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GIFT OF
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OALIFOENIA STATE SERIES
ADVANCED
AKITHMETIC
BY
J. W. McCLYMONDS and D. R, JONES
I (
EBVI8KD BY THE
STATE TEXT-BOOK COMMITTEE
ANI> APPROVED BT THE
STATE BOAKD OF EDUCATION
SACRAMENTO
Friend Wm. Richardson, Supt. State Printing.
Copyright, 1910, by
THE PEOPLE OF THE STATE OF CALIFORNIA.
Copyright, 1907, by
J. W. McCLYMONDS and J). E. JONES.
lyt
iiiUi^^'nim .i?^^
Jn the compilation of thin book certain matter from
an " EmentiaU of Arithmetic'''' by J. W. McClymondt
and I). It. Jones has been used. All such matter it
protected by the copyright entries noted above.
12th Ed.— 25.000— Oct., 1914.
PREFACE
This text is designed for use in the grammar grades,
following the completion of the Elementary Arithmetic
of the same series. In the preparation of this text the
authors have aimed (a) to secure skill in numerical com-
putations and (6) to develop the power necessary to the
solution of any practical problem that may arise in the
common experiences of life.
The following are some of the distinguishing features
of this text:
1. The text contains an unusually large number of
exercises that are designed to give facility in numerical
computations.
2. In the presentation of each topic an effort has been
made to stimulate thought and to develop self-reliance on
the part of the pupils. Whenever the nature of the work
admits, it calls for action on the part of the pupils, as in
making measurements, engaging in business relations with
others in the class, etc.
3. The scope of the work is restricted to the needs of
the majority of persons in the common experiences of life.
Traditional materials that make no contribution to the
mastery of the essentials of arithmetic have been carefully
eliminated. All of the work prescribed in the text proper
is easily within the capacity of pupils in the grammar
grades. Certain topics that are prescribed in some courses
of study but purposely ^j^yo^^fipm other courses have
4 PREFACE
been presented in an Appendix, so that they may be used
or omitted, as desired in each case, without destroying the
continuity of the other work.
4. The problems of the text have been drawn from the
common field of everyday experience. The necessary
arithmetical training is had from dealing with practical
problems within the experience of the pupils. No unreal
problems, or problems dealing with artificial situations,
or problems treating of situations remote from the experi-
ences of the average pupil in the grammar grades, are
introduced. The text aims to teach arithmetic only.
5. The text contains an unusual amount of oral work,
including oral problems under every topic treated. The
oral problems are everywhere related to the written work.
No additional text in " mental " arithmetic need be used
m conjunction with this text.
6. The methods of the text are those commonly em-
ployed in business life.
7. The work in fractions and compound numbers is
limited to the practical needs of life. Special attention is
given in fractions to the use of those fractions which
pupils must handle later on as the fractional equivalents
of certain per cents. Commission, Taxes, Insurance, etc.,
are made part of the work in Percentage and are not
treated as separate topics. The work in Interest has been
considerably reduced, and but one method of finding
interest is recommended.
8. A constant review of all previous work is maintained
throughout the text.
Finally, the aim of the authors has been to present
a course in arithmetic that will secure a thorough knowl-
edge of the essentials of this subject.
CONTENTS
PART I
Review of Integers and Decimals
PAGES
The Decimal System — Notation and Nuix sration — Addition —
Subtraction — Multiplication — Bills and Accounts — Divi-
sion by Measurement and Partitio i — Comparison — Meas-
urements — Divisibility of Numbers 7-89
PART II
Fractions
Objective Fractions — Ratio — Reduction — Addition — Sub-
traction — Multiplication — Division — Scale Drawing —
Aliquot Parts — Measurements 90-165
PART III
Percentage
Percentage — Profit and Loss — Commission — Insurance —
Taxes — Customs and Duties — Trade Discount — Interest
— Promissory Notes — Partial Payments — Compound Inter-
est—Bank Discount— Present Worth . . . 166-220
PART IV
Form^ and Measurements
Lines — Angles — Surfaces — Solids — Longitude^ and Time —
Ratio r . . 221-243
5
6 CONTENTS
PART V
Powers and Roots
PAOX8
Powers — Square Root — Right-angled Triangles — Similar Sur-
faces and Solids 244-255
PART YI
Appendix
Corporations, Stocks, and Bonds — Commission and Brokerage
— Trade Discount — Partial Payments — Interest Table -
Exact Interest — State and Local Taxes — Customs and
Internal Revenue — Banking — Life Insurance — The Equa-
tion — Proportion — Surfaces and Solids — Measurement of
Public Lands — IMetric System — Tables of Denominate
Measures — Table of Compound Interest . . . 256-320
Index 321-324
ESSENTIALS OF ARITIJMETiq,:^
PART I ..^i' -'^'-'•''
REVIEW OF INTEGERS AND DECIMALS
1, The Decimal System.
1. A unit is a single thing, or a group of things
regarded as a single thing, as a book, an apple, a box of
apples, etc. A unit is represented by the least whole
number, one (1).
2. Point to several units of the same thing in your
schoolroom. Can you think of a way by which you could
tell your parents how many children there are in your
room without using number?
3. Any definite quantity used to measure quantity of
the same kind is called a unit of measure.
The unit of 6 is 1 ; of 6 cows is 1 cow ; of 9 ft. is 1 ft.
The inch, foot, yard, rod, and mile are units used to meas-
ure length or distance. Name the units used to measure
areas. What is the unit of 10? of 1^10? In finding the
number of hats at $2 each that can be bought for $10, the
unit of measure is 82. What is the unit of measure in
finding the number of 4-ft. shelves that can be made from
a board 12 ft. long?
4. Name the units used to measure liquids; time;
weight.
8
REVIEW OF INTEGERS AND DECIMALS
5. Ih the number 111, the 1 at the right denotes some
unit, and the 1 next toward the left denotes a unit ten times
as great, and the 1 at the left denotes a unit ten times the
second unit, or one hundred times the first unit. This
aaay-be shp\Yij thus :
one hundreds' uhit
one tens' unit
one unit
100
10
6. In 236, the 6 represents 6 units ; the 3 represents
3 units, each of which is ten times each of the units repre-
sented by 6 ; and the 2 represents 2 units, each of which
is ten times each of the units represented by 3, or one
hundred times each of the units represented by 6.
7. Tell what each figure represents in 125, 47, 352.
8. In 30, the shows that there are no units of ones ;
and the 3 represents 3 units of tens. What does each figure
represent in 60, 600, 405, 530, 203, 478, 700, 520?
9. In 324, the units represented by 4 are called units
of the first orders or of unM order ; the units represented
by 2 are called units of the second order^ or of tens'* order ;
and the units represented by 3 are called units of the
third order ^ or of hundreds* order,
10. Our number system is a decimal system. Decimal
means tens, A decimal system is one in which ten units of
one order are equal to one unit of the next higher order.
The decimal system is believed to have had its origin in the prac-
tice of using the fingers for counting.
DECIMAL SYSTEM 9
11. Beginning at the left of 111, the 1 in the third
order represents some unit ; the 1 in the second order
represents a unit one tenth as great ; and the 1 in the
first order represents a unit one tenth as great as that
represented by a unit of the second order. A unit one
tenth as great as that represented by the 1 in the first
order may be represented by 1 written to the right of a
decimal point (.) placed to the right of units' order, thus :
.1 (111.1). A unit one tenth as great as this last unit
may be represented by 1 written in the second place to
the right of the decimal point, thus : .01 (111.11).
12. .1 is read one tenth; .01 is read one hundredth ; .11
is read eleven hundredths ; 1.1 is read one and one tenth;
A is read/(92^r tenths. Read 6.7; 8.05; 56.25.
13. The decimal point is placed after the figure that
represents whole units. The figures to the right of the
decimal point represent decimal parts of units. The parts
thus represented are tenths, hundredtlis, thousandths,
etc. ; and are called decimals.
14. A whole number is called an integer. Write an
integer. On which side of the decimal point are integers
written ?
15. What is the meaning of the word decimal? Why
is our number system called a decimal system?
16. What does each 2 in 222.222 represent?
17. Write the following so that units of the same order
are below one another: 45.5, 214.25, 347, 4.315, 17.
18. Compare the value of 2 in 24 with the value of 2
in 240 ; with the value of 2 in .24.
19. Is the system of United States money a decimal
system? Explain your answer.
10 REVIEW OF INTEGERS AND DECIMALS
NOTATION AND NUMERATION OF INTEGERS AND
DECIMALS
2. 1. Numbers are commonly expressed by means of
figures (or digits) as 5, 10, etc. ; by means of words, a,sfive,
ten, etc. ; and by means of letters, as V, X, etc. The art
of writing numbers by means of symbols is called notation.
The word digit means finger. Why were the figures called digits ?
2. The figures 1, 2, 3, 4, 5, 6, 7, 8, 9, 0, are called Arabic
numerals, as they were introduced into Europe by the
Arabs, who borrowed most of them from the Hindus.
The system of denoting numbers by means of figures is
called Arabic notation.
3. The figure is called naught, cipher, or zero. It has
no value. It is used to fill out places that are not oc-
cupied by other figures. Using figures, write six; six
tens ; six hundreds.
4. The art of reading numbers is called numeration.
5. Integers of more than three places are read more
easily when the figures are separated by commas into
groups of three each, beginning at the right. The groups
are called periods, and each period is named after the
order of the right-hand figure in the group.
6. The names of the first four periods, and the orders
in each, are as follows:
'S
-s
%
•s
Oedbes :
1
S
^
1
CO
a
a
Tens
Units
s
Tens
Units
7_
8
9,
6
5,
8
4 2,
3 1
Peeiods :
Billions
Millions
Thousands
Units
a
b
c
1.
3,625*
35,205
825,380
2.
8,017
20,007
308,016
3.
9,008
45,500
950,025
4.
6,303
12,012
404,040
5.
What is
the
) name of the first perio(
. NOTATION AND NUMERATION 11
3. Reading Integers.
To read an iiiteger of more than three figures^ begin at the
right of the number and point off periods of three places each.
Read the part occupying the left-hand period as though it
stood alone^ and add the name of the period ; then read the
part occupying the next period as though it stood alone, and
add the name of the period. Continue until units' period is
reached ; there omit the name of the period.
Read the following :
d
7,125,380
6,000,150
8,040,075
5,505,050
? of the second
period ? of the third period ? of the fourth period ?
6. How many periods are there in the numbers in
column a? hi c? d?
7. How many places do the numbers in column c
occupy ? in column d ?
8. What is the name of the left-hand period in the
numbers in column a?
9. Kead the left-hand period of the first number in
column d. Read the middle period of the same number.
Read the number.
10. Read the third period of the fourth number in
column d. Read the second period. Read the number.
11. When a number consists of three periods, how
many places must there be in the lirst period ? in the
second? How many places may there be in the third period ?
* 625 is read six hundred twenty-five.
12 REVIEW OF INTEGERS AND DECIMALS
4. Writing Integers.
To write a number in figures^ begin with the higheg,
period and write it as though it stood alone^ and add a
comma; then write the next highest period as though it
stood alone, and add a comma; continue until units' period
lias been written, thus: 5,807,050.
Write in figures :
1. Three thousand, two hundred four.
2. One hundred two thousand, eight hundred ninety.
3. Twelve million, eight hundred seven thousand,
eighty-four.
4. Seven hundred two million, sixteen thousand.
5. Write numbers dictated by your teacher.
Read aloud the following statements :
6. The area of Rhode Island is 1,250 sq. mi. ; of Mas-
sachusetts is 8,315 sq. mi.; of Illinois is 56,650 sq. mi.; of
California is 158,360 sq. mi. ; of Texas is 265,780 sq. mi.
7. In 1900 the population of Rhode Island was 428,556 ;
of Massachusetts was 2,805,346 ; of Illinois was 4,821,550 ;
of California was 1,485,053; of Texas was 3,048,710.
8. The total number of votes cast for president in 1900
was 13,964,812. The five states polling the largest num-
ber of votes were : New York, 1,548,042 ; Pennsylvania,
1,173,210; Illinois, 1,131,894; Ohio, 1,040,073, and Mis-
souri, 683,656.
9. The grain production of the United States in 1902
in measured bushels was as follows: Indian corn,
2,523,648,312; wheat, 670,063,008; oats, 987,842,712;
barley, 134,954,023 ; rye, 83,630,592 ; buckwheat,
14,629,770.
ROMAN NOTATION 18
ROMAN NOTATION
5, 1. The letters used in Roman notation are:
I V X L C DM
1 5 10 50 100 500 1000
2. The above letters are called Roman numerals.
Other numbers are represented by combinations, thus :
a. Repeating a numeral repeats its value. XXX de-
notes 30, CCC denotes 300. The numerals V, L, and D
are not repeated. Why ?
h. If a numeral is followed by another of less value,
the sum of their values is denoted. XXVI denotes the
sum of 10, 10, 5, and 1.
c. If a numeral is followed by another of greater value,
the difference of their values is denoted. XC denotes the
difference of 10 and 100, or 90; CD is 500-100, or 400.
d. A bar glaced over a numeral increases its value
1000 times. V denotes 5000 ; IX denotes 9000.
3. Read the following and tell which of the above
rules is illustrated in each: LX, XL, CIX, MDCC, IV,
MDCCCCVI, LXXIV, MMDXL, MDCXX, XLII.
4. Write 1776 in Roman numerals.
Model : 1776 may be divided into the parts 1000 — 700 — 70 — 6.
These parts expressed in order, beginning at the left, are M — DCC —
LXX — VI. 1776 is written MDCCLXXYI.
5. Write in Roman numerals : 18, 27, 68, 1492, 1907.
6. Write in Arabic figures XCVI, XLVII, XIX,
LXXIV, MDCCCXII.
Roman numerals are frequently used to designate chapter numbers
in books, the hours on the clock face, dates on monuments and pub-
lic buildings, etc. The M is used to designate a thousand feet of
lumber.
14 REVIEW OF INTEGERS AND DECIMALS •
UNITED STATES MONEY
6. 1. The units of United States money are decimal
units. The standard unit of value is the dollar. The
other units are derived from it. The dime is one tenth part
of the dollar, and the units that represent dimes are there-
fore written in the first place to the right of the decimal
point. The cent is one hundredth part of the dollar, and
the units that represent cents are therefore written in the
second place to the right of the decimal point.
2. Dimes are written as cents. Two dollars and four
dimes is written thus: 12.40. This is read two dollars
and forty cents.
3. The unit one dollar is written $1. The unit one
dime is written $.10. The unit one cent is written $.01.
The unit one mill is written f.OOl.
7. Reading United States Money.
Read the following
:
1. 1425.15
5. $30,755
9.
$8340.05
2. $301.08
6. $ 7.057
10.
$9015.807
3. 1220.20
7. $10,105
11.
$7200.50
4. $100.10
8. $ 4.005
12.
$1306.065
8. Writing United States Money.
Write the following in columns :
1. Six dollars and seventy-five cents.
2. Twenty-five dollars and fifty cents.
3. Eighty-five dollars and six cents.
4. Three hundred forty dollars and eighty cents.
5. One hundred dollars and fifty-two cents.
6. Eight cents. 7. Thirty-five cents and eight mills.
READING AND WRITING DECIMALS 16
9. Reading Decimals. |
•a
00 +3 -^ w "1" 2
Okders: '^wiS :S'«s^'«.2
9 13.452 876
Integers Decimals
1. Memorize the number of decimal places required
for each of the first six orders.
Tenths (first) . .' . . . . . . .5
Hundredths (second) 45
Thousandths (third) 367
Ten-thousandths (fourth) 6745
Hundred-thousandths (fifth) 62789
Million ths (sixth) 346329
To read a decimal, read the number without reference to
the decimal point, and add the name of the order of the
right-hand figure.
2. .375 is read three hundred seventy-five thousandths.
3.08 is read three and eight hundredths. Read: .125, .875,
4.625, 37.075, 670.005, 3.1416, 2150.42, .7854.
10. Writing Decimals.
1. Write sixty-two thousandths. As thousandths is
the name of the third order to the right of the decimal
point, three figures will be required in writing the num-
ber. Two figures are necessary to denote sixty-two ; so
one cipher must be supplied. To write sixty-two thou-
sandths, first write the decimal point, then write 0, and
then write 62 (.062).
2. Write the following: Sixty-nine ten-thousandths^
forty-eight hundred-thousandths ; thirteen thousandths.
16 REVIEW OF INTEGERS AND DECIMALS
ADDITION OF INTEGERS AND DECIMALS
11, 1. A number that is not applied to any particular
thing, as 6, 43, etc., is called an abstract number.
2. A number that is applied to some particular thing,
as 6 ft., 43 lb., etc., is called a concrete number.
3. Quantities that are expressed in the same unit of
measure, as 3 lb. and 6 lb., are called like quantities.
4. Quantities that are expressed in different units of
measure, as 5 lb. and 4 hr., are called unlike quantities.
5. Write two abstract numbers; two concrete num-
bers; two unlike quantities. Like quantities can be
combined and expressed as a single quantity. 3 ft. and
2 ft. may be combined and expressed as 5 ft. Can the
unlike quantities 5 lb. and 4 hr. be combined and ex-
pressed as a single quantity?
6. Units of the same order may be combined and
expressed as single numbers. 3 tens and 2 tens are
5 tens.
7. When two or more numbers are combined and ex-
pressed as a single number, this number is called their
sum, or amount.
8. The process of finding the sum of two or more
numbers is called addition. The numbers that are added
are called addends.
9. The sign of addition is + and is read plus.
10. This sign = is the sign of equality, and when
placed between two numbers is read equals or is equal to^
thus: 6 = 4 + 2 means that 6 is equal to the sum of 4
and 2.
ADDITION 17
12. Oral Exercises.*
To each number in Exs. 1-4, add in succession 3, 2, 7,
6, 9, 4, 8, 5.
1. 23 35 84 69 26 88 82 57 47 60
2. 39 76 48 87 65 74 33 22 81 30
3. 52 86 49 61 73 95 40 18 67 94
4. 90 66 38 17 41 93 55 74 12 99
13. Add each column as written. Add each column,
increasing the number at the bottom of the column by 10,
by 20, etc., to 90 ; thus for column a, having increased
the number at the bottom of the column by 20: 22, 25,
29, etc.
a b c defghijklmn
98958797778896
3
7
8
8
7
9
3
3
7
9
9
9
5
8
6
5
9
2
5
7
9
6
8
8
9
5
6
4
4
6
3
7
4
6
5
8
6
6
5
9
4
9
3
8
9
6
8
9
5
2
4
9
1
8
8
8
8
9
7
5
9
6
8
7
8
7
7
7
3
9
7
7
6
4
5
8
5
5
6
8
1
9
4
2
7
7
3
5
7
6
9
9
4
4
8
7
3
7
2
4
5
9
8
8
7
8
5
9
4
6
9
6
3
7
2
3
7
9
6
5
8
8
2
8
5
5
4
3
9
5
6
4
9
7
9
7
6
3
7
7
3
8
9
2
9
6
6
7
8
5
8
2
7
7
2
6
8
4
4
5
7
9
6
2
8
9
7
4
* If the pupils require a more extended drill upon addition than is pro-
vided in the above exercises, the method indicated in the elementary text
should be followed.
AR. — 2
18 REVIEW OF INTEGERS AND DECIMALS
14. Written Exercises.
Numbers to be added or subtracted must be written
so that units of the same order are directly below one
another, units under units, tens under tens, and tenths
under tenths, etc. Why ?
When numbers are written so that the decimal points are
directly below one another, units of the same order are
directly below one another. Explain.
Add:
1. 2. 3.
$ 345.67 $ 58.06 $ 9.045
84.075 275.936 590.
650. 83.07 5.15
70.004 342.457 69.075
572.806 34.08 610.75
6.605 8.125 57.246
852.451 64. . 540.375
^ ^ W " Kead aiduii 6ach of ihe aqove.' ^
7. Write the above from dictation.
8. Add 74.06 mi., 6.8 mi., 320.45 mi., 17.04 mi.
9. Add 64.5 A., 79.14 A., 160.75 A., 321.15 A.
10. Add 60.5 cu. in., 352.24 cu. in., 80.125 cu. in.
11. Add 168.05, 1107.98, 1730.04, f 9.75, 1894, $80,
1740.40, 1375.15, f486.75, 1836.95, i.95.
12. Add six and nine hundredths, thirty-seven and
six tenths, eighty-five thousandths, seven hundredths.
13. Find the sum of nine hundred eighty and five
tenths, seventy and seven hundredths, one hundred and
five thousandths, six hundred twenty-five.
14. Write five addition exercises similar to Exs. 1-5
above and add each. Read each answer.
4.
5.
$405.27
$ 68.
73.435
125.87
487.50
45.369
50.258
845.075
250.50
8.75
.375
100.
62.50
58.268
ADDlTIOiSr 19
15.
Oral Exercises
.
Add
a
h
c
d
e
/
9
h
i
1. 40
60
130
120
60
150
90
140
80
50
20
90
30
140
40
30
60
20
2. TO
40
140
50
60
80
70
20
70
25
54
63
139
42
59
96
192
.36
3. 29
55
79
56
54
89
46
92
m
90
80
70
90
80
70
80
90
60
4. 23*
43
64
36
59
54
68
94
39
89
52
95 .
94
43
46
43
36
27
5. Frank weighs 95 lb. and his little brother weighs
34 lb. How much do they both together weigh ?
6. A farmer has 46 sheep and his neighbor has 54
sheep. How many have both together?
7. A man paid f 94 for a wagon and 836 for a harness.
How much did both cost him ?
8. Mr. White had 23 head of cattle and bought 39
more. How many had he then ?
9. A girl spent 50 ^ for cloth and 45/ for lace. How
much did she spend for both ?
10. A boy placed 60/ into his bank one week and 46/
the next week. How much did he put into the bank in
the two weeks?
11. A girl spent 20/ for stamps, 25/ for some meat,
and 50 / for sugar. How much did she spend for all ?
12. Make and solve ten oral problems in addition.
♦ Add : 89, 109, 112.
20 REVIEW OF INTEGERS AND DECIMALS
SUBTRACTION OF INTEGERS AND DECIMALS
16. 1. Like quantities, such as 5 marbles and 9 marbles,
may be compared, and the difference between them found,
thus:
9 marbles: •••••••••
5 marbles: • • • • •
2. If there is added to 5 marbles a quantity that will
make it equal to 9 marbles, how much is added ? This
amount is the difference between the two quantities.
3. If that part of 9 marbles that is equal to 5 marbles
is taken from 9 marbles, liow many will remain ? This
remainder is the difference between the two quantities.
4. How does the difference as found in Ex. 3 compare
with the difference as found in Ex. 2 ?
5. The difference between the two quantities may be
found by answering either of the following questions:
a, 5 marbles and how many marbles are 9 marbles ?
h. 5 marbles from 9 marbles leaves how many marbles ?
In either case, the answer is known by recalling that
the sum of 5 marbles and 4 marbles is 9 marbles.
6. The difference between two numbers is the number
which when added to one number makes the other number.
7. The process of finding the difference between two
numbers is called subtraction.
8. The number to which the difference is added is
called the subtrahend.
9. The sum of the subtrahend and difference is called
the minuend.
Or the subtrahend is the number which is subtracted, and the
minuend is the number from which the subtrahend is taken.
SUBTRACTION 21
17. Oral Exercises.
a b c
1. 6 and — are 11 8 and — are 12 9 and — are 16
2. 9 and — are 14 7 and — are 13 8 and — are 14
3. 8 and — are 11 4 and — are 11 7 and — are 11
4. 7 and — are 12 9 and — are 15 5 and — are 14
5. 5 and — are 13 5 and — are 11 6 and — are 15
6. 4 and — are 12 3 and — are 12 9 and — are 13
7. 8 and — are 15 5 and — are 12 8 and — are 16
8. 7 and — are 16 6 and — are 14 7 and — are 14
9. 9 and — are 17 8 and — are 13 8 and — are 17
10. 6 and — are 12 7 and — are 15 9 and — are 11
11. 9 and — are 12 9 and — are 18 6 and — are 13
9
12. _ r is read 5 and how many are 9? Or, ^ from 9 leaver
how many? Use the form with which you are familiar.
13. The sign of subtraction is ~, and is called minus.
It indicates that the number that follows it is to be sub-
tracted from the number that precedes it. 7 — 4 is read
seven minus four,
18. Explanation of Subtraction.
1. Find the missing addend.
(one addend) The other addend may be
2874 (one addend) ^^^"^ ^7 a^d^^S *° *^« g^^^^
rnor« y £ ^ i i i n addeiid the number that will
ozdo (sum 01 two addends) . ^, ^, ^ j «
^ ^ give the sum, thus : 4 and 2 are
6 ; 7 and 6 are 13 ; carry 1 to 8, making it 9 ; 9 and 3 are 12 ; carry
1 to 2, making it 3; 3 and 2 are 5. Missing addend, 2362.
22 REVIEW OF INTEGERS AND DECIMALS
2. From 5236 subtract 2874.
Model a : 5236 -A-fld to the subtrahend the number that will
2874 give the minuend, thus : 4 and 2 are 6 ; 7 and
23(32 6 are 13 ; carry 1 to 8 as in addition, making it
9; 9 and 3 are 12; carry 1 to 2 as in addition,
making it 3; 3 and 2 are 5. AVrite the answer as in the model.
This is known as the Austrian, or additive, method.
Model b : Subtract thus : 4 from 6 leaves 2 ; as 7 tens cannot be
taken from 3 tens, 1 hundred is " borrowed " from 2 hundreds and
called 10 tens; 10 tens and 3 tens are 13 tens; 7 tens from 13 tens
leaves 6 tens ; as 1 hundred was borrowed from 2 hundreds, there is left
1 hundred; as 8 hundreds cannot be taken from 1 hundred, 1 thousand
is borrowed from 5 thousands and called 10 hundreds ; adding 10 hun-
dreds to 1 hundred gives 11 hundreds ; 8 hundreds from 11 hundreds
leaves 3 hundreds ; as 1 thousand was taken from 5 thousands, there
are left 4 thousands; 2 thousands from 4 thousands leaves 2 thousands.
Model c : If the same number is added to both the minuend and
the subtrahend, the difference remains unchanged. Subtract thus:
4 from 6 leaves 2 ; as 7 tens cannot be taken from 3 tens, add 10 tens
to 3 tens, making 13 tens ; 7 tens from 13 tens leaves 6 tens ; as 10 tens
were added to the minuend, the same number must be added to the
subtrahend, so 1 hundred (10 tens) is added to 8 hundreds, making 9
hundreds; as 9 hundreds cannot be taken from 2 hundreds, 10
hundreds are added to 2 hundreds, making 12 hundreds; 9 hundreds
from 12 hundreds leaves 3 hundreds; as 10 hundreds were added
to the minuend, the same number must be added to the subtrahend,
so 1 thousand (10 hundreds) is added to 2 thousands, making 3
thousands ; 3 thousands from 5 thousands leaves 2 thousands.
19. Written Exercises.
Solve :
1. 38,256-21,359 6. 1,106,800-289,060
2. 40,175-19,688 7. 4,083,453-613,757
3. 85,430-41,856 8. 3,256,845-465,868
4. 93,950-17,275 9. 4,741,242-572,847
5. 97,204-57,240 lO. 2,814,004-935,940
SUBTRACTION 23
20. Oral Exercises.
Subtract :
a h c d e f g hi
1. 40 140 150 100 120 150 120 90 110
20 30 20 40 30 60 90 40 50
2.
95
40
126
90
83
50
142
60
149
80
155
70
153
20
129
40
124
80
3.
124*
92
109
44
138
85
96
13
75
24
139
68
136
45
88
16
99
44
4.
75 1
38
142
96
34
19
57
29
83
68
74
18
42
27
36
19
52
29
5. Harry bought 120 yd. of string and used 85 yd. for
a kite string and gave the rest to George. How many
yards did he give to George ?
6. A farmer had 52 head of cattle and sold 29. How-
many had he left ?
7. Mary read 87 pages in a book that contained 124
pages. How many more pages must she read to complete
the book ?
8. There are 38 pupils in Room A and 47 in Room B.
How many pupils are there in both rooms ? How many
more pupils are there in Room B than in Room A?
9. The frontage of a certain city lot is 40 ft. and its
depth is 135 ft. Find the difference between the depth
and frontage of the lot.
* Suggestion. The difference iDetween 92 and 124 Is 30 and 2, or 32.
t Suggestion. The difference between 38 and 75 is 80 (38 to 68) and
7 (68 to 76), or 37 ; or 40 less 3, or 37.
24 REVIEW OF INTEGERS AND DECIMALS
21, Before solving, represent each by a diagram.
1. Two boys started from the same place. One boy
rode east 32 mi. and the other boy rode west 24 mi. How
far apart were they then?
W.
2^ mi. S 32 mL
From S. to E. is 32 mi. and from S. to W. is 24 mi.
From E. to W. is the sum of 32 mi. and 24 mi., or ^Q mi.
2. Two boys started from the same place. One rode
east 32 mi. and the other rode east 24 mi. How far apart
were they then ?
3. How far apart are two places, if one is 40 mi. north
of the center of a certain city, and the other is 65 mi.
south of the center of the same city ?
4. Mary lives 8 blocks east of the schoolhouse, and
Ethel lives 14 blocks west of the schoolhouse. How far
apart do the girls live ?
5. Two trains left a certain station at the same time,
going in opposite directions. How far apart were they at
the end of 2 hours, if one traveled at the average rate of
42 mi. an hour, and the other at the average rate of 36 mi.
an hour ?
6. How far apart would the trains mentioned in Prob.
5 be at the end of 2 hours, if both traveled in the same
direction ?
7. In a bicycle race Frank and Henry rode around a
park 400 ft. long and 200 ft. wide. When Frank had
ridden once around tlie park, Henry hud gained 200 ft.
on him. At tlie same rate of gain, how many times will
Frank ride around the park before Henry overtakes him ?
SUBTRACTION 25
22. United States Money.
Write units of tlie same kind below one another. Do
not supply unnecessary O's.
1. Subtract: a. $12.75 from 137.25; h. $12 from
137.25; c. 112.75 from $37.
Model a: $37.25 Model &: $37.25 Model c: $37.
12.75 12. 12.75
$21.50 $25.25 $24.25
Solve :
2. $307.57 -$200.69 6. 120.375-93
3. $925.07 -$570.80 7. 690.125-209
4. $700.40 -$180.05 8. 542-45.78
5. $860.455 -$280 9. 640-70.65
10. Read aloud each of the above amounts.
11. Write the above amounts from dictation.
23. Decimals.
Subtract :
1. 2. 3. 4.
320.564 450.125 35.7 600.
206.7 86.75 6.875 57.375
5. A man owned 158.15 acres of land. He sold 79.5
acres. How many acres had he left ?
6. If it is 844.7 mi. from San Francisco to Ogden and
1004.7 mi. from Ogden to Omaha, how far is it from San
Francisco to Omaha? How much farther is it from Ogden
to Omaha than from San Francisco to Ogden ?
7. A cubic foot of rain water weighs 62.5 lb. and a
cubic foot of petroleum weighs 54.875 lb. How much
heavier is a cubic foot of rain water than a cubic foot of
petroleum (kerosene) ?
26 REVIEW OF INTEGERS AND DECIMALS
24. 1. Show the effect, if any, upon the difference :
(a) of adding the same number to both minuend and
subtrahend; (6) of subtracting the same number from
both minuend and subtrahend. Illustrate each with
several exercises.
2. Write ten exercises in subtraction of decimals and
solve each.
25. Oral Exercises.
1. Name five combinations whose sums are 10.
When these combinations occur in a column, they should
be treated as 10. Exercise a below may be added : 15,
25, 32, 42, 48, 58, 66, 16. Add the following exercises in
a similar manner :
abcdefgh i j k I m
57841574 5 87 89
5326.9526573 98
8697648567712
r5 3841347638 97
15 726976348246
68798866766 64
r5 7269782145 6 b
l5 384132898 559
75586997723 21
r5 3269541379 85
15 7841569891 79
66789999977 83
989786 5 29688^4
2. Write ten columns, in each of which some of the
five combinations wliose sums are 10 occur several times.
Add these columns.
SUBTRACTION
26. Written Exercises.
27
Nbw England States
Area in
Sy. MiLBs
Gkbat Lakes
Area in
Sq. Miles
33,040
9,305
9,565
8,315
1,250
4,990
Superior
Huron
Michigan
Erie
Ontario
31,200
23,800
22,450
9,960
7,240
New Hampshire
Vermont . . .
Massachusetts .
Rhode Island .
Connecticut . .
1. Find the combined area of the New England states;
of the Great Lakes.
2. Find the difference between the combined area of
the New England states and of the Great Lakes.
3. Compare the area of Vermont with the combined
area of Massachusetts and Rhode Island.
4. Find the difference between the area of Lake
Ontario and the combined area of Rhode Island and
Connecticut.
5. Find the difference between the area of Lake Su-
perior and the combined area of Lakes Huron, Erie, and
Ontario.
6. Compare the area of Maine with the area of Lake
Superior.
7. Find the difference between the area of Maine and
the combined area of the other five New England states.
8. The area of Missouri is 69,415 sq. mi. Compare
the area of Missouri with the combined area of the New
England states.
28 REVIEW OF INTEGERS AND DECIMALS
27. Written Exercises.
In solving a problem, follow these steps in the order
given :
a. Read the problem carefully^ if convenient, aloud.
h. State what facts are given in the problem and what
fact you are asked to find.
c. Determine what relation the given facts have to one
another, and state what operation you must use in finding
the facts that are asked for, — whether you must add or
subtract, etc.
d. Make an estimate of the answer. When you have
found the answer, compare it with this estimate.
1. A man bought a house for $2400 and sold it for
$3000. Find the amount gained.
2. A farmer sold his farm for $7500, which was $1800
more than it cost him. How much did he pay for the
farm ?
3. A farmer bought a farm for $6250 and sold it at a
gain of $1200. How much did he get for the farm?
4. By selling a farm for $4500, a farmer received $900
less than it cost him. How much did it cost him?
5. A room is 24 ft. long and 18 ft. wide. Find how
many feet of picture molding it will require for the room.
6. After drawing out $2300 from a bank, a merchant
had $760 left in the bank. How much had he on deposit
in the bank?
7. A merchant had $1600 on deposit in a bank on
Jan. 1, 1907. On Jan. 2 he drew out $200. On Jan. 5
he deposited $750. On Jan. 15 he drew out $2000.
How much had he left in the bank?
SUBTRACTION
29
28. Written Exercise*.
1. The total production of corn in the United States in
1899 was 2,666,440,279 bu. In 1889 tiie total production
was 2,122,327,547 bu. How much had the production
increased during the decade (10 years)?
2. In 1899 the production of corn in Illinois was
398,149,144 bu. The production in 1889 was 289,697,256
bu. Find the increase in production during the decade.
3. From the amounts given in Frobs. 1 and 2, find the
total number of bushels produced in all states other than
Illinois in 1899.
4. The total production of rice in the United States in
1899 was as follows :
Louisiana .
South Carolina
Hawaii
Georgia
North Carolina
Texas
Florida
Alabama .
Mississippi
Arkansas .
Virginia
172,732,430 lb.
47,360,128 lb.
83,442,400 lb.
11,174,562 lb.
7,892,580 lb.
7,186,863 lb.
2,254,492 lb.
926,946 lb.
739,222 lb.
8,630 lb.
4,374 lb.
a. Read aloud the above quantities.
h. Write the above from dictation.
c. Find the total number of pounds produced.
d. Was the amount produced by Louisiana more or less
than that produced by all others combined, and how much ?
e. Compare the amount produced in South Carolina
with the total amount produced in Hawaii and Georgia.
30 REVIEW OF INTEGERS AND DECIMALS
29. Written Exercises.
1. Ill 1890 the population of San Francisco was 298,997,
and in 1900 it was 342,782. Find the increase in popula-
tion between 1890 and 1900.
2. In 1898 the population of London was 4,504,766,
and' in 1900 the population of New York was 3,437,202
and of Chicago was 1,698,575. Find the difference be-
tween the population of London and the combined popu-
lation of New York and Chicago.
3. The area of the earth's surface is about 196,940,000
sq. mi. Of this, 141,486,000 sq. mi. is covered with
water. Find the area of the land.
4. A cattle dealer bought some cattle, for which he paid
$ 380. He paid out f 67 for feed and care of the cattle.
He then sold them for ef 500. How much was his net
profit, that is, the profit after deducting all expenses ?
5. A real estate dealer bought a city lot for 11750.
He built a house on it that cost $ 3275 and then sold the
property for $ 6000. Find the amount of his gain or loss.
6. The total area under broom-corn cultivation in the
United States in 1899 was 178,584 acres. In 1889 it was
93,425 acres. How much was the increase in the area
under cultivation during the decade ?
7. The appropriation for the maintenance of the navy
for 1907 was $ 98,773,692, for the military $ 72,305,270, for
pensions $ 143,746,106. How much was appropriated for
these three purposes ? How much more were the combined
appropriations for the navy and military than for pensions ?
8. The total appropriations of the government for 1907
amounted to $5 701,551,566. Find the appropriations for
all purposes other than the navy, military, and pensions.
MULTIPLICATION 31
MULTIPLICATION OF INTEGERS AND DECIMALS
30, 1. Find the sum of a column of four 2's.
2. The sum of a column of four 2's is . In this
column the addend 2 is repeated 4 times. Four times 2
are . 2
X 4
3. Four 2's are 8 may be indicated thus : - • Here
2 is taken 4 times, or is multiplied hy 4. The 2 is the
addend that is repeated, and the 4 tells the number of
times this addend is repeated.
% 82
4. The sum $32 taken 4 times may be ^ oo
found by addition, thus: ^ g2
It may also be found by ^ ^^
multiplication, thus : ^Toq $128
Since four 2's are 8 and four 3's are 12, four |32's are $128.
5. Find the cost of 3 cows at $48 each by addition;
by multiplication. Which method is the shorter?
6. A man paid the following amounts for three horses :
$120, $85, and $100. Can the cost of the three horses be
found by multiplication ? Give a reason for your answer.
7. Multiplication is the process of taking one number
as many times as there are units in another.
8. The number that is multiplied is called the multi-
plicand; the number by which we o^^c^oa i^- t j
u- 1 • ^^ A ^x. 1.- 1- ^284, multiplicand,
multiply IS called the multiplier; « if r
and the result obtained is called ^^t^ , .
,, , ^ $702, product,
the product.
^284 . . ^^.^.
9. Express « as an exercise in addition.
32 .REVIEW OF INTEGERS AND DECIMALS
10. Regard the multiplicand as a repeated addend and
tlie multiplier as the number of times the addend is
repeated.
11. The product and the multiplicand are always like
quantities. Why ?
12. The multiplier is always an abstract number. It
tells how many times the multiplicand is to be taken, or
how many times the addend is to be repeated.
13. The sign of multiplication is x . It indicates that
the number before it is to be multiplied by the number
after it. $3 x 4 is read 83 multiplied by 4. The sign
( X ) is sometimes used in place of the word times in such
an expression as 2 times f 5.
14. Express each of the following in the form of addi-
tion : $4x6; 5 lb. X 3; 4 times 6 yd. ; 7 in. x 8; 9x4.
31. Law of Commutation.
1. In the following diagram there are 3 rows of
squares, with 4 squares in each row. Or, there are 4
rows of squares, with 3 squares in each
row. There are in all 12 squares. We
see that 3 times 4 squares and 4 times
3 squares are the same number of
squares.
2. Find the sum of three 4's and of
four 3's. Since the sum of four 3's is the same as the
sum of three 4's, the product of 3 and 4 is the same,
without regard to which is multiplier and which is mul-
tiplicand.
3. Show by the addition of columns that the sum of
five 6*s equals the sum of six 5's. Show by a diagram
that 5 times 6 squares equals 6 times 5 squares.
MULTIPLICATION
33
32. Remembering that the multiplicand is the same as
the repeated addend, answer the following :
1. Can you multiply 16 by 3? 3 by |6? 6 ft. by 3 ft.?
2. Can you find 8 lb. X 2 ? 2x8? 2 ft. x 3 ft. ?
3. When the multiplicand is some number of yards,
what is the product?
4. Can the multiplier ever be concrete ? Why ?
5. Which is more, |6 x 3 or |3 x 6?
33. Table of Products and Quotients.
For reference only.
1
2
3
4
5
6
7
8
9
10
11
12
2
4
6
8
10
12
14
16
18
20
22
24
3
C
9
12
15
18
21
24
27
30
33
36
4
8
12
16
20
24
"28
32
36
40
44
48
5
10
15
20
25
30
35
40
45
50
55
60
6
12
18
24
30
36
42
48
54
60
6Q
72
7
14
21
28
35
42
49
56
63
70
77
84
8
16
24
32
40
48
56
64
72
80
88
96
9
18
27
36
45
54
63
72
81
90
99
108
10
20
30
40
50
60
70
80
90
100
110
120
11
22
33
44
55
66
77
88
99
110
121
132
12
24
36
48
60
72
84
96
108
120
132
144
Note. If the pupils are not thoroughly familiar with the facts
of multiplication and division, a thorough mastery of these facts
should precede the attempt to use them in the exercises that follow.
For a systematic method of teaching these facts, see Elementary Arith-
metic of this series.
AR.— 3
84 REVIEW OF INTEGERS AND DECIMALS
34. Oral Exercises.
Supply products for x^ and add to each product the
numbers above the column, as in a (products 12, 16, 35,
etc.); adding 2: 14, 18, 37, etc. ; adding 6: 18, 22, 41, etc.
a
h
c
d
^x2 = x
3, 5
9x3 = 2;
4, 7
7x7 = 2;
8, 9
7 x4 = 2;
4x4 = 2;
8x5 = 2;
8x9 = 2;
7x5 = 2;
5 X 1 = x
4x9 = 2;
6x6 = 2;
4x5 = 2;
8x4 = 2:
6x7 = 2;
7x9 = 2;
9x8 = 2;
8x7 = 2;
9x5 = 2;
6 x4 = 2;
5 x5 = 2;
9 x4 = 2;
6 x8 = 2;
9x6 = 2;
4x8 = 2;
5x6 = 2;
9x9 = 2;
8x8 = 2;
3x9 = 2;
9x3 = 2;
7 X 8 = 2;
8x7 = a;
8x6 = a;
35. Written Exercises.
1. Multiply 25,435 by 304.
Model: 25435 Explanation: 25435
304 304
101740 101740 = 4 times 25435
76305 7630500 = 300 times 25435
7732240 7732240 = 304 times 25435
Multipl}^ by 4; multiply by 3 (that is, 300), placing the product
of 3 and 5 directly below the 3, or in hundreds' place. Add.
Solve :
2. 435,450x504 7. $386.50x527*
3. 978,689x450 8. $868.75x689
4. 230,302 X 800 9. $768.30 x 843
5. 967,843 X 769 10. $364.97 x 107
6. 845,397x896 11. $426.87x489
♦ Point off two places for cents in the answer.
MULTIPLICATION 35
36. Oral Exercises.*
1. How much will 4 sheep cost at $5 each?
Model for oral recitation : Since 1 sheep costs $5, 4 sheep will cost
4 times $5, or 120.
2. How much will 4 chairs cost at $3 each ?
3. At f 2 each, how much will 6 books cost?
4. There are 4 quarts in a gallon. How many quarts
are there in 5 gallons ?
5. Make and solve ten similar problems in multiplication.
37. Written Exercises.
1. Find the cost of 32 acres of land at $75 an acre.
Model ; Statement : 1 75 X 32 = a;
Work: $75, cost of 1 acre.
32
150
225
$2400, cost of 32 acres.
2. If a boy attends school 180 days each year for 12
years, how many days will he attend in 12 years ?
3. Find the cost of 25 cows at $45 each.
4. If a man earns $1.75 a day, how much will he earn
in 24 days ?
5. Make and solve ten problems in multiplication simi-
lar to those a clerk in a grocery store has to solve.
* Drill should be given upon these and similar problems until the pupils
are familiar with the language forms used in the analysis. The written
form should be taken up after the oral form has been mastered. Apply
this form of analysis (or some suitable form) to similar problems on the
succeeding pages of the text. When tlie form has been mastered, the
pupils should be permitted to reply briefly, thus for Prob. 1 : Four times
five dollars, or twenty dollars.
86 REVIEW OF INTEGERS AND DECIMALS
REVIEW — FARM PROBLEMS
38. 1. A man bought a farm of 160 acres at i 75 an
acre. Find the cost of the farm.
2. During the first year he expended the following
sums for improvements : repairing fences, $165.80; dig-
ging a well, $95; building a carriage house, $640; re-
shingling the barn, $124.35; draining a marsh, $60.
Find the cost of the improvements.
3. The farm was divided into 5 fields of 20 acres
each, 3 fields of 10 acres each, 10 acres of orchard, 5 acres
for yards and garden, and the rest was timber land. How
many acres of timber were on the farm ?
4. To stock up the farm, the farmer bought 14 head of
cattle at an average of $ 37 apiece, 5 horses at an average
of $120 apiece, 24 sheep at $4.50 apiece, 6 hogs at $5.25
apiece, and 30 chickens at $.35 apiece. Find the cost of all.
5. The following amounts were received from the sale
of milk for one year: Jan., $60.80; Feb., $68.17; March,
$70.30; April, $71.90; May, $79.25; June, $80; July,
$72.35; Aug., $66.10; Sept., $63.28; Oct., $59.37; Nov.,
$50.40; Dec, $54.30. Find the amount received during
the year.
6. The farmer employed one man for 8 months, paying
him $35 a month, and another man for 3 months, paying
him $ 38 a month. Find the amount expended in wages.
7. Two of the 20-acre fields were sown in oats, and
the yield was 45 bu. to the acre. If oats were worth 34^
per bushel, find the value of the crop.
8. During the month of April the farmer sold $15.75
worth of eggs. At the same rate, how much would the sale
of eggs amount to in one year?
MULTIPLICATION 37
39. Multiplication and Division by 10, 100, etc.
1. Compare the value of 2 in 20 and in 2; in 200 and in
20 ; in 2000 and in 200.
2. What effect upon the value of a figure has (a) mov-
ing it one place to the left ? (J) moving it two places to
the left ? (c) moving it three places to the left ?
3. Annexing a cipher to an integer has the effect of
moving the digits each one place to the left. This mul-
tiplies the number by 10. Annexing two ciphers has
what effect upon the places occupied by the digits ? State
a short method of multiplying an integer by 100 ; by 1000.
4. Using the short method, multiply each by 10; by
100; by 1000: 6, 47, 390, 20, 475, 8, 600, 72, 25, 64, 640.
5. State how an integer may be multiplied by 10; by
100; by 1000.
6. Moving a figure one place to the right has what
effect upon its value ? Compare the value of 6 in 60 and
in 6; in 600 and in 6; in 6000 and in 6.
7. Dropping the cipher at the right of 60 changes the
number to 6. What change does this make in the value
of the number ? State a short method of dividing an in-
teger ending in a cipher by 10.
8. What change is made in the value of 400 by drop-
ping the two ciphers ? State a short method of dividing
a number ending in two ciphers by 100.
9. Divide each by 10 ; by 100: 4500,700,400, 3700.
10. An integer that does not end in a cipher may be
divided by 10 by placing a decimal point to the left of
the right-hand figure, thus: 87 divided by 10 is 8.7. Is
this the same as moving each digit one place to the right ?
88 REVIEW OF INTEGERS AND DECIMALS
11. An integer that does not end in two ciphers may be
divided by 100 by placing a decimal point at the left of
the figure in tens' place, thus: 475 divided by 100 is 4.75.
12. Give the quotient of each divided by 10; by 100:
325, 560, 4582, 4500, 48, 4, 2, 10, 5, 50.
13. Write integers and divide each by 10 ; by 100.
* 40, Short Methods. ( To be used in subsequent work. )
1. What part of 100 is 25 ? Compare 25 times a num-
ber with 100 times the number.
2. To multiply by 25, multiply by 100 and divide by 4.
Multiply $489 by 25.
Model : $ 489 Explanation : Write as in multiplication.
25 Mentally multiply $489 by 100. Divide the
$ 12225 product by 4.
3. Multiply by 25: 1680,11225, 5280 ft., 231 mi.,
187.56,1247.82.
4. Multiply 7865 by 369.
Model: 7865
Explanation : First multiply by 9. As
70785 36 is 4 times i), multiply 70786 by 4, writ-
283140 ^"S ^^® product as in the model. Add.
2902185
5. In multiplying by 84, first multiply by 4 ; then mul-
tiply this product by 2, writing the first figure of the prod-
uct in tens' place. State how you would multiply by 63;
by 126 ; by 246 ; by 729; by 279. Illustrate each.
6. Multiply 6840 by 248 ; by 328 ; by 648 ; by 168.
7. Multiply 1840 by 287 ; by 147 ; by 637 ; by 639.
MULTIPLICATION 39
4:1. Multiplication of Decimals.
1. Find the sum of 5.2 mi., 5.2 mi., and 5.2 mic This
sum is the same as the product of 5.2 mi. x 3. How
many decimal places are there in this product? Why?
2. Find the sum of 5.2 mi., 5.2 mi., 5.2 mi., 5.2 mi., and
5.2 mi. Multiply 5.2 mi. by 5. How many decimal places
are there in the product? Why?
3. Write 6.08 x 3 in the form of addition and find the
sum. Multiply 6.08 by 3. How many decimal places are
there in the product? Why?
4. Write 6.08 X 5 in the form of addition and find the
sum. Multiply 6.08 by 5. Compare the results. How
many decimal places are there in the result? Why?
5. State a short method of multiplying an integer by 10.
6.25 may be multiplied by 10 by moving the decimal point
one place to the right. 6.25 x 10 is 62.5. Has moving
the decimal point one place to the right the same effect as
moving the digits one place to the left? Compare 2.2
with 22. Compare 75.25 with 752.5.
6. State a short method of multiplying a decimal by
100. How many places to the right must the decimal
point be moved to multiply by 10? by 100? by 1000?
7. Compare 25 with 2.5. Here the digits have been
moved one place to the right. Compare 2.5 with .25.
Here the digits have been moved another place to the
right. This has been done by moving the decimal point
one place to the left. Moving the decimal point one
place to the left has what effect upon the value of a
decimal? upon the value of an integer?
8. State a quick way of dividing a decimal by 10 ;
by 100. Illustrate with integers and with decimals.
40 REVIEW OF INTEGERS AND DECIMALS
42. Oral Exercises.
1. Divide each by 10 and by 100: 450, 25.74, 45, .4,
4.5, 346.2.
2. What is 6 times 4? 1 times 4? J of 4 ? ^ of 4 ?
3. What is meant by 4x6? 4x1? 4xJ? 4x.l?
4. 4 X .1 is the same as 4 divided by v^^hat number?
4 X. 1 = 37. 4x.2 = a:. 4x.3 = a;.
5. Divide 4 by 100. What is meant by 4 x .01 ? 4 x .01
= x. 4 X ,06 = x.
6. What is meant by .4 x 2? .4 x 13? .4 x .1? .4 x .1
is the same as .4 divided by 10. .4 -s- 10 = a;. .4 x .1 = a;.
.4 X .2 — x. .4 X .5 = x,
7. Multiply 82.30 by 10. Multiply 12.36 by 100.
Multiply 12.30 by .1. Multiply $24.50 by .01; by .1.
Multiply 145.75 by 10 ; by .1 ; by 100 ; by .01.
\J 8. To multiply by .1 is the same as to divide by 10.
What change made in the place of the decimal point in
the multiplicand divides it by 10 ?
n
43. Multiply each by 10 ; by .1 ; by 100 ; by .01.
1.
$37.50
5.
625 ft.
9.
2240 lb.
13.
3.1416
2.
$2500
6.
1726 yd.
10.
2000 lb.
14.
24 cwt.
3.
14.525
7.
5280 ft.
11.
625 lb.
15.
4.75 cvvt.
4.
17500
8.
2150.42
12.
630 lb.
16.
20 T.
^ 17. 624 X .001 = a:. 2000 lb. x .001 = a:.
18. Divide by 100: 4632 lb.; 3000 lb. ; 2160 mi.;
37.40 mi.; $234.50; .425 mi.; .03 mi.; 3.1416 ft.; $60.
19. At $5 per hundredweight, how much is sugar
worth per pound ? at $4.75 per cwt. ? at $4.50 per cwt. ?
20. Multiply each in the shortest way : 5280 ft. X 25 ;
1728 X 25 ; 987,647 by 648 ; 7,389,675 by 369.
MULTIPLICATION 41
44, Written Exercises.
1. Multiply 6.23 by 4.2.
Model : Explanation :
6.23 6.23 First multiply 6.23 by .2. This is equiva-
. ^ . ^ l^iit to dividing 6.23 by 10 and multiplying
-^ ^ ^ the quotient by 2. 6.23 -- 10 = .628 ; .623
1246 1.246 ^2 = 1.246. Next, multiply 6.23 by 4.
2492 24.92 6.23 x 4 = 24.92. Add the products.
26.166 26.166
Notice that in the above the number of decimal places
in the product is the same as the sum of the number of
decimal places in the multiplicand and multiplier. As
this is always true, the following method may be employed :
To multiple/ decimals^ multiple/ as in integers and point
off in the product as many decimal places as there are in
both multiplicand and multiplier.
Solve. Estima^ each result b efore mu ltiplying:
2.
59.786
X8.97
r-
.0056 X 385.07
3.
487.69
53.008
x.479
X 7.086
\-
7.0758 X 67.09
4.
9.
.07854 X 8.0065
5.
.69387
X 6.9075
10.
46,897x4.008
6.
13.006
X 3.1416
11.
785.06 X 6300
45. Written Exercises.
1. If a train travels at an average rate of 43.5 mi.
an hour, how far will it travel in 24 hours ?
2. The circumference of a circle is 3.1416 times its
diameter. Find the circumference of a circle that is 9.5
in. in diameter.
3. If it costs a boy $.50 a week to keep a pony, how
much will it cost to keep it for 1 year (52 wk.)?
42
REVIEW OF INTEGERS AND DECIMALS
BILLS AND ACCOUNTS
46. A Receipted Bill.
Los Angeles, Cal., May 31, 1907.
Mr. James J. Davies, 2217 Vine St.
In account with S. D. James & Co.
May
3
Z lb. coffee $.40
2 lb. tea M
6 bars soap .05
$1
1
20
30
30
(4
5
3 cans corn .08
1 doz. lemons .15
24
15
((
6
10 lb. sugar .06
Received Payment^
60
$3
79
S. D. James ^ Co.
1. The person who buys on account is called the debtor,
and the person who sells on account is called the creditor.
2. Each purchase, or payment, is an item. How many
items of debit are there in this bill ? of credit ?
3. A bill must show the date of each transaction. "When
was the above bill made out, or rendered?
4. A bill must also name the debtor and the creditor
and the several items of debit and credit. Who is the
debtor named in the above bill ? Who is the creditor ?
5. When a bill is paid, the creditor writes " Paid " or
" Received payment " on the bill and signs his name below.
Has the above bill been paid?
6. Should a person make a practice of keeping receipted
bills ? Why ?
MULTIPLICATION 43
47. Written Exercises.
Make out and receipt the following bills. Supply all
necessary data not contained in the problems:
1. Mr. J. S. White, residing at 234 First Street, bought
of the grocery firm of Allen and Baker the following :
May 25, 1907, 2 lb. tea @ 60^; 3 lb. coffee @ 45/; 2 lb.
bacon @ 20 /; 1 lb. butter @ 30 /. On May 31, 4 cans
tomatoes @ 8/; 2 doz. eggs @ 18j^; 1 lb. cheese @ 20 j^.
2. Mrs. Harry Smith, residing at 1450 Jackson Street,
bought of Cole Bros, the following : April 24, 1907, 9 yd.
silk @ $1.50 ; 2 yd. dress lining @ 25^; 4 spools silk @
10/; 1 bolt skirt binding, 15/; dress trimmings, $1.50.
The bill was rendered April 30.
3. Insert your own name as purchaser of the following
bill of hardware: 1 garden rake, 45/; 1 shovel, 60 /; 3
lb. nails @ 6/; 12 yd. wire netting @ 30 /; 2 lb. staples
@ 5/; 1 lawn mower, 12.50.
4. Dr. C. L. Ward employed a schoolboy to take care
of his horses, for which he agreed to pay him $6 per
month, with extra pay for additional services. During
the month of May the boy mowed the lawn twice, for
which he was to receive 50 / each time ; and he also
worked 12 hr. in the garden, at 10 /. per hour. At the
end of the month Dr. Ward asked the boy to render
his bill for services during the month. Make out the
above bill, using your own name or the name of some boy
in your school as the creditor. -
5. Make out a bill for 8 music lessons at $1.50 each.
6. Make out a bill for purchases at a furniture store.
7. Make out a bill for purchases at a meat market.
8. Make out a bill for purchases at a dry goods store.
k
44 REVIEW OF INTEGERS AND DECIMALS
RECEIPTS
48. 1. Explain the meaning of the following :
Oakland, Cal., May 1, 1907.
Received of Mr. C. W. Smith twenty-five dollars
($ 25) in full for rent of house at 704 Logan Avenue for
the month of May, 1907. D. S. Stone.
2. E. M. Day bought a sewing machine of P. Orr
for |;30. Write a receipt for the payment of this amount.
3. Mr. J. E. Thomas rented a farm of D. R. James for
$ 300 per year. Write out a receipt for the payment of
rent for the year beginning March 1, 1905.
4. The manager of an athletic club received $7.50 from
the treasurer of the club. Write the receipt.
49. Oral Exercises.
Whenever the partial sum is 10, 20, etc., take the next
two numbers together, as in <2, 10, 19, 26, 30, 39, etc.
a
b
c.
d
e
/
9
h
i
J
A;
Z
6
8
4
6
8
7
7
9
5
4
7
8
2
3
6
7
8
6
5
9
8
9
8
5
3
3
3
9
5
7
8
5
4
7
9
4
8
7
9'
8
9
6
5
7
3
8
7
3
4
7
6'
3
8
7
7
9
8
5
5
8
5
3
6
8
8
4
8
9
5
7
8
9
4
3
8'
9
4
3
5
2
7
6
6
8
7
4
6
3
8
6
7
9
5
7
2
3
4
3
6
8
8
7
5
9
8
8
2
8
5
3
4
9
6
9
8
7
4
6
9
6
4
2
7'
7
7
6
8
8
9
8
4
7
6
8
9
3
7
5
9
5
7
8
7
\)
^
DIVISION 45
DIVISION OF INTEGERS AND DECIMALS ^
50. Factors and Multiples.
1. 4 times 3 are 12. 4 and 8 are each a factor of 12.
Name two other factors of 12; two factors of 14; of 20.
2. 2 and 8 are each a factor of . 3 is a factor of
. 5 and are each a factor of 10.
3. The factors of a number are the integers which, when
multiplied, make the number. A number may have sev-
eral pairs of factors, thus : the pairs of factors of 24 are 4
and 6, 3 and 8, 2 and 12. Some numbers have only a
single pair of factors, thus : the factors of 15 are 8 and 5.
4. Name all the pairs of factors of each of the follow-
ing : 4, 6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 24, 27, 28, 30.
5. As 3 times 5 are 15, 15 is a multiple of both 3 and
5. Name a multiple of both 4 and 5; of both 4 and 7.
6. The number obtained by multiplying together two
integers is called a multiple of either integer. Name a
multiple of 4; of both 6 and 5.
7. In multiplication, two factors are given to find their
product. In division, the product and one of two factors
are given to find the other factor.
61. 1. The process of finding the other factor, when
the product and one of two factors are given, is called
division.
2. Since three 5's are 15, the number of 5's in 15 is 3.
3
This may be indicated thus : 5)15 Here 5 is the given
factor, 15 is the given product, and 3 is the other factor.
3. The given factor is called the divisor, the given
product is called the dividend, and the factor found is
called the quotient.
46 REVIEW OF INTEGERS AND DECIMALS
62. 1. Name all the multiples of 3 to 30 ; of 4 to 40 ;
of 5 to 50; of 6 to 60; of 7 to 70; ofStoSO; of9to90.
2. In each of the following, name the higliest multiple :
a, Of 2 in 17, 11, 13, 9, 5, 7, 15, 19, 3, 10, 14, 21, 17, 19.
h. Of 3 in 10, 8, 14, 19, 57, 16, 20, 29, 23, 26, 17, 22.
c. Of 4 in 10, 17, 7, 15, 38, 31, 22, 27, 35, 19, 25, 30.
d. Of 5 in 17, 23, 7, 12, 28, 34, 42, 39, 48, 27, 33, 19.
e. Of 6 in 16, 10, 29, 22, 38, 45, 51, 34, 57, 43, 53, 41.
/. Of 7 in 52, 46, 17, 33, 25, 68, 57, 39, 13, 30, 44, 53.
ff. Of 8 in 52, 15, 28, 20, 35, 46, 69, 75, 58, 30, 60, 44.
7i. Of 9 in 20, 78, 34, 42, 47, 37, 59, 68, 11, 29, 53, 16.
3. Repeat Ex. 2, naming the highest multiple and the
difference between it and the given number, thus for g :
48 and 4 ; 8 and 7 ; 24 and 4, etc.
4. Repeat Ex. 2, giving the quotients and the remain-
ders, thus for g : 6 and 4 over ; 1 and 7 over ; etc.
The sign of division is -^. It indicates that the num-
ber before it is to be divided by the number after it.
6 ^ 3 is read 6 divided hy 3. Division may also be indi-
cated thus : 3)^; or thus : |.
53. Oral Exercises.
Supply the value of x in each of the following :
a h c d
24-^6 = a: 58^8 = a; 67^9 = a; 76-f-8 = a?
30-j-8 = a: 76^9 = 2: 69-^8 = a; 61^-9 = a;
42-^9 = ic 40-^7 = 2; 60-j-7=.r 45-!- 6 = a;
18-r-7=a; 23^9 = a: 52 - 6 = a: 39^ 7 = a:
25-5-7=2; 70-8 = 2: 43-^9 = 2; 26-^3 = 2:
36-*- 8 = 2; 52-7 = 2: 59-r-6 = 2? 22-6 = 2:
51-1-6 = 2; 28^6 = 2: 39^-5 = 2: 50-5-8 = 2:
DIVISION
47
54, Written Exercises.
Use as successive divisors
the numbers above the
columns. Solve
by short division.
a
h
c
d*
6,T,8
9,5,4
3,8,7
2,9,6
1. 672,458
327,459
836,594 ^
$4387.24
2. 237,400
574,063
683,127
$3058.26
3. 946,305
508,342
427,060 i
$9576.30
4. 375,268
970,654
738,967 i $4256.75
5. 834,008
624,307
520,380
$5687.50
6. 463,925
207,193
946,425
$7495.38
7. 927,384
423,075
315,017
$8250.15
55. Written Exercises.
Use as multipliers the numbers above the columns :
a
789
6
465
c
305
d
890
1.
$8796.50
7568.93
975.864
432.501
2.
$4578.69
835.769
586.097
34.2056
3.
$9760.50
58.7964
4376.89
Q^Q4.M
4.
$6389.75
95,687.3
605.008
5046.32
5.
$4975.86
2456.78
789.645
265.423
6.
$7869.45
697.583
83.7956
123.456
7.
$3698.70
4309.58
3945.78
45.0635
8.
$8970.56
37.6895
687.905
5643.06
9.
$6875.09
709.608
'201.003
326.504
.0.
$3204.56
854.076
4567.98
Q^MbQ
* Place a decimal point in the answer above the decimal point in the
dividend.
48 REVIEW OF INTEGERS AND DECIMALS
56. Measurement and Partition.
1. All division is either measurement or partition.
2. 6 ft. 6 ft.
_4 6 ft.
24 ft. when expressed as addition is 6 ft.
6 ft.
24 ft.
3. The quantity 24 ft. may be measured by the
quantity 6 ft. The measure 6 ft. is contained in 24 ft.
4 times. This may be indicated thus : ^ „ .r—-
^ 6 ft.)24 It.
4. The process of finding how many times one number
or quantity is contained in another is called division by
measurement.
5. In Ex. 2 the quantity 6 ft. is taken 4 times to give
the quantity 24 ft. Therefore, one fourth of 24 ft. is 6 ft.
(K f '■
This may be expressed thus : A^^^oTTr
6. The process of finding one of the equal parts of a
number or quantity is called division by partition.
7. In every problem in division you will be required
to find either (a) how many times some unit of measure
is contained in a quantity to be measured, or (5) to find
a given part of some quantity to be divided.
67. Measurement.
1. The number of 4 cubes in 12 cubes may be found
by using the unit 4 cubes as a measure to measure the
quantity 12 cubes. 4 cubes, the unit of measure, is con-
tained in 12 cubes, the quantity to be measured, exactly
3
3 times. This may be indicated thus: 4 cubes) 12 cubes
DIVISION 49
2. Show with objects or by a diagram how many
3 apples there are in 12 apples. Indicate this in the form
of division. Write the quotient.
3. Show by a diagram or by actual measurement how
many 2 ft. there are in 12 ft. Indicate this in the form of
division. Write the quotient.
4. Show with objects or by a diagram what is meant
by each: 2 books)10 books; 5 pencils)15 pencils;
4 boys) 12 boys; 6 books) 12 books; 3 ft.)9lt7; 5^)20/.
Write a problem for each.
5. In finding the number of 4 hr. there are in 24 hr.,
what is the unit of measure ? What is the quantity to be
measured?
6. Show by using books (a) how many 3 books there
are in 9 books ; (6) how many 3 books there are in 10
books; (<?) how many 8 books there are in 11 books.
Indicate each in the form of division, with quotient, and
remainder, if any.
7. Show with objects that the number of 4 apples in
10 apples is 2, with 2 apples remaining.
8. Find the number of $4 there are in |16 ; of 3 yd.
there are in 15 yd. ; of 7 da. there are in 21 da. Indicate
each in the form of division.
9. In division by measurement the divisor is always
like the dividend. The quotient is always an abstract
number, since it tells how many times the unit of measure
is contained in the quantity measured.
10. Find by measuring how many times a 1-ft. meas-
ure must be applied to measure 4 ft. (4 ft. ^ 1 ft. = a;) ;
a J-ft. measure to measure 4 ft. (4 ft. -r- ^ it. = x) ; a. J-ft
measure to measure 4 ft. (4 ft. -t- J ft. = a;).
60 REVIEW OF INTEGERS AND DECIMALS
58. Oral Exercises — Measurement.
1. At $ 4 each, how many desks can be bought for fi 12 ?
The unit of measure is f 4 and the quantity to be meas-
ured is 112.
Model for oral recitation : Since 1 desk costs $4, as many desks
can be bought for $ 12 as there are $ 4 in ^ 12, or 3.
In each of the following, name the unit of measure and
the quantity to be measured:
2. At f 3 each, how many chairs can be bought for $ 15 ?
3. At 1 4 a pair, how many pairs of shoes can be bought
for 1 24?
4. If berries cost 6 ^ a box, how many boxes of berries
can be bought for 30 ^ ?
5. How many yards of ribbon at 8 ^ a yard can be
bought for 40^?
6. If a boy earns 1 9 a month, in how many months will
he earn f 45 ?
7. Write ten additional problems in measurement.
69. Written Exercises — Measurement
1. At $ 9 a ton, how many tons of hay can be bought
for 1 216?
2. A man bought sheep at $6 each. He paid $96 for
all. How many sheep did he buy ?
3. A farmer divided a farm containing 160 acres into
10-acre fields. Find the number of fields and the value of
each field at $ 80 per acre.
4. How many 9's are there in 1728 ? Is this measure-
ment?
5. Write five problems in division by measurement, and
solve each.
DIVISION 51
60. Partition.
1. One fourth of 12 cubes may be found by dividing
12 cubes into 4 equal groups or parts.
One fourth of 12 cubes is 3 cubes. This may be in-
dicated thus :
3 cubes
4)12 cubes
2. Show with objects and by diagrams what is meant
by one third of 12 objects ; by one fourth of 8 objects ; by
one third of 9 objects. Indicate these in the form of
division, and write the quotients.
3. Show with objects and by a diagram what is meant
by each of the following :
4)8 apples 2)10 in. 5)10 in. 3)15 ft. 4)16 books.
4. In division by partition the quotient is always a
part of the dividend. Therefore, the quotient is always
like the dividend, — concrete when the dividend is con-
crete, and abstract when the dividend is abstract. The
divisor is always an abstract number. Why ?
61. Oral Exercises — Partition.
1. If 2 chairs cost f 8, what is the cost of 1 chair ?
Model for oral recitation : If 2 chairs cost $ 8, 1 chair will cost
one half of $ 8, or $ 4.
Name in each the quantity to be divided and the num-
ber of parts into which it is to be divided :
2. If 2 tables cost $ 12, what is the cost of 1 table ?
3. If 3 stoves cost 115, what is the cost of 1 stove ?
4. At $ 12 a ton, what is the cost of one half ton of hay?
5. Write ten additional problems in partition.
62 REVIEW OF INTEGERS AND DECIMALS
62. 1. Show by a diagram or with objects the mean-
ing of :
2)f6 4)12 da. 3)9l5r i3)f9 .f45fl2
2. Write a problem for each :
3)il8 4j20? 2)10 yd.
$6)112 I7)$21 5)10 yd.
3. 4) 1 2 may be either partition or measurement. State
what is meant by 4)12 (a) when it is partition;
(5) when it is measurement.
4. Is the divisor ever concrete in partition ? Is the
quotient ever concrete in measurement ? Give reasons.
5. In division by measurement the quotient is always
what kind of a number ?
6. When the divisor is a concrete number, is the divi-
sion partition or measurement ?
7. Make ten problems in division, and tell which are
partition and which are measurement.
8. Make problems for each of the following. Tell
which are partition and which are measurement.
$2)fl0 25110 4)16 yd.
8 wk.)16 wk. 3^115 55W
63. Oral Exercises.
■^3. indicates that 12 is to be divided by 3. Solve each :
1. :^ 6. Ap- 11. ^Q- 16. ^
2. ■*/ 7. -^8. 12. ^ 17. -figL
3. J^I 8. ^ 13. ^ 18. ^
4. Y 9. ^ !*• \^ 19- ^
5. Y 10. ^^ 15. 5g^ 20. -V-
DIVISION 53
64. Written Exercises.*
1. At $S each, how many tables can be bought for
8128?
Model for measurement:
16, number bouglit for $ 128
cost of 1 table, $ 8)-|128
2. A man spent 1 216 in 6 mo. What was the av-
erage amount spent each month ?
Modp:l for partition : _^^ .• i
^ $86, spent m 1 mo.
6) $216, spent in 6 mo.
Tell which of the following are partition and which are
measurement, and solve:
3. If a boy saves 85 a month, in how many months
will he save 1120?
4. How many tons of coal at #7 a ton can a man buy
for 8161?
5. Five boys agreed to share equally the expenses of
a camping trip. The trip cost them 821.70. What was
each boy's share ?
6. A girl bought 8 yd. of cloth for 82.56. How much
did the cloth cost her per yard ?
7. A hardware merchant bought some stoves at 89
each. His bill amounted to 8198. How many stoves
did he buy ?
8. If a boy worked 65 problems correctly in 1 school
week (5 days), what was the average number worked
correctly each day ?
9 A dealer bought 6 copies of a certain book. His
bill amounted to 87.50. What was the price of the book ?
* Give the oral analysis of Probs. 1-9.
7C
64 REVIEW OF INTEGERS AND DECIMALS
65. Ratio or Comparison.
1. How many times must the measure 2 ft. be applied in
measuring 6 ft. ? The number 3 expresses the ratio, or
relation, of the quantity 6 ft. to the unit 2 ft.
2. In measuring 6 ft. by 2 ft. the quantity to be meas-
ured is 6 ft., and the unit of measure is 2 ft. The ratio of
6 ft. to 2 ft. is found by dividing 6 by 2.
3. What is the ratio of 8 ft. to 4 ft.? of 12 ft. to 3 ft.?
4. What is the ratio of 6 da. to 3 da.? of 24 hr. to 6
hr.? of 25^ to 5^? of fl to 1.25? of 75^ to 25^?
5.. What is the ratio of 100 to 50 ? to 25 ? to 10 ?
to 20 ? to 5 ?
6. In measuring 2 ft. by 4 ft. the unit of measure is
4 ft., and the quantity to be measured is 2 ft. The meas-
ure 4 ft. is applied J time ; that is, one half the measure
is applied in measuring 2 ft. The ratio of 2 ft. to 4 ft. is ^.
66. Draw on the blackboard lines the length of the
quantities to be measured. Make measures the length
of the units of measure to be used in measuring each line.
By applying the measure to the line to be measured, de-
termine the ratio of the following:
1. Of 2 ft. to 1 ft. 7. Of 1^ ft. to 3 ft.
2. Of 2 ft. to I ft. 8. Of J ft. to 1 ft.
3. Of 1 ft. to J ft. 9. Of f ft. to 2 ft.
4. Of 3 ft. to J ft. 10. Of IJ ft. to 2 ft.
5. Of J ft. to 1 ft. 11. Of f ft. to J ft.
6. Of I ft. to 2 ft. 12. Of I ft. to ^ ft.
13. What part of 14 da. are 7 da.? What is the ratio
of 7 da. to 14 da.? of 14 da. to 7 da. ?
DIVISION 65
14. 3 in. is what part of 6 in.? What is the ratio of
3 in. to 6 in. ? The ratio tells the number of times the
unit of measure must be applied to measure the given
quantity. A 6-in. measure must be applied times to
measure 3 in. A 3-in. measure must be applied times
to measure 6 in. The ratio of 6 in. to 3 in. is .
The ratio of 3 in. to 6 in. is .
15. What part of the measure 12 in. must be applied to
measure 3 in.? What is the ratio of 3 in. to 12 in.? of 12
in. to 3 in.?
16. The ratio of 3 yd. to some quantity is ^. What is
the quantity ?
17. The ratio of some quantity to f 2 is 4. What is
the quantity ?
18. If 4 is the ratio gf some amount to 8 20, what is
the amount?
19. A J-ft. measure was used 12 times in measuring
the length of a line. How long was the line ?
20. Draw a line of such length that a 6-in. measure
will be applied IJ times in measuring it. Prove your
work by applying the measure.
21. What is the ratio of 2 to 8? of 6 to 2 ? of 20 to
5 ? of 5 to 30 ? of 8 to 48 ? of 40 to 8 ?
22. Two tons of coal will cost what part of the cost of
8 tons ? of 6 tons ? of 12 tons ?
23. 3 yd. of cloth wdll cost what part of the cost of 9
yd.? of 15 yd.? of 6 yd.? of 12 yd.?
24. If the cost of 24 yd. of cloth is given, how may the
cost of 8 yd. be found ? of 6 yd.? of 4 yd. ? of 12 yd.?
25. If the cost of 6 sheep is $24, what is the cost of
18 sheep? of 12 sheep? of 3 sheep? of 2 sheep?
^
66 REVIEW OF INTEGERS AND DECIMALS
67. Oral Exercises.
1. If 5 desks cost 120, how much will 7 desks cost?
The quantities 5 desks and 7 desks are measured by
the common unit 1 desk. In solving this problem, first
find the cost of the unit 1 desk. Next find the cost of
the required number of units.
Model for oral recitation : If 5 desks cost $ 20, 1 desk will cost ^
of $ 20, or $4. Since 1 desk costs $ 4, 7 desks will cost 7 times $ 4, or
^28.
2. If 4 tons of hay cost $82, how much will 6 tons cost ?
3. If 7 tablets cost 35^, how much will 4 tablets cost ?
4. At the rate of 5 for 25^, how much will 8 spelling-
blanks cost ?
5. A girl paid 30^ for 5 yards of ribbon. How much
would 8 yards have cost at the same rate ?
6. Make ten additional problems similar to the above.
68. Written Exercises.
1. A farmer raised 220 bu. of oats on 4 acres of land.
How much at the same rate would a 7-acre field have
produced ?
2. If a train travels 138 mi. in 3 hr., at the same
rate, how far will it travel in 8 hr. ?
3. From a farm of 160 acres 8 acres were sold for
$500. At this rate, what was the value of the entire
farm ?
4. 7 men picked 210 boxes of prunes in 1 day. At the
same rate, how many boxes would 15 men liave picked ?
5. The expenses of a family amounted to $328.75 for
5 ino. At the same rate, what would the expenses
amount to in 1 yr. (12 mo.)?
DIVISION 67
69. Oral Exercises.
Finding a part of an amount, when the amount is given :
1. Find I of 12 ft.
i2ft
I I 1
V 3ft. 3 f t 3ft J 3 ft
To find I of 12 ft., divide 12 ft. into 4 equal parts. Then | of 12
ft. will be 3 of these parts.
2. Show by a diagram that ^ of 12 ft. is 4 ft. and that
f of 12 ft. are 8 ft; that | of 8 yd. are 6 yd.
3. Show with objects that -| of 10 objects are 6 ob-
jects; that f of 6 objects are 4 objects.
4. Show by a diagram that if a board is 8 ft. long, |
of the length of the board is 6 ft.
5. Whatisf of $12?
Model for oral recitation: Since \ of $12 is $3, | of $12 is 3 times
$3, or $9.
6. What is f of $ 15 ? of 124 ? of $30 ? of 812 ?
7. What is f of 10 mi.? of 25 mi.? of 35 mi.?
8. What is I of 18 lb.? of 30 lb.? of 42 lb.? of 60 lb.?
9. At 20^ a pound, how much will | of a pound of
cheese cost?
10. How many months are there in | yr.? in f yr.?
11. If Fred worked 15 problems and John worked | as
many, how many did John work ?
12. A girl worked 16 problems, and | of them were
correct. How many of them were correct ?
13. How many inches are there in |^ of a foot ?
14. At $8 a ton, how much will -I of a ton of coal cost?
15. Make ten additional problems similar to the above.
58 REVIEW OF INTEGERS AND DECIMALS
70. Written Problems.
1. There are 2000 pounds in a ton. How many pounds
are there in | of a ton of hay ?
2. There are 320 rods in 1 mile. How many rods are
there in | of a mile ?
3. A girl read a book containing 210 pages. How many
pages had she read when she had read | of the book ?
4. Two boys, Henry and Frank, bought out a news-
paper route that cost them §4.50. Frank paid | of the
cost of the route and Henry paid J the cost. Kow
much did each pay ?
5. A man had 320 acres. He rented | of his land.
How many acres did he rent ?
6. There are 6280 feet in 1 mile. How many feet
are there in | of a mile?
71. Dividing by 20, 30, 40, 200, 300, etc.
1. State a short method of dividing a number by 10 ;
by 100.
2. Divide 476 by 40.
Model :
11.9 First place a decimal point in the quotient above and
40^476 between 7 and 6. Then divide by 4.
3. State how you would divide a number by 50 ; by
400 ; by 4000.
Solve. Before dividing, estimate each quotient :
4. 324 -H 40 8. 1260 -f- 400 12. 4860 -i- 40
5. 1728-*- 60 9. 5280 -f- 600 13. 3600 + 900
6. 1720-!- 80 10. 7854-^-500 14. 4240 -^ 800
7. 320 + 20 11. 6250 + 500 15. 1240+ 30
DIVISION 59
72. Oral Exercises.
Finding an amount, when part of the amount is given :
1. When I" of the length of a board is 8 ft., what is
^ of the length of the board ?
8ft
If I of the length of a board is 8 ft., J of the length of
the board is what part of 8 ft. ? If J of the length of a
board is 4 ft., what is | of the length of the board ?
2. Show by a diagram that if \ of the length of a line
is 6 ft., I of the length of the line is 2 ft. If \ of the
length of a line is 2 ft., what is the length of the line ?
3. Show by a diagram that if | of a line is 6 ft. long,
\ of the line is 2 ft. long and the line is 8 ft. long.
4. Using 12 objects, show that since \ of 12 objects is
4 objects, I of 12 objects are 8 objects.
5. Show that since | of 12 objects are 8 objects, \ of
12 objects is \ of 8 objects.
6. Draw a diagram to show the length of a room, if |
of the length of the room is 9 ft.
7. If 112 is I of the cost of a suit of clothes, what is
\ of the cost of the suit? What is the cost of the suit?
Model for oral recitation : If $12 is | of the cost of a suit, \ of the
cost of the suit is \ of $12, or $4. Since $4 is | of the cost of a suit,
the cost of the suit is 4 times $4, or $16.
8. If ^20 is I of the cost of a cow, what is the cost
of the cow?
9. If I of the cost of a book is 40^, what is the cost of
the book?
60 REVIEW OF INTEGERS AND DECIMALS
10. Two boys together bought a baseball. One boy-
paid ^.60, which was | of the cost of the ball. How much
did the other boy pay ? What was the cost of the ball ?
11. A boy spent 90/, which was | of the whole amount
of money he had. How much money had he ? How much
money had he left ?
12. Fred weighs 100 lb. This is f of George's weight
How much does George weigh ?
13. Make ten additional problems similar to the above.
73. Oral Exercises.
Two addends whose sum is 10 or less may be taken as
a single addend. Exercise a below may be added : 13, 23,
30, 47, 54, 62, 70. Add h-m in a similar manner :
a
b
c
d
e
/
S'
Ti
1
J
fc
I
m
8
8
8
7
2
3
8
7
3
2
3
2
4
4
6
1
6
5
6
5
4
6
7
3
1
7
4
1
3
2
2
7
3
5
2
7
7
5
2
2
3
2
2
5
6
8
8
9
2
3
6
4
5
2
4
2
2 ■
3
5
5
7
5
5
8
8
8
4
5
5
5
3
3
4
6
2
4
3
3
9
2
2
4
2
6
5
7
2
7
3
4
1
3
3
6
7
5
8
9
4
9
8
3
2
4
4
6
7
8
7
6
5
5
2
7
8
o
-j
6-
2
2
6
2
3
3
7
6
2
3
7
1
4
3
2
3
5
9
8
9
7
6
3
8
2
5
8
7
7
3
3
3
5
2
2
9
3
3
8
6
9
4
4
T)
5
4
6
7
8
7
6
Write ten columns in which two addends whose sum is
10 or less may be taken as a single addend. Add your
columns.
Add the columns in Sec. 13.
REVIEW 61
REVIEW
74. Multiplication.
Name the multiples of 2 to 24 ; of 3 to 36 ; of 4 to 48 ;
of 5 to 60 ; of 6 to 72 ; of T to 84 ; of 8 to 96 ; of 9 to 108.
Multiply the numbers in each column by the number at
the head of the column, and add to each product the num-
ber in parentheses :
a
h
c
d
e
/
9
^
2(9)
3(2)
4(3)
5(4)
6(5)
7(6)
8(7)
9(8)
7
6
5
4
9
8
2
8
3
3
8
2
4
4
6
2
9
2
4
5
7
5
9
9
4
7
2
8
2
7
3
3
8
9
9
3
3
6
4
7
5
8
7
6
8
2
8
4
9
4
6
9
6
9
T
6
2
5
3
7
5
3
5
5
6
7
8
9
7
6
9
7
8
9
6
6
9
8
8
6
6
6
9
8
6
7
6
8
7
8
7
7
8
9
7
9
75. Write the following in the forms of bills, and find
the amounts in each. (See p. 42.)
1. Jan. 2, 1907 : 3 bars of soap at 6^ each ; 4 lb. of
prunes at 8^ per pound; 85 lb. of potatoes at 2^ per
pound. Jan. 5, 1907 : 2 lb. of coffee at 33^ per pound;
2 lb. of cheese at 18^ per pound ; 3 lb. of tea at ^^^ per
pound.
2. Jan. 9, 1907: 7 yd. dress cloth at $1.20 per yard ;
1 doz. handkerchiefs at 11.40 per dozen ; 3 shirts at $1.75
each. Jan. 10, 1907 : 5 pair socks at $.25 each ; 1 um-
brella at $2.40 ; 1 pair scissors at $.75. i
62 REVIEW OF INTEGERS AND DECIMALS
76. Division.
1. Name the highest multiple of tlie number at the head
of the column in each number in the column, thus for
column a : 18, 9, 15, etc.
a h G d e f g
3 4 5 6 7 8 9
20
17
38
39
40
30
40
11
10
17
28
25
46
78
16
23
26
11
58
37
81
8
25
14
22
13
22
86
25
31
23
58
32
76
68
14
14
49
46
65
53
23
23
19
33
33
39
60
59
17
27
43
52
52
29
60
28
38
24
16
18
69
51
19
29
39
40
60
18
34
22
34
28
50
34
59
48
13
21
34
55
68
14
70
29
39
63
63
87
97
75
24
26
32
70
48
50
89
32
43
54
57
29
17
110
2. Name the highest multiple of the number at the head
of the column in each number in the above columns, and
give the difference between the multiple and the number
in the column, thus for column a ; 18 and 2 ; 9 and 2; 15
and 1, etc.
3. Divide the numbers in each column by the number
above the column, giving the quotient and remainder,
thus for column a : 6 and 2 over ; 3 and 2 over, etc.
4. Divide 21,487,249 by 6; 459,738,795 by 8; 59,482,395
by 7 ; 708,718,907 by 9.
LONG DIVISION 63
LONG DIVISION
77, When all the steps in division are written, the
process is called long division. Long division is generally
used with divisors of two or more places.
For the purpose of finding the quotient figure, all divi-
sors are classified into two general cases, as follows :
Case I. All divisors in which the second figure* is
the same as or less than the first figure, as Q6, 65^ 832.
Case II. All divisors in which the second figure is
greater than the first figure, as 15, 68, 271, 795.
Write in order all numbers from 13 to 100 that come
under Case I, as 21, 22; all that come under Case II,
as 13, 14, 15.
Case I
78. When the second figure of the divisor is the same as
or less than the first figure, a trial quotient figure may be
found, as in the following :
Divide 2792 by 65.
It will take three places at the left of
42 2792 to contain 65 at least once.
Model : 65)2792 ^i^P 1- 6 is contained in 27 four times,
260 with 3 remainder. Is 5 contained in 39
^Qfj as many as 4 times ? Yes. Try 4 as a quo-
tient figure. Place 4 in quotient above 9.
Step 2. Multiply the divisor 65 by 4, writing the product below 279.
. Step 3. Subtract 260 from 279.
Step 4. Bring down the next figure in the dividend. The new
dividend is 192.
Repeat Step 1. 6 is contained in 19 three times, with 1 remainder.
Is 5 contained in 12 as many as 3 times? jN"o. Try 1 less than 3, or
2, as a quotient figure.
Complete the work, writing the remainder as in short division.
* In 66 regard 6 as the first figure and 5 as the second figure.
64 REVIEW OF INTEGERS AND DECIMALS
With divisors of Case I, the trial quotient figure found
as in the model is always the correct quotient figure when
the divisor contains two places, as 42, 87, etc.
It is also the correct quotient figure when the divisor
contains three or more places, whenever the second figure
of the divisor is not contained in its dividend as many
times as the first figure is contained in its dividend.
With divisors of Case I, when the first figure of tlie
divisor is contained in its dividend ten times, the correct
quotient figure is 9, thus : 9
532)52678
5 is contained in 52 ten times. Take 9 as the quotient
figure. It is unnecessary to test the second figure of the
divisor.
79. Written Exercises.
Tell to which case each divisor in the following belongs :
Solve only the exercises in which the divisors are of
Case I.
1.
75,679 by 42
13.
15,672 by 29
25.
57,606 by
21
2.
73,496 by 24
14.
71,896 by 83
26.
40,000 by
20
3.
12,500 by 64
15.
83,678 by 95
27.
59,684 by
84
4.
62,847 by m
16.
79,678 by 88
28.
85,678 by
96
5.
95,438 by 27
17.
53,678 by 61
29.
45,672 by
63
6.
18,245 by 29
18.
29,678 by 54
30.
456,783 by
15
7.
10,000 by 75
19.
38,006 by 47
31.
648,739 by
16
8.
60,000 by 85
20.
47,608 by 96
32.
457,820 by
14
9.
35,640 by 44
21.
64,896 by 88
33.
426,789 by
13
10.
11,045 by 19
22.
52,873 by 68
34.
480,068 by
96
11.
16,712 by 18
23.
49,678 by 79
35.
124,530 by
144
12.
27,672 by 28
24.
68,368 by 72
36.
231,672 by
772
LONG DIVISION 65
Case II
80. For the purpose of finding a trial quotient figure
that will seldom vary much from the correct quotient
figure, divisors of Case II are classified into three groups.
Group a. When the second figure of the divisor is 7,
8, or 9, as 17, 18, 19 ; 578, 588, 598, etc.
Group h. When the first figure is more than 1 and the
second figure is 3, 4, 5, or 6, as 23, 24, 35, 46, etc.
Group c. When the first figure is 1 and tlie second
figure is 3, 4, 5, or 6, Le, 13, 14, 15, 16.
Write tlie numbers from 13 to 100 that come under
Group a. Case II ; that come under Group 6, Case II ;
that come under Group c. Case II.
81. Group a. With divisors of Group a, the trial quotient
figure may be found by using as a divisor 1 more than the
first figure of the divisor, as in the following :
1. Divide 379,868 by 476.
Model :
79 It will take four places to contain 476 at least
4-7fiy^7Q8^ once. 5 (1 more than 4) is contained in 37 seven
times. Try 7 as a quotient figure. Complete the
??£^ division.
4666
The trial quotient figure found as in the model will
sometimes be 1 less than the correct quotient figure.
2. Solve all exercises in Sec. 79 in which the divisors
are of Group a. Case II.
3. Write and solve five exercises in division, using
divisors of Group a. Case II, and five exercises using
divisors of Case I.
MOCL. & JONES'S ESSEN. OF AB. — 6
66 REVIEW OF INTEGERS AND DECIMALS
82. Group b. With divisors of Group 6, Case II, the trial
quotient figure may be found by using as a divisor 1 more
than the first figure of the divisor and adding 1 to the quo-
tient, as in the following :
1. Divide 11,678 by 24.
4
Tv/r^T.T,r . o i \-iior7o I* will *^^® three places to contain 24
Model: 24)llb<8 -i . o ^i xi. ^.x •
at least once. 3 (1 more than 2) is con-
*^^ tained in 11 three times. Try 4 as a
207 quotient figure. Complete the division.
The trial quotient figure found as in the model will
sometimes vary 1 from the correct quotient figure.
2. Solve all exercises in Sec. 79 in which the divisors
are of Group 5, Case II.
3. Write and solve ten exercises in division, using di-
visors of Groups a and 5, Case II.
4. Write and solve five exercises in division, using
divisors of Case I.
83. Group c. With divisors of Group c, Case II, the trial
quotient figure may be found by using 2 as a divisor, and
adding 2 to the quotient, as in the following :
1. Divide 12,678 by 142.
It will take four places in the divi-
dend to contain 142 at least once. 2 is
Model: 142)12678 contained in 12 six times. Try 8 as a
quotient figure. Complete the division.
The trial quotient figure found as in the model will
seldom vary more than 1 from the correct quotient figure.
2. Solve all exercises in Sec. 79 in which the divisors
are of Group c, Case II.
3. Write and solve ten exercises in division, using
divisors of Case II.
LONG DIVISION 67
84. With 1367 as a dividend, find the first trial quo-
tient figure, using each of the following as divisors, and
explain how each is found : 32, 16, 327, 48, 59, 24, 375,
698, 166, 426, 276, 149, 137, 161.
85. Written Exercises.
1. Write and solve ten exercises in division, using as
divisors numbers of Case I between 100 and 1000.
2. Write and solve ten exercises in division, using as
divisors numbers of Groups a, 5, and (?, Case II, between
100 and 1000.
86. Written Exercises.
1. How many dozen eggs at 18^ a dozen must be sold
to pay for 1 lb. of tea at 60/, and 1 lb. of coffee at 30/?
2. A man's yearly salary is f 1860. Find his salary
per month. What is the amount of his salary per week?
3. If a train travels at an average rate of 45 mi. an
hour, in how many hours will it travel 2000 mi. ?
4. A ton is 2000 lb. How many pupils of your own
weight will it take to weigh IT.?
5. A bushel of wheat weighs 60 lb. How many
bushels will it take to weigh IT.?
6. The monthly rental of an apartment house amounted
to ^144. The average rental of an apartment was $24.
How many apartments were there in the house?
7. The total annual expenditure of the United States
government for the year ending June 30, 1905, amounted to
1532,122,762.47. What was the average daily expenditure?
8. The number of school children in a certain city is
1640. If the average number of pupils to each room is 40,
how many schoolrooms are there in the city?
68 REVIEW OF INTEGERS AND DECIMALS
DIVISION OF DECIMALS
87. 1. How many $2 are there in $4? How many 2
tenths are there in 4 tenths? .4 -j- .2 = x?
2. How many 3 qt. are there in 9 qt. ? How many
3 hundredths are there in 9 hundredths?
9 qt. ^3 qt. =x? .09 -t- .03 = a;?
3. How many times are 4 yd. contained in 8 yd. ? 4
tenths in 8 tenths? 4 hundredths in 8 hundredths? 4
thousandths in 8 thousandths?
4. What is the quotient in each of the following :
4)8; AyS; .04)708; .004:)Am? Prove the correct-
ness of your answer by multiplying the divisor by the
quotient and comparing it with the dividend.
5. If the divisor contains tenths, tenths of the divi-
dend may give a whole number in the quotient.
3 36
.4)1.2 .6)21.6
6. If the divisor contains hundredths, hundredths of
the dividend may give a whole number in the quotient.
9 59 8 54
.05).45 .04)2.36 .04)34.16
Place the decimal point in the quotient above and after the
figure in the dividend occupying the same order as the lowest
order in the divisor. Divide as in integers.
, 7. Divide 21.66 by 6.
Model : As the lowest order in the divisor is units,
6)21.66 place the decimal point in the quotient
above and after the figure occupying units' order in the dividend.
Divide as in integers. When the divisor is an integer, the lowest
order in the divisor is units, and the decimal point in the quotient is
directly above the decimal point in the dividend.
LONG DIVISION 69
a Divide 43.38 by .8.
Model- 8 ^4S S8 ^^ *^® lowest order in the divisor is
^ * tenths, place the decimal point above and
after the figure occupying tenths* order in the dividend. Divide.
9. Divide 2.4 by .006.
Model ; 006^2 400 ^'^ *^® lowest order in the divisor is
thousandths, supply two ciphers in the
dividend to make the lowest order thousandths. Place the decimal
point above and after the figure in the dividend occupying thou-
sandths' order. Divide.
10. With 45.06 as a divisor, the decimal point will be
placed in the quotient above and after the figure in the
dividend occupying hundredths' order. State where the
decimal point should be placed with each of the following
as divisors: 5.05; 67; 3.15; 3.1416; .008; 26.1;
.0045; 50; .05; 2150.42; 6.
88. Arrange as in the models and fix the decimal points
in the quotients :
a bed
1. .05-5-2.5 .04-^.002 180 -4-. 006 3^4
2. .6-^.3 2.5-^50 36-^750 20^50
3. 1.44 -f- .12 .0048 -5- .6 .27-^-3 1.75 -f- .025
4. 2.7 -f- 9 3.6-^.12 .007^3.5 .075-^2.5
5. .48-1-8 .16^20 14 -.007 120 -5- .004
6. .1--.005 2.4-5-12 22.5 -T- .15 4.8 -5- .0012
7. .024^.8 .012-^.03 2-f-lO .065-^3.25
8. 3.6-^.006 .005 -4- .1 45-^-90 64 -f- .0008
9. 1-4-.045 4-4.56 100^.1 .75-^.6
10. 10-?- .01 .01^10 .001 -5- .01 101 -i- 1.01
70 REVIEW OF INTEGERS AND DECIMALS
89, 1. Divide 75.51 by 60.4, and carry the result to
two decimal places.
1.25 +
Model: 60.4)75.51 ^^^ quotient, carried to two deci-
' mal places, is 1 .25. The sign ( + ) is
^ ^ placed after the quotient to indicate
15 11 that the division is not exact.
12 08
3 03
Arrange as in the model, fix the decimal point, estimate
the result, then divide. When the division is inexact,
carry the quotient to three decimal places.
2.
6.25-25
12.
500-^.005
3.
.1728 -^ .12
13.
512.16-^64.02
4.
720.405 ^ 3.15
14.
68.045-^-42,125
5.
250-^.75
15.
12.5 H- .0375
6.
1210.605-^6.05
16.
1000^.875
7.
.0045 -f- .045
17.
12.75^3.1416
8.
37.806-^8.7
18.
458,766 -f- 2150.42
9.
48.312-^3.1416
19.
8.05 -^ 40.25
10.
1000-^.0025
20.
8790 ^ 2150.42
11.
.1224-^2.04
21.
100 -.125
22.
Read the decimals in the above exercises.
23. When a number is divided by .4, is the quotient
greater or less than the number ?
24. When a number is divided by .2, the quotient
obtained is how many times the dividend?
25. Estimate the quotient of 12 divided by each: .1,
.25, .5, .4.
26. State a short method of dividing by 10 ; by 100 ;
by 1000 ; by .1 ; by .01 ; by .001.
REVIEW 71
REVIEW
90. 1. Solve exercises in Sec. 88.
2. Write five exercises in division of decimals, and
solve each.
3. Write five exercises in multiplication of decimals,
and solve each.
4. Write five exercises in Case I in long division, and
solve each.
5. Write five exercises in Group a of Case II in long
division, and solve each.
6. Write five exercises in Group h of Case II in long
division, and solve each.
7. Write five exercises in Group c of Case II in long
division, and solve each.
8. Write five columns in addition, and add each as in-
dicated in Sec. 73.
91. Add the following :
1.
2.
3.
$786.45
$578.04
% 16.45
97.08
35.16
8.12
300.90
900.
947.
7.87
80.47
32.76
46.59
570.09
6.58
807.98
98.17
300.
345.56
315.40
97.26
96.
9.98
1.95
4.75
405.56
647.15
400.
58.08
45.
57.09
930.
780.35
815.35
40.76
46.27
72 REVIEW OF INTEGERS AND DECIMALS
92. 1. Give the number of pints in a quart ; of quarts
in a gallon ; of pints in a gallon.
2. How many ounces are there in a pound of siigar?
in 5 lb. ? How many pounds are there in a ton?
3. There are 2000 lb. in a short ton, and 2240 lb. in
a lonff ton. A company imported 11,200,000 lb. of coal,
paying for it by the long ton. The company sold the
coal by the short ton. How many more tons did it sell
than it imported?
A long ton (2240 lb.) is used sometimes in weighing coal and in
weighing certain materials imported into the United States.
4. In dry measure 2 pints are 1 quart, 8 quarts are
1 peck, and 4 pecks are 1 bushel. How many quarts are
there in 1 bu. ? in 1 pk. and 3 qt. ? in 1 bu. 2 pk.?
5. How many months are there in 1 yr. ? in 7 yr.?
How many days are there in 1 yr. ? in 1 leap year?
6. The depth of the sea is measured in fathoms, A
fathom is 6 ft. Express 1728 ft. in fathoms.
7. The circumference of a circle is 3^ (3.1416) times
its diameter. Show that this is correct by comparing the
diameter and the circumference of some circle (top of
barrel, pail, stovepipe, etc.).
8. A denominate number is a concrete number in
which the unit of measure has been established by law
or custom, as 4 ft., 12 gal., etc. Such expressions as 10 ft.
6 in., 2 yr. 7 mo. 6 da., etc., are called compound denomi-
nate numbers.
Tables of denominate numbers are found on pp. 312-319.
9. Mr. Davis's expenses for January, 1907, were
$121.45. What were his average daily expenses?
REVIEW 73
93. 1. The exports for the first ten months of 1906
amounted to $1,425,184,757, while the exports for the
corresponding period in 1905 amounted to $1,256,924,354.
At the same rate of increase, by how much would the
exports for 1906 exceed the exports for 1905 ?
2. The imports for the first ten months of 1906
amounted to $1,066,462,295, while the imports for the
first ten months of 1905 amounted to $979,917,437. At
the same rate of increase, by how much should the im-
ports for 1906 exceed the imports for 1905?
3. The population of Massachusetts was 2,805,346 in
1900. The area of Massachusetts is 8315 sq. mi. Find
the average population for each square mile.
4. The area of Texas is 265,780 sq. mi. and of Iowa is
56,025 sq. mi. How many states of the size of Iowa can
be made of Texas?
5. Four places, A, B, C, and D, are located on a line
running due east and west. B is 16 mi. east of A, C is
12 mi. west of A, and D is 8 mi. west of C. How far
apart are B and C? A and D? B and D? (Draw
a diagram.)
6. Mr. Wright of Chicago is employed by a wholesale
house and receives $125 per month and necessary expenses
while traveling. During the month of January Mr.
Wright paid $69 for railroad fare, $86 for hotel bills,
and $18.65 for other expenses. How much did the com-
pany owe Mr. Wright for the month, including salary?
7. Charles deposited $2.75 in a savings bank on Oct.
15, and $3.45 on Oct. 23. He drew out $4.10 on Oct. 29.
He deposited $4.80 on Dec. 1. How much had he then
in the bank?
74
REVIEW OF INTEGERS AND DECIMALS
94. Reading a Railroad Time Table.
SAN FRANCISCO — LOS ANGELES
20
Shore
Line
Limited
22
The
Coaster
18-8
Los
Angeles
Passen-
ger
10
Sunset
Ex-
press
ai
8T1TI0I8
17
San
Fran-
cisco
Passen-
ger
9
Sanset
Ex-
press
19
Shore
Line
Umited
21
The
Coaster
READ DOWN
Lv.
SAN FRANCISCO Ar.
READ UP
8.00
8.80
3.15
8.00
9.16
10.15
9.30
11.45
9.25
9.55
4.45
9.30
51
Lv.
SAN JOSE Lv.
7.86
8.45
8.05
10.15
6.19
8.15
4.85
7.30
871
Lv. SANTA BARBARA Lv.
8,20
11.00
11.15
12.10
9.30
11.45
8.45
11.00
475
Ar.
LOS ANGELES Lv.
4.00
7.30
8.00
8.80
Light-face figures, a.m. ; dark-face, p.m.
95. Answer the following from the above table :
1. How many passenger trains leave San Francisco for
Los Angeles each day over this route ? How many leave
Los Angeles for San Francisco ?
2. What is the distance from San Francisco to Los
Angeles ?
3. What is denoted by the light-face figures ? by the
dark-face figures ?
4. Which is the first train in the morning from San
Francisco to Los Angeles? from Los Angeles to San
Francisco ?
5. How many hours does it take each train to make
the run ? Which are the fastest trains?
6. Find the average number of miles per hour of train
No. 20, of train No. 22, and of the Sunset Express, on
the run from San Francisco to Los Angeles, and on the
run from Los Angeles to San Francisco.
7. Find the distance from Los Angeles to Santa
Barbara; from Los Angeles to San Jose; from San Jose
to Santa Barbara.
REVIEW
75
96. Reading a Meter.
The amount of water, gas, and electricity consumed
is usually measured by instruments called meters. These
instruments are furnished with dials, on which the
amounts consumed are indicated in the decimal scale, as
shown in the picture.
97. Dials of a Gas Meter.
CUBIC
The unit dial at the top is used for testing the meter.
For every 100 cu. ft. of gas that passes through the
meter, the hand on the first (right-hand) dial moves over
one of the divisions, as from to 1 ; for every 1000 cu. ft.
consumed, it makes a complete revolution, the hand on
the second dial moves over one division, and the hand on
the third dial moves over -^^ of one division.
Ten revolutions of the hand on any dial produce one
revolution of the hand on the dial of the next higher
order.
The first dial is now recording 300 cu. ft. How much
is the second dial recording ? How much are the three
dials recording ? The dials should be read from left to
right as you would read a number, thus : 68,300 cu. ft.
The cost of the gas would be stated for each 1000 cu. ft.
76 REVIEW OF INTEGERS AND DECIMALS
MEASUREMENT OF LENGTH
98. 1. Length and distance are commonly measured in
inches, feet, yards, rods, and miles. The yard is the stand-
ard unit of length. The other units are derived from it.
2. Draw on the blackboard a line 1 in. long. Draw
a line 1 ft. long. Draw a line 1 yd. long. Using a
yard stick, test the correctness of your drawings. Prac-
tice drawing these lines until you can estimate an inch, a
foot, and a yard without much error.
3. Estimate in inches the length and the width of
each : your desk top ; your book cover ; a window pane.
4. Estimate in feet the length and the width of each :
your schoolroom ; the blackboard ; the window ; the door.
5. Estimate the length of your room in yards ; of your
school yard ; of the blackboard.
6. A rod is 16 J ft., or 5| yd. Measure off a rod on the
floor of your schoolroom or on the school yard. Estimate
the length and width of your school yard in rods.
7. Determine some place that is 1 mi. from your
schoolhouse.
99. Table of Linear Measure.
12 inches (in. or "^ = 1 foot (ft. or ')
3 ft. =1 yard (yd.)
5^ yd., or 16J ft. = 1 rod (rd.)
320 rd., or 5280 ft. = 1 mile (mi.)
1. How many feet are there in 3 mi. ? in 5 mi. ?
2. How many rods are there in 2 mi. ? in J mi.? in \
mi.?
3. Change to rods : \ mi., \ mi., J mi.
4. How many inches are there in 1 yd. 6 in.?
MEASUREMENT OF LENGTH 77
5. The lengths of three pieces of blackboard in a school-
room were measured by the pupils and found to be 18 ft.
6 in., 14 ft. 9 in., and 6 ft. 4 in., respectively. Find the
combined length of the three boards.
Model :
18 ft. 6 in. 4 in. and 9 in. and 6 in. are 19 in., or 1 ft. and 7
14 ft. 9 in. ^^- Write 7 in. in the answer as shown in the
6 ft. 4 in. ^odel, and carry 1 ft. to the column of feet.
39 ft. 7 in.
6. Find the combined length of the blackboards in
your schoolroom.
7. Find the distance around your schoolroom.
8. From 8 ft. 4 in. subtract 4 ft. 10 in.
Model : ^^ ^^ ^"- ^^® more than 4 in., the sum of 10 in.
8 ft 4 in ^^^ ^^® number of inches in the answer is 1 ft. 4
A jfi^ 1 r\ ' ill- Subtract thus : 10 in. and 2 in. are 1 f t. : 2 in.
4 ft. 10 in. , . . ^ . ,nr .^ o . . ^.
. — and 4 in. are 6 m. Write 6 in. in the answer as
o It. o m. s]iown in the model. Carry 1 ft. to 4 ft., making
5 ft. 5 ft. and 3 ft. are 8 ft.
9. From a board 9 ft. 6 in. long a carpenter sawed a
shelf 3 ft. 10 in. long. How long was the piece of board
that was left ?
10. From a piece of cloth 4 yd. 8 in. long a woman cut
a piece 1 yd. 9 in. long. How long was the piece of cloth
that was left ?
11. Find how much longer the length of your school-
room is than its width.
12. On Jan. 1, 1903, a boy's height was 4 ft. 7 in., and
on Jan. 1, 1906, it was 5 ft. 2 in. How much taller was
he on the second date ?
13. How many feet are there in 1 mile? How many
yards are there in 1 mile ?
78 REVIEW OF INTEGERS AND DECIMALS
BIEASUREMENT OF SURFACES
100. 1. The number of square units in any surface
is called its area.
2. The area of surfaces is commonly measured in
square inches, square feet, square yards, square rods, acres,
or square miles.
3. Using a ruler, draw on the board a square whose
side is 1 foot. This is called a square foot. A square
foot is a square whose side is 1 foot.
4. Using a ruler, draw upon the board a square whose
side is 1 inch. This is called a square inch. A square
inch is a square whose side is 1 inch.
5. Divide a square foot into square inches. How many
square inches are there in 1 square foot?
6. Using a yard stick, draw a square whose side is 1
yard. What is this square called? Divide a square yard
into square feet. How many square feet are there in 1
square yard?
7. What is the purpose of having several different
units for measuring length and area? In what unit
should you express the area of the cover of this book?
of the top of your desk ? of the surfaces of the walls in
your schoolroom ?
8. Mark out a square rod on the school yard.
101. Table of Square Measure.
144 square inches (sq. in.) = 1 square foot (sq. ft.)
9 sq. ft. = 1 square yard (sq. yd.)
30 J sq. yd. = 1 square rod (sq. rd.)
160 sq. rd. = 1 acre (A.)
640 A. =1 square mile (sq. mi.)
ANGLES 19
ANGLES
RightAngle
102. 1. The difference in direction of two lines is called
an angle. The name angle is used also to denote the
opening between two lines that meet.
2. When two straight lines meet
and form two equal angles, the
angles are called right angles, and
the lines are said to be perpendicular RighfAngle
to each other.
3. The lines that form an angle are called the sides of
the angle.
4. Angles whose sides are not perpendicular to each
other are called oblique angles. Oblique angles are either
acute or obtuse. /\ \ /
5. An angle that is less than a / \ v
right angle is called an acute angle. Acute angles.
6. An angle that is greater than a
right angle but less than two right /^"^ ^^^^^
angles is called an obtuse angle. Obtuse angles.
7. Draw a right angle ; an obtuse angle ; an acute
angle.
8. Stand your pencil upon the top of your desk, per-
pendicular to the desk top. What kind of angles are
formed by the pencil and the desk top ?
9. Keeping the pencil resting at the same point on
the top of the desk, move the top of the pencil to the left.
Does the pencil now form any right angles with the
desk top ? any obtuse angles ? any acute angles ?
10. Point to surfaces in your schoolroom that meet at
right angles. Are there any that meet so as to form
obtuse angles ? acute angles ?
80 REVIEW OF INTEGERS AND DECIMALS
RECTANGLES
103. 1. Lines that extend in the same direction and
are everywhere the same distance apart are
— called parallel lines. If parallel lines are
Parallel Lines, extended, will they ever meet ?
2. Point to two surfaces in the schoolroom that are
parallel to each other ; that are perpendicular to each
other ; that meet at right angles. Are there any sur-
faces in your schoolroom that meet at obtuse or acute
angles ?
3. A figure having four straight sides and four right
angles is called a rec-
tangle. Find by drawing
figures how many sides a
figure must have in order
Rectanglks. t^a^ all Its angles may be
right angles.
4. A rectangle whose sides are equal is called a
square.
5. What is a square inch ? Draw a square inch;
a square foot. What is a square yard ? a square mile ?
6. All rectangles are either square or oblong. Point
to surfaces in the schoolroom that are rectangles.
7. Draw a 2-inch line. Build a square upon it. This
is a 2-inch square. Divide it into square inches. How
many square inches are there in a 2-inch square ?
8. Draw a 3-inch square. Divide it into square
inches. How many square inches are there in a 8-inch
square ?
9. Draw a rectangle 2 in. wide and 8 in. long. Di-
vide it into square inches.
RECTANGLES
81
■
■
•
«
104. Area of Rectangles.
1. Draw a rectangle 6 in.
long and 1 in. wide. Divide
it into square inches.
A rectangle 6 in. long and 1 in. wide contains 6 sq. in.
2. Draw a rectangle 6 in. long and 3 in. wide. Divide
it into square inches.
As both dimensions are
given in inches, the unit in
which the area of the rectangle
is to be expressed is 1 sq. in.
In a rectangle 6 in. long and
1 in. wide there are 6 times 1 sq. in., or 6 sq. in. Since
the given rectangle is 3 in. wide, it contains 3 times
6 sq. in., or 18 sq. in.
Or, a rectangle 3 in. long and 1 in. wide contains 3
times 1 sq. in., or 3 sq. in. Since the given rectangle
contains 6 such rectangles, its area is 6 times 3 sq. in., or
18 sq. in.
3. The number of square inches in the given rectangle
may be found by multiplying 6 by 3. 6 times 3 is 18, the
number of square inches in the rectangle. Never multiply
inches by inches. Why ?
105. The following are dimensions of rectangles.
State the unit in which the area of each is to be found.
Represent 1-4 by drawings. Give the area of each.
1. 8 in. by 5 in.
2. 12 yd. by 9 yd.
3. 8 ft. by 6 ft.
4. 4 ft. by 3 ft.
5. 14 ft. by 10 ft.
6. 20 ft. by 15 ft.
7. 20 rd. by 10 rd.
8. 40 rd. by 40 rd.
9. 6 mi. by J mi.
10. 25 ft. by 100 ft.
82 REVIEW OF INTEGERS AND DECIMALS
106, 1. Compare the size of a 2-inch square with 2
square inches ; of a 3-inch square with 3 square inches.
2. A 4-inch square is how many times 4 square inches ?
Compare a 5-inch square with 5 square inches.
3. Draw an inch square. Divide it into 4 equal
squares. How long is the side of a square that contains
one quarter of a square inch ?
4. Draw a square that contains one sixteenth of a
square inch.
5. Draw a rectangle containing 18 square inches,
making it 6 inches long ; 9 inches long.
6. Which is greater, an inch square or a square inch ?
a half-inch square or one half of a square inch? Show
by drawing.
7. How long is the perimeter of a rectangle 6 ft. long
and 4 ft. wide? (Perimeter means distance around.)
8. How long is the perimeter of a 9-inch square ? of a
4-inch square ? of a square inch ?
9. How wide is a rectangle that is 8 in. long and con-
tains 8 sq. in. ? 16 sq. in. ? 24 sq. in. ? 32 sq. in. ?
10. By what number must T sq. in. be multiplied to
give 28 sq. in. ? If the area of a rectangle is 28 sq. in.
and its length is 7 in., how wide is the rectangle ?
11. When the area of a rectangle and one dimension
are given, how may the other dimension be found ? Illus-
trate with several examples.
12. How many dimensions has a rectangle ?
13. A surface that has the same direction throughout,
as the surface of a blackboard, a window pane, etc., is
called a plane surface. The surface of a globe is not a
plane surface. Why ?
RECTANGLES 83
107. In each of the following the area of a rectangle
and one dimension are given. Find the other dimension :
1. Area, 20 sq. ft., length, 5 ft.
2. Area, 48 sq. yd., width, 6 yd.
3. Area, 100 sq. in., length, 10 in.
4. Length, 45 ft., area, 900 sq. ft.
5. Width, 50 ft., area, 6500 sq. ft.
6. Area, 1728 sq. in., length, 144 in.
108. 1. Find the area of a garden 10 rd. long and
8 rd. wide.
2. Find the number of acres in a field 40 rd. by 20 rd.
3. At $85 an acre, find the value of a farm 80 rd. by
40 rd. ; 160 rd. by 80 rd.
4. A farm containing 80 acres is 80 rd. wide. How
long is it ?
5. How long is a 10-acre field, if its width is 40 rd.?
20 rd.?
6. How long is a 20-acre field, if its width is 40 rd. ?
20 rd. ?
109. 1. Find the number of square feet of window
space in your schoolroom.
2. Estimate the number of square inches in the cover
of this book. Test the correctness of your estimate.
3. Estimate the area of the floor of your schoolroom in
square feet. Test the correctness of your estimate.
4. Estimate the number of square rods in your play-
ground. Test the correctness of your estimate.
5. Are there as many as 60 sq. yd. of surface in the
ceiling of your schoolroom? Test your answer.
84 REVIEW OF INTEGERS AND DECIMALS
CUBIC MEASURE
llO, 1. Describe a rectangle ; a square ; an oblong.
Draw each.
2. How many dimensions has a rectangle ? a plane
surface ?
3. How many dimensions has a book ? a block ? a box ?
4. Any object that has length, breadth, and thickness
is called a solid. Is a book a solid ? Name other solids.
5. A solid having six rectangular faces is called a rec-
tangular solid. Is a brick a rectangular solid? Isame
objects which are rectangular solids.
6. A solid having six equal square surfaces is called a
cube. How many edges has a cube?
^ 7. A cube whose faces are each a foot
llHli^ square is called a cubic foot. Describe a
^^B ill cubic inch ; a cubic yard.
8. How many inch cubes will form a
solid 12 in. long, 1 in. wide, and 1 in. thick?
12 in. long, 12 in. wide, and 1 in. thick?
12 in. long, 12 in. wide, and 2 in. thick?
12 in. long, 12 in. wide, and 3 in. thick?
12 in. long, 12 in. wide, and 6 in. thick?
12 in. loner, 12 in. wide, and 12 in. thick?
Rectangular °
Solids. 9. What name may be given a solid
formed by placing inch cubes 12 deep on a surface 1 ft.
square ? There are cu. in. in 1 cu. ft.
10. How many foot cubes will form a solid 3 ft. long,
3 ft. wide, and 1 ft. thick ? 3 ft. long, 3 ft. wide, and 3 ft.
thick?
11. How many cubic feet are there in a rectangular
solid 8 ft. long, 4 ft. wide, and 4 ft. thick?
CUBIC MEASURE 85
As the dimensions are all expressed in feet, the cubic
contents will be found in cubic feet. In a rectangular
solid having the same base, 8 ft. by 4 ft., but only 1 f t . in
height, there are 1 cu. ft. x 8 x 4, or 32 cu. ft. Since the
given rectangular solid is 4 ft. thick, it contains 4 times
32 cu. ft., or 128 cu. ft. 1 cu. ft. x 8 x 4 x 4 = 128 cu. ft.
When all the dimensions of a rectangular solid are ex-
pressed in like units, the contents of the solid may be
found by multiplying the number of units in the length by
the number of the units in the width and the product by
the number of units in the thickness, and calling the result
cubic units of the given dimension. Thus, the number of
cubic feet in a rectangular solid 8 ft. by 4 ft. by 4 ft. is
8 X 4 X 4, or 128.
12. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high
is called a cord of wood. How many cubic feet are there
in a cord of wood ?
13. The number of cubic units in a solid is called its
volumCo
111. Table of Cubic Measure.
1728 cubic inches (cu. in.) = 1 cubic foot (cu. ft.)
27 cu. ft. = 1 cubic yard (cu. yd.)
128 cu. ft. = 1 cord of wood (cd.)
Find the volume of rectangular solids of the following
dimensions :
1. 6' x4^ x2^ 4. 12^' X 9''x 8''
2. 10' X 8' X 5' 5. 14" X 12^' X 10''
3. 9"x6"x4" 6. 24' xl8' x 16'
7. Find the number of cubic feet of air in a room 14'
by 12' by 9'.
86 REVIEW OF INTEGERS AND DECIMALS
8. If each pupil requires 35 cu. ft. of fresh air per
minute, how many cubic feet of fresh air per minute will
50 pupils require ?
9. Find the number of cubic feet of air in a school-
room 32' X 24' X 12'.
10. A watering trough is 12 ft. long, 2 ft. wide, and
2 ft. deep. How many gallons will it contain ? (231 cu.
in. = 1 gal.)
11. Find the number of measured bushels in a bin 6 ft.
by 4 ft. by 4 ft. (2150.42 cu. in. = 1 bu.)
12. The length of a rectangular solid is 8 ft., and its
width is 5 ft. If its volume is 160 cu. ft., what is its
thickness ?
13. How many cubic yards of dirt must be removed in
excavating a cellar 36 ft. long, 24 ft. wide, and 8 ft. deep ?
Find the cost of making this excavation at 30^ per cubic
yard.
14. How many cubic yards of dirt must be removed in
digging a trench for a sewer, if the trench is 3 ft. wide,
6 ft. deep, and 120 ft. long ?
15. A contractor's bid for excavating a basement 60 ft.
by 36 ft. and 9 ft. in depth is $216. How much is this
per cubic yard ?
16. How many cords of wood in a pile 24 ft. long,
4 ft. wide, and 4 ft. high ?
17. How many cords of wood in a pile 8 ft. long, 2 ft.
wide, and 4 ft. high ?
18. At $6 per cord, find the cost of a pile of wood
20 ft. long, 4 ft. wide, and 6 ft. high.
19. State how you would find the capacity of a box car.
DIYISIBILITY 87
DIVISIBILITY OF NUMBERS
112. 1. Count by 2's to 20. Numbers that are ex-
actly divisible by 2 are called even numbers. All numbers
ending in 0, 2, 4, 6, and 8 are even numbers. Name the
even numbers from 40 to 60.
2. Numbers that are not exactly divisible by 2 are
called odd numbers. Name the odd numbers from 20 to 40.
3. Which of the following are even numbers : 18, 27,
31,46,50,65,123,2456?
4. Some numbers are not exactly divisible by any
whole number except themselves and 1. Such numbers
are called prime numbers. Write the prime numbers
below 80. Numbers that are exactly divisible by whole
numbers other than themselves and 1 are called composite
numbers.
5. The factors of 15 are 3 and 5. These two numbers
when multiplied together give 15. Some numbers have
many factors. 2, 3, 4, 6, 8, and 12 are each a factor of 24.
By what must each be multiplied to give 24 ?
A number which when multiplied by another number
makes a given number is called a factor of the given
number.
6. Name the factors of 6 ; of 12 ; of 16 ; of 30 ; of
36. Has a prime number factors ?
7. The prime factors of a number are the prime num-
bers which when multiplied together make the number.
The prime factors of 24 are 2, 2, 3, 2.
8. Name the prime factors of 18 ; of 20 ; of 36.
9. A factor is always an exact measure of a number.
What numbers are exact measures of 21 ? of 30 ? of 16 ?
of 12 ? of 48 ? Is 24 an exact measure of 48 ?
88 REVIEW OF INTEGERS AND DECIMALS
10. All the prime factors of a number may be fovind
by dividing the number by one of its prime factors, and
dividing each quotient in turn by one of its prime factors
until the quotient is a prime number. The prime factors
of 60 may be found by dividing 60 by 2, and dividing the
quotient (30) by 2, and this quotient (15) by 3. The
last quotient (5) is a prime number. The prime factors
of 60 are the divisors, 2, 2, 3, and the last quotient, 6.
11. Find the prime factors of 48 ; of 72 ; of 80.
12. Which of the following are prime numbers : 27, 17,
39, 51, 29, 91, 53, 89, 77, 57 ?
13. Count by 5's to 35, beginning with 5. Numbers
ending in 5 and in are exactly divisible by 5.
14. What numbers are exactly divisible by 10 ? by 2 ?
by 5?
15. Write a number the sum of whose digits is 3. Is
the number exactly divisible by 3? Is 51 exactly di-
visible by 3 ?
16. Write a number the sum of whose digits is 6, 9, 12,
or some other multiple of 3. Is the number exactly di-
visible by 3 ? Show by several illustrations that the fol-
lowing statement is correct :
A number is exactly divisible by S if the sum of its digits
is exactly divisible by 3.
17. Which of the following are exactly divisible by 3 :
54, 177, 81, 52, 819, 57, 69, 71, 213, 105, 86, 1612 ?
18. Write a number the sum of whose digits is divisible
by 9.
A number is exactly divisible by 9 if the sum of its digits
is exactly divisible by 9.
19. Which of the following are exactly divisible by 9 :
54, 504, 522, 711, 827, 218, 745, 891, 5375, 457?
DIVISIBILITY 89
20. "Which of the following are exactly divisible by 2 ?
by 3 ? by 5 ? by 9 ? by 10 ? 45, 61, 360, 207, 783, 53,
540, 117, 102, 107, 37, 97, 201, 855, 732, 380, 4320, 105 ?
21. A number that is exactly divisible by 9 is exactly
divisible by 3. Why ?
22. All multiples of even numbers are even numbers.
Why?
23. The number denoted by the two right-hand figures
of 216 is 16. Will 4 exactly divide 16 ? 200 ? 216 ?
24. Write a number of three or more places in which
the number denoted by the two right-hand figures is some
multiple of 4. Is the number exactly divisible by 4?
A numher is exactly divisible hy 4 if the number denoted
by its two right-hand figures is exactly divisible by 4.
25. Which of the following are exactly divisible by 4 :
112, 202, 420, 532, 514 ?
26. Centennial years that are divisible by 400 (1200,
1600, etc.) and other years divisible by 4 are leap years.
Which of the following will be leap years: 1910, 1912,
1908, 1926, 1924, 1960, 1990, 2000, 2100 ?
27. Write 10 numbers that are exactly divisible by 3.
113. The three pairs of factors of 24 are 2 and 12, 3
and 8, 4 and 6. Name all the pairs of factors of each of
the following, naming no factor larger than 20 :
4
15
25
34
44
56
77
100
6
16
26
35
45
57
80
108
8
18
27
36
48
60
81
110
9
20
28
38
49
64
84
120
10
21
30
39
50
m
90
121
12
22
32
40
51
70
96
132
14
24
83
42
54
72
99
144
PART II
FRACTIONS
114, 1. Draw a line 4 inches long. Divide it into
two equal parts. What is each part called ?
2. If 6 pupils are separated into three equal groups,
what part of the pupils will each group contain?
3. Draw a line. Divide it into eight equal parts.
What is each part called ?
i
1
2
*■
i
i
■k
i , i
i
^
1
8
i
i
i ^
1 l^\ 1 1
16 \ie\/6\ 16
/il/6
76 \ 76
1 1
76 \76
76 1 76
/ /
16 1 /6
T6\l6
B
4. A represents the line undivided. B represents
the line as divided into two equal parts. What does
C represent? i>? J57? What name is given to each part
of the line in jB? in (7? in i>? in ^? In each case, how
many of the parts does it take to equal the entire line?
5. The length of the line is represented in turn by
1? f ^ \'> f 1 and \\. 1 of the line = | = | = ^^g of the line.
6. I of the line = f = ^^^ of the line.
7. \ of the line + \ of the line = f of the line. \ of
the line — J of the line = f of the line.
8. What is the sum of \ of the line and \ of the line?
What is the difference between J of the line and J of the
line? \ of the line and | of the line?
00
FRACTIONS 91
9. I of the line is longer than f of the line. -^^ of the
line is longer than | of the line, f of the line is longer
than I of the line.
10. Using 8 objects, show that |^ of 8 objects is the
same as | of 8 objects, and that | of 8 objects is the same
as "I of 8 objects.
34. Show by dividing circles that | of a circle is equal
to I of a circle ; that | of a circle is equal to | of a circle ;
that J of a circle plus |^ of a circle is equal to |- of a
circle ; that f of a circle is equal to ^| of a circle.
12. Show by dividing rectangles that |^, |, -|, -j^, and
-^ of a rectangle are equivalent parts.
13. Using objects, show that ^ of 12 objects is the same
as I of 12 objects ; that J of 12 objects is the same as |
of 12 objects.
14. Show by folding paper that i = |== | =« ^g- ; that J
=- 1 ^ 3 _ _4_
6 9 ~ 12*
116. Ratio.
A
S .
c .
1. Line A is what part of line B ? what part of line 0?
2. If B is called 1, what isA?0? If (7 is called 1,
what is ^ ? ^ ? If (7 is called 6, what is ^ ? J5 ?
3. If A is called 3, what is ^? (7? If J. is called J,
what is J5? 0?
4. The ratio of line A to line B is ^, What is the
ratio of line A to line (7? of line B to line ^ ? of line B
to line a? of line O to line B ? of line O to line A ?
FRACTIONS
n
R
116. 1. The surface A is what part of the surface
Bl oiC^ oiD? oiU?
2. The ratio of ^ to ^ is ; of A to (7 is ; of
-A to 2> is ; oi A to U is .
3. The surface B is what part of the surface C? of D ?
of ^? The ratio of B to (7 is ; of J5 to i) is ;
of 5 to ^ is .
4. What is the ratio of C to U? If (7 represents 40 A,
what does U represent ?
5. What is the ratio of ^ to ^ ? oi Oto A? of i> to
A7 of ^ to A? of toB? oiEto (7? of i>to^? of ^
to 5?
6. The ratio of (7 to D is |, or § ; oi D to C is f, or |.
7. What is the ratio oiDtoEl of ^ to i>? oiUtoC?
8. If A represents 10 acres, what does B represent ?
C? B? B?
9. If B represents 40 acres, what does A represent ?
C? B? JEJ?
10. If the cost of the land represented by B is f 100,
what is the cost of the land represented hy A? C? B? B?
11. If the area represented by B is 640 acres, what is
the area represented hy 0? B? A? B?
12. Draw two lines such that the ratio of one to the
other is -J; |; 2; 5.
13. Draw oblongs such that the ratio of one to the other
FRACTIONS 93
117. 1. The unit of 3 is 1 ; of 3 da. is 1 da.; of 3 mi.
is 1 mi.
2. The unit 1 mi. may be regarded as composed of
equal parts, as of 2 half miles, of 4 quarter miles, of 8
eighth miles, etc. If the unit 1 mi. is regarded as com-
posed of 4 equal parts, each part is expressed as ^ mi. ; 3
such parts are expressed as | mi. A unit may be re-
garded as composed of 2 or more equal parts.
3. A fraction is one or more of the equal parts of a
unit, as |, |, etc.
4. In the fraction |, 4 is the denominator. It shows
the number of equal parts into which the unit has been
divided. It names the equal parts. 3 is the numerator.
It shows the number of the equal parts of the unit that
have been taken to make the fraction |. | denotes 3 of
the 4 equal parts of the unit 1.
5. When a unit is divided into two or more equal
parts, each of these parts becomes in turn a unit. Such
a unit is called a fractional unit. \, \, ^, etc., are frac-
tional units. The unit of ^ is \, What is the unit of
each of the following : |, |, |, |- yd., -f^ yr. ?
6. Draw a line 1 ft. long. Divide it into 4 equal parts.
Show the part that is expressed by \ ft. ; by | ft. ; by
^ ft. The ratio of 1 part of the line to the whole line is J.
What is the ratio of 2 parts of the line to the whole line ?
of 3 parts ? What is the ratio of the line to 1 part ? to 2
parts ? to 3 parts ?
7. Draw a line 8 in. long. Let it represent 1 mi.
Show the part that represents f mi.; | mi.; | mi. Show
the part whose ratio to the whole line is \^ ■^, |^,
|, |. Show the part to which the ratio of the whole
line is 2; 8; 4; j; j; f
94 FRACTIONS
118. 1. f f wk., f yd., f gal., IJ yr., |, J;^ lb., f,
I mi., -Jj^, 3%.
«. Read aloud each of the above fractions.
h. Tell into how many parts the unit in each has been
divided.
c. Name the unit in which each is expressed.
d. Tell how many of these parts are expressed in each
fraction.
e. Read the denominator of each fraction.
/. Read the numerator of each fraction.
g. Draw a line to represent the unit. Mark on this
line the parts expressed in each fraction.
2. The numerator and denominator are called the
terms of the fraction.
3. A fraction whose numerator is less than the denomi-
nator is called a proper fraction, as |, ^, etc. Name ten
proper fractions.
4. A fraction whose numerator is equal to or greater
than the denominator is called an improper fraction, as
J, |, etc. Name ten improper fractions.
5. When a number is composed of an integer and a
fraction, it is called a mixed number. Q\ is a mixed num-
ber. Its value is expressed in two different units. The
6 is expressed in units of ones; the -J is expressed in units
of one jifths. Name ten mixed numbers.
6. The value of 1, expressed in the fractional unit ^,
is f ; of 2 is ^ ; of 3 is f ; of 4 is f ; of 5 is f ; of 8 is f.
7. What kind of a number is 5|? In what unit is 5
expressed? In what unit is | expressed? The value of
6| may be expressed in the fractional unit J. There are
I in 1. In 5 there are 5 times |, or ^, ^ and | are ^.
What kind of a fraction is ^?
REDUCTION 96
REDUCTION
119. Changing Mixed Numbers to Improper Fractions.
Change 4|- to an improper fraction.
Model : 5 times 4 is 20 ; 20 and 3 are 23 ; write 23 over the de-
nominator, thus : \^.
To change a mixed number to an improper fraction^ mul-
tiply the integer hy the denominator of the fraction^ add the
7iumerator, and write the sum over the denominator of the
fraction.
120. Oral Exercises.
Change the following to improper fractions : *
a
b c
d
e
/
ff
^ * J
ifc
1.
H
n 8f
H
5|
4f
H
2| 1| 8i
8f
2.
9|
5| 7f
^
n
9*
H
2| H H
7i
3.
5f.
5i If
^
2f
8*
H
7f 4| 6f
5f
4.
2f
9f 7^
41
n
5i
6f
3f 8f 2f
9^
5.
2|
9f 7f
6t
H
7|
6f
4f 2A 7f
s*
6.
6*
2*31
^
n
3i'j
9t%
7t^ 5^ 8i«J
6*
7.
4i^
8fi 9i\
3f
7|
6t\
7i\
5A 7A 4^
5A
8. Write ten mixed numbers and change them to im-
proper fractions.
9. Express the value of the following integers in the
fractional unit ^: 3, 5, 7, 6, 9, 2, 8, 10, 12.
10. Write ten proper fractions. State what the frac-
tional unit is in each.
U. Change to improper fractions : 3| yd., 4| in., 8|- mi.
* This exercise contains practically all the combinations in addition
and multiplication. It should be used frequently as a review exercise.
96 FRACTIONS
121. Changing Improper Fractions to Whole or Mixed
Numbers.
1. What kind of a fraction is ^? What is the unit
in which its value is expressed? How many of these frac-
tional units does it take to make the unit 1 ? How many
units of 1 are there in ^? in ^? in J^? in -1^3.? in J^?
2. What does the denominator of a fraction show?
Which term of the fraction tells the number of the frac-
tional units it takes to make a unit?
To change an improper fraction to a whole or a mixed
number, divide the numerator hy the denominator,
122. Oral Exercises.
Change the following to whole or mixed numbers : *
a
b
c
d
e
/
9
A
i
y
A
1.
¥
V-
V
Y
¥
¥
¥
¥
¥
¥
¥
2.
¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
3.
¥
¥
h^
¥
¥
¥
¥
¥
¥
¥
¥
4.
V
¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
5.
Y
¥
¥
¥
¥
¥
¥
¥
¥
¥
¥
6.
¥
\l
¥
¥
¥
M
¥^
\l
^
¥i^
If
7.
M
W
¥i^
¥
¥
a
!!
\\
!f
!i
\{
8. Write ten improper fractions and change them to
whole or mixed numbers.
9. Write ten mixed numbers and change them to im-
proper fractions.
♦ This exercise contains nearly all of the facts of division and subtrac-
tion. It should be used frequently as a review exercise.
ADDITION AND SUBTRACTION 97
ADDITION AND SUBTRACTION OF FRACTIONS
123. Oral Exercises.
1. What is the sum of 2 books and 3 books and 1
book ? Are these quantities expressed in the same unit
of measure ?
2. Name three quantities that are not expressed in the
same unit of measure. Can their sum be found ?
3. Are the following fractions expressed in the same
unit of measure : fi f ? i ? Fractions that are expressed
in the same unit of measure are said to be similar
fractions. Similar fractions have the same denominator.
Only fractions that are expressed in the same unit of
measure can be added.
4. The sum of ^, |, |, and | is |, which is equal to 2^.
5. Add 2| ft., 5^ ft., and 6J ft.
Modi
!l: ^
ft.
Add the fractions first
: J ft. and J ft. and 1 ft.
5J
ft.
are % ft.,
which
are equa!
1 toli
ft. Write } ft.
6i
14^
ft.
below the column of fractions,
and carry :
Lft.to
ft.
the column of whole numbers.
6.
Add the following :
a
h
G
d
e
/
ff
h
2
6i
3
H
4f
H
H
6|
7^
5t%
8
7*
9f
9f
U
6
n
5
6if
3f
^
4
^
H
H
H
6^
8t%
H
4*
7i
2
H
4f
^
4t^
7x^
5i
n
8*
8+
6i
2i
^
3^
3t^
4|
6|
5i
1\
5J
7f
9|
8iV
m
8
2*
6|
^
»i
n
6|
7*
9^
98
FRACTIONS
124:, Oral Exercises.
1. } ft. - J ft. = f ft. I mi. - I mi. = f
-A yj*- = 1% y^' f da. -I da. = f da.
2. 6f ft. -4 ft. = ft. i8|-$5 = $-
mi.
Hyr-
mi.
2mi. -
mi.
3. Subtract
> the
fractions first
and then
the whole
numbers :
••
a
8f wk.
If wk.
h
HI
s/yd.
3| yd.
d
^ yr.
6|lb.
4flb.
/
8f wk.
5 wk.
3-A
in.
in.
h
t9|
24| yd.
17i yd.
29| yd.
19| yd.
4. Find, the sum of each of the above.
5. Subtract 3J ft. from 6 ft.
Since there is no fractional part in the minu-
Model : 6 ft.
end, the
suna of the fraction of the subtrahend
SI ft.
and the fraction of the difference is :
I ft. i ft.
2|ft.
and I ft.
are 1 ft.
Carry 1
ft. to 3 ft. and sub-
tract tht
) integers.
4 ft. and 2 ft. are (
3 ft.
6. Subtract :
8 hr. 1 da.
19
16
5 ft.
8 lb.
9 mi.
4| hr. 2J da.
$H
$2|
If ft.
311b.
6|mi.
7. Subtract:
7fyd. gjyr.
8 A.
6|mi.
5f wk.
6 yd.
8|hr.
2Jvd. 6 yr.
4J A.
3|mi.
3f wk.
3fvd.
4 hr.
8. If a boy attended school 3| da. in a certain week,
how many days was he absent ?
ADDITION AND SUBTKACTION 99
9. A girl who was taking lessons on the piano prac-
ticed as follows during one week : Monday, 1 J hr. ; Tuesday
morning, 1 hr. ; Tuesday afternoon, | hr. ; Wednesday,
Ifhr.; Thursday, I hr. ; Friday, 1^ hr. ; Saturday morn-
ing, If hr. ; Saturday afternoon, | hr. How many hours
did she practice during the week ?
10. A boy had 10 mi. to travel. If he traveled 3|
mi. on foot and rode the remainder of the distance, how
far did he ride ?
125. Oral Exercises.
1. Subtract 6| from 9f .
Model : 91 The sum of | and the fraction of the difference
p4 is 1§. Find what must be added to f to make 1 and
93 add it to I . f and ^ are 1. | and f are f. Carry 1
to 6. 7 and 2 are 9.
The nnmerator of the fraction in the difference may be found by
subtracting 4 (the numerator of the fraction in the subtrahend) from
5 (the denominator of the fraction in the minuend^, and adding 2
(the numerator of the fraction in the minuend). Explain why this
method will give the correct result. Use this method* in subtracting.
2. Subtract without the use of a pencil :
a
b
c
d
e
/
ff
h
I
6|
n
n.
H
9H
Vj
m
6if
n
21
6f
3i
H
4if
sji
m
ii!
^
6t^
H
n
m
^
8i^
9t^
6t^
m
?i
M
M
m
H
2H
14
4^^
m
3. Find the sum of each of the above exercises.
4. A dressmaker had two pieces of cloth containing
8| yd. and 6| yd., respectively. She used lOJ yd. in
making a dress. How much cloth was left?
100
FRACTIONS
REDUCTION
126. Changing to Higher and Lower Terms.
w.
«
M
»
s
^
^
1. If A represents a unit divided into 2 equal parts,
what does ^ represent ? C? i>? ^? J'?
2. The fractional unit of ^ is \\ what is the fractional
unit oi CI Bl El Fl What part of the fractional unit
of A is the fractional unit of ^? (7? i>? ^? ^? How
many of the fractional units of B does it take to make one
of the fractional units of -4. ? How many of (7? of 2>?
of El of Fl
3. The fractional unit \ is what part of the fractional
unit J ? I = |. I is what part of J ?
4. The denominator of the fractional unit ^ is 2 times
the denominator of the fractional unit \. It shows that
the unit has been divided into twice as many equal parts.
It will therefore take 2 of the fractional units sixteenths to
make one of the fractional units eighths. i%=|-
5. The fractions J^ f ^ f ^ ^ are the same in value. They
differ in form. Changing the form of a fraction without
changing its value is called reduction.
6. The fraction | is equal to the fraction ■^. Compare
their numerators. 8 is times 4. Compare their de-
nominators. 16 is times 8. How may ^ be derived
from I ? How may } be derived from ^ ?
REDUCTIOlSr-''' • " • ' ' 101
' •'• * 1 •.* ' ''*»? * V, '. A
7. Compare in a similar way tlie tet'nis of the' fractions
I and ^2 5 \ ^"^ 1^5 f ^^^ A* What effect upon the
value of a fraction has multiplying both terms by the same
number ?
8. A fraction is an indicated division. | is the same
as 6}^. The denominator of the fraction is the divisor, and
the numerator is the dividend. What effect upon the
quotient has multiplying both the dividend and the divisor
by the same number ? Is multipl3dng both the numerator
and the denominator of a fraction by the same number
the same as multiplying both the dividend and the divi-
sor by the same number ? | = | = ^^ = j^.
9. The fraction \^ is equal to the fraction |-. Compare
their numerators. 5 is what part of 10 ? Compare their
denominators. 8 is what part of 16 ? Compare in a sim-
ilar way -5^2 with |; -^^ with |. What effect upon the
value of a fraction has dividing both terms by the same
number? 3-^ = | = |=f.
10. What effect upon the quotient has dividing both
dividend and divisor by the same number ? Is dividing
both numerator and denominator of a fraction by the same
number the same as dividing both dividend and divisor by
the same number ?
Multiplying or dividing both terms of a fraction hy the
same number does not alter the value of the fraction.
11. Change the form of the following without changing
their value : f , ^, f , if, ^, If, |^, ^.
12. By what must the terms of the fraction ^ be multi-
plied to reduce the fraction to lOths ? to 15ths ? to 20ths ?
13. How many 12ths are there in 1 ? in ^ ? in | ?
in|^
102 FRACTIONS
127. Written Exercises.
1. Change | to 12ths. As the denominator 12 is 4
times the denominator of |, the numerator of the required
fraction must be 4 times the numerator of |.
Model: ^^f^- 3 is contained in 12 four times.
4 times 2 is 8. | = -j^.
Another method. 1 = H i i = i of [f , or ^^2 ; | = 2 tunes ^^^ or ^.
2. State how you would find the number that 3 must
be multiplied by to change | to 20ths.
3. Change to 12ths
4. Change to 18ths
5. Change to 24ths
6. Change to 20ths
7. Change to 36ths
8. Change to 30ths
2' 3' t' 6' 3' I' h 6' f •
2' 3' 6' 9' 6' 9' 6' 9' "S*
f' I' '2' 6' 8' 1^' t' A*
2' I' h A' I' 10' h tV-
I' h i I h {h tV h i'
iV ^u^ h h h h \h h A-
9. Write eight fractions and change them to 48ths.
10. When the terms of a fraction have been made larger
by reduction, the fraction is said to have been reduced to
higher terms.
128. Oral Exercises.
1. Express each in a different form without changing
the value: f, 8|, ^ J, 2, 2^, |, f, 7^, J^, |f, 1.
2. Find the sum of 3J, 2|, and 7.
3. Find the difference: gf^?! |?| S|
4. Show that J of 2 yd. is equal to | of 1 yd.; that
J of 8 ft. is the same as | of 1 ft.
REDUCTION 103
129. Oral Exercises.
1. Show by a diagram that f of a line is equivalent to
f of the line ; that J f t. = f ft. ; that | ft. + J ft. = | ft.,
or 11 ft.
2. Show with objects that | of 12 objects = 3^2 of
12 objects; that 1 of 12 objects = ^^2 ^^ ^^ objects; that
f of 12 objects + i of 12 objects = -^^ of 12 objects + -^^ of
12 objects, or \^ of 12 objects.
3. Why is it necessary to change | and | to 12ths be-
fore finding their sum ?
4. Show by a diagram that the difference between J ft.
and \ ft. is ^^2 ^^'
5. If I of a group of objects contains 6 objects, show
the number of objects in | of the group.
6. Show with objects that if | of a group of objects is
6 objects, the whole group contains 9 objects.
7. Show by a diagram that if | of the length of a line
is 4 ft., the entire length of the line is 10 ft.
8. If 6 objects represent | of the number of books on
a certain shelf, represent by objects ^ of the number of
books on the shelf. Represent all the books.
9. What is the least number of boys that may be
separated either into groups of 3 boys or of 4 boys ?
10. What is the least number of equal parts into which
a rectangle can be divided so that either ^ or -J of the
rectangle may be shown ?
11. What is the least number of girls that can be sepa-
rated into groups containing as many girls as are indicated
in the denominator of any one of the following : J, f , f , f ?
104 FRACTIONS
130. Common Factors.
1. The exact measures of 12 ft. are 1 ft., 2 ft., 3 ft., 4
ft., and 6 ft. Name the exact measures of 18 ft. Which
are exact measures of both 12 ft. and 18 ft.? 1 ft., 2 ft.,
3 ft., and 6 ft. are common measures of 12 ft. and 18 ft.
2. A number that is a factor of two or more numbers is
called a common factor, or a common measure, of the numbers.
3. Name the common factors of 18 and 24. Name the
largest common factor of 18 and 24.
4. The greatest common factor of two or more numbers
is the largest number that will exactly divide each of
them. This is also called the greatest common measure^ or
the greatest common divisor^ of the numbers.
5. Draw a line 12 in. long and another line 18 in. long.
What is the longest measure that can be applied without
a remainder in measuring both of these lines ? What is
the greatest common measure of 12 ft. and 18 ft.?
6. Show with objects or by a diagram all the common
measures of 8 objects and 12 objects.
131. Name the exact measures of :
1.
10 ft.
6.
30 mi.
11.
50 rd.
16.
$80
2.
16 gal.
7.
36 yd.
12.
27 pt.
17.
190
3.
20 da.
8.
40 yr.
13.
28 da.
18.
$60
4.
18 hr.
9.
481b.
14.
64 ft.
19.
175
5.
24 in.
10.
45 qt.
15.
72 mi.
20.
$84
132. Name the greatest common factor of :
1. 6 and 8 4. 12 and 18 7. 12 and 48 10. 20 and 60
2. 8 and 12 5. 24 and 30 8. 15 and 30 11. 18 and 36
3. 9 and 12 6. 24 and 36 9. 30 and 36 12. 14 and 28
REDUCTION 105
133. Oral Exercises.
1. The fractional unit J must be repeated how many
times to equal the fractional unit ^ ? The fractional unit
^^ must be repeated how many times to equal the frac-
tional unit -^^ ?
2. By what must the terms of the fraction |^| be divided
to give the fraction ^2 • i M M ^ 2 ^
3. When the terms of a fraction have been made smaller
by reduction, the fraction is said to have been reduced to
lower terms. A fraction is in its lowest terms when the
terms have no common factor.
4. The fraction \^ is not in its lowest terms, as both
terms are exactly divisible by 5. Which of the following
are in their lowest terms : |, f, |, |, 1|, if, IJ. |f ?
5. Dividing both terms of a fraction by a common fac-
tor is called canceling the common factor.
In reducing a fraction to its lowest terms, cancel in turn the largest
factors that are seen to be common to both teims. Canceling the
greatest common factor of both terms reduces the fraction to its low-
est terms .
134. Oral Exercises.
Reduce to lowest terms ;
1.
1^2' tI' iV ^2' A' it
9.
II. If. If. If. M. 11
2.
A' A' ^¥^ A' A' A
10.
A. ^. If. M. 1 2' H
3.
M^if^Maili^il
11.
If. f I. ^i. M. If. ft
4.
/o^ A' -io' A' ro' A
12.
fi.fMi.if.ii.H
5.
il ih il' ih ih u
13.
if. 1!. ih If. If. If
6.
ih /r B^6^ A' ih Jl
14.
if. Ih ih f«. If. iV\
7.
U^ M^ 1% iV A' A
15.
it 1%. If. li. II. H
8- M. H. if. if. H- It 16. if, if if H, T%, ,vk
106 FRACTIONS
135. Multiples.
1. 2 is a factor of 4, 6, 8, 10, etc. Each of these num-
bers is a multiple of 2. Name the multiples of 3 to 27.
2. A number that is exactly divisible by a given num-
ber is called a multiple of the given number. Name a mul-
tiple of 6 ; of 8 ; of 7.
3. Write the multiples of 3 to 27 and of 4 to 36.
Which of the numbers written are multiples of both 3 and
4 ? These numbers are common multiples of 3 and 4.
Which is the least multiple common to 3 and 4 ?
4. A number that is a multiple of each of two or more
numbers is called a common multiple, and the least number
that is a common multiple of each of two or more numbers
is called the least common multiple of the numbers.
5. Write all the multiples of 4 to 36 and of 6 to 54.
Name the multiples common to 4 and 6. Which of these
is the least common multiple of 4 and 6 ?
6. Name the least common multiple of 3 and 4 ; of 2,
3, and 4. Since 4 is a multiple of 2, the least common
multiple of 2, 3, and 4 is the same as the least common
multiple of 3 and 4. Tlie least common multiple of 2, 3,
4, and 6 is the same as the least common multiple of 4 and
6. Why?
7. Write four numbers such that the least common
multiple of the numbers is the same as the least common
multiple of some two of the numbers.
8. In finding the least common multiple of 2, 3, 4, and
9, which numbers need not be considered, and why ?
9. Find the least common multiple of 3, 5, and 7. Show
by several illustrations that the least common multiple of
two or more prime numbers is their product.
MULTIPLES 107
136. Name the least common multiple of :
1. 4 and 5 7. 8 and 12 13. 10 and 12
8. 6 and 9 14. 12 and 9
9. 4 and 10 15. 3 and 7
10. 5 and 10 16. 2, 3, and 6
11. 10 and 4 17. 3, 4, and 8
12. 5 and 8 18. 4, 5, and 15
2. 4 and 6
3. 3 and 4
4. 6 and 8
5. 5 and 7
6. 4 and 8
137. 1. Find
and 48.
Model : ^
48
Find the least common multiple of 8, 10, 18,
As 48 is a multiple of 8, cancel 8. As the
factor 2 is common to 10 and to 48, cancel this
factor of 10, leaving the factor 5. As 6 is a
factor common to 18 and 48, cancel this factor
of 18, leaving the factor 3. 5 x 3 x 48, or 720,
is the least common multiple of 8, 10, 18, and
48.
2. In finding the least common multiple of 3, 4, 6, 9,
and 12, which numbers may be canceled? From which
number may a factor be canceled?
3. In finding the least common multiple of 4, 12, 7, and
35, which number may be canceled because it is a factor of
12 ? Which may be canceled because it is a factor of 35 ?
138. Find by inspection the least common multiple of:
7. 10, 15, 25, 40 13. 7, 35, 45, 90, 70
8. 36, 48, 60, 72 14. 5, 14, 42, 60
9. 12, 18, 24, 36 15. 6, 7, 8, 9, 84
10. 3, 5, 30, 45 16. 20, 24, 30, 100
11. 4,18,27, 72 17. 4,9,20,54
12. 8, 12, 15, 60 18. 6, 15, 24, 86
1.
4, 5, 8, 24
2.
3, 12, 15, 30
3.
5, 8, 25, 40
4.
2, 6, 15, 45
5.
7, 21, 49, 84
6.
9, 15, 36, 60
108 FRACTIONS
ADDITION AND SUBTRACTION
139. Written Exercises.
1. Change to 12ths and add |, |, f .
Model :
l=*
2.
3.
4.
5.
6.
7.
l = A
I
6
1
f
i*
§
A
S = M
i
1
i
i
i
f
!J = 2^J=2J
i
6*
1
i
8. Add the fractions in Exs. 3-8, Sec. 127.
9. Change to 24ths and add 3f, 4 J, 6l|, 7|.
Model: 3f = 31f
41 __ 4_4 The sum of the fractions is f |, which
f\ll _ r2 2 reduces to 2|. Write f as the frac-
12 ~ ^21 tional part of the answer. Carry 2 to
*t ~ ^2ir the column of integers.
22f
10. Change to 12ths and add 4f, 3f, 4f , 5J, 6^\.
11. Change to 18ths and add 8f, 7|, 9J, 6^, 51^.
12. Change to 24ths and add 7|, 6^^, 9J, 8|, 4J.
13. Change to 36ths and add 311 7i|, sj, 51 9f .
14. Change to 48ths and add 7 J, 9f, 3^, 5||, 8^^.
15. Change to 72ds and add 8^, 9f, 7^, 18^^, 3|.
16. Reduce to lowest terms: ||, ||, -||, |-|.
17. Change to improper fractions: 7|, 9|, 8|, 7|.
18. Change to mixed numbers : -^, -j^, -^1, -^.
19. Add 4f , 6f 3f 8f 9f , 7f
20. Change 4 to 12ths ; 3 to 18ths ; 5 to 20ths.
21. Write ten fractions and reduce them to lower terms.
ADDITION 109
140. Oral Exercises.
1. What is the least common multiple of 2, 3, and 4 ?
of 3, 5, and 6 ? of 4, 5, and 6 ? of 4, 6, and 8 ? of 3, 6, and
9 ? of 5, 8, and 12 ?
2. Can you add the following fractions without first re-
ducing them : |^, |^, |-? Are they expressed in the same
fractional unit ?
3. Can you add the following fractions without first re-
ducing them : |, |, and J ? Are they expressed in the
same fractional unit ? Only fractions that are expressed
in the same fractional unit can be added.
4. What is the unit of measure in f ft. ? | ft. ? -^ ft. ?
These fractions may be expressed in the same unit of
measure, -^ ft. | ft. = f^ ft. f ft. = f^ ft.
5. Can the following be expressed in the same unit of
measure : -J ft., | da., and ^ gal.? Can the following :
1 ft., f ft., and iV ft.?
6. What is the least common multiple of the denomi-
nators of the fractions |^, f , |, and ^ ? The least common
multiple of the denominators of two or more fractions is
called their least common denominator.
141. Reduce the fractions to fractions having the least
common denominator, and add ;
a
h
c
d
e
/
9
h
i
H
n
H
3A
4iJ
3^
5!
^ii
6f
6f
H
6f
4|
n
8|
8A
m
8f
n
^
7^1
n
6f
6-11
^
n
4i^
4A
6il
8i
H
n
^
5i
H
6«
n
n
5i
7f
5A
6J
n
8f
n
110 FRACTIONS
u
2. Written Exercises.*
1.
From 5f subtract a§.
Model :
5| =
8| =
5-5^ Reduce
3^ the least
2^1^ tract.
Subtract
:
a
b
i
2.
6*
^
8|
5f
6f
7f
8fl
Reduce the fractions to fractions having
the least common denominator, and sub-
e f g h
2^ 9| 23^ 43f
■"""'
"~~"
"""■
3. 9|
6|
4|
8i
6f
6i
li
5|
2f
6|
fi
4. 5i
4f
6f
4t^
7^
2|
9*
18?
7|
19|
9|
12i
8t\
15i
8|
B. 21f
10|
80|
9
20
9!
26f
4|
43|
7
16
101
18|
81
29|
9^
6. 87^
64f
79^
90A
3TA
68J,
19t\
74|
20
13^f
20
9H
4f
8£
143. 1. Add each exercise in Sec. 142.
2. The lengths of the bhicl^boards in a certain school-
room are 12J ft., 14J ft., 8| ft., and GJ ft., respectively.
Find the combined length of the four blackboards.
3. Find the difference in the weight of two turkeys, if
one of the turkeys weighs 22J lb. and the other 17| lb.
« See Sec. 125.
ADDITION AND SUBTKACTION 111
U4. Review Sec. 134.
Reduce to lowest terms :
1- Ih If ih il Ih If 5- If. t\\' AV 1%' m
2- 100' TOO"' TOO' T0T7' Too ®- T2' "50' yS' ■g'O' QO"' 80
3 70 JLO__ _9JL JL5_ _4 5_ 7 100 JTS _2_5_ _5JL __4 0_
**• 100' 100' 100' 100' 100 '' 150' 150' 150' 200' 200
*. 1^%' ^¥t' 1^5' T^A' ilt 8. ^, ^%' Hi if^' ii
145. Review Sec. 120.
Reduce to improper fractions :
1. 14|, 30J, 161, 331 66f 4. 17f mi., 51 yd., 8f wk.
2. 16f, 37f , 111, 9_i_, 28f 5. 231 ft., 8f lb., 9f wk.
3. 38^^, 5f, 16f , 25|, 671 g. 6^^ yd., 4J mi., 811 A.
146. Review Sec. 122.
Reduce to integers or mixed numbers :
1. 101 iQjQ., loii^ i.Qii, loii 5. ^ yd., ■^<I gal., ^ wk.
2. i|ii, ian, ij(^(i, i_o^o., j^^ 6. |j^^, 1^8., iJ^, 1^, 1^
3. W. W, -W' V.^-' ¥/ 7- ¥ da., ^/ mi., ^2. gal.
4- i^' 4^ ¥' ¥' ¥' ¥' 8. ^^1 T., 1/ A., ^g<l bu.
147.
Add:
«
5
e?
c?
e
/
5'
43f
64J
l^
7Tf
m
39^
66|
691
58f
79J
24|
67J
34|
76J
73-1
74|
83i
661
58|
65i
84f
87tV.
''56J
95^
78*
49f
47^
59|
2
FRACTIONS
148. Subtract:
a h
1. 190f 265|
137| 124f
c d
398f 443^^
154f 217|
178§
25|
/
296J
180|
9
467f
337|
2. Add each of the above exercises.
149. Review Sec. 137.
Give the least common multiple of :
1.
2,3,4
8.
4,5, 8
15.
T,
8, 9
2.
3,4,5
9.
3,4, 7
16.
3,
4, 6, 8,12
3.
4, 5, 6
10.
6,7, 8
17.
5,
7, 8,12, 4
4.
2,3,5
11.
3,7, 9
18.
3,
5, 6, 8, 15
5.
3,4,7
12.
4,7, 9
19.
7,
9, 12, 14, 21
6.
4,6,8
13.
6, 8, 12
20.
8,
10, 12, 15, 20
7.
5,7,8
14.
5, 8, 12
21.
4,
6, 10, 14, 20
150. When the least common multiple of the denomi-
nators cannot be found readily by inspection, use the fol-
lowing method :
1. Find the least common denominator : ^^, ^^, y"^, |J.
Model: 2)>24 50 72 80 Find the least common multi-
pie of 24, 50, 72, 80. Cancel 24
as it is a factor of 72. Select a
prime number that is a factor of
two or more of the remaining
5 9 2 numbers. Divide the multiples
of this number by the number used as a divisor, and write the
quotients and the numbers that are not exactly divisible as shown
in the model. Continue the division until no two numbers bro\ight
down have a common factor. The product of the several divisors
and numbers remaining is the least common multiple of the
denominators.
1. C. m.=2x2x 2x 5x Find the least common multi-
R w Q vx o oc*c\f\ pl® of the same numbers by the
method explained m Sec. 137.
2)
25
36 40
2)
25
18 20
5)
25
9 10
MULTIPLICATION AND DIVISION 113
MULTIPLICATION AND DIVISION OF FRACTIONS
151. Multiplying a Fraction by an Integer.
1. In the fraction |, which term tells the number of
equal parts into which the unit has been divided ? How
many of the equal parts are expressed in the fraction ?
Write a fraction expressing twice as many equal parts.
2. Draw a diagram to show what part of a mile is ex-
pressed in I mi. Show the part that represents f mi.
Compare the part | mi. with the part | mi.
3. What is the sum of |, |, and | ? 3 times | = f •
4. Write I four times as an addend and find the sum.
State how the sum was found.
5. If I is written five times as an addend, what is the
sum ? If f is multiplied by 5, what is the product ?
6. State how a fraction may be multiplied by a whole
number. Compare the results thus obtained with the re-
sults obtained by addition.
7. Multiply. Reduce all products to their simplest
forms: | by 5; | by 3 ; f by 6 ; f by 7 ; | yd. by 4.
8. If I is multiplied by 3 by multiplying the numera-
tor by 3, the result will not be in its lowest terms. Why?
I may be multiplied by 3 by dividing the denominator 9
by 3. |x3 = f
9. Dividing the denominator of a fraction by a whole
number has what effect upon the value of the fraction ?
10. Multiply I by 12.
As the factor 4 is common to both 12 and 8, it is canceled before mul-
tiplying. Canceling 4 in 12 leaves 3 ; canceling 4 in 8 leaves 2. 3 times |
AR. — 8
114 FRACTIONS
152. Oral Exercises.
Solve each in the shortest way :
a
h
c d
e
1. 1 x5
Hx5
f i X 10 i| X 9
Hx6
2. 1 x4
i%x3
I|x7 -Hx8
i|xl2
3. f x3
Ax7
11 X 8 If X 3
Bx6
4. 1 x5
i|x4
«x6 11x7
Mx4
5. 1 x20
1 xl2
41x24 1 xl6
If x48
6. 1 x24
i^.x7
^^ X 30 f X 42
Iix3
7. 11x18
f x5
t\x8 ||x16
}fx28
8. If XIO
i\x5
ft X 75 If x 24
Hx60
9. ^x36
H X 28
Mx7 /,x8
fix 17
153. Written Exercises.
1. Multiply
■43|by 8.
Model: 43|
o
^ First, multiply | by 8. Next, multiply 43 by 8.
Add the products.
344
350
Solve. Perform the cancellation and reductions with-
out the use of a pencil :
a
h
c d
e
2. 47f x5
64f x7
82fxl4 749i-Jx5
708f x 30
3. 68fxT
74f x9
65-1 x 15 896i| x 7
580f J X 15
4. 96fx3
59f x6
94f X 12 780i| X 3
496Jix48
5. 78fx9
76^x9
70f X 18 973f X 9
573| X 25
6. 56fx9
38|-x8
27f X 24 587| x 2
609f X 42
7. Write ten mixed numbers and multiply
them by in-
tegers.
MULTIPLICATION AND DIVISION 115
154:, Multiplying an Integer by a Fraction.
1. What is the meaning of 4 ft. x 2? of 4 ft. x 1? of
4 ft. X I? Name the multiplicand and the multiplier in
each, and tell what each shows.
2. 4 ft. X I is the same as J of 4 ft. How may |^ of a
number be found? How may ^ oi sl number be found?
When you know what J of a number is, how can you find
J of the number?
3. Show with objects what is meant by | of 9 things ;
of 12 things ; of 6 things.
4. I of 24 yd. means 5 of the 6 equal parts of 24 yd.
Draw a line to represent 24 yd. Divide it into 6 equal
parts. Show the part that represents f of 24 yd.
5. Show by a diagram what is meant by | of 12 in. ;
by I of 1 mi. ; by | of 6 mi. ; by | of 10 mi.
6. How many thirds of 18 are equivalent to 18? Are
I of 18 more or less than 18? If 18 is multiplied by |,
will the answer be more or less than 18? Why?
7. Read each of the following, name the multiplicand
and the multiplier in each, and tell what each shows :
120 X f ; 16 yr. X f ; 25 mi. x f ; 18 mo. x | ; 24 lb. x f .
8. Compare | of 120 with ^ of 3 times 1 20. Compare
■| of 25 mi. with ^ of 4 times 25 mi.
9. f of 8 ft. is the same as J of ft. | of $5 is
the same as ^ of $ .
10. Show by a diagram that | of 1 yd. is the same as
i of 3 yd.
11. 5 divided by 7 may be indicated IJS, or ^. Indi-
cate ^ of 3 ; J of 2 ft. ; J of 5 mi. ; | of 5 mi.
12. The products of -^ x 18 and of 18 x | are the same.
116
FEACTIONS
155. Written Exercises.
1. Multiply 36 by J|.
3
11
Model : ^^ x ^ = .^ = 16 J.
12 is a factor common to
36 and 24. Cancel the con*-
mon factor. 3 times 11 is
33; -8^=16i.
Solve :
a
h
c
rf
e
2. 30 x|
Sxif
25 xf
144 xf
100 xf
3. 48xiJ
Txf
30x^5^
54 X J
100 xf
4. 36 X ^
8xi\
27 X il
60 Xt^
100 x|
5. 21 xf
9xi|
45 xf
16xJ|
100 xf
6. 4xf
6xf
72x|J
86 x|
100 xf
156. Written Exercises.
1. Multiply 845 by 4f .
Model : 845
f of 845 is 120f ; ^ of 845 are 3 times 120f,oP
^T 362f 4 times 845 are 3380. Add the products.
120f (I of 845)
3621
(f of 845)
3380
(4 times 845)
3742|
Multiply :
a
h
c
d
2.
60x8f
64 X 45f
827 X 47f
801 X 84f
3.
36x9f
81x47^2^
459 X 75f
153 X 46f
4.
55 X 8^^
72 X 67f
693 X 68|
360x48|
5.
27x6|
96 X 87-1
745 X 47|
578x96f
6.
33x4|
48 X 541 J
584 X 37|
609x24f
7.
45x8f
75 X 49^
144 X 35{j.
586 x 27|
MULTIPLICATION AND DIVISION 117
157. Dividing a Fraction by an Integer.
1. Divide 12 by 3. Find ^ of 12. -l of a number may
be found by dividing the number by 3. State how |^ of a
number may be found.
2. Draw a line 12 inches long. Show -^^ of the line.
Show \ of -j^ of the line. Divide -f^ of the line into 3
equal parts. How does each of these parts compare
with I of -^^ of the line? J of -^^ is the same as -^^ -f- 3.
3. What is \ of I? of I? of if? of 1^?
4. State how a fraction may be divided by an integer,
when the numerator is exactly divisible by the integer.
5. Solve: f^-^3; l|-^7; lf-^6; ||-^8; ^f-^5.
6. Draw a line 12 in. long. Divide it into 12 equal
parts. Each part is -^^ of the whole line. Divide each
part into 3 equal parts. Each of these smaller parts is what
part of the entire line ? To divide -^ of the line into 3
equal parts, each of the 8 parts must be divided into 3
equal parts and ^ of these taken. Show that J of -^ of
the line is ^ of the line. This result may be found by
multiplying the denominator of ^^2 t)y 3.
7. Divide \^ by 4 by dividing the numerator by 4 ; by
multiplying the denominator by 4, Compare the results.
158. Oral Exercises.
Divide, using the shortest method for each :
1. -rV^yS 5. if by 2 9. |T. by2
2. f^bylO 6. If by 6 10. $tby7
3. I by 6 7. if by 7 11. 1| mi. by 5
4- M by 8 8. 2§ ft. by 3 12. f da. by 3
118 FRACTIONS
159. Written Exercises.
1. Divide 65| bj 8.
TVT . ^S^ ^ ^^ contained in 65 eight times, with 1
J.M.I
^^^^- 8)65|
over ; ^ of If is ^ of
h or ij.
2.
37^^ 7
5
325|^4
c
1321 H- 5
3.
62|-j-9
423f^6
836f^4
4.
87f-f-8
7561^8
456^ ^ 2
5.
46|-^9
637|-f-7
3871-*- 3
6.
90-1- ^ 8
436^^9
7261^14
7.
Divide 645f by
24 ; 1645| by 3^
; ; 195| by 27.
8.
Divide 347f by
46; 73|byl4;
4721 by ^c>.
160, Written Exercises.
1. A carpenter sawed a board 9 ft. 4 in. (9 J ft.) long
into 4 equal parts to make shelves for a bookcase. How
long was each shelf?
2 ft. 4 in.
Or, divide 9 ft. 4 in. by 4 thus: 4)9 ft. 4 in. ^ of 8 ft. is 2 ft.;
I of 16 in. (1 ft. 4 in.) is 4 in. Ans. : 2 ft. 4 in.
2. The perimeter of a square flower bed is 14 ft. 8 in.
(14| ft.). How long is each side?
3. If a train travels at an average rate of 45 mi. per
hour, how far will it travel in 4 hr. 45 min. (4| hr.) ?
4. At 5 ^ per pound, how much will 11| lb. of sugar cost ?
5. In a magazine of 160 pages, 45 pages were devoted
to advertisements. What part of the magazine was de-
voted to advertisements?
6. Two boys caught 8 fish. The combined weight of
the fish was lOJ lb. What was their average weight?
MULTIPLICATION AND DIVISION 119
161. Multiplying a Fraction by a Fraction.
1. Show by a diagram or with objects what is meant
by f of 6 ft. ; by f of 1 ft. ; by f of i ft. | of ^ ft. is
what part of 1 ft. ?
2. Draw a line and divide it into 5 equal parts. What
is each part called ? Show ^ of one of these parts. | of
^ of the line is what part of the line ?
3. Draw a line and divide it into 4 equal parts. Show
■J of one of these parts. |^ of J of a line is what part of
the line ?
4. Draw a line and divide it into 3 equal parts.
Show ^ of one of these parts. Show f of one of these
parts. -J^ of |- of the line is what part of the entire line ?
5. How much is I of 6 da. ? J of 6 sevenths ?
6. How much is -J of ^g- ? of -^j ? of 1| ?
7. When you know what |^ of a number is, how can
you find | of the number ? When you know what ^
of a fraction is, how can you find | of the fraction ?
8. State how you would find :| of i| ; J of 1| ; J of if.
9. State how you would find J of f ; | of | ; ^ of | ;
I off.
10. Divide rectangles to show | of | ; | of | ; | of |^ ;
f of ih
11. Show by a diagram that J of |- is ■^. Since | of J
is ^2' f o^ i ^s ^^w many times -^^ ^
12. Divide rectangles to show that ^ of J is equivalent
to J off
13. Divide rectangles to show | of |^ and |^ of | ; f of f
and f off; I of 1 audi off.
120 FRACTIONS
14. Draw a line. Show ^ of the line. Show | of the
line. Show J of ^ of the line. What part of the line is
J of I of the line ? Show J of | of the line. J of f of
the line is what part of the line ? Show | of | of the
line, f of |- of the line is what part of the line?
l—J . 1 . . 1 U—i \ . . I . ■ I
J of ^ of the line = ^ of the line.
L-j—j I I I I I I I I I \ ■ ■ 1
I of the line = -^ of the line.
I of f of the line = ^^ of the line. ^ of f = ^^.
1 1 I ^ ' ■ \ » * \ t I I ' ^
V >
I of I of the line = -^ of the line, f of | = i^*
15. Draw a line. Show | of the line. Show | of the
line. Show J of ^ of the line. J of J of the line is what
part of the line ? Show J of f of the line. | of f of the
line is what part of the line ? Show | of | of the line.
I of I of the line is what part of the line ?
16. Finding |- of a number is the same as multiplying
the number by J. Finding f of a number is the same as
multiplying the number by |. Examine the illustrations
under Ex. 14, and tell how J of J of a number is found;
J of I of a number ; |- of | of a number.
17. Since J of ^ is ^, J of f is ^^ and | of f is ^.
In each case the product of the numerators is the numer-
ator of the answer, and the product of the denominators is
the denominator of the answer.
To multiply a fraction hy a fraction, multiply the numera-
tors for the numerator of the product and the denominators
for the denominator of the product. Before midtiplyitig^
cancel factors common to both terms.
MULTIPLICATION AND DIVISION 121
162. Written Exercises.
1. Multiply fl by f .
Model :
3 1 8 is a common factor of 24 and 8. 5 is a com-
?^,^_3
nion factor of 5 and 25.
Cancel these common
n^"^
factors and
multiply.
5 1
Solve. Before multiplying, cancel
common
factors :
a
b
c
d
e
2- fxf
f|xf
II X 11
h^xi
Mxl!
3- ixf
II X il
f>'l
|x|
Ifxi|
4. |X|
\\ X \
l>-f
Jxii
^Xt^j
5. fxf
if X 1
¥xf
^x^
Wx|
163. Written Exercises.
Change the mixed numbers to improper fractions and
solve :
a
6
c
d
1.
Hx^
fx2i
2|x4f
l|x3i
2.
4ix|
|x6|.
6|x4|
4Jx2J
3.
Sfxf
|x4|
7|x4|
6|x4|
4.
5^x1
T%x5i
9fx6|
8|x2J
164. Written Exercises.
Review Sees. 152, 153, and 156.
1. Multiply 45f by |.
Model : 45| ^ of 45§ is 7ii; f of 45f = 5 times 7H.. or
5 38^1,.
6
1{1 (^of45|)
5
38 J^ (iof45|)
122 FRACTIONS
Solve without reducing the mixed numbers to improper
fractions :
a b c d e
2. 34|x| 84fxf 546| x| 654f xf 840f x f
3. ISfxf 55|xf 385| xf 235| xf 468f x |
4. 72fx| 40|x| 463J xf 900,9^ x | 479| x |
5. 48fx| 38Jx| 847A x| 783f x J 673f x f
6. 96fxf 94^ X I 170^\x| 680^ x ^^ 574| x |
7. 481 x^ 63Jx| 4311 x| 598f x^ 650J x |
165. Written Exercises.
1. Multiply 349f by 3f ,
Model : 349| First multiply 349f by f . Next multiply 349|
ga by 3. Add the products.
mi
3
(I times 349f)
209t
1049
12581
(1 times 349f)
(3 times 349f)
Solve without reducing the mi^ed numbers to improper
fractions :
abed
2. 645fx6f 584^^ x4f 963J x 7J 642f x 7^
3. 867fx5f
982f x8f
333ix6§
789^ X 7f
4. 694fx7i
648| x5J
78Hx9f
537^ X 6 J
5. 748^ x5f
457f x7|
450f x 7f
521^ x2|
6. 384fx4f
926| x2f
467J X 9J
830| x4f
7. 412|x6|
726| x8J
940^ X 3f
590^ X 7J
8. 240|x4f
948| xl^
640^ X 7f
810f x3f
MULTIPLICATION AND DIVISION 123
166. Review Questions.
1. What is a proper fraction? an improper fraction?
a mixed number ?
2. Write 5 proper fractions; 5 improper fractions;
5 mixed numbers.
3. What is a fractional unit? How many fractional
units are expressed in |?
4. What is meant by a factor of a number? Illustrate.
5. What is meant by a multiple of a number ? Illustrate.
6. When is a fraction said to be in its lowest terms?
Write five fractions that are in their lowest terms.
7. How may a fraction be reduced to its lowest terms ?
Write five fractions and reduce them to their lowest terms.
8. How may an improper fraction be reduced to a
whole or a mixed number? Write five improper fractions
and reduce them to whole or mixed numbers.
9. How may a mixed number be changed to an im-
proper fraction? Write five mixed numbers and change
them to improper fractions.
10. What effect upon the value of a fraction has multi-
plying or dividing both terms of the fraction by the same
number? Illustrate.
11. What is cancellation? Illustrate.
12. State two ways in which a fraction may be multi-
plied by an integer. Illustrate. When a proper fraction
is multiplied by an integer, is the product greater or less
than the multiplicand? Why? Is the product greater or
less than the multiplier? Why? Illustrate.
13. When an integer is multiplied by a proper fraction,
is the product greater or less than tlie multiplicand?
than the multiplier? Why? Illustrate.
124 FRACTIONS
167. Dividing by a Fraction,
1. Draw a line 4 ft. long. Make a measure J ft.
long. Apply this measure to the line. How many times
must the measure J ft. be applied to measure a 4-ft. line?
2. After finding how many times the measure | ft.
must be applied to measure 1 ft., how may you find, with-
out performing the actual measurement, how many times
the measure must be applied to measure a 4-ft. line ?
3. Repeat Exs. 1 and 2 above, using a J-ft. measure to
measure a 6-ft. line.
4. As the measure ^ ft. must be applied 2 times to
measure 1 ft., to measure any given number of feet it
must be applied as many times 2 as the number of feet to
be measured. To measure 24 ft., it must be applied 24
times 2, or 48 times. How many times must the measure
J ft. be applied to measure 6 ft. ? 10 ft. ? 12 ft. ?
5. To measure a line J ft. long, the measure J ft.
must be applied J times 2, or J times. That is, one half
of the measure must be applied.
6. The expression J ft. -4- J ft. indicates that a line J
ft. in length is to be measured by a measure J ft. in
length. What is meant by each of the expressions:
6ft.^Jft.? |ft.+Jft.? Jjft.-i-Jft.?
7. Draw a line J ft. long. Determine how many
times each of the following measures must be applied to
measure it: J ft., J ft., ^ ft.
8. The measure | ft. must be applied how many times
to measure a 1-ft. line? 1J= J. It must be applied |
times. How many times must it be applied to measure a
4-ft. line? a 6-ft. line? a 15-ft. line? What is meant by
the expression 18 ft. -*- f ft. ?
MULTIPLICATION AND DIVISION 125
9. If a |-ft. measure must be applied | times to
measure a 1-ft. line, to measure a |-ft. line it must be ap-
plied I times |, or | times.
10. To measure a 1-ft. line, a 2-ft. measure must be ap-
plied I time. That is, one half of the measure must be
applied. How many times must the measure 3 ft. be ap-
plied to measure a 1-ft. line?
11. Determine how many times the measure in each
must be applied to measure 1 ft., and solve each:
5 ft.-^^ ft.; 3 ft.^J ft.; | ft. -^ 2 ft.; J ft. -i- 3 ft.;
4 ft. -^ 1 ft.
12. To measure a 1-ft. line, the measure ^ ft. must be
used 2 times. 2 is the reciprocal of J. To measure a
1-ft. line, the measure 2 ft. must applied J times. \ is the
reciprocal of 2.
13. The reciprocal of 4 is | ; of 3 is J ; of ^ is 3 ; of f
is |; of 1^ is |. If I is used as a measure to measure 1,
the quotient is |. Multiply | by |. The product is 1.
14. When the product of two numbers is 1, the num-
bers are said to be reciprocals of each other.
15. The reciprocal of the number used as divisor shows
the number of times the divisor is contained in a unit,
thus : The divisor ^ is contained in 1 three times. The
divisor J is contained in 1 f times. Hence the following
rule :
To divide hy a fraction^ miultiply the reciprocal of the di-
visor hy the dividend.
16. What is the reciprocal of each: |? ^? |^? y^^? ^?
17. Compare the terms of a fraction with the terms of
the reciprocal of the fraction. When the terms of a frac-
tion are interchanged, the fraction is said to be inverted.
56 Fl
168. Written Exercises
MCI
IONS
1. Divide 6| by f .
3
Model: 6| -f- ^ = ?I >
6 i
^5
Solve : 2
2. 3| yd. -1- f yd.
10.
6f^6f
18.
H^f
3. 5iyd. ^|yd.
11.
4| ^ 6|
19.
3J^2J
4. 6f wk. -i- f wk.
12.
5|-2,V
20.
81J^9^
5. 8| yd. H- 1 yd.
13.
7f^8J
21.
m^^^
6. |7f-i-||
14.
f^l
22.
16|^14f
7. 3y\ in. ^ 1 in.
15.
1%-^T%
23.
6|^12|
a 8|-f-4|
16.
A^l
24.
7J-^6|i
9. 7^^5f
17.
A^?
25.
24|^8
26. Divide 100 by 331 ; by 66f ; by 371; by 87 J.
27. Divide if by 4 ; if by 5 ; 316 J by 8 ; 435| by 27.
28. Multiply 635f by 8; 315f by 8; 80f by 9.
29. Add 8f, 4|, 31 6^, 8^.
30. Take 32f from 96|. From 80| take 19f .
31. Divide 8.125 by .04; 180.40 by .05; 725 by 1.25.
32. Multiply 3.1416 by 4|; .7854 by6J; 2150.42 by
60|.
169. Oral Exercises.
1. Divide each by 100 : |43, 3.14, 60.75, .9, 2000.
2. Add J and \ ; J and \ ; \ and \. State a short
method of getting the sum of two fractions whose nu-
merators are 1 and whose denominators are prime to each
other.
MULTIPLICATION AND DIVISION 127
170. 1. How many strips of carpet 1 yd. wide will it
take to cover a room 7 yd. wide ?
2. How many strips of carpet | yd. wide will it take
to cover a room 9 yd. wide ? 6 yd. wide ? 3 yd. wide ?
3. Draw a diagram of a room 24 ft. long and 18 ft.
wide. Show on the diagram the number of strips of
carpet | yd. wide that are needed to cover the floor, the
strips running lengthwise of the room,
4. What is the length in yards of each strip (Prob.
3)? How many yards of carpet are needed to cover
the room, making no allowance for matching the strips ?
5. At 75^ per yard, how much will it cost for carpet
for a room 28 ft. long and 18 ft. wide, the carpet being
27 in. wide, the strips running lengthwise of the room ?
6. How many ribbons each -| yd. long can be made
of 8 yd. of ribbon? of 12 yd.? of 18 yd.? of 6 yd.?
7. At 5/ a pound, how many pounds of sugar can
be bought for 40^?
8. How many pounds of sugar can be bought for 1 5,
at 4^^ a pound ? at 4|^ a pound ? at 5| ^ a pound ?
9. How many strips of matting 42 in. (IJ yd.) wide
will it take to cover a room 21 ft. (7 yd.) wide?
10. If a certain lamp consumes | pt. of oil each even-
ing, how long will a gallon of oil last ? 5 gal. ?
11. What part of 1 yd. is 1 ft.? 30 in. ? 32 in. ? 27 in. ?
171, 1. How many times must 2| T. be written as an
addend so the sum of the column will be 24 T. ?
2. If a boy earns <l | a day, in how many days will he
earn |15 ?
128 FRACTIONS
3. 1 yd. of cloth will cost how many times the cost of
I yd.?
4. If a dealer charges f 6 for | T. of coal, what is the
price of the coal per ton ?
5. Find the cost of 1 yd. of lace if 2J yd. cost 75 ^.
6. At $ 1 J per yard, what will be the cost of 8 J yd. of
silk?
7. Find the area of a rectangle H^ in. by 6| in.; of a
square whose side is 2| ft.
8. Hqw many pounds of meat at 12 J^ a pound can be
bought for 75^?
9. If 3| lb. of coffee are sold for $ 1, how many pounds
can be bought for $ 6 ? for 1 3 ? for $ 9 ? for $ 12 ?
10. If 1 lb. of tea costs $ |, how many pounds can be
bought for f 2f ? f or $ 6 ? for $ 12 ?
11. A tailor used 2| yd. of cloth for each pair of trou-
sers. How many pairs can be made from 22 yd. ?
12. Change to feet : 6 in. ; 9 in. ; 10 in. ; 3 in. ; 4 in.
13. Change to inches : | ft. ; | ft. ; | ft. ; J f t. ; ^ ft.
14. Change to months : J yr. ; J yr. ; § yr. ; | yr. ; f yr.
15. The atmosphere presses equally in all directions
with a pressure of about 15 lb. to the square inch. Find
the pressure on the top of your desk.
16. If the circumference of the wheel of a bicycle is 6^
ft., how many times will the wheel turn in going 1 mi.?
17. The cost of laying a concrete sidewalk at 11 J ^ per
square foot was $34.50. Find the area of the sidewalk.
If the walk was 6 ft. wide, how long was it ?
MULTIPLICATION AND DIVISION 129
172. Review Exercises.
1. Find the perimeter of a rectangle 8 ft. 6 in. wide
and 12 ft. 9 in. long. Find its area.
2. Find the number of square feet of blackboard sur-
face in the schoolroom.
3. The diameter of a cylindrical tank is 6.5 ft. Find
its circumference, (circum. = diam. x 3|.)
4. A farmer asked his two boys, George and Frank, to
figure out the number of posts necessary to build a fence
28 rd. long, the posts to be placed J rd. apart. George
said it would take 56 posts, and Frank said it would take
67. Was either boy's answer correct ?
5. The farmer (Prob. 4) asked the boys to find the
number of posts necessary to build a fence around a gar-
den 6 rd. by 8 rd., the posts to be placed J rd. apart.
Both boys said it would take 57 posts. Was the answer
correct ?
6. If a man had 60 sheep and sold | of them, how
many did he sell ? How many did he have left ?
7. There are 640 acres in 1 square mile. How many
are there in | of a square mile ?
8. If f of the distance between two cities is 15 mi.,
how far apart are the cities ?
9. Mary's age is 12 years. She is | as old as Ethel.
How old is Ethel ?
10. The cost of 15 T. of hay was $112.50. What
was the cost per ton?
11. At 75^ a yard, how many yards of cloth can be
bought for $15?
130 FRACTIONS
12. Find the value of the potato crop on 50 acres, if
the average yield is 125 sacks to the acre and the potatoes
are worth il.20 per sack.
13. Draw three dials, as in Sec. 97, and fix the hands
to read 84,500 cu. ft. ; 32,800 cu. ft. ; 47,000 cu. ft.
14. Is the height of any mountain given in your geog-
raphy text ? If so, express the height of some mountains
in miles.
15. Draw a line 20 in. long. Test it with a ruler.
16. If the height of your schoolroom is 12 ft. 6 in., find
the distance from the highest point on your desk to the
ceiling.
17. Find the area of your desk top.
18. Find the area of a blackboard in your schoolroom.
19. Draw a diagram to show the ratio of 2 in. to 4 in. ;
of 4 in. to 6 in. ; of 8 in. to 12 in.
20. What number expresses the ratio of 3 in. to 6 in.?
of 6 in. to 3 in.? of 8 in. to 4 in.? of 4 in. to 12 in.?
21. Express the value of 5| in the fractional unit J ; of
4f in the fractional unit ^ ; of 8| in the fractional unit J.
22. From a piece of cloth containing 22J yd. a tailor
used 17| yd. How many yards remained in the piece?
23. . How much heavier is Mary than Ethel, if Mary
weighs 101 J lb. and Ethel weighs 92| lb.? Find the com-
bined weight of the two girls.
24. What effect upon the value of a fraction has multi-
plying or dividing both terms by the same number?
Illustrate, using ^.
25. Show by a diagram that f of 3 ft. is the same as J
of 2 times 3 ft.
MULTIPLICATION AND DIVISION 131
173. Finding what Fraction One Number is of Another.
1. What fraction of a dollar is 33 J^?
Model: 33^^ is —r^ of a dollar. Performing the indicated divi-
sion : 33^ H- 100, or ife x ik = i.
To find what fraction one number is of another^ take the
number denoting the part for the numerator and the number
denoting the whole for the denominator. Express the result
in its simplest form.
What fraction of :
2.
8 is 5?
9.
100 is 75?
16.
100 is 121?
3.
15 is 10?
10.
100 is 60?
17.
100 is 37| ?
4.
20 is 24?
11.
100 is 125?
18.
100 is 1371?
5.
16 is 10?
12.
100 is 175?
19.
100 is 1331?
6.
36 is 30?
13.
100 is 120?
20.
100 is 66f ?
7.
100 is 25?
14.
100 is 40?
21.
100 is 166f ?
8.
100 is 150 ?
15.
100 is 87J?
22.
100 is 1121?
23. What fraction of a dollar is 75^? 50^? 25 i^? 121^?
20^? 60^? 371^? 331/? 66|/? 87|/? 62|/? 80^? 70^?
40/? 150/? 125/? 175/? 14f/ 16|/? 112|/? 140/?
6J/?
24. If a train travels at an average rate of 40 mi. per
hour, in what part of an hour will the train travel 15 mi. ?
25 mi. ? 60 mi. ? 75 mi. ? 100 mi. ?
25. If a 20-acre field produced $600 worth of wheat in
a certain year, at the same rate what part of this amount
would a 10-acre field have produced ? a 30-acre field ? a 15-
acre field? a 25-acre field? a 60-acre field?
132
FRACTIONS
DRAWING TO A SCALE
174. 1. Lines and surfaces are frequently represented
by drawings. As most lines and surfaces are too large to
be drawn full size, they have to be drawn on a reduced scale.
2. By letting | in. represent 1 ft., a line 8 ft. long
may be represented thus:
8 ft.
Scale i in. = 1 ft.
3. Using the scale J in. = 1 ft., represent a line 12 ft.
long ; 5 ft. long ; 20 ft. long ; 16 ft. long.
4. Using the scale J in. = 1 ft., represent a square
whose side is 4 ft.; 6 ft.; 12 ft.
5. Using the scale J in. = 1 yd., represent a line 12
yd. in length and a line 8 yd. in length.
6. Using a convenient scale, represent a rectangle 12
ft. by 10 ft.; a garden 16 ft. by 12 ft.
7. Using the scale ^ in. = 1 ft., represent a school
garden 24 ft. by 16 ft. Find the area of this garden.
8. Using a convenient scale, represent a sidewalk 6 ft.
wide and 48 ft. long. Find the area of this walk.
9. Below is a diagram of a school yard, drawn to
the scale ^ in. = 20 ft. Find the dimensions of the yard.
Find its area.
SCALE DRAWING 133
176. 1. Find the scale to which each of the following
lines has been drawn :
^ 60ft
/? 90 f I
C
D
£
/soft
4Sfl
/20yd
2. Find the dimensions of the floor of your schoolroom.
Using a convenient scale, draw a floor plan of your
schoolroom.
3. Find the dimensions of the school grounds. Using
a convenient scale, draw a plan of the school grounds.
Show the ground plan of the schoolhouse properly located
and in correct proportions.
4. The relative areas of the several oceans are repre-
sented by the lines below, as follows : A^ Arctic ; jB,
Antarctic ; (7, Indian ; i), Atlantic ; H^ Pacific. The num-
bers above the lines indicate the number of millions of
square miles in each.
23
3±
7/
5. Using a convenient scale, represent the relative
population of Asia, Europe, Africa, North America, South
America.
184
FRACTIONS
d
c —
e
a
b
176, 1. This figure represents a section of land. Find
the dimensions of each division ; the num-
ber of acres in each ; the cost of the sec-
tion; and of each division at $45 per acre.
(Section = 1 sq. mi.= 640 A.)
2. A field containing 10 A. is 40 rd.
long. How wide is it? Draw it to a
scale.
3. This figure represents a garden. What is the scale ?
Draw the plan of this garden on a scale twice as large as
the figure, viz. \ in. = l rd.
Divide the garden into
three rectangles. Find
the area of each rec-
tangle. Find the area of
the garden ; the perime-
ter. Find the cost of
fencing this garden at
11.75 per rod.
4. Draw to a scale the side of your schoolroom and
locate the openings. Mark the dimensions on your drawing.
5. Draw lines to represent the population of New York
City, Chicago, Philadelphia, London, and Paris, using the
same scale for each.
•g 6rd.
4-rd.
1
>.
16 rd.
erd
111. City Lots.
1. The figure on p. 135 is a diagram of a city block.
The dimensions are expressed in feet. Using your ruler,
determine the scale used in making this diagram. Find
the width of East Avenue.
SCALE DRAWING
135
_j
Grove St
I
0)
1.
too
^
o
^
-^
^
l\)
Co
Cr>
Co
Co
•li^
Oi
6
7
8
9
/O
^5
60
40
30
30
40
East Av.
~^
186 FRACTIONS
2. Find the length and the width of the block.
3. Find the area of each of the lots.
4. Lot 7 was sold at $35 per front foot. Find the
selling price of the lot.
5. At the same price per front foot, what is the value
of each of the other lots ?
6. At $45 per front foot, what is the value of Lot 6
(60 ft. front) ?
7. Lot 1 was sold for $2250. How much was this
per front foot?
8. Lot 10 was sold for $ 1600. How much was this
per front foot ?
9. The selling price of Lot 5 was $ 750. How much
was this a front foot ?
10. From its location in the block, which should be
worth the more per front foot. Lot 7 or Lot 10 ? Lot 4
or Lot 5 ?
11. Find the cost of laying a 6-ft. cement sidewalk in
front of Lot 9 at 12 ^ per square foot.
12. Find the cost of laying a 6-ft. cement sidewalk in
front of Lot 2 at 13 ^ per square foot.
13. Mr. Thomas bought Lot 8 at $35 per front foot.
He sold it for $1200. Did he gain or lose, and how
much?
14. Mr. Brown paid $ 35 per front foot for Lots 3 and
4. He sold both lots for $2500. Did he gain or lose,
and how much ?
15. Mr. Newton paid $35 per front foot for Lot 7. He
had a 6-ft. cement sidewalk laid in front of the lot, cost-
ing 12^ per square foot. He afterward sold the lot for
$1500. Did he gain or lose, and how much ?
REVIEW 137
REVIEW
178. Written Exercises.
1. Add 34.125, 4.36, 180.006, .67, 3.1416, 10.07,
2. Solve: 326.87-83.65; 9.82785 - 4.003 ; 346.85 -
184.
3. Multiply: 32.064 by .045 ; $465.73 by .08 ;
$2456 by .06.
4. Divide: 13.046 by 1.8; 143.78 by.06; $120.78 by
1.06.
5. Write five improper fractions, and change each to
a whole or a mixed number.
6. Write five mixed numbers, and change each to an
improper fraction.
7. Write five proper fractions that are not in their
lowest terms, and change each to lowest terms.
8. Add 6f , 4|, 71, 21, 14^3_.
9. From 28^\ subtract 9y\. From 834f take 186f
10. Multiply 8| by 6| ; 683| by 3| ; 31 1 by 18.
11. Divide 654| by 9 ; 195J by 7 ; 200 by 3J.
12. Divide f by I ; 6f by 71; 8| by 7|.
^ n 1 A ' vf 12x14x21 X 9
13. Cancel and simplity : -— — -- — -— — — - •
^ ^ 16 X 20 X 15 X 30
14. Reduce to lowest terms : |^, -5^, ||f , l^\, ^^*
T. A 4. 1 w 75 66^ 331 40 12J
15. Reduce to lowest terms: — , ^, ^, — , ^,
100' 100 ■
$8
FRACTIONS
179. Oral Exercises.
Solve :
1. 12 xj
10. 1 off 25
19.
f of 21 da.
2. 24 x|
11. 1 of 140
20.
■^ of 84 yd.
3. 15 X J
12. 1
of 21 ft.
21.
1 of 48 mo
4. 10 Xf
13. f
of 64 mi.
22.
Jf of 30 da.
5. 18 x|
14. 1
of 36 in.
23.
f of 120
6. 14 Xf
15. 1
of 18 in.
24.
1 of $120
7. 30 x|
16. f
of 35 yd.
25.
J of 160
8. 16 xf
17. f
of 24 lb.
26.
f of 170
9. 27 X 1
18. 1
of 45 lb.
27.
f of $100
180. Oral Exercises.
Find the quotient of :
1. 1 ^4
S.
1^6
9. 1,\
-r-4
13. 2J-i-5
2. lf-^6
6.
1^10
10. 6f
^3
14. 31-*. 8
3. {i^7
7.
1^2
11. m
-^20
15. 4j-*-4
4. if^5
8.
1^8
12. If
-4-10
16. 9f-i-4
181. Oral Exercises.
Find the product of :
•
1. ioii
'• f
off
13
■ *xi
2- ioii
8- f
of|
14.
|x|
3. foff
9- 1
off
15.
?xi
4- fofj
10. J, Of H
16.
|xf
5. -|0fA
u. 1
off
17.
*xi .
6. fofi
12. 1
off
18.
JxH
REVIEW 139
182, Oral Exercises.
1. If I of a ton of coal costs $6, what is the cost of a
ton?
2. If f of the cost of a farm is f 2400, what is the cost
of the farm ? What is | of the cost of the farm ?
3. If I of the cost of a carriage is $80, what is the
cost of the carriage ? What is ^ of the cost of the
carriage ?
4. A farmer sold -| of his crop of oats for f 160. At
the same rate, how much was the entire crop worth ? ^ of
the crop ?
5. Some men entered into partnership. One man
contributed $800, which was f of the capital invested.
How much capital was invested ? How much was con-
tributed by one of the partners who furnished J of the
capital ?
6. A man sold | of his land for f 1200. At this rate,
what was the value of all his land ?
7. A poultry dealer sold 80 turkeys and then had ^ of
his stock left. What part of his stock of turkeys did he
sell ? How many turkeys had he at first ?
8. After spending 'S 18 for an overcoat, a man had $6
left. What part of his money did he spend ? What part
of his money did he have left ?
9. After traveling 24 miles, a man still had f of
his journey to travel. Find the length of the entire
journey.
10. Mary had | as much money as Ethel. If Mary had
60 ^, how much did Ethel have ?
140 FRACTIONS
11. If George has f as many books as Walter, and
George has 12 books, how many books has Walter ?
12. 120 is f of ^ of $20 =
13. $35 is ^ of fof$35 =
14. After increasing his farm by buying | as many
acres as his farm contained, a farmer owned 120
acres. How many acres did he own before making the
purchase ?
15. Ethel weighs J more than Edna. Ethel's weight is
105 lb. What is Edna's weight ?
16. Thomas solved | more problems than Henry. He
solved 6 more problems than Henry. How many problems
did Henry solve ? How many did Thomas solve ?
17. $80 is If (f ) times what amount ?
18. $120 is 2| times what amount ?
19. $200isfofic. $60 is IJ of a;.
20. What amount less | of itself equals $100 ?
21. What amount less | of itself equals $60 ?
22. $1200 is 2| times x, ^ of some amount is $160.
What is the amount ?
23. After gaining -J of his capital, a merchant had
$14,000. Find the amount of his capital at first.
24. After buying 3 books, a girl had 8 books. The
number of books bought was what part of the number she
previously had ?
25. $80 is f of . $120 is J of . $90 is} of .
26. $60 is 1 J times $150 is 1 J times .
27. $6 -4- $.75. $9-<-$1.50. $50 + $1.25.
flEVIEW 141
183. Oral Exercises.
1. If I of the cost of a pair of skates is 60 ^, the cost of
the pair of skates is how many times 60 ^ ?
2. If J of the cost of a desk is $3, the cost of the desk
is how many times $3 ?
3. Compare | with |. Show by a diagram that | is
IJ times |, or ^ times |.
4. If I of the cost of a table is $9, the cost of the table
is how many times $9 ?
5. I of the cost of a clock is $S, In finding the cost
of the clock we may find ^ of its cost, and then J of its
cost. Show that multiplying $8 by J is the same as find-
ing first ^ of the cost, and then f of its cost.
6. If I of the value of a horse is $60, what is its value?
7. If 30 sacks of oats is | of the yield per acre, what is
the yield per acre ? Find the answer in two ways.
S. A man sold | of his crop of apples for $120. At
the same rate, what was the value of his entire crop ?
9. If I c' a ton of coal costs 86, what is the cost per
ton ? Are these two solutions identical in character ?
A. 3)16, 'B. $2
f 2, cost of { ton. ^^ >^ i = S8
__4 ^
$8, cost of 1 ton. 1
10. If I of the cost of a farm is f 6000, what is the cost
of the farm ? What is ^ the cost of the farm ?
11. After selling | of his sheep, a farmer had 60 sheep
left. How many had he at first ?
12. A boy sold 16 papers, which was J of all he had.
How many papers had he at first ?
142 FRACTIONS
13. After solving 8 problems, a girl had | of her prob-
lems yet to solve. How many problems had she to solve
at first ?
14. By selling an article for 45^, a merchant gained J
of the cost. Find the cost of the article.
15. By selling a horse for '1^90, a man lost J^ of its cost.
For what part of the cost did he sell the horse ? Find
the cost of the horse.
16. By selling a book for 60^, a boy lost | of its cost.
For what part of its cost did he sell the book ? Find the
cost of the book.
17. Two boys bought a sled in partnership, one paying
I of its cost and the other paying | of its cost. The boy
who paid | of its cost paid 70^. Find the cost of the
sled.
18. After having his salary increased by J, a boy re-
ceived $)20 a month. What was his salary before it was
raised ?
19. A dealer advertised second-hand books at f of
their ordinary price. At what price does he sell a book
that costs 60^ when new? What is the price when new
of a book which he sells for 90^?
184. Written Exercises.
Find the whole when the part is given :
1.
112 is 1
7.
160 A. is {
13.
3| mi. is J
2.
120 is 1
8.
320 rd. is |
14.
H gal. is A
3.
75 mi. is |
9.
$42.50 is f
15.
81 ft. is 5^
4.
lis}
10.
36 yd. is 1
16.
$1.20 is 1
5.
fT.isf
11.
90 ft. is f
17.
3|isA
6.
$6400 is f
12.
5280 is 1
18.
^44 is f
' f
of the
; 1
of the
t¥o
of the
REVIEW 143
185. Oral Exercises.
1. Find the whole amount when | of the amount is
$60; is $18; is $54; is $90; is $240; is $1500.
2. Find the whole amount when $ 36 is | of the
amount; | of the amount; -f^ of the amount; -^^ of the
amount; | of the amount.
3. Find the whole amount when $120 is 1^ times the
amount; IJ times the amount; 1^ times the amount; IJ
times the amount; If times the amount.
4. Find the whole amount when $240 is
amount; f of the amount; | of the amount
amount; |^ of the amount; ^ of the amount.
5. Find the whole amount when $600 is
amount; 1|^ of the amount; ^|gof the amount; -^-^^ of the
amount; -^^^ of the amount; \^ of the amount.
6. A boy walked 2 blocks, which was ^ of the distance
from his home to the schoolhouse. How many blocks
must he walk in going to and coming from school each
day, if he goes home for lunch ?
7. Charles weighs -^ less than Albert. The difference
in their weight is 11 lb. How much does each weigh?
8. Margaret wrote 7 more words than Emma, which
was ^ more words than Emma wrote. How many words
did each write ? How many did both together write ?
9. A collector charged ^ of the amount of a certain
bill for collecting it. Find the amount of the bill, if the
creditor received $24.
10. After selling 60 acres, a farmer had | as much land
left. How many acres had he before making the sale ?
11. $80 is I of . 90 mi. is f of
144 FRACTIONS
REVIEW
186. Written Exercises.
1. 47f-14| 9. 87 -66f 17. 2|x3|
2. 93|-52f 10. 47|+62f is. 41- f
3. 48fx84f 11. 19fx38| 19. 3f-2|
4. 9|-4J 12. 2fx6f 20. 7f+8J
5. 324|x| 13. 96|x7| 21. | = 2%
6. 453|-h5 14. 30Jx45| 22. 87 = f
7. 526|^f 15. 897|4-6 23. |x|xf
8. 736fx5| 16. 7301 x I 24. 3^x4|
187. Written Exercises.
1. Find the value of | of a farm of 160 A. at f 85 per
acre.
2. A man sold | of his farm for $4800. At the same
rate, what was the value of his entire farm?
3. If 3| yd. of cloth cost #2.25, what is the cost per
yard?
4. Find the cost of 8| yd. of silk at $1.14 per yard.
5. Find the cost of a roast of lamb weighing 4| lb. at
IQj^ per pound.
6. A turkey weighing 9^ lb. was bought for f 1.90.
Find the price paid per pound.
7. Express in cents and find the sum of the following :
^. $|, $i, I J, ^. li, $1 $i, H. ^h H^ ^h H'
8 Write ten improper fractions and change them to
whole or mixed numbers.
9. Write ten mixed numbers and change them to im-
proper fractions.
NUMBER RELATIONS 145
NUMBER RELATIONS
188. Oral Exercises.
Express all fractional parts in their lowest terms.
1. What part of 10 is 5 ? of 6 is 3 ? of 8 is 2 ? of 12 is
4 ? of 20 is 5 ? of 30 is 6 ?
2. What is the ratio of 5 to 15? of 6 to 12? of 8 to
24? of 9 to 81? of T to 56? of 20 to 4? of 28 to 7? of 42
to 6?
3. What part of 4 is 3 ? of 8 is 5 ? of 9 is 7 ? of 11 is
3? of 10 is 6?
4. What is the ratio of 3 to 5 ? of 5 to 3 ? of 3 to 11 ?
of 11 to 3 ? of 7 to 9 ? of 9 to 7 ?
5. What is the ratio of 6 sacks of oats to 18 sacks of
oats ? 6 T. of coal will cost what part of the cost of 18
T. ? 18 T. will cost how many times the cost of 6 T. ?
6. If 5 sacks of flour cost $7.50, how much will 10
sacks cost ?
7. If a boy earns i3 in 4 da., how much will he earn
in 16 da. ?
8. If a boy rides at the rate of 7 mi. in 2 hr., how far
at the same rate will he ride in 6 hr. ?
9. If 12 pads cost 1.60, how much will 36 pads cost?
10. If 3 T. of coal cost $24, how much will 9 T. cost ?
11. What number expresses the ratio of 4 lb. to 8 lb. ?
of 5 lb. to 20 lb. ? of 15 yd. to 5 yd.? of 20 A. to 4
A.? of $20 to $30? of $24 to $36? of 18 bu. to 24 bu.?
of 36 ft. to 21 ft.? of 48 mi. to 36 mi.? of $25 to $50?
12. What fraction expresses the ratio of 5^ to 25^? of
10^ to 50^? of 20^ to 100^? of 25^ to 100^? of 25^ to
75^? of 10^ to 40^? of 5^ to 45^? of 20)^ to 50^?
iK. — 10
146 J'ractionS
13. What fraction expresses the ratio of 3 qt. to 4 qt.?
of 5 mi. to 8 mi.? of 4 lb. to 6 lb.? of 8 bu. to 12 bu.? of
12 yd. to 9 yd. ? of 20 mi. to 15 mi.? of 2b^ to 40^? of 18
yr. to 12 yr.? of 6 mo. to 9 mo.? of 9 mo. to 12 mo.?
14. What part of $1 is 25^? 50^? 75^? 40^? 60^?
70^? 80^? 90^? 5/? lOj^? 20/? 30/?
15. What part of |1 is 12|/? 37|/? 621/? 87^/?
33-1-/? 662^? 162/? 831/? 142/9 8J/?
16. What part of 1 ft. is 2 in.? 3 in. ? 4 in. ? 5 in. ? 6 in. ?
7 in.? 8 in.? 9 in. ? 10 in.? 11 in. ?
17. What part of 1 yd. is 2 in.? 3 in.? 4 in.? 6 in.?
8 in.? 9in. ? 12 in.? 18 in.? 20 in. ? 24 in.? 30 in.? 13
in. ? 21 in. ?
18. Whatpartof lib. isJoz.?4oz.? 8oz.? 12 oz.? 7oz.?
' 19. What part of 1 yr. is 2 mo. ? 3 mo. ? 4 mo. ? 5 mo. ?
6 mo.? 7 mo.? 8 mo.? 9 mo.? 10 mo.? 11 mo.?
20. What part of 1 da. is 2 hr. ? 3 hr. ? 4 hr. ? 5 hr. ?
6 hr.? 8 hr.? 10 hr. ? 12 hr. ? 15 hr.? 16 hr. ? 18 hr.?
20 hr.?
21. What part of 1 hr. is 5 min. ? 10 min. ? 15 min. ?
20 min. ? 25 min. ? 30 min. ? 35 min. ? 40 min. ? 45 min. ?
50 min. ? ^^ min. ? 17 min. ?
22. What part of 1 mi. is 10 rd. ? 20 rd. ? 40 rd. ?
80 rd.? 60 rd.? 90 rd. ?
23. What part of 1 T. is 1000 lb. ? 500 lb. ? 250 lb. ?
200 lb. ? 100 lb. ?
24. What part of 1 section of land (1 sq. mi., or 640 A.)
is 320 A.? 160 A.? 80 A. ? 40 A.? 20 A.?
ALIQUOT PARTS 147
189. Aliquot Parts.
1. Name several amounts that are exactly contained in
136.
2. How many times is each of the following contained
in$l: 50^? 25^? 12^^? 10)^? 20^? 5/? 4^? 2^?
3. A part of a number or a quantity that will divide
it without a remainder is called an aliquot part. Name
several aliquot parts of 100.
4. What part of $1 is each: 50/? 25^? 10^? 20)^?
6/? 121/? 331^? 16|/? Hi/? 6|/? 2/? 4/? 14^/?
5. If 40 sheep can be bought for $100, how many sheep
can be bought for 120 Q of |100)? for $25? for $12.50?
for 110? for $5?
6. If 100 sacks of potatoes cost $80, how much will 25
sacks cost? 50 sacks? 10 sacks? 5 sacks? 20 sacks?
7. How much will 30 yards of cloth cost at $1 a yard?
at 25/ a yard? at 12^/ a yard? at 20/ a yard? at 16|/ a
yard? at 33J/ a yard? at 8^/ a yard?
8. From the cost of any number of articles at $1 each
how may the cost of the same number of articles at 25/
each be found? at 50/ each? at 20/ each? at 121/ each?
at 33^/ each?
190. Memorize the following fractional parts of 1 :
.50= i
.20 = i
^^^ = i
.40 =1
.25= I
.83J= i
.37^ = 1
.60 =1
•W=rV
.75= 1
.66|= 1
.05 =^V
•62J = f
•87|=J
•16f = J
•14? = f
From the above table construct a table showing the
same fractional parts of $1 ; of 100 ; of $100 ; of 1000.
1.
At 50 ^ each.
2.
At 25 ^ each.
3.
At 75 ^ each.
4.
At 121^ each.
5.
At 331^ each.
6.
At 66|^each,
148 FRACTIONS
191. Oral Exercises.
From the cost of 120 articles at f 1 each find the cost:
7. At 20 ^ each.
8. At37|^each.
9. At 621^ each.
10. At 40 ^ each.
11. At 60 ^ each.
12. At 80 / each.
192. Written Exercises.
Solve eacli by the shortest method.
1. Find the cost of 24 yd. of cloth at 37^^ per yard.
Suggestion : At $1 per yard the cloth would cost $24.
2. Find the cost of 24 yd. of cloth at 87 J i per yard.
Suggestion : 87^^ per yard is \ less than $1 per yard.
3. Find the cost of 24 yd. of cloth at 66|^ per yard.
4. Find the cost of 16 articles at 1 25 each; at $250
each (1 of $1000) ; at $125 each (^ of $1000) ; at $75 each
(J less than $100) ; at $37.50 each.
Suggestion : At $100 each the 16 articles would cost % 1600.
5. How many articles can be bought for $48 at $1
each? at 25^ each? at 331^ each? at 66|j^ each? at Vl\^
each? at 20)^ each? at 37| j^^ each?
193. Short Methods.
Solve each, using the shortest method:
1. $40x25 5. $2040x.l2J 9. 400 1b. x. 625
2. $120x25 6. 640A. x37J lo. $8.60x75
3. $80x250 7. 240 mi. X. 125 ii. $5.60x750
4. 60mi. x33i 8. 36 ft. x 125 12. $4.64x12.5
ALIQUOT PARTS 149
194. 1. Divide by 25: 12; $16; 640 A.
To divide hy 25, divide hy 100 and multiply the quotient
by 4:,
2. State a short method of dividing a number by 250;
by 50 ; by SSJ ; by 66| ; by 371 ; by 12| ; by 375 ; by 75 ;
by 125; by .25; by .125; by 12.5; by 2.5; by 62.5; by
625 ; by 500.
ivide :
3. $400 by 25
10.
2240 lb. by .25
4. $ 300 by 250
11.
2000 lb. by 2.5
5. $600 by 50
12.
5280 ft. by 37.5 ft.
6. 320 rd. by .125
13.
1728 by 250
7. 640 mi. by 12.5
14.
$400 by 87.5
8. 540 ft. by 331
15.
$ 3200 by f 625
9. 120 yr. by 66f
16.
$1500 by 12.50
195. 1. What is the cost of 24 yd. of cloth at 50 / per
yard? at 12| / per yard? at 16|)^ per yard? at 75/ per
yard ? at 87^ ^ per yard ? at 37| / per yard ?
2. How many yards of cloth can be purchased for $12
at 25 / a yard ? at 12| / a yard ? at 61 / a yard ? at 37^ /
a yard ? at 33^ / a yard ? a\; 66 J / a yard ? at $1.50 a yard?
at $1,331 a yard? at $1.20 a yard?
3. If 40 acres of land cost $ 2000, how much will 50
acres cost at the same rate ? 60 acres ? 100 acres ? 45 acres ?
55 acres (40 acres + | of 40 acres -4- J of -J of 40 acres) ?
4. George White paid Thomas Evans $ 12 for the loan
of some money for 60 da. At the same rate, how much
must he pay for the use of the same sum for 90 da. ? for
30 da. ? for 75 da. (60 da. + I of 60 da.) ? for 120 da. ?
for 70 da.? for 50 da. ? for 20 da. ? for 80 da. ?
150 FRACTIONS
REVIEW
196. 1. Draw a square and show the following parts
of it : .50, .25, .75, .121 .37^, .871, .33J, .66|.
2. Draw a square, and divide it into as many equal
parts as are necessary to show either J or J of the square.
3. What is the 1. c. m. of 2, 3, 9? of 2, 3, 4, 6?
4. Draw a square, and divide it into as many equal
parts as are necessary to show all of the following parts :
h h h h h S^C)w on the square the parts |, |, |, |.
5. Describe a cubic foot. Think of a box whose inside
measure is 1 ft. by 1 ft. by 1 ft. How many bricks do
you think the box will contain? How can you find the
exact number it will contain? If this box is watertight,
how many gallons will it contain? (1 gal. = 231 cu. in.)
6. A boy made a bookcase. The top of the lower shelf
is Q" from the floor. The space between the lower shelf
and the top of the case is 3' 6''. The case contains four
shelves. The space between the two lower shelves is 12''.
The other shelves are placed so that the distance between
them is exactly equal. How far apart are they if the
shelves are each ^" in thickness? The width of the case
is 2' 6", and the depth 1'. Draw a diagram, using the
scale 2'' = 1'.
197. 1. Divide each by 10 : 47; 138.40; f.80; 3.1416.
2. Divide by 100: 22001b. ; 5280 ft.; 1760 yd.; 1457.50.
3. State a short method for multiplying by 10 ; by
100; by 25; by 33^; by 66|; byT5; by 12|; by .25;
by .87J; by 37|. Give several illustrations of each.
4. Explain what is meant by the reciprocal of any
number. What is the reciprocal of |? of 8? of ^? of 5^?
REVIEW 161
5. Divide f ft. by ^\ ft.; 6.2 by .02; if by 6 ; if
by 3.
6. State how to multiply and how to divide a fraction
by a fraction ; a whole number by a fraction ; a fraction
by a whole number; a mixed number by a whole number
or a mixed number. Illustrate each.
7. State how to determine the place of the decimal
point in the quotient in division of decimals. Fix the
decimal point in: .002)4.8368; 2.36)13:4; 34)4.275.
8. The question, How many square inches are there
in 1 square foot? is answered by the number 144. Ask
a similar question that is answered by each of the follow-
ing: 3, 12, 9, 4, 2, 60, 24, 51 144, 7, 320, 231, 1728, 5280,
365, 640, 27, 52, 160, 128, 2150.42, 8, 32, 100, 16, 2000, 2240.
9. The question. What is ^ of 100? is answered by
the number 121. Ask a question concerning a fractional
part of 100 that is answered by each of the following : 20,
75, 25, 121, 871, 5, 621 qqi^ 871 331, 40, 10, 80, 16|.
10. The question. What is the ratio of 10^^ to |1? is
answered by the number -^q. Ask a question concerning
the ratio of some quantity to $1 that is answered by
each- 1 -^ 4 4 1 4 ^ X 4 i ^ J- -1- 4 ^ 4^ 2 -5^ 4
Ctn^u . 2^ 3? ^1 ^5 6' 1' 5' 8' ^' ^' 5' 10' 2 0' 1' 2' 4' "^^ 3' F*
198, The first number is the product of two numbers.
The second number is one factor. Find the other factor.
1.
36, 6
7.
1.5, .5
13.
816.40, 1
2.
30,1
8.
$.15, 3
14.
$20, .331
3.
15, .5
9.
1.15, 5^
15.
96 ft., .121
4.
l'2
10.
11.80, .06
16.
96 ft., 1
5.
f'i
11.
$30, .05
17.
$34.40, .08
6.
.5,10
12.
125.60, .08
18.
$10, .04
152
FRACTIONS
199. Oral Exercises.
Fill in the amounts omitted under each heading
Cost
Selling Price
Gain
Loss
Part Gained
Part Lost
1. 140
150
$10
i
2. 125
130
X
X
3. 160
$50
X
X
4. $80
X
$20
X
5. X
$100
$25
X
6. X
$150
$50
X
7. $75
X
$50
X
8. X
$75
X
\
9. $150
X
X
\
10 X
$110
X
tV
11. Express as hundredths the part gained or lost in
each of the above exercises, as \ gain= .25 gain.
12. Write and solve ten exercises similar to Exs. 1-10
above.
13. Example 5 above may be stated as a problem:
Illustration: A man sold a horse for $100 at a gain of $25.
Find the cost of the horse and what part the gain is of the cost.
14. State Exs. 1-4 and 6-10 as problems.
200. 1. If .04 of some amount is $10, what is the
amount?
2. By what must $240 be multiplied to produce $12?
3. Multiply: $600 by .06; $300 by .04; $80.50 by .07.
4. Find .06 of $360; of $4 ; of $24; of $30; of 80 mi.
5. If 4 times some amount is $16, what is the amount?
REVIEW 153
201. Oral Exercises.
1. If 12 articles cost $ 36, how may the cost of 6 articles
be found without finding the cost of 1 article ?
2. What part of the cost of 12 articles must be added
to the cost of 12 articles to give the cost of 18 articles ?
of 15 articles ? of 14 articles ? of 16 articles ? Illustrate.
3. What part of the cost of 6 articles must be added to
the cost of 6 articles to give the cost of 9 articles ? of 8
articles ? of 7 articles ? Illustrate each.
4. What part of the cost of 6 articles must be sub-
tracted from the cost of 6 articles to give the cost of 5
articles ? of 4 articles ? Illustrate each.
5. When the cost of 6 articles is known, how may the
cost of 3 articles be found ? of 2 articles ? of 1 article ? of
12 articles ? Illustrate each.
6. When the cost of 50 articles is known, how may the
►st of 12J articles be
articles ? of 75 articles ?
cost of 12J articles be found ? of 37J articles ? of 62^
202. 1. If .06 times some amount is $12, what is
the amount? If .04 of an amount is $20, what is the
amount ? If .03 of an amount is $ 24, what is the amount ?
2. 145 is .09 of what amount? 175 is .15 of what
amount ? f 1.60 is .08 of what amount?
3. $ 15 is 1.25 of what amount ? is .20 of what amount ?
4. How much is 1.75 of $80? of $200? of 640 A.?
5. $ 40 is what part of $ 80 ? J = yf ^.
6. $20 is how many hundredths of $40? of $80?
7. .6 of 600 is .12 of what number?
8. .9 of 800 is .3 of what number ?
154 FRACTIONS
203. Written Exercises.
Keep each result until all the problems have been solved.
1. A farmer rented a field 60 rd. long and 40 rd.
wide. Find the number of acres in the field.
2. The annual rent of the field was $8.75 per acre.
Find the rent of the field for 1 yr.
3. The farmer planted the field in broom corn, which
yielded J T. to the acre. Find the total yield of broom
corn. Each acre of broom corn yielded 1 T. of seed,
valued at $ 16 per ton. Find the value of the seed.
4. The farmer sold the broom corn at $80 a ton.
Find the value of the crop.
5. The farmer paid a commission merchant $4 a
ton for selling the broom corn. Find the commission.
6. The commission merchant paid $2.50 per ton
freight and f .75 per ton cartage on the broom corn.
How much should he remit to the farmer, after deduct-
ing these expenses and his commission ?
7. The expense of seed and of planting and harvest^
ing the crop amounted to $ 15 per acre. How much was
the farmer's net profit per acre from the crop ?
8. If 25 lb. of broom corn are used in making 1 doz.
brooms, how many dozen brooms can be made from the
yield of 1 A.?
9. If the manufacturer sells the brooms for $2.50 per
dozen, how much does he receive for the brooms made
from the yield of 1 A.?
10. How much did the broom corn cost per pound at
$80 per ton?
'■^'^ 11. How much is the cost of the broom corn used in
making 1 doz. brooms ?
REVIEW 155
12. If the cost of labor and of the material other than
the broom corn is f .80 for each dozen brooms, how much
do the brooms cost the manufacturer per dozen ?
13. How much is the manufacturer's profit on each
dozen brooms ? The manufacturer's profit on each dozen
brooms is what part of the cost of a dozen ?
14. A wholesale merchant bought'the brooms from the
manufacturer at f 2.50 a dozen and sold them to retail
dealers at a profit of -^^ of the cost. What was the price
of the brooms per dozen to the retail dealer?
15. The retail dealer sold the brooms to his customers
at a profit of ^ of the cost to him. Find the price paid
to the retail dealer for each broom.
16." If a retail dealer's net profit on each dozen brooms
is ^ of the gross profit, how much was his net profit on the
sale of 1 doz. brooms ?
17. Find the difference between the cost of 1 doz.
brooms to the manufacturer and the cost to the consumer.
204. Written Exercises.
1. .06 of some amount is f 30. Find .03 of the same
amount.
Suggestion : .03 is one half of .06. Take one half of $ 30.
2. .06 of an amount is $30. Find .09 of the amount.
Suggestion: .09 is one half more than .06.
Follow the above suggestions in the solution of each :
3. Find .06 of 16400. From the answer find .02 of
16400; .03of|6400; .09of 16400; .08 (1 more than. 06)
of 16400 ; .04 (i less than .06) of 16400 ; .05 (J less than
.06) of $6400; .07 of $6400,- J.2 of ^6400.
156 FRACTIONS
205. Changing Decimal Fractions to Common Fractions.
1. What is the numerator in .375? What is the
denominator?
2. How many decimal places are there in .375? Write
.375 as a common fraction. Compare the number of
decimal places in .375 with the number of O's in -f^.
3. Change .875 to a common fraction.
Model: .875 = ^V^ = f
To change a decimal fraction to a common fraction^ write
the numerator of the fraction over the denominator of the
fraction. Reduce to lowest terms.
Change to common fractions :
4. .1, .2, .3, .4, .5, .6, .7, .8, .9.
5. .10, .20, .30, .40, .50, .60, .70, .80, .90.
6. .12, .15, .25, .35, .45, .55, Sb, .75, .85, .95.
7. .125,' .375, .625, .875, .025, .075, .04, .05, .02.
8. Reduce to mixed numbers : 4.25, 26.5, 8.75, 15.375,
45.125, 7.875, 12.625, 8.20, 35.60, 2.04, 5.40.
9. In the following the cents and mills are expressed
decimally as fractions of a dollar. Write with the cents
expressed as common fractions; thus, |56.40 = 'f 6|: $5.25,
$8.20, $4.10, $3.50, $7.80, $2.75, $6.60, $4,125, $9,625,
$8,375, $4,875, $25.30, $15.05, $4.01, $7.90.
10. What decimal is equivalent to J? i? |? i? |? |?
F V f? I? 1^? A? T%?
11. Express each as an improper fraction: 1.25, 1.10,
1.20, 1.125, 2.40, 1.375, 1.625, 1.875, 3.6, 1.80.
12. Add as decimals: .25, .125, .4, .875, .2,4.75,6.07.
Change to common fractions and add.
Common to DfeciMAt 157
S06. Changing Common Fractions to Decimal Fractions.
1. Change | to a decimal fraction.
3 -7- 4 is the same as -I- A fraction is an indicated
division, in which the numerator is the dividend and the
denominator is the divisor. The division indicated by |
may be performed by placing the decimal point after 3
and dividing, thus :
.75
I = . ' . The fraction | has been reduced to a decimal fraction.
2. Perform the indicated division. Continue the
division until there is no longer a remainder. I, |, |^, ^,
h h h h h
To change a common fraction to a decimal fraction^
divide the numerator hy the denominator.
3. The prime factors of 10 are 2 and 5. Name all the
numbers to 30 which have no other prime factors than 2
or 5. Find by trial whether any fraction whose denom-
inator has any prime factors other than 2 or 5 can be
changed to an exact decimal. Which of the fractions in
Sec. 207 can be reduced to exact decimals ?
207. Written Exercises.
Change the following to decimals. Where the decimal
is inexact, continue the division to three places.
1.
f
7.
1
13.
A
19.
2|
2S.
$H
2.
f
8.
f
14.
^
20.
m
26.
|7t
3.
\
9.
1
6
15.
f
21.
Hs
27.
»8fV
4.
i
10.
\
16.
1
22.
H
28.
I7f
5.
f
11.
f
17.
i
23.
82V
29.
f5i
6.
i
12.
1
18.
A
24.
^2\
30.
i6T-L
158 FRACTIONS
208. Changing Fractions to Hundredths.
1. Change | to a fraction whose denominator is 100.
4 X
Model: r— ^j — 7" ^ i^ contained in 100 twenty times.
^ 1^^ Multiply both terms of i by 20. i = ^%.
Since 1 is equivalent to {^^, ^ is equivalent to ^q%, and ^ is equiva-
lent to j%.
2. Change to fractions whose denominators are 100 :
1' h h h h i ft,' sV- #5' h h h h h h h h T^-
3. Express as decimal fractions each of the fractions in
Ex. 2.
4. Memorize the following :
1 = ?^ = .334
8 100 ^
3 100 ^
8 100 '
? = ?li = .37i
8 100 ^
'^ = ^^ = .621
8 100 ^
8 100 *
6 100 ^
l = lil = .14?
7 100 ^
5. Express as common fractions in lowest terms: .25,
.20, .40, .50, .60, .70, .75, .80, .05, .02, .04, .10, .90, .01.
6. Write with the fractional part expressed as a deci-
mal : 7^, 4f , 6f , 8J, 9|, 4^, Z^, S^, 12^, 9f
7. Express as dollars and cents and add t8J, i3|,
$9J, f4fj, $7^, *8^, il2|. |15f, IWjV, i72i5.
1 =
.ioo_
100
1
1_
2
60
100
.50
1_
4
25
'ioo~
.25
3_
4
75
100
.75
1_
5
20
100
.20
2 =
6
. 40 _
100
.40
3_
5
60
100
.60
4_
5
80 _
100
.80
1 _
20"
5
100
.05
1 _
25
4
100
.04
1 _
50
2
100
.02
1 _
12
100
.08^
1
10
_ 10 _
"100
.10
3 _
10
- 30 _
"100
.30
7
10
70 _
"100
.70
9 _
10"
. 00 _
"100
.90
REVIEW 159
209. Written Exercises.
1. There are 2000 lb. in a ton. How many tons of
hay are there in 5400 lb. ?
2. At $8.50 per ton, how much will 6500 lb. of hay
cost?
3. At f 10.50 per ton, how much will 950 lb. of coal
cost?
4. How many hundredweight (100 lb.) are 575 lb. ?
At $5.60 per hundredweight, how much will a farmer
receive for some hogs weighing 3750 lb. ?
5. At $37.50 per ton, how much will a farmer receive
for 12,400 lb. of wheat ?
6. At $1.25 each, how many hats can be bought for
$20?
7. How much will 45.75 A. of land cost at $65.50 per
acre ?
8. The circumference of a circle is 3.1416 times its
diameter. Find the diameter of a tree, the circumference
of which is 7.75 ft.
9. Find the circumference of a cylindrical tank, the
diameter of which is 4 ft. 9 in. (4.75).
10. At $38.50 each, how much will 14 cows cost ?
11. The area of a rectangle is 42.625 sq. in. Its length
is 7.75 in. How wide is the rectangle ?
12. If a train travels at an average rate of 46.75 mi.
per hour, in how many hours will it travel 390.6 mi. ?
13. At 5J^ ($.0525) per pound, how many pounds of
sugar can be bought for $4.20?
14. When hay is worth $7.50 per ton, how many tons
can be bought for $90?
160 FRACTIONS
LUMBER MEASURE
210. 1. The unit used in measuring lumber is the
board foot, which is the equivalent of a piece of board 1 ft.
long, 1 ft. wide, and 1 in. thick.
A board 12 ft. long, 12 in. (1 ft.) wide, and 1 in. or less in thick-
ness contains 12 times 1 board foot, or 12 board feet In measuring
lumber, boards less than 1 in. thick are considered inch boards. The
name board foot is generally shortened to " foot." The Roman
jiumeral "M" is used to denote a thousand feet.
2. How many board feet are there in a piece of board
1 ft. long, 12 in. (1 ft.) wide, and 2 in. thick? 1 ft. long,
6 in. (^ ft.) wide, and 2 in. thick ?
3. What part of a board foot is there in a piece of board
1 ft. long, 6 in. Q ft.) wide, and 1 in. thick ? 1 ft. long,
8 in. (I ft.) wide, and 1 in. thick? 1 ft. long, 4 in.
(1^ ft.) wide, and 1 in. thick?
4. A piece of board 1 ft. long, 6 in. wide, and 2 in. thick
contains 2 times | board foot, or 1 board foot. Explain.
How many board feet are there in a piece of board 1 ft.
long, 6 in. wide, and 3 in. thick ?
To find the number of hoard feet in a piece of lumber^
multiply the number of board feet in one foot of the length by
the number of feet in the length of the piece,
5. Find the number of board feet in 16 pieces of 3" by
4^', each 12 ft. long.
Model- ^^^^ either 3" or 4" as
the width. Taking 4" as the
16 X ?;2 X 3 X 4 board feet = f^^.' *''t"T'f ' f ^"^
^^ yp feet m 1 ft. of the length is
192 board feet. ^ ^ i^ ^^^^^ ^^®*5 and in 1
piece 12 ft. long, 12 x 3 x ^^
board feet ; and in 16 pieoes, 16xl2x3x^ board feet, or 192
board feet.
LUMBER MEASURE 161
211. Oral Exercises.
Find the number of feet in a piece of lumber of the
following dimensions:
1. V^ X 12'', 10 ft. long 5. 4" X 4'^ 12 ft. long
2. 2'^ X 12'^ 1 ft. long 6. 4'^ X 8'', 15 ft. long
3. 2" X 6", 14 ft. long 7. 6'^ x 6'', 18 ft. long
4. 2'^ X 4'^ 16 ft. long 8. 1'^ X 16'^ 15 ft. long
212. Written Exercises.
1. Find the number of feet in 120 pieces of lumber, each
2" by ^" by 16'.
2. Measure various pieces of lumber.
3. Find the cost of lumber for a bridge 10 ft. long, if
planks 3'' x 12'' x 14' are laid over four timbers 8" x 8"
xl4'. Boards costing 118 per M; timbers 1 20 per M.
4. Find the cost of the lumber for a 5-board fence
around an orchard 160 ft. by 240 ft. The boards used
are 6 in. by 1 in. by 16 ft. and cost 1 14 per M. The
posts, 8 ft. in length, are set 8 ft. apart, and are made of
pieces 4 in. by 4 in. by 16 ft., costing f 16 per M.
213. Flooring.
1. When tongued and grooved^ a board 3 in. wide is
2^ in. wide when laid. The part of the board thus lost is
^ of the width covered by the board after it has been fitted.
Explain. If 168 ft. of flooring 3 in. wide are needed for a
certain floor, ^ as much must be added if tongued and
grooved flooring 3 in. wide is used. Why?
2. Find the number of feet of flooring needed for a
room 24 ft. wide and 30 ft. long, if tongued and grooved
flooring 3 in. wide, | in. thick, and 12 ft. long is used.
What is the cost of the flooring at $40 per IVf"
di).. XB. — 11
162 FRACTIONS
3. What part of 5|- in. is J in.? of 2J- in, is J in.?
Having found the number of feet of lumber needed to
floor a certain room with boards 6 in. wide, how may
the number of feet needed to floor the same room with
boards 6 in. wide that have been tongued and grooved
be found ?
214. Shingling.
1. The unit of shingling is a square, which is an area
of 100 square feet.
2. When shingles have been laid, about 4 inches of
their length is exposed to the weather. The average
width of a shingle is about 4 inches. Consequently the
exposed surface of one shingle is about 16 square inches,
or about J square foot. It will thus take about 900 shingles
to cover a square. Allowing for waste, 1000 shingles are
estimated for a square. A hunch of shingles contains 250
shingles. How many bunches should be allowed to each
square ?
3. Find the cost of the shingles necessary to cover both
sides of a roof, if each side is 24' by 40', at f 2.25 per
thousand shingles.
Model : 24 x 40 x 2 x .01 x 1 2.25 = x.
The number of square feet in both sides of the roof is 24 x 40 x 2,
and the number of squares is .01 times this product. The cost of the
shingles is ^2.25 multiplied by the number of squares. Why?
4. Find the cost of the shingles necessary to cover both
sides of a roof, if each side is 36' by 48', and the shingles
cost $2.50 a thousand.
5. Estimate the cost of shingles to cover the roof of
your schoolhouse, at $2 a thousand.
yr.
1907
mo.
1
da.
4
1893
5
26
DIFFERENCE BETWEEN DATES 163
215. Difference between Dates.
1. Walter Harris was born May 26, 1893. How old
was he on January 4, 1907?
Model : Write the later date as the minuend and
the earlier date as the subtrahend. It is
evident that some number of days added to
26 da. equal 1 mo. and 4 da. Subtract
thus: 26 da. and 4 da. are 1 mo. 4 da.
13 7 8 and 4 da. (in the minuend) are 8 da.
Carry 1 mo. to 5 mo., as in subtraction of integers. 6 mo. and 6 mo.
are 12 mo. ; 6 mo. and 1 mo. are 7 mo. Carry 1 yr. tp 1893. Com-
plete the subtraction.
2. Find your age by subtraction.
3. Find the time from the landing of Columbus in
America to the date when the Declaration of Independence
was signed.
4. Frank Thomas borrowed 1 750 of Charles Gray on
Oct. 8, 1902, and paid it on July 2, 1903. How long did
he have the money?
5. When the exact number of days between two dates
that are less than a year apart is required, it is necessary
to take account of the number of days in each month in-
cluded, as in the following : Find the exact number of
days from Jan. 4, 1907 to April 11, 1907. There are 27
full days left in January, 28 days in February, 31 days in
March, and 11 days in April (including April 11), or
27 da. + 28 da. + 31 da. + 11 da., or 97 da.
6. Find the exact number of days from the Fourth
of July to Christmas; from Christmas to May 1.
7. Mr. Jenkins borrowed a team of Mr. Slate on
Aug. 21 and returned it on Nov. 15. At il.50 a day,
how much did he owe for the use of the team?
164
FRACTIONS
o
1}
216. Review Exercises.
1. Make a drawing to represent a city lot 40 ft. front
and 120 ft. deep, using the scale 1 in. = 20 ft. Using the
same scale, represent at the back of the lot the space occu-
pied by a barn 20 ft. by 30 ft.
2. A man bought a tract of land 160 rd. long and SOi^
rd. wide. How many ' "^
acres did it contain ?
The tract was di-
vided as shown in the
^ , figure. Find the area
of each field. Find
the cost of fencing the
tract as shown in the
/60rd.
40ra/.
40rcf.
eOrd.
1
40rd.
40rd
1
\
aord.
4/t
figure at f 1.25 per rod.
3. A field containing 20 A. is 40 rd. wide. How long
4. Mr. James bought Lot 2 (p. 135) for 140 per front
foot. After paying for a 6-ft. cement sidewalk costing -/ ^VS
12)^ per square foot, he sold the lot at a profit of $320.
How much did he receive for it ?
5. After selling 60 acres a farmer had | of his land i / /; ^
left. Kow many acres had he before making the sale ?
6. If 80 A. of land cost 14000, how much at the same^/ .,^
price per acre will 320 A. cost? ,■
7. If hay is worth $12 a ton, how much is 500 lb. of^jL
hay worth ? How much is 400 lb. worth ? '^; • '
8. If a boat traveled 120 mi. during the first 8 hr.
after leaving port, how far at the same rate will it travel
in 1 da.? in 2 da.? ^^-^ ^^^
9. At $1 per yard, what is the cost of J yd. of silk?,
off yd.? ofiyd.?, 7
REVIEW 166
10. If a man's expenses for 3 mo. amount to 8135, at
the same rate, how much will his expenses amount to in
lyr.? ?5'^'C>
11. If a horse is fed 1 bu. of oats in 5 da., how many-
bushels will be necessary to feed it for 1 mo. (30 da.)? (>
12. If it costs $10 to pasture 6 horses for 1 month, how ; /) <
much will it cost to pasture 9 horses for the same length \
of time ?
13. At 1 7.25 per ton, how much will 6. 75 T. of coal cost ? Vy . ^^
14. How much more sugar is received for $ 1 by buying
at 16 J lb. for a dollar rather than at 6|^ per pound ? T ^
15. On the morning of March 7 a ship captain announced
that he had on board enough provisions to last 80 da.
Give the date on which the provisions would give out.
16. A boat that was due in port on Dec. 25 arrived on
Jan. 6. How many days was she overdue ? '
17. Change to decimals ; |, J, -J, |, -J, ^, |, -J, |, -^, /^
., li, H, li, 2f, If.
/^ 18. Change to common fractions : .125, .375, .25, .875,
.6, .625, .8, .40.
19. Find .05 of $200; .06 of $18.75; 1.04of|80. ^
20. $ 282 =.94 of ^-^; $375 =.75 of ; $60 =|
of — . '^^;^.ixv
21. 160 = 1^ times ^-^; $60 = 1^ times
= 1.20 of .
22. Change to lOOths : i, f , i, |, |, iV
23. If 4 of the cost of a city lot is $ 1200, how much is
the cost of the lot ? ' -^ ^
24. How many tons of hay at $12 a ton will equal in
value the cost of laying a concrete sidewalk 40 ft. long and
6 ft. wide at 12^^ per square foot? z. / '
X
PART III
7-
PERCENTAGE
217. Hundredths as Per Cents.
1. Read each : ^% .60, .10, ^, .06, .85, \^, .05.
2. ^1^, or .05, may be written b^fo- It is then read 5
per cent. Per cent means hundredths. 5 per cent means
5 of the 100 equal parts. The sign (%) is called the per
cent sign.
3. The unit 1 is equivalent to how many hundredths ?
to how many per cent ?
4. Read the following: 4%, 8%, 25%, 40%, 75%»
100%, 150%, 200%, 6J%.
5. Express as per cents : y^, y^^, ^-f^, ^, -^^ \^^.
6. Express as per cents : .01, .03, .12, .18, .50, .90,
.99, 1, 2, .125 (121%), .375, .625, .875.
7. Write as common fractions : 7%, 2%, 40%, 85%,
45%, 4%, 100%.
8. Write as decimal fractions : 1%, 5%, 7%, 30%,
3%, 75%, 80%, T%, 100%, 37J%, 33^ %, 14f %.
218. Finding some per cent of a number.
1. 4% of $500 is the same as 1500 multiplied by .04.
Find 4% of $500; of $250; of $45.50; of $875.
To find any per cent of a number^ multiply the number by
the required per cent expressed as a decimal fraction.
2. Find 5% of $860; of $60; of $100; of $840.25.
106
HUNDREDTHS AS PER CENTS 167
3. Find 12% of 1400; of 1350; ofUlOO; of $247.25;
of 11300.
4. Find 45% of 650 mi. ; 80% of 640 A. ; 62% of 400
bu.; 1% of $400.
5. Find 100% of $500. Compare 100% of $500 with
$500.
6. 125% means -ifl, or 1.25. Find 125% of 300 mi.
7. Name a per cent of $600 that is the same as $600;
that is less than $600 ; that is more than $600.
8. Is 80% of a number more or less than the number?
What per cent of a number is equivalent to one half of
the number?
9. A man owes 8% of $700. How much does he
owe?
10. A man borrowed $800 and agreed to pay 8% of the
amount borrowed for the use of it for one year. How
much did he pay for the use of $800 for a year?
11. A man borrowed $700 and agreed to pay 8% of
tlie amount borrowed for the use of the money each year.
How much did he pay for the use of $700 for 1 year?
for 2 years? for 3 years?
12. Money paid for the use of money is called interest.
13. A man borrowed $400 and agreed to pay 6% in-
terest each year. How much interest did he pay in 1
year? in ^ year ? in 1^ years ? in 2 years? in 2^ years ?
14. Find the interest on $600 for 2 years at 6%.
15. A real estate agent sold a city lot for Mr. Thomas
for $1500. He received for his services 5% of the selling
price of the lot. How much did he receive for selling the
lot?
D
168 PERCENTAGE
16. A real estate agent sold a city lot for Mr. Brown
for 12000. He received a commission of 6% of the selling
price for his services. How much did he receive for sell-
ing the lot? How much did Mr. Brown receive for the
lot, after paying the commission?
17. A farmer shipped 25 tons of hay to a commission
merchant in a city, who sold it for $8 per ton. The com-
mission merchant received for his services 2% of the
amount of the sale. Find the amount of his commission.
18. A commission merchant received a car of broom-
corn containing 8 tons, which he sold at $120 per ton.
He received a commission of 5% for selling it. Find the
amount of his commission.
19. A farmer shipped 40 tons of hay to a commission
merchant who sold it for f 10 per ton. He received a
commission of 6%. Find the amount of his commission.
How much did the farmer receive for the hay, after de-
ducting the commission?
20. A farmer had 160 acres of land. He sold 40% of
it. How many acres did he sell? What per cent of the
land did he have left? If he received 1 85 per acre for
the land sold, how much did he receive for it?
21. A farmer had 320 acres of land. He sold 60% of
it for $80 per acre and the remainder for $75 per acre.
How much did he receive for the land?
22. Mr. Evans borrowed $250 of Mr. White and paid
7% interest. At the end of the year, how much should
Mr. White receive from Mr. Evans, if he received the
interest and the money loaned ?
23. Find 50% of the number of children in your school-
room.
FRACTIONS AS PER CENTS 169
219. Fractions as Per Cents.
1. The unit 1 is equivalent to how many hundredths ?
to how many per cent ?
2. What per cent of a number is equivalent to the
number ? to J of the number ? to | of the number ? to
^^ of the number ? to 2 times the number ? to 5 times
the number ?
3. State how a common fraction may be reduced to a
decimal fraction.
4. Change f to per cent.
go __ gQ of Change the fraction to a decimal,
Model • t^q aq extending the reduction to two deci-
mal places. Express hundredths as
per cent.
Another Method: Since 1 is 100%; f is f of 100%, or 60%.
20% „
Work: W%x| = 60%.
5. Change | to a decimal fraction. .375 is the same
as .37 J, which is the same as Zl\ %.
6. State how a common fraction may be changed to
hundredths, expressed as a decimal. What is the "mean-
ing of per cent ?
To change a common fraction to per cent^ divide the numera-
tor hy the denominator^ and carry the reduction to two deci-
mal places in the quotient. Express the quotient as per cent,
7. Change to per cents : |, |, |, |-, |, ^9_ 1,
8. Change to per cents : 11, if, -j^^ |f if, |f.
9. Change to per cent : 1|.
1.80 = 180 (fo
Model: 1-| = |. 5)9.00
la Change to per cents : IJ, If, 1\, IJ, If, 1|.
ro
PERCENTAGE
11.
Memorize the following :
1 =
100%
1 = 60 %
T%= 30%
i = 12J%
1 =
50%
1=80%
T^= 70%
f = S7^%
i =
25%
2V= 5%
A= 90%
f=62J%
1 =
75%
2V= 4%
iV= 8i%
i = 87J %
i =
20%
3V= 2%
i = S3i %
i = 16|%
1 =
40%
tV = io%
| = 66|%
| = 14f%
220. Oral Exercises.
1. Findl6| % of $480.
16f % of $480 may be found by multiplying $480 by .16f, or it
may be found by taking ^ of $ 480. Solve by both methods. Which
method is the shorter ?
2. Certain per cents of quantities may be found more
easily by the use of fractional equivalents. One of these
is 33| %. Name others.
Solve, using fractional equivalents : *
3. 16| % of 24 hr. 12. 62i % of 640 A.
4. 331 % of 1 15. 13. 87|- % of 320 rd.
5. 142 % of 1 35. 14. 16f % of $ 7.20.
6. 66f % of 36 mi. 15. 14f % of 184.
7. 37| % of 48 yd. 16. 66| % of 60 bu.
8. 25 % of 320 rd. 17. 33 J % of 12.10.
9. 50 % of 11.60. 18. 75 % cf 1400.
10. 8J % of 360 da. 19. 80 % of 1 25.
11. 12| % of 80)2^. 20. 16| % of 30 ft.
* This exercise should be supplemented with oral drills until the pupils
are able to find the above per cents readily by the use of their fractional
equivalents. The fractional equivalents of the above per cents should be
used in subsequent exercises.
S.
DISCOUNT 171
221. Oral Exercises.
1. At a sale the following discounts were advertised.
(a) Find the amount of reduction and (5) the selling
price :
a. 16f % off on carpets marked 90^ per yard.
h. 33^ % off on bric-a-brac marked f 6.
c, 14f % off on ladies' hats marked $14.
d, 66| % off on damaged cloth marked 30^ per yard.
e, Ill % off on tables marked $16.
/. 371 % off on cloaks marked $16.
2. At what price should the following be marked :
a. Cloth that cost 80^ per yard, to make a profit of
2.5%?
h. Suits that cost $15, to make a profit of 33^ % ?
c. Hats that cost $2.40, to make a profit of 25% ?
d. Shoes that cost $3 per pair, to make a profit of
331%?
g. Silk that cost $1.50, to make a profit of 50 % ?
/. Overcoats that cost $16, to make a profit of 37^%?
g. Lace at 60 ^ per yard, to make a profit of 16| % ?
3. What per cent of a number remains after subtract-
ing 25% of it? 20%? 40%? 75%? 5%? 66|%? 33i%?
50 % ? 60 % ? 10 % ? 2 % ? 100 % ? 90 % ?
4. What fractional part of a quantity remains after
subtracting 50% of it? 20%? 25%? 30%? 40%? 75%?
33|-%?80%?66f%?10%? 16f%? 142%? 5^^? 121%?
100%? 90%? 37J%? 15%?
5. How much remains of $24 after deducting 50% of
it? 25%? 75%? 16|%? 331%? 66|-%? 12i%? 37|%?
6. How much remains of $36 after deducting 25% of
it? 50%? 75%? 33J%? 66f%? 16|%? 100%?
172 PERCENTAGE
7. Find 1 of 36; 50% of 80; ^ of 90; 83J% of 76;
^ of 200; 75% of 200; 20% of 15 • |- of 60; 40% of
120; ^ of 40 ; 12J% of 72 ; 66^% of 90 ; f of 64.
8. A merchant bought silk at $1.80 per yard and sold
it at a profit of 33J % . How much did he make on each
yard ?
9. A man bought hay at 1 8 per ton and sold it at a
profit of 25 % . What was his profit on each ton ? "What
was the selling price per ton ?
10. A grocer bought tea at 40^ per pound and sold it
at a profit of 50 %. What was the selling price?
11. A suit of clothes marked $ 20 was sold at a reduc-
tion of 20 %. Find the amount of the discount and the
selling price of the suit.
12. A wagon that cost $72 was sold at a profit of
16| %. What was the selling price of the wagon ?
""^^^ 13- A merchant advertised a reduction of 25 % on all
goods. Find the reduction on suits marked $ 30 ; on
shoes marked $4 ; on hats marked $2; on cloth marked
80 ^ per yard ; on rugs marked $ 6.
14. A house owned by Mr. West was rented to Mr.
James by a real estate firm for one year at $ 30 per month.
If the firm received as commission 10 % of the first
month's rent, what was the amount of the commission ?
15. A hardware merchant invested $ 5000 in his busi-
ness. He cleared 15 % on the investment in one year.
What was the amount cleared during the year ?
16. In the catalogue of a carriage manufacturer a cer-
tain carriage was listed at $ 150. It was bought by a re-
tail dealer at a discount of 20 % from the list price. How
much did the carriage cost the retail dealer ?
PERCENTAGE 178
17. A farmer shipped 50 boxes of apples to a commis-
sion merchant, who sold them at 90 i per box. The com-
mission merchant charged a commission of 5 % for his
services. Find the amount of his commission. He paid
freight charges amounting to $3.50. How much should
he remit to the farmer after deducting for commission
and freight ?
18. Mr. A bought a cow for 140 and sold it at a
profit of 20 % . What was the selling price of the cow ?
19. Mr. A sold a cow for |- of the cost. He received
% 48 for the cow. Find the cost of the cow.
20. Mr. A sold a cow at a profit of \ of the cost. His
profit was % 8. Find the cost.
21. A real estate dealer bought a lot for % 600. After
five years he sold it at a profit of 100 % of the cost. What
per cent of the cost of the lot did he receive for it ?
(22 J A merchant's stock of goods valued at % 4500 was
damaged by fire. He was obliged to dispose of the goods
for ^^\ % of their former value. What fractional part of
their value did he receive for them ? How much did he
receive for his stock ?
23. A dealer was asked the price of a certain carriage.
He replied that he would sell the carriage for % 200 and
allow the purchaser 60 days in which to make the pay-
ment, or that he would allow a discount of 2 % for cash
payment. Find the cash price of the carriage.
24. Mr. James pays \\% taxes on $4000. Find the
amount of his tax.
25. A man bought a lot for f 1600. He sold it for
% 1800. How much did he gain on the lot ? His gain
was what part of the cost \
174 PERCENTAGE
■f"
26. Frank Thomas borrowed $ 1200 for 1 yr. at 6%
interest. How much did he pay for the use of the
money ?
222. Finding the number of which a given number is a
certain per cent.
1. If 4 % of a sum of money is $12, what is 1 % of it?
If 1 % of a sum of money is $3, what is 100 % of it?
2. When 5% of a selling price is f 80, what is 1 % of
the selling price ? What is the selling price ?
3. If 8 % of a number is 160, what is 1 % of the num-
ber? What is 2 % of the number? What is the number?
4. When 6 % of a number is given, how may 1 % of it
be found? How, then, may the number be found? Any
number is equivalent to how many per cent of itself?
5. To find a number when a certain per cent of it is
given, find what 1 % of it is, then find 100 % of it. By
this method find the number of which 24 is 8 % .
6. When the multiplier and the product are given,
how may the multiplicand be found?
7. Some number when multiplied by 4 is 216, what is
the number? Some number when multiplied by .08 is
24, what is the number?
8. To say that 7 % of a number is 161, is the same as
to say that some number when multiplied by .07 gives
161 as a product. The number may be found by divid-
ing 161 by .07.
To find the number of which a given number is a certain
per cent, divide the given number by the given per cerU
expressed as a decimal.
PERCENTAGE 175
In each of the following, name the multiplier and the
product, and state how the multiplicand may be found :
9. 15 % of a number is 60. 12. 8 % of some land is 25.6 A.
10. 9 % of a number is 135. 13. 6 % of some money is $ 108.
11. 18 % of a number is 81. 14. 45 % of a crop is 135 bu.
223. Written Exercises.
1. $11.49 is 12% of what amount?
Model • S345 75 Some amount when multiplied by
'. — .12 gives $41.49 as a product. The
.l^^-lt>4i.4y amount is found by dividing the
product (141.49) by the multiplier (.12).
2. 240 mi. is 12 % of how many miles ?
3. 128 tons is 8 % of how many tons?
4. A man paid $32 interest for the use of some money
for one year at 8 % . What was the sum borrowed ?
5. A farmer received 40% of a crop as rent for his
land. His share of the wheat amounted to 400 bu. in
one year. What amount of wheat was raised on the farm
in that year?
6. A man received |80 interest on some money which
he loaned for a year at 10%. Find the amount of the
loan.
7. On a certain day 4 of the pupils in a school were
absent. This was 8% of the number enrolled. How
many pupils were enrolled in the school?
8. During one season a baseball team lost 14 games,
which was 40% of the number of games played. How
many games did the team play?
O
X
176 PERCENTAGE
9. Two men entered into partnership in a retail hard-
ware store. One agreed to furnish 40% of the capital
and the other 60 % of the capital. The partner who con-
tributed 40% of the capital invested $2400. Find the
whole amount of the capital.
10. A farmer sold 120 acres of land, which was 30%
of his entire farm. How many acres had he before mak-
ing the sale? What per cent of his farm did he still own?
224. Oral Exercises.
1. A number is how many times 20% of itself? If
20 % of a number is 8, what is the number ?
2. A number is how many times 33 J % of itself? If
33 J % of a number is 25, what is the number?
3. What part of a number is each of the following
per cents of the number : 25 %, 37^ %, 50 %, 62| %, 66|%,
75%, 871%, 16|%, 14f %, 8J%, 12|%, 20%, 33J%?
4. A number is how many times each of the following
per cents of itself: 14f%, 75%, 25%, 16|%, 50%, 81%,
20%, 371%, 12|%, 331%, 62J%, 871%, 66|%?
5. If 50 % of the amount of money a boy has is f 12,
how much money has he? How much money has he if
25% of his money is |5? if 10% of his money is f 3 ?
if 33^% of his money is $8? if 16^ % of his money is
1 10 ? if 66| % of his money is $ 24 ?
6. $ 5 is 25 % of —. $8 is 33^ % of — . 4 mi. is 20 %
of — . 6 gal. is 50 % of — . 12 yd. is 75% of — . 160
rd. is 10 % of — .
7. Find the number of which 16 is 25 % ; 30 is 20 % ;
18 is 66^ % ; 40 is 200 % ; 60 is 300 % ; 15 is 37J % ;
25 is 50 % ; 50 is 62^ % ; 70 is 33J % ; 75 is 100 %.
11
r
PERCENTAGE 177
8. When 14| % of a number is given, how may the
number be found? 14 1 % of a farm is 25 acres. How
many acres are there in the farm ?
9. A man sold 45 head of cattle, which was 25% of
the number he had. How many head of cattle had he?
10. A merchant sold goods at a discount of 16| % from
the cost price and lost $60. What was the cost?
11. In 37 1 % of a farm there are 90 acres. How many
acres are there in the farm?
12. A merchant made 12| % on the cost of some goods
by selling them at a profit of $6. Find the cost of the
goods. Find tlie selling price of the goods.
13. A number is how many times ^ of itself? |- of
itself? I of itself? f of itself?
14. If 66|% of a number is 120, what is the number?
15. If 87 J % of a number is 70, what is the number?
16. In a spelling test a boy spelled correctly 30 words,
which was 75 % of the number of words in the test. Find
the number of words in the test.
17. Mr. A sold a cow at a profit of 25%. His profit
amounted to f 10. Find the cost.
18. Mr. A sold a cow at a loss of 25 % of the cost.
For what part of the cost did he sell the cow? He re-
ceived $ 30 for the cow. Find the cost.
19. A fruit grower planted 120 apple trees. 20 of
them died. What per cent of the trees died ?
20. If 12 trees are 25 % of the number planted by a
fruit grower, how many trees did he plant ?
21. Eight pupils were absent from school on a certain
day, which was 20 % of the pupils enrolled. How many
pupils were enrolled in the school ?
AK. — 12
D
178 PERCENTAGE
225. Written Exercises.
1. A f iirin was sold for f 6000, which was 25 % more
than it cost. Find the cost of the farm.
The fractional equivalents of per cents should be used whenever
the work can be made easier or shorter by their use.
Model A : f of the cost of the farm = $ 6000.
$1200
I of the cost of the farm = i of ^jim, or $ 4800.
Since the farm was sold for f (125%) of its cost, the cost of the
farm is | of the seUing price.
Since $6000 is 125% of the cost of
f 48 00. the farm, the cost of the farm may be
Model B : 1.25)^6000.00 found by dividing 1 6000 by 1.25. (See
Sec. 223.)
2. A city lot was sold for $1200, which was 20 % more
than it cost. Find the cost of the lot.
3. After increasing his stock 33J%, a merchant found
that he had $12,000 invested. Find the amount of his
investment before the increase.
4. A sum of money was borrowed for a year at 8 %
interest. At the end of the year the money borrowed and
the interest amounted to $432. What per cent was this
of the sum borrowed ? Find the sum borrowed.
5. If the population of a certain city in 1905 was 81,250,
and this was an increase of 25 % over the population in
1895, what was the population in 1895?
6. A dealer sold a carriage for $96, at a loss of 20 %.
What per cent of the cost of the carriage did he receive
for it ? How much did the carriage cost him ?
7. A firm sold a carriage to a retail dealer for $119,
which was at a discount of 15% from the list price of
the carriage. Find the list price of the carriage.
-^•^
PERCENTAGE 179
226. Finding what per cent one number is of another.
1. Each of the following fractions is equivalent to what
per cent ; J, -J, |, |, |, f , -|, -J, |, J, f J, f , |, |, ^j, -^q, -^,
r5^ A' 2^' A ^
2. Each of the following is equivalent to what per cent :
ij, li, i{, H, If, If, H, li, H, i^v If. If' If If i| ?
3. 8 is what part of 16 ? What is the ratio of 8 to 16 ?
1^ of a number is what per cent of the number ? 8 is what
per cent of 16 ?
4. 24 is what part of 36 ? 24 is what per cent of 36 ?
5. 12 is what per cent of 36 ? of 24 ? of 48 ? of 60 ?
6. Express as a common fraction the ratio of 6 to 8 ;
of 20 to 25 ; of 25 to 20 ; of 30 to 35 ; of 40 to 60.
7. Express as hundredths in decimal form the ratio of
3 to 5 ; of 5 to 8 ; of 4 to 5 ; of 24 to 30.
8. 16 is what per cent of 20 ?
^6 = |. Reduce f to a decimal and
.80 = 80 % carry the reduction to two decimal
Model A : 5)4.00 places in the quotient. .80 is the same
as 80%.
Some per cent of 20 is 16. 20 is
the multiplicand and 16 is the prod-
^- ^" '° uct. The multiplier may be found
Model B : 20)16.00 ^^ ^.^.^^^ ^^^ p^^^^^^ ^^g^ ^^ ^^^
multiplicand (20). The multiplier
is .80, which is the same as 80 %.
To find what per cent one number is of another, express as
a common fraction the ratio of the one to the other, and reduce
the fraction to a decimal, carrying the reduction to two
decimal places in the quotient. Express the result as per
cent.
3.
50 is 20.
4.
25 is 50.
5.
18 is 15.
6.
48 is 60.
7.
54 is 27.
8.
240 mi. is 180 mi.
9.
360 bu. is 600 bu.
180 PERCENTAGE
227. Written Exercises.
1. What per cent of 120 mi. is 90 mi. ?
2. 145 is what per cent of $50 ?
Find what per cent of :
10. 320 rd. is 80 rd.
11. 640 A. is 120 A.
12. 360 da. is 30 da.
13. 5280 ft. is 1760 ft.
14. 5000 ft. is 1000 ft.
15. 2000 mi. is 6000 mi.
16. 2000 lb. is 750 lb.
17. A man owned 320 A. of land. He sold 80 A.
What per cent of his land did he sell ? What per cent of
it did he have left?
18. A coal dealer bought 240 tons of coal. He sold
160 tons. What per cent of it did he sell? What per
cent of it did he have left ?
228. Oral Exercises.
1. 1^ times a number is what per cent of the number ?
If 1^ times a number is 36, what is the number? If
133 J 9^ of a number is 48, what is the number ?
2. What per cent of a number is 1 J times the number ?
If 150 % of a number is 12, what is the number ?
3. If If times a number is 20, what is the number ? If
166^% of a number is 60, what is the number?
4. If 6 % of a certain amount is $30, what is 1 % of the
amount ?
GAIN AND LOSS 181
229. Per Cent of Gain or Loss.
1. Mr. A bought a cow for $40 and sold it at a gain of
$S, $8, the gain, is what per cent of f 40, the cost ?
2. Mr. Clark bought a cow for $40 and sold it for $48.
Find the gain. The gain is what per cent of the cost ?
3. Mr. Brown bought a horse for $120 and sold it for
$100. Find the amount of his loss. His loss is what per
cent of the cost of the horse ?
4. When the cost price and the selling price are given,
how is the amount of the gain or loss found ?
5. The per cent which the amount of gain or loss is of
the cost is called the gain or loss per cent. The gain or
loss per cent is always some per cent of the cost.
To find the gain or loss per cent, find what per cent the
amount of gain or loss is of the cost.
6. A furniture dealer bought some rocking-chairs for
$4 each and sold them for $6 each. How much did he
make on each chair ? What was his gain per cent ?
7. A bicycle that cost $40 was sold for $30. What
was the loss per cent ?
8. A fruit dealer bought berries at 6^ per box and
sold them at 10 ^ per box. What was his gain per cent ?
9. A man bought a cow for $30 and sold it for $40.
What was his gain per cent ?
10. A newsboy bought papers for Sfi each and sold them
for 5 ^ each. What was his gain per cent ?
11. A newsboy bought papers for 1^ each and sold
them for 2 ^ each. What was his gain per cent ?
182
PERCENTAGE
230. Oral Exercises.
Find the gain or loss per cent
Cost
Selling Price
Cost
Gain
Loss
1.
$10
$15
6. $16
$4
2.
$15
$10
7. $12
•4
3.
$25
$30
8. $15
$3
4.
$30
$25
9. $20
$2
5.
$40
$45
10. $25
«5
231. Written Exercises.
Find the value of x in each
Cost
Selling Price
Gain
Loss
Gain%
Loss%
1. $80
$100
X
X
2. $75
X
$25
X
3. X
$120
$30
X
4. $50
X
$5
X
5. X
$60
$20
X
6. X
$4.80
$1.20
X
7. $20
X
X
8%
8. $36
X
X
hfo
9. X
$80
X
20%
10. X
■ $24
X
20%
11. Ex. 1 above may be stated in the form of a prob-
lem, thus : A man bought a horse for $ 80 and sold it for
$100. Find the gain or loss per cent. State problems
for Exs. 1-10 above.
12. A certain baseball team won 6 games out of 10.
What per cent of the games did the team win ?
GAIN AND LOSS 188
232. Written Exercises.
1. A real estate agent bought a city lot for 11200 and
sold it for §1500. What was the gain per cent ?
2. A merchant disposed of a stock of goods valued at
18000 for $6000. What was the loss per cent?
3. An agent received $40 for selling hay at a com-
mission of 5%. Find the selling price of the hay.
4. The interest on a sum of money for one year at
6 % was $ 72. On what amount was interest paid ?
5. A farmer lost 45 % of his wheat crop by fire. His
loss amounted to 600 bushels. What was the amount of
his entire crop?
6. After suffering a loss of 35% of the value of his
stock of goods, a merchant found that the remainder of
his stock was worth $13,000. What was the value of his
stock before the loss ?
7. A stock of goods valued at $4500 was partly de-
stroyed by fire. After the fire the stock was estimated
to be worth $3000. What was the per cent of loss ?
8. Mr. Thomas bought a farm for $5250. He rented
the farm for $420 a year. His rent amounted to what
per cent of his investment ?
9. Mr. Bunker bought a lot for $1500 and built a
house on it costing $3000. He rented his property for
$300 a year. His rent amounted to what per cent of his
investment ?
10. A business block in a city was advertised for sale
for $75,000. This block rented for $500 per month.
The income from the rent amounted to what per cent of
the price asked for the property ?
>
184 PERCENTAGE
REVIEW
233. Oral Exercises.
1. By selling land at $25 per acre more than it cost
him, a farmer gained 20 % of the cost of the land. Find
the cost of the land.
The gain, or $ 25 per acre, amounts to 20 % of the cost, or | of the
cost. Since $ 25 per acre is ^ of the cost, the cost is 5 times ^25 per
acre, or $ 125 per acre.
2. By selling a carriage for $15 more than it cost
him, a dealer gained 12J % of the cost of the carriage.
Find the cost of the carriage.
3. A city lot increased $200 in value, which amounted
to an increase of 33 J % of its cost. Find the cost of the
lot.
4. A gain of 66^% of the cost amounted to a gain
of $120. Find the cost.
5. A horse was sold for $150, which was 120 % (|) of
the cost. Find the cost of the horse.
6. By selling an overcoat for $35, a merchant made a
profit of 16| % of the cost. What fraction expresses the
ratio of the selling price to the cost ? Find the cost.
7. A boy sold a pony for $6 more than it cost him.
His profit amounted to 16| % of the cost of the pony.
Find the cost and the selling price.
8. After selling 80% of his land, a farmer had what
per cent of it left ? After selling 80 % of his land, a
farmer had left 40 acres. How many acres had he before
making the sale ?
9. By selling a cow for $32, a farmer lost 20% of the
cost price. What fraction expresses the ratio of the sell-
ing price to the cost ? Find the cost of the cow.
Y-
REYIEW 186
10. A liveryman made 40 % on the cost of a horse by
selling the horse for $140. What fraction expresses the
ratio of the selling price to the cost ? Find the cost.
11. By selling a lot for 1640, a dealer lost 20% of the
cost price. The selling price was what fraction of the
cost of the lot ? Find the cost of the lot.
12. A field of wheat was damaged by floods to the ex-
tent of 25 % of the expected yield. The yield amounted
to 30 bushels of oats to the acre. This was what
fractional part of the expected yield? What was the
expected yield ?
13. A watch that cost $25 was sold for 200 % of the
cost. Find the selling price of the watch.
14. A painting that cost f 60 was sold for 33^ % less
than it cost. It was sold for what fractional part of its
cost ? Find the selling price. V"
15. A merchant made a profit of 25 % of the cost of
silk by selling it for | . 80 per yard. Find the cost of the
silk per yard.
16. A sum of money loaned at 7 % yields f 42 interest
each year. Find the sum loaned.
17. $ 20 is what part of % 100 ? A carriage that cost
% 100 was sold for $120. It was sold for what per cent
of the cost price ?
18. A stove that costs $40 is sold for 1 36. The loss
is what part of the cost of the stove ? The loss is what
per cent of the cost of the stove ? The selling price is
what per cent of the cost price ?
19. A farm that costs % 60 per acre is sold for $ 70
per acre. The gain on each acre is what part of the cost
per acre ? The gain is what per cent of the cost ?
"b
18« PERCENTAGE
234. Written Exercises.
1. Hay that cost $40 for 5 tons was sold at $ 9 a ton.
What was the profit on each ton? the gain per cent?
2. A clothing merchant advertised a reduction of 20 %
on all goods. Find the amount of reduction and the sale
price of suits marked $35, hats marked §2, suspenders
marked 50^, shoes marked $3.50, neckties marked 25^,
overcoats marked $20, cuffs marked 20^ per pair, collars
marked 2 for 25^.
3. A jeweler sold a watch for $26, which was at a
profit of 33^%. Find the cost of the watch.
4. Goods damaged by fire were sold for $2400, which
was at a loss of 40 % . What was their original value ?
5. What per cent of his earnings does a man save who
earns $ 80 a month and saves $300 each year ?
6. A farmer paid $4000 for a farm and sold it for
$ 4200. Find the gain per cent.
7. A man's yearly income from a farm valued at $ 6000
is $ 1500. The income is what per cent of the value of
the farm?
8. By selling a carriage for 12 J % more than it cost
him, a dealer made a profit of $15. How much did the
carriage cost him ?
9. Two men entered into partnership to purchase a
boat that cost $300. Each contributed one half of the
capital. One of the men sold his share of the boat for
$120. Did he gain or lose, and what per cent?
10. A house that was valued at $ 2400 was rented so
that the yearly rent amounted to 12 % of the value of the
property. What was the monthly rent of the house ?
^
REVIEW 187
11. By selling a cow for $15 more than it cost him, a
farmer gained 33^ % of the cost of the cow. Find the cost
of the cow. Find the selling price.
12. Tea that was sold at 60 f^ per pound was sold at a
profit of 33J%. Find the cost of the tea.
13. A piano dealer sold two pianos for $240 each. On
one he made a profit of 20 % and on the other he lost 20 %,
How much did each of the pianos cost him ? Did he gain
or lose on the two pianos ?
14. How should goods that cost f 1.20 per yard be
marked to sell at a profit of 20% ? 25% ? 33^% ? 50% ?
15. Three men bought some land for $ 3600. One fur-
nished $1500, another $1200, and the third 1 900. They
sold the land for $4200. What per cent of the capital
did each furnish? What per cent of the profit should
each receive ? What was each man's share of the profit ?
16. A man had $800 in a bank. He drew out first
$200 and then $300. What per cent of his money did he
draw out ? What per cent was left in the bank ?
17. The salary of a clerk was increased from $60 per
month tp $75 per month. What per cent of increase was
made in his salary ? The increase would amount to how
many dollars in two years ?
18. The population of Los Angeles was 50,300 in 1890
and 102,479 in 1900. The population in 1900 was what
per cent of the population in 1890 ? What was the in-
crease in population from 1890 to 1900 ? What was the
per cent of increase in the ten years ?
19. A newsboy sold 25 papers at 5^ each, which had
cost him h^ each. What was the amount of his profit?
What was his gain per cen<^ ^
188 PERCENTAGE
235. Oral Exercises.
1. Find 25%, 50%, and 75% of each of the following:
$100; ISO; |120; 40 A.; 36 in.; 2000 lb.; 16 oz.; 12
mo.; 24 hr.; 360 da.; 144 sq. in.
2. What is 33i % and 6G| % of each of the following:
$120? $1200? 360 da.? 36 in. ? 180 mi.? 27 ft. ? $1500?
12 mo. ? 60 min. ? 24 yd. ?
3. Find 121%, 371%, 62| %, and 871% of each of the
following: 24 hr.; $240; $1; 20 mi.; 2000 1b.; 144 sq.
in.; 360 da.; $1200; 640 A.; 72 yd. ; 16 oz.; 216 cu. in.;
320 rd.; $4000; $.48.
4. Express each in per cent : ■^, J, |, |, f , J, f , ^^, |,
h h h h h h h h h h ^^ H^ H' If' H^ 21, H, If, 2, 8,
10.
5. Express each as a common fraction in lowest terms:
80%, 50%, 33J%, 25%, 125%, 20%, 120 %, 70%, 40 %,
150%, 66|%, 14f %, 75%, 133J-%, 175%, 180%, 12J%,
140%, 160%, 112J%, 16|%, 90%, 37^%, 87J%, 62J%,
1371%, 187J%, 110%, 130%.
6. Find 125%, 150%, 175%, 112J%, 1371%, 162J%,
and 1871% of each of the following: 24 hr.; 320 rd.;
640 A.; 360 da,; 16 oz.; 20001b.; $1200; $4000; $80.
7. Find 133J%, 120%, 166f %, 140%, and 160% of
each of the following : $150; 30 da.; 360 da.; $6000;
120 rd.; $250; 60 ff; $1.80.
8. Find the number of which 30 is 33^%; 60 is 25 %;
20 is 40%; 36 is 66^%; 35 is 125%; 120 is 120%; 48 is
37J%; 90 is 150%; 180isl2l%; 180isll2}%; 42 is
175%; 50is200%; 24isl60%; 200is40%; 80ia66J%;
15 is 166f %; 24 is 4%; 30 is 5%; 18 is 6% ; 45 is 9%.
38 is 110%; 12 is 2%; 130 is 200%.
PERCENTAGE 189
236. Written Exercises.
1. A sum of money borrowed, together with the in-
terest on it for one year at 7%, amounted to $909.50.
This was what per cent of the money borrowed? Find
the sum borrowed.
2. A boy spelled correctly 45 words in a test of 50
words. What per cent should he receive as his standing
in the test ?
3. A girl missed 4 problems in an arithmetic test con-
taining 10 problems. What per cent of the problems did
she miss ? What per cent did she have correct ?
4. 5 % of a certain amount is $20. Find the amount.
5. Find the amount when 8 % of the amount is $240 ;
$80.
6. A farmer had 24 cows and sold 16 of them. What
per cent of the cows did he sell ? What per cent did he
have left ?
7. A house and lot was advertised for sale for $8000.
This property was rented for $32.50 per month. The
rent amounted to what per cent of the price asked for the
property ?
8. If a man spent 60% of his savings in building a
barn and had $400 left, how much had he saved ?
9. A liveryman made 40 % on the cost of a horse
by selling it at a profit of $36. Find the cost of the horse.
10. An article that cost a retail merchant $ 14 was sold
to a customer at a profit of 14| %. How much did the
customer pay for the article ?
11. The total enrollment in a certain school was 180
pupils. On a certain day 150 pupils were present. The
number present was what per cent of the enrollment ?
190 PERCENTAGE
237. Oral Exercises.
1. By selling a horse for 20 % more than it cost him a
liveryman gained 130. How much did the horse cost him?
For how much did he sell it ?
2. What is 2| o/o of $400 ? 31 % of $60 ? 5 % of $1400?
6% of $250? 10% of 2000 lb.? 5^% of $200? 7% of
$150? 8% of $2500?
3. What is the difference between 1 % and .1 % ? be-
tween ^ of a number and | % of a number ?
4. What is \ % of $200 ? .2 % of $400 ? ^ % of $8000 ?
5. 8 is what part of 24 ? 8 is what per cent of 24 ?
20 is what per cent of 25 ? $20 is what per cent of $30 ?
$40 is what per cent of $30 ?
6. A boy missed 1 word in a spelling lesson of 20
words. At the same rate, how many would he have
missed in a lesson of 100 words ?
7. After having his salary raised $10 a month, a
clerk's yearly salary amounted to $1620. What was his
monthly salary before receiving the increase ?
8. A carriage that cost $120 was sold for $80. The
sale price was what per cent of the cost ?
9. A man's monthly salary was raised from $60 to $75.
What per cent was his salary increased ?
10. 25^ is what per cent of 30^? The cost of 3 bars of
soap when bought at 3 bars for 25^ is what per cent of
the cost when bought at 10^ a bar ?
11. What per cent of profit is made when articles are
bought at 40^ a dozen and sold at 5^ apiece?
12. What per cent of profit is made when articles are
bought at 10^ a dozen and sold at 2 for 5^?
PEROENTAQE 191
238. Oral Exercises.
1. If I of the value of a piece of property is $1500,
what is the value of the property ?
2. If I of a man's yearly salary is f 1200, what is his
yearly salary ?
3. A clerk saved $40 a month, which was | of his
monthly salary. What was his monthly salary ?
4. What part of his income does a man save who saves
$60 a month from an income of $1200 a year ?
5. Frank has a certain sum of money and James has
I as much. They both together have 60^. How much
money has each?
The moDey of both together is how many thirds of Frank's money ?
6. Two boys took a piece of work to do for $6.
One boy worked twice as many hours as the other boy.
How much should each receive ?
7. A man gave Henry $3 as many times as he gave
Walter $4. He gave $14 to the two boys. How much
did each receive ?
8. Separate $45 into two amounts in the ratio of 5
to 4 ; $36 into three parts in the ratio of 2, 3, and 4.
9. In a school of 120 pupils there were 5 girls to
every 3 boys. Find the number of boys and girls.
10. Rob, Fred, and Ada together received $2.40 from
their father. For every 15^ that Rob received Fred re-
ceived 10^, and Ada 5^. How much did each receive ?
11. A newsboy wished to make an estimate of his
yearly earnings, so he kept account of his earnings for 3
weeks and found that he earned $6 the first week, $4 the
second week, and $5 the third week. At the same rate,
how much would he earn in a year ?
192 PERCENTAGE
2f39. Commission.*
A person who transacts business for another frequently
receives as his pay a certain rate per cent of the amount
involved in the transaction. This is known as his com-
mission. One who buys or sells for another on commis-
sion is called a commission merchant, a broker, or an agent.
240. Written Exercises.
1. Find 2% of 12400.
2. A commission merchant sold $2400 worth of hay
for a farmer and charged 2 % for his services. Find the
amount of his commission. How much should he remit
to the farmer, after deducting his commission and $300
for freight charges and $150 for cartage ?
3. An agent received $16 as his commission for sell-
ing a bill of goods at a commission of 5%. Find the
amount of his sales.
4. A farmer shipped 40 sacks of potatoes to a commis-
sion merchant, who sold them at 95^ a sack. After de-
ducting his commission of 5%, how much should he remit
to the farmer ?
5. A merchant's profits averaged 15%. His total
sales for January, 1906, amounted to $13,800. Find the
cost of the goods sold. Find the profits for the month.
6. A farmer shipped 18 tons of hay to a commission
merchant, who sold it at $9.50 per ton. How much did
the merchant remit to the farmer, after deducting his com-
mission of 5 % and freight and cartage charges amounting
to $1.75 per ton?
*For a more extended treatment of Commission, see Appendix,
pp. 262-264.
PERCENTAGE 193
7. Find the net proceeds of the sale of 860 lb. of
butter at 18^ per pound, commission 6%.
8. A real estate agent received a commission of 5%
for selling a city lot. Find the sale price of the lot, if
the agent's commission amounted to $62.50.
9. If the salary of a traveling salesman is $20 a week
and a commission of l^ % on the amount of his sales, how
much does he earn in a week in which his sales amount to
i 2254. 75?
10. A carriage dealer offered to sell a certain carriage
for $250 on two months' time, or to allow a discount of
2 % for cash. Find the cash price of the carriage.
11. If a collector retains 10 % of the amount of a cer-
tain bill for collecting it, what per cent of the amount of
the bill does the creditor receive ? A collector remitted
to a creditor $126 as the net proceeds of a collection, after
retaining his commission of 10%. Find the amount of
the bill collected.
12. After deducting his commission of 4%, an agent
remitted $79.20 to a shipper. Find the amount of the
sales.
13. The amount received by a shipper, after a commis-
sion of 5 % has been deducted, is what per cent of the
amount of the sales? A shipper received $60.80 as the
net returns of a sale of some potatoes, after paying a com-
mission of 5 % . Find the amount of the sale.
14. A dairyman shipped 1250 lb. of butter to a com-
mission merchant, who sold it at 22^ per pound. If the
cost of shipping was $2.40 and the cartage amounted to
$1.75, how much did the shipment net the dairyman, after
paying a commission of 4 % ?
AB. — IS
194 PERCENTAGE
241. Oral Exercises.
1. If a boy sells 1 newspaper for what 2 papers cost
him, what per cent of profit does he make?
2. If a baker sells 2 pies for what 3 pies cost him,
what per cent of profit does he make?
3. A merchant sold 5 yd. of cloth for what 6 yd. cost
him. What per cent of profit did he make?
4. What per cent of profit does a grocer make who
buys canned tomatoes at the rate of 3 cans for 25^ and
sells them at the rate of 2 cans for 25/?
5. A dealer marked his goods so that he would make
30 % profit on them. In order to dispose of his goods, he
was obliged to sell them at a discount of 10 % from the
marked price. What per cent of profit did he make?
6. A collector was allowed a commission of 20% on
a bill of $80. What amount did the creditor receive ?
7. A dealer marked his goods at 20 % above cost. In
order to close out his stock, he was obliged to sell the
goods at a discount of 25%. Did he gain or lose, and
what per cent?
8. There are 4 boys and 8 girls in a class in arithmetic.
What per cent of the pupils in the class are girls?
9. The enrollment of pupils in a class was 25 in 1905
and 30 in 1906. What was the per cent of increase ?
10. On a certain day a boy missed 3 words out of 15 in
spelling. What was the per cent of words correctly
spelled?
11. An agent received a commission of 5 % for selling
a lot for $1500. Find the amount of his commission.
12. An agent's commission of 5% for selling a city lot
amounted to $60. For what amount did he sell the lot?
INSURANCE 195
242. Insurance.* i. Owners of buildings, merchandise,
etc., generally protect themselves against loss by fire by
having such property insured. Insurance of property
against loss by fire is called fire insurance, against loss by
sea marine insurance. What is life insurance? accident
insurance? Name other forms of insurance.
2. The written agreement between the insurance com-
pany and the person protected is called a policy. Examine
a fire insurance policy. The amount paid for insurance is
called the premium. The rates of insurance are expressed
as a rate per cent on the face of the policy, or as a speci-
fied sum for each |100, or for each 11000, of the face of
the policy.
243. Written Exercises.
1. Mr. Wilson insured his store for $6000. The in-
surance cost him 1J%. Find the premium.
2. Mrs. Hardy insured her house, valued at $8000, for
I of its value. Find the amount of the face of the policy.
The insurance cost her $1.40 on each $100 and extended
for three years. How much did the insurance cost her?
3. If 90 % of a sum is $28.80, what is the sum ?
4. For what price was a city lot sold if the agent's
commission of 5% amounted to $87.50? How much did
the owner receive ?
Find the premium on the following amounts of in-
surance at the rates given :
5. $4000 at 1\%. 8. $5600 at $1.20 per $100.
6. $2400 at If %. 9. $1400 at $1.35 per $100.
7. $12,000 at 11%. 10. $4250 at $1.80 per $100.
* For a more extended treatment of Insurance, see Appendix,
pp. 278-283.
196 PERCENTAGE
11. Mr. Rogers built a house that cost him $4500. It
cost him 11800 additional to furnish it. To protect him-
self against the complete loss of his property by fire, he
insured his house for $3000 and his household goods for
$1200. The insurance for three years cost him !{% of
the face of the policy.
a. Find the cost of the insurance.
h. If the house and contents were destroyed by fire,
how much insurance would he receive ?
c. What would be the amount of his loss, including
the amount paid for insurance ?
d. If the house were damaged to the extent of $400,
how much would he receive?
12. Two men own a store in partnership. One has
$16,000 invested in it, and the other has $10,000. What
part of the store does each own? If the store were sold
for $39,000, what part of this amount would each re-
ceive? How much would each receive? If the store
were damaged by fire to the extent of $13,000, how
much would each lose?
13. A hotel valued at $80,000 was insured for $50,000
in one company and for $25,000 in a second company.
How much would each company be liable for (a) if the
hotel were totally destroyed ; (h) if it were damaged to
the extent of $12,000? of $30,000?
14. What was the amount of commission received by an
architect who charged a commission of 5 % for drawing
the plans and supervising the construction of a house that
cost $4500, exclusive of the architect's fees?
15. Write five insurance problems based on conditions
in your community.
PERCENTAGE 197
244. Oral Exercises.
Express the part and the per cent the first quantity is of
the second :
1. 130, 140. 6. 12.50, $3. 11. 11200, 11500.
2. $40, 150. 7. 80 A., 160 A. 12. 45 T., 60 T.
3. 20 mi., 25 mi. 8. 10 yd., 16 yd. 13. 80 A., 320 A.
4. 40 ft., 60 ft. 9. $4, 124. 14. 2000 ft., 2200 ft.
5. 1100,1120, 10. 60 lb., 100 lb. 15. |12, |200.
16. Express the ratio of the second quantity to the first
in each of the above in the form of a fraction in lowest
terms and in per cent.
245. Oral Exercises.
1. A collector's commission of. 5% amounted to $30.
Find the amount of the bill collected.
2. After deducting his commission of 20%, a collector
remitted f 24 to the creditor. Find the amount of the
bill collected.
3. Mr. Wright has $4500 out on interest at 6 %. His
annual taxes on the money are $20. What is his net
annual income from the $4500?
4. The yield from a certain field was 30 bu. of oats to
the acre in 1904 and 40 bu. to the acre in 1906. What
was the per cent of increase in the yield in 1906 over the
yield in 1904?
5. The enrollment in a certain school in 1906 was 36
pupils, which was an increase of 20 % over 1905. What
was the number of pupils enrolled in 1905 ?
6. 40% of the pupils in a certain school are boys.
There are 24 girls in the school. How many pupils are
there in the school?
198 PERCENTAGE
246. Taxes. * l. What are some of the expenses of a city
government? of a state government? of the national gov-
ernment? The money necessary for the maintenance of
state and local governments is derived mainly from taxes
levied upon persons, property, and business.
All movable property, such as household goods, money, cattle,
ships, etc., is called personal property. Immovable property, such as
lands, buildings, mines, etc., is called real estate, or real property.
Both forms of property are subject to taxation.
2. For the purpose of taxation, the value of all taxable
property is estimated by a public officer called an assessor.
Property is not generally assessed at its full value.
3. The rate of taxation is expressed as a per cent on the
assessed valuation, or as a specified sum on each $1, or on
each ^100, of assessed valuation. Thus, a tax rate of 1^ %
may be stated as a tax of 1|^ (on each |1), or of $1.50
(on each $100).
4. The national government is supported mainly by rev-
enues derived from taxes levied upon goods imported from
other countries, called duties, or customs, and from internal
revenues, which consist chiefly of taxes levied upon the
manufacture of liquors and tobacco products.
Some imports are admitted without duty. These are said to be on
the free list. Nearly all imports are subject either to an ad valorem
or a specific duty^ or both.
5. An ad valorem duty is a tax of a certain rate per cent
on the cost of the goods.
6. A specific duty is a tax of a specified amount per pound,
yard, etc., without reference to the cost of the goods.
7. Customhouses have been established at all ports wliere
vessels are authorized to land cargoes. The revenues are
collected by federal officers stationed at ports of entry.
* For a more extended discussion of Taxes, see Appendix, pp. 269-273.
TAXES 199
247. Written Exercises.
1. A man had 16000. He invested 11500 in a city
Ipt. What per cent of his money did he invest ?
2. A certain city had an assessed valuation of
#8,000,000. The amount needed to defray the expenses
of the city for a year was estimated at 1100,000. The
amount needed for expenses was what per cent of the
assessed valuation?
3. The assessed valuation of a certain city is $ 12,000,000
and the amount to be raised by taxation is 1180,000. What
rate of taxation is necessary in order to raise this amount?
4. What is the amount of an agent's commission for
selling real estate for $150,000 at a commission of 1^ % ?
5. What is the amount of a man's taxes on property
assessed at 16000 if the tax rate is 11.20 on each 1 100?
6. A real estate agent received #84 for selling a piece
of property at a commission of 2%. Find the selling
price of the property.
7. The assessed valuation of a certain farm is -$3600.
This is 40 % less than the actual value of the farm. Find
the value of the farm.
8. What per cent on his investment did a boy make
who bought a pony for $40 and sold him for $50?
9. The assessed valuation of the property in a county
'm $42,000,000, and $672,000 is to be raised by taxation.
Express the rate of taxation in three ways.
10. Find the rate of taxation on:
a. Valuation, $450,000; taxes, $6000.
h. Valuation, $275,000; taxes, $2475.
c. Valuation, $360,000; taxes, $6300.
200 PERCENTAGE
11. What rate of commission was charged by a col-
lector who charged $15 for collecting a debt of $225 ?
12. The premium on an insurance of $4500 is $60.
What is the rate of premium ?
13. The premium received for insuring a store at IJ %
was $105. What was the amount of insurance?
14. At the rate of 1| %, how much is the tax on prop-
erty assessed at $4500 ?
15. When the valuation and the rate of taxation are
given, how may the tax be found ? Find the tax on :
a, $12,000 at 1|%; at .8% ; at 1.4%; atlf%.
h. $10,000 at $1.20 per $100; at $.80 per $100.
c. $6000 at 8 mills on a dollar ; at 7.6 mills on a dollar.
d. $3600 at $.007 on a dollar ; at $.014 on a dollar.
16. If a broker received a commission of IJ % for his
services, find the amount of his brokerage for buying
2450 cwt. of wheat at $1.34 per cwt. If this wheat was
bought for a milling company, what was the total cost of
♦^^he wheat to the company ? the cost per cwt. ?
17. If a traveling salesman sells on an average $400
worth of goods every week, which of the following offers
should he accept from the wholesale firm : (a) a salary
of $25 per week and expenses; (5) a salary of $15 a
week and expenses, and a commission of 5 % on all sales
over $300 per week; (c) or his expenses and a commis-
sion of 8 % on all sales ?
18. When the tax and the rate of taxation are given,
how may the valuation be found? Find the valuation:
a. Tax, $60; rate of taxation, 1J%.
h. Tax, $120 ; rate of taxation, 8 mills on a dollar.
(?. Tax, $96 ; rate of taxation, $1.20 per $100.
INSURANCE AND TAXES 201
19. What was the amount of insurance if the premium
received for insuring a house at $1.40 per $100 was $49 ?
20. Furniture yalued at $600 was insured for $400.
For what part of its value was the furniture insured?
The premium paid for 3 years was $8. What was the
rate of premium paid ?
21. The pupils of the advanced arithmetic class in a
certain school were told that the school building was
insured for | of its estimated value, and that the annual
premium at 1 % was $75. They were asked to find the
estimated value of the building. One pupil found the
value to be $5625. Was his answer correct ?
22. A certain school district voted $12,000 to erect a
new schoolhouse. The assessed valuation of the property
in the district was $600,000. Find the rate of taxation.
23. If a tax collector in a certain city receives a com-
mission of 2 % for collecting taxes, what per cent of the
amount collected does the city receive ? Find the amount
of taxes that must be levied in order that a city may
receive $19,600, after allowing a collector a commission
of 2% for collecting.
119,600 = 98% of the sum levied.
24. Property worth $9000 was assessed at $6000. The
rate was $1.50 for each $100 of assessed valuation. Had
this property been assessed at its full value, what rate of
taxation would have yielded the same amount of taxes ?
25. Examine a tax receipt. Is a separate entry made
for taxes on personal property and on real property?
Is there an entry for school taxes ?
26. Make and solve five problems in taxes, using when
possible the actual rates in your county or city.
202 PERCENTAGE
248. Customs and Duties. The following rates of cus-
toms are from the schedule adopted by Congress in 1897,
commonly known as the Dingley Tariff :
Newspapers, periodicals, free.
Hay, % 4 per ton.
Coffee, free.
Carpets (velvet), 60^ per sq. yd.
Musical instruments, 45 %
and 40% ad. val.
ad. val.
Table knives, 16^ each and 15%
Potatoes, 25 f per bu.
ad. val.
Tea, free.
Paintings, 20% ad. val.
249. Written Exercises.
1. Find the duty on 60 sq. yd. of velvet carpet worth
$1.50 per square yard.
2. What is the duty on 45 tons of hay ?
3. What is the duty on a violin worth 180 ?
4. A painting valued at 12500 was purchased in Italy
and brought to the United States. Find the amount of
customs on it.
5. Find the amount of the duty on 6 doz. table knives
worth $1.80 per dozen.
6. Why are tea and coffee on the free list, while a duty
of 25^ per bushel is placed upon potatoes ?
7. What were the net proceeds of an auction sale, if the
sales amounted to 11215.40, and the auctioneer received
a commission of 10 % ?
8. After deducting his commission of 5% and $12.50
for freight and cartage, a commission merchant remitted
$633.50 to the shipper. Find the amount of the sales.
9. A city lot that cost $1600 was sold for $1800.
Find the gain per cent.
PERCENTAGE 203
250, Oral Exercises.
1. What is the price of coal a ton when it is selling at
i.25 a hundredweight ?
2. When hay is selling at $12 a ton, what is its price
per hundredweight ?
3. If I of the length of a certain bridge is 240 ft.,
how long is the bridge ?
4. If the interest for one year at 5% is $80, what is
the sum on which the interest is paid ?
5. A boy shot 10 times at a target and hit it 8 times.
Express as per cent the ratio of the number of accurate
shots to the number of shots taken.
6. On a certain day a girl missed 3 out of 12 words in
a spelling lesson. What per cent of the words did she
spell correctly ?
7. A baseball team played 8 games and lost 3 of them.
What per cent of the games played did the team win ?
8. A girl was absent from school 4 days and present
16 days during a school month. What per cent of the
time was she present ?
9. A man paid a tax of 1J% on property valued at
$4000. Find the amount of his tax.
10. A commission merchant received $20 for selling
$1000 worth of produce. What was his rate of com-
mission ?
11. If a spelling lesson consists of 25 words, what per
cent of the lesson is each word ? What per cent of the
words does a boy spell correctly who misspells 4 words ?
12. A boy caught a ball 6 times and missed it 2 times.
The number of times he caught the ball is what per cent
of the number of chances he had to catch it ?
204 PERCENTAGE
251. Trade Discount.* i. Manufacturers and wholesale
dealers issue catalogues describing articles sold by them
and giving their list prices, A discount from the list
price is made to retail dealers and sometimes to other
customers, particularly when goods are purchased in large
quantities. Such a discount is generally known as trade
discount, or commercial discount.
2. Several successive discounts are sometimes allowed.
Thus, an article may be sold subject to discounts of 25 %,
10 %, and 5 % ; that is, a discount of 25 % is made from
the list price, and a second discount of 10 % is made from
the price after making the discount of 25 %, and a third
discount of 5 % is made on the price after the two dis-
counts have been made. A separate cash discount is
usually allowed when payment is made within a specified
time after the purchase of the goods.
252. Written Exercises.
1. Find the net cash price to a retail hardware mer-
chant of a stove listed at $45, trade discounts of 20 %
and 10 (fo^ and a cash discount of 5 %.
Model : $ 45, list price. ^^^ ^'^^ discount is 20%
^ n . 1. . of $45, or $9. The price
_9, first discount. ^^^^^ ^^^^.^^ ^^^^ ^.^^^^^^
% 36, second price. is $ 45 - $ 9, or $ 36. The
3.6 0, second discount, second discount is 10% of
$32.40, third price. ^36, or $3.G0. The price
1.62, cash discount. ^^^^ ^^^"^g *^« ««^^"^
discount is $ 36 - 1 3.60,
$30.78, net price. " d^on <n -n xu- '
' ^ or $32.40. From thw a
cash discount of 5 % is deducted, leaving the net price $30.78.
Instead of deducting each discount separately, the sum of the
several discounts may be stated as a single discount. This may be
* For a more extended discussion of Trade Discount, see Appendix, p. 865.
TRADE DISCOUNT 206
found thus : A discount of 20 % from, the list price reduces the cost
to 80% of the list price; and the second discount reduces it to 90%
of this, or to 90% of 80% of the list price, which is 72 % of the list
price, and the cash discount reduces it to 95% of 72% of the list
price, or to 68.4 % (68| %) of the list price * Compare 68| % of $ 45
with the answer found.
In figuring discounts, use the shortest method in every
part of the problem.
2. Which is the greater discount, a single discount of
25 % or a discount of 20 % and a second discount of 5 % ?
3. If the hardware merchant (Prob. 1) sold the stove
for $45, how much was his profit if he paid out $ 2.50 for
freight and cartage ? What per cent profit did he make ?
4. What was the net cash price to a jeweler of a watch
listed at $35, discounts 30 %, 15 %, 5 %, and a cash dis-
count of 2 % ? If the jeweler sold the watch for $ 35,
what per cent profit did he make on it ?
5. A piano firm bought a piano listed at $350 and
received discounts of 40 %, 20 %, and 10 %. How much
did the firm make on the piano by selling it at $350?
What per cent profit was made by the firm ?
6. How much does a dealer make on a carriage listed
at $120, if he buys it at a discount of 20 %, 5 %, and
takes advantage of a cash discount of 2 %, and sells it
at the list price ?
7. How much less does a dealer pay for a wagon listed
at $150, if he is allowed a single discount of 35 % than if
he is allowed successive discounts of 15 %, 15 %, and 5 % ?
8. What per cent of $48 is $60 ? of $50 is $30 ? of $120
is $100 ? of $100 is $120 ? of $150 is $200 ?
* The single equivalent discount may be found by adding together
20%, 10%of 80%, or8%,and5%of 72%, or 3.6%. 20%+8%+3.6% =
31.6 %.
206 PERCENTAGE
253. Oral Exercises.
1. A man bought a house for f 4000 and sold it at a
gain of 20 %. For how much did he sell it ?
2. A farmer sold 200 sacks of potatoes, which was 80 %
of his entire crop. How much was his entire crop ?
3. If the price of steak was raised from 12 ^ a pound
to 15 ^ a pound, what was the per cent of increase in
price ?
4. The total enrollment in a certain school was 40
pupils, and the average number in daily attendance was
35. The average daily attendance was what per cent of
the enrollment ?
5. An agent received f 6 for collecting a bill of $ 30.
What was his per cent of commission ?
6. A man bought a farm for $4000 and sold it for
1 5400, what was his per cent of gain ?
7. At what price must a dealer sell carriages that cost
$ 120 to make a profit of 33J % ? 20 % ? 25 % ? 12^ % ?
10%?
8. After increasing his capital by $1200, a merchant
had $ 4200 invested in his business. What amount had he
invested before increasing his capital ? By what per cent
of itself was the original capital increased ?
9. By selling a city lot for $ 1500, a man gained 25 %.
Find the cost of the lot.
10. Property worth $ 6000 was assessed for purposes of
taxation at i 4000. For what per cent of its value was the
property assessed ?
11. What monthly rental must a man get from property
valued at $3000 to yield a net income of 6 %, if it costs
him $ 60 a year to maintain his property ?
INTEREST 207
INTEREST
264. 1. On July 1, 1907, Charles H. Thomas borrowed
of Joseph R. White i 300, which he promised to return in
one year, with interest at 6 per cent. As an acknowledg-
ment of his indebtedness and as a promise to pay, Mr.
Thomas gave Mr. White his note, of which the following
is a copy:
$300. fSe.\.k£.t£Af, (^clL, fioUf f, 190/.
€yL& if&av after date, for value received,... J...
promise to pay to jao^eyfsA R. -W^kite^, or order,
100
zvith interest thereon at 6% per annum from date until
paid.
^/icuvt&Q^ /if. ^kayyuao^.
2. The sum specified in a note is called its face, or the
principal.
3. What is the principal and the rate of interest speci-
fied in the above note ? When and where was the note
executed ? On what date did the above note fall due ?
4. The person to whom, or to whose order, the amount
named in a note is to be paid is called the payee, and the
person by whom the note is to be paid is called the payer.
In the above note who is the payer and who the payee ?
What is the meaning of the words or order? per annum?
5. Has any provision been made in the above note for
the payment of interest beyond the period of one year ?
208 PERCENTAGB
6. Where in the note did Mr. Thomas acknowledge
that he had received of Mr. White something of the value
of $ 300 ?
7. As the note was made payable to Mr. White, or
order^ it is said to be negotiable ; that is, it may be passed
from one person to another, and it becomes payable to
the person to whom it is ordered paid. Six months
after the note was executed, Mr. White bought a city
lot of J. C. Anderson, and as part payment for the lot,
he transferred the note to Mr. Anderson. In making
the transfer, Mr. White wrote on the back of the note,
over his own signature, "Pay to J. C. Anderson, or
order." By this indorsement, the note was made payable
to Mr. Anderson.
8. Find the interest on the note to July 1, 1908.
9. The sum of the principal and interest is called the
amount.
10. What was the amount of the note on July 1, 1908 ?
255. Method of Aliquot Parts.
1. The unit of time for which interest is computed is
usually one year. The interest on a given note for three
years is how many times the interest for one year? The
interest for six months is what part of the interest for
one year?
2. What part of the interest for one year is the interest
for 3 months? for 2 months.? for 4 months? for 1 month ^
3. The interest for 1 month is what part of the in-
terest for 6 months ? for 4 months ? for 3 months? for 2
months ?
4. The interest for 5 months is the interest for 4
months plus the interest for what part of 4 months?
INTEREST 209
5. Find the interest on 1400 for 1 yr. and 6 mo. at 7%.
Model: |400
.07
$28.00 = interest for 1 yr.
4 mo. = J yr. 9.33 = interest for 4 mo.
1 mo. = J of 4 mo . 2.33 = interest for 1 mo.
139.66 = interest for 1 yr. and 5 mo.
Explanation. First find the interest for 1 yr. by multiplying
the principal by .07. Next find the interest for 4 mo. by dividing the
interest for 1 yr. by 3 ; then find the interest for 1 mo. by dividing
the interest for 4 mo. by 4. The sum of these three amounts is the
interest for 1 yr. and 5 mo. In dividing drop all fractions of cents.
6. From the interest for one year, the interest for any
number of months may be found by taking the following
parts;
1 mo. = ^ yr. 3 mo. = ^ yr.
2 mo. = I yr. 4 mo. = | yr.
6 mo. = I yr.
5 mo. (4 mo. and 1 mo.) = ^ yr. plus | of | yr.
7 mo. (6 mo. and 1 mo.) = ^ yr. plus | of | yr.
8 mo. (4 mo. and 4 mo.) = ^ yr. plus J yr.
9 mo. (6 mo. and 3 mo.) = | yr. plus ^ of ^ yr.
10 mo. (6 mo. and 4 mo.) = I yr. plus } yr.
11 mo. (6 mo. and 5 mo.) = I yr. plus i yr. plus | of i yr.
Into what other suitable parts for finding interest may each be
divided : 9 mo. ? 8 mo. ? 10 mo. ?
256. Written Exercises.
Find the interest and the amount of :
1. 8250, 1 yr. 6mo., 8%. 6. 1260, 1 yr. 10 mo., 8%.
2. 8560, 2 yr. 4 mo., 6%. 7. |720, 1 yr. 3 mo., 5%.
3. 8875, 3 yr. 5 mo., 7%. 8. 81200, 10 mo., 6%.
4. 8100, 1 yr. 9 mo., 7%. 9. 82400, 2 yr. 3 mo., 6%.
5. 8620, 7mo., 4J%. lo. 8500, 1 yr.'8 mo., 4%.
AB. — 14
210 PERCENTAGE
257. Interest for Years, Months, and Days.
1. It is sometimes necessary to find the interest for
years, months, and days, in which case thirty days are
usually regarded as one month.
2. When the interest for one month is known, how may
the interest be found for 15 da. ? for 10 da. ? for 6 da. ?
for 5 da. ? for 3 da. ? for 1 da. ? When the interest for
6 da. is known, how may the interest be found for 1 da. ?
3. Find the interest on |150 for 1 yr. 7 mo. 14 da. at 8 %.
Model: $150
m
$12.00 = interest for 1 yr.
6 rao. = I yr. 6.00 = interest for 6 mo.
1 mo. = ^ of ^ yr. 1.00 = interest for 1 mo.
10 da. = I rao. .33 = interest for 10 da.
3 da. = fj mo. .10 = interest for 3 da.
1 da. = I of 3 da. .03 = interest for 1 da.
$19.46 = interest for 1 yr. 7 mo. 14 da.
4. From the interest for one month, the interest for any
number of days may be found as in the following :
22 da. (10 da. and 10 da. and 2 da.) = | mo. plus | mo. plus ^ of ^ mo.
18 da. (6 da. and 6 da. and 6 da.) = \ mo. taken 3 times. Is this
easier than to separate 18 da. into the parts 15 da. and 3 da., or
into the parts 10 da. and 6 da. and 2 da. ? Explain and illustrate.
5. Determine for each number of days, from 1 to 29,
how the interest can be found most readily, when the in-
terest for one month is known. Compare your results
with those determined by other pupils, to see who has the
best method. Test each method by taking some amount
as the interest for one month.
6. Write a note, naming some pupil as payee and your-
self as the maker, and find the amount of the note for
1 yr. 5 mo. 12 da.
INTEREST 211
258. Written Exercises.
Find the interest on:
1. $250 for 1 yr. 9 mo. 15 da. 'dtl%,
2. $700 for 2 yr. 8 mo. 21 da. at 5%.
3. 1684.50 for 7 mo. 25 da. at 6 %.
4. $1200 for 3 yr. 4 mo. 14 da. at 5| %.
5. $300 for 1 yr. 10 da. at 8 %.
6. $45.75 for 2 yr. 8 mo. 12 da. at 4 %.
7. $2500 for 9 mo. 13 da. at 8%.
8. $560 for 2 yr. 3 mo. 23 da. at 7 %.
9. $645.40 for 2 yr. 7 mo. 26 da. at 9%.
10. $820.15 for 1 yr. 5 mo. 20 da. at 4%.
11. $125 for 11 mo. 17 da. at 8%.
12. $214.45 for 4 yr. 2 mo. 19 da. at 6 %.
13. $750 for 1 yr. 6 mo. 28 da. at 7 %.
Find the interest on each of the following at 6%; at
6%; at 7%; at 4|%:
14. $800 from Oct. 1, 1904 to May 10, 1906.
15. $475 from June 11, 1903 to Nov. 18, 1904.
16. $240.60 from April 8, 1904 to Feb. 21, 1905.
17. $350 from Jan. 1, 1904 to Nov. 20, 1904.
18. $1340 from June 8, 1903 to Dec. 29, 1904.
19. $26.48 from Sept. 12, 1905 to Aug. 10, 1906.
20. $1700 from March 24, 1905 to Aug. 15, 1906.
21. $48.62 from Nov. 18, 1902 to July 20, 1904.
22. $5000 from Sept. 7, 1903 to Dec. 23, 1903.
23. $467.89 from April 4, 1904 to July 26, 1905.
212 PERCENTAGE
259. Sixty Days Method.
-1. Money loaned for less than one year is usually
loaned for 90 da., 60 da., or less. The best unit of
time to use in finding the interest is 60 da., and the
best rate is 6%, as the interest at 6 % for 60 da. is 1 %
(.01) of the principal, found by moving the decimal
point.
2. Find the interest on $2700 for 60 da. at 7%.
Model : $ 27 = interest at 6 % for 60 da.
4.50 = interest at 1 % for 60 da.
$ 31.50 = interest at 7 % for 60 da.
3. What part of 60 da. is 30 da.? 10 da.? 20 da.?
5 da. ? 15 da. ? 12 da. ? 6 da. ? 3 da. ? 2 da. ?
4. From the interest for 60 da., how may the interest
be found for 30 da. ? for 15 da. ? for 6 da. ? for 20 da. ?
for 10 da.? for 5 da. ? for 90 da. ? for 120 da. ?
5. Find the interest on 14000 at 6 % for 60 da. ; for
90 da. ; for 30 da. ; for 20 da. ; for 120 da.
6. From the interest for 30 da., how may the interest
be found for 15 da.? for 45 da. ? for 10 da. ? for 5 da. ?
7. From the interest at 6%, how may tlie interest be
found at 7 %? at8%? at 9 % ? at5%? at 5J % ?
8. Find the interest on $500 for 90 da. at 7%.
Model : 15= interest at 6 % for 60 da.
2.50 = interest at 6 % for 30 da.
$ 7.50 = interest at 6 % for 90 da.
1.25 = interest at 1 % for 90 da.
9 8.75 = interest at 7 % for 90 da.
Find the interest on :
9. $600 at 6 % for 30 da. 11. $1200 at 5 % for 60 da.
10. $1000 at 7 % for 90 da. 12. $10,000 at 7% for 45 da.
INTEREST 213
260. Cancellation Method. (For problems, see Sec. 258.)
1. What part of 360 da. are 90 da. ? The interest for
90 da. is what part of the interest for 1 yr. (360 da.)?
2. Find the interest on $600 for 90 da. at 7 %.
$150 ^0
Model : $ ^^^ x .07 x f^ = $ 10.50.
i
The interest for 1 yr. is $ 600 x .07, and the interest for 90 da. is
/^, or ^ of 1600 X .07.
3. What part of 1 year's interest is the interest for
6 mo. ? for 2 mo. ? for 8 mo. ? for 9 mo. ? for 10 mo. ? for
15 da. ? for 45 da. ? for 1 mo. 15 da. ? for 3 mo. 20 da. ?
for 1 yr. 3 mo. ? for 1 yr. 6 mo. ? Find the interest on
$600 at 5% for each of these periods.
261. Six Per Cent Method. (For problems, see Sec. 258.)
1. Interest is sometimes calculated by a method
commonly known as the Six Per Cent Method. By this
method, the interest at 6 % for the given time is found,
*and from this the interest at the required per cent.
2. As 1 mo. is -^^ of 1 yr., the rate of interest for 1 mo.
is y^ of 6 %, or I % (,005); and as 1 da. is -^-^ of 1 mo., the
rate of interest for 1 da. is -^^ of .005, or .000|^.
The interest at 6% for 1 yr. = .06 of the principal.
The interest at 6% for 1 mo. = .005 of the principal.
The interest at 6% for 1 da. = .000^ of the principal.
3. Find interest on $350 for 2 yr. 7 mo. 21 da. at 7 %.
Model : Rate for 2 yr. at 6 % 12
Rate for 7 mo. at 6 % 035
Rate for 21 da. at 6 % .0035
Rate for 2 yr. 7 mo. 21 da 1585
$350 X .1585 = $ 55.475 = int. on $350 for 2 yr. 7 mo. 21 da. at 6 %.
9.245 += int. on $350 for 2 yr. 7 mo. 21 da. at 1 %.
$64.72 = int. on $350 for 2 yr. 7 mo. 21 da. at 7%.
214 PERCENTAGE
263. Promissory Notes.
1. A written promise to pay a definite sum of money at
a specified time is called a promissory note.
A promissory note is usually called a note.
2. Compare the promissory note on p. 207 with the fol-
lowing. The note on p. 207 is a time note, as the time
of payment is specified in it.
A Demand Note
/ci'^^ €aUoinci, ^aC, fam,. ^, 1907.
€.^vci&yyuhncl, for value received <J promise to pay to
fo^&^iA R. W^kite^, or order, tkv&& k^L'ylclA^&ci cLo-ttoAj^,
mith interest thereon ai) ^% per annum from date
until paid.
^kcL'i.t&Q^ //. 5k(yYyux^.
A Joint Note
$600 ^taltU, y>a^k., futif /6, 1906.
^ioctAf dai^ after date, '^^, o-v &ttk&^ o-^ loQy, promise
to pay to S^uin^ S^. /CenA^cyyv, or order, Q^Coc kicr^iecC
dattaiO', with interest thereon at ^% per annum from
date until paid.
Value received. W^cult&v f. BityvkeA,,
fo-kn, R. Jbav^^e^.
3. Each maker of a joint note is liable for its payment
in full.
4. The following should appear in a note :
a. The time and place where the note was executed.
This is usually written at the top and toward the right
PROMISSORY NOTES 215
h. The sum to be paid, including the rate of interest, if
any is paid.
The face of the note is usually written in figures at the top and
toward the left and in words in the body of the note.
c. The signature of the maker or makers.
d. The time of payment.
When no time of payment is specified, the note is payable on
demand.
e. Notes usually contain the words value received.
Answer each concerning the two notes in Sec. 262.
5. When was the note executed ? Where ?
6. When is the note payable ?
7. What is its face ?
8. Who is the maker ? the payee ?
9. Is the note negotiable ?
10. When a note becomes due it is said to mature. In
some states a note matures three days after the time
specified in the note. The three additional days are
called days of grace. Days of grace have been abolished
in most states, and are not computed in the answers given
in this book.
11. Each state has fixed its own legal rate of interest,
which is the rate allowed on claims drawing interest when
no rate of interest has been otherwise arranged. What
is the legal rate in the state in which you live ?
12. Many states have fixed a maximum rate of interest
that can be collected by agreement. A higher rate than
that authorized by law is called usury. Is there a law
against usury in the state in which you live ?
13. Write a demand note ; a time note ; a joint note.
216 PERCENTAGE
263. Partial Payments. — Mercantile Rule.* i. On
Jan. 1, 1906, James Smith of Los Angeles, Cal., borrowed
of Frank Adams $ 1000 for one year at 6 %, giving his note
for this amount. Write the note.
When the loan was made, it was agreed that if James Smith made
any payments on the note before its maturity, he would be credited
with each partial payment and would be credited with the same rate
of interest as he was paying, 6 %, from the date of each payment until
the time of final settlement.
2. On July 1, 1906, James Smith paid Frank Adams
$400. Indorse this payment by writing " July 1, 1906,
f 400 " on the back of the note. Final settlement was
made Jan. 1, 1907.
Under these conditions James Smith had the use of $1000 bor-
rowed of Frank Adams for 1 yr., and Frank Adams had the use ox
$400 paid by James Smith for 6 mo. (from July 1, 1906, to
Jan. 1, 1907).
At the time of final settlement James Smith owed Frank Adams
$ 1000, with interest for 1 yr., at 6%, or $ 1060 ; and Frank Adams owed
James Smith $ 400, with interest for 6 mo. at 6 %, or $ 412. In settling
the note, James Smith paid Frank Adams $1060 - $412, or $648.
Notes and accounts which do not run for more than one year, on
which partial payments are made, are often settled by business men
as above.
Mercantile Rule. 1. Find the amount of the faee
of the note at the time of settlement,
2. Find the amount of each payment from the date of
payment to the date of settlement,
3. Subtract the sum of the amounts of the payments from
the amount of the face of the note,
3. Write a note, naming some pupil as payee and your-
self as payer. Make three partial payments and have
them indorsed to your credit. Settle the note.
• For the United States Rule of Partial Payments, see Appendix, p. 266.
COMPOUND INTEREST 217
264. Compound Interest.*
1. When the unpaid interest is added to the principal,
as it becomes due, to form a new principal on which interest
is computed, the interest is called compound interest.
Interest may be added to the principal annually, semiannually,
quarterly, etc., according to agreement.
The payment of compound interest cannot usually be enforced by
law, but if the debtor is willing to pay compound interest, it may be
collected without violating the law against usury.
2. Savings banks generally pay interest semiannually.
When it is not collected by the depositor, it is added to
his deposit and he is paid compound interest.
3. If interest is collected when due and reinvested at
once at the same rate of interest, the result is the same as
when compound interest is received.
4. Find the amount of |600 for 2 yr. 6 mo. at 8%,
interest compounded annually. Find the difference be-
tween the compound interest and the simple interest.
Model : f 600 = principal for first year.
48 = interest for first year.
$648 = amount, or principal, for second year.
51.84 — interest for second year.
$699.84 = amount, or principal, for third year.
27.99 = interest for 6 mo.
$727.83 = amount for 2 yr. 6 mo. at 8%.
Compound interest = $727.83 - $600 = $127.83.
Siniple interest = 120.
Difference = $ 7.83.
5. If a man invests $1000 at compound interest at 6%
when he is 30 years of age and keeps it earning at the
same rate until he is 50 years of age, what will be the
amount of the f 1000 at that time ? (Use table, p. 320.)
* For table of compound interest, see Appendix, p. 320.
218 PERCENTAGE
265. Bank Discount and Proceeds.
1. Banks usually collect interest in advance on sums
loaned. Thus, if George White borrows $100 at a bank
for 60 da. at 6%, his note will be made out for flOO,
and the bank will deduct from this amount the in-
terest on $100 for 60 da. at 6%, or $1. Mr. White wiU
receive $99. At the end of 60 da. he will pay the bank
the face of the note, or $100.
2. If interest is collected in advance, how much money
will a person receive at a bank on a note for $2000 for 60
da., if the bank charges 8 % interest ?
3. On April 8 J. J. Dow bought $ 600 worth of goods
of D. C. Brown, on 90 da. time, giving his note for the
amount without interest. On the same day D. C. Brown
sold the note to a bank, the bank deducting 6 % interest
for the term of the note (90 da.). Find the amount
received for the note by D. C. Brown.
4. Interest paid in advance upon the amount due on a
note at its maturity is called bank discount. Bank dis-
count is computed from the date of the purchase of the
note by the bank to the legal date of maturity.
Some banks include both the day of purchase and the day of
maturity in the discount period. "When days of grace are allowed,
these are included in the discount period.
5. The sum paid for a note when sold is called the
proceeds of the note. The proceeds on a note is the
amount due at maturity, less the bank discount.
6. C. W. Smith held a note against R. E. Orr for
$4000 for 60 da. without interest. After 20 da., he sold it
to a bank at a discount of 6 % ; that is, the bank deducted
6 % int. on the note for the 40 da. between its purchase and
expiration. Find the bank discount and the proceeds.
BANK DISCOUNT 219
7. On April 24, 1906, James J. Hall sold a horse to
G. M. Bruce for f 150, taking in payment his note for 1
year with interest at 6 % . Find the amount of the note
at maturity.
8. Mr. Hall (Ex. 7) needed money, so he sold the note
to a bank on the same day, the bank discounting it at
6 % . How much did Mr. Hall receive for the note ?
Model : Face of note = $ 150
Interest for 1 yr. at 6 % = 9
Amount at maturity = $ 159.
Discount 1 yr. at 6 % = 9.54 (computed on $ 159)
Proceeds = $ 149.46
9. If the note (Probs. T and 8) had been discounted
at 8 % instead of 6 % , what amount would Mr. Hall have
received ?
10. If the note (Probs. 7 and 8) had been discounted
three months after date of issue, or on July 24, 1906, the
bank would have deducted interest on the amount due at
maturity (1 159) for the exact number of days from July
24, 1906 to April 24, 1907 (7 da. 4- 31 da. + 30 da. + 31
da. 4- 30 da. + 31 da. + 31 da. + 28 da. + 31 da. + 24
da.), or for 274 da. Find the amount which Mr. Hall
would have received.
11. A 90-da. note for $ 500, without grace, dated Aug.
5, 1905, with interest at 5%, was discounted at a bank on
Aug. 25 at 6%. Find the day of maturity, the amount
at maturity, the bank discount, and the proceeds.
12. A man borrowed flOOO of a bank for 1 yr. at 6%,
paying interest in advance. 6% interest in advance on
$1000 is equivalent to what rate paid at the end of the
year ?
MO PERCENTAGE
266. Review.
1. During a certain school month a boy worked 209
problems, of which 194 were correct. Find the per cent
of correct work.
2. A baseball team won 43 games and lost 15 games
one season. Find the per cent of games won.
3. The rent of a house was raised from $ 30 a month
to I 35 ; this was an increase of what per cent ?
4. A person bought a house for $6000. The taxes,
insurance, repairs, and other expenses connected with the
property amounted to $120 a year. For how much a
month must the property be rented to net 6% on the
investment ?
5. A man built two flats costing him $ 5000 on a lot
which cost $2000. He rented one of the flats for $40 a
month and the other for $35. The expenses connected
with the property amounted to $200 a year. The net
income amounted to what per cent on the investment ?
6. An electric light meter registered 80,000 watt
hours on Oct. 9, and 106,000 watt hours on Nov. 9. Find
the amount of the bill for the month at 9^ for each 1000
watt hours.
7. A gas meter registered 29,800 cu. ft. on April 24,
and 31,800 cu. ft. oh May 24. Find the amount of the
bill for the month at $ .90 per 1000 cu. ft.
8. The population of a certain city was 47,235 in 1900,
and 60,624 in 1910. Find the increase per cent.
9. At 66 ff a sack (100 lb.), what is the price of coal
per ton ?
10. Find the interest on $2800 from June 8 to Jan. 16
at 6 % per annum.
PART IV
FORMS AND MEASUREMENTS*
267. 1. Lines are vertical j , horizontal — , and
oblique \ /.
2. These are right angles. | | [ |
3. A rectangle has four right angles. QJ | [
4. These are right triangles. \ yx\
5. These are acute angles. /\ v
6. These are acute-angled triangles. ^ \J
7. These are obtuse angles. ^-^"-\ ^^..^
8. These are obtuse-angled triangles. ^-^^ ^^^Z^
9. Perpendicular (p) means at right . |^^
angles to. -1^ l^
10. These figures have a base (5) ^i i^ y^a A
and an altitude (a). ^-t "^""^ b"
11. These lines are parallel.
12. These are quadrilaterals. V~\ /~~J / \ ^ |
13. These quadrilaterals are parallelograms. I I I 7
13 ^Pl^]
14. These are rectangular
prisms.
♦ With complete reviews.
221
222
FORMS AND MEASUREMENTS
15. These are triangular
prisms.
16. Circumference, diameter, and radius belong
to the circle.
17. These are cylinders.
B ^
circle.
268. Relation of Forms.
Study the relation of these forms
L
Right Angle.
Rectanqle. Rectangular Prism.
L [X
Right Angle. Right Tbl^u^gle. Right Triangular Prism.
A A
Acute Angle. Acute Triangle. Acute Triangular Prism.
Obtuse Angle. Obtuse Triangle. Obtuse Triangular Prism.
O
Cl&CLiB.
m
Sphere.
Ctlindbb.
LINES AND ANGLES 223
269. Lines.
1. Lines that extend in the same direction and are the
same distance apart are called parallel
lines. —
2. Suspend a weight by a string. p^^allel Lines.
When the weight is at rest, the line represented by the
string is called a vertical line.
3. The surface of the water in a tank or a pond is said
to be level, or horizontal. A slanting line is called an
oblique line.
A vertical line is represented on a page by a line parallel to the
sides, and a horizontal line by a line parallel to the top and bottom.
4. Hold your pencil in a vertical position ; in a hori-
zontal position ; in an oblique position,
5. Point to surfaces in the schoolroom that are hori-
zontal, vertical, oblique, parallel.
6. Draw two vertical parallel lines on the blackboard;
two horizontal parallel lines; two oblique parallel lines.
270. Right Angles.
1. Two lines that meet form an angle, Z. When two
lines form a square corner, the angle between them is
called a right angle.
2. Draw four right angles. . . . — — .
3. Point to surfaces in your school- — - — ' ' ^
room that meet at right angles.
4. Two lines that form a right angle are said to be per-
pendicular to each other. Draw perpen- i \_p_ p/
dicular lines. — ■— ' ^
Perpendiculab
5. Point to lines or surfaces in the Lines.
schoolroom that are perpendicular to each other.
224 FORMS AND MEASUREMENTS
n
n
Square. Ilurizontal. Vertical.
Bectanolbs.
271. Rectangles.
1. A figure whose angles
are all right angles is called
a rectangle. Reel means
right. Rectangle means
having right angles.
2. A rectangle whose sides are all the same length is
called a square.
A rectangle having two opposite sides longer than the other two
opposite sides is sometimes called an oblong.
3. Draw a square ; a vertical rectangle ; a horizontal
rectangle. Point to surfaces in your schoolroom that are
rectangles. Are any of these squares?
4. How many sides has a rectangle ? How many angles
has a rectangle? Are the sides of a rectangle parallel?
Name surfaces not in your schoolroom that are rectangles.
272. 1. Draw a square whose side is 1 foot. This is
called a square foot. Draw and name a square whose side
is 1 inch.
2. Draw a square yard. Divide it into square feet.
How many square feet are there in a square yard?
3. Divide a square foot into square inches. How many
square inches are there in a square foot ?
4. A square 16 J feet each way is called a square rod.
Mark out a square rod on the school grounds.
5. Draw a square whose side is 2 inches. Divide it
into square inches. How many are there ?
6. Draw 3 inch squares and a 3-inch square. Compare
the size of a 3-inch square and 3 square inches.
7. The number of square units in a surface is called
its area.
RECTANGLES 225
273. Areas of Rectangles.
1. Repeat the Table of Linear Measure (§ 99). Re-
view p. 81. Repeat the Table of Square Measure (§ 101).
2. Using the scale J in. = 1 rd., make a drawing to
represent a rectangle 16 rd. long and 10 rd. wide. Ex-
press the area of the rectangle in acres.
3. Find the area of a flower bed that is 12 ft. 9 in.
long and 8 ft. 4 in. wide.
4. How many acres are there in a tract of land 80 rd.
by 80 rd. ?
5. A farm that contains 80 A. is J mi. wide. How
long is the farm ?
6. Find the number of square yards of surface in the
walls and ceiling of your schoolroom, deducting for the
doors and windows.
7. Find the value of a field 40 rd. long and 20 rd. wide
at f 85 an acre.
274. Written Exercises.
1. Reduce 2 yd. 2 ft. 7 in. to inches.
2. Find the sum of 8 ft. 6 in., 7 ft. 4 in., 9 ft. 11 in.,
and 6 ft. 5 in.
3. Find the perimeter of a rectangle whose length is
24 ft. 8 in. and whose width is 15 ft. 10 in.
4. How many rods of fence are required to inclose a
rectangular 20-acre field whose length is 80 rd. ?
5. How many bundles of shingles are necessary to
shingle a surface 50 ft. by 16 ft., if the shingles are laid
4 in. to the weather ?
6. How many yards of carpet are necessary to cover a
floor 16 ft. by 12 ft., if the carpet is 27 in. in width ?
AB. — 15
226 FORMS AND MEASUREMENTS
275. Right Triangles.
1. Draw a right angle. The point at which the lines
meet is called the vertex of the angle.
2. Draw a rectangle. Draw a straight line joining the
vertices of the opposite angles of the rec-
tangle. This line is called the diagonal of
the rectangle. The diagonal divides the
rectangle into two equal triangles.
3. A figure having three angles is called a triangle.
Tri means three. Triangle means having three angles.
4. A triangle having one right angle is called a right
triangle.
276. Area of Right Triangles.
1. The base of a figure is the side on which it is as-
6 sumed to rest, and the altitude is the
perpendicular distance between the top
a and the base, or the base produced.
Consider the length of a rectangle as its
and the width as its altitude.
6
RiaHT Triangles.
2. The area of a right triangle is what part of the area
of a rectangle having the same base and altitude?
The area of a right triangle is equal to one half the prod-
uct of its base and altitude.
The work is sometimes shorter if the altitude is multiplied by one
half the base, or if the base is multiplied by one half the altitude.
Dimensions must be expressed in like units. By the product of
the lines is meant the product of the numbers denoting them.
3. Find the area of right triangles of the following
dimensions: base, 12 in., altitude, 8 in. ; base, 6J ft., alti-
tude, 9 J ft. ; base, 40 rd., altitude, 80 rd.
Parallelogram.
PARALLELOGRAMS 227
277. Parallelograms.
1. A plane (flat) figure bounded by four straight lines
is called a quadrilateral. Quadri means four and lateral
means sides. Quadrilateral means having four sides.
2. A quadrilateral whose opposite sides are parallel is
called a parallelogram. Is a rectangle a parallelogram?
Draw a quadrilateral that is not a parallelogram.
3. This figure represents a city lot. Is the form of the
lot a parallelogram ? The
form of the lot may be
regarded as composed of
a rectangle, a^ and two
right triangles, b and o.
Since the figure is a par-
allelogram, triangle h is the same size as triangle c. If
triangle h were cut off and placed alongside triangle c,
with the side fd on the side ge^ what change, if any,
would this make in the size of the lot ? What change, if
any, would it make in the form of the lot? What would
be the dimensions of the resulting lot? How should you
find its area?
4. The sum of the areas of triangles h and c is equiva-
lent to the area of a rectangle having the same base and
altitude as the triangles. Therefore the number of square
feet in the lot may be found by multiplying 120 by 40.
The area of a parallelogram is equal to the product of its
base and altitude.
5. Cut several parallelograms out of paper, and show by
the method of Prob. 3 that each parallelogram has the
same area as a rectangle having the same base and altitude.
6. Draw several parallelograms. Assign the dimen-
sions, and find the area of each.
"5-
" K
26rd.
Fig. 1.— Tkapkzoid.
228 FORMS AND MEASUREMENTS
278. Trapezoids.
1. Fig. 1 represents a field whose area is to be found. It
may be regarded as composed of a rectangle, a, and a right
2Q , triangle, h. Its area is the sum
of the areas of these two parts.
The number of square rods in the
rectangle is 10 x 20, and in the tri-
angle is 10 X 3. The number in
the entire field is 10 x 23. Explain.
2. What is the average length of the two sides of the
field (Fig. 1) ? The field has the same area as a field of
the same width whose length is one half of the sum of
20 rd. and 26 rd., or 23 rd. Explain.
3. Draw five figures similar to Fig. 1. Assign the
dimensions, and find the area represented by each.
4. Figure 2 represents a cross section of a foundation
wall. It may be regarded as composed of a rectangle, a,
and two right triangles, h and c. What is the combined
length of the bases of the two triangles ? If triangle h
were cut off and placed in an inverted position alongside
triangle c, what change would it make in the form of the
figure ? What would be the dimensions of the resulting
figure ? How would you find its area ?
5. The area of triangle h (Fig. 2) is equivalent to the
d 4ft e area of a rectangle of the same altitude
whose base is one half of the base of
the triangle. Is the same true of the
area of triangle c 2 The area of the sur-
face represented by the entire figure is
equivalent to the area of a rectangle of
Fio. 2. the same altitude but whose base is one
half the sum of the two bases de and fg.
TRIANGLES
6. A quadrilateral that has only two sides parallel is
called a trapezoid. See Figs. 1 and 2, p. 228.
The area of a trapezoid is equal to the product of its alti-
tude and one half the sum of its bases.
7. Draw five figures similar to Fig. 2, p. 228. Assign
the dimensions, and find the area of each.
The area of any quadrilateral may be found by resolving it into
triangles or into rectangles and triangles.
279. Area of Triangles.
1. An angle that is less than a right
angle is called an acute angle. Acute
means sharp or pointed.
2. An angle that is greater than a
right angle is called an obtuse angle.
Obtuse means dull or blunt.
3. A triangle whose angles are
all acute is called an acute-angled
triangle.
4. A triangle one of whose an-
gles is obtuse is called an obtuse-
angled triangle. Obtuse-anqled triangles.
5. Can you draw a triangle having more than one ob-
tuse angle ? Draw an acute-angled triangle; an obtuse-
angled triangle.
6. Any triangle may be divided into two right triangles.
Acute Angles.
Obtuse Angles.
AV
Acute-angled TblAuNGles.
7. Triangle Imn in each of the above figures is one half
of the rectangle opln. Explain.
^-80
FORMS AND MEASUREMENTS
8. Any triangle
may be considered as
one half of a parallelo-
a c gram of the same base
Fig- 1- and altitude. Triangle
ahe in Fig. 1 is one half of the parallelogram ahdc.
9. Draw five triangles. Show by the methods given
in Probs. T, 8 that the triangles are each one half of a
parallelogram having the same base and altitude.
The area of a triangle
is equal to one half the
product of its base and
altitude.
10. Draw five trian-
gles. Assign their di-
mensions and find the
area of each.
The area of any figure
may be found by resolving
it into triangles.
11. Figure 2 repre-
sents a farm 80 rd. by
160 rd. Find the area
of each field.
12. The two sides of
a field, one 132 rods,
the other 152 rods in
length, are parallel.
The perpendicular dis-
tance between the
two sides is 80 rods.
FiQ. 2. Diagram and find area.
r-'^
£Ord
40rd.
^"^
t
eOrd.
"P
^
^ c ^
E
CU
J
40 rd.
t
/
^
f
05
^
? //
t
5
25 rd.
/
^ D
*ti
V.
/
^
/
15 rd
/
I'/
40 rd.
O)
'^ /
. ^ ^
/
^'
^
§ J
B
?
40rd
/ ^
^
^
/ 80 rd
<\j
CIRCLES 231
280. Circles. .
1. Draw a circle. Mark the center ((y) (^ "^
of the circle. The line bounding the circle
is called the circumference. Draw a line
from the center of the circle to the cir-
cumference. This is called a radius of
the circle. Draw a straight line through the center of
the circle from circumference to circumference. This is
called the diameter of the circle. Compare the length of
the diameter with the length of the radius.
2. Measure the diameter of a circle. Measure the cir-
cumference of the same circle. Divide the circumference
by the diameter. The answer should be nearly 3.1416
(3|). This is the ratio of the circumference to the
diameter. This ratio is commonly denoted by the symbol
TT, which is a Greek letter named pi.
3. The circumference of a circle is 3.1416 times the di-
ameter. How can you find the circumference when the
radius is given ? When the circumference is given, how
can you find the diameter ? the radius ?
4. If a wagon wheel is 3| ft. in diameter, what is its
circumference ? How many times will it turn in going 1
mile ?
5. Find the circumference of a circle whose diameter is
24 in. ; 7J in. ; 2 ft. 6 in. ; 40 mi. ; 80 rd.
6. Find the circumference of a circle whose radius is
2 in. : 2 ft. 4 in. ; 6 ft. ; 40 rd. ; 3J yd.
7. Find the diameter and radius of a circle whose cir-
cumference is 24 ft. ; 12 J in. ; 1 mi. ; 36 rd.
8. Find the equatorial diameter of the earth if its
equatorial circumference is 24,900 mi.
232 FORMS AND MEASUREMENTS
281. Area of a Circle.
1. A circle may be regarded as com-
posed of an infiuite number of triangles,
the sum of whose bases is the circumfer-
ence of the circle and whose altitude is
the radius of the circle. Therefore the
area of a circle is the area of the triangles
composing it.
The area of a circle is equal to one half the product of its
circumference and radius.
2. The circumference of a circle is 3.1416 times the
diameter, or tt times 2 times the radius = 2 7rr.
Area of circle (Prob. 1) = ^lli2ij:.
Substituting 2 irr for Cir.,
Area of circle = - — = tt x r x r, or tt r^, read pi r square.
The area of a circle is equal to irr^ (3.1416 X r x r).
3. Find the area of a circle whose radius is 2 in. ; 12
in. ; 24 ft. ; 4.5 rd. ; 64 yd. ; 18 ft. ; 3 ft. 7 in.; 8| in.
4. Find the area of a circle whose circumference is 24
in. ; 2 ft. 4 in. ; 40 rd. ; 28 yd. ; J mi. ; 1 mi.
5. The diameter of a circular flower bed is 6 ft. What
is its area ?
6. The atmospheric pressure is about 15 lb. to the
square inch. Find the pressure on a surface of a watch
crystal \\ in. in diameter.
7. Over how many square feet of surface can a horse
graze when tied with a rope 20 ft. long ?
8. Which has the greater surface, a rectangular table
top that is 3' 6" by 3' 6'' or a circular table top that is 3'
9" in diameter?
PRISMS AND CYLINDERS
233
282. Volume of Prisms and Cylinders.
1. Draw a rectangle. A rectangle has two dimensions;
namely, length and width. If it is given a third dimension,
thickness, it becomes a rectangular solid or a rectangular
prism.
2. Anything that has length, breadth, and thickness is
called a solid.
/^
hi
Rectangles.
Rectangular Prisms.
3. How many rectangular faces has a rectangular
prism ? Name some rectangular prisms that jo\i have seen.
4. A rectangular prism whose faces are all equal squares
is called a cube.
5. Construct rectangular prisms out of cardboard or
paper.
283. 1. Draw a triangle. How many dimensions has a
triangle ? If it is given a third dimension, it becomes a
triangular prism. The ends of a triangular prism are tri-
angles and the sides are rectangles.
Triangles.
Triangular Prisms.
2. How many faces has a triangular prism ? How many
of the faces are triangles ? How many are rectangles ?
3. Construct a triangular prism out of cardboard or
paper.
234
FORMS AND MEASUREMENTS
284. 1. Draw a circle. If a circle is given three di-
mensions, it becomes a cylinder. Mention some cylindrical
objects.
Cylinders.
2. Construct a cylinder out of paper.
3. The number of cubic units that a solid contains is
called its volume, or capacity.
s ^
9 square units.
Fig. 1.
9 cubic units.
Fig. 2.
8 times 9 cubic unlta.
Fig. 3.
285. 1. The area of the end, or base, of a prism or a
cylinder tells how many square units that surface contains
(Fig. 1). There are as many cubic units in one unit of
length as there are square units in the surface of the end,
or base (Fig. 2). Explain. There are as many cubic
units in the prism or cylinder as the product of the
number of units in the area of the end, or base, and the
number of units in the length or altitude of the prism or
cylinder (Fig. 8). Explain.
2. The volume of a prism or a cylinder is equal to the
product of the area of the end, or base, and the length,
or altitude.
PRISMS AND CYLINDERS 236
3. Find the capacity of a cylindrical tank whose diameter
is 18 in. and whose height is 4 ft. 6 in.
Number of square feet in area of base = 3.1 416 x | x f(| X | = r'^).
Number of cubic feet in capacity = 3.1416 x | x | x 4|.
4. There are 231 cubic inches in a gallon. How many
gallons will the tank (Prob. 3) hold ?
5. Find the number of cubic feet in a bin 8 feet long,
4 feet wide, and 6 feet deep.
6. There are 2150.42 cubic inches in a measured bushel.
How many bushels will the bin (Prob. 5) hold ?
7. Find the number of cubic feet of air in a room 16
feet long, 10 feet wide, and 9 feet high.
8. Find the number of cubic yards of earth that must
be removed in excavating a basement 8 feet deep, 36 feet
long, and 24 feet wide.
286. Surfaces of Prisms and Cylinders.
1. How many surfaces has a cube? a rectangular
prism ? a triangular prism ? Construct each out of card-
board. State how the area of the combined surfaces of a
prism may be found.
2. Find the area of the surfaces (excluding the ends) of
a timber 12" by 12" and 16' in length. What name is
given to such a solid ?
3. Bring together the ends of a sheet of paper so that
the sides form circles. What name is given to the fig-
ure formed by the sheet? The length of the sheet be-
comes the circumference of the base of the cylinder and
the width of the sheet becomes its altitude. The area of
the cylinder (excepting the bases) is therefore the area of
the rectangle forming its convex surface.
4. Find the convex surface of a cylindrical tank whose
diameter is 6 feet and whose altitude is 8 feet.
286
FORMS AND MEASUREMENTS
287. Written Problems.
1. If it takes .98 cu. yd. of crushed stone, .47 cu. yd.
of sand, and 1.56^hbl^of cement to make 1 cu. yd. of
concrete, how much di each will make 100 cu. yd. of
concrete ?
\y 2. How much crushed stone (Prob. 1), sand, and
cement will be required to build a concrete wall 6 ft.
high, 18 in. thick, and 60 ft. long, if 3j cu. ft. of cement
is a barrel? (Express each answer as a whole number,
since a_fractional part of these units cannot be purchasedL)_^-
3. If the sand costs 80^ per cubic yard, the cement $2
per barrel of 3| cu. ft. each, the stone $1.50 per cubic
yard, and the labor for building the wall 80^ per cubic
yard, find the cost of the wall (Prob. 2).
4. Mr. Adams owns a 50-ft. lot on which he has re-
cently built a house. He wishes to have a 6-ft. cement
sidewalk laid along the front of the lot 2 ft. in from the
D
(
^
curb, and a 4-ft. cement sidewalk laid
from the curb to the front steps of the
house, 20 ft. in from the curb. Find
the cost of the walks at $.12 J per square
foot. First make a diagram of the
walks.
95'
73
5. The figure represents a lot owned
by Mr. Morse. Find the area of the lot.
Find the cost of excavating a basement
8 ft. deep on the property at $1.25 per cubic yard.
6. How much will it cost at $ .32 a cubic yard to re-
move 1 ft. of dirt from a lot 100 ft. by 150 ft. ?
7. At $5.50 a cord, how much will a pile of wood
24 ft. long, 4 ft. wide, and 8 ft. high cost?
CIRCULAR MEASUREMENTS
237
MEASUREMENT OF CIRCLES AND CIRCUMFERENCES
288. 1. For the purpose of measurement circumfer-
ences of circles are considered to be divided into 360
equal parts, called degrees (°).
2. A portion of a circumference is called an arc.
3. What portion of the circumfer-
ence of the larger circle is arc a5?
What portion of" the circumference of
the smaller circle is arc dV ?
4. How many degrees are there in
arc ah ? in arc ac ? in arc aJW ? in
arc a! c^ ?
5. If the circumference of the larger circle is 24,90^0.
mi., how long is each degree on the circumference ? If
the circumference of the smaller circle is 6000 mi., how
long is each degree on the circumference ?
6. As the angle at 0^ the common center of the two
circles, increases or diminishes as fast as the arc suspended
by its sides increases or diminishes, the angle is also
measured in degrees. Thus, when the arc between two
radii is 90°, the angle formed at the center of the circle
by the radii is an angle of 90°, or a right angle.
7. How many arcs of 90° are there in a circumference ?
How many right angles can be formed at the center of a
circle ?
289. Arc and Angle Measure.
60 seconds ('') = 1 minute (')
60' = 1 degree (°)
860° = a circumference
360° = 4 right angles (rt. -4)
238
FORMS AND MEASUREMENTS
29t^. Latitude and Longitude.
1. Using a map, point to the equator ; to a meridian.
The equator is midway between what two points? Do
the meridians extend around the earth or only from pole to
pole ? What is a meridian ? How must two or more places
be loc^,ted to have the same meridian ? How must two or
more places be located so as not to have the same meridians?
Are the meridians of all places shown on the map ?
2. Places on the earth's surface may be located by two
measures taken from
two lines intersecting
at right angles.
3. The lines taken
for locating places on
a map are the equa-
tor and some selected
meridian, called the
prime meridian. The
meridian of the Royal
Observatory at Green-
wich (near London),
England, is taken by
most nations as the
prime meridian.
4. The distance in degrees north and south from the
equator is called latitude ; and the distance in degrees east
and west from the prime meridian is called longitude.
Places north of the equator are in north latitude. What
places are in south latitude ? in east longitude ? in west
longitude ?
5. In the figure, point to a place located 60° west
longitude and 45° north latitude ; 90° west longitude and
LATITUDE AND LONGITUDE 239
45° north latitude ; 30° east longitude and 45° south lati-
tude ; 45° west longitude and 45° south latitude. Give
the latitude and longitude of each point at which lines
intersect in the figure.
6. What is the difference in degrees between two
places, one 60° west longitude and the other 15° east
longitude? one 105° west longitude and the other 30°
west longitude ? one 45° north latitude and the other 60°
south latitude ?
7. How many degrees is it from the equator to the
North Pole ? from the equator to the South Pole ? from
the North Pole to the South Pole ? How many degrees
is it from any point on the equator halfway around the
earth ? one fourth way around the earth ?
8. The equatorial circumference of the earth is 24900
mi. What is the length in miles of a degree on the equator ?
9. What is the latitude of your home ? If the polar
circumference of the earth is 24800 mi., how far do you
live from the equator ? from the North Pole ?
10. Which is longer, a degree on the Arctic Circle, on
the Tropic of Capricorn, or on the equator ? Explain.
11. What is the greatest latitude that a place can have ?
12. What is the greatest longitude that a place can
have either east or west from the prime meridian ? Why ?
13. Where must a place be located to have a latitude of
0° ? Where must a place be located to have a latitude
of 0° and a longitude of 0° ?
14. If the prime meridian in the figure on p. 238 is
the meridian of Greenwich, locate on the figure your own
home ; the city of New York ; Chicago ; San Francisco ;
Rio Janeiro : Berlin.
7
-A
240 FORMS AND MEASUREMENTS
291. Longitude and Time — Local Time.
1. What part of the earth's surface receives the light
of the sun at any one time ? Why does the sun appear
to move from east to west?
2. How many hours does it take the earth to rotate
once on its axis ? Through how many degrees does any
meridian pass during each rotation ?
3. How many degrees of longitude pass under the
sun's rays during 24 hours ? during 1 hour ?
4. Which passes under the sun's rays first, the meridian
of your home or that of a place 15° east of you ? west of
you?
5. When the vertical rays of the sun fall on any part
of the meridian of your home, it is noon by sun time at
all places on the meridian. Is it then before noon or
after noon by sun time at places east of your home ? west ?
6. Since the earth rotates through 360° in every 24
hours, it must rotate through 15° each hour. Therefore
the difference in sun time between places 15° apart, in an
east and west line, is 1 hour.
7. When it is noon by sun time on the prime meridian,
what is the time at a place 15° E.? 15° W.? 30° E.?
30° W.? 45° W.? 60° E.? 75° W.? 90° W.? 105° W.?
^ tr 6 7
8. When it is noon by sun time on the principal
meridian, what is the longitude of a place at which it
is 11 A.M.? 1 P.M.? 10 A.M.? 2 P.M.? 9 A.M.? 4 P.M.?
9. What is the difference in sun time between places
30° W. and 30° E. ? 45° W. and 60° E. ? 30° W. and
60° W.? 60° W. and 105° W.?
10. When it is noon on the prime meridian, where is it
midnight? 9 a.m.? 9 p.m.? 6 a.m.? 6 p.m.?
LONGITUDE AND TIME
941
292. Standard Time.
To avoid the confusion that would arise if every place
used its own local time, in 1883 the railroads of the
United States and Canada agreed upon a system known
as standard time. Under this system the United States is
divided into four time belts, each approximately 15° in
width, and each having the local time of its central me-
O'idian, which is some multiple of 15°. These divisions are
named after the sections of the country embraced by
them as follows : Eastern, having the time of the meridian
of 75° W. ; Central, having the time of the meridian of
90° W. ; Mountain, having the time of the meridian
105° W. ; and Pacific, having the time of the meridian
120° W.
5 A.M. 6 A.M. 7 A.M. 8 A.M.
STANDARD TIME
BELTS
105
While the time belts are theoretically 15° in width, they are
actually wider or narrower than 15°. The irregularities of the
divisions are due to the fact that the railways find it convenient to make
the changes in time at the division termini that are nearest to 7|*
east or west of the central meridians. ~" *
^
242 FORMS AND MEASUREMENTS
293. Map Questions.
1. When it is noon in Philadelphia, what time is it in
Chicago? in Denver? in San Francisco? in New York?
2. When it is 9 A.M. in Chicago, what time is it in
San Francisco ? in Washington ? in New Orleans ? in
Denver? in Seattle? in New York?
3. A telegram was sent from Washington at 2 p.m.
and was received in San Francisco at 11 A.M. of the same
day. Explain.
4. At 10 P.M. the people of Los Angeles, Cal., were
reading the election returns of New York, which had
f been compiled at 11 p.m. of the same day. Explain.
5. The passengers on a west-bound train arrived in
Sparks, Nev., at 6.05 A.M. and after a stop of 10 min.
started on their journey at 5.15 a.m. Explain.
6. On leaving North Platte, Neb., the passengers on an
east-bound train found that their watches were all 1 hr.
behind time. Explain.
7. How many times must a person reset his watch in
traveling from Boston to San Francisco, if he wishes to
have correct time on the journey?
8. If the telegraph office in Chicago, 111., closes at 6
p.m., what is the latest time a message can be sent from
San Francisco in time to reach this office before it closes,
allowing 30 min. for delays in transmission ?
9. When it is noon by standard time in western
Iowa, is it earlier or later than noon by local time?
Name some place where it is 6 P.M. by standard time
before it is 6 p.m. by local time.
RATIO 243
RATIO ^
294. 1. What is the ratio of 4 ft. to G ft. ? Compare
the ratio of 4 ft. to 6 ft. with the ratio of 2 ft. to 3 ft.,
and with the ratio of 8 ft. to 12 ft.
2. What effect upon the ratio of two quantities has
(a) multiplying both terms by the same number ?
(6) dividing both terms by the same number ?
3. Name two quantities whose ratio is -|. Name two
other quantities having the same ratio. (
4. Name numbers whose ratio is expressed by the
fraction f, f, |, |, f , |, J.
5. Name whole numbers whose ratio is as | to 2 ;
as 3 to J ; as | to 6 ; as 6 to |.
6. Write ratios equivalent to the following, but with
one or both terms a fraction : 1 to 2 ; 1 to 4 ; 2 to 3 ; 2
to 1 ; 3 to 7.
4 X
7. Supply the number in place of x: -=-— . The
fractions are stated as equivalent fractions. Compare
their terms to find the number in the place of x.
8. The ratio between two numbers may be stated in
the form of a fraction. In | = | we have an equality of
ratios. The equality between ratios is called a proportion.
Qi 2_4 6_x 5_10 ^_12 ^_10
^ ''^- 5~x' i0~30' 7~ X ' 8~16' 4""l2'
10. If 20 bbl. of flour cost $ 80, how many barrels of
flour can be bought for $120 ? ($120 is one lialf again as
much as $80.)
11. In a certain city the ratio of the number of school-
census children to the total population is 1 to 4J. If the
school census is 20,000, what is the population of the city ? .
/ ;
PART V
POWERS AND ROOTS
295- 3x3 = 9. 3 is used twice as a factor to give 9.
9 is called the second power of 3. What number is the
second power of 2 ? of 4 ? of 5? of 10 ? of 1 ? of 12 ?
2. The second power of a number is called its square, as
the number of units in the area of a square surface is found
by taking the second power of the number denoting the
length of a side of the square.
3. The square of 3 may be indicated thus : 3^. Indi-
cate the square of 4 ; of 5 ; of 1 ; of 10 ; of 12. Give the
value of each : 1\ 82, 2\ 62.
The small figure written at the right and above indicates
how many times the number is to be taken as a factor and
is called the exponent of the number.
4. 3 X 3.x 3 = 27. 27 is the third power, or cube, of 3.
What number is the cube of 1 ? of 2 ? of 4 ? of 5 ? of 6 ?
of 10? of 12?
5. The cube of 3 may be indicated by an exponent,
thus : 33. Indicate the cube of 7 ; of 8 ; of 9. Give the
value of each : 1^, 2^, lO^, 123.
5* is read the fourth power of 5, or 5 to the fourth power ; it means
5x5x5x5. Read and tell meaning of : 6*, 3^ 2^.
6. Find the volume of a cube whose edge is 5 in. Find
the cube of 5.
7. Give the square of each of the numbers from 1 to 12.
8. Square J, |, j, .5, 1.5, .04, ^, 2J.
9. Find and memorize the cubes of 1, 2, 3, 4, 5, 6, 10, 12.
244
POWERS AND ROOTS 246
The process of finding a power of a number is some-
times called involution.
10. A number that is the square of some integer or
fraction is called a perfect square. Thus, 25 (5 x 5) and
II (5 X I) are perfect squares. Is 24 a perfect square ?
11. Square each : 20, 30, 40, 50, 60, 70, 100.
12. Is the square of 2 plus the square of 3 the same as
the square of 5 ?
296. 1. Which is the more and how much, 20^ + 5^ or
252?
2. The square of any number composed of tens and
units may be found thus :
20 + 5 The square of 25
20 + 5 is seen to be the
100 + 25 (20 + 5)x5 square of the tens,
^ ' y plus twice the
400 + 100 (20 + 5) X 20 ^..^uct of the
400 + 2(100) + 25 = 202+ 2(20 X 5) + 52 tens and the units,
plus the square of
the units.
3. Square as above : 23, 47, 105 (100 + 5).
4. The figure represents a square whose side is 25
units. The square whose side is 20 units
contains 400 square units. The two rec-
tangles 20 by 5 contain 100 square units "^
each. The square is completed by the
addition of the small square 5 by 5, o
containing 25 square units. The area
of the square is (400 + 2(20 X 5) + 25), ^^
or 625 square units.
20
5
too
25
•^
400
100
5. Construct a square whose side is 10 + 5 units.
>(
246 POWERS AND ROOTS
297. Roots.
1. Since 9 is the square of 3, 3 is the square root of 9 ;
that is, it is one of the two equal factors of 9. What
number is the square root of 4 ? of 25 ? of 64 ? of 36 ? of
49 ? of 16 ? of 144 ? of 100 ? of 81 ? of 121 ? of 1 ?
2. Since 27 is the cube of 3, 3 is the cube root of 27.
What is meant by the cube root of a number? What
number is the cube root of 1 ? of 125 ? of 8 ? of 1000 ? of
1728?
3. The sign ( V ') is called the radical, or root sign, and
is placed over a number to show that its root is to be
taken. The root to be taken is indicated by a small figure,
called an index, written in the radical thus, VST, which
is read the cube root of 27. The index 2 for square root
is usually omitted.
4. Read and give the roots : V64, "v^, V49, VTOO,
a/125, V81, V36, V144, -v^.
The process of finding the root of a number is some-
times called evolution.
298. Finding Roots by Factoring.
Roots of perfect squares may be found by factoring.
1. Find the square root of 324.
By factoring, 324 = 2 x 2 x 3 x 3 x 3 x 3.
Arranging the factors into two like groups,
324 = (2 X 3 X 3) X (2 X 3 X 3).
V324 = 2 X 3 X 3, or 18.
2. Find the cube root of 2744.
Factor 2774. Group the factors into three like groups. The
product of one of these groups is the cube root.
3. The square root of a fraction is the square root of
its numerator over the square root of its denominator,
thus : V|"= |.
SQUARE ROOT 247
299. Find the roots indicated : t^
5 4. ^3875 7. ^/^OOO 10. V1296
Wo 5. Vl29,600 8. Vff 11. V15,625
3. Vll,664 6. ^/'5T2 9. V^ 12. V6|
300. 1. Compare Vl = 1, Vlp = 10, and Vl|00|00
= 100. Notice that there is one figure in the square root
for each period of two figures each into which the square
can be separated, beginning at units. The period at the
left may contain only one figure. By separating any
number into such periods, the number of figures in the
square root may be told.
2. How many figures are there in the square root of
each: 11,664? 129,600? 11,025?
3. 1.22 = 1.44 ; 9.92 = 98.01; 1.222 = 1.4884. Notice
that there are two decimal places in the square for each
decimal place in the root.
4. How many decimal places are there in the square
root of each : 4.1616 ? 1190.25 ? 2550.25 ?
301. Square Root.
a. Find the square root of 529. b. Find the side of a
square whose area is 529 square units.
As the square root of some numbers cannot be found
by factoring, another method of finding the square root
of numbers is necessary. From Sec. 296 we see that the
square of a number is the square of the tens, plus twice
the product of the tens and units, plus the square of the
units; and from Sec. 300 we see that the number of fig-
ures in the square root of any number is the same as the
248
POWERS AND ROOTS
number of periods of two places each, beginning with
units into which the number can be separated.
Model: 5'29|23
202 = 4
2x20 = 40 129
(40+3) X 3=129
a. As 529 can be separated into two
periods, its square root consists of tens
and units. Since the square of tens is
hundreds, 5 hundreds must include the
square of the tens of the root. The
largest perfect square in 5 hundreds is 4
hundreds. The square root of 4 hundreds
is 2 tens. Write this in the answer at the right. The square of 2
tens is 4 hundreds. Subtract 4 hundreds from 529. The remainder
is 129. This remainder must be twice the product of the tens and
the units, plus the square of the units. Twice 2 tens is 40. The
units' figure of the root is found by taking 40 as a partial divisor.
40 is contained in 120 (omitting the 9, as it is evidently the square
of the figure in units' place, or a part of its square) three times.
Write 3 as the units' figure of the root. Use 43 as the complete
divisor. 3 x 43 = 129, which exhausts the remainder.
20 B
60
A
AGO
20
D3
h. As the largest perfect square in
5 hundred square units contains 4
hundred square units, its side is 20
units {A). 129 square units remain to
be added in such form as to keep the
figure a square. It is evident that
these units must be added along two
adjacent sides, as B and C, and at the
corner, as D. The combined length of
the two rectangles, B and C, is 40
units. Their width may be deter-
mined from the fact that their com-
bined areas, plus the area of the small
square D, is 129 square units. Omitting the 9, as it evidently is the
number (or a part of the number) of square units in the small square,
120 square units -^ 40 square units = 3, the number of units in
the width of the rectangles, and also in the side of the small
square.
SQUARE ROOT 249
302. To extract the square root of a number :
1. Separate the number into periods of two figures
each, beginning at the decimal point.
2. Find the greatest square in the left-hand period,
and write its root for the left-hand figure of the required
root.
3. Subtract the square from the left-hand period,
and bring down the next period to form the complete
dividend.
4. Double the part of the root already found, and
place it at the left of the dividend for a partial divisor.
Disregarding the right-hand figure of the dividend, divide
by the partial divisor. The quotient (or quotient di-
minished) will be the next figure of the root.
5. Annex the root figure last found to the partial divi-
sor for a complete divisor. Multiply the complete divisor
by the root figure last found. Subtract the product from
the dividend, bring down the next period to form the
complete dividend, and continue as before.
303. Written Exercises.
Square roots of numbers that are not perfect squares may be ap-
proximated by annexing periods of two decimal ciphers and continu-
ing the process to several decimal places in the roots.
Extract the square root of each :
1. 841 / 4 56.25 ^771600
;2. 104,976 5. .6724 8. 10.24
3844 6. 160 U^. 007225
10. Find the side of a square whose area is 256 sq. rd.
11. Find the side of a square field whose area is 10 A.
12. Find the perimeter of a square 40-acre field.
\
250
POWERS AND ROOTS
D
304. 1. Draw a right angle. Draw a rectangle. Draw
a diagonal through the rectangle. Into how many equal
triangles does the diagonal divide the rectangle? What
kind of triangles are they ?
2. The longest side of a right triangle is called its
hypotenuse, and the other two sides are called its legs.
3. Draw a right triangle whose legs are 6 in. and 8 in.
Measure the length of the hypotenuse. Construct squares
upon each of the three sides, and divide them into square
inches. Compare the number of square inches in the
square on the hypotenuse with the number in the other
two squares together.
4. The figure represents a
right triangle whose legs are
3 units and 4 units and whose
hypotenuse is found to be 5
units. Compare the number
of units in the square upon
the hypotenuse with the
number of units in the sum
of the squares upon the other
two sides.
The square of the hypotenuse
of a right triangle is equal to
the sum of the squares of the other two sides.
Answer the following from the figure:
5. If the number of squares in A and B are given, how
may the number in C be found ?
6. If the number of squares in B and C are given, how
may the number in A be found ?
7. If the number of squares in A and C are given, how
may the number in B be found ?
^/<>'
xyi 3
ri "
B
A
-<-
RIGHT TRIANGLES
261
305. Written Exercises.
1. Find the length of the third side of each
15ft
12ft
20ft eOrd. 9ft
2. Find the length of the diagonal of the floor of a
rectangular room 14 ft. by 16 ft., to the nearest thousandth
of a foot.
3. A boy stood on the ground 45 ft. from the foot of
a tree 60 ft. in height. How far was it in a straight line
from the boy's feet to the top of the tree ?
4. How much less is the distance along a diagonal path
across a rectangular field 40 rd. by 80 rd. than the dis-
tance around two sides of the field ?
5. How long must a rope be to reach from the top of a
60-ft. pole to a point on the ground 30 ft. from the foot of
the pole ? ^
6. Find the diagonal of a square field whose side is
40 rd.
7. Find the side of a square whose diagonal is 60 ft.
8. Find the diagonal of a rectangular room 20 ft. by
26 ft. If the ceiling of the room is 10 ft. from the floor,
what is the distance from one of the lower corners of the
room at one end of the diagonal on the floor to the upper
corner at the other end ?
9. If A is 85 mi. south of B, and C is 75 mi. west of
B, how far is it from A to C ?
10. One side of a rectangular field is 40 rd. The diag-
onal is 50 rd. Find the other side.
11. Find the diagonal of a 10-acre square field.
252 POWERS AND ROOTS
306. Laying off a Rectangle.
1. When the two sides of a rectangle are in the ratio
of 3 units to 4 units, the diagonal is one fourth more than
the longer side, thus : If the sides of a rectangle are 18 ft.
and 24 ft., the diagonal is 24 ft. plus 6 ft., or 30 ft.
Prove that this is correct and that it holds with various
rectangles when the ratio of the sides is as 3 to 4.
2. A farmer asked two schoolboys to lay off a rec-
tangle 16 ft. by 24 ft. to mark the foundation of a car-
riage house. The boys used two
pieces of cord and a measure. They
tied the cords to two stakes so that
they crossed at ^, a corner of the
rectangle, and extended one cord in
the direction of cd and tlie other in
the direction of ce. To make the
angle at <? a right angle and thus to get the positions of
the two sides of the rectangle, they measured off from c
15 ft. on one cord and 20 ft. on tlie other cord. With
these as the legs of a right triangle, they adjusted the
position of the cords so as to make the distance hi 25 ft.
Having determined the direction of the sides cs and ce^
they measured 24 ft. on cs and 16 ft. on ce and marked
the corners « and i. From « they extended a line in the
direction so, and from i a line in the direction io. They
measured 16 ft. on the line so and 24 ft. on the line io.
They marked the point o where the two measured lines
met. They tested their work, finding that the diagonals
CO and si were equal.
3. Using cords and a measure, lay out on the school
grounds rectangles 12 ft. by 16 ft. ; 20 ft. by 30 ft.
4. Lay off a baseball " diamond " whose side is 60 ft.
SIMILAR FIGURES 268
SIMILAR SURFACES AND SOLIDS
307, 1. Draw squares whose sides are 1 incli; 2 inches;
3 inches. Find their areas. The areas of the three squares
are to each other as 1^, 2 2, and 3 2.
2. Express the ratio of the areas of a 4-inch square and
a 6-inch square.
3. Draw circles whose diameters are 4 inches ; 6 inches.
As the area of a circle is Trr^, the ratio of the areas of these
two circles is as 2^ is to 3 2. Explain.
4. Is the ratio of the squares of the diameters of two
circles the same as the ratio of the squares of their radii ?
5. Is the ratio of the squares of the diagonals of two
squares the same as the ratio of the squares of their sides ?
6. Figures that are of exactly the same shape are called
similar figures. Draw two similar figures. Are similar
figures necessarily the same in size ?
The areas of similar plane figures are proportional to the
squares of their corresponding lines,
7. Find the volume of a 2-inch cube ; of a 3-inch cube.
Their volumes are in the ratio of 2^ to 3 3.
The volumes of similar solids are proportional to the cubes
of their corresponding lines,
8. Compare the volumes of two spheres, one 5 inches in
diameter and the other 10 inches in diameter.
As the diameter of the larger sphere is twice the diameter of the
smaller, the volume of the larger is 2* times the volume of the smaller.
Explain.
9. Compare the weights of two solid iron spheres of the
same density, if one is 2 inches in diameter and the other
is 4 inches in diameter.
254 MISCELLANEOUS EXERCISES
MISCELLANEOUS EXERCISES
308. 1. A merchant gained -f 28 by selling some goods
at a profit of 20 % . Find the cost of the goods.
2. An agent who canvassed for a book received 40%
of the amount of the sales for selling and delivering the
books. Find the amount of his commission for selling
and delivering 60 copies at 11.25 per copy.
3. If the amount received for goods sold averages
20 % more than the cost of the goods, find the net profit
for a month when the sales amounted to $24,000 and the
expenses for the month amounted to f 3000.
4. 20 % of the pupils enrolled in a certain school were
absent one stormy day. Twenty-four pupils were present.
Find the number enrolled.
5. A commission merchant sold $6000 worth of prod-
uce at a commission of 1|%. Find the amount of his
commission.
6. If a collector received 20 % commission for collect-
ing a bill of $17.75, what was the amount of his commis-
sion? What per cent of the amount of the bill did the
creditor receive?
7. A man paid $42 taxes when the tax rate was 2%.
What was the assessed valuation of his property ?
8. If a person pays $50 tax on property when the rate
is 1-| %, what is the assessed valuation of his property ?
9. A miller bought a ton of wheat through a broker,
who charged a commission of 2 %. What was the amount
paid for the wheat if the cost of the wheat and the broker-
age amounted to $25.50 ?
The cost of the wheat plus 2% of the cost of the wheat, or 102% of
the cost of the wheat, was $25.50. Prove your answer.
MISCELLANEOUS EXERCISES 265
10. A fruit grower shipped 25 boxes of apples to a
commission merchant, who sold them at 85^ per box,
charging 4 % commission. He was directed to invest the
proceeds in groceries, after deducting a commission of 2 %
for making the purchase. Find the amount expended
for the groceries.
The net proceeds of the sale of the apples was $20.40, which was
102 7o of the amount invested in groceries. Prove your answer.
11. After selling 25 % of his interest in a flour mill, a
man considered his remaining interest worth $ 9000. At
this rate, what was the value of his interest before making
the sale ?
12. A real estate broker bought two 60-ft. lots, adjoin-
ing, for $ 1500 apiece, and divided them into three lots of
equal frontage which he sold for #1200 apiece. What
per cent of profit did he make?
- 13. At $ 9 a ton, find the cost of two sacks of coal, each
weighing 100 lb.
14. A man bought a lot for $ 1200 and sold it for 1 1400.
He bought it back for $1500 and resold it for $1600.
How much did he make on the lot ?
15. A man bought four 50-ft. lots and divided the land
into 40-ft. lots, which he sold at the same price per lot as
he had paid. Find his gain per cent.
16. At 22 ^ a square foot, find the cost per front foot of
paving a street 60 ft. wide. Find the cost per front foot
to a property owner who pays for half the width of the
street. Find the cost to the property owner of paving the
street in front of a 45-ft. lot.
17. If 27 tons of coal cost 1 243, how many tons can be
bought for 1 189 ?
PART VI
APPENDIX
CORPORATIONS, STOCKS, AND BONDS
309. 1. Corporations. A large business enterprise frequently
requires more capital than one person may care to invest in
it. Provision is made in the laws of the various states
whereby a number of persons may organize a company, called
a corporation, to engage in business as one body. Sometimes
all the necessary capital is subscribed by the persons who organ-
ize the corporation, but often the organizers of a company sub-
scribe only a part of the capital.
The laws regulating the incorporation of companies differ considerably
in the several states. Frequently a corporation intending to transact busi-
ness in one state will incorporate in another state, because of certain ad-
vantages to be derived thereby.
2. Railway companies, mining companies, express com-
panies, oil companies, and banking institutions are among the
largest business corporations.
310. Shares of Stock. 1. Each corporation is capitalized for
a special amount, as $ 25,000, $ 50,000, $ 1,000,000, etc. The
capital is divided into shares, usually of $ 100 each or of $1
each. Thus, a corporation that is capitalized for $ 100,000 may
issue 1000 shares of the face value of $ 100 each, or 100,000
shares of the face value of $ 1 each, etc. These shares are
bought by persons who invest in the enterprise. Each person
who owns one or more shares of stock is called a stockholder.
The several stockholders constitute the corporation.
266
CORPORATIONS, STOCKS, AND BONDS 257
2. Every stockholder receives a certificate of stock, showing
the number of shares he owns and the face value, or par
value, of each. These certificates are negotiable, and a
record of their transfer is usually made on the books of the
corporation.
The affairs of a corporation are managed through a board of directors,
elected by the stockholders, each stockholder having as many votes as the
number of shares of stock he owns.
311. Value of Stock, l. The price at which stocks are
bought and sold in the stock market is called their market value.
When the market value of stock is the same as its face value,
the stock is said to be at par. Stock is said to be at a premium^
or above par, when its market value is more than its face value,
and at a discount, or below par, when its market value is less
than its face value.
2. Examine the stock quotations in a newspaper. Can you
tell from the quotations what the face value of the stock is ?
Tell which stock is at par, above par, below par.
312, Dividends. The net earnings of a corporation, after
a surplus sufficient to cover the probable needs has been re-
served, are divided among the stockholders according to the
number of shares owned by each. These divided profits are
called dividends. Dividends are computed on the par value of
the stock, and are declared annually, semiannually, quarterly,
etc.
Illustration. If the amount of capital stock is $ 500,000
and the amount to be divided among the stockholders is
$25,000, the rate of dividend is $ 25,000 --$ 500,000, or 5%.
A stockholder who owns stock of the face value of $10,000
will receive 5% of $10,000, or $500.
Stock that regularly pays a large dividend is usually regarded as a good
investment, and is likely to be above par. When the dividends are not
equivalent to a fair rate of interest on the investment, the stock is likely
to be below par.
MccL. & Jones's essen. of ar. — 17
258 APPENDIX
313. Corporations sometimes issue two kinds of stock, called
preferred and common. When both kinds of stock are issued,
the holders of common stock are not entitled to participate in
the profits until a fixed rate of dividend has been paid to
holders of preferred stock.
314. Stock Brokers. 1. A person who is engaged in buying
and selling stocks for others is called a stock broker. Stocks are
usually bought and sold through brokers, generally at a regu-
lar meeting place for transacting such business, called a stock
exchange.
2. The commission of a broker is called brokerage. The rate
of brokerage varies in different parts of the country from
1% of the par value for buying and also for selling, to ^% or
more. A minimum amount is fixed for making small sales
and purchases.
The standard amount of stock bought and sold is 100 shares, although
a smaller amount may be negotiated. As a rule, fractions of a share
cannot be bought.
In stock quotations fractions are always expressed in halves, fourths,
and eighths. See quotations, p. 259.
315. Assessments. When the funds of a corporation are not
sufficient to carry on its business, an assessment is sometimes
levied upon the stock. The assessment is usually some number
of cents per share, and the failure to pay it is generally pun-
ishable by the forfeiture of the stock.
316. Examine a newspaper for stock quotations. What
is meant by par value ? premium ? dividend ? market value ?
brokerage ? Name some of the corporations engaged in busi-
ness in the state in which you live. How many shares of
stock of the par value of $ 100 each are issued by a corporation
tliat is capitalized for ^ 500,000 ? for $ 2,000,000 ? What is a
stockholder ?
CORPORATIONS, STOCKS, AND BONDS 259
Stock Quotations
317, New York Stock
San Francisco Stock .
A.ND
Exchange
Exchange Board
Mar. 9,
190T
Jan. 22,
1907
Mar. 9,
190T
Mining
Jan. 22,
1907
Adams Express
295
300
Caledonia
.43
.67
American Express 223
242
Confidence
1.05
1.30
C. & N. W.
154|
190^
Combin. Frac.
4.45
5.50
Denver & Rio G.
31f
39^
Jumbo Exten.
1.85
1.90
Denver & Rio G. ,
pfd. 72
81
Mustang
.27
.21
Illinois Central
147|
166
Ophir
2.65
3.10
Northern Pacific
137 1
155^
Red Top Exten.
.50
.37
Southern Pacific
84|
95|
Silver Pick
1.35
1.45
Pullman Car Co.
165
173|
St. Ives
1.82
.93
Union Pacific
1561
176|
Utah
.06
.08
Do. preferred
88
921
Vernal
.20
.25
Wells Fargo Ex.
280
300
West End
1.40
1.90
Western Union
80f
83|
Yellow Jacket
1.05
1.15
The par value of the stock quoted in the left-hand column is $100 per
share, and in the other column is $1 per share. On Jan. 22 the mar-
ket was high, and on March 9 low. Which of the stocks quoted are above
par ? below par ?
At the above prices, find the cost without brokerage, on
March 9, of 100 shares of Wells Fargo Express stock; of
100 shares of Chicago and Northwestern Railway stock ; of 100
shares of Western Union stock ; of 100 shares of stock in the
Ophir mine; of 100 shares of stock in the Silver Pick mine.
Find the same for the prices given for Jan. 22.
318. Written Exercises.
1. Find the cost of 100 shares of Western Union stock at
30|, including \ % brokerage.
Market price of each share, $80|.
Brokerage on each share, ^ \.
Cost of each share (including brokerage), $80|, or $80.76.
Cost of 100 shares (including brokerage), $8075.
260 APPENDIX
2. Find the cost of 100 shares of Northern Pacific stock as
quoted for March 9, including | % brokerage.
3. Find the net proceeds of the sale of 100 shares of Southern
Pacific stock as quoted for Jan. 22, allowing |% brokerage.
Market price of each share, $95f.
Brokerage on each share, $ J.
Net proceeds on each share, $95|, or $95.25.
Net proceeds on 100 shares, $ 9525.
4. Find the net proceeds from the sales of 400 shares of
Illinois Central stock as quoted for March 9, allowing J%
brokerage.
5. How much would a man clear by buying 100 shares of
stock at 142^ and selling them at 150|, allowing ^ % broker-
age both for buying and for selling ?
6. If the Pullman Car Company declares a dividend of 10 %,
how much will a person receive who owns 100 shares of the
stock ?
Par value of 100 shares, $10,000. His dividend will amount to 10% of
$10,000.
7. If the Union Pacific Railway declared a dividend of
8%, how much did a person receive who owned 500 shares of
the stock ?
8. Find the cost of 100 shares of stock at $1.20 a share, in-
cluding J % brokerage. If a 10 % dividend is declared on this
stock, how much does a person receive who owns 100 shares of
the stock ?
9. What per cent does a person receive on the amount in-
vested who buys 100 shares of stock at $1.20, paying ^ %
brokerage, and receives a dividend of 10 % ?
The dividend amounts to 10% of $ 100. This amount is what per ceot
of $ 120.50, the cost of the stock ?
CORPORATIONS, STOCKS, AND BONDS 261
10. A man bought 100 shares of Bed Top Extension stock
at 37 cents, paying $ 1 brokerage. Find the cost of the stock.
If he received a dividend of 5 %, what per cent did he receive
on his investment ?
$5, the dividend, is what per cent of $38, the amount invested ?
11. What per cent did a man make on his investment who
bought 100 shares of stock at 45 ^ and sold them at 55/, pay-
ing $ 1 brokerage both for buying and for selling ?
Cost of the shares, $46; received for the shares, $64; net profit, $9.
$9 is what per cent of $46 ?
319. Corporation Bonds. The promissory note of a corpora-
tion, issued under seal, is called a bond. The bond of a cor-
poration is secured by a mortgage.
320. A city or an incorporated village is called a municipal
corporation.
321. 1. Governments, states, cities, counties, etc., are some-
times obliged to issue bonds to meet urgent demands for money
and to provide needed improvements. Such bonds are not
secured by mortgages. The integrity of the government
issuing a bond is accepted as sufficient security for its pay-
ment. The bond of a government is a certificate of in-
debtedness, with a promise to pay a certain sum to the holder
of the bond, with a fixed rate of interest at a specified
time, as at the expiration of five years, twenty years, fifty
years, etc.
2. Under what conditions is it sometimes necessary for the
United States government to borrow money ? For what pur-
poses are cities frequently bonded ? What is a bond election ?
Which is the better security, the note of an individual or the
bond of the national government ?
322. Bonds that are registered by number and in the name
of the holder are called registered bonds. Bonds having interest
certificates attached in the form of coupons are called coupon
bonds.
262 APPENDIX
Bond Quotations
323. Examine a newspaper for bond quotations. The fol-
lowing quotations are from a newspaper report of prices on
the New York Stock Exchange.
Japan 6's 99| Southern Pacific 4's . . 84
Mexican Central 4's . 83 U. S. New 4's reg. . . . 129^
Northern Pacific 4's . 100| Do. coupon .... 129^
The par value of the bonds quoted is $100 each. Bonds are bought
and sold in the market in the same manner as stocks. Government
bonds are exempt from taxation.
Find the cost of Mexican Central bonds of the face value of
$1000, including | % brokerage.
COMMISSION AND BROKERAGE
324, Producer and Consumer. A person who grows agri-
cultural products or manufactures useful articles out of crude
materials' is called a producer. A person who uses up products
is called a consumer. Name some producers of foods, of cloth-
ing, of fuel, of building materials. Name some products that
are consumed by nearly every one. Name some classes of
persons who are consumers but are not producers.
In early times there was comparatively little buying and selling, and
the exchange of products was very limited. Each family produced nearly
everything that it consumed. At that time, most of the trade was directly
between the producer and the consumer. With the invention of machin-
ery and with improved means of transportation, cities increased rapidly
in number as centers of manufacture and trade. People began to devote
themselves more particularly to special lines of work. As trade con-
ditions grew more complex, it became more difficult for the producer to
trade directly with the consumer. Wholesale and retail establishments
developed as agencies for marketing products.
Middleman. A person who deals between the producer and the con-
sumer is known in trade as a middleman. Products are often handled by
several middlemen before they reach the consumers. The middlemen are
generally persons who make buying and selling products their special
occupation. Is the retail dealer a middleman ? Which of the following
COMMISSION AND BROKERAGE 263
are middlemen : farmers ? stock buyers ? hardware merchants ? carpen-
ters ? shoemakers ? hay and grain dealers ? What are some of the con-
ditions that make it inconvenient for persons living in large cities to buy
agricultural products directly from farmers? What are some of the
conditions that make it inconvenient for farmers to buy their clothing and
tools directly from the manufacturers ? Which contributes to the vs^ealth
of a country : the producer, the consumer, or the middleman ?
325. Commission. A person who transacts business for an-
other frequently receives as his pay a certain rate per cent on the
amount involved in the transaction. This is known as his com-
mission. One who buys or sells produce for another, receiving as
his pay a certain rate per cent on the cost of the products bought
or on the selling price of the products sold, is called a com-
mission merchant. A commission house is an establishment con-
ducted by a commission merchant, where products are received
and sold to retail dealers or consumers. Why are the commission
houses located in the cities ? If your home is in a city in which
there are commission houses, tell in what section of the city they
are located. If your home is in the country, tell what products
are shipped from your community to commission merchants.
Farmers frequently dispose of their products by shipping them to
commission merchants in the cities, who sell the products at the market
prices and retain as their pay a certain per cent of the selling price.
Commission merchants do not usually buy the products shipped to them,
but they act merely as the agents of the shippers in receiving and selling
the goods. The freight charges for shipping and the cartage charges for
hauling the goods are generally paid by the commission merchants from
the proceeds of the sale of the products. After deducting their commis-
sion and the charges for freight and cartage, the commission merchants
remit to the shippers the balance of the sum received for the products sold.
The entire amount received from the sale of goods, before any deduc-
tion is made for expenses, etc., is called the gross receipts of the sale.
The amount remaining from the sale of goods after all expenses have been
deducted is called the net receipts of the sale.
Where do city retail dealers buy their supplies of fruits, vegetables,
etc. ? Farmers frequently sell or exchange small quantities of agricul-
tural products at the general merchandise stores in small cities and villages.
How do these merchants dispose of the products ?
264 APPENDIX
326. Market Reports. Newspapers publish market reports^
giving the prices at which grain, live stock, dairy products,
fruits, etc., were sold on the previous day. Of what use are
such reports ? Read a recent market report. The price of
produce is affected by the supply offered for sale and the
demand for it. What effect upon prices has an increase in
supply and a decrease in demand ? How is the price affected
by an increase in demand and a decrease in supply ?
327. Brokerage. 1. One who acts as an agent for others to
contract for the purchase or sale of goods, receiving as his
pay a certain rate of commission, is called a broker. The com-
mission of a broker is called brokerage. Commission mer-
chants usually take possession of the goods bought and sold
by them, while brokers merely contract for the sale or purchase
of goods in the name of the person buying or selling, without
taking possession of the goods. Brokers deal in stocks, bonds,
grain, etc.
2. What is real estate ? One who buys and sells lands, ex-
changes and leases property, etc., is called a real estate agent,
or a real estate broker. He usually receives as his commission
a certain rate per cent on the selling price of property, or on a
month's rent when property is leased by the month. The
rates of commission charged for selling and renting property
vary in different communities. Find what rate of commission
is charged by agents for selling and renting property in your
community. What are some of the conditions that make it
inconvenient for each person to sell or rent his own property ?
When a person wishes to rent a house in a city, how does he
find out what houses are for rent ?
3. Traveling salesmen, store clerks, auctioneers, insurance
agents, etc., frequently receive as pay for their services a com-
mission on the amounts of their sales. Can you name any
other business in which those employed receive a commission
for their services ?
TRADE DISCOUNT 265
TRADE DISCOUNT
328. 1. Manufacturers and wholesale dealers issue cata-
logues and price lists in which the articles manufactured by
them are described and their prices given. Most manufacturers
sell their goods to wholesale dealers, who supply, in turn, the
retail dealers. The prices quoted in catalogues and price lists
are commonly known as the list prices of goods. A discount
from the list prices is generally made to retail dealers, and
sometimes to others when goods are bought in large quanti-
ties. This discount is reckoned as a certain rate per cent of
the list price. Such a discount is generally known as trade
discount.
2. Often several discounts are allowed. Thus, an article
may be sold subject to discounts of 25 %, 10 %, and 5 % from
the list price. In such a case, the first discount is from the list
price, and the second discount is from the price after deducting
the first discount, and the third discount is from the price
after deducting the second discount. Usually the amount of
discount allowed is varied as the market prices change.
3. A special discount is usually made when cash is paid for
goods. The amount of cash discount varies considerably in
different lines of trade, ranging generally from i % to 6 % of
the amount of the bill after deducting the trade discount, and
averaging about 2 %. Generally the retail dealer is given
until about the tenth day of the following month in which to
make cash remittances. Instead of allowing a special discount
for cash payments, sometimes an arrangement is made by
which the retail purchaser is given 30 days, 60 days, 90 days,
6 months, or even a longer time in which to make his payment.
The bill then becomes due at the end of the specified time, and
in case it is not paid when the time has expired, the purchaser
of the goods agrees to pay a specified rate of interest upon
the amount of the bill from the time the bill is due until it is
paid.
266 APPENDIX
PARTIAL PAYMENTS
329, Instead of paying the whole amount of a note at one
time, the maker sometimes pays it in two or more parts. Such
payments are called partial payments. A record of each pay-
ment is indorsed on the back of the note.
330. United States Rule. The method of computing in-
debtedness when partial payments have been made, illustrated
in the following problem, was adopted by the Supreme Court
of the United States, and is commonly known as the United
States Rule. This method is the legal one in most states.
In states where other methods are legal, teachers should follow thera.
1. A note for $ 2000 dated May 1, 1906, at 6%, was indorsed
as follows: July 25, 1906, $150; Dec. 16, 1906, $40; Feb. 12,
1907, $ 100. Find the amount due May 1, 1907.
Principal, May 1, 190a ^2000
Interest on $ 2000 to July 25 (2 mo. 24 da.) . . . . 28
Amount, July 25, 1906 $2028
First payment (July 25, 1906) 150
New Principal, July 25, 1906 § 1878
Interest on $ 1878 to Dec. 16, 1906 (4 mo. 21 da.) . 44.13
Second payment, which is less than the interest due,
$40
Interest on $ 1878 from Dec. 16, 1906, to Feb. 12, 1907
(1 mo. 26 da.) 17.53
Amount, Feb. 12, 1907 $ 1939.66
Third payment, $100, which is to be added to the
second, $40 140
New Principal, Feb. 12, 1907 § 1799.66
Interest on $1799.66 to May 1, 1907 (2 mo. 19 da.) . 23.69
Amount due May 1, 1907 $1823.35
If the second payment, $40, which was less than the interest due, had
been deducted from the amount due at the time the payment was made,
and if the remainder had been regarded as a new principal, the effect
would have been to increase the amount on which interest was paid.
Hence, the interest must be reckoned to the time of the third payment.
INTEREST
267
BuLE. Fi7id the amount of the principal to a time when a
payment, or the sum of two or more payments, equals or exceeds
the interest due.
Subtract the payment or payments from the amount.
Treat the remainder as a new principal and proceed as before.
2. Write a note for $ 800, naming the teacher as payee and
yourself as maker. Make three partial payments and have
them indorsed on the note. Find the amount due at the time
of settlement.
3. Write a note for $ 1000, naming some pupil as payee and
yourself as maker. Make two payments, such that the first is
less than the interest to date, but the sum of both exceeds the
interest to time of the second payment. Find the amount due
at time of settlement.
INTEREST
331. Bankers' Table of Days Betweex Dates
Jan.
Feb.
Mar.
Apr.
May
June
JUL^
Aug.
Sept.
Oct.
Nov.
Dec.
Jan.
365
31
59
90
120
151
181
212
243
273
304
334
Feb.
334
365
28
59
89
120
150
181
212
242
273
303
Mar.
306
337
365
31
61
92
122
153
184
214
245
275
Apr.
275
306
334
365
30
61
91
122
153
183
214
244
May
245
276
304
335
365
31
61
92
123
163
184
214
June
214
245
273
304
334
365
30
61
92
122
153
183
July
184
215
243
274
304
335
365
31
62
92
123
153
Aug.
153
184
212
243
273
304
334
365
31
61
92
122
Sept.
122
153
181
212
242
273
303
334
365
30
61
91
Oct.
92
123
151
182
212
243
273
304
335
365
31
61
Nov.
61
92
120
151
181
212
242
273
304
334
365
30
Dec.
31
62
90
121
151
182
212
243
274
304
335
365
The exact number of days from any day of one month to the same
day of another month within a year is found thus : Find the number of
days from April 12 to Aug. 12. Starting from April, in the left-hand col-
unm, pass the pencil across to the column headed August. The number
122 in the column headed August denotes the number of days from any
day in April to the corresponding day in August. Hence it is 122 days
from April 12 to Aug. 12.
268 APPENDIX
1. How many days must be added to 122 days if you are
required to find the number of days from April 12 to Aug.
14 ? to Aug. 20 ? How many days must be subtracted from
122 days if you are required to find the number of days from
April 12 to Aug. 8 ? to Aug. 3 ?
2. What change would be made in the number of days
given in the table if Feb. 29 of a leap year should intervene
between dates ?
332. Written Exercises.
Using the table in Sec. 331, find by the 60-day method
(p. 212) the interest on :
1. $ 2000 from April 1 to July 4 at 6 %.
2. $ 4500 from Dec. 20 to Feb. 15 at 7 %.
3. $ 20,000 from May 12 to Aug. 5 at 5 %.
4. $ 80,000 from April 16 to June 4 at 6 %.
5. $25,000 from July 4 to Sept. 16 at 8 %.
6. $4370 from Aug. 4 to Dec. 7 at 6 %.
EXACT INTEREST
333. 1. In the methods of computing interest in Sees. 255-
261, 360 days were taken as one year in reckoning the interest
for periods less than a year. Interest computed on the basis
of 365 days to the year is called exact interest.
2. Exact interest is found by taking such a fractional part
of a year's interest as the exact number of days is of 365 days.
3. Find the exact interest on $ 400 from March 2 to June
15 at 8%.
Find the exact number of days from March 2 to June 15.
Do not count March 2, but count June 15. From March 2 to
June 15 there are 29 da. + 30 da. + 31 da. + 15 da. or 105 da.
Model : 21
$400 X .08 X W = e9.20+.
m
78
TAXES 26d
4. Compare the interest on $ 400 for 105 da. at 8 % com-
puted in the ordinary way with the exact interest.
5. Exact interest for whole years is the same as interest
computed by the ordinary method. For periods of less than a
year, the exact interest is |f-| of the interest computed by the
ordinary method, or -^-^ (y^g) less than the ordinary interest.
Compared with interest computed by the ordinary method, is
exact interest favorable to the person loaning money or to the
person borrowing it ?
6. Exact interest is computed by the United States govern-
ment and sometimes by others.
7. How much interest does the United States government
save by paying exact interest rather than ordinary interest on
$ 5,000,000 from Jan. 1 to April 1, 1907, at 3 % ?
8. Find exact interest on each of the amounts in Sec. 332
for the time specified.
TAXES
334. 1. Every person in the United States derives some
benefits from one or more of the following divisions of govern-
ment : national (also called federal), state, county, township, school
district, city, and village. Explain the purpose of each of these
divisions of government and name some of the benefits you
derive from each. The maintenance of these several forms
of government necessarily involves the expenditure of money.
This money is derived from taxes levied upon the persons,
property, incomes, or business of individuals. Mention some
purpose for which money is expended by each division of
government named above.
2. Direct taxes are sums of money levied upon persons, prop-
erty, incomes, or business of individuals for the support of
state or local governments. They are called direct taxes be-
cause they are levied directly upon the persons, lands, build-
ings, etc., of individuals and are payable directly to public
officials authorized to collect taxes.
270 APPENDIX
3. Indirect taxes are sums levied upon goods imported from
other countries, upon certain products manufactured in our
own country, and upon the privilege of engaging in certain
pursuits, as selling liquors, etc. They are called indirect taxes
because they are paid indirectly by the consumers.
4. Ascertain how the money is raised for building school-
houses, for repairing roads or streets, etc., in the community
in which you live.
STATE AND LOCAL TAXES
335. Poll Tax. Each male inhabitant of voting age (generally
with certain exceptions) is usually taxed a fixed sum annually,
regardless of the ownership of taxable property. This is called a
poll tax. Poll means head. Is a poll tax a direct or an indirect
tax ? It is the only form of personal tax levied in the United
States. What are the provisions relating to poll tax in your state ?
336. 1. Property Tax. Property is classified either as per-
sonal property or as real property. All movable property, as
household goods, money, cattle, ships, etc., is called personal
property. Immovable property, as lands, buildings, mines, etc.,
is called real property. Both forms of property are subject to
taxation.
2. For the purpose of taxation, the value of all taxable prop-
erty is estimated by a public officer called an assessor. This
officer prepares a list of all taxable property in his district,
showing the names of the owners, the location of the property,
and its assessed valuation. Such a list is commonly called an
.assessment roll. The assessed valuation of property is gen-
erally less than its actual value.
3. Taxes are usually collected by a public officer called a
tax collector, who deposits the amounts collected with another
public officer called a treasurer.
4. What is public property ? What properties, other than
public property, are exempt from taxation in your state ?
CUSTOMS S71
5. The total assessed valuation of the property in any state
is the sum of the assessed valuation of the property in the
several counties of the state. How is the assessed valua-
tion of the county in which you live determined ? The state
and local taxes are usually levied and collected together. As-
certain how state, county, and local taxes are levied and col-
lected in your state.
6. Examine a receipt for the payment of taxes on real
property. How is the location of the property described?
337. Written Exercises.
1. The assessed valuation of a certain county is $8,000,000.
Find the rate of taxation that will yield revenue sufficient to
pay for a new courthouse costing $ 100,000. How much must
Mr. Thomas pay toward the building of the courthouse if his
taxable property is assessed at $4000?
2. Find the value of Mr. White's property if his taxes
amount to $67.50 when the tax rate is $.01|- and his property
is assessed at | of its value.
3. The rate of taxation in a certain county is $.01| when
the assessed valuation is f of the actual value of the property.
What would the rate of taxation have been to yield the same
amount had the property been assessed at its full value ?
4. Make and solve five problems in taxes, using if possible
the actual tax rates of the community in which you live.
CUSTOMS AND INTERNAL REVENUE
338. 1. Funds for the maintenance of the national gov-
ernment are derived chiefly from customs and internal revenue.
2. Customs, or duties, are taxes levied by the national govern-
ment upon goods imported from other countries. Internal
revenue is a tax levied by the national government upon the
manufacture or sale of certain articles in the United States.
272 APPENDIX
339. The following tables show the receipts and expendi-
tures of the national government during the fiscal year ended
June 30, 1906:
Revenues, Fiscal Year 190G
Customs $300,251,877.77
Internal Revenue 249,150,212.91
Lands 4,879,833.05
Miscellaneous 40,172,197.34
Total, exclusive of Postal .... ^594,464,121.67
Expenditures, Fiscal Year 1906
Civil and Miscellaneous $159,823,904.50
War Department 119,704,113.09
Navy Department 111,166,784.35
Indians 12,746,859.08
Pensions 141,034,561.77
Interest 24,308,576.27
Total, exclusive of Postal .... $508,784,799.06
The following table shows the receipts from the principal
objects of internal taxation during the same fiscal year :
Internal Revenue Receipts, Fiscal Year 1906
Distilled Spirits . . $ 143,394,055.00
Tobacco 48,422,997.88
Fermented Liquors 55,641,858.56
Oleomargarine 570,037.93
Mixed Flour 2,567.23
Adulterated Butter 9,258.43
Renovated Butter 138,078.09
Playing Cards 489,347.26
Penalties 283,901.62
Collections 150,494.88
340. 1. Customs. A list or schedule of goods with the rates
of import duties adopted by Congress is called a tariff. Under
our tariff laws some imported articles are admitted without the
payment of duties. These articles are said to be on the free
list. Articles not on the free list are subject to an ad valorem
duty, a specific duty, or to both.
CUSTOMS 273
2. An ad valorem duty is a duty reckoned according to the
value or cost of the goods in tlie country from whicli they are
imported. Thus, the duty on jewelry is 60 % of the value.
3. A specific duty is a tax of a specified amount on each
pound, yard, gallon, bushel, etc., regardless of the cost of the
goods. Thus, the duty on onions is 40)^ a bushel.
4. Certain ports are designated sls ports of entry, where duties
on cargoes are payable. A customhouse has been established
at each port of entry for the collection of customs. The col-
lection of customs at each port is under the direction of a
government officer called the collector of the port.
5. The following rates of custom are from the schedule
adopted by Congress in 1897, and commonly known as the
Dingley Tariff.
Cheese, 6^ per lb. Paintings, 20 % ad val.
Hay, $4: per T. Penholders, 25 % ad val.
Coffee, free. Carpets (Axminster), 60/ per
Apples, Dried, 2/ per lb. sq. yd. and 40 % ad val.
Tea, free. Carpets (Brussels), 44/ per sq.
Bacon and Hams, 5/ per lb. yd. and 40% ad val.
Honey, 20 / per gal. Oats, 15/ per bu.
Soap (Castile), Ij/ per lb. Wheat, 25/ per bu.
Musical instr'ts, 45% ad val. Hops, 12/ per lb.
341. Written Exercises.
1. How much is the duty on a violin worth $ 80 ?
2. A painting costing $2500 was purchased in Italy and
brought into the United States. Find the amount of the duty
charged for its admission.
3. What is the duty on 200 sq. yd. of Brussels carpet valued
at $1.20 a square yard ?
4. What per cent of the expenditures of the national gov-
ernment were defrayed from customs receipts in 1906 ? from
internal revenue receipts ?
AR.— 18
274
APPENDIX
BANKING
342. Give the names of some banks that you know of. Of
what use to a community are banks ? Why is a bank a safer
place in which to keep money than a house or an office ? If a
person wishes to make a payment to another, he may do so by
giving him the money. Do you know of any other method
commonly used in making payments ?
343. Savings Banks. Savings banks are banks organized
under the laws of the different states for the purpose of re-
ceiving and investing the savings of people. Their capital
consists of the money put into the bank by the depositors, and
Deposit Slip. their profits are divided
among the depositors in pro-
portion to the amount that
each has on deposit. The
profits are paid to the depos-
itors in the form of interest,
usually ranging from 3% to
4% annually, which is paid
monthly, quarterly, or semi-
annually. If the interest is
not collected by the depositor
when it becomes due, it is
entered to his credit on the
books of the bank and it
thereafter draws interest in
the same manner as the
ordinary deposits. Do sav-
ings banks pay compound
interest ?
344. Savings Bank Ac-
counts. Any reliable person
who wishes to deposit money
for safekeeping and invest-
ment may open an account
SAVINGS DEPOSIT
No. f^8S Bal. f 3/6.^6
deposited with
Union Savings Bank
FOR ACCOUNT OF
Lo8 Angklbs, Cal., March 22, 1907.
Gold ....
Silver . . .
Currency . .
Checks . . .
<<
«
DOLLARS
10
BANKING 275
with a savings bank. On opening an account, the depositor
is usually required to answer certain questions and to leave his
signature with the bank, to protect the bank against fraudulent
demands upon his accounts. The depositor then hands the
"receiving teller" or the "cashier" the money which he
wishes to deposit, together with a deposit slip, showing the
amount of his deposit. He is then given a savings bank book,
usually bearing a number, with the amount of his deposit
credited to his account. This bank book must be presented
whenever the depositor wishes to make a deposit or to draw
against his credit in the bank. The smallest amount received
for deposit is usually one dollar.
When a depositor withdraws money from a savings bank, he is
required to give the bank a receipt for the amount he has
withdrawn.
Receipt
SAVINGS ACCOUNT
Los Angeles, Cal., {Z^ll f6, /(^OJ No. 1^88
Received from the UNION SAVINGS BANK
c^(:/^^K. o^^ f^-....^.^..-..-^.--^.--^ f 15.00
Balance, f^85./S. .
As the usage of banks differs considerably in the various
sections of the country, no attempt is made here to give
details. Pupils should familiarize themselves with local
customs. Deposit slips, receipt blanks, or check blanks, sam-
ple bank books, note forms, etc., should be examined by the
pupils and their use explained to them. Where possible, a
visit to a neighboring bank should be made during banking
hours.
276
APPENDIX
345. Illustrative page from a savings bank book, showing
entries of deposits, withdrawals, and interest:
Statement
Date
Withdrawn
DEPOsrrED
Balance
Dec. 6, 1907 ....
45
50
128
75
Dec. 21, 1907 . . .
15
143
75
Dec. 24, 1907 . . .
35
108
75
Int. to Jan. 1, 1908 .
4
74
113
49
Jan. 3, 1908 ....
50
163
49
Feb. 2, 1908 ....
74
50
237
99
March 7, 1908 . . .
100
137
99
March 25, 1908 . . .
80
217
99
346. Banks of Deposit. Banks other than savings banks are
sometimes called haiiks of deposit. They are variously known
as national banks, state banks, commercial banks, private banks,
etc. These banks are organized for the purpose of receiving
deposits, making loans, etc. National banks are organized
under the national law and are under the direct supervision of
the Comptroller of Currency, who is appointed by the President
of the United States. In addition to carrying on a general
banking business, national banks have authority to issue paper
money, called bank notes. The payment of these notes is
secured by government bonds deposited with the Secretary of
the Treasury. All other banks are organized under the laws
of the states in which they are located.
347. Deposits. Accounts are opened with banks of deposit
in much the same manner as with savings banks. Business
men and others who. wish to keep money on hand for the pay-
ment of bills, etc., usually have an account with a bank against
which they may draw checks whenever they wish. Such ac-
counts usually do not draw interest. Each depositor receives
a hank book in which all deposits must be entered.
BANKING
277
348. 1. Checks. When the account is opened, the depositor
receives a check hook which he nses in making demands against
his accoimts.
Stub
Check
San Francisco, Cal., ?]1a.if ^', /^OS No. 68
UNION NATIONAL BANK
Pay to the order of
<S^kvityY-iiV90 am^cL ■p-N.^N^N^-s.r>^'-v/-%^DoLLAKS
2. Checks are indorsed in the same manner as promissory-
notes.
3. To cash this check, James E. Thomas will indorse it and
present it at the bank, or will deposit it to his credit at the
bank where his accounts are kept. The check will finally be
returned to R. E. Davies after it has been paid. It will then
serve as a receipt from James E. Thomas, since it bears his
indorsement.
4. A depositor may draw money for himself from his bank
by making his check payable to " Cash," in which case no in-
dorsement is necessary.
349. Certificate of Deposit. When a person who does not
intend to become a regular customer of a bank makes a deposit,
he is given a certificate of deposit, showing the amount of his
deposit. Such a deposit is not subject to check. The amount
may be withdrawn upon the return of the certificate with the
proper indorsement. Certificates of deposit are usually issued
to persons who deposit money with banks when interest is paid
upon the deposit.
278 APPENDIX
350. Drafts. Banks usually keep money on deposit in
some bank, called a correspondence bank, in Boston, New York,
Chicago, San Francisco, or other financial center, against which
to draw checks. When a person wishes to make a payment in
a distant place, he may purchase the check of his local bank on
a bank of correspondence, which will honor this check when
presented for payment. Such a check is called a bank draft.
Thus, Mr. A, in Seattle, Wash., may wish to pay Mr. B, in
Peoria, 111., $45.60. Mr. A may purchase a draft of a bank
in Seattle on a bank in Chicago, which will honor the draft
when presented either by Mr. B, or by some bank which has
purchased it from Mr. B. The bank in Seattle may charge
Mr. A a small amount for making this exchange of money.
351. Clearing House. Daily settlements of accounts between
banks are made through an association called a clearing house.
At a fixed hour each day representatives from each bank that
is a member of the clearing house visit the clearing house
and settle the accounts of their bank with other banks.
All large cities have clearing houses and nearly all banks in
these cities are members.
By means of the clearing house the American Exchange National
Bank one day transacted a business of $ 18,000,000 in checks, with a bal-
ance of only 12 cents to pay. The clearing house sheets showed that
$9,049,255.40 in checks, drawn by the depositors of the bank, had been
turned in by other institutions. Against these the bank had .$ 9,049,255.28
in checks of other banks belonging to the clearing house, which had been
deposited with the American Exchange National Bank. The clearance
was made by a payment by the bank of 12 cents to the clearing house.
LIFE INSURANCE
352. Personal Insurance. There are various kinds of per-
sonal insurance. Of these the most common are: Accide^it
insurance, which is an indemnity for injuries sustained by
accident ; health insurance, which is an indemnity for loss
of time caused by illness ; life insurance, the principal forms
of which are discussed on p. 279.
LIFE INSURANCE 279
353. 1. Persons who insure their lives usually do so to pro-
vide for those who are dependent upon them. The cost of life
insurance depends (a) upon the age of the person insured;
and (b) upon the kind of policy taken out. Life insurance
premiums are always stated at so much on each $1000 of
insurance. The most common kinds of policies are :
2. Ordinary Life Policies, called also straight life policies and
life policies. The insured pays a premium, usually annually,
at the beginning of each year from the time he insures his life
until his death. At his death, the company pays the face of
the policy to the person (or persons) named in the policy as
his beneficiary.
3. Limited Payment Life Policies. The insured pays a pre-
mium for a limited number of years, as 20 years, at the expira-
tion of which the policy is said to be paid tip. The face of the
policy is paid to the beneficiary at the death of the insured.
4. Endowment Policies. Premiums are paid for a period of
years, as 10, 15, or 20 years, and the face of the policy is paid
at the end of the period specified, or at death if the insured
should die before the expiration of the period.
5. Term Policies. The insurance extends for a specified
period, as for 10, 15, 20 years, etc., at the expiration of which
the insurance ceases. The face of the policy is paid if the
insured dies within the period specified.
The amount of the annual premium on $1000 of insurance for a life
policy, a limited life policy, and for an endowment policy, ages 20 years
to 40 years, is given in the table on j). 280.
6. Which is the more likely to live twenty years longer, a
person twenty years of age or a person forty years of age?
Statistics have been carefully compiled showing the ages at
which persons die. From these statistics insurance companies
are able to determine the average number of years a healthy
person of a given age may be expected to live. The rates of
annual premiums are based upon these statistics.
280
APPENDIX
Tablb or Annual Premiums for $ 1000 (Ages 20 years to 40 years)
Policies Non-fobfbitable and Participatinq
Premiums may also be paid half-yearly or quarterly ; and if desired, may be
paid iu 10, 15, or 20 years instead of during the whole term.
Ordinary Life
20
Endowments
J
Payment
Life
Age
Yearly
10 Year
15 Year
20 Year
20
$17.30
$24.16
$99.27
$62.34
$44.10
21
17.80
24.00
99.40
62.40
44.25
22
18.30
25.10
99.50
62.45
44.40
23
18.70
25.70
99.00
62.50
44.55
24
19.30
26.20
99.75
62.G0
44.70
25
19.80
26.75
99.90
62.70
44.82
26
20.30
27.30
100.00
62.80
44.95
27
20.90
27.90
100.05
62.90
45.10
28
21.50
28.50
100.10
63.05
45.25
29
22.10
29.10
100.20
63.20
45.45
30
22.70
29.70
100.30
63.34
45.63
31
23.40
30.35
100.40
63.i30
45.85
32
24.10
31.00
100.50
63.70
46.05
33
24.80
31.72
100.60
63.90
46.25
34
25.60
32.50
100.75
64.05
46.45
35
26.50
33.28
100.90
64.20
46.70
36
27.40
34.10
101.15
64.40
46.85
37
28.30
34.96
101.45
64.65
47.05
38
29.30
35.88
101.75
64.95
47.25
39
30.40
36.84
101.95
65.30
47.45
40
31.50 j
37.84
102.14
65.67
48.64
364. Dividends. The premium charged represents the esti-
mated cost of insurance and is based upon conservative assump-
tions as to future death rate, the rate of interest which the
company may expect to receive for loans, etc. The actual cost
of insurance is determined by experience from year to year.
The difference between the estimated cost and the actual cost
J 8 called the profit. Policy holders are usually allowed to
LIFE INSURANCE 281
participate in the profits, either by having them applied to
reduce the yearly premiums or by having them accumulate in
the possession of the companies until the expiration of the
term of insurance. An insurance policy in which it is stipu-
lated that no dividend shall be paid until the close of the term
of insurance is called a tontine policy.
Examine a life insurance policy. Kead all its provisions.
355. Use the table in answering the following :
1. How much will it cost annually to carry an ordinary life
policy for $ 1000, if it is taken out at the age of 20 ? at the
age of 25 ? at the age of 35 ?
2. How much will it cost annually to carry a 20-payment
life policy for $2000, if it is taken out at the age of 20? at
the age of 27 ? at the age of 40 ?
3. How much will it cost annually to carry a 10-year en-
dowment policy for $ 5000, if it is taken out at the age of 20 ?
at the age of 30 ? at the age of 40 ?
4. Suppose that a young man 20 years old takes out a 20-
payment life policy for f 1000 and dies after paying 8 annual
premiums. Find the net cost of the insurance, if dividends
amounting to $40 were applied to reduce the premiums. How
much would the beneficiary named in the policy receive at his
death ?
5. How much will it cost to carry a 20-year endowment
policy for $ 1000 for the term of the policy, if it is taken out
at the age of 30 ? How much would the insured receive from
the insurance company at the end of the term, not including
the dividends ?
6. If the insured (Prob. 5) died after paying 15 premiums,
how much more than the amount paid as premiums would
the beneficiary receive ?
7. What is meant by a non-forf citable and participating
policy ? by a tontine policy ?
282
APPENDIX
Table of Loak and Surrender Values
356. The following table shows the loan and surrender
values on a 20-payment life policy for $ 1000 taken out when the
insured was 25 years of age, the annual premium being $ 26.95 :
At End of
Loan
Cash
Value
Pald-up
Insurance
Extended
Insurance
Years Days
3d
«150
4 342
4th
200
6 291
5th
$54
$60
250
8 232
6th
68
76
300
10 317
10th
130
145
500
19 17
15th
224
249
750
26 134
19th
315
351
950
31 111
20th
342
380
Policy full-paid
Answer the following from the above table :
1. If the insured wished to borrow money, how much would
the company loan him at the end of the 10th year, if he
assigned to the company his policy as security ? how much at
the end of the 15th year ?
2. If the insured surrendered his policy at the end of the
5th year, how much would the insurance company pay him
for his policy ? How long would they continue his insurance
without the payment of premiums ?
3. Find the amount of the annual premiums for 20 years.
What is the cash value of the policy at the end of 20 years ?
If the dividends average $ 6.50 a year, how much will they
amount to in 20 years ? What is the sum of the cash value
and dividends at the end of the insurance term ? How does
this sum compare with the total cost of the insurance for the
term ?
THE EQUATION 283
4. Using the compound interest table on p. 320, find the
amount of $ 26.95 (the premium) for 20 years. If money is worth"
6%, find the total amount of the premiums paid at the end of
20 years, the premium being paid at the beginning of each year.
5. If the insured should die at the age of 40, how much
would the beneficiary receive ?
THE EQUATION
357. 1. The relation of the quantities involved in some prob-
lems can be stated in a simpler and clearer way by the use
of the equation. In an equation, the vahie of the unknown
quantity is usually represented by the letter x. Thus, in
6 -f 4 = 0^, X is called the unknown quantity, and the expres-
sion 6 -h 4 = ^' is called an equation.
2. An equation may be compared to a balance scale. In
an equation the quantities on the two sides are equivalent —
they balance one another.
3. If a package weigh-
ing 4 lb. is placed in one
pan of a balance scale,
what weight must be
placed in the other pan to
make the scale balance ?
4. A package weighing 5 lb. was placed in the pan on the
right of a balance scale and a 2-lb. weight was placed in the
pan on the left. What additional weight must be placed in
the pan on the left to make the scales balance ?
5. Would the scales as represented in the figure still bal-
a!ice if a 5-lb. weight were added to the weights in each pan ?
Would they balance if a 5-lb. weight were added to the weight
in one pan ?
6. Would the scales as represented in the figure still bal-
ance if 2 lb. were removed from both pans ? Would they
balance if 2 lb. were removed from only one pan ?
284 APPENDIX
7. Would the scales as represented in the figure balance if
the weight on both sides were doubled ? Would they balance
if the weight on only one side were doubled ?
8. Would the scales as represented in the figure balance if
one half of the present weight were removed from each pan ?
Would they balance if one half of the weight were taken out
of only one pan ?
9. What is the value of the unknown quantity in the
equations lb. =5 lb. + a;? in 6 lb. + a; = 10 lb. ? inZlb. — « =
2 1b.?
10. What is the value of x in 9 + a? = 12 ? Finding the
value of the unknown quantity in an equation is called solving
the equation.
11. If 3 is added to both sides of the equation a; + 4 = 0,
the result is a; + 7 = 12. How does the value of a; in a; + 4 = 9
compare with the value ofa;ina;-l-7 = 12?
12. Write 5 equations. Add some number to both sides of
each of the equations. Compare the value of x in the result-
ing equation with the value of x in the original equation.
13. State what effect adding the same number to both sides
of the equation has upon the value of x in the equation.
Prove the truth of your statement.
14. If 2 is subtracted from both sides of the equation
a; -I- 4 = 9, the result is a; + 2 = 7. How does the value of x
in a; -f- 2 = 7 compare with the value of a; in a; + 4 = 9 ?
15. Write 5 equations. Subtract some number from both
sides of each equation. Compare the value of x in each of the
resulting equations with the value of x in the original equation.
16. State what effect upon the value of x in any equation
subtracting the same number from both sides of the equation
has upon the equation. Prove the truth of your statement.
17. If 4 is subtracted from both sides of the equation
jc -f 4 = 9, what is the result ?
THE EQUATION . 285
18. If 3 is subtracted from both sides of the equation x -\- S
— 8, the result is a? = 5. If 2 is subtracted from both sides of
the equation 9 = 2 + a;, the result is 7 =x. What number
must be subtracted from both sides of each of the following
equations to leave x bj itself on one side: a? 4- 5 = 8?
a; + 7 = 15? x + 6 = 9? S = 6-\-x? 12 = x + o? 9-f-ic=15?
74-aJ = 10?
19. If 4 is added to both sides of the equation a; — 4 = 5,
the result is a; = 9.
20. What number must be added to both sides of each of
the following equations to leave x by itself on one side :
iB-5 = 8? a;-7 = 10? a;-4 = ll? x-0 = 2?
12 = a;-15? 10 = a;-5? 4 = a;-G? 2 = a;-3?
21. The value of a; in a; + 4 = 9 may be found by subtract-
ing 4 from the left side of the equation and indicating the sub-
traction of 4 from the other side, thus : a; = 9 — 4. Find the
value of X in each of the following equations : a; -f- 6 = 15 ;
fl; + 8 = 15; a; + 9 = 16; a;-f-20 = 45; a;-f345 = 670.
22. The value of a; in a; — 4 = 10 may be found by adding 4
to the left side of the equation and indicating the addition of 4
to the other side, thus : a; = 10 + 4. Find the value of x in
each of the following equations : a; — 7 = 13 ; a; — 5 =18 ;
a;-12 = 20; a;-14 = 17; a;-46 = 35; a;-80 = 120.
23. Write each of the following equations with x by itself on
the left side of the equation : a? + 3 lb. = 10 lb. ; 5 f t. + a' = 13 ft. ;
24 yd. +a; = 45yd.; $ 7.50 + aj= $12.75; a; + $15 = $80;
a; -12 ft. =20 ft.; a; - $3.45 = $1.20.
24. Compare 2 + 3 = 5 with 5 = 2 + 3. Compare a; + 4 = 9
with 9 = a; + 4. State what effect, if any, writing the equation
with the sides changed has upon the equation.
25. Write each of the following equations so the side con-
taining X is on the left : 45 ft. + 33 f t. = a; ; $ 2.45 = a; - $ 1.20 ;
14 yr. = 9 yr. + a; ; 10 yr. = a; — 7 yr.
286 APPENDIX
358. Solve each of the following without using x. Then
write the equation for each, using x^ and find the value of x :
1. If 45 is added to a certain number, the sum is 73. What
is the number ?
Model : Let x = the unknown number
a; + 45 = 73
a = 73 - 46
a; = 28
2. If 27 is subtracted from a certain number, the remain-
der is 56. What is the number ?
3. If a certain number is increased by 347, the result is
591. What is the number ?
4. If a certain number is diminished by 274, the result is
483. What is the number ?
5. A boy deposited ^ 17 in a savings bank. He then had
% 61 in the bank. How much money had he in the bank before
depositing the $17?
6. After drawing out $35 from a savings bank a boy
had left $ 7.45 in the bank. How much money had he in the
bank before drawing out the $35 ?
7. After gaining 7 lb. a girl weighed 103 lb. How much
did she weigh before gaining the 7 lb. ?
8. George and Frank together have as much money as
Walter. George has $2.15 and Walter has $4.10. How
much money has Frank ?
9. A man owns three farms amounting together to 240
acres. Two of the farms contain 80 acres and 120 acres re-
spectively. How many acres are there in the third farm ?
10. A house and lot together cost $4500. The lot cost
$ 1500. Find the cost of the house.
11. The sum of two numbers is 238. One of the numbers
is 79. What is the other number ?
THE EQUATION ^87
12. The sum of the three sides of a triangle is 24 in. One
of the sides is 8 in. and another is 9 in. What is the length
of the third side ?
13. After selling 40 sheep a farmer had 236 sheep. How
many sheep had he before selling the 40 sheep ?
14. Write 3 problems similar to each of Probs. 1-13 and
write the equation for each.
359. 1. If ^ is added to both sides of the equation
7 — aj'= 2, the result will be 7 = 2 + «• What will be the
result if x is added to both sides of the equation 10 — a; = 7 ?
2. If X is added to both sides of the equation 14 = 25 — a;,
the result will be 14 + a; = 25. The value of x is found by-
adding to both sides of the equation some number that will
leave x by itself on the left side.
3. Write each of the following equations so that x will be
by itself on the left side of the equation. First, add x to both
sides of the equation, then write the equation so that the side
containing x will be on the left. 18 = 43 — iK ; 21 = 72 — a; ;
60-0^ = 37; 33-aj = 19; 54 = 62-a;; 68-ic = 28.
Write each of the following statements so that x will be
by itself on the left side of the equation, and solve :
4. 167-a; = 100. 8. 74-a; = 18.
5. a;-$36 = |75. 9. $ 45.75- a? = $30.50.
6. 15 lb. = 25 lb. -a.-. 10. 65.4 -a; = 18.45.
7. 78 ft. = 135 ft. -X. 11. 125 da. - a; = ^h da.
12. Write 10 equations and find the value of x in each.
360. 1. The sum of x and x and x, or 3 times x, is written
Zx. Write the sum of x and x. Write the product of 4 times
x\ of 5 times a.
2. If X is 4, what is the value of 2 a; ? Compare a; = 4 and
2aj = 8. What must both terms of a; = 4 be multiplied by to
give 2 a; = 8? Multiply both terms of a; = 3 by 5. Has this
changed the value of x in the equation ?
288 APPENDIX
3. State what effect multiplying both sides of the equation
by the same number has upon the vahie of x in an equation.
Prove the truth of your statement.
4. Divide both sides of the equation Go; = 18 by 2; by 3;
by 6. Has this changed the value of x in the several equations ?
5. State what effect dividing both sides of tlie equation by
the same number has upon the value of x in an equation.
Prove the truth of your statement.
6. If 2a; + 4 = 21, what is the value of 2a; ? of a; ? of 3a; ?
7. If x = 6, what is the value of 7a; ? of 3a; ? of 5a; ?
Find the value of x in each of the following equations.
Where the equation shows an unknown quantity to be sub-
tracted from one side of the equation, add this unknown
quantity to both sides of the equation ; then write the equation
with the unknown quantity on the left side and solve :
8.
2a; -45 = 69.
13.
146 -3a; = 83.
9.
24 = 78 -2a;.
14.
$35.40 - 3 a; = f 10.95.
10.
45 ft. -4a; = 13 ft.
15.
f.85 = $.40-f-3a;.
11.
345 -4 a; =135.
16.
$90-6a;=$48.
12.
240 A. = 880 A. -4a;.
17.
24yr. -3a;=6yr.
361. Solve each of the following without using x. Then
solve each, using x.
1. If 3 times a certain number, plus 25, is 55, what is the
number ?
2. If 4 times a certain number, less 20, is 40, what is the
number ?
3. Mary is 20 years old. This is 2 years more than twice
Edna's age. What is Edna's age ?
4. Walter has $45. This is $13 more than 4 times the
amount of money James has. How much money has James?
5. If 4 times a certain number, plus 3 times that number,
is 28, what is the number ? (4a; -|- 3a; = 7a;.)
THE EQUATION 289
6. If 6 times a certain number, plus 4 times that number,
is 160, what is the number ?
7. If 4 times a certain number is the same as 6 times 18,
what is the number ?
8. A man bought three railroad tickets, each costing the
same amount, and paid $1.50 for bus rides. He paid out
$ 6.90 in all. Find the price paid for each ticket.
9. The sum of two numbers is 48, and one number is 5
times the other. What are the numbers? (Let x and 5x
represent the numbers.)
10. A man bought two carriages. For one he paid twice
what he paid for the other. Both carriages cost him $210.
Find the cost of each.
11. Write problems similar to each of the above, and state
the equation for each.
12. Draw an oblong whose length is twice its width. Let x
represent its width. What will represent its length ? its
perimeter? If the perimeter of the oblong is 30 in., how
wide is it ? How long is it ?
13. Draw two lines, one of which is 3 times the length of
the other. If the sum of their lengths is 24 ft., how long is
each line ?
14. Two men together own 540 acres of land. One owns
twice as much as the other. How many does each own ?
15. A man offered to divide $10 between two boys in pro-
portion to their ages, provided the boys could tell how much
each should receive. The boys were 12 years and 13 years
respectively. After solving the problem the boys stated that
the younger should receive $4.50 and the older $5.50. Did
they solve it correctly ? If not, what is the correct answer ?
AK. — 19
290 APPENDIX
16. A man offered some boys $ 1.50 for weeding his garden.
The boys found that they could not all work at the same time,
so the man agreed to pay each boy the same wages per hour
for the work done. One boy worked 7 hours, another worked
5 hours, and the third worked 3 liours. How much of the
money should each boy receive ?
17. A man wished to leave $ 3500 to his three sons so that
the second son would receive twice what the youngest received
and the eldest would receive 4 times what the youngest re-
ceived. How much should each son receive ?
362. 1. The expression f is used to denote ^ of x. Write
the expression that denotes J of a; ; -J- of x; -J of a; ; ^ of a; ; -f
of X.
2. By what number must f be multiplied to make x?
By multiplying both sides of the equation f = 3 by 4, the
equation is changed to a; = 12.
3. Multiply f by the number that will give x as the re-
sult. Multiply ^ by the smallest whole number that will give
a whole number of a;'s as the result.
4. What is the smallest number that both sides of the
equation f = 12 can be multiplied by to leave only whole
numbers in the equation? Multiplying both sides of an
equation by some number that will leave the equation without
fractional quantities is called clearing the equation of fractions.
5. Clear the following equations of fractions : f = 8 ;
1=14; ¥ = 12; |t = 36; 45=^; 60 = VV ; -¥--8 = 1;
72-^^ = 56.
6. Clear the following of fractions and find the value of x :
Solve each of the following without using x. Then solve
each, using x :
7. Divide 60 into two numbers such that the first is J of
the second.
THE EQUATION 291
8. Separate 36 into two parts whose ratio is -|.
9, Divide $2.10 into two amounts whose ratio is the same
as the ratio of 15^ to 20 /.
10. If ^ of a certain number, plus | of it, is 39, what is the
number ?
11. Solve : 6 times 8 = 12 times x ; 4 times 9 = 6 times x.
1 9 Snl VP • 6 a;, a — 12. « — 42
J.4S. OOlVe . 9- — -3 , -g- — T6 J T — 4 9"-
13. The equation J = ^ may be cleared of fractions by mul-
tiplying both sides of the equation by the least common mul-
tiple of the denominators. This is 2 x. The equation is thus
changed to the form 12 = a;.
14. Solve: i = ii; ,\ = i; V"=l«; I = Il-
ls. Solve: w = J^; M = ¥; A=^; ^=^-
16. Write ten exercises similar to exercises 8-12 and find
the value of x in each.
363. Proportion.
Solve each without using x. Solve each, using x :
1. The shadow of a post 5 ft. high is 3 ft. 6 in. long.
How high is a telephone pole whose shadow is 28 ft. long ?
a. The height of the post is -^ times the length of its
shadow. The height of the telephone post is -^ times 28
ft. Explain.
b. The length of the shadow of the post is in the same ratio
to the height of the post as the length of the shadow of the
telephone pole is to the height of the pole. The equality of
these ratios may be expressed thus :
3^^28
5 X
Solve to find the value of x, the number of feet in the height
of the telephone pole.
c. The equality of the two ratios may be expressed thus :
3.5 ft. : 5 f t. : : 28 ft. : x, which is read, 3.5 ft. is to 5 ft. as
28 ft. is to X. The first and last terms (as 3.5 ft. and x) of a
292 APPENDIX
proportion are called the extremes, and the two middle terms
the means. Tlie product of the extremes in a proportion is
always equal to the product of the means. Hence, 3.5 times x
= 5 times 2S, or 3.5 a; = 140. Solve to find the value of x, the
number of feet in the height of the telephone pole. This
method of solving a proportion differs only inform of expres-
sion from the method (b) given on p. 201.
2. How high is a tree whose shadow is 34 ft. 6 in., if the
shadow of a boy whose height is 4 ft. 9 in. is 3 ft. 3 in. ?
3. If the distance traveled by a trail* in 1 hr. 45 min. is 80
mi., how long, at the same rate of speed, xWU it take the train
to travel 475 mi. ?
4. Find by the method used in solving Probs. 1 and 2 the
height of objects near the schoolhouse.
MEASUREMENT OF SURFACES AND SOLIDS
364. Areas of Surfaces.
1. Draw a vertical line; a horizontal line; an oblique line.
2. Draw a line perpendicular to another line ; parallel to
another line.
3. Draw a right angle ; an acute angle ; an obtuse angle.
4. Draw a rectangle. Is a rectangle a parallelogram?
Draw a parallelogram that is not a rectangle.
5. How many dimensions has a rectangle ? Is a rectangle
a quadrilateral ? Draw a quadrilateral that is not a parallelo-
gram.
6. State how the area of a parallelogram is found. Find the
area of a parallelogram whose base is 20 ft. and whose altitude
is 18 ft.
7. A quadrilateral that has only two parallel sides is called
a trapezoid.
8. State how the area of a trapezoid is found. Draw a
trapezoid. Assign its dimensions and find its area.
MEASUREMENT OF SURFACES AND SOLIDS 293
9. TVTiat is a triangle ? Draw a right triangle ; an acute-
angled triangle ; an obtuse-angled triangle.
10. State how the area of a triangle is found. Draw a tri-
angle. Assign its dimensions and find its area.
11. Make a drawing to show the relation of the area of a
triangle to the area of a parallelogram having the same base
and altitude.
12. Draw a parallelogram. Draw its diagonals. Do they
cross at the middle of the parallelogram ?
13. What is meant by the perimeter of a figure? Find the
perimeter of your schoolroom.
14. Draw a circle. Draw its radius ; its diameter. Point to
its circumference.
15. State how the circumference of a circle is found when
the length of its radius is known. State how the diameter of
a circle is found when the length of its circumference is known.
16. State how the area of a circle is found. Assign the
necessary dimensions and find the area of a circle.
17. State how the area of the convex surface of a cylinder
is found. Find the area (including the ends) of a cylinder whose
diameter is 6 ft. and whose length is 8 ft.
365. Regular Polygons.
1. Mention a surface that is a plane surface. A plane figure
bounded by straight sides is called a polygon. A polygon
whose sides are all equal and whose angles are all equal is
called a regular polygon.
o o
Triangle . Square Penta^'ou Hexagon
Rbgdlak Pulygons
294 APPENDIX
2. A regular polygon of three sides is called an equilateral
triangle; of four sides, a square; of five sides, a pentagon; of
six sides, a hexagon ; of seven sides, a heptagon ;
^'^N. of eight sides, an octagon. Draw an octagon.
^ C-'^ ^* ^ straight line from the center of a
\ f ' / regular polygon to any vertex is called its
\ ; / radius (r).
4. The perpendicular from the center of a
regular polygon to any side is called its apothem (a).
5. The area of a regular polygon is the sum of the areas of
the triangles formed by its radii and sides.
The apothem is the altitude of each of the
triangles, and the perimeter is the sum of
the bases of the triangles. Hence,
The area of a regular polygon is equal to one
half the product of its perimeter and apothem,
6. Draw a pentagon. Assign its dimensions and find its
area.
7. Draw a hexagon. Assign its dimensions and find its
area.
8. Draw an octagon. Assign its dimensions and find its
area.
9. The area of a circle is one half the product of its radius
and circumference. Compare the method of finding the area
of a regular polygon with this method of finding the area of a
circle.
366. Solids.
1. How many dimensions has a plane surface? Name
them.
2. How many dimensions has a solid ? Name them.
3. What name is given to a solid whose faces are all rec-
tangles ? to a solid whose faces are equal squares?
MEASUREMENT OF SURFACES AND SOLIDS 295
4. What name is given to a solid whose ends are triangles
and whose sides are rectangles ?
5. State how the volume of a prism is found. Draw a
prism. Assign its dimensions and find its volume.
6. Name solids that are rectangular prisms.
7. State how the volume of a cylinder is found. Draw a
cylinder. Assign its dimensions and find its volume. Find
its area, including the ends.
367. Pyramids and Cones.
1. A solid whose base is a polygon and whose faces are
triangles meeting at a point (vertex) is called
a pyramid.
2. The area of the surface of a pyramid is
the sum of the areas of the triangular faces.
3. The perpendicular distance from the
base to the vertex of a pyramid is called its
altitude (vb).
4. The altitude of one of the triangular faces of a pyramid
is called its slant height (vs).
5. Construct a pyramid of cardboard. Which is the
greater, the altitude of a pyramid or its slant height ? The
apothem of a polygon forming the base of a pyramid may be
regarded as the base of a right triangle (bs), the altitude as the
other leg (vb), and the slant height as the hypotenuse (vs).
How may the altitude be found when the apothem of the base
and the slant height are given ?
6. Draw a regular polygon. Draw its radius and the
apothem of an adjacent side. The figure formed by the radius,
apothem, and one half of the adjacent side is what kind of a
triangle ? if the radius and side of a regular polygon are
given, how tnay the apothem be found ?
296
APPENDIX
7. If the altitude of a pyramid, the radius
of its base, and the adjacent side are given,
how may the slant height be found ?
8. A solid whose base is a circle and
which tapers to a point called the vertex or
apex, is called a cone.
9. A cone may be regarded as a pyramid
whose surface is an infinite number of narrow triangles. Its
altitude and slant height correspond to the altitude and slant
height of a pyramid.
The area of the surface of a pyramid or a cone is equal to one
half the product of its slant height and the perimeter of its base.
Prism
Pyramid
Cylinder
Cone
10. The volume of a pyramid is equal to one third the
volume of a prism of the same base and altitude, and the
volume of a cone is equal to one third the volume of a cylinder
of the same base and altitude. Hence,
TJie volume of a pyramid or a cone is equal to one third the
product of its altitude and the area of its base.
11. A cj^indrical granite stone 3 ft. in diameter and 4 ft. in
height was cut down into a cone of the same base and altitude.
What part of the stone was cut away ?
368. Spheres.
TJie area of the surface of a sphere is four times the area of a
great circle (irr') of the sphere.
1. As (2?-)', or 4 r^ is equal to d^, 4 irr'^ is equal' to tk?.
PUBLIC LANDS 297
77i« area of tlie surface of a sphere ia equal to the square of the
dmineter x vy or trd^,
2. Whicli is the greater and lio"w much, the area of a cube whose
side is 1 ft. or the area of a sphere whose diameter is 1 ft. ?
3. A sphere may be divided into an infinite number of
figures that are essentially pyramids. The combined volume
of these pyramids is the volume
of the sphere. The convex sur-
face of the sphere may be re-
garded as the sum of the bases
of the pyramids and the radius
of the sphere as the altitude of
the pyramids. Hence,
Tlie volume of a sphere is equal to one third the product of its
radius and its convex surface, or | ttt^ (| of r x 4 trr^).
4. As d^, or (2iry is equal to 8r^, ^irr^ is equal to ^ird^.
Hence,
To find the volume of a sphere, multiply the cube of its diameter
by. 5236 a of ^).
5. The earth is how many times the size of the moon, if the
diameter of the earth is 8000 mi. and the diameter of the
moon is 2000 mi.?
Volumes of spheres are to each other as the cubes of their
like dimensions. The ratio of the earth and moon is 8^ (8000^)
to 2^ (2000^), or 4:^ to 1^.
MEASUREMENT OF PUBLIC LANDS
369. 1. At the time the colonial settlements were made, no
uniform system of measuring lands was used. Generally, each
settler was permitted to occupy whatever lands he wished, and
the boundary lines were often designated by such convenient
natural objects as rocks, streams, trees, hilltops, etc. Later
these boundaries were recorded as the legal " metes and
bounds '' of their several possessions. These tracts of land
298
APPENDIX
were often so irregular in shape as to make it difficult to fix
their exact boundaries and to determine their exact areas.
2. Shortly after the close of the Kevolutionary War, the
Continental Congress appointed a committee, of which Thomas
Jefferson was chairman, to draw up some plan for the survey
of public lands. This committee reported a plan which, after
being slightly amended, was adopted by Congress in 1785, and
thus became the government system of measuring public lands.
3. In accordance with this system, all public lands, except
"waste and useless lands," have been laid out in tracts 6 miles
square called townships. The exact location of each township
is determined by north and south lines called principal meri-
dians, and by east and west lines called base lines.
Study the following diagram :
Standard.
Base.
Uj
6^
B
T. 5n.
_ Parol lef
T. 4 N.
T. 3 N.
T. 2 N.
T. I N.
— Line
T I S.
T 2 S.
4. In surveying a tract of land, a prominent point that is
easily identified and is visible for some distance is established
astronomically, and is known as the initial (beginning) point.
In the figure, the initial point is at 0.
PUBLIC LANDS 299
5. A line extending north, or south, or both north and south,
from the initial point is taken as a principal meridian. The
principal meridian is the true meridian at the initial point.
Locate the principal meridian in the figure.
6. A line extending either east or west, or both east and
west, through the initial point, or a line perpendicular to the
principal meridian, is taken as a base line. The base line is
always a true parallel of latitude. Locate the base line in
the figure.
7. East and west lines 6 miles apart, called town lines, are
run parallel to the base line, and north and south meridian
lines 6 miles apart, called range lines. These lines divide the
tract into townships 6 miles square. Point to the township
lines in the figure. How far apart are these lines ? Point to
the range lines. How far apart are these lines ?
8. Point to a township in the first tier of townships north
of the base line. Point to a township in the second tier of
townships north of the base line. Point to a township in the
first tier of townships south of the base line.
9. A township in the third tier of townships north of a base
line is said to be in township 3, north (T. 3 N.). A township
in the second tier of townships south of a base line is said to
be in township 2, south (T. 2 S.).
10. Point to the first north and south row of townships, east
of the principal meridian. These townships are said to be in
range 1, east (K. 1 E.). Point to a township in range 3, east-,
in range 2, west.
11. The township marked A is numbered township 4 north,
range 2 east (T. 4 N., R. 2 E.). Describe the location of town-
ships B, C, D, E, and O. Write the description of each, using
abbreviations.
12. Locate in the figure each of the following described town-
ships : T. 2N., R. 3E.; T. 4N., R. 5E.; T. IK, R. IW.;
T.2S.,R.4E.; T.1S.,R.3W.; T.4K,R.2W.; T.2S.,R.2E.
300 APPENDIX
la Draw a diagram showing a principal meridian, a base
line, and townships and ranges as in the figure on p. 298. In
your diagram, locate the following : T. 4 S., K. 1 E. ; T. 6 K.,
R. 5 W.; T. 6K, R. 6 E.
14. Locate on a map a principal meridian and a base line
from which ranges and townships in your state are numbered,
if the land has been measured by this system.* Give the
number of the township in which you live. Can you tell the
width of the state in which you live from the number of town-
ships along the base line? Is there any similarity between
the method of locating townships by means of principal me-
ridians and base lines and the method of locating places on the
earth's surface by means of degrees of longitude and latitude ?
15. The lands of Florida, Alabama, Mississippi, of the states
west of Pennsylvania and north of the Ohio River, and of all
states west of the Mississippi River, except Texas, have been
surveyed in the manner described. Can you tell from your
study of United States History why the lands of the other
states were not surveyed in this manner ?
16. The initial points are located somewhat arbitrarily.
Sometimes they are located on the east or west boundaries of
states, at other times they are located at the junction of rivers,
or on the summits of elevations. They are at irregular intervals
apart. Consequently, the land in a single state may be meas-
ured from more than one principal meridian, or a single me-
ridian may be used for measuring the land in several states.
Much care is taken to preserve the exact location of all initial
points.
"An initial point sliould have a conspicuous location, visible from
distant points on lines ; it should be perpetuated by an indestructible
monument, preferably a copper bolt firmly set in a rock ledge ; and it
should be witnessed by rock bearings, without relying on anything
perishable like wood." Manual of Surveying Instructions, 1902.
• Unmounted land maps of the various states may be purchased from
the Department of Interior, Washington, for a few cents.
PUBLIC LANDS
301
17. As the lines that bound the ranges on the east and west
are true meridians, they converge as they extend north from a
base line. As a result, townships are not true squares. To
correct the effect of the convergeucy of the meridians, standard
parallels (formerly called correction lines) are established at
regular intervals (now 24 miles apart) from the base line,
and new meridians are established 6 miles apart on the stand-
ard parallels. Guide meridians are also established at inter-
vals (now 24 miles apart), east
and west of the principal meridi-
ans, to correct inaccuracies in
measurement.
370. Townships.
1. A township is a tract of land
6 miles square. It contains 36
square miles of land. A square
mile of land is called a section.
The sections of a township are
numbered as shown in the dia-
gram. The sections of a town-
ship are numbered, respectively, beginning with number 1
in the northeast section and numbering west and east alter-
nately. Draw a township and number the sections.
2. Section 16 of each township in the state was granted by
Congress to the states for educational purposes. This section
is therefore commonly known as the school section, and all
moneys derived from the rent or sale of these sections is
placed in the public school fund of the state. States that
have been organized since 1852 have been granted two sections
in each township for the support of public schools, sections 16
and 36.
Owing to the convergency of the meridians that bound the
townships on the east and west, a township is never exactly 6
miles from east to west, and does not therefore contain 36 full
6
5
4
3
2
1
7
8
9
10
n
12
18
17
B
15
14
13
19
20
2/
22
23
24
30
29
28
27
26
25
31
32
33
34
35
■
A Township divided into
Sections.
302
APPENDIX
sections of G40 acres each. The survey of the sections in each
township is begun in the southeast corner of the township, and
all sections except those along the western and northern bound-
aries of the township are 1 mile square, and contain 640 acres
each. All excess or deficiency is added to or deducted from
the sections along the western and northern boundaries of the
township. These sections generally contain less than 640
acres. The sections along the western boundary of a township
often contain less than 630 acres. Section 6 is frequently re-
duced to about 620 acres.
371. Sections.
1. A section is subdivided into quarter sections, and these are
again subdivided into quarters, etc., as shown in the diagram.
2. The part of the section marked A is described as the west
one half (W. ^) of the section, and contains 320 acres. The part
marked 5 is described as the south-
east quarter of the section, and
contains 160 acres. C is the west
one half of the northeast one fourth
of the section (W. | of N.E. J).
How many acres does it contain ?
The part marked F is described
as the S.E. \ of the S.E. \ of the
N.E. J of the section. How many
acres does it contain ?
3. Describe the part marked O
and tell bow many acres it contains.
4. Describe the part marked E and tell how many acres it
contains.
5. Describe the part marked D and tell how many acres it
contains.
6. Draw a section and subdivide it to show the following
and give the number of acres in each :
7. N.W. \ of the N.E. J. 8. S.W. J of the N.W. J.
D
.\G
\F
A Sbction Subdivided.
PUBLIC LANDS 303
9. E. I of the S.W. J. 10. W. I of the N.W. f
11. S.E. -1- of the S.W. i of the KE. \.
12. S. i of the S.E. i of the N.E. i.
372. Using the scale 1 in, = 1 mi., draw a plot to represent
a township, say T. 6 'N., K. 4 E. ; locate and find the area of
each of the following :
1. E. i of the S.E. ^ of Sec. 9, T. 6 N., E. 4 E.
2. N. W. 1 of the S.E. J of Sec. 22, T. 6 N., K. 4 E.
3. S.E. 1 of the S.W. i of the S.E. Jof Sec. 32, T. 6 N., R.4 E.
4. E. i of the S.W. i of the N.E. i of Sec. 24, T. 6 N., R.
4 E., which is a farm owned by Mr. Thomas.
5. S.E. 1 of the K.W. ^ of Sec. 18, T. 6 N., R. 4 E., which
is the description of a piece of property on which Mr. White
pays taxes.
373. Review.
1. The unit of land measure is the township, which is theo-
retically 6 miles square. The word town is commonly used
for township.
2. What are initial points ? principal meridians ? base
lines ?
3. What is a range ? How many sections are there in a
township ? How are they numbered ?
4. How many acres are there in a section ? in a quarter
section ?
5. Public lands are generally sold in sections, half sections,
quarter sections, and in half quarter sections. What part of a
section is 80 acres ? 40 acres ? 20 acres ?
6. How many acres are there in a full township ? in a full
section ?
7. What are standard parallels, or correction lines ?
8. Which are the school sections ? Why are they so called ?
804 APPENDIX
9. Cau you tell from your study of United States history
why some uniform system of surveying public lands was
necessary soon after the close of the Revolutionary War ?
10. "What sections generally contain less than 640 acres ?
Why?
11. Locate the principal meridian and the base line used in
measuring the land in which your schoolhouse is located.
12. A new standard parallel is located at intervals of 24
miles north or south of the base line, and a new guide meridian
is located at intervals of 24 miles east and west of a principal
meridian. Make a diagram showing these lines.
THE METRIC SYSTEM OF WEIGHTS AND MEASURES
374. 1. The system of denominate units of measure in com-
mon use in the United States is practically the same as that in
use in Great Britain, with the exception of the units used in
measuring value. Nearly all the other civilized nations use a
decimal system of denominate numbers, called the metric
system. The metric system has been legalized by the United
States and Great Britain, and has been adopted as the sys-
tem for use in the Philippines and Porto Rico. It is exten-
sively used in scientific work.
2. A little more than a century ago the French government
invited the nations of the world to a conference to consider an
international system of weights and measures. Later, the
French government appointed a committee to devise a conven-
ient system of denominate units. The committee originated
what is known as the Metric System of Weights and Meas-
ures. The metric system includes measures of length, surface,
capacity, volume, and weight. The primary unit of linear
measure is the meter The primary unit of each of the other
measures is based upon the meter.
3. One ten-millionth part of the distance from the e(]na<^or
to the North Pole, measured on the meridian of Paris, \\</»
METRIC SYSTEM 305
selected as the primary unit of linear measure. This unit is
called the meter. Meter is the French word for measure. The
meter is a little longer than the yard. As it is based upon a
measurement of the earth's polar circumference, the meter
is a fixed natural unit.*
4. An International Bureau of Weights and Measures has
been established in Paris, and is now supported by the contri-
butions of more than twenty nations. A standard meter,
made from an alloy of platinum and iridium, is carefully pre-
served by this bureau. All the nations of the world have been
furnished with duplicates of this standard meter. These
duplicates are made of the most durable and least expansible
metals known. The United States Bureau of Standards has
fixed the legal equivalent of the meter as 39.37 inches.
5. The metric system is a decimal system. Units larger
than the primary units are 10 times the primary units, 100
times the primary units, and 1000 times the primary units;
units smaller than the primary units are -^^ the primary units,
Y^^ the primary units, and y^Vir ^^® primary units. Units of
any given denomination are therefore reduced to units of a
larger denomination by dividing by 10, by 100, and by 1000 ;
and units are reduced to units of a smaller denomination by
multiplying by 10, by 100, and by 1000. Quantities are not
generally expressed in terms of two or more units, but in some
single unit, parts of the unit being expressed as a decimal of the
unit, as 6.35 meters.
6. Names of units larger than the primary units are formed
by prefixing to the names of the primary units prefixes de-
rived from the Greek words meaning ten, one hundred, and
one thousand, etc.; and names of units smaller than the
primary units are formed by prefixing to the names of the
primary units prefixes derived from the Latin words meaning
ten, one hundred, and one thousand, as follows ;
* Subsequent calculations have shown that the meter is not exactly a
ten-millionth part of the distance from the equator to the North Pole.
AR. — 20
306 APPENDIX
Greek Prefixes
dekay meaning 10; dekameter, meaning 10 meters.
hekto, meaning 100; hektometer, meaning 100 meters.
kilo, meaning 1000; kilometer, meaning 1000 meters.
myria, meaning 10,000 ; myriameter, meaning 10,000 meters.
The prefixes deka and hekto are sometimes written dcca and hecto.
Latin Prefixes
deci, meaning 10 ; decimeter, meaning .1 meter.
ceyiti, meaning 100 ; centimeter, meaning .01 meter.
milli, meaning 1000 ; millimeter, meaning .001 meter.
Very small linear measurements are expressed in mikrons. Mikron
is a Greek word meaning small.
375. Measures of Length.
In the following exercises use a meter stick on which the
centimeters and millimeters are marked off. Practice drawing
these units until you can estimate their lengths quite accu-
rately. Test all estimates by actual measurements.
1. Draw on the blackboard a line 1 meter long ; 2 meters
long ; 3 meters long.
2. Fix two points on the floor 1 meter apart; 2 meters
apart ; 3 meters apart ; 4 meters apart.
3. Estimate the length, width, and height of your school-
room in meters.
4. Measure the length of a blackboard in meters. Express
fractional parts as a decimal of a meter, thus : if the black-
board is 4 meters and 12 centimeters long, its length may be
stated as 4.12 meters.
5. Estimate the length and width of the school yard in
metevQ.
'^. Draw a line 1 decimeter in length. Name some object
in the schoolroom that is one decimeter in lengtli, width, or
thickness.
METRIC SYSTEM 307
7. Draw a line 1 centimeter in length, 2 centimeters in
length, 3 centimeters in length.
8. Measure the length and width of this book in centi-
meters. Express fractional parts as a decimal of a centimeter.
9. Measure the thickness of this book in millimeters.
How many millimeters make a centimeter ? a decimeter ? a
meter ?
10. A kilometer is 1000 meters. It is equivalent to about |
of a mile. Select some place that is about 1 kilometer from
the schoolhouse.
11. Using rulers on which the units are marked off, com-
pare the millimeter with -^^ of an inch.
12. Which is the longer, a centimeter or an inch ?
376. Reduction of Linear Units.
1. A meter is how many decimeters ? how many centi-
meters ? how many millimeters ?
2. 67 centimeters may be expressed as a decimal of a
meter, thus : .67 meter. Express as meters : 34 centimeters,
15 centimeters, 76 centimeters.
3. A decimeter is what part of a meter ? 4 decimeters may
be expressed as a decimal of a meter, thus : .4 meter. Ex-
press as meters : 3 meters and 4 decimeters ; 7 meters and 32
centimeters ; 9 meters, 2 decimeters, and 4 centimeters.
4. Write a millimeter as a decimal of a meter. Write 8
millimeters as a decimal of a meter. Write 3 centimeters and
8 millimeters as a decimal of a meter.
5. Write 2 kilometers as meters. Write 2 kilometers and
430 meters as meters. Write as meters : 24.5 kilometers; 4.25
kilometers.
6. Reduce to meters: 304 centimeters; 2.467 kilometers;
245.376 kilometers ; 30 centimeters.
308 APPENDIX
377. Table of Measures of Length.
The following is the complete table of linear measure. The
units most commonly used are the millimeter, centimeter,
meter, and kilometer.
1000 mikrons (/a) = I millimeter (mm.)
10 mm. = 1 centimeter (cm.)
10 cm. = 1 decimeter (dm.)
10 dm. = 1 meter (m.)
10 m. = 1 dekameter (Dm.)
10 Dm. = 1 hektometer (Hm.)
10 Hm. = 1 kilometer (Km.)
10 Km. = 1 myriameter
Abbreviations of the names of the units that are multiples of the pri-
mary unit are written with capital letters to distinguish them from the
abbreviations of the names of the units that are parts- of the primary
unit.
378. Measures of Surface.
1. Draw on the blackboard a square whose side is 1 meter
in length. This is called a square meter.
2. Divide a square meter into square decimeters. How
many square decimeters are there in a square meter ?
3. Divide a square decimeter into square centimeters.
How many square centimeters are there in a square decimeter?
4. How many square millimeters are there in a square
centimeter ?
5. How many square centimeters are there in a square
meter ?
6. In what square unit should you express the area of the
surface of the cover of this book ? of the floor of your school-
room ?
7. Draw on the school grounds a square whose side is 10
meters. This is called an are. It is the primary unit of land
measure.
The are is equivalent to 119.6 square yards.
METRIC SYSTEM 309
8. A square whose side is 100 meters is called a hektare.
9. A square whose side is 1 kilometer is called a square
kilometer.
The area of gardens, etc. is usually given in ares ; of fields, etc. in
hektares ; and of countries, etc. in square kilometers.
10. Estimate the number of square meters in tho surface of
the floor of your schoolroom. Test your estimate.
11. Estimate the number of ares in the school yard. Test
your estimate.
12. How long is the side of a hektare ? of a square kilo-
meter ?
The hektare is nearly 2| acres.
379. Table of Measures of Surface.
100 square millimeters (qmm.) =1 square centimeter (qcm.)
100 qcm. = 1 square decimeter (qdm.)
100 qdm. = 1 square meter (qm.)
100 qm. ■ =1 square dekameter (qDm. )
100 qDm. = 1 square hektometer (qHm.)
100 qHm. = 1 square kilometer (qKm.)
380. Table of Land Measure.
100 centares (ca.) = 1 are (a.)
100 a., =1 hektare (Ha.)
381. Measures of Volume.
1. From a piece of cardboard construct a cube whose edges
are each 1 decimeter. This is called a cubic decimeter.
2. How many cubic decimeters are there in 1 cubic meter ?
3. From a piece of cardboard construct a cubic centimeter.
Kstimate the capacity of a crayon box in cubic centimeters.
4. Estimate the number of cubic meters of air in your
pchoolroom. Using a meter stick, make an approximate test
(•f your estimate.
810 APPENDIX
5. The primary unit of volume is the cubic meter.
The cubic meter is equivalent to 1.308 cubic yards.
6. The primary unit of wood measure is the stere, which is
a cubic meter.
382. Table of Measures of Volume.
1000 cubic millimeters (cu. mm.) = 1 cubic centimeter (cu. cm.)
1000 cu. cm. r= 1 cubic decimeter (cu. dm.)
1000 cu. dm. = 1 cubic meter (cu. m.)
Units higher than the cubic meter are seldom used.
383. Measures of Capacity.
1. The primary unit of capacity for both liquid and dry
measure is the liter, which contains 1 cubic decimeter. Using
the measures, compare the capacity of a liter and a quart.
The liter is equivalent to 1.0567 liquid quarts or .908 dry quart.
2. How many cubic centimeters are equivalent to 1 liter?
3. 100 liters are 1 hektoliter. The liter is used to measure
comparatively small quantities ; the hektoliter is used to meas-
ure grain, produce, etc., in large quantities.
The hektoliter is equivalent to 2.8377 bushels.
4. Mention some things that are bought or sold by the
quart, dry measure ; by the quart or gallon, liquid measure.
Where the metric system is used, these are bought and sold by
the liter, or by the hektoliter if the quantities are large.
5. How many liters of water will a tank hold whose inside
dimensions are 3.45 m. by 80 cm. by 60 cm.?
846 X 80 X 60
1000
-, number of liters in the tank. Explain.
6. Find the capacity in liters of a cylindrical tank whose
diameter is 2.85 m. and whose altitude is 3.68 m.
METRIC SYSTEM 311
384. Table cf Measures of Capacity.
10 milliliters = 1 centiliter (cl.)
10 cl. = 1 deciliter (dl.)
10 dl. = 1 liter (1.)
10 1. =1 dekaliter (Dl.)
10 Dl. =lhektoliter (HI.)
385. Measures of Weight.
1. The primary unit of weight is the gram, which is the
weight of 1 cu. cm. of pure water at its greatest density.
2. Heft a gram weight. How many grams does a liter of
pure water at its greatest density weigh ?
3. The weight of 1000 cubic centimeters of water (a liter) is
called a kilogram, or a kilo. Heft a kilogram weight.
A kilogram is equivalent to 2.2046 pounds avoirdupois. How many-
grams are equivalent to an ounce avoirdupois ?
4. The gram is used in weighing precious metals, medicines,
etc. ; the kilogram in weighing meat, groceries, etc. Express
your weight in kilograms, calling 2.2 pounds 1 kilogram.
5. 100 kilograms are 1 metric quintal, and 1000 kilograms
1 metric ton.
A metric ton is equivalent to 2206 pounds or 1.1023 tons.
6. Express as grams: 2.125 Kg. ; 3.4 Kg. Express as kilo-
grams : 245 g. ; 28 g. ; 362 M. T. ; 4.25 M. T. ; 4 Kg. 72 g.
386. Table of Measures of Weight.
10 milligrams (mg.) = 1 centigram (eg.)
10 eg. =1 decigram (dg.)
10 dg. = 1 gram (g.)
10 g. =1 dekagram (Dg.)
10 Dg. = 1 hektogram (Hg.)
10 Hg. = 1 kilogram (Kg.)
10 Kg. = 1 myriagram (Mg.)
100 Kg. = 1 metric quintal (Q.)
1000 Kg. = 1 metric ton (M. T.)
312 APPENDIX
387. Equivalents of Metric Units.
The following equivalents are given for comparison and for
reference :
Mktric to Common
Common to Mktkk-
1 m.
= 39.37 in., or 1.0936 yd.
1 yd. = .9144 m.
IKm.
= .62137 mi.
1 mi. =1.60935 Km.
1 sq. m
. = 1.196 sq. yd.
1 sq. yd. = .836 sq. m.
1 Ha.
= 2.471 A.
1 A. = .4047 Ha.
1 cu. m
. = 1.308 cu. yd.
1 cu. yd. = .765 cu. m.
11.
= .908 qt. (dry)
Iqt. (dry) =1.1012 1.
11.
= 1.0567 qt. (liquid)
1 qt. (liquid) = .94636 1.
IHl.
= 2.8377 bu.
1 bu. = .35239 HI.
Ig.
= 15.43 gr. (troy)
loz. (troy) = 31.10348 g.
IKg.
= 32.1507 oz. (troy)
1 lb. (av.) = .45359 Kg.
IKg.
= 2.2046 lb. (av.)
1 M. T.
= 1.1023 T.
1 T. = .90718 M. T.
TABLES OF DENOMINATE MEASURES
(For Reference)
388.
Measures of Time.
60 seconds = 1 minute
365 days =1 year
60 minutes = 1 hour
366 days = 1 leap y^ar
24 hours = 1 day
10 years = I decade
7 days = 1 week
100 years = 1 century
1. The day is the primary unit of time measure. It is the
time taken by the earth to make one rotation on its axis. Is
it a natural or an artificial unit ? The earth revolves around
the sun in 365 days 5 hours 48 minutes 46 seconds (nearly
365\ days). This period is the solar (sun) year.
2. As the exact period taken for the earth to make a revolu-
tion around the sun is a little less than 365 J^ days, an extra day
(Feb. 29) is added to the common year once in four years (leap
year), except in centennial years not exactly divisible by 400.
3. Centennial years dioisihh by 400 and other years divisible
by 4 «^<? leap years.
^Vas 1700 a leap year ? Will 2000 be a leap year ?
TABLES OF DENOMINATE MEASURES 313
4. More than four thousand years ago the Chaldeans, a people
living in the valley of the Euphrates, calculated the length of
the year to be 360 days. They believed that the sun traveled
around the earth in a circle in this period. They therefore
divided the circular path of the sun into 360 equal parts,
called degrees — one for the part traversed each day. Hence
there are 360 degrees in a circle. They observed twelve clusters
of stars (constellations) in the zone in the heavens (zodiac) in
which the paths of the sun and planets lie, and the occurrence
of twelve full moons in successive parts of the zodiac each
year. They therefore divided the course of the sun into
twelve equal parts, one for each constellation. Hence there
are twelve months in a year. The exact length of the lunar
month is 29.53059 days. The Chaldeans divided the day into
twelve " double hours.'^ The number 60 was used by them as
a unit, and they therefore divided the hour and the degree
into 60 minutes ; and the minute into 60 seconds.
5. Seven days were made to constitute a unit of time meas-
ure (week), either in accordance with the Mosaic law or from
the fact that seven planets were known to the ancients. The
days of the week were originally named after seven heavenly
bodies. The English names of the days of the week are derived
from the Saxons, a Germanic people who invaded and con-
quered England in the fifth and sixth centuries. The Saxons
borrowed the week from some eastern nation and substituted
the names of their own divinities for those of the Grecian
deities.
Names of the Days of the Week
Latin
Saxon
ENGLISn
Dies Solis (Sun)
Sun's day
Sunday
Dies Lunae (Moon)
Moon's day
Monday
Dies Martis (Mars)
Tiw's day
Tuesday
Dies Mercurii (Mercury)
Woden's day
Wednesday
Dies Jovis (Jupiter)
"Thor's day
Thursday
Dies Veneris (Venus)
Frlga's day
Friday
Dies Saturni (Saturn)
Seterne*s day
Saturday
314 APPENDIX
6. Until the time of Julius Caesar (46 b.c.) the calendar was
in almost constant state of confusion, owing to the fact that
the number of days allowed for a year was more or less than
the actual number of days taken for one revolution of the
earth iu its orbit. As a result of this error, in the time of
Julius Caesar the winter months had been carried back into
autumn, and the autumn months into summer. To correct the
error, Caesar decreed that 90 days should be added to the year
to restore the time of the vernal equinox, and that the year
should consist of 365J days. He ordered that the common
year should thereafter consist of 365 days and that every
fourth year should consist of 366 days. The extra day was
added to February, which at that time had 29 days. This
arrangement is known as the Julian Calendar, or Old Style.
The month of July was named after Julius Caesar.
7. Augustus Caesar ordered that the month following that
which bore the name of Julius (July) should be named after
himself; and in order that the month bearing his name should
have as many days as the month bearing the name of Julius,
he ordered that one day be taken from February and added to
the month which should bear his name. Hence the eighth
month is named August and consists of as many days as July.
8. The year established by the Julian Calendar (365J days)
was .00778 of a day longer than the actual time taken for one
revolution of the earth in its orbit. This error had amounted
to 10 days by 1582, when Pope Gregory XIII undertook the
correction of the calendar. To adjust the time of the vernal
equinox, Pope Gregory ordered that ten days be skipped, from
October 5th to the 15th, and that only centennial years that are
exactly divisible by 400 and other years that are exactly divis-
ible by 4 be made leap years. This arrangement is known
as the Gregorian Calendar, or New Style, and is the one in
common use. Kussia still follows the Julian or Old Style.
The error in the Gregorian Calendar will amount to one day in
about 5000 years.
TABLES OF DENOMINATE MEASURES 315
389. Measures of Length.
12 inches = 1 foot
3 feet = 1 yard
16| feet (5^ yd.) = 1 rod
320 rods = 1 mile
1 mile = 1760 yards = 5280 feet
1. The yard is the primary unit of length. All the other
units of lengtli are derived from it.
2. A furlong is \ mile. It is little used at the present time.
3. A hand, used in measuring the height of horses at the
shoulder, is 4 inches.
4. A fathom, used in measuring the depth of the sea, is 6 feet.
5. A knot, or nautical mile, used in measuring distances at sea,
is 6080.27 feet, or approximately 1.15 (about 1^) miles. The
speed of vessels is expressed in knots. A vessel that travels
18 knots an hour travels about 21 miles an hour (18 mi. plus |
of 18 mi.).
6. For the supposed origin of the inch, foot, fathom, etc.,
consult a dictionary or an encyclopedia.
390. Measures of Surface.
144 square inches = 1 square foot
9 square feet =1 square yard
30J square yards = 1 square rod
160 square rods = 1 acre
640 acres = 1 square mile
1. A square acre is 208.71 + feet on a side.
2. A tract of land 1 mile square is called a section. A town-
ship is a tract of land 6 miles square and consists of 36 sections.
3. 100 square feet of flooring, roofing, or slating is called a
square.
391. Measures of Volume.
1728 cubic inches = 1 cubic foot
27 cubic feet = 1 cubic yard
316 APPENDIX
1. A pile of wood 8 feet long, 4 feet wide, and 4 feet high,
or 128 cubic feet of wood, is called a cord. For the origin of the
name, consult the dictionary. Stonework is sometimes meas-
ured by the cord.
2. In measuring stonework, a pile of stone 16^ feet long, 1^
feet wide, and 1 foot high, or 24| cubic feet of stone, is called
a perch.
392. Surveyors* Measures of Length.
100 links (l.)=l chain (ch.)
80 chains = 1 mile
The chain in common use is called Gunter's chain. It is
4 rods, or 66 feet long. A link is .66 foot. Links are written
as hundredths of a chain, thus : 30 chains 45 links is written
30.45 chains.
393. Surveyors' Measures of Surface.
10 square chains = 1 acre
640 acres = 1 square mile
Square chains are reduced to acres by moving the decimal
roint one place toward the left. Explain.
394. Avoirdupois Weight.
16 ounces = 1 pound
100 pounds = 1 hundredweight
2000 pounds = 1 ton
1. The English ton, known in the United States as the long
ton, is 2240 pounds. It is used in United States custom-
houses and in weighing coal and mineral products at the mines
and sometimes in retailing coal.
2. The smallest unit of weight is the grain. A i)ound avoir-
dupois is 7000 grains. Consult a dictionary for an explanation
of the origin of the name.
TABLES OF DENOMINATE MEASURES 317
3. Grains, vegetables, etc., are commonly sold by weight or
measure. The weight of 1 bushel of the most common of these
articles is as follows :
wheat - 60 lb. oats = 32 lb.
beans = 60 lb. barley = 48 lb.
peas = 60 lb. sweet potatoes = 55 lb.
clover seed = 60 lb. rye = 56 lb.
Irish potatoes = 60 lb. shelled corn = 56 lb.
395. Troy Weight.
Troy weight is used in weighing precious metals.
24 grains = 1 pennyweight
20 pennyweights = 1 ounce
12 ounces = 1 pound
A pound troy is 5760 grains. It is f^f ^ pound avoirdupois.
Precious stones and pearls are weighed by the carat. A
carat equals 3^ grains troy. The term carat is used also to
express the proportion of gold in an alloy. It then signifies a
twenty-fourth part. Thus, gold that is 18 carats fine is ^|, or
I pure gold.
396. Apothecaries' Weight.
Consult a dictionary for the meaning of the word apothecary.
This system of weights is used to some extent in filling pre-
scriptions. The pound, ounce, and grain are the same as in
troy weight, but the ounce is subdivided differently.
20 grains (gr.) = 1 scruple . . . sc. or 3
3 scruples = 1 dram . . . dr. or 3
8 drains = 1 ounce . . . oz. or ^
12 ounces = 1 pound . . .lb. or lb
397. Apothecaries' Liquid Measures.
60 drops (gtt.) or minims (TTL) = I fl»i<l dram . . . /3
8 fluid drams = 1 fluid ounce . • • / 5
16 fluid ounces = 1 pint O.
8 pints = 1 gallon Cong.
318 APPENDIX
39o. Liquid Measures.
4 gills = 1 pint
2 pints = 1 quart
4 quarts = 1 gallon
QuaH means one fourth. A quart is one fourth of a gallon.
A gallon is 231 cubic inches. A gallon of water weighs about
8^ pounds. A cubic foot of water (about 7^ gal.) weighs about
62^ pounds. In measuring the capacity of cisterns, etc., 31^
gallons are called a barrel.
399. Dry Measures.
This system is but little used in some parts of the United
States. Where it is not used, articles are usually sold by
W^^Slit- 2 pints = 1 quart
8 quarts = 1 peck
4 pecks = 1 bushel
The dry quart contains 67.20 cubic inches, the fluid quart
57.75 cubic inches. A bushel contains 2150.42 cubic inches.
The standard bushel in the United States is the Winchester
bushel. It is the volume of a cylinder 18. V inches in internal
diameter and 8 inches in depth.
400. Measures of Angles and Arcs.
GO seconds (") = 1 minute (')
60 minutes = 1 degree (°)
360 degrees = 4 right angles, or 1 circumference
90° of angle = 1 right angle ; 90"^ of arc = 1 quadrant
For an explanation of the origin of 3G0 degrees in a circum-
ference, etc., see Measures of Time, p. 313.
401. Counting Table.
2 units = 1 pair 20 units = 1 score
12 units = 1 dozen 12 dozen = 1 gross
12 gross = 1 great gross
TABLES OF DENOMINATE MEASURES 319
402. Measures of Value — United States Money.
10 mills = 1 cent 10 dimes = 1 dollar
10 cents = 1 dime 10 dollars = 1 eagle
The standard unit of value is the gold dollar. A gold dol-
lar (no longer coined) contains 23.22 grains of pure gold and
2.58 grains of alloy. A silver dollar contains 371.25 grains of
pure silver and 41.25 grains of alloy. The symbol for dollar
..is $, which is taken from U.S.
The coins of the United States are bronze, 1^ ; nickel, 5 ^ ;
silver, 10^, 25^, 50/, $1; and gold $2i, $5, $10, and $20.
The mill is not coined. These are coined at mints located in
Philadelphia, New Orleans, Denver, and San Francisco.
The paper currency is issued in the denominations of $1,
$ 2, $5, $ 10, $20, $50, $100, $500, and $ 1000. Paper cur-
rency consists of bank notes, silver certificates, and gold cer-
tificates. Examine some paper currency. The provision made
for the redemption of each piece of paper currency is printed
on each bill.
Paper currency issued by national banks is commonly called bank notes.
Their payment is guaranteed by deposits of government bonds with the
national government.
403. Values of Common Coins.
Country
Monetary Unit
Valttk in Terms
OF U.S. Gold
Dollar
KOTIQH
Equivale:
Austria-Hungary
Crown
$ .203
J$.20
Britisli Possessions,
N. A. (except New-
foundland)
Dollar ($)
$ 1.000
$1.00
France
Franc (F.)
$ .193
$ .20
German Empire
Mark (M.)
$ .238
$ .25
Great Britain
Pound Sterling (£)
^ 4.8661
$5.00
Italy
Lira (L.)
$ .193
% .20
Japan
Yen (y.)
$ .498
$ .50
Mexico
Peso
$ .498
$ .60
Philippine Islands
Peso
1 .500
$ .50
Russia
Ruble
$ .515
$ .60
320
APPENDIX
404. Table of Compound Interest.
Aiiwunt of^l, at various rates, interest compounded annually.
Yeaus
1%
lV,7o
2%
2%%
3%
3Va%
1
1.010000
1.015000
1.020000
1.025000
1.030000
1.0.35000 1
2
1.020100
1.030225
1.040400
1.060625
1.060900
1.071225
3
1.030301
1.045678
1.061208
1.076891
1.092727
1.108718
4
1.040604
1.061364
1.082432
1.103813
1.126509
1.147523
5
1.051010
1.077284
1.104081
1.131408
1.159274
1.187686
6
1.061520
1.093443
1.126162
1.169693
1.194052
1.229265
7
1.072135
1.109845
1.148686
1.188686
1.229874
1.272279
8
1.082857
1.126403
1.171659
1.218403
1.266770
1.316809
9
1.093685
1.143390
1.195093
1.248863
1.304773
1.362897
10
1.104622
1.160541
1.218994
1.280085
1.343916
1.410599
11
1.115668
1.177949
1.243374
1.312087
1.384234
1.459970
12
1.126825
1.195618
1.268242
] .344889
1.425761
1.511069
13
1.138003
1.213552
1.293607
1.378511
1.468634
1.66.3956
14
1.149474
1.231756
1.310479
1.412974
1.612690
1.618695
15
1.160969
1.250232
1.345868
1.448298
1.567967
1.676349
16
1.172579
1.268986
1.372786
1.484506
1.604706
1 '33986
17
1.184304
1.288020
1.400241
1.521618
1.652848
1.794676
18
1.196148
1.307.341
1.428246
1.56i)659
1.702433
1.867489
19
1.208109
1.326951
1.456811
1.598660
1.753506
1.922601
20
1.220190
1.346855
1.486947
1.638616
1.806111
1.989789
Years
4%
4V2%
6%
G%
7%
8%
1
1.040000
1.045000
1.060000
1.060000
1.070000
1.080000
2
1.081600
1.092025
1.102500
1.123600
1.144900
1.16(5400
3
1.124864
1.141166
1.167626
1.191016
1.225043
1.259712
4
1.169859
1.192519
1.215506
1.262477
1.3107{>6
1.3(50489
5
1.216653
1.246182
1.276282
1.338226
1.402562
1.469328
6
1.265319
1.302260
1.340096
1.418519
1.500730
1.586874
7
1.315932
1.360862
1.407100
1.503630
1.606782
1.713824
8
1.368569
1.422101
1.477456
1.693848
1.718186
1.850930
9
1.423312
1.486095
1.651328
1.689479
1.838459
1.991HXI5
10
1.480244
1.652969
1.628895
1.790848
1.967151
2.168925
11
1.539454
1.622863
1.710339
1.898299
2.104852
2.331639
12
1.601032
1.095881
1.796856
2.012197
2.252192
2.618170
13
1.665074
1.772196
1.886649
2.132928
2.409845
2.719(524
14
1.731676
1.851946
1.979932
2.260904
2.578634
2.937194
15
1.800944
1.935282
2.078928
2.396668
2.75iK)32
3.172169
16
1.872981
?.022370
2.182875
2.640352
2.952164
S. 426943
17
1.947901
2.113377
2.292018
2.692773
3.168816
3.700018
18
2.026817
2.208479
2.406619
2.864339
3.379932
3.9J)6020
19-
2.106849
2.307860
2.526950
3.026600
3.616528
4.316701
20
2.191123
2.411714
2.653298
3.207136
3.869684
4.660967
INDEX
Abstract number, 16.
Accident insurance, 278.
Accounts, 42, 43, 274.
Acute angle, 79, 221, 229.
Acute-angled triangle, 221, 229.
Ad valorem duty, 198, 273. •
Addend, IG.
Addition, of denominate numbers 77.
effractions, 97-99, 108-112.
of integers and decimals, 16-19.
Additive method of subtraction, 22.
Aliquot parts, 147, 208.
Altitude, 221, 226, 295.
Amount, in addition, 16.
in interest, 208.
Angle measure, 237, 318.
Angles, 79, 221, 229, 318.
Apothecaries' measures, 317.
Apothem, 294.
Appendix, 256-320.
Approximate ansvrers, 28.
Arabic numerals, 10.
Arc, 237, 318.
Are, 309.
Areas, 78, 80, 225-232, 235, 292.
Assessed valuation, 198, 270.
Assessors, 198, 270.
Austrian method of subtraction, 22.
Avoirdupois weight, 316, 317.
Bank, of deposit, 276.
savings, 274.
Bank accounts, 274.
Bank discount, 218.
Bank notes, 276.
Banking, 274-278.
Base, 221, 226.
Base line, 299.
BiUs, 42, 43.
and receipts, 44.
Board foot, 160.
Bonds, 261.
Broker, 192, 258, 264.
Brokerage, 258, 262-264.
Calendar, 812-314.
Cancellation, 105.
Cancellation method, 213.
Capacity, measures of, 233, 234, 310, 811.
Capital, 256.
Carat, 317.
Cash discount, 204.
Certificate of deposit, 277.
Check, 277.
Cipher, 10.
Circle, 231, 232, 237.
area of, 232.
Circular measure, 237, 318.
Circumference, 222, 231, 237.
City lot, 134.
Clearing house, 278.
Coins, value of, 319.
Collector, of the port, 273.
of taxes, 270.
Commercial discount, 204.
Commission, 192, 193, 262-264.
Common divisor, factor, or measure, 104.
Common multiple, 106.
Common stock, 258.
Composite number, 87.
Compound denominate numbers, 72, 77.
Compound interest, 217.
table of, 320.
Concrete number, 16.
Cone, 295, 296.
Consumer, 262.
Corporation, 256.
Corporation bond, 261.
Correspondence bank, 278.
Counting measure, 318.
Coupon bond, 261.
Credit, creditor, 42.
Cube (rectangular prism), 233.
Cube of numbers, 244.
Cube root, 246.
Cubic measure, 84, 85, 233, 309, 310, 815.
Customhouse, 198, 273.
Customs and duties, 198, 202, 203, 271-278.
Cylinder, 222, 234, 235.
— 21 321
322
INDEX
Dates, difference between, 168, 267.
Days of grace, 215.
Debit, debtor, 42.
Decimal point, 9.
Decimal system, 7.
Decimals, addition of, 18.
division of, C8-70.
multiplication of, 89-41.
notation and numeration of, 15.
reduction of, 156-158.
subtraction of, 25.
Degree, 237.
Denominate numbers, 72, 70-86.
tables of, 312-319.
Denominator, 93, 109.
Deposit, bank of, 276.
Deposit slip, 274.
Diagonal, 226.
Diameter, 222, 231.
Difference, 20.
Direct taxes, 269.
Discount, 171.
bank, 218.
cash, 204.
trade or commercial, 204, 265.
true, 220.
Dividend, in division, 45.
in insurance, 280.
in stocks, 257.
Divisibility tests, 87.
Division, effractions, 117, 118, 124-131.
of integers and decimals, 45-70.
Divisor, 45.
Draft, bank, 278.
Dry measure, 318.
Duties, 198, 202, 208, 271-278.
Endowment policy, 279.
Equation, 283-292.
Equator, 238.
Equivalents, 312.
Even number, 87.
Evolution, 246.
Exact interest, 268.
Exponent, 244.
Extremes, 292.
Face of note, 207.
Factor, 45, 87, 104.
Factoring, 89, 104.
roots found by, 246, 247.
Farm problems, 86.
Fire insurance, 195.
Flooring, 161.
Foreign money, 819.
Forms, 221-248.
Fraction deflued, 98.
Fractional unit, 93.
Fractions, 90-165,
addition and subtraction of, 97-99, 108-112.
multiplication and division of, 113-131.
reduction of, 96, 96, 100-105, 166-168.
Gain and loss, 181-183.
Government expenses, 272.
Gram, 311.
Greatest common divisor, factor, ormeasure,
104.
Gregorian calendar, 314.
Health insurance, 278.
Heptagon, 294.
Hexagon, 294.
Horizontal lines, 221, 228.
Hypotenuse, 250.
Improper fraction, 94.
Index of roots, 246.
Indirect taxes, 270.
Insurance, 195, 196, 278-283.
Integers, defined, 9.
Integers and decimals, 7-89.
Interest, by aliquot parts, 208.
cancellation method, 218.
compound, 217, 320.
defined, 167.
exact, 268, 269.
simple, 207-213.
six per cent method, 218.
sixtv dav method, 212.
tables, 267, 320.
Internal revenue, 198.
Joint note, 214.
Land measure, 297, 809.
Latitude, 288.
Law of commutation, 32.
Least common denominator, 109.
Least common multiple, 106.
Life insurance, 278-283.
Life policies, 279.
Like quantities, 16.
Linear measure, 76, 808, 816.
Liquid measure, 318.
List prices, 204.
Liter, 310.
Local taxes, 198, 270.
Long division, 63-67.
Long measure, 76, 808, 816.
Longitude and time, 240-242.
Loss and profit, 181-188.
Lowest terms, 105.
Lumber measure, 160-162.
INDEX
323
Maker of note, 214.
Marine insurance, 195.
Market reports, 264.
Maturity of note, 215.
Means, 292.
Measurement, division by, 48.
Measurements, 76-86, 160-165, 221-243,
292-297.
Measures, 312.
Merchants' rule for partial payments, 216.
Meridian, 238, 299.
Meter, 804.
Meter reading, 75.
Metric system, 304-312.
Minuend, 20.
Mixed number, 94.
Model bill, 42.
Multiple, 45, 106.
Multiplicand, 31.
Multiplication, 31.
effractions, 113-116, 119-123.
of integers and decimals, 81—44.
Multiplier, 31.
Municipal corporation, 261.
Negotiable note, 208.
New style calendar, 314.
Notation and numeration, 10-15.
Note, 207, 208, 214.
Number relations, 145, 146.
Numerals, 10.
Numeration, 10-15.
Numerator, 98.
Oblique angle, 79.
Oblique line, 221, 223.
Obtuse angle, 79, 221, 229.
Obtuse-angled triangle, 221, 229.
Octagon, 294.
Odd number, 87.
Old style calendar, 814.
Par of stock, 257.
Parallel lines, 80, 221, 223.
Parallelogram, 221, 227.
Partial payments, 216, 266.
Partition, 48, 51.
Payee, 207.
Payer, 207.
Pentagon, 294.
Percentage, 166-220.
Perfect square, 245.
Periods, 10.
Perpendicular lines, 79, 221, 223.
Personal insurance, 278.
Personal property, 198, 270,
Pi {T), 231,
Policy, 195, 279.
Poll tax, 270.
Polygons, regular, 293-294.
Powers and roots, 244-251.
Preferred stock, 258.
Premium, on policy, 195.
stock at, 257.
Price lists, 265.
Prime meridian, 238.
Prime number, 87.
Principal, 207.
Prism, 221, 222, 283-235.
Proceeds, 218.
Producers, 262.
Product, 81.
Profit and loss, 181-188.
Promissory note, 207, 214.
Proper fraction, 94.
Property, 198, 269.
Property tax, 270.
Proportion, 243, 291.
Public lands, 297-299.
Pyramid, 295, 296.
Quadrilateral, 221, 227.
Quotient, 45.
Radical, 246.
Radius, 222, 231.
Railway time table, 74.
Range lines, 299.
Rate, of dividend, 257.
of interest, 215, 274.
of taxation, 198.
Ratio, 54, 91, 145, 243.
Real estate or real property, 198, 264,
270.
Receipts, 44.
Reciprocals, 125.
Rectangle, 80, 81, 221, 224, 252.
Rectangular solid, 84, 221, 224, 225, 233,
Reduction of fractions, 100.
Registered bond, 261.
Reviews, 71-75, 137-144, 150-155, 184-187,
254, 255.
Right angle, 79, 221, 223, 237.
Right-angled triangle, 221, 226, 250, 251.
Roman notation, 13.
Roots, 244-251.
Savings banks, 274.
Scale drawing, 132-136.
Section, 301, 802.
Shares of stock, 256.
Shingling, 162.
Short methods, 88, 148.
Similar figures, 253.
324
INDEX
Similar fractions, 97.
Similar surfaces and solids, 253.
Six per cent method, 213.
Sixty day method, 212.
Slant height, 295.
Solar year, 812.
Solids, 84, 294-297.
Specific duty, 198,273.
Sphere, 29C, 297.
Square, rectangle, 80, 224, 294,
second power, 244.
Square measure, 78, 315.
Square root, 247-250.
Standard time, 241.
State taxes, 198, 270.
Stere, 310.
Stock quotations, 259,
Stocks and bonds, 250-202,
Subtraction, of denominate numbers, 77.
effractions, 97-99, 108-112.
of integers and decimals, 20-30.
Subtrahend, 20.
Sum, 10.
Surface measure, 78, 308, 809, 815.
Surveyors' measure, 81G.
Tariff, 272.
Tax collector, 270.
Tax rate, 198.
Taxes, 198-201, 269-271.
Term policy, 279.
Terms effraction, 94.
Time measure, 812, 318.
Tontine policy, 281.
Town lines, 299.
Townships, 298, 801.
Trade discount, 204, 205, 265.
Trapezoid, 228, 229, 292.
Triangle, 221, 226, 229, 230, 294.
Triangular prisms, 222, 233.
Troy weight, 817.
Unit, 7.
Unit, fractional, 93.
Unit of measure, 7.
United States money, 14, 25, 819.
United States rule, 2CG.
Unknown quantity, 283.
Usury, 215.
Value, table of, 319.
Vertex, 226.
Vertical line, 221.
Volume, 233-235, 309, 810, 315, 816.
Weight, measures of, 811, 816, 317.
re 35874
ivi55974 ^^
Mp:
THE UNIVERSITY OF CALIFORNIA LIBRARY