 in. -250, 2, 11.
17. 73 ml. 2 fur. 15 po. -38, 7, 10.
18. 47 po. 3 yds. 1 ft. - 38, 1, 2.
19. 56 fur. 3 po. 2 yds. -27, 15, 3.
20. 18 po. 2 yds. 1 ft. -7, 3, 2.
21. 290 yds. 1 ft. 11 in. - 49, 2, 11.
EXERCISE VIII.
1. 37 yds. 2 ft. 8 in.
2. 90 ml. 7 fur. 24 po.
3. 82 po. 5 yds. 1 ft.
4. 25 fur. 20 po. 4 yds. 1 ft.
5. 13 po. 2 yds. 1 ft. 2 in.
6. 64 yds. 2 ft. 11 in.
7. 160 yds. 2 ft. 2 in.
8. 129 ml. 7 fur. 20 po.
9. 137 po. 4 yds. 2 ft. 6 in.
10. 68 fur. 29 po. 0 yd. 1 ft. 6 in.
11. 1 fur. 35 po.
12. 292 yds. 1 ft. 6 in.
x 6, 49, 315.
x 9, 24, 482.
x 56, 800, 493.
x 81, 720, 848.
x 18, 350, 925.
x 36, 450, 637.
-+2, 22, 118.
H-4, 15, 110.
~6, 10.4, 236.
-M3.4, 210, 375.
.5-22$, 150, 561.
4-204J, 405, 914.
1. 14 ml. 5 fur. to yds.
2. 7 fur. 25 po. to ft.
3. 4 yds. 2 ft. to in.
4. 29 po. 3 yds. to in.
5. 30 ml. 25 po. to ft.
6. 17 po. 4 yds. 2 ft. to ft.
7. 5 fur. 39 po. 2 yds. to yds.
8. 58 fath. 3 ft. to ft.
9. 13 hands 2 in. to in.
EXERCISE IX.
10.
11.
12.
13.
14.
15.
16.
17.
18.
2597 yds. to fur.
9256 po. to ml.
29738 ft. to ml.
2500 ft. to fath.
375960 yds. to ml.
593 in. to yds.
63 in. to hands.
2570085 ft. to fur.
47268 in. to po.
COMPOUND RULES. 73
80. Cloth Measure — Table VI.
EXERCISE X.
1. 2. 3. 4. 6.
yds. qr. nl. ells qr. n: yds. qr. in. ells qr. nl. yds. qr. nL
256 2 3 73 4 2 64 2 7 192 4 2 74 2 1
93 1 2 19 2 1 29 0 8* 96 4 3 47 3 2
80 3 1 156 1 3
158 0 3 90 0 2 72 1 51 yds. qr. in.
100 22 118 4 1 28 3 6i (5.) 315 2 1 (7.) 85 3 7A
56 1 3 54 2 2 58 0 7£ 196 43 47 3 8j
8. 384 yds. 2 qr. 1 nl. +79, 1, 3 + 74, 3, 2 + 17, 1, 2.
9. 97 ells, 4 qr. 3 nl.+17, 3, 2 + 39, 1, 1 + 14, 3, 2.
10. 114 yds. 3 qr. 7 in. +200, 1, 2^ + 74, 3, 4£ + 92, 1, 6J.
11. (1.) 94 ells 3 qr. 3 nl. -47, 4, 1. (2.) 173, 4, 1-85, 4, 3.
12. (1.) 83 yds. 2 qr. 2 nl. - 29, 3, 1. (2.) 92, 3, 6^ in. - 56, 2, 7J in.
EXERCISE XI.
1. 16 yds. 3 qr. 3 nl. x 6, 15, 87. 4. 17 yds. 2 qr. 2 nl.^-6, 47, 71.
2. 38 ells 4 qr. 2 nl. x 9, 24, 123. 5. 141 ellsS qr. 3 nl.^-7, 81, 156.
8. 73 yds. 2 qr. 8 in. x 12, 63, 274. 6. 74 yds. 2 qr. 6 in. 4-8, 84, 121.
EXERCISE XII.
1. 7 yds. 3 qr. to nl. 7. 756 nl. to yds.
2. 56 ells to nl. 8. 1000 nl. to ells.
3. 3 qr. 2 nl. 1 in. to in. 9. 250 in. to yds.
4. 240 ells to yds. 10. 680 yds. to ells.
5. 37 ells 3 qr. to yds. 11. 2764 nl. to ells.
6. 18 yds. 2 qr. 7 in. to in. 12. 1296 in. to yds.
81. Land Measure— Table VII.
EXERCISE XIII.
1. 2. 3. 4.
ml. fur. eh. Ik. ml. far. ch. fur. eh. Ik. ml. fur. ch. Ik.
3 7 8 50 13 6 7 6 8 70 63 7 8 70
19 5 9 64 49 2 8 5 9 60 25 7 9 83
72 1 6 36 25 3 7 3 7 54
25 6 7 90 19 7 9 7 6 26 5.
11 2 5 46 10 0 5 4 9 30 171 6 9 45
7 3 8 54 766 1 5 20 83 7 8 93
6. 8 fur. 7 ch. 60 Ik. +7, 8, 45 + 5, 3, 28 + 2, 5, 73.
7. 7 ml. 2 fur. 8 ch. 50 Ik. + 19, 7, 5, 60 + 25, 3, 9, 80.
8. 34 ml. 7 fur. 6 ch. 40 Ik. -15 ml. 7 fur. 8 ch. 90 Ik.
9. 154 ml. 2 fur. 5 ch. 85 Ik. - 76 ml. 6 fur. 5 ch. 86 Ik.
WEIGHTS AND MEASURES.
EXERCISE XIV.
ml. fur. ch. Ik. nil. fur. ch. Ik.
1. 0 7 8 45x5,36,120. 4. 0 8 0 64-4,21,56.
2. 3 2 9 60x7,96,213. 5. 15 3 9 84—8,63,123.
3. 7 6 8 56x9,132,260. 6. 16 5 9 50-10,19,235.
EXERCISE XV.
1. 150 ml. 47 ch. to Ik. 5. 2596 Ik to fur.
2. 7 fur. 3 ch. to Ik.
3. 25 ml. 70 ch. to ch.
4. 347 ml. to Ik.
6. 9000 Ik. to ml.
7. 586 ch. to ml.
8. 8256 Ik. to ch.
82. Square Measure— Table VIII.
In the surface A B c D, let its length A D be 12 inches, and
its breadth AB be 12 inches;
then the surface contains 144
parts, 1-inch long and 1-inch
broad, or, as they are called,
144 square inches. The area
is found by multiplying the
length by the breadth.
The figure A B c D, being one
foot long and one broad, is one
square foot, which measure there-
fore contains 144 sq. inches.
Any area containing 144 square inches is regarded as one
square foot ; e.g.,, a figure 18
inches long by 8 inches broad. A D F
Observe that, whilst one square
foot means one foot measured
every way, or one foot square,
any other number of square feet
does not mean the same number
of feet square. Thus, in Fig 2,
if A B c D is one square foot, and
D c E F is another part equal to
it, then the whole A B E F is two
square feet. But in Fig. 3, if the
length A F is two feet, and the
breadth AG also two feet, the
figure is two feet each way, or
two feet square, which, as we see,
contains four square feet.
COMPOUND RULES.
75
EXERCISE XVI.
1.
IMS. ro. po.
39 3 29
57 2 20
93 0 15
64 2 39
12 1 12
27 3 17
2.
ac. ro. po. yds.
172 2 34 24
85 0 27 20
276 1 11 15
93 3 7 8
57 2 18 17
190 1 22 10
3.
ro. po. yds. ft.
3 25 8 8
2 17 21 5
3 3 7 7.4
1 28 27 6j
2 30 19 4|
0 14 22 3j
4.
po. yds. ft. in.
24 20 8 47
20 18 1 98
17 5 3 87
36 27 6 79
8 9 2 24
24 25 4 110
5.
yds. ft. in.
25 7 130
18 8 94
7 6 56
30 8 104
15 5 85
9 2 118
6.
ac. ro. po.
93 2 18
47 3 20
7.
ac. ro. po. yds.
256 3 36 25
98 3 38 29
8.
ro. po. yds. ft.
3 27 21 7
1 19 26 8
9.
po. yds. ft. in.
36 25 1 93
19 26 0 100
10.
yds. ft. in.
25 7 110
18 8 129
11. 24 ac. 2 ro. 27 p. + 194, 3, 30 + 98, 1, 25+100, 1, 36.
12. 25 yds. 8 ft. 100 in. +11, 7, 94 + 56, 2, 120 + 62, 5, 85.
13. 2ro. 25 po. 26 yds. +1, 19, 13 + 3, 11, 14 + 1, 39, 29.
14. 36 po. 25 yds. 3 ft. +17, 13, 4 + 28, 30, 6 + 19, 26, 4.
15. 156 ac. 3 ro. 26 p. 7 yds. -98 ac. 2 ro. 27 po. 18 yds.
16. 258 ac. 2 ro. 0 po. 15 yds. - 89 ac. 1 ro. 5 po. 26 yds.
17. 125yds. 8 ft. 56 in. - 16 yds. 8 ft. 100 in.
18. 25 po. 30 yds. 6 ft. -16 po. 30 yds. 7 ft.
EXERCISE XVII.
1. 136 ac. 3 ro. 27 po. x 7, 24, 73.
2. 16 ac. 2 ro. 3p. 16 yds. x8, 91.4, 540.
3. 2 ro. 14 po. 25 yds. 6 ft. x 11, 84, 837.
4. 25 po. 18 yds. 3 ft. 110 in. x 35, 270, 492.
5. 7 yds. 0 ft. 93 in. x 16, 105, 308.
6. 7 ac. 3 ro. 35 po. —3, 25, 85.
7. 8 ac. 1 ro. 11 po. 21 yds. 2 ft. 36 in.— 4, 93 j, 256.
8. 1 ro. 6 po. 1 yd. 7 ft. 72 in. —12, 33, 324.
9. 22 yds. 2 ft. 6 in. —96, 300, 849.
10. 205 yds. 7 ft. 72 in. -72, 185|, 620.
EXERCISE XVIII.
1. 27 sq. yds. 8 sq. ft. 90 sq. in. to sq. in. 10.
2. 191 ac. 3 ro. 31 po. to po. 11.
3. 25 ac. 1 ro. to sq. yds. 12.
4. 75 sq. ml. to ac. 13.
5. 84 ac. to sq. Ik. 14.
6. 19 p. 25 yds. 8 ft. to sq. in. 15.
7. 101 ac. 27 per. to sq. yds. 16.
8. 93 ac. 1 ro. 21 per. 9 yds. to sq. yds. 17.
9. 7 sq. yds. 8 sq. ft. 120 sq. in. to sq. in. 18.
50000 sq. in to sq. yds.
97326 sq. yds. to ro.
858 po. to ac.
29682 sq. ft. to ro.
8256000 sq Ik. to ac.
84720 sq. in. to po.
27384 sq. yds. to per.
139286 sq.in. tosq.yds.
76536 ac. to sq. ml.
Cubic or Solid Measure — Table IX.
A piece of wood or brick 1 inch long, 1 inch broad, and 1
inch thick (or deep), that is, 1 inch every way, is called 1 cubic
TO WEIGHTS AXD MEASURES.
inth. If in Fig. 1, sect. 82, each square inch had 1 cubic incK
placed upon it, the square foot would contain 144 such inches ;
another layer would make 2 times 144, a third 3 times 144,
and so on, till 12 layers would give 12 times 144 such inches,
or 1728. But 12 such layers would reach 12 inches, or 1 foot
high, and the figure would now be 1 foot every way, or 1 cubic
foot, which therefore contains 1728 cubic inches. Similarly, u
1 yard contain 3 feet, 1 cubic yard will contain 3 feet every
way ; that is, 3x3x3, or 27 cubic feet.
EXERCISE XIX.
1. 2. 3. 4. 5.
c. yd. c. f. c. in. yds. ft. in. yds. ft. in. yds. ft. in. yds. ft. in.
7 13 356 136 16 460 36 9 300 173 20 892 250 17 800
8 26 938 282 9 200 42 8 154 49 24 900 92 18 948
5 15 701 325 8 154 29 18 408
6 10 1263 482 21 938 54 24 293 6. 7.
9 7 1564 254 17 1628 18 17 567 538 25 1130 814 19 710
4 22 842 123 4 801 62 6 482 299 26 1628 298 20 1260
8. 248 c. yd. 14c. ft. 309 c. in. + 159, 24, 560 + 78, 15, 914, + 82, 3, 284.
9. 19 yd. 16 ft. 847 in. +34, 19, 936 + 22, 23, 1000 + 36, 19, 100.
10. 247 yd. 19 ft. 560 in. - 198, 24, 700. 11. 72, 25, 1608 - 27, 26, 1700.
EXERCISE XX.
c. yd. ft. in. c. yd. ft in.
1. 17 21 1500x6, 56, 13i, 156. 4. 26 2 1488-3, 7i, 87, 420.
2. 23 24 900x9, 84, 26.J, 632. 5. 160 2 1584-5, l2i, 36, 191.
3. 79 11 372 x 132, 720, 365, 800. 6. 92 24 576-56, 120, 43, 321.
EXERCISE XXI.
1. 69 cub. ft. to cub. in. 9. 2576 cub. ft. to cub. yd.
2. 75 cub. yd. to cub. ft. 10. 14850 cub. in. to cub. ft.
3. 291 c. yd. 19 c. ft. to c. in. 11. 235 cub. ft. to B.b.
4. 17 lo. rough to cub. ft. 12. 1250 cub. ft. to lo. rough.
5. 189 T. sh. to cub. ft. 13. 6728 cub. ft. to T. sh.
6. 34 lo. hewn to cub. ft. 14. 9362 cub. ft. to lo. hewn.
7. 129 B.b. to cub. ft. 15. 795 cub. ft. to B.b.
8. 457 T. sh. to cub. ft. 16. 15728 cub. in. to cub. ft.
84. Measure of Capacity— Table X.
5.
qrs. Ira. pk. gal
114 6 3 1
84 4 2 0
74 7 0 1
56 5 3 1
90 1 2 0
28 6 3 1
EXERCISE XXII.
1.
2.
3.
4.
qts.
pt.
gj.
galls.
qt.
pt.
bu.
pk.
gall.
qrs.
1m.
pt
17
1
3
48
3
1
120
3
1
78
7
3
6
0
2
27
2
1
59
2
0
84
5
1
9
1
3
36
0
1
78
3
1
93
6
2
7
1
1
42
3
0
24
1
1
27
I
2
14
0
2
20
2
1
143
3
1
56
4
3
13
1
3
7
3
0
84
2
0
49
2
3
C'UMrOUiN'D itULES. 7 i
6. 7. 8. 9. 10.
qts. pts. gt. galls, qts. pts. bu. pk. gal. qrs. bu. pk. qrs. bu. pk. gal.
42 02 161 3 0 130 2 1 275 62 314 5 2 1
27 1 3 89 3 1 65 3 1 118 7 3 283 6 3 1
11. 94 galls. 3 qr. 1 pt. 2 gi. + 47, 2, 0, 3 + 84, 3, 1, 2.
12. 82 bu. 2 pk. 1 gall. + 56, 1, 1 + 70, 3, 1 + 62, 2, 0.
13. 156 qrs. 7 bu. 2 pk. 1 gall. +273, 6/1, 0 + 193, 4, 3, 1.
14. 83 gal. 1 qt. 0 pt 1 gi. -65, 1, 1, 3. 16. 72 galls. 3 qt. -25, 3, 1.
15. 64 bu. 1 pk. 1 gall. - 38, 2, 1. 17. 125 qrs. 6 bu. 2 pk. 1 ga. - 84,7,3.
EXERCISE XXIII.
1. 25 galls. 2 qt. 1 pt. x5, 14J, 36, 93.
2. 64 bu. 3 pk. 1 gall. x 7, 24|, 63, 124.
3. 5 qr. 4 bu. 2 pk. x 9, 42, 85, 250.
4. 9 qts. 1 pt. 3 gi. x!2, 70, 101, 339.
5. 156 qrs. 4 bu. 3 pk. 1 gall, x 11, 108, 700, 413.
6. 52 galls. 2 qt. 4-6, 15, 84, 91.
7. 229 bu. 2 pk. -$-4, 27, 17, 130.
8. 508 qrs. 6 bu. -f-8, 55, 37, 185.
9. 2569 qts. 1 pt. 3 gi. 4-3, 77, 89, 400.
10. 1855 qrs. 7 bu. 3 pk. 1 gall.-f-7, 71, 239, 372.
EXERCISE XXIV.
1. 5 galls. 3 qts. to pts. 10. 2572 gi. to qts.
2. 2 qts. 1 pt. 3 gi. to gi. 11. 593 pts. to galls.
3. 14 pk. 1 gdl. to pts. 12. 1876 qts. to pks.
4. 7 bu. to galls. 13. 705 galls, to bu.
5. 12 qrs. 6 bu. to pks. 14. 193 pks. to qrs.
6. 13 galls. 1 pt. 2 gi. to gi. 15. 628 pts. to pks.
7. 9 pks. 1 gall. 3 qts. to qts. 16. 3000 gi. to galls.
8. 5 bu. 3 pks. to galls. 17. 484 pks. to bu.
9. 284 qrs. 5 bu. 3 pks. to galls. 18. 1608 galls, to qrs.
35. Measurement of Time— Table XI.
EXERCISE XXV.
1.
2.
3.
4.
5.
ho. min.
sec.
da.
ho.
min.
•wk. da.
ho.
yrs.
\vk. da.
ho.
min.
sec.
19 41
50
84
10
30
41 6
20
28
10 4
18
48
42
3 25
45
150
14
24
25 4
10
94
30 6
27
13
21
17 9
11
79
20
18
43 1
9
15
45 3
9
35
17
10 30
8
148
13
49
9 3
18
8
24 1
24
8
9
9 6
29
64
8
27
17 5
15
49
36 5
19
51
37
12 58
37
293
15
11
50 4
16
11
9 4
36
29
1
6. 7. 8. 8. 10.
ho. min. sec. da. ho. min. -wk. da. ho. yrs. wk. da. ho. min. sec.
36 41 29 93 12 54 36 4 12 152 41 4 27 32 18
19 50 30 29 17 58 18 6 19 76 49 6 8 45 30
78
WEIGHTS AND MEASURES.
11. 11 yr. 27 wk. 6 da. +25, 36, 4 + 30, 40, 3 + 15, 26, 1.
12. 5 ho. 40 min. 36 sec. +20, 36, 51 + 11, 25, 0 + 17, 0, 54.
13. 27 da. 14 ho. 46 min. +93, 10, 0, 31 + 87, 0, 0, 47 + 59, 0, 10.
14. 21 ho. 36 min. 14 sec. -19 ho. 45 min. 20 sec.
15. 93 yr. 28 wk. 4 da. - 57 yr. 36 wk. 6 da.
16. 13 ho. 46 sec. - 9 ho. 11 min. 17. 17 yr. - 8 yr. 10 wk. 3 da. 1* sec.
EXERCISE XXVI.
1. 3 ho. 42 min. 7 sec.
2. 17 wk. 3 da. 11 ho.
3. 64 yr. 27. wk. 1 da.
4. 24 da. 20 ho. 16 min.
5. 13 wk. 1 da. 19 ho.
x7, 8.i, 19, 73.
x3, ISA, 47, 132.
x 10, 132, 400, 89.
x8, 63, 120, 247.
x 11, 42, 160, 303.
6. 75 yr. 8 wk. 4 da.
7. 15 ho. 36 min. 9 sec.
8. 36 wk. 5 da. 13. ho.
9. 250 da. 14 ho. 58 min.
10. 293 yr.
—5, 24, 19, 31.
—9, 42, 220, 93.
—6, 7i, 84, 140.
—11, §i, 34, 215.
—12, 108, 71, 168.
1. 29 wk. 6 da. to ho.
2. 25 yr. 79 da. to da.
3. 3 wk. 5 da. 19 ho. to min.
4. 23 ho. 48 min. to sec.
EXERCISE XXVII.
9.
10.
11.
12.
5. 39 wk. 5 da. to da. 13.
6. 115 yr. 140 da. to ho. 14.
7. 14 ho. 36 min. 50 sec. to sec. 15.
8. 7 yr. 10 wk. to wk. 16.
29650 sec. to ho.
78928 min to da.
5620 ho. to wk.
1795 da. to yr.
805 wk. to yr.
290786 ho. to yr.
8349250 sec. to da.
93078 min. to wk.
86. Angular Measure— Table XII.
In the circle (see Fig. A B c D), the whole circumference is
supposed to be divided into 360
equal parts, called degrees (°), each
of which is divided into 60 minutes
('), and each minute into 60 seconds
("). Each quarter A B, B c, CD, and
D A, contains therefore 90 degrees.
The angle BOG has the same mea-
surement as the quarter of the circle
(B c) opposite to it ; it is 90°, and is
called a right angle. It is the angle at which a wall stands to
the ground. If the angle E o c be half of B o c, it is 45° ; as is
the arc E c.
A familiar example of measurement on the circle is the
reckoning of the position of places on the earth's surface by
latitude and longitude.
MISCELLANEOUS EXERCISES. 79
EXERCISE XXVIII.
1. 2. 3.
42<> 49' 57"
36° 28' 29"
17° 19' 36"
28° 15' 25"
93° 50' 10"
57°
260
18°
39°
19°
29'
13'
47'
21'
15'
45"
29"
50"
36"
42"
72°
48°
28'
39'
46"
49"
37°
19°
19'
20'
25"
36"
41°
9°
5.
26'
38'
37"
42"
28°
18°
6.
0'
7'
17"
29"
87.
7. 24° 36' 49" + ll° 25' 43" + 29° 37' 59"+45° 28' ll" + 37° 41' 5"
8. 36° 18' 24" + 56° 49' 10" + 1° 18' 11" 4- 40° 5' 17" + 71° 53' 19"
9. 72° 45' 54" - 38° 53' 59" (10.) 51° 24' 27" - 27° 36' 48"
EXERCISE XXIX.
1. 36° 27' 36" x 6, 21, 29, 97. 4. 186° 24' 15"^5, 33, 12|, 83.
2. 24° 38' 42" x 8, 45, 82.i, 130. 5. 829° 30'^7, 108, 200, 158.
3. 55° 10' 28" x 11, 37, 59£, 242. 6. 732° 35' ^4, 150, 149, 236.
EXERCISE XXX.
1. 37° to " 7. 56280" to °
2. 48° 50' to " 8. 2794' to °
3. 910 42' 25" to " 9. 324000" to L's
4. 1 © to ; 10. 6896' to o
5. 73° to ' . 11. 4287' to °
6. 2 L'S to ' 12. 98756" to °
EXERCISE XXXI.— MISCELLANEOUS.
1. A ship delivered a cargo of 18 qr. 7 bu. 3 pk. barley ; half as much
again of flour ; and as much wheat as both barley and flour together :
how many quarters were there in all ?
2. A railway has three stations. The first is 3 ml. 5 fur. 21 po. 3 yds.
from the terminus ; the second half as much from the first ; the third
4 ml. 20 po. 2 yds. from the second ; and the distance thence to the
terminus is the average of the three distances mentioned. Find the
length of the railway.
3. A footrule, 3 ft. long, was broken through at 17 in. 4 tenths : how
long was the other part ?
4. A cart with coals weighed 1 ton 15 cwt. 3 qr. 25 lb, and the coals
alone 19 cwt. 26 K> : what was the weight of the cart ?
5. Find the weight of 15 sugar loaves, each 25 lb 11 oz. ?
6. What distance is travelled in 37 da., each 8 ho., at 3 ml. 2 fur.
per hour ?
7. What is the weight of 10000 sovereigns, each 5 dwt. 3 gr. ?
8. How many 6-lb packages may be made out of a hhci. sugar,
weighing 5 cwt. 3 qr. 8 K> ; the tare, or weight of the cask, being 2 qr.
9. How many shirts may be made of 243 yds. cotton, each requiring
10. Twenty-five carts of coals weigh 23 tons 15 cwt. : find the aver-
age weight of each.
80 WEIGHTS AND MEASURES.
11. How many sq. yds. in a court 138 ft. long by 64 broad?
12. How many chests of tea, 40 Ib each, were required to distribute
1 oz. to each of 2000000 poor people ?
13. Three lots silver plate were exposed at a sale ; one weighing
35 oz. 17 dwt. 9 gr., the second 19 oz. 16 dwt. 15 gr., and the third
14 oz. 6 dwt. : what cost the whole at 4/6 per oz. ?
14. How many hours had a boy, who was born Jan. 1, 1848, lived,
when he was 10 yr. 7 wk. and 3 da. old ?
15. Out of 16 cwt. butter a grocer sells 27 Ib. daily for 5 days : how
much has he still on hand?
16. 1000 Ib sugar are sold, each 1 oz. short in weight : find the real
weight of the whole.
17. If the " lona" steamer sails 19 miles per hour, how far will she
sail in 3 days, 6 hours ?
18. How much cloth will clothe a regiment of 560 men, if each suit
takes 4 yds. 1 qr. 3 ill. ?
19. Twenty carts carried each 7 bars lead, and each bar weighed
2 cwt 25 Ib : find the total weight of lead.
20. A railway 47 ml. 4 fur. has 9 stations : what is their average
distance ?
21. If 4 Ib gold are coined into 187 sovereigns, what is the weight
of a sovereign ?
22. A square field is 93 yds. long : how many acres has it ?
23. If the circumference of the earth is 24899 ml., how many yds.
of cotton-thread would reach round it ?
24. If I am 10 yr. 7 wk. and 3 da. old ; my elder brother 3 yr. 17
wk. 5 da. older than I am, and my younger brother 1 yr. 42 wk. 4 da.
younger : what will be the average of our ages 3 yr. 5 wk. hence?
25. Two places on the same meridian are respectively 37° 45' N. and
22° 59' s. : find their difference in latitude.
26. What is the whole area of a farm, of which one field has 19 ac.
3 ro. 29 po. ; the second 27 ac. 36 po. ; the third 36 ac. 2 ro. 18 po. ;
and the fourth 56 ac. 1 ro. 19. po. ?
27. A and B start from points 28 ml. 6 fur. 18 po. apart, to meet
each other. When A has walked 4 ho. at 3^ ml. per ho., and B at
3J ml. per ho., how far are they still apart ?
28. A was born 28th Jan. 1844, and B 16th Nov. 1845, what is
the difference between their ages ? If C was born 3 y. 45 da. later,
what was his birthday ?
29. The daily supply of bread to an hospital is 47 Ib loaves : what
weight of bread is sent per annum ?
30. Find the weight of 279 cub. ft. water, if 1 cub. ft. weighs 62 K>
7 oz. 4 dwt. Find also the weight of 1 cub. in.
31. The ship " Ino" landed a cargo of cotton in 960 bales, weighing
in all 210 tons 15 cwt. : what was the weight of each bale ?
32. A line of 29 yds. 1£ ft. is told off 18 times for sounding : what
is the depth of the sea there in fathoms ?
33. How many miles of rails in a double line of railway 29 ml. 3
fur. 29 po. long ?
34. How many cub. ft. of air in a hall 60 ft. long, 21 broad, and 18
high?
35. How many chains would measure a road 17 ml. 4 fur. long?
36. If sound travels at the rate of 1116 ft. per second, in what time
will the sound of a cannon-shot be heard 6 miles off?
MISCELLANEOUS EXERCISES. 81
37. Divide a hhd. ale, containing 63| gall., into an equal number of
one gallon, one quart, and one pint measures.
38. England is 50000 sq. miles in area : how many acres is that ?
39. A baker uses 6 qr. 5 bu. 3 pk. wheat weekly : how much is that
in a year ?
40. Gold of the value of £500000 arrives from California : what is
its weight avoirdupois, the price being £3, 18s. per oz. Troy ?
41. A ship sailing due north passed through 3° 30' : how many
nautical miles was that ?
42. A road 17 ml. 7 fur. 20 po. 3 yds. is repaired by 23 men : what
share of the work falls to each ?
43. How often will the forewheel of a carriage, 5 ft. 8 in. round,
revolve in a journey of 45 miles ? and how much oftener than the hind-
wheel, which is 7 ft. 6 in. round ?
44. How long will it take to count a million of penny-pieces at 100
a minute ?
45. A watch gains 3' 25" daily ; if it starts from true time on Monday
at 1 o'clock, what time will it show that day and hour three weeks ?
88. EXEECISE XXXII.— MISCELLANEOUS— continued.
1. Three parcels of paper contained respectively 3 re. 14 qu. 20 sh.,
5 re. 16 qu. 10 sh., 7 re. 17 qu. 23 sh. : how much was there in all ?
2. What will remain of a piece of cloth 7 ells, 1 qr. 2 na., if 2 yds.
3 qr. are cut off for a coat ?
3. If a silver spoon weighs 3 oz. 4 dwt. 10 gr., what is the weight
of 3^ dozen ?
4. What cost the gilding of a box, 6 in. long, 4 broad, and 4 deep,
at 4^d. per sq. inch ?
5. The great bell at Moscow is said to weigh 443772 lb : how much
does it exceed the weight of that of St. Paul's, which is 5 ton 2 cwt.
Iqr. 22 lb?
6. If a puncheon of whisky contain 84 gall. 2 qt., how many dozen
bottles will be required to draw it off, counting 6 bottles to the
gallon ?
7. A journey of 87| ml. is performed in 2 ho. 12 min. 40 sec. : what
is the rate per min. ?
8. Five stones butter are to be made up into parcels of £ lb, 1 lb,
and 2 lb, the same number of each : what is that number ?
9. If a soldier takes 75 steps a minute of 2 ft. 8 in. each, in what
time should a regiment march 7f ml. ?
10. To reach the bottom of a pit 12 ladders are required, each
with 23 steps 1| ft. apart : how many fathoms deep is the pit ?
11. If the sovereign weighs 5 dwt. 3 gr., but becomes lightened by
4 gr., what is the weight of 100 sovereigns ?
12. If a class uses 36 sheets paper 5 days a week for writing, how
much will it use in a year, allowing 6 weeks for vacation ?
13. Find the difference between 18 square feet and 18 feet square.
14. How many geographical miles between London, 51° 30' N., and
Saragossa, 41° 46', which are nearly on the same meridian of longitude ?
15. If the weight of I/ be 3JT dwt., how many may be counted out
of 3 bars silver, each 7 lb 9 oz. 14 dwt. ?
16. If 8 qr. 4 bu. 3 pk. weigh 18 cwt. 2 qr. 26 lb, what is the weight
of a peck ?
82 WEIGHTS AND MEASURES.
17. A field 400 feet long by 175 broad is intersected along its length
by a stream 21 links wide : what is the area of the field ?
18. From a cistern containing 2560 galls, are drawn on Monday 859
galls. 2 qts., on Tuesday 384 galls. 3 qts., on Wednesday the difference
Between these quantities. How much must be drawn on each of the
three remaining days of the week to empty the cistern ?
19. At 5/3 per ounce, what is the value of silver plate weighing 3 Ib
9 oz. 10 dwt. ?
20. A piece of cloth was 56 yards long : how much more would have
given 7 yds. 2 qrs. 3 nls. to each of 8 persons ?
21. How much pure gold in spoons weighing in all 2 Ib 6 oz., 18
carats fine, i.e., 18 parts out of 24 being pure, and the rest alloy?
22. A journey of 127 miles 6 fur. is made by rail, at the rate of
26 miles 6 fur. per hour. That is only \ of the time a steamer would
take, and f of the time a stage-coach would take. How much less time
does the railway take than the steamer, and than the stage-coach ?
23. How many solid feet of earth ill a mass 15 yds. long by 11 broad
and 9 thick *
24. Four hhd. sugar weighed 7 cwt. 2 qr. 1 Ib, tare 2 qr. 16 Ib ;
6 cwt. 1 qr. 18 Ib, tare 2 qr. 5 Ib ; 5 cwt. 3 qr. 20 Ib, tare 1 qr. 26 Ib ;
7 cwt. 3 qr. 24 Ib, tare 2 qr. 18 ft> : what is the net (or neat} weight ?
25. How many coils of rope, each 5^ ft., will stretch half a furlong?
26. What is the girth of a wheel which revolves 1365 times in 2 ml.
1 fur. 36 po. ?
27. Find the weight of water in a tank whose bottom is 10 ft. square,
and depth 14 ft. (See No. 30, Ex. xxxi.)
28. If 1 acre yield 3 bu. 2 pk., what is the produce of 193 ac. 3 ro. ?
29. How many hurdles, each 4£ ft. long, will enclose a park 9 chains
long by 5.^ broad ?
30. If you waste 10 minutes daily, how much time do you lose in
3 years (counting 1 leap year) ?
31. If I walk 8 yds. 2 ft. 6 in. a minute faster than my neighbour,
in what time will I be a mile ahead of him ?
32. A milkmaid, carrying a pitcher of milk containing 3 gallons, to
be delivered equally to 16 families, loses 3 pints by leakage ; what
must each family get ?
33. How many benches, 8 feet long, would seat a class of 42 pupils,
allowing 1 ft. 4 in. to each ?
34. If the breadth of an oblong field is 17 ch. 56 Ik., and its length
twice and a half as much, how many yards does a person walk who
goes round the field three times ?
35. Light travels at 186,000 miles per second ; how long does the
sun's light take to reach the earth, the distance being 95 millions of
miles ?
36. The Jewish silver shekel weighed 9 dwt. 2^ gr., and its value
was 2/3£ : find the weight and value of the talent, which was 3000
shekels. Find also the weight and value of the silver vessels (" one
silver charger of 130 shekels, and one silver bowl of 70 shekels")
offered by the twelve princes of Israel, Num. vii. The present of
each also included one golden spoon of ten shekels : what was the
weight and value of the spoons, the shekel of gold being estimated at
£1, 16s. 6d. ?
BILLS OF PARCELS. 83
89. BILLS OF PARCELS.
Ex. — Mrs. Wilson bought of William Dixon, grocer : 4 ft>
tea at 3/6 per lb ; 12 lb sugar at 5d. per lb ; 7 lb butter at
1/1 J per lb ; 10J lb rice at 3|d. per lb ; and 2 lb -urrants at 8d.
per lb : what has she to pay ?
The grocer presents her with an account drawn up in the
following form.
Sometimes, for ready money payment, a deduction or dis-
count of £5 for every £100 of account — or, as it is called, 5 per
cent. — is allowed. That is ^th part of the whole, and is
1 calculated with sufficient accuracy by allowing I/ for every
pound, and 3d. for every 5/ besides.
Mrs. WILSON,
Bought of WILLIAM DIXON, Grocer.
U March. 4 ft) tea @ 3/6, . . . . .£0 14 0
12 lb sugar @ 5d., . . . 050
7 lb butter @ 1/1 1, . . . 07 lOi
10^ ft) rice @ 3^d., . . . 0 3 0|
2 ft) currants @ 8d., . . . 014
£1 11 3i
Discount, . . 016
£1 9 9jr
The tradesman discharges the account by writing below it,
when he receives payment, "Received payment, William
Dixon."
These accounts are called " Bills of Parcels."
90. EXERCISE I.
Make up the following accounts, allowing discount at 5 per
cent. : —
1. Mr. John Thomson bought of William ITendry, May 15, 1864 :
17 yds. lace at 6/6 per yd. ; 14 yds. cambric at 10/8 per yd. ; 58^ yds.
calico at lOd. per yd. ; 36 yds. muslin at 7/4| per yd. ; 6f yds. linen
at 3/8.}.
2. Mr. William Simson bought of Thomas Adams, June 14, 1864 :
7 \ lb beef at 8d. per lb ; 9 lb mutton at 7d. per lb ; 2 spring chickens
at 2/9 each. July 3. 12 lb beef at 8^d. per ft ; 6^ lb pork at 6d. per
lb ; 2 hares at 2/9 each ; 1 pair of pigeons at l/4d.
3. Mr. Alexander Cooper bought of Alfred Garland, Aug. 7, 1859 :
28 yds. Brussels carpet at 4/8^d. per yd. ; making do. 4/ ; 18 yds.
stair carpet at 2/9 per yd. ; 3 hearth rugs at 17/8 each ; 2 pieces floor-
cloth, 15| yds. each, at 2/10 per yd.; men's time laying down, 3/6.
4. Mr. "Henry Wallis bought of Robert Mapleton, January 1, 1563 *
84 BILLS OF PARCELS.
41 yds. super, black cloth at 16/6 per yd. ; 6 yds. tweed at 5/6 per
yd. ; making and mounting 17/8 ; 2 black ties at 3/6 each ; 6 silk
pocket handkerchiefs at 3/9 each ; 2 pair kid gloves at 3/.
5. Mr. Philip Chorley bought of Andrew Stewart & Cc. : 8 quires
note-paper at 7d. per quire ; 5 reams foolscap at 22/6 per ream ; 12
boxes steel pens at 1/3 per box ; 3 bos. black ink at 2/6 per bo, ; 3
account books at 4/9 each.
6. W. H. Wilson, Esq., bought of Dalton & Tait : 4 dozen sherry
at 35/; 6 dozen do. at 48/; 8 dozen port at 56/ ; 3 galls. Scotch
whisky at 18/6 ; 6 dozen ale at 6/ ; 2 dozen claret at 63/ ; carriage
paid, 11/4.
7. Mrs. Barton bought of Hobkirk & Son : 33 lb Cheshire cheese at
8|d. per lb ; 15 lb butter at 1/2 per lb ; 1 Belfast ham, 13i lb, at lid.
per lb ; 3-J lb bacon at 9d. per lb ; 6 dozen eggs at 7d. per dozen ; 2
pks. fine flour at 1/2 per pk.
8. Mr. Robert Beaton bought of John Gardiner : Feb. 1, 1860, 5 lb
tea at 4/4, 12 lb loaf-sugar at 5|d ; Feb. 28, 8 lb rice at 3^d. per lb,
3 It) currants at 5|d ; March 4, 4 fb coffee at 1/8, 6 lb loaf-sugar at
5^d., 3 lb table raisins at 1/2, 4 dozen oranges at 1/6 per dozen.
9. Wilson & Hill bought of Farmer Brothers, June 6, 1861 : 36
qrs. barley at 25/ ; 58 qrs. wheat at 52/ ; 16 qrs. of oats at 35/ ; 17
bushels pease at 3/8.] ; 19 bushels tares at 1/9|.
10. Mr. David Hodson bought of Thorn & Maclean : 4 chests tea,
180 lb each, at 3/8 ; 9 hhcls. sugar at £4, 19s. 3d. per hlid ; 7 bags of
rice, each 1£ cwt., at £1, 7s. 6d. per c\vt. ; 8 bags coffee, each 84 lb,
at 1/5 per lb ; 64 sugar loaves, each 12^ lb, at 4|d. per lb.
11. Mr. John Smith bought of Thomas Rogerson, Fek 15 : 12 yds.
flannel at 1/5 ; 30 yds. calico at 6^d. ; 18 yds. linen at 3/10 ; 6 pairs
stockings at 1/2 ; 6 pairs blankets at 14/6 per pair.
12. Mr. R. Thompson bought of Hancock Brothers : 16 yds. broad
eloth at 27/6 per yd. ; 18 seconds do. at 14/9 ; 13 yds. brown cloth at
11/10 ; 14J yds. scarlet at 24/4 ; 62£ do. at 9/8.
PRACTICE.
Ex. 1. — Find the price of 385 yards at 7^d. per yard.
If we find the price at (3d. and then at l|d., and add the two
prices, we shall have the price at 7Jd.
Now, the price of 385 yards at Is. per yard being 385s., the
price at 6d. will be one-half of that, or 192s. 6d. ; and the price
a.t l^d. will be £ of the price at 6d., that is 48s. l^d. Adding
the two together, we find the price at 7^d. to be 240s. 7|>d.,
that is, £1%, Os. 7jd.
The working is written down thus : —
y:ls. s. d.
Price of 385 at Is. 385 0
Price „ @ 6d.
Price „ @ l|d.
Therefore, Price „ @ 7jd.
XT2~CT
PRACTICE. 85
The parts of the whole price chosen are 6d. and 1 Jd. because
these are even, or, as they are called, aliquot parts of the next
highest name, that is, a shilling.
The answer to this question might be got by compound
multiplication ; but the process is longer. The method of find-
ing prices by aliquot parts is therefore commonly practised ;
hence it is called " Practice."
Rule. — Take aliquot parts of the next highest name, and
find the prices at these ; add the several results, and reduce
the sum to pounds.
92. Ex. 2. — Find the price of 385 J yards at lOjd. per yard.
Here the price at 1 s. is 385s. 6d.
Then we might take as aliquot parts of a shilling, 6d., 3d.,
and l^d., and add the results ; but in cases like this, whei .
the price differs from the next highest name by an aliquot part
exactly, it is shorter to take that difference as an aliquot part,
and subtract the result from the price at the next highest name.
Thus :—
yds. s. d.
Price of 385j- at Is. 385 6
Price „ @ l^d.
Therefore, Price „ @ 10 Jd.
EXERCISE I.
*»* The pupil should first be exercised in the aliquot parts to be taken in
the following examples, till he can state them readily.
1. 73 yds. @ If, J, 2f, 3.J, 4J, 5J, 6J, 7J, 8Jd.
2. 294 @ 2i, 1J, 44, 3J, 4f, 5f, 6i, 7f, fyd.
3. 596$ @ 3$, lid, 2|, 6f, 4.J, 5|, 8±, 7i, 9fd.
4. 7384 @ loi, Sf, 64, 9^, 11^ 11.J, f, 5"f, 9jd.
5. 8036J @ 7i, 2i, 9.i i, 11*, lO.i, 3f, 8^, 4Ad.
6. 5690 @ Hi, lOi, 9^ |, Il|, 8f, 7|, lOJ, fd.
7. 6853
83. -^or workincr questions in which the price is shillings, the
pupil must be familiar with the following tables : —
1. ALIQUOT PARTS OF £1.
10s. Od. is £A 4s. Od. is ££ 2s. Od. is £^ Is. 3d. is £^
6s. 8d. £j 3s. 4d. ££ Is. 8d. £fa Is. Od. £^
5s. Od. £^ 2s. 6d. £i Is. 4d £fr Os. 6d. £^
86
PRACTICE.
2. ALIQUOT PARTS OF THESE PARTS.
*. d.
s. d.
d.
d.
s. d. d.
2 6 is i
of 10 0
10 is A
of 10 0
7i is J of 1 3
4
2 6
1 8 j
5 0
- 10 0
10
10 \
6 8
5 0
6" ^ 10 0
6 A 5 0
4
3
1 8 1
I 5 0
10
3 4
6 140
3
1 4 ;
4 0
10 \
2 6
6 * 2 6
3
1 3 5
10 0
10
1 8
6 I 1 0
3
1 3 ;
5 0
8 -h
, 10 0
5 A 5 0
1 3 ..
2 6
8 i
4 0
5 f 3 4
8 1
3 4
5 £ 2 6
2^
8
t 2 0
5 \ 1 8
8 i
1 4
4 A 4 0
l)
7\ '*
5 0
4 A- 3 4
2 6
4 £ 2 0
EXERCISE II.
1. Of one pound, what is £, |, J, fc J, i,
• 2. Of one pound, what is |, f , f , f , |, &,
3. Of one pound, what is f , f , &, rf,, ££,
4. Of ten shillings, what is $, i, |, ^, ^,
5. Of five shillings, what is $, ^, i, J, |,
6. Of four shillings, what is \, \, \, J, \,
7. Of half-a-crown, what is \, |, ^, J, ^,
8. Of two shillings, what is \, \, J, |,
9. Of one and eightpence, what is }, \, ^
10. Of one and threepence, what is -£, £,
0 6
A, A> A» A, A, A* A ?
, &, A, A. A» A, etc. ?
^, A, ii, M> etc. ?
, ^, f , T%, ^, etc. ?
, ^, |, f , f , etc. ?
&, X, ^, §, f, f, etc. ?
^, ^, |, f , ^, etc. ?
T\, f , |, f , etc. ?
|, ^, ^ f , |, f, ^, etc.?
, ^, ^, f, f, f, ^, etc. ?
Q A Ex. 3. — Find the price of 385 yards at 11/10^ per yard.
" The price here being shillings, we take aliquot parts of the
next highest name, the pound.
If we find the price at 10/, 1/8, 2d., Jd., and add these seve-
ral prices, we shall have the price at 11/10^. Thus :
yds. £ s. d. £ s. d.
Price of 385 @ 1 0 0 385 0 0
Price „ @ 0 10 0
Price „ @ 0 1 8
Price „ @ 0 0 2
Price „ @ 0 0 0
192 10 0
32 1 8
342
8 0^
Therefore, Price
@ 11 10 J
228 3 10 J
The aliquot parts should always be chosen so as to give the
smallest number possible. Thus, if the price were 3/9, and if
we took the largest aliquot part, viz., 3/4, we should still have
PRACTICE. 87
two to take, viz., 4d. and Id. ; whereas, taking 2/6 and 1/3 aa
the aliquot parts, we need only two.
Ex. 4. — Let the price of the 385 yds. be 17/6 per yard.
yds. £ s. d. £ s. d.
Price of 385 @ 100 385 0 0
48 2 6
Price
Therefore, Price
0 2 6 |
0 17 6
336 17 6
EXERCISE III.
93 © 2/10, 5/8, 3/9, 13/4, 18/4, 7/9.
118 @ 4/6, 6/8, 6/3, 15/10, 19/4, 1/5.
I/7L 4/3|, 6/21, 8/7|. 11/24, 14/32.
5/104, 17/8, r
3/72, 8/52, 1!
6/10, 16/11, 12/6'f , 15/i'4, 5/lll, 17/72.
2/1, 15/6, 13/41, 8/52, 3/81, 19/12.
9/54, 10/0 j, 15/6 J, 17/6, 14/2$, 4/T
548
805
724
327
2937
5608
7890
8000
},
1,
|, 8/7i, 9/2|, 15/6J, 7/
95. ^n ^ne following examples, which involve fractional numbers
of articles, it is better to work first for the value of the whole
number, and then add the fraction of the price of one article.
Ex. 5.— What cost 397f at 10/6 ?
yds. £ s. d. £ s. d.
Price of 397® 100 397 0 0
Price
>j
@
0
10
0
I
198
10
0
Price
11
@
0
0
6
^T
9
18
6
Price
of
1
@
0
10
6
0
7
0
Price of 397
1. 236£ (
2. 693* (
3. 600A(
4. 1594f (
5. 7240|(
6. 94364 (
0 10 6
EXERCISE IV.
208 15 6
2/3, 4/7*, 8/11, 13/4J, 18/2f, 7/6.
10/9, 14/8, 17/2J, 9/3^, 12/81, 1/7J.
17/8, 13/1J, 15/yj, 3/10.^, 4/8, 11/5J.
3/3, 7/9f^l3/2|, 5/10^ 18/6, 19/5._
i 14/1, 10/6, 10/8J, 15/7i 18/9, 2/5|.
\ 17/4, 6/10J, 9/3, 18A1, 3/4J, 15/2.
9 D . When the price of each article is a number of shillings exact,
the shortest way to find the price of the whole is by multiplica-
tion.
88
PRACTICE.
Ex. 6.— What cost 278 yds. at 9 sh. and at 14 sh. ?
sh. sh.
278 price at I/. 278 price at I/.
9 7
2(0 ) 250(2 price at 9/. 10) 194(6 price at 14/.
£125, 2s. £194, 6s.
What has been done in the latter case, in which the number
of shillings is even, is to multiply by ^ instead of J£ ; which is
shorter, and comes to the same thing.
EXERCISE V.
1. 742 ® 6/, 14/, 16/, 5/, 9/, 13/. 4. 1894 ® 14/, 6/, 8/, 117, 13/.
2. 913 @ 3/, 7/, 8/, 12/, ll/, 19/. 5. 3565| @ 17/, 2/, 9/, 3/, 12.
3. 296i @ 16/f i8/> 5^ 7/; 127, 157. 6. 7924 ® 13/, 7/, 15/, ll/, I/.
97. Ex. 7.— Find the price of 385| yds. at £3, 11s. 10|d.
Here we find the price at £3 separately, and proceed for the
rest as before.
yds. £ s. d. £ s. d.
Price of 385 @ 1 0 0 385 0 0
3
Price
» ®
3
0
0
1155
0
0
Price
w ©
0
10
0
1
192
10
0
Price
99 @
0
1
8
1
32
1
8
Price
» @
0
0
2
JL
3
4
2
Price
0
0
01
|
0
8
oj
Price
of f <§T
3
11
iol 2
13
10J
1
Price of 385 J@ £3 11 10£ £1385 17 8j f
Rule. — When the price consists of £ s. d.t multiply by the
number of £ ; take aliquot parts for the rest, and add tho
several results.
EXERCISE VI.
£ 5.
d.
£ s.
d. £
s.
d.
£ s.
d.
£
s. d.
1. 235 @
8 4
10
4 17
3 5
18
10
15 6
2i
11
7 11
2. 486 @
9 2
3^
7 10
6 24
7
2
30 18
9
18
2 11
3. 592 @
1 5
6
9 15
10
20
10
1
25 1
7£
3
16 lOa
4. 3560 @
2 10
6
5 7
3
9
2
6}
7 2
3|
1
9 6
5. 5986 @
7 13
4.i
11 10
6
23
10
9
14 13
0^
2
18 10
6. 7852 @
5 9
31
8 0
7
36
2
5J
9 17
3
5
2 11J
7. 8194 @
12 10
9
20 15
7
41
8
7
28 5
6i
17
13 8£
8.2936 @
]4 5
3$
2 19
9 6
14
2i
34 17
4f
100
19 4
9. 23%1 @
50 7
6
43 1
9.1, 11
5
8}
17 9
3
7
2 11
10. 7852^ ©
120 10
9A
84 2
I" 70
16
2i
24 11
5
1
9 6J
11. 9324,} (&
55 1
11"
72 18
2i 24
5
01
36 2
0}
3
0 104
12. 56498 OZJ
27 17
0
39 18
0 46
8
OA
10 0
9i
92
14 6j
PRACTICE.
89
98. E'j. 8. — Find the value of 9 cwt. 3 qr. 7 tb at .£4, 13s. 6d. per
cwt. Here the quantity is compound. It comes to the same thing
whether we multiply the quantity by the price of one, or the
price of one by the quantity ; and we choose the latter as the
simpler in this case. Thus we multiply the price of 1 cwt. by 9,
which gives the price of 9 cwt., and take aliquot parts of a cwt.
for the rest of the quantity.
cwt.
Price of 1
qr.
0
ft £
0 4
s.
13
d.
6
9
Price of
»
»
>»
9
0
0
0
0
2
1
0
0
0
0
7
1
42
2
1
0
1
6
3
5
6
9
4*
ion
Price of 9 3 7 ^£45 17 5j
Rule. — When the quantity is compound, multiply the price
by the highest name of the quantity, and take aliquot parts for
the remainder of the quantity.
Note. — In the above example, the price of one of the highest
name is given. If the price of one of the lowest be given,
reduce the quantity to that lowest name, and find the value by
a previous rule.
EXERCISE VII.
%* In this and the following Exercise, the exact calculation of the fractions of
pence may be left till the pupil studies the chapter on that subject.
1. 5 cwt. 2 qr. 9 ft (
2. 9 tons 16 cwt. 1 qr. (
3. 25 ft 12 oz. 7 dr. (
4. 27 ft 8 oz. 9 dwt. (
5. 64 yds. 3 qr. 2 nl. (
6. 144 yds. 2 qr. 1 nl. (
7. 15 qrs. 5 bu. 3 pks. <
8. 36 bu. 2 pks. 1 gall. (
9. 7 pks. 1 gall. 3 qts. (
10. 7 qts. 1 pt. 1 gi. (
11. 18 qrs. 7 bu, 2 pks. (
12. 96 ac. 1 ro. 20 po. (
13. 144 ac. 2 ro. 16 po. (
14. 27 po. 20 sq.yd. 8 sq.ft. (
15. 49s.y. 5s.f. 100 sq. in. (
16. 10 rea. 7 qu. 6 sh. <
17. 42 yds. 2 ft. 9 in. (
18. 63 yds. 1 ft. 7 in. (
19. 25c.y. 18c.ft.144c.in. <
£ s. d.
£ s. d. £ s. d.
£ s. d.
217 6
415 3 911 3
15 14 10A
314 5
7 9 10J 12 8 5] 13 17 llj
1 15 6
2 7 84 5 8 fa
* 9 0 1J
711 5
914 7j 8 2 3J
u 7 ;r
1 3 6
2 5 4* 873^
409
2 3 71
3 7 8j 4 911;
562
317 6
414 6j 513 9;
7 6 OA
1 15 8^
2 310 5 6 45
6 211
105
110 11A 276^
2 8 OJ
2 7 81
1 9 5;r 3 0 7J
1 17 8j
318 9j
624.
i. 2 7 3^
1 19 Ilk
24 7 6
15 8 2.-
36 18 OJ 12 10 10"
8 5111
978;
71311
10 5 61
238
345;
416 2.t 5 S10J
1 2 6*
2 311]
1173} 2 15 14
158
1 13 11
213 9
2 010
516 2i
319 5;
11 2 1J
L 9 7 8
2 13 61
219 O.J
3 7 3j
562
4 Oil
319 2^ 49 3.:
18 7 0
90
PRACTICE.
£ 5. d.
20 8 5
11 14 81
21610
118 5
21 01U
218 2
£
16
10
5
1
27
1
8. d
9 3^
9 4^
311;
0 9"
1 6i
15 6;
£ 5. d.
14 8 0£
13 7 3
710 92
318 If
14 BIO!
314 4j
7
6
5
4
1
5
s.
16
12
0
9
18
9
d
4
8
6
3
6
Oi
20. 4 ml. 7 fur. 12 po.
21.84ml. 6 fur. 10 po.
22. 4 yds. 2 ft. 6 in.
23. 25 da. 10 ho. 30 min.
24. 7 wk. 5 da. 16 ho.
25. 22 ho. 48 miu. 36 sec.
99. Ex. 9.— Find the value of 527 cwt. 3 qrs. 7 ft) at £5, 11s. 8d.
per cwt. — To avoid the long process of multiplying by 527, we
may in this example first find the value of 527 cwt. at the
given price ; and take then 3 qrs. 7 lb at the given price ; and
add the results. Thus
*. A
10 0
1 8
l
527
5
price of 527 cwt. at £5 per cwt.
„ 10s. „
„ ls.8d. „
„ 2 qr. at £5, 1 1, 8 p.cwt.
" i£r-
i » 7 ib „ „
£2635
263
43
2
1
0
0
10
18
15
7
6
0
0
4
10
H
2 qr.
Iqr.
7ft
L
£2946 19
This method really combines two questions in Practice, and
may be used with advantage whenever the aliquot parts for the
shillings and pence in the price are few.
EXERCISE VIII.
1. 239 cwt. 2 qrs. 16 tb
2. 412 tons, 14 cwt. 1 qr.
3. 193 tb 4 oz. 8 dwt.
4. 4541b7oz.ll dwt.
5. 325 ac. 3 ro. 29 po.
6. 1426 ac. 3 ro. 3 po.
7. 724 qrs. 6 bu. 2 pk.
8. 95 bu. 1 pk. 1 gall.
9. 713 galls. 2 qt. 1 pt.
10. 315 qts. 1 pt. 3 gi.
11. 347 yds. 2 qr. 3 na.
12. 250 yds. 1 qr. 2 na.
13. 536 yds. 2 ft. 8 in.
14. 115yds. 1ft. 6 in.
15. 425 mi. 3 fu. 8 po.
16. 374 mi. 7 fu. 16 po.
17. 723 da. 6 ho. 14 mi.
18. 117 wk. 5 da. 15 ho.
7 010;
317 L
209
11 7
21410
5 211
784
314
411
36 2
2918 5;
309^
21 7
PRACTICE. 91
100 EXERCISE IX.— MISCELLANEOUS.
1. What does a contractor pay weekly for wages who employs 546
labourers at 15/9 each ?
2. Find the price of 8 pieces cloth, each 36 yards, @ £1, 3s. 6d per
yard.
3. Bought soap at 4|d. per K> : what is that per cwt. ?
4. Bought 56 hampers apples @ 16/2 £ each : how much cost the
whole ?
5. What cost 236^ gross bottles @ 1/3^ per dozen ?
6. A farmer rents 129 acres @ £3, 17s. ~6^d. : what is his total rent ?
7. What is my nett income, if my taxes are 1/10^ on my gross in-
come of £320 ?
8. What is the freight of 7 trucks, each 6 tons, 3 cwt. 2 qrs. @ £1,
6s. 7£d. per ton ?
9. What is the expense of making 147 miles 3 fur. 24 po. railway
@ £2345, 10s. 6d. per mile ?
10. What did I pay for 3 cwt. 2 qrs. butter @ 15/4 per stone ?
11. What cost 937 yards ribbon @ 3|d. per yard?
12. A farmer sold 39 oxen @ £14, 10s. 6d. each, and 256 lambs @
£1, 2s. 6d. each : how much money did he get ?
13. In a railway train there were 79 first-class passengers @ £2,
17s. 6d. each ; 193 second-class @ £1, 14s. 8d. ; and 256 @ 19/8 each :
what were the receipts from the train ?
14. What will I make by selling 26 stones starch @ 11 |d. per Ib ?
15. A bankrupt paid 9/10£ a pound on a debt of £2456, 17s. 6d. :
what were his assets ?
16. Find the price of 123 yards 3 qr. 3 na. cloth @ 3/8 per yard ?
17. If a dollar is 4/2^, how many pounds are in 798 dollars ?
18. The cost of a vessel was 758296 francs : what is that sum in
sterling money, if the franc is 9|d. ?
19. If a sovereign weighs 5 dwt. 3 gr., what is the weight of 25000
sovereigns ?
20. Bought 27| yds. flannel @ 1/5 ; 18 doz. pairs of stockings @
1/10 per pair ; 156£ yds. linen @ 2/6| ; 596 yds. calico @ 7|d. per yd. :
what was the amount of the bill ?
21. What cost 172 doz. and 80 bo. sherry at 42/ per doz. ?
22. What cost 3 qr. 17 Ib sugar @ £3, 18s. 6d. per cwt. ?
23. What did I pay in all for 25^ R> beef @ 9^d. ; 16 R> 8 oz. cheese
@ lO^d. ; 23^ loaves bread @ 6£d. ; and 2 R> 11 oz. tea @ 4/9.
24. Find the duty on a puncheon (84 galls.) whisky @ 9/6^ per gall.
25. Find the nett weight of 217 cwt. 2 qr. 25 Ib, allowing 13 Ib per
cwt. as tare (deduction for weight of package).
26. Find tne value of a silver tea-service weighing 325 oz. 6 dwt. @
5/6 per oz.
27. If I spend 13/8^ daily, what do I save out of an annual income
of 350 guineas ?
28. A company of seven miners find a nugget of gold weighing 23
R> 8 oz. : what does each make by it, if gold sells @ £3, 16s. lO^d.
per oz. ?
29. If a labourer's wages are 17/6 a week, what does he earn in
6 weeks 4 days ?
30. An apprentice, whose wages are £60 a year, dies after a service
of 7 weeks 3 days : how much falls to be sent to his friends ?
92 RULE OF THREE.
RULE OF THREE.
101 1. By Multiplication and Division.
Ex. 1.— If 4 ft> tea cost 16/, what cost 16 It) ?
Here the price of a certain quantity is given, and we wish to
know the price of so many times that quantity. 16 ft) is 4
times 4 ft), therefore the price of 16 lb will be 4 times the
price of 4 Ib ; that is, 4X 16/, which is 64/, or ,£3, 4s.
Questions of this sort, in which the quantity whose price is
sought is so many times the quantity whose price is given, are
solved by multiplication.
In all such questions there are three numbers given, two
being of the same kind, and the third of a different kind ;
hence the name applied to the solution is the " Rule of Three."
A fourth quantity is in all cases sought, which is of the same
kind with the third given.
The Rule of Three is chiefly useful for the finding of prices ;
but it will be seen from the examples that it is applied also to
questions in which money is not involved.
EXERCISE I. ;
1. If 3 yds. cost 17/, what cost 18 yds ?
2. If 16 tb sugar cost 6/, what cost 48 tb ?
3. If 2 quarts cost 1/6, what cost 2 galls. ?
4. If a labourer earns 25/in 13 days, what will he earn in 13 weeks ?
5. A coach goes 19^ miles in 3 hours, what distance will it go in 15
hours ?
6. If 45 men can build a wall in 18 days, in what time will 9 men
do it ?
7. If 3 books cost 3/9, how many may be bought for 18/9?
8. If 6 acres produce 2t> bu., what will 43 acres of4 same land pro-
duce?
9. If 44 acres rent for £17, what will be the rent of 34 acres ?
10. How many yards of cloth at 3/6 are worth 27 yds. at 14/ ?
11. If ^ tb tea cost 2/3, what cost 8 tb?
12. If | tb cost 7/1 A, what will be got for £3, 16s. ?
10 4 Ex. 2.— If 12 yards cost .£2, 8/, what is the price of 4 yards \
Here the quantity whose price is sought is an even part o
that whose price is given.
Since 4 yards is the third part of 12 yards, the price of 4 yards
will be the third part of that of 12 yards.
Now the £d of .£2, 8/ is 16/, which is the answer.
Such examples of the Rule of Three are wrought by Division.
EXERCISE II.
1. If the cost of printing 128 pages be £17, 4/8, what cost the print
Ing of 32 pages ?
RULE OF THREE. 93
2. If 32 cwt. cost £36, what cost 4 cwt. ?
3. If 1 lb tea cost 4/6, what is that per oz. ?
4. If 3 dozen oranges are bought for 2/, what could I buy 4 oranges
for?
5. If 7 men reap a field in 4 days and 2 hours, how many days will
28 take ?
6. If a railway train goes 200 miles in 8 hours, in what time will it
go a journey of 40 miles ?
7. I read a book of 365 pages in 15 days ; how many pages do I get
over every 3 days ?
8. A ship with a crew of 160 men was provisioned for 84 days ; how
long would these last a crew of 960 ?
9. If 21 yards cost £6, Os. 9d., what will 5£ yards cost ?
10. A firm expends on wages £61, 12/ a week for 75 men ; what does
another expend, employing only 25 men at same rate ?
, 11. If 63 qrs. of wheat weigh 18 tons, find the weight of 7 qrs.
12. A farmer rents 275 acres for £400 : if he adds 55 acres at same
rate, what was his rent then ?
103 Ex. 3.— If 8 ft cost 28/, what cost 11 ft) ?
Here the quantity whose price is sought neither contains, nor
is contained in, the quantity whose price is given, an even
number of times.
We therefore find the price of 1 ft, as an intermediate step,
the number 1 being contained in both the quantities.
Thus, since 8 lb cost 28/, 1 ft cost J of 28/ ; and
since 1 ft cost J of 28/, 11 ft cost ^ of 28/ ;
that is, £1, 18s. 6d.
Such examples of the Eule of Three are wrought by Multi-
plication and Division combined.
EXERCISE III.
1. If 7 sheep cost £8, 15/, what should be paid for 11 ?
2. If 5 books cost 6/8, what cost 9 ?
3. For 7 doz. wine I paid £9, 19s. 6d. ; what would I have got 3 doz.
for?
4. If I walk 17£ miles in 5 hours, in what time shall I walk 267^
miles ?
5. If an apprentice earns 3/9 in 5 days, how long must he work foi
£3, 7s. 6d. ?
6. My income is £120, and I pay taxes £3, 4s. 7d. : what should an
income of £420 pay ?
7. If 13 men mow 7 ac. 2 ro. 25 po. in a given time, what will 19
men do in same time ?
8. If 29 sheep are worth 3 oxen, how many sheep are worth 21 oxen ?
9. If 11 silver spoons weigh 9| oz., what is the weight of 2^ dozen ?
10. If 12 lb butter cost 5/6, what cost 3 st. 3 lb ?
11. When wheat is at 37/6 per qr., what should I get 5 bushels for?
12. If the carriage of a parcel by railway for 17 miles be 10d., what
should it be for 32 miles ? and how far should it go fo* 2/11 ?
104
105
94 RULE UF THREE.
Ex. 4. — If 8 books cost 18/, what cost 25 ? and what cost 23 f
This is a case like the last, and could be wrought by multi-
plying by 8, and dividing by 25 and 23 respectively. But since
25 contains 8 three times and 1 over, the price of 25 may be
found by taking three times the price of 8, and adding to it the
price of 1.
Thus : price of 24 is 3Xl8/= £2 14 0
price of 1 is £ of 18/= 2 3 therefore
price of 25 is £2163
Again : to find the price of 23, since 23 is 24—1, say,
price of 24 is 3X 18/ = £2 14 0 and
price of 1 is J of 18/ = 2 3 therefore
price of 23 = £2 11 9
EXERCISE IV.
1. If 8 oz. of tea cost 2/8, what cost 9 oz. ?
2. If 3 yds. of cloth cost 17/3, what cost 7 yds. ?
3. For 8 chairs I paid £5, 4s. : what cost 15 ?
4. 9 cwt. sugar cost £16, 16s. : what cost 26 cwt. ?
5. 16 reams of paper go to 300 copies of a book : how many copies
may be got out of 36 reams ?
6. If I pay £8, 5s. for 45 yds., what is the price of 50 yds. ?
7. If a coach go 42 miles in 7 hours, how many miles will it go in
10A hours ?
8. If 35 gall, cost £20, 3s. 6d., what cost 10 gall. ?
9. If 4 lb coffee cost 6/8, what should I get 27 R> for ?
10. If 2 tons 3 cwt. cost £4, 10s., what cost 16 tons 2 cwt. 2 qr. ?
11. If 4 ac. 2 ro. are rented for £15, what is the rent of 23 ac. 2 ro.
20 po. ?
12. If a box, 2 ft. sq. and 3 ft. deep, contain 100 oranges, how many
should be in one 2 ft. long, 3 broad, 7 deep ?
2. By Proportion.
Ex. 1. — What number contains 9 as often as 12 contains 6 ?
Since 12 contains 6 twice, and 18 contains 9 twice, 18 is the
number sought.
The number of times that one number contains another is
called the ratio of the two numbers ; thus the ratio of 12 to 6
is 2, and of 18 to 9 is 2.
In the question, the ratio of 12 to 6 is equal to the ratio of
18 to 9 ; and the four numbers are on that account said to be
proportional, and to form a simple proportion.
That 12 contains 6 as often as 18 contains 9 is usually ex-
pressed more shortly thwa *
PROPORTION.
95
As 12 is to 6, so is 18 to 9. Or 12 is to 6, as 18 to 9.
By symbols :— 12 : 6 : : 18 : 9.
The four numbers of a proportion are called its terms.
EXERCISE V.
State the term wanting to the following proportions, and read each
proportion as completed : —
1. 4:
2. 6:1
2 :
} •
8:
. 5. 36 :
. 6. 21 :
12 : : 2
7: :
7:( ). 9. 18:
9 : ( ). 10. 7:3
6::1
5 • •
27:
8:
3. 24:
4 49 •
3 :3
1 '6
6:
3 •
. 7. 8:
8 4 •
16 : :1
20 : :
2 : ( ). 11. 14 : 4
5 : ( ). 12. 108 : 1
2 : :
?'• '
2:
P •
13. 3:
14. 2 :
15. 6 :
16.7:
25.4:
26. 3 :
27. 6 :
28. 8:
37. ( )
38. ( )
12::
8::
i8 :
53:
I
: 6
• 8
. c
11
|
7
'•}
:
i
:
.
9 :
1 •
16. 17.
20. 18. 1
8. 19.
27. 20. 1
18. 29. '
48. 30. !
25. 31. 11
35. 32. 1<
18. 41.
44. 42.
9 :36 :
3:52:
9:3:
2: 4:
|
( : 16. 21. 36
: 32. 22. 56 •
( : 8. 23. 84 :
( : 7. 24. 42 :
: 13 : 39. 33. 21 :
: 8 : 32. 34. 12 :
: 18 : 3. 35. 54 :
: 63 : 9. 36. 35 :
)•:: 8:40. 45. ( )
J:: 3:36. 46. ( )
12
7
12
6
: 4
i
3
8
10
: 2
:: $
):
0:
4:
3:
8:
JO:
7 •
2.
1.
7.
12.
10.
7.
12.
4.
5
1
39. ( )
40. ( )
•42
7:
9:
21. 43.
63. 44.
\ . i
5 : : 18 : 6. 47. ( )
*::54: 6. 48. ( )
: 8
::1J
: : i
$2:
)6 :
12
12
Ex. 2. — What sum of money contains £7, as often as 12 oz.
contains 3 oz. 1
Since 12 oz. contains 3 oz. four times, and .£28 contains £7
four times, £28 is the sum sought ; thus : —
12 oz. : 3 oz. : : £28 : £7.
Quantities may be proportional as well as numbers.
But the two terms in each ratio must be of the same kind,
as only such can contain each other.
EXERCISE VI.
10.
11.
12.
17.
18.
19.
20.
9 lb : 3 ft> : : £12 : ( )
18 cwt. : 6 cwt. : : £60 : ( )
16 yds. : 4 yds. : : £4 : ( )
15 ac. : 3£ ac. : : 16s. : ( )
2o yds. : 5 yds. : : ( ) : 6s.
9 ac. : 36 ac. : : ( ) : 12s.
ml. : 22 ml : : (' ) : lOd.
>. ::( ):.
15 ho. : 60 ho.
:£36.
7 ml : ( ) : : £8 : £24.
8 wk. : ( ) : : 6/8 : £2.
8 in. : ( ) : : 32/ : 4/.
45 galls. : ( ) : : £63 • £7
5. 6 da. : 1£ da. : : 3 gs. : ( )
6. 10 ml. : 2 ml. 4 f. : : lOd. : ( )
7. 7 gall. : 3£ gall. : : £10, 10 : ( )
8. 8 cub. ft. : 1 c. ft. : : £9 : ( )
13. 24 in. : 6 in. : : ( ) : 9d.
14. 5 qrs. : 30 qrs. : : ( ) : £72.
15. 27 lb : 3 lb : : ( ) : 4/6.
16. 10 dwt. : 25 dwt. : : ( ) : 20/.
21. 4 years : ( ) : : £600 : £150.
22. 32 qrs. : ( ) : : £5 : £1, 5.
23. 96 yds. : ( )::£!,!: 14/.
24. 72 tons : ( ) : : £132 : £12.
96 RULE OF THREE.
: 16 ml. : : 2$ : lOd. 29. (
: 23 qrs. : : 3 A : 14<L 30.
27. ( ) : 3 ac. : : 6/8 : £3. 31.
28. ( ) : 2 days : : 15/ : 3/9. 32.
: 1 qt. : : £27 : £9.
: 2 pk. : : £8, 10s. : £2, 2/6.
: 1 cwt. : : £100 : £5.
8 oz. : : 7£d. : 2£d.
JLU f If 12 oz. contains 3 oz. as often as £28 contains £7, then
3 oz. is contained in 12 oz. as often as £7 in £28,
also,
£28 contains £7 as often as 12 oz. contains 3 oz., and
£7 is contained in £28 as often as 3 oz. in 12 oz.
Every proportion may thus be expressed in four ways : —
either of the two ratios may be put first, which gives two ways ;
and either of the two terms of each ratio may be put first,
which gives two ways for each ratio ; making four ways in all.
Thus :—
(1.) 12 oz. : 3 oz. : : £28 : £7. (3.) £28 : £7 : : 12 oz. : 3 oz.
(2.) 3 oz. : 12 oz. : : £7 : £28. (4.) £7 : £28 : : 3 oz. : 12 oz.
EXERCISE VII.
Read each of the proportions in Ex. v. and vi. in its four different
orders.
108 In any proportion, such as 12 : 6 : : 8 : 4, the 1st and 4th
terms are called the extremes, the 2d and 3d are called the
means.
. The product of the means is 6X8=48. Now, as the first
extreme is as many times greater than the first mean, as the
second extreme is less than the second mean, the product of
the extremes must be the same as that of the means ; which it
is, for 12X4=48.
This is a second way, therefore, in which a proportion may
be tested ; the product of the numbers whicn form its extremes
is equal to the product of the numbers which form its means.
EXERCISE VIII.
Test each of the proportions in Exercises v. and vi, proceeding
thus :— 4 : 2 : : 8 : 4. " Four times 4 are 16, product of the extremes ;
two times 8 are 16, product of the means."
109 Since the lstX4th = 2dx3d, it follows that when three
terms of a proportion are given, we can find the remaining
term. Thus : —
, , 2dX3d 0, Istx4th „, 1st X 4th , , 2dx3d
lst==--' 2d=-~' 3d= --'4th--"
PKOPOKTION.
97
Ex. 3. — What is the first term in the proportion :
( ) : 15 : : 12 : 20 ?
15X12_0
lst=
20
l.( ):
2. ( ):
7. 45 : |
8. 55 : i
13. 14:
14. 15 :
19.2:1
20. 4 : i
21. 12 :
22. 21 :
EXERCISE IX.
) : 12 : : 15 : 9. 5. ( ) : 60
) : 9 : : 30 : 22.}. 6. ( ) : 64
:: 792: 990. 9. 25 : ( ) : : 175 :49. 11. 143 :
: : 495 : 594. 10. 572 :( ) : : 360 : 95. 12. 252 :
: : ( ) : 400. 15. 60 : 45 :
; : ( ) : 15. 16. 177 : 59
23. 37 : 150 : ;
24. 10 : 19 : :
<?n Qfi • K9 • •
: : : 8 : 20.
' : : 14 : 63.
:985:
:100:
: 84 : 210.
; 162 : 288.
) ::11:9.
i:21. 17. 234 : 252 : : ( ):28,
:37i 18. 19:32::( ):428,
: ( ) 27. 18 : 29
: ( ) 28. 48 : 41
i. 96 :52 ::18 :( ) 29.113:84
26. 20 : 24 ;
30. 912 : ]
234 : (
576 : (
:3289:(
i::57:(
JLU The first and second terms must not only be of the same
kind, but, when compound, must be reduced to the same name ;
the third term, when compound, must also be reduced to its
lowest name.
The fourth term, when found, is of the same name as that
to which the third has been reduced ; and must be brought
back, when necessary, to its own highest name.
1.
2.
3.
4.
6.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
EXERCISE X.
4 cwt. 2 qrs. 8 ft : 9 cwt. 2 qr. 24 lb : : £10, 3s. : (?)
3 ml. 3 fur. 18 po. : 1 fur. 21 po. : : 10/10.4 : ( ? )
16 el. 0 qrs. 2 nl. : 25 el. 1 qr. : : £1, 11s. 7Jd. ( ? )
1 cub. ft. 36 cub. in. : 189 cub. in : : £2, 18s. lid. ( ? )
£1, 14s. 4d. : £38, 3s. lid. : : 4 sq. ft. (?)
7 qrs. 7 bu. 3 pk. : 4 bu. 1 pk. : : 20 guineas ( ? )
22 ft tr. 2 oz. : 1 ft 7 oz. : : £2, 13s. 8d. ( ? )
97 yds. 2 ft. 10 in. : 17 ft. 1 in. : : £22, 2s. 6Jd. ( ? \
1 ton. 6 cwt. : 1 ton 16 cwt. : : £1, 9s. 3d. (?)
22 yds. 8 in. : 1 ft 5 in. : : £1, 13s. 4d. (?)
12 sq. ft. 72 sq. in. : 72 so. in. : : £1, Is. 4jd. ( ? )
13 ac. 2 ro. 36 po. : 9 ac. 3 ro. 6 po. : : £5, Is. 8d. ( f *
1 ho. 48 min. : 2 da. 6 ho : : 8/4. ( ? )
38 lb : 4 st. 10 :: £1, 3s. 9d. (?)
26 ft : 12 qr. 2 lb : : 4^d. ( ? )
£1, 3s. O^d. : lOJd. : : 34 wk. 5 da. (?)
20 galls. 2 qts. 1 pt. : 5 galls. : : £4, 14s. 6d. (?)
15 qu. paper 20 sh. : 19 sh. : : £3, Is. 8d. ( ? )
39 qrs. 6 bu. 3 pk. : 2 qrs. 5 bu. 1 pk. : : £15. ( ? )
2yrs. 20 da. : 25 da. : £54. (?)
3bu. 2pks. : 52.} bu. :: 12/9. (?)
1 ton 13 cwt. 3 qrs. : 2 tons 3 cwt. 3 qrs. : : £2, 16«. ( ?)
9 oz. 9 dwt. : 1 ft 12 dwt. : : 7/10i. ( ? )
•
.
98 RULE OF THREE.
Ill Ex. 4.— If 6 ft) butter cost 5/3, what cost 3 st. 3 ft) ?
This is a question in the Eule of Three, and may be wrought
by multiplication and division, as already shown.
It may be solved, however, by means of a statement of propor-
tion, which in many cases facilitates the applying of the two
processes.
Thus 6 ft) are contained in 3 st. 3 ft) as often as the price of
6 ft) is contained in that of 3 st. 3ft> : hence
lb st. Ib s. d.
6 : 3 3 : : 5 3 : 14 st. 6 ft).
14 12
4Mb 63
4th term=45*63 =gJg=472^d. = 0£l, 19s. 4J.
u 2
Bule for statement. — For the third term place that
•which is of the same kind as the answer. Find out from the
question whether the answer is to be greater or less than the
3d term ; if it is to be greater, place the greater of the other
two terms second ; if less, place the less second.
Ex. 5. — If 12 men build a wall in 8 days, how many will
build it in 6 days ?
Here the answer is to be so many men ; place 12 men in
3d place. Then, as it will take more men to build the wall in
6 days than in 8, the 8 must be placed second, and the 6 first,
da. da. men.
112
Thus, 6 : 8 : : 12 : answer i-*P=16 men.
EXERCISE XL-MISCELLANEOUS.!
1. If 4 pints of gooseberries cost 1/8, for how much may a party of
12 persons get a pint and a half each ?
2. If 16 copies of an arithmetic cost 5/, how much will it take to
supply a class of 50 ?
3. If 4 galls, ale cost 28/, what cost 1 hhd ?
4. A farmer was offered 65 sheep for. £73, 2s. 6d., but he took only
20 : what did they cost him ?
5. Two dozen oranges cost 2/6, now many may I buy for 7£d. ?
6. Six pairs shoes cost 34/6, what cost 6^ dozen pairs ?
7. What cost 48 yds. cloth, if 72 yds. cost £26, 5s. ?
8. What cost 15 casks sugar, at the rate of £5, 10s. for 4 ?
9. If | ft tea cost 1/1$, how much may be had for 27/?
10. If 8 oz sugar cost 2^d., what cost 7 ft.?
1 To be wrought by the direct application of the elementary rules, or by
means of a statement of proportion, at the teacher's discretion.
ntoroiiTiON. 90
11. If 11 men do a piece of work in 17 days, in what time will 25
men do it ?
12. A farmer has 36 lb bread for his 24 reapers, but he engages
other 6 : how much does each get ?
13. If 2 lb 6 oz. meat cost 1/10, what is that per lb?
14. My railway fare is 6/3 for 30 miles, what will it be for 92 miles ?
15. If 35 oz. silver-plate cost £9, 3s. 9d., what cost 34 oz. ?
16. In a bundle of 26 sovereigns, 1 lose 9|d. on three, how much
may I expect to lose in all ?
17. If I walk 9 ml. 6 fur. in 3 hours, in what time shall I walk 37
ml. 5 fur. at the same rate ?
18. If f stone of salt cost 3^d., what cost 1| cwt. ?
19. I have 1/6 to spend on apples, how many shall I get at 3 for 2d. ?
20. If a class writes 6 lines per lesson, in how many lessons will
they fill a copy-book of 12 leaves, each having 9 lines ?
21. How many yds. linen can be had for £24, 18s. 8d., if 17 yds.
cost £3, 2s. 4d. ?
22. If 22 ac. 2 ro. 30 po. let for 65 guineas, what is the rent of
173 ac. 3 ro. 30 po. of the same land ?
23. Eight men are engaged in ditching at 12/4 a day for the whole ;
if 5 more are employed, what will be the daily expense ?
24. If a ton of coals cost 19/6, what is that per ^ cwt. ?
25. What is the weekly rent of a room at 19/10 for 17 days ?
26. A field of 38 ac. was ploughed in 16£ days : in what time would
a farm of 175 ac. 20 po. be ploughed ?
27. If £100 gain £4, 10s. in a year, what is the gain for the same
time on £22, 10s. ?
28. A tradesman's wages are £168 per year : how much is that for
25 days ?
29. I can go a journey of 63 miles in 18 hours, which the coach
goes in 7£ hours : how many times faster can I go by coach than by
walking ?
30. How long will a field, which has pasture for 6 score sheep for
15 days, graze 4 score and 5 ?
31. Two slates have the same area. The one is 8 inches by 14 in.,
the other is 10 in. broad : what is its length ?
32. A parcel is carried 15 miles for 2/6 ; if I pay 19/3, how far should
it be carried ?
33. A parcel is carried 15 miles for 3/6 : for what should it be carried
172 1 miles ?
34. The railway van delivers a parcel weighing 1 ton 4 cwt. 3 qr.
for 5/6 : what is the weight of a parcel for which the charge is
17AOJ!
85. If 4 tons 14 cwt. cost £9, 5s. 6d., what cost 14 tons 16 cwt. ?
36. I bought 28 lb sugar at 5d. per lb : wishing to change it for
sugar at 5<]d., what quantity should I get?
37. If 4 guineas weighed 21 dwt. 12 gr., what is the value of 10 oz.
14dwt. 6 gr. of gold?
38. The current of a river runs at the rate of 1\ miles in 3 hours :
in what time will a boat drift to the sea 245 miles off?
39. If the current reaches the sea, 150 miles down, in 45 hours, in
what time will it pass a village 24 miles down ?
40. What cost 1000 pencils at 5 for 3£d. ?
113
100 RULE OF THREE.
EXERCISE XII.— MISCELLANEOUS- -continued.
1. A floor is covered with carpet, 25 yards of 5 qr. wide : liow many
yds., 3 qr. wide, would be required ?
2. If the sixpenny loaf weighs 4 Ib when wheat is 32/ per quarter,
what should it weigh when wheat is 27/ ?
3. If the 4 Ib loaf sells for 7.^d. when wheat is at SO/, what should
it sell at when wheat is 34/ ?
4. How many yds. cloth at 10/6 per yd. should I receive in exchange
for 4 pieces of 7 yds. each at 8/6 per yd. ?
5. What is the price of a pipe of wine at 41/3 for three gallons?
6. A bankrupt pays £250 on a debt of £350 : if my claim is £15,
how much shall I get ?
7. Bought linen at 7 yds. for £1, 2s. UJd. : how much may be bought
for £4, Os. 11 id. ?
8. A pipe fills a cistern of 280 gall, in 31 hours : in what time will
it fill a 20 gall, cask ?
9. How many pounds sugar at 4|d. should I receive in exchange for
2 casks butter, each of 25.^ Ib, at 10£d. per K> ?
10. What cost 24 bales cotton, each 90A Ib, bought at the rate of
I/a for 7 ft 1
11. If a tree 20 feet high cast a shadow 30 feet long, how much
longer will be the shadow of one 50 feet high ?
12. If 60 men can build a house in 90^ days, how much longer will
it take 15 men to build it ?
13. What weight Avoirdupois is equal to 9 Ib 6 oz. Troy, if an oz.
Troy is to an oz. Avoirdupois as 192 to 175 ?
14. A. and B. contribute to the capital of a business £1955 and
£1720 respectively. The profits for the first year are £910 : B. receives
a salary of £175 for management ; how should the rest be divided ?
15. A goods train runs 10 miles an hour for 7 ho. 30 min. : in what
time will the express train make the distance at 36 miles per hour?
16. A tradesman has a yearly wage of 75 guineas : if he leave his
situation on 18th August, how much has he to receive ?
17. If the diameter of a circle is to its circumference as 113 to 355,
what length of plank will measure across a tank 45 feet round ?
18. Two plots of ground are equal in area : if the one is 28 feet long
by 22 broad, and the other 77 feet long, what is its breadth ?
19. What cost 3 Ib raisins at £6, 7/6 per barrel of 100 Ib ?
20. How many yards cloth at 15/6 per yard should be exchanged
for 230 yards at 17/6 per yard ?
21. If a bankrupt owes £279, 10/, and his estate realizes £200, how
much will he pay per pound ?
22. A man takes 7 steps for a lad's 13 : in a distance of 8 miles, if
the lad takes 19,500 steps, how many more does he take than the
man?
23. A pound Troy, silver, is coined into 66 shillings : what is the
value of 1 Ib Avoirdupois ?
24. Bought iron at £37, 10/ per ton : how must it be sold to gain
the price of 1 ton upon 15 ?
25. If £10 sterling are equal to 256 francs, how many francs should
be got for 36 guineas ?
26. At the rate of 16/8 sterling for £1 Canadian currency, what is
the sterling value of £1000 currency }
COMPOUND RULE OF THREE. 101
27. Find the rent of 106 ac. 2 ro. 30 po. of land at the rate of £25
for 23 ac. 1 ro. 27 po.
28. A ditch is measured with a line supposed to be 22 feet long, and
found to be 530 yards long ; but the line is discovered to be 23 feet
6 inches long : what is the true length of the ditch ?
29. Of 75 men engaged for 24 days to do a piece of work, only 65
make their appearance : how much longer time will be required ?
30. £29, 17s. 6d. was collected as rate on a property at 9^d. per
pound : find the rental.
31. How many yards of carpet, f yard wide, will be required for a
hall, now covered with 87^ yards l| yard wide ?
32. If 6 men or 10 women can do a piece of work in 12 days, in what
time will 3 men and 7 women do it together ?
33. Twenty-five printers can do a job in 12 days ; but three are sick :
how long does that delay it ?
34. What length of board, !
, 9| inches wide, will make a square foot ?
35. How many gallons water must I add to a puncheon of whisky
which cost £39, 4/, to sell it at 9/ per gallon, and neither gain nor
lose?
COMPOUND RULE OF THREE.
Ex. — If 16 cwt. are carried 45 miles for .£3, 10s., how far
ought 36 cwt. be carried for ,£15, 15s. ?
Here the distance required depends on two things —
(1.) The weight to be carried, and (2.) the price to be paid.
The question therefore resolves itself into two others :
(1.) If 16 cwt. are carried 45 miles for a certain price, how
far ought 36 cwt. to be carried 1 which gives the statement —
36 cwt. : 16 cwt. : : 45 ml. ; and
(2.) If .£3, 10s. pay the carriage for 45 miles, for wbat dis-
tance should .£15, 15s. pay? which gives the statement —
,£3, 10s. : ,£15, 15 : : 45 ml.
These statements are accordingly combined, thus :
36 cwt. : 16 cwt. : : 45 ml.
£3, 10 : £15, 15.
The third term in the combined statement being 45, the first
term of the whole is the product of the two several first terms ;
the second of the whole the product of tbe two several second
terms ; and the fourth term = 45Xl6x315 = go ^
36X70
A statement of proportion, consisting of two or more state-
ments in simple proportion, is said to be in compound pro-
portion ; and the questions solved by it are sometimes said — •
because involving more than one question in Rule of Three —
to be in Compound Rule of Three.
102 COMPOUND RULE OF THREE.
Rule.— Let the third term be of same kind as the answer.
Arrange each pair of similar terms as in simple Rule of Three.
Multiply all the first terms together for the compound first term,
and all the second terms together for the compound second
term ; then work as in simple Rule of Three.
H5 EXERCISE I.
1. If 24 tons are carried 68 miles for £8, 10s. 6d., for what money
will 14 tons be carried 120 miles ?
2. If I travel 144 miles in 4 days of 9 hours, in what time will I
travel 560 miles, walking 8 hours a day ?
3. And liow far will I travel in 15 days of 10 hours daily ?
4. If 27 persons consume 252 lb bread in 6 days, how many pounds
will 146 persons consume in 5 days ?
5. And how many persons will consume 1 ton weight in 14 days?
6. If 20 men can mow a field of 81 ac. in 9 days, how many men
can mow 36 ac. in 10 days ?
7. If 16 men can build a wall 60 ft. long, in 45 days of 8 hours
each, in what time will 35 men build a wall 145 ft. long, working 10
hours a day ?
8. A household of 18 persons spends £35, 15s. in 6 weeks : at same
rate what should maintain a household of 14 persons for a quarter of
a year?
9. If 5 labourers earn £12, 18s. 9d. in 16 days, how much will 12
labourers earn in a week of 6 days ?
10. If 18 yds. cloth, yard wide, cost £25, 7s. 6d., what cost 11 yds.
5 quarters wide ?
11. If £250 gain £28 in two years, in what time will £675 gain 150
guineas ?
12. If the fourpenny loaf weighs 72 oz. when wheat cost 77/ per
quarter, what should the threepenny loaf weigh when wheat is at 81/
per quarter ?
13. And what is the price of wheat when the sixpenny loaf weighs
4 lb?
14. If it cost £3, 12s. 6d. to paper a room 21 ft. square and 14£ high,
what is the cost to paper one 15 it. square and 9 ft. high ?
15. I borrowed £50 for 6 months when money was at 5 per cent. :
how much should I lend for 10 months in return at 4 per cent. ?
16. It costs 16/6 to supply 14 men with bread for 9 days when wheat
is at 12/6 per bushel : what is the price of wheat to furnish 25 men
for 7 days at same cost ?
17. To fill a tank of water, 30 ft. long, 8 ft. deep, and 24 broad,
cost £3, 2s. 6d. : what cost it to fill another 6 ft. greater every way?
18. A wheel, 15 feet round, makes 52 revolutions a minute, and
goes 75 miles in 5 hours : what distance will be traversed in 8 hours
by another, which is 17| feet round, and revolves 45 times a minute?
19. If a boatful of herrings, containing 5000, be sold for £17, 13s. 6d.,
what should a boatful of 8560 bring, three of the latter being equal to
in value to five of the former ?
20. A block of stone, 9 ft. x 6 ft. x 4J ft. weighs 5^ cwt, what is the
weight of a stone -|d longer each way f
21. If the wages of 18 workmen amount to £78, 14s. for 22 days,
MEASURES AND -MULTIPLES. 103
*hat will the wages of 25 apprentices coine to in 45 days, if 7 work-
men receive as much as 13 apprentices ?
22. If £100 gain £4 in one year (1.) what sum will gain £36 in four
years? (2.) what will £500 gain in six years? (3.) in what time will
£400 gain £2 ?
23. A besieged town of 2500 men had provisions for 60 days at the
rate of 30 oz. daily for each man : (1. ) how long will the stock last if
they are reinforced by 500 men, and receive 24 oz. daily? (2.) what
rate of distribution will make it last for 40 days ?
24. A family of 5 persons spend £18, 15s. in 3 weeks : (1.) what will
it cost to maintain, at the same rate, a household of 18 persons for a
year? (2). for what time will £225 maintain a family of 6 persons?
(3.) how many persons will 1000 guineas maintain for 5 yrs. 20 wks. ?
. 25. If 45 men cast a ditch, 48 feet long 9 broad 6 deep, in 24 days
of 9 hours each —
(1.) What length of drtch will 75 men cast in 15 days 10 hours each,
6 feet broad and 8 feet deep ?
(2.) How many men would cast a ditch, 6 feet broad 20 long and 8
deep, in 15 days of 8 hours each ?
(3.) In how many days of 8 hours each, would 15 men cast a ditch
12 feet long by 8 broad and 4 deep ?
(4.) How many hours daily, for 10 days, would 16 men take to cast
a ditch 20 feet long by 8 broad and 4 deep ?
LI 6 MEASURES AND MULTIPLES.
Prime Numbers.
A number which cannot be broken up into factors is called
a prime number ; but 1 is not counted as a factor.
A number which can be so broken up is called a composite
number.
Thus 3, 5, 7 are prime numbers ; and
4, 6, 8 are composite numbers.
To find whether a number is prime or composite, we divide it
by any of the simple numbers which we think it will contain ;
bearing in mind the rules already given for this purpose in
sect. 53.
EXERCISE I.
"Resolve the following numbers into prime factors : —
1. 27, 46, 64. 4. 156, 240, 236. 7. 1456, 1728, 2445.
2. 95, 56, 121. 5. 198, 432, 560. 8. 2384, 5408, 7280.
8. 87, 115, 140. 6. 375, 820, 972. 9. 6372, 4116, 4716.
10. Write down the prime numbers in order from 1 to 50.
11. Do. from 50 to 100. • (12.) Do. from 100 to 150.
104 MEASURES AND MULTIPLES.
117 Greatest Common Measure.
One number is called a measure of another when it is con-
tained in it evenly ; thus 6 is a measure of 12.
One number is called a common measure of two or more
numbers, when it is contained in them evenly ; thus 3 is a
measure of 18 and 24.
One number is called the greatest common measure of two
or more numbers, when it is the greatest number which is con-
tained in them evenly ; thus 6 is the greatest common measure
of 18 and 24.
The greatest common measure can often be found by inspec-
tion ; when it cannot, we proceed as follows : —
Ex. — Find the greatest common measure of 237 and 395.
Divide the greater (395) by the less 237)395(1
(237) ; then the first divisor (237) by the 237
first remainder (158) ; then the second 158)237(1
divisor (158) by the second remainder 15g
(79) ; which is a final divisor. — 79)1 W2
Now 79 is the greatest common measure 158
of 79 and 158. But whatever measures
any two numbers measures their sum ; therefore 79 is the
greatest common measure of 158 and 237. On the same prin-
ciple, the greatest common measure of 158 and 237 is the
greatest common measure of 237 and 395, the given numbers.
Therefore 79, the last divisor, is the greatest common measure
required.
Rule. — Divide the greater of the two numbers by the less,
the first divisor by the first remainder, the second divisor by
the second remainder, and so on. The last divisor is the
greatest common measure required.
If there be three or more numbers given, find the greatest
common measure for the first two, then for that greatest com-
mon measure and the third number, and so on.
EXERCISE TI.
Find the greatest common measure of the following numbers : —
1. 122 and 427. 5. 2387 and 2563. 9. 5705 and 6559.
2. 148 and 185. 6. 2002 and 2509. 10. 18996 arid 29932.
3. 285 and 465. 7. 2145 and 3471. 11. 5415 and 30105.
4. 576 and 744. 8. 6465 and 7335. 12. 34789 and 85937.
13. 36, 84, 720. 17. 1241, 1428, 1853.
14. 45, 63, 108. 18. 925, 1475, 5680.
15. 256, 372, 522. 19. 1092, 1716, 2940.
16. 244, 472, 636, 20. 3081, 5451, 6255, 8703.
MEASURES AND MULTIPLES. 105
Least Common Multiple.
One number is a multiple of another when it contains it
evenly, and a common multiple of two or more numbers when
it contains them all evenly.
One number is the least common multiple of two or more
numbers, when it is the least number that contains them evenly.
If the numbers are prime to each other, the least common
multiple is got by finding their product ; thus, the least common
multiple of 11 and 13 is 11 X 13 = 143. But, if they have some
common factor, the least common multiple is got by throwing
out the greatest common factor and multiplying the remaining
factors together ; thus, the least common multiple of 24 and
36 is, by throwing out the factor 12 from one of the numbers,
24X3 or 2X36 = 72.
Ex.— Find the least common multiple of 15, 24, 30, 42, 72.
15, 24, 30, 45, 72
15, 45, 36
~~
Arrange the numbers in a line. 2
Strike out 15 and 24, since any 3
multiple of 30 and of 72 will be 0
i, • 1 i r j.1 O
a multiple also of them.
Divide by the first prime factor '
of more than one of the numbers,
that is, by 2; writing the quo- 2X3X3X5x4 = 360 Lc.m.
tients, and any of the numbers
not divisible by 2, in a line below ; thus, 15, 45, 36.
As 2 is not a common measure of these, divide by the next
prime, which is 3 ; giving 5, 15, 12.
Divide again by 3, which is a common measure ; giving
5, 5, 4.
Divide by the next prime which is a common measure, that
is 5 ; giving 1, 1, 4. These numbers have no common measure.
The divisors and the remaining factors are the only necessary
factors of the least common multiple required.
Rule. — To find the least common multiple of two num-
bers : — Divide one of them by the greatest common measure,
and multiply the other by the quotient.
To find the least common multiple of more than two num-
bers : — Throw out any number which is a measure of another ;
divide the rest by 2 as often as it is a common measure,
bringing down the undivided numbers into the line of quotients ;
then" by 3 and the other primes in order, till the division is
exhausted ; multiply the divisors and the remaining quotients
for the least common multiple.
106 VULGAR FRACTIONS.
EXERCISE III.
Find the least common multiple of the following numbers : — •
1. 16, 20. 6. 63, 108. 11. 14, 21, 28. 16. 288, 360, 1728.
2. 32, 48. 7. 98, 156. 12. 18, 32, 56. 17. 720, 336, 1736.
3. 56, 64. 8. 391, 659. 13. 32, 44, 52. 18. 6, 9, 15, 18, 20.
4. 120, 144. 9. 703, 1036. 14. 17, 29, 53. 19. 1, 2, 3, etc., to 9.
5. 72, 132. 10. 1548, 2537. 15. 90, 100, 125. 20. 5, 7, 9, 12, 15.
21. 25, 60, 72, 35. 23. 3, 7, 8, 9, 11, 49, 55.
22. 14, 54, 63, 81. 24. 12, 16, 24, 36, 48, 72, 144.
119 VULGAR FRACTIONS.
Notation of Fractions.
Any part of a whole number is called a fraction.
Thus, if the line A D be divided into three equal
parts, any one part, as A B, is one-third of the whole,
denoted J, and any two parts, as A c, are two-thirds
of the whole, denoted §.
The lower of the two numbers is the name, or, as
it is called, the denominator of the fraction, and shows
the size of the parts into which the whole is divided ;
the upper is the number, or, as it is called, the nume-
rator of the fraction, and shows the number of these
parts which the fraction contains.
Fractions denoted by a numerator and denomi-
nator, like J or §, are called vulgar fractions, to dis-
tinguish them from a certain kind of fractions, which
—as will be noticed further on — may be denoted in
a different way.
Any part of a number larger than one is also a
fraction. Let the whole line AC be composed of
AB = 1 and BC = 1, and therefore be = 2. Divide
A c into three equal parts, A D being one : divide A B
into three equal parts, A D will be found to be
two. Thus § of 1 is equal to J of 2 ; and both are
denoted f .
EXERCISE I.
Read the following fractions (1.) as fractions of unity ; and (2.) as
fractions of their own numerators.
i. f, *, T9*> &, &, H. 3. y, ft, v°, to m-
2. fc H, ifc to i*> T*I- 4- to Ik tt> tt> *&•
--B
VULGAR FRACTIONS. 107
A whole is equal to the sum of all its parts ; thus the line
A D above is equal to its three thirds together, A B-J-B c-f-c D.
Fractions, whose numerator and denominator are the same
number, such as f , f, or £ , etc., denote one unit, broken up
into 4, 5, or 6 parts respectively ; and therefore they are all
equal in value to one another.
>Q Improper Fractions and Mixed Numbers.
If we have two whole numbers, and divide each into three
equal parts, we have six parts each the third part of one. If
we take four of these, we have a fraction made up of one whole
number and one-third more, and denoted f. A fraction
greater than one, and whose numerator is consequently greater
than its denominator, is called an improper fraction.
An improper fraction may always be resolved into a whole
number, or into a whole number and a fraction, which is called
a mixed number ; and so a whole number, or a mixed number,
may always be resolved into an improper fraction.
Ex. 1. — How many whole numbers in 2^ ? and in %f 1
Since seven sevenths are one whole number, there will be
as many whole numbers in *p as there are sevens in 28, that
is, 4 ; which is the answer.
And as many whole numbers in %f as there are sevens in
29, that is 4}- ; which is the answer.
Rule. — To change an improper fraction to a whole, or to
a mixed number, divide the numerator by the denominator,
EXERCISE II.
Change to whole or mixed numbers : —
i. v, ¥, v, v, if- 4. v, ft, is, w, m-
2. ¥*2, W, W, W- 5- w, W, W, **•
3. V, It, ffc II &• 6. v9i3, <W, ft, Hi-
Ex. 2. — How many sevenths in 5 ? and in 5 J ?
Since there are seven sevenths in 1, there will be 5 times
as many in 5, that is 3^.
To 35 sevenths add the 6 sevenths already given in 5|, and
the total number will be ^.
Rule. — To change a whole number, or a mixed number, to
an improper fraction, multiply the whole number by the given
108 VULGAR FRACTIONS.
denominator, and (in a mixed number) add the given numera-
tor. Place the denominator below.
EXERCISE III.
1. How many ninths and elevenths in 3, 6, 8, 9, 13, 16?
2. Write with 8 and 12 as denominators :— 4, 6, 7, 10, 12, 15 ?
Change to improper factions :
3. 24, 8ft, 9J, 120$. 5. 17ff, 29|, 45&, T2&.
4. 8ft, 73^, 125tf, 200ft. 6. 342£, 74^, 2ft£f . ;
121 Equal Fractions of Different Denominators.
Let the line A c be divided into three equal parts,
A B being $. Let each of these third-parts be sub-
divided into two, making six parts in all : A B will
contain £. Thus f — ^ ; and a fraction is not altered
in value, if its terms be multiplied or divided by the
same number.
Rule. — To change a fraction to higher terms,
multiply both by the same number : to change it to
lower terms, divide both by a common factor.
Note. — In all operations with fractions, they should, as a
rule, be reduced to their lowest terms. The common factor to
be taken as divisor may often be found by inspection ; if not,
find the greatest common measure, and divide both by it.
EXERCISE IV.
1. How many ^ths in $, \, I, \, TV, TV, •& ?
2. How many ^ths in fe f, f , £, £ , |, &, A, &, ft, ft, f|, ft ?
3. How many T^ths in \, f, f, f, f, ft, ^, ft, ft, ft, f|?
4. Reduce to their lowest terms— &, £f , f-f, |f, ^^ ft|.
5. Also, T^, ftf , Ml, ftft, Hft, mi, MM, iV^«
6. Also, H|, flfl, |Mi, m I IM,
-B
Common Denominators.
^x. 1. — Change the series of fractions J, |, J, ^ to another
series having the same denominator.
The least common multiple of the denominators is the new
denominator required.
It is obvious that 12 contains them all ; so that all the frac-
tions have to be brought to ^t
VULGAR FRACTIONS. 109
Multiplying the terms of the first by 6, of the second by 4,
of the third by 3, we have, as the series of fractious required,
T2> T2> V2"> T2*
Ex. 2. — Change f , J, £ to fractions of the same denominator.
The denominators being prime to each
other, the common denominator is f X 4 X 5 = |§
3X4X5 = 60. kx3x5 = 45
Then, multiplying the terms of each ^ x 3 x 4 = 48
fraction, so as to bring it to ^th, we have
Jtule. — To change fractions to a common denominator : find
the least common multiple of the given denominators for the
new denominator, and multiply each numerator by the number
of times the new denominator contains the old.
EXERCISE V.
1. !, *, A; *, t, A; i, i A, H; *, &, If
2. *,f,i; f,*,f; *,*»*; *>*,*; f,*,A-
3. A, H, tt ; §> ^ A, A ; A, H, H ; i, I, H, if.
4 £ 7 8 11 13. 5 JT 13. 56 29 _4\ 194 8 S4
*• 6* l~8> IT> 1TO; 60 > 8J ~tt) "24 > TDIT^ TJlfOJ TGU^ BTTd^ 8TTO'
I Addition and Subtraction.
Ex. — Find the sum and the difference of J and f .
The fractions must first be brought to the same name : |-J
Thus, as 21 sh.+16 sh. are 37 sh., so f J + Jf = JJ, or 1
And, as 21 sh.-16 sh. are 5 sh., so
^Tbte. — The denominator is not changed, since adding or sub-
tracting quantities does not change their name.
Rule. — To add or subtract fractions, change them to a
common denominator, add or subtract their numerators, and
place the common denominator below.
EXERCISE VI.
l. i + 1 + A* f + A + &> f + i + 8, 1 + f + I + H-
2- A + A + A* TS* + A + H> A + A + M, M + *f + H-
3. 2| + 44 + 6|, 7f + 9A + l^ii, If + »A + 1
*. I - f . A - A, H - A, H - H» T9^ - A-
5. 8* - 21, 9ft - 7|, 42^ - 264, 158£f i 79§f-
: Multiplication of Fractions.
Ex. 1.— Multiply A- ^7 3.
Since 4 X 3 = 12, ^5- X 3 = i^ = If ^ |.
This is also the result for 3 X r§-.
110 YULG All F 11 ACTION S.
Ex. 2.— Multiply ^ by f,
T% X 3 = ^ ; but, as the multiplier is only the fifth part
of 3, this product is 5 times too great, and the product required
*'B£'«*
Rule. — To multiply a fraction by a whole number, multiply
the numerator, or divide the denominator, by the whole number.
To multiply a fraction by a fraction, multiply the numerators
together, and also the denominators together, cancelling when
possible. If one of the terms be a mixed number, change it
to an improper fraction.
Note. — ^ of f is another way of denoting the fraction got
by multiplying T^ by f ; and is called a compound fraction.
EXERCISE VII.
1. |f x 4, 7, 8, 14, 18, 6, 24, 48, 150.
2. f£ x 12, 16, 4, 42, 7, 26, 36, 21, 84, 256.
3. 288 x f, If, &> H, *, A, ft, Hi tt> ffl.
5. Ifxf, 2{fxlfc, 7fx6§, lO&xlOf,
6. 5x£of£, 4fx|of^, f of2£x|,
*** In connexion with this and the following section, the pupil may revisf
sections 52 and 53.
Division.
Ex. 1.— Divide £ by 2.
Since 4-=-2 is 2, -f-s-2 is f. If the numerator does not con-
tain the divisor, multiply the denominator instead ; thus -J-v-2,
being equal to f , is also equal to •£$.
Ex. 2.— Divide £ by J.
A_i_2=_:L. But, as we are to divide only by the third
part of 2, we have divided by a number 3 times too great, and
therefore the quotient is 3 times too small. The quotient re-
4x3
quired will therefore be g^, that is, f 'or If.
Rule.— To divide a fraction by a whole number, divide the
numerator, or multiply the denominator, by the number ; to
divide a fraction by a fraction, invert the divisor, and multiply
the fractions.
EXERCISE VIII.
1. T4&-M2, 4, 16, 8, 48, 96, 192, 14, 36, 40.
• 2. tf-H. *Ki, flH-H> tm > 2J-M& 5H-tf.
3. * of I-M of |, -I of 5H-& of 3$, i|^f of 9f .
4- 34!-^, A of f|-l of fc T«T of SS-5-t, H of
VULGAR FRACTIONS. Ill
.26 Reduction.
Fractions of quantities often require to be reduced to a
higher or a lower name.
Ex. 1. — What part of a shilling is ^-th of a pound ?
Pounds are reduced to shillings by multiplying by 20,
therefore £^f = |J sh. or f sh.
Ex. 2. — What fraction of a pound is f of a shilling ?
Shillings are reduced to pounds by dividing by 20 ; so
Ex. 3.— What is the value of
Pounds are reduced to shillings by multi- 4
plying by £0 and then by 12. Multiplying 20
by 20, we get 5 for the shillings ; multiply- 15j80^ 5
ing the remainder of shillings by 12, we get 75
4 for the pence. ^~
Ex. 4.— Express 2/3J as a fraction of a 12
pound. ^ )6C)(4
Farthings are reduced to pounds by divid- 60
ing by 4+12+20, or 960. The number of £J± ^Jf
farthings in 2/3J- being 109, the required
fraction therefore is «
Rule. — To reduce a fraction of a quantity from one name to
another : — If to a lower, multiply by the number of times the
lower name is contained in the higher ; if to a higher, divide
by that same number.
Note. — Fractions of different names must be reduced to the
same name, before they can be added or subtracted.
EXERCISE IX.
1. What part of a shilling & of a penny is £T\, £&, £^, £-^S) £^3.
2. Reduce to oz. and to dwt. £ lb tr., fib, £ lb, -*f lb, ^ lb.
3. Express as fractions of a foot and of an inch, T% yd., £ yd., -fa yd,
4. Reduce to quarts and gills, £ gall., ^ gall., ^ pk., ^ pk.
5. What part of a £ is f sh., 3T% sh., f cr., £f cr., ^ sixp., ^ hlfcr.
6. Reduce to quarters £ bu., ^ bu., -|| bu., ££ pk., £f pk.
7. Express as fractions of a yard f qr., | nl., £f qr., & nl.
8. Reduce to cwts. V oz., %$ lb, 3^6 lb, 2 | qrs., 72 ^ lb.
9. Reduce to acres 2| ro., f ro., 25| po., T\ po. 25f sq. yds., & yds
10. Find the value of £|f, £££, ^ cr., ^ flo., f half-gum., ^ guin.
11. Also of ^ yr., ^ day, ^ wk., ^ ac., T% ml., ^ ^ tr.
1 1 2 VULGAR FRACTIONS.
12. Also of ^ fur., f§| cwt., f £ gall., £ f- sq. yd., ££ cub. foot.
13. Express 7/10$ as fraction of £, also as fraction of 17/6.
14. What part of an acre is 2 ro. 27 po., also 15 po. 20 yd. 6 sq. ft. f
15. Reduce 16 wk. to years, and 5 bu. 3 pk. to qrs.
16. Reduce 7 oz. 16 dwt. 20 gr. to ft), and 15 cwt. 1 qr. 25 tb to tons.
127 EXERCISE X.-MISCELLANEOUS.
1. Find the difference between 3£ guineas and £4£.
2. Multiply the sum of £ and £ by their difference.
3. From§ bu. were given away ^ pk. and £ gall. : what remained?
4. How often will the price of 4§ ells Eng. exceed that of 4§ yds ?
5. If 3 of a loaf is divided equally among 12 children, what share
of the whole loaf does each get ?
6. What is the difference between £ of ff and £ of & ?
7. What number has 16 for its £ths ?
8. What number added to i+| + & will make 2 ?
9. What number is contained in £ three times ?
10. I read £ of a book in an hour : when shall I finish it?
11. If a train goes 74 miles in 2 hours, what is that per minute?
12. What part of 5 guineas is 3§ of 5| half-crowns ?
13. A £-cwt. tea in J-lb packages is further subdivided into 6 equal
parts each package : what part of the whole does a family get which
receives 7 shares ?
14. How many 3.} lb loaves are required to give 100 poor people
each Ulb of loaf? "
15. What part is 13/6f of £3, 7s. 10M., and of 3 guineas ?
16. If my property is only ^ of my debts, what is that per pound ?
17. From 340 yards cloth take away |, and then § of remainder :
how much is left?
18. What number is reduced to 64 when $ of it are taken away?
19. Divide the sum of 2$ and 3.4 by their difference.
20. What number multiplied % 3£ will give 2 ; and what number
divided by it will give § ?
21. Bought 45 shares at £105^, and sold them at 106 £ : find the
gain.
22. What part is a square of If inches a side,of one of 3| inches ?
23. In a bag of 1000 sovereigns, each is light by fa dwt., find their
total value. (See § 88, qu. 11.)
24. If I hold f of a house, whose value is £2760, 10s., and sell | of
my share, what value remains to me ?
25. What number will multiply § of llf so as to give 1 ?
26. Find the price of 35f- stones sugar @ 12/4£ per 2^ stone ?
27. If f tt> cost 25/6, what cost 3| cwt. ?
28. What is the breadth of an acre of land 47^ yards long ?
29. In a school of 100 pupils, of whom f are boys, 7 boys and 4
girls are absent : what part of each is present ?
30. If beer is distributed at the rate of 4^ gallons to 9 persons, what
will a family of four persons get ?
31. What part of 14 days 10 hours is § of 2^ days ?
32. How much of a mile remains if 150|y fathoms be cut off?
33. What part of £1 is £ of 6/10, -| of f crown, and f florin ?
34. Add together £ inches + f foot + £ yard.
VULGAR FRACTIONS. 113
35. What number has 9^ for ^ of its eighth part ?
36. If I gain £^ in a day, what part of a crown do I gain per hour ?
37. How many cub. ft. in a box 4^ ft. long, 3£ broad, and 7^ deep ?
38. After walking 15^ miles I had^still £ of my journey before me :
what was its entire length ?
39. If a stopcock empty a cistern in 6 hours and another in 9 hours,
in what time will both together do it ?
40. And in what time will both do it, if No. 2 begins to run after
No. 1 has run for 2 hours ?
41. Divide the quotient of ^fa and j-jjW by their product.
42. If ^ of a property is worth £7g > what is ^ worth ?
43. A shepherd said that if he had as many more sheep, and half
as many more, and quarter as many more, his flock would number
132 : what was its actual number?
44. If 2 persons buy 1 Ib tea, and one pays 2/11 £ for f of it, how
many oz. does the other get, and what does he pay ?"
45. If a labourer can mow a field in 7f$ days, how much of it can
he mow in 1 day ?
46. What number is that of which the third part of its quarter is
47. How many steps of 2f feet each are in a quarter of a mile ?
48. And how many more steps, if each is only 2£ feet ?
49. If from £1, I give away 5s. 9d., then 2/3£, and then -fa of the
£1, what part of it remains to me ?
50. What number is that to f of which if 9 be added, there will
result 19 ?
51. What remains of 1000 Ib troy, after subtracting £, £, £, and
| of it ?
52. If 56 labourers get each -| florin per hour, how much do they
all get together in 6 days 8 hours, working 10 hours daily ?
53. What part of 4 da. 5 ho. 20 min. is £ of 3 da. 16 ho. 10 min. ?
54. What fraction is 1 ton of 3 cwt. 1 qr. 16 Ib and of 3 ton 17 cwt.
1 qr. ?
55. What cost 27| yards at llf d. per yard ?
56. A. can collect a given sum in 6 da., B. in 8, C. in 9, and D. in
10 : in what time will they do it together ?
57. A field 47$ yards x 27f yards is equal to another of 29£ yards
long : what is its breadth ?
58. If a farm of 276££ acres is rented at £478, 10/, what is that
per acre ?
59. If I can walk 20 miles in 5 hours, and my friend can do it in
6 hours : starting from opposite ends at the same time, how far are
we from each other after 1 hour? and in what time from starting
should we meet ?
60. Divide ^ acres among a family of 7 persons, giving to the four
oldest jt of a share each more than to the three youngest.
61. What is the difference between the fourth proportionals to |,
62. From a certain field its third part was cut off, but 8 acres were
added, making it now 172 ac. 2 ro. : what was its original size ?
63. A tradesman bought 13| Ib tea for 2| guineas, and retailed it
at 2|d. per oz. : what did he gain or lose ?
%* Work the questions in Practice, Exercises vii. and viii.
H
1&I9
114 DECIMAL FRACTIONS.
DECIMAL FRACTIONS.
Notation.
In any number, as 348, the first place is units, the second
tens, and the third hundreds ; each place being ten times the
value of the place to its right. If the notation were extended
to numbers having places of lower value than units, the place
to the right of units would be tenths, the next hundredths, the
next thousandths, and so on.
Such a notation is actually in use. Let there be figures after
the 348, marked off from it, for distinction's sake, by a point,
thus, 348-888 ; the 8 immediately after the point denotes A,
the next 8 denotes yg^, and the next 8 denotes 10900. The
figures after the point are therefore really fractions, whose
numerator alone is written, and whose denominator, 10 or
powers of 1 0, is understood.
The value of each place is known by its position from the
point, that is, from the unit's place ; so that each place must
always be represented, if not by a figure, then by a cipher.
Thus, if T% is denoted by '8, y^ (which is j%+tfer) is denoted
by -08, and j^ (which is J^+^+j^) is denoted by '008.
But ciphers occurring after figures are of no use ; thus, 8 or ^
is the same as "80, or I%+TOTT-
These fractions are called decimal fractions, from the deno-
minators being either 10 or powers of 10.
Rule. — The number of ciphers in the denominator of a
decimal fraction is equal to the number of figures after the
point.
EXERCISE I.
1. Write with denominators : 7, '07, 3'06, '009, -209, '5763.
2. 36-2008, -7064, -0009, '0101, -006001, -090005.
3. -061, -250, -300145, '007201, -000051, -0000001.
4. Write without denominators : ^, ffo, T^, f $, *£$, -ft?.
**• itTo* Tiro1 • 1000* looflo* Tooo* loooo • 1 oo o 5 o> loooooo*
"• IffOOO* TITOS TuTTO • 1000000* 100000* 1000000*
Equivalent Vulgar and Decimal Fractions.
Ex. 1. — Express f as a decimal fraction.
If we add ciphers to both terms till the denominator may be
cancelled, we have t=f$$==TT5ifo = "375 ; which is just the
quotient we should get, by adoling ciphers to the numerator,
and dividing by the denominator.
DECIMAL FRACTIONS. 115
Rule. — To change a vulgar fraction to a decimal, add ciphers
to the numerator, and divide by the denominator.
EXERCISE II.
Change to decimal fractions.
1- f> t, I, A, A, & H-
o 9 o 69 15 93 • 1 83
*• TWV> fSt 3Tff> l¥ff> S&OOJ ^~617> 64'
Ifo. 2. — Change "78 to a vulgar fraction.
•78 = /& = f&.
Rule. — To change a decimal to a vulgar fraction, write the
decimal denominator below, and reduce to lowest terms.
EXERCISE III.
Change to vulgar fractions :
1. 0.25, -875, -68, -36, -780, -375, -008, '02, -068.
2. -072, -100, -144, -00628, '0560, -0081, 2-00125, 6-00408.
.30 Interminate Decimals.
Ex. 1. — Change i to a decimal fraction.
Adding ciphers to the numerator by rule, and dividing by deno-
minator, J = "222222, etc., the 2 repeating itself for ever. Such a
decimal is called a repeating or recurring decimal, and is denoted
by a point over the repeating figure ; thus : f == '2.
A recurring decimal, when changed to a vulgar fraction, will
therefore have 9, instead of 10, for its denominator.
EXERCISE IV.
1. Change to decimal fractions, I, tf, J, I, V V.
2. Change to vulgar fractions, '3, '1, '4, 1'6, 3'3, 4'S.
Ex. 2. — Reduce /T to a decimal fraction.
By the rule TTT = 636363, etc., the 63 repeating itself. Such a
form of decimal fraction, where more than one figure repeats itself,
is called a circulating decimal, and is denoted by a point over the
first and last figures of the part which repeats itself ; thus :
A = '63.
Rule. — A circulating decimal is changed to a vulgar fraction by
writing for denominator as many nines as there are repeating figures ;
thus: GSssSfssA
116 DECIMAL FRACTIONS.
EXERCISE V.
1. Change to decimal fractions, T9T, §?, A, !?, *V,
2. To vulgar fractions, -64, '024, -72S, '0198, 2 57,
.Re. 3. — Reduce if to a decimal fraction.
By the rule, i|='43181S18, etc., the 18 repeating. Such a deci-
mal, where part only repeats itself, is called a mixed decimal, and is
denoted by a point over the first and last figures of the part repeat-
ing ; thus iJ=-43i8.
To reduce '43i& back to a vulgar fraction :— -43i&=-43+'OOl$.
Now, -43=i*oV, and -OOite^o of '18 or of {&, that is, =9J8ij.
Therefore,
43x99+18 43(1 00-1)+ 13 4300-43 + 18 4318 -43^
- -
Rule.— Subtract the finite part of the^ decimal from^the whole
decimal given, and below the difference write as many 9's as there
are figures repeating, with as many ciphers as there are figures in
the finite part.
EXERCISE VI.
1. Change to decimal fractions : —
i. u, m, n, if, Ttir, ill..
2. Change to vulgar fractions : —
2. '272, '025, -0045, "0286$, '3666, '3666.
\* For practical purposes, interminate decimals are little used, as all
necessary accuracy may be secured by carrying out the fraction a few places.
We shall, therefore, exhibit the rules for operation with finite decimals alone,
Addition and Subtraction.
9. — To add or subtract decimal fractions, (1.)
write the numbers so that places of the same name 293*406
shall be under each other, and proceed as in whole 29'06
numbers. 7'093
Note. — Any decimal may be extended by the !59?i
addition of ciphers to the right ; but if it be a re- 430'0674
peating or circulating decimal, the repeating part
is used for that purpose. <2.)
lOl Ol
24-078
37-542
EXERCISE VII.
1. 72-093+391-7+805'006+r094+48'0008+730-0514.
2. 63'904-flO'09-j-240-099+381-0001+l-0904+51'280i.
8. 79-6+82-7214+301-73+293-§+26'64+31-125+-0004.
DECIMAL FRACTIONS. 1 1 7
4. 83*7024+36-620l44*0001+7*3+29-2i+25-23+7'0108.
5. 256'704-f2*0093+47-6002+39'0804+2*09+3*014.
6. 7-30S2+31*0041+7'0001+38-009+25'4+42*72+-i.
7. 94-7—48-08, 74-002— 39'C09, 41-1—28-601.
8. 37-0004—19-071, '00098— '000041, 2'7041 — '0047.
9. 24-6— 7-3, 125-1 — 76-009, '00821— -000047.
L 3 2 Multiplication.
Ex.— Multiply (1.) -75 by -5, and (2.) '075 X '5.
•75 X -5=TVb X Tsn=vVD50=*375.
•075 X -5=1^ X ^s=Ig^0=-0375.
Rule. — To multiply decimals, multiply as for whole numbers;
and point off in the product as many decimal places as there are in
both factors together, prefixing ciphers if necessary to make up the
number.
Note. — A decimal is multiplied by 10, 100, or 1000, by carrying
the point to the right one, two, or three places respectively.
EXERCISE VIII.
1. 730.x -84, -093, '006. 7. 17827 X '00006, '0905, 3'0075.
2. 4-709 X *38, 1-72, -0024. 8. 73'04 x 27'02, 56*009, '4056.
3. 36-001X76, -076, 7'006. 9. 684*6 X2'56, '784, '003.
4. 84-008X1000, 3003, '093. 10. 2*847x10000, 100, *001, '64.
5. 258X-075, 3'005, 24'01. 11. 'OOOSX'Oo, *7, '009, *732.
6. 1824X182'4, '0002, '195. 12. 'C00091X-004, -71, 7000, 1*4.
Division.
Ex. 1.— Divide 19'305 by -65.
It will not alter the quotient if both divisor and dividend be multi-
plied by the same number. Multiply both by 1000 ; then
19 -305 _ 19305
~~765 ~650~~i
proceeding with the division as in whole numbers, and pointing the
quotient when the fractional part of it occurs.
Ex. 2.— Divide -000042 by -007.
Multiply both terms by 1,000,000 to remove the fractions ; then
"We may check the correctness of the position of the point in the
Quotient, by observing that the dividend should contain as many
ecimal places as the divisor and quotient together.
Rule. — To divide decimals multiply both divisor and dividend
by the larger of the two denominators, and divide as in whole numbers.
118 DECIMAL FRACTIONS.
EXERCISE IX.
1. 256-72—36, -174, -006 7. 1-^*76, '009, 2-56, -1.
2. 3-4894-1-84, 12'62, -0007. 8. 2'708-h*33, 5*07, 40'602.
3. '00063—9, '09, '009, '0009. 9. 853-096-^-037, lOOO'l, -298
4. '00063—90, 900, 90'09. 10. 7'9-7-39*68, '85, '0027.
5. 5000 —1, -05, -0025. 11. 305*081-5-3456, '29, '528.
6. *85642-*74, *96, '0056. 12. 3476-h*0008, '094, 3476'07.
Reduction.
Decimal fractions of quantities often require to be reduced to a
higher or to a lower name.
Ex. 1.— What part of I/ is £'025 ?
Pounds are reduced to shillings by multiplying by 20 ; therefore
£-025=-025 X 20 sh.s='5 sh.
Ex. 2. — What part of a pound is *37o sh.?
Shillings are reduced to pounds by dividing by 20 ; therefore
Ex. 3.— What is the value of £-0875 ? *0875
90
Pounds are reduced to shillings and pence by multi-
plying by 20 and by 12. Multiplying by 20 we get I/ 1'7500
and a remainder ; multiplying the remainder by 12, 1%
we get 9d. Answer 1/9. 9*0000
Ex. 4. — Express 2/3 J as decimal of a pound.
Farthings are reduced to pounds by dividing by 960 or (4 X 12 X 20).
The number of farthings in 2/3$ being 109, the sum is £$g$, which,
reduced to a decimal, is £'113573 nearly.
Rulo. — To reduce a decimal fraction of a quantity from one
name to another : — If to a lower, multiply by the number of times
the lower is contained in the higher ; if to a higher, divide by that
number.
Note. — Decimal fractions of different names must be reduced to
the same name before they can be added or subtracted.
EXERCISE X.
1. What part of a shil. & of a penny is £'75, £-296, £'0085, £'54.
2. Reduce to oz. and dwt. '396 ft tr., *094 ft, '11875 ft, *0792 ft.
3. Express as parts of a foot and inch -294 yd., '0576 yd., -0075 yd.
4. As parts of a quart & of a gill, '0375 gall., '0063 pk,, -1859 gall.
5. What part of a pound is '275 sh., '945 cr., 6'275 flo., 9'736 hfl-cr.
6. What part of a quarter is '98 bu., -095 bu., 8' 625 pk., '986 pk.
7. What part of a yard is -825 qr., 2-76 nl., -0856 qr., *125 nl.
8. What part of a cwt. is 6'75 oz., 15'375 R>, -8930 qr., -0824 ft.
9. What part of an acre is 1'36 ro. '86 ro., 1S'32 po., 12*96 sq. yd.
DECIMAL FRACTIONS . 119
10. Find the value of £-784, £2 "0086, '98 cr., '656 hf.-sov., '8 guin.
11. Also of '0872 year, -3768 wk., '175 ho., 4-085 da., '756 min.
12. Also of -279 fur., *936 cwt, '785 galls., '0025 tons, '6248 cu. ft.
13. What part of a pound, and of 13/9, is 7/6, 8/9, 5/10 J, 17/6, 12/9*. ?
14. ,, of an acre, &of5Jac., is 3 ro. 7po., 14 po. 15yd., 5yd. 6 s.ft. ?
15. „ oflyr.,&ofl yr. 175 da.,is 10da.6ho., 27 wk. 5da.,5h. 10m.?
16. „ of Icwt.,&oflc.2qr.6ft>,isl7ft>6oz.,3qr.l5ft>, 10c.lqr.4ib?
L35 EXERCISE XL— MISCELLANEOUS.
1. Add £'375+*675 guin. +'792 cr.+'125 fl.
2. Divide f of 8*236 by -138 of A.
3. My age is 1/075 of my brother's ; if I am 30, what is he ?
4. What number is that of which '45 is 25 ?
5. If | yards cost £1-235, what cost 3'7896 yards?
6. What is the area of a grass plot which is '296 yds. less than
a pole ?
7. What decimal of 2} yds. is 1 ft. 7 in. ?
8. What part of a gallon is '08935 quarter?
9. Find the price of 28*6 st. butter at 16/9* for 1'75 st.
10. Find the area of a field '0876 miles by '0056 miles.
11. Find the weight of four packages, of which one is "276 ton,
another -025 cwt,, a third 75'8 ft), and the fourth 1'96 qr.
12. From 3*285 of 16/5, take 1'3 of 17/6.
13. Reduce 3 da. 6 ho. 30 min. to decimals of a week and of a year.
14. How many imperial acres in a farm which measures 295*65 ac.
Scotch, if the ac. Scotch be 1-261183 of the acre imp.?
15. Bought 3-75 cwts. for £-0125 per R> : find the whole price.
16. Find the number which, taken twelve times, is '1728.
17. Divide the average of 3'079, 4-276, 5*60548 by '006.
18. If I walk 3*789 miles an hour, how far will my friend walk in
5 hours, if he goes 1*075 miles for my one ?
19. The French metre is 39*37079 inches: how many in 147*895
yards ?
20. What decimal is 2 galls. 3 qts. 1 pt. of 14'576 pks. ?
21. How much carpet 1*25 yds. wide will cover a floor 22'3x 19*45
ft., and having 2 oriel window spaces, each 4J feet by 2|?
22. If I pay 9jd. per £ as income-tax, what part is that of my
whole income ? and what part of my income should I save by a re-
duction of 2 Jd. per £?
23. A regiment of 560 men has on its sick-list '295 of the whole :
how many men are fit for service ?
24. By what fraction of itself does '00125 ac. fall short of 7'86 sq.
yards ?
25. What decimal fraction multiplied by f of 71 gives J of I of \ ?
26- In a town of 240756 inhabitants, it was found that *0475 of
the whole could not read, and only '575 of those able to read could
write : how many were there of each ?
27. From 3'265 of 17/6 take 1*3 of 2778.
1 2 0 DECIMAL FRACTIONS.
23. St. Peter's Cathedral is 437 ft. high : what fraction is that oi
St. Paul's, which is 340, and of the cathedral of Strasbourg, which
is 574?
29. What is the average length of the first four months of the
year ? and by what fraction of a day does it differ from the average
length of the second four ?
30. If I bought 2 cwt. 2 qrs. 16 ft> sugar at £3*0296 per cwt., and
sold it at £'035 per Ib, what did I gain or lose on the whole, and on
each pound ?
31. How often is £2*375 contained in *6 guineas?
32. What fraction is 1 Ib troy of 1 Ib avoir., and vice versa ?
33. How much is | of -00295 of £8, 2s. 6d.?
34. From Hamburgh to Bremen is22J German miles, or 109 J English
miles : what fraction is a German mile of an English one ?
35. If the cost price of a book is 2/3i, and the selling price 3/4,
what fraction of the former represents the profit?
36. In a school of 100 boys, 80 girls, and 58 infants, there waa
absent 12 boys, 9 girls, and 8 infants : what fraction of each was
absent, and what fraction of the whole school? Also, how many
per cent, of the school were absent ?
37. Find the value of 4r- x 2 of '9*5 + ^T^ x rb-
*07o 84 4*07
38. The French litre is '220097 gallons: express the bushel in
litres.
39. At £3 '875 per acre, what is the rent of a farm which is equal
to one field 193*85 yards square ?
40. If the diameter of a circle is to the circumference as 1 to
3*1416, what is the difference in length between the rims of two
wheels whose diameters are 5 and 5f feet respectively ?
41. And how much oftener will the former revolve in a journey of
3*65 miles than the latter?
42. What ratio does the fourth proportional of 3, 4'75, 5'095 bear
to that of -6, '063, and -0005 ?
43. Divide twice the sum of 1*0006 and 1-0606 by 5 times their
difference.
44. A. walks 2*5 feet each step, B. 2*785: when B. has gone a
mile, what part of a mile has A. still to go?
45. What fraction is "6 of 2| ells of 1;07 of 3£ yards ?
46. In multiplying any number by *36, what is the difference in
the product (expressed as a vulgar fraction of the multiplicand) if we
multiply by 2 and by 4 decimal places respectively ?
47. What cost 8 rounds of beef weighing in all 2*64 cwt., at 1/4 J
for 2-25 Ib?
48. Divide 576*58 guineas among 3 men and 4 women, giving each
man 1*75 of a woman's share.
49. To give 7 persons 1/8J each out of half a guinea, what frac-
tion of a crown do I want ?
50. A cubic foot of water weighs 62 Ib 7 oz, 4 dr. : what is the
INTEREST. 121
weight of water in a cistern 6'2 feet long, 4*5 broad, and 3*75 deep ?
and how many quarts may be filled of it, if a quart weighs 2 '25 Jfo
avoir. ?
51. Gold coined has ^th alloy: the weight of a sovereign being
•02139 lb, what would be the value of a purse of 670 sovereigns were
the gold pure ?
• 36 INTEREST.
Simple Interest.
If I borrow ,£50 from the bank, I have to pay so much for
the use of its money ; if I pay .£50 into the bank, I receive so
much from it for the use of my money.
The sum which produces this profit is called the Principal ;
what is paid for the use of the principal is called the Interest ;
and the principal, with the interest added to it, is called the
A mount.
Interest is calculated at so much every year for every £100,
or per cent. (%), as it is called. Thus
5 per cent. =£5 for £100, or .£1 for £20.
4 „ =£4 ,£100; £1 £25.
3 „ =£3 £100, £1 £33 J.
2J „ = £2^ £100, £1 £40.
2 „ =£2 £100, £1 £50.
Ex. 1. — Find the interest on £275 for a year at 4 per cent.
This is a question in the Rule of Three ; thus, if 100 gain 4,
what will 275 gain ?
£100 : £275 : : £4 : Answer=£ll.
Ex. 2. — Find the interest of the same sum at the same rate
for three years.
This will be three times the interest for one year ; or, stating
it in double Rule of Three —
£100 : £275 : : £4 / A ,QQ
I , g > Answer =£33.
If the time given be less than a year, express it as a fraction
of a year ; thus, had it been five months, the second statement
would have been 1 : ^ ; if it had been 239 days, it would
have been 1 : f^f .
It is here supposed that the principal remains the same
during the five years ; not accumulating by the addition of
the interest each year. Interest given on this supposition is
called Simple Interest.
122 INTEREST.
Questions in Interest can always be solved by Rule of Three,
or by the following rule derived therefrom : —
Rule. — To find the simple interest of a,ny sum, multiply the
principal by the rate per cent., the number of years or part of
a year, and divide by 100.
EXERCISE I.
1. What is 4 per cent, of £150, £250, £375, £20, £1000, 25/. £3 15s ?
2. „ 5 per cent, of £275, £60, £400, £3, £6, 10s., £36 ?
3. ,,2 per cent, of £10, £300, £2, 10s., £75, £875, £2000 ?
4. Find the interest of £256, 10s. for 1 year at 3 per cent, per ann
5. Of £4562, 17s. 6d. for 1 year at 4.J per cent, per ann.
6. Of £675, 19s. 4d. for 3 years at 5 per cent, per ann.
7. Of £89, 14s. 8Ad. for 8 years at 2 per cent, per ann.
8. Of £560, 15s. for 9 months at 3 per cent, per ann.
9. Of £849, 13s. 6d. for 15 weeks at 4 per cent, per ann.
10. Of £2000, from March 15 to November 18, at 5 per cent.
11. Of £1625, 9s. 8Jd. for 189 days at 5 per cent.
12. Find the amount of £125, 10s. for 8 years at 4 per cent
13. Of £97, 16s. 2d. for 8 months at 2i per cent.
14. Of £87, 15s. lOd. for 12 weeks at 5 per cent.
15. Of £216, 9s. 3d., from January 16, to May 30, at 4 per cent.
J.O I Ex. 1. — What principal, invested for 10 years at 4 per cent.,
will bring in a sum of .£240 ?
In the given time, and at the given rate, £100 of principal
will bring interest ,£40, so that the question is equivalent to
this :— If £100 give £40, what will give £240, same time and
rate ? And the statement will be
£40 : £240 :: £100 : Answer=£600.
Ex. 2. — What sum will amount to £1250 in 8 years at 4 per
cent, per annum ?
In the given time, and at the given rate, £100 will amount
to £132, so that the question is equivalent to this : — If £100
amount to £132, what will amount to £1250, same time and
rate ? And the statement will be
£132 : £1250 :: £100 : Answer=£946, 19s. 4jff.
Ex. 3.— At what rate per cent, will £900 amount to £1116
in 6 years ?
In the given time £900 gains £216, and therefore in one
year £36, so that the question is equivalent to this : — If £900
gains £36 in one year, what will £100 gain ? And the state-
ment will be
£900 : £100 : : £36 : Answer=4 per cent
INTEREST. 123
Ex. 4. —In what time will £1500 amount to £1980 at 4
per cent. 1
At the given rate £1500 will gain £60 interest in one year,
so that the question is equivalent to this : — If £1500 gain £60
in one year, in what time will it gain £480, the sum required
to make up the amount ? And the statement is
£60 : £480 :: 1 year. Answer =8 years.1
EXERCISE II.
1. What sum must be lent at 4 per cent, on April 1, to bring in for
interest £3, 17s. 4d. on May 28 ?
2. What principal lent at 4^ per cent, for 7 mo. will yield interest
£17, Is. 9£d. ?
3. What capital sum at 5 per cent, will bring a yearly income of
£250?
4. What sum must be lent at 6 per cent, for 1 year 2 months to
amount to £56, 3s. 6d. ?
5. What principal will amount to £1109, 11s. 3d. in 3 years at £2,
15s. per cent. ?
6. What sum will amount to £3376, 8s. 11 Jd. in 9 years 7 months
at £4, 11s. 8d. per cent, per annum?
7. If £350 gain £60 simple interest in 4 years, what has been the
rate per cent, gain ?
8. A capital of £780 brings a return of £126, 15s. in 5 years, simple
interest, find the rate per cent. gain.
9. In what time will £25 become £27, 3s. 9d. at 5 per cent, simple
interest ?
10. How long must £7200 be lent at 4 per cent, simple interest to
amount to £9760 ?
11. If I lodge £180 in bank, how long must I let it lie at 3 per cent.
to gain £24 ?
12. What rate of interest must I receive on a sum of money which
I wish to double itself in 12 years ?
1 If I denote interest, P principal, r, rate, and tt time, the rule for find-
ing Simple Interest may be expressed thus : —
r Pxrxt
100
From that formula, other formula may be devised for working each of the
four cases now given, thus : —
/ X 100
1. Where Principal is sought, its interest being given, P=
2. Where Principal is sought, its amount being given, P=
3. Where rate is sought,
100+rx*
JxlOQ
Pxt
4. Where time is sought, t= p—
124 INTEREST.
138 Compound Interest.
Ex. — If, however, I lodged £275 in the bank for 3 years at
4 per cent., I should get more than the simple interest as cal-
culated, sect. 136. For at the end of the first year the interest
would be added to the principal, and the amount would be the
principal for the second year. At the end of the second year
the interest on the amount would again be added, and the new
amount would be the principal for the third year, and so on.
Interest calculated in this way is called compound interest.
To find it, we have to work three questions in simple
interest.
(1.) Int. of £275 for 1st year at 4 p. c. = £11.
(2.) Int. of £286 for 2d year at 4 p. c. = £1 1, 8s. 9|d.
(3). Int. of £297, 8s. 9|d. for 3d year at 4 p. c.=£ll, 17s. llrf^d.
And the amount I should then draw from bank would be
£309, Gs. 9y|^d.
The simple interest of the sum was £33, showing a difference
of £1, 6s. 9yf Tfd.
And if the interest were paid half-yearly, as it is sometimes,
I would receive still more ; for at the end of the first half-year,
the interest would be added to the principal, which wrould
make the interest for the first year greater than before, and
consequently the amount for the second year, and so on.
Rule. — To find compound interest, find the amount for the
first year, and make that the principal for the next ; then find
the amount for the second, and make that the principal for
the third ; and so on for the number of years.
EXERCISE III.
1. Find the compound interest of £65, 14s. for 3 years at 4 per cent.
2. Of £378, 10s. 7d. for 2 years at 5 per cent.
3. Of £100 for 1 year at 4 per cent, payable half-yearly.
4. How much more would it be if interest be paid quarterly.
5. Find the amount of £750, 10s. for 1 year 9 months at 2^ per
cent, compound interest.
6. Find the amount of £1250 for 2 years 100 days at 4 per cent,
compound interest.
139 The computation of compound interest may be effected more
simply thus : —
The interest of £1 for 1 year at 4 per cent, being £'04, the
amount will be £1*04.
Then as £1 is to its amount for a year, so is any other prin-
cipal to its amount for a year ; hence —
DISCOUNT. 125
£1 : £1'04 : : «£1'04 : «£1'042 amount for 2 years.
£1 : ^1'042 : : £i'04 : £r043 „ 3 years.
etc. etc.
Then as £1 : £275 : : £l'Q^ : ^l-04?x275, the amount of
.£275 for 4 years at 4% = £309, 6s. 9^(1.
Interest for the time= £309, 6, 9Tfir-£275=«£34, 6, 9y£T.
Rule 1. — To find amount at compound interest, raise the
amount of £l for one year to the power denoted by the number
of years, and multiply by the principal ; to find the interest,
subtract the principal from the amount.
From that follows immediately —
Rule 2. — To find the principal that will yield a given
amount at compound interest, divide the amount by the amount
of £1 for the number of years.
Thus - i-04a T7*^ = -^75, principal giving the amount.
Note. — If the interest is paid half-yearly or quarterly, calcu-
late as above, with half-yearly or quarterly terms respectively.
EXERCISE IV.
1-6. Perform by this method the questions in Ex. iii.
7. What sum will amount to £1019, 10s. 2d. in 6 yr. at 5^ per cent. ?
8. What sum will amount to £1351, 7s. 10£ ^ in 3 yrs. at 4 per
cent, interest, payable half-yearly ?
9. I invest £5000 for 3 years at £5 per cent. : how much more
should I receive if the interest be paid half-yearly than if paid
annually?
.40 DISCOUNT.
Merchants' accounts are not generally paid ready money, but
either after a certain time (six or twelve months), or by bill,
in which the buyer promises to pay at a date agreed upon. The
merchant may take this bill to a banker, who will — if the
buyer's credit be good— give ready money for it, charging a
certain percentage for paying the money before it is due. This
charge is the banker's discount, and he is said to discount the
bill The ready money received for the bill is called its present
value.
Ex. — What is the present value of £400, payable 9 months
hence, interest being 4 per cent. ?
126 DISCOUNT.
£100, at 4 per cent., would amount in 9 months to ,£103 ; in
other words, ,£100 is the present value of £103, payable 9
months hence, interest 4 per ceut.
Therefore, as £103 is to £4'K), so is the present value of
£103 to that of £400 :—
£103 : £400 : : £100 : Answer = £3887 nearly.
Rule. — As the amount of £100 for the time is to the sum,
so is £100 to the present value required.
To find the discount, subtract the present value from the
sum : £400 -£388, 7s. =£11, 13s. ; or, making a statement in
Proportion, we have the following : —
£103 : £400 : : £3 : Answer = £11, 13s.
Rule. — As the amount of £100 for the time is to the sum,
BO is the discount of the former to that of the latter.
In practice, however, discount is never reckoned in this
way, but exactly as interest ; thus the discount on £400, pay-
able 9 months hence, at 4 per cent., would be the interest
on that sum for the given time. The common discount is
therefore somewhat greater than the true discount.
In discounting bills, bankers calculate interest for the num-
ber of days the bill has still to run ; but three days are allowed,
called days of gracey after a bill is nominally due, before it be-
comes legally due.
EXERCISE.
1. Find the common and the true discount on £450, due 8 months
hence, at 4 per cent.
2. On a bill of £536, 10s., drawn and discounted March 14, payable
Sept. 10, at 3 per cent.
3. On a bill of £279, 15s. 3d., drawn March 1 at 9 months, discounted
April 20, at 3 per cent.
4. On a bill of £1000, dated May 15, for 8 months, and discounted
August 24, at 4 per cent.
5. Bought sugar at £2, 18s. per cwt. and 6 months' credit : what
should be allowed for present payment per cwt. at 3^ per cent. ?
6. I engage to pay £750, 14s. lOd. 10 months hence : find its present
worth at 5 per cent.
7. What sum at 10 per cent, will amount to £325 in 2 years ?
8. A bill of £472, 16s., drawn May 8 at 3 months, is discounted
June 19 at 6 per cent.# what is its present worth ?
9. Find the present worth of £875, 5s. 8d., drawn Feb. 25 at 7
months, discounted June 4, at 5 per cent.
STOCKS. 127
L41 STOCKS.
The sum required for constructing a railway or any other
public work is called the capital. This is raised by means of a
Joint-Stock Company, as it is called, in so many shares, it may
be of £5, £10, or .£50, according to the amount of the capital.
The value of the share, as fixed at the outset of the scheme, is
called par.
If the scheme turn out a good one, many will want shares
in it for the sake of the high interest yielded ; so that those
who hold shares may sell them, if they choose, at a higher price
than they paid for them. A ,£50 share, e.g., may sell for £60 ;
in which case the value is said to be £10 above par. On the
other hand, if the scheme turn out a bad one, nobody will care
to buy the shares ; and those who hold them, if they sell out,
will have to do so at a loss. A share ,£50 may have to be sold
for £40, that is, £10 below par.
Joint-Stock Companies are formed in the same way for other
purposes, e.g., banks, mining operations, and the like. And
the shares or stock of these companies are regularly sold in the
market, at a higher or lower price according to the opinion
entertained at the time of their value.
There is a different kind of stock, called Government Stock,
or the Funds. The Government has sometimes to borrow large
sums of money for public purposes. As it cannot expect to get
the loan from one person alone, it divides the amount into £100
shares ; and to each person who gives it £100, it binds itself
to give a certain rate of interest, say 3 per cent. The rate of
interest is not high, but that is compensated by the certainty
of its payment. The person who lends £100 really buys a £100
share ; he cannot claim his money again, but he has got the
right to £3 annual interest instead. He can sell his share, how-
ever, in this 3 per Cent. Stock, or the 3 per Cents, as it is called.
The selling price depends on circumstances, and is constantly
varying. Thus, if public affairs are in such a state of peace and
prosperity as to offer other safe means of investment and a
higher rate of interest, the price will be lowered ; if otherwise,
the price rises.
Ex. 1. — What should be paid for £1790 in the 3 per cents.
at94|?
i.e., if £100 bring £94|, what will £1790 bring?
£100 : £1790 : : 94| : Answer =£1689, 6s. 3d,
128 BROKERAGE.
Ex. 2. — How much stock at 94 may be purchased ftr .£2573
brokerage Jth per cent ?
i.e., if £94J buy a £100 share, what will £2570 buy ?
£94J : £2570 : : £100 : Answer=£2730, 8s. 2J
Ex. 3.— What interest will ;>9 got by buying 4 per cents. »t
87^ ?
i.e., if £87£ bring interest £4, what will £100 bring ?
£87i : £100 : : £4 : Answei = £4f5£ per cent, or £4'584.
EXERCISE.
1. What should be paid for £576. 10s. in the 3 per cents, at 941
and at 89$ ?
2. Find the buying price of £295 stock in the 3£ per cents, at 77g
and at 84§, brokerage J per cent. ?
3. What cost £475 in 4 per cents, at 86$, and at 91 £ ?
4. How much stock at 98$, and at 96§, may be bought for £1280,
brokerage £ per cent. ?
5. What amount of stock at 83$ and at 89$ may be purchased for
£2000, brokerage £ per cent. ?
6. If I invest £600 in stock at 107$ and at 111$ (brokerage & per
cent.), what amount will I hold ?
7. What interest will be got by buying India stock 9A per cent, at
179.1, and at 168$ ?
8f Do., by buying 3 per cents, at 934, and at 96$ ?
9. Do., stock giving 6 per cent, at 124$, and at 136£?
10. Bought 3 per cents, for £1000 at 94$ and sold at 96$, what did
I gain ?
11. What annual income is got by investing £2500 in 3 per cents.
90$, and at 101 J ?
12. How much must be invested in the 3$ per cents, at 96$ and at
99 to yield an annual income of £350, brokerage £ per cent. ?
142 BROKERAGE.
The selling of stock, both Government stock and that of
private companies, is a regular business carried on by Brokers
or Stockbrokers ; their market is called the Exchange or Change ;
and their charge for transacting business Brokerage. This is
generally Jth per cent., that is 2s. 6d. for every £100 value
bought or sold.
Similarly, buying and selling of goods of all kinds is very
largely carried on upon commission. Thus a merchant may
import provisions from a firm or company of merchants abroad,
and sell them on commission ; or a manufacturer may have
INSURANCE. 129
agents through the country selling his goods on commission.
The rate of commission is much higher than that of brohrage ;
varying from 2j per cent, to as high as 10 per cent.
EXERCISE.
1. Find the commission on £1495, 10s. at 3] per cent.
2. On £295, 17s. 6d. at 1£ per cent.
3. On £793, 18s. 9d. at £ per cent.
4. At $ per cent, what is the brokerage on insuring goods to the
value of £1560, 13s. 4d. ?
5. Sold by auction goods which fetched £375, 10s. 6d. ; what sum
would be paid to the owner by the auctioneer after deducting his com-
mission of 5 per cent. ?
6. What is the cost of collecting debts to the amount of £794, 16s.
allowing 25/ per cent, to the collector ?
INSURANCE.
Insurance is the name applied to the percentage paid on
property to secure it against damage. Thus there is Fire In-
surance, to secure it against risk from fire ; Hail and Storm
Insurance, to secure the farmer's crops against damage from
storms ; Insurance against sea-risk, to secure vessels and their
cargoes against shipwreck ; and Life Insurance, to secure a
man's family or friends against risk of poverty at his death.
The percentage is paid annually in all these cases, except
sea-risk, and is called the Premium of Insurance. On this a
Government duty is charged of so much per cent. ; all sums
intermediate between hundreds being charged as the higher
number of hundreds. The deed of agreement between the
person who insures and the Insurance Company is called the
Policy of Insurance.
The premium for fire insurance is Jth per cent., or 2/6 for
.£100 ; but higher rates are charged on property exposed to
more than average risk.
The premium of life insurance increases with the age of the
person insured, its amount being fixed by calculations based
on the average duration of life.
EXERCISE.
1. What must be paid for insurance of £1495 at 2f per cent. ?
2. For insurance of £790 at 4£ per cent. ?
3. For insurance of £2485, 17s. 6d. at 2/6 per cent. ?
4. What is the expense of insuring a ship worth £3000 at 3 guineas
per cent., duty 3/6 per cent. ?
130 PROFIT AND LOSS.
5. A person aged 32 insures his life for £800, what is the expense
at £3, 13s. 6d. per cent. ?
6. What sum will insure a cargo valued at 1790 guineas at 7£
guineas per cent. ; duty, 3/ per cent. ; commission £ per cent. ?
Ex.— What sum must be insured to cover £470 at 2^ guineas pel
cent. ; duty, 2/6 per cent. ; commission, A per cent. ?
The expenses of insurance are £2, 12s. 6d. +2/6 + 10/ per £100;
BO that every £100 insured produces £100- £3, 5s. ; that is £96, 15s.
Then £96, 15s. : £470 : : £100 : £485, 10s. 7£ T6w-
7. What must be insured to cover £2570 at 4£ guineas per cent. ;
duty, 3/6 per cent. ; brokerage, ± per cent. ?
8. To cover £4875, 10s. at 5^> guineas per cent. ; duty, \ per cent. ;
commission, | per cent. ; extra charges 1 per cent. ?
9. What sum insured will cover total loss of house property worth
£2765, 10s. at 3$ guineas per cent. ; duty, 3/6 per cent. ; commission,
^ per cent. ?
144
PROFIT AND LOSS.
Ex. 1. — Bought tea at 3/6 per lb, and sold it at 4/ per ft) :
what is the gain per cent. ?
As the cost price is to ,£100, so is the gain upon the cost price
to the gain upon ,£100 ; i.e.,
3/6 : £100 : : 6d. : to the gain per cent. = £14, 5s. 8fL
Ex. 2. — Bought tea at 3/6 per ft), what must I sell it at to
gain 10 per cent. ?
As £100 is to the cost price, so is the selling price of £100
to the selling price required ; i.e.,
£100 : 3/6 : : £110 : selling price, or 3/10 J.
EXERCISE.
1. Bought butter at £3, 15s. per cwt., which I sold at £4, 4s. 6d. :
what was the gain per cent. ?
2. Bought cloth at 4/9 per yard, and sold it at 5/2 : find the gain
per cent.
3. Sold sugar at 4^d. per lb, for which I paid £1, 17s. 4d. per cwt. :
what was the gain per cent. ?
4. Bought coffee at £6, 15s. 6d. per cwt., what should I have sold
it for to gain 1\ per cent. ?
5. Bought linen at 3/6 per yard, and lost 4 per cent, on it : find the
selling price.
6. How must wine, which cost 21/6 per gall., be sold to gain 4.^
per cent. ?
7. Bought 74 yards carpet at 2/10 per yard, and sold the whole for
£27, Us. : find the gain or loss per cent.
8. If 10 yards are found to be damaged, at what per cent, profit
must the remainder be sold so as not to lose upon the whole ?
SQUARE HOOT. 1 3 1
9. Sold oranges at 1/1 per dozen, with a gain of 3 per cent. : what
did they cost ?
10. Find the cost price of a book on which a bookseller gained 9
per cent, by selling it at 25/6.
11. Bought paper at £1, 10s. 6d. per ream, which I sold at 1/9 per
quire : what was gained on 15 reams, and what was the gain per cent. ?
12. If 2^ per cent, be lost by selling bacon at 6£d. per tb, what did
it cost per cwt. ?
AC SQUARE BOOT.
The square of any number is that number multiplied by
itself ; thus 25, being equal to 5 X 5, is the square of 5, sect. 38.
The square root of any number is that number which, when
multiplied by itself, has the given number for product ; thus
5 is the square root of 25.
As the square of 5 denoted by 52, so the square root of 25 is
denoted by 25* or v'25.
The square root of any number, if it be any of the first 12
numbers, is known from the multiplication table ; thus since
72=49, ^49=7.
Ex. — Find the square root of 576 ?
When the square root of a number exceeds 12, the method
of finding it depends on the principle that " the square of any
number is equal to the square of its two parts, together with
twice their product ;" thus, 242=202-f 2X20X4+42 which is
the same as 202+(2x20+4)x4.
To find the square root of 576, therefore, we separate it into
two parts, the one containing the hundreds, 500, the other the
units, 76.
The nearest square number to 500 is 400, 2 5,76(24
which gives 20 for the first part of the root. 2 4
Subtract the 400 and 176 remains. By the 44) 176
above formula, if this be divided by twice the 176
first part of the root increased by the second
part, the quotient will be the second part. But the second or
units part of the root is so much smaller than the first, that
twice the first part of the root may be taken as the approxi-
mate divisor. Double the 2 tens, giving 4 tens or forty ; this is
in 176, 4 times ; add this 4 to the 4 tens for complete divisor,
making 44 ; 4 times 44 is 176. Complete root 24 ; as may be
proved by multiplying it by itself.
If the root consist of more than two places, divide the given
number into periods of two figures, beginning with the units*
place ; annex one period to each remainder for the successive
132 MENSURATION.
dividends ; and for each trial divisor add the part of the root
last found to the divisor preceding it. Thus the next trial
divisor in the above example would have been 48.
The square root of decimal fractions is found precisely as
that of whole numbers ; only the periods of two figures are
reckoned from the point. And if the square root is wanted of
a number which has no even square root, it may be found ap-
proximately by adding ciphers, and finding the root as of a
whole number and a decimal fraction.
The square root of a vulgar fraction is found by extracting
the root of its numerator and of its denominator ; if these have
no even root, it should first be reduced to a decimal fraction.
EXERCISE.
Find the square root of —
1. 289.
9. 5184.
17. 6-25.
25. -06.
2. 361.
3. 484.
10. « $f
11. 20736.
18. -0121.
19. -001156.
26. 6J.
27. 6|.
4. 676.
12. 55225.
20. 65-1249.
28. 2.
5. ttf.
13. 126736.
21. 2067844.
29. §.
6. 1849.
14. 316969.
22. -00053361.
30. 6-5536.
7. 3136.
15- IWfti-
23. -030976.
31. 655-36.
8. 3969.
16. fftft.
24. 6.
32. 65-536.
MENSURATION.
Ex. 1. — What is the area of an oblong plot 4 feet 9 in. long
by 3 feet 6 in. broad ?
The area being found by multiplying the length by the
breadth (sect. 82), we multiply 4 feet 9 in. by 3 feet 6 in.
If feet multiplied by feet give square feet, feet multiplied
by inches will give twelfths of feet, and inches multiplied by
inches will give one-hundred-and-forty-fourths of feet, or square
inches.
Multiplying first by 3 feet : 4 9
3 times 9 are 27 ; 3 and carry 2. _3 6
3 times 4 are 12, and 2 are 14. 14 3
Then multiplying by the 6 inches : 246
6 times 9 are 54 ; 6 and carry 4. -— — ^7-77
6 times 4 are 24, and 4 are 28 ; 4 and carry 2. '
Adding the partial products, we have 16 7' 6", i.e., 16 sq. ft.,
7 twelfths of a square foot, called primes, and 6 one-hundred-
and-forty-fourths of a sq. foot, or sq. inches, or seconds, as they
are called, to distinguish them from the primes.
Reducing the primes to sq. inches, the answer may also be
expressed as 16 sq. feet 90 sq. in.
MENSURATION. 133
The next lower names to primes and seconds are thirds and
fourths, denoted by '" and .
Because of the carrying by tivelves, this multiplication for
areas is often called duodecimal.
Rule. — To find area, multiply the greater dimension by the
highest name of the less, carrying by twelves ; then by the
lower names in their order, setting the first result :n each pro-
duct one place to the right : add the several products.
To find solid content, multiply the product of the two greater
dimensions by the least.
EXERCISE I.
V* Express the answers (1.) in sq. feet, primes, seconds, etc. ; (2.) in sq. feet
and sq. inches.
ft. in. ft. in. ft. in. li. ft. in. li.
1. 36 4x7 6 7. 745x276
2. 9 8x6 10 8. 936x187
3. 87x52 9. 18 62x740
4. 11 10 x 9 1 10. 25 10 9 x 6 0 3
5. 24 6 x 18 5 11. 16 80x789
6. 29 3 x 10 11 12. 14 61x985
13. How many sq. feet in a floor 26 feet 8 in. x 17 feet 6 in. ?
14. rina rhe area of a field 136 feet 8 in. x 78 feet 10 in.
15. There are two windows in ray room, each containing 12 panes,
each pane 1 foot 7 in. x 10£ in. Find the price of the whole at 1/10|
per sq. foot.
16. What is the expense of carpeting a floor 21 feet 6 in. x 18 feet
9 in., at 3/7 per sq. yard ?
17. What will it cost to paper a room 11 feet 6 in. high and 86 feet
8 in. in circuit, at 8£d. per sq. yard ?
18. A square plot of grass measures 18 feet 8 in. long, but in its
centre is a square flower-bed 4 feet 6 in. long : what is the extent of
the grass surface ?
19. What is the length of a paved court containing 186 sq. feet
10 sq. in., which is 9 feet 6 in. broad ?
20. The carpet of a room at 5/ per sq. yard cost 10 guineas. If the
room was 17 feet 6 in. broad, find its length.
21. Find the content of a wooden cube 4 feet 8 in. of edge. Express
this and the following answers in cubic feet and inches.
22. A block of granite measured 10 feet 8 in. long by 7 feet 4 in.
broad, and 5 feet 6 in. thick : what was its content ?
23. How many cubic feet of air in a room 60 feet long by 25 feet
6 in. broad, and 14 feet 8 in. high ?
24. Find the depth of a cistern 4 feet 6 in. square, to <. ontain 84
cubic feet of water.
25. Find the expense of painting a chest (exclusive of the bottom)
6 feet 3 in. long by 4 feet 4 in. broad, and 4 feet 2 in. deep, at 1/2^
per sq. yard.
26. A school-room is 45 feet 8 in. long by 27 feet 6 in. broad : what
134 MENSURATION.
must be its height to accommodate 220 pupils, allowing 80 cubical feet
for each pupil ?
27. How many bricks 8i in. long, 3 in. broad, and 2£ thick, would
be required to fill a cubical space measuring 18 feet 8 in. long ?
28. What will it cost to remove a mass of earth 36 feet 4 in. long
by 25 feet broad, and 14 feet deep, at 7£d. per cubic yard ?
29. Find the weight of water in a tank 25 feet long by 16 feet 8 in.
broad, and 12 feet 6 in. deep, if a cubic in. of water weighs 252*458 gr.
30. How many galls, water in a cistern 7 feet 6 in. long, 5 feet 4 in.
broad, and 6 feet deep, one imp. gall, being equal to 277*274 cub. in. ?
14cO Ex. 2. — What is the area of a rectangular field, whose length
is 7 chains 40 links, and breadth 8 ch. 30 Ik. ?
Reducing the dimensions to links, the area is 740X830 Ik.
= 614200 sq. Ik. =6 ac. 0 ro. 227 po.
Note 1. — If the field is not rectangular, but has its opposite
sides parallel, multiply the length by the perpendicular
breadth.
Note 2. — If only two of its opposite sides are parallel, multi-
ply half the sum of these by the perpendicular breadth.
Note 3. — If the field is triangular, multiply the base of it
by half the perpendicular height.
Ex. 3. — Close by a wall 8 ft. high flows a stream 6 ft. broad :
what length of plank will just reach from the edge of the
stream to the top of the wall ?
In any right-angled triangle, the square of the * \
side opposite the right angle (A B) is equal to the
sum of the squares of the other two sides (B o
and AC).
If then A B2=A c2+c B2, A B= VA cH-c B2 ; i.e.,
the length of the plank =V82+62=VlOO= 10 ft.
Note. — In such a triangle, one of the shorter sides will be
equal to the square root of the difference between the squares
of the other two.
Ex. 4. — What is the circumference of a circle whose diameter
is 3£ feet?
The circumference is the line containing the circle ; the
diameter, any line drawn through the centre, and bounded by
the circumference. Any line drawn from the centre to the
circumference is called the radius, and is half the diameter.
The circumference is 3'1416 times the diameter. If, there-
MENSURATION. 135
fore, -the diameter is 3j feet, the circumference is 3^X3*1416
= 10-995 feet.
Ex. 5. — Kequired the area of the same circle 1
The area of any circle =3*1416 times square of the radius.
The area required is therefore l£2X31416 = 9*62115 sq. ft.
Note. — If the area were given, the radius would be found by
dividing the area by 3*1416, and taking the square root of the
quotient.
Ex. 6. — The trunk of a tree which was cut down measured
4 feet across one of its ends, and 12 feet in length : what was
its content ?
A body of this shape is called a cylinder. Content of a
cylinder=area of base X length.
Therefore area required is 22X 31416X12 = 1507968 ft.
Ex. 7. — A glass globe is 14 inches in diameter : what is the
area of its surface ?
Area of surface=circumferenceXdiameter=3'1416Xi>2.
Required surface is therefore 142X3*1416=4 sq. ft. 387 in.
Note. — If the area of surface were given, the diameter would
be found by dividing the area by 3*1416, and extracting the
square root of the quotient.
Ex. 8. — What is the content of the same globe ?
The content of a globe ='5236 times the cube of the dia-
meter.
The content required is therefore 143X '5236 = 14367584
cubic inches.
L49 EXERCISE II.
1. What is the area of a rectangular field 22 ch. 45 Ik. by 17 ch.
29 Ik. ?
2. A room is 12 ft. long by 9 broad : what length of line will stretch
between its two opposite corners ?
3. A circular plot measures 27 '65 feet across its widest part : what
length of netting would enclose it ?
4. Find the area of a rhomb-shaped field, 28 '6 poles long and 25
poles in perpendicular breadth.
5. The spoke of a cart-wheel measures 3^ feet from centre to riia :
how many revolutions will the wheel make in a mile ?
6. A line 75 ft. long, attached to the top of a flag-staff 60 ft. high,
was fastened in the ground : at what distance was it from the bottom
of the staff?
7. Find the breadth of a rectangular field of 16 ac. 2 ro. 21 po-
whose length is 26 ch. 61 Ik.
136 MENSURATION.
8. A circular field mecisures 729 sq. yds. in area : what is the length
of the side of a square field of the same area ?
9. Find the area of a circular plot, diameter 20 yds.
10. Find the area of a quadrilateral plot, having 2 parallel sides of
15 and 21 feet respectively, and 12 feet in perpendicular breadth.
11. An oblong field, 18 ch. 75 Ik. long by 25 ch. 30 Ik. broad, is
under potatoes and turnips, and the potatoes extend over '375 of its
length : find the area under both.
12. What is the radius of a circular pond 7854 sq. ft. in surface ?
13. If a square field is laid off on a base of 12 ch. 40 Ik., what is its
area ? and if a circle is traced within it on a diameter of the same
length, what area is lost in the corners ?
14. A diamond-shaped grass plot was found to contain 272 sq. ft. ;
if its length was 17 ft., what length of iron fence would divide it into
two parts broadways ?
15. How many cubic feet of air in a balloon 2 ft. 7 in. in diameter?
16. What is the length of a moor containing 2£ sq. miles in area,
and 17 ch. 44 Ik. broad ?
17. How long a tether will give a cow an acre of pasture ?
18. Two fields, one oblong, the other square, are measured and
found to contain the same area. The length of the one is 32 po., and
its breadth 18 po. : what is the length of a side in the latter?
19. A pillar, 14 ft. high, was 2£ feet across its end : find its solid
content.
20. Find the surface of a circular table-cover, the table measuring
5 ft. 3 in. across by its centre, and the cover hanging 4 inches over the
edge.
21. A four-sided field contained 3 ac. 1 ro. 38 po. in area, and its
two parallel sides were respectively 25 and 37 poles : find its breadth.
22. A carriage-wheel revolves 900 times in a distance of 2 '5 miles,
find the length of a spoke.
23. What cost the gilding of a ball for the vane of a church
spire, 3 ft. 6 in. in diameter, at 4^d. per sq. inch ?
24. How many feet of plank would cover a well whose mouth mea-
sured 4*75 feet across its centre, leaving a round hole one foot across
for air ?
25. A cheese measured 2 ft. 8 in. across one of its ends ; if it was
9 in. in depth, what was its content ? and its weight, allowing 10.^
cub. in. per lb ?
26. What superficial content of paper would be required to cover
a pair of 21-inch globes ?
27. An oblong garden, 3 ch. 25 Ik. broad by 3 ch. 60 Ik. long, is
surrounded by a walk 10 ft. 6 in. broad : find the expense of paving
the walk at 2/7 per sq. yd.
28. A point at the end of one of the sails of a wind-mill is distant
from the centre 27 ft. 9 in. : through what distance will it travel in an
hour, at the rate of 2| revolutions a minute ?
29. A spherical metal boiler was 10 ft. 3 in. in radius : how many
galls, water will it contain, if one gall. =277 '274 cub. in. ?
30. And if the watt r it contains just fills a circular pond 2 ft. 6 in.
in depth, what is the diameter of the pond ?
150
MISCELLANEOUS EXERCISE. 137
MISCELLANEOUS EXERCISE.— I.
1. Express in words (1) the sum of one hundred and seven millions
five hundred and eighty-four thousand and twenty ; one hundred and
ten thousand five hundred and two ; thirty-seven thousand and five :
(2) the difference between nine hundred and sixteen thousand and
nine, and fifty-six millions and three.
2. A man spends £155, 5s. 7d. per year : how much will he lay by
in 37 years out of £200 per annum ?
3. Find the value of (1.) 7 tons 14 cwt. 2 qr. 25 Ib hay, at £3, 10s. 6d.
per ton ; and (2.) 2 tons 7 cwt. 1 qr. 15 Ib, at £1, 3s. 4^d. per cwt.
4. If 24 oxen require 6 acres turnips to supply them for 10 weeks,
how many acres would supply 6 score sheep for 15 weeks, if 3 oxen
eat as much as 10 sheep ?
5. Divide 35 by -0175, and -0175 by 35.
6. A bill of £760 is due 7 months hence : find its present value at
5 per cent, per annum.
7. Find the interest on £189, 16s. 6d. for 341 days at 34 %.
8. If I gain 16 per cent, by selling 98 yards of cloth for £23, 13s. 8d.,
what was the buying price per yard ?
9. Find the tare on 84 hhd. sugar at 30 K> per hhd.
10. Find the square root of 10-624 to 3 dec. places.
11. How much carpet will cover a room 12 ft. 6 in. long by 14 ft.
9 in. broad ; and what will be the cost at 5/6 per square yard ?
O 'Jo
12. Find the sum, difference, product, and quotient of -4- and —1 .
13. How many yds. cloth would be needed for the clothing of 10000
soldiers, if each coat took If yds., a pair of trousers 1£ yds., and a
waistcoat f yds. ?
14. If the thirteenth part of 5 yds. 2 qr. 3 na. cloth be divided by
\, what results ?
15. How many parcels of sugar of 2 K>, 1 ft>, £lb, | Ib, can be made
out of a cask containing 8 cwt. 2 qr., the number of each being the
same?
16. Exchanged 40 yds. muslin, worth 2/6 per yd., for 30 yds. linen :
what was the linen valued at per yd. ?
17. If 1 K> weight standard gold were worth £46, 14s. 6d., how much
should one sovereign weigh ?
18. What is the rent of 1200 ac. 3 ro. at £1, 8s. 6d. per acre ?
19. If 2 cwt. 3 qr. 21 Ib sugar cost £12, 3s. 4d., what is the value of
17 cwt. 2 qr. 14 Ib ?
20. Reduce $ of 16/4| to the decimal of £1, 9s. 10|d.
21. Find the amount of £5433, 13s. ll^d. for 5 years 5 months at
2J per cent.
22. A circular tank, eight feet in depth, contains 10000 galls. : find
its diameter.
i n i OK K o r>i
23.
24. What sum will purchase £820 stock in the five per cents,
at 108?
25. Divide £2850 between A, B, and 0, giving ^ of B.'s share to A,
and to c £300 more than to A and B together.
138 MISCELLANEOUS EXERCISE.
II.
1. Multiply f of -175 by -285714, and divide the result by -00425
2. If 7| yds. cost £7, 18s. 4d., what cost 49A yds. ?
3. Make out the bill for—
40 chests of cloves at 2/1 each ; 35 bags coffee at £2, Os. 6d. per
bag ; 71 bags saltpetre at £1, 5s. 6d., per bag : and 5 casks sugar at
£2, 6s. 6d. per cask.
4. How many yards of stuff 3 qrs. wide will line a cloak 5 A yds.
in length, and 1 J yds. wide ?
5. What rate of income-tax will yield £38. Is. lid. on an income
of £570, 16s. 6Jd.f
6. Find the value in inches and fractions of an inch of '0003551 13G
of a mile.
7. In a school of 360 children, 10 pay 6d. a week, 80 pay 4d., 104
pay 3d., 75 pay 2d., 91 pay l£d. : what is the average sum paid by
each child per week ?
8. How much per cent is 7/6 of 4 guineas ?
9. Find the square root of 7 to 4 decimal places, and multiply the
result by^j^..
10. In what time will the interest of £325 at 31 % per annum
pay a debt of £67, 12s. ?
11. What length must be taken from a rectangular field 66 yds.
broad to cut off from it two acres ?
12. How long would it take to count a million of sovereigns at the
rate of 80 a minute, for 12 hours each day ?
13. I exchange 4375 yards for pieces of 3 qr. 2 nl. : how many
should I receive ?
14. Find the value of— (1.) 3068 articles at £1. 15s. 7fd. ; (2.) 217£
at £5, 19s. 6Ad.
15. If 120 bushels oats serve 14 horses for 56 days, how many days
will 90 bushels serve 6 horses ?
16. Find the sum of the greatest and the least of the fractions ^-,
H> M> £ 5 the sum of the other two ; and the difference of these sums.
17. A bill of £894 is drawn February 16, 1860, at 7 months' date,
what will be the immediate discount at 5 per cent. ? and what the
discount on 1st June same year ?
18. \Vhat sum will amount to £162, 8s. in 5 years at 4 per cent,
simple interest ?
19. Find the sum of -09 of £1, 3s. 2d., and -51 of £19 : and what
part of £5 that sum is.
20. If a loaf weighing 48 oz. cost 8p. when wheat is at 60/ per qr.,
what should be the price of wheat when a 6d. loaf weighs 38 oz. 8 dr. ?
21. Find the interest on £215, 12s. for 3 years 73 days at 4£ % per
annum.
22. A invests £1000 in the 3 per cents, at 84 ; B the same sum in the
4 per cents, at 110 : find their respective incomes.
23. A circular plate of gold, 3 inches in diameter and ^-inch thick,
is extended by hammering so as to cover 5 square yards : find its
present thickness.
24. Bought oranges at 20 for a shilling, and sold them at Id. a
piece : what was gained per cent. ?
25. Find to 4 decimal places the square of the sum of the square
roots of -25, -025, 3'6, and 14 -4.
MISCELLANEOUS EXE11CISE. 139
III.
1. Find the value of (2346784 x 53-4-583) x (107298 4-18 x 79).
2. A person's quarterly income is £135, 10s. and his daily expendi-
ture £2, 5s. : how much will be his debt for the two years and a half
ending June 30th ?
3. Find the least number that will contain 225, 255, ^89, 1023, and
4095, without remainder.
4. If 3 men or 4 women can do a piece of work in 56 days, in what
time will one man and one woman (working together) do it ?
5. What must be added to '356 of £2, 17s. 6d. to make up J| of
£8, 9s. 7^d. ?
6. If you have £1000 money in the three per cents, at 83£, and ex-
change it into the security of shares at £233 each, on which a dividend
is paid annually of £7, 13s. 4d. : what difference will it make on your
income ?
7. Required the amount of £400 in 3 years 35 days at 3f °/o Per
annum.
8. Five tubes have an internal diameter of 1 inch, 1-2 inches, 1'4
inches, 1/6 inches, 1*8 inches respectively: how high will a pint of
water stand in each, a pint containing 35 cub. in. ?
9. When will an investment of 1000 gs. at 6 per cent, double itself ?
10. Find the discount (true and common) of £132 payable at the
end of 3 months at 3£ %•
11. What is the price of the 3| per cents., when £3930 invested in
them produces £130 per annum ?
12. The pavement of a street is 15 ft. broad ; and from a point in
the street 9 ft. beyond the pavement, a ladder 40 ft. long just reaches
to the top of a house : what is the height of the house ?
13. If £25, 11s. 3|d. pay the carriage of 15 tons 16 cwt. 14 K> for
240 miles : what weight should be carried 180 miles for the same sum ?
14. What will £19, 13s. 9d. a day amount to in a solar year of 365
days 5 ho. 48 min. ?
15. What number is that from which if you deduct £- £, and to the
remainder add the quotient of J7 by 5J, the sum will be Hf + 10^r?
16. What is the value of the recurring decimal 3*4545 ?
17. A piece of cloth when measured with a yard measure two-thirds
of an inch too short, appears to be 10£ yds. long : what is its true
length?
18. How many pieces of gold leaf, 4 inches square, must be bought
to cover one face of a diamond-shaped kite, 2 feet broad and 4 feet
long (diagonally) ?
19. A quadrilateral field has two parallel sides ; one is 67 chains,
the other 5*8 chains, and the perpendicular distance between them 7'4
chains : find the acreage of the field.
20. To what other pairs of numbers is the mean proportional be-
tween 6 and 24 also a mean proportional ?
21. The diameter of a circular enclosure is 370 yards : what will 9
wall going round it cost at 9/6 per yard ?
22. A field of grass is rented by two persons for £27. The one
keeps in it 15 oxen for 10 days, the other 21 oxen for 17 days : find
the rent to be paid by each.
23. What fraction of if of 5 tons 17 cwt. 6 K> is 1 ton 2 cwt. 6 lb?
1 4 0 MISCELLANEOUS EXERCISE.
24. Three merchants make a stock of £700, and their profits are
respectively £231, Os. 5fd., £64, 3s. 3f^d., and £39, 8s. 7Ad. : how
much did each contribute ?
25. A and B exchange goods, A gives B 15 cwt. of hops, the retail
price of which is 58/ per cwt., but which he reckons at £3, 3s. per
cwt. ; B gives A 12 barrels of beer, retail value 1/2 a gallon, but the
value of which he raises in proportion to the increased value of the
hops : how much must be paid in money ?
IV.
1. Find the square root of 20££|.
2. The diameter of a well is 375 ft. and its depth 22'5 ft. : what
did it cost in sinking at 3/7 £ per cub. yd. ?
3. Bought at £193, 12s. and sold for £216, 13s. 4d. : find the gain %.
4. Find the interest on £1199, 19s. 6d. from April 1, 1858, to Jan.
9, 1859, at 3J per cent.
5. If I add f of a pound to ^ of a guinea, into how many shares,
each 4 of a shilling, can the sum be divided ?
6. If 1 oz. tea cost 375d., what cost 17'28 lb ?
7. Find the following bill :— 20 doz. copybooks at 15/ per dozen ;
1000 quills at 5/6 per hundred ; 125 inkstands at 7jd. each ; 24 doz.
lead pencils at 2/9 per doz. ; 64 boxes steel pens at 1/8 per box.
8. If 40 men require £20 worth of bread in 10 days, when wheat is
at 63/ per qr., how long would £90 worth serve 54 men, when wheat
is at 56/ per qr. ?
9. How many pounds of tea at 5/6 per lb must be exchanged for
293 yds. silk at 3/4£ per yd. ?
10. The annual deaths in a town being 1 in 45, in the country 1 in
50 ; in how many years will the number of deaths out of 18675 persons
in the town, and 79250 persons in the country, amount to 10000 ?
11. What is the value of a quarter of oats if 17g qr. cost £33^ ?
12. Reduce iVi%& t° its lowest terms, and divide it by ^ of 4f .
13. A person who began business 5| years ago has increased his
capital at the rate of 15 per cent, per annum simple interest, and it
now amounts to £5960 : what had he at first ?
14. The length of a room is 20 ft. 6 in., its breadth 15 ft. 9 in., and
its height 10 ft. 6 in. ; what will it cost for plastering, the ceiling
at 8d. a yd., and the rendering (on the walls) 3d. a yd. ? Allow for a
door 6 ft. 9 in. by 4 ft. 2 in., and a fireplace 5 ft. 6 in. by 5 ft, 3 in.
15. What is the side of a square field of 48 ac. 10 po. 22£ yds. 49 ft. ?
16. Find the nett weight of 64 hhds. sugar, each 5 cwt. 2 qr. 10 lb ;
tare 5 lb per cent.
17. A person's weekly expenditure is £15, 5s. : what must be his
daily income, so that at the end of eleven years he may have saved
£425, 18s. 8d., supposing that the first is leap year?
18. Find the value of— (1.) 15 reams 9 qu. 6 sh. paper at £1, 6s. 9d.
per ream ; (2.) 6 tons 7 cwt. 2. qr. at £3, 10s. 7£d. per cwt.
19. A ship, with a crew of 32 men, has provisions for 45 days, at a
daily allowance of 2 lb per man. It picks up another crew of 16
men : what allowance will make the provisions last 40 days ?
20. Divide the sum of (11^- -35)-:-(-05-^) and T^+6'007 by the
difference between f (1-35- 72) and 5'0004.
21. A merchant has teas worth 4/6 and 3/6 per tt> respectively, which
MISCE LL ANEOUS EXERCISE. 1 4 1
I
he mixes in the proportion of 2 Ib of the former to 1 Ib of the latter.
He sells the mixture at 4/4 per Ib : what does he gain or lose per cent. ?
22. If I invest £1200 in the 3 per Cents, at 72, what is my income ?
and how much per cent, do I get for my money ?
23. Find the cost of covering with gravel, at 7§d. per sq. yd., a path
3 ft. wide, round the outside of a bed whose diameter is 9 It.
24. If I am liable for a bill of £380 due 3 months hence, and I pro-
pose to pay at once, partly in cash, and partly with a bill for £152
due 4 months hence, what sum must I pay down, interest 4 %• ?
25. A, B, and c join in an enterprise to which they each contribute
in the proportion of 3, 3£, 3£ respectively. A pays down £220, 10s.,
B £205, 4s., and c £213, 5s. : what must each pay to the others, or
receive from them, to make the proportion of capital accurate ?
V.
1. The population of New York in 1830 was 203007 ; in 1840,
312710 ; and in 1845, 371102 : find the rate per cent, of increase each
interval, and on the whole period.
2. The interest on a certain sum lent for 85 days at £4, 6s. 8d. per
cent, per annum is £3, 13s. 8d. : what is the principal ?
3. What is the present worth of £120, payable thus :— £50 in 3 mo.,
£50 in 5 mo., and the rest in 8 mo., discount at 5 per cent, per ann. ?
4. Divide the square root of -00093636 by 2^.
5. A reservoir is 56 ft. 8 in. long by 17 ft. 6 in. broad : how many
cub. feet of water must be drawn off to sink the surface 2 ft. 6 in. ?
6. A number of men proceed on an expedition, with provisions for
nine days, at the rate of 1 Ib 2 oz. for each man per day. The quan-
tity furnished was 5062 Ib 8 oz. : required the number of men.
7. If A can do as much work in 5 hours as B in 6 hours, or as c in 9
hours, how long will it take c to complete a piece of work, one half
of which has been done by A working 12 hours and B working 24 hours ?
8. (1.) Multiply 2^ by 16£, and divide the lesult by f of 2f- ; and
(2.) Reduce p x -JZ to its simplest form.
9. (1.) What number of shillings and pence is equivalent to '6 of
£3, 5s. 8d. ? and (2.) what decimal of £99, 3s. 4d. is ff | of £233, 9s. 8d. ?
10. If 204 men build a wall of 306 feet long, 8 feet high, and 3 feet
thick, in 42 days of 6 hours each, in how many days of 8 hours each
will 188 men build a wall 6 feet high and 18 inches thick, round a
rectangular enclosure whose length is 319 feet and breadth 97 feet ?
11. Make out an account for : — 1 piece flannel 28^ yds., at 3/4 per
yd. ; 35 yds. calico, at 5.^(1. per yd. ; 3^ doz. pairs stockings, at 18/6
per doz. ; 7 pairs gloves, at 3/3 per pair ; 12£ yds. Irish linen, at 5/6
per yd. ; 4 pairs muslin curtains, at 12/8 per pair.
12. If 29,040 copies of a paper be printed, each of 3 sheets, and each
sheet 3.^ feet by 2 feet, how many acres will the edition cover ?
13. A tradesman marks his goods with two prices ; one for ready
money, and the other for one year's credit, allowing discount at 5 per
cent. If the credit price be 12/3, what ought to be the cash price ?
14. A person loses 10 per cent, by selling cloth at 15/ a yard : how
should it have been sold to gain 20 per cent. ?
15. If I lay out £1270 in the 3 per Cents at 92.} ; and, after allow-
ing the simple interest to accumulate for two years, I sell out at 93,
142 MISCELLANEOUS EXERCISE.
aud invest the sum in debentures at 104 paying 4£ per cent. : neglect-
ing all fractions of a penny, what is my income ?
16. A triangular field contains 10 ac. 2 ro., and measures 8 cli. 75 Ik.
along its base : what is its perpendicular height ?
17. A bankrupt had £214, 17s. 6d. of good debts, and the following
bad debts :— £340, 8s. 4d., £60, 13s. 6d., and £19, 4s. 6d., for which he
receives respectively 8/, 4/, and 16/ per pound : his liabilities amount
to £1200 : how much can he pay in the pound ?
18. Find the difference between | °[ ^ and £ (ft-^)+W + Jft).
19. A person bequeaths an annuity of £100 a year : what sum must
he invest in 3£ stock at 97 to do so ?
20. A semi-circular area whose base is 15 yards is covered with
carpet 2 feet wide, what will it cost, at 3/6 per yard, allowing 11 -6
sq. yards for waste ?
21. If a family of 9 persons spends £300 in 8 months, how much
money will serve 17 persons for 11 months at the same rate of expen-
diture ?
22. A cubic inch of water weighs 252 '458 grains, and the weight of
an imp. gallon is 10 Ib av. Find (to 3 dec. places) the number of
cub. in. in an imp. gall., there being 7000 grains in the Ib av.
23. What sum will amount to £194, 16s. 10. iu2| yrs., at 4% per an.?
24. Goods are purchased at £28. 10s. 6d. per cwt. ; trade profits are
15 per cent, on invested capital ; the income-tax due on these at 9d.
per pound amounts to £24, 3s. 6d. : how many cwt. were purchased ?
25. If a lump of iron 16 cwt. 1 qr. 5 Ib 5 oz. be rolled into a
cylindrical bar 12 ft. long, find the diameter of the bar (to three places
of decimals). A cubic foot of iron weighs 7788 oz.
VI.
1. Divide £1175 into 4 shares, which shall have the proportions of \, f , f , £.
2. Find the compound interest of £2500 in 4 yrs. at 4%.
3. Two ships sail, one due north at the rate of 12 miles an hour, and the
other due east at the rate of 5 miles an hour : how far apart will they be in
6 hours ? and when will they be 69 miles apart ?
4. A boy buys a suit of clothes for which he will pay in a year at so much
per week. There are 2£ yds. cloth at 11/4 per yd., the trimming costs 7/> and
the making 15/3 : what has he to pay per week?
5. By selling goods at £3, 14s. 6d. a cwt., which cost me 50/ per cwt., I
gained 2 guineas : what quantity did I buy ?
6. For the rent of a farm of 27 ac. 3 ro. 27 po. at £7, 10s. 8d. per acre, 139|
yds. velvet at £1, 19s. 4d. a yard were taken : what money was returned?
7. A ship's company take a prize of £1001, 19s. 2d. which is divided accord-
ing to their pay and time on board. The officers and midshipmen have been
on board 6 months, the sailors 3 months ; the officers receive 40/, the mid-
shipmen 30/, the sailors 22/ a month. There are 4 officers, 12 midshipmen,
and 110 sailors : what will be each man's share?
8. In the year of the Great Exhibition of 1851, London was supplied with
butter by 215000 cows, whose produce was 17210 tons 15 cwt. 3 qr. 23 ib 8 oz.
for the year : what did a single cow produce?
9. The carpeting of a room 32 ft. 2 in. long at 5/9 per sq. yd. came to £20,
find the breadth of the room.
10. The price of bread is 8£d. a loaf, and of butter 1/1 \ a pound : how many
loaves of bread are equivalent to 5J cwt. butter?
11. The population of Dundee in 1821 was 30575; in 1821-31 it increased
per cent. ; in 1831-41. 38£S§2 per cent. : what was it in 1831 aud 18411
MISCELLANEOUS EXERCISE. 143
12. (a.) A'S income, after deducting income-tax at 1/2 per pound, is £1000 :
what was it before deduction? (b.) B'S income, after paying income-tax on
half his income at 9d. per pound, and on the other half at 10d., was £576, 5s.:
what was it before ? (c.) The income-tax is raised from 7d. to 1/4 per pound ; if
c's clear income after paying the tax is £500 before it is raised, what is it after ?
13. Divide -14 by 7, 140 by '07, and "014 by 7000 ; and give the sum of the
quotients as a vulgar fraction.
14. The length of a hollow iron roller is 3 feet, the exterior diameter 2 feet,
and the thickness of the metal f of an inch : find its content, and how often it
will turn from end to end of a gravel walk 65 yds. long.
15. (a.) If 4 per cent, is lost by selling linen at 2/9 per yard, how must it be
sold to gain 10 per cent. ? (6.) By selling cheese at £3, 6s. 6d. per cwt., 12 per
cent, was gained : what was the prime cost ?
16. The interest on £754, 6s. 8d. for 8 mo. 10 da. is £23, 5s., what is the
rate per cent, per ann. ? (Reckon 28 days, one month.)
17. A marble slab, 6 ft. 3 in. long, by 2 ft 8 in. broad, and 4 in. thick,
weighed 8 cwt. 1 qr. 20 Ib, and cost £4, Os. 6|d. : how much was the cost
per cub. foot, and what is the weight of a cub. foot of marble?
18. When the 3J per cents, are at par, the 3 per cents, at 92£, and a stock
which pays 3| per cent, is at 104, which is the best, and which the worst ?
19. A barters sugar with B for flour worth 2/3 per stone, but uses a false
weight of 13£ Ib to the stone : what value should B set on his flour that the
exchange may be fair?
20. A bankrupt owes £900 to three creditors, and his whole property
amounts to £675 ; if the claims of two of the creditors are £125 and £375
respectively, what will the remaining creditor receive for his dividend?
21. If 3 men, working 11 hours a day, can reap 20 acres in 11 days, how
many men working 12 a day will reap a rectangular field, 360 yds. long and
320 wide, in 4 days ?
22. A watch, which is 10 min. fast on Tuesday at noon, loses 2 rain. 11 £
sec. per day : what time will it show at 5 a.m. the following Saturday?
23. What sum will amount to £194, 16s. l^d. in 2| years at 4 % comp. int. ?
24. The present worth of a sum due 11 mo. hence, when discounted at
4 % per ann., is £2212 : what is its present worth, discounted at 5% per ann. ?
25. Find the product of (f of f of H) *y (f of $f of f£) : add the result
to the difference between '014 and -^ ; and express the result decimally.
VII.
1. (a.) Whether is heavier, a pound of gold or a pound of feathers, also an
ounce of gold or an ounce of feathers, the one being troy and the other avoir-
dupois weight? In both cases express the one as a decimal of the other.
(&.) What must be insured at 4£ per cent, on goods worth £2450, so that in
case of loss the worth of the goods and of the premium may be recovered ?
2. If 1 Ib Tr. is coined into 46 £ sovereigns, and 1 Ib av. into 48 halfpence,
what is the difference in weight between a sovereign and a halfpenny ?
3. One man can do a piece of work in 5§ days, a second in 7£, and a third
in S^, in how many days can they perform it when working together ?
4. (a.) What is the difference in gain per cent, between selling goods at 2d.
which cost l£d., and selling goods at 2£d. which cost 2d. ?
(5.) If a person, selling cloth at 15/6 per yard, gain £26, 10s. per cent, on
outlay, what does he lose per cent, when he sells the same at ll/ per yard ?
5. The assets of a bankrupt estate are just sufficient to pay a dividend
of 16/7 per pound, but the expense of realizing amounts to £1100, which
reduces the dividend to 14/9 per pound : what was the debt?
6. A reservoir is supplied by one pipe and emptied by another. The sup-
ply-pipe would fill it in 5 hours, and the escape pipe would empty it in 5
hours ; required the time of filling it when both are opened together.
7. Of every 24 oz. of gold, only 18 oz. in jewellery and 22 oz. in sovereigns
are really gold. A sov. weighs 123 gr. When a goldsmith offers chains for
their weight in soy., what does he charge per oz. for workmanship?
8. Siippose a railway train, proceeding at the rate of § of a mile in a minute
144 MISCELLANEOUS EXERCISE.
to be audible at a distance of 2J miles, how long exactly will its noise pre-
cede it — sound travelling at the rate of 1130 feet per second?
9. The rise of interest from 3J to 4 per cent, increase:? a person's nett in-
come (after deducting income-tax of 7d. per pound) by £485, 8s. 4d. : what is
the principal sum from which his income is derived ?
10. A person sells out £1725, 3 per cents., at 92J, and buys 3J per cents, at
95J, brokerage fcth per cent : what is the alteration on his income ?
11. Standard silver contains 37 parts of silver and 3 of alloy. Now, 5/6 just
weighs an oz. Tr. : what weight of pure silver is in £100 ?
1 2. The weight of water being 1000 oz. av. per cub. ft. , what weight of water
will an inch (area) pipe discharge in a day, flowing at the rate of 2 ml. pr. ho. ?
13. Find the number of cubic inches in a cube of which the edge is 2 ft.
5 in. long : find also the length of the diagonal of the cube.
14. Add f of ^ to & of 3-$ ; multiply the sum by the difference between
§ and ^f » and divide the product by 15 times the difference of -^fg- and -5 J^.
15. A tunnel ^ of a mile long is excavated at the rate of ^ of a yard per
day : in how many years will it be completed ?
1C. (a.) If cloth be bought at 15/3 per ell of 5 qrs., and sold at 15/6 per yd.,
what is gained on an outlay of £47, 12s. 6d? (&.) If sugar be bought at £2,
Is. 9d. per cwt, and sold at 6Jd. per ft, what part of every £100 is gain ?
17. The following bill was paid by a number of persons contributing £1, 16s.
7d. each: how many were there? 23 articles at £9, 2s. lid. per score ; 54
at 11/8 per score ; 37 at 12/6 per dozen ; 15 at 4Jd. each ; and 11 at 7Jd. each.
IS. A manufacturer employs 60 men and 45 boys, who work respectively
10 and 8 hours per day during 5 days of the week, and half the time on the
remaining day ; if each man receives 6d. per hour, and each boy 2d. per hour,
what is the amount of wages paid in the year of 52 weeks ?
19. A wall 700 yds. long was to be built in 29 days. At the end of 11 days,
IS men had built 220 yds. : how many additional men had to be engaged to
work at the same rate, that the wall might be completed in the given time ?
20. A square, whose side is 500 feet, has a circular garden within it 400 feet
in diam. : what will it cost to pave the part outside the garden at 2/ per sq. yd. ?
21. I buy a set of watches at 50/ each. I sell them at a profit of T^th prime
cost, but for ready money deduct 5 per cent ; for every 8 sold for ready money,
6 are sold without discount : find the gain per cent, on the money invested ?
22. (a.) It is half-past 3 ; at what hour will the hands of the watch first
meet ? (6.) At what time between 1 and 2 are the hands together, in opposite
directions, and at right angles respectively ?
23. Find the cost of covering a roof with lead at IS/ per cwt on the follow-
ing data :— the length of the roof is 43 feet ; the breadth 32 feet ; the gutter-
ing is 57 feet long and 2 feet wide; the former requires lead at 9 '831, the
latter at 7 373 ft> to the square foot.
24. A cistern 12 feet long, 2 ft. 4 in. wide, and 9 in. deep, contains pulp for
making paper. If half its volume is lost in drying, how many sheets 8 in.
by 6 in. will be obtained, if 300 sheets in thickness go to the inch ?
25. A man has an income of £400 a year, and the income-tax is 9d. a pound.
A duty of l£d. per Ib is taken off sugar :_what must be the yearly consumption
of sugar in his family that he may -jus* save Iris income-tax ?
/r^C/ 0# ••'";
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