LIBRARY
OF THE
UNIVERSITY OF CALIFORNIA
Accession o b 1 9 8 Class
COMPLIMENTS
AMERICAN BOOK CO,
A. F. GUNN, Gen'l Ag't,
204 PINE STRKET,
SAN FRANCISCO.
GRAMMAR SCHOOL
ARITHMETIC
BY
A. R. HORNBROOK, A.M.
NEW YORK : CINCINNATI : CHICAGO
AMERICAN BOOK COMPANY
HORNBROOK'S MATHEMATICS.
HORNBROOK'S PRIMARY ARITHMETIC.
Number Studies for the Second, Third, and Fourth
Years.
HORNBROOK'S GRAMMAR SCHOOL ARITHMETIC.
A Course for the Last Four Years.
HORNBROOK'S CONCRETE GEOMETRY.
An Introduction to Geometry.
COPYRIGHT, 1900, BY
A. R. HORNBROOK.
GUAM. SCU. ABITH.
W. P. I
DISTINCTIVE FEATURES
THIS Arithmetic is designed for use in the last four years
of the grammar schools. The method of presentation is the
result of long and close observation in the schoolroom, and
conforms to the order and manner in which mathematical
concepts are most naturally developed in children.
Practical work has been so combined with work of a purely
disciplinary character that each reenforces and enhances the
value of the other. In business arithmetic, where the practi-
cal demands the greater emphasis, the most simple and direct
methods of computation are presented. Applications of per-
centage which are little used in business, but which have a
value as a stimulus to thought, are introduced at the point
where they will afford the best discipline. Some subjects
that have neither a practical nor a high disciplinary value,
though found in many text-books, are designedly omitted from
this book. The time saved by the omission of such matter is
devoted to more fruitful drill on practical exercises.
A carefully planned and continuous system of reviews runs
through the book. These reviews take the form, first, of an
excursion at the end of each chapter over all the ground thus
far traversed, and, second, of a constant correlation of acquired
knowledge with concepts about to be developed.
No hard and fast line is drawn between mental and written
work. Economy of time and effort is the sole basis of distinc-
tion, and this is a self-regulating principle.
Rules and definitions are given as guides in the preliminary
stages of acquirement. They are not to be formally memo-
3
86198
4 DISTINCTIVE FEATURES
rized; and, when clear ideas of their contents have been
gained, they are to be superseded by rules and definitions of
the pupil's own framing.
Constructive work with simple geometrical forms is intro-
duced at intervals whenever the numerical relations of those
forms offer valuable material illustrative of arithmetical princi-
ples. Exercises to test and develop the pupil's power of visualiz-
ing are inserted in every chapter. The pupil's activity is further
brought into play by a series of exercises in which he is called
upon to supply the conditions for the problems as well as their
solution.
Problems involving unknown quantities, which are solved
arithmetically only by most complicated processes, are deferred
until familiarity with some of the principles governing the
use of literal quantities may suggest simpler methods of
procedure.
The' aim throughout has been to secure a ready skill in deal-
ing with numbers and to develop thought power adequate to
the attack of any arithmetical problem that may arise in
practical life.
CONTENTS
CHAPTER p AGE
I. INTEGERS AND DECIMALS ....... 7
Fundamental Operations and Proofs 16
Addition of Decimals .27
Subtraction of Decimals 30
Multiplication of Decimals ...... 39
Division of Decimals ........ 47
Miscellaneous Exercises 51
II. PROPERTIES OP NUMBERS ....... 62
Multiples and Factors . ... . .62
Composite Numbers ........ 64
Prime Numbers ......... 65
Prime Factors 68
Least Common Multiple 69
Divisibility of Numbers . 74
Common Divisors .77
Powers and Roots 79
Miscellaneous Exercises 84
III. RATIO 95
Miscellaneous Exercises . . . . . . .102
IV. FRACTIONS 108
Addition and Subtraction of Fractions . . . .116
Multiplication of Fractions 126
Division of Fractions 134
Miscellaneous Exercises 142
V. DENOMINATE NUMBERS 150
Miscellaneous Exercises ....... 185
5
6 CONTENTS
CHAPTER PAGE
VI. ALIQUOT PARTS . . . 199
Miscellaneous Exercises 210
VII. PERCENTAGE 216
Merchandising 228
Commission 230
Trade Discount 233
Interest 236
Promissory Notes . . . . . . . . 244
Partial Payments 247
Bank Discount 250
Insurance .......... 254
Taxes .259
Miscellaneous Exercises . . . ... . . 265
VIII. BONDS AND STOCKS 280
Bonds 280
Stocks 285
Miscellaneous Exercises 290
IX. LITERAL QUANTITIES ........ 296
Miscellaneous Exercises . 316
X. INVOLUTION AND EVOLUTION ...... 322
Miscellaneous Exercises 333
XI. PROPORTION 340
Proportional Parts 348
Miscellaneous Exercises 351
XII. MEASUREMENTS AND CONSTRUCTIONS ..... .366
Lines and Surfaces 366
Solids. . . . ' 383
Arcs and Angles 397
Longitude and Time 408
Miscellaneous Exercises , 412
GRAMMAR SCHOOL ARITHMETIC
CHAPTER I
INTEGERS AND DECIMALS
1. Write an integer of three places.
2. Read: 235
. 235 Read "two hundred, thirty five."
2/or Do n t use "and" in reading an
124,235
3. How many figures are used to express the last number
in Ex. 2 ?
4. For what are figures used ? Explain.
5. Express a number of two places by the figures 5 and 3.
Express another number by the same figures. Which is the
greater, and how much ?
6. What is the largest integer that can be expressed by
using once all the figures 3, 7, and 5 ? The smallest integer ?
Find their difference.
NOTE TO TEACHER. Strictly speaking, the largest integer would be
53", but in the exercises in this chapter powers of numbers are excepted.
7. Find the difference between the largest integer and the
smallest integer that can be expressed by using once all the
figures 5, 1, and 8.
8. There are six different integers that can be expressed
by using once all the figures 1, 2, and 3. Write these numbers
in the order of their size and find their sum.
7
8 INTEGERS AND DECIMALS
9. Can numbers be expressed without figures ?
10. Write in words the number represented by 105.
11. Express in good English the number represented by
228,427. By 699,108.
12. Name all the figures that are used to express number.
13. When the figure stands alone, does it express number?
In the sentence " John has marbles," what does express ?
14. is called naught, zero, or cipher. The other nine
figures used to express numbers in Arabic notation are called
digits. What is the tens' digit of the number 75 ? Of 235 ?
What is the thousands' digit of the number 8421 ? Of 29834 ?
Of 127446 ?
15. In the number 815, which is greater, the hundreds' digit
or the tens' digit ? How much ? What is the sum of all the
digits of that number ?
16. Write a number the sum of whose digits is 10.
17. Write a number of four places the sum of whose digits
is 12.
18. Bead: 3
30
300 What is the ratio of 30 to 3 ? Of
o 00(] 300 to 30 ? Of each number in the
' list to the one j ust before it ?
300000
SUGGESTION TO TEACHER. If pupils are not familiar with the terra
" ratio," substitute the question, " 30 is how many times 3 ? "
19. Write a digit and place at the right of it. The result
equals how many times the original digit ?
20. Placing ciphers at the right of a digit is called annexing
ciphers to the digit. Annex two ciphers to 5 and state how
many times 5 the result equals.
INTEGERS AND DECIMALS 9
21. The easiest way to multiply an integer by 10 is to annex
one cipher to it. What is the easiest way to multiply an inte-
ger by 100 ? By 1000 ?
22. Give at sight the following values :
a 35 multiplied by 10 e 1000 times 16 e 10 times 30000
b 100 times 71 d 10 times 3000 / 1000 times 50
23. Give at sight the quotient of :
a 40-hlO c 4370 --10 e 15000 -r- 1000
b 420 -T- 10 d 2500 H- 100 / 28000 -h 1000
24. Give at sight the following values :
a T Vof520 c T i^ of 2300 e T ^ of 4000
b T ^ of 600 d T i of 3100 / T^TT of 18000
25. CLASS EXERCISE. may name a number ending in
three ciphers, and the class may give ^ of it. y-^ of it.
unr <7 of it-
26. Multiply 1000 by 1000. A thousand thousands equal a
Million. How many figures are required to express a million ?
27. Kead 8,636,448.
Bead, 8 million (not millions), 636 thousand (not thousands),
448.
28. Eead: 9,240,827. 31,676,201. 125,475,042.
29. Why is it useful to separate a number into periods of
three figures each before reading it ?
30. Separate into periods and read: 8347621. 98470245.
616823146. 47825001.
31. CLASS EXERCISE. - may write 9 figures on the
board in a horizontal line, and another pupil may tell what
number they represent.
32. Write in figures, placing a comma after millions and
also after thousands : 5 million, 323 thousand, 471. 81 mil-
lion, 175 thousand, 241. 815 million, 278 thousand, 924.
10
INTEGERS AND DECIMALS
Millions
Thousands
Units
00
a
-a
C
1
CO
00
'S
1
1
73
c
o
s
a
o
!
T3
1
I
00
TJ
a
o3
CO
j
w
a>
C
3
g
H
1
a
s
1
O
H
3 'S
W H &
4
7
6
8
2
1
023
33. CLASS EXERCISE. Copy on the board the above dia-
gram, placing different figures in the spaces and reading the.
numbers thus expressed.
34. Write and read a number of seven places, having 3 in
the millions' place, 8 in the thousands' place, 4 in the tens'
place, and in all the other places.
35. Write and read a number of 8 places, having 2 in the
ten-millions' place, 7 in the millions' place, 4 in the units'
place, and in the other places.
36. When numbers are expressed in figures they are said to
be written in Arabic Notation. Write in Arabic notation :
a 323 million, 224 thousand, 24
b 27 million, 960 thousand, 7
c 169 million, 201 thousand, 25
d 41 million, 41 thousand, 41
e 75 million, 75 thousand, 76
/ 121 million, 3 thousand, 3
37. Write a number of 7 places whose units' figure is 5.
Find $ of it. | of it. -|- of it.
INTEGERS AND DECIMALS 11
38. Write the largest number that can be written with 4
places. With 6 places. With 9 places. Give the sum of
the digits of each of them.
39. How many can you count in a minute ?
SUGGESTION TO TEACHER. Find by trial the rate of speed at which
different pupils count, timing them by the watch.
40. At your rate of counting, how many could you count in
an hour ? In a day of 10 hours ?
41. Mary Wallace, a little girl living in Philadelphia,
counted 75 in a minute. At that rate, how many whole
minutes would it take her to count a million? How many
whole hours ? How many days if she counted 10 hours a day ?
42. CLASS EXERCISE. may report the number which
he can count in a minute. The class may find how many
minutes would be required for him to count a million at that
rate. How many whole hours. How' many days of 10 hours
each.
43. Write in Arabic notation :
1st. 435 million, 347 thousand, 526.
2d. The number that is 2 million greater than the 1st.
3d. The number that is 3 thousand less than the 2d.
4th. The number that is 300 thousand more than the 3d.
5th. The number that is 3 more than the 4th.
6th. The number that is 20 thousand less than the 5th.
7th. The number that is 30 million more than the 6th.
8th. The number that is 200 million more than the 7th.
9th. The number that is 40 more than the 8th.
44. Find difference between 1st and 9th number in Ex. 43.
SUGGESTION FOR CLASS EXERCISE. A pupil may write on the board a
number containing millions, and the other members of the class may
direct modifications as in the previous examples. When the pupil at the
board blunders, another pupil may take up his work.
12 INTEGERS AND DECIMALS
45. Beginning at 2, count by twos to 10. How many num-
bers did you name ?
46. What is the sum of 4 twos ? 6 twos ? Numbers which
are the sum of a number of twos are called Even Numbers.
47. What is the first even number after 20? How many
twos does it equal ?
48. Write all the even numbers that can be expressed by
one digit.
49. What is the 8th even number ? The 12th even number ?
50. Divide 1,735,328 by the 7th even number.
51. Can you write an even number which does not end with
0, or 2, or 4, or 6, or 8 ?
52. Write an even number the sum of whose digits is 9.
Find 1 of it. Find of it. Find Jg- of it.
s
53. Write an even number consisting of millions, thousands,
and units. Divide that number by 32. By 102. By 104.
54. In 1895 the expenses of the United States government
were $ 356,195,298. The revenues of the government for that
year were $ 313,390,075. How much did the amount spent
exceed the amount received ?
55. Mention some of the things for which the United States
government spends money, and make an example in addition.
56. The cost of the United States army in the year 1895 was
$51,804,759. The cost of the navy was $28,797,796. How
much did they both cost ?
57. Africa contains 11,514,000 square miles, North America
6,446,000 square miles, South America 6,837,000 square miles,
Asia 14,710,000 square miles, Australasia 3,228,000 square miles,
Europe 3,555,000 square miles, the Polar Kegions 4,888,800
square miles. How many square miles of land does the whole
world contain ?
INTEGERS AND DECIMALS 13
58. The total exports of the United States in 1895 amounted
to $ 807,538,165 ; the imports amounted to $ 731,969,965. How
many more dollars' worth of goods were sold to foreign coun-
tries than were bought from them ?
59. The earth is about 92,800,000 miles from the sun; the
planet Mars about 140,000,000 miles from the sun. How much
nearer to the sun is the earth than Mars ?
60. Multiply a million by a thousand by annexing ciphers.
61. A thousand millions equal a Billion. How many figures
are required to express a billion ?
62. Point off and read :
a 414141414141 c 232648648648 e 58914367281
b 673673673673 d 827345827345 / 42781632512
63. CLASS EXERCISE. may write twelve figures on
the board in a horizontal line, and others may tell what number
they represent.
64. Write in Arabic notation :
1st. 427 billion, 338 million, 484 thousand, 521.
2d. The number that is 4 billion less than the 1st.
3d. The number that is 2 billion, 7 million, 20 thousand less
than the 2d.
4th. The number that is 1 billion, 1 million, and 1 thousand
more than the 3d.
5th. The number that is 13,013,013,013 more than the 4th.
65. CLASS EXERCISE. may write on the board a num-
ber containing billions, and the class may direct changes of it
as in Ex. 64.
66. Write and read an even number consisting of billions,
millions, thousands, and units.
14 INTEGERS AND DECIMALS
67. Write:
a 98 billion, 348 million, 693 thousand, 207
b 15 billion, 279 million, 427 thousand, 48
c 216 billion, 849 million, 348 thousand, 7
d 821 billion, 326 million, 475 thousand, 75
e 2 billion, 2 million, 2 thousand, 2
/ 21 billion, 21 million, 21 thousand, 21
g 78 billion, 78 million, 78 thousand, 78
68. CLASS EXERCISE. may write on the board num-
bers consisting of billions, millions, thousands, and units which
are given to him by the class.
69. To count a billion takes how many times as long as to
count a million ?
70. From the time of the establishment of our govern-
ment in 1789 till 1896 there -had been spent for pensions
$ 1,950,403,063 and for interest on public debts $2,791,537,714.
How much more had been spent for interest than for pensions ?
To whom are pensions given ? Why ?
71. In 1881, the public debt of the United States was
$ 2,077,389,253 and in 1882 it was $ 1,926,688,678. How much
was the debt decreased during the year ?
72. In 1894, Europe produced 897,231,061 Ib. of wool, North
America 342,210,712 Ib., South America 397,970,000 Ib., Cen-
tral America 2,000,000 Ib., Australia 663,600,000 Ib., Asia
258,000,000 Ib., Africa 131,925,000 Ib. How many pounds of
wool were produced that year ?
73. Africa has about 127,000,000 inhabitants, North America
89,250,000, South America 36,420,000, Asia 850,000,000, Aus-
tralasia 4,730,000, Europe 380,200,000, Polar Regions 300,000.
What is the entire population of the world ?
74. Write the largest number that can be written with 12
figures.
INTEGERS AND DECIMALS 15
75. Write a number of 15 places and find from the follow-
ing note how to read it.
The period of figures next higher than billions is called trillions, the
next quadrillions, then come quintillions, sextillions, septillions, octillions,
nonillions, decillions.
76. Write a number larger than 999 trillions and read it.
Why do we seldom use such large numbers ?
77. Write the largest number that can be written with 7
places. Find 1 of it. f of it. ^ T of it.
78. What people in ancient times used letters to express
numbers ? .
79. Copy the Roman numerals and write under each the
corresponding Arabic numeral.
I V X L C D M
80. In Eoman notation when a letter is repeated its value
is repeated. Eead : XX. COG. MMMM. Write in Eoman
notation: 3. 30. 300. 3000. 50. 500.
81. V, L, and D are not repeated. Can you see why ?
82. When a letter of less value is placed after a letter of
greater value the sum of their values is represented. Eead
VIII. XVI. LXVI. CLV. MDCL. MDCCCC. Write in
Eoman notation : 28. 36. 53. 75. 125. 381. 722. 1605.
1620. 1905.
83. When a letter of less value is placed before a letter of
greater value, the difference of their values is represented.
Eead : IV. IX. XL. XC. MXCIX. Write in Eoman nota-
tion: 14. 49. 99. 144. 579. 714. 1239. 1569. 1889. 1909.
84. A line over a letter denotes that its value is multiplied
by 1000. Eead: MVI. VDC. Write in Eoman notation:
10051. 5525. 10630. 4324. 8956. 5427. 6385.
85. Write the following numbers in Arabic notation and
find their sum : MDIII. MDCCCIV. MDCXX. MDCCCCI.
MMDLXV. MDLXII. MDCCCCIV. MDCCCCXIX.
16 INTEGERS AND DECIMALS
86. Write the following numbers in Arabic notation and
find their difference : MCCXCIX and MDCCCXLV.
87. The poet Longfellow was born in MDCCCVII. How
many years old was he at the breaking out of the Civil War
in MDCCCLXI ?
88. How many years elapsed between July 4, MDCCLXXVI,
the date of the Declaration of Independence, and July 4th of
the present year ?
89. Write in Arabic notation, MDC and find y^- of it.
90. Write in Roman notation the following dates :
a The present year.
b 25 years hence.
c 100 years before you were born.
d The year in which our present president was elected.
e The year in which your state was admitted to the Union.
/ The year of Dewey's victory in the Philippine Islands.
SUGGESTION FOR CLASS EXERCISE. Let the pupils suggest important
dates to be written in Roman notation by the class.
FUNDAMENTAL OPERATIONS AND PROOFS
91. A statement that two quantities are equal is called an
Equation, as 60 minutes = 1 hour, 14 days = 2 weeks, 8 -r- 4 = 2.
Write an equation, using the numbers, 7, 5, and another number.
92. Numbers that are added are called Addends. In the
equation 3 + 4 = 7, which numbers are addends ?
93. Fill out the following equations and name the addends.
5 + 7 = ? 6+2i = ? 1+3 = 9
94. Give two addends whose sum is 15. 27. T 8 T .
95. Give three addends whose sum is 14. 20. -f.
96. Give three equal addends whose sum is 27. 30,
FUNDAMENTAL OPERATIONS AND PROOFS 17
97. When John has caught 5 more fish he will have caught
7 fish. How many has he caught ?
98. If Mr. Reed had $ 325 more, he could buy a farm cost-
ing $ 2168. How much money has he ?
99. When the sum of two addends is 29, and one of them
is 4, what is the other ? Give the missing addend when the
sum is 29, and the known addend is 21.
100. CLASS EXERCISE. Think of two addends and their
sum. Then give the sum and one of the addends to the class.
The class may find the other addend.
101. In adding 25 and 18 James carelessly wrote 45 as the
answer. If either of his addends were subtracted from the
number he wrote, would the other addend be found ? Explain.
102. Add 13 and 26. If your work is correct, and if one
addend is subtracted from your answer, what will be left ?
103. Complete these equations. Illustrate with small num-
bers.
Addend -f- Addend = Sum Addend =
104. Add 124 and 354 and prove your work.
To prove the correctness of the addition of two numbers subtract one
addend from their sum. If the work is right, the remaining number will
equal the other addend.
105. Find sums and prove :
abed e f
628 949 639 457 1639 1854
354 848 728 622 2136 237
106. Write an example in subtraction and show which
number is the minuend. The subtrahend. The difference.
107. What is a minuend ? A subtrahend? A difference?
108. From 728 Add subtrahend and difference. If the
take 516 work is correct, the result will equal the
minuend.
HORN. GRAM. SCH. AR. 2
18 INTEGERS AND DECIMALS
109. From 824 Subtract the difference from the minu-
take 512 end. If the work is correct, the result
will equal the subtrahend.
110. Complete these equations. Illustrate.
Minuend Subtrahend = Mm. Dif. ==
Dif. + Sub. =
111. Find the number for which x stands in the following :
Minuend Subtrahend Difference
a 240 x 160
b x 16 30
c 40 x 10
d 60 50 x
112. CLASS EXERCISE. - may give a minuend and a
difference. The class may find the subtrahend.
SUGGESTION TO TEACHER. Every pupil should be required to bring to
the class his contribution to the class exercise carefully prepared.
113. CLASS EXERCISE. - may give a subtrahend and a
difference, and the class may find the minuend.
114. How can you prove an example in subtraction ?
115. Subtract and prove :
a b c d e f g
849 623 814 338 599 451 2148
321 517 276 124 378 239 1939
116. Multiply 123 by 3. By 30. By 300. Find the sum
of the products. Compare the sum of the products with the
product of 123 multiplied by 333.
117. Multiply 275
by 137
I n this example of what two num-
bers is 1925 the product ? 8250 ?
275QO 27500 ? How is 37675 obtained ?
37675
FUNDAMENTAL OPERATIONS AND PROOFS 19
SUGGESTION TO TEACHER. Show that in multiplying by a number of
two or more places we are finding the sum of the several products of the
multiplicand and the number expressed by each figure of the multiplier
in its present position ; and that in practice the naughts are omitted for
the sake of convenience.
118. Multiply a number of 3 places by a number of 4 places,
writing out the full partial products. Why are the full prod-
ucts not usually written out ?
119. What is a multiplicand ? A multiplier ? A product?
Illustrate.
120. Use 25 as a multiplicand and 17 as a multiplier.
121. If the product of 25 and 17 is divided by 17, what
result will be obtained ? If the product of 25 and 17 is divided
by 25, what result will be' obtained ?
122. When the product of two numbers is divided by one
of the numbers, what result is obtained ? Illustrate.
123. The product of two numbers is 35, the multiplicand
is 7. Find the multiplier.
124. Take Ex. 123, substituting in turn for 35 the numbers
84, 42, 77, 91, 112.
125. Find the values of x.
Product Multiplicand Multiplier
a 75 x 15
b x 14 3
c 70 x 10
d 60 6 x
126. Complete the equations. Illustrate.
Multiplicand x Multiplier = Product -r- Multiplier =
Product -r- Multiplicand =
127. How can you prove an example in multiplication ?
128. Multiply and prove :
a 18 x 20 b 13 x 14 c 15 x 16 d 14 x 25
20 INTEGERS AND DECIMALS
129. Multiply and prove :
a b c d e f
836 457 791 625 927 654
125 243 348 244 238 289
130. What is a dividend? A divisor? A quotient?
Illustrate.
131. Find quotients of 800 -f- 2, 60^2, and 8-5-2. Add
the quotients and compare the sum with the quotient of
868 -r- 2.
SUGGESTION TO TEACHER. Show that in dividing 868 by 2, we divide
800 by 2, then 60 by 2, and then 8 by 2.
132. Divide by 3 first 900, then 60, and then 6, and find the
sum of the quotients. Show a shorter way of dividing the
sum of those numbers by 3.
133. Divide 400 and 80 and 4 each by 4. Show the usual
way of dividing the sum of these numbers by 4.
134. 764 -f- 2 = ?
In dividing 764 by 2 we divide 600 by 2, then 160 by 2, and then 4 by 2.
135. 976 -i- 4 = ? (800 -v- 4).+ (160 - 4) + (16 -h 4) = ?
136. Divide 765 by 5, and show how many hundreds, how
many tens, and how many units are used in the separate
divisions.
137. Divide 5468 by 4, and show how many thousands, how
many hundreds, how many tens, and how many units are used
in the divisions.
138. Divide 3765 by 5, and show what parts of the number
are used in each division.
139. What number divided by 8 will give 2 for a quotient ?
140. The quotient of a certain number divided by 8 is 7.
What is the dividend ?
FUNDAMENTAL OPERATIONS AND PROOFS 21
141. Complete the following equations. Illustrate.
Dividend -+- Divisor = Dividend -*- Quotient =
Divisor x Quotient =
142. Find the values of x.
Dividend Divisor Quotient Dividend Divisor Quotient
a 72 4 x c x 21 5
b 99 x 11 d 32 x 4
143. Of what number is 11 both divisor and quotient ?
What number has for divisor and for quotient 7 ? 12 ? 13 ?
15? 17? 21?
144. Use 21 as a divisor and 378 as a dividend. Multi-
ply divisor by quotient, and compare the result with the
dividend.
SUGGESTION TEACHER. Using small numbers, show that as multi-
plication and division are reverse processes, they may be used to prove
each other.
145. Make a rule for proving an example in division when
there is no remainder.
146. Divide each of the following numbers by 23, and prove
your work :
322 391 575 759 874 943 1288
147. Use 41 as a divisor and 618 as a dividend. Multiply
divisor by quotient, add the remainder, and compare the result
with the dividend.
148. Make a rule for proving an example in division when
there is a remainder.
149. Divide each of the following numbers by 25, and prove
your work :
625 879 579 824 386 758 1028 981
22 INTEGERS AND DECIMALS
150. Divide 26103 by 12 as follows :
12)26103(2175
24000
2103
1200 How many times is 12 contained in
24,000 ? How much, remains to be di-
vided by 12 after 24,000 is subtracted
from 26,103?
63
60
SUGGESTION TO TEACHER. Show the process of dividing each re-
mainder after the successive subtractions, and call attention to the fact
that it is more convenient to omit ciphers and that it gives the same result.
151. Divide some numbers by others by long division, writ-
ing out all the work. Why is it better usually to omit some
of the work ?
152. A schoolboy brought this example to his teacher and
121)87493(723 told her that he had discovered that
. if the numbers here printed in heavy
tvpe were added in the order in which
27Q
' * they stand, the result would equal the
*^* dividend. He proved his problems in
373 long division in that way. Take an
example in long division and prove it
10 in the same way.
153. Can you see why adding all the subtrahends and the
remainder in long division will give a result equal to the
dividend ?
SUGGESTION TO TEACHER. Show that in this example 700 times 121
= 84,700, 20 times 121 = 2420, 3 times 121 = 363, and that the sum of
these numbers, plus the remainder 10 must equal the dividend.
154. Divide a number by another number that it contains
exactly 8 times. Double your dividend and see how the quo-
tient is changed.
FUNDAMENTAL OPERATIONS AND PROOFS 23
155. Work with small numbers and show the truth of the
following principles :
PRIN. 1. Increasing the dividend increases the quotient.
PRIN. 2. Decreasing the dividend decreases the quotient.
PRIN. 3. Increasing the divisor decreases the quotient.
PRIN. 4. Decreasing the divisor increases the quotient.
156. Some of a milkman's customers buy a pint of milk
at a time, and some buy a quart. How many of the customers
who buy a pint will together dispose of a gallon ? How many
who buy a quart ?
157. How many pints of water can be drawn from a 20-
gallon tank ? How many quarts ? How many gallons ?
158. A 40-gallon tank contains how many times as many
pints as a 20-gallon tank ?
159. Illustrate the following principles with small numbers :
PRIN. 5. Multiplying both dividend and divisor by the same
number does not change the quotient.
PRIN. 6. Dividing both dividend and divisor by the same
number does not change the quotient.
160. Find the product of 8 and 10 and divide that product
by 4.
If the following expression of the problem were used,
8 x 1Q , and if before multiplying, both 8 and 4 were divided
2
by 4 as follows, $ x 10 = 20, would the result be the same ?
F
SUGGESTION TO TEACHER. Let pupils prove by trial with many small
numbers that canceling common factors in dividend and divisor does not
change the quotient.
24 INTEGERS AND DECIMALS
161. Find the values of the following, canceling when you
can:
a b c
4 x 8 x 6 _ 6 x 9 x 18 21 x 4 x 6
2x4x2 3x3x6~ 7x2x3
d e /
40 x 8 x 10 3 x4 x 5 16 x
= 9
~
20x4x5 7x8x10 8x8
21 x28x4 2 x 8 x 12 30 x 7 x 5
_ 9 _ 9
7x7x6 5x4x6 60 x 21 x 15
^ k I
80 x 4 x 25 = ? 24 x 7 x 10 = 9 35 x 6 x 9 = 9
16 x 20 x 5 12 x 14 x 20 18 x 7 x 5 x 3 "
Can you show how the process of cancellation depends upon
Prin. 6 ?
162. Cancel and find values :
a b c
5 x 8 x 10 x 27 7 x 9 x 56 x 65 21 x 84 x 6 x 8
54x4x20 26 x 49 x~32 x 5 12x49x18
d e f
25 x 6 x 7 x 30 16 x 9 x 28 42 x 15 x 10 x 6
50x9x5 8x36x2 25x14x5
163. CLASS EXERCISE. With 24 x 30 x 35 as the dividend,
- may make a cancellation exercise, and the class may
solve it.
164. CLASS EXERCISE. With 28 x 12 x 36 as a divisor,
may give a cancellation exercise for the class to solve.
165. Copy upon the board and read 1,111,111.
166. Point out the figure that expresses a hundred thousand.
How many hundred thousand make a million ?
FUNDAMENTAL OPERATIONS AND PROOFS 25
167. What part of a million is a hundred thousand ?
168. How much is y 1 ^ of one hundred thousand? Show
the figure that expresses it.
169. How much is y 1 ^- of ten thousand ? Show the figure
that stands for it.
170. Show the figure that stands for y^- of a thousand, y 1 ^
of a hundred, y 1 ^ of ten.
171. Copy 111.1. The point after the units is called a
Decimal Point.
172. The mtmber that is y L of 1 is written .1. Eead 111.1.
This should be read 111 and 1 tenth.
173. 1 at the right of the tenths' place means y 1 ^ of yL or
yi . How many tenths and hundredths in .11 ?
174. 1 at the right of the hundredths' place means y^- of
j-^. or 10 1 00 . How many tenths, hundredths, and thousandths
in .111 ?
175. What does 1 at the right of the thousandths' place
mean? What does 1 in the next place to the right mean?
In the next ?
176. Numbers written at the right of the decimal point are
called Decimals or Decimal Fractions. They decrease in value
at a tenfold rate from left to right just as integers decrease.
A decimal of one place expresses tenths. What does a decimal
of two places express ? Of three places ?
177.' Eead:
.1 .2 .9 .07 .007 .09 .009
178. Eead .19.
This is read 19 hundredths (^%). A decimal is read like an integer,
and then the name of the last decimal place is added to show what kind
of fractional parts it represents.
179. Eead:
12.1 4.3 8.01 7.02 9.021 .003 .009 .03 .006
26 INTEGERS AND DECIMALS
180. Write and read a decimal of two places. Of three
places. Of four places.
181. Write in words :
.123 4.5 .4 41.41 .103 19.2
.15 3.75 8.6 41.041 21.109 11.025
182. Eead:
576.137 432.25 57.41 32.06 75.37 45.81
183. Write in one number an integer of 4 places and a
decimal of 3 places. Read it.
184. Write and read a number consisting of an integer of
6 places and a decimal of 2 places.
185. Write and read a number in which there is an integral
part of 3 places and a decimal part of 2 places.
186. Write and read a number in which there is an integral
part of 5 places and a decimal part of 3 places.
187. Which is greater and how much, the integer 1 or the
fraction ^ ?
Fractions whose denominators are 10, 100, 1000, or 1 with any number
of ciphers annexed, may be written as decimals. This way of expressing
such fractions is convenient because when thus expressed they may be
added, subtracted, multiplied, and divided in the same way as integers.
188. Write as common fractions :
.12 .029 .125 .17 .27 .013 .049 .019
Notice that the denominator of a decimal is not written, but is indicated
by the number of places it occupies. It is always 1 with as many ciphers
annexed as there are places in the decimal. Thus in .32 the denominator
is 100.
189. Write as decimals:
190. Write as decimals :
A
TTnT A TTOF ifo TTfr T5~G
ADDITION OF DECIMALS 27
191. Express T 9 ^, using only one figure. Express y^ by
two figures, y-^ by three figures.
192. Write as common fractions :
.6 .06 .027 .125 .0004 .00025 .000005
193. If you cut a string into ten equal parts, in how many
places must you cut it ? What is each part called ? Express
it as a decimal.
194. How ma^iy times must a string be cut to divide it into
100 equal parts ? Into 1000 equal parts ?
195. Show on a ruler .1 of 10 in. .1 of 5 in.
196. If 20 pupils were in a class and .1 of them were dis-
missed, how many would remain ?
197. How much is .1 of 100 ? .01 of 100 ? Why can you
not easily show on a ruler .001 of an inch ?
ADDITION OF DECIMALS
198. Read and add:
a be
462.001 321.12 725.375
25.01 56.37 409.003
63.475 81.07 361.1
181.312 73.22 448.0035
692.436 195.87 772.6
199. In adding decimals why is it best to arrange them so
that the decimal points are in a vertical line ?
200. Write and add :
a 57 and 123 thousandths, 181 and 28 hundredths, 49 and
3 tenths.
6 167 and 4 tenths, 2128 and 4 hundredths, 396 and 4
thousandths.
28 INTEGERS AND DECIMALS
c 821 and 47 hundredths, 526 and 47 thousandths, 2936
and 1 tenth.
d 674 and 37 hundredths, 25824 and 128 thousandths,
1948 and 4 tenths.
e 666,666 and 6 tenths, 7,777,777 and 77 hundredths,
88,888,888 and 888 thousandths.
201. Add:
a 7.3 in. and 4.7 in.
b 8.4 sq. yd. and 5.6 sq. yd.
c 3.35 sq. ft. and 4.95 sq. ft.
202. .1 is how many times as great as .01 ?
203. In the expression 333.333, which 3 expresses the great-
est value ? The least value ?
204. 3 in the first integral place expresses how many times
as much as 3 in the first decimal place ?
205. 5 in the first decimal place expresses how many times
as much as 5 in the third decimal place ?
206. Which figures in a decimal stand for the greater value,
those near the decimal point or those far away from it ?
' I
TJ "O
w ^ fl T3 13 3 3 73
1 . 1 f I ' I a. *- ' : .1 I I I I- I
1 I | a I I I : H S 1 a I I 1
207. GLASS EXERCISE. Copy the diagram, placing a figure
in each decimal place, and read the numbers thus expressed.
208. How much is T J of .01 ? T V of .001 ? ^ of .0001 ?
of .00001 ?
ADDITION OF DECIMALS 29
209. Give values of x:
a x = 1 in the 4th decimal place.
b x = 1 in the 6th decimal place. ^
c x = 4 in the 5th decimal place.
d x = 7 in the 4th decimal place.
e q = 6 in the 6th integral place.
/ x = 2 in the 6th decimal place.
210. Add, and write the sum in words :
a b c
1.235 24375 325685
123.5 2.4375 .325685
.1235 .24375 3.25685
211. Write and read a decimal of 4 places. Of 5 places.
Of 6 places.
212. Which integral place is occupied by millions ? Which
decimal place by millionths ?
213. Give the places of the following: Thousands, thou-
sandths, ten-thousands, ten-thousandths, hundreds, hundredths,
hundred-thousands, hundred- thousandths.
214. The expression .3 shows that some unit is divided into
10 equal parts, and that 3 of those parts are taken. What
does .003 show ? .0003 ? .00003 ? .000003 ?
215. When we have .3 of an inch, what unit has been
divided into 10 equal parts ? Explain the expression " .7 of
a foot." .17 of a dollar."
216. If A and C were each 3 in. from It,
1 ' ' how far apart would they be ?
217. If A and Owere each 122.57 mi. from B, how far apart
would they be ?
218. If A is 7.7 in. from B, and C is
"J ? 3.8 in. from B, how far apart are J^and <7?
UNIVERSITY
30 INTEGERS AND DECIMALS
219. CLASS EXERCISE. Let pupils give different lengths
(with decimals) to AB and BC, and find distance from A to (7.
220. By rail the distance from Nashville, Tenn., to Evans-
ville, Ind., is 155.07 mi., and from Evansville to Chicago, 111.,
287.15 mi. Mary Allen lives in Nashville. How far will she
travel in going from her home through Evansville to Chicago
and returning by the same route ?
221. How many cents equal 1 hundredth of a dollar ? .17 ?
222. Add:
a
b
c
d
$48.33
$ 75.25
$ 81.39
$ 813.45
76.48
38.60
47.50
425.15
13.15
49.76
86.72
327.40
SUGGESTION TO TEACHER. Call the attention of pupils to the fact that
they have been using decimals of a dollar in their work with dollars,
dimes, and cents.
SUBTRACTION OF DECIMALS
223. Subtract:
a b c d e
446.35 674.37 821.42 123,478.008 964,821.88
29.78 338.49 365.17 1,939.981 283,464.79
224. Which is greater and how much, .6 of a dollar or
225. Which is greater, .1 or .10? .4 or .40? .50 or .5?
.7 or .700?
226. Write a decimal. Annex a cipher to it, and tell how
the value of the decimal is affected.
Without changing values,
227. Change to hundredths : .7. 2.1. 45.3.
228. Change to thousandths : .25. .4. 8.1. 2.56.
229. Change to ten-thousandths : .125. 2.4. .17.
SUBTRACTION OF DECIMALS 31
230. Change to hundred-thousandths : .4758. 3.56. .417. .9.
231. Change to millionths: .85674. 18.35. 42.7. .489. .9249.
232. Write the expression .3. Place a cipher between the
decimal point an<J the figure 3. How is the value of the
expression .3 changed by placing the cipher?
233. .5 of a dollar equals how many cents ? .05 of a dollar
equals how many cents ? Find their difference.
abed
234. From 175.5 691.15 436.4 827.3
take 20.35 420.615 125.25 121.125
235. Mr. Adams had 75.1 acres of land and sold 13.4 acres.
How many acres had he left ?
236. From 195.35 sq. yd. take 37.15 sq. yd.
237. Find the values of :
a 1 - .04 c 2 - 1.8 e 800 - .390
b 300 - .08 d 6001 - 40.683 / 8602 - 304.407
238. Write decimally 11 tenths. 4125 thousandths.
239. 11 - 11 tenths = ? 113 - 113 tenths = ?
240. 117 134 thousandths = ? 8 tenths 436 thou-
sandths = ?
241. 297 - 4138 thousandths = ? 480 thousand - 483 thou-
sandths = ?
242. From one take three hundred seventy-one thousandths.
243. From two and three hundred forty-seven thousandths
take eight hundredths.
244. From ten thousand take ten thousandths.
245. From ten millions take ten hundredths.
246. From five hundred take five hundredths.
32 INTEGERS AND DECIMALS
247. From eight hundred thousand take eight thousandths.
248. From five tenths take five hundredths.
249. /Use 12.75 as a minuend with 3.50 as a subtrahend and
read the difference.
250. A merchant bought goods for $ 89.35 and sold them
for $ 125.75. How much did he gain ?
How much is gained on goods :
251. Bought for $ 129.37, sold for $ 178.12 ?
252. Bought for $ 363.48, sold for $ 429.95 ?
253. Bought for $ 428.35, sold for $ 516.81 ?
254. Bought for $ 596.47, sold for $ 731.97 ?
255. Bought for $ 1028.50, sold for $ 1296.75 ?
256. Bought for $ 1534.81, sold for $ 2346.55 ?
257. Bought for $ .17, sold for $ .23 ?
How much is lost on goods :
258. Bought for $ 275.37, sold for $ 179.33 ?
259. Bought for $ 186.38, sold for $ 175.47 ?
260. Make problems about buying and selling.
261. Add 875.15 to itself.
262. Add 324.75 to the number that is 4 more than 324.75.
Add 324.75 to the number that is .2 more than 324.75. Add
324.75 to the number that is .25 more than 324.75.
263. Draw the line XZ 4 in. long.
x T" z Mark the point Y, 1 in. from Z. How
long is the line XY?
264. If XZ were 8.8 inches and YZ were 2.2 inches, how
long would the line XFbe ?
SUBTRACTION OF DECIMALS 33
265. If XZ were 10 in., and XY were 7 in., how long
would YZ be ?
266. If XY were t.8 in., and XZ were 10.9 in., how long
would YZ be ?
267. Harold stands 7.8 rd. directly east of a certain point,
and his brother Stanley stands 15.6 rd. directly west of it.
How far apart are the boys ? Represent by lines.
268. Stanley and Harold measured lines on the " floor.
Harold started in a corner and measured 3 ft. along by the
side wall. Stanley measured 5 ft. from the same corner in
the same direction. How far apart were the ends of their
lines ?
SUGGESTION TO TEACHER. Let two boys take the parts of Harold and
Stanley for the benefit of those who cannot imagine the conditions.
269. In problem 268 if Harold's line were 7.6 ft. long and
Stanley's 9.8 ft. long, what would be the distance between the
ends of 'their lines ?
270. Stanley was at one end of a side wall of a room 21 ft.
long, and Harold was at the other end. Stanley walked 7 ft.
in the direction of Harold, and Harold walked 2 ft. toward
Stanley. How far apart were they then ?
SUGGESTION TO TEACHER. For a class exercise let children find length
or width of the schoolroom, imagine or enact movements like those in the
previous problems, and find distances.
271. A room is 30 ft. long. If Stanley walked from one
corner of it 7.1 ft. toward Harold who is at the other end of
the same side wall, and Harold walked 7.9 ft. toward him, how
far apart would they be ?
272. High-water mark at a certain town on the Ohio Elver
was 38.3 ft. one year and 42.1 the next year. How much
higher did the river rise the second year than the first ?
HORN. GRAM. SCH. AR. 3
34
INTEGERS AND DECIMALS
273. Find the cost of the materials for a Thanksgiving
dinner at the house of Mr. Smith. Turkey $ 1.75 ; oysters
$ .55 ; potatoes $ .05 ; other vegetables $ .10 ; bread $ .05 ;
pickles $.10; jelly $.10; plum pudding $.40; mince pies
$.20; milk $.10; coffee $.05; salt, pepper, sugar (estimated)
$.05; nuts and raisins $.30. The family consisted of Mr.
and Mrs. Smith and six children. What was the average
cost for each person ?
274. Plan an ordinary dinner and its cost.
NOTE TO TEACHER. For the following
work pupils must be provided with rulers
showing the decimeter, centimeter, and
millimeter.
275. How many centimeters in 3
decimeters ? In 5^ decimeters ?
276. How many millimeters in 5
centimeters ? In 7| centimeters ?
277. How many millimeters in a
decimeter ?
278. Draw on the board a line 10
dm. long. Its length is 1 meter. A
decimeter equals what part of a
meter? A centimeter equals what
part of a meter ?
279. The Latin word "centum"
means 100. How many cents make
a dollar ? How many centimeters
make a meter ?
SUGGESTION TO TEACHER. Let some
pupils make meter sticks, marking the sub-
divisions of 1 dm. and of 1 cm. Let others
mark off a meter and its subdivisions on
ribbon or tape. Keep the best of these a's
a part of the school apparatus.
SUBTRACTION OF DECIMALS 35
280. What'is meant by the perimeter of a figure ?
Draw a square centimeter. How many centimeters in
its perimeter ? How many millimeters ?
281. Draw a square decimeter. How many decimeters in its
perimeter ? How many centimeters ? How many millimeters ?
282. Draw a line on the board 1 decimeter and 6 centimeters
long. Lengthen it 4 centimeters. How many decimeters
long is it now ? How many centimeters ?
283. A millimeter is what part of a centimeter ? Of a deci-
meter ? Of a meter ?
284. M. stands for meter; dm. for decimeter; cm. for centi-
meter, and mm. for millimeter. Can you see why ?
SUGGESTION TO TEACHER. Let pupils find in metric measurements
the length of room, book, desk, writing tablet, pencil, penholder, door,
blackboard, or any other object.
285. Draw a line 1.3 cm. long. How many millimeters long
is it?
286. How many millimeters in 3 cm. ? In 7 cm. ? 9 cm.
and 4 mm. ? 2 dm. ? 3 dm. and 4 cm. ? 5 dm. and 2 cm. ?
The metric system is a very convenient way of measuring, because a
unit of each denomination is ^ of a unit of the next higher. As it is
used in government service, every child should learn it.
287. Compare 1 cm. with 1 in. 1 dm. with 4 in. 1 m. with
lyd.
288. Draw the rectangle ABCD, making the base 8.4 cm.
A B and the altitude 5.6 cm. How long is the
perimeter ? Describe a rectangle.
289. Draw in your rectangle the line
AC. A line drawn from one angle of a
figure to another angle that is not next to
FIG. 1. it is called a Diagonal. Draw as many
diagonals as you can in your rectangle ABCD. Which diago-
nal is the longer ?
36
INTEGERS AND DECIMALS
Can you draw a diagonal of the
290. Draw a triangle,
triangle ? Explain.
291. Two boys were in diagonally opposite corners of a
room. The length of the diagonal was 35 ft. If each walked
3.5 ft. toward the other, how far apart would they be ?
SUGGESTION TO TEACHER. Let children find the length of the diago-
nal of the floor. Two pupils may stretch a string from opposite corners
at a convenient height parallel to the floor, and hence perpendicular to the
intersection of the side walls, and then measure the string. Use the terms
"diagonal," "parallel," and "perpendicular," and let children measure
and adjust until parallelism is secured. Let pupils give to the class prob-
lems similar to Ex. 291.
292. By measuring, find the length of the diagonals of the
cover of your arithmetic. Of the top of your desk.
8.4 4.2 293. How long is the perimeter of
Fig. 2 if the dimensions are centimeters ?
NOTE TO TEACHER. The expression " Fig. 2 "
is used for the sake of brevity instead of the
more exact expression, " The figure repre-
sented by Fig. 2." This contraction is used
throughout the book.
294. How long is the perimeter of
the triangle ABC ? The measurements
are given in centimeters.
295. Triangles that have two sides
equal are called Isosceles Triangles.
Draw two lines of the same length
meeting at a point. Join the ends of
the lines by a straight line. What
FIG. 3. kind of a triangle have you drawn ?
296. Draw an isosceles triangle whose equal sides are each
7 cm. long.
297. The side of a triangle upon which it is supposed to
stand is called its Base. One of the equal sides of Fig. 3 is
how much longer than its base ?
n
TjJ
<
**
4.2
16.8
FIG. 2.
SUBTRACTION OF DECIMALS
37
10.2 cm.
FIG. 4.
1.08 cm.
Fia. 6.
298. What is the sum of the equal sides
of Fig. 3 ? The sum of the equal sides is
how much more than the base ?
299. How long is the perimeter of Fig. 4 ?
The sum of the equal sides is how much
more than the base ?
300. How long is the perimeter of Fig. 5 ?
What kind of a triangle is it ? Why ?
301 . How long is the perimeter of Fig. 6 ?
Is it an isosceles triangle ? Explain. The
sum of the two shorter sides is how much
more than the longest side ?
302. The side AB is how much longer
than the side AC? Than the side BC?
The sum of AB and AC is how much more
than BC? The sum of AB and BC is
how much more than AC? The sum of
B AC and BC is how much more than
AB?
303. Name the denominations in order from a millimeter to
a meter.
304. Express in decimal form the part which one unit of
each lower denomination is of one meter.
305. In the perimeter of a square decimeter, how many
centimeters ? How many millimeters ?
306. If there are 5 sq. cm. in a row, how many square centi-
meters are there in a rectangle composed of 3 rows ? 7 rows ?
307. If there were 5 sq. cm. in a row, how many rows would
it take to make a perfect square ?
308. If there were 5 sq. cm. in a row, how many rows
would it take to make a rectangle containing 30 sq. cm. ?
40 sq. cm. ?
38 INTEGERS AND DECIMALS
309. How many rows would it take to make a perfect square
if in each row there were 3 sq. cm. ? 4 sq. cm. ? 6 sq. cm. ?
310. How many square millimeters in a square centimeter?
311. Draw a square decimeter. How many square centi-
meters in it ? How many square millimeters ?
312. Draw on the floor a square meter and divide it off into
square decimeters. Divide one of the square decimeters into
square centimeters. Can you easily divide square centimeters
on the floor into square millimeters ? Explain.
313. In a square meter, how many square decimeters ? How
many square centimeters ? How many square millimeters ?
314. Express in decimal form the part which one unit of
each lower denomination is of one square meter.
315. John had 10 cents. He spent 6 cents for a ball and
3 cents for a top. How much had he left ? His father had
$ 537.84. He bought a horse and a carriage for $ 300. and a
set of harness for $ 19.75. How much had he left ?
316. One day a bank cashier paid out seven thousand-
dollar bills. On the next he paid out seven hundred-dollar
bills. How much more did he pay out on the first day than
on the second ?
317. A coat that cost $9.75 was sold for $12.50. How
much was gained ?
318. A coat that cost $14.75 was sold for $13.50. How
much was lost ?
319. A ball that cost 6 cents was sold so as to gain 1 cent.
For how much was it sold ? A horse that cost $ 115 was sold
so as to gain $ 17.35. For how much was it sold ?
320. By selling a horse for $475.50, Mr. Smith gained
$87.75. How much did the horse cost? Make similar
problems.
MULTIPLICATION OF DECIMALS 39
321. Mr. Cox spent $237.38 in May, and $348.31 in June.
How much more did he spend in June than in May ?
322. Mr. Ward deposited $ 89.25 in the bank on Monday,
and on Tuesday $ 48.55. On Wednesday he drew out $ 105.35.
How much remained to his credit in the bank ?
323. Have you ever seen a bank ? If so, describe it.
324. I had two notes due me, one of $ 420 and another of
$ 266.66. How much was still due me after $ 389.50 was paid ?
SUGGESTION TO TEACHER. Show promissory note and explain its use.
325. Mr. Gage bought a piece of land of Mr. Wood and
gave him his note for $ 700* When the interest on the note
amounted to $ 38.?5, Mr. Gage paid $ 500 on it. How much
did he still owe ?
326. When $56.25 interest was due on the amount Mr.
Gage then owed he paid $ 175.25. How much did he still owe ?
MULTIPLICATION OF DECIMALS
327. Multiply 1.2 by 3.
When 2 tenths are multiplied by 3, the result is 6
tenths, just as 2 units multiplied by 3 are 6 units, or
T^ as 3 times 2 oranges equal 6 oranges.
328. When a decimal is multiplied by a whole number,
there are as many decimal places in the product as there are
in the multiplicand. Multiply 4.75 by 5. By 20. By 30.
329. Give rapidly the products obtained by multiplying
each of the following numbers in succession by each integer
between 1 and 13 : .6. .8. .9. 1.2. .12. .012.
a
b
c
d
e
330.
Multiply
478.37
21.175
9.35
2.3
24.7
by
6
11
36
24
81
331. What is the cost of a dozen hats at $ 3.75 apiece ?
40
INTEGERS AND DECIMALS
FIG. 7.
332. Multiply 1.28415 by the third even number.
333. All numbers that are not even are called Odd Numbers.
Write in order the first eight odd numbers and find their sum.
334. Every odd number ends with one of 5 digits. Name
them.
335. How many of the first 19 numbers are odd ?
336. Multiply 8.8571 by the fourth odd number.
337. Multiply 16.754 by the seventh odd number.
338. An Equilateral Triangle is a tri-
angle all of whose sides are equal, as
Fig. 7. If each side of Fig* 7 were 8.75
in. long, how long would its perimeter be ?
SUGGESTION TO TEACHER. Show the follow-
ing method of constructing an equilateral tri-
angle. Draw a line of convenient length for
base BC as in Fig. 8. With B as a center and
BC as a radius, draw an arc. With C as a
center and CB as a radius, draw a second arc
intersecting the first at A. Draw A B and A G.
Erase construction lines.
339. Construct an equilateral triangle
each side of which is 4 in. How long
is its perimeter ? How long would the
perimeter be if each side were 8.25 in. ?
4.875 in. ?
340. In the triangle ABC each of the equal
sides is twice as long as the base. What kind of
a triangle is it ? How long is the perimeter ?
341. Construct an isosceles triangle each of
whose equal sides is twice the base.
SUGGESTION TO TEACHER. Let pupils use the same
method of construction as is used for equilateral triangles
except that each arc should be drawn with a radius twice
as long as the base.
FIG.
4.79
FIG. 9.
MULTIPLICATION OF DECIMALS 41
342. In the triangle ABC, AB and AC are
each 3 times as long as BC. How long is the
perimeter ? The sum of the equal sides is how
much more than the base ?
343. Construct an isosceles triangle each of
whose equal sides is 3 times as long as the base.
The perimeter is how many times as long as the
base ?
344. Construct an isosceles triangle whose base
J - : - J, is 4.5 inches and each of whose equal sides is
Fia. 10. 6.5 in. The perimeter is how much longer than
the base ?
345. Construct an isosceles triangle whose base is 3 in. and
the sum of whose equal sides is 9 in.
346. How much will a dozen knives cost at $ .87 apiece ?
347. John had $ .15 which was i of what he needed to buy
a music book. What was the price of the book ?
348. .00256 is | of what number ? ^ of what ?
349. If a dozen knives are bought at $ .67 apiece and sold
for $ 1.00 apiece, how much is gained ?
350. The rent of a house is $ 17.50 per month. How much
is the rent for a year ?
351. A man spends on the average $ .25 a day for cigars.
How much does he spend in a leap year ?
NEW YORK, Sept. 12, 1898.
352. MR. WM. H. MORSE
Bought of THOMAS D. LONG,
25 Ib. Sugar ..... @ $ .05 .... $ 1.25
17 Ib. Coffee ..... @ .25 .... 4.25
6 Ib. Tea ...... .87 .... 5.25
$ 10.75
Received Payment,
THOMAS D. LONG.
42 INTEGERS AND DECIMALS
353. Make a bill similar to Ex. 352, in which the price of
the sugar is 6^ per lb., the coffee 30^, and the tea 95^.
354. Imagine that you are a clerk in a store where a cus-
tomer buys the following bill of goods. Make out the bill and
receipt it.
8 yd. Gingham @ $ .371 per yd.
9 yd. Binding @ .07 per yd.
11 yd. Percale @ .11 per yd.
2 Fans @ .75 each.
SUGGESTION TO TEACHER. Get bill heads from merchants and let
them be copied in the following exercises.
355. Imagine yourself to be a clerk in a grocery store.
Make out and receipt a bill of goods bought by Mr. James
Jones.
356. Make out and receipt a bill of goods bought in (a) a
dry goods store. (6) A shoe store, (c) A music store, (d) A
toy store, (e) A clothing store.
357. Make out a meat bill for Mr. Walter Smith for the
week ending Saturday, Sept. 8, 1900.
358. When 4.8 is changed to 48, by what is it multiplied ?
By what must 4.8 be multiplied to make it 480 ?
SUGGESTION TO TEACHER. Show the method of multiplying by any
power of 10 by moving the decimal point to the right, and of dividing by
any power of 10 by moving the decimal point to the left.
359. Multiply 1.357 by 10. By 10000. By 100000.
360. Divide 125.7 by 10. By 100. By 1000. By 100000.
361. Add 1.25 to 100 times itself.
362. Add to 3.25 the number that is y^ of it.
363. Subtract from 875 the number that is -fa of it.
364. Subtract .213 from 1000 times itself.
365. 3.78 is how much less than 1000 times itself?
MULTIPLICATION OF DECIMALS 43
366. How do you multiply an integer or a decimal by 10 ?
By 100 ? By 1000 ? By any number expressed by 1 with one
or more ciphers annexed ?
367. How is an integer or a decimal divided by 10? By
1000? By any number expressed by 1 with one or more
ciphers annexed?
368. How much is .1 or of 30? .2 of 30?
369. How much is .01 or T ^ of 300 ? .02 of 300 ?
370. How much is .01 of 375 ? .02 of 375 ?
371. Write an integer of three places and find .03 of it.
372. Write an integer of four places and find .7 of it.
373. When an integer is multiplied by a decimal there are
as many decimal places in the product as there are in the mul-
tiplier. Multiply 325 by .7. By .13. By .125.
a b c d e
374. Multiply 275 283 413 671 1289
\A ^17 1.01 _.21 .001
375. How much is .25 of a square 8 in. in dimensions?
376. With 375 as a multiplicand and .31 as a multiplier,
what is the product ?
377. With 145 as a multiplicand and .41 as a multiplier,
what is the product ?
378. How many places and in which direction must the
decimal point be moved in order to divide 125.7 by 100, or to
find y-^ of it?
379. Find .01 of 217.25. Of 365.7. Of 412.137.
380. How much is .01 of 225.7? .03 of it? .08 of it?
Compare the number of decimal places in the products with
the number of decimal places in multiplicand and multiplier.
381. When an integer is multiplied by a decimal, how must
the product be pointed off ?
44 INTEGERS AND DECIMALS
382. How must the products be pointed off when a decimal
is multiplied by an integer ?
383. When a decimal is multiplied by a decimal, the product
contains as many decimal places as there are decimal places in
both multiplicand and multiplier. Multiply .05 by .5.
384. Find products :
a 1.57 x .3 / 84.2 x .43
b 14.5 x .7 g 1.32 x 4.1
c 41.42 x 6 h 6.71 x .11
d 2.42 x 1.21 t 3.41 x .701
e 3.43 x 6.41 j 1.2 x .41
385. If there are not as many figures in the product as
there are decimal places in both multiplicand and multiplier,
ciphers must be prefixed to the product before pointing it off.
Explain.
386. Multiply .15 by .3. .35 x .07. .002 x .7. .021 x .008.
387. To square a number is to multiply it by itself. Square :
1.5. .16. 2.3. .009. .18. 1.9. 3.2. .051. 2.8. 4.08. .025.
388. Draw a square whose dimensions are 1.5 dm. How
many square decimeters in it ?
389. How many square decimeters in a rectangle 1.3 dm.
long and 1.2 dm. wide ? Represent.
390. How many square feet in a square whose dimensions
are 1.25 in. ? Eepresent.
391. How many square inches in a rectangle 7.5 in. long
and 3.5 in. wide ?
392. A lot cost $ 687.50, and the house which stood upon it
cost 4.5 times as much. How much did the house cost? How
much did both cost ?
MULTIPLICATION OF DECIMALS
45
393. Kate drew a rectangle 4.5 in. long and 2.75 in. wide.
Anna drew a rectangle 3.5 times as large as Kate's rectangle.
How many square inches in Anna's rectangle ?
394. One hundredth of anything is called 1 per cent of it.
Per cent is written %, as 4% means .04. Write as per cent :
.17. .07. .03f ^ .50. ^ .61.
395. What per cent of
Pig. 11 is shaded ? Un-
shaded ?
396. BOARD WORK.
Draw a square decimeter
and mark it off into square
centimeters. Shade 13%
of it. What per cent of
it is unshaded ?
397. Shade ^ of it.
What per cent is un-
FIG. 11. Shaded?
398. Shade the following parts of the figure and tell in each
case what per cent is unshaded :
\ .3 39% .4 i 57% .6
.7 | 78% f .9 97% 100%
399. What per cent of a dollar is 1 cent? 3 cents?
21 cents?
400. George had a dollar and lost 5% of it. How many
cents had he left ? How many cents had he left when he had
spent another 5% of the dollar?
401. Of 100 words that John wrote in a spelling test, 13
were wrong. What was his per cent on that test ?
46 INTEGERS AND DECIMALS
402. When your record in an examination is 99%, how
many himdredths of your work are correct ? How many
hundredths are wrong?
403. CLASS EXERCISE. - may name a number less than
100, and the class may tell what per cent it is of 100, and how
many per cent of 100 it lacks of being 100.
404. As 6% of anything is .06 of it, we may find 6% of any
number by multiplying it by .06. Find 6% of 44.
405. In the same way find 6% of: 28. 39. 63. 144. 135.
406. How would you find 1% or any other per cent of a
number ? Illustrate.
407. A company of soldiers consisted of 100 men. 7% of
them were mustered out. How many soldiers remained ?
408. Find 5% of: 14. 24. 75. 1.83. 6.44. 3.72. 8.49.
409. Mr. Miller bought $960 worth of goods, and, in selling
them, gained 15%. How much did he gain? How much did
he receive for them ?
410. Mr. Low bought $3125.50 worth of goods and sold
them at a loss of 2%. How much did he lose ? How much
did he receive for them ?
411. Make problems about buying goods and selling them at
a certain per cent of gain or of loss.
412. How many per cent of anything is the whole of it ?
i- of it ? i of it ? f of it ?
413. Fill out the following and learn:
The whole = 100% \ = -- %
414. Find 25% of 64 by multiplying it by .25. Find 25%
of 64 by taking \ of it.
DIVISION OF DECIMALS 47
415. Find the values of each of the following in two ways,
first by multiplying by the decimal fraction which the per cent
equals, and then by a common fraction which it equals. 25%
of 48. 50% of 12. 75% of 8. 75% of 24. 75% of 32.
416. Find in the shortest way how many men equal 50% of
14 men. How many bu. in 75% of 12 bu. ?
417. How many inches in 50% of a foot? 25% ? 75% ?
How many quarts in 25% of a gallon ? How many quarts in
75% of a peck? How many ounces in 25% of a pound?
50% ? 75% ?
DIVISION OF DECIMALS
418. Divide .75 into 3 equal parts. If 75 cents are divided
equally among 3 persons, will the "25 " which each receives be
25 cents or 25 other things ? If .75 of anything are divided
into 3 equal parts, one of these parts will be 25 what ?
419. When a decimal is divided by an integer, there are as
many decimal places in the quotient as there are in the divi-
dend. Divide 9.24 by 7.
In dividing a decimal by an integer by short division place the decimal
7">0 24 P oint of the Q uotient directly under that of the dividend as
' 9 soon as it is reached.
In this case the quotient of 9 units divided by 7 is 1 unit
with a remainder of 2 units. The decimal point should be placed after
the 1 unit before the division is continued.
420. By 7 divide: 2.583. 1.0332. 4.1328.
421. Find values of x:
a b c d
19.64 38.82 5.76 _ 21.60
X = X = X = X
4686
e ' f 9 *
x = 343.7 x = 13.25 g = 848.8 x = 1.989
48 INTEGERS AND DECIMALS
422. Find the length, of one side of an equilateral triangle
whose perimeter is 7.5 in.
423. By 9 divide: 8.811. 34.569. 672.3. 4712.31
424. $ 12,384.75 were divided among 5 heirs. How much
did each receive ?
425. In one week Mr. A. earned $ 123.66. What were his
average earnings for each working day of the week ?
426. Find J of .0076.
4). 0076 Queries. How many tenths in \ of .0 ? How many
.0019 hundredths in \ of .00 ? How many thousandths in \
of .007 ? How many ten-thousandths in \ of .0036 ?
427. Find 1 of .008. Of .016. Of .246.
428. By 8 divide : .01728. .002016. .12102. .025832.
429. Divide .12 by 9, carrying the division to three places
of decimals.
9). 120 Annexing a cipher to .12, we have .120, which is
.0131 equal in value to .12. .120 divided by 9 equals .
430. How many ten-thousandths in the quotients of the
following ?
1.34 87.1 .128 .542 76.4
69783
431. Divide to three places of decimals: H. Sfi. 5if.
4 o 5
432. Divide 22.75 by 13.
1.75
13)22.75 In dividing a decimal by an integer by long division,
13 write the quotient above the dividend and place the
~~97 decimal point of the quotient above the decimal point
?J_ of the dividend as soon as it is reached.
65
433. By 21 divide: 8.82. 26.04. 10.353. 4.1349.
434. By 32 divide: 5.44. 1.632. .11424. 20.48.
DIVISION OF DECIMALS 49
435. By 24 divide: .3456. .5184. .241584. .5544.
436. Divide 55.44 by 44. By 28. By 77.
437. If $ 3.15 were divided among 15 boys, how much
would each receive ?
438. If hats are bought at $ 8.64 a dozen, how much does
one hat cost ?
439. If $ 250 were divided equally among 3 men, how many
dollars and cents would each man receive ?
440. If the following sums of money were divided equally
among five persons, how many dollars and cents would each
person receive ?
$124 $661 $946 $12823 $67847
441. What is | of 18.24 ? Of 17.52? Of 86.25?
442. If 768.32 acres of land were divided equally among
16 men, how many acres would 3 men receive ?
443. If 11 doors cost $19.25, how much would 2 doors
cost?
444. Multiply 549.36 by 3. By 4J. By 81 By 12J.
445. 46.125x21 = ? 46.125 x3fc=?
446. How much will one knife cost at $ 9.00 a dozen ? At
10.50 per dozen ? At $ 15.00 per dozen ?
447. How much is gained on each hat by buying hats at
$ 20 a dozen, and selling them at $2.00 apiece ?
448. How much is gained on each quart of milk :
a Bought at $ .28 a gallon, sold at $ .08 a quart ?
b Bought at $ .25 a gallon, sold at $ .07 a quart ?
c Bought at $ .30 a gallon, sold at $ .09 a quart ?
449. In buying milk at $ .20 a gallon and selling it at $ .06
a quart, how many quarts must a milk dealer sell to gain
$ 1.00 ?
HORN. GRAM. SCH. AR. 4
50 INTEGERS AND DECIMALS
450. In buying balls at $ 1.00 a dozen and selling them for
$ .10 apiece, how much is gained on each ball ?
451. When 45 yards of calico are bought for $ 1.35, and
sold at $ .05 a yard, how much is gained on each yard ?
452. Divide 8.64 by 2. If both dividend and divisor were
ten times as large as they are, what would the quotient be ?
SUGGESTION TO TEACHER. The principle, " Multiplying both dividend
and divisor by the same number does not change the quotient," should
be thoroughly reviewed and illustrated before the following work is done.
453. Divide 3.76 by .2.
If both dividend and divisor are multiplied by 10, we have 37.6 ~ 2.
This is similar to previous problems.
454. By the following rule perform this example in division
of decimals, and give reason for the rule. 1.96 -r- .4.
To divide by a decimal
Move the decimal point of the divisor to the right until the
divisor is an integer. Move the decimal point of the dividend
an equal number of places to the right, annexing ciphers if
necessary. Divide, and point off as many decimal places in the
quotient as there are then in the dividend.
455. Divide each of the following by .09 :
1.125 12.33 43.119 62.91 4.815
456. Use .06 as a divisor with the following dividends :
221.4 13.2 54.6 91.2 .636 5940 2100
457. Find values of x :
-
a
78.3
/
168
.35
b
x = Ws
9
x- 7S
X ~T5
c
*- 49 - 7
d
x- 37 - 5
e
x- SSA
' .14
k
X - 65A
"12
".15
i
^_165
~ .08
i
a- 89 - 1
~^06
MISCELLANEOUS EXERCISES 51
458. A music teacher earned $ 100 in a month, giving
lessons at $ 1.25 each. How many lessons did she give ?
459. At 75^ a yard, how many yards of lace can be bought
for $ 12.75 ? For $ 23.25 ?
460. Find quotients :
Dividend Divisor Dividend Divisor
a 2.25 1.5 / 2.057 12.1
b 2.75 2.5 g 3.144 1.31
c 137.5 1.25 h 539.6 14.2
d 396 1.2 i 114.92 .221
e 4.84 1.1 j 603.2 .232
461. How long is a rectangle which is
a 5 ft. wide, and contains 35 sq. ft.?
b .3 ft. wide, and contains .75 sq. ft.?
c .7 ft. wide, and contains .77 sq. ft.?
d .5 cm. long, and contains .125 sq. cm.?
e .9 in. long, and contains .72 sq. in.?
MISCELLANEOUS EXERCISES
1. Add, 1248.375, 115.67241, 3935.5428, and 138.463249.
2. From 13 thousand and 21 thousandths take 11 hundred
and 4 hundredth s.
3. Multiply .246 by .89.
4. Divide 243.26647 by .98.
5. Write in Arabic notation and find the sum : MI, MV,
MX, ML, MC, MD.
6. Find the sum of all the numbers less than 100 that are
expressed in Eoman notation by 2 letters. By 3 letters. By
4 letters. By 5 letters. By 6 letters. By 7 letters.
7. Subtract from 100 the number less than 100 that is
expressed in Eoman notation by 8 letters.
52 INTEGERS AND DECIMALS
8. Express decimally and add : 137 and 17 hundredths, 23
thousand 67 and 19 ten-thousandths, 38 thousand 5 and 11
millionths.
9. From 256 thousand 17 and 15 thousandths take 128 and
129 ten-thousandths.
10. How many square feet in a rectangle 1.75 ft. long and
1.25 ft. wide ? How long is its perimeter ?
11. How wide is a rectangle that contains 1.92 sq. in. and is
1.6 in. long? How long is its perimeter ?
12. 1.44 is how many times .0012 ?
13. A merchant bought $2125.75 worth of goods, and sold
them so as to gain 12% of the cost. How much did he gain ?
14. Mr. Duncan bought goods that cost him $ 1226.35, and
sold them so as to gain 16%. For how much did he sell
them?
15. A man died, leaving $ 12,000. He willed 50% of it to
his wife, 30% to his daughter, and the rest to a library. How
much did each receive ?
16. Thomas bought a dime's worth of ice cream, which was
only 50% of the amount he wanted. How many cents' worth
of ice cream did he want ?
17. Jennie has 7 cents, which is 25% of her sister's money,
and 50% of her brother's money. How many cents has each
of them ?
18. $ 45.75 is 25% of how many dollars ?
19. How many pounds in 25% of a ton ? In 10% ? 20% ?
20. Mr. Wade invested $870, and gained 10% on it in a
year. How much had he at the end of the year ?
21. Mr. Brooks invested $9000, gained 10% on it in the
first year, and added the gain to his capital. He gained 10%
MISCELLANEOUS EXERCISES 53
on. that amount in the second year, and added it to his capital.
During the third year he increased his capital by 10%. Find
how much he had at the end of each year. How much more
than his original investment had he at the end of the third
year?
22. How much is 100% of 2 watermelons ? Of 4 chairs ?
23. A chair that cost $ 3 was sold at a gain of 100%. For
how much was it sold ?
24. What number is as much greater than 10 as 10 is
greater than 8 ?
25. What number is as much more than 20 as 20 is more
than 17 ? As much less than 20 as 20 is less than 21 ?
26. What is the average of 10 and 16, or what number is as
much greater than 10 as it is less than 16 ?
To find the average of two numbers, divide their sum by 2. To average
three numbers, divide their sum by 3. To average four numbers, divide
their sum by 4, etc.
27. What is the average of 18 and 20 ? Of 4 and 50 ? Of
9, 21, and 24 ? Of 8, 12, and 25 ?
28. If you stand 98% on an arithmetic test, 95% on
a spelling test, and 92% on a geography test, what is your
average per cent ?
29. Joseph worked 9 problems on Monday, 12 on Tuesday,
and 12 on Wednesday. How many problems a day did he
average ?
30. What is the average of 8.48, 10.24, and 4.96 ?
31. Mr. Harris earned $25.37 in one week, $38.75 the
next week, $ 31.25 the next week, and $ 40.50 the next week.
How much were his average earnings during the four weeks ?
32. High-water mark at a certain town on a large river was
48.3 ft. one year, 50.5 ft. the next, and 47.6 ft. the next. WJiat
was the average ?
54 INTEGERS AND DECIMALS
33. What was the average height of a river for four suc-
cessive days, if on the first day it was 33.9 ft. high, on the next
34.3 ft., on the next 34.9 ft., and on the next 35.1 ft. ?
34. Mr. Howe invested $ 36,000 in business. At the end of
8 years his capital was $ 64,000. What was his average gain
per year ?
NOTE TO TEACHER. The following work requires a Fahrenheit ther-
mometer and an explanation of its use.
35. How many degrees are there between the freezing point
and the boiling point on the Fahrenheit thermometer ?
36. The temperature at noon for four successive days in
July was 90, 91, 92, and 84. Find the average temperature.
37. The temperature at noon for five successive days in
January was 21, 19, 15, 18, and 27. Find the average
temperature.
38. The temperature of our bodies is about 98. How much
above freezing point is that ? Below the boiling point ?
39. How many degrees does the temperature fall when it
changes from 57 to 3 below the freezing point ?
* 40. What is the decrease in temperature from 20 to 1
above ? From 20 to 1 below ? From 15 to 5 below ?
41. The temperature at Minneapolis one winter day was 11.
Before night it fell 20. What was the temperature then ?
42. The next day it rose 15. What was the temperature?
43. The next morning it was 4 below 0. How much had
it fallen ?
44. Mrs. A. bought 19^ worth of groceries and offered $ 1.00
in payment. The clerk gave her 1^ and said, " Twenty."
Then he gave her a nickel and said, " Twenty-five." He then
gave her a quarter saying, " Fifty." He ended by giving her a
half doliar and saying, " One dollar." In the same way find
how that amount of change could be given with different coins.
MISCELLANEOUS EXERCISES 55
45. CLASS EXERCISE. may tell a story of a purchase
made and payment offered. Members of the class may show
different ways of making change.
46. Draw two horizontal lines and two vertical lines.
47. Lines which lie in the same direction are called Parallel
Lines. Find parallel lines in the room. In your book. Name
some capital letters that have parallel lines when printed.
48. Think of your own name printed in capitals. Can you
see any parallel lines in it ?
49. How many pairs of parallel lines has a rectangle ? Are
there any parallel lines in a triangle ?
A 4.5 B 50. A four-sided figure that has only 2
^ ^\ parallel sides is called a Trapezoid. AB
c and CD are parallel. How long is the
perimeter of the trapezoid ABCD, the
measurements representing inches ?
51. The sum of the parallel sides of Fig. 12 is how much
more than the sum of the non-parallel sides ?
52. Draw trapezoids of different shapes.
53. Two lines meeting at a point form an Angle, Z. The
point where the lines meet is called the Vertex of the angle.
Draw an angle and mark its vertex A.
A 54. When one straight line meets another
straight line so as to make two equal angles,
the angles are called Right Angles. What
letter is at the vertex of each angle in
Fig. 13?
Right 55t Pl ace two pencils so as to show two
Right
Angle
Angle right angles.
56. Fold a strip of paper so that the
FIG. 13. line of the fold makes right angles with
1 the edge.
56 INTEGERS AND DECIMALS
57. Cut out a paper circle and fold it into fourths. What
kind of angles are made by the folds ?
58. Find right angles made by lines in the surfaces of the
room or of objects in it.
59. An angle less than a right angle is called
an Acute Angle. Draw an acute angle.
SUGGESTION TO TEACHER. As children naturally
FIG. 14. Judge of the size of an angle by the length of the lines
that form it, pupils should draw and cut out a right
angle, and by applying it to given angles, find out whether they are
acute, right, or obtuse.
60. Draw a trapezoid and mark the acute angles.
61. An angle greater than a right angle
is called an Obtuse Angle. Draw an obtuse
angle.
FIG. 15. 62. Draw a trapezoid and mark obtuse
angles and acute angles.
63. What kind of angles has a rectangle ? An equilateral
triangle ?
64. Draw a trapezoid that has two right angles. Name the
other two angles.
65. In the printed words "ADMIRAL DEWEY," how many
right angles are there ? Acute angles ? Obtuse angles ?
66. If the name of the county in which you live were
printed in Gothic type like the words " ADMIRAL DEWEY,"
how many right angles would there be in it? How many
acute angles? How many obtuse angles?
SUGGESTION TO TEACHER. Splints or toothpicks are useful in the
following exercises.
67. With 3 lines make 2 right angles; 2 obtuse angles;
2 acute angles. Show the vertices of the angles.
MISCELLANEOUS EXERCISES
57
68. With 2 lines make 4 angles, and tell of what kind they
are.
69. With 3 lines make 12 angles, and tell their kinds. Make
10 angles. 9 angles.
70. With 4 lines make 16 angles. 20 angles. 24 angles.
71. With 5 lines make 4 angles. 5 angles. 20 angles.
72. CLASS EXERCISE.
may tell how many angles he
can make with a certain number of lines, and the class may
make them.
73. A triangle that has a right angle is called a Right
Triangle. Draw a right triangle.
74. Draw an isosceles triangle. The angles at the base are
equal. What kind of angles are they ?
75. Draw an isosceles triangle on paper. Cut it out and
fold it so that the equal sides coincide. Cut along the line of
the fold, and you have two equal triangles. What kind of tri-
angles are they ?
76. How long would the perimeter of one of these right tri-
angles be if the base were 20 in., perpendic-
ular 11 in. longer than the base, and hypot-
enuse 8 in. longer than the perpendicular ?
<<
Base
FIG. 16.
77. Place together the two triangles
you have made so that they form a
rectangle. If the area of that figure
were 28 sq. in., what would be the area
of each right triangle ?
78. A figure drawn upon a flat surface is called a Plane
Figure. Can you draw a plane figure on the surface of a ball ?
Of a slate ? Of a piece of gas pipe ?
58
INTEGERS AND DECIMALS
FIG. 17.
79. A plane figure bounded by five straight
lines is called a Pentagon. When (as in
Fig. 17) the lines are all equal and make
equal angles, the figure is called a Regular
Pentagon. What kind of angles has a
regular pentagon?
80. Find the length of the perimeter of
the pentagon represented by Fig. 17.
81. How long is one side of a regular pentagon whose perim-
eter is 9.15 in. ?
82. Draw a pentagon that is not
regular.
83. The pentagon in Fig. 18 is
o divided into triangles by equal lines
drawn from its center to the vertices
of its angles. What kind of triangles
are thus formed ? How many of them ?
Each triangle is what part of the penta-
gon? What %?
FIG. 18.
84. What % of the pentagon is the figure ABCO ? AEDO ?
BCDEO? CDEABO?
85. Figure 19 differs from Fig. 18 in
having the lines Og, Oh, etc., drawn
from the center of the pentagon to
the middle point of each side. They
are perpendicular to the sides. Each
right triangle thus formed is what part
of the pentagon ? What % ?
E i D
FIG. 19.
86. What % of the pentagon is AOB?
EOi? BOhC? ABCh?
87. Give the outlines of a figure which is 70% of the
pentagon.
MISCELLANEOUS EXERCISES 59
88. A butcher bought a hog weighing 375 Ib. at $ .03 a Ib.
How much did it cost ?
89. He sold 15 Ib. of it at 2^ per Ib., 50 Ib. at 5^ and the
rest at 12^ per Ib. How much did he receive for it ?
90. A farmer sold 15 doz. eggs at 18^ a dozen, receiving for
them sugar at 6^ a pound. How many pounds of sugar did
he receive?
91. A fruit dealer buys 29 doz. oranges for $8.70. How
much does he pay for each orange ?
92. If he sells the oranges at the rate of -5^ apiece, how
much does he gain on each orange ? How much on all the
oranges ?
93. What number multiplied by 9 will give the same prod-
uct as 12 multiplied by 6 ?
94. Mr. Hale had $ 5728 and paid 25% of it for a farm.
How much did the farm cost ? He sold the farm for $ 1200.
How much did he lose ? ^ ;
IVA A '
^ 95. He left the other 75p of his money in the bank until it
had gained $472 interest. How much money had he then,
including the money he received from his farm ?
96. A grocer bought 185 barrels of flour at $ 3.75 a barrel,
and sold it all for $ 740. How much did he gain ?
97. A miller bought 35 bu. of wheat for $ 22.75, and sold it
at $ .61 a bushel. Row much did he lose ?
98. A farmer had an orchard of 276 trees. One year they
averaged 13 bu. of apples to each tree. What was the value
of that season's crop at $ .75 a bushel ?
99. The next year the trees averaged 9 bu. per tree, and
the apples brought $ .80 a bushel. What was the value of the
crop that year ?
60 INTEGERS AND DECIMALS
100. A merchant's profits in January, 1899, were $ 1428.75.
In January, 1900, his profits were 20 % less. What were his
profits in January, 1900 ?
101. Mr. Strong had $975.85 in a bank; he drew a check
on the bank for $ 625.47. How much money had he remain-
ing in the bank ?
SUGGESTION TO TEACHER. Show bank checks. Explain their use and
let pupils copy and fill them out for use in imagined transactions.
102. If you had $ 65.87 in a bank, and should draw a check
for $ 38.45, how much of your money would be left in the bank ?
103. Mr. Gale had $1225 in a bank. He drew $12.25
every Saturday night for 10 weeks. How much had he left in
the bank ?
104. Aline deposited $ 11.75 in a savings bank in February.
She drew out $ 3.25 in March and $ 2.95 in April. She de-
posited $ 14.45 in May. How much had she then in the bank ?
105. Mr. Davis bought a stove worth $ 18.75. The dealer
allowed him 2^ a pound for an old stove, weighing 195 Ib. He
gave a check on the bank for the balance. What was the
amount of the check ?
106. There were 276 houses on the street. A postman
delivered 3 letters at 28 of the houses, 2 letters at 41 of the
houses, and 1 letter at 105 houses. At how many houses were
no letters delivered ?
107. There were 559 books in a school library, which was
an average of 13 to each pupil. How many pupils were there
in the school ?
108. In February of a common year, Mr. Fisk's family
burned a ton of coal in 14 days. At $ 8.50 per ton, what was
the cost of the coal for that month ?
109. A lot is in the form of a trapezoid. One of the parallel
sides is 16.8 rd. long, and the other is twice as long. Of the
MISCELLANEOUS EXERCISES 61
sides that are not parallel, one is 19.7 rd. long, the other is
15.4 rd. long. How long a fence is required for the whole lot ?
110. Mr. Lee started to Denver with $ 300. He paid $ 47
for railroad fare, his hotel bill was $ 4 a day for a week, other
expenses $ 7.50, and his return ticket was $ 47. How much
did he spend ? How much had he left ?
111. Passengers were first carried on railroads in the United
States in the year MDCCCXXVIIL How many years have
we had railroads ?
112. Square: 13. 1.5. 1.7.
113. Ella had a flower bed a yard square. She divided it
into square feet and placed a rosebush in the middle of each
square foot. How many rosebushes had she ? Represent.
114. Draw a square decimeter and show into how many
square centimeters it can be divided. Into how many square
inches can a square foot be divided ?
115. How many square inches in 5 squares whose sides are
each 11 in. long ?
116. How long is the perimeter of a square, a side of which
is 3.1 in. long ? What is its area ?
117. Draw a right triangle. If its base were 7.5 in., its
perpendicular 10 in., and its whole perimeter 30 in., how long
would the hypotenuse be ? What would be its area ?
118. Find the length of the perpendicular of a right triangle
whose perimeter is 90 in., its base 22.5 in., and hypotenuse
37.5 in. Find the area of the triangle.
CHAPTER II
PROPERTIES OF NUMBERS
1. When an integer can be divided by another number
without a remainder, it is said to be divisible by that number.
Is 9 divisible by 5 ? Give a reason for your answer.
2. Choose an even number and illustrate this statement :
An even number is an integer that is divisible by 2.
3. Choose an odd number and illustrate this statement:
An odd number is an integer that is not divisible by 2.
4. Name the first even number after 10. How many twos
does it equal ?
5. Square the third odd number. Square the sixth even
number. Multiply the seventh even number by the fifth odd
number. Find the difference between the sixth odd number
and the eighth even number.
6. Find | of the fifth even number. Find 7% of the fourth
odd number.
MULTIPLES AND FACTORS
7. A Multiple of a number is the product obtained by
multiplying it by an integer. Thus 5 is the first multiple
of 5, 10 is the second multiple of 5. Give quickly the first
twelve multiples of
3 4 5 6 7 8 9 10 11 12
8. Figure 1 represents two lots of land owned by Mr. Smith
and Mr. Brown. The fence between the lots is 150 ft. long
MULTIPLES AND FACTORS
63
and cost 7^ a foot. How much of this expense should each
man pay ? Give reasons for your answer.
. , _ 9. Mr. Smith and Mr. Brown decided
to take away the fence and leave a strip
10 ft. wide on each side of the line where
it had stood. This strip is used as a com-
mon playground by the children of both
families. How many square feet in their
common playground ?
10. A fine park in Boston is called Bos-
ton Common. What does the word " com-
mon " mean in this case ? " 20 is a com-
What does that statement mean ?
FIG. 1.
mon multiple of 10 and 5."
11. A number which is a multiple of two or more numbers
is called their Common Multiple. Give several numbers that
are common multiples of 2 and 3. Of 3 and 7. Of 4 and 5.
12. Of what two numbers besides itself and 1 is 15 a mul-
tiple? 10? 35? 21? 22? 33?
13. Write all the numbers of which 6 is a common multiple.
8. 14. 16. 12. 24. 40. 36.
14. Give two numbers which multiplied together make the
product 18. 20. 27.
15. The numbers that make a product are called the Factors
or Divisors of that product. 12 has 3 pairs of factors, 1 x 12,
2x6, and 3x4. Give all the pairs of factors of 24, except
the pair of which 1 is the least factor.
SUGGESTION TO TEACHER. Let pupils find factors of a number by
using as a trial divisor each number in succession, beginning with 2.
Lead them to see that as soon as the quotient which they obtain is less
than the divisor they use, it is unnecessary to try" any more numbers, as
they will merely get the same pairs of factors stated in* reverse order.
16. Give all the pairs of factors of 45. 28. 36. 60. 72.
17. Give all the numbers of which 30 is a multiple. 66.
64 PROPERTIES OF NUMBERS
18. CLASS EXERCISE. may give a number which is a
multiple of some other numbers, and the class may find all
its factors.
COMPOSITE NUMBERS
19. A number that is the product of two or more integers
is called a Composite Number. Give three, composite numbers
and their factors.
20. What number is composed of the factors 2 and 11 ? 3
and 11 ? 2, 3, and 11 ? 7 and 7 ? 2, 3, and 7 ? 2, 2, and 3 ?
21 . What factors compose 77 ? 40 ? 18 ? 42 ?
22. Name an even composite number, and give factors that
compose it.
23. Name an odd composite number, and give factors that
compose it.
24. Name a composite number that is a multiple of 5, and
give its other divisors.
25. Make and keep a list of all the composite numbers less
than 41.
26. "Write a composite number whose tens' digit is 2, and
give its factors.
27 . Write the following numbers and their factors :
a A composite number whose units' figure is 5.
I) The first composite number after 26.
c A composite number between 30 and 40 that is not a mul-
tiple of 5.
d A composite number between 30 and 40 that is not a mul-
tiple of 2 nor of 5.
e A composite number between 20 and 30 that is not a mul-
tiple of 2, 5, nor 7.
28. Divide .00168 by the 3d composite number.
PRIME NUMBERS 65
29. Find 5% of the 9th composite number.
30. Multiply the 8th composite number by .009.
PRIME NUMBERS
31. A number that has no integral factors except itself and
1 is called a Prime Number. Think of each of the numbers from
2 to 10 and tell which of them are prime.
2 is the first prime number, as 1 is considered neither prime nor
composite.
32. No prime number of more than one place ends in 2, 4, 6,
8, 0, or 5. Can you tell why ?
33. Make a list of all the prime numbers less than 50 in
their order.
SUGGESTIONS TO TEACHER. Show pupils how to find prime numbers
less than 50 by examining each number to see if it can be exactly divided
by 2, 3, 5, or 7. Develop the fact that there is no need of dividing even
numbers because they are all divisible by 2 ; nor numbers that end in 5,
because they are divisible by 5. Lead pupils to see that if a number will
not contain 2, it will not contain 4, 6, 8, or any number of twos ; that if
it will not contain 3, it will not contain 9, 15, 21, or any number of threes,
and so on.
' 34. Group the prime numbers less than 50 as they are found
in each ten numbers, as :
1st ten
2dten
11
13
17
Keep the list.
19
35. Find the sum of all the prime numbers that are ex-
pressed by one digit.
36. Divide 13.5 by the 2d prime number.
37. Divide the 3d prime number by .8.
38. Find the difference between 26.4 and the llth prime
number.
HORN. GRAM. SCH. AR. 5
66 PROPERTIES OF NUMBERS
39. Multiply the 8th prime number by .0004.
40. Find 6% of the 9th prime number.
41. Write the first 30 numbers in two lists, one of prime
numbers, the other of composite numbers. Leave out the
number 1.
42. Find the sum of all the composite numbers less than 15.
43. Find the sum of all the primes less than 20.
44. What prime number is nearest to 20 ?
45. What two prime numbers are near to 12?
46. 15 is half way between two prime numbers. What are
they ?
There are three pairs of these numbers.
47. What prime number is nearest to the 2d multiple of 5 ?
To the 8th multiple of 5 ?
SUGGESTION FOR CLASS EXERCISE. Let children select prime numbers
and give a clew to them, and let the class identify them.
48. Find the difference between the prime number nearest
to 20 and the prime number nearest to 8.
49. CLASS EXERCISE. may name a number larger than
50 which he thinks is prime, and the class may see if he is right.
NOTE TO TEACHER. The following method of finding prime numbers
less than 100 is very useful :
Write the first hundred numbers as on p. 67, omitting 1 because it is con-
sidered neither prime nor composite. Counting from 2, the first prime
number, strike out as composite every second number because it is a
multiple of 2 ; counting from 3, strike out as composite every third
number. Lead the pupils to discover why it is unnecessary to strike out
the multiples of 11 or of any larger primes in finding the prime numbers
less than 100.
This device, which is an adaptation of the well-known "sieve of
Eratosthenes," may be used to any limit to separate prime from compos-
ite numbers by writing additional columns of numbers and striking out all
the multiples except the first multiple of those primes whose squares are
included within the limit.
t ' n f\ pr
PRIME NUMBERS 67
.-
11 2; 31 41 $1 61 71 2J 91
2T9 99 319 AW 39 ft ' 79' S9 Q9
A^ ^ P^ rr Pr rr l*r Pr PP
3 13 23 33 43 53 03 73 83 ?3
7 17 27 37 .47 57 67 77 27 97
1% ?$ 32 2 ^2 02 72 22 R2
? 19 29 3? 9 59 0? 79 89 99
50. A Greek mathematician named Eratosthenes, who was
born 275 B.C., devised this plan of finding prime numbers.
Instead of marking out the composites, he cut them out.
Can you see why the table of primes that was left was called
" Eratosthenes' sieve " ?
51. How many and what prime numbers in the 2d ten
numbers ? In the 10th ten ? In the 5th ten ?
52. Give the primes less than 100 whose units' digit is 1.
3. 7. 9.
53. Find the sum of all the primes in the 3d ten. In the
6th ten. In the 4th ten. In the 7th ten.
54. Name all the prime numbers less than 100 whose tens'
digit is 2. 4. 1. 3. 5. 7. 9. 6. 8.
NOTE TO TEACHER. This adaptation of "Eratosthenes' sieve" may be
made helpful in studying composite numbers, divisors, and multiples.
Let the columns of numbers be written on the board in large figures.
Instead of striking out multiples, draw a circle or square around each of
them, using crayon of the same color to inclose the multiples of a partic-
ular number. If, for instance, the multiples of 2 are inclosed in blue,
those of 3 in red, those of 5 in green, those of 7 in yellow, 30 shows itself
at once by its motley framing as a multiple of 2, 3, and 5 ; 42 by its
slightly different framing as a multiple of 2, 3, and 7 j 66 as a multiple of
2, 3, and 11.
68 PROPERTIES OF NUMBERS
The following questions are based upon a diagram of that kind and
refer to numbers less than 101.
55. Point out the multiples of 3 whose units' digit is 2. 3.
5. 6. 7. 8. 9. 4.
56. Point out the multiples of 7 whose units' digits is 1. 2.
4. 9.
57. What figure ends the expression of all the multiples of
5 that are odd numbers ? Even numbers ?
58. Point out all the numbers that are multiples of 2 and
also of 5, beginning with the least.
59. Show the common multiples of 2 and 3, beginning with
the least common multiple.
60. Show the common multiples of 2 and 7, beginning with
the least common multiple.
61. Beginning with the least common multiple, show all the
common multiples of 3 and 7. Of 3 and 5. Of 5 and 7. Of
2, 3, and 5. Of 2, 3, and 7. Of 2, 5, and 7.
62. CLASS EXERCISE. The teacher or a pupil pointing to a
number in the diagram, members of the class tell of what
numbers it is a common multiple, and whether or not it is
their least common multiple.
PRIME FACTORS
63. Those factors of a number that are prime, are called
Prime Factors. What are the prime factors of 4 ? Of 6? Of 8?
64. Find the prime factors of 42.
2 1 42 To find the prime factors of a number divide it by the smallest
3 "21 prime number of which it is a multiple. Then divide the quo-
4r tient by the smallest prime number of which it is a multiple.
Continue dividing until the quotient is prime. In this case 42
divided by 2 gives a quotient of 21, 21 divided by 3 gives the prime quo-
tient 7, Hence the prime factors of 42 are 2, 3, and 7.
LEAST COMMON MULTIPLE 69
65. Find the prime factors of:
10 15 20 24 27 32 35 39 44 48
12 16 21 25 28 33 36 40 45 49
14 18 22 26 30 34 38 42 46 50
66. Find the prime factors of all the even numbers greater
than 49 and less than 59. Of all the composite odd numbers
between those limits.
67. Find the prime factors of all the even numbers between
59 and 69. Of all the composite odd numbers.
68. Find the prime factors of all the multiples of 5 between
69 and 91. Of all the multiples of 3 between those limits. Of
all the composite numbers between 90 and 101.
69. Resolve into prime factors :
a 336 c 1225 e 639 g 3105 t 1470 fc 1296
b 3456 d 2214 / 560 h 888 j 810 I 1488
70. CLASS EXERCISES. may give to the class a number
that is the product of several small prime numbers, and the
class may find its prime factors.
71. Separate the first 100 numbers into two lists, one of
prime numbers, the other of composite numbers. Write oppo-
site each composite number the prime factors of which it is
composed.
SUGGESTION TO TEACHER. Class drill upon these lists should be given
frequently until their contents are learned.
LEAST COMMON MULTIPLE
72. Forty contains how many more fives than eights ?
73. CLASS EXERCISE. '- may name a number which is
a common multiple of two or more numbers. The class may
give the numbers and tell how many times their common
multiple contains each of them.
70 PROPERTIES OF NUMBERS
74. The Least Common Multiple of two or more prime
numbers is their product ; the next common multiple is twice
their product ; the next is three times their product. What is
the next ? Write the first six common multiples of 2 and 5,
and underscore the least.
75. Write the first four common multiples of 3 and 5. Give
the least common multiple and find how many times 3 and 5
are each contained in it.
76. Write the first three common multiples of 2 and 7.
Find how many times 2 and 7 are each contained in their least
common multiple.
77. Find the least common multiple of 2, 3, and 5. How
many threes does it contain ? How many twos ? How many
fives?
78. Find the least common multiple of 2, 3, and 7. How
many sevens in it ? Twos ? Threes ?
79. The abbreviation for least common multiple is 1 .c. m.
Find the 1. c. m. of 2, 5, and 7. How many fives in it ?
Sevens ? Twos ?
80. Find the 1. c. m. of 5 and 9. Use the following method
of finding the 1. c. m. mentally : Think of the multiples of
the larger number in order until one is found which is a
multiple of the smaller. For instance, in finding the 1. c. m.
of 5 and 9, think of the multiples 9, 18, 27, 36, until the first
that contains 5 is reached.
81. Find the 1. c. m. of :
a 10 and 3 10 and 5 10 and 8 10 and 12 10 and 15
b 6 and 9 4 and 9 7 and 9 11 and 9 8 and 9
c 8 and 3 8 and 5 8 and 6 8 and 10 8 and 11
82. Find the 1. c. m. of 6, 5, and 3. Can a number be a
multiple of 6, without being also a multiple of 3 ?
LEAST COMMON MULTIPLE 71
83. Find the 1. c. m. of :
a 6, 2, and 3 10, 5, and 2 20, 10, and 5 7, 5, and 6
b 10, 5, and 6 8, 4, and 12 5, 10, and 15 8, 4, and 3
c 2, 3, 4, and 6 3, 4, 8, and 6 4, 8, and 7 3, 6, and 8
84. 12 is the 1. c. m. of 2, 3, 6, and 4. Make similar state-
ments about the numbers 18, 20, 24, 25, 30, 36, 35, and 48.
85. Which of the first 12 multiples of 3 are common mul-
tiples of 12 and 3 ?
86. Can you find the greatest common multiple of 3 and 4?
Of any two other numbers ? Explain.
SUGGESTION TO TEACHER. Some of the pupils inay discover that the
search for the greatest common multiple leads into infinity.
87. Draw a line 18 in. long, and show how many times a
3-in. line can be laid off upon it. How many times can a 6-in.
line be laid off upon it ?
88. How long is the shortest line that can be laid off into
2-in. lines or 7-in. lines? 3-in. lines or 7-in. lines? 7-in.
lines or 5-in. lines ? 5-in. lines or 11-in. lines ?
89. How large is the smallest number that can be divided
into groups of 2 and of 7 ? Into groups of 3 and of 7 ? Into
groups of 7 and of 5 ? Into groups of 5 and of 11 ?
90. CLASS EXERCISE. may name a composite num-
ber, and others may show the different groups into which it
may be separated.
91. A company of children were playing games. At first
they played games which required them to be divided into
groups of three. Afterwards they played in groups of four.
Every child played all the time. What is the least number of
children there could have been in the company ?
SUGGESTION TO TEACHER. Illustrate, by grouping children, for the
benefit of those pupils who cannot imagine clearly.
72 PROPERTIES OF NUMBERS
92. How many roses must a girl have, to be able to divide
them into bunches of 3 roses or bunches of 5 roses ? How
many bunches may she have of 3 roses ? Of 5 roses ?
93. A teacher has just enough pupils to divide into groups
of 7 pupils or groups of 4 pupils. How many pupils has she ?
How many groups of each kind can she have ?
94. I have just enough books to be arrangedon a number
of shelves, 12 books on a shelf, or by using more shelves, 9
books on each shelf. How many books have I ? How many
shelves would be needed under the first arrangement ? Under
the second ? t
95. What is the least number of gallons that can be
exactly measured by either of two casks, one holding 6 gal.,
the other 8 gal.? How many times can the smaller cask be
filled by them ? The larger cask ?
96. What is the smallest sum of money that can be en-
tirely spent in buying books at 15 ^ apiece, or in buying books
at 9^ apiece? How many of each kind of books could be
bought?
97. Ho.w long is the shortest piece of ribbon that can -be
cut without remainder into lengths of 2 yd., 3 yd., or 5 yd.
each ? How many lengths of each kind could be made ?
98. What is the least number of bananas that a mother
can exactly ivide between her 2 sons, or among her 4
daughters, or among all her children ? How many bananas
would each child receive in each case?
NOTE TO TEACHER. Written methods of finding the 1. c. in. and
g. c. d. are useful, because convenient, but the reasons for the processes
are beyond the comprehension of ordinary children in the grade for
which this work is designed. Hence the methods should be presented
as convenient rules that have been discovered by mathematicians. The
reasons for these rules should be learned later.
' 9
LEAST COMMON MULTIPLE 73
99. By the following rule find the 1. c. m. of 8 and 10 4
To find the least common multiple of several numbers
Arrange the numbers in a horizontal line, leaving out all
numbers that are factors of any of the other numbers. Find the
smallest prime number that will exactly divide anyjwo^ofjhejn^
and divide by it all the numbers of which it is a factor, placing
the quotients and undivided numbers below. Continue this pro-
cess until no prime number will divide two o the numbers in the
last horizontal line. Find the product of the divisors, quotients,
ojid undivided numbers.
100. Find the I.e. m. of 12, 16, and 18.
SOLUTION. 2
12, 16, 18 2 x 2 x 3 x 4 x 3 = 144 1. c. m.
6, 8, 9
3, 4, 9
1, 4, 3
' 101. Find by the written method the 1. c. m. of:
a 25, .60, 72, 35 c 63, 12, 84, 72 e 54, 81, 14, 63
b 24, 16, 15, 20 d 16, 12, 24 / 15 9, 6, 5
102. Find 1. c. m. of 5, 6, 18, 15.
3)?, 0, 18, 15
6, 5 3 x 6 x 5 = 90. Ans.
Why may the 5 and 6 be crossed out and not considered in finding the
1. c. m. of 5, 6, 18 and 15 ?
103. Find 1. c. m. of:
a 1, 2, 3, 4, 5, 6, 7, 8, 9 d 4, 8, 12, 24, 48, 84
b 8, 12, 16, 24, 36, 48 e 5, 10, 15, 20, 30, 40
c 4,3,6,7,8,16,9, / 7,28,35,14,70
104. CLASS EXERCISE. may name four composite num-
bers, and the class may find their 1. c. m.
105. CLASS EXERCISE. may name three prime num-
bers, and the class may find their 1. c. m.
106. Numbers that are multiples of any given number are
said to be divisible by that number. Is 7 divisible by 3 ?
Name several numbers that are divisible by 10.
74 PROPERTIES OF NUMBERS
DIVISIBILITY OF NUMBERS
107. Illustrate the following principle:
PRINCIPLE 1. A number that ends in 2, 4, 6, 8, or is divisi-
ble by 2.
108. Tell without dividing which of the following numbers
are not divisible by 2, and what the remainder is in each case :
8906. 2127. 2139. 2111. 2145. 1898.
109. Name in order the first fourteen multiples of 5.
110. When a multiple of 5 is expressed in figures, what
figures may represent the units' digit?
PRINCIPLE 2. A number that ends in 5 or is divisible by 5.
111. Without dividing, select from the following the num-
bers that are not divisible by 5, and tell what the remainder is
in each case: 75. 120. 81. 22500. 393. 920.
112. When a number is divisible by 2 and by 5, with what
figure must its expression end ? By what other number is it
divisible ?
113. Among all the prime numbers less than 100, can you
find any the sum of whose digits is 9 ?
114. Write a number of two places the sum of whose digits
is 9. Find how many times 9 is contained in that number.
115. Write a number of three places the sum of whose digits
is 9, and find how many times 9 is contained in that number.
116. Write a number of four places the sum of whose digits
is 9, and find how many times that number contains 9.
117. Write a number the sum of whose digits is 18, and see
whether it contains 9 exactly.
118. Write numbers the sum of whose digits is 9 or some
multiple of 9, and divide those numbers by 9 until you see the
truth of the following principle :
DIVISIBILITY OF NUMBERS 75
PRINCIPLE 3. Any number is divisible by 9 if the sum of its
digits is a multiple of 9.
119. Tell without dividing which of the following num-
bers are not divisible by 9, and give the remainder in each
case: 2025. 105. 117. 112. 2114. 189. 207. 1026. 4154.
120. Write several large numbers of which 9 is a divisor.
121. Write ten multiples of 3 no one of which is less than
36. Find the sum of the digits of each of them, and see if
that sum is a multiple of 3.
PRINCIPLE 4. A number is divisible by 3 if the sum of its
digits is a multiple of 3.
122. Tell at sight which of the following numbers are not
divisible by 3, and give the remainder in each case : 213. 411.
6951. 343. 1125.
123. Can you find a multiple of 9 that is not a multiple of 3 ?
Name a multiple of 3 that is not a multiple of 9.
124. Would it be possible for a number to be a multiple of
10 and not a multiple of 2 and 5 ? Explain.
125. Choose numbers ending in 0, and show what factors
they have besides 2, 5, and 10.
126. Can a number be divisible by 6 without being divisible
by 3 and by 2 ?
127. Write an even number the sum of whose digits is
divisible by 3. As the number is divisible by 3 and by 2, it is
divisible by 6. Write three other numbers divisible by 6.
128. Write three numbers each of which is divisible by 3
and by 5. Find how many times each of them contains 15.
129. Write three numbers ending in the sum of whose
digits is divisible by 3. Find how many times each of them
contains 30.
76 PROPERTIES OF NUMBERS
130. Write three numbers divisible by 2 and by 9. Divide
each of them by 18.
131. Write three numbers divisible by 5 and by 9. What
number between 40 and 50 is a factor of each of them ? How
can you tell ?
132. Count by 4's to 100.
133. Add some multiples of 4 to 100, and see if the num-
bers thus obtained are divisible by 4.
134. Add to 100 some numbers that are not multiples of 4,
and see if the resulting numbers are divisible by 4. Explain.
PRINCIPLE 5. A number is divisible by 4 if the number
expressed by its two right-hand figures is divisible by 4.
135. Tell without dividing the whole number which of the
following numbers are not divisible by 4, and give the remain-
der in each case: 127. 244. 365. 782. 728. 496. 338. 2672.
SUGGESTION TO TEACHER. Lead pupils to see that as 100 is a multiple
of 4, any number of hundreds is a multiple of 4, and if there is added to
any number of hundreds a number which is an aggregation of fours, the
result will be a still greater aggregation of fours.
136. Write a number of four places. Let the number
expressed by the two right-hand digits be a multiple of 4.
Let the sum of the digits of the whole number be a multiple
of 3. Find how many times 12 is contained in the number.
137. Write three numbers divisible by 4 and by 5. What
multiple of 10 besides 10 is a factor of each of them ? Explain.
138. Write three numbers divisible by 4 and by 9. What
number between 30 and 40 is a factor of each of them?
Explain.
139. How many eights in 1000 ?
140. To 1000 add 3 eights or 24. How many eights in the
number thus formed ?
COMMON DIVISORS 77
141. Add to 1000 a number which is not a multiple of 8.
Is the sum divisible by 8 ? Explain.
PRINCIPLE 6. A number is divisible by 8 if the number ex-
pressed by its three right-hand figures is divisible by 8.
142. Tell without dividing the whole number which of the
following numbers are not divisible by 8, and give remainders :
3640. 5728. 9076. 4126. 5345. 1724. 8638. 1124. 10008.
143. Make some numbers which are divisible by 8 and by 5,
and tell how you make them. With what figure do they end ?
What two multiples of 10 besides 10 are contained in each of
them?
144. Tell how to compose numbers that are divisible by 8,
and also by 9 and hence by 72.
COMMON DIVISORS
145. A number which is a factor of each of two or more
numbers is called their Common Divisor. Illustrate.
146. Turn to the diagram on page 67 and find all the num-
bers in it of which 11 is a common divisor.
147. Name all the numbers less than 100 of which 8 is a
common divisor. Give all the numbers less than 100 that
have as a common divisor : 9. 10. 12. 6.
148. What common divisor have all even numbers ?
149. Give a common divisor of 14, 21, 28, 35, 42, 49, 56,.
and 63.
150. Name three multiples of 11, and give a common divisor
of them.
151. Make a list of sets of numbers that have one or more
common divisors, and write the Greatest Common Divisor of
each set.
78 PROPERTIES OF NUMBERS
SUGGESTION FOR CLASS EXERCISE. Let a pupil name two or more
numbers that have a common divisor, and let the class discover the
divisor.
152. What divisor is common to the 7th even number and
the llth odd number ?
153. 2 is a common divisor of 10 and of 20. Is it the
greatest common divisor of these two numbers ?
What is the greatest common divisor of 24 and 36 ?
154. Give at sight the greatest common divisor of :
a 10, 20, and 40 g 70, 80, and 90
b 15, 30, and 45 h 60, 72, and 84
c 18, 27, and 45 i 63, 72, and 90
d 16 and 24 ./ 28, 32, 40, and 44
e 50, 75, and 100 k 15 and 25
/ 25, 30, and 35 I 12, 18, and 30
155. A candy manufacturer filled some boxes with choco-
lates, and some others of the same size with bonbons. There
were 24 Ib. of chocolates and 28 Ib. of bonbons. What is
the largest number of pounds each box can contain ?
156. A boy wishes to divide two ropes, one 42 ft. long, the
other 56 ft., into pieces of equal length, each as long as possible.
How long will each piece be after this division, and how many
pieces will there be ?
157. Mr. Allen has three strips of land. The first contains
10 acres, the next 12 acres, the next 14 acres. He wishes to
lay them off into the largest possible equal lots. How many
acres will there be in each lot, and into how many lots can
each piece be divided ?
158. The abbreviation for greatest common divisor is g. c. d.
What is the g. c. d. of 35 and 65 ?
POWERS AND ROOTS 79
The g. c. d. of two numbers may be easily found by the following
process of continued division. It is to be used with numbers which are
so large that their divisors cannot be readily found by inspection.
159. By the following rule find the g. c. d. of 8 and 10 :
To find the greatest common divisor of two numbers
Divide the greater number by the less. If there is a remainder,
use it as a divisor of the preceding divisor, and continue until there
is no remainder. The last divisor is the greatest common divisor.
160. Find by continued division the g. c. d. of 49 and 168.
49)168(3 Using 49 as a divisor of 168, the quotient is 3 and
147 remainder 21. Using 21 as a divisor of 49, the quo-
21,)49(2 tient is 2 and remainder 7. Using 7 as a divisor of
42 21, the division is exact, hence 7 is the last divisor, or
7)21(3 the g. c. d. of 49 and 168.
161. Find by continued division the g. c. d. of the following :
a 24 and 132 i 77 and 847 q 198 and 252
b 36 and 120 ./ 18 and 243. r 176 and 242
c 35 and 105 k 96 and 224 s 361 and 431
d 49 and 140 I 85 and 187 t 288 and 536
e 64 and 480 m 125 and 175 u 84 and 154
/ 72 and 252 n 105 and 195 v 189 and 405
g 30 and 735 o 135 and 245 w 960 and 204
h 44 and 242 p 795 and 1105 x 236 and 576
162. CLASS EXERCISE. Let compose two large num-
bers having a common divisor, and let the class find this
common divisor by continued division.
POWERS AND ROOTS
163. The product obtained by multiplying a number by itself
one or more times is called a Power of that number. Illustrate.
80 PROPERTIES OF NUMBERS
164. The product of two equal factors is the Square of each
factor. 2 x 2 is expressed 2 2 , and read 2 square, or 2 to the
second power. Find values of : I 2 . 30 2 . 50 2 . 120 2 . 15 2 . 20 2 .
165. Give quickly in order the first 12 numbers that are
perfect squares. Learn them.
166. 9 2 equals how many times 3 2 ? 16 2 how many times
8 2 ? 8 2 equals how many times 2 2 ?
167. What two perfect squares less than 100 have 6 for
their units' digit ? 9 ? 4 ? 1 ?
168. Can you find a perfect square less than 100 whose tens'
digit is 9? 7? 5? 3?
169. How much is T V of 8 2 ? Of II 2 ?
170. Multiply 6 2 by the first prime number after 31.
171. The product of three equal factors is called the Cube of
each factor. 2 x 2 x 2 is expressed 2 3 , and read 2 cube, or 2 to
the third power. Find values of : 2 3 . 3 3 . 4 3 . 5 3 . I 3 .
172. Continue the following table through 12 3 . Learn the
table. I 3 = 1
173. Name a perfect cube whose units' digit is : 1. 2. 3.
4. 5. 6. 7. 8. 9. 0.
174. Multiply the cube of 8 by .07. By .125.
175. Find 6% of :4 3 . 9 3 . 7 3 . II 3 . 50 3 . 12 3 . GO 3 .
176 *=* *=* ^=9 *=* i*=?
' 43 33 6 3 53 4 3
SUGGESTION TO TEACHER. By the following work lead pupils to dis-
cover the relations between a solid that is a cube and the third power of the
number that measures one of its dimensions. Inch cubes should be used
in this work until pupils are able to image the solids clearly without them.
POWERS AND ROOTS
81
177. How many cubic inches does a 2-inch cube contain ? A
3-inch cube ? A 4-inch cube ? A 5-inch cube ? A 6-inch
cube ? A 7-inch cube ? An 8-inch cube ? A 9-inch cube ?
178. One of the boys may draw a square yard on the floor
in one corner of the room. How many cubic blocks 1 ft. in
dimensions would cover the square yard ?
179. One of the girls may show how high she thinks the
blocks must be piled to make a cubic yard. Another member
of the class may measure with the yard stick and see how
nearly right she is.
180. How many cubic feet in the lowest layer of blocks?
How many layers would it take to make a cubic yard ? . How
many cubic feet in a cubic yard ?
181. How many layers of inch cubes would be required to
cover a square foot ? How many layers of the inch cubes to
make a cubic foot ? How many inch cubes in a cubic foot ?
182. Into how many 2-inch cubes can a 4-inch cube be
divided ?
183. How many 6-inch cubes can be packed into a box
whose inside dimensions are each
1ft.?
6 Square Centimeters
FIG. 2.
184. Copy Fig. 2 on paper or
pasteboard, making each square
1 sq. cm. Cut out the copy and
fold and fasten it so that it will
inclose a cubic centimeter. How
many such cubes would be re-
quired to cover a square deci-
meter? How many layers of them
would be required to make a cubic decimeter ?
185. A cubic decimeter is called a Liter,
centimeters does it contain ?
HORN. GRAM. SCH. AR. 6
How many cubic
82 PROPERTIES OF NUMBERS
186. Copy Fig. 2 on paper or pasteboard, making each
square 1 sq. dm. Cut, fold, and fasten to make a cubic deci-
meter or a liter.
SUGGESTION TO TEACHER. The most perfect liters and cubic centi-
meters made by the children should be kept as a part of the school
apparatus.
187. How many square centimeters in all the surfaces of a
liter ?
188. In the metric system the liter is the measure that cor-
responds very nearly to the quart in the English liquid
measure. How much will 12 liters of oil cost at 65 cents
per liter?
189. Place a cubic centimeter upon a cube that holds a liter.
How many cubic centimeters in the figure thus formed ? How
many square centimeters in all its surfaces ?
SUGGESTION TO TEACHER. It should be explained that a liter is a unit
of measure, and not a fixed form.
190. How many liters will be contained in a box that is 3
dm. long, 2 dm. wide, and 5 dm. high?
191. About how many liters of wheat can be put into a peck
measure ?
192. 2 x 2 x 2 x 2 is 2 4 , which is read 2 to the fourth power.
Raise to the fourth power each of the first five numbers.
193. Kaise to the fifth power each of the first three odd
numbers.
194. Raise to the sixth power each of the first two even
numbers.
195. Raise 10 to the seventh power.
196. Give the number whose prime factors are 2, 2, and 3.
2, 3, and 3. 2, 5, and 5.
POWERS AND ROOTS 83
197. Find values of x.
a x = 3 2 x 5 e x = 2 s x 3 i x = 7 2 x 13
6 ^ = 2 2 x 7 / x = 2 6 x 5 j x = 2 3 x 5 x II 2
c a; = 2 3 x 3 4 g x = 3 3 x 11 A; a; = 2 s x 3 x 5 2
d = 2 4 x 3 3 h x = 5 3 x 11 J a> = 3 4 x 5 2 x 7
198. Each of the two equal factors that compose a perfect
square is called a Square Root of that number. Give the
square root of 9. 25. 16.
199. V is used as the sign of square root. Vl6 = ?
= ? V144=? Vl21 = ? VI = ? V6l=? VlOO = ?
200. Divide .63 by Vl9. By V9. By V81.
201. Divide .36 by V9. By Vl6. By V4. By V8l.
202. Divide 5^44 by V1. By V36. By Vl21. By V49.
By V64. By Vl6. By V9. By Vl44.
203. How long is one side of a square whose area is 9 sq.ft.?
49 sq. ft. ? 100 sq. ft. ? 81 sq. ft. ? 25 sq. ft. ?
204. How much is 50% of VlOO? 25% of V64? 75%
ofV64? 25%ofVl44?
205. Multiply Vl44 by .3. By .05. By .007.
206. Find 6% of Vlll. Of V64. Of V8l.
207. How much is 3 times V9? 4 times V9?
208. "5V9" is read "5 times the square root of 9." Find
the value of the expression.
209.
210. 3 V16 = ? 4 VlOO = ? 2 V81 = ? 6V64 = ? 3 Vlll = ?
211. How long is the perimeter of a square containing
49 sq. in. ?
84 PROPERTIES OF NUMBERS
212. At $ 1.25 per rod, how much will it cost to fence a
square lot containing 25 sq. rd. ? Eepresent.
SUGGESTION TO TEACHER. For oral "quick work" exercises similar
to the following are useful : "Think of the 3d multiple of 6, subtract 2,
take the square root, add 1, square, add 5, take ^, take j 1 ^, add 1, square,
add 5, take square root." Allow children to lead the work, letting them
prepare their numbers beforehand to read to the class, until they are able
to extemporize.
213. Each of the three equal factors that compose a number
that is a perfect cube is called a Cube Root of that number.
Give the cube root of : 8. 512. 64. 1000. 729. 1331. 1728.
125. 216. 343.
214. Give the cube root of a perfect cube whose units' digit
is: 1. 2. 3. 4. 5. 6. 7. 8. 9. 0.
215. How long is one side of a cube that contains 8 cu. in. ?
27cu.in.? 1728 cu. in.? 64cu.in.? 729cu.in.? 1000 cu. in. ?
512cu.in.? 216cu.in.? 125cu.in.? 343cu.in.?
216. ^/ is read "The cube root of." How much is V64?
v/343? -v/729? -v/1728 ?
217. Multiply the -\/8 by the first prime number after 40.
218. Multiply -v/125 by .001. -^512 by .75.
219. Find 50% of A/64. Of ^/1728.
220. -\/729--v/2l6 = ? -^1000 -^- A/125 =? A/1331 X A/343 =?
MISCELLANEOUS EXERCISES
1. Write in decimal form and add : 1 ten-thousandth,
1 hundred-thousandth, y^fo, T H^ 1 millionth,
1 00000*
2. If you have a string a foot long and cut one inch from
each end, how long is the string that is left?
MISCELLANEOUS EXERCISES 85
3. When a line 3.4 ft. long is cut from each end of a line
that is 1 rd. or 16.5 ft. long, how long is the line that is left ?
4. The diagonal of a certain schoolroom is 35.1 ft. John
makes a mark on the diagonal 7 ft. from one corner, and James
makes a mark 9 ft. from the opposite corner. If each boy
stands at the mark he has drawn, how far apart are they ?
SUGGESTION TO TEACHER. In most classes there are some pupils who
fail to visualize. Select two of these to take the parts of John and James
in illustrating this and similar problems.
5. Find the sum of 81.375 and the prime number nearest
to 24.
6. Find the difference between 21.84 and the largest prime
that can be expressed by two digits.
7. When a decimal of 3 'places is multiplied by an integer,
how many decimal places should be pointed off in the product?
Illustrate.
8. When an integer is multiplied by a decimal of 2 places,
how many decimal places should be pointed off in the product ?
Illustrate.
9. How many decimal places in the square of .007 ?
10. 8.283-*- 3 = ? 45.6 --12=? .286-^22=?
11. Tell how you divide a decimal by a decimal.
12. .12-j-.4=? .15 -f-. 005=? .75 -=-.5=? .84 -.12=?
13. When one decimal is divided by another decimal of the
same denomination, how many decimal places are there in the
quotient ?
14. At $ .05 per pound, how many pounds of sugar can be
bought for $ .45 ? For $ .75 ? $ 1.25 ? $ 2.50 ? $ 8 ?
15. Harriet has some money in the bank, the interest of
which is $ 1.30 every year. How long must the money stay in
the bank that the interest may be $ 5.85 ?
86 . PROPERTIES OF NUMBERS
16. 1.728 -s-. 0012=?
17. Give the prime factors of the first odd composite num-
ber after 81.
18. The largest prime factor of 66 is how many times the
smallest prime factor of 66 ?
19. Find the 1. c. m. of 3, 8, 4, 9, 6, 12.
20. Find the g. c. d. of 44 and 66. Of 128 and 144.
21. Divide 7235.2 by the 1. c. m. of 4 and 7.
22. Divide 4.725 by the g. c. d. of 45 and 105.
23. Find by cancellation the value of x :
a
7x8x4
b
3x7x9
c
64x21
14x32
d
25x21
21 x 18 x 5
e
48x63
42 x 8 x 8
/
16 x 25 x 36
35x30
g
49 x 63
36 x 24 x 18
h
48 x
200 x 18 x 6
i
56
21x84
24x36
21 x 16
24. What is one of the two equal factors of 121 ?
25. Name a perfect square whose units' digit is 9, and give
its square root.
26. Give quickly the first 12 numbers that are perfect cubes.
27. How long is one edge of a cube that contains 1000 cu.
in. ? 1728 cu. in. ?
28. Give one of the three equal factors of 216. Of 729.
29. Find the difference between O 2 and I 2 . I 2 and 2 2 . 2 2
and3 2 . 3 2 and 4 2 . 4 2 and 5 2 . 5 2 and 6 2 . 6 2 and 7 2 . 7 2 and8 2 .
8 2 and 9 2 .
MISCELLANEOUS EXERCISES 87
30. Write these differences in a column and tell whether
they are even numbers or odd numbers.
31. Find the sum of the first 7 odd numbers. Compare that
sum with the square of 7.
32. Compare the sum of the first 8 odd numbers with the
square of 8. The sum of the first 5 odd numbers with the
square of 5. Of the first 9 odd numbers with the square of 9.
33. Find the sum of the first 5 even numbers. Subtract the
square of 5 from that sum.
34. Find how much the sum of the first 7 even numbers
exceeds the square of 7.
35. Take Ex. 34, substituting other numbers for 7.
36. Ella's record on an arithmetic test was 75%. What
fractional part of her work was right and what part wrong ?
37. Mr. Hudson had $ 8000 in bank and took out 20% of it.
How much did he take out ? How much had he left ?
38. Edward buys oranges at the rate of 4 for 25 ^, which is
just one half of what he receives for them. What is the selling
price of each ?
39. A milkman's horse ran away with a wagon containing
4 gal. of milk, and 25% of it was spilled. How many quarts
of milk were spilled ? If the milk was worth 6 ^ per quart,
what money value was lost ?
40. From a liter of oil 13% was spilled.
How many cubic centimeters of oil re-
mained ?
41. Draw a circle. What is a radius ?
Diameter ? Circumference ? The radius of
a circle equals what part of the diameter ?
42. How long is the diameter of a circle
FlG - 3 ' whose radius is 5 in. ? 3J in. ? 7.5 in. ?
88
PROPERTIES OF NUMBERS
43. What is the diameter of the largest circle that can be
cut from a piece of paper 3 in. square ?
44. The surface passed over in 1 hr. by the minute hand of
a clock is what figure ? The minute hand of a clock in a tower
is 2-J- ft. long. How long is the diameter of the circle it passes
over every hour ?
45. A plane figure bounded by
six straight lines is called a Hexa-
gon. When the sides are all equal,
and the angles are all equal, as in
Fig. 4, the hexagon is called a
Regular Hexagon. What kind of
angles has a regular hexagon ?
46. If each side of a regular
hexagon is 6.75 in. long, how long
is the perimeter of the hexagon ?
FIG. 4.
47. Draw a hexagon that is not regular.
SUGGESTION TO TEACHER. Show the following method of drawing
a regular hexagon : Draw a circle with a radius of any convenient
length. Beginning at any point of the circumference, lay off the radius
as a chord six times consecutively. Erase the circle.
48. Draw a regular hexagon whose sides are each 3 in. long.
How long is the perimeter ? How dees the side of a regular
hexagon compare with the radius of the circle in which it is
inscribed ?
49. By drawing diagonals the regular hexagon may be di-
vided into 6 equilateral triangles. Draw them, and find how
long each diagonal is. How long is the perimeter of each
equilateral triangle ?
50. If the perimeter of the hexagon were 32.4 in., how long
would the perimeter of each equilateral triangle be ?
51. How many triangles in 50% of the hexagon ?
MISCELLANEOUS EXERCISES
89
FIG. 5.
52. A plane figure bounded by four equal straight lines,
and having no right angles, is called a Rhombus. What kind
of angles has a rhombus ?
53. Draw a rhombus by the following
method :
Draw the line AB of any convenient length.
With AB as a base construct an isosceles triangle
CAB, making AC greater than f of AB. With
AB as a base construct an isosceles triangle ADB,
making AD equal to AC. Erase the construction
line AB. (A construction line is a line forming no
part of a figure, but used simply to help in its con-
struction.)
54. Construct a rhombus each of whose
sides is 5 in.
55. Mr. Jones laid out a flower bed in the
shape of a rhombus, each side of which was
4.75 ft. long. How long was the entire edge
of the flower bed ?
56. If the entire edge had been 28.8 ft. long, how long would
have been one side of the flower bed ?
57. Draw a rhombus and the long diagonal of the rhombus.
Into what kind of triangles is the rhombus divided ?
58. If the side of a rhombus is 7.5 in., and its longer diago-
nal is 10.875 in., how long is the perimeter of one of the tri-
angles into which the long diagonal cuts the rhombus ?
59. Draw a circle and inscribe a hexagon.
Join the vertex of each alternate angle with
the center of the circle. Into what kind of
figures is the hexagon divided ? How long
would the perimeter of each of the figures
be, if the radius of the circle were 8 cm. ?
FIG. 7. 12 cm. ?
60. Each rhombus is what fractional part of the hexagon ?
90 PROPERTIES OF NUMBERS
61. Finish the following course of reasoning:
Since the whole of anything equals 100% of it,
.1 = 33J% of it.
f = % of it.
62. Write in each rhombus the per cent which it is of the
hexagon. Shade one rhombus and tell what per cent of the
hexagon is unshaded.
63. How much is 331% of 12? 21? 24? 30? 45? 48?
64. How much is 66f% of 15? 27? 18? 36? 33? 6?
65. 331% of a school of 48 pupils are boys. How many
girls are there ?
66. How much is 33^% more than $ 15 ? $ 300 ? $ 600 ?
67. How much is 66f % more than $ 900 ? $ 1200 ? $ 1800 ?
68. $ 3000 - 66f % of 3000 = ? 2100 - 66f % of 2100 = ?
69. Each of the equal sides of an isosceles triangle is 33^%
longer than the base, which is 15 in. long. How long is the
perimeter of the triangle ?
70. A merchant found that some of his goods were shopworn
and marked them at a reduction of 25% of their cost. How
were goods marked that cost 12^? 20^? 40^? $1.00?
$1.60? $10.00?
71. Find the selling price of goods marked at the following
prices, which are to be reduced in price 33^% on account of
being out of style. Cloaks costing $ 7.50, bead trimming $ 1.50
per yard, lace ruffling $ .57 per yard.
72. A grocer bought goods at the following prises. For how
much must they be sold to gain 33 J% ? 25% ?
a Tomatoes @ 12^ per pound. d Oranges @ 24^ a dozen.
b Raisins @ 6^ per pound. e Bananas @ 18^ a dozen,
c Molasses @ 36^ a gallon. / Potatoes @ 30^ a bushel.
MISCELLANEOUS EXERCISES 91
73. At the end of a season a merchant decided to reduce
prices 33J% on all of the following goods whose prices were
over $1.00 and to reduce them 66f % on all those whose prices
were less than a dollar. Find the new selling prices.
a Lace @ $ 1.80 per yard. d Silk @ $ 2.70 per yard.
b Ribbon @ $ .75 per yard. e Velvet @ $ 1.68 per yard.
c Calico @ $ .06 per yard. / Alpaca @ $ .60 per yard.
74. William is 15 years old. His age is 33|% of his father's
age. How old is his father ?
75. Mr. Gage had $ 396.66 in a bank and took out 33^% of
it. How much remained in the bank ?
76. Mrs. Wallace lent Mr. Brown $ 1200 until the interest
amounted to 66f % of the principal. How much was the inter-
est ? How much did Mr. Brown then owe, including principal
and interest ?
77. Write the following fractions in a column and opposite
to each its value in % : \. \. f . \. }. f f f . f .
78. How many minutes in 33^% of an hour? In 66f% ?
25%? 20%? 40%? 50%?
79. How many hours in 50% of the time from 9 A.M. Mon-
day to 9 A.M. Tuesday ? In 331% o f it ? In 75% ? 20% ?
80. How many square centimeters in 80% of a square deci-
meter ? In 25% ? In 331% ?
81. Draw the equilateral triangle
ADC, one of whose sides represents
3 in. With DC as a base line construct
another equilateral triangle ACB.
Erase AC. What kind of a figure is
ABCD ? How long is its perimeter ?
82. With either side of the rhombus as a base line, construct
another equilateral triangle. Erase the base line. What kind
of a figure have you drawn ? How long is its perimeter ?
92
PROPERTIES OF NUMBERS
83. Continue adding equilat-
eral triangles until you have a
regular hexagon. Complete the
following reasoning.
84. Since the whole of any-
thing = 100%,
iofit= 16|%
fofit=-%?
FIG. 9. 85. On your copy v of Fig. 9
write in each equilateral triangle the % which it is of the
hexagon.
86. If 16f % of the hexagon were shaded, what
would be unshaded ?
of it
87. What % of the hexagon is the figure AOCB? AODCB?
ABCDEFO? CDEFAO? '
88. How long is the perimeter of the six-pointed star repre-
sented by Fig. 10 if each side is 3.5 in. ?
What kind of angles are those whose ver-
tices are at the points of the star ?
89. Make a six-pointed star.
TJraw a regular hexagon, and construct an
equilateral triangle upon each of its sides.
Erase the sides of the original hexagon. The
star may also be made by prolonging the sides
of the hexagon until they meet.
90. Divide your star into 6 equal rhom-
buses. Write in each rhombus the %
which it is of the star.
91. Put a letter at the center and one
at each angle of your copy of Fig. 11 and
tell what figure is 33% of it. 831%.
FIG. 11. 50%. 66|%.
MISCELLANEOUS EXERCISES 93
92. What is 16f% of 12? Of 24? Of 72? Of 84?
Of 120? Of 144?
93. What is 83% of 18 ? Of 30? Of 48? Of 66? Of 144?
94. To make a profit of 16f % for what price must goods
be sold that cost 6^? 18^? 15^? 30^? 54^?
95. What must be the selling price of the same goods to
allow a profit of 831% ?
96. Select from the following list the per cents which are
most easily used by reducing them to common fractions in their
lowest terms, and give the equivalent fractions :
33J% 11% 16f% 25% 831% 50% 17%
20% 3% 40% 75% 9% 66|%
97. James had a dollar and lost 17 cents. What per cent
of his money was left ?
98. Mary has only a dollar. Can she lose 101% of it?
Explain.
99. 20 equals what part of 30 ? Express it in per cent.
100. CLASS EXERCISE. may give a number, and the
class may give 33|% of it. 16f %. 66J%. 831%.
101. CLASS EXERCISE. - may give a number, and the
class may give the number of which his number is 16J%.
331%. 25%.
102. Draw a right triangle whose base is 3 in. and perpen-
dicular 4 in. If your drawing is correct, the hypotenuse will
be 5 in. Each side, of the triangle equals what part of its
perimeter ?
103. Draw a right triangle whose base is 6 in. and perpen-
dicular 8 in. Its hypotenuse is just twice as long as the hypot-
enuse of the triangle given in Ex. 102. Each of its sides
equals what part of its perimeter ?
94 PROPERTIES OF NUMBERS
104. Draw a square 3 in. in dimensions. If you drew a
larger square, having each of its sides 1 in. from the corre-
sponding side of the first square, how long would its per-
imeter be?
105. Separate the following into two lists, one of odd
numbers, the other of even numbers. How many are there of
each ?
874; MDCCCLXXXVIII ; the square of 7 ; the fifth multiple
of 4 ; the product of 7 and 8 ; the quotient of 84 divided by 2 ;
the difference between 81 and 18 ; the sum of 85 and 37 ; the
largest numbe*r that can be expressed by two figures ; the
largest factor of 12 except itself ; the number that is 5 greater
than 212 ; the largest number that can be expressed by three
figures ; the smallest number that can be expressed by three
figures ; the number that means a dozen ; the number that tells
how many days in May; the integer between 17,345 and
17,347 ; one of the equal factors of 25 ; the factor that helps 7
to make 77 ; the square root of 100 ; the number that shows
how many quarts in a peck; the denominator of the fraction
T \; the greatest common divisor of 6 and 8 ; the remainder
after dividing 25 by 11 ; the smallest multiple of 7 that will
contain 5 ; the least common multiple of 4 and 7 ; the number
that is just half way between 30 and 50; the smallest prime
number greater than 25 ; the largest prime number less than
25; the numerator of the fraction |^; the number that tells
how many square inches in a square foot ; the number that is
just as much less than 15 as it is greater than 11; the average
of 19, 20, and 21 ; the first composite number ; the number
that shows how many pounds in a ton ; the number that shows
how many cubic inches in a cubic foot ; the number that shows
how many sides a pentagon has ; the largest prime number that
can be written with two figures ; the smallest prime number
that can be written with three figures ; the quotient of 13.14
divided by .06; 50% of 862; the largest prime factor of 102 j
the number that shows how many millimeters in a meter.
CHAPTER III
RATIO
SUGGESTION TO TEACHER. Review ratio as given in Hornbrook's
"Primary Arithmetic." See notes on pp. 117 and 118 and tables on
pp. 145, 160, 174, 183, of that book.
1. A 3-inch line equals what part of a 4-inch line, or what
is the ratio of a 3-inch line to a 4-inch line ?
2. What is the ratio of a pint to a quart ? Of a quart to a
gallon ? Of an inch to a foot ? Of a foot to a yard ? Of 2 ft.
to a yard ? Of an ounce to a pound ? Of 8 oz. to a pound ?
Of 15 oz. to a pound ?
3. CLASS EXERCISE. may name a number less than
100, and the class may give its ratio to 100.
4. 6 is how many times 3, or what is the ratio of 6 to 3 ?
5. What is the ratio of a yard to a foot? Of a foot to an
inch ? Of a foot to 3 in. ? Of a foot to 6 in. ? Of a foot to
7 in. ? 9 in. ? 11 in. ?
6. CLASS EXERCISE. may name some number greater
than 10, and the class may give its ratio to 10.
7. The ratio of two numbers is the quotient of the first of
those numbers divided by the second. Thus the ratio of 10 to
5 is 10 -f- 5, or 2. The ratio of 7 to 5 is 7 -^ 5, or If. What
is the ratio of 4 to 5 ?
8. Draw a rectangle 4 in. long and 1 in. wide. A rectangle
3 in. long and 1 in. wide equals how many fourths of the first
rectangle ? A rectangle 8 in. long and 1 in. wide equals how
many fourths of the first rectangle ? What do j equal ?
95
96
RATIO
The ratios indicated by " parts " and " times " are really of the same
kind. They both express the quotient of one quantity divided by another
of the same kind.
9. What is the ratio of a second to a minute ? Of a year
to a month ?
10. Build from inch cubes or draw right prisms like the
following :
3x2x1
5x2x1
7 / /
2x2x2
3x3x1
a rn
IIT
I
fir
5x1x1
7x2x1
I
1
1
1 1 . Find ratios of :
a to b a to c
b to c b to d
c to b c to d
FIG. 1.
& to e
e to d
btof
ftod
c to/
c to e
d to c
/to a
a to e
12. Mr. Jones works every day from 8 until 12 o'clock, and
from 1 until 5 o'clock. At 9 o'clock in the morning, what is
the ratio of the work he has done to the work he still has to
RATIO 97
do that day ? What is the ratio of the work he has done to
his whole day's work ?
13. At ten o'clock, what is the ratio of the work he has done
to a day's work? At 12 o'clock? At 1 o'clock? At 3 o'clock?
At 5 o'clock ?
14. What is the ratio of a rod to a mile ?
15. If your home is a mile from the schoolhouse, how many
rods must you travel each school day of two sessions, if you
go home at noon ?
16. Joseph rode a mile on his bicycle. When he had
ridden a rod, what was the ratio of the distance he had ridden
to that which he afterward rode ?
17. Ella walked to the home of her cousin, who lived a
mile away. What was the ratio of the distance she had walked
to the remaining distance after she had gone 16 rd. ? 32 rd. ?
80 rd. ? 120 rd. ?
18. What is the ratio of an ounce to a pound ?
19. Margaret had half a pound of candy and gave away all
of it except one ounce. What was the ratio of what she had
left to what she had at first ?
20. What is the ratio of a pound to a ton ?
21. Just after a ton of hay was weighed in market, a horse
ate one pound of it. What was the ratio of what he ate to
what was left ?
22. Ratio is expressed by a colon. Give ratios of : 15 : 3.
3:15. 16:2. 2:16. 3:18. 18:3. 5:20.
23. Give quickly the ratio of 2 to each of the first ten mul-
tiples of 2. Give the ratio of the 2d multiple of 2 to each of
the first ten multiples of 2. Do the same with the 3d multiple
of 2. With the 4th, 5th, 6th, 7th, 8th, 9th, and 10th.
110KN. GRAM. SCH. AR. 7
98 RATIO
24. What is the ratio of the 2d multiple of any number to
its 3d multiple ? Of its 2d multiple to its 4th ? Illustrate.
25. What is the ratio of 2471 to 17 ?
26. What is the ratio of 1.422 to 1.8 ?
27. The ratio of 3 to 6 is -J ; the ratio of 6 to 3 is 2. These
two ratios between the numbers 3 and 6 are called reciprocal
ratios. Give the reciprocal ratios between the following num-
bers : 2 and 3. 3 and 5. 8 and 4. 9 and 12. 18 and 20.
28. 19 equals how many twentieths of 20 ? 20 equals how
many nineteenths of 19 ?
29. John is 8 yr. old, and his sister Mary is 16 yr. old.
What is the ratio of Mary's age to John's ? Of John's age to
Mary's age ?
30. When is the ratio of one number to another number an
integer ? Illustrate.
31. The line A B rep-
ACT) JP f' B
T I I I I 1 resents a distance of
Fl - 2 - 35 mi. divided into 5
equal parts. How much is the distance AE ? CB ? AF ? EB ?
32. Find the ratio of AF to AB. CF to CB. AB to AD.
AB to DB. AD to CF. AB to CF.
33. What is the ratio of the first composite number after 19
to the first composite number after 29 ? Of the first odd
number after 20 to the first odd number after 5 ? Of the first
prime number after 7 to the first even number after 20 ?
34. Draw a 2-inch square and a 4-inch square and find the
ratio of each square to the other.
35. Find the reciprocal ratios of a 3-inch square and a
4-inch square. Of an 8-inch square and a 6-inch square.
RATIO 99
36. What is the ratio of 3 V4 to 2 V25 ? 2V9:6Vl6 = ?
37. 2V49:3Vl21 = ? 3V36 : 2 VI44 = ?
38. 5Vl6:4V9 = ? 2 V81 : 3 V64 = ?
39. 3 VlOO : 5 Vl6 = ? 2\/25:5V36 = ?
40. Build with cubes and find the following ratios : An
inch cube to a 3-inch cube. An inch cube to a 4-inch cube. A
2-inch cube to a 3-inch cube.
41. If a 5-inch cube is cut into inch cubes, what is the ratio
of one of the small cubes to the large cube ?
42. What is the ratio of a cube an inch in dimensions to a
6-inch cube ?
43. 3 3 :4 3 =? 40 3 :60 3 = ? 11 3 :9 3 =? 6 3 :20 3 =?
44. Give quickly the cube root of : 27. 64. 8. 216. 512.
729. 343. 1000. 125.
45. A/125 : ^/lOOO = ? ^/729 : \/2l6 = ? -\/1728 : -J/512 = ?
46. Image the following figures and tell the ratio of one side
of each figure to its perimeter. A square. A regular hexagon.
A rhombus. A regular pentagon. Express the ratios in / .
47. What is the ratio of the perimeter of a square yard to
the perimeter of a square foot ? Of the perimeter of a square
centimeter to the perimeter of a square decimeter ?
48. What number is that whose ratio to 8 is f ? Or what
is I of 8 ?
49. How much is | of 12 ? f of 49 ? f of 15 ? \ of 21 ?
SUGGESTION TO TEACHER. In order to insure correct reasoning on the
part of pupils, they should occasionally be required to explain the steps
by which they arrive at results, as : since \ of 21 is 3, f of 21 are 4 times
3 or 12. After this is thoroughly understood, the habit of mental cancel-
lation should be encouraged. For instance, in finding ^ of 21, children
may be led to visualize the expression and mentally to cancel the terms.
100 RATIO
|
50. Find values :
| of 36 | of 56 of63 ^ of 77 J of 72 f of 36
51. If a boy earns $ 77 in 11 wk., how much would he earn
in 3 wk. ? 5 wk. ? 9 wk. ?
52. When 9 yd. of calico cost 72^, what is the cost of 2 yd. ?
5 yd. ? 7 yd. ? 8 yd. ?
53. Goods that cost 8^ a yd. are sold for f of their cost.
What is the selling price ?
54. Give quickly the selling price of goods :
a Bought at $ 0.12 and sold at f of the cost.
b Bought at f 0.18 and sold at more than cost,
c Bought at $ 0. 20 and sold at \ more than cost.
d Bought at $ 0.40 and sold at ^ more than cost.
e Bought at $0.50 and sold at more than cost.
/ Bought at $ 0.60 and sold at ^ more than cost.
g Bought at $ 0.80 and sold at \ less than cost.
h Bought at $ 1.00 and sold at f of the cost.
i Bought at $ 0.40 and sold at f of the cost.
SUGGESTION TO TEACHER. Before the following work is taken up,
pupils should be drilled in finding reciprocal ratios of pairs of numbers
until they readily see the truth of the first statement in the solution of
Ex. 55.
55. 6 is f of what number ?
SOLUTION. 6 is f of the number that is f of 6. % of 6 is 2. f of
6 are 8.
56. 8 is | of? f of? of? I of?
57. 12 is | of? * of? fof?
58. Find the values of x.
a 10 = of x. f of x. | of x. f of a. | of x.
b 12 = | of x. | of x. f of x. |- of x. f of x. T 2 T of x.
c 15 = | of x. f of x. f of x. f of x. | of x. f of x.
KATIO 101
d 18 = f of ar. T % of x. T 6 T of . -& of x. f of a. f of a?,
e 20 = J of a;. ^ of x. if of aj. -V- of x - V of ^ V of aj -
/ 5 = | of a;. 10 = fofx. 7 = |ofa?. 9 = $ of a;.
59. Anna's age is f of Mary's age. What is the ratio of
Mary's age to Anna's age. If Anna is 12 years old, how old
is Mary ?
60. James has 24 marbles. He has % as many as John.
How many marbles has John ?
61. Land in one part of a certain county in Illinois is worth
$20 an acre, which is only of the price of land in another
part of the county. What is the price of the better land?
What is the value of the land owned by Mr. Baxter, who has
40 acres of each kind ?
62. Mr. Walker sold gingham at 8^ a yard, which was f of
what it cost him. How much did it cost ?
3. Find the cost of goods :
a Sold at $ 0.09, which was f of the cost.
b Sold at $0.20, which was f of the cost,
c Sold at $ 1.98, which was f of the cost.
d Sold at $ 2.97, which was f of the cost.
e Sold at $ 1.47, which was -J of the cost.
64. 24 marbles will cost how many times as much as 12
marbles of the same kind ?
65. What is the cost of 24 marbles when 12 marbles cost
$.25? $.08? $.60?
66. What is the ratio of the price of 10 hats to the price of
1 hat ? To the price of 2 hats ? 5 hats ? 7 hats ?
67. What will be the cost of 10 hats when 5 hats cost $3 ?
$7? $9? $7.50?
Use ratio. 10 hats cost how many times as much as 5 hats ?
102 RATIO
68. What will be the cost of 10 hats when 2 hats are worth
$3? $4? $5.25? $7.65?
69. If 3 articles of the same kind cost $.17, how much will
12 such articles cost ? 18 articles ? 21 articles ? 30 articles ?
70. If 5 things cost $19, how much will 15 things of the
same kind cost ?
71. Take Ex. 70, substituting another number for 5 and for
15 some multiple of the number that you have substituted.
72. If 10 acres of land are sold for $375, how much would
80 acres cost at the same rate ? 60 acres ? 100 acres ?
73. Find the cost of 48 oranges when 5^ are paid for 6
oranges. For 8 oranges. For 4 oranges. For 12 oranges.
74. If 15 marbles are sold for 9^, how much do 5 marbles
cost ? 3 marbles ? 20 marbles ?
MISCELLANEOUS EXERCISES
1. Add the square of 7.9 to 7.9.
2. Subtract the cube of 1.3 from 10.
3. Divide 14 2 by .007.
4. A regular hexagon is inscribed in a circle whose radius
is 8 in. How long is the perimeter of the hexagon ? Eepresent.
5. How long is one side of a regular hexagon whose perim-
eter is 5.4 cm. ?
6. How long is one side of a regular pentagon whose perim-
eter is 8.45 cm. ?
7. Is it correct to say that an inch line is \ of a 4-inch line ?
If the inch line were in Boston and the 4-inch line in New York,
would the shorter line be a part of the longer ? What part of
the longer line would the shorter line equal ?
MISCELLANEOUS EXERCISES 103
8. What is the ratio of an hour to a day ? Of a week to a
day?
9. Give the ratio of 21 to each of the first 12 multiples
of 7.
10. What is the ratio of a square whose side is 3 ft. to a
rectangle 9 ft. by 8 ft. ?
11. Find the prime factors of 546. Of 495.
12. Find the ratio of the largest prime factor of 35 to the
largest prime factor of 39. Of the largest prime factor of 49
to the largest prime factor of 15. Of the smallest prime factor
of 49 to the smallest prime factor of 15.
13. 60 3 : 30 3 = ? 6 3 : 5 3 = ? 7 3 : 5 3 = ? 8 3 : 12 3 = ?
14. 7V4:3V49 = ? 4V25 : 2 VTOO = ?
15. How long is the shortest line that can be divided into
either 8-inch lines or 10-inch lines ?
16. Divide the 1. c. m. of 2 and 5 by the 1. c. in. of 3 and 5.
17. Divide the g. c. d. of 36 and 45 by the g. c. d. of 12 and 3.
18. Which power of 6 is 216 ?
19. Which power of 2 is 16 ? 64? 256?
20. Which power of 10 is the denominator of the decimal
.11? .0125? .6? .345? .00^004? .000009?
21. AB represents a distance of 320 rd.
J- 1 or 1 mi., which is just f of the distance
from A to G. How far is it from A to C ?
r IG. o.
From B to C?
22. Mary had some money in a toy bank. She took out 35^
which was |- of it. How much was left ?
23. Lizzie spent 15^ for a singing book, which was J of the
price of her arithmetic. The price of the arithmetic was -f of
the price of her story book. How much did they all cost ?
104 RATIO
24. On Monday John rode on his bicycle 21 mi., which was
f of the distance he rode during the rest of the week. How
far did he 'ride during the whole week ?
25. A boy had 12 agates, for which he paid 60 He sold
them for 6^ apiece. How much did he gain on each ?
26. Mr. Cooper paid $64, which was 50% of what he owed.
How much does he still owe ?
27. When a man hires a house to live in, he is said to pay
rent for it. When he hires money to use, he is said to pay in-
terest for it. If you have deposited $ 100 in a bank that pays
4% interest, how much interest will you receive each year?
28. How much would you receive each year if you had $ 100
at6%? 3%? 7%?
29. If a man borrows $25 and pays .06 of $25 as interest
for 1 yr., how much interest does he pay ?
30. At 6% what is the interest of $14 for a yr. ? Of
$16? $30? $40? $60?
31. At 8% what is the interest each year of $ 7 ? $ 11 ?
32. At 1% what is the interest each year of $ 9 ? $12 ?
33. At 8% what is the interest of $12 for a year? For
2yr.? 6yr.? 8 yr. ? 12 yr.?
34. At 8% what is the interest of $ 7 for a year ? For | yr. ?
Forl^yr.? 2 yr. ? 3 yr. ? 1J yr. ? 3J- yr. ? 1\ yr. ?
35. At 6% what is the interest of $8 for a yr. ? For \ yr. ?
For3yr.? 4yr.? 5 yr. ? 6J yr. ?
36. CLASS EXERCISE. may tell how many dollars he
would like to have at interest at 6%, and the class may tell
how much interest he would have each year from it. How
much in 2 yr. 3 yr. 4|- yr. 5 yr. 6 yr.
MISCELLANEOUS EXERCISES 105
37. Mrs. Ware lent $800 @ 4% interest, $900 @ 3%, and
$2500 @ 5%. How much interest did she receive each year
from those loans ?
38. What is the ratio of a 5-inch square to an oblong 13 in.
long and 5 in. wide ?
39. Draw on paper a rectangle 4 in. long and 3 in. wide.
Draw a diagonal. Cut out the rectangle and divide it along
the diagonal. Into what kind of triangles is a rectangle
divided by a diagonal ? What is the area of the rectangle ?
Of each triangle ?
40. Show the truth of the following statement :
The area of a right triangle equals one half the area of a
rectangle which has the same base and altitude.
41. What is the area of a right triangle whose base is 8 in.
and altitude 5 in. ?
42. Give directions for finding the area of a right triangle.
43. Find the area of a right triangle whose base is 8 cm.
and altitude 9 cm. Base 27 in., altitude 13 in.
44. Find the area of a right triangle whose base is 12 in.
and whose altitude is 25% of the base.
45. Can you bisect a rectangle and place the two parts so as
to form an isosceles triangle ? Kepresent.
46. Bisect a rectangle and place the
parts as in Fig. 4. Show two horizontal
parallel lines ; two oblique parallel lines.
47. A four-sided plane figure whose
opposite sides are parallel, and whose
angles are not right angles, is called a
FIG. 4. * Rhomboid. Draw a rhomboid.
106
RATIO
48. How long is the perimeter
of a rhomboid whose short sides are
each 7 in. and long sides each 10 in. ?
FIG. 5.
49. How long is the perimeter of
a rhomboid two of whose sides are
each 1.75 ft. long, and the other two each 2.5 ft. long ?
50. Represent and find perimeters of rhomboids having :
A long side, 18 in. ; short side, 33^% of a long side.
A short side, 12 in. ; long side, 25% longer than a short side.
A short side, 1 ft. 4 in. ; long side, 50% longer than a short
side.
51. What per cent of the angles of a rhomboid are obtuse ?
52. Find a rhombus in your book, and see whether it agrees
with the definition of a rhomboid.
A rhombus differs from other rhomboids in having all its sides equal.
53. Draw and cut out two equal
equilateral triangles. Cut them in
two, and arrange the right triangles
thus formed as in Fig. 6. How long
would the perimeter of Fig. 6 be if
each side of the equilateral triangles
were 6 cm. long? 16 cm.? 25 cm.?
54. Place the four triangles as in
Fig. 7. What kind of a figure is
formed ? How long is its perimeter
if each side of the original triangles
is 10 in. long ? 11 in. long ?
55. By changing the position of
one triangle, change the figure into
a rhomboid. Find length of perim-
FIG. 6.
FIG. 7.
eter if each side of the original triangles is 7 in. long.
MISCELLANEOUS EXERCISES
107
56. Place the four triangles as in
Fig. 8, and name the figure. How
long would the perimeter of Fig. 8
be if each side of the two equilateral
triangles were 6 in. long? 1 ft. 3
FIG. 8. in - lon ?
57. Place the four triangles so as to form a rectangle.
58. Draw a 2-inch square. Draw its
diagonals. With the point where the
diagonals meet as a center and a radius
of 1 in., draw a circle. At how many
points do the sides of the square touch
the circle ? At how many points do the
diagonals cut the circumference ? Draw
lines AB, BC, etc., between these points
as in Fig. 9.
59. Erase all but the part shown in
Fig. 10. Such a figure is called an Octagon.
How many sides has an octagon? How
long would the perimeter of your octagon
be if each side were 7 in. ? 9 in. ?
60. Fold the octagon in various ways,
and see whether the angles are all equal,
and whether the sides are all equal. If
they are, what kind of an octagon is it ?
61. What kind of angles has a regular octagon ?
62. Draw diagonals of the octagon as in
Fig. 11. Into how many isosceles triangles
is the octagon divided? Each triangle is
what part of the octagon ?
63. If the area of the octagon were 60.48
sq. in., what would be the area of one of the
FIG. 11. isosceles triangles ?
FIG. 10.
CHAPTER IV
FRACTIONS
1. Draw a line an inch long and divide it into halves and
quarters. How many halves of an inch are there in an inch ?
How many fourths of an inch ? How many eighths ? How
many thousandths of an inch ? How many millionths ?
A Fraction is an expression of one or more of the equal
parts into which a unit is divided.
2. In the expression J, 4, the denominator of the fraction,
shows that some unit is considered as separated into 4 equal
parts ; 3, the numerator, shows how many of those parts are
expressed. What is meant by the expression of an inch ?
3. In the fraction |f which number is the denominator?
What does the 16 show ? What name is given to the number
above the line ?
4. What is meant by |- of an inch ? | of an apple ?
5. Make a mental picture of what each of the following
expressions represents, and tell how much each lacks of a unit
of its own kind : |- of a pie ; J of an apple ; T 7 ^ of a foot ; |- of
a square yard ; f of a regular pentagon ; of a regular hexa-
gon ; J of a 2-inch cube.
6. A fraction whose numerator is less than its denominator
is called a Proper Fraction. Give some proper fractions and
tell how much each lacks of being equal to the whole of which
it is a part.
7. A fraction whose numerator is equal to or greater than
its denominator is called an Improper Fraction. Give some
improper fractions and tell how much each exceeds one unit.
108
FRACTIONS 109
8. Separate the following fractions into two lists, one of
proper fractions, the other of improper fractions :
f I * A
H tt -3 V& 101%
9. Write a proper fraction whose terms are 5 and 7. Write
an improper fraction with the same numbers as terms.
10. A number that consists of an integer and a fraction is
called a Mixed Number ; as 3J. How many halves of a circle
in 31 equal circles ? Illustrate. Does the following explana-
tion seem to you to be true ?
As there are 2 halves in 1 whole, in 3 wholes there are 3 times 2 halves,
or 6 halves, 6 halves -f 1 half = 7 halves.
11. Change 2| to an improper fraction and explain the
process.
12. Change to equivalent improper fractions :
7* Si 8i 16| Sf 7J of 20J
8* 15f 21f 6f 5* 1$ 6J llf
13. Tell how a mixed number is changed into an equivalent
improper fraction.
14. Change to equivalent improper fractions :
7 5lf 12f
4f 6}
15. Write a mixed number whose fractional part is f.
Change it to an equivalent improper fraction.
16. Write a mixed number whose integral part is 7. Change
it to an equivalent improper fraction.
17. CLASS EXERCISE. - may give a mixed number, and
the class may reduce it to an improper fraction.
110 FRACTIONS
18. How many wholes are there in f ? f ?
SUGGESTION TO TEACHER. Lead pupils to express in their own way
the evident fact that since it takes 2 halves to make a whole, there will
be as many wholes in any number of halves as there are groups of 2 in
that number.
19. To reduce a fraction is to change its form without
changing its value. Reduce the following improper fractions
to mixed numbers :
f -- tt W W
20. Give directions for reducing N an improper fraction to a
whole or a mixed number.
21. Reduce to integral or mixed numbers:
m Hi W-
22. CLASS EXERCISE. may give an improper fraction,
and the class may change it to a mixed number or to an integer.
23. A fraction is an expression of division, if- equals how
many units ? In the expression -^ which number is the divi-
dend ? Which is the divisor ? What is the quotient ? Show
the same with regard to the expression - 1 - -. With -^k With - : 7 5 -.
SUGGESTION TO TEACHER. Lead pupils to see that proper fractions
also express division. indicates that one unit is divided by 3. Let
lines be drawn and divided to illustrate such facts as that ^ of 2 yd. or of
6 ft. equals f of a yd. , that f of a ft. or 9 in. equals % of 3 ft.
\
24. Draw a line 3 in. long. Divide each inch into fourths
and show that f of 1 in. equals of 3 in.
25. When the numerator and denominator of a fraction are
made to change places, the process is called inverting the frac-
tion, as f inverted is f. The fraction resulting from this inver-
sion of a fraction is called the Reciprocal of the original fraction.
Thus -f is the reciprocal of f . What is the reciprocal of the
fractionf? ? T 9 T ? y~ ?
26. Which is greater, f or its reciprocal ? ^ or its reciprocal ?
FRACTIONS
111
FIG. 1.
27. The ratio of Mr. A's money to Mr. B's money is -f . What
is the ratio of Mr. B's money to Mr. A's money ? If Mr. A's
money is $ 8, how much has Mr. B ?
28. Draw a regular octagon. Divide it
by diagonals into 8 equal isosceles trian-
gles. Divide each isosceles triangle into
equal right triangles as in Fig. 1. Each
isosceles triangle equals what part of the
octagon ? Each right triangle equals what
part of an isosceles triangle ? Of the octa-
gon ? i of i = ?
4SuGGESTioN TO TEACHER. Let a large copy of Fig. 1 be drawn upon
the board as a basis for the following exercises.
9. Find from the figure the values of the following:
iofj i'afj ifj ioff
ofi |off |of| JofJ
30. A fraction of a fraction is called a Compound Fraction.
What is the value of the compound fraction 1 of ^ when
expressed in simple form ?
31. How many inches equal J of \ of a foot?
32. See if the following reasoning is true :
Since of =>
of = 2 tiroes or
33. The following rule is founded upon the same reasoning:
To find the simple form of the value of a compound fraction
Find the product of the numerators for the numerator of the
simple fraction and the product of the denominators for its
denominator. Cancel if possible.
By the same reasoning find the value of % of |.
112 FRACTIONS
34. Simplify :
a b c d
f of | of f * of ^ of f| of | of 72 f of | of | of ^f
e f g h
|of of A<* foffiof*
< j A; I
f off of if foffofi^ fof2 3 5of2i foffiofS
m no
A of H of 7J fof^ofGi | of | of 8i
35. How many square centimeters in ^ of ^ of a square
decimeter ?
36. Mr. King owned 1 of a farm and sold f of his share.
What part of the farm did he sell ? If there were 200 acres
in the farm, how many acres did he sell and how many had
he left ?
37. Six boys divided a number of marbles equally among
themselves. Edward Wells, one of the boys, gave ^ of his
share to his younger brother. What part of the marbles did
Edward keep ? If there were 54 marbles, how many did he
keep?
38. Mr. Hubbard owned | of a mine and sold f of his
share. If the mine was worth $80,000, how much did he
receive ?
39. How many cubic feet in f of f of a cubic yard ? In f
of f of a cubic yard ?
40. How many minutes in f of f of an hour ? In | of of
an hour ?
41. -I of f of 4 of 4 is how much less than 1 unit ? Than 2
o 4 y /
units?
42. f of f of 2f. is how much more than 1 unit ? How much
less than 3 units ? Than 8 units ?
FRACTIONS 113
43. Find, from the octagon on page 111, the values of x in
the following equations :
i=T Z 6 t = f i = f i = TF t = f f = T<r
fx 3 z 1 z 3 g 5 * 7 *
= Tff T T "8 Tff ~ T6" t ~ TF
44. In changing to 16ths, by what number are the
numerator and the denominator multiplied? How do you
find it?
45. Change i to a fraction of equal value, having a larger
denominator, and show the truth of the following prin-
ciple :
PRINCIPLE 1. Multiplying both terms of a fraction by the
same number does not change the value of the fraction.
46. Change to 6ths. f to 6ths.
r
47. Change 1, f, f, and to 12ths.
48. Change f, f, ^ to 20ths. J, |, f to 18ths.
49. To 24ths change T %, |, |, f, f, 1.
50. Write 10 fractions whose value is \, but whose denomi-
nators are all different. What is the ratio of the numerator
of each fraction to its denominator ?
SUGGESTION TO TEACHER. Develop the idea that the value of a frac-
tion does not depend upon the magnitude of the numbers by which it is
expressed, but upon their ratio.
51. Write 7 fractions whose value is J, and whose denomi-
nators are all different. What is the ratio of the numerator
of each fraction to its denominator ?
52. Tell how you change fractions to equivalent fractions
expressed in higher terms ; that is, by larger numbers.
53. Change \ to hundredths and write it as per cent.
54. Change to hundredths and write as per cent :
*********
HORN. GRAM. SCH. AR. 8
114 FRACTIONS
55. When T 6 3- is expressed as J-, what has been done to each
term of the fraction T 6 ^ ?
56. Choose some fraction whose value is J, although ex-
pressed differently, and show the truth of the following
principle :
PRINCIPLE 2. Dividing both terms of a fraction by the same
number does not change the value of the fraction.
57. What number will divide both terms of the fraction T 6 ^ ?
What fraction results from the division ?
58. In this way reduce to lowest terms : -J-J. T 5 7 . T 9 T . -}-J.
59. Reduce ||- to lower terms, but not to its lowest terms.
60. What is meant by the phrase, " reducing a fraction to
lowest terms " ?
61. In reducing a fraction to its lowest terms, by which of
the common divisors of both terms is it best to divide them ?
Why?
62. Reduce to lowest terms and tell what common divisor
you use: ft. . ff if if A- if.
63. Reduce to lowest terms:
. H H
tt If A H i A H
A *f H , H if H
64. Tell how a fraction is reduced to its lowest terms.
65. Find by continued division the greatest common divisor
of 147 and 196, and reduce the fraction ||-J to lowest terms.
66. Reduce to lowest terms :
m m m m
T8TF "252"
FRACTIONS 115
67. How many inches in |f ft. ? In f f ft. ? In ff J ft. ?
68. As John was studying fractions one evening, his uncle
said to him, " If you will go on an errand for me I will give
you f||f of a dollar." John did so, and received the money.
How much was it ?
69. If John's uncle had given him fffffff ^ a ^ ^ ar ? now
many cents would he have received ?
70. What is the use of reducing fractions to lower terms ?
71. CLASS EXERCISE. - may give a fraction which the
class may reduce to lowest terms.
72. Can you reduce f to lower terms ? Give reason for your
"Yes" or "No."
73. Write a fraction whose numerator and denominator are
both prime numbers. Can you reduce it to lower terms ? Give
reason for your answer.
74. Numbers that have no common factor are said to be
prime to each other, although they may be composite numbers.
f is a fraction whose terms are prime to each other. Are the
terms prime numbers or composite numbers ?
75. Write 5 different fractions, each having for its terms
composite numbers which are prime to each other.
76. Write a fraction with 7 for a numerator and the square
of 7 for the denominator, and reduce it to lowest terms.
77. Write a fraction with the square of 8 for the numerator
and the cube of 8 for the denominator, and reduce it to lowest
terms.
78. A fraction whose denominator is 10, or any other power
of 10, may be expressed as a decimal or as a common fraction.
Express T \ as a decimal. .50 as a common fraction.
116 FRACTIONS
79. Express in decimal form and in common form each of
the following : Thirteen hundredths ; one hundred sixty-seven
thousandths; two thousand six hundred seven ten-thousandths;
forty-three thousandths; six hundred fifty -one hundred-thou-
sandths; forty one rnillionths.
80. Write as common fractions and give in lowest terms:
.5. .25. .125. 99%. 75%. .072. .064. 5%. 40%. 80%.
90%.
81. CLASS EXERCISE. - may give a decimal fraction,
and the class may express it as a common fraction in its
lowest terms.
ADDITION AND SUBTRACTION OF FRACTIONS
82. A man dying left 3 sevenths of his property to his wife,
3 sevenths to his children, and the remainder to a library.
What share of it did the library receive ?
83. 1 ninth -f- 4 ninths + 2 ninths = ?
84. How are fractions added when they have the same de-
nominator ?
85. Fractions that have the same denominator are called
Similar Fractions. Write three similar fractions and find their
sum.
86. Give four proper fractions that express 13ths and find
their sum.
87. From T 6 T take T V T T-T 3 T = ?
88. Find from the octagon on page 111 the values of x :
1 i 3 _ se 1 3 __ a: 3 _ 5 .
Y + T3" T5" 1 ~~ TS TF ? IT TS"
+ A = TS t ~~ T 5 B" = fV f ~~ A = T Z 6
90.
ADDITION AND SUBTRACTION OF FRACTIONS 117
91. + y 7 ^ of the octagon needs how much to complete the
figure ?
92. In adding 1 and 1, how must i be changed? Why ?
93. By what number must each term of the fraction \ be
multiplied to change the fraction to 12ths ? How did you find
that number ?
94. Can the terms of the fraction \ be multiplied by an
integer that will change it to llths?
95. Give five different numbers that can be used as de-
nominators for fractions whose value is J.
SUGGESTION TO TEACHER. Lead pupil to see that in changing a frac-
tion to an equivalent fraction having a larger denominator the new
denominator must be a multiple of the old.
96. Change % to 15ths by the following rule :
To change a fraction to an equivalent fraction having a larger
denominator
Multiply both terms of the fraction by the quotient obtained by
dividing the new denominator by the old denominator.
97. Change J to an equivalent fraction having the denomi-
nator 21. 27. 24. 33. 36. How many lOths equal f ?
How many 30ths ? 20ths ? 35ths ? 50ths?
98. Copy the following equations, substituting for x its
value :
1 x 5 x 5 _ * 2 x 5 *
T = T2" T T2" 2T T2 TT ~ T2" 6" ~ VI
99. Find the least number that is a multiple of 4 and 7.
Change f and -f to equivalent fractions having that number for
their denominators. Add those fractious.
100. Change | and f to equivalent fractions having for their
denominators the least common multiple of 8 and 5. Add
those fractions.
118 FRACTIONS
101 . Change |- and ^ to equivalent fractions having the
least integral denominator that they can both use. Add the
fractions.
102. Change ^ and -J to equivalent fractions having for their
denominator the least common multiple of their present de-
nominators. Then add them.
103o The least common denominator that two or more frac-
tions may have is the least common multiple of their denomi-
nators. Express the following fractions with their least
common denominator: 4. f. . J^.. 4-. f-
4 O o 12 - >
104. Change ^ and ^ to equivalent fractions having their
least common denominator, and explain your method of
changing them.
105. Express with least common denominator and find
values: +f. f + f + A- i + A-
106. Find values :
i-i c t-i e *-* ? t-t
b i-i-V d |-i /|-i h }-|
Add:
a 6 c d
107. i i i i, |, } i i, i i i, f
108. i, ^ | i, |, -f- ^ ^ * i, |, f
109. 1, ^ i i, |, f ^ i, T V i, f, H
no. i, *, A i i A i i- i i i f
in. f, i I i i, A f> i A i i f
112. i, |, i |, $, i i, i i> i t
113. i, |, A i f A i T> H i i A
114. f, f, A i, i, i I, f, f f, fc A
SUGGESTION TO TEACHER. For additional practice let the class sub-
tract the second fraction from the sum of the first and third in Examples
107-114.
ADDITION AND SUBTRACTION OF FRACTIONS 119
116. How much greater than ^ is ^ ? ^ ? ? ? |?
117. & is how much more than -ft ? fa ? -fa ? fa ? fa?
118. CLASS EXERCISE. may give two fractions, and the
class may find their sum or their difference as may be directed.
119. Draw a line AB and mark two points in it, C and D.
If AC represents \ a mile, CD \ of a mile, and DB -fj- of a
mile, what distance is represented by AB ?
120. John spent ^ of his money for a top, J of it for a
ball, and of it for candy. What part of it had he left?
121. How much money did John spend for each article and
how much had he left if he had at first 12^ ? 48^ ? 24^ ?
122. f + T \ is how much more than f T \ ?
123. The sum of -I and 4 is how much more than their
o o
difference ?
124. Express in lowest terms the ratio of 15 to 20 and of
16 to 30. Find their sum. Their difference.
125. Find the sum of -f- and its reciprocal. Of f and its
reciprocal.
126. Find the sum of | and its reciprocal. Find the differ-
ence between f and its reciprocal.
127. What is the square of 10? The 5th power? 3d
power ?
128. Which power of 10 is the denominator of the decimal
.01? .0015? .003? .00008? .000009?
129.
Add -j 2 ^, -^
jfa>, TOTP Add
nr<p &> ii
130.
Add:
a
6
c
rf
.07
.165
.06
.485
.018
.2145
.016
.6
.5
.31
.07
.21
120 FRACTIONS
131. When several fractions have denominators that are
powers of 10, is it easier to add them as common fractions or
as decimals ? Why ?
132. Write in decimal form and add :
jflT TflTFTF nfinnF T%
133. Write in decimal form and find values :
TO" ~~ T^TJ" Tuinr ~~~ TTIF TTRT ~o ~~~ TTrluo"
134. Which is greater, 3 or 3.00 ? 7 or 7.000 ?
135. Change f to a decimal.
3 reduced to lOOths equals 3.00. As 3 equals 3.00, 3 -r- 4 equals
3.00 H- 4. 3.00 -r- 4 equals .75. Hence f = .75.
136. Change f to a decimal by the following rule. Explain.
To reduce a common fraction to a decimal
Annex ciphers to the numerator and divide by the denominator.
137. Change to decimals and add :
t i * A it i A
138. Change -^ to a decimal, stopping at lOOOths.
Common fractions cannot be reduced to exact decimals, if when re-
duced to their lowest terms their denominators contain any factors other
than 2 and 5.
139. Reduce the following fractions to decimals, not carry-
ing the work beyond ten-thousandths :
i r' . , jt i -.H ' .. ...t .. if -
m H If A A A A A W
140. Reduce the following to decimals of not more than 3
places, and add them. If the division is not exact, make the
remainder the numerator of a common fraction :
f* TV I I A A f A
ADDITION AND SUBTRACTION OF FRACTIONS 121
141. Tell at sight which of the following fractions can be
reduced to exact decimals :
A $ A I TT M 'fr T M
Eeduce the fractions to decimals of not more than three
places, and find their sum.
142. Divide 2 by each number larger than itself that is
expressed by one digit. Express the quotient as a decimal of
not more than three places.
143. Divide three by each number between 10 and 20, and
express the quotient as a decimal of not more than three places.
144. Change to decimals, stopping at hundredths :
* t i i * * * * t I
145. How many lOOths or per cent equal $ ? f? |? f ? |?
146. How many per cent equal ^ ? -f^ ? ft? |$ ?
147. How many per cent equal ft ? ft? ^? ft? ft ? | ?
148. What per cent of anything is \ of it ? % ? f I ? -fa?
149. Add 2| and 7$:
7$ f + = | or 1|, which added to the sum of 2 and 7 = 10.
10J
150. Add:
a b c d e
7f lOf 8| 6f 2|
3f 6 3^ 8| 6ft
8f 5^ 4f 7j% 8J
151. Tell how the following mixed numbers are added:
abed
8 16f 66|
18f 8J
GJ: 12$ 16|
122 FRACTIONS
152. Add:
91| 41f 93f 68f
153. CLASS EXERCISE. may give three mixed num-
bers, and the class may find their sum.
154. A farmer used 77^- acres of land for wheat, 40 J acres
for corn, If acres for vegetables, 29^- acres for pasturage, and
acres for an orchard. How many acres were in the farm ?
155. Mr. White has 3 fences on his farm; one is 168^- rd.
long, another 456 T 4 T rd. long, and another 328 T 5 T rd. long.
How many rods of fencing has he in all ? How much did his
fence cost at 75^ a rod ?
156. After selling 3f acres, a farmer had left 123 j acres.
How much land had he at first ?
157. How long is it from half past eight A.M. to noon?
From quarter before nine to half past eleven ? From quarter
past two to quarter to six? From half past ten to quarter
past one ? From a quarter of eleven to half past three ?
158. From 14 We subtract \ from 1 of the units of the minu-
take 7-i- en( ^' '^^ ie rema ' n der is \. As 1 unit has been
^ subtracted from the 4 units 3 units are left. 7 from
6 13 leaves 6.
a b c d e f
159. From 8 9 18 25 16 20
take 6 5| 4f 3| 11 4
160. Mary's aunt sent her 6 yd. of cashmere for a dress.
5| yd. were used. How many yards were left ?
161. In a jumping match Thomas jumped 3 ft. and his
brother jumped 2-jAj- ft. How much farther did Thomas jump
than his brother ?
162. Make problems in which a mixed number is subtracted
from an integer.
ADDITION AND SUBTRACTION OF FRACTIONS 123
163. Subtract :
a b c d e f g
2$ 3& 2| 2j_ 3| 2| 3i
164. A piece of string 3^- ft. long was cut from a piece
ft. long. How much was left ? 5^ ft. were cut from the re-
mainder. How much then remained ?
165. From 41 take -f.
4 i
2 As | cannot be subtracted from |, we subtract f from 1|.
_Z This leaves only 3 units in the minuend.
3f
166. Find difference :
a b c d e f g
8| 6i 8J 6i 8f 6$ 7i
1 J J J J J J
167. Find difference :
a b c d e f g
h i j k I m
S$ 9i 7f 8i 4| 6i
3| 2| 4f 2| 2| 2|
168. Write two mixed numbers whose fractional parts have
the same denominator. Let the mixed number whose integral
part is the larger have the smaller fractional part. Find dif-
ference between the mixed numbers.
169. Find difference:
a b c d e f g
3J or -| 7J 8^
2f
124 FRACTIONS
170. Write two mixed numbers the fractional parts of which
have different denominators. Let the mixed number that has
the larger integral part have also the larger fractional part.
Find their difference.
171. CLASS EXERCISE. may give two mixed numbers
like those described in Ex. 170, and the class may find their
difference.
172. Find difference :
a b c d e f g
91 97
"5" 9
I m n
6 7 4|
2J 2f 21 4^ 4 T \ C 2\ 2*
173. Write two mixed numbers whose fractional parts have
different denominators. Let the mixed number whose integral
part is the greater have the smaller fractional part. Find
their difference.
174. If a line is 3J in. long, how many inches must be
added to make it 5 J in. long ? Represent.
175. 3^
176. Find values of x:
a 10-7i = x. e 9 -3% = x.
b 6 - 4 = x. f 10 - 5| - x.
c 7 -3% = x. g 7$-2=x.
4 = z. h 12 -7=a.
177. A weighs 148 J lb., B 157f lb., C 1611 lb., D 175| Ib.
How much do they all weigh ?
178. What is the difference between the weights of A and
C? AandB? AandD? B and D ? B and C ? CandD?
ADDITION AND SUBTRACTION OF FRACTIONS 125
179. Mr. Otis rode 23f miles on Monday, llf miles on Tues-
day. On Wednesday he rode as far as on Monday and Tuesday.
How far did he ride in the three days ?
180. Mr. Carr rode on his bicycle to a city 91f miles dis-
tant. The first day he rode 16 miles ; the next day he rode
3^ miles more than on the first day. On the third day he rode
21 miles more than on the second day. How far did he ride in
those three days ? How many miles more did he ride before
he reached the city ?
181. Mr. Grey planted 75J acres in wheat, 45f acres in
corn, and 7-J acres in oats. How many acres of grain did he
cultivate ?
182. In a township containing 23,039 T ^ acres, the roads
occupy 345f acres, and the rest is divided into farms. How
many acres in the farms of that township?
183. A stove burned 180f Ib. of coal in one week, 175 J Ib.
in another week, and 205^ Ib. in another week. How many
Ib. did it burn in the three weeks ?
184. A, B, and C own a mine. A owns T 5 ^ of it, B owns f of
it, and C owns the rest. How much does C own ?
185. If the mine is worth f 248,400, what is the value of
each man's share ?
186. A has 75f acres of land, B has 13| more acres than
A and 4| acres less than C. How many acres has B ? C ? A
and B ? A and C ? B and C ?
187. A farmer has a field in the form of a trapezoid. One
of the parallel sides is 71 T 4 T rd. long, the other is 68| rd. long.
Of the non-parallel sides, one is 53f rd. and the other is 54^- rd.
Represent and find length of perimeter of the trapezoid.
126 FRACTIONS
MULTIPLICATION OF FRACTIONS
188. Multiply by .
To multiply any number by ^ is to take of it.
189. By 1 multiply :
f f ! T G T I I f
190. Make a rule for multiplying a fraction by a fraction.
Multiply :
abed
1 Q 1 4 \/ 7 .2.1 V 8 1 5 y 49 6 y 5 g
1 qo 4 5 v 1 5 v 1 1 1 8 v 7 27 v 8
IVA. -Q-f A YT 77 A TJ-J T5T A IF 32 * "9
194. 14 x 44 M x -rV M x -U *- x M
195. CLASS EXERCISE. may give two fractions and
the class may find their product.
196. CLASS EXERCISE. may give a proper fraction
and an improper fraction and the class may find their product.
197. CLASS EXERCISE. may give three fractions of
such a kind that cancellation may be used in finding their
product and the class may find the product, canceling wherever
possible.
198. Multiply | by itself.
199. Square :
2 4 5 5 10 12 3 151917 25 41
3 o 7 9 11 la 7 16 20 18 30 53
200. What part of a square inch is a rectangle that is J-
of an inch long and ^ of an inch wide ? Draw a figure and
prove your work.
201. What part of a square yard is a rectangle ^ of a yard
square ? How many square feet in it ?
202. Multiply f by the square of f .
MULTIPLICATION OF FRACTIONS 127
203. Cube:
t I * * I I I A T 4 T A
204. Draw a rectangle whose length is of a foot and width
i of a foot. What fraction of a square foot is its area ? Prove
by reducing the fractions of a foot to inches.
205. What fraction of a square foot is a rectangle f of a foot
long and ^ of a foot wide ? How many square inches in it ?
206. How long is the perimeter of a rectangle of a foot
long and ^ of a foot wide ? Give the area of the rectangle
in fractions of a square foot, and also in square inches.
207. How many square feet in a square f of a foot in
dimensions ? How many square inches ?
208. How many square feet in a square f of a foot in dimen-
sions ? How many square inches?
209. Add the product of ^ x f to the product of f x 7.
210. Subtract the product of f x from the product of
I x I-
211. The product of several numbers is called their con-
tinued product. What is the continued product of 3, 5, and 7 ?
Of i |, and f? Of , 2 4 T , and f ? Offhand}?
212. Multiply | xf
Observe that the denominator of the fraction f may be omitted without
changing the result.
Multiply :
a b c d e
213. f x 24 ^ x 35 5 x 27 J x 64 T 2 T x 33
214. ^ x 100 f x 18 f x 21 ^ x 18 | x 24
215. 36 x fV 48 X I 32 x A 39 x A 60 x ii
216. 77 x T 4 T 42 x f 63 x 2^ 54 x ^ 65 x fV
217. Multiplying the numerator of a fraction by an integer
has what effect upon the value of the fraction ? Illustrate.
128 TRACTIONS
218. Dividing the denominator of a fraction by an integer
has what effect upon the value of the fraction ? Illustrate.
219. Give either of two ways by which a fraction may be
multiplied by an integer.
220. Multiply 12 by a proper fraction. Is the product
greater or less than 12 ?
221. Multiply 12 by an improper fraction. Is the product
greater or less than 12 ?
222. When will the product of an integer and a fraction
be greater than the integer ?
223. Multiply 64 by 3J.
64
8*
192 = the product of 64 and 3.
16 = the product of 64 and .
208 = the product of 64 and 3J.
Multiply :
a
b
c
d
224.
55
164
164
125
4 i
21
5 t
5
225. 72 81 208 144
226. 81 115 64 343
H
227. 66 201 192 512
51 11} 4f 7|
228. 172 981 111 169
21 101
MULTIPLICATION OF FRACTIONS 129
a b c d
229. 385 243 78 408
230. Multiply an integer by a mixed number and explain
your method.
231. At 60^ a yard, what is the cost of 7 yd. of silk ? 8
yd.? lOfyd.? 12f yd. ? 14| yd. ? 15| yd. ? 18f yd. ?
232. 160 square rods equal 1 acre. How many square rods
in | of an acre? In | A. ? ^ A. ? |f A. ? ^ A. ? A. ?
233. Mr. Hill has a lot 40 rd. long and 3 rd. wide. What
fractional part of an acre is it ?
234. How many square rods in 2| A. ? 5| A. ? 7 T % A. ?
A.?
235. Multiply 16 $ by 6.
Multiply :
a
16f
6
96 = the product of 16 and 6
4 = the product of f and 6
d
100 = the product of 16f and 6
b c
236. 15J
8
92J 39|
24 12
122|
22
237. 124f
16
72| 164|
15 4
109J
27
238. 24|
9
119f 81$
15 16
98|
5
HORN. GRAM. 8CH. AR. 9
130 FRACTIONS
239. Tell how a mixed number is multiplied by an integer.
240. At 18f ^ per yard, what is the cost of 4 yd. of gingham ?
7yd.? 8yd.? 10yd.? 12yd.?
241. How long is the perimeter of an equilateral triangle, a
side of which is 4 T 7 2 ft. long ?
242. If each side is 4 T 9 ^ ft. long, how long is the perimeter
of a rhombus ? Of a regular octagon ? Of a regular hexa-
gon ? Of a regular pentagon ?
243. At 331 ^ per yard, what is the cost of 8 yd. of dress
goods ? 9 yd. ? 12 yd. ? 14 yd. ? 15 yd. ? 18 yd. ?
244. Multiply 2^ by 3^.
Before small mixed numbers are multiplied together, they should be
reduced to improper fractions.
fx = Y = 8i
Multiply :
245. a 21 by 3J d 5| by 12J
b 3J by 6J e 18f by 11|
c 4f by 6^ / 7i by 66|
246. Square:
2| H 3i 2} 6i If If 2f
47. Cube:
11 H 34 31 4i 6J 3f 2^
248. CLASS EXEKCISE. may give two small mixed
numbers, and the class may find their product.
249. CLASS EXERCISE. may give a small mixed num-
ber, and the class may find its square.
250. Draw on the floor a square 5 yd. in dimensions. Each
side is 1 rd. long. Find the number of square yards in a square
rod.
MULTIPLICATION OF FRACTIONS 131
251. How many feet long is a rod? Find the number of
square feet in a square rod.
252. Each of the short sides of a rectangle is 7-f in. The
long sides are 9| in. each. Find the area of the rectangle.
253. One side of a rectangle is llf in., and each of its
adjacent sides is 3 in. shorter. Find the area of the
rectangle.
254. Draw on paper or pasteboard a circle whose radius is
3 in. Cut it out. By measuring the circumference with a
tape measure, it will be found to be nearly 22 in. %f- is
considered the ratio of the circumference of a circle to its
diameter. What is the ratio of a diameter to the circum-
ference ?
255. Find the circumference of a circle whose diameter is 7
in. 21 in. 14 in.
256. How long is the circumference of a circle whose radius
is 21 in. ? 3 in. ? 10 in. ? 17J- in. ?
257. How long is the circumference of the largest circle that
can be cut from a piece of paper 4-| in. square ?
258. A round flower bed 14 ft. across has a border of pinks,
set 6 in. apart. How many pinks in y^ of the border ?
Represent.
259. Mrs. Smith's wash bench is 4 ft. long and If ft. wide.
A tub is set upon it in such a way that the lowest hoop of the
tub touches the front edge and also the back edge of the bench
without extending over either edge. What is the circumference
of the hoop ?
260. A round tin pail with straight sides is 8 in. across and
10 in. high. How long is the diameter of the largest plate that
can be placed on the bottom of it ? The circumference ?
261. If a ball is cut into two equal parts by one cut, what is
the shape of the flat surface of each part ?
132
FRACTIONS
262. A round apple 4 in. in diameter was cut into halves.
One of the halves was laid with its flat side down upon a
plate, in such a way that no part of the cut surface of the
apple extended beyond the plate. Find the diameter and the
circumference of the smallest plate that could be used for that
purpose.
263. A part of a circumference is called an Arc. Draw a
circle and divide its circumference into several arcs.
264. If the radius OA (Fig. 2) is 3 in.,
how long is the diameter AC? How long
is the circumference ? .
265. What part of the circumference is
the arc AC? How long is it ?
266. The arc AB is % of the circum-
ference. How long is it?
FIG. 2. 267. What part of the circumference is
the arc BC ? How long is it ?
268. If the radius of a circle is 5 in., how long is the
diameter? The circumference? An arc which is J T of the
circumference ? of the circumference ?
269. Find the length of an arc which is -^
ference of a circle whose radius is 5 in. 12 in.
of the circum-
37 in.
270. The circumference ABCD is
divided into how many equal arcs?
If the diameter of the circle ABCD is
7 in., how long is the arc AB? The
arc ABC? The arc BCD? The arc
ABCD?
271. An arc which is of a circum-
ference is called a Quadrant. If the
diameter of the circle ABCD were
28 cm., how long would a quadrant be?
MULTIPLICATION OF FRACTIONS 133
272. Multiply 124f by 6J.
124f
744 = the product of 124 by 6.
62 = the product of 124 by J.
4 = the product of f by 6.
= the product of f by .
810 = the product of 124| by 6.
273. This method of multiplying mixed numbers together is
useful when the numbers are large. Can you see why ?
Multiply :
a b c d
274. 441 344f 288f 456f
16f
abed
275. 819f 64| 816^
9f 28^
276. Square 64f. 32|. 24f 22 T 2 T . 36J.
277. Cube 12J. 16J. 16J. 14|.
278. At the rate of 17f mi. per hour, how far will a steam-
boat go between nine o'clock Monday morning and half past
ten on Tuesday morning ?
Find the cost :
279. Of 51 yd. of cloth at $ 4| a yard.
280. Of 7-J- A. of land at $ 24^ an acre.
281. Of 7| T. of hay at $ 21 J a ton.
282. Of 10J yd. of ribbon at $ .23 a yard.
283. Multiply ^ by fa and write the product in decimal
form.
284. Find the product of -}- and -f^ and express it as a
mixed number.
134 FRACTIONS
285. Express fj and T 8 ^ in decimal form and find their
product.
286. If asked for the product of two fractions having for
denominators some power of 10, would you find it easier to
multiply them as common fractions or as decimals ? Why ?
287. Give a rule for pointing off the product of two decimals.
288. Let a = .04, b = .02, c = .005, d = .0007, e = .00002,
/= .3. Find the value of ab, ac, ad, ae, af, be, bd, be, bf, cd, cf.
DIVISION OF FRACTIONS
289. How many times is 1 fourth of an inch contained in 3
fourths of an inch ? 1 of anything in of it ? |-^-i=? |-=-i=?
290. 6 sevenths -v- 2 sevenths =? f -j- -f = ? 10 elevenths
^- 5 elevenths = ? |f -*- ^ = ?
291. A-nA=?
T 8 T contain ^ as many times as 8 units of any kind contains 3 units of
the same kind. 8 -j- 3 = f or 2|.
292. What are similar fractions ? Illustrate.
293. Divide |f by a similar fraction.
294. CLASS EXERCISE. may give two similar fractions,
and the class may divide the greater by the less.
295. How is a fraction divided by a similar fraction ?
296. Use |- as a dividend and % as a divisor.
297. Multiply | by its reciprocal.
298. Multiply several fractions by their reciprocals and
compare the results.
299. How many times is f of anything contained in f of
the same thing ? inj? |in|?
300. Which is greater, } -j- f or j x | ? }-5-'{orJx^?
DIVISION OF FRACTIONS 135
301. $dividedbyf=? f multiplied by the reciprocal of f=?
302. Think of two similar fractions. Divide the larger frac-
tion by the smaller. Compare the result with the result
obtained by multiplying the larger fraction by the reciprocal
of the smaller. Think of two fractions that are not similar.
Reduce them to similar fractions and make the same com-
parison. Continue this until you see the reason for the fol-
lowing rule :
To divide a fraction by a fraction
Multiply the dividend by the reciprocal of the divisor.
303. By this method divide -^ by ^.
Divide
: a
b
c
d
304.
A by f
& by \
&byf
|by |
305.
A b y T
by*
if by A
iby |
306.
A by |
f by f
ifby |
Hby -
307.
ft b y i 7
ft b> A
ffby |
Mby
308.
by ff
M by f
1 4 Vv V 7
"2TT J 10"
Hby
309.
1 b y TT-
Mby.fr
M by A
if by
310
Wby |
tf by A
ff by ||
liby -
311. How do you find the ratio of one number to another ?
Find the ratio of f f to ^-. To ^-. To -fa.
Find ratios :
abed
312. A : f tt'f H : li tt : H
313. &* H : A A : f H^
314. |f:A If^A *:* if I
315. Multiply the numerator of a fraction by an integer.
Is the value of the fraction increased or decreased?
136 FRACTIONS
316. Find by trial how the value of a fraction is changed by
multiplying its denominator. By dividing its numerator. By
dividing its denominator.
317. Which gives the greater quotient, 16 divided by 8, or
16 multiplied by | ?
318. |f -5-8 = ?
We may write 8 as f . The question is then f -4- f = what ? This is
solved by the rule for dividing one fraction by another.
Find values of x :
a b c
319. f-=-2 = a; | T -j- 3 = a; T 6 7 -7- 12 = a
321. f -r- 4 = a; y 2 ^-8 = aj fj -j- 11 = a
322. Divide 28 by -f.
Consider 28 as *.
323. Divide 24 by: f f T T . f. f |f f
324. Divide 41 by : f . f . f f . f . ||. |f
Divide after reducing mixed numbers to improper fractions :
a b c d
325. 3|-T-2i- 7J-*-18f 8J-f-lf 5f-r-
326. 2^-f-3J 13i-f-4| 6|-
327. 3-i-6J 12 ^-5f 37|-
328. CLASS EXERCISE. may give a small mixed
number for a dividend and another small mixed number for
a divisor, of such a kind that cancellation may be used in
finding the quotient. The class may find their quotient.
329. At a picnic 5^- pies were divided equally among 44
persons. What part of a pie did each receive ?
DIVISION OF FRACTIONS 137
330. Mr. Tod has 13^ acres devoted to celery, which is
just four times as much as his brother has. How many acres
of celery has his brother ?
331. The top of a newel post is an octagon whose perimeter
is l|i ft. How long is one side of the octagon ?
332. The circumference of a circle is 9f in. How long is its
diameter ? Its radius ? Eepresent.
333. Find the length of the diameter of a circle, the circum-
ference of which is 11 in.
334. A quadrant of a circle is 4^ in. How long is the cir-
cumference ? Diameter ? Eadius ?
335. For 75^, how many yards of lace' can be bought at 21 f
per yard? At 3^? At 6^? At8i^? At 12^? At 16.}??
At 37J^? At 621^? At66|^? At 83^? At
336. If a coat costs 3 j dollars, how many coats may be bought
for 62 dollars ?
337. If 1J yd. of cloth are required for a coat, how many
coats can be made from 87-J- yd. ?
338. If Jerry walks 2^ mi. an hour, in how many hours will
he walk 7-J- mi. ? 11 mi. ? 1 mi. ? | mi. ? J mi. ? 1 mi. ?
339. What number divided by 3 will give 5 for a quotient ?
What fraction divided by 3 will give f for a quotient ? Prove.
340. What mixed number multiplied by 3 will give
181? 6f?
341. A fraction that has a fraction in one or both of its terms
1 2~ 7
is called a Complex Fraction ; as, |, =f> ^T. Write a complex
fraction whose numerator is a mixed number, and denominator
a whole number.
| is read "| divided by i v
138 FRACTIONS
342. Write and read complex fractions as follows :
a Numerator an integer, denominator a simple fraction.
b Numerator a simple fraction, denominator a mixed num-
ber.
c Numerator an integer, denominator a mixed number.
d Numerator and denominator both simple fractions. Both
mixed numbers.
343. A complex fraction, like other fractions, is merely an
expression of the division of the numerator by the denominator,
and it is reduced to a simple fraction by performing that
division; as,
f=H r I x H-
3
Reduce to their simplest form the following complex fractions:
a b c d e f
4} U 3 1} 3f 17
3J 4| T 5| 13 8j
344. Simplify:
a b c d e f g h
^12J16J41|2|-3J6J7J
33! 83j 87j 91| 20 30 50 60
345. CLASS EXERCISE. may give the hardest complex
fraction that he wrote in Ex. 342, and the class may reduce
it to a simple fraction.
346. The product of two numbers is 15. One of them is 3.
What is the other ? How is it found ?
347. The product of two fractions is -fa. One of them is
f . What is the other ?
348. CLASS EXERCISE. may give the product of two
fractions and one of the fractions. The class may find the
other fraction.
DIVISION OF FRACTIONS 139
349. Divide:
liv ~^ iV T7TO"0~ ~*~ TO" 10000 "*" TOO" 1000 ~*~ TOTF
350. Reduce the same fractions to decimal form and find
the quotients. Which is the easier way of dividing in this
case ? Why ?
351. Leta=.4, 6 = .08, c=.032, d=.0016. Find values of :
a a a b b 6
b c d a c d
. - f? - -
a b d a b c
352. Change f and ^f to decimals and divide the greater by
the less.
353. Change to decimals and divide : f by J. f by |.
354. If 4.5 yd. of silk cost $ 6.75, how much will 1 yd. cost ?
3.7 yd. ? 6.75 yd. ?
355. Mr. K bought a lot in Washington, D.C., for $4500,
paying $1.875 per square foot. How many square feet in the
lot?
356. If 1.7 yd. of cloth is used to make -a coat, how many
coats can be made from 81.6 yd. ?
357. How much cloth at $ .75 a yard can be bought for
$ 45.75 ?
358. At the rate of 8.25 mi. an hour, in how many hours
will a stage run 125 mi. ?
359. If a barrel of beef costs $ 14.25, how many barrels can
be bought for $ 798 ?
360. Traveling 215.6 mi. a day, in how many days will a
steamer go 1000 mi. ?
361. If a dollar gains 5 i interest each year, in how many
years will it gain another dollar ?
140 FRACTIONS
362. In how many years will $1.00 double itself at 4% ?
363. One third of John's money is if. How much has he ?
364. Thomas spent f of his money and had 5 f left. How
much had he at first ? Explain.
365. How much money has a boy who can spend J of his
money and have left 70? 90? 20? 120? $1.00? $8.00?
366. How much money has a boy who after spending 30
will have left f of his money ? J ? T 9 7 ? ? f ? } ?
367. Three fourths of John's money is 150. How much is
of it ? How much is the whole ?
368. 6= f of what number ?
SOLUTION BY ANALYSIS. As 3 fifths of the number = 6
1 fifth of the number = 2
5 fifths of the number = 10
369. 8 is f of what number ? Analyze as above.
370. Analyze. 12 is f of what number ? 4. of what ?
371. Find values of x. Analyze.
a -ff of x = 14 e f of x = 15 i f of x = 18
b | of a? = 10 / | of x = 16 J V of x = 20
c f of a = 25 g f ofx = 28 fc J of a; = 28
d f of a; = 24 h f of a? = 30 / | of a = 75
372. How long is a line f of which is 9 in. long ?
373. A man rode 16 mi. on Monday, which was f of the dis-
tance he rode on Tuesday. How far did he ride on Tuesday ?
374. f of John's money is 210. How much will he have
left if he gives away $ .05 ?
375. f of James's money is $20. How much will he have
left if he gives away of his money ?
DIVISION OF FRACTIONS 141
376. How much money must a boy have that he may lose |
of it and have 12^ left? 30^? 42^? 60^?
377. How much money must a man have so that after gain-
ing i as much, he may have $ 700 ? $ 210 ? $ 441 ? $ 7.70 ?
378. Mary has 12^. Her money equals y 3 ^ of Florence's
money. How much has Florence ?
379. William has 8 marbles. He has f as many marbles as
James has. How many marbles has James ?
380. Alice has 14^, which is just -f of the money she needs
to buy her geography. What is the price of the geography ?
381. Make problems in which a certain number is a fractional
part of the number which is to be found.
382. John gave away -- of his marbles and then had 30
marbles left. How many had he at first?
383. Susie gave -| of her money to her sister, and found that
she had 16 cents left. How much had she at first ?
384. Harry gave away -| of his pigeons and sold f of them.
He had 15 pigeons left. How many had he at first ?
385. In a storm, a ship's crew threw overboard 30 bbl. flour,
which was y\ of the whole cargo. How much was the
whole ?
386. A man owned f of a mine. He sold J of his share for
$ 6000. How much was the whole mine worth ?
387. Mr. Buchanan sold J of his share of a store for $ 2000.
What was his share worth? His share was f of the whole
value. What was the whole value ?
388. Owning | of a quarry, Mr. Harris sold ^ of his share
for $ 6000. What was the value of the quarry ?
142 FRACTIONS
389. Mr. Madison owned \ of an Indiana gas well. He sold
f of his share for $ 1500. What was the value of the whole
well?
390. yL of the pins in a cushion were crooked, and there were
66 straight pins. How many were there in all, and how many
were crooked ?
391. If 8f yd. of tape cost $ .70, how much will 1 yd. cost?
9|yd.?
392. If 16f yd. of rope cost 100 cents, how much will 1 yd.
cost ? 23f yd. ?
393. If 6 bu. of seed cost $15, how much will 19f bu. cost?
394. If | of a quart of seed cost $ .18, how much will 1 pk.
cost?
MISCELLANEOUS EXERCISES
1. Divide .0096 by .12. By .008. By 24. By 3.2. By .16.
2. Divide .000048 by .012. By .4. By .0024. By 2.4.
3. Divide .144 by .04. By 48. By 1.6. By .0003.
4. What is the ratio of 889.44 to .102 ? To .105? To .108?
5. Multiply 7* by the 4th prime. 9 2 by the 6th prime.
6. f of -f-g of y\ of 24 hr. is how much less than a day ?
7. 24 sheets of paper make a quire. How many sheets in
| of | of | of a quire ?
8. How many sheets in y 7 ^ of ^ of -f of a quire ? In |- of
of -fe of a quire ?
9. Mrs. Smith is f as old as Mr. Smith, who is 48 yr. old.
Their daughter Alice is % as old as her mother. How old is
Alice ?
10. Add yV, &, A, iV Add A A> A> A-
11. From L- take --. From - take .
MISCELLANEOUS EXERCISES 143
12. Multiply the first prime number after 9 by -|.
13. Multiply the largest prime factor of 330 by 2^-.
14. Multiply 64 by the largest prime factor of 390.
15. Multiply 8f by the 1. c. m. of 5, 6, and 10.
16. Multiply the 1. c. m. of 8, 6, 9, and 12 by 3^.
17. Multiply the g. c. d. of 36, 48, and 60 by 3|.
Leto = f; & = fi; c = 10; d = 5; e = f Find values of :
18. a x b or ab ac ad ae be bd cd ce de
19. a-f-c a + e a + d b + c b + a d + e
20. c a d a e a c b c d d b
Let a = |; & = 3J; c = l^; d = 2f; e = f Find values of :
^ a a a a b b 6cd
6cdeeda6e
22. f of Anna's money is $ .50. How much will she have
after giving away 7 f ?
23. Of what number is 21 three fourths ? f ? f ? ^ ?
24. Of what number is 16 four sevenths ? | ? f ? ^ ?
25. If ^ of the price of a house is $ 400, what is the price
of the house ? How much will five such houses cost ?
26. If 1 apple costs |^, how much will 4 doz. apples cost ?
27. If f of the price of an orange is 3^, how much will a
dozen oranges cost ? 6 doz. ?
28. At 16|^ a yard, how many yards of ribbon can be
bought for$l? $2? $3? $5?
29. 12 doz. make a gross. When buttons are bought for 25 1
a gross, what is the cost of 1 button ?
30. If 500 pins cost 10 ^, how much will 1 pin cost ?
144 FRACTIONS
31. If a gross of pencils cost 50^, how much will 1 pencil,
cost?
32. Mrs. Norton paid 5^ for a box of toothpicks, in which
there were 2000 toothpicks. How many did she get for a cent ?
What was the cost of 1 toothpick ?
33. She paid a nickel for a box of matches. What was the
price of each match if there were 500 matches in a box ?
34. A gross of glass vials cost 48 ^. How much did 1 vial
cost ?
35. Find the average receipts of a peanut stand for 6 days.
Monday, $ 1.37 ; Tuesday, $ 2.11 ; Wednesday, $ 1.87 ; Thurs-
day, $ 1.04 ; Friday, $ 1.75 ; Saturday, 9 3.10.
36. If the average cost of keeping up the stand was $ 1.25
per day, what were the owner's average gains per day ?
37. Suppose a pie to be exactly round, and 101 in. in diam-
eter. If it were cut into 6 equal pieces, how long would the
curved edge of each piece be ?
38. The surface which is bounded by
an arc and two radii is called a Sector.
Show five sectors in Fig. 4.
You may remember the figure of a sector
more easily if you recall the way in which pies,
waffles, and round cakes are usually cut.
39. How long is the perimeter of a sec-
tor of a circle whose radius is 4 in., if the
arc of the sector is 5^ in. ? Represent.
40. Draw a circle whose radius is 3 in. Divide it into 4
equal sectors. Write the word " sector " in each. Write upon
each line of the perimeter of a sector the length of the line
and find the length of the perimeter of a sector.
41. What kind of an angle is the angle of a sector which is
J of a circle ? Less than \ ? Greater than 1 ?
MISCELLANEOUS EXERCISES 145
42. Draw a circle and apply its radius six times as a chord.
What regular polygon have you drawn ? Each arc thus cut off
is what part of the circumference? If the radius is 2f in.,
how long is the circumference ? Each arc ?
43. Draw lines from the ends of each arc to the center of the
circle. What are these lines called ? What kind of angles do
they make ?
44. Erase the chords. Find length of the perimeter of each
sector, supposing the radius of the circle to be 2-f in. 5^ in.
10 in.
45. Erase radii so as to leave the circle divided into three
equal sectors. Find length of perimeter of each sector, assum-
ing the radius to be -j- in. T 9 T in. l T ^ r in.
46. If the radius of a circle is 2f in., how long is the perim-
eter of a sector which is J of the circle ?
47. Find the perimeter of the sector which remains when
a sector that is of a circle is subtracted from the circle.
48. If a strip of paper 5 in. long were curled around so that
its edge inclosed a circle, how long would the circumference of
that circle be ?
49. The circumference of the wheel of a toy wagon is 20 in.
How far does the wagon run when the wheel turns around
once ? 3 times ?
SUGGESTION. Let pupils roll a coin, button, or other circular objects,
as a help in realizing the conditions of these problems.
50. How far will a hoop 2^ ft. in circumference run in
turning 7 times ? 9 times ?
51. How many times will a wheel 2 ft. in circumference
revolve in running 8 ft.?
52. How many times will a wheel 6 ft. in circumference
revolve in running 12 ft.? 8 yd.?
HORN. GRAM. SCH. AR. 10
146 FRACTIONS
53. A mile is 5280 ft. If the front wheels of a wagon are
each 6 ft. in circumference, and the hind wheels are 8 ft., how
many times will each wheel revolve in running a mile ?
54. Draw a rhomboid whose long sides are each double a
short side. How long would its perimeter be if each short
side were 8 in.? 12 J in.?
55. How many square feet in a lot 30 ft. wide, and 150 ft.
deep ? If the owner uses \ of the lot for a house, and 1 for a
chicken yard, how many square feet remain ?
56. A house is 48 ft. long, and the distance from the ridge-
pole to the eaves on each side is 23 ft. How many shingles
will be required to cover it if 6 shingles are required to cover a
square foot?
57. John is 8^ yr. old, and his sister is 6 T 7 T yr. old. What
is their average age ?
58. A grocer bought 3 cheeses, one weighing 32|- lb., another
28J lb., another 41| lb. How many pounds were there in all ?
59. Which is greater, and how much, -f- x f , or -f- -=- J ?
60. What is the area of a rectangle 3f ft. long and If ft.
wide ?
61. What is the area of a right triangle whose base is 4
in., and altitude 3 in.?
62. What is the use of reducing fractions to a least common
denominator ?
63. Write the fraction that expresses the ratio of the first
composite number after 18 to the first composite number after
30, and reduce the fraction to its lowest terms.
64. When Arthur was a year old his father placed $50
in the bank as money to be used in sending him to college.
He put $ 50 in the bank on every birthday until, at the age
MISCELLANEOUS EXERCISES 147
of 18, Arthur was ready for college. How much money had
been placed in the bank for him ?
65. Arthur's expenses at college for the first year were
$218.75; for the second year, $310.50; for the third year,
$ 365.25. How much of the amount was left at the end of the
third year ?
66. Arthur's expenses for the last year were $ 410.90. He
received $465.67 as interest. How much was left of the
money when he had finished his college course ?
67. A man bought a lot for $ 2000, built a house upon it for
$ 2500, and sold the property so as to gain $ 100 on his invest-
ment. For how much did he sell it ?
68. Charles bought a ball for $.08, and sold it for $.12.
The gain equaled what part of the cost ? What per cent ?
69. A man's salary is $ 2400 a year. He saves J of it one
year, 1 of it the next yea^ and -| of it the next year. How
much has he saved at the end of the third year ?
70. A gentleman had 1200 books in his library, and gave
away - of them. He lost -^ of the remainder. How many
books were left ?
71. He added 200 more volumes to the library, and then
gave away f of it. How many had he left ?
72. The base of an isosceles triangle is 3 ft. The ratio
of one of the equal sides to the base is -J. How long is the
perimeter of the triangle ? Kepresent.
73. How long is the perimeter of an isosceles triangle whose
base is 14 in. and each of whose equal sides is 5 in. longer than
the base ?
74. How long is the perimeter of an isosceles triangle whose
base is 21 in. and each of whose equal sides is 33|% longer
than the base ?
148
FRACTIONS
75. The perimeter of a certain isosceles triangle is 25 in.
and one of the equal sides is 9 in. How long is the base ?
76. The base of an isosceles triangle is 11 in. and the
perimeter 35 in. How long is each of the equal sides ?
FIG. 5.
77. Draw a rectangle 4 in. long and 3
in. wide. Draw a diagonal of it. Into
what kind of figures does a diagonal
divide a rectangle? If the angles of
your figure are exact right angles, and if
your lines are exactly drawn, the diag-
onal will be just 5 in. long.
SUGGESTION TO TEACHER. Let pupils find by trial that if 3 in. be
measured off upon one of the lines about a right angle and 4 in. upon the
other line, the joining line will be 5 in. long.
A 78. In the right triangle ABC how long is
the hypotenuse AC if the numbers represent
inches ?
79. If BC and AB were each twice as long
as they are, AC would be twice as long as it
is. If AB is 8 in. and BC 6 in., how long is
AC? Prove by measuring.
80. If the perpendicular sides of a right
triangle are in the ratio of 3 to 4, the ratio
of the hypotenuse to the less side is f, and the ratio of the
hypotenuse to the other side is f . In a right triangle whose
base is 30 and altitude 40 how long is the hypotenuse?
Represent.
81. How long is the hypotenuse of a -right triangle whose
perpendicular sides are 9 in. and 12 in. ? 21 and 28 ? 15 and
20 ? 33 and 44 ?
82. The first steamship crossed the Atlantic Ocean in
MDCCCXIX. Tor how many years has it been possible for
Americans to go to Europe in a steamship ?
3
FIG. 6.
MISCELLANEOUS EXERCISES 149
83. Imagine a block of ice 1 yd. in dimensions. How many
square feet are there in all the surfaces ?
84. Imagine the same figure with one cubic foot cut out of
one corner of it. How many square feet in all its surfaces ?
85. Imagine a cubic yard of ice, and suppose a cubic foot of
it to be cut from the middle of one side. How many square
feet in all the surfaces of the solid that is left ?
86. How many square feet in all the surfaces of the solid
that would be left, if the cubic foot were put back in its place
and the cubic foot above it were taken away ?
87. If a box 1 yd. in dimensions were packed f full of
groceries, how many cubic feet of space would be left ?
88. What part of a cubic yard is a cube which is f of a yard
in dimensions ? How many cubic feet are there in it ?
89. A coal dealer bought 1246 tons of coal at $ 4J- a ton, and
sold it for $ 6 a ton. What was his gain on each ton ? On
the whole ?
90. A man bought $ 88^- worth of furniture, paying in
weekly installments of $ 14^ each. In how many weeks did
he pay for the furniture ?
91. A grocer bought strawberries at the rate of 4 boxes for
a quarter, and sold them, at the rate of 3 boxes for a quarter.
How much did he gain on each box ? On a dozen boxes ? On
a gross of boxes ?
92. Mr. Jones worked |- of a day on Monday, f of a day on
Tuesday, and a whole day on Wednesday, on Thursday, and
on Friday. On Saturday he worked % a day. At $ 3 per day,
how much did he earn in the week ?
CHAPTER V
DENOMINATE NUMBERS
1. How many feet equal a yard? How many pints equal
a quart? How many ounces equal a pound?. How was it
decided in these cases how many units of a certain denomina-
tion should make one of the next higher denomination?
SUGGESTION TO TEACHER. Let facts concerning the origin of our systems
of measuring be obtained from encyclopedias and other sources of informa-
tion and brought to the class. Pupils should understand that the value of
a unit in terms of lower denominations is an arbitrary value, varying in
different kinds of measurements.
2. Numbers that show measurements whose values are set-
tled by custom or law are called Denominate Numbers, as 5
bushels, 2 hours, 1 dollar. Denominate numbers that consist
of more than one denomination are called Compound Denominate
Numbers. Write a compound denominate number whose larg-
est denomination is bushels. Hours. Tons. Miles. Acres.
Gallons. Dollars. Meters.
3. The denominations of United States money are mills (m.),
cents (), dimes (d.), dollars ($) and Eagles (E.).
UNITED STATES MONEY
One dollar is the standard
1 eagle = 10 dollars
1 dollar
1 dime = ^ of a dollar
1 cent = jfa of a dollar
1 mill = j^Vo of a dollar
150
DENOMINATE NUMBERS 151
4. Name each denomination of the following : $ 5875.
$10,125. $20,705.
5. Express 3 dollars as cents. As dimes. As eagles.
6. A 10-dollar gold piece is called an Eagle. A 20-dollar
gold piece is called a Double Eagle. A 5-dollar gold piece is
called a Half Eagle. What is the value of a Quarter Eagle ?
7. Name the silver coins. What other coins are there?
8. How much money has a man who has 2 double eagles, an
eagle, 3 half eagles, a quarter eagle, 2 dollars, 3 dimes, 2
nickels, and 3 cents ?
9. CLASS EXERCISE. may name a certain number of
different kinds of coins, and the class may find the amount
of money which their sum equals.
10. Treasury or bank notes are also used as money. If a
man has eight $ 100 bills, seven $ 20 bills, a $ 10 bill, a $ 5
bill, a $ 2 bill, and three $ 1 bills, how much less than $ 1000
has he ?
11. How many mills in a dollar? In a half eagle? Why
is there no coin to represent a mill ?
12. The denominations of liquid measure are gills (gi.),
pints (pt.), quarts (qt.) and gallons (gal.).
LIQUID MEASURE
424
gal. qt. pt. gi.
Over the abbreviation of each denomination above you will find the
number of units that equal a unit of tha.next higher denomination.
Give the table of liquid measure, beginning with the units
of the lowest denomination.
13. Fill the blank in the following table of equivalent values :
1 gal. = 4 qt. = 8 pt. = gi.
SUGGESTION TO TEACHER. Pupils should make actual measurements, so
far as is practicable, in connection with the study of each table, and should
learn to change rapidly units of one denomination into units of another.
152 DENOMINATE NUMBERS
14. Illustrate each of the following statements :
a As 4 gills equal a pint, any number of pints equals 4
times as many gills as pints.
b As 2 pints equal a quart, any number of quarts equals
twice as many pints as quarts.
c As 4 quarts equal a gallon, any number of gallons equals
4 times as many quarts as gallons.
15. Express -f gal. as quarts, qt. as pints, f pt. as gills.
16. Express % gal. as quarts, -- qt. as pints. T 8 T pt. as gills.
17. Express .75 gal. as quarts. .5 qt. as pints. .625 pt. as
gills.
18. Express .375 gal. as quarts. As pints.
19. Express 5|- gal. as quarts. As pints. As gills.
20. Express 1\ gal. as pints.
21. Express 2| gal. as gills.
22. Express 3^ gal. as quarts. 3 gal. and 2 qt. as quarts.
23. How many quarts in 5 gal. 2 qt. ? 7 gal. 1 qt. ?
24. How many pints in 7 qt. 3 pt. ? In 1 gal. 3 pt. ?
25. How many gills in 1 pt. 3 gi. ? 1 qt. 3 gi. ? 3 qt. 1 gi. ?
26. 1 gi. equals what part of 1 qt. 1 pt. ?
1 qt. 1 pt. = 5 pt., which equal 20 gi.
1 gi. = Js of 20 gi.
27. 1 gi. equals what part of 2 qt. 1 pt. ? Of 3 qt. 1 pt. ?
Of 1 gal. 1 pt. ?
28. Illustrate each of the following statements :
a As 4 gills make a pint, any number of gills equals \ as
many pints as gills.
DENOMINATE NUMBEKS 153
b As 2 pints equal 1 quart, any number of pints equals as
many quarts as pints.
c As 4 quarts equal 1 gallon, any number of quarts equals
as many gallons as quarts.
29. Express 32 gi. as pints. As quarts. As gallons.
30. Express 40 pt. as quarts. As gallons.
31. Express 25 gi. as pints. Ans. 6| pt.
32. Express 7 pt. as quarts. 9 qt. as gallons.
33. Express 11 pt. as quarts. As gallons.
34. Express 13 pt. as quarts and pints. Ans. 6 qt. 1 pt.
35. Express 15^ pt. as quarts, pints, and gills.
36. Express 17^ qt. as gallons, quarts, and pints.
37. Express 19 pt. as gallons, quarts, pints, and gills.
38. 1 gi. equals what part of a pint ? Of a quart ? Of a
gallon ?
39. 3 gi. equal what part of a quart ? Of a gallon ?
40. Express If pt. as quarts. As gallons.
41. Express 1 pt. 3 gi. as pints.
42. Express 2 qt. 1 pt. 2 gi. as pints. As quarts. As gallons.
43. Which is greater and how much, 2 gal. 1 qt. 3 pt. or
22 pt. ?
44. Express 1 pt. as a decimal of a quart. 3 qt. as a deci-
mal of a gallon.
45. At 6 cents a quart, how much will a gallon of cider
cost ? 31 gal. ? 4f gal. ? 1 gal. 3 qt. ? 1 pt. ? 3 qt. 1 pt. ?
46. Name several articles that are measured by liquid
measure.
154 DENOMINATE NUMBERS
47. Add:
gal qt pt i We find the sum of the gills to be 7 git 7 &*
equal 1 pt. 3 gi. We place the 3 gi. under the col-
2312 urnn of gills and add the 1 pt. to the number of pints.
7103 The sum of the pints is 3 pt., equal to 1 qt. 1 pt.
4212 The * pt> is P^ aced un der the column of pints, and the
1 qt. is added to the number of quarts. The sum of
14 3 1 3 the quarts is 7 qt., equal to 1 gal. 3 qt. The 3 qt.
are placed under the column of quarts, and the 1 gal.
is added to the number of gallons, making 14 gal. 3 qt. 1 pt. 3 gi.
Add:
gal. qt. pt. gi. gal. qt. pt. gi. gal. qt. pt. gi.
48. 1 3 1 3 49. 5 2 1 1 50. 8 3 3
3302 7313 9302
gal. qt. pt. gi. gal. qt. pt. gi. gal. qt. pt. gi.
51. 6 3 1 2 52. 11 3 1 1 53. 15 2 1 1
7213 5213 11 3 1 3
54. A milkman leaves 25 gal. 3 qt. 1 pt. of milk at one
hotel, and 33 gal. 2 qt. 1 pt. at another. How much does he
leave at both hotels ?
gal. qt. pt. gi.
55. From 7313 Subtract each number in the subtra-
, -i 1 1 2 nen d from the corresponding number in
. the minuend.
2201
56. CLASS EXERCISE. may write a compound denomi-
nate number consisting of gal., qt., pt., and gi. The class may
use it as a minuend, making every number in the subtrahend
less than its corresponding term in the minuend.
gal. qt. pt. gi. As 3 gi. cannot be taken from 1 gi.,
57. From 7111 we reduce * P*- * S 1 - to S ills > which gives
take 2303
5 gi. 5 gi. minus 3 gi. equal 2 gi. As
the 1 pt. has been taken from the column
4202 of P ints and reduced to gills, there are no
pints left in the minuend, from which
pt. are to be taken. As 3 qt. cannot be taken from 1 qt., we reduce
1 gal. to quarts, which, with the 1 qt., equal 5 qt. 3 qt. from 5 qt. leave
2 qt. As 1 gal. has been taken from the column of gallons and reduced
to quarts, only 6 gal. remain. 6 gal. minus 2 gal. equal 4 gal. Hence the
difference is 4 gal. 2 qt. pt. 2 gi.
DENOMINATE NUMBERS 155
Subtract :
gal. qt. pt. gi. gal. qt. pt. gi.
58. 17 3 3 59. 16 3 1 1
11 111 8103
gal. qt. pt. gi. gal. qt. pt. gi.
60. 13 1 1 2 61. 15 2 1
6311 9303
62. 17 gal. 1 qt. 1 pt. of oil were in a tank. 11 gal. 2 qt.
1 pt. were drawn out. How much remained ?
63. Multiply 3 gal. 1 qt. 1 pt. 2 gi. by 9.
gal. qt. pt. gi. 9 times 2 gi. equal 18 gi., which equal 4 pt. 2 gi.
3112 ^ e ^ &* are wr ^ten under the gills. 9 times 1 pt.
p plus the 4 pt. already found equal 13 pt., which
equal 6 qt. 1 pt. 9 times 1 qt. plus the 6 qt. already
30 3 J 2 found equal 15 qt. 15 qt. equal 3 gal. 3 qt. 9 times
3 gal. plus the 3 gal. already found equal 30 gal.
Hence the product is 30 gal. 3 qt. 1 pt. 2 gi.
gal. qt. pt. gi. gal. qt. pt. gi
64. Multiply 5312 65. 10 113
by 6 9_
66. Multiply 7 gal. 2 qt. 1 pt. 3 gi. by 3. By 5. By 7.
67. A milkman sold 99 gal. 3 qt. of milk on Monday. If
he were to sell the same amount every day for a week, how
much milk would he sell ?
68. At 24^ a gallon, how much would he receive for the
milk ? If the whole cost of the milk was $ 135, how much
would he gain ?
69. How many times is 3 gal. 3 qt. 1 pt. contained in 19 gal.
1 qt. 1 pt. ?
Express both dividend and divisor in the same denomination before
dividing.
156 DENOMINATE NUMBERS
70. How many bottles each containing 1 pt. 2 gi. can be
filled from a flask containing 3 gal. ?
71. How many bottles each containing 1 pt. 2 gi. can be
filled from a 6-gallon tank ?
72. Divide 21 gal. 3 qt. 1 pt. 3 gi. by 6.
In dividing denominate numbers, if there is a remainder after dividing,
it is the custom to reduce that remainder to the next lower denomination
instead of writing the quotient as a mixed number. In this way, frac-
tions are avoided in all the denominations of the quotient except the
lowest.
Dividing 21 gal. by 6, we have 3 gal. for the quotient and 3 gal. for the
remainder. 3 gal. or 12 qt. plus 3 qt. equal 15 qt. 15 qt. divided by 6
give 2 qt. for a quotient and 3 qt. for a remainder.
gal. qt. pt. gi. 3 ctf" or 6 Pk P lus ! P fc - e( J ual 7 P*- 7 ?* divided
6 1 21 3 1 3 ky 6 give 1 pt ' for a ^ uotient and 1 P*- for a
' - - - remainder. 1 pt. or 4 gi. plus 3 gi. equal 7 gi.
^" 7 gi. divided by 6 equal 1 gi. Hence the quotient
is 3 gal. 2 qt, 1 pt. 1 gi.
gal. qt. pt. gi. gal. qt. pt. g
73. Divide 2)8 2 1 2 74. 3)6 3 1 2
75. Divide 9 gal. 3 qt. 1 pt. 1 gi. by 4. By 5. By 6. By 8.
76. How many half-pint bottles can be filled from a 10-
gallon can of milk?
77- The denominations of dry measure are pints (pt.),
quarts (qt.), pecks (pk.), and bushels (bu.).
DRY MEASURE
482
bu. pk. qt. pt.
Give the table of dry measure, beginning with the units of
the lowest denomination.
78. Fill the blanks: 1 bu. = pk. = qt. = pt.
79. Give the ratio of 1 pt. to a unit of each denomination
of dry measure.
DENOMINATE NUMBERS 157
80. Express 3 qt. 1 pt. as pints. As quarts. As pecks.
81. Express 2 pk. 5 qt. 1 pt. as pt. As qt. As pk. As bu.
82. Name several articles that are measured by dry measure.
83. At 20^ a peck, how much does a bushel of tomatoes
cost ? 3 qt. 1 pt. ? 5 qt. 1 pt. ?
84. At 25^ a quart, how much does a pint of strawberries
cost ? 3 qt. 1 pt. ? 5 qt. 1 pt. ?
85. At 121 ^ a quart, how much does a peck of potatoes
cost ? 1 pk. 2 qt. ? 1 bu. ?
86. Express -f bu. as pk. % pk. as qt. -J qt. as pt. f bu. as qt.
87. Express f pk. as bu. T 8 T qt. as pk. ^ pt. as qt. |- qt. as bu.
88. Express .125 pk. as qt. .875 bu. as pk. 55 qt. as pt.
89. Express .375 bu. as pk. As qt. As pt.
90. Express 4 qt. as a decimal of a peck. Of a bushel.
91. How many pints in -J of ^y of J of a bushel ?
92. By selling apples at f .40 a peck, Mr. Allen doubled
his money. How much did they cost him per bushel ?
93. Express in pecks 7% of a bushel. Express in quarts
51 % of a peck.
Add:
bu. pk. qt. pt. bu. pk. qt. pt.
94. 20 3 5 1 95. 21 3 1 1
4111 33 201
5261 48 371
bu. pk. qt. pt. bu. pk. qt. pt.
96. 6 2 5 1 97. 10 3 6 1
3340 8151
6171 12 240
158 DENOMINATE NUMBEKS
Subtract :
bu. pk. qt. pt. bu. pk. qt. pt.
98. 18 3 2 1 99. 40 1 5
14 1 6 1 17 3 2 1
bu. pk. qt. pt. bu. pk. qt. pt.
100. 8360 101. 19 1 2 1
1241 14 361
Multiply :
bu. pk. qt. pt. bu. pk. qt. pt.
102. 4261 103. 8371
5 7
bu. pk. qt. pt. bu. pk. qt. pt.
104. 7151 105. 8371
6 8
106. Multiply 6 bu. 3 pk. 3 qt. 1 pt. by 4. By 6. By 8.
107. A grocer has 3 bins, each holding 4 bu. 3 pk. 2 qt.
of potatoes. How much do they all hold ?
108. How much wheat is there in 10 bins, if each bin con-
tains 40 bu. 2 pk. 6 qt. ?
109. At 5 1 a quart, how much will a bushel of walnuts cost ?
bu. pk. qt. pt. bu. pk. qt. pt.
110. Divide 5)6 1 7 1 111. 6)8 2 5 1
112. Divide 9 bu. 2 pk. 7 qt. 1 pt. by 2. By 3. By 4.
113. How many boxes containing 2 bu. 3 pk. of sawdust
can be emptied into a bin which will hold 13 bu. 3 pk. ?
114. Place a cubic centimeter upon each of the corners of
the upper surface of a cube that holds a liter, and find how
many square centimeters there are in the surface of the figure
thus formed.
DENOMINATE NUMBERS 159
115. The standard unit of metric measure of capacity is
the Liter, equal to about .9 of a quart dry measure and 1.05
quarts liquid measure.
METRIC MEASURE OF CAPACITY
1 kiloliter (Kl.) = 1000 liters
1 hectoliter (HI.) = 100 liters
1 decaliter (Dl.) = 10 liters
1 liter (1.)
1 deciliter (dl.) = .1 of a liter
1 centiliter (cl.) = .01 of a liter
1 milliliter (ml.) = .001 of a liter
Write table of equivalents.
1 Kl. = 10 HI. = Dl. = 1. = dl. = cl. = ml.
116. Learn to give quickly, forward and backward, the de-
nominations of this table, and the meaning of each prefix to
the word "liter."
To help you remember the value of the units observe that D, H, and
K, the abbreviations for Deca, Hecto, and Kilo, follow one another in
alphabetical order.
117. Give the meaning of the following prefixes: Kilo,
milli, Hecto, centi, Deca, deci. Of d, H, D, c, K.
118. Bead as a decimal 1235.576 1. Give the denomination
of each figure.
119. How many centiliters in 5.37 1. ? In 8.4 1. ? 10.251.?
6.875 1. ?
120. How are liters reduced to centiliters ?
121. Express 7 kiloliters as liters. As hectoliters. As
centiliters. As milliliters.
122. Express 2 HI. 5 Dl.'as 1. As cl. As Kl. As dl.
123. Reduce 2345.248 1. to units of each of the other de-
nominations.
124. CLASS EXERCISE. may name a number of liters,
and the class may reduce them to dl. To cl. To ml.
160 DENOMINATE NUMBERS
125. Write in one number, 5 Kl. 2 HI. 5 Dl. 2 1. dl. 7 ml.
126. Write and add : 3 Kl. HI. 2 Dl. 5 1. dl. 2 cl. 7 ml.
2 HI. 4 Dl. 1. 2 dl. 7 cl. 9 Kl. 3 HI. 2 Dl. 6 1. 5 dl. 2 cl. 8 ml.
127. From 8 Kl. 2 HI. 7 Dl. 4 1. 6 dl. 2 cl. 9 ml.
take 6 HI. 4 Dl. 2 1. 3 dl. 8 cl. 5 ml.
128. 825.346 1. - 27.59 1. = ?
129. Multiply 125.275 1. by 5. By 8. By 12.
130. Multiply 341.626 1. by 10. By 100. By 1000.
131. Divide 239.268 1. by 4. By 6. By 12.
132. Under which system of denominate numbers is it
easier to add, subtract, multiply, and divide, the metric system
or the English system ? Why ?
133. How many liters in 7% of 132.5 1. ? Of 178.7 1. ?
134. 17 1. +3% of 17 1. = how many liters? How many
dl.? How many cl.?
135. How much will 7.5 1. of wine cost at $ 1.25 a liter ?
136. What is the cost of a Dl. of wheat, at $ 7.25 per HI.?
137. What is the cost of a hogshead of wine containing
225 1. at $ .15 per liter ?
138. What is the cost of 25 1. of vinegar at $ 15 per HI.?
139. How much wheat is contained in 125 sacks, each hold-
ing 1 HI. 2 Dl. ?
140. The denominations of avoirdupois weight are ounces
(oz.), pounds (lb.), hundredweights (cwt.), and tons (T.).
AVOIRDUPOIS WEIGHT
20 100 16
T. cwt. lb. oz.
Fill out the following table of equivalents.
1 T. = cwt. = lb. = oz.
DENOMINATE NUMBERS 161
141. 1 oz. is what part of a pound ? Of a hundredweight ?
Of a ton ?
142. Express 5 Ib. 8 oz. as ounces. As pounds. As hun-
dredweights. As tons.
143. Express 3 T. 10 cwt. 25 Ib. 12 oz. in each denomination
of avoirdupois weight.
144. Express -fa T. as cwt. f cwt. as Ib.
145. Express f oz. as Ib. -f Ib. as cwt. f Ib. as cwt.
146. Express .7 T. as cwt. .17 cwt. as Ib. .125 Ib. as oz.
.75 T. as Ib.
147. Express .625 T. as cwt. As Ib. As oz.
148. Express 15 Ib. as a decimal of a hundredweight.
149. At 20^ a pound, how much will 2 Ib. and 8 oz. of
butter cost ? 3 Ib. 4 oz. ? 5 Ib. ? 12 oz. ? 7 Ib. 2 oz. ?
150. How many pounds in 21 T. ? In 3% of a ton ?
151. A farmer brought a ton of hay to market and sold
25% of it. How many pounds had he left ?
152. How many cwt. in 5% of a ton ? In 45% ?
153. 1 Ib. f of a pound = how many ounces ?
Add:
T.
cwt.
ib.
oz.
T.
cwt.
Ib.
oz.
154.
7
15
75
8
155. 10
19
67
5
4
12
55
12
12
14
25
13
6
17
80
15
15
16
84
11
T.
Ib.
oz.
T.
Ib.
oz.
156.
8
425
7
157. 16
875
10
9
375
8
14
985
11
7
425
5
6
435
13
HORN. GRAM. SCH. AR. 11
162 DENOMINATE NUMBERS
T. cwt. Ib. oz. T. cwt. Ib. oz.
158. From 9 3 30 4 159. 10 5
take 27 41 7 6728
T.
Ib.
oz.
T.
Ib.
oz.
160.
From
8
201
8
161. 18
700
12
take
4
175
12
4
900
8
162. From 30 T. 800 Ib. of hay there were sold 7 T. and
900 Ib. How much was left ?
163. Mrs. Harris brought 20 Ib. 8 oz. of butter to market,
and sold 18 Ib. and 12 oz. How much remained unsold ?
164. Name articles that are weighed by avoirdupois weight.
165. John weighs 87 Ib., Thomas 92 Ib., William 97f Ib.
How much do they all weigh ? Give answer in pounds and
ounces.
SUGGESTION FOR CLASS EXERCISE. Find sums and differences of
weights of pupils.
T. cwt. Ib. oz. T. cwt. Ib. oz.
166. Multiply 5 10 40 10 167. 15 15 75 15
by 8 12
168. Multiply 15 T. 7 cwt. 25 Ib. 8 oz. by 3. By 5. By 8.
By 9.
Divide :
T. cwt. Ib. oz. T. cwt. Ib. oz.
169. 4)9 15 20 8 170. 5)16 14 50 10
171. Divide 20 T. 12 cwt. 48 Ib. 12 oz. by 6. By 8. By 12.
172. One Christmas day 40 T. of coal were equally distrib-
uted among 11 poor families. How many tons, hundred-
weights, and pounds did each receive ?
173. Formerly 2240 Ib. were considered a ton, and that
standard is sometimes used now. In that case the ton was
called a "long ton." How many pounds in 4J long tons ? In
of along ton?
DENOMINATE NUMBERS 163
174. A coal dealer buys 150 T. of coal, 2240 Ib. each, at
$ 4.50 per ton. He sells it at $ 4.75 per ton, giving 2000 Ib.
per ton. How much does he gain ?
175. The standard unit of metric measure of weight is a
Gram, equal to about 15^- Troy grains.
METRIC MEASURE OF WEIGHT
1 kilogram (Kg.) = 1000 grams
1 hectogram (Hg.) = 100 grams
1 decagram (Dg.) = 10 grams
1 gram (g.)
1 decigram (dg.) = .1 of a gram
1 centigram (eg.) = .01 of a gram
1 milligram (mg.) = .001 of a gram
Fill blanks :
1 Kg. = Hg. = Dg. = g. = dg. = eg. = mg.
176. The weight of a cubic centimeter of water is a gram.
How many grams does a liter of water weigh?
177. What part of a kilogram is a decagram? Decigram?
Milligram ? Hectogram ? Centigram ?
178. Name each denomination of the expression 1978.347 g.
179. Write in one number, making the gram the unit 4 Kg.
7 Hg. 6 Dg. 5 g. 2 dg. 6 eg. 3 mg.
180. Express 75 Kg. as grams. As eg. As mg. As Dg.
As dg. As Hg.
181. Express 186 eg. as grams. As mg. As Dg. As dg.
As Hg. As Kg.
182. Add 325 g., 55 Kg., 75 Dg.
183. How many grams in 20% of 425 g. ?
184. 126 g. - 4% of 126 g. equal how many Dg. ? dg. ?
eg. ? mg. ?
185. A nickel weighs 5 g. $5.00 in nickels weigh how
many kilograms?
164 DENOMINATE NUMBERS
186. A kilogram is equal to about 21 Ib. Find the approxi-
mate cost of a kilogram of a drug that costs $ .60 a pound.
187. What is the cost of 2242 g. of coffee at $ .60 a kilogram ?
188. If a kilogram of wool costs $ 1.75, how much will 6.5
kilograms cost ?
189. The denominations of time measure are seconds (sec.),
minutes (min.), hours (hr.), days (da.), and years (yr.).
TIME MEASURE
100 365 24 60 60
century yr. da. hr. min. sec.
Till out the following table of equivalents :
1 century = yr. = da. = hr. = min. = sec.
190. Find the ratio of one hour to a unit of each denomina-
tion of time measure.
191. Express 2 hr. 10 min. as sec. As min. As hr.
192. Express f yr. as da. T 5 g- da. as hr. f hr. as min.
193. Express ^ of a year as da. As hr. As min. As sec.
194. Express |f min. as hr. 2| hr. as da. 100 sec. as min.
195. Express .12 yr. as da. .33^ da. as hr. .75 hr. as min.
.66| min. as sec. .17 centuries as yr.
196. Express 108 sec. as a decimal of a minute. Of an hour.
197. How many minutes in 5% of an hour? In 65%? 95%?
198. 1 hr. + 15% of an hour = how many minutes ?
199. At 6% now much interest will $300 gain in 1 yr. ? In
1| yr. ? 3 yr. 6 mo. ? 2 yr. 9 mo. ?
200. Express in years, 1 yr. 1 mo. 15 da. Find the interest
of 400 for that time at 6 % At 8 % At 5 % .
DENOMINATE NUMBERS 165
201. Usually every fourth year has 366 da., and is called a
leap year. In leap year the month of February, which usually
has 28 da., has 29 da. How many hours in February of a leap
year ?
202. Every year whose number is divisible by 4 is a leap
year, excepting the years whose number ends in two ciphers, as
1500, 1700, 1800. Such years are leap years only if their
number is divisible by 400, as 1200, 1600, 2000. Which of the
following are leap years ?
1848 1862 1892 1900 1904 2000 2108 2200 1000
203. At a dollar a day, how much could a man earn, working
6 da. in a week, in the month of February, 1896, which began
on Saturday ? How much in February, 1898 ?
204. A $ 1.50 per day, how much could a man earn in
February of a leap year, in which the 1st of February fell on
Sunday ?
205. A man's heart beats at the rate of about 72 beats in a
minute. At that rate how many times does it beat in an hour ?
In a day of 24 hr. ? In a common year ?
206. At that rate how many times would a man's heart beat
in a lifetime of 80 yr., - of which are leap years ?
207. When a person born in 1883 is 20 yr. old, how many
years of his life have been leap years ?
208. The time in which the earth passes once around the sun
is 365 da. 5 hr. 48 min. 46 sec. That is how much more than
365 da. ? To how much will the extra time amount in 4 yr. ?
How much does it lack of being a whole day ?
209. In adding the whole day to every fourth year or leap
year, how many more minutes and seconds are added to the
year than rightly belong to it ?
166 DENOMINATE NUMBERS
210. Find how nearly the error caused in a century by this
arrangement is corrected by omitting the extra day in February
at the end of the century.
211. Find the amount of error at the end of the fourth cen-
tury, and see how nearly it is corrected by restoring the extra
day in February of that year.
212. Henry rose at 5.50 A.M. and went to bed at 8.20 P.M.
How long was his day ?
213. CLASS EXERCISE. may give a time for rising
and a time for going to bed, and the class may find the length
of the included day.
214. How many hours were there in the year 1800 ?
" Thirty days hath September,
April, June, and November. "
215. Learn the above rhyme and remember the fact that all
the rest of the months except February have 31 days.
216. Write the names of the months in order, beginning with
January, using abbreviations. Write opposite the name of
each month the number of days it contains.
217. Find the number of days from May 1st to June 7th.
There are 30 days after the 1st in May, which, with the 7 days in June,
make 37 days.
Find the number of days between the following dates :
218. May 7, 1896, July 4, 1896.
219. Jan. 1, 1900, Mar. 1, 1900.
220. Sept. 28, 1899, Nov. 5, 1899.
221. Dec. 15, 1899, Jan. 31, 1900.
222. How many days from to-day until next Christmas?
Next 4th of July ?
DENOMINATE NUMBERS 167
'
223. CLASS EXERCISE. may give the date of his next
birthday, and the class may find the number of days intervening.
224. Thirty days from the 4th of July, 1876, was what date ?
225. What was the date 60 days after Oct. 14th, 1492?
Christmas, 1897 ?
226. On the 17th day of June, Mr. Herbert borrowed from
a bank $ 100 to be paid in 60 days. When was it due ?
227. What will be the date 30 days after to-day ? 60 da. ?
228. What will be the date 33 days from next Monday ?
63 da. ? 93 da. ? 105 da. ?
229. What was the date 30 days before the first of May,
1891 ? 60 days before the 1st of March, 1892 ?
230. How many days since the 1st of January of this year ?
231. CLASS EXERCISE. - may give the date of his last
birthday, and the class may find how many days have passed
since then.
Add
da.
hr.
min.
sec.
da.
hr.
min.
sec.
232.
17
20
30
40
233. 19
19
45
30
20
16
40
10
25
20
15
30
40
18
50
20
16
12
30
30
Add:
da. hr. min. sec. da. hr. min. sec.
234. 19 14 30 45 235. 15 2 30 30
70 20 45 15 30 10 45 25
6 18 15 15 40 6 15 35
4 2 30 30 9 8 15 15
236. Mr. Cox earns $ 2.00 for each day of 10 hours that he
works. On Monday he worked 8 hr. 30 min. ; on Tuesday, 9
hr. 10 min. ; Wednesday, 7 hr. 40 min. ; Thursday, 8 hr. 30 min. ;
Friday, 7 hr. 50 min ; Saturday, 4 hr. 30 min. How much did
he earn in that week ?
168 DENOMINATE NUMBERS
237. At $ 2.00 per day of 8 hours, how much would Mr. Cox
have earned ?
Find the differences :
da. hr. min. sec. da. hr. min. sec.
238. 47 18 2 10 239. 62 10 30 15
25 20 1 40 48 20 19 45
240. Which is the 7th month ? 12th ? 3d ? 5th ?
241. Which month is February? September? June?
November ?
242. Bead the following dates : 9/5/99. 10/5/98. 7/4/76.
243. Find the difference of time between March 5, 1898, and
Oct. 21, 1902.
Write as below and subtract :
1902 10 21
1898 3 5
In finding the difference between two dates in years, months, and days,
we assume that 30 days = a month.
American Authors
Ralph Waldo Emerson, born May 25, 1803 died April 27, 1882.
John G. Whittier, born Dec. 17, 1807 died April 27, 1892.
Henry W. Longfellow, born Feb. 27, 1807 died March 24, 1882.
James Russell Lowell, born Feb. 22, 1819 died Aug. 12, 1891.
English Authors
Alfred Tennyson, born Aug. 6, 1809 died Oct. 6, 1892.
Charles Dickens, born Feb. 7, 1812 died June 9, 1870.
244. Find the number of years, months, and days since each
of the authors mentioned above was born. Since each died.
245. Find the age of each author at his death.
246. Find the time between the birth of Emerson and that
of each of the other authors.
DENOMINATE NUMBERS 169
247. Find the time between the death of Dickens and that
of each of the other authors.
da. hr. min. sec. da. hr. min. sec.
248. Multiply 7 18 20 10 249. 11 15 8 30
by 7 9
250. Multiply 5 da. 10 hr. 20 min. 30 sec. by 5. By 6. By 8.
251. If you spend 4 hr. 30 min. in school every day, how
many hours and minutes do you spend in a school week ? In
a school month of 4 wk. ? In a school year of 10 mo. ?
Divide :
da. hr. min. sec. da. hr. min. sec.
252. 4)21 16 2 40 253. 6)13 13 13 30
254. Divide 15 da. 12 hr. 40 min. 30 sec. by 5. By 6. By 8.
255. The denominations of linear measure are inches (in.),
feet (ft.), yards (yd.), rods (rd.), and miles (mi.).
LINEAR OR LONG MEASURE
320 5| 3 12
mi. rd. yd. ft. in.
Fill out the table of equivalent values.
1 mi. = rd. = yd. = ft. = in.
256. How many inches in 2 yd. 1 ft. 7 in. ? 5 yd. 3 ft. 7 in. ?
257. Express 2 yd. 1 ft. 6 in. as in. As ft. As yd.
258. Express 2 mi. 20 rd. as mi. As rd. As yd. As ft.
259. 1 ft. is what part of a yard ? Of a rod ? Of a mile ?
Reduce complex fractions to simple fractions.
260. Express 1 ft. 6 in. as yd. As rd. As mi.
261. Express 1 yd. 1 ft. 9 in. as yd. As rd. As mi.
262. Express T 5 ^ mi. as rd. T 8 T rd. as yd. % yd. as ft. -| ft.
as in. rd. as ft.
170 DENOMINATE NUMBERS
263. Express f in. as ft. ^ ft. as yd. | yd. as rd. 11$ rd.
as mi. If ft. as rd. If in. as ft. 4 ft. as yd.
264. Express .875 mi. as rd. 3.6 rd. as yd. 5.5 yd. as ft.
.9 ft. as in. 1.66| yd. as ft. .64 rd. as yd.
265. Express 7.2 in. as a decimal of a foot. As a decimal
of a yard.
266. Express 115.5 ft. in yd. In rd. In mi.
267. How many rods in 5% of a mile ? In 15% ? 35% ?
Add:
yd. ft. in. yd. ft. in.
268. 2 1 Hi- 269. 6 2 7
5 1 2j 3 2 llf
mi. rd. yd. ft. in. mi. rd. yd. ft. in.
270. 6 200 2 1 10 271. 10 20 1 2 10
8 120 1 2 6 7 300 1 1 8
mi. rd. yd. ft. in. mi. rd. yd. ft. in.
272. 8 150 4 2 11 273. 10 180 4 2 3
4 100 1 1 5 7 40 2 1 11
mi. rd. yd. ft. in. mi. rd. yd. ft. in.
274. 16 200 3 2 4 275. 18 2 4 2 10
14 150 1 1 10 5 319 2 1 7
mi. rd. yd. ft. in.
276. 15 3223
10 319 1 1 9
277. Mr. Smith's lot is 1 yd. 1 ft. shorter than Mr. Brown's
lot, which is 30 yd. long. How long is Mr. Smith's lot ?
278. An elm tree is 32 ft. 9 in. high. How high is a fir
tree that is 6 ft. 10 in. less in height ?
DENOMINATE NUMBERS 171
279. Find the difference between the height of John, who
is 5 ft. 3 in. tall, and his sister, who is 3 ft. 11 in.
A B c 28 - ^ ^ were ^Of miles from A to
C, and 5J- miles from A to J5, how far
would it be from B to C ?
281. If it were 20 mi. 20 rd. from A to C, and 5 mi. 80 rd.
from A to J3, how far would it be from BtoC?
Multiply :
yd. ft. in. mi. rd. yd. ft. in.
282. 312 283. 52109
3 4
mi. rd. yd. ft. in. mi. rd. yd. ft. in.
284. 7 80 2 6 285. 10 160 4 1 11
2 7
286. Multiply 2 mi. 240 rd. 3 yd. 5 ft. 6 in. by 2. By 3.
287. How long is the perimeter of a regular pentagon, each
of whose sides is 3 yd. 1 ft. 9 in. long ?
288. A summer house is built in the shape of a regular
hexagon, each side being 2 yd. 1 ft. 6 in. What is the entire
distance around it ?
289. How long is the edge of the border of a flower bed in
the shape of an octagon, if each side is 1 yd. 1 ft. 8 in. long ?
290. Divide 3 mi. 16 rd. 2 yd. 1 ft. 8 in. by 2. By 3. By 4.
291. Divide 19 mi. 10 rd. 5 yd. 2 ft. 6 in. by 5. By 6. By 7.
292. Divide 7 mi. 160 rd. 4 yd. 1 ft. 8 in. by 2. By 3. By 4.
293. Henry measured a cornstalk and found it to be 5 ft.
6 in. long. How many such stalks laid in a continuous line
would it take to extend a mile ?
172 DENOMINATE NUMBERS
294. Harry has two dogs, Don Quixote and Sancho Panza.
Don measures 4 ft. 3 in. from the end of his nose to the tip of
his tail, weighs 77 Ib. 12 oz., and is 3 yr. 7 mo. 9 da. old.
Sancho measures 3 ft. 8 in., weighs 53 Ib. 14 oz., and is 2 yr.
11 mo. 28 da. old. Find the difference of the lengths of the
dogs. Of their weights. Of their ages.
295. Harry's dog, Sancho Panza, chased a rabbit 40 yd. and
then gave up the chase. The rabbit had 10 yd. the start of the
dog and ran twice as fast as he ran. How far apart were the
animals when Sancho Panza gave up the chase ? Represent.
296. If from a string 2 yd. 2 ft. long, 2.5 ft. is broken off at
one end, and 3.7 ft. at the other, how long a string is left ?
297. The denominations of square measure are square inches
(sq. in.), square feet (sq. ft.), square yards (sq. yd.), square
rods (sq. rd.), acres (A.), and square miles (sq. mi.).
SURFACE OR SQUARE MEASURE
640 160 30 9 144
sq. mi. A. sq. rd. sq. yd. sq. ft. sq. in.
Write a table of equivalent values.
1 sq. mi. = A. = sq. rd. = sq. yd. = sq. ft. = sq. in.
SUGGESTION TO TEACHER. Let a diagram of the square rod be drawn on
the floor, a square yard in one corner of the square rod being subdivided
into square feet, and one of the square feet into inches, so that literally each
square unit may be a part of the units of higher denominations. Let
pupils make many practical problems upon the figures.
298. 1 sq. ft. equals what part of a sq. yd. ? Of a sq. rd. ?
299. How many square feet in 2 sq. rd. ? 3 sq. rd. ? 5 sq.
rd. ? 7 sq. rd. ? 10 sq. rd. ? 12 sq. rd. ?
300. How many square inches in 5 sq. ft. 60 sq. in. ? In
4 sq. ft. 20 sq. in. ? In 2 sq. ft. 80 sq. in. ?
301. How many square feet in 4 sq. yd. 6 sq. ft.? In 20
sq. yd. 7 sq. ft. ? In 18 sq. yd. 5 sq. ft. ?
DENOMINATE NUMBERS 173
302. How many square yards in 8 sq. rd. 15 sq. yd. ? In 10
sq. rd. 4f sq. yd. ? In 24 sq. rd. 19 sq. yd. ?
303. Express 1 sq. mi. 320 A. 80 sq. rd. in sq. rd. In A.
In sq. mi.
304. Express 10 sq. yd. 7 sq. ft. 72 sq. in. in sq. in. In sq.
ft. In sq. yd.
305. Express 2 sq. rd. 15 sq. yd. in sq. yd. In sq. rd. In
sq. ft. In A.
306. Express ^ sq. mi. as A. -^ A. as sq. rd.
307. Express T 8 T sq. rd. as sq. yd. ^ sq. yd. as sq. ft. -ff sq.
ft. as sq. in. - 1 / sq. in. as sq. ft.
308. Express .625 sq. mi. as A. .375 A. as sq. rd. .48 sq. rd.
as sq. yd. .175 sq. yd. as sq. ft. .7 sq. ft. as sq. in. 4.32 sq.
in. as sq. ft. .18 sq. ft. as sq. yd.
309. Express 345.6 sq. in. as a decimal of a square foot.
Add:
sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in.
310. 21 2 100 311. 36 3 70
786 15 7 60
16 5 40 20 7 20
Find difference :
sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in
312. 17 8 85 313. 21 6 100
4 3 75 6 8 75
Multiply :
sq. yd. sq. ft. sq. in. sq. yd. sq. ft. sq. in.
314. 24 3 140 315. 16 4 96
6 7
316. Multiply 2 A. 40 sq. rd. 10 sq. yd. 4 sq. ft. 20 sq. in.
by 3. By 4. By 5.
174 DENOMINATE NUMBERS
317. How many acres in 5% of a square mile ? 7% ? 15% ?
318. 1 sq. ft. 37|% of a square foot = how many square
inches ?
319. Divide by 3, 48 sq. yd. 7 sq. ft. 4 sq. in.
320. Divide by 5, 25 sq. yd. 6 sq. ft. 2 sq. in.
321. Divide 20 A. 80 sq. rd. 20 sq. yd. 4 sq. ft. 72 sq. in. by
4. By 8.. By 6.
322. How many square feet in a square f of a foot in
dimensions ? How many square inches ?
323. How long is the perimeter of a square f of a foot in
dimensions ? What fraction of a square foot is its area ? How
many square inches in its area ?
324. Mr. Gilbert owns 400 A. 120 sq. rd. of land in Gibson
County, 225 A. and 10 sq. rd. in Warrick County, and 14 A.
40 sq. rd. in Vanderburgh County. How much does he own in
those counties ?
325. A farmer had 80 A. 50 sq. rd. of land. After selling
30 A. 10 sq. rd., how much had he left ?
326. Mr. Carter owns three times as much land as his
cousin, who owns 120 A. 80 sq. rd. How much land do both
own?
327. A garden 180 ft. long. 150 ft. wide is surrounded by a
tight board fence 6 ft. high. How much will it cost to paint
the fence on both sides @ 12 ^ per square yard ?
SUGGESTION TO TEACHER. Pupils who cannot imagine the conditions
of this problem may be required to inclose a surface on their desks by
a strip of paper folded so that its divisions represent the parts of the
fence.
328. From each corner of a square, a side of which is 2 ft.
5 in., a square measuring 5 in. on a side is cut out. Represent
and find the area of the remainder of the figure.
DENOMINATE NUMBERS 175
329. Find the area of the walls of a room 12 ft. long, 10 ft,
wide, 8 ft. high.
Find the areas of the walls and ceiling of rooms of the
following dimensions, and the cost of plastering them at 20
cents a square yard, no allowance being made for openings.
a b
Length Width Height Length Width Height
ft. ft. ft. ft. ft. ft.
330. 20 18 10 40 30 12
331. 30 25 9 21 20 9
332. 25 21 9 30 18 8
333. 18 15 8 4 20 9
334. 15 12 8 16 15 8
335. Which is greater, a rectangle 12 in. by 12 in. or a
rectangle 16 in. by 9 in.? What is the difference in the length
of their perimeters ?
336. Give dimensions of several rectangles, each equal to a
square foot. Compare the lengths of their perimeters.
337. If two rectangles have equal areas but different shapes,
which will have the longer perimeter, the one which is more
nearly square or the other ? Illustrate.
338. Find the area of a square whose perimeter is 24 in.
28 in. 36 in. 40 in.
339. Find the perimeter of a square whose area is 9 sq. in.
340. The denominations of cubic measure are cubic inches
(cu. in.), cubic feet (cu. ft.), and cubic yards (cu. yd.).
CUBIC MEASURE
27 1728
cu. yd. cu. ft. cu. in.
Write table of equivalent values.
1 cu. yd. = cu. ft. = cu. in.
176 DENOMINATE NUMBERS
341. How many cubic inches in 5 cu. ft. 192 cu. in. ? In
1 cu. yd. 624 cu. in. ?
342. How many cubic inches in .875 cu. ft.? In .625 cu. ft.?
343. John may draw a square yard on the floor in a corner
of the room. How many blocks of ice 1 foot in dimensions
would it take to cover that square yard ?
344. If another layer of cubic feet of ice were laid upon the
first, how many cubic feet of ice would there be ? How high
would the ice be piled ?
345. If a third layer of cubic feet of ice were placed upon
the other two, how many cubic feet of ice would there be ?
346. What name is given to a solid which is 3 ft. long, 3 ft.
wide, and 3 ft. high ?
347. In the square yard which John drew William may set
a yard stick upright at that corner which is not against a wall.
Two other boys may place sticks in such a position that a cubic
yard is outlined in the corner of the room.
SUGGESTIONS TO TEACHER. Devote a certain space in the room to the
imaginary cubic yard. See that every member of the class images a cubic
yard in that particular place. Let pupils show divisions of the cube by
outlining them with their hands in the space devoted to it.
Let pupils model before the class with inch cubes the figures given in
the following exercises.
348. Image a cubic yard with one cubic foot cut from the
upper layer at a corner that is not against a wall. Model the
figure. Outline in the cubic yard in the corner the part taken
out. What is the ratio of the part taken out to the whole cubic
yard ? What is the ratio of the part taken out to the part left ?
349. Take two more cubic feet from the upper layer, one on
each side of the vacant space. Model. Outline in the cubic
yard. Tell the ratio of the vacant space to the whole cubic
yard. Of the vacant space to the filled space.
DENOMINATE NUMBERS 177
350. Take out the two cubic feet that were directly under
the cubic foot first removed. Model. Outline. Find the ratio
of the vacant space to the cubic yard. Of the filled space to
the cubic yard. Of the vacant space to the filled space.
351. Restore the whole cubic yard. Take away the middle
cubic foot on each side of the upper layer. Model. Outline.
Tell the ratio of the vacant space to the cubic yard. Of the
filled space to the cubic yard. Of the vacant space to the filled
space.
352. CLASS EXERCISE; may give directions for taking
away parts of the cubic yard. The class may tell the ratios
of the spaces to one another. Some members may model the
figures, and some one else may show the outline of the space
imaged as vacant in the cubic yard.
353. How many feet in the sum of all the edges of a cube 1
yd. in dimensions ?
354. How much will a cubic yard of building stone cost at
$ 2.50 a cubic foot ?
355. How many cubic feet in a right prism 6 ft. long, 2 ft.
wide, and 1 ft. high ? Model the prism.
356. How many cubic feet in a right prism,
a 4 ft. long, 2 ft. wide, 2 ft. high ?
b 5 ft. long, 3 ft. wide, 1 ft. high ?
c 8 ft. long, 2 ft. wide, 2 ft. high ?
d 10 ft. long, 4 ft. wide, 2 ft. high ?
357. How many cubic feet in a tank 14 ft. long, 10 ft. wide ;
5 ft. high ?
358. How many cubic yards in a wall 81 ft. long, 3 ft. thick,
and 9 ft. high ? In a wall 30 ft. long, 6 ft. high, and 3 ft. thick ?
HORN. GRAM. SCH. AR. 12
178 DENOMINATE NUMBERS
359. A monument is in the shape of a right prism, 7 ft.
long, 4 ft. wide, and 3 ft. high. How much will it cost at
$ 3.50 per cubic foot ?
360. If the engraving on it costs $ 62.50, what will be the
entire cost of the monument ?
361. If a cake 3 in. long, 3 in. wide, and 3 in. high, has
icing all over it except on the under side, how many square
inches of icing has it ?
362. To cut the cake into inch cubes, how many cuts would
be necessary ? How many inch cubes would there be ?
363. How many of the cubes would have icing on three
sides ? On two sides ? On one side ? On no side ?
364. How many inch cubes can be placed on a square foot ?
How many layers of those cubes would it take to make a cubic
foot ? How many cubic inches in a cubic foot ? What is the
ratio of 1 cu. in. to a cubic foot ?
365. Imagine a cubic foot of marble with 1 cu. in. cut from
each of the upper corners. What would be the ratio of the
part cut out to the part left ?
366. Which is greater, a right prism 12 in. by 12 in. by 12
in., or one 24 in. by 12 in. by 6 in. ? Compare their surfaces.
367. Give dimensions of several right prisms, each of which
equals a cubic foot. Compare their surfaces.
368. How many cubic feet in a stick of timber 12 in. wide,
9 in. thick, and 24 ft. long?
369. How many cubic feet in a cistern 5 ft. square and 6 ft.
deep ? How many cubic inches ? How many gallons will the
cistern hold ? (231 cu. in. = 1 gal.)
Add:
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
370. 5 10 1700 371. 3 5 1400
4 8 129 10 24 300
DENOMINATE NUMBERS 179
Add:
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
372. 9 11 1720 373. 4 15 1600
6 18 10 8 20 200
Subtract :
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
374. 120 13 1700 375. 41 10 1634
65 15 1125 25 18 1507
Subtract :
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
376. 81 3 208 377. 16 4 800
40 20 125 4 21 525
Multiply :
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
378. 20" 5 1160 379. 15 10 989
3 4
380. Multiply 2 cu. yd. 20 cu. ft. 1000 cu. in. by 2. By 4.
Divide :
cu. yd. cu. ft. cu. in. cu. yd. cu. ft. cu. in.
381. 5) 6 20 72 382. 8) 10 4 36
383. Divide 10 cu. yd. 15 cu. ft. 180 cu. in. by 3. By 5.
384. Imagine a cubic rod of marble. Why do we have no
such measurement as a cubic mile ?
385. A pile of wood 8 ft. long, 4 ft. wide, and 4 ft. high is a
cord of wood. How many cubic feet in a cord of wood ?
386. Bepresent a cord of wood by drawing or by placing
blocks or toothpicks.
387. How many cords in a wood pile 16 ft. long, 8 ft. wide,
and 8 ft. high?
180 DENOMINATE NUMBERS
388. At $ 5.00 per cord, what is the value of a pile of wood
20 ft. long, 4 ft. wide, and 4 ft. high ? Of a pile 18 ft. long,
8 ft. wide, and 8 ft. high ?
389. The standard unit of metric linear measure is a Meter,
which is 39.37 in. This length was obtained by calculating
one ten-millionth of the distance from the equator to a pole of
the earth. A kilometer is about f of a mile.
METRIC LINEAR MEASURE
1 kilometer (Km.) = 1000 meters
1 hectometer (Hrn.) = 100 meters
1 decameter (Dm.) = 10 meters
1 meter (m.)
1 decimeter (dm.) = .1 of a meter
1 centimeter (cm.) = .01 of a meter
1 millimeter (mm.) = .001 of a meter
"Write table of equivalents :
1 Km. = Hrn. = Dm. = m. = dm. = cm. = mm.
390. Express 42 m. as centimeters. As decimeters. As
decameters. As hectometers.
391. Express 375 m. as kilometers. As decameters. As
decimeters. As millimeters.
392. Express 4287 m. as kilometers. As decimeters. As
hectometers.
393. How many centimeters in 11% of 12 m. ? Of 25 dm. ?
394. 7 % of 192 m. = how many meters ? Decameters ?
SUGGESTION TO TEACHER. Let pupils find in meters and decimals the
length and width of room. Length of blackboards. Length of diagonal
of room or blackboard. Heights of pupils.
395. About how many inches in a decameter ? In a deci-
meter ?
396. Find approximately the number of inches in a kilo-
meter. In 4 Hm. In 12 Dm. In 7 dm.
DENOMINATE NUMBERS 181
397. What is the cost of 12 m. of cloth at $.75 per meter ?
Is the cloth cheaper or dearer than at $ .75 per yard ?
Explain.
398. At the rate of 36 Km. per hour, how far will a train run
in 3 hr. 30 min. ?
399. What is the value of a decameter of silk at $ 1.65 per
meter ?
400. How many centimeters long is the perimeter of a regu-
lar octagon, one side of which is 8 mm. ?
401. Find the length in decimeters of one side of a regular
pentagon whose perimeter is 75 cm.
402. How long is the base of an isosceles triangle whose
perimeter is 4 dm. and whose equal sides are each 12 cm. ?
Represent.
403. How long is each of the equal sides of an isosceles
triangle whose perimeter is 3 dm. and base 8 cm. Construct.
404. A kilometer is about what fraction of a mile ?
405. Find approximately the number of miles in 40 Km.
In 72 Km. In 3.2 Km. In 6.72 Km.
406. Find the approximate number of miles in 9288 m.
SUGGESTION. Express 9288 m. as kilometers before finding its equiva-
lent in miles.
407. Find approximately the number of miles in 45864 m.
In 63824 dm. In 59888 Dm. In 71848 Hm.
408. Find approximately the number of kilometers in 75 mi.
In 235 mi. In 84.5 mi.
409. How many square millimeters in a rectangle 1 cm. long
and 1 cm. wide ?
410. In a square decimeter, how many square centimeters ?
Square millimeters ?
411. In a square meter how many square decimeters?
Square centimeters ? Square millimeters ?
182 DENOMINATE NUMBERS
412. A square decameter equals how many square meters ?
Square decimeters ? Square centimeters ? Square millimeters ?
413. A square hectometer equals how many square deca-
meters ? Square meters ? Square decimeters ?
414. A square kilometer equals how many square hecto-
meters ? Square decameters ? Square meters ? Square deci-
meters ? Square centimeters ? Square millimeters ?
415. In long measure, under the metric system, what is the
ratio of a unit of each denomination to a unit of the next
higher denomination ?
416. In square measure, metric system, what is the ratio of
a unit of each denomination to a unit of the next higher
denomination ?
417. Write a table of square measure, metric system. Write
a table of equivalents of units of square measure, metric
system.
418. Express 3 sq. Km. 2 sq. Dm. 50 sq. m. in square
meters. In square decameters. In square kilometers.
419. Express 7 sq. m. 20 sq. dm. 30 sq. cm. in square
millimeters. In square centimeters. In square decimeters.
420. Express 1 sq. m. 2 sq. dm. 3 sq. cm. in sq. mm. In
sq. cm. In sq. dm. In sq. m. In sq. Dm. In sq. Hm. In
sq. Km.
421. What is the area of a square whose perimeter is 24
cm. ? 20 cm. ? 40 mm. ?
422. How long is the perimeter of a square whose area is 81
sq. cm.? 49 sq. cm.? 64 sq. dm.?
423. What is the area and the perimeter of a rectangle
which is 35 cm. long and ^ as wide as long ?
424. A land measurement, 10 meters square, or its equivalent,
is called an Are (a.). How many square meters in an are ?
How long is the perimeter of an are in the form of a square ?
DENOMINATE NUMBERS 183
425. What is the cost of f a. of land at $ 12.50 per are?
426. The standard unit of metric land measure is an Are,
which is equal to a square decameter or approximately to ^
of an acre.
METRIC LAND MEASURE
1 hectare (Ha.) = 100 ares
1 are (a.)
1 centare(ca.) =.01 are
Notice that the final vowel of "hecto," and "cento" is dropped be-
fore the word " are."
427. How many meters of fence would be required to
inclose a hectare in the form of a square ?
428. Draw on the floor a square containing a centare. How
long is its perimeter ?
429. How many square decimeters in a centare? In an are ?
In a hectare ?
430. What is the cost of 24.7 Ha. at $425 a Ha.? Of
63.25 Ha. at $1032 a Ha.?
431. Approximately, how many acres in 280 a. ? In 160 a. ?
In 240 a. ? In 120 a. ?
432. Find the approximate value in ares of 30 A. 75 A.
17| A. 8 A. 120 sq. rd. 12 A. 80 sq. rd. 6 A. 350 A. 500 A.
40 sq. rd.
433. Image a cubic centimeter and a cubic decimeter.
How many cubic centimeters are equal to the cubic decimeter ?
434. Draw a square meter in one corner of the room. Imag-
ine it covered with a layer of cubic decimeters or liters. How
many are there ?
435. With a meter stick outline a cubic meter. How many
layers of cubic decimeters are there in it ? How many cubic
decimeters ?
184 DENOMINATE NUMBERS
436. In cubic measure, metric system, how many units of
each denomination make one unit of the next higher denomina-
tion ?
437. Write the table of cubic measure, metric system.
438. A cubic meter, or its equivalent, is called a Stere (s.).
Image a stere of ice, of cubical form. How many square deci-
meters in all its surfaces ?
439. Image a stere of marble, 2 m. long and 1 m. wide.
How high is it? How many decimeters in all its edges?
Represent with blocks.
440. A box which holds a stere is full of packages of Break-
fast Food, each of which holds a liter. How many packages
are there?
441. At 15^ a liter, what is the value of the contents of the
box?
442. At 2^ a liter, what is the cost of 3 s. of wheat?
4726 s. ? 8347 s. ?
443. The standard unit of metric wood measure is a Stere,
which is a little over ^ of a cord.
METRIC WOOD MEASURE
1 decastere (Ds.) = 10 steres
1 stere (s.)
1 decistere (ds. ) = . 1 of a stere
444. A pile of wood 7 m. long, 6 m. wide, and 5 m. high
contains how many steres ? How much is it worth at $ 1.50
a stere ?
445. At 70^ a stere, what is the value of a pile of wood
4.5 Dm. long, 3.5 m. wide, and 300 cm. high ?
446. About how many cords are there in 20 s. ? In 32 s. ?
In 42 s. ? In 12.8 s. ? In 6.36 s. ?
DENOMINATE NUMBERS 185
447. How many cubic feet in a cord? About how many
cubic feet in a stere ?
448 v Approximately how many steres in 5 cd. ? In 7 cd. ?
In 9J cd. ? In 3 cd. 64 cu. ft. ? In 6 cd. 32 cu. ft. ? In
12 cd. 16 cu. ft. ? In 4 cd. 8 cu. ft. ? In 11 cd. 4 cu. ft. ? In
24 cd. 8 cu. ft. ?
449. Which is greater and how much, a stere or a kiloliter?
Explain.
450. About how many liquid quarts equal a liter?
(See page 159, Ex. 115.)
451. About how many quarts are there in a decaliter? In
a hectoliter ? In a kiloliter ?
452. About how many liters in 7.7 qt. ? In 132 qt. ? In
39 qt. ? In 17 qt. ?
453. A cask of oil containing 187 1. was bought at 20^ a
liter and sold at 25^ a quart, a liter being counted as 1.1 qt.
How much was gained ?
454. How much is gained by buying 209 1. of wine at 30^
a liter and selling them, at 40^ a quart, counting a liter as
1.1 qt. ?
455. Image a milliliter. What else is it called ?
(See page 81.)
456. The weight of a cubic centimeter of pure water at its
greatest density is called a Gram (g.).
457. Image a glass vessel of cubical shape containing a
liter of pure water. How many grams would it contain ?
458. Approximately 1000 g. equal 2| Ib. What is the
approximate equivalent of a gram?
MISCELLANEOUS EXERCISES
1. Divide 62.5 by .0025.
2. Find the g. c. d. of 567 and 637.
186 DENOMINATE NUMBERS
3. Reduce to lowest terms : -J-JJ.
4. Find the 1. c. m. of 24 and 57.
5. Add -fa and T 4 T .
6. From T 7 -j take -fa-
7. Kesolve into prime factors 26,460 and 60,060.
8. Write the improper fraction that expresses the ratio of
the first prime number after 40 to the first prime number after
20, and reduce it to a mixed number.
9. How many 6ths in J of 10 ? In J of 7 ?
10. One eighth of 88 is how many times 3 ? 4-J- ? 5 ?
11. If 13 is a divisor and 39 a dividend, what is the
quotient ? If both divisor and dividend are multiplied by 4,
what is the quotient ?
12. From 49J subtract a number which is ^ as large.
13. A minuend is 15, and a subtrahend 11. What is the
difference ? If 3 is added to both minuend and subtrahend,
what is the difference ?
14. If 3^ is added to both minuend and subtrahend in the
preceding question, what is the difference ?
15. Find difference between % of ft of 40 and f of Jf of 3.
16. Square f f. f ft.
17. What fraction multiplied by itself will give f ?
18. What is the square root of ff ? T W ? ft ? ft ? yinr ?
19. CLASS EXERCISE. may give a fraction that is a
perfect square, and the class may give its square root.
20. How many cubic yards of earth must be removed to
make a reservoir 120 ft. long, 44 ft. wide, and 9 ft. deep ?
21. How much will it cost to dig a cellar 36 ft. long, 18 ft.
wide, and 6 ft. deep, at $ 2.50 a cubic yard ?
MISCELLANEOUS EXERCISES 187
22. How many cords of wood in a pile 36 ft. long, 4 ft. wide,
and 8 ft. high ? At $ 3.50 a cord, how much would it cost ?
Find the cost of plastering ceilings of the following rooms
at 20 $ a square yard :
23. 18 ft. x 20 ft. 27. 10 ft. x 13| ft.
24. 16 ft. x 17 ft. 28. 15 ft. x 18 ft.
25. 141 ft. x 20 ft. 29. 27 ft. x 36 ft.
26. 9 ft. x 16J- ft. 30. 3 yd. x 16 ft.
31. Estimate the cost of plastering the ceiling of your
schoolroom at 25^ a square yard.
Find the cost of plastering the walls and ceiling of rooms of
the following dimensions :
length width height
32. 6m. 5m. 2.8m.
33. 8m. 7m. 3m.
34. 6.5m. 5m. 3m.
35. 7m. 6.5m. 3m.
The price is 25 $ per square meter, and no allowance is made
for openings.
36. An arc which is \ of a circumference is 1 yd. 1 ft. 3 in.
long. How long is the circumference ? The diameter ? The
radius ?
37. How many acres in a field 56 rd. long and 40 rd. wide ?
38. How much will it cost to pave a walk, 60 ft. long and
15 ft. wide, at $ 1.25 a square yard ?
39. How many trees can be planted on 3 A. of ground if
only 1 tree is planted on each square rod ?
40. How many cubic feet in a pile of wood 24 ft. long, 3 ft.
wide, and 8 ft. high ? How many cords ?
188 DENOMINATE NUMBERS
41. Express in grams the weight of the following measure-
ments of pure water at its greatest density :
1 cu. dm. 3 1. 15 cu. cm. 1 ins. 1 ml. Is. 4 cu. m.
42. Express the measurements above in kilograms.
43. Taking 2^ Ib. as the equivalent of a kilogram, what is
your weight in kilograms?
44. Express the following in avoirdupois on the basis of
2 Ib. to the kilogram :
75 Kg. 88 Hg. 15 Dg. 175 g. 395 eg.
45. Express the following in avoirdupois weight:
33 Kg. 275 g. 924 mg. 99 Hg. 16 Dg.
46. How much is gained by buying a barrel of flour (196 Ib.)
for $ 6.00 and selling it at 7^ a kilogram ?
47. How much is gained by buying 99 Ib. of sugar at 5 f a
pound and selling it at 13 ^ a kilogram ?
48. How much is gained or lost by buying 440 Ib. of dried
fruit at 10 ^ a Ib. and selling it at 22 f a Kg. ?
49. How much is gained by buying 100 Kg. of coffee at 50^
a Kg. and selling it at 30^ a Ib. ?
50. How much is gained by buying 500 Kg. of raisins at
12 j a Kg. and selling them at 8 ^ a Ib. ?
51. Imagine a cubic decimeter cut from each corner of the
upper layer of a cubic meter, and find the surface of the figure
thus formed.
52. A sector whose arc is a quadrant was cut from a circle.
If the area of the whole circle was 4| sq. in., what was the
area of the part that was left ?
53. A California woman took 300 Ib. of honey from her
hives in a month. What was its value at $ 5.00 per hundred-
weight ?
MISCELLANEOUS EXERCISES 189
54. Earning $ .75 per day, how long will it take a boy to
earn enough to buy a $ 12.00 watch ?
55. Mr. Taylor bought 3 prize pigs whose respective weights
were 3 cwt. 73 Ib. 12 oz., 4 cwt. 99 Ib. 15 oz., 5 cwt. 12 oz.
How much did they all weigh ?
56. How many baskets, each holding 2J pk., can be filled
with 10 bu. of apples ?
57. A garden containing 1089 sq. yd. is 49^- yd. long. How
wide is it ?
58. A fisherman had a line 24 yd. 2 ft. long. A fish broke
off 3 yd. 1 ft. 6 in. of it. How much was left ?
59. A dealer bought 2 T. 3 cwt. of carpet tacks in 8-oz.
papers. How many papers of tacks were there ?
60. How long is one side of an equilateral triangle whose
perimeter is 5 yd. 1 ft. 3 in. ? Of a regular pentagon having
an equal perimeter ? Of a regular octagon of equal perimeter ?
61. A string, 4 yd. 2 ft. 6 in. long, was used to outline a
regular hexagonal flower bed. How long was each side ?
62. A farmer sold 5 loads of hay, each Containing 17 cwt.
85 Ib. How much did he sell ?
63. How far will a man walk who begins walking at 9 A.M.
and walks until 3.30 P.M., at the rate of 5 mi. an hour ?
64. A family started to go in a wagon to St. Louis from a
town 132 miles away. They rode 24 miles a day for 5 days.
On the morning of the sixth day, they started at 9 o'clock to
ride the remaining distance at the rate of 6 miles an hour. At
what time did they reach St. Louis ?
65. Mr. A has a lot 40 rd. square, and Mr. B has a lot con-
taining 40 sq. rd. How many more square rods in Mr. A's lot
than in Mr. B's ?
190 DENOMINATE NUMBERS
66. A flight of stairs in Mrs. Long's house consists of 18
steps, each 1 ft. wide and 8 in. high. How much will the stair
carpet cost at $ .75 per yard, if 3 in. is allowed at each step
for the turning in of the carpet ?
67. How much can be earned in two weeks by a person who
earns $ 2.34 every working day ?
68. General McClellan was born Dec. 3, MDCCCXXVI, and
died Oct. 29, MDCCCLXXXV. How old was he when he
died?
69. April 4th, 1898, was Monday. At the close of that day,
Kuth Mayo found that there were 8 weeks and 3 days left of
the school term. On what day did the term close ?
70. She entered college Sept. 14, 1898. The first term
closed Dec. 21. How long was it ?
71. Her expenses for the term were $ 95.75. What was the
average per week ?
72. The Thanksgiving vacation began Nov. 24 and ended
Nov. 28, and there was no other vacation in the term. She
attended a Saturday class. How many working days had
she in that term ?
73. Her second term began Jan. 3, 1899, and ended March
25, 1899. How long was it ?
74. Her expenses for that term averaged $ 7.50 per week,
and she earned $ 25 during the term by outside work. Her
expenses were how much more than her earnings ?
75. The floor of Mrs. Beed's dining room, which is 15 ft.
long and 14 ft. wide, is laid with parquetry flooring. How
much did it cost at $ .621 per square yard ?
76. The wainscoting is 3 ft. high. There are 4 doors, each
3 ft. wide. Two windows, each 3 ft. wide, extend down into
the wainscoting 1 J ft. There is a fireplace 4. ft. wide. How
many square yards in the wainscoting?
MISCELLANEOUS EXERCISES
191
77. Her dining table is 6 ft. long and 4 ft. wide. How many
square yards in the top of it ?
78. A rug under the table is 12 ft. long and covers 12 sq. yd.
of the floor. How wide is it ?
79. Make a problem about the dimensions of a room.
80. Fourteen cords of wood are piled evenly on an open car
28 ft. long and 8 ft. wide. How high is the wood piled ?
81. If a leaf of a book is 12 cm. long and 9 cm. wide, how
many square centimeters in the surfaces of both sides of the leaf ?
82. If your schoolroom were 36 ft. long and 30 ft. wide,
how many square yards could be drawn on the floor, provided
no two overlapped ?
83. In Pig. 1 the angles are all right
angles. How long is the line repre-
sented by GH? HA?
84. Copy Fig. 1, making the dimen-
sions inches or centimeters. Draw a
H G construction line BE. How long is it ?
Find the area of Fig. 1 by finding the
sum of the areas of the two rectangles
that are thus formed.
85. Copy Fig. 2. Find its area by
drawing a construction line from C per-
pendicular to FE and finding the area
of the two rectangles that are thus
FIG. 2. formed.
86. Find the area of Fig. 2 by drawing a construction line
A BE F from O perpendicular to AF and finding
the area of two rectangles thus made.
87. Copy Fig. 3, making AB 6 in.,
EC 3 in., CD 3 in., DE 3 in., EF
3 in., FG 8 in. Find length of GH
FIG. 3. and HA.
FIG. 1.
10
E
192
DENOMINATE NUMBERS
88. Find area of Fig. 3 by drawing construction lines that
will divide it into three rectangles and finding the area of the
rectangles. Show different ways of dividing it.
89. Find area of Fig. 3 by finding area of a rectangle
AFQH and subtracting the square BE DC.
90. Copy Fig. 4, making AB 5 in.,
BC 3 in., CD 4 in., DE 6 in., EF 2 in.,
FG 4 in. GH=? HA= ?
91. Show four different ways of di-
viding Fig. 4 into three rectangles.
Find its area.
jT~ a 92. Beginning at a point marked A,
FIG. 4. raw to t h e rig]^ 4 in<? down 3^ to tne
right 3, down 4, to the left 3, down 3, to the left 4, up 3, left 3,
up 4, right 3, up to A. Find perimeter of the figure. Find
its area.
93. Beginning at A, draw down 7, to the right 3, up 3, right
3, down 3, right 4, up 4, left 2, up 2, left 3, up 1, left 5. Show
several different ways of dividing the figure into rectangles.
Find its area.
94. CLASS EXERCISE.
may give directions to the class
for drawing a figure which has only straight lines and right
angles. The class may divide the figure by different construc-
tion lines and find their length and the area of the figure.
95. ABCD is a square 9 in. in dimen-
sions, and EFGH is a square 5 in. in
dimensions. How many square inches
in the surface lying between the perim-
eter of the squares ?
96. The frame of a mirror is 28 in.
long and 20 in. wide on the outside edge.
The glass in the center of the frame is
Eepresent. What is the width
FIG. 5.
20 in. long and 12 in. wide.
MISCELLANEOUS EXERCISES
193
of each side of the frame ?
surface of the frame ?
How many square inches in the
97. A rug 12 ft. long and 9 ft. wide was laid on the floor,
leaving a margin 3 ft. in width all around the rug. What was
the area of the floor ? Of the rug ? Of the uncovered part ?
98. A picture 18 in. long and 15 in. wide has a frame each
side of which is 6 in. wide. How many square inches in the
surface of the frame ?
99. A door 7 ft. high and 3 ft. wide has a 6-inch casing
around it. How many square inches in the surface of the
casing ?
100. Measure a door and the width of its casing and find the
number of square feet in the surface of the casing.
101. ABCD represents a square 15 in.
in dimensions. The altitude and the base
of each triangle is 4 in. Eind the area
of the octagonal figure left when the
triangles are cut away.
102. Two lines are respectively 6 in.
and 10 in. What is their average length ?
103. Reproduce the trapezoid ABCD,
making AB 4 in., BC 8 in., and AD 6 in.
Let x be the middle point of AB, and
y the middle point of DC. xy repre-
sents the average length of the parallel
FIG. 7. sides of the trapezoid. How long is xy ?
104. Through the point y draw the line EF parallel to AB.
EF is the altitude of the trapezoid.
Find the area of the rectangle AEFB.
Cut off the triangle yFC and apply it
to the triangle yED. How does the
^ area of the rectangle AEFB compare
with the area of the trapezoid ADCB ?
FIG. 8.
HORN. GRAM. SCH. AR. 13
194 DENOMINATE NUMBERS
105. Draw another figure and show the reasons for the fol-
lowing rule.
To find the area of a trapezoid
Multiply the average length of the parallel sides by the altitude.
106. Draw a trapezoid whose parallel sides are respectively
9 in. and 5 in., and whose altitude is 6 in. Find its area.
107. A board is 16 ft. long, 2 ft. wide at one end and 1 ft.
wide at the other, tapering gradually. How many square feet
in the surface of the board ?
108. A farmer has a field in the shape of a trapezoid. One
of the parallel sides is 40 rd. long, the other is 24 rd. long.
The distance between them is 25 rd. Find area of the field.
Make diagrams to illustrate the following problems :
109. Mrs. Hall's parlor has a bay window, the floor of which
is in the shape of a trapezoid. The longer of the parallel sides
is 12 ft., the shorter 9 ft., and the distance between them is
4 ft. How much will it cost to cover it with parquetry floor-
ing at $ 1.25 per square yard ?
110. Her sitting room is 18 ft. long and 15 ft. wide. How
many strips of carpet 1 yd. wide, running lengthwise of the
room, will be required to cover the floor ? How many yards in
each strip ? How much will the carpet cost at $ ,87-J- a yard ?
111. How many strips of carpet 1 yd. wide, running length-
wise, will be required for a floor 24 ft. wide ? If the floor is
30 ft. long, how many yards will be required ?
112. How much will it cost to carpet a room 30 ft. long 24
ft. wide with ingrain carpet at $ .75 per yard, if a margin of
^ yard is left uncovered ?
113. Mrs. Eoss covered the floor of her parlor, 20 ft. long
and 18 ft. wide, with velvet carpet 27 in. wide. How many
strips were used ? What was the cost of the carpet at $ 1.65
per yard ?
MISCELLANEOUS EXERCISES 195
114. Her sitting room, 16 ft. by 12 ft., is carpeted with in-
grain carpet at $ .67 per yard. In order to match the figures,
the carpet layer was obliged to cut off or waste a piece of car-
pet ^ of a yard long from each strip except the first. How
much did the carpet cost?
115. Allowing ^ of a yard to be turned in or cut off from
each strip except the first in order to match the figures, how
much will it cost to carpet a room 21 ft. by 18 ft. with carpet
27 in. wide, worth $ 1.35 per yard ?
SUGGESTION FOR CLASS EXERCISE. Let pupils give dimensions of
floors and estimate cost of covering them with carpet of different widths.
116. What is the volume of an 8-inch cube ?
117. What number cubed equals 27 ? 216.?
118. Find value of x : or 5 = 64. y? = 125. 0^ = 343.
119. Image a box in the shape of a cube whose volume is
8 cu. in. Suppose the box to be entirely covered with blue
velvet. How many square inches of the velvet are there ?
120. When a 3-inch cube is built of inch cubes, how many
inch cubes are there that cannot be seen from the outside in
whatever position the cube may be placed ?
121. How many of the cubic centimeters that make a cubic
decimeter have any of their surfaces on the outside of the
figure ? How many have one surface ? Two surfaces ? Three
surfaces ? No surface ?
122. If a pipe discharges 245 gal. 2 qt. 1 pt. in 1 hr., how
much will it discharge in the time from Tuesday, 6 P.M., to
Wednesday, 11 A.M.?
123. A school that uses 12 crayons in a week will use how
many gross of crayons in 40 weeks ?
124. A stationer bought 8 gross of lead pencils and sold
50 dozen of them. How many lead pencils had he left?
196 DENOMINATE NUMBERS
125. If he bought them at $ 3 per gross and sold them at
5 $ apiece, how much did he gain on each pencil ? On 50 doz.
pencils ?
126. At 10^ a square yard, how much will it cost to sod a
lawn 40 ft. long and 36 ft. wide ?
127. If you had $23.70 in the bank at 4% interest, how
much interest would it yield you each year ?
128. Mr. Thomas Kepler bought of Mr. Prank Barton an
overcoat and a vest, the price of which was $ 27. Mr. Kepler
gave in payment a check for that amount on the First National
Bank of Denver, Colorado, of which the following is a copy :
SUGGESTIONS TO TEACHER. Explain the method of using checks to make
payments. Get a check book from a bank. Select some pupil to act as
banker and let pupils make imaginary purchases from one another, giving
checks for the necessary amounts.
129. What is the advantage of keeping money in a bank and
drawing it out as it is needed ?
130. Mr. Dow borrowed $ 800 at 5% from Mr. Howe, kept
it 3 yr., and then gave Mr. Howe his check for the amount
due. Make out the check.
131. Mr. Ford had $ 427 in bank. He drew out $ 135.87,
deposited $ 77.50, then drew out $ 35.25. How much remained
to his credit in the bank ?
MISCELLANEOUS EXERCISES 197
132. Mr. Arnold had $ 1200 in the bank. On Monday he
drew out $ 60. On Tuesday he drew out $ 30 more than on
Monday. On Wednesday he drew out $ 90 more than on
Tuesday. How much had he left in the bank ?
133. Mr. Monroe lent $ 100 at 5% interest. At the end of
7 yr. the principal (that is the sum lent) and the interest were
both paid. To how much did they both amount ?
134. What amount will Mr. Day receive from $ 228
which he lent 2 yr. ago at 6%, if both principal and interest
are paid ? How much if the rate of interest is
135. CLASS EXERCISE. - niay name a sum of money
supposed to be lent at 3|-% for 2 yr., and the class may find
the interest and the amount of principal and interest.
136. Mr. Shaw borrowed $ 750, kept it until it had gained
$ 78.75 interest, and then paid $300. How much did he
still owe ?
137. Mr. Shaw borrowed $ 600 and gave his note for it, due
in 2 yr. with 6% interest. How much was due at the end of
the two years ? At that time he made a partial payment of
the note, paying only $ 200. How much did he still owe ?
138. Mr. Shaw borrowed '$ 700, giving his note at 5% in-
terest. At the end of 2 years how large a partial payment
must he make that only $ 500 may be due ?
139. Mr. Shaw borrowed f 900 at 6%. At the end of the
first year he paid $ 154. How much was still due ? That
sum went on gaining interest until the end of the second year ;
then he paid f 148. At the end of the third year he paid all
that was due. How much did he pay ?
140. CLASS EXEKCISE. - may name a sum of money and
a number of years for which it was borrowed. Other members
of the class may suggest partial payments to be made at dif-
ferent times, and the class may find the amount due after each
payment.
198 DENOMINATE NUMBERS
141. Make out a bill for the following goods and receipt it:
C. H. Wilson bought from J. G. Cooper & Co., at Columbus,
Ohio, on the tenth day of June, 1875: 13 Ib. coffee @ 30^;
4 Ib. butter @ 35 f ; 10 Ib. flour @ 6 ^ ; 12 Ib. dried beef @ 24 f ;
25 Ib. sugar @ 18 ^j 3 Ib. starch @ 20 /.
142. From a field containing 400 sq. rd., the owner sold a
piece of land 15 rd. square and another piece containing 15
sq. rd. How many square rods had he left ?
143. Mr. Eay sold his house and a farm of 75 A., receiving
$ 7500 for both. If the house was worth $ 2000, how much
did he receive per acre for the land ?
144. If Mr. Eudd earns $ 15 a week and spends $ 7, in how
many weeks will he earn $ 100 ?
145. Mrs. Hall's sitting room has a picture rail extending all
around it 1^ ft. from the ceiling. The room is 18 ft. long and
15 ft. wide. How much did the picture rail cost at 7 J ^ per
foot?
CHAPTER VI
ALIQUOT PARTS
1. Numbers, either integral or fractional, by which a given
number is divisible are called Aliquot Parts of that number.
For example, 5 and 21 are aliquot parts of 10.
Give three numbers which are aliquot parts of 100.
2. Draw three vertical lines each 10 in. long
and divide them into lengths each 2J in. long.
Number the lengths consecutively as in the
diagram. How many 2J in. lengths in 10 in. ?
In 20 in. ? In 30 in. ?
3. Beginning with 21-, count quickly by inter-
10-1 20-J 30-J vals of 2 i to 30 - Count back from 30 to 2 by
intervals of 2^.
NOTE TO TEACHER. The following exercises are for rapid drill, which
should be given frequently until pupils learn the ratios of the smaller ag-
gregations of 2^ to one another. This kind of work leads to expertness
in business calculations.
4. How many times 2| is 7? 17|? 27? 12 J? 25?
15? 22? 10? 20? 30?
6. Give quickly the 4th multiple of 2J. The 7th. 10th.
5th. 3d. 6th. 9th. llth. 8th. 12th. 2d.
6. Which multiple of 2J is 7|? 15? 25? 10? 17?
224? 30? 121? "274? 20? 5?
7^ How many times 2 must be added to 7-J- to make 20 ?
30? 121? 17? 27J? 221? 15? 25? 10?
8. How many times 2^ must be taken from 27^- to leave
15? 7? 22^? 12^? 2? 10? 5? 17|? 25? 20?
199
200
ALIQUOT PARTS
9. With 15 as a starting point find how many times 2J
must be added to it or subtracted from it to equal 27. 12.
2 30. 20. 7 22. 10. 17 5. 25.
10. Learn to give quickly the ratio of 5 to each of the mul-
tiples of 2J that are less than 32^.
11. Take Ex. 10, substituting for 5 each multiple of 2J that
is greater than 5 and less than 32^-.
12. CLASS EXERCISE. - may name two multiples of 2^-,
and the class may give the ratio of the greater to the less, then
the ratio of the less to the greater.
Cancel :
10 x
'
x 6
7fxl5
= 9
7x2ixl2i
16
'
25 x 15 x 3
12^x22^x6 =
25 x 9 x 21 x 7
17. At 21 $. per yard what is the cost of 8 yd. of lace ? 12
yd. ? 7 yd. ? 9 yd. ? 6 yd. ? 11 yd. ?
18. At 21^ per yard, how many yards of
lace can be bought for 10 f? 20 f ? 30^ ?
50^? 121^? 25^?
36%-
10 J
40-
19. Draw four vertical lines each 10 in.
long and divide them into lengths of 3^- in.
Number the lengths consecutively. 40
in. equals how many times 3J in. ?
to 40 by intervals of 3.
similar to those in Exs.
20. Learn to count quickly from
From 40 to by intervals of 31
NOTE TO TEACHER. Give exercises upon
4-12 upon 2|.
21. At 3-^ per yard, what is the cost of 7 yd. of lace ?
10 yd. ? 8 yd. ?
ALIQUOT PARTS 201
22. At 3J^ per yard, how many yards of calico can be bought
for 20^? 40^? 10^? 30^?
23. How many more yards of ribbon can be bought for 40 f
when the price is 2J ^, than when it is 3^ $ ?
Cancel :
31x7^x5 =9 21x61x121-^
' 6fx3x2ix7 10x3-ix9
12ix6xlO =9 13|x2jx2 =9
'25x7^x11 6fx7ix5
28. A merchant bought goods at 8 ^ a yard, and sold them
for 10 1 a yard. How much did he gain on each yard ? What
is the ratio of the gain to the cost ?
29. Find ratio of gain to cost of goods :
a b c d e f g li
Bought at 6^ 8^ 9? 5/ 1\j 10 X 10^
Sold at 9^ 12^ 12^ 7J^ 10 f 12J^ 15^
30. When goods are bought for 9 ^ a yard and sold for 6^ a
yard, how much is lost ? What is the ratio of the loss to the
cost?
31. Find ratio of the loss to the cost of goods:
a b c d e f g 7i
Bought at 8^ 10^ 15^ 20^ 18^ 1\f 10^
Sold at 6^ 8^ 10 # 15^ 15 5^ 7|^
32. Find ratio of gain or loss to cost of goods :
a b c d e f g h
Bought at 45^ 20^ 25^ 36^ 22^ 33^ 36^ 60^
Sold at 50^ 15^ 30^ 40^ 33^ 22^ 32^
33. Find ratio of gain to cost of goods :
a b c d e f
Cost, 10^ 20^ 30^ 6f^ 16f^ 26|^
Selling price, 13^ 23$ f 33^^ 10^ 20^ 30^
202 ALIQUOT PARTS
34. How much is gained on each apple bought at the rate
of 2 for 5 ^, and sold at 3 f apiece ?
35. On the 4th of July, Andrew bought lemons at the rate
of 3 for a dime. He used one lemon and f of a cent's worth
of sugar to make each glass of lemonade. How much did he
gain on each glass of lemonade sold at 10^ a glass? How
much on l^ doz. glasses ?
36. For $ 1.00 William bought ice cream enough to fill 36
ice cream plates. He sold it at 10 f a plateful. How much
did he gain on each plateful of ice cream ? He sold 1^- doz.
platefuls. How much did he gain ?
37. The rest of his ice cream was unsold and was wasted.
Did he gain or lose, and how much, on his whole trans-
action ?
38. Beginning at 16f, count quickly to 100 and back to
by intervals of 16|.
39. How many times is 16| contained in 66J ? In 33 ?
In 100? In83J?
40. How many times 16J must be added to 50 or subtracted
from it to make 831 ? 100 ? 66| ? 16| ?
41. Give the ratio of 16| to each of its multiples that is
less than 101.
42. Learn to give quickly the ratio of each multiple of 16|
that does not exceed 100, to every other multiple of 16| that
does not exceed 100.
43. CLASS EXERCISE. may name two multiples of 16 j,
and the class may give the ratio of the less to the greater, and
of the greater to the less.
44. At 16f $ per yard, what is the cost of 3 yd. of lawn ? 5
yd. ? 7 yd. ? 2 yd. ? 6 yd. ? 4 yd. ?
45. At 16 \$ per yard, how many yards of lawn can be
bought for $1? $2? $7? $10? 33^? 83^?
ALIQUOT PAKTS 203
46. Beginning with 8^, count quickly to 100 and back to
by 8Vs.
SUGGESTION TO TEACHER. Exercises should be given upon 8| similar
to those in Exs. 39-43 upon 16f .
47. CLASS EXERCISE. may name two multiples of 8J,
and the class may give their reciprocal ratios.
48. 8^ = 25% of what ? 50% ? 10% ?
49. At 8J f per yard, what is the cost of 6 yd. of muslin ?
3 yd. ? 12 yd. ? 4 yd. ? 8 yd. ? 10 yd. ? 5 yd. ? 9 yd. ?
50. At 8 J f per yard, how many yards of lace can be bought
for $1.00? $.83f? $1.08J? $.33J? $ .66} ? $.50?
61. Find gain and ratio of gain to cost.
a b c d e f g
Cost, 25? 50^ 16}^ 33 ^ 66|^ 75^ 91}^
Selling price, 33^ 66|^ 25^ 50^ 75^ 83^ $1.00
52. How much would Mr. Lee gain by buying 12 yd. of
cassimere, at $ .66| a yard, and selling it at $ 1.00 a yard ?
53. Would he gain or lose, and how much, by buying 12 yd.
of silk at $ .33^ per yard, selling half of it at $ .50 per yard,
and the rest at $ .25 ?
54. -^of 100%=?
55. Complete the following table and learn it :
= - 58|% = -
= =
25% = 50% = 75% = 100% =
56. Copy the six-pointed star given on page 92 and divide
it into 6 equal rhombuses.
57. If the perimeter of the star were 100 in., how long
would the perimeter of one of the rhombuses be ?
204
ALIQUOT PARTS
58. What is the ratio of each rhombus to the star ? Express
the ratio in per cent. What per cent of the star would remain
if one rhombus were erased?
59. Draw a line from each point of the
star to the center of the figure as in Fig. 1.
Into what kind of figures is each rhombus
divided ?
60. What is the ratio of each isosceles
triangle to the rhombus of which it is a
part ? To the star ?
61. If one triangle were shaded, what
per cent of the star would be unshaded ?
FIG. 1.
62. Place letters at the center and at the vertices of the
angles of the star and name a figure that is 50% of the star.
75%. 81%. 16|%. 331%. 66|%. 831%.
63. Kate drew a square foot on the board and marked it off
into inches. She erased 81% of it. How many square inches
were left ?
64 o How many cubic inches in 8^% of a cubic foot ?
65. To gain 81%, at what price must goods be sold that cost
48^? 60^? 30^? 45^? 72^? 75^? $1.08?
66. Mr. Barnett drew out of the bank $ 56, which was 16f %
of the money he had in the bank. How much had he in the
bank before he drew any out ? How much afterward ?
67. A line a yard long was lengthened
it then ?
How long was
68. How many times is 81 contained in the second multiple
of!6|? In the 5th? 3d? 6th?
69. How many times is 3J contained in the second multiple
of!6|? In the 3d? 6th? 4th?
Selling Price
Cost
Selling Price
$.66f
/ '
$.831
$.66f
.50
9
.66|
33i
1.00
h
.33J
.16|
.66|
i
.66|
1.00
.50
j
.331
.66|
h is i of 100 ?
f? .*?
J ?
ALIQUOT PARTS 205
70. Find ratio of gain or loss to cost of goods at the follow-
ing prices. Express ratios in per cent.
Cost
a $.50
6 .331
c .831
d 1.00
e .66|
71. How m
72. Beginning at 121, name all the multiples of 121 that are
less than 101. Write them in a vertical column.
SUGGESTION TO TEACHER. Exercises should be given upon 12|
similar to those in Exs. 39-43 upon 16f .
73. Name all the multiples of 12J, less than 101, that are
also multiples of 8 l s , and tell how many times they contain 8^ ;
also how many times they contain 121.
74. If 6 yd. of calico are required for Mary's dress, how
much more would it cost at 12J^ per yard than at 8J^ ?
75. Jennie has $ 1.25 to spend in ribbon. How many more
yards can she buy at 8|^ per yard than at 121^?
76. Find the amount of gain on each article, and the ratio
of the gain to the cost of the goods at the following prices :
a b c d e f 9
Cost, 25^ 50/ 62$ j 75? 12j^ 50^ 62$ f
Selling price, 37$ f 62$ t 87^ S7$t 25^ S7$f 750
77. How much did Mr. Luce gain on 14 yd. of silk bought
at $ .871 per yd. and sold for f 1.25 per yd. ?
78. Count from to 100 and back from 100 to by inter-
vals of 6J.
SUGGESTION TO TEACHER. Exercises should be given upon 6 similar
to those in Exs. 39-43 upon 1C|.
206 ALIQUOT PARTS
79. CLASS EXERCISE. The multiples of 6J being written
upon the board, points to one of them, and the class give
quickly its ratio to 6J.
80. At 6J cents a yard, what is the cost of 8 yd. of calico ?
12yd.? 3yd.? 5yd.? 7yd.? 9yd.? llyd.? 6yd.?
81. At 6J cents a yard, how many yards of calico can be
bought for 25^? 50^? $1? $.12|? $.37|?
82. Find gain and ratio of gain to cost :
a bed e f
Cost, 121^ 25^ 25^ 37^ 50^
Selling price, 18|/ 3lf 37^ 50^ 56^
83. Find loss and ratio of loss to cost :
a be d e f
Cost, 18|^ 25^ 37^ 50 f 75? 68f^
Selling price, 121^ 18|^ 31 J^ 43f^ 68|^ 62^ 66 J^
84. How many times is 6J contained in the 4th multiple of
12 ? In the 7th multiple of 12 J ? In the 3d multiple of 12 J ?
85. How many more yards of goods can be bought for $ 1.00
when the price is 6^ than when it is 121^?
86. How much did Mr. Colton gain on 8 yd. of denim,
bought at $ .12 a yard and sold at $ .18f ?
87. How much did he lose on 20 yd. of calico, bought at
$ .06J a yard and sold at $ .05 a yard ?
88. Find loss or gain and its ratio to cost :
a b c d e f g
Cost, 66^ 37 J^ 62^ 75 f 25? 43f^
Selling price, 62J^ 43|^ 68fX 500 12^50^
89. How much did Mr. Hale gain on 8 yd. of ribbon, bought
at f .37J a yard and sold at $ .50 a yard ?
ALIQUOT PARTS 207
90. Write all of the first 12 multiples of 8J that are also
multiples of 6J, and tell how many times each of them con-
tains 8^, and how many times it contains 6^.
91. How many more yards of ribbon can be bought at 6J^
than at 8J if the sum spent is $ 1.00 ? $ .50 ? $ .75 ? $ .25 ?
92 8 ? x6 i = ? 93
'
25 x 25 16| x 10 x 25
2ix3jx6jx8i_ ? 9g 8txl2jx2t ?
' 5xlOx25xl()| ' 33x6ix7i
12*x8i-x2^_ ? 8jx25xl2j-_ 9
7x25x6i ' 25x75x62~
75x25x2j ? Q9 871x5x3 _ ?
62^x6x5 ' 14x12^x2^
100. Draw the line AB of any con-
venient length and bisect it.
To bisect the line AB. With A as a
center and any convenient radius
greater than one half of AB, describe
ah arc. With B as a center and the
same radius describe an arc cutting
the first arc at C and D. Join O
and D. The point E where CD cuts
AB is the middle point of the line
AB. CD is perpendicular to AB.
101. Bisect AE and mark its middle point F. Bisect EB
and mark its middle point G.
102. Mark on each new division what per cent it equals of
the line AB.
103. What per cent of the line AB is AF? AG? GB?
EB? FG? FB?
AxFyEzGmB
104. Bisect each division of the line
on each division its per cent of the whole line AB.
208 ALIQUOT PARTS
105. What per cent of the whole line is Ay? Am? Az?
Fz? xy? xm? Fm? xB? yB? ym? xz?
106. Draw a regular octagon and divide it into 8 equal
isosceles triangles. Each triangle is what per cent of the
octagon ?
107. Divide each triangle into 2 equal right triangles.
Each triangle is what per cent of the octagon ?
108. Complete the following table and learn it:
= 87i%=
18f%= 43|%=- 68f%= 93f%=-
25%= 50%= 75%= 100%=-
109. How many square inches in 12 1% of a rectangle 1 ft.
long and J ft. wide ?
110. How much is 37-1% of 72** ? Of $ 1.44 ?" Of $ 4.80 ?
111. To gain 121%, how must goods be sold that cost 8^ ?
40^? 60^? 50^? $1.20? $1.60? $3.20? $6?
112. $5 is 12^-% of Charles's weekly salary. How much
does he receive each week?
113. Mr. Owen sold 15 acres of land, which was 12-J% of his
farm. How large was his farm before the sale ? After the sale ?
114. In Mrs. Abbot's parlor there is a rug 6 ft. long and 5 ft.
wide, which covers 12^% of the floor. How many square feet
are there in the floor ?
115. What is the ratio of the uncovered part of the floor to
the whole floor? Express that ratio in per cent.
116. Find loss or gain and ratio to cost:
Cost Selling Price Cost Selling Price
a $.25 $ .37 e $1.00 $ .75^
b .50 .62 / 1.00 .62
c .75 .87* g .75 .62*
d .87i 1.00 h .37* .12J
ALIQUOT PARTS 209
By observing some aliquot parts of numbers we may find short methods
of multiplying.
117. Annex two ciphers to 48 and divide the result by 4.
Compare the result with the product of 48 x 25.
Observe that by annexing two ciphers to any integer we multiply it by
100, which gives a result 4 times as great as when the integer is multiplied
by 25.
118. Take Ex. 117, substituting other numbers for 48.
What general truth connected with that work can you state ?
119. Illustrate the following rules and give reasons for
them:
To multiply by 3 J. Annex a cipher to the multiplicand and
divide the result by 3.
To multiply by 33^. Annex two ciphers to the multiplicand
and divide the result by 3.
120. Tell how you would multiply a number by 333J by the
method of aliquot parts.
121. Multiply 72 by 25. By 33. By 333J.
122. Annex a cipher to 36 and divide the result by 4. Why
is the quotient thus obtained equal to the product of 36 and 2|- ?
123. If you annex a cipher to 24 and divide the result by 3,
the quotient thus obtained equals how many times 24 ? Why ?
124. By 2J multiply 84. 56. 128. 31.
125. By 3J multiply 27. 81. 43. 28. 15.
126. Give a rule for finding the product of 2J and any
integer without using either 2J or the integer as a multiplier.
Give reasons for the rule.
127. Give a similar rule for finding the product of 3J and
any integer. Give reasons.
128. How many ciphers must be annexed to an integer, and
by what must the result be divided, in order to multiply the
integer by 16 J? By 12? Illustrate.
HORN. GRAM. 8CH. AR. 14
210 ALIQUOT PARTS
MISCELLANEOUS EXERCISES
1. What number is 6 more than 1^
2. Square 2.1. 3.2. 1.25. 1.02. 2.003.
3. Cube .02. .8. 1.2. 1.03. 1.05.
4. Change \ and f to decimals and find their product.
5. Change to decimals and find their continued product :
a b c
* i i if* iti
6. Find sum of all the prime numbers between 20 and 50.
7. How many times 4.37 equal 17.48 ?
8. What number besides 137 will exactly divide 11,371 ?
9. | of 8J x | X ff X if of if of rffr = ?
10. Reduce :
15 10 16| 33| 40 13J
H 3* 3J 3J 3J 3
11. A man worked 5 da. at the rate of $ 9f a week (6 da.).
How much did he receive for the 5 da. work ?
12. 6|xifxAxH=? 5|- T V-ll
13. Give a fraction whose reciprocal is less than the fraction
itself. Is the original fraction proper or improper? What
kind of a fraction is the reciprocal ?
14. Find 33^% of the largest odd number that can be
written with two figures.
15. Find 11% of the only prime number between 89 and 101.
16. Find 16f % of 24 2 .
17. Find 25% of V64. Of Vl44.
18. Find 66f % of 6 3 . Of 8 3 .
19. Find 50 % of the cube root of the following numbers :
27 64 343 125 216 729 1728
MISCELLANEOUS EXERCISES 211
20. Divide twenty-four thousandths by sixteen millionths.
21c A lady buys a dollar's worth of soap at 6J^ a bar. If
she uses 8 bars in 1 mo., how long will it last ?
22. How long is the circumference of a circle whose diameter
is 1^- cm. ? The arc that is T 4 T of that circumference ?
23. At 12|^ a qt., how much will 2 gal. of molasses cost?
8 gal. ? 3 gal. 3 qt. ? 1 pt. ? 2 qt. 1 pt. ? 1 gi. ? 3 gi. ?
24. It costs 6 \ ? to make a gill of a certain kind of medicine.
If it is sold for a dollar a quart, how much is gained on a quart ?
25. Express in per cent the ratio of a peck to a bushel.
26. Express in per cent the ratio of 7 m. to a dm. 6 m.
to a Hm. A square meter to a hectare. 9 sq. dm. to a sq. m.
700 cu. dm. to a stere.
27. Reduce to a common fraction in its lowest terms .075.
.875. .5625.
28. Write five fractions whose numerators are each 6^ and
whose denominators are multiples of 6^, and reduce them to
lowest terms.
29. Write six fractions whose numerators are each 8J and
whose denominators are multiples of 8J, and reduce them to
lowest terms.
30. Divide 50% of 216.48 by 33.
177.76-25% of 177.76 _ 9
1.2
32. Change to hundredths expressed as a decimal:
A I A
33. Express in per cent J. }. f . f. f. f . J. f .
34. A regiment in marching takes 128 steps in a minute,
each step 2| ft. long. How many feet does it advance in an
hour?
212 ALIQUOT PARTS
35. How many rods of fencing will it take to inclose a lot
45 rd. 3 yd. long and 30 rd. 1 yd. wide ?
36. A room is 17f ft. long and 12f ft. wide. What will be
the cost of a molding around it at 3|^ per foot ?
37. Divide the snni of J and J by -J-.
38. A farmer's wife sold to a grocer 30 doz. eggs at 18f ^
per dozen, receiving in payment a barrel of flour at $ 5.50 per
barrel, and the balance in cash. How much cash did she
receive ?
39. A grocer bought apples at $ 1.50 per bushel, and sold
them at 50^ a peck. How much did he gain on each bushel ?
40. What is the cost of excavating 437.24 cu. yd. of earth
at $ 1.65 a cubic yard ?
41. A man sold J of his farm of 216 A. to one neighbor,
and f of it to another. How many acres were left ?
42. Four men, A, B, C, and D, together bought a ship for
$ 16,256. A paid $ 4756, B paid $ 763 more than A, and C
paid $ 256 less than B. How much did D pay ?
43. How many quarts of water can be poured into a tin box
that is 11 in. long, 6 in. wide, and 7 in. deep? (231 cu.
in. = 1 gal.)
44. Seven boys pick 4 bu. 3 pk. 7 qt. of berries and share
them equally. What is each boy's share ?
45. How many more pounds of candy can be bought for $ 1,
at 6^ per pound, than at 33^ ?
46. How many axes, each weighing 3 Ib. 8 oz., can be made
from a ton of iron ?
47. If a man walks 65 ft. in 1 min., how many miles will
he walk in 10 hr.?
48. How many steps, each 2 ft. 6 in. long, will a boy take
in going around a lot 5 ft. square ?
MISCELLANEOUS EXERCISES 213
49. A man has 285 bu. 3 pk. 6 qt. of grain, which he wishes
to take to market in 15 equal loads. How much must he
put into each load ?
50. A worker in a cotton mill weaves 6 cuts of cloth in a
day, receiving 16f ^ a cut. How much does she earn in a
week?
51. The circumference of a bicycle wheel is 3 ft. How
many times does it turn in running 18 ft.? 15 yd.? 4 rd.?
52. If a horse eats 2 bales of hay, costing 60^ a bale, and
2 bushels of oats, costing 30^ a bushel, in 1 week, how much
does it cost to feed him 1 year of 52| weeks ?
53. An automobile ran 67 f mi. in one day, and 1J times as
far the next day. How far did it run in both days ?
54. If Lucy washes dishes 3 times a day, how many times
will she wash dishes in the winter months, beginning Dec. 1,
1903?
55. One of the parallel sides of a trapezoid is 9 T 5 ^ ft. long.
The other is 3^ ft. longer. Each of the non-parallel sides is
5_^. ft. long. How long is the perimeter of the trapezoid ?
56. Multiply 26 by 11 by a short method.
This can be done by writing the sum of the digits between them.
Eight is the sum of the digits 2 and 6. This written between them gives
the number 286, which is the product of 26 and 11.
57. By 11 multiply :
33 72 54 81 45 71 60 22 70
58. Think of a number the sum of whose digits is less than
10. Multiply it by 11 by the ordinary written method, and
try to discover why the short method gives the same result.
59. Try to discover how the product should be written when
the sum of the digits is more than 9.
60. By 11 multiply :
36 48 79 85 91 87 58 69 75
214 ALIQUOT PARTS
61. A man had two plots of land fronting a street. The first
piece was 600 ft. wide, the second 900 ft. wide. He divided
them into house lots of the greatest possible equal width.
How wide was each lot ? How many lots were there ?
62. A man hoed a piece of land in 9| da., hoeing f of an
acre each day. How many acres were there in the piece ?
63. If 18 suits of clothing can be made from 101 yd. of
cloth, how many yards will be needed for 30 suits ?
64. Mr. Wade, dying, left $ 9600 to be divided equally
among his three sons, Arthur, Henry, and Joseph. How
much did each receive ?
65. Mr. Arthur Wade bought a lot for $ 1000, built a house
costing $ 2200, and rented it for $ 24 a month. The taxes on
the property for the first year were $ 75. The insurance was
$ 8. The house was vacant two months. At the end of the
year he sold the property for $ 2800. Did he gain or lose, and
how much ?
66. Mr. Henry Wade bought $ 3200 worth of mining stock,
on which his taxes were $ 13.75. He visited the mine at an
expense of $ 22.50. His yearly profits were $ 173.75. At the
end of the year he sold his mining stock for $ 3000. Did he
gain or lose, and how much ?
67. Mr. Joseph Wade bought a farm of 50 A. at $40 an
acre, with a house worth $ 1200. The taxes on the farm
were $ 41.50. He rented the farm, receiving as rent of the
value of the crops. The crops sold for $ 935.75. At the end
of the year the farm was worth as much as at the beginning.
Did he gain or lose, and how much ?
68. Scott & Co. of St. Louis bought a bill of goods in New
York amounting to f 836.75. They bought them with the
agreement that they need not pay for them for three months,
MISCELLANEOUS EXERCISES 215
but that if they chose to pay for them at once, 5% of the bill
would be taken off. They chose to pay at once. How much
did they pay ?
69. The same firm bought another bill of goods amounting
to $ 1573.84, and were allowed a discount of 5% for cash
payment. How much did they pay ?
70. Kinkle, Barbour & Co. of Springfield, 111., sold a bill of
goods to Luther Johnson of Jackson, Tenn., as follows :
1 doz. pairs Men's Shoes @ $ 1.50 per pair.
1 case (12 pairs) Children's Oxfords @ $ 75.
4 gross of Leather Laces at $ .50 per gross.
2 cases Children's and Misses' Oil Grain @ $ 1.00 per pair.
1 doz. Men's Kangaroo Calf @ $ 1.60 per pair.
% doz. Misses' Sandals @ $ .90 per dozen.
Make out the bill and receipt it, allowing 5% off for
cash.
71. Mr. Lang borrowed $700 at 7%. At the end of 3}
years how much did he owe ?
72. Mr. Lang borrowed $800 at 5% and at the end of 2
years made a payment of $ 200. How much did he still owe ?
73. Mr. Davis invested $625 in a mine, gained 7% on his
investment, and paid $ 5.25 in taxes on it. How much of his
profit remained ?
74. Find loss and ratio of loss to cost of goods :
a b c d e f g hi
Bought at 6^ 12? 8^ 8^ 16^ 20^ 30^ 25^
Sold at 5^ 9^ 7^ 5^ 14^ 15^ 25^ 20^
Express in per cent.
CHAPTER VII
PERCENTAGE
1. The Latin phrase "per centum," meaning "by the hun-
dred," is shortened to "per cent," and is represented by the
sign " %."
Express 15% as a common fraction. As a decimal.
2. Arrange the following in the order of their size, placing
the smallest first : T 3 ^. .33. 34%.
3. Express 21% in three different ways.
4. Find 7% of $ 20 by the following rule :
To find any per cent of a number
Multiply the number by the per cent expressed as a decimal.
5. Give reasons for the rule.
6. By this rule find 11% of $48. Of 96.
7. By the rule you have been using find 50% of 8. Show
an easier, way of finding 50% of 8.
8. Whenever a given per cent can be expressed as a common
fraction, a convenient method of finding that per cent of a
given number is to multiply the number by that fraction ex-
pressed in its lowest terms.
In this way find 16f % of 1200. Of 2400. Of 72.6. Of f
9. Give the value of each of the following fractions in per
cent:
216
PERCENTAGE 217
10. How much is 100% of a dollar? Of a day? If you
had only a dollar, could you spend 150% of it ? Explain.
11. What is 100% of 5 books ? 100% of 2 watermelons ?
12. The number or quantity of which a per cent is taken is
call the Base of percentage.
With 72 as a base find 111%. 444%. 88|%. 16f%.
13. Of a school of 40 children 20% were absent. How
many pupils were absent ? How many were present ? What
per cent of them were present ?
14. When you have written 40% of a spelling lesson of 20
words, how many words have you written ? How many are
yet to be written ?
15. Think of $1000; decrease it 10%, then increase the
remainder 33^%, decrease that result 25%, decrease $ 100,
increase 50%, decrease $200, increase 50%. What is the
last result?
16. How many different bases of percentage were given in
the preceding problem ? Name each.
17. Mr. Smith had $500, gained 20%, then lost 50% of
what he had, gained 33%, lost 25%, gained $300, lost 16f %.
How much had he then ?
18. Mr. A. had $600, lost 16|%, gained 25%, lost $25,
gained 33J%, gained 121%, l os t 331%, gained 10%, lost $160,
gained 25%. How much more had he than at first^?
19. Make a problem similar to the above.
20. A laborer was earning $ 1.60 a day when his wages were
cut 25%. What were his daily wages then ? After a time his
wages were raised 25%. How much less wages did he receive
then than he received before the reduction ?
21. If the rate of decrease and the rate of increase were
each 25%, why do they not balance ?
218 PERCENTAGE
22. Tell which of the following per cents are most easily
found by reducing to a common fraction :
a b c d e f g
31 % 16f% 16J% 37i% 11|% 35% 25%
h i ' j k I m n
871% 26 % 27 % 55f% 831% 85% 88f%
23. Use 144 as a base with each of the rates given in
Ex. 22.
24. Find 101% of 135. Of 146. Of 168. Of 179.
25. We may find 10% of 75 by placing a decimal point be-
tween the 7 and the 5. Explain.
26. Find in this way 10% of 25. Of 88. Of $ 17.50.
27. A cavalry regiment took 960 horses into a battle. If
8J% were killed, 121% were wounded, and 16f % were caught
by the enemy, how many were left for use ?
28. Mr. Ellis willed 25% of his property to his son, 62%
to his daughter, and the remainder to a library. After the
estate was settled there remained $ 153,600. How much did
each legatee receive?
29. Express each of the following ratios, first as a common
fraction, then as hundredths in decimal form, and then as
per cent:
3:6 7:10 8:40 1:11 9:72 6:24 24:36 36:48
30. What per cent of 35 is 18 ?
SOLUTION., 18 equals $f of 35. 35 = 100% of 35. f of 100% =
i|xW = ^,or51f Ans. 61%.
P& i
7
31. Illustrate the following rule :
To find what per cent one number is of another
Express the ratio of the one to the other as a common fraction,
and multiply that fraction by 100.
32. What per cent of 72 is 12 ? 9 ? 24 ? 36 ? 48 ? 72 ?
PERCENTAGE 219
33. Mary is 7 yr. old, and her brother is 21 yr. old. Mary's
age equals what per cent of her brother's age ?
34. Find what per cent each number in the first
column is of each number in the second column.
35. Find what per cent each number in the second g 25
column is of each number in the first column. g 4Q
36. What per cent of 25 is each number greater 10 18
than 10 and less than 20 ? 4 16
37. Find what per cent 9 is of each number between 20 and
30. Between 2 and 12.
38. John bought and sold peanuts, gaining 4? on every 6?
that he invested. What per cent did he gain?
39. What per cent is gained on goods bought at 12? and
sold ut 14?? Sold at 15?? 16?? 18?? 20?? 21?? 22??
40. What per cent is gained on goods :
abode f 9
BoughtatlO^? 11?? 8?? 15?? 5?? 16?? 16??
Sold at 15?? 14?? 15?? 18?? 71?? 25?? 32??
41. What per cent is lost on goods :
a b c d e f g
Bought at 20?? 18?? 25?? 15?? 7?? 10?? 10??
Sold at 15?? 16?? 20?? 10?? 5?? 7?? 6??
42. Some pupils were finding out facts about evaporation.
They placed a quart of water in a shallow pan and left it in the
sun until 3 gi. of it had evaporated. What per cent was left ?
43. A pan with straight sides was filled with water to the
depth of 6 in. The water was left to evaporate until it was
only 5 in. deep. What per cent of the water evaporated?
44. One evening Mrs. Eaton prepared batter for buckwheat
cakes. She set it to rise in a jar 10 in. high, which it half filled.
The next morning the batter filled T 9 ^ of the space in the jar.
What was the per cent of increase ?
220 PEKCENTAGE
45. Russell caught 12 fish, and Walter caught 8. What per
cent of all the fish caught did each boy catch ?
46. Two boys formed a partnership to mend bicycle tires.
The material cost $5.00, and they received $7.50. What per
cent was gained ?
47. At 8 o'clock one morning Mr. Field started on a trip of
60 mi. on a train running 25 mi. an hour. What per cent of
the distance had he traveled by 9 o'clock ?
48. Vincent and Fred bought a rabbit house. Vincent paid
$ 1.50, and Fred $ .50. What per cent of it does each boy own ?
49. In playing ball, John caught the ball 23 times out of 25.
What per cent of the throws did he miss ?
50. Two boys, wishing to know how much water dry bricks
would absorb, put 30 Ib. of brick into water. When the bricks
were taken out they weighed 32 Ib. What per cent of that
weight was water ?
51. Milk was poured into a straight glass jar to the height
of 10 in. The next morning the cream in the jar was ^ in.
thick. What per cent of the fluid was cream ?
52. Mr. Bates owed Mr. Weber $ 7.50 and gave him a $ 10
bill in payment. What per cent of the ten dollars was returned
in change ?
53. What per cent of a cubic foot are 512 cu. in. ? 576 cu.
in. ? 824 cu. in. ? 1024 cu. in. ? 504 cu. in. ? 1584 cu. in. ?
54. What per cent of $ 3.75 is $ 1.50?
If the decimal point in each term of the fraction ^ is moved two
O.75
places to the right, the value of the fraction is unchanged. |f can then
be reduced to its lowest terms, \.
20
f x W = 40. Ans. 40%.
P
It is often more convenient not to reduce the fraction to its lowest
termS ' aS 375)15000(40
1600 Ans. 40%.
PERCENTAGE 221
55. What per cent of $62.50 is $12.50? $37.50? $18.75?
$31.25?
56. Mr. Hull bought some goods for $ 87.50. How much
would he gain by selling them for $112.50? $93.75?
118.75 ? $131.25 ? $ 121.875 ?
57. Mr. Gordon bought groceries costing $ 218.75. What
per cent would he gain or lose by selling them for $ 243.75 ?
$193.75? $206.25? $212.50? $256.25? $225?
58. What per cent of 133 is 16| ?
59. Write 5 numbers that are multiples of 8, and find what
per cent each one is of 116J .
60. Write 5 numbers that are multiples of 3, and find what
per cent each one is of 150.
61. What per cent is gained by buying goods at $ .16 J a
yard, and selling them at $ .25 per yard ? $ .30 ? $ .331 ?
62. What per cent is lost on goods by buying them at $ .75
and selling them at $.70? $.62? $.66f? $.50? $.56J?
$.45?
63. After a line 5 in. long was increased 20%, what per
cent of 1 ft. was its length ?
64. Edwin lost 7 ^, which was 33J % of his money. How
much had he at first?
Consider 33|% as \.
65. Five cents is 25% of what sum of money ? 12|% ?
16f%?
66. Find the number of which 15 is 66|%.
Consider 66f% as f.
67. 66 J% of a railroad is in Missouri. If that part is 252
miles long, what is the length of the railroad ?
68. 35 is 87^% of what number ? 83^% of what number ?
222 PERCENTAGE
69. 6 is 3% of what number?
SOLUTION
MlfeM
then ^ = 2
and fl$ = 200 Ans.
70. By the same reasoning find the number of which 8 is
4%. The number of which 21 is 3%.
71. Tell of what number I am thinking if 35 is 7% of it.
If 4 is 3% of it.
72. CLASS EXERCISE. may think of a number and tell
the class how much a certain per cent of it is. The class may
find the number.
73. Can you see the reason for the following rule ?
To find the number of which a given number is a certain per
cent
Multiply the given number by 100 and divide the result by the
number of per cent.
74. Sixteen equals 8% of what number? 40% of what
number ?
75. Find the number of which $ 24 is 12%. 50%. 15%.
76. Find the number of which $45.75 is 5%. 15%. 25%.
77. How much money has a man if 7% of it is $3500?
$ 6216 ? $ 3110.80 ? $ 4498.20 ?
78. In a certain city, 75% of the telegraph wires are under
ground. If 300 mi. are under ground, how many miles are
above ground?
79. Mr. Allen sold Mr. Cummings 1240 bu. of corn, which
was 62% of his entire crop. How much was his entire crop?
80. During a storm at sea 600 bu. of grain were thrown over-
board. How many bushels were there on board before the
storm, if the number of bushels thrown overboard equaled
24% of the whole number ?
PERCENTAGE 223
81. Nine hundred acres of a Florida plantation are marsh.
How many acres are there in the whole plantation if the marsh
is 75% of it?
82. Mr. Leeds lost in a business transaction $ 1850.50, which
was 15% of his entire property. What was the value of his
property before his loss ? After his loss ?
83. Sixty years is 150 % of Mr. Harvey's age. How old is he?
84. Eighteen months is 75% of Edgar's age. How old is he?
85. Forty per cent of a load of hay was clover. If 1 T.
was clover, what was the weight of the load ?
86. A rectangle 9 in. long and 8 in. wide equals 3% of
another rectangle. What is the area of the larger rectangle ?
87. A rectangle 8 in. by 6 in. equals 4% of a rectangle
which is 6 ft. long. How wide is it ?
88. Selling a bicycle for $30 more than I gave for it, gives
me a gain of 25%. How much did I give for it ?
89. I gained 20% by selling a watch for $5.75 more than I
paid for it. How much did I pay for it ?
90. I sold some goods at a profit of $ 418, which was
of their cost. How much did they cost me ? For how much
did I sell them ?
91. By selling a house for $ 100 less than it cost I lost 5%.
How much did the house cost me ?
92. By selling a stove for $ 5.80 less than cost, I lost 29%.
What was the cost ?
93. I gained $340 by selling wheat at an advance of 20%.
How much did I pay for the wheat ?
94. A merchant sold silk at $ 2.20 per yard, which was 110%
of what it cost him. How much did it cost him ? What per
cent did he gain on it ?
224 PERCENTAGE
95. What is the cost of goods sold at 18^ a yard, which
is 20% more than the cost ?
SOLUTION. As the selling price is 20% more than the cost, it is
120% of the cost. If 18 jZ is |$ of the cost, the cost is jf$ or f of 18ft
which is 15 p.
96. Find the cost of tea sold at 85^ per pound, which
was 25% above cost.
97. Find the cost of goods sold at the following prices,
which are 25% above cost:
a Calico, @ $ .05 per yard. d Velvet, @ $ 1.85 per yard.
b Ribbon, @ .65 per yard. e Lace, @ 2.75 per yard,
c Flannel, @ 1.35 per yard. / Silk, @ 4.75 per yard.
98. Thirty per cent less than the cost of anything equals
what per cent of its cost ? Illustrate.
99. A wheel was sold for $ 28, which was 30% less than
cost. Find the cost.
100. A stove sold for $ 6 brought 40% less than cost. Find
the cost.
101. Arthur had $4.20, which was 40% more than his
brother had. How much had his brother ?
102. Mr. X raised 350 bushels of wheat this year, which is
16f % more than the number of bushels that he raised last
year. How much wheat did he raise last year ?
103. A grocer has 180 Ib. of dry sugar. In drying, it lost
10% of its weight. What was its weight before drying?
104. There are 240 A. in Mr. King's farm, which is 20%
less than the number of acres in his uncle's farm. How many
acres has the latter ?
105. A merchant had 380 yd. of cloth after it had shrunk.
What was its length before shrinking, if the shrinkage was 5% ?
PERCENTAGE 225
106. Twenty per cent of a pole was broken off. The part
remaining was 16 ft. long. What was the length of the pole
before it was broken ?
107. After using 10% of a load of coal I find there are
900 Ib. left. How many pounds were in the load ?
108. In Mr. Crosby's orchard there are 336 fruit trees;
16f % of them are plums, 12-J-% of them are pear trees, and the
remainder are peach trees. How many are there of each kind ?
109. A man whose wages are $14 per week, spent 8% of his
earnings last year for music lessons. How much did he spend
for the lessons if he worked 50 weeks ?
110. A man buys a horse for $ 200 and sells it for $ 46 more
than it cost him. What per cent does he gain ?
111. A bicycle is bought for $75 and sold at a gain of
$ 13.50. What per cent is gained ?
112. A watch costing $25 is sold at an advance of $4.75.
What per cent is gained ?
113. A pile of wood 50 ft. long, 6 ft. wide, and 5 ft. high was
25% oak, 33^% maple, 21% beech, and the rest walnut. How
many cords were there of each ?
114. The average lung capacity of men is about 320 cu. in.
If your lungs hold 75% as much, how much will they hold?
If your clothing were so tight as to reduce your lung capacity
40 cu. in., what per cent would the reduction be ?
115. The amount of food required by a man is about 51 Ib. a
day. If you ate 80% as much, in how many days would you
eat one ton and two pounds of food ?
116. A man's heart weighs about 11 oz. If your heart
weighs 70% as much as that, how much does it weigh?
117. Of a certain fence, 200 rd. were wire fencing, and the
rest rail fencing. If 60 % was rail, what was the length of the
fence ?
HORN. GRAM. SCH. AR. 15
226 PERCENTAGE
118. Forty-five miles of country road is macadamized. If
this is 20% of the whole road, how much remains to be
macadamized ?
119. How much will it cost to plaster a room 20 ft. 4 in.
long, 18 ft. wide, 8 ft. high, at $ .20 a square yard, if 12% of
the surface is covered with wainscoting ?
120. A certain school enrolls 23 boys and 22 girls. The
girls equal what per cent of the school ?
121. The same school has an average attendance of 42.
What per cent of the enrollment is the attendance ?
122. What per cent of the pupils in your school are boys ?
123. If 3 pupils belonging to your school were absent every
day, the attendance would be what per cent of the enrollment ?
124. A boy having broken his bicycle sold it for $ 20, which
was 33 1 % of the cost. What was the cost ?
125. I buy goods at $ 1.20 and sell them at a gain of 10%.
What is the selling price ?
126. I sell goods at $1.32, gaining 10%. Find the cost.
The gain.
127. I sell goods at a gain of $ 12, which is 10% of the cost.
Find the cost.
128. I buy goods for $ 120 and sell them for $ 12 more than
they cost me. What per cent do I gain ?
129. I buy goods for 1 120 and sell them for $ 132. What
per cent do I gain ?
130. Your age in years is what per cent of the age of a per-
son who is 4 yr. older than you ? 6 yr. older ? 10 yr. older ?
131. A man sold two horses for f 150 each, gaining 20% on
one and losing 20% on the other. Did he gain or lose by the
two transactions, and how much ?
132. An inkstand that cost 50 fi was sold at a gain of 100%.
What was the selling price ?
PERCENTAGE 227
133. If berries that cost 5^ a quart are sold for 10^ a quart,
what is the per cent of gain ?
134. How long will it take a dollar to gain 100% if it is at
interest at 5% ? At 6% ? 8% ? 9% ? 12% ?
135. Walter and Thomas sold lemonade at a fair. Walter
furnished $ .25 worth of lemons, and Thomas $ .50 worth of
sugar. To what per cent of the profits was each boy entitled ?
If they took in $ 2.64, how much was each boy's share of the
gain?
136. Three men raised a fund for charity. One man gave
$ 15, another $> 20, another $ 25. What per cent of the whole
did each man give ?
137. Mr. Low had his money invested in three houses as
follows : in the first $ 1620, in the second $ 8100, in the third
$3240. What per cent of his money was invested in each
house ?
138. Mr. Eves insured Mr. Croft's building for $ 1600, which
is 75% of its value. What is its full value ? If Mr. Croft
paid 1% upon the amount insured, how much did he pay ?
139. Mr. Jones's yearly income from a mine is $ 4000, which
is 15% of the sum he invested in it. How much did he
invest ?
140. A store is rented for $ 60 a month. The yearly rent is
8-J% of the value of the property. What is its value?
141. A lawyer collected some money for his client, receiving
for his services $ 80, which was 5% of the sum collected. How
much did he collect and how much did he pay over to his
client ?
142. Mr. Eoy gained 8% by selling his cow for $20 more
than it cost him. For how much did he sell it ?
143. Mr. Litch received $ 500 a year rent for one of his
houses, which was 7 % of its value. What was its value ?
228 PERCENTAGE
144. After Mr. Lane had paid 37% of his debts, he found
that $3568 would pay the remainder. What was his total
indebtedness ?
145. Mr. Bingham sold two houses for $ 7000 each; for the
first he received 12|% less than its value and for the second
16f % more than its value. What was the value of each ?
146. Mr. Allen bought for $ 3200 a store which had depre-
ciated 40% in value. What was the original value of the
store ?
147. When the Twentieth Century Novelty Company failed
in business they had $ 3350, which paid 67 % of their debts.
What was the amount of their debts ?
148. A grocer sold his stock for $ 2000, which was at a loss
of 10%. How much did the stock cost ?
149. Mr. Drew bought 5 doz. oranges. Of these 30% were
not good. How many oranges were good ?
150. A florist used in January 4000 bu. of coal. This
quantity was 36% of the number of bushels used during the
winter. How many bushels were used in all ?
151. Mr. M. receives $150 each ye.ar as the interest on a
sum of money which pays 6%. What is the sum of money ?
152. How much money must one have at interest that he
may receive from it $ 750 a year when the rate of interest
is 5% ? 6% ? 8% ? 4% ? 3% ?
MERCHANDISING
153. Buying and selling goods for profit is called Merchan-
dising. Those who carry on merchandising are called mer-
chants. Mention several lines of merchandising.
154. A merchant bought $ 6000 worth of dry goods, and in
the first year gained 20% of his capital. How much did he
gain that year ?
MERCHANDISING 229
155. He used $ 900 that year for living expenses, and put
the rest into his business. What was his capital at the begin-
ning of the second year ?
156. In the second year he cleared 33-^% on his capital.
What was his gain ?
157. The second year he used $ 1200 for his living expenses,
and put the balance into his business. What was his capital
at the beginning of the third year ?
158. In the third year he gained 37%% on his capital. His
expenses outside of his business that year were $ 1800. The
rest of his gain went into the business. What was his capital
at the end of the third year ?
159. In the fourth year he gained 44|-% on his capital, and
used $ 1700 for outside expenses. What was his capital at the
end of the fourth year ?
160. January 1, 1899, Mr. A. went into business with a capi-
tal of $10,000. At the end of the year he found that the
amount of his sales for the year had been $ 18,448 ; that his
business expenses, such as rent, clerk hire, advertising, etc.,
had been $ 6000 ; that he had spent $ 10,625 in replenishing
his stock of goods. How much had he gained that year ? His
living expenses were $1000. What was his capital Jan. 1,
1900, including the amount invested in goods ?
161. In 1900 he netted 10% on his capital, and his living
expenses were $ 1500. What was his capital Jan. 1, 1901 ?
162. In 1901 his profits were 7% on his capital, and his
living expenses were $ 1300. What was his capital Jan. 1,
1902?
SUGGESTION TO TEACHER. Lead pupils to realize some of the condi-
tions of merchandising, and let them make similar problems, tracing the
course of simple business ventures.
163. A merchant buys broadcloth at $ 2.40 per yard. How
shall he mark it that he may sell it at a gain of 40% ? 33-|% ?
35%? 50%?
230 PERCENTAGE
164. How much would he gain by selling 20 yd. at each of
those rates of advance ?
165. Mr. B. bought 40 yd. of novelty goods at $ 1.60 per yard.
He sold ^ of it at an advance of 75%, ^ of it at an advance of
50 /o i and the rest at 25 % advance. How much did he gain on
the whole piece ?
166. Mr. C. bought 60 yd. of goods at $ 1.20 per yard. He
sold i of the piece at a gain of 60%, -i- of the rest at a gain of
50%, 14 yd. at $ 1.30 per yard, and the rest at $.75 per yard.
How much did he gain on that piece ?
167. Mr. Evans bought goods at $.75 a yard and marked
them to sell at an advance of 33J%. Later in the season he
sold them at a reduction of 10% from the marked price. What
was the actual selling price ?
Find actual selling price, and amount of gain or loss on goods :
Bought Marked Sold
168. $.60 50% above cost 10% below marking
169. .80 25% above cost 30% below marking
170. .90 33^% above cost 16f % below marking
171. 1.75 $ 2.00 above cost 12J% below marking
172. 1.60 25% above cost 12^% below marking
173. CLASS EXERCISE. may mention prices at which
goods might be bought, marked, and sold, and the class may
find the amount of gain or loss.
COMMISSION
174. Instead of buying and selling for themselves, some
merchants buy or sell goods for others at a given per cent of
their value. The money which they receive for buying or sell-
ing is called their Commission, and they are called commission
merchants, agents, or brokers.
What is the commission on $ 900 at 10% ?
COMMISSION 231
175. Mr. Ward sold a piano for $ 575, receiving a commis-
sion of 10% on the selling price. How much was his com-
mission, and how much money should he send to the owner of
the piano ?
176. Mr. Clark is an agent for the Elliott Piano Co. which
pays him a commission of 25% on all his sales. During the
month of January, 1899, he sold one piano at $325, one at
$ 400, two at $ 250, and three at $ 200. How much were his
commissions, and how much should he return to the Elliott
Piano Co. ?
177. The expenses of his store for rent, heating, lighting
advertising, clerk hire, etc., for January, were $ 178.50. How
much did Mr. Clark make that month above expenses ?
178. His living expenses for that month were $ 109.75.
How much did he save ? Would it be safe to reckon his
yearly income on this basis ? Why ?
179. A cotton broker receives a shipment of cotton from
Alabama consisting of 400 bales. He sells it at $ 20 a bale,
receiving 11% commission. How much is his commission on
that sale, and how much should he return to the owners of the
cotton ?
180. John sold his sister's sled for her, receiving $ 1 for it.
When he gave his sister the dollar, she handed him a dime for
selling the sled. What per cent was his commission ?
181. Mr. Adams sold a wagon for the Melvin Wagon Co. at
$ 200, and after taking out his commission, sent the Wagon
Co. -$180. How much was his commission? What per cent
of -the value of the wagon ?
182. Mr. Gordon buys chickens for a Baltimore firm, re-
ceiving a commission of 15% on all the money spent in buying
them. If he buys $ 12,000 worth of chickens in a year, what
is his income for that year ?
232 PERCENTAGE
183. Mr. Wilson travels in the South, selling stoves. He
receives a commission of 10% on sales. When he has sold
$ 18,000 worth of stoves, how much has he earned ?
184. Mr. Wood buys hogs for a pork-packing establishment,
receiving a commission of 3% on the amount spent for the
hogs. If he buys $ 7500 worth in a month, what are his earn-
ings for that month ?
185. A commission merchant bought $ 5000 worth of pea-
nuts in Tennessee, and shipped them to a Chicago firm. The
cost of hauling them to the depot in drays was $ 2.75, the
freight charges were $ 47.50. The buyer's commission was
4% on the amount paid for the peanuts. What was the
amount of his commission ? How much did the peanuts cost
the Chicago firm ?
186. A commission merchant in Kentucky bought for a
New York firm 3000 Ib. of pecans at $ .06^ per pound. How
much was his commission at 2% ?
187. CLASS EXERCISE. may fill out the following, and
the class may find the amount received as commission:
!1 worth of r
sold . . ^
, , , I receiving % com'
g J mission.
188. People are sometimes hired to collect money due to
other people. These collectors receive a percentage on the
amount collected.
How much does Mr. Cox earn by collecting a bill of $ 600 at 2 % ?
189. Mr. Ho well gave Mr. Scott a list of bills to be col-
lected, 5% of the amount of which Mr. Scott was to receive as
commission. How much did he earn by collecting the follow-
ing bills, and how much did he turn over to Mr. Howell ?
John Andrews, Groceries, $ 28.75.
Charles Stockton, 21 Ib. Sugar, @ $ .05.
George Baldwin, 12 Ib. Turkey, @ $ .15.
Peter Garrison, 10 gal. Molasses, @ $ .45.
TRADE DISCOUNT 233
190. CLASS EXERCISE. - may fill out the following,
and the class may find the collector's receipts :
A collector collects , receiving for his work %.
191. A collector undertook to collect $2125 worth of bills
due a physician. He collected 67% of them, receiving 10%
as his fee. How much did he collect ? How much did he
keep ? How much did he send to the physician ?
192. Mr. Blair sold a carriage for the Hale Carriage Co. at
$450. His commission was 15%. The purchaser paid $300
cash, and Mr. Blair was obliged to pay a collector 10% on the
balance for collecting it. How much did Mr. Blair gain on
the sale ?
TKADE DISCOUNT
193. A reduction in the selling price of goods is called a
Discount.
Mr. Reed found that he could buy the bicycle that he wanted
for $ 70, on 3 months' time, or for $ 66.50 cash. How much
was the discount ?
194. How much is paid for a bill of goods invoiced at
$ 18.75, with a discount of 30%?
195. The employees of a certain large dry goods store are
allowed to buy goods from it at a discount of 10%. What
amount was paid for a bill of $ 17.25 bought by an employee ?
196. A man received a bill for $ 75. Printed on the bill
head were the words, "5% discount if paid within 30 days."
The bill was paid at the end of 4 weeks. What amount was
paid?
197. Hale & Co., Springfield, 111., bought of D. W. Lamont
& Co., St. Louis, Mo., 3 doz. plain gold rings, @ $ 20 per doz. ;
4 gold rings, diamond settings, @ $ 50 ; 6 gold watches, @
$ 15 ; 4 sets teaspoons, @ $ 6. Make out the bill, allowing a
discount of 25%.
234 PERCENTAGE
198. Mr. K. sold Mr. D. furniture for his home to the
amount of $ 75, on 60 days 7 time ; but Mr. D. accepted the
offer of 3J% off for cash. How much did Mr. K. receive for
the furniture ?
Many manufacturers and wholesale dealers have a fixed price for their
goods, called the list price. They sell to the retail dealers at a discount
from the list price. For instance, a manufacturer of carriages sends out
a catalogue containing a list of the different kinds of carriages he makes,
and their prices, with the discounts he gives to retail dealers.
199. Howard, Cowperthwait, & Co., who sell carriages,
select from a manufacturer's list a carriage whose list price
is $ 750. The discount on this carriage is 55%. How much
do Howard, Cowperthwait, & Co. pay for the carriage? If
Mrs. Douglas buys the carriage for $ 700, how much do they
make on it ?
200. At the same rate of discount, how much will be gained*
by Howard, Cowperthwait, & Co. if they buy four carriages
whose list price is $ 600, and sell them at the list price ?
201. Frequently more than one discount is given upon the
same purchase. If Howard, Cowperthwait, & Co. buy a $ 1000
carriage at 55% off, its cost is $ 450. If they get an ad-
ditional discount of 10%, it is reckoned on the $ 450 and
deducted from it. In that case, how much will the carriage
cost them ?
Observe that when there is more than one discount, each successive
discount is made upon a smaller sum of money than the preceding.
202. How much is left of $800 when it is discounted 25%,
then 33|%, then 25%, then 33% ?
Find cost of the following :
List price Discount List price Discount
203. $ 175 45 and 10 off 206. $ 500 20, 10, and 5 off
204. $ 370 35 and 5 off 207. $ 600 40, 30, and 2 off
205. $ 350 15 and 10 off 208. $ 800 50, 10, and 5 off
TRADE DISCOUNT 235
209. If you were buying goods should you prefer discounts
of 15% and 10%, or a straight discount of 25% ? Find the
discount on different sums at these rates, and state which terms
are more advantageous to the buyer, and why.
210. A dealer buys wagons whose list price is $ 60, at a
discount of 30%, 20%, and 5% off for cash. How much do
these wagons cost him ? He sells them at the list price with
5% off for cash. How much does he make on a cash sale ?
On a time sale ?
If a time sale is made in which the time exceeds a certain limit, the
buyer gives his note for the amount and pays the interest upon it. This
time limit varies with different firms, being usually not less than 60 da. ,
nor more than 6 mo.
211. How much is paid for a bill of goods invoiced at
$37.25, discounts 10, 15, and 5 off. How much is gained if
the goods are sold 10 % above the list price ?
212. Ball & Co., piano dealers, select from the manufac-
turer's catalogue, 3 pianos listed at $400, $450, and $700,
and order 6 pianos of each kind. The trade discount is 60, 10,
and 5. What is the cost of that shipment of pianos ?
213. Mrs. Fox pays cash for one of the $ 400 pianos sold at
list price, and gets a discount of 5%. How much do Ball &
Co. make on that sale ?
214. Mr. Shaw buys a $700 piano at list price, pays $75
cash, and gives his note for the balance, payable in 6 mo. At
the end of that time, finding himself unable to pay for it, he
returns the piano to Ball & Co., who spend $ 2 in polishing and
tuning it, and then sell it for $ 650. How much do Ball & Co.
make on that piano ?
215. Mr. King buys one of the pianos whose list price is
$ 450, on the installment plan, paying $ 10 a month. Ball &
Co. charge him $ 500 for the piano. How much do they make
on that sale ?
236 PERCENTAGE
216. Mrs. Lee buys from Ball & Co. a piano listed at $ 700.
She agrees to pay for it $750, in monthly payments of $15
each. After making 7 payments she returns the piano. Ball
& Co. spend $ 5 for repairs and sell it for $ 625. What is
their gain on it ?
SUGGESTION TO TEACHER. Require the pupils to bring to class similar
problems, describing business occurrences. Let them find out facts which
will enable them to keep the conditions of their problems within the range
of probabilities.
Find gain on the following goods bought at discounts of 10,
20, and 5 off, and sold at an advance of 20% on list price :
Items List price
217. 1300 yd. Carpet @ $ .75
218. 800 yd. Drapery Silk @ $ 3.50
219. 100 prs. Lace Curtains @ $6.75
220. On a bill of $ 675, what is the difference between
discounts of 40% and 20%, and a straight discount of 60%?
221. Mr. Dow, a merchant in Kentucky, goes to New York
twice a year to buy goods. He had been getting discounts of
30, 20. and 5% off for cash, but on his last trip he found that
he could get discounts of 30, 40, and 5%. If he bought goods
to the amount of $ 9000, how much was he benefited by the
change of discounts ?
222. The list price of an article with three different houses
is $400. One house offers discounts of 20, 10, and 5% ; the
second 5, 10, and 20% ; the third 10, 5, and 20%. Which is
the best offer, and why ?
INTEREST
223. Money paid for the use of money is called Interest.
The sum on which interest is paid is called the Principal.
If $ 250 is loaned at 6 % for 1 yr., how much is the interest ?
What sum is the principal ?
INTEREST 237
224. When you know the interest of a sum of money for 1
yr., how may you find the interest of the same sum at the same
rate for 2 yr. ? 7 yr. ? 31 yr. ? ^ of a year ? 2 yr. 1 mo. ?
Find interest :
Prin. Rate Time Prin. Rate Time
225. $ 200 6% 4yr. 232. $1800 5% 1 yr. 8 mo.
226. $ 700 8% 21 yr. 233. $ 400 7% 2 yr. 9 mo.
227. $ 300 7% 21 yr. 234. $ 600 4% 1 mo.
228. $ 800 8% 2 yr. 6 mo. 235. $ 800 6% 7 mo.
229. $ 900 5% 3yr. 4 mo. 236. $1200 3% 11 mo.
230. $ 600 4% 2yr. 1 mo. 237. $1800 9% 5 mo.
231. $5000 6% 3yr. 2 mo. 238. $1500 8% 10 mo.
There are many ways of calculating interest, all depending
upon this fact. Principal x Rate x Time = Interest. A 6%
method and a cancellation method are given in this book.
/Six Per Cent Method
239. From the first equation reason out the equations which
follow it. Learn them.
At 6% the interest of $ 1 for 12 mo. = $ .06
the interest of $ 1 for 2 mo. = $ k 01
the interest of $ 1 for 1 mo. = $ .005
the interest of $ 1 for 6 da. = $ .001
the interest of $ 1 for 1 da. = $ .0001
At 6 % what is the interest of $ 1.00 for :
yr.
mo.
da.
yr.
mo.
da.
240.
1
2
6
246.
8
10
18
241.
2
1
6
247.
9
3
1
242.
3
1
12
248.
7
5
5
243.
4
3
6
249.
8
9
7
244.
5
3
12
250.
6
11
8
245.
7
8
12
251.
7
7
9
238 PERCENTAGE
252. CLASS EXERCISE. may give a number of years,
months, and days, and the class may find the interest on $ 1
for that time at 6%.
253. Find the interest of $ 1 at 6% for 2 yr. 6 mo. 24 da.
and then find the interest of $ 2 for that time. $3. $7. $ 25.
254. Find the interest of $ 25.75 for 5 yr. 8 mo. 12 da. at 6%.
25.75
.342 The interest of $ 1 for 5 yr. 8 ino. 12 da., at 6%, is $.342.
5150 The interest of $25.75 is 25.75 times $.342. In practice it is
10300 more convenient to multiply 25.75 by .342. Of course the
7725 result is the same.
$8.80650
Find the interest of $ 125.37 at 6% for:
yr. mo. da. yr. mo. da.
255. 346 260. 297
256. 7 9 12 261. 3 10 8
257. 8 2 18 262. 659
258. 7 5 24 263. 5 5 10
259. 6 11 7 264. 11 2 11
Find interest at 6% :
265. $ 175.25 for 1 yr. 6 mo. 7 da.
266. $ 210.60 for 3 yr. 2 mo. 9 da.
267. $ 625.48 for 2 yr. 9 mo. 9 da.
268. $ 330.27 for 5 yr. 7 mo. 9 da.
269. $ 45.60 for 8 yr. 3 mo. 10 da.
270. $ 910.75 for 1 yr. 6 mo. 13 da.
271. $ 712.25 for 2 yr. 8 mo. 14 da.
272. $ 861.60 for 3 yr. 8 mo. 14 da.
273. $ 520.40 for 7 yr. 4 mo. 14 da.
INTEREST 239
274. What is the interest of $100 for lyr. at 6%? At 3% ?
275. What is the ratio of the interest of a sum of money for
a given time at 3%, to the ratio of the same sum of money for
the same time at 6% ?
276. Find the interest of the following amounts for 1 yr.
8 mo. 24 da., first at 6% and then at 3% :
$276 $24.76 $13.25 $417 $625
277. Find by the 6% method the interest of $318 at 5%
for 2 yr. 7 mo. 12 da.
SOLUTION. The interest of $ 318 for 2 yr. 7 mo. 12 da. at 6 % is $ 49.926.
At 1 % the interest is - as much or $ 8.321. At 5 % the interest is 5 times
as much as at 1 % or 41.605.
Required interest :
Prin. Rate Time
278. $ 28.35 6% 1 yr. 6 mo. 16 da.
279. $ 49.36 7% 2 yr. 9 mo. 12 da.
280. $ 30.75 8% 6 yr. 15 da.
281. $ 252.00 6% 4 yr. 7 mo. 27 da.
282. $ 160.00 7% 8 mo. 26 da.
283. $ 72.00 5% 6 yr. 8 mo. 13 da.
284. $ 75.00 4% 8 yr. 3 mo.
285. $ 112.00 5% 6 yr. 7 mo. 22 da.
286. $ 46.75 3% 2 yr. 11 mo. 20 da.
287. If you borrowed $ 100 and paid it back at the end of
one year, with the interest on it at 6%, how much would you
pay?
288. The sum of the principal and interest is called the
Amount.
What amount must be paid back when $ 200 is borrowed at
6% interest and kept 2 yr. 10 mo. 18 da. ?
240
PERCENTAGE
Prin.
Kate
289.
$900
6%
290.
$800
3%
291.
$600
4%
292.
$144
5%
293.
$672
7%
294.
$ 145.36
7%
295.
$816.35
5%
296.
$696
8%
297.
$ 216.25
4%
298.
$625
7%
299. CLASS EXERCISE.
Find the amounts of the following:
Time
7 yr. 4 mo. 6 da.
3 yr. 5 mo. 7 da.
11 yr. 2 mo.
8 mo. 12 da.
3 yr. 8 mo. 15 da.
2 yr. 2 mo. 1 da.
3 yr. 3 mo. 3 da.
5 yr. 7 mo. 13 da.
4 yr. 18 da.
9 yr. 9 mo. 9 da.
may mention a sum of money,
and the class may find the amount of it for any length of time
and at any rate which he may decide.
300. Find the time from Jan. 1, 1898, to July 7, 1899.
301. Mr. Monroe borrowed $ 300 Jan. 1, 1897, at 6%. What
was the interest March 1, 1898 ? Sept. 1, 1899 ?
302. On Nov. 7, 1896, what amount was due on $600 bor-
rowed May 1, 1892, with interest at 6% ?
303. Find the amount of $376.25 borrowed July 1, 1883,
and paid Nov. 13, 1887, with interest at 6%.
At 6%, what is the amount of $ 700?
304. Borrowed Sept. 7, 1898 Paid April 19, 1899
305. Borrowed June 15, 1895
306. Borrowed Dec. 12, 1891
307. Borrowed Aug. 6, 1880
308. Borrowed Feb. 29, 1896
Paid Oct. 3, 1898
Paid May 15, 1894
Paid May 12, 1885
Paid June 30, 1897
INTEREST 241
Cancellation Method
309. If the interest of a sum of money for a certain time is
$ 72, what will be the interest of that sum for J of that time ?
f of that time ?
310. The interest of a certain sum of money for a certain
time is $ 96. Find by cancellation the interest of that sum for
| of that time, f 1 of it. of it.
311. Find the interest of $ 400 for 2 mo. 20 da. at 3%.
SOLUTION. The interest of $ 400 for 1 yr. at 3 % is 400 x jfo. The
interest of the same sum for 2 mo. 20 da., or 80 da., is -gfe as much.
Hence the interest = ^ of $ 12.
4 2
Canceling we have x -|- x * = | = $2.66|
J0J5 jJpjJ 6
m
3
Find the interest of the following by the cancellation method :
Prin.
Rate
Time
312.
$ 276
12%
3 mo. 9 da.
313.
$ 184.50
4%
5 mo. 27 da.
314.
$1200
6%
1 mo. 21 da.
315.
$1400
5%
3 mo. 15 da.
316.
$1800
7%
3 mo. 20 da.
317.
$ 625
8%
90 da.
318.
$ 800
1%
63 da.
319.
$ 900
8%
100 da.
320.
$1100
6%
33 da.
321.
$2175
3%
93 da.
322.
$4150
6%
63 da.
323. Find the interest of $840 at 5% for 1 yr. 3 mo.
840 x T ^ x |f. When no days are given, find the number ol months
and express them as twelfths of a year.
HORN. GRAM. SCH. AR. 16
242 PERCENTAGE
Find interest :
Prin. Rate Time
324. $560 3% lyr. 8 mo.
325. $218.64 6% 2 yr. 1 mo.
326. $175.25 8% 1 yr. 6 mo.
327. $165.36 4% 4 yr. 2 mo.
328. $500 5% 7 mo. 6 da.
When days are given, reduce the whole time to days and express as
360ths of a year.
Find interest of :
Prin. Rate Time
329. $700 2% 1 yr. 1 mo. 6 da.
330. $750 6% 3 yr. 1 mo. 15 da.
331. $420 1% 2 yr. 1 mo. 10 da.
332. $800 9% 2 yr. 3 mo. 6 da.
333. $875 10% 1 yr. 8 mo. 20 da.
334. $630.25 7% 1 yr. 4 mo. 20 da.
335. There are some special rules for calculating interest
which are derived from the principles of the cancellation
method as,
To compute interest at 8 %
Multiply the principal by the number of days, move the decimal
point of the product two places to the left, and divide the result by 4$-
Find interest of $200 for 53 da. at 8%.
By cancellation method By special rule
2 53
45)1060(2.35+
90
It will be seen that dividing by 45 gives 160
the same result as multiplying by 8 and 135
dividing by 360.
225
INTEREST 243
336. Give the reason for the special rule for computing
interest at 8%.
337. Show how the following rules are derived from the
cancellation method :
(a) To compute interest at 5 %
Multiply the principal by the number of days, move the deci-
mal point of the product two places to the left, and divide the
result by 72.
(b) To compute interest at 6%
Multiply the principal by the number of days, move the deci-
mal, point of the product three places to the left, and divide the
result by 6.
338. Give a similar rule for computing interest at 9%. At
4%. At 12%. At 10%. At 3%.
Exact Interest
339. Usually 360 da. are considered one year, but some-
times calculations of interest are made, in which a year is
considered as 365 da. This is called Exact Interest. To find
the exact interest of a sum of money, use the cancellation
method, expressing the exact number of days as 365ths of
a year.
Find the exact interest of $900 from Dec. 1, 1898, to Feb. 12,
1899, at 8%.
SOLUTION. The exact number of days from Dec. 1, 1898, to Feb. 12,
1899, is 73 da., or ^ of a year.
Find exact interest of :
340. $300, 1 yr. 1 mo. 1 da. at 9%.
341. $240, 11 mo. 25 da. at 4%.
342. $336, 1 yr. 2 mo. 10 da. at 6%.
343. $430 from Oct. 15, 1897, to Jan. 11, 1898, at
244 PERCENTAGE
PROMISSORY NOTES
344. July 1, 1900, Mr. James Allen bought a horse of
Mr. William Brown for $125, paying $25 cash and giving
a promissory note, like the following, for the balance. The
note was paid in full when due. What amount was paid ?
ilntfi, neHUbi nT, A-i/x, njpn. n?/nl,
jpn, n/n/ruj/m,
REVENUE
STAMP
O.HW.
345. A Promissory Note is a written promise to pay money.
Who is the maker of the above note?
346. Who should keep the note until it is paid? What
should be done with it after the money is paid ?
347. Write a note promising to pay Robert Ruskin $300
with interest at 6% in one year from date.
348. The person to whom the note is to be paid is called
the Payee. Who is the payee of the note you have just
written ?
349. The sum mentioned in the note is called the Face of
the note. What is the face of your note ?
350. The date at which the note becomes due is called the
date of Maturity. What is the date of maturity of your note ?
351. Is it an interest-bearing note ?
SUGGESTION TO TEACHER. Procure blank forms of promissory notes
of different kinds, and let the differences between them be discussed in
class.
352. What is the amount of a note for $425 that matures
in 4 mo., interest being 5% ?
PROMISSORY NOTES 245
353. In some states the law allows 3 days more than the
specified time for the payment of a note, but interest is
exacted for these 3 days, called Days of Grace. If a note is
made payable Aug. 1, on what day is it really due, when grace
is allowed ?
In the problems of this book, days of grace are not to be considered
unless mentioned.
354. A note for. $ 400 is dated March 1, 1896, and made
payable May 1, 1896, with grace. It is said to mature May 1/4,
and interest is computed to May 4. What is the interest
at 6% ?
355. A note for $ 500, dated July 1, 1900, is made payable
in 3 mo. with grace. When does it mature, and what is the
interest at 6% ? At 5% ? At 7% ?
356. How much must be paid for the use of $ 625 from
June 1, 1897, to July 1, 1898, with grace, at 6% ?. At 3% ?
357. How much must be paid for the use of f 500 from
Dec. 1, 1899, to March 1, 1900, with grace, at 6% ? At 4% ?
358. What is the interest on a note for $ 300 at 6%, dated
Aug. 31, 1895, and made payable in 30 da., with grace ?
359. June 17, 1897, Mr. Kent gave a note for $500 at 6%,
payable in 60 da., with grace. When was the note due?
What was its face ? Its amount at maturity ?
360. Notes, being promises, may be varied to suit the inten-
tions of the parties concerned. Some notes draw interest from
date, some after maturity, and some not at all. Some are made
payable at a specified time, and are called Time Notes. Some
are made payable upon the demand of the holder for payment,
and are called Demand Notes. Some are made in such a way
that they can be sold or transferred to other persons, and are
called Negotiable Notes. Some are made payable only to a
certain person, and are called Non-negotiable notes.
What is the special advantage of a negotiable note? Of
a non-negotiable note ?
246 PERCENTAGE
Demand Note
$300.75. BOSTON, MASS., Sept. 20, 1900.
On demand, I promise to pay William D. Owen three
hundred and -^-fa dollars, with interest at 6 %. Value received.
EDWARD M. ARLINGTON.
361. What is due Jan. 19, 1901 ?
362. If the note above were not paid until May 20, 1902,
how much would be due?
363. How much would be due on the above note if it were
paid Sept. 25, 1902 ? January 11, 1903 ? March 1, 1903 ?
364. How does a demand note differ from a time note?
365. Write a time note for $ 500, due in 3 mo., at 4%.
Negotiable Note
$175.50. LOWELL, MASS., Sept. 15, 1899.
One year after date, I promise to pay to Henry Scott, or
bearer, one hundred and seventy-five and -ffa dollars, with
interest at 6%. Value received. MARY GREEN
366. What two words in the above note make it a negotiable
note?
367. If the negotiable note given above was paid when it
was 6 mo. past due, how much was paid ?
368. Write a negotiable time note for $600, interest 6%,
and find the amount of it when due. When 3 mo. past due.
11 mo. past due.
$ 1000. SAN FRANCISCO, CAL., Nov. 12, 1901.
One year after date, I promise to pay to the order of Ellen
Eames, One Thousand Dollars, with interest.
JAMES PORTER.
369. When a note includes the words "with interest/' but
gives no specified rate, interest is computed at the rate legal in
PARTIAL PAYMENTS 247
the state in which it is dated. Copy the above note, dating it
at the place where you live, and find the amount of it at
maturity under the laws of your state.
370. Write a note for $ 700 due in 3 mo., with interest after
maturity. Find the amount due on it 8 mo. after its date
under the laws of your state.
PARTIAL PAYMENTS
371. May 7, 1896, Mr. James Smith gave Mr. John Brown
a note for $ 700 payable on demand, with interest at 6%. How
much was due May 7, 1897 ? At that time Mr. Smith made a
partial payment of the amount due by giving Mr. Brown $ 442.
On how much money ought Mr. Smith to continue to pay
interest? How much was due May 7, 1898? At that time
Mr. Smith made another partial payment, giving $ 218. How
much was due on that note Nov. 7, 1898 ?
SOLUTION
May 7, '96, Mr. Smith owed Mr. Brown $ 700 Prin.
Int. on prin. from May 7, '96, to May 7, '97 . . . . 42 Int.
May 7, '97, Mr. Smith owed Mr. Brown $742 Am't.
May 7, '97, Mr. Smith paid Mr. Brown 442 Pay't.
Mr. Smith still owed Mr. Brown $ 300 New prin.
Int. on new prin. from May 7, '97, to May 7, '98 . . 18 Int.
May 7, '98, Mr. Smith owed Mr. Brown $318 Am't.
May 7, '98, Mr. Smith paid Mr. Brown 218 Pay't.
Mr. Smith still owed Mr. Brown $ 100 New prin.
Int. on last prin. to Nov. 7, '98 3 Int.
Nov. 7, '98, Mr. Smith owed Mr. Brown $103 Am't due.
372. When a partial payment is made, the holder of the
note writes upon the back of it the amount of money paid
and the date of payment. The writing is called an Indorsement,
and serves as a receipt for the amount paid. What were the
indorsements that Mr. Brown wrote ?
373. Aug. 1, '93, Mr. John Dow gave to Mr. Frank Rand
his note for $800 at 6%. Aug. 1, '94, he paid $218. Feb.
1, '95, he paid $ 223.90. How much did he owe Aug. 1, '95 ?
248 PERCENTAGE
SUGGESTION TO TEACHER. Let the pupils in one section of the class
enact the part of Mr. Dow in writing the note, and those of another
section take the part of Mr. Rand, making the indorsements upon the
notes written by the others. Let class discuss justice of the settlements.
$ 800. NEW ORLEANS, March 1, 1898.
For value received, 60 da. after date, I promise to pay to the
order of Amos Butler, Eight Hundred Dollars, with interest
at 6%. HOWARD CURTIS.
374. On the back of this note these indorsements were
written by Mr. Butler : Dec. 1, 1898, $ 300. June 1, 1899,
$ 222.08. How much was due March 1, 1900 ?
375. Make a problem in which you suppose that you give
a note for $ 900 due in 3 yr., with interest at 8%.
What amount would you owe at the end of 3 yr. if you made no pay-
ment before that time ? On what principal would the yearly interest be
reckoned ?
But suppose that instead of waiting until the end of the three years
you made a payment of $ 12 at the end of the first year. The interest
then due would be $ 72.
If, now, in this case, as in the previous problems, the payment $ 12
were deducted from the amount $ 972, and if the difference, $ 960, were
regarded as a new principal, observe that simply because you had made
a payment on the note, you would be charged interest on a greater prin-
cipal. Would that be just ?
376. To prevent injustice in such cases, the Supreme Court
of the United States has adopted the following rule :
UNITED STATES RULE. When the payment is less than the interest due
at the time of payment, no change of principal shall be made at that time,
but the interest shall be computed upon the same principal until the sum
of the payments shall equal or exceed the interest due.
Make a problem in which the first payment on a note is less
than the interest due when the payment is made.
$ 900. CINCINNATI, OHIO, Sept. 30, 1896.
One year from date, I promise to pay Henry Moore, or order,
Nine Hundred Dollars, with interest at 8%. Value received.
MARTIN CAMPBELL.
Indorsements : March 30, 1897, '9 16; Sept. 30, 1897, $ 56.
PARTIAL PAYMENTS 249
377. How much was due June 1, 1898?
SOLUTION. The first payment, $ 10, is less than the interest, $36, that
has accrued at the time this payment is made (int. of $ 900 for 6 mo. at
8% = $36). Therefore, we compute interest to the time of the second
payment. The interest of $900 for 1 year at 8% is $ 72, and the amount
is $972. Subtracting from this amount the sum of the payments
($16 + $56 = $72), we find that the new principal on Sept. 30, 1897,
is $900. The interest on $900 from Sept. 30, 1897, to June 1, 1898
(9 months), is $ 54. Therefore, the amount due June 1, 1898, is $ 954.
378. A note of $280 was dated June 25, '94, interest 6%,
indorsed $ 20, Jan. 25, '95. How much was due June 25, '95 ?
379. Face of note, $ 700. Date, July 2, '95. Kate, 8%. In-
dorsed, Jan. 2, '96, $ 225. Find the amount due Oct. 2, '96.
380. Face of note, $200. Date, Sept. 7, 1896. Rate, 7%.
Indorsed, March 7, 1897, $30. June 7, 1897, $40. Sept. 7,
1897, $ 60. Find the amount due Dec. 7, 1897.
381. When settlement is made within a year, the following
rule is generally used :
MERCANTILE RULE. Find the amount of the principal from date to
time of settlement. Find the amount of each payment from its date to
the time of settlement. Subtract the amounts of the payments from the
amount of the principal.
Find by this rule the amount due at the end of a year on
a note of $500 with interest at 6%, if a payment of $200
is made 4 months before settlement.
382. A note for $1000 was given Feb. 7, 1898. Kate, 6%.
Settlement was made 63 da. later. A payment of $200 was
made 30 da. before settlement, and 15 da. before settlement
$ 300 was paid. How much was paid at settlement ?
SOLUTION. The amount of the principal, $ 1000, from date of note to
time of settlement (63 da.), is $1010.50. The amount of first payment,
$200 (30 da.), is $201 ; the amount of second payment, $300 (15 da.), is
$300.75. Subtracting the amounts of the payments, $501.75, from the
amount of the principal, $1010.50, there remains to be paid $508.75.
Find amount paid at settlement applying Mercantile Rule.
383. Face of note, $60. Date, June 20, '85. Rate, 8%.
Indorsed, $ 20, July 6, '85. Settled, Aug. 23, '85.
250 PERCENTAGE
384. Face of note, $ 80. Date, Nov. 5, '91. Rate, 7%. In-
dorsed, $ 30, Dec. 5, '91. Jan. 5, '92, $ 25. Settled, Feb. 5, '92.
385. Face of note, $ 120. Date, Aug. 9, '93. Bate, 6%. In-
dorsed, Sept. 15, '93, $ 48. Oct. 1, '93, $ 45. Settled, Oct. 9, '93.
BANK DISCOUNT
386. Mr. James Gage sold a carriage to Mr. John Lyman,
price $ 600, terms $ 100 cash, and the balance by a note due
in 6 mo. without interest. As Mr. Gage wished to use the
money in his business, he took the $ 500 note immediately to
Mr. Peter Reed, who discounted it at 8 % ; that is, in exchange
for the note, he gave Mr. Gage what remained after the
interest of the $ 500 at 8 % for 6 mo. had been deducted from
the $ 500. How much did Mr. Gage receive for the note ?
At the end of the 6 mo. to whom should Mr. Lyman pay the
$ 500 ? How much did Mr. Reed make by the transaction ?
387. When a note is discounted, the payee indorses it, mak-
ing it payable to the one who discounts it. The payee is
then responsible with the maker of the note for its payment.
Mr. Gage wrote on the back of the note when he transferred it
to Mr. Reed,
Pay to the order of Peter Reed.
JAMES GAGE.
If when the note became due Mr. Reed should be unable
to collect the amount of it from Mr. Lyman, to whom could
he look for payment ?
SUGGESTION TO TEACHER. Let three pupils enact the parts of Mr. Gage,
Mr. Lyman, and Mr. Reed, one making, signing, and giving the note ;
another receiving, indorsing, and transferring it ; the third discounting it.
Let the class discuss the purpose and the justice of each step in the
transaction.
388. To discount a note is to take from its face the simple
interest on it for the time between the date of discounting and
the date of maturity.
BANK DISCOUNT 251
At 5%, what is the discount on a non-interest-bearing note
for $ 700 due in 60 da. ?
389. Discount that is found by computing interest for a
certain time is called Bank Discount.
How does it differ from trade discount ?
390. If you had a note which promised to pay you $ 300 at
the end of a year's time without interest, would it be worth
$300 now? If it were discounted at 8%, how much would
the discount be, and how much would you receive for the note
now?
391. The difference between the bank discount and the face
of the note discounted is called the Proceeds of the note.
What are the proceeds of a non-interest-bearing note for
$400 due in 6 mo. discounted at 10%?
392. Hale & Co. sold a carriage for $ 400 ; terms $ 50 cash,
balance by note, payable in 60 da., without interest. As they
wished to use the money at once, they sent the note to a bank
where it was discounted at 8%. What were the proceeds of
the note ? How much did Hale & Co. really receive for the
carriage ?
Find bank discount and proceeds of non-interest-bearing
notes for the following amounts:
393. $ 250, due in 90 da., discounted at 6%.
394. $450, due in 30 da., discounted at 9%
395. $900, due in 60 da., discounted at 8%.
396. $ 750, due in 60 da., discounted at 9%.
397. $900, due in 30 da., discounted at 6%.
398. $ 650, due in 100 da., discounted at 7 % -
399. A note for $800 due in 4 mo. was discounted at 6%
3 mo. before it was due. What were the proceeds ?
As the note had only 3 mo. more to run, it was discounted for 3 mo.
252 PERCENTAGE
Find proceeds of non-interest-bearing notes discounted at 8%
400.
Am't
$600
Date of note
Feb. 1, '97
Date of discount
Mar. 1, '97
Date of maturity
Apr. 1, '97
401.
$500
Feb.
15,
'96
Mar.
31,
'96
May 1,
'96
402.
$870
Sept.
1,
'95
Oct.
10,
'95
Dec. 1,
'95
403.
$660
June
4,
'97
July
I,
'97
Aug. 4,
'97
404.
$745
Apr.
6,
'84
June
3,
'84
July 9,
'84
405. Find proceeds of a note given April 15, 1894, due in
60 da., discounted May 15, 1894, at 7%.
Find proceeds of the following non-interest-bearing notes
discounted at 8% :
Face Date of note Time Date of discount
406. $ 38 June 7, '95 60 da. July 1, '95
407. $ 900 May 10, '84 90 da. June 10, '84
408. $ 850 Sept. 7, '90 60 da. Sept. 21, '90
409. $ 750 Mar. 15, '87 4 mo. May 1, '87
410. $ 1500 Oct. 2, '93 60 da. Nov. 1, '93
411. If you had a note promising to pay you $ 400 in one
year with interest at 6%, how much would it be worth at the
end of the year ? If that amount were discounted at 8%, what
would the proceeds be ?
As national banks do not usually discount long-time notes, if you
wished to obtain the money in advance on this note you might apply to a
savings bank or to one of those persons who deal in money and are called
capitalists, money lenders, brokers, or loan agents. Their rates of
discount and their ways of computing it vary. In the case of interest-
bearing notes, the discount is sometimes reckoned upon the face of the
note, sometimes upon the amount due at maturity, and sometimes the
face of the note is discounted at a rate per cent equal to the difference
between the rate of interest and the rate of discount.
412. A note for $ 1500 due in one year with interest at 7%
was taken to a money lender, who deducted 8% from the
BANK DISCOUNT 253
face. What were the proceeds? At the end of the year,
how much did the money lender receive ? How much did he
gain?
413. A note for $ 1500 due in one year with interest at 7%
was taken to another money lender, who calculated the amount
due at maturity and discounted that amount at 8%. What
were the proceeds ?
414. A note for $ 1500 due in one year with interest at 7%
was taken to another money lender, who agreed to discount it at
8%. He found the difference between the rate of interest and
the rate of discount, 1%, and took 1% of the face. What were
the proceeds ?
Find the proceeds of the following notes, discounted at 8%
by the method explained in Ex. 414 :
In these problems, the notes are supposed to be discounted on the
days on which they are dated.
Face of note Time Rate
415. $ 750 2 mo. 5%
416. 650 3 mo. 6%
417. 1700 1 mo. 6%
418. 1200 4 mo. 7%
419. 1000 60 da. 6%
420. Mr. Ashby bought an automobile for $ 1000 and sold
it for $ 1250, receiving $ 1000 cash and the balance in a note
due in 6 mo., with interest at 6%. On the day of the sale the
note was discounted at 8 % by the plan given in Ex. 414. How
much did Mr. Ashby gain by the sale ?
421. Mr. Day sold three bicycles, each of which cost him
$75, 011 the following terms: For the first he received $50
cash and a note for $50 due in 4 mo., interest 6%. For the
second he received $ 75 cash and a note for $ 25 due in 6 mo.,
interest 5%. For the third he received $ 99 cash. The notes
were discounted at date at 8% by the method used in Ex. 414.
Compare the profits on the three sales.
254 PERCENTAGE
INSURANCE
422. Mr. Adams has a house worth $ 7000. He has made
an agreement with the agent of an insurance company by
which, if the house is destroyed by fire, he will receive from
the company $ 5000, or, if it is injured but not destroyed, he
will receive a sum in proportion to the damage done. For this
insurance against loss by fire, he pays the company every year
1% of the sum for which the house is insured. How much
does he pay for insurance ?
423. The money paid for insurance is called a Premium.
Mr. Green, wishing to provide for his wife in case of his
death, has taken, for her benefit, an insurance of $ 5000 upon
his life. The company has agreed to pay her $ 5000 upon
proof of his death. For this he pays a premium of $27.35
a year, for each thousand dollars. If he pays premiums for
25 yr., what amount will he pay to the company ?
424. Mr. Strong, wishing to provide for his future, has taken
out what is called an endowment policy. This agreement
provides that if he is alive at the end of ten years, he shall
receive $3682.25, and in case of his death at any time during
the ten years, his heirs shall receive $ 2500. For this he pays
an annual premium of $ 347.47. To how much will his premi-
ums amount in the ten years?
425. There are two kinds of insurance. Property Insurance
and Personal Insurance.
To which kind of insurance does each of the three preceding
problems refer?
There are many kinds of property insurance, as insurance against loss
by fire, tornadoes, shipwreck, theft, unpaid debts, etc.
426. Mr. Campbell has a house worth $1800. If it were
insured for f of its value at 1 % each year, what would be the
annual premium? If the house were destroyed by fire, how
much insurance would Mr. Campbell receive ?
INSURANCE 255
427. What would be Mr. Campbell's annual premium if his
house were insured for j of the value, and the rate of insurance
were 1J%? How much would the premium be if the rate were
1^%, and the insurance covered of the value of the house?
428. A stock of goods invoiced at $ 10,500 was insured for
f of its value at l-j-%. How much premium was paid ?
429. A ship worth $ 75,000 was insured for f of its value at
If % . The cargo, valued at $ 7500, was insured for of its value
at 24%. Find amount of premiums.
430. Insurance companies generally insure property for a
period of years, as 1, 3, or 5 yr., charging a certain number of
cents on each hundred dollars insured. They also charge a
certain amount, usually $ 1, for the written agreement to pay
the insurance in case of loss. This written agreement is called
a Policy.
An insurance company insured Mr. Allen's house, worth
$ 1600, for f of its value, for a period of 3 yr., charging $ 1.30
for each $100, and $1 for the policy. How much did the
insurance cost him?
SUGGESTION TO TEACHER. Let an insurance policy be brought into
the schoolroom to be discussed and examined by the pupils. After solv-
ing the following problems let pupils compose similar ones.
431. Mr. Stevens takes out an insurance policy of $7000
for a period of 3 years. The 3 yr. rate is twice the annual rate,
which is $ .65 for $ 100. Policy, $1. Find cost of insuring.
Find cost of insuring the following property at the above
rates ; the policy costing $ 1 in each case.
432. Barn, $ 600. Hay, $ 300. f value insured, 1 yr.
433. House, $ 2700. Stable, $ 600. value insured, for 3 yr.
434. Stock of goods, value $6000, |- insured, 1 yr.
435 . House, $ 5000, $ 4000 worth of insurance taken for 3 yr.
436. House, $ 9500, $ 7000 worth of insurance taken for 3 yr.
256 PERCENTAGE
437. Mr. Eice takes $3000 worth of insurance from the
Helena Fire Insurance Co. for a period of 3 yr. at the rates
given above. Policy, $1. Six months later his house is
damaged by fire to the extent of $ 1000, for which amount he
receives a check from the company. How much better off
is Mr. Eice than he would have been had he taken no
insurance ?
438. Mr. Wood insured his house five times successively
for 3 yr. periods at $1.25 a hundred, each policy costing $1.
During the first period the house was insured for $ 8000, and
during the next period for $ 7500. He continued to reduce the
amount of insurance $ 500 each time he renewed it. The house
was never injured by fire. How much did he pay out for in-
surance on the house during those 15 yr. ? What did he
receive in return for his payments?
439. Mr. Charles Olney insures his furniture for $500 for 3
yr. The annual rate is $ .55 per hundred. The rate for 3 yr.
is twice the annual rate. What is the premium ?
440. John Gibson took out two policies: $3000 on dwell-
ing, and $12oO on furniture. Term, 5 yr. Annual rate, 40^.
5 yr. rate three times the annual rates. What was the
amount of both premiums ?
441. The trustees of Perry Township hold a policy on a
school building for $3250.00. Term, 5 yr. What premiums
have they paid, the rate being as in the previous problem ?
442. West & Co. insure their stock against wind storms, for
3 yr., for $20,000. Eate, 40^ a hundred for 3 yr. What is
the amount of premiums?
443. George Brown takes out a policy for 3 yr. : $3000 on
his dwelling and $500 on his furniture. Eate, $.90 for 3 yr.
Policy fee, $1. What is the cost of his insurance?
444. A house valued at $ 3000 and insured for f its value
was struck by lightning. The adjuster for the insurance com-
INSURANCE f TJ
pany estimated that it was damaged 37% of its value, and paid
that per cent of the amount insured. How much did the
owner of the house receive from the insurance company ?
There are several kinds of personal insurance, as Life Insurance, Acci-
dent Insurance, Endowments, Annuities, etc.
445. Mr. Blake took out a life insurance policy of $5000
for the benefit of his wife, upon which he paid $33.30 per
thousand, yearly premium. He lived 20 yr. How much
more was paid to his widow than he had paid to the insurance
company ? What would have been the difference between the
amount paid to the company by Mr. Blake and the amount
received from it by Mrs. Blake, if he had taken an insurance
of $ 10,000 ?
' Successful insurance companies take the small savings of those who are
not able to invest them to advantage, and, massing them, invest them profit-
ably. The insured loses the interest of the money which he pays to the
company, but receives an assurance that those for whose benefit he is
insured will receive the full amount in case of his death at any time.
How much more or less is received from the insurance com-
pany than is paid to it in the following cases ?
Amount insured Yearly premium per $ 1000 Years
446. $7,000 $30.70 20
447. 6,000 44.84 40
448. 10,000 28.28 21
449. 30,000 27.38 50
450. 50,000 35.65 5
451. 200,000 39.00 15
452. Mr. Bland takes out a life insurance policy for $ 4000,
paying $ 24 per thousand. What is the annual premium ? If
he pays it for 50 years, how much more does he pay than his
heirs receive ?
453. Mr. Corlen insured his life for $ 13,000 paying his first
premium of $ 57.50 per thousand on Jan. 1, 1901. He died
Mar. 1, 1901. If the company paid the agent a commission of
HORN. GRAM. SCH. AR. 17
258 PERCENTAGE
15% on the first premium for insuring Mr. Corlen's life, how
much did it lose by the insurance ?
454. Mr. Hill took out a life insurance policy for $ 25,000
at the rate of $32.75 per thousand. He died three months
after paying the first premium. How much more did his heirs
receive than he paid ?
455. Many insurance companies divide a part of their earn-
ings among those who are paying premiums, giving each one
a certain per cent on the premium he pays. This amount is
called a Dividend.
Find the value of a dividend of 5% on a premium of $ 250.
456. For 25 yr. Mr. Field paid an insurance premium on
a life policy for $ 7000 at the rate of $ 27.50 per thousand.
Ten per cent of the amount of the premiums was returned to
him in dividends. How much did his insurance cost him ?
457. Mr. A., who is a traveling salesman, carries an accident
policy. When he had paid $ 205.75 in premiums, he was acci-
dentally injured and received an allowance of $25 per week
for 7 weeks. How much more or less did he receive from the
company than he had paid to it ?
458. At 25 yr. of age Mr. B. took out an endowment policy
by which he will receive $ 5000 when he is 45 yr. old. An-
nual premium, $ 240.38. How much more will he receive from
the company than he pays to it ? How can the company afford
to do business in that way ?
SUGGESTION TO TEACHER. Let pupils make problems under various
imaginary conditions, getting facts about insurance from agents or circu-
lars in order that their problems may approximate to the actual.
459. Neil & Co., agents for the Westchester Insurance Co.,
insured the following risks for periods of 3 yr. at $ 1.30 per
hundred. Their commission was 15% on the premiums, and
they received from the insured a policy fee of $ 1 in each case,
which they retained. How much did Neil & Co. earn and how
much did they send to the insurance company ?
TAXES 259
Dwelling, value f 1,800, f value taken
Store, value 15,000, -| value taken
Stock of goods, value 17,000, % value taken
Opera house, value 75,000, f value taken
460. Miss Otis bought from an insurance and annuity com-
pany a yearly annuity of $ 100, paying for it $ 1382.50. How
much more or less would she receive from it than she paid for
it if she lives 20 yr. ? 7 yr. ?
461. Mrs. Green owns a house from which she receives a
monthly rental of $ 25. The insurance on it for the year 1898
was $ 8 and the taxes were $ 36.75. It was vacant three
months. How much was her net income from it?
TAXES
462. It is necessary for all governments to tax the people
to pay public expenses. Taxes upon property are calculated
at a certain per cent of the assessed value.
"At 1% how much are the taxes upon a piece of property
worth $ 9000 ?
SUGGESTION TO TEACHER. Explain the duties of the assessor and tax
gatherer. Procure copies of a part of an assessor's list of taxables and
let pupils compute the taxes.
463. How much are the taxes upon Mr. Hudson's property
which is assessed at $ 8000, where the rate of taxation is
464. A fixed sum assessed, without regard to their property,
upon male citizens who are at least 21 yr. of age, is called a
Poll Tax.
If Mr. Hudson lived where the poll tax was $1.50, how
much would all his taxes be ?
465. Mr. Howe, who pays a poll tax of $ 2, owns property
assessed at $ 6000, in a city where the rate of taxation is $ .75
per $ 100, and $ 2500 in another city where the rate of taxation
is 2^%. What are his taxes ?
260 PERCENTAGE
466. What are the taxes of Mr. Hearn who owns real estate
assessed at $ 11,375, and other property valued at $ 2500, the
rate of taxation being l-g-%, and his poll tax $ 1 ?
467. Property is considered to be of two kinds : Real Estate,
as lands, houses, stores, factories, mines, and other immovable
property ; and Personal Property, such as money, notes, furni-
ture, and other property which can be carried from place to
place.
Name a piece of real estate. Name different kinds of per-
sonal property.
468. Mrs. Kent owns real estate assessed at $ 5600, and per-
sonal property assessed at $ 1000. The rate of taxation is
$ 1.20 on $ 100. What is the amount of her tax bill ?
469. A penalty of a certain per cent of the amount of the
tax is sometimes enforced if the taxes become delinquent ; that
is, if they are not paid at the time required. If Mrs. Kent
allows her taxes to become delinquent, and the penalty is 10%,
what will be the amount of her tax bill ?
470. In a certain city the rate of taxation is $ 1.35 per $ 100,
and the poll tax is $ 1. How much are the taxes of an
adult male citizen whose real estate is valued by the assessor
at $ 1540, and personal property at $ 300 ?
471. What is the tax of an adult female citizen whose real
estate is valued at $ 1500, and personal property at $ 650 ?
472. Mention several things which are paid for with the
money raised by taxation.
473. Some property, as churches, government bonds, and
public property of all kinds, is exempt from taxation. Why
are public school buildings not taxed ?
474. A certain town raised $ 21,845 by taxation, a part of
which was the assessment of 721 polls at $ 2 each. How much
tax was raised from property ?
TAXES 261
475. Taxes to the amount of $ 24,704.35 were raised in the
town of Nalasco, of which $ 23,456.35 was raised from property,
and the rest from polls at $ 2 each. How many citizens paid
poll taxes in the town ?
476. In a town where there are 1236 polls assessed at $ 1.50
each, it was decided to raise $ 46,854 by taxation. How much
must be raised from the property ? If the property valuation
of the whole town was $ 3,600,000, how much must each dollar's
worth of property yield ?
What would be the tax of each of the following residents of
that town ?
477. Mr. A., 31 yr. old, whose realty is $ 7000, personals
$ 1375.
478. Mr. B., 19 yr. old, whose realty is $ 938, and who has
$ 4500 worth of personal property, of which $ 3000 is in
government bonds.
479. Mrs. C., whose realty is $ 7500, personal property $ 635.
480. Mr. D., 43 yr. old, who has no property.
481. Miss E., who has no real estate and $500 worth of
personal property.
482. The money which defrays the public expenses of cities,
counties, and states is raised by direct taxation upon property
or person. Money for the expenses of the national government
is raised by indirect taxation, of which there are two kinds,
Internal Revenue and Duties or Customs.
The internal revenue is mostly derived from taxes on the
manufacture of liquors and tobacco products and from the sale
of stamps which the government requires to be placed upon
certain legal documents and articles sold.
To defray the expenses of the war with Spain in 1898, a law
was passed by Congress requiring among other provisions that
a one-cent stamp should be affixed to every telegraphic message
or express receipt, a two-cent stamp to every bank check,
UNIVERSITY
262 PERCENTAGE
sight draft, etc., two cents for every hundred dollars, or frac-
tional part thereof, named in the face of time drafts, prom-
issory notes, etc., and stamps of different values upon patent
medicines, proprietary articles, insurance policies, contracts,
leases, etc.
What should be the value of a stamp affixed to a promissory
note for $ 2500 ? For 9 275 ? For $ 39.50 ?
483. A drug firm sold in one week 1216 bottles of patent
medicines, each requiring a stamp whose value is f of a cent,
1172 packages each requiring a stamp costing 1^ ^, and 298
packages each requiring a stamp costing 21 ^. The firm sent
15 telegrams and 137 express packages. Fifty-one checks
were given by the firm. How much revenue accrued to the
government from the sale of stamps necessary for the business
of that firm for that week ?
484. The taxes levied by the government upon imported
goods are called Duties or Customs. All goods which come into
the country must be brought in at certain places called Ports
of Entry. At these places the government maintains custom
houses, with officers who collect the duties.
There are two kinds of duties, Specific and Ad Valorem.
A duty of a certain per cent of the amount at which the
goods were invoiced in the country from which they were im-
ported is called an Ad Valorem Duty. The Latin phrase ad
valorem means " according to value."
Find the ad valorem duty of 100 yd. silk invoiced at $ .50
per yd., duty 55%.
At the rates given, how much ad valorem duty would be
paid by a firm of importers upon the following goods ?
485. 50 yd. of silk, invoiced at $ 1.25 per yd., duty 55%.
486. 500 pieces of ribbon, 10 yd. in a piece, at 75^ per yd.,
duty 40%.
487. 50 yd. of lace, at $ 2.25 per yd., duty 60%.
TAXES 263
488. At 20%, what is the duty on 75 bales of wool, 400 Ib.
each, invoiced at 25 ^ per pound ?
489. At 25%, what is the duty on 500 boxes of raisins, each
containing 40 Ib., costing 6J cents per pound ?
490. A duty levied upon a certain quantity of goods, with-
out reference to their value, is called a Specific Duty.
If the specific duty is $ 2.25 per dozen pairs, how much is
that duty on 600 pairs of gloves invoiced at 50 f a pair ? If
they were invoiced at 75^ a pair, what would be the specific
duty ?
491. Sometimes both specific and ad valorem duties are
levied upon the same article.
What is the duty on 30 pieces of carpet, 25 yd. each, in-
voiced at $1.75 per yard, the specific duty being 25^ per
yard, and the ad valorem duty 40% ?
492. What is the duty upon 800 Ib. of cigars, invoiced at
$5 per pound, which pay a specific duty of $ 4.50 per pound
and 25 % ad valorem ?
493. An importation of. silks from France was invoiced at
9324 fr. At 60% ad valorem, how much is the duty in American
currency, $ 1 being considered equal to 5.18 fr. ?
494. What duty is paid by an American importer upon
600 doz. pairs of gloves invoiced at 60 fr. per dozen, if there
is a specific duty of $2 per dozen pairs and an ad valorem
duty of 40% ?
495. Persons are allowed to bring from abroad a limited
amount of goods for their own use without paying duties upon
them. An American lady brought home from Europe a silk
dress pattern, upon which the duty was $ 31.75 ; ^ doz. pairs
of kid gloves, upon which the duty was $ 2.25 per dozen pairs ;
20 yd. of lace worth f 2 per yard, upon which the duty was
ad valorem; and 12 yd. Irish linen at 60^ per yard, ad
264 PERCENTAGE
valorem duty 35 %. As these goods were for her own use, they
were passed in duty free. How much less did the goods cost
her than they would have cost had the duty been collected ?
496. If an importer buys 700 yd. of velvet at $1.50 per
yard, pays an ad valorem duty of 60%, and sells it at $ 4 per
yard, how much does he gain ?
497. How much is gained by an importer who buys 20 pieces
of matting, 40 yd. in a piece, at $ .10 per yard, pays a duty of
25%, pays for transportation $60, and sells the matting at
35^ per yard?
498. A list of articles upon which duties must be paid, with
the special duty upon each, is called a Tariff. Tariffs are
changed from time to time by acts of Congress.
An importer brought through the custom house $ 80,000
worth of cut glass when the duty was 35% ad valorem. He
sold | of it at a profit of 25% upon invoice price plus the duty.
The tariff upon glass was raised after his purchase to 60%. He
sold the other half of his stock at a profit of 25% upon invoice
price plus the new duty. How much more did he gain on the
last half of his stock than on the first half ?
499. A New York firm imported goods invoiced at $64,000,
upon which there was a duty of 12^% ad valorem. For how
much must these goods be sold to give a profit of 20% ?
500. Soon after those goods were bought, the duty on that
class of goods was changed to 25% ad valorem. Another firm
imported $ 64,000 worth of the same kind of goods under the
new tariff, and sold their goods at a profit of 20%. How much
did they receive ? If the first firm sold their goods for the
same amount as the second firm, how much more did they gain
than the second firm gained ?
501. Mr. Gilman imported $ 100,000 worth of goods, the
duty upon which was 30% ad valorem. If, after he sold J of
them at a profit of 20%, this class of goods was put on the
MISCELLANEOUS EXERCISES 265
free list, for how much could his competitors in business
buy an amount of goods equal to what he had left on hand ?
If he sold the rest of his goods for 20% more than that
sum, would he gain or lose on the whole transaction, and
how much ?
502. Make a problem in which an importer's business is
injured or benefited by changes in the tariff.
MISCELLANEOUS EXERCISES
1. Resolve into prime factors 6750. 7920.
2. Find the g. c. d. of 235 and 685.
3. Find the 1. c. m. of 8, 10, 12, 16, 18, 20.
4. Divide .012261 by 2.01.
5. 3V64 + 7V81=?
6. 2A/64 + 4^/125 = ?
7. If 1 qt. of nuts costs 11^, how many bushels can be
bought for $ 13.20 ?
8. Find the interest of $ 1240 for 5 yr. 9 mo. 27 da. at 3%.
9. A wheel of a bicycle is 7 ft. in circumference. How
many times does tjie wheel turn in going 10 rd. 1 yd. ?
10. John weighs 115 lb., and his cousin weighs 110 Ib.
John's weight is what per cent of the sum of their weights?
11. Write the following decimally and as common fractions
in their lowest terms: 13%. 18%. 22|%. 158%. 875%.
12. How much is 30% of 40 minus 16|% of 66 ?
13. Find 11% of 24 2 . 37^% of 16 2 . 62i% of 64 2 .
14. Find 12|% of 12 3 . 66f % of 9 3 . 87J% of 12 3 .
15. Square: f. .3. 1.2. 2f .06. ?i.
266 PERCENTAGE
16. Name four numbers between 100 and 200 that are
perfect squares and give their square roots.
17. What number between 100 and 200 is a perfect cube ?
18. Every prime number greater than 10 must end with
either 1, 3, 7, or 9. Give the reason.
19. The arc AB is 24 in. long. BC is 50% longer than AB.
CD is 33% longer than BC. DA is 25% shorter than DC.
How long is the circumference ? Diam-
eter ? Radius ? Perimeter of the sector
DO A? BOG'! DOC? AOB?
c 20. A circumference which is 65 in.
long is divided into two arcs, the smaller
arc of which is 13 in. long. The smaller
arc is what per cent of the greater ?
21. How many coins an inch in diam-
eter could be placed in rows touching one another on a rectangle
4 in. by 3 in. ? Represent.
22. When the hour hand of a clock is at 3, what per cent of
one revolution around the clock face has it made since 12 ?
23. At 4 P.M. the time past noon is what per cent of the
time before midnight ?
24. The time past noon is what per cent of the time to mid-
night at 2 P.M.? 8 P.M.? 1.30 P.M.?
25. Thomas wished to add the fractions ^, , and i. He
first multiplied each term of each fraction by the product of
all the denominators except its own. How were the three
fractions then expressed? He then added these fractions
and reduced their sum to its lowest terms. Was his process
correct? Can you show a better way to find the sum of
these fractions ?
26. Take Ex. 25, substituting the fractions, , -f, and Jy.
MISCELLANEOUS EXERCISES 267
Fill blanks and solve :
27. A house worth dollars was insured for of its
value, at per cent. What was the annual premium ?
28. Mr. A. held a life insurance policy for $ 2000, on
which he paid an annual premium of $ 52. He was insured
March 1, 1890, and died June 1, 1900. How much more did
his heirs receive than he had paid out in premiums ?
29. A room 36 ft. long and 24 ft. wide is to be covered with
carpet f yd. wide, at $ 1.10 per yard. How much will it cost
if the strips run lengthwise of the room and each strip is
turned in 4 in.?
30. Advance the following goods 15% in price: Caps at
30 t, coats at $ 8, shoes at f 1.25, gloves at 78 f, ties at 15
31. A fisherman caught herring enough to fill 500 barrels.
He sold 35% of the catch, and kept the rest for a rise in price.
How many barrels of herring did he keep ?
32. How many quarts of berries at 12-^ a quart would be
required to pay for 9 yd. of cloth at 16^ a yard?
33. Two men traveled from the same point, one east, 45^
mi. ; the other west, 92f mi. How far apart were they ?
34. Two men started from the same point, and traveled in
opposite directions. One man traveled at the rate of 7^ mi.
per hour, the other at the rate of 6J mi. per hour. How far
apart were they at the end of 1 hr.? Of 3 hr.? Represent.
35. John's uncle showed him a half eagle one morning, and
promised to give him at night 25% of all of it that was not
spent. At night his uncle reported that 100% of the money
had been spent, but he gave him 75 $ instead. How much
more or less would John have received if his uncle had spent
only 50% of the value of the half eagle ? 40% ?
36. Nine is how much greater per cent of 144 than of 288 ?
268 PERCENTAGE
37. How much is gained on each tablet bought at the rate
of $ 1 per dozen, and sold at 10 $ each ?
38. Mr. Hall earned $ 125 in one month, which was 62
of his earnings the next month. How much did he earn in
both months ?
39. An apple tree bore 21 bu. of apples, which was 87 -J%
of what the tree next to it bore. What was the difference in
the yield of the two trees ?
40. An automobile started from New York, and ran 60 mi.
the first day. On the next day its speed was 33 \] greater
than on the first day, and on the third day it was 25% greater
than on the second. How far was the automobile from New
York at the end of the third day ?
41. What number plus 1% of itself equals 909 ? 2424 ?
42. Forty-five is 50% more than what number? 50% less
than what number ?
43. A man had $ 654 in bank. He drew out 33% of it,
and afterward drew out 25% of the remainder. How much
had he left in bank ?
44. A man sold a wagon for $ 180, and gained 25%. What
was the cost of the wagon ?
45. A man sold a wagon for $180 and lost 25%. What
was the cost of the wagon ?
46. Near the close of summer the price of goods costing
$1.10 per yard was cut to 95^ a yard. What per cent was
lost?
47. Fifty yards of cloth were bought for $30. For what
price per yard must they be sold to gain 25% ?
48. A house valued at $8000 was insured for f of its value
at 1J%. What was the premium ?
49. On the day before Christmas Mary counted at a cer-
tain corner 37 ladies who were carrying packages, and 13 who
MISCELLANEOUS EXERCISES
269
had no packages. What per cent of the ladies that she counted
had no packages?
50. Thirteen children were transferred from a class of 42.
What per cent of them remained ?
51. In making peach marmalade, Mrs. Harland boiled 4 Ib.
of peaches and 3 Ib. of sugar in a quart of water. Each pint
of water weighed a pound. If 1 pt. of the water evaporated in
cooking, what per cent of the marmalade was sugar ? Peaches ?
52. How wide is a rectangle 20 cm. long and equal to f of a
square decimeter ? Represent.
53. In the rhomboid ABCD the
line BC represents 10 ft. AB is
50% longer than BC. How long is
the perimeter ? What per cent of
FIG. 2. the perimeter is AD ? DC?
54. AB and DC are parallel. DC
= 48 ft. AB = 25% of DC. BC =
250% of AB. AD = 100% of BC.
Find the perimeter of the trapezoid.
55. A farm is in the shape of a
trapezoid. The shorter parallel side
is 16 rd. long. The longer parallel side is 121% longer. One
of the non-parallel sides is 10 rd. and the other is 20% longer.
Represent. Find the cost of fencing the farm at 75^ per rod.
56. What is a trapezoid? How does it
differ from a rhomboid?
57. A four-sided plane figure which has no
two sides parallel is called a Trapezium.
Draw a trapezium.
58. How long is the perimeter of a trape-
zium of which the side AB is 3J- in., the
side BC 5 in., the side CD 6J in., and DA
FIG. 4. 7|in?
FIG. 3.
270 PERCENTAGE
59. A garden is fenced in the form of a trapezium. One
side is 4 rd. 3 yd. 2 ft. 8 in. long, another side is 5 rd. 1 ft.
10 in. long, another side is 4 rd. 5 yd. 4 in. long. The other
side is 6 rd. 2 yd. 6 in. long. How long is the fence ?
60. How long is the perimeter of a trapezium, the shortest
side of which is 12 in. long, the next side 2 in. longer than the
first, the next side 3 in. longer than the second, and the last
4 in. longer than the third ?
B 61. In the trapezium ABCD, AD re-
presents 8 ft. DC represents 50% more
than AD. CB represents 33^% more than
DC. BA represents 121% more than CB.
How long is the perimeter ?
62. John had a kite frame in the
shape of a trapezium having two short
FlG 5 equal sides and two long equal sides.
If a long side was 3 ft. long and a short
side 33|% as long, what was the combined length of the sticks
that made the frame, allowing -J- an inch for lapping the sticks
at each angle ?
63. Plane figures bounded by four straight lines are called
Quadrilaterals.
You have learned six different kinds of quadrilaterals. Draw
one of each kind and write its name upon it.
64. How long is the perimeter of a rhombus whose sides are
each 1.7 in. ?
65. Find the perimeter of a rhomboid whose long sides are
each 9.9 in. and whose short sides are each 5 in. less than a
long side.
66. How long is the perimeter of a trapezoid if one of the
parallel sides is 7.65 in., the other 8.45 in., and each of the
non-parallel sides is 4.7 in. ? Kepresent.
67. Plane figures bounded by straight lines are Polygons.
Name four kinds of polygons.
MISCELLANEOUS EXERCISES 271
68. What name is given to a polygon of 3 sides ? 5 sides ?
6 sides? 8 sides? 10 sides?
69. Is a sector a polygon ? Explain.
70. Mr. K. bought 9 doz. pencils for $ 2.16. He sold them
at $ .03 apiece. What per cent was gained ?
71. 500 sacks of coffee were bought for $300. At what
price per sack must they be sold to gain 10% ?
72. 350 bottles of ink were bought for $ 21. For how much
per bottle must they be sold to gain 66f % ?
73. 280 penknives cost $ 70. For how much apiece must
they be sold to gain 25% ?
74. A dealer paid $ 24 for 300 slates. In selling them he
gained 50 % . What was the selling price of each ?
75. Two gross of handkerchiefs were bought for $28.80.
At what price apiece must they be sold to gain 30% ?
76. Mr. Fowler's salary was $ 1800 a year. Last year he
paid $ 1175.50 for household expenses, $ 22.50 for life insur-
ance, $ 15.75 for taxes, $ 178.25 for clothing, and $ 23.75 for
incidentals. What per cent of his salary did he save ?
77. A grocer sold 630 heads of cabbage, which was 66|% of
what he had. How many had he left ?
78. It cost $ 15 to build a certain fence, and $ 10 to paint
it. The cost of painting was what per cent of the whole cost ?
79. A quart of water was added to 6 gallons <ff cider. What
per cent of the mixture was water ?
80. Eight pounds of river water, when distilled, furnished
7f Ib. of pure water. What per cent was removed by distilling ?
81. In making brown bread, Mrs. Goodwin mixed one cup-
ful of white flour, one cupful of Graham flour, and four cupfuls
of corn meal. What per cent of each was in the mixture ?
272 PERCENTAGE
82. Ten years ago Mr. D. paid $ 4000 for a house and lot,
which has increased in value 18%. What is its present
value ?
83. By the sale of a horse, a man gained $ 10, which was
12% of what he gave for it. For how much did he sell it?
84. A house worth $4000 depreciates in value to $3280.
What per cent does it depreciate ?
85. If I buy 2000 bu. of wheat at 85 cents a bushel, and sell
it for $ 340 more than I paid for it, what per cent shall I gain ?
86. A boy buys papers for 2 cents each, and sells them at
a gain of 150%. What price does he get for them ?
87. A book agent receives a commission of 28% on all his
sales. If he gets orders for 125 books at $ 2.50 each, how
much does he gain ?
88. If I send orders amounting to $ 28.75, getting a discount
of 25% with 5% off for cash, how much money must I send ?
89. About 80% of a human body is water. At that rate, if
a man weighs 165 lb., how much of his body is water?
90. What per cent of 75 is each of the first 12 multiples
of 6}?
91. What per cent is gained or lost by buying goods at
$ .33 per yard and selling them at $ .25 ? $ .50 ? $ .66| ?
92. What per cent is gained by buying apples at the rate
of 3 for a cent and selling them at the rate of 2 for a cent ?
93. Mr. Chapman built a house which cost $ 3600. He had
$ 3250, and borrowed the rest June 1, 1895, giving his note at
6%. June 1, 1896, he paid the interest due and $ 100 of the
principal. June 1, 1897, he paid the interest and $ 50 of the
principal. How much was due June 1, 1898 ?
94. Mr. S. built a house for Mr. M., on which he made a
profit of 25%. He received in payment $2000 and two lots
MISCELLANEOUS EXERCISES 273
at $ 500 each. The lots depreciated in value $ 50 each before
he sold them. How much was his real profit on the house ?
What per cent ?
95. A wholesale dealer pays $ 260 for a car load of bananas
containing 520 bunches, and sells them at $ .60 a bunch. What
per cent does he gain ?
96. A retail dealer pays 60 f for a bunch of bananas contain-
ing 8 doz. If he sells them at 15^ a doz., what per cent does
he gain ? What per cent does he gain on the whole, if \ of
them spoil before he sells them ?
97. Mr. Malone, a traveling salesman, sold Mr. J. W. Smith
of Salem, 111., $360 worth of staples, and $170 worth of
notions. He was allowed a commission of 2% on the staples,
and 6% on the notions. How much commission did he
receive ?
98. Mr. Perry brings 49 bu. of wheat to the Melrose Mill to
exchange for flour. If he gets 36 Ib. of flour for every bushel
of wheat, how many sacks of flour, each weighing 98 Ib., will
he get ?
99. What number increased by 62 J-% of itself equals 3214 ?
100. Mr. Taylor bought 6 loads of hay, each weighing 1J T.,
at $ 10 a ton. He sold all of it for $ 99. What was the gain
per cent ?
101. A farmer had a field 21 rd. square. Three rows of
wire fencing were put around it, costing 2^ a foot, 5% off for
cash. If he paid cash, how much did the fence cost ?
102. 33^-% of the fence was blown down. The price of wire
having risen 50%, and the cash discount being the same, how
much will it cost to replace the fence if he pays cash ?
103. Mr. Gibson's salary is $ 6000 a year. Last year he
saved 20% of the first three months' salary, 40% of the next
three months' salary, and 30% of the last six months' salary.
How much did he save during the year ?
HORN. GRAM. 8CH. AR. 18
274 PERCENTAGE
104. W. H. Small & Co. bought 10 loads of hay, weighing
2700 Ib. each, at $ 10 per ton, and sold it immediately to Mr.
Knox at a profit of 20%. Mr. Knox gave in payment a note
payable in 60 da. If it was discounted in bank at date of issue,
at 7%, how much did Small & Co. gain by the transaction?
105. Arthur and Edward bought a paper route, paying $ 10
for it. Arthur put in $ 4, and Edward the remainder. What
fractional part of the route belongs to Arthur ? To what per
cent of the profits is he entitled ? In one week the earnings
were $ 7.50. How much is each boy's share ?
106. Mr. A. and Mr. B. hire a pasture for $ 50. Mr. A. puts
in 5 cows, Mr. B. 15 cows . Mr. A.'s cows are what per cent
of the whole number of cows ? How much ought he to pay
as his share of the cost of the pasture ?
107. Mr. C. owns f of a business, the profits of which last
year were $ 16,000. What was his share of the profits ?
108. Mr. Davis owns 51% of a business which year before
last lacked $ 1000 of paying expenses. How much money was
he obliged to advance in order to keep the business running ?
109. Last year the receipts of the business were $ 1248.75
more than the expenses. How much did Mr. Davis receive
from it ?
110. If 10 hr. are considered a day's work and $ 2 a day's
pay, what are the weekly earnings of a man who works 9 hr.
on Monday, 8J hr. on Tuesday, 11 hr. on Wednesday, 7 hr.
on Thursday, 6|- hr. on Friday, and 5 hr. on Saturday ?
111. What would be his weekly earnings for the same num-
ber of hours' work at $ 2.50 per day ?
112. What would be his weekly earnings for the same num-
ber of hours' work at $ 2 a day, 8 hr. being considered a day's
work ?
113. At $ 2 a day for an 8-hour day, calculate the weekly
earnings of each man in the following time sheet :
MISCELLANEOUS EXERCISES 275
Mon.
Tues.
Wed.
Thurs.
Fri.
Sa
a
Mr.
Cox,
7
*i
7}
6|
7
7:
b
Mr.
Dow,
8
8 i
6
7 i
7 i
6
c
Mr.
Lee,
n
8|
9
9
6 i
5
d
Mr.
Van,
5
6 i
8
8}
7 i
6
e
Mr.
Gay,
7
6
7*
8
8
7
114. Julia saved money to buy a typewriter, the price of
which was $80. When she had saved $48, what per cent
of the price was lacking ? When she had paid $ 16 more,
what per cent of the price was still unpaid ?
115. Three boys bought a wagon, John paying $ 1, James
$ 2, and Charles $ 3. They sold the wagon for $ 7.50. How
much was each boy's share of the gain ?
116. A man received a legacy of $ 5000. He bought a
house and lot with 40% of it, purchased $ 1500 worth of bank
stock, and invested the remainder in business. What per cent
of it did he invest in business ?
117. At $ 1.50 per cord, how much will a man earn by
sawing a pile of wood, 16 ft. long, 8 ft. wide, and 4 ft. high ?
118. Find the amount of the following bill :
4 rms. sandpaper @ $ 3.50, discounts 30% and 10%.
3 doz. packages tacks @ $ 5.50, discounts 50% and 5%.
12 car jacks @ $ 6.00, discounts 40%, 10%, and 5%.
2000 tin rivets, $ 1.44, discounts 40% and 10%.
119. A man received a legacy. After investing 66f % of it
and spending 25% of it, he had left $ 1728. How much was
the legacy ?
120. A man having $ 4800 in a bank, drew out 12-J-% of it
and then deposited a sum which was 75% of what he had
drawn out. How did his bank account stand then ?
276 PERCENTAGE
121. Mr. Boyd sold a watch for $ 45, which was 25% more
than he gave for it. For how much must he have sold it to
gain 66f % on it ?
122. Simplify:
abed
^ JQff 1* iij:
A tf* 1* i + i
123. How long is the circumference of a circle whose radius
is 5$ in. ?
124. A burial lot in the form of a circle, 40 ft. in diameter,
is inclosed by a fence. Find the cost of the fence at $ .12J a
foot.
125. A calf is tied to a tree by a rope 10 ft. long. The rope
can slip around the trunk of the tree, which is 1 ft. in diameter.
As the calf runs around the tree, how long is the circumference
of the largest circle he can make ?
126. Reduce 5 qt. 1.725 pt. to decimals of a bushel.
127. 14 x ( ) x 2.3 x -f = 3.91. Find the missing factor.
128. How many square centimeters are there on the surfaces
of 7 1. ?
129. In a park there is a circular fountain 35 ft. in diameter,
bordered by a gravel walk 7 ft. wide. What is the distance
around the outer edge of the walk ?
130. A swimming pool is 60 ft. long, 20 ft. wide, and the
water in it is 7 ft. deep. How many cubic feet of water are
in it?
131. Draw a trapezoid, making one of the parallel sides 6
in. long, the other 8 in. long, and the shortest distance between
them 4 in. Find the area of the trapezoid.
132. A farmer has a field in the shape of a trapezoid. One
of the parallel sides is 90 rd. long, the other is 120 rd. long,
and the shortest distance between them is 30 rd. How many
acres in the field ?
MISCELLANEOUS EXERCISES 277
133. One of the sides of a rhomboid is 12 in., and a side
adjacent to it is 8 in. Eepresent and tell what per cent of the
perimeter each side is.
134. One of the parallel sides of a trapezoid is 10 in., which
is 83|-% of the other parallel side. Each of the non-parallel
sides is 9 in. Represent and find what per cent of the perimeter
each side is.
135. A pupil drew a square and shaded 11% of it. If the
area of the part shaded was 44 sq. in., what was the area of the
whole square ?
136. What per cent of the perimeter is each side of a
rhombus? Of a regular hexagon? Of a regular octagon?
Of a regular decagon?
137. A jeweler sold a watch for $ 90, gaining 20%. What
per cent would he have gained if he had sold it for $ 100 ?
$110? $120?
138. Mr. Baker bought $ 100 worth of goods and sold f of
them for what ^ of them cost. What per cent did he gain ?
139. What per cent is gained or lost by buying $ 100 worth
of goods and .selling \ of them for what f of them cost?
140. An importer buys in France 1000 Ib. of perfumery
invoiced at $ 2.00 per pound. Specific duty 60^ per pound, ad
valorem duty 45%. He sells it at $ 5 per pound. How much
does he gain and how much does the government gain by the
transaction ?
141. On Aug. 1, 1890, a man insured his life to the amount
of $15,000 in favor of his wife, paying an annual premium of
$ 29.30 per $ 1000. He died Oct. 17, 1897. How much more
than he had paid did his widow receive?
142. A farmer brought to market 24 doz. eggs, 21 chickens,
and 20 Ib. of butter. He sold -f of the chickens at 25^ apiece,
all the butter at 15^ a pound, and 75% of the eggs at 10^ a
278 PERCENTAGE
dozen. How many eggs and chickens did he have left ? He
took groceries to the amount of | of his sales. He bought 10
yd. of calico at 7^ a yard, and a pair of shoes for $ 2.75. How
much money had he left?
143. A boy made a chicken coop, using a bundle of laths
which cost 15^, and 5^ worth of nails. He sold the coop for
$ .75. What was his gain per cent?
In the following shipments find the amount belonging to
each railroad company :
144. From Trent's Landing, Ky., to Milwaukee, Wis., via
A. & B., B. & E., and E. & M.: Empty kegs, 20,000 lb.; rate,
11^ per hundredweight. The A. & B. receive 25% of the
freight charges. Of the remainder, the B. & E. receive 45%,
and the E. and M. 55 <
145. From Mather, 111., to Cincinnati, Ohio, via M. W. & E.
and A. & C. : 1 car hogs, 18,000 lb. ; rate, 20^^ per hundred-
weight. The M. W. & E. receive 25%, and the A. & C. the
remainder.
146. From Leeds, Ky., to Eawley, Minn., via L. & H., E. &
C., Chicago & St. Paul, St. P. & R.: 25 % tons pig iron; rate,
$2.75 per ton. 15^ a ton goes to the Ohio Bridge Co.; the
lines south of Chicago receive 55% of the remainder. The
lines north of Chicago receive the balance. The L. & H. re-
ceives 25% of the amount paid to the southern lines, and the
St. P. & R. 25% of the amount paid to the northern lines.
147. Mr. Ward bought a lot for $600 and built on it a
house worth $ 2400. The property was assessed at f its value,
and the tax rate was 1^-%. The insurance was for f of the
value of the house, and the rate was 40 ^ a hundred. He rented
the house for a year at $ 30 a month. How much more did he
receive the first year from his investment of $ 3000 than he
would have received by putting it at interest at 6%?
148. The second year the house was vacant one month and
required $ 10.65 worth of repairs. Insurance and taxes were
MISCELLANEOUS EXERCISES 279
the same as the first year. How much more were the net re-
ceipts than 6 % of the amount invested ?
149. ' The third year the rent was raised to $35 a month.
The repairs cost $ 48.75, and the house was vacant two months.
Compare the net receipts with 6% on the investment.
150. Make problems showing the expenses and receipts of
Mr. Ward's house for successive years.
151. At the end of 7 yr. Mr. Ward sold the house and lot
for $4500. If he had received as rental during the 7 yr.
$2000 more than the expenses of the house for that time,
what average yearly per cent had he gained on his investment ?
152. By selling a house for $ 1964, 20% was lost. What
would have been the selling price if only 5% had been lost ?
153. Room No. 5 in Baker School has 44 desks in it. If
$ 77 was 87%% of the cost of all of them, how much did one
desk cost ?
154. A bought 640 A. of woodland at $38 per acre. He
sold the timber for $ 19,600, and the land for $ 13 per acre.
Find the per cent of gain.
155. An insurance policy for $2200 cost $17.60. What
was the rate of premium ?
156. An agent for an oil company sells in three weeks
$48,000 worth of oil at a commission of l-g-%. If his expenses
are $ 47.50 per week, how much are his net earnings for that
time?
157. 62|% of 200 = how many times 121 ? 6J ?
158. The sum of all the edges of a cube is 36 in. What is
the volume of the cube ?
CHAPTER VTII
BONDS AND STOCKS
BONDS
1. When a government or a private corporation is in need
of money it sometimes borrows it and issues its bonds for the
amount. A Bond is a promissory note issued under the seal of
a government or a corporation.
Mr. A. has a bond issued by the United States government
which promises to pay $ 1000 at a certain time, with interest
at 4%. What is his yearly income from this bond?
2. In 1898 when the government wished to raise money to
carry on the war with Spain, it issued bonds payable in 20 yr.,
with interest at 3%. People who wished to lend their money to
the government made application for bonds. When their appli-
cations were granted, they sent to the Treasury Department of
the United States the payment for the bonds, and received in
return bonds for that amount. If you had sent $4000 asking
for $ 500 bonds, how many bonds would you have received ?
3. In September, 1898, Mr. Koss bought a $ 500 bond of the
issue of 1898 and gave it to his daughter Julia. The bond
contained these words :
The United States of America are indebted unto the bearer in the sum
of Five Hundred Dollars. This bond is issued under authority of an Act
of Congress entitled "An Act to provide ways and means to meet war
expenditure," and is redeemable at the pleasure of the United States
after the first day of August, 1908, and payable August 1, 1918, in coin
with interest at the rate of three per centum per annum payable quarterly
in coin on the first day of November, February, May, and August in each
year. The principal and interest are exempt from all taxes or duties of
the United States as well as from taxation in any form by or under State,
municipal, or local authority.
When did the first payment of interest become due ? How
much was paid ?
280
BONDS 281
4. If Miss Ross keeps her bond until it becomes payable,
how much interest will she receive ? How many interest pay-
ments ?
5. Mr. Eoss has 85 $100 bonds, 375 $ 200 bonds, and 875
$ 500 bonds, all of the issue of 1898. What is his income each
quarter from these bonds ? What is the yearly income ?
6. Many persons wish to invest their money in United
States bonds. Can you see why?
7. As there was a great demand for the 1898 bonds, and the
issue was limited, many people who applied for bonds could not
get all they wanted from the government. They therefore
tried to buy them from the holders, and the bonds soon rose in
price. They were then said to be above par, or at a premium.
When bonds are at a premium of 4%, a $ 100 bond costs $ 104.
When bonds are at a premium of 4%, how much will 30 $100
bonds cost ?
8. At 104, or at a premium of 4%, how many $100 bonds
can be bought for $ 936 ? For $ 2600 ?
9. How much will 75 $ 100 bonds cost at 105 ?
10. Find the cost of 30 $ 500 bonds at 2% premium.
11. When bonds are offered at a lower price than their face
value they are said to be below par, or at a discount. When
bonds are at a discount of 2%, one $100 bond can be bought
for $98. At 2% discount, how much would 50 $100 bonds
cost?
12. At 98 how much must be paid for 20 $100 bonds ?
13. At 98 how many $100 bonds can be bought for $1274?
For $1666?
14. September 1, 1898, Mr. Kane bought at par a $5000
bond of the issue of 1898. He held it until March 1, 1899, and
sold it at a premium of 7%. How much did he gain ?
Observe that Mr. Kane would receive the interest that fell due while
he held the bond.
282 BONDS AND STOCKS
In the following problems the value of a bond is $ 100, unless otherwise
stated.
15. What is the value of 183 bonds of a city corporation at
92%? If the purchaser holds them long enough to realize two
4% interest payments on them, and then sells them at 90%,
how much does he gain by the transaction?
16. In order to build a court house the county of Accalama
issued 2500 bonds bearing 5% interest. One half of them were
sold at par, and the rest at 102. How much money did the
county receive for its bonds ?
17. Mr. Harvey bought $15,000 worth of the Accalama
bonds at par, and held them seven years, receiving his interest
annually. At the end of the eighth year the county refused to
pay the interest on them. The bonds fell to 70%, at which
price Mr. Harvey sold them to Mr. Norton. Did Mr. Harvey
gain or lose, and how much?
18. The next month after Mr. Norton bought the bonds, the
county of Accalama redeemed them at 95%. How much did
Mr. Norton gain by the transaction?
19. In order to raise the money to redeem the 5% bonds, the
county of Accalama issued the same amount of bonds bearing
3% interest. The new bonds sold at 93. Mr. Norton bought
150 bonds at this price, held them 8 yr., and sold them at 97%.
How much did he gain on this investment ?
20. How much interest did the county save each year by
substituting 3% bonds for 5% bonds?
21. Persons who buy and sell bonds for others are called
brokers. They are paid a certain per cent on the par value of
the bonds bought and sold. This percentage is called Brokerage.
If you were to pay a broker -|-% for buying 40 $100 bonds
for you, how much brokerage would you pay ? If the bonds
were at par, how much would they cost you, including broker-
age ? If the bonds were below par would the brokerage be less ?
BONDS 283
22. Imagine yourself a broker receiving %<f for buying or
selling bonds for others. If you were to sell 70 $ 100 bonds
and to buy 90 $ 500 bonds, how much brokerage should you
receive ?
23. How much brokerage should you receive if you sold 30
$ 100 bonds, 40 $ 1000 bonds, and a $ 5000 bond ?
24. A broker sold 20 United States $ 100 bonds, 300 Cass
County bonds, par value $ 50, and 200 City Improvement bonds,
par value $25 each. His brokerage was \%- To how much
did it amount ?
25. A broker bought for a client 40 railroad bonds at 87,
and 40 United States bonds at 109. His brokerage was %.
From which transaction did he receive the more brokerage,
and how much more ?
Observe that brokerage, premium, discount, and interest are all com-
puted on the par value.
26. A broker bought for Mr. X. 20 bonds at 103, charging
\Jo brokerage. How much did the bonds cost Mr. X. ?
Since the market value of each bond was $ 103, and the brokerage on
each bond was \ of a dollar, the purchasing price of each bond was $ 103.
20 bonds would cost 20 times $ 103, or $ 2062.50.
27. How much must be paid for 80 railroad bonds quoted at
77, brokerage J% ?
28. Through his broker Mr. S. invested $ 2619 in bonds at
109, paying J% brokerage. How many of these bonds did he
buy?
29. Buying United States 4's at 111-J-, and paying |%
brokerage, Mr. S. invested $ 100,046.25. How many bonds
did he receive ? What was his annual income from them ?
The expression " United States 4's " means United States bonds paying
4% interest.
284 BONDS AND STOCKS
30. If brokerage is %, how much money would be needed
to make the following investments ?
a 80 United States bonds at 105.
b 70 A. and X. Eailroad bonds at 72.
c 19 Memphis bonds at 89.
31. Mr. X. owned 20 bonds. When they were quoted at 103,
a broker sold them for him, charging him 1%. How much
did he receive for the bonds ?
In this case shall the brokerage be added to the market price of the
bond, or subtracted from it ? Why ?
32. Mr. B. had 90 bonds issued by the M. and Q. E. R. Co.
When they were quoted at 79, his broker sold them for him.
How much did he receive, brokerage being %% ?
33. Mr. N. ordered the purchase of 90 shares at 87-J-. When
they had fallen to 86J, he ordered their sale. Brokerage being
\/o for buying and \f for selling, how much did he lose ?
34. Mr. James obtains 30 bonds quoted at 85, paying a
broker -J-% for buying them. The same broker sells them for
Mr. James at 86 \. Brokerage |%. How much does Mr.
James gain, and how much does the broker receive for his
work ?
35. Mr. A. bought 20 United States 4's at 108. How much
did they cost him ? How much interest did he receive from
them each year ?
36. Mr. A. invested $7800 in bonds which were selling at
104. How many bonds did he buy ? If they paid 3%, what
yearly income would he receive from them ?
37. When United States 4's were selling at 111| Mr. A.
invested $ 10,704 in them. How many bonds did he buy, and
what income did he receive from them ?
38. Find what yearly income can be derived from the fol-
lowing investments :
STOCKS 285
a $8240 invested in 5% bonds at 103.
b $ 6755 invested in 3% bonds at 96f
c $ 1584 invested in 5% bonds at 99.
d $20,400 invested in 4% bonds at 102.
STOCKS
39. Across a certain river in Ohio there was formerly a toll
bridge on which the fare for foot passengers was 15^. John
Smart, a schoolboy, thought it would be a profitable scheme to
buy a $10 skiff, and to carry passengers across the river
during his vacation, for 10 $ each. As he had not enough
money to pay for the skiff, he formed a plan similar to a busi-
ness enterprise called a stock company, in which his father
was interested. John induced several of his friends to join
him in buying the skiff. They agreed that John should row
the passengers across the river every day in the week, except
Sunday, and should retain in return for his labor 50 ^ a day
from the gross receipts. The rest of the money received was
to be divided among the owners of the skiff in proportion to
the amount each had invested. The first week 55 passengers
were carried across. After deducting John's salary, how much
remained to be divided among the owners of the boat ? What
per cent was that of the whole capital ?
40. At that rate how much was received by Albert Blake,
who had put in $ 5 ? By Edgar Howe, whose share of the
capital was f 3 ? By Fred Lee, whose investment was $ 1 ?
41. John had put in $1. How much did he receive from
his investment and his salary ?
42. Find the per cent of gain on capital, and the amount
received by each boy at the end of the week in which 32 pas-
sengers were carried across. Of the week in which 69 passen-
gers were carried across.
43. Through John's carelessness the boat was overturned
one day, and the skiff route became unpopular. The receipts
286 BONDS AND STOCKS
for the week in which the accident occurred were only $2.
How much did each boy have to contribute to pay John's
salary of 50 $ a day ?
44. The next week they sold the skiff for 60% of what it
cost. What per cent of his investment should each boy receive
from the sale ? How much money ?
SUGGESTION TO TEACHER. Before taking up the study of stocks each
pupil should comprehend fully the principles involved in the problems
about John and the skiff. Let pupils make similar problems by imagin-
ing different happenings to John and his companions.
45. As large business enterprises require more capital than
is usually owned by one man, it is common for many persons
to unite and form what is called a Stock Company. The money
with which the company carries on business is called its
Capital. Each member of the company is called a Stock-
holder.
That part of the earnings of a company which is divided
among the stockholders is called a Dividend. Dividends are
computed at a certain per cent on the par value of the capital.
The par value of A's stock is $ 5000. Find his dividend at 6 % -
46. A company whose capital stock is $ 500,000 distributes
$ 20,000 in dividends. What is the rate of dividend ? What
is the rate of dividend When it distributes $ 30,000 ?
47. Mr. Smart is a stockholder in a stock company called
the Ohio Transportation Co., which runs a line of steamers.
The capital stock is $ 100,000, and it is divided into 1000
shares of $ 100 each. Mr. Smart owns 10 shares ; Mr. Howe,
20 shares; Mr. Blake, 500 shares; Mr. Lee, 70 shares; and
the remaining shares are owned by others. In the first year
of its existence the earnings of the company, after paying all
the expenses, were $ 7000. What per cent of the capital were
the earnings? How much should Mr. Howe receive? Mr.
Blake ? Mr. Lee ?
STOCKS 287
48. Mr. Smart is the superintendent of the line of steamers,
and receives a salary of $ 5000 a year. What was his income
in the first year from his stock and his salary ?
49. The next year the Ohio Transportation Co. paid a divi-
dend of 9%. How much was received by the four men
mentioned in Ex. 47 ?
50. How much would be received by each of those gentle-
men if the dividend were 12% ? 5% ? 7% ?
51. When a company pays a large dividend, there arises a
demand for its stock, and its shares sell above par, or at a
premium.
When the stock of the Ohio Transportation Co. reached
104, Mr. Blake sold 400 shares of it. How much did he gain
by the sale ?
52. The next year the dividends of the Ohio Transporta-
tion Co. fell to 2%. How much dividend was received by Mr.
.Howe ? Mr. Blake ? Mr. Lee ?
53. How much did Mr. Smart receive from the company
that year ?
54. The next year, through some unfortunate management,
the company was unable to pay dividends. Instead, an assess-
ment of 5% was made upon each share, in order to pay the
running expenses of the business. How much was paid by
Mr. Howe ? Mr. Blake ? Mr. Lee ?
55. Mr. Smart's salary was lowered 10%. How much did
he receive from the company that year ?
56. The price of the shares of the Ohio Transportation Co.
had fallen to 67. Mr. Blake bought 600 shares at that price.
Mr. Smart, Mr. Howe, and Mr. Lee bought 50 shares each at
the same price. At the end of the year the dividend was
10%. What was the amount of each man's dividend?
Remember that the dividend is always reckoned on the par value
whatever may be the quotation in the market.
288 BONDS AND STOCKS
57. March 1, 1895, Mr. Eeed bought 40 shares of A. & B.
R. R. stock at 87. The shares paid a semiannual dividend of
3%. He sold them March 1, 1896, at 88. How much did he
receive in dividends ? What was his profit from the advance
in price ?
In the following problems the par value of a share of stock is assumed
to be f 100.
58. If you were to receive 100 shares in a mining company
which pays an average semiannual dividend of 4%, what
would be your yearly income from those shares ?
SUGGESTION TO TEACHER. Get blank certificates of stock. Let pupils
form themselves into an imaginary stock company.
59. How much must be paid for 90 shares of A. & C. R. R.
stock at 97-J- ? If these shares pay a semiannual dividend of
4%, how much yearly income will be derived from them ?
60. The A. & B. Belting Co., whose capital stock was
$ 500,000, distributed $ 50,000 among its stockholders. What
was the rate of dividend ? How much was received by Mr.
Smith, who owned 37 shares ?
61. At the time the above dividend was declared, banks
were paying 2% interest on long time deposits. Would the
shares of the A. & B. Belting Co. be likely to be at par, at a
premium, or at a discount ?
62. How much must be paid for 70 shares of the A. & B.
Belting Co., at 105 ?
63. Mr. A. bought 900 shares of the Unity Coal Mine at 47.
He held them until he had received two semiannual dividends
of 3%, two of 3%, and three of 4%. He then sold the shares
at 71. How much did he gain, including the dividends ?
64. How much must be paid for 315 shares of the W. U.
Telegraph stock at 137 ? If these shares yield a semiannual
dividend of 11 what is the annual income from them ?
STOCKS 289
65 o How many shares of N. & St. L. stock quoted at 93 can
be bought for $ 7440 ? If these shares pay an annual dividend
of 5%, what is the annual income from them ?
SUGGESTION. Since one share costs $ 93, how many shares will $ 7440
buy ? On what is the dividend reckoned ? What is the par value of the
stock ?
66. How many shares at 95 can be bought for $ 7600 ?
What is the annual income from them if they pay 6%
dividend ?
67. Find annual income from $ 7254 invested in D. & H.
R. R. stock at 78, the semiannual dividend being 4% ?
68. The K. & X. R. R. Co. paid a dividend of 8% in Janu-
ary, and another of 7% in July. What was the yearly
income of a stockholder who owned 750 shares ?
Find annual income from the following :
69. $ 22,464 invested in stocks at 108, which pay 12%.
70. $ 6965 invested in stocks at 99, which pay 7%.
71. $ 12,390 invested in stocks at 88, which pay
72. Mr. A. bought a share of stock at 80, which paid 8%.
What per cent did he gain on his investment ?
What amount of dividend did Mr. A . receive ? An $ 8 dividend is
what per cent of an $ 80 investment ?
Find what per cent is gained annually on the following
investments :
73. Stocks bought at 50, paying 2%.
74. Stocks bought at 70, paying 3%.
75. Bonds bought at 102, paying 3%.
76. Stocks bought at 87-j-, paying 7%.
77. Stocks bought at 41f, paying
HORX. GRAM. SCH. AR. - 19
290 BONDS AND STOCKS
78. In the case of a person who does not have to pay brok-
erage, which pays the better per cent, and how much, 4%
bonds at par or 6% shares at 90 ?
79. Mr. E. bought 75 U. S. 4's at 110 and 75 shares C. &
L. E. E. stock at 90 without brokerage. The stocks paid 5%
dividend. The bonds cost how much more than the stocks ?
How much more income did he receive from his stocks than
from his bonds ? What per cent did he make on each invest-
ment?
Brokerage being |% for buying and the same for selling,
how much is gained or lost on :
80. 90 shares bought at 70, sold at 83 ?
81. 113 shares bought at 64, sold at 59 ?
82. 27 shares bought at 58, sold at 57} ?
83. 800 shares bought at 41 j, sold at 52 J ?
84. 70 shares bought at 112J, sold at 113 ?
85. 900 shares bought at 102, sold at lOlf ?
MISCELLANEOUS EXERCISES
1. What is the largest prime number that can be expressed
by three figures ?
2. Eesolve 54 into prime factors. What per cent of them
are 3's ?
3. If 4- of the price of a ship is $ 12,000, how much is the
whole ship worth ?
4. A man owning f of an estate sells f of his share for
$ 2400. At this rate, how much is the estate worth ?
5. Jane is 8 yr. old, and Lucy 13. The sum of Jane's and
Lucy's ages less 7 yr., is the age of Mary. How old is Mary ?
MISCELLANEOUS EXERCISES 291
6. A farmer had two fields of wheat ; the first yielded 840
bu., which was -f% of the amount yielded by the second. How
many bushels did he get from both fields ?
7. A man bought a firkin of butter for $ 17, a crock of lard
for $ 8, and a barrel of flour for $ 9. To pay for them he
needed $ 7.50 more than he had. How much money had he ?
8. By what must 1.7 be multiplied to make 5.95 ? 6.46 ?
9. John rode 7-J miles on his bicycle in one hour, 6 T 7 ^ in
the next hour, and 6| in the next. How far did he ride in
all ? How much farther in the first hour than in the second ?
Than in the third ?
10. Harry walked 7.64 miles, and James walked twice as far.
How far did they both walk ?
11. A merchant bought a barrel of sugar for $ 28.50, and a
barrel of flour for $ 7.50. He sold the two for $ 40. What
per cent did he gain ?
12. How many millimeters in the circumference of a circle
whose diameter is 7 centimeters ?
13. The circumference of a wheel is 2.6 m. How many
times will it revolve in rolling 33.8 m. ?
14. How many square decimeters in the surface of a stere ?
15. How many steres of wood in a pile 17 m. long, 8 m.
wide, and 2 m. high?
16. Find the cost of digging a cellar 7 m. long, 5 m. wide,
and 2 m. deep, at 20 ^ a stere.
17. Image a cubic centimeter of water. How much does
it weigh ? 5 liters of water weigh how many grams ?
Kilograms ?
18. How many kilograms will 7 liters of alcohol weigh if
alcohol is % as heavy as water ?
19. If ice weighs 94% as much as water, how many kilo-
grams of ice are there in a block of ice 9 dm. long, 5 dm. wide,
and 4 dm. high ?
292 BONDS AND STOCKS
20. How many kilograms do 2 liters of mercury weigh,
mercury being 13.5 times as heavy as water?
21. The product of two numbers is f. One of the numbers
is 2J. What is the other number ?
22. A landowner divided 1\ A. of land into city lots 55 ft.
in front and 132 ft. deep, first taking out for streets and alleys
108,900 sq. ft. How many lots were there ? He sold them at
an average of $ 40 a front foot. The land had cost him $ 100
an acre 20 yr. before. He had paid an average of $ 300 a year
in taxes upon it, and the expense of platting and selling it was
$ 315. How much did he gain by holding the land ?
23. How many bricks 8 in. by 4 in. will be required to pave
a yard 168 ft. long and 60 ft. wide ?
24. If Mr. A. were to lose 33^% of his money, he would have
$ 2000 left. How much money has he ?
25. What is the value of a pile of wood 30 ft. long, 8 ft.
wide, and 4 ft. high at f 3.75 a cord?
26. \ of 8 is what per cent of of 20 ?
27. Mrs. A. has at interest $800 at 6% and $1000 at 5%.
What is her yearly income from both investments ?
28. If a piano which cost $ 260 is sold at $ 325, what per
cent is gained?
29. What is the interest of $ 4270 from May 1, 1895, to
Aug. 1, 1901, at 5% ?
30. Mr. A. borrowed $ 7000 at 5%. At the end of each of
the first two years he paid $ 1000. At the end of the third
year he paid all that was due. How much did he pay ?
31. A note for $782.50 payable in 60 da. with grace was
discounted at 6%. What were the proceeds?
32. Mr. 0. failed in business, owing $60,000 and having*
$30,000 with which to pay. What per cent of the amount
could he pay ? How much would a creditor receive to whom
he owed $ 1800 ? A creditor to whom he owed $ 2456.65 ?
MISCELLANEOUS EXERCISES 293
33. If by selling fruit at 9^ a pound a grocer gains
how much will he gain by selling it at 11 $ a pound ?
34. What per cent would a jeweler gain by selling a watch
at $80, if by selling it at $ 75 he gains 50% ?
35. A merchant bought goods at $ 1.6.0, marked them to sell
at an advance of 37-J%, and sold them at a reduction of 25% on
the marked price. At what price were they sold and what per
cent was gained on them ?
36. Goods costing $ 864 were marked at an advance of 50%
and sold at a discount of 16 f % from the marked price. How
much was gained on them ? What per cent ?
37. Find the net cost of a bill of goods amounting to
$375.50 with discounts of 60%, 40%, and 5%.
38. Make a problem which involves trade discount.
39. An agent sold 250 bbl. of flour at $ 3.80 per barrel, com-
mission 3%. What was his commission, and how much was
sent to the owner of the flour ?
40. A steamer valued at $ 750,000 was insured for f of its
value at 1%, in two companies, one company taking \ of the
risk and the other the remainder. What was the amount of
premium for each company ?
41. Mr. A. has real estate assessed at $ 20,000 and personal
property to the amount of $ 8224. He pays a poll tax of
$1.50. What is the amount of his taxes when the rate of
taxation is 37^ mills on a dollar ?
42. What is the duty on 20 casks of wine, each cask con-
taining 56 gal., invoiced at $ 2.35 per gallon, if 12^% is allowed
for leakage and if there is an ad valorem duty of 45% ?
43. Find the cost of 80 shares 1ST. Y. C. E. E. stock at 112,
brokerage, 1%.
44. How much must one pay for 65 U. S. 4's at 108, broker-
age,
294 BONDS AND STOCKS
45. How much would one receive from selling 30 shares of
mining stock at 91, brokerage, -J% ?
46. How much does the owner receive from the sale of 95
shares of stock sold at 103, brokerage, %%?
Brokerage being -J%, for buying and for selling, how much is
gained or lost by
47. Buying 72 shares at 89 and selling them at 90?
48. Buying 40 shares at 71 and selling them at 70?
49. Buying 35 shares at 59 and selling them at 58 J?
50. At 57|- how many shares of stock can be bought for
$7156.25, brokerage, 1% ? If they pay a semiannual divi-
dend of 4%, what income is derived from them ?
51. What income is derived from $ 15,300 invested in stocks
at 95, brokerage, -J%, if the stocks pay a semiannual dividend
of 3% ?
52. A blacksmith's price for shoeing a horse was 50^ a shoe,
but he allowed a discount of 10 /o to any person bringing him
10 or more horses at the same time. Mr. Boyd's horses were
shod there one day at a cost of $ 13.50. How many horses were
there?
53. A room 18 ft. by 15 ft. was covered with matting 1 yd.
wide, at a cost of $10.50. To lay the matting cost 5^ a yard.
What was the price per yard of the matting?
54. The Troy Edge Tool Works sold 12 doz. sledge ham-
mers, weighing 5 Ib. each, at 10^ per lb., and
9 doz. hammers at 50^ apiece. Discount,
1 . Make out the bill.
55. A triangle whose sides are all unequal
is called a Scalene Triangle.
Draw a right triangle whose base is 3 in.
and altitude, 4 in. Is it scalene ? Give rea-
sons for your answer.
MISCELLANEOUS EXERCISES 295
56. Construct an equilateral triangle. An isosceles tri-
angle. A scalene triangle.
57. Find the perimeter of a scalene triangle, of which one
side is 4 in., another side is 6| in., and the third side is 8f in.
58. Find perimeter of the scalene triangle of which the side
AB is 5 in., BG is 2 in. longer than AB, and CA is 3 3 in.
longer than BC. Represent.
59. How long is the perimeter of the triangle ABC, when
AB is 12 in., BC is 33% longer than AB, and CA is 25%
longer than BC ?
60. Tell how a line is bisected. In the same way bisect an
arc.
61. If the base of an isosceles triangle is 16 in., and if each
of the other sides equals 87% of the base, how long is the
perimeter ?
62. At $3 per day for board, how many days can a man
board at the seashore for $28?
63. A man had three lots, each containing 6 A., which he
redivided into building lots of f of an acre each. How many
building lots did he have ?
64. Mrs. A. wishes to cover the floor of a room 16 ft. long
and 12 ft. wide with ingrain carpet 1 yd. wide. The carpet
will cut to the best advantage if the strips are laid lengthwise.
One pattern requires that the shortest possible strips shall be
16 ft. 8 in. long. This carpet costs 73^ per yard. Another
pattern requires that the strips shall be only 16 ft. 2 in. long,
but it costs 75^ a yard. What is the difference in the cost of
the carpets ?
65. How much more will it cost to cover a floor 21 ft. long
and 18 ft. wide with Brussels carpet 27 in. wide at $ 1.25 per yard
than with a yard wide ingrain at 85 ^, if the Brussels requires
only 1 in. to be turned in at the end of the strips and the
ingrain requires 6 in. ? The strips run lengthwise.
CHAPTER IX
LITERAL QUANTITIES
1 . If x = 10, how many days in x weeks ? How many
minutes in x hours ? How many cents iff x dollars ?
2. If x = 48, how many yards in x ft. ? Years in x mo. ?
Ounces in x Ib. ? Pecks in x qt. ? Gallons in a? qt. ?
3. If 6 represents 3, how much is 3 times b or 3 b ?
4. If a = 8, how much is 3a ? Jaor-? 2J times a?
5. If x = 18, how much is .5 x ? .7 x? fa;? 33|% of a?
6. If a = 12, how much is 25% of a ? 16f % of a ? 12J%
of a?
7. When a; = 20, how much is ? ? -? ?
10 30 5 5
8. x dollars + y dollars = how many dollars, when x = 8 and
9. If a = 10 and 6=3, how much is a + b? 3 a + ob?
6b-a? Sb-2a? a + b? b + a?
10. Give some values to x and y that will make the follow-
ing equations true : # + ?/ = 19. x-\-y = 15. x y = \.
11. If an orange costs 3^, how much will x oranges cost,
when x = 10 ? When x = 4 ?
12. How many oranges in a; doz. when $ = 7 ?
13. If ic = 25, how many weeks in the number of days repre-
sented by a; + 3 ? a + 10? z-4? 2x-l?
296
LITERAL QUANTITIES 297
14. If we let x stand for 35, how many gallons are there in
the number of quarts that are represented by x -f 1 ?
15. When x represents 14, how many feet in the number of
yards represented byoj-f-1? a? -f 3 ?
The expression ab means a times 6, just as 4 b means 4 times b.
16. If a = 7 and b = 5, how much is ab? Which is the
greater, 7 times 5 or 5 times 7 ? a times b or b times a ?
17. If a = 11 and 6 = 2, how much is 3 ab ? 3ba ?
18. How much will x apples cost at y$ a piece, if x = 4 and
y = 2 ? If z = 8 and # = 3 ?
19. If we represent the cost of one apple by y and a number
of apples by aj, how shall we represent the cost of them all ?
20. Give some values to x and y that will make the follow-
ing equations true. #i/ = 48. xy = 35. xy = 70.- xy = 98.
21. When x = 50 and y = 2, how many oranges can be bought
for xtf, if one orange costs ytf? How many when # = 30 and
22. If we represent the cost of one orange by y and the cost
of a number of oranges by x, how shall we represent the num-
ber of oranges ?
23. If we represent the cost of one orange by x and the cost
of a number of oranges by y, how shall we represent the num-
ber of oranges ?
24. Give some values to x and y that will make the follow-
ing equations true :
2.7. 2 = 4. * = 6. Z = 9 . 5 = 3.
y y a; 2/
25. 3(5 + 4) = ?
This expression means "3 times the sum of 5 and 4." Quantities
inclosed in a parenthesis are to be considered as one quantity.
298
LITERAL QUANTITIES
26. 2(7 + 2) = ? 2(7-2) = ? 8(6 + 4) = ? ( 7 + 3 ) = ?
o
27. If a =7 and 6 = 2, how much is 2 (a + 6)? (a + 6) 2 ?
(a-6) 2 ? 2(o-6)? (a + 5) ? ( a ~ 6 )?
3 5
28. When a = 4 and 6 = 3, how much is 5 (a + 6) ?
2(a-6)? a(a + 6)? 6(a + 6)? (a + 6) 2 ? (a-6) 2 ?
29. When = 10 and u = 3, how much is
3* + 5w? w(2* + u)? t 2 + 2tu + u 2 ? (t + u) 2 ?
30. If a; is 3, then 9 x 5 x = what number ?
Finish the following equations, supposing x to equal 7 :
35.
: 36. 44 5x =
37. In the broken line ABODE
the part BC is twice as long as
AB, CD is 3 times as long as AB,
and DE is 4 times as long as
AB. How long is the entire line
if x represents 7 in.? 2 ft.?
5 yd.?
38. Draw a broken line consisting of two parts in which one
part is 4 times as long as the other. Let x stand for the length
of the smaller part. What will represent the length of the
other part ? How long is the entire line if x = 2 in. ? 8 in. ?
A 3# B 39. How long is the perimeter of
V~ ~A^ the rhomboid AB CD if x = 5 in. ? If
FIG. 1.
FIG. 2.
40. Draw a rhomboid, making a longer
LITERAL QUANTITIES
299
side twice as long as either of its adjacent sides. Mark a
short side x and the other sides accordingly. How long would
B the perimeter of the rhomboid be if
x=Sin.? 1.1 in.? 2J in. ?
41. In the circle whose center is 0,
the arcs are in the ratios represented
in Fig. 3. How long is the circumfer-
ence if x = l in. ?
ft.?
42. How long is the perimeter of
the trapezoid DEFG, if x = 10 in.?
3f in. ?
43. 8 a 6a + 3a = how many a's ?
8a-6a = 2a; 2 a + 3 a = 5 a.
Express in one term
44. 96-76 + 46.
45.
46.
47.
c-8c.
+ 2x-5x-l3x.
48.
49.
50.
51.
4Sx-3x-llx-20x.
/D 52. In the broken line, ABCD,
the part BC is twice as long as
AB, and CD is 3 times as long
as AB. The entire line is how
many times as long as AB ?
If we know the length of the
entire line, we may find by equa-
tions the length of each part. If the entire line is 12 in.
long, we have,
x + 2 x + 3 x = 12 in.
then 6 x = 12 in.
and x = 2 in., length of AB.
and 2 x = 4 in., length of BC.
and 3 x = 6 in., length of CD.
300 LITERAL QUANTITIES
53. Find the length of each part of the broken line repre-
sented in Fig. 5, if that line is 30 in. long. 54 in.
long.
54. In the isosceles triangle, ABC, each of the
equal sides is twice the base. The perimeter is
45 in. How long is each side ?
55. Construct an isosceles triangle, in which
each of the equal sides is 3 times the base. Let
FIG 6 x = the base, and find how long each side would be
if the perimeter were 56 in. 105 in. 147 in.
56. In the trapezoid, ABCD, the non-parallel sides are
equal, the upper base is 3 times as long
^ 5 as either of its adjacent sides, and the
2_ \ lower base is 4 times as long as either
of its adjacent sides. How long is each
side, if the perimeter is 45 in. ? 153 in. ?
57. Turn to Fig. 4, page 299, and find the length of each
side of the trapezoid, if its perimeter is 42 in. 9 ft. 4 in.
58. In the trapezium, ABCD, the sides have
to one another the ratios expressed in Fig. 8.
How long would each side be if the perimeter
were 80 in. ? 75 in. ?
59. Mr. Morton spent some money on Monday,
3 times as much on Tuesday, and 5 times as much
on Wednesday. If he spent $ 36 in all, how
much did he spend each day ?
Let x = the number of dollars spent on Monday.
60. John has 4 times as many marbles as James, and they
both have 75 marbles. How many has each ?
61. The sum of two numbers is 21, and one of them is 6
times the other. What are the numbers ?
LITERAL QUANTITIES 301
62. Ella's mother is 3 times as old as Ella. Her father is
4 times as old as Ella. The sum of all their ages is 96 years.
How old is each ?
63. I am thiirking of two numbers, one of which is 5 times
the other. Their sum is 18. Find the numbers.
64. CLASS EXERCISE. - may think of two numbers, one
of which is a multiple of the other. He may give the sum and
the ratio of these numbers and the class may find them.
65. The circumference represented is
54 ft. The arc AB is twice the arc BC,
and the arc CA is three times the arc
\c BC. How long is each arc ?
66. Turn to Fig. 3, page 299. Find the
length of each arc when the circumfer-
FIG. 9. ence is 96 ft. When it is 8 ft. 4 in.
67. Ida set out a number of geraniums, twice as many roses
as geraniums, and three times as many pansies as roses. There
were 27 plants in all. How many were there of each ?
Let x = the number of geraniums,
then 2x= " " " roses,
and 6x= u " " pansies.
68. There are three numbers whose sum is 80. The second
is 3 times the first, and the last is 4 times the second. Find
them.
69. Ida, Frank, and Henry paid $ 10 to have a tennis court
prepared. Frank gave three times as much as Ida, and Henry
gave twice as much as Frank. How much did each give ?
70. On the day that Euth Owen was 22 years old she re-
ceived a bunch of roses consisting of one rose for each year of
her life. There were twice as many pink roses as red roses,
302 LITERAL QUANTITIES
and four times as many white roses as pink roses. How many
were there of each ?
71. Make similar problems.
72. A certain number plus itself equals 320. What is the
number ?
x + x = 320.
73. Find the number which added to itself equals 258. 237.
74. A certain number plus twice itself equals 396. Find the
number.
75. Find the number which added to twice itself equals 297.
76. Find the number which added to four times itself equals
195. 275. 300. 177.
77. Separate 18 into two parts, one of which is 8 times the
other.
78. Separate 30 into two parts, one of which is 5 times the
other. One of which is 9 times the other. One of which is 14
times the other.
79. Separate 24 into two parts, one of which is twice the
other. 3 times the other. 5 times the other.
80. CLASS EXERCISE. - may name a number which he
can separate into two parts whose ratio is a whole number. He
may give the ratio of those parts. The class may find the parts.
81. Two brothers, Messrs. Arthur and Philip Owen, paid
$ 18,000 for a piece of land. Mr. Arthur Owen paid 5 times
as much as his brother. How much did each pay ?
82. What number is that to which, if 4 times itself, and 6
times itself be added the sum is 77 ?
83. A farmer sold a cow and a pig for $ 30, receiving 9 times
as much for the cow as for the pig. What was the price of
each?
LITERAL QUANTITIES 303
84. Separate 60 into three parts such that the second is 4
times the first, and the third is 5 times the first.
85. What number added to six times itself equals 147 ?
86. Albert has 3 times as many marbles as James. Roy
has as many marbles as both the other boys have. They all
have 72 marbles. How many has each ?
87. Thomas caught 3 fish. The largest fish weighed as
much as the other two. One of those weighed twice the other.
The weight of all was 6 Ib. How much did each weigh ?
88. One hundred can be separated into 3 integers, of which
the second is 4 times the first, and the third is equal to the
sum of the first and second. What are the numbers ?
x + 4x + (x 4- 4z) = 100.
89. In the same way separate 120. 150. 600.
90. AB is a diameter. The arc EC
is 3 times the arc AC. The arc BA is
72 in. How long is the arc AC? BC?
The circumference ?
91. If the circumference of the circle
in Pig. 10 were 112 in., how long would
FiiTTo. be the arc AC? AB?
92. Seven times a certain number, minus three times that
number, equals 24. What is the number ?
Let x = the number,
then Ix 3x = 24.
93. Six times a certain number 4 times that number = 10.
What is the number ?
94. If a certain number is multiplied by 7, and if the same
number is also multiplied by 5, the difference between those
products is 16. Find the number.
95. The difference between the 8th multiple and the 5th
multiple of a certain number is 21. Find the number.
304
LITERAL QUANTITIES
96. CLASS EXERCISE.
may think of a number and of
two of its multiples. He may tell the class which multiples
they are and the difference between them. The class may
find the number.
97. John picked 3 times as many quarts of berries as his
sister picked. He picked 8 more quarts than she did. How
many quarts did each pick ?
98. Mr. Bond drew a sum of money from the
bank on Friday, and five times as much on Satur-
day. He drew $ 128 more on Saturday than on the
r previous day. How much did he draw each day ?
99. The sides of the triangle ABC are in the ratios
expressed in Fig. 11. The sum of AB and BC is
FIG. 11. 77 in. more than AC. How long is each side ?
s 100. In the rhomboid ABCD, the side
AB is three times the side BC, and it is
10 in. longer than BC. How long is the
FIG. 12. perimeter?
101. The arc BC is twice as long as the
arc AB, and it is 8 in. longer than AB.
How long is AB? BC? AOC is a right
angle formed by radii. How long is the
circumference ? The diameter ?
102. In the trapezium ABCD, AB = BC and
CD = AD. AD = 3 times AB, and it is 6 in.
longer than AB. Find the perimeter of the
trapezium.
FIG. 14.
LITERAL QUANTITIES 305
103. In the trapezoid ABCD, the side
DC equals 4 times AB, and is 9 in. longer
than AB. If each of the non-parallel
D c sides is 7 in., how long is the perimeter ?
104. If John's shoes cost twice as much
as his hat, and they both cost $ 3.60, how much will each
cost?
In solving this problem it will be more convenient to let x equal the
number of dollars that John's hat costs than to let x equal the number
of dollars that his shoes cost. Do you see why ?
105. What number subtracted from 3 times itself gives for a
remainder 14 ? 26 ? 32 ?
106. Mary has twice as many books as Alice, and together
they have 36 books. How many has each ?
107. John and William together have eighty marbles, and
John has 7 times as many marbles as William. How many
has each ?
108. Albert walks 3 times as far east from a certain point as
John walks west from the same point. They are then 80 ft.
apart. How far does each walk ?
109. A pole 12 ft. long is sunk in the water so that the part
below the surface is 3 times as long as the part above. How
much is below the surface ?
110. A tree 60 ft. high is broken so that the part which has
fallen down is 5 times as long as that which remains standing.
Find height of the stump.
111. Mr. Colton had a sum of money at interest at 5% and a
sum twice as large at 6%. In all he had $600 at interest.
Find how much he had at each rate and his yearly income from
both principals.
112. 3 x = 8 + 7. Find the value of x.
HORN. GRAM. SCH. AR. 20
306 LITERAL QUANTITIES
113. If we have the equation "4 a? + 7 a; = 52 + 3," how is
the equation " 11 x = 55 " obtained from it ?
Uniting quantities of the same kind on the same side of the
equation is called collecting the terms.
Find value of the literal quantity in each of the following :
114. 2x + 5z-7
115. 5
116. 6y-4y = 21-6-6.
117. 9 x H- 2 x = 50 -4 + 20.
118. 18 y - 5 y + 7 y = 30 - 10 + 24 + 36.
119. The terms written before the sign of equality form the
first or left-hand member of the equation. Those written after
the sign of equality form the second or right-hand member of
the equation.
How many terms are there in the second member of the
equation in Ex. 118 ? In the left-hand member ?
120. The number which shows how many times the literal
quantity is taken is called the Coefficient of that quantity. In
the expression 11 x, 11 is the coefficient.
Supply missing coefficients in the following equations,
assuming that x = 3. ?o? = 24. ?o? = 21: a?+?cc = 24.
121. Make an equation containing a literal quantity whose
coefficient is 5.
122. If a = b, is it true that a + 7 = b + 7? Illustrate.
An equation is like a pair of scales and the sign of equality is like
the beam of the scales. If a pound is added to one side, what must be
added to the other side, in order to keep it balanced ? If the amount
on one side is doubled, what must be done to keep the balance ?
123. Add 7 to both members of the equation 40 = 40. Is the
resulting equation true ?
124. If 7 is added to one member of an equation and 9 to
the other member, is the resulting statement true ? Illustrate.
LITERAL QUANTITIES 307
125. Write an equation. Subtract the same quantity from
both members of it and show whether or not the members are
still equal.
126. Multiply both members of the equation 8 = 8 by the
same quantity. Are the members still equal ?
127. If both members of an equation are divided by the
same quantity, how is the equation affected ? Illustrate.
128. If a 5, is it true that a 2 = 25? What has been done
to each member of the original equation ?
129. Illustrate by numbers the truth of the statement, "If
the same operation is performed upon each member of an equa-
tion the members are still equal."
130. Find the value of x when x 2 = 6.
SOLUTION
x - 2 = 6.
Adding 2 to each member -2 + 2=6 + 2.
Hence, x = 6 + 2.
or x = 8.
What was the purpose of adding 2 to the left-hand member ? To the
right-hand member ?
Compare the first and third equations in the above solution. It will be
seen that in the first equation "2" is written in the left-hand member
and has the minus sign, while in the third equation "2 " is in the right-
hand member and has the plus sign.
131. Find the value of x when x +- 3 = 12.
SOLUTION
x + 3 = 12.
Subtracting 3 from each member x + 3 3 = 12 3.
Hence, x = 12 3.
or x = 9.
Why should we here subtract, instead of adding 3 to each member ?
Compare the first and the third equations. What change has been
made in the first to produce the third ?
308 LITERAL QUANTITIES
132. Changing a quantity from one side of an equation to
the other is called transposing the quantity. Illustrate.
133. Study the solutions of Exs. 130 and 131 until you see
the truth of the following principle :
A quantity may be transposed from one side of an equation to
the other if the sign prefixed to the quantity is changed from plus
to minus or from minus to plus.
When no sign is prefixed, the plus sign is understood.
134. In the equation 2 a? + 7 = 28 x, if x is transposed
what sign will it have ? What sign will 7 have if transposed ?
What is the purpose of transposing quantities ?
It may help you to remember to transpose correctly if you repeat
" When I change the side I change the sign."
Find the values of the literal quantities :
135. x 7 = 23. 139. 3 x + 7 = 25.
136. x - 8 = 21. 140. 7 x - 3 = 67.
137. 5 x 5 = 50. 141. 4 y + 3 = 51.
138. 5^-4 = 35. 142. 12x + 7 = 67.
143. CLASS EXERCISE. - may think of some number,
call it x, and make an equation like the above for the class to
solve.
144. Finding the value of the literal quantity in an equa-
tion is called solving the equation.
Solve
Transposing, we have 12 x 8 se = 31 7.
Collecting the terms, we have 4 x = 24.
Dividing, we have x = 6.
Solve the equations:
145. 5a + 9 = 3x + 17. 148.
146. 10x + 8 = 32-2x. 149. 9-7a+4 = .x-f 8.
147. 7/-9 = 5 + 7, 150.
LITERAL QUANTITIES
151. 15y-21 = 14y-37. 153.
152. 15a+9-12x=25-2a. 154. 23^-24=48- 11 a?.
155. Complete the equation 4 x + ? = 35, when as = 8.
156. Complete the equation 7 ?/ + 5 = 30 + ?, when 2/ = 9.
157. If a? = 10, is the equation 5 x + 7 = 54 true ?
158. Substituting the value of the unknown quantity in an
equation and thus proving the truth of the equation is called
verifying the equation.
Solve and verify the equation x + 9 = 15.
Solve and verify :
159. a + 7 = 21. 162. 7a-2z + 8 = 33.
160. 3a?-5 = 19. 163. 5x - 3x + 21 = x + 34.
161. 5cc = x + 36. 164. lla + 1 = a; + 91.
165. One of two numbers is 3 more than twice the other.
Their sum is 15. What are the numbers ?
Let x = the less number,
then 2 x + 3 = the greater number,
then x + 2 x + 3 = 15.
166. There are two numbers whose sum is 17. The greater
is 2 more than 4 times the less. Find the numbers.
167. In the broken line ABC, BC
represents a distance which is 7 ft. more
than twice AB. If the entire line rep-
resents 31 ft., how much does AB rep-
resent? BO?
168. Each of the equal sides of an
isosceles triangle is 4 ft. longer than
the base. The perimeter is 29 ft. How long is each side?
Represent.
169. Fred is seven years older than his brother, and the
sum of their ages is 23 years. How old is each ?
310 LITERAL QUANTITIES
170. CLASS EXERCISE. may think of the ages of two
persons, and tell the class the sum of those ages and the dif-
ference between them. The class may find the ages.
171. Mr. Lee has a watch which is worth $20 more than
the chain. The watch and chain together are worth $50.
How much is each worth ?
172. Two boys bought a skiff for $ 8. The older boy gave
$ 2 more than the younger. How much did each give ?
173. A bootblack earned 30^ more on Tuesday than on
Monday. His earnings for the two days were $ 1.70. How
much did he earn on each of these days ?
174. A farmer, who had 100 acres of corn and wheat, had 20
acres more of wheat than of corn. How many acres of corn
had he ? How many acres of wheat ?
175. The sum of two numbers is 72. The greater is 8 more
than the less. Find the numbers.
176. The sum of two numbers is 90, and the greater is 26
more than twice the less. Find the numbers.
177. A traveled north from the Chicago post office, and B
traveled south from that point. When they were 50 ini.
apart, A had traveled 10 mi. more than B. How far was
each from the Chicago post office ?
178. The length of a rectangular lot is 70 ft. more than its
width. Its perimeter is 220 ft. Represent and find the length
and the area.
179. Edwin went fishing. If he had caught 10 times as
many fish as he did catch and 40 fish more, he would have had
100 fish. How many did he catch ?
180. Find three numbers such that the first is 10 more than
the second, the second is 5 more than the third, and their sum
is 47.
LITERAL QUANTITIES 311
Let x = the smallest or third number,
then x 4- 6 = the second number,
and x 4- 5 + 10 = the first number.
181. The perimeter of the triangle which
ABC represents is 36 in. The side AB is 3
in. longer than the side BC, and the side AC
is 3 in. longer than the side AB. Find length
of each side.
Represent the following :
B ^+ G 182. In the scalene triangle ABC, the side
AB is 12 in. longer than the side AC. The
side AC is 8 in. longer than the side BC. The perimeter is
73 in. Find each side.
183. The side XYof the triangle XYZis 11 in. longer than
the side YZ, and the side XZ is 17 in. longer than the side YZ.
The perimeter is 88 in. Find each side.
184. In the triangle DEF, the side DE lacks 8 in. of being
twice as long as the side EF. The side DF lacks 17 in. of
being three times as long as EF. The perimeter is 65 in.
Find each side.
185. 4 times a certain number = that number -f 21. Find
the number.
186. Separate 27 into two parts such that the greater is 9
more than the less.
187. There were four brothers, the sum of whose ages was
32 yr. Each boy was 2 yr. older than his next younger brother.
How old was each ?
188. If 3 yr. were subtracted from 4 times John's age, the
remainder would equal his father's age, which is 45 yr. How
old is John ?
312 LITERAL QUANTITIES
189. 32 boys voted for the president of their club. John
received 6 more votes than the other candidate. How many
votes did each candidate receive ?
190. Mr. A pays $ 13 more in taxes than Mr. B. Mr. C
pays $ 7 more than Mr. B. Mr. D pays $ 8 more than Mr. C.
They all pay $ 69. How much does each pay ?
191. An importer received three shipments of goods from
Germany. The duty on the second shipment was $ 3000 more
than on the first, and the duty on the last was $ 2500 more
than on the second. The duty 011 all the shipments amounted
to $ 9500. How much was paid on each ?
192. In a game of football the successful team scored 3
times as many points as the other. The difference in the
scores was 18. What was the score of each team ?
193. Make a problem to be solved by equations.
194. The profits of a farm in 3 yr. were f 2300. The profits
for the second year were $ 100 more than for the first year.
The profits for the third year were $ 300 less than for the
second year. Find the profits for each year.
195. Mr. Eowe gave 3 notes to a collector, who collected
$ 7 more on the second note than on the first, and on the third
$ 3 less than on the second. The sum of the collections was
$ 40. How much was each ?
196. The senior partner in a firm has $ 20,000 more in
the business than the junior partner. The whole capital is
$ 80,000. What is the capital of each partner ?
197. Mr. A owes Mrs. B $ 21 more than he owes Mr. C.
Both debts amount to $ 225. How much is his debt to Mr. C ?
If he pays $ 7 a week to Mrs. B, in how many weeks will he
have paid his debt to her ?
LITEKAL QUANTITIES 313
198. A chord AB divides a circumference into two arcs,
the greater of which is 30 ft. longer than twice the less. The
circumference is 120 ft. How long is each arc ?
199. Multiplying all the terms of an
equation by the same number has what
effect upon the equation? Illustrate
with numbers.
200, The expression | means - of x; -^
2 3 o o
means - of x. If you multiply both terms
FIG. 18. 3
of the fraction ^ by 3, to what integral expression is the result
equal ? Why ? 3
201. Multiply by the same number both terms of the equa-
tion - = 7 and solve the equation.
3
202. Multiplying the terms of a fractional equation by a
quantity that causes the terms to become integral is called
clearing the equation of fractions. How did you clear of frac-
tions the equation in Ex. 201 ?
203. Solve - + - = 7.
3 4
Multiplying all the terms by 3 we have x + = 21. Multiplying all
4
the terms of that equation by 4 we have 4 x + 3 x = 84. It would have
been a shorter process to multiply all the terms by 12 at once instead of
by the 3 and 4 separately. Hence we use the method given below.
204. Solve Ex. 203 by the following rule :
To clear an equation of fractions
Multiply each term of the equation by the least common mul-
tiple of the denominators.
205. Clear of fractions the equation - -f - = 5.
6 4
Multiplying each term by 12, the 1. c. m. of 6 and 4, we have
|x!2 = 2z ^x!2 = 3z 5x12 = 60. Hence, 2 x + 3 x = 60.
o 4
314
LITERAL QUANTITIES
Solve :
206. ^ + ^=24. 207. - + - =
57 69
209. +^=19f 215.
7 5
210. 2z + ^4-^ = 43 T i ir . 216.
208. ^f + ^ =
3 15
o . QC r r*
218. 4 =
213.
214.
219. _.:= .
4 7 14
220. _4:=
of it = 24. Find
of it are
3 6
221. One fifth of a certain number +
the number.
222. Find a number such that if $ of it and
added to it, the sum will be 28.
' A 223. The parts of the broken
line ABCD are in the ratios given
c in Fig. 19. How long is each
part if the entire line is 30 in.?
75 in.? 74- in.?
224. A circumference which is
FI G- 19- 1 yd. in length is divided into 2
arcs, one of which is % of the other. How long is each arc ?
Represent.
225. Find a number such that if 15 is subtracted from 3
times the number, the remainder will be 2 times the original
number.
226. The perimeter of a given isosceles triangle is 286 ft.,
and the base is T 4 r of one of the equal sides. Find the length
of its sides.
LITERAL QUANTITIES 315
227. The perimeter of a rectangle is 1254 ft., and the width
is j^- of the length. Find width, length, and area.
228. A certain number 4- 2 times itself + 7 = 37. Find the
number.
229. Draw a right-angled scalene triangle. If the altitude
were 1|- times the base, the hypotenuse 2^ times the base, and
the perimeter 40 in., how long would each side be ?
230. Separate 45 into two parts, one of which is J of the
other.
Let x = the greater number.
231. 42 is the sum of two numbers whose ratio is f . What
are they ?
232. Separate 90 into two parts whose ratio is -J.
233. CLASS EXERCISE. may give the sum of two num-
bers whose ratio is a fraction, and the class may find the
numbers.
234. How would you divide 75 $ between two boys, giving
one boy % as much as the other ?
235. The session of a certain school is 4J- hr. a day. How
many hours and minutes are given to recitation if the recitation
periods take f as much time as that devoted to other purposes ?
236. There are three numbers whose sum is 108. The first
is f of the second, and the third is twice the first. Find the
numbers.
Let x = the second number.
237. 4- of a certain number minus -J- of it = 2. What is the
number ?
238. There are three numbers whose sum is 84. The second
number is 1^- times the first, and the third number is ^ of the
second. Find each number.
239. The profits of a business during its second year were
1 times the profits during its first year, and the profits for the
316 LITERAL QUANTITIES
third year were 1-^- times those for the second year. The profits
for the 3 yr. were $ 8250. Find the profits for each year.
240. Draw a trapezoid making the upper base the lower
base. If one of the non-parallel sides is J of the lower base,
the other non-parallel side -f^ of the lower base, and the per-
imeter of the trapezoid is 54 in., how long is each side ?
241. How long is each side of a rhomboid whose perimeter
is 14 ft. 6 in. and whose short sides are each as long as a
long side ?
MISCELLANEOUS EXERCISES
1. 7% of 7% of $825 = ?
2. A lawyer collected $ 1275 for a client. He charged
10% for collecting. He gave 60% of his fee to his wife. How
much money was received by the client ? By the lawyer ? By
his wife ?
3. Mr. and Mrs. Shaw and two children took a trip on a
lake steamer. The fare was $ 9.00, children half price. Meals
on the steamer were $ 1.00 each. The family took supper,
breakfast, and dinner on board, and paid $ 5.00 for a state-
room. What was the cost of the trip ?
4. Find the g. c. d. of the first composite odd number after
39 and the first composite odd number after 57.
5. How much is gained by buying a $ 500 bond at 105,
keeping it until 2 yr. interest at 3% has been received, and
selling it at 109 ?
6. In a certain city, the highest temperature in July was
100. The highest temperature in December was 75. The
difference in temperature was what per cent of the highest
temperature in July ? In December ?
7. Solve 9 a - 25 -f 3x = 7x-5.
8. Solve 2lx-20 = 7x- 15^ + 67.
9. Three times a certain number equals 148 minus the
number. What is the number ?
MISCELLANEOUS EXERCISES 317
10. What number doubled and increased by 4 equals 188 ?
11. If Mary were 15 years older than twice her present age,
she would be as old as her cousin, who is 37 years old. How
old is Mary ?
12. Find a number such that the difference between ^ of it
and -j- of it is 2.
13. The perimeter of a rhomboid is 70 dm. Its long sides
are each 5 dm. longer than the sum of its short sides. Find
the length of each side.
14. CLASS EXERCISE. may write an equation having 90
for its second member and a prime number for the coefficient
of the unknown quantity. The class may solve the equation.
15. Solve .6^ = 120.
Clear the equation of fractions by multiplying each term by the denom-
inator of the decimal .6.
16. Solve .06 x = 24, .03 x = $ 240, 8 % of x = 32.64.
17. Solve .7 x = 2800, .016^ = 32, 4% a = 48.
18. Solve .9 x = 540, 11 % x = 33, .012 x = 720.
19. An agent who charged 7% for collecting a sum of money,
received $ 210 as his commission. How much did he collect ?
Let x equal the number of dollars collected, then .07 x equal 210.
20. What amount must be collected that the fee for collecting
it may be $70 when the rate is 5% ? 7% ? 2% ? 10%? S%?
21. Express in terms of x the interest of $ x for 1 yr. 6 mo.
at 6%.
The interest of $ 1 for 1 yr. 6 mo. at 6 % is 9 ^. The interest of $ x is
x times 9^ or 9x^.
22. Express in terms of x the interest of $ x at 6% for 2 yr.
6 mo. 6 da. For 5 yr. 8 mo. 12 da. For 7 yr. 10 mo. 24 da.
23. What principal will gain $ 157.50 in 3 yr. 6 mo. at 6% ?
Let x = the number of dollars in the principal. $ 1 in 3 yr. 6 mo. at
6% will gain .21. x dollars will gain x times $.21 or $.21. Then
$. 21 x = $157.50.
318 LITERAL QUANTITIES
24. By similar reasoning find the principal which will gain
$ 19.75 in 2 yr. 3 mo. at 6%. When you have found it, see if
the interest upon it at 6% for 2 yr. 3 mo. is $ 19.75.
SUGGESTION TO TEACHER. Pupils should prove these problems until
they realize that each of them is merely a reversed case of the ordinary
problem in which the interest is required to be found.
Find the principal which will gain :
25. $240 in 3 yr. at 5%.
26. $ 360 in 4 yr. 6 mo. at 6%.
27. $780 in 5 yr. at 8%.
28. $ 175 in 6 yr. 3 mo. at 4%.
29. $ 200 in 3 yr. 2 mo. 15 da. at 8%.
30. $ 250 in 2 yr. 8 mo. at 6%.
31. In what time will $ 500 gain $ 34 at 6% ?
Let x = the number of years. The interest of $ 500 at 6 % for 1 yr. is
$ 30. For x yr. the interest will be x times $ 30 or 30 x dollars. Hence
30 x 34 and x ^ 1^ yr. or 1 yr. 1 mo. 18 da.
In what time will :
32. $560 gain $106.40 at 8% ?
33. $ 750 gain $ 192 at 6% ?
34. $ 187.50 gain $ 37.50 at 5% ?
35. $ 65 gain $ 2.60 at 6% ?
36. $216 gain $122.22 at 10%?
37. At what per cent will $ 400 gain $ 35 in 2 yr.
Let x = the number of per cent. The interest of $ 400 for 2 yr. at
1 % is $ 10. At x % the interest will be x times $ 10 or 10 x dollars.
Hence 10 x = 35 and x = 3 \ %.
At what per cent will :
38. $ 700 earn $ 63 in 2 yr. 3 mo. ?
39. $ 600 earn $ 45 in 1 yr. 6 mo. ?
MISCELLANEOUS EXERCISES 319
40. $ 225 earn $ 49.50 in 2 yr. 9 mo. ?
41. $ 500 earn $ 105 in 7 yr. ?
42. $ 600 earn $ 125 in 8 yr. 4 mo. ?
43. What principal will amount to $ 532 in 3 yr. 8 mo. at 9% .
Let x = the number of dollars in the principal. The amount of $ 1 for
3 yr. 8 mo. at 9 % = $ 1.33; the amount of x dollars = 1.33 x dollars.
44. What principal will amount to $ 94.50 in 2 yr. 6 mo.
at 5% ?
45. $ 155 in 3 yr. 4 mo. at 6% ?
46. $ 85 in 1 yr. 8 mo. at 8% ?
47. $ 168.36 in 2 yr. 5 mo. at 7% ?
Find the missing term in the following :
Prin. Rate Time Int.
48. $600 5% 2yr. x
49. $400 x 3 yr. $48
50. x 7% lyr. $31.50
51. $500 6% x $50
52. A man wishes to set aside a sum of money, the interest
of which will furnish his daughter a yearly income of $ 1000.
If 6% can be obtained for it, how much shall he invest for her?
How much would be necessary to invest if only 3-|-% could be
obtained for it ?
53. Which is the greater price for an article, $ 100 cash or
$ 108 due in 1 yr., without interest, when the customary interest
is 8%. Why?
54. The Present Worth of a sum of money due at a given
time is that smaller sum of money which, when put at interest
at the usual rate, will amount to the given sum in the given
time.
What is the present worth of $ 770 due in 2 yr., when
money is worth 5% ?
The above question really asks, what sum put at interest at 5% will in
2 yr. amount to $ 770 ?
320 LITERAL QUANTITIES
55. When money is worth 6%, what is the present worth of
$224, due2yr. hence?
When you have found the present worth, prove your work by comput-
ing the interest upon it for the given time and rate to see whether the
principal and interest will amount to $ 224.
56. The difference between the present worth and the
amount due at maturity is called the True Discount.
Supposing money to be worth 6%, find the present worth
and the true discount of $ 4720, due 3 yr. hence.
True discount must be distinguished from bank discount, which is
merely the simple interest on the face of the note.
Find present worth and true discount of : ,
57. $199.80, duel yr. 10 mo. hence.
58. $307.50, due 5 mo. hence.
59. $ 143.75, due 2 yr. 6 mo. hence.
60. There are two ways of finding the true discount after
the present worth is known. What are they ?
61. CLASS EXERCISE. may think of a sum of money and
find how much it will be worth at a given future time, the cus-
tomary rate of interest being / . He may report to the class
the amount of that sum, and the time and rate, and the class
may find the original sum.
Find the difference between the bank discount and the true
discount of the following, the rate of interest being 4% :
62. $856.00, due in 1 yr. 9 mo.
63. $817.50, due in 2 yr. 3 mo.
64. $712.50, due in 3 yr. 6 mo.
65. $ 626.00, due in 1 yr. 1 mo.
66. $ 987.00, due in 2 yr. 5 mo.
67. Kesolve 18 into two factors, one of which is a perfect
square.
MISCELLANEOUS EXERCISES 321
68. Resolve 108 into two factors, one of which is the largest
possible square.
69. Find the area of a right triangle whose base is 20 in.
and altitude 1 ft.
70. Find the area of a trapezoid whose upper base is 10 in.,
lower base 14 in., and altitude 5 in.
71. Find the interest of $1000 for 3 yr. at 6%.
72. If Mr. Brown puts $1000 at interest at 6%, how much
interest will be due him at the end of one year ? If, instead
of collecting this interest, he adds it to the principal, and loans
the $1060 for a second year at 6%, what will be the second
year's interest? If this second year's interest, $63.60, is
added to the second principal, $ 1060, and the sum, $ 1123.60,
is put at interest for the third year, what will be the amount
at the end of that year ?
73. How much greater is that amount than the original
$ 1000 ? That increase is the Compound Interest. How much
greater is the compound interest than the ordinary simple
interest of $ 1000 for 3 yr. at 6%?
74. Find by the method shown in Exs. 72 and 73 the com-
pound interest of $ 1000 for 4 yr. at 10%.
75. Money put at compound interest gains more rapidly as
the number of years increases. Can you see why ?
76. One dollar put at compound interest at 7% will amount
in 25 yr. to $ 5.427. To how much will $ 2000 amount in that
time at that rate of compound interest ? How much of that
sum is interest ? Find the simple interest of $ 2000 for 25 yr.
at 7%, and find how much less it is than the compound interest.
77. On his eighteenth birthday John Smith deposited $ 100
in a savings bank, which paid 4% compound interest. He did
the same on his next two birthdays. How much had he to
his credit in that bank on his twenty-first birthday ?
HORN. GRAM. SCII. AR. 21
CHAPTER X
INVOLUTION AND EVOLUTION
1. What is meant by the power of a number ? Illustrate.
2. Raise to the 3d power 19. . .3. 2. 1.2. .000007.
3. The process of raising a quantity to a higher power is
called Involution.
Involve 7 4 . .2 5 . (4i) 3 . .6 3 . (l.l) 4 . () 2 .
4. 9 3 :3 5 = ? 8 3 :2 7 = ? 6 4 :4 7 = ?
5. When a = 2, how much is 3 a 5 ?
6. Give quickly the squares of the first 12 numbers.
7. Give the squares of 20. 30. 40. 80. 70.
8. Give quickly the cubes of the first 12 numbers.
9. Give the cubes of 20. 40. 60. 80. 50.
10. Name a perfect square that is a factor of 72. Of 50.
11. Name a perfect cube that is a factor of 250. Of 24.
12. What is meant by the root of a number ? Illustrate.
13. The process of finding any root of a given quantity is
called Evolution. It is the exact opposite of involution.
What is the square root of 121 ? Of 144 ?
14. Draw a square, and illustrate the following statement :
The number of units of length in one side of a square is the
square root of the number of corresponding units of square
measure in the area of the square.
322
INVOLUTION AND EVOLUTION
323
15. Copy Fig. 1, making ABCD a 4-inch square, and AE
and CO each 2 in. How many square
inches in the square, ABCD? In
HFKB? In the rectangle, BKCG?
In the rectangle EHBA? How many
square inches in all the additions to the
square ABCD ?
16. How long is one side of a square
which contains 4 sq. ft. Represent it.
Draw the additions which would be
needed to make the figure represent a square yard.
17. Draw a square which contains 81 sq. cm. Add squares
to it until it is a square decimeter. How many square centi-
meters are there in the two rectangles and little square which
were added ?
18. How many square inches are there in the two rectangles
and the small square which, when added to a 5-inch square,
will change it to an 8-inch square ?
19. How many square inches are there in the two rectangles
and small square which, when added to a 10-inch square, will
change it to a 14-inch square ?
additions on two sides. How wide
is each addition ? How many
square "inches are there in the
sum of all the additions?
21. If the square ABCD in
Fig. 2 contains 100 sq. in., how
long will the line DC be ? If the
sum of all the additions is 69 sq.
in., how long will the line CG
be?
20. A 10-inch square
gr HI
n
FIG. 2.
324
INVOLUTION AND EVOLUTION
10x10
H
B K
10x10
FIG. 3.
FIG. 4.
SOLUTION. CG- in Fig. 4 represents the width of the additions which
will change the square ABGD (Fig. 3) to the square EFGD (Fig. 2).
E
n K
1 10 B B 10
FIG. 5.
G
"When the two rectangles in Fig. 4 are placed side by side as in Fig. 5,
they form a rectangle whose length is 10 ft. + 10 ft., or 20 ft. We know
that its area plus the area of the square needed to complete Fig. 4 is
69 sq. ft. We wish to know its width. Dividing 69, the number of square
feet in all the additions, by 20, the number of feet in the base of the two
rectangles, we have the quotient 3, which shows that the probable width
of the rectangles is 3 ft., and that a side of the small square needed to
complete Fig. 4 is 3 ft.
H K
G JIS F
A. 10 B B 10 C J5 8 K
- FIG. 6.
When the small square is added to the sum of the rectangles as in
Fig. 6, a rectangle is formed whose length is 10 ft. + 10 ft. -f 3 ft., or
23 ft. Assuming that its width is 3 ft., its area is 69 sq. ft., which exactly
equals all the square feet which were to be added to the 10-inch square.
INVOLUTION AND EVOLUTION 325
22. How many square feet would be represented by Fig. 4
if the square were completed ? How long would one side of
it be ? What, then, is the square root of 169 ?
23. A lot 30 ft. square was increased by the addition of
61 sq. ft. The additions were made on two sides, and in such'
a way that the lot when increased was also in the form of a
square. Eepresent and make additions as in the solution of Ex.
21. How wide was each addition ? How long was a side of the
square after the additions were made ? How many square feet
were in the completed square ? What is the square root of 961 ?
24. A square contains 100 sq. in. If 96 sq. in. are added
to it, as in Fig. 2, how wide will the addition be ? How long
will one side of the whole square be? What is the square t
root of 196 ?
25. How wide must be the additions that will change a
square containing 400 sq. in. to a square containing 441 sq. in ?
What is the length of one side of a square containing 441 sq. in. ?
What is the square root of 441 ?
26. If you had 576 sq. ft. of boards to be arranged in the
form of a square, how long would one side of the square be ?
First make a square containing 400 sq. ft., then apply the remaining
176 sq. ft. on two sides.
27. Find in the same way the side of a square containing
625 sq. in. What is the square root of 625 ?
28. 1 2 = 1. 9 2 = 81. 10 2 = 100. 99 2 = 9801.
How many figures are there in the expression of the square
of a number less than 10 ? In the expression of the square of
a number greater than 9 and less than 100 ?
To find how many figures there are in the integral root of a given
number separate the number into groups of two figures each by placing
an arc over the units' and tens' figures, and also over each succeeding group
or part of a group, as 12544. The number of arcs equals the number of
figures in the root.
326 INVOLUTION AND EVOLUTION
29. How many figures are there in the integral root of a
number whose expression takes three places ? Six places ?
Nine places ?
30. When t = 40 and u = 5, how much is (t + u) 2 ?
31. When = 30 and u = 2, how much is (t + u) 2 ?
Z 2 + 2 tu + u 2 ?
32. Find the square of 25 in terms of its tens and units.
25 = 2 tens and 5 units.
20 + 5 Beginning at the right, and multiplying units
_ 20 + 5 by units, we have 5x5, expressed 5' 2 . Multiply-
(20 x 5) + 5 2 ing tens by units we have (20 x 5). Multiplying
20 2 + (20 x 5) 5 units by 20 we have the equivalent of another
20 2 + 2 (20 x 5) + 5 2 (20 x 5), which is written under the first. 20 x 20
is expressed 20 2 . The sum of all these products
is 20 2 + 2 times (20 x 5) + 5 2 .
33. Is there any difference between the value of 25 2 and of
20 2 + 2(20 x 5) + 5 2 ?
34. Illustrate the following principle:
The square of any number consisting of tens and units is equal
to the square of the tens plus twice the product of the tens and
units plus the square of the units.
This principle may be expressed by the following formula:
(t +- u*) 2 = t 2 +- 2 tu +- u 2 . By the use of this formula we may
readily find the square root of any number.
35. Find the square root of 5329.
^^ Beginning at the right and separating 5329
& + 2 tu + w 2 = 5329 |73 i nto groups of two figures each, we find that
t 2 = 4900 the root will consist of tens and units. We
2 1 + u = 143) 429 find that 4900 is the largest square of a
u (2 1 + w) = 429 multiple of ten that 5300 includes. Here,
4900 = 2 , and t = 70. We write the 7 as the
tens' figure of the root. Subtracting J 2 , or 4900, from 5329, we have
remaining 429, which equals 2 tu + w 2 . As t equals 70, 2 t equals 140.
We wish to find the value of u. We use 140 as a trial divisor, and
place the quotient 3 as the unit figure of the root. We add the 3 to
INVOLUTION AND EVOLUTION 327
140 to find the true divisor. Multiplying the true divisor 143 by the
quotient 3 and subtracting the product from 429, we find that there is no
remainder. Hence the number 5329 is a perfect square, and 73 is the
square root.
NOTE TO TEACHER. It is suggested that when this explanation is fully
understood, the process be shortened by adopting the following method
for the work, after the tens' figure of the root has been found, and its
square subtracted from the left hand period.
"Double the root already found and place the result as a trial divisor.
Cover the right-hand figure of the dividend, and see how many times the
trial divisor is contained in what remains. Place the quotient beside the
tens' figure as the next figure of the root, and annex it also to the trial
divisor, thus making the true divisor. Multiply the true divisor by the
quotient and subtract the product from the dividend," etc.
36. Find the square root of 1225.
Extract the square root of the following :
37. 625 43. 1764 49. 2809
38. 484 44. 9216 50. 2025
39. 289 45. 6889 51. 1089
40. 1681 46. 2601 52. 6561
41. 4356 47. 8649 53. 1156
42. 2209 48. 5184 54. 5041
55. Find the square root of 324.
Although 2 is contained in 22 eleven times, yet nothing larger than 9
can be used for a root figure.
56. V792i = ? V2401 = ? V~348l = ?
57. Of what number is 15 the square root ? 20 ?
58. Give a perfect square whose root is a prime number.
59. CLASS EXERCISE. may give to the class the square
of a number of two places. The class may find the number.
60. Find the square root of 12544.
There are three figures in the root. After finding the hundreds' and
tens' figures double the root already found and proceed as before.
328 INVOLUTION AND EVOLUTION
Find the square root of the following numbers :
61. 245,025 66. 249,001 71. 511,225
62. 375,769 67. 546,121 72. 811,801
63. 784,996 68. 56,169 73. 674,041
64. 776,161 69. 23,409 74. 70,756
65. 938,961 70. 529,984 75. 119,716
76. How long is a side of a square whose area is 1681 sq. in. ?
961 sq. cm. ? 729 sq. ft. ? 1024 sq. m. ?
77. How long is the perimeter of a square whose area is
1089 sq. in. ? 8836 sq. ft. ? 5476 sq. cm. ? 3969 sq. dm. ?
78. A field contains 3600 sq. rd. How much will it cost to
fence it at 80 $ a rod ?
79. At $ .60 a rod how much will it cost to fence a square
field containing 1849 sq. rd ? 1369 sq. rd ?
80. Find the square root of -J. Of -f^.
81. Find the square root of : -fff \ 9 . yf^. T 8 ^.
82. Keduce 1-j- to an improper fraction and find its square
root.
83. Find the square root of : 2J. 4ff 3^. 2^. 6J|.
84. CLASS EXERCISE. may give to the class the square
of a small mixed number. The class may find the number.
85. Find the square root of the decimal .000625.
In dividing a decimal into groups, begin at the decimal point and group
to the right as .000625. There will be as many decimal places in the
root as there are groups in the original decimal.
86. Find the square root of: .0049. .0081. .0025. .000036.
87. Find the square root of: .000529. .000121. 29.16.
88. Give one of the two equal numbers whose product is:
50.41. 60.84. 94.2841. " 38.3161.
89. Which is greater, 124 or 124.0000 ?
90. Find the approximate square root of 124 to two places
of decimals.
Annex naughts to 124.
INVOLUTION AND EVOLUTION
329
91. Find to two places of decimals the square root of:
27. 24. 85. 1.35. 2.7. 3.81. 4.09.
92. Find to one decimal place the length of the side of a
square that contains 45 sq. in. 75 sq. in. 95 sq. in.
93. Find value of x, when y? = .014641. When tf = .3969.
94. Draw a right triangle mak-
ing the base 3 in. and altitude 4
in. If your drawing is correct how
long will the hypotenuse be ?
95. Construct a square on each
side of the triangle as in Fig. 7 and
compare the square of the hypote-
nuse with the sum of the squares of
YIG. 7. the shorter sides.
96. Geometry shows the reasons for the following fact,
called the "Pythagorean Theorem" after Pythagoras, the
Greek philosopher who gave it to the world.
In a right triangle the square of the hypotenuse is equal to
the sum of the squares of the other two sides.
How long is the hypotenuse of a right triangle whose alti-
tude is 12 in. and base 5 in. ?
Find the hypotenuse of each of the following triangles :
Base. Altitude. Base. Altitude.
97. 7 24 100. 15 20
98. 8 15 101. 60 11
99. 40 9 102. 24 45
103. A traveled east 120 miles and B south 160 miles from
the same point. How far apart were they then ?
104. How long is the diagonal of a rectangle which is 24 in.
long and 10 in. wide ?
105. How- long is the longest straight line that can be
drawn on a blackboard 12 ft. long and 5 ft. .wide ? :
330 INVOLUTION AND EVOLUTION
106. If your schoolroom were 36 ft. long and 27 ft. wide,
what would be the length of the longest straight line that could
be drawn upon the floor ?
107. Draw a vertical line AB 8 in. long,
and mark the middle point E. Draw CD, a
horizontal line, 6 in. long, whose middle point
is also E. Draw AD, DB, BC, and CA.
~~ D What kind of a figure do they outline?
How long is its perimeter?
108. The diagonals of every rhombus bi-
B sect each other at right angles. If the
diagonals of the rhombus you have drawn
were 16 in., and 12 in., how long would a side of the rhombus
be?
109. How long is the perimeter of a rhombus whose long
diagonal is 2 ft., and short diagonal 1J ft. ?
110. A gate 8 ft. long and 6 ft. wide has a diagonal cross-
bar. How long a piece of wood was required for the crossbar ?
111. A window is 12 ft. from the ground. How long a
ladder is required to reach it if the foot of the ladder is 16 ft.
from the foundation of the house ?
112. John's kite is caught in the top of a tree 69 ft. high,
and John, who is standing 156 ft. from the tree, holds the kite
by the end of the string. His hand is 4 ft. from the ground.
Represent the conditions of this problem and find the length of
the string.
113. How long is the perimeter of a right triangle whose
base is 21 ft. and altitude 72 ft. ?
114. In a right triangle whose base is 80 cm. and altitude
18 cm. the hypotenuse is how much less than the sum of the
other two sides ?
115. What distance is saved by walking diagonally across
a vacant lot 300 ft. long and 160 ft. wide instead of walking
along two sides of it ?
INVOLUTION AND EVOLUTION 331
116. By taking the diagonal path across a rectangular lot
instead of walking between the same points along the edge of
the lot, what fractional part of the distance is saved when the
lot is 42 ft. long and 40 ft. wide ? 75 ft. long, 40 ft. wide ?
80 ft. long, 60 ft. wide ?
117. Since the square of the hypotenuse equals the sum of
the squares of the shorter sides, the square of either short side
is equal to the square of the hypotenuse minus the square of
the third side. Let b stand for base, a for altitude, and h for
hypotenuse. Show how the following equations are derived :
6 2 + a 2 = h 2
6 2 = 7i 2 - a 2
a 2 = 7* 2 - 6 2
Use the equations of Ex. 117 in finding the missing side of
triangles which have the following measurements.
Base. Alt. Hyp. Ease. Alt. Hyp.
118. 18 x 30 121. 48 x 50
119. x 12 26 122. x 72 75
120. 7 12 a 123. # 9 20
124. The center pole of a round tent is 18 ft. high. A rope
stretches from a point 3 ft. from the top of the pole to a point
on the ground 36 ft. from the base of the pole. How long is
the rope ?
125. Draw a right-angled isosceles triangle. If each of the
equal sides were 9 in., how long would the hypotenuse be ?
126. How long is the diagonal of a square each of whose
sides is 10 ft. ?
127. How long is the diagonal of a square whose area is
64 sq. ft. ?
128. The diagonal of a square is 15 in., how long is one side
of the square ?
*2 + X 2 = 152
332 INVOLUTION AND EVOLUTION
129. A chimney 60 ft. high, casts a shadow 144 ft. long.
What is the distance from the end of the shadow to the top of
the chimney ?
130. Try to find an integer whose square is equal to the
sum of the squares of two other integers. With the numbers
make a problem about finding one of the sides of a right
triangle.
The following is an easy way of finding three integers, the square of
one of which is equal to the sum of the squares of the other two. Take
any two unequal numbers. The sum of their squares will represent the
hypotenuse of a right triangle, the difference of their squares one of its
sides, and twice their product the other side. Taking 2 and 1,
22+12 = 5, 22-12 = 3, and2(2xl)=4.
131. Find three integers which may represent the length of
the three sides of a right triangle, by combining 3 and 1 as di-
rected in the previous note. Combine 4 and 1, 3 and 2, 1 and
1, 5 and 2. Prove your work.
In the following problems, when the square root is not exact, extract
the root to one decimal place.
132. In a room 20 ft. long, 15 ft. wide and 10 ft. high, a
beetle crossed the floor diagonally and then crawled up to the
ceiling. How far did he travel ? What was his shortest dis-
tance then from the point at which he started ?
Below are given the dimensions of several rooms. Find dis-
tances from an upper corner to the opposite lower corner.
Length. Width. Height.
133. 24 ft. 18 ft. 9 ft. .
134. 17ft. 15ft. 10ft.
135. 25ft. 24ft. lift.
136. A tank 8 ft. long and 6 ft. wide has water in it to the
depth of 5 ft. How long is the longest straight stick that can
be wholly in the water if the stick is pointed at both ends ?
MISCELLANEOUS EXERCISES 333
MISCELLANEOUS EXERCISES
1. A party of excursionists, consisting of 9 adults and 4
children, went from St. Louis to Washington, D. C., and
returned. Eound trip tickets were $ 25.45, children half price.
The party occupied 7 berths in the sleeping car in going, and
the same number in returning. The price of each berth was
$4.75 each way. Meals in the dining car cost $1.00 each.
The party left St. Louis at 4 P.M., on Monday, and arrived in
Washington in time for breakfast Wednesday morning. During
the trip the whole party took each meal in the dining car.
They took an equal number of meals on the cars on their return
trip. What were the expenses of the whole party on the trip?
2. Multiply nine hundredths by three ten-thousandths and
divide the product by three hundred seventy-five hundred-
thousandths. Then multiply the quotient by six hundredths
and subtract the result from seven thousand eight hundred fifty.
3. What is the 1. c- m. of the three prime numbers next
after 19 ?
4. How is the least common multiple of several- prime
numbers found?
5. The 1. c. m. of 6, 8, and 12 is what per cent of the fourth
multiple of 12?
6. What is the largest integer that will exactly divide 8, 12,
and 16?
7. The g. c. d. of 18 and 24 is what per cent of the third
multiple of 8?
8. Find four numbers between 1 and 101, each of which
can be resolved into two factors, both of which are perfect
squares whose roots are greater than 1.
9. The divisor of a certain number is 189, the quotient is 22,
and the remainder is 14|. What is the number ?
10. Find the interest of $17.35 for 2 mo. 3 da., at \/o per
month.
334 INVOLUTION AND EVOLUTION
11. Mr. Rowe bought a bill of goods amounting to $500,
receiving a discount of 40 % and 60 % - How much did he pay ?
12. Mr. E. sold $150 worth of goods to Mr. W., taking
Mr. W.'s note for the amount on 3 mo. time. The note was
discounted at 8%. How much did Mr. K. receive?
13. Mr. A. bought 120 U. S. 3's at 102, brokerage \%.
These bonds are non-taxable. How much more or less would
he gain from them each year, than from the same amount of
money loaned at 6% in a locality where the tax rate is 37 mills
on a dollar?
14. Mr. N. bought 1000 shares of Gas and Electric Light Co.
stock at 105. He kept the shares until a semi-annual dividend
of 7J% and another of 9% had been paid, and sold them at 110.
How much did he gain?
15. Mr. Walker bought a horse for $ 75 and sold it at a gain
of 33^%. He took a note for the amount, due in one year, and
had the note discounted at 10%. What per cent did Mr.
Walker realize ?
16. A boy had 80% of a dollar, spent 80% of what he had,
and lost 50% of what remained. What per cent of the dollar
did he then have ?
17. A lawyer having a debt of $1346.50 to collect, compro-
mised by taking 80 % . His fee was 5 % of the amount collected.
What was his fee and how much should he return to his client ?
18. A farmer sold 25% of a tract of land containing 120 A.,
at $.50 a square rod. How much did he receive for it?
19. The largest bell in the world is in Moscow. It weighs
216 T. If 77% of it is copper, and the rest tin, how many tons
of each are in the bell ?
20. A bell in Burmah weighs 117 T. If it contains the
same proportion of tin as the Moscow bell, how much tin is
there in it ?
21. A bell in Pekin containing the same proportions of
copper and tin weighs 53 T. How much copper is there in it ?
MISCELLANEOUS EXEfcCISES 335
22. 25 bu. of lime were bought for $6.25. At what price
per peck must it be sold to gain 66f % ?
23. In a war the Brazilians lost 43,365 men, which was 35%
of their army. How many were left in the army ?
24. Mr. C.'s agent in N. Y. bought for him a bill of goods
amounting to $ 2575, charging him a commission of 3%%. How
much must Mr. 0. remit to pay all expenses?
25. An agent's commission at 3% upon a sale was $99.
For how much was the property sold? How much did the
owner receive?
26. How much must an importer pay as duty on 7 casks of
wine, each containing 42 gal., 2% being allowed for leakage,
and there being a specific duty of $1.25 per gallon?
27. 3 gal. 1 qt. 2 gi. is 16f % of how much?
28. A sold goods for $70, making a profit of 16f %. What
per cent would he have made by selling them for $ 72 ?
29. After retaining 3% for selling my potatoes, my agent
sends me $523.80. For how much did he sell them?
30. The smaller of two numbers is 359.7 and the difference
is 28 J. What is the larger number?
31. Multiply the square of ^ by the reciprocal of -J-.
32. Multiply f by the square of the reciprocal of f.
33. Multiply (|) 2 by the cube of the reciprocal of j.
34. How many times is the square of 5 contained in the
square of 10 ?
35. How many times is the square of 6 contained in the
square of 12 ?
36. How many times is the square of any number contained
in the square of twice that number ? Illustrate.
37. How many times is the cube of 3 contained in the cube
of 6 ? The cube of 5 in the cube of 10 ?
38. How many times is the cube of any number contained in
the cube of twice that number ? Illustrate.
336 INVOLUTION AND EVOLUTION
39. How many times does a cube of a number contain the
cube of half that number ? Illustrate.
40. Make a true equation about the number 10 with two
terms in each member. Let one of the coefficients be 7.
Solve:
41. 8z-5o;-4 = 31. 42. 9# - 5z + 30 = 58.
43. 3x + Sx + 6x-5 + 2x 8 = 31-x.
44. What number increased by 5 times itself equals 42 ?
45. Each of the sides of an isosceles triangle is -f as long as
the base. The perimeter is 34 in. How long is each side ?
46. The perimeter of a trapezium is 54 in. The first side is
J as long as the second, the third is 1J times the second, the
fourth is 1 J times the second. Find the length of each side.
47. What principal will gain $ 108 in 3 yr. at 5% ?
48. In what time will $ 800 gain $ 108 at 6% ?
49. At what rate will $ 900 gain $ 78 in 2 yr. 9 mo.
50. What principal will amount to $ 629.20 in 3 yr. 6 mo.
at 6% ?
51. What is the present worth of $635.60 due in 2 yr. 3
mo., when money is worth 6% ?
52. What number is that to which if its ^ and its be added
the sum is 33 ?
53. From a rectangular field 20 rd. long and 16 rd. wide, a
man bought a square lot which was ^ of the field. What was
the cost of fencing the lot at $ 4.95 per rod ?
54. In the Webster School building there is a flight of stairs
in which each step is 6 in. high and 10 in wide, each step pro-
jecting 2 in. over the edge. There are 13 steps in the flight.
Find the length of the handrail.
55. How much would it cost to carpet the same flight of
stairs with carpet at $ .90 per yard, allowing 4 in. for turning
under at top, the same at the bottom, and 3 in. for each turn at
the edge of the steps ?
MISCELLANEOUS EXERCISES
337
56. How long is the perimeter of a right triangle whose
base is 60 in. and altitude 61 in. ?
A F
E
N
H
M L
FIG. 9.
57. ABCD is a 12-inch square. AF,
BH, CL, and DN are each 4 in. EA,
OB, KG, and MD are each 3 in. How
long is the perimeter of the irregular
octagon EFGHKLMN? What is its
area?
58. The diagonal of a square is 12 in.
Find its perimeter.
59. The diagonal of a rectangle is 51 in., and the width of
the rectangle is 24 in. Find the area and the perimeter of the
rectangle.
60. How long is a quadrant of a circumference whose
diameter is 21 in. ?
61. ABCD is a 10-inch square. With the
vertex of each angle as a center and with
a radius of 31 in. an arc is drawn. How
long is the perimeter of each sector formed
by the arc and the parts of the sides of the
square ? How long is the perimeter of the
irregular figure left after the sectors are
cut from the square ?
62. A chord is drawn across a circle in
such a way as to divide the circumference into two arcs, one of
which is four times the other. If the circumference is 85 cm.,
how long is each arc ?
When a body falls from an elevated place, if it is not hindered by the
air or other obstructions, it falls 16 T ^ ft. in the first second, and 3 times as
far in the second second. In the third second of time it falls 5 times as far
as in the first second. In the fourth second it falls 7 times as far as in the
first second, and so on, the ratio of the distance passed over in the first
second to the distance passed over in any given second being equal to the
ratio of 1 to the corresponding odd number.
HORN. GRAM. SCH. AR. 22
FIG. 10.
338 INVOLUTION AND EVOLUTION
63. Develop the following table to the fifth second of time :
Distance passed over in 1st sec. equals 16^ ft.
Distance passed over in 2d sec. equals 3 times 16 T L ft.
Distance passed over in 3d sec. equals 5 times IGyL ft.
64. Through what distance will a falling body pass in the
fourth second of its fall? In the fifth ?
65. How far does such a body fall in the first three seconds?
In the next three seconds ?
66. 1024 sq. ft. can be arranged either as a perfect square or
as a rectangle 64 ft. long, or 128 ft. long, or 256 ft. long. What
would be the length of the perimeter in each case ?
67. 1296 sq. ft. can be arranged either as a perfect square or
as a rectangle 72 ft. long, or 216 ft. long, or 432 ft. long, or
648 ft. long. Find the length of the perimeter in each case.
It will be seen that when a given area is arranged in the form of a
square, it has a shorter perimeter than when it is arranged in the shape
of any other rectangle, and that as the ratio of the length to the width
increases, the perimeter of the figure also increases. Illustrate this fact.
68. A farmer has a rectangular field 160 rd. long and 40 rd
wide, which he wishes to fence and divide by fencing into 4
equal lots. If the fencing costs $ .60 a rod, what will be the
difference between the expense of running his dividing fences
parallel with the long sides of the field, and the expense of
running them parallel with the short sides of the field?
69. Mrs. Wood has a rectangular hall 35 ft. long, and 14 ft.
wide, the floor of which is laid with parquetry flooring.
The border is 3 ft. wide. At $ 1.25 per square yard, what is
the cost of the floor surface inside the border?
70. The floor of Mrs. Wood's sitting room, which is square,
is covered with the same kind of flooring, and has a border of
the same width. The surface inside the border in the sitting
room is equal in area to the surface inside the border of the
hall floor. If the cost of the border is $2.75 per linear yard,
MISCELLANEOUS EXERCISES 339
measured around the edge of the room, how much more will the
border in the hall cost than that in the sitting room?
71. Find the area of the largest rectangle that can be
inclosed by a line 48 in. long.
72. In a box, each of whose inside measurements is 9 in.,
there were packed blocks enough to make 3 cubes. One was
a 6 in. cube, another was an inch cube. How long was the
edge of the third cube ?
73. How many cubic inches are there in a cube whose
edge is i^ of a foot?
74. A pan in the shape of a rectangular prism, 11 in. long,
and 7 in. wide, was out doors during a rain storm. After the
storm the pan was found to contain a gallon of water. How
deep was the water in the pan?
231 cu. in. = 1 gal.
75. Extract the square root of 2704. Of 60516. Of 1 ^_.
Of ff Of 44.89.
76. How much is x when 3 y? = 75 ? 432? 243 ?
77. If a rectangular lot is twice as long as it is wide, and its
width is represented by x, how is its area represented ? If the
area is 72 sq. in., what are the dimensions of the lot?
Represent.
78. Figure 11 is composed of
three equal squares so placed that
BC= BF and DE = DH. If the
area of the whole figure is 147 sq.
in., how long is its perimeter?
' 79. A rectangular lot is 5
times as long as it is wide, and it
contains 80 sq. rd. . How many
trees can be placed on its edge, the distance from center to
center of each tree being 1 rod? Represent.
80. How long is the radius of the largest circle that can be
drawn on a piece of paper 8 in. long and 6 in. wide ?
CHAPTER XI
PROPORTION
1. What number has the same ratio to 6 that 1 has to 3?
That 2 has to 3 ? That 4 has to 3 ?
2. Supply the missing terms :
3 _ x 5 _ x x _ 5 7 _ x
5~10 9~27 4~10 15~90*
3. Which of the following statements are untrue ?
3:6 = 5:10. 8:4 = 12:6. 9:3 = 7:5. 1:3 = 3:15.
4. An equation which states the equality of two ratios is
called a Proportion.
Write a proportion.
5. Write a proportion with the terms 4, 6, 8 and 12, and
show that it is true.
6. Can you write more than one true proportion with the
terms 4, 6, 8, and 12 ?
7. Write a proportion with the terms 15, 20, 10, and 30.
With 4, 24, 7, and 42. With 7, 9, 21, and 27.
8. Substitute values for x and y that will make true pro-
portions.
10:5 = *:y. 8 : 12 = x:y. =- 80 : 40 = x : y.
Zi y
9. The terms of each ratio form a Couplet of which the first
term is called the Antecedent and the second term the Consequent.
In the proportion 8 : 16 = 9 : 18, which are greater, the ante-
cedents or consequents ?
340
PROPORTION
10. Write a proportion in which the antecedent of the first'
couplet is 12 and the antecedent of the second couplet is 8.
11. Write a proportion in which the consequent of the first
couplet is 10 and the consequent of the second couplet is 30.
12. Can you write a true proportion in which the antecedent
of the first couplet is greater than its consequent and the
antecedent of the second couplet is less than its consequent ?
Illustrate and explain.
13. Of how many terms must a proportion consist ?
14. The first and fourth, or outside terms of a proportion,
are called the Extremes. The second and third, or inner terms,
are called the Means.
In the proportion 10 : 5 = 14 : 7 find the product of the
means and the product of the extremes and compare them.
15. Write some proportions and compare the product of the
means with the product of the extremes until you see the truth
of the following principle :
In a proportion the product of the means equals the product of
the extremes.
When three terms of a proportion are given the other term is easily
found by this principle, as :
3 : 6 = 25 : x. The product of the means is 3 x. The product of the
extremes is 6 x 25. As these products are equal we have the equation
3 = 6 x25.
Find the missing term in each of the following proportions :
16. 27:40 = 9:#. 19. 144:^ = 12:1.
17. 24 : 3 = 48 : x. 20. 9 : 6 = x : 12.
18. 25:z = 35:7. 21. 11:77 = ^:42.
A double colon is sometimes used between the ratios instead of the sign
of equality, as 6 : 4 : : 3 : 2. This is read 6 is to 4 as 3 is to 2.
342 PROPORTION
22. Find the value of x in the proportion x : 9 : : 10 : 18.
Find the values of x :
23. x : 36 : : 3 : 12. 30. 2| : 10 : : 8J : x.
24. a : 40:: 18: 80. 31. 6: 25 : : 12|- : a.
25. 21:aj::26:8J. 32. 6J : 12} : : 76 : a.
26. 28: 35:: 16: a. 33. 16| : 33J : : 66| : x.
27. 6$: 33$:: 12: a. 34. 6} : a : : 6 : 12.
28. 7}: 22}:: 5: a. 35. 3J : 50 : : a : 9.
29. 3J:7}::3:a;. 36. 16f : 12} : : a : 4.
37. What is the ratio between 30 min. and 10 hr. ? Be-
tween 3 Ib. and 1 Ib. 8 oz. ?
Observe that a ratio is only possible between quantities of the same
denomination.
Substitute numbers for x and y that will make the following
proportions true:
38. x : 2 : : 12 : y. 40. x : 6 : : 3 : y.
39. x:8::8:y. 41. x : 1 : : 1 : y.
42. A proportion in which the means are equal is called a
Mean Proportion, and the number which each mean represents
is called a Mean Proportional between the other two numbers.
Write a proportion in which each of the means is 6.
43. In the mean proportion 1 : 4 = 4 : 16, what number is
the mean proportional ? In the proportion 2 : 6 = 6 : 18, 6 is
a mean proportional between what numbers ?
Find value of x :
44. 3 :#::#: 12. 46. 2 : x : : x : 8. 48. 7 : x : : x: 28.
45. !:::: 121. 47. 2 : x : : x : 98. 49. 3 : x : : x : 27.
50. Write several proportions in which each of the means
is 12.
PROPORTION 343
51. If 3 hats cost $ 11, how much will 6 hats cost?
Let x = the number of dollars paid for 6 hats. The greater the number
of hats the greater the number of dollars paid for them. 3 hats are to 6
hats as $ 1 1 (the price of 3 hats) are to x dollars (the price of 6 hats) or
3:6 : : 11 :X.
52. If 5 hats cost $ 7, how much will 10 hats cost ?
In each problem it is assumed that the articles considered are of the
same kind and of equal value.
53. If 8 apples cost 15^, how much will 16 apples cost ?
54. If 9 apples cost 17^, how much will 3 apples cost?
55. If 10 oranges cost 53^, how much will 5 oranges cost?
56. If 7 hats cost $ 25, how much will 21 hats cost ?
57. If 3 vases cost $ 26, how much will 5 vases cost ?
58. If 9 pairs of opera glasses cost $36, how much will 11
pairs cost ?
59. If 51 yd. of lace cost $ 17, how much will 16 yd. cost ?
60. If 10 yd. of silk cost $ 23, how much will 13^ yd. cost ?
61. If 80 yd. of cloth cost $375, how much will 5f yd. cost ?
62. If 20 yd. of calico cost $ 1.35, how much will 6f yd. cost?
63. If 15 yd. of jet cost $7.50, how much will 8 yd. cost?
64. In the proportion 5 : 10 : : 12 : 24, if both terms of the
first couplet are divided by 5, is the proportion still true ?
By what number may each term of the second couplet be
divided without destroying the proportion?
65. If 9 men can earn $ 23 in a day, how much can 18 men
earn ?
66. If 15 men earn $ 37 in a day, how much will 25 men
earn?
67. If 30 men earn $ 175 in a day, how much will 20 men
earn ?
344 PROPORTION
68. When $19 are paid for 8 hats, how much will 24
hats cost ?
69. If a train runs 85 mi. in 3 hr., how far will it run in
21 hr.?
All problems in proportion can be solved by analysis, as "If a train
runs 85 mi. in 3 hr., in 21 hr. it will run -^ 1 - times 85 mi., or 595 mi."
Solve the following problems by analysis as well as by pro-
portion :
70. John rode on his bicycle 5 mi. in 45 min. At the same
rate, how far would he ride in 90 min.?
Find the cost of 10 articles of the same kind :
71. When 3 fans cost $ 3.50. 74. When 4 bags cost $ .75.
72. When 6 maps cost $.25. 75. When 6 caps cost $ 1.
73. When 8 tops cost $ .25. 76. When 4 books cost $ 5.
77. Mr. A. collected $ 125 in the first four days of the week.
During the rest of the week he collected at the same rate.
How much did he collect in the whole week ?
78. From 80 A. of land Mr. Porter gathered 1680 bu. of
corn. At the same rate, how many bushels could he gather
from 167 A. of corn land ?
79. Polygons which have the same shape are said to be
Similar. Their corresponding angles are equal, and their cor-
responding sides are proportional.
There are two similar rectangles. The larger rectangle is
18 in. long and 12 in. wide. The smaller rectangle is 9 in.
long. How wide is it ? Represent.
80. Draw a rectangle 8 in. long and 2 in. wide. Draw a
similar rectangle 4 in. long. Find the ratio of the perimeters
of the rectangles.
PROPORTION
345
12
5 B D
FIG. 1.
10
81. The right triangles ABC and DEF
are similar. Find the side BC. Then
find by proportion the sides DF and FE
of the larger triangle. Find the ratio of
the perimeters of the triangles.
82. Find the hypotenuse of a right
triangle whose shorter sides are 7 in. and
24 in. Find the length of each side of a
E similar triangle whose shortest side is 21
in. Find the ratio of the perimeters.
83. The perimeter of an isosceles triangle, whose base is 5
in., is 19 in. How long is each of the equal sides ? Find the
length of each side of a similar triangle whose base is 20 in.
Represent. Find ratio of perimeters.
84. Find the length of each side of a
trapezoid similar to ABCD, but larger,
each of the non-parallel sides of the larger
trapezoid being 8 in. long. What is the
ratio of the perimeter of the greater trape-
zoid to the less ?
4 in.
6 in.
FIG. 2.
Observe that the ratio of the perimeters of two similar polygons is the
same as that of anj^ pair of their corresponding sides.
85. There are two similar trapeziums. The perimeter of
the smaller is 18 in., and its shortest side is 3 in. Find the
perimeter of the greater trapezium, its shortest side being 15
in. Represent.
86. The rectangle AB CD is
twice as long as the similar
rectangle EFGH, and all the
corresponding lines of the two
rectangles are proportional.
If AB is 24 in., EF 12 in., and
DB 26 in., how long is HF ? If BC is 10 in., how long is FG ?
Find the area of each rectangle and the ratio of their areas.
346
PROPORTION
87. There are two similar rectangles, whose perimeters are
respectively 36 in. and 9 in. If the base of the larger rec-
tangle is 11 in., how long is the base of the smaller rectangle ?
If the altitude of the larger rectangle is 7 in., what is the
area of the smaller rectangle ?
88. The upper base of a certain trapezoid is 5 in. Its per-
imeter is 17 in. How long is the upper base of a similar
trapezoid whose perimeter is 51 in.? If the altitude of the
smaller trapezoid is 4 in., what is the altitude of the larger
trapezoid ?
89. There are two similar rhomboids, a base of one being 12
in., and a corresponding side of the other 6 in. If the alti-
tude of the larger rhomboid is 4 in., what is the altitude of the
smaller ? Eepresent.
90. The pentagon FQHJK is similar to the pentagon
ABODE. Find the length of the sides which are unmarked.
91. The perimeter of an irregular hexagon is 330 in. Its
longest side is 72 in. Find the length of the longest side of a
similar irregular hexagon, whose perimeter is 55 in.
92. If 6 men can do a piece of work in 15 da., how long will
it take 2 men to do the work ?
PROPORTION 347
In cases like the foregoing the proportion is inverse. The less the
number of men employed, the greater the number of days required.
6 men : 2 men, not as 15 da. : x da., but as x da. : 15 da. We have, there-
fore, the proportion
men men da. da.
6 : 2 : : x : 15
We know that in this proportion x stands for a number greater than 15
because 6, the antecedent of the first couplet, is greater than 2, the conse-
quent of that couplet, and the ratio of x to 15 is the same.
93. If 30 men can do a piece of work in 40 da., how many
men are required to do the work in 20 da. ? In 10 da. ? In 5
da. ? In 8 da. ? In 4 da. ?
In solving a problem by proportion it is necessary first to determine
whether the proportion is inverse or direct. In the following problems
write the word " Direct " or " Inverse " at the beginning of each solution.
94. A pole 40 ft. high casts a shadow 8 ft. long. How long
is the shadow of a 10-ft. pole at the same time and pface ? Of
a 20-f t. pole ? Of a 16-f t. pole ?
95. Twelve days are required for a piece of work, if the men
work 10 hr. a day. How many days will be required if they
work 8 hr. a day ? 9 hr. a day ? 6 hr. a day ?
96. The taxes upon a piece of property valued at $ 7800 are
$ 195. At the same rate, what is the amount of taxes upon
property valued at $ 5600 ?
97. When a factory runs 18 hr. a day, a piece of work is
finished in 20 da. How many hours a day must the factory
run to finish the work in 15 da. x ? In 24 da. ? In 30 da. ?
. 98. Goods costing $ 700 are sold for $ 800. At the same
rate what is the selling price of goods costing $ 1050 ? $ 1750 ?
99. If 63 gal. of vinegar cost $ 12.60, how much will 7 gal.
cost ? 9 gal. ? 23 gal. ? 10 gal. ?
100. From a debt of $8000, Mr. A. collects $ 7000. At the
same rate how much will he collect from a debt of $ 4800 ?
Of $1600? Of $560? Of $60?
348 PROPORTION
101. If a 5^ loaf of bread weighs 12 oz. when flour is $ 6 per
barrel, how much should it weigh when flour is $ 4 per barrel ?
$ 8 per barrel ? $ 3 per barrel ?
102. If it costs $ 60 to cover a floor with carpet costing $ 1.25
per yard, how much would it cost to cover it with carpet at
$ 2 per yard ?. $ 1.75 per yard ? $ 1 per yard ?
103. The carpet which covers a room 20 ft. long costs
$ 88.60. What would be the cost of covering with carpet of
the same quality a room of the same width but 5 ft. longer ?
4 ft. shorter ?
104. A stock company paid a semiannual dividend of 5%.
Mr. B.'s dividends were $ 800. How much would he have
received if the rate had been 1% ? 9% ? 10%
105. TJie amount of duty upon a certain importation was
$ 216.64. The rate was 40 % ad valorem. If the rate had been
55 /o ad valorem what amount of duty would have been paid ?
106. Mr. L.'s yearly income from bonds paying 3% is $ 1275.
What would be his yearly income from those bonds if they
were 5% bonds?
107. Mr. K. has a sum of money invested at 6%. 12 years'
interest on that sum paid for his house and lot. How many
years' interest would have been sufficient to pay for the house
and lot if the rate of interest had been 8% ?
PROPORTIONAL PARTS
108. John and Harry hired a boat to go fishing. John paid
30^. Harry paid -f as much as John. How much did both
pay ? What part of the whole expense did each pay ? They
sold their fish for $ 1. How ought the money to be divided ?
109. If 20^ are divided between two boys so that the first
boy's share is to the second boy's share as 2 is to 3, how much
will each boy receive ?
PROPORTIONAL PARTS 349
If 20 were divided into 5 equal parts, ought not the first boy to have 2
of these parts, or f of the whole, and the second boy 3 of the parts or f of
the whole ?
110. Separate 27 into 2 numbers in the ratio of 4 to 5.
Combining 4 and 5 we have 9 equal parts to which 27 is equal. Each
number equals how many 9ths of 27 ?
111. Divide 12 in the ratio of 5 to 1. In the same way divide
36. 54. 84.
112. How would you divide 14^ between 2 boys, giving
one boy 6 times as much as the other ?
113. How must $9.9 be divided among 3 men that their
shares may be in the proportion of 1, 3, and 5 ?
114. Three men are to receive $54 for papering a house.
The first man has worked 5 da., the next 6 da. and the next
7 da. How much shall each receive ?
115. The bill for the labor of shingling a house is $48.
How shall that be divided among the three workmen A, B, and
C, if A has worked 6 da., B 4 da., and C 2 da. ?
116. Two painters received $ 51 for painting a house. The
first worked 10 da., and the other worked 7 da. How ought
the money to be divided ?
117. Mr. A., Mr. B., and Mr. C. sent some poor children to
spend a week in the country. Mr. A. paid for 5 children, Mr.
B. for 8 children, and Mr. C. for 11 children. The whole
expense was $ 54.72. How much did each man pay ?
118. In a pasture there were 40 cows of which Mr. D.
owned 7 cows, Mr. E. 21 cows, and Mr. F. the remainder. If
the cost of pasturage was $ 70 a month, how much did each
owner of the cows pay each month ?
119. Two men contract to do a piece of work for $ 122.50.
The work is finished in 20 days, but one man is unable to work
for 5 days. How shall the amount paid be divided ?
350 PROPORTION
120. A and B sent in a bid for the plumbing of a house for
$ 150, but afterwards reduced the price $ 14. The work was
finished in 16 da. after its commencement. They hired a
helper at $ 2.50 per day who worked every day. A worked
every day except one, and B was absent 7 da. How shall the
amount be divided ?
121. Gunpowder is made of 75 parts of saltpeter, 15 of
charcoal, and 10 of sulphur. How many pounds of each are
there in 1 T. of gunpowder?
122. Two young men hired a boat for $ 1.25. One invited
two friends, and the other invited one friend to go in the boat
with them. How should the young men divide the expense ?
123. Divide 150 into 3 parts in the ratio of 3, 4, and 8.
124. Mrs. A. had three children, one 5 yr., one 7 yr., and
one 13 yr. old. She proposed to divide a bag of nuts among
the children in the ratio of their ages, if the eldest would tell
correctly how they should be divided. In that case, how many
nuts would each child receive, if there were 75 nuts in the bag ?
125. Mr. Allen and Mr. Ward formed a partnership, Mr.
Allen putting in $ 1200, and Mr. Ward $ 2400. They gained
$ 2700 the first year. How ought the money to be divided ?
126. The next year Mr. Ward put in $ 600 more, and the
profit was $ 3000. Find each man's share.
127. The next year Mr. Allen took out $ 200, and Mr. Ward
put in $ 1000. The profit was $ 3500. As Mr. Allen's share
of the profits was so small and he was giving much time to
the business he was paid $ 500 from the profits. How much
did each man receive ?
.128. Two brothers bought a $5000 United States bond at
105, the younger brother furnishing J as much of the money
as the elder. How much did each pay ?
MISCELLANEOUS EXERCISES
351
129. Mr. A. and Mr. B. hired a pasture for $ 76. Mr. A.
put in 8 cows for 9 weeks, and Mr. B. 10 cows for 8 weeks.
How much ought each man to pay ?
130. CLASS EXERCISE. may make a~ problem in which
a number of dollars are unequally divided among a number of
persons, and the class may solve it.
MISCELLANEOUS EXERCISES
1. 9 2 :3 2 = 6 2 :?
2. 7V529 = ? 4Vl521 = ? 8V1369 - 6-
3. V2025 : VT225 = ?
4. V3025 : V4225 ::!!:?
V2304 : ?
3V784:?
5. V1764 : V2401 :
6. 4Vl96:7V576
7. 8 3 :16::4 8 :?
8. V:
9. V729 : 3V2916 : :
1296 : ?
10. How many cubic inches in a cube whose surface contains
96 sq. in.
11. How many acres of land in a road 10 mi. long and 55 ft.
wide?
Use cancellation.
12. A lake, whose area is 45 A. is covered with ice 3 in.
thick. Find the weight of the ice in tons, if a cubic foot of
ice weighs 920 oz.
13. A room 10 ft. high contains 30,000 cu. ft. How much
will it cost to carpet it at 75 ^ per square yard ?
14. Sunset Park contains 115 A. of land. How much is it
worth at $ .12| a square foot ?
15. A man bought a cow and paid $ 20.25 cash, which was
90% of the cost. How much did the cow cost?
352 PROPORTION
16. Change .15 and .025 to common fractions in their lowest
terms.
17. Eeduce to lowest terms |f J.
18. Eednce 5 to a fraction having 11 for a denominator.
19. Change .37 and .0016 to decimals having common
denominators.
7.5 x .25 x 3.6 x .18 =?
' .009 x .08 x 5 x .125 x .3
21. Divide the product of 13.5 and 1.8 by 8.1.
22. A farmer sold 8J cords of wood at $ 5 per cord, and
received an equal amount of money from selling apples at
$ 1.25 per bushel. How many bushels did he sell ?
23. Mr. Brown bought 47 T. of coal at $ 6f a ton. He
paid cash $ 175, and gave his note for the balance payable in
3 mo. with grace. What were the proceeds of the note dis-
counted at 6% ?
24. Make out and receipt a bill for the following : May 1,
1898, James Bentry of Ft. Wayne, Ind., bought of H. A. Cook
& Son, 16 Ib. of tea at $.85 per pound, 36 Ib. of coffee at $ .18}
per pound, 8 packages of macaroni at $ .12J per package,
3 gal. strawberries at $ .25 per gallon, 25 loaves of bread at
$ .05 per loaf.
25. Find the l.c.m. of 18, 24, 15, 30.
26. Find the prime factors of 342.
27. The sum of two numbers is 219.5, and one of the num-
bers is 96.875. What is the other number ?
28. If a table costs as much as two chairs, and five chairs
cost $ 43.75, how much does the table cost ?
29. Draw a representation of a cord of wood, marking its
dimensions.
30. What per cent is gained by buying peanuts at $ 2 a
bushel and selling them at 5^ a pint ?
MISCELLANEOUS EXERCISES
.
*
31. How mucli will it cost to fence a rectangular l
long, containing 425 sq. rcL, if the fence costs 20^ a foot ?
32. The dimensions of a bin are 1 ft. 3 in. by 3 ft. 4 in. by
9 ft. 4 in. How many bushels of wheat will it hold ?
There are 2150.4 cu. in. in a bushel.
33. At 15^ a square yard, how much will it cost to paint
the walls and ceiling of a room 36 ft. long, 24 ft. wide, and
12 ft. high, having a baseboard 9 in. high. No allowance for
openings.
34. When 5 children had left a class, 75% of the class re-
mained. How many children belonged to the class at first ?
35. If envelopes are bought at the rate of 15 cents for a
package of 25 and sold for a cent apiece, what per cent is
gained ?
36. From a box containing a dozen packages of envelopes,
100 envelopes were used. What per cent of the number was
left?
37. After a battle, the number of soldiers who answered to
their names at roll call was only* 612, which was 60% of the
number that went into battle. How many went into battle ?
38. There were 72 bananas in a bunch that cost 85^. 12-^-%
of them were sold at 3^ apiece, 33^% at 15^ a dozen, and
S^% of them were spoiled. The rest were sold at 2^ each.
What was the gain on the whole bunch ?
39. From a school of 48 pupils, 8 J% were absent on a rainy
day. 25% of those present went out of the room to a recita-
tion. How many remained in the room ?
40. A bar of fresh soap weighed 3 Ib. 6 oz. When dry, it
weighed 33^% less. How much did it weigh then ?
41. Walter bought a knife for 50^, and exchanged it for a
school book worth 67^. What per cent did he gain ?
HORN. GRAM. SCH. AR. - 23
354 PROPORTION
42. A jeweler bought some pins at the rate of $24.00 a
dozen. The cost of each was 66 f% of the price for which
he sold them. What was the selling price of each pin?
What per cent of profit did he make ?
43. A shoe dealer sold a pair of shoes for $ 4.00, gaming
33^%. How much did they cost ?
44. On the 4th of July Alfred had half a dollar and James
had 30 cents. They put their money together, and spent 30%
of it for firecrackers, 25% of the remainder for candy, and the
rest for lemonade. How much did they spend for each ?
45. John's arithmetic when new cost $ .60. When he had
finished the study of arithmetic, he sold the book for $ .40.
What was the per cent of reduction ?
46. Out of 3 dozen trees that a gardener set out, only 24
trees lived. What per cent died ?
47. Cranberries sold at 15^ a quart brought a gain of 20%.
How much did they cost per bushel ?
48. A man who had $ 675 spent 21% of it for a coat, and
put 66f % of the remainder in bank. Find price of coat,
amount of money in the bank, and money left.
49. A housekeeper used 8^ Ib. from a bag of flour leaving
66f % of it. How many pounds were left ?
50. 20% of Mr. Ward's farm is planted in corn and 50%
in wheat. The rest, which is 36 acres, is pasture land. How
many acres are there of corn ? Of wheat ?
51. Mr. A. bought a horse for $ 350 and sold it to Mr. B. at
a gain of 10%. Mr. B. sold it to Mr. C. at a gain of 5%. How
much more did Mr. C. pay for the horse than Mr. A. paid ?
52. 100 yd. of carpet which cost $60 to manufacture were
sold by the manufacturer to the wholesale dealer at a profit of
The wholesale dealer sold them at a profit of 25%.
MISCELLANEOUS EXERCISES 865
The retail dealer sold them to a customer at a profit of 25%.
What was the retail price per yard ? How much more per yard
did the customer pay than the manufacturer received ?
53. A dealer lost 16f% by selling goods for $500. What
was the cost ?
54. A boy sold his bicycle for $31.25, gaining 5%. How
much did the bicycle cost him ?
55. By selling a bicycle for $40 a dealer gained 33|%.
How much would he have gained by selling it for $ 50 ?
56. 21 is what per cent of 1\ ? 25 ? 8J ? 16| ?
57. What per cent is gained by buying berries at the rate of
24 qt. for a dollar and selling them at the rate of 4 qt. for
25^?
58. What per cent is gained by buying fans at the rate of $ 1
a dozen and selling them at 10 ^ apiece ?
59. Mr. X. bought a stock of groceries for $ 1200. He sold
J of them at a profit of 33|%, of them at a profit of 25%,
and lost 10% on the remainder. Did he gain or lose on the
whole, and how much ? What per cent ?
60. A lawyer collected 60% of a debt of $2400, receiving
10% for collecting. He afterward collected 75% of the re-
mainder, for which he was paid at the same rate. How much
of the debt was paid to the creditor, how much did the lawyer
receive, and how much did the debtor fail to pay ?
61. A collector who received 5% for his services earned in
one day $ 17.50. How much did he collect ?
62. A store valued at $ 25,000 was insured for of its value
at | % . What was the premium ?
63. The premium for insuring a house for ^ of its value at
1 % was $ 16. What was the value of the house ?
64. The premium for insuring a building for of its value
at 1% was $ 60. What was the value of the building ?
356 PROPOKTION
65. Mr. S. has real estate assessed at $ 8000 in a city where
the tax rate is $ .033 on the dollar, and the poll tax is $ 2.
How much are his taxes ?
66. The taxes of Mr. T., who lives in the same city, are
$ 233, of which $ 2 is his poll tax. For how much is his prop-
erty assessed?
67. What is the duty on 750 yd. of cloth invoiced at 5
francs a yard, a franc being 19 T \^, the duty being 30% ad
valorem ?
68. An importer paid $ 7000 in duties upon an importation.
The duty was 25% ad valorem. How many dollars' worth of
goods did he import ?
69. A man failing in business paid $ .40 on the dollar. He
owed Gay & Co. $8756, W. H. Reed $10,857, the First
National Bank $ 5000, and Hall & Co. $ 4221. How much did
each receive ?
70. A man failed in business owing $24,000, and having
assets $ 12,000. What per cent of his debts could he pay ? How
much would a creditor receive to whom he owed $ 1785.50 ?
71. How many cents on a dollar can each of the bankrupts
in the following list pay ?
Liabilities Assets Liabilities Assets
Mr. Low, $84,000 $46,200 Mr. Dow, $125,600 $37,680
Mr. Van, $96,800 $33,880 Mr. May, $242,850 $97,140
72. Mr. Wright is a creditor of Mr. Low to the amount of
$7000. Mr. Van owes Mr. Wright $6400. Mr. Dow owes
him $ 475, and Mr. May owes him $ 2821. How much will he
receive in the settlement of those debts ?
73. Mr. L. paid $ 800 for a lot, and f as much for another
lot. He sold the higher priced lot at a gain of 62i%, and the
cheaper lot at a loss of 10%. How much did he gain on the
whole investment ? What per cent ?
MISCELLANEOUS EXERCISES 357
74. Goods invoiced at $ 837 were discounted 30% and 20%.
What was the net price ?
75. The list price of a bill of hardware was $ 90. The dis-
counts were 60% and 25%. The goods were sold at 40% below
list price. What per cent was gained ?
76. After a discount of 30% had been taken off, the net
price of some goods was $ 140. What was the list price ?
77. A merchant was offered a bill of goods for $ 1000, with
10%, 15%, and 5% off for cash. He offered $ 800 cash for the
goods, which was accepted. Who, if either, was the loser, the
buyer or the seller ? How much ?
Solve the equations :
78. 9^-12 = 8^-5.
79. 8^-24-4^ + 16 = 32.
80. 21a-35-8z + 14 = 2a + 89.
4 4
82. 4 times a certain number + 5 times the number + 10 =
82. What is the number?
83. An agent sold a sewing machine for $ 50 and sent to the
company $ 10 less than 4 times the amount of his own com-
mission. How much was his commission ? How much did he
send to the company ?
84. In three camps there were 4000 soldiers. In the second
camp there were 3 times as many men as in the first camp, and
in the third camp 4 times as many as in the first camp. How
many soldiers in each camp ?
85. The perimeter of an isosceles triangle is 32 inches. Each
of the equal sides is 7 inches longer than the base. Find length
of each side.
86. Find a number which when multiplied by 7 and divided
by 8 gives 42 for a quotient.
358 PROPORTION
87. The perimeter of a rectangle is 110 cm. Its length is 15
cm. greater than its width. Find width and length. How
many square centimeters in each of the triangles formed in it
by its diagonal ?
88. Two squares, one containing 144 sq. in. and the other
121 sq. in., are placed side by side so that their base lines form
one continuous line. Represent and find perimeter of the sur-
face which they cover.
89. Divide 39 into two parts such that one part shall be 12
times the other.
90. John caught three times as many fish as James. Wil-
liam caught twice as many as John and James both caught.
They all caught 120 fish. How many fish did each catch ?
91. Tom, Fred, and Will whitewashed both sides of the fence
around a circular lot 60 ft. in circumference. Fred whitewashed
three times as much as Tom, Will whitewashed four times as
much as Tom. How many feet of the fence did each white-
wash ?
92. The circumference of a given circle is 20 in. How long
is the circumference of a circle whose radius is 4 times as long
as the radius of the given circle ?
93. A merchant bought dress goods at 60^ a yard and
marked them to be sold at an advance of 33 \%. They were
sold at a reduction of 12% from the marked price. What was
the selling price ? What per cent was gained ?
94. CLASS EXERCISE. may name a cost price for goods,
a marking price, and a selling price, which is a certain per cent
of reduction from the marked price. The class may find the
per cent of gain or loss on the goods.
95. The marked price of some silk was $ 1.60 per yard.
It was sold at a reduction of 5% from the marked price. The
selling price was 27 ^ more than the cost price. What was the
per cent of gain ?
MISCELLANEOUS EXERCISES 359
96. When lace marked $ 1.50 per yard was sold at a reduc-
tion of 16f % from the marked price, a gain of $ .25 per yard
was made. Find cost price and per cent of gain.
97. A coat marked at $ 7.50 was sold at a reduction of
33^%. The selling price was 125% of the cost price. Find
cost.
98. A suit of clothes marked $ 25 was sold at a reduction
of 20% from the marked price. If the selling price was 25%
above the cost price what was the cost price ?
99. Goods marked at 40 ^ per yard were sold at a reduction
of 12 % from the marked price. The selling price was 75%
above cost. Find the cost.
100. Some goods were sold for $ .80, which was 25% above
the cost price. How much did they cost ? The selling price
was 20% below the marking price. How were they marked ?
101 . Find the cost and the marked price of goods sold at $ .75,
which was a reduction of 25% from their marked price and an
advance of 50% upon their cost.
102. Find the cost and the marked price of goods sold at
$ 1.20, the selling price being an advance of 25% upon cost
and a reduction of 25% from the marked price.
103. CLASS EXERCISE. may suppose himself to be a
merchant buying goods at a certain price and marking them
to sell at any price he may select. Let him lower the marked
price by a certain per cent and then find the per cent of gain or
loss which the selling price is on cost price. Let him give the
selling price to the class and tell them what per cent that is
below the marked price and above or below the cost price.
Let the class find the marked price and the cost price.
104. How shall goods that cost 80 ^ per yard be marked that
a reduction of 10% from the marked price may be made and
the goods sold at a gain of
360 PROPORTION
SOLUTION. Let x = the number of dollars in the marked price. Since
a reduction of 10 % from the marked price is to be made, the actual selling
price is .90 x. The actual selling price is 80^ + 12% of 80^, which is
90. Therefore
Clearing of fractions, 90 x
x = $ 1.00
Prove this and the following problems.
105. I marked goods which cost me $ 1.44 per yard so that
I could deduct 10% from the marked price and still make 15%
profit. What was the marked price ?
106. How shall I mark goods that cost $ 72, so as to deduct
10% from the marking price and yet gain 12-J-% ?
107. A buys goods for $ 12. He wishes to make 33^% after
discounting 20% from the marked price. How shall he mark
them?
108. How shall goods costing 75^ be marked that 10% may
be deducted from the marked price and the goods still be sold
at a profit of 20% ?
109. How shall goods that cost $ 1.20 be marked that a dis-
count of 25% from the marked price may be made and the
goods sold at a profit of 25% ?
110. After a merchant had marked goods that cost 90^, a
clerk by mistake sold them at a reduction of 40% from the
marked price. This selling price was 20% below cost. What
was the marked price ?
111. The list price of some goods is $1.60. A merchant
buys them at a discount of 40%, marks them to be sold at a
profit of 25% on the cost price, and discounts them to the cus-
tomer 10% from marked price. How much does the merchant
gain ? What per cent ?
112. The list price of granite soup kettles is $ 1.20 and the
discounts are 50% and 30%. If the merchant who buys them
MISCELLANEOUS EXERCISES 361
marks them at an advance of 16f % on list price, and dis-
counts them to his customer 40% from marked price, how
much does he gain on each ? What per cent ?
113. A merchant buys knives at a discount of 80% and 10%
from list price, which is $ 18 per dozen. He marks them at a
price 40% below the list price, and sells a knife to a- boy at a
reduction of 10% from the marked price. How much and
what per cent does he gain on that sale ?
114. Find the length of the hypotenuse of a right triangle
whose base is 40 ft. and altitude 75 ft.
115. The hypotenuse of a right triangle whose base is 40
ft. and altitude 42 ft. is how much less than the sum of the
other two sides ?
116. If the hypotenuse of a right triangle is 95 ft. and alti-
tude 57 ft., how long is the base ?
117. How long is the perimeter of a right triangle whose
base is 42 ft. and hypotenuse 150 ft. ?
118. How long is the longest stick that can be carried
through a doorway 6 ft. high and 2 ft. wide, the stick being
sharply pointed at both ends ?
119. The sum of all the edges of a cube was 132 in. What
was the volume of the cube ?
120. What is the volume of a cube if the area of all its
faces is 384 sq. in. ?
Find the missing term in each of the following proportions :
121. 6:8 = 54:x. 126. |:J = 40:aj.
122. 5:3 = aj:21. 127. 2 : 17 = 3 : x.
123. 7: = 35:45. 128. 8J : 35 = x : 20.
124. a;: 61 = 2: 122. 129. 3f:75 = x:24.
125. i:f = 6:o?. 130. -J r : of f = 42 : x.
362 PROPORTION
Solve by proportion and by analysis :
131. If 7 T. of hay cost $ 77, how much will 5 T. cost ?
132. If 10 men earn $ 25 in one day, how much will 3 men
and a boy earn provided the boy receives half as much as a
man?
133. If a man travels 124 mi. in 4 da., how far will he travel
in 9 da. ?
134. The perimeter of a right triangle is 8 ft. The hypote-
nuse is ^ ft. 4 in. and the base 2 ft. 8 in. Find the sides of a
similar triangle whose base is 3 ft. 8 in.
135. The circumference of a given circle is 12.78 in. Find
the circumference of a circle the radius of which is ^ that of
the given circle.
136. If 8 bbl. of flour can be made from 40 bu. of wheat,
how many barrels of flour can be made from 70 bu. of wheat ?
137. If 75 United States bonds can be bought for $ 7725,
how much will 60 of the same bonds cost? At 3% what will
be the income from the 60 bonds ?
138. If the grocer's bill for a family of 5 persons is $ 15 per
week, what will be that of a family of 7 persons at the same
rate ?
139. $ 40 pays the board of 10 persons for a week at a farm-
house. If the price of board were doubled, how many persons
could obtain board for the same time for the same amount ?
140. If 15 men can do a piece of work in 40 da., how long
will it take 12 men to do it ?
141. If 30 men can do a piece of work in 16 da., how many
men can do the same work in 15 da. ?
142. Mr. A. holds stock in a company which last year paid
12% dividends. His dividends were $ 192. If this year the
rate of dividend is 9%, how much will he receive from that
investment ?
MISCELLANEOUS EXERCISES 363
143. Mrs. C. has a yearly income of $ 1876 from an invest-
ment which pays 8%. If it paid 6%, how much would she
receive from that investment ?
144. When Mr. D.'s house is rented at $ 40 a month the rent
is 8% of the cost of the house. What per cent of the cost of
the house does he receive when the tenant pays $ 35 a month ?
145. A quantity of wheat was shipped 1400 mi. For the
first 800 mi. the charges were $ 60. For the rest of the distance
the rate per mile was twice as great. What was the cost of
transportation for the whole distance ?
146. How would you divide $ 90 among three persons in the
ratios of 2, 3, and 4 ?
147. A and B formed a partnership, A's capital being ^ as
much as B's. Their profits were $ 8000 the first year. How
much ought each to receive ?
148. Mr. Ball owned twice as many shares in a mining com-
pany as his brother. The sum of their dividends was $ 1800.
How much dividend should each receive ?
149. Three heirs, James, Lucy, and Henry, own a farm
which rents for $ 2100 a year. James owns 3 times as many
acres as Lucy, and Lucy owns twice as many as Henry. How
shall the rent be divided ?
150. The distance from Alta to Eeed's Crossing is 84 mi. by
the A. & M. E.E., and from Eeed's Crossing to Doane 96 mi.
by the C. & D. E.E. The freight charges on some merchandise
carried from Alta to Doane are $ 29.16. How much shall each
railroad receive from it ?
151. The freight charge on merchandise shipped 824 mi.
over three railroad lines is f 206. To how much is the A. & X.
B.K. entitled which carried it 378 mi. ? What sum belongs to
the C. & Y. E.E. which carried it 212 mi. ? To the third
railroad ?
364 PROPORTION
At 5% what is the interest of
152. $ 725 from May 9, 1892, to Sept. 11, 1899 ?
153. $638 from June 17, 1873, to May 21, 1880 ?
154. $525 from Aug. 12, 1884, to June 17, 1893?
155. $631 from Jan. 21, 1887, to June 20, 1892 ?
156. $375 from Oct. 13, 1899, to Feb. 3, 1905?
157. How long is the side of a square whose area is equal
to that of a rectangle 36 in. long and 4 in. wide ? The perim-
eter of the square equals what per cent of the perimeter of
the oblong ?
158. How long is the side of a square whose area is equal
to that of a rectangle whose length is 343 ft. and width 7 ft. ?
The perimeter of the square equals what per cent of the perim-
eter of the oblong ?
159. A agreed to dig potatoes for B, taking of the potatoes
for his pay. He dug 5 bu. of potatoes and set them aside for
his employer. The next bushel he set aside for himself, and
so continued. When he had dug 600 bu., how many bushels
that belonged to him had he failed to get because of his
ignorance of ratio?
160. Two paper hangers finished papering a house. They
were to receive $ 22.60 for their work. The first man was to
receive $1.40 more than the other; they divided the $22.60
into two equal parts ; then the second man gave the first man
$1.40. Was the division right? If not, how may it be
corrected ?
161. Find the sum of the squares of two numbers, as 9 and
7. Find the difference of their squares. Add the difference
of their squares to the sum of their squares. Divide the num-
ber thus found by 2, extract the square root, and the result
will be the greater number. Take several pairs of unequal
numbers and see if this holds true.
MISCELLANEOUS EXERCISES 365
162. VF+2^? Vl 3 +2 3 +3 3 =? Vl 3 +2 3 +3 3 +4 3 =?
163. Find the square root of the sum of the first 5 perfect
cubes. Of the first 6 perfect cubes.
164. Mr. Wilson is building a house which has a roof 45 ft.
long with rafters 15 ft. long. How many shingles will be
used to cover the two sides of the roof if 900 shingles 4 in.
wide and exposed 4J in. to the weather are required to cover
1 square 10 ft. in dimensions ? How much will the shingles
cost at $2.75 a thousand? At 3^ per pound, how much
will the shingle nails cost if 5 Ib. are required for each square ?
How much must be paid to the carpenter for shingling the roof
at $ 1.25 per square ? What is the entire cost of shingling the
roof?
165. Think of a number, multiply it by 6, add 9, divide the
result by 3, subtract 3, divide the remainder by 2, and you will
have the original number. Try this with several numbers, and
then try to find why this is true whatever the original number
may be.
166. What is the ratio of the perimeter of an equilateral
triangle a side of which is 3 in. to the perimeter of an equi-
lateral triangle a side of which is 6 in. ?
167. Take Ex. 166 substituting "hexagon" for " triangle."
168. Take Ex. 166 substituting "pentagon" for "triangle."
169. How many right angles can a trapezium have? A
trapezoid ? A rhomboid ?
CHAPTER XII
MEASUREMENTS AND CONSTRUCTIONS
LINES AND SURFACES
1. A plane figure bounded by straight lines is called a
Polygon.
Draw a polygon of five sides.
A polygon of three sides is a Triangle ; of four sides, a Quadrilateral ;
of five sides, a Pentagon ; of six sides, a Hexagon ; of seven sides, a Hep-
tagon ; of eight sides, an Octagon ; of nine sides, a Nonagon ; of ten sides,
a Decagon ; of twelve sides, a Dodecagon.
2. A polygon having all its sides equal and all its angles
equal is called a Regular Polygon.
Draw a regular polygon.
Polygons which approximate very closely to regular polygons may be
constructed by a method of which the following
construction of an approximately regular heptagon
is an illustration : Draw AB 7 units long, and
mark the divisions of units. Draw a circle of
which AB is the diameter. With A as a center
and AB as a radius draw an arc. With the same
radius and with B as a center draw an arc inter-
secting the first arc at C. Draw (72 passing
through the second division of the diameter,
and prolong it until it meets the circumference
at D. AD will lie seven times consecutively as a
chord.
366
LINES AND SURFACES 367
A polygon of any required number of sides may be constructed in the
same way by making the number of divisions in the diameter equal to
the required number of sides. The line CD must always pass through
the second division of the diameter.
3. Draw a heptagon by the method described in the note
and change it to a seven-pointed star. In the same way draw
a five-pointed star and a nine-pointed star.
4. Triangles are classified with regard to their sides as,
Equilateral, having three equal sides; Isosceles, having two
equal sides ; and Scalene, having no two sides equal.
Construct an equilateral triangle, an isosceles triangle, and
a scalene triangle, each having a perimeter of 15 in.
5. Triangles are classified with regard to their angles as Right
triangles, Obtuse-angled triangles, and Acute-angled triangles.
Draw a triangle containing a right angle.
A
6. A Right triangle is one that has a
right angle.
Draw a scalene right triangle. An
isosceles right triangle.
7. An Obtuse-angled triangle is a triangle
that has an obtuse angle.
Draw a scalene obtuse-angled triangle. An
isosceles obtuse-angled triangle.
8. An Acute-angled triangle is a triangle
in which all the angles are acute.
Draw a scalene obtuse-angled triangle. An
isosceles acute-angled triangle.
9. Into what kind of triangles is a rhombus divided by its
long diagonal ? By its short diagonal ?
368
MEASUREMENTS AND CONSTRUCTIONS
10. Quadrilaterals are of three kinds, trapeziums, trapezoids,
and parallelograms :
Trapezium. No parallel
sides.
Trapezoid. Two parallel
sides.
Quadri-
laterals.
Parallelo-
gram.
Opposite
sides par-
allel.
Kectangle.
All angles
right angles.
Square. Equilateral
rectangle.
Rhomboid.
Angles oblique.
Ehombus. Equilateral
rhomboid.
Draw four kinds of parallelograms and write the name of
each upon it.
11. State the difference between a square and any other
rectangle. Between a rectangle and a rhomboid. Between a
square and a rhombus. Between a rhombus and any other
rhomboid. Between a trapezoid and a trapezium. Between
a trapezium and a rectangle.
LINES AND SURFACES 369
12. Find the area of a rectangle whose base is 7 in. and alti-
tude 4 in. Find the area of a right triangle having the same
base and altitude.
13. Represent a trapezoid whose lower base is 8 in. and
upper base 6 in., the bases being 4 in. apart. Find the area of
the trapezoid.
14. The parallel sides of a field of trapezoidal shape are
120 rd. and 80 rd. long; they are 90 rd. apart. How many
acres are there in the field ?
15. What is the altitude of a trapezoid whose area is 825
sq. ft. and whose bases are 60 and 90 ft. ?
Let x = the number of feet in the altitude.
16. Which is greater, and how much, a trapezoid whose
parallel sides are 10 in. and 4 in., and altitude 5 in., or a
trapezoid of the same altitude whose parallel sides are 8 in.
and 6 in. ? Represent.
17. The sum of the parallel sides of a trapezoid is 16 in. and
the altitude is 3 in. What is the area ?
18. Draw and cut out the rhom-
boid ABCD. Draw AE perpendicu-
lar to DC (Fig. 1). Cut off the tri-
angle ADE and place it on the other
side of the rhomboid so that AD and
BC unite (Fig. 2). The figure thus
formed will be a rectangle having the
same base and altitude as the rhom-
boid. If the base DC of the rhomboid
is 7 in. and altitude AE 3 in., what is
the area of the rhomboid ?
19. Draw and cut out a rectangle 6 in. by 4 in. Cut and
arrange its parts into a rhomboid of equal area. If you had
another rectangle 6 in. by 4 in., could you make another rhom-
boid equal to the first but of different shape ? Explain.
HORN. GRAM. SCH. AR. 24
370
MEASUREMENTS AND CONSTRUCTIONS
20. Make problems to illustrate the following rule :
To find the area of a rhomboid
Multiply the base by the altitude.
21. Find the area of a rhomboid whose longer sides are 9 in.
apart, and are each 48 in. long.
Supply the missing numbers in the measurements of the
following rhomboids :
Alt.
5 in.
10 in.
Base
22.
40 in.
23.
x in.
24.
25 in.
25.
13.8 in.
26.
x in.
27.
5ft.
Area
x sq. in.
60 sq. in.
175 sq. in.
55.2 sq. in.
10.2 sq. in.
2 sq. ft. 12 sq. in.
x in.
x in.
251 in.
x in.
28. The perimeter of a certain rhomboid is 60 in. The base
is 20 in. The altitude is 6 in. Represent and find area.
29. A base of a rhomboid is 5 in. longer than the altitude,
which is 16 in. What is the area of the rhomboid ?
30. The altitude of a rhomboid is sometimes
represented by a line that falls outside the rhom-
boid. BE, the altitude, is the perpendicular dis-
tance between the side AB and the opposite side
produced. If DC is 7 in. and BE 8 in., what is
the area of the rhomboid ABCD ?
31. LB and ED are
parallel. Which is the
greater rhomboid, OFKL
or GFBC? Explain.
32. Reproduce LB and
ED, making them 5 in.
apart. Let GF be 4 in.
Draw several rhomboids whose base is GF and whose side
parallel to the base is a part of LB. What is the area of each ?
K
F
FIG. 4.
LINES AND SURFACES 371
33. A farmer who had a rectangular lot 300 ft, long and
80 ft. wide, sold a strip of it to a railway company for a right
of way. The agreement was that, beginning at the south-
west corner of his lot, the width of the strip should be
measured off 80 ft. along the southern boundary of the lot,
that the strip should thence cross the lot, bounded by straight
parallel lines, and that the railroad company should pay 5 ^ a
square foot for it. How much did the farmer receive for the
land?
34. The shaded part of Fig. 5 shows the portion of land
that the farmer intended to sell. The shaded part of Fig. 6
E shows the portion of land that the
company bought. AD being 80 lit., is
the strip in accordance with the agree-
ment ?
35. The farmer, thinking that the
company had taken more land than
belonged to it, consulted a lawyer,
who proved to him that the two strips
were equal, by drawing diagrams like
Fig. 5 and Fig. 6, and showing him
that if the two triangles AHB and DCG in Fig. 6 were cut
out and applied to the rectangle CEFD in Fig. 5, they would
exactly equal it. How could you determine whether or not
the strips were equal ?
36. Construct a rhomboid, cut and rearrange its parts in
such a way as to make a trapezoid of equal area. A rectangle
of equal area.
37. What name is given to a rectangle whose base and
altitude are equal ?
38. Construct a rhomboid whose base and altitude are each
3 in. Cut and rearrange its parts into a square.
372
MEASUREMENTS AND CONSTRUCTIONS
39. Find the area of the rhom-
bus ABCD if AB is 8 in. and EB
7 in.
40. Find the area of a rhombus
whose perimeter is 4 ft. 4 in., and
whose altitude is 6 in. less than
FIG. 7. one side.
41. Find the area of a rhombus whose perimeter is 40 in.
and whose altitude is f of a side.
42. Is a rhombus a regular polygon ? Give reasons for
your answer.
43. Draw a square whose sides are each 2 in. long, and a
rhombus whose sides are each 2 in. long. Which has the
greater altitude? The greater area?
44. Can you draw a rhombus in which the base and altitude
are equal ? Explain.
45. If the rectangle AEDC were 8
in. long and 5 in. wide, what would be
the area of the triangle ABC? Give
reasons.
46. What would be the area of ABC
if AEDC were 9 in. long and 4 in.
wide? 50 in. long and 18 in. wide?
6 ft. 3 in. long and 1 ft. wide?
47. Draw a figure and show the truth of the following
statement:
The area of a triangle is equal to one half the product of its
base and altitude.
48. Find the area of a triangle whose base is 18 in. and
altitude 5 in.
49. Find the area of a triangle whose base is 17 in., and
whose altitude is 5 in. greater than the base.
50. What is the area of a triangle whose base is 2 ft. 6 in.,
and altitude as much?
H
FIG. 8.
LINES AND SURFACES
373
51. If AD were 10 in., and CE 5
in., what would be the area of the
Of the triangle
rhomboid ABCD?
ADC ? ABC ?
FIG. 9.
52. If AD were 24 cm., and CE
were half as long as AD, how many
would the rhomboid contain? Each
square ' centimeters
triangle ?
53. Either side of a triangle may be considered a base. The
angle opposite the base is called the Vertical Angle. The
Altitude is the perpendicular distance from the vertical angle
to the line of the base.
B is a right angle. If AC is
B
considered the base, what angle
is the vertical angle? What line
is the altitude ? If AB is the base,
what angle is the vertical angle?
What line is the altitude? If AB
is 6 in., and BC is 10 in., what is
Which ^ B x
Fio. 10.
the area of the triangle?
AC x BD
is greater,
or
The altitude of a triangle is sometimes represented by a line that falls
outside the triangle.
54. If BC is considered the base of
the triangle ABC, what line is the alti-
tude? If BC is 1 in., and AD 6 in.,
what is the area of the triangle ?
55. Reproduce the triangle ABC, and
draw a line to represent the altitude when
BC is considered the base. When AB is
the base. When CB is the base.
56. If CB were 10 in., and the corresponding altitude 6 in.,
what would be the area of the triangle ? If AB were 12 in.,
how long would the corresponding altitude be ?
FIG. ii.
374
MEASUREMENTS AND CONSTRUCTIONS
58.
59.
60.
61.
62.
Base
10 in.
6 in.
20ft.
60yd.
15 rd.
Alt.
Tin.
9 in.
30ft.
40yd.
13rd.
63.
64.
65.
66.
67.
Base
12ft.
13ft.
x in.
x ft.
20 rd.
Alt.
X ft.
ft.
10 in.
50ft.
x rd.
36
65
200
750
300
Area
sq. ft.
sq. ft.
sq. in.
sq. ft.
sq. rd.
57. The longest side of a triangular field is 120 rd. The
perpendicular distance from the opposite corner to that side
equals 33J% of the side. Find the area of the field in square
rods. In acres.
Find the missing measurements of the following triangles :
Area
x sq. in.
x sq. in.
x sq. ft.
x sq. yd.
x sq. rd.
68. The base of a triangle is 45 in., and the altitude is twice
the base. What is the area?
69. The altitude of a triangle is 14 in., and the base is 3
times the altitude. What is the area?
70. Find the area of a triangle whose base is 8 in. and alti-
tude 9 in. How long is the side of a square whose area equals
that of the triangle ?
71. Find the perimeter of a square equal in area to a tri-
angle whose base is 20 in., and altitude 10 in.
72. How wide is a rectangle 18 in. long, and equal in area to
a triangle whose base is 12 in. and altitude 9 in. ?
73. Draw an isosceles right triangle. If each of the equal
sides were 12 in. long, what would be its area? If a rectangle
equal in area to the triangle is 3 in. wide, what is its length?
74. By folding an isosceles triangle in
such a way that the equal sides coincide, it
will be seen that a line drawn from the ver-
tical angle to the middle of the base divides
the triangle into two equal right triangles.
How long is the altitude AD of the isosceles
triangle, of which the base CB is 10 in., and
the equal sides are each 13 in. ?
FIG. 12'.
LINES AND SURFACES 375
75. Find the altitude of an isosceles triangle whose base
is 30 in., and whose equal sides are each 39 in. Find the area of
the triangle.
76. Find the altitude of an isosceles triangle whose perim-
eter is 50 in. and base 16 in. Find the area.
77. The area of an isosceles triangle whose base is 18 in.
is 108 sq. in. What is its altitude ? The length of one of its
equal sides ? Its perimeter ?
78. Given an isosceles triangle whose base is 40 in. and
area 300 sq. in. Find its altitude. Find one of its equal
sides. Find its perimeter.
79. Given an isosceles triangle whose altitude is 42 in. and
area 1680 sq. in. Find the base. Find the length of each of
the equal sides.
80. Find to one place of decimals the altitude of an equilat-
eral triangle whose side is 10 in. Find its area.
81. Find the altitude and the area of an equilateral triangle
whose side is 8 in.
82. Find the area of an equilateral triangle whose perimeter
is 60 in.
83. Arrange 6 equilateral triangles so as to form a regular
hexagon, and find the area of the hexagon, supposing each
side of the triangles to be 12 in.
84. Construct a regular octagon, and draw a line from its
center to the vertex of each angle.
85. Into how many and what kind of triangles is a regular
octagon divided by lines drawn from its center to the extremi-
ties of its sides ?
86. The distance from the center of a regular polygon to the
middle point of one of its sides is called the Apothem of the
polygon.
Draw a regular polygon and a line to show its apothem.
376
MEASUREMENTS AND CONSTRUCTIONS
87. How does the apothem of a polygon compare with the
altitude of one of the isosceles triangles into which a regular
polygon may be divided by lines drawn
from its center to the extremities of the
sides ?
88. If a side of the regular pentagon
BCDEF were 8 in., the apothem OA
would be 5.44 in. What would be the
area of the triangle OBC? Of the
whole pentagon ?
89. If a regular pentagon were cut into 5 equal isosceles
triangles, and arranged as in Fig. 14, the sum of the bases of
the triangles would equal what line ?
E D
FIG. 13.
FIG. 14.
90. Since every regular polygon may be divided into as
many isosceles triangles as it has sides, we may find the area
of a regular polygon by the following principle :
The area of a regular polygon is equal to one half the prod-
uct of its perimeter and apothem.
What is the area of a regular pentagon if the perimeter- is
80 in. and the apothem 10.88 in. ?
91. The ratio of the apothem to the side of a regular
pentagon = .68 ; of a regular heptagon = 1.03 ; of a regular
octagon = 1.20 ; of a regular decagon = 1.86. The apothem of
a regular hexagon can be easily found by the Pythagorean
Theorem.
If the perimeter of a regular pentagon is 60 in., what is the
apothem? The area?
92. Find the area of a regular decagon, one side of which is
8 in.
LINES AND SUKFACES
377
Given one side 14 in. :
93. Find the area of a regular pentagon.
94. Of a regular hexagon. 96. Of a regular decagon.
95. Of a regular octagon. 97. Of a regular heptagon.
98. What is the area of a flower bed in the shape of a
regular hexagon whose perimeter is 96 ft.?
99. At $ 1.25 per square foot, what is the value of a park
in the shape of a regular octagon, each side of which is 40 ft.?
100. In decorating a schoolroom 15 six-pointed stars were
used. Each star was made by combining 12 triangles, each
side of which was 4 in. Draw a figure to show how the stars
were made. Find the cost of
gilding all of them at 5 ^ a square
foot.
101. Figure 15 represents a
"block" of patchwork from a quilt.
Find the area of the whole block
if the perimeter of the inner hexa-
gon is 12 in.
102. What is the difference
between the area of a square
whose perimeter is 24 in. and the area of a regular hexagon
whose perimeter is 24 in. ?
103. Construct the regular hexagon
ABODE 'F, each side being 4 in., and
divide it into equal trapezoids by a line
represented by AD. What is the area
of each trapezoid?
104. Draw the square GHJK. What
is its area? How long is the perimeter
of the irregular figure ABCDJH? What
is its area?
FIG. 15.
378
MEASUREMENTS AND CONSTRUCTIONS
105. If the apothem of a regular pentagon is 20.4 in., how
long is one side?
See table of ratios, p. 376.
106. Find the area of a regular decagon whose apothem is
B 130.2 in.
107. Find the area of a regular
octagon whose apothem is 13.2 in.
108. Find the area of a regular
heptagon whose apothem is 175.1
in.
109. Find the area of the
trapezium ABCD, if the line AC
is 8 in., BE perpendicular to AC
4 in., and FD perpendicular to
AC 3 in.
In finding the area of an irregular figure it is customary to divide it into
triangles or trapezoids and find the area of each part separately.
110. A surveyor found the area of a piece of land repre-
sented by the irregular pentagon ABODE. The diagonal AD
was 50 ch., of which AH was 10 ch. and HF 28 ch.
o
17.
FIG. 18.
The perpendiculars BH, CF, and EG were respectively 12
ch., 18 ch., and 15 ch. A chain measures 4 rd. How many
acres were there ?
It will be observed that HBCF is a trapezoid and that the angles at H
and at F are right angles.
LINES AND SURFACES 379
111. CLASS EXERCISE. may draw an irregular polygon
upon the board, and the class may show different ways of divid-
ing it into triangles or trapezoids to find its area.
112. A polygon is said to be inscribed in
a circle when the vertex of each angle of
the polygon is in the circumference of the
circle. The circle is said to be circumscribed
about the polygon.
Inscribe a regular hexagon in a circle as
5*Ti9. in Fig. 19.
113. Can a rhombus be inscribed in a circle ? Explain.
114. Can a hexagon whose sides are each 4 in. long be drawn
within a circle whose radius is 5 in. ? Can it be inscribed in
the circle ? Explain.
115. If the side of a regular hexagon is 5 in., how long is
the radius of the circumscribed circle ? How long is its cir-
cumference ?
116. The part of a circle between an arc and its chord is
called a Segment.
How long would be the perimeter of each segment of Fig. 19,
if the radius of the circle were 24^- in. ?
117. Bisect each arc of your copy of
Fig. 19. Join the middle point of each arc
with the extremities of its chord as in
Fig. 20. How many sides has the regular
polygon you have thus formed ? How
would its area be found if the length of
one side and the apothem were known ?
FIG. 20.
If in the same way a polygon of 24 sides were
inscribed in the circle, and then another of double that number of sides,
and so on, we should soon have a polygon whose sides were so small that
the perimeter of the polygon could not be distinguished from the circum-
ference of the circle, and the polygon and the circle would appear to be
the same.
880 MEASUREMENTS AND CONSTRUCTIONS
118. A circle may be considered as a polygon of an infinitely
great number of sides, its circumference being the perimeter of
the polygon. To what would the radius of the circle correspond ?
119. Cut out a small circle and fold
it in halves. Fold it again, and continue
folding until many small sectors are
made by the folds. Cut along the folds
and place the circle as in Fig. 21.
To what is the sum of the bases of
these sectors equal ? To what is their
altitude equal ?
120. If a circle is considered as a polygon of an infinitely
great number of sides, the circumference is the sum of those
sides and the radius of the circle is the apothem of the poly-
gon. Can you see the reason for the following fact ?
The area of a circle is equal to one half the product of its cir-
cumference and radius.
This may be expressed by the formula A=C x - or A= , in which
"-4" stands for "area of circle," "C"' for "circumference," and "r"
for "radius."
121. Find the area of a circle whose radius is 7 ft.
Find the areas of circles of the following dimensions :
122. Radius 10 ft. 125. Circumference 77 ft.
123. Circumference 110 ft. 126. Eadius 1 ft. 9 in.
124. Diameter 18 in. 127. Circumference 5 ft. 10 in.
128. Measure or estimate the diameter of the face of a clock
or watch and find its area.
129. The minute hand of a clock in a tower is 3 ft. 6 in.
long. What is the area of. that part of the clock face over
which the exact middle line of the hand passes in 15 min. ?
130. What is the area of a sector which is ^ of a circle, if the
arc of the sector is 55 in. ? Represent.
131. A cow is tied to a stake in a field by a rope 20 ft. long.
What is the area of the surface over which she can move ?
LINES AND SURFACES
381
132. A statue whose base is 8 ft. square is placed in the
center of a grass plot having a circumference of 1254 ft. How
many square feet of the grass plot are around the statue ?
133. AB, a diameter, is 14 in. How
long is one side of the inscribed square ?
What is its area ? Find the area of each
segment cut off by the square. Find the
length of its perimeter.
134. Turn to Fig. 19, page 379. If the
radius of the circle is 8 in., what is the area
of the circle ? Of the hexagon ? Of each
segment cut off by the inscribed hexagon ?
135. A circle 10 in. in diameter is cut from a square 1 ft. in
diameter. What is the area of the remaining surface ?
136. The circles in Fig. 23 have the
same center. The diameter of the smaller
circle is 16 in. and the diameter of the
larger circle 28 in. Find the area of each
circle. Find the area of the circular ring
which is left when the smaller circle is cut
from the larger.
137. Circles which have a common
center are called Concentric Circles.
Draw two concentric circles, one with a radius of 5 inr., and
the other with a radius of 7 in. Find the area of the circular
ring which lies between their circumferences.
138. If the larger circle in Fig. 23
were 16 in. in diameter and the circular
ring were 2 in. wide, what would be the
G diameter of the inner circle ? Its area ?
139. Draw a circle, and circumscribe
a square about it as in Fig. 24. If the
radius of the circle is 3 in., how long is
FIG. 24. one side of the square ?
FIG. 23.
382
MEASUREMENTS AND CONSTRUCTIONS
140. If a side of the circumscribed square is 8 ft., how
long is the circumference of the circle? The perimeter of
the irregular figure FAE ? What is the area of FAE ?
The area of a circle may be easily found by a formula derived from
the formula A = C = x 2 r, or r. Substituting this value in
27 7
the formula A = -, we have A = r x -, or A - r 2 . Thus, to find
of r 2 .
of 25 =
the area of a circle whose radius is 5 in., we take
Ans. 78^ sq. in.
141. Find by the formula given in the note the area of a
circle whose radius is 9 in. If in. 12 in. 3i- in. I in.
T: Z o
142. Each member of a geometry class numbering 24 pupils
constructed a pasteboard cylinder and a pasteboard cone.
Supposing the diameter of each base of those figures to be
7 cm., how many square inches of pasteboard were in the
bases of the figures ?
143. A grass plot in the form of a semicircle, whose straight
edge is 15 ft., has within it a round bed of pansies 5 ft. in
diameter. Represent and find the area of the grass plot not
occupied by the pansies.
144. A garden 28 ft. square has flower
beds arranged as in Fig. 25. Each semi-
circle is 12 ft. in diameter, and the small
circle is 4 ft. in diameter. Find the area
of the ground space not within the flower-
beds.
145. Turn to Ex. 61, page 337, and find
the area of the irregular figure which is
left when the sectors are subtracted from
the square.
146. Four circles whose centers are A,
B, C and D, and whose radii are 7 in., are
placed as in Fig. 26. What is the area of
FIG. 26. the square ABCD ?
O
FIG. 25.
SOLIDS
383
147. Find the area
between the circles.
148.
which
of the surface which is included
FIG. 27.
The radius of the large circle
is 14 in. equals the diameter of
each of the small circles. Find the area
of a small circle. Of the large circle. Of
the irregular figure ABCDEF.
Observe that ABCDEF is one half of the
space remaining when the small circles are sub-
tracted from the large circle.
SOLIDS
NOTE TO TEACHER. All the solids treated here are right solids, and
the bases of the figures are regular polygons. Pupils
should model these solids. The problems that follow
require that models should be used as an objective basis
for work until the pupils are able to visualize the forms
accurately.
149. A Prism is a solid whose bases are poly-
gons and whose sides are rectangles.
Quadrangular
p r i sm . Mention some objects that are in the form
Bases Squares. o f a prism.
150. Copy Fig. 28 on paper or cardboard, cut it out, and
fasten its parts together so as to
make a quadrangular prism.
151. If the base of the prism
were 8 in. square and the altitude
of the prism 10 in., what would be
the area of all the surfaces of the
prism ? How many inch cubes
would equal it ?
152. What are the cubic con-
tents of a drawer which is 18 in.
square and 4 in. deep ?
FIG. 28.
384
MEASUREMENTS AND CONSTRUCTIONS
153. Give the reason for the following rule for finding the
cubic contents of a prism.
Multiply the area of the base by the altitude.
The volumes of all prisms and cylinders are found in
the same way.
154. Approximately 1J cu. ft. equal 1 bu.
How many bushels of wheat can be stored in a
bin 20 ft. long, 8 ft. wide, and 10 ft. deep ?
Find the value of the apples that fill a
, 6 ft. wide, 44- ft. deep, at $.75
lar
Bases Triangles.
155
bin 8
a bu.
-T IG.
156. From the outline given in Fig.
29 construct a triangular prism.
157. If each base edge of the prism
you constructed were 4 in. and each
lateral edge were 8 in., how many
inches would there be in all the edges
of the prism ? Find the area of its
lateral or side surface. Find the area
of its entire surface. Find its volume.
158. Find the area of the lateral surface of a triangular
prism whose altitude is 9 in. and each side of whose base is
5 in. Find its entire surface. Find its volume.
159. Find the entire surface and the volume of a triangular
prism each side of whose base is 6 in. and
whose altitude is 4.5 in.
160. Given a pentagonal prism, the perim-
eter of whose base is 15 in. and whose altitude
is 8 in. Find the area of a base. (See table
of ratios, p. 376.) Find the entire surface.
Find the volume. Find the sum of all its
edges.
Bases Pentagons.
SOLIDS
385
161. Construct a hexagonal prism, the perimeter of whose
base is 18 in. and altitude 7 in. Find its entire surface and
volume.
162. Given a hexagonal prism the perimeter of a face of
which is 24 in. and altitude 10 in. Find the sum of all its
Its entire surface. Its volume.
163. Turn to Ex. 103, p. 377. Suppose the square GHJK
to represent the base of a quadrangular prism of wood 10 in.
high. Suppose the wood to be cut away until a hexagonal
prism of the same height remains, whose base is represented
by the hexagon ABCDEF. Find the number of cubic inches
cut away.
164. Eeproduce the square ABCD.
With D as a center, and DO, which is \
the diagonal, as a radius, draw the arc
LOM. With A, B, and C as centers and
with radii equal to DO, draw equal arcs.
If the side of the square is 8 in., how
long is DO? DL? AL? DK? AE?
165. What is the area of each of the
triangles cut off from the square ?
What is the area of the octagon remaining ?
It can be proved by geometry that if the corners of a square are cut off
by the method given in Ex. 164, a regular octagon will be left. This
method is used by carpenters in marking off the end of a square piece
of lumber in order to change it to an octagonal form.
166. Find the area of the largest possible regular octagon
that can be cut from a 16-in. square.
167. A piece of lumber 4 ft. long and 16 in. square was
changed into an octagonal prism and used as a newel post.
The newel post was as large as it could be made from the piece
of lumber. How many cubic inches of wood were cut away ?
How many cubic inches were in the newel post ?
HORN. GRAM. SCH. AR. 25
386
MEASUREMENTS AND CONSTRUCTIONS
Cylinder.
Bases Circles.
168. As the number of sides of a regular
prism is increased, the base approaches more
nearly to a circle, and the prism more nearly
to a cylinder.
Draw a rectangle and construct two circles
whose circumferences are equal to a side of
the rectangle as in Fig. 31. Cut out and
combine the figures so that they inclose a
cylinder.
169. If the diameter of
each circle were 14 in.
and the shorter sides of
the rectangle were 16 in.,
what would be the area of
the entire surface of the
cylinder ? If the area of
the base of a figure is 154
sq. in., how many cubic
inches of sand would be
required to cover it to the
depth of 1 in.? 8 in ?
What would be the vol-
ume of this cylinder ?
170. Find the entire
surface of a cylinder 10 in. high, the diameter of whose base
is 1 ft. 2 in. Find its volume.
171. How is the area of the entire surface of a cylinder
found? How is the volume of a cylinder found?
172. How many cubic feet in a circular cistern 4 ft. in diam-
eter and 7 ft. deep ? How many gallons will it hold ?
231 cu. in. = 1 gal.
173. How many square inches of tin are there in a dozen
tin pails of cylindrical shape, the diameter of each being 8 in.
and the height 10 in. ?
FIG. 31.
SOLIDS
387
174. A cylindrical tank 10 in. in diameter and 28 in. in
height is full of water. How many gallons will remain in it
when a pail of similar shape 5 in. in diameter is filled from it ?
175. A cylinder 7 in. in diameter and 8 in. in height, outside
measurement, was placed within another cylinder 14 in. in di-
ameter and 8 in. in height, inside measurement. How many
cubic inches of space were between the two cylinders ?
176. A grindstone 28 in. in diameter was worn off until it
was 21 in. in diameter. If the grindstone was 4 in. thick, how
many cubic inches were worn off ?
177. A solid whose base is a polygon and
whose sides are triangles meeting at a com-
mon point is a Pyramid.
Construct an equilateral triangle, and with
each side as a base construct an isosceles
triangle as in Fig. 32. Cut out the figure
and bring the isosceles triangles together in
such a way that their vertices meet in a common point and
a triangular pyramid is formed.
178. The point where all the faces
of a pyramid meet is called the Apex
of the pyramid.
A perpendicular from the apex to
the base meets the base at its center.
That perpendicular is the Altitude of
the pyramid.
A line from the apex to the middle
point of a side of the base is perpen-
It is called the Slant Height of the
Triangular Pyramid
Base a Triangle.
FIG. 32.
dicular to that side,
pyramid.
If a side of the equilateral triangle which you have con-
structed were 10 in., what would be the area of the base of the
pyramid ? If each of the equal sides of the isosceles triangles
were 13 in., what would be the distance from the vertical
388 MEASUREMENTS AND CONSTRUCTIONS
angle of each triangle to the middle point of each side of the
base ? What would be the area of the lateral surface of the
pyramid ? Of the entire surface ?
179. If a side of the base of the quadrangu-
lar pyramid whose apex is A, is 10 in., how
long is the distance BC from center of the
base to middle point of a side ? If the alti-
tude AB is 12 in., how long is the slant height ?
What is the area of the lateral surface of the
Quadrangular pyr- . ,
amid Base a pyramid?
180. If a side of a base of a quadrangular
pyramid were 18 in., and the slant height were 12 in., how long
would a lateral edge be ? Find the sum of all the edges of the
pyramid. Find its entire surface.
181. Given a side of the base of a quadrangular pyramid 40
in., the altitude 21 in., find the slant height. Find the area of
all the surfaces of the pyramid.
182. Given a quadrangular pyramid whose base is 20 in.
square, the slant height 26 in. Find the entire surface of the
pyramid. Find the altitude of the pyramid.
183. Given a hexagonal pyramid, one side of the base being
16 in., and a lateral edge 17 in. Find the slant height of the
pyramid. Find its lateral surface. Find the sum of all its
184. It can be proved by geometry that the volume of a
pyramid equals -J- of the volume of a prism having the same
base and altitude.
Find the volume of the pyramid described in Ex. 178. Ex.
179. Ex. 180. Ex. 181. Ex. 182. Ex. 183.
185. Find the contents of a pyramid whose base is a trian-
gle, each side of which is 8 ft. and whose altitude is 21 ft.
186. If a prism of wood whose base is a square 18 in. in
dimensions and whose altitude is 10 in., be cut away until
SOLIDS 389
a pyramid is left having the same base and altitude as
the prism, how many cubic inches of wood must be cut
away ?
For data for the following problems see table of ratios, p. 376.
Given a side of a base 6 in. and the altitude 10 in., find the
volume of :
187. A pentagonal pyramid.
188. An octagonal pyramid.
189. A hexagonal pyramid.
190. A pyramid whose base is a decagon.
191. The perimeter of the base of a hexagonal pyramid is
54 in., and a lateral edge is 15 in. Find a side of the base.
Find the distance from the center of the base to the vertex
of an angle of the base. Find the altitude of the pyramid.
Find its volume.
192. Given a quadrangular prism and a quadrangular pyra-
mid. The perimeter of the base of each solid is 5 ft. 6 in.
The altitude of the prism equals the slant height of the
pyramid, which is 1 ft. 8 in. The lateral surface of the prism
equals how many times the lateral surface of the pyramid ?
Which has the greater altitude, the quadrangular prism or the
quadrangular pyramid ?
193. At 5^ per square foot, what would be the cost of
painting the sides of a steeple which is an octagonal pyra-
mid, each side of the base being 8 feet and the slant height
being 75 feet?
194. The great pyramid of Gizeh was
originally 480 feet high, with a square base
764 feet on each side. How many cubic
feet of masonry were there in it ?
195. As the number of sides of a pyra-
Cone. Base a circle. mid ig increased; the base of the pyram i d
390 MEASUREMENTS AND CONSTRUCTIONS
approaches more closely to a circle and the pyramid to a
cone.
Draw and cut out a sector, and also a circle whose circum-
ference equals the arc of the sector as in Fig. 33. With them
construct a cone.
196. What would be the area
of the convex or curved surface
of the cone if the radius of the
sector from which it were made
were 4 in. and the arc 11 in. ?
What would be the circumfer-
ence of the circle which forms
the base of the cone ? The diam-
eter ? The area?
r ]<;. oo.
197. Would it be possible to make a cone by using as a base
a circle exactly equal to the circle from which a sector is cut to
form the curved part of the cone ? Explain.
198. What is the area of the curved surface of a cone if the
circumference of the base is 4 ft. 8 in. and the slant height is
2 ft. 9 in. ? What is the area of the entire surface ?
199. Can a cone be constructed having the diameter of its
base 10 in. and the slant height 4 in. ? Explain.
200. How many square inches of tin are needed to make a
funnel in the shape of a cone, the circumference of the base
being 5 in. and the slant height 4 in. ?
201. Four conical towers, each having a diameter of 4 ft. and
a slant height of 12 ft., ornament a pavilion in a park. Find
the cost of gilding them at 15 ^ per square foot.
202. Find the entire surface of a cone the radius of whose
base is 24 in. and the slant height of which is 4 ft. 2 in.
203. A line from the apex of a cone to the center of its base
is perpendicular to the base. It is the altitude of the cone.
What is the slant height of a cone whose altitude is 24 cm., if
SOLIDS 391
the diameter of the base is 14 cm. ? What is the area of the
entire surface of the cone ?
204. As a cone is a pyramid of an infinitely great number , ^
of sides, its volume is equal to J that of a cylinder whose alti- ^
tude and base are respectively equal to those of the cone.
Find the volume of a cone whose diameter is 5 in. and alti-
tude 10 in.
Find volumes of cones having the following dimensions :
205. Kadius 10 in., altitude 2 ft.
206. Diameter 15 in., altitude 11 in.
207. Circumference 5 ft. 6 in., altitude 1 ft. 10 in.
208. Radius 5 in., slant height 1 ft. 1 in.
209. What is the volume of the largest possible cone that
could be cut from a prism 1 ft. long, whose base is 8 in. square ?
210. A cylinder whose diameter is 8 in. and altitude 10 in.
is cut entirely across, parallel to its base at a distance of 3 in.
from its base. What kind of solids are formed ? What is
the area of a base of each ?
211. When a solid is cut entirely through in such a way
that two plane surfaces are formed, the surfaces are called
Sections.
Represent a section made by cutting across a cylinder
in such a way that the section is not parallel to the base.
212. Sections parallel to their bases were made of (a) a
quadrangular prism, (6) a hexagonal pyramid, (c) a cone, (d) a
triangular pyramid. What was the shape of each section ?
213. Represent a section not parallel to the base of each of
the above figures.
Frustum of Pyramid. Frustum of Cone.
392 MEASUREMENTS AND CONSTRUCTIONS
214. If a pyramid or a cone is cut by a plane parallel to its
base, the part below the plane is called a Frustum of the pyra-
mid or of the cone. See illustrations on page 391.
Construct a frustum of a cone or of a pyramid.
215. Each of the bases of the frustum of a triangular pyra-
mid is what figure ?
216. If each side of the lower base of a triangular pyramid
is 10 in., each side of the upper base is 8 in., and the slant
height is 7 in., what is the area of the lateral surface ?
If we find the area of one of the trapezoids that compose the lateral
surface of the frustum of a pyramid and then multiply that area by the
number of trapezoids we shall have the area of the lateral surface, but
it is more convenient to find that area by multiplying the average length
of the perimeters of the upper and lower bases by the slant height.
217. Find by each method the area of the lateral surface of
the frustum of a triangular pyramid of the following dimen-
sions, and then try to find why the results are the same:
Edge of lower base 5 in., edge of upper base 3 in., slant height
8 in.
218. The area of the upper base of the frustum of a square
pyramid is 100 sq. in., the area of the lower base 144 sq. in.,
and the slant height. 10 in. Find the entire surface.
219. A quadrangular pyramid, each side of whose base is
16 in., is cut by a plane so that each side of the upper base of
the frustum is 11 in. long. The slant height of the frustum is
10 in. What is the entire surface of the frustum ?
220. What is the lateral surface of a frustum of a hexagonal
pyramid, the perimeter of the lower base being 42 in., that of
the upper base 24 in., and the slant height being 5 in. ?
221. Find the lateral surface of the frustum of a hexagonal
pyramid if a side of the lower base is 17 in., a side of the upper
base 15 in., and the slant height is 1 ft.
SOLIDS 393
222. Find the convex or curved surface of a frustum of a
cone, the upper base of which is 40 in., the lower base 60 in.,
and the slant height 10 in.
Remember that the frustum of a cone is a frustum of a pyramid of an
infinitely great number of sides.
Find missing measurements in frustums of cones :
Circurn.
upper base
Circum.
lower base
S. height
Convex surface
223. 8 in.
1ft.
Tin.
9
224. 2 ft. 3 in.
3 ft. 2 in.
8 in.
9
225. 11 in.
15 in.
?
78 sq. in.
226. 1 ft. 4 in.
2 ft. 6 in.
9
1 sq. ft. 86 sq. in.
227. How many square feet of tin are used in constructing a
tin pail in the shape of a frustum of a cone whose smaller base
is 9 in. in diameter, upper base 1 ft., and whose slant height is
1 ft. 2 in., no allowance being made for overlapping at the
seams ?
228. If a perfectly round ball 7 in.
in diameter were cut into two hemi-
spheres, A and B, what would be
the area of each plane surface of the
hemispheres ?
229. It can be proved by geometry
that the curved surface of a hemi-
Hemispheres
sphere is exactly twice as great as its
plane surface. What, then, would be the area of the outside
surface of the ball mentioned in Ex. 228 ?
230. Find the surface of a sphere whose diameter is 1 ft.
9 in. 1 ft. 51 in.
231. Find the surface of a sphere whose radius is 8 in.
232. A flagstaff is surmounted by a ball 1 ft. in diameter.
Find the cost of gilding it at 30 4 a square foot.
394 MEASUREMENTS AND CONSTRUCTION
233. What is the entire curved' surface of a hemispherical
dome whose height is 35 ft. ?
234. If the earth were an exact sphere 8000 miles in
diameter, what would be the area of its surface ?
235. Solids that have the same shape are said to be
Similar Solids.
Think of two boxes of the same shape, each dimension of
the larger box being twice the corresponding dimension of
the smaller. If the larger box is 8 in. by 4 in. by 2 in., what
are the dimensions of the smaller ? What is the area of the
surfaces of each ? What is the volume of each ? What is the
ratio of their areas ? Of their volumes ?
236. In the case of similar figures, every line of one figure
has a corresponding or Homologous Line on the other figure,
and every angle on one figure has a Homologous Angle on the
other figure.
Take two similar right triangles of different dimensions and
point out the homologous lines and angles.
SUGGESTION TO TEACHER. Similar solids should be handled and ex-
amined by the pupils. The magnitudes of their homologous angles, lines,
surfaces, and volumes should be compared until the following principles
are realized.
1. On similar solids, homologous angles are equal.
2. On similar solids, any two homologous lines are to each other
as any other two homologous lines.
3. On similar solids, homologous surfaces are to each other as
the squares of their homologous lines.
4. The volumes of similar solids are to each other as the cubes
of their homologous lines.
237. A. side of a base of a quadrangular pyramid is 6 in.
The altitude of the pyramid is 4 in. What is the slant height?
The sides of the base of a similar pyramid are each 12 in.
What is the altitude ? The slant height ?
SOLIDS 395
238. Find the area of one of the sides of the smaller
pyramid. Of one of the sides of the larger. Find the ratio
of their areas. Find the volume of the smaller pyramid.
Of the larger. Find the ratio of their volumes.
239. Image two similar quadrangular pyramids whose
homologous lines are in the ratio of 1 to 3. What is the ratio
of the areas of their bases ? If a lateral edge of the smaller
pyramid is 8 in., what is the lateral edge of the larger?
Assume dimensions for the figures, and find the areas of a
triangular side of each solid, and also the ratio of those
areas.
240. The slant height of the frustum of a hexagonal pyramid
is 12 in. A side of the lower base is 8 in. A side of an upper
base is 6 in. Find the perimeters of the bases of a similar
frustum, the slant height of which is 3 in.
241. Find the lateral surface of each of those frustums.
Find the ratio of the surfaces.
242. If a bucket 8 in. in diameter holds 3 gal., how many
gallons can be poured into a bucket of similar shape whose
diameter is 16 in. ?
243. If it takes 110 sq. in. of tin to make a milk can 5 in.
in diameter, how many square feet of tin will be required to
make a similar can 20 in. in diameter ?
244. If the volume of a cone whose altitude is 6 in. is
54 cu. in., how many cubic inches are there in a similar cone
whose altitude is 10 in. ?
245. A solid iron ball weighs 7 Ib. What would be the
weight of a similar ball whose diameter is twice as great as
that of the first ?
246. The cost of gilding a vase was $ 1.60. What would
be the cost of gilding a larger vase of the same shape, the
396 MEASUREMENTS AND CONSTRUCTIONS
larger vase being twice as high as the smaller ? If the cubic
contents of the larger vase were 328 cu. in., what were the
cubic contents of the smaller ?
247. Two similar cylinders are respectively 2 in. and 8 in.
in diameter. If -a section is made parallel to the base of each,
what is the ratio of the area of the section of the greater
cylinder to that of the less ?
248. A pyramid whose base is 6 ft. square and whose alti-
tude is 4 ft. is cut by a plane parallel to its base and 2 ft. above
it. The pyramid above the cut equals what part of the origi-
nal pyramid ?
249. If a coal bin 4 ft. long holds 20 bu. of coal, how many
bushels can be put into a bin of similar shape which is 8 ft.
long?
250. If a city lot one side of which is 80 ft. is worth
$ 12,000, what is the value of a lot of the same shape, the
corresponding side of which is 40 ft. ?
251. 16 sq. ft. of galvanized iron were used in making a
water tank for a stove. How many square feet must be used
to make a tank of the same shape, each edge of which is
twice that of the first tank ? If the first tank held 30 gal.,
how many gallons would the second tank hold ?
252. A certain freight car contains 24,000 cu. ft. of space.
How many cubic feet of space will be contained in a model of
this car, which is -^ as long as "the original ?
253. What is the ratio of a diagonal of a face of a liter to
a diagonal of a face of a stere ?
254. What is the ratio of the sum of the surfaces of a liter
to the sum of the surfaces of a stere ? What is the ratio of
a liter to a stere ?
255. If each of the sides of a polygon is trebled, the result-
ing polygon equals how many times the original polygon ?
ARCS AND ANGLES 397
256. If each of the lines of a frustum of a hexagonal pyra-
mid were made 5 times as long, the volume of the resulting
solid would be how many times the original solid ?
257. If a straw stack 5 ft. high contains 3 tons, how many
tons are there in a stack of similar shape 10 ft. high ?
258. How many balls of lead 2 in. in diameter will weigh
as much as a ball of lead 8 in. in diameter ? What is the
ratio of the sum of all the surfaces of the 2-inch balls to the
surface of the 8-inch ball ?
ARCS AND ANGLES
259. Circumferences of circles are considered to be divided
into 360 equal parts called degrees, marked . How many
degrees in a semicircumference ? In a quadrant?
260. Over how many degrees does the minute hand of a
clock pass in 15 min. ? In 45 min. ?
261. How many degrees are described by the hour hand of
a clock in 4 hr. ? In 2 hr. ? In 5 hr. 30 min. ?
262. If a circumference is divided by a chord in such a
way that the greater arc is 4 times the less, how many degrees
are there in each arc ?
263. If a regular hexagon is inscribed in a circle, how many
degrees are there in each arc cut off by a side of the hexagon ?
264. How many degrees are there in each arc subtended by
a side of a regular inscribed octagon ? By a side of a regular
inscribed decagon ? Dodecagon ? Heptagon ? Nonagon ?
265. An arc 20 ft. long equals how many degrees of a cir-
cumference 160 ft. long ? Of a circumference 240 ft. long ?
266. A horse trotted 1760 ft. on a circular race track 1 mi.
in length. Over how many degrees of its circumference did
he pass ?
267. If a circumference is 24 in., how long is an arc of 60 ?
90 ? 30 ? 150 ? 120 ? 45 ? 75 ? 108 ?
898 MEASUREMENTS AND CONSTRUCTIONS
268. Find the length of an arc of 110 of a circumference
which is 20 ft. Of a circumference whose radius is 7 ft.
269. How long is an arc of 45 in a circle whose diameter
is 6 ft. 5 in. ? In a circle whose radius is 4 ft. 8 in. ?
270. How long is an arc subtended by the side of a regular
pentagon inscribed in a circle whose circumference is 40 ft. ?
In a circle whose diameter is 10 ft. ?
271. In order to make calculations more exact, a degree is
divided into 60 equal parts called minutes, marked ', and a
minute is divided into 60 equal parts called seconds, marked ".
TABLE OF ANGULAR MEASURE
60 seconds (") = 1 minute (')
60 minutes = 1 degree ()
Do not confound minutes and seconds that are measures of arcs and
angles with minutes and seconds that are measures of time.
An arc which is one minute is what part of a circumference ?
An arc which is one second is what part of a circumference ?
272. How many minutes in 25 30' ?
273. How many minutes in 4 ? In 720" ? In 5 55' ?
274. How many seconds in 35' 25" ? In 8 10' 20"? In
90 15' 25" ?
275. Express 50 15' 30" as seconds. As minutes. As de-
grees.
Express in each denomination of angular measure :
276. 7 10' 20". 281. 9' 20".
277. 8 10". 282. 10 45".
278. 4' 30". 283. 8 6' 6".
279. 7 20' 40". 284. 4' 50".
280. 15 15' 15". 285. 12 12' 12".
286. Express in degrees, minutes, and seconds, 42784 sec.
31125 sec. 57241 sec.
ARCS AND ANGLES 399
287. Express in higher denominations, 97860 sec. 77825 sec.
Add:
288. 8 5' 50"
6 11' 27"
289. 6 7' 24"
20 37' 48"
290. 8 29' 33"
17 31' 47"
Find difference :
291. 8 7' 25"
3 8' 16"
292. 24 16' 38"
7 19' 49"
293. 85 21' 36"
17 27' 54"
Multiply :
294. 41 17' 25"
8
295. 16 17' 19"
9
296. 23 28' 39"
11
Divide :
297. 15)18 36' 45" 299. 24)13 19' 28"
298. 11)19 36' 48" 300. 15)7 8' 43"
301. The sum of two arcs, one 49 1' 28", the other 16 38' 59",
is how much less than the whole circumference ?
B 302. How many degrees, minutes, and seconds in $
of a circumference ?
303. Think of the position of the hands of a clock
' 34 at 12 o'clock. Imagine two equal lines BO and AO
in the same position (Fig. 34). Suppose AO to remain
fixed, and that BO makes a complete revolution around the
point and returns to its former position, the point B de-
scribing a circle. At the different stages of its revolution BO
makes different angles' with AO, corresponding in the number
of degrees to the arcs described. As the angle of the
whole revolution is considered an angle of 360, the
angle formed by the two lines, when \ of the revo-
lution has been made, equals 90, or a right angle.
When BO has made of a revolution, and the two
lines are so placed that each is a continuation of the
other (Fig. 35), the angle formed by them is an angle
of 180, or a straight angle.
How many right angles form a straight angle ?
400 MEASUREMENTS AND CONSTRUCTIONS
304. How many degrees are there in the sum of
the angles a and b ?
305. -Reproduce Fig. 36 several times, changing the
dividing line between the angles to make it lie in dif-
ferent directions. The sum of the angles is always
how many degrees ?
306. Two angles, whose sum is equal to 180, are
said to be Supplements of each other.
How many degrees are there in the supplement of an angle
of 100 ? Of 179 ? Of 3 ? Of a right angle ?
307. How many degrees are there in the supplement of an
angle which is | of a right angle ?
308. How much greater or less than its supplement is an
angle of 80? 90? 130? 170? 500? 75?
309. How many degrees are there in an angle whose sup-
plement is twice the given angle ?
310. How many degrees are there in an angle which is twice
its supplement?
311. What is the ratio of an angle of 110 to its supple-
ment ? To a straight angle ? To a right angle ?
312. A fan, which when opened was semicircular in shape,
was opened -| of its extent. What angle was formed by the
outside edges of the sticks ?
313. A branch of a tree made an angle of 45 with the
trunk of the tree. What was the supplement of that angle ?
314. How many degrees are there in the angle formed by
the hands, of a clock at 2 o'clock ? 4 o'clock ? 6 o'clock ?
315. How many degrees are there in the supplement of an
angle formed by the hands of a clock at 1.30 P.M. ?
316. What kind of an angle is the supplement of an acute
angle ? Of a right angle ? Of an obtuse angle ? Explain.
ATCCS AND ANGLES
401
FIG. 37.
317. When is an angle greater than its supplement? Less
than its supplement?
318. How many degrees are there in
the sum of all the angles, a, 6, c, d,
formed at a given point and 011 the same
side of a straight line ? Explain.
319. An angle formed by radii of a
circle contains just as many degrees as
the arc which is included between the
ends of the radii.
is the center of the circle of which
the arc AB is 60. How many degrees
are there in the angle AOB ? In BOG ?
In the straight angle AOC?
In naming an angle by three letters, the let-
ter at the vertex is placed between the other two
letters.
320. The angle ABC is a right angle.
ABD is 10 more than DBG. QBE is 4
E
times EBF. How many degrees in
" F DBG? In ABD? In EBF? In EBC?
InDBE? In ABE?
321. Three angles, a, 6, and c, are formed at the same point
and on the same side of a straight line, a = 3 times b } and
c = 4 times b. How many degrees are there in each ?
Let x = number of degrees in angle b.
322. Four angles, a, b, c, and d, are formed on one side of a
straight line at the same point, b has 10 more than a, c has
10 more than &, and d has 10 more than c. How many de-
grees are there in each ?
323. How many degrees are there in the sum of an angle
of 17 3' 15" and an angle of 17 4' 21"?
324. How many degrees, minutes, and seconds are there in
the supplement of an angle of 105 15' 20" ?
HORN. GRAM. SCH. AR. 26
B
FIG. 39.
402
MEASUREMENTS AND CONSTRUCTIONS
325. How many degrees, minutes, and seconds are in the sup-
plement of an angle which is 4 times an angle of 7 40' 50" ?
326. Give the measurement of an angle which is 5 times
the supplement of an angle of 150 10' 24".
327. Give the measurement of an angle whose supplement
is 6 times an angle of 10 20' 30".
328. The figure below represents a Protractor, a device for
measuring or constructing angles. The point at the center
marked A in this figure must always be placed at the vertex
of the angle that is measured or constructed. The degrees
corresponding to the angle are marked upon the arc.
Make a protractor. Lay off by means of it an angle of 20.
329. Draw an angle and find by the protractor the number
of its degrees.
330. Find by the protractor how many degrees there are in
an angle of an equilateral triangle. Of a regular hexagon.
SUGGESTION TO TEACHER. Let pupils measure angles found in decora-
tive designs, as in wall paper, carpet, parquetry, also the angles formed by
branching stems, the angles of crystals, and of other natural objects.
331. How many degrees are there in each angle of a square ?
In all the angles of a square ?
ARCS AND ANGLES
403
332. Draw a square. With each corner of the square as a
center and with a radius less than ^ the side of the square draw
arcs inside the square ending in its sides.
(See p. 337, Fig. 10.) Cut out the 4 sectors
and place them so that their vertices are
at the same point. What figure is formed ?
333. Plow many degrees are there in
all the angles a, &, c, d t e formed about a
common point in Fig. 40 ?
334. Angle 1 (Fig. 41) is 89 10', angle
2 is 48 30', angle 3 is 90 b 5'. How many
degrees in angle 4 ?
335. How many degrees are there in
each of 8 equal angles whose vertices are
Fl( , 41 .afr-the same point ?
336. There are 4 angles a, 6, c, and d around a common point,
a = 3 times b, c = twice 6, and d = twice a. Find the number
of degrees in each. Represent.
337. Construct around a common point an angle of 75, an
adjacent angle of 65, and another adjacent angle of 80.
How many degrees are there in the remaining angle ?
338. How long is the perimeter of a sector of 60 of a circle
whose radius is 15 in. ? What is the area of the sector ?
339. If a sector of 45 were cut from a circle whose diameter
is 20 in., how long would be the perimeter of the remaining
figure ? What would be its area ?
340. Find the perimeter and the area of a sector of 40 of
a circle whose radius is 18 in.
341. What is the area of the figure that remains when a
sector of 36 is cut from a circle whose radius is 3 ft. 6 in. ?
How long is the perimeter of the figure ?
404
MEASUREMENTS AND CONSTRUCTIONS
342. Construct an angle of 50, an angle of 60, and an
angle of 70. Cut them out and place them side by side so that
their vertices are at a common point. What kind of an angle
do they form ?
FIG. 42.
343 . Draw and cut out a triangle, mark-
ing its angles 1, 2, 3, as in Fig. 42.
Cut off the corners 1 and 3 as in
Fig. 43. Place them beside corner 2 as
in Fig. 44.
The angles will have their vertices at
a common point and will cover all the
surface around that point on the same
side of a straight line. Hence they are
equal to two right angles.
FIG. 44.
344. Kepeat the experiment with triangles of different
shapes until you see the truth of the following:
The sum of the angles of a triangle is equal to two right angles,
or 180.
By geometry this is proved to be true in all cases.
345. How many degrees are
there in angle A of Fig. 45?
346. If A were 83 and B were
75, how many degrees would angle
C be?
347. If A were 91 and C 41,
how many degrees would angle B
be?
348. If one angle of a triangle is a right angle, how many
degrees are there in the sum of the other two angles ? What
kind of angles are they ?
FIG. 45.
ARCS AND ANGLES 405
349. If one of the angles of a right triangle is 38, how many
degrees are there in the other acute angle ?
350. At one extremity of a line construct an angle of 35.
Construct an angle of 75 at the other end of the line. Prolong
the added lines until they meet. How many degrees are there
in the angle formed by their meeting ?
351. Fold an isosceles triangle so that the equal sides coin-
cide. Can you see that the following statement is true ?
In an isosceles triangle the angles opposite the equal sides are
equal.
352. How many degrees are there in each angle
of the isosceles triangle ABC ? Explain.
353. How many degrees are there in each angle
of an isosceles triangle in which a base angle is
80 ? Explain.
354. Draw two equal lines forming a right angle and join
their extremities. What kind of a triangle is
formed? How many degrees are there in
each angle?
355. Find the measurement of each angle
of an isosceles triangle in which a base angle
is 3 times the vertical angle. Represent.
356. How many degrees are there in
angle c ? How many degrees in each angle
of the isosceles triangle in Fig. 47 ?
357. An angle formed by a side of a polygon and a pro-
longation of an adjacent side is called an Exterior Angle.
Show two exterior angles in Fig. 47, and tell how many
degrees in each.
358. Reproduce Fig. 47 and make an angle exterior to b.
How many degrees are there in it ? Can an isosceles triangle
have two exterior angles at the base unequal ? Explain.
406 MEASUREMENTS AND CONSTRUCTIONS
359. How many degrees are there in each angle
of the isosceles triangle ABC (Fig. 48), whose base
is BC?
360. The triangle ADO (Fig. 49) is isosceles.
A, the vertical angle, is 40. CB is perpendicular
to AD. How many degrees are there in each angle
of the triangles ABC and DBC?
361. The triangle ABC (Fig. 50) is isosceles.
AE bisects the angle A. Angle DBC is 100.
Find each angle of the triangles AEB and AEC.
362. In a scalene triangle ABC, A is a right
angle, and the angle B is 8 times the angle C.
How many degrees are there in each angle of the
triangle ? Represent.
363. How many degrees apart are the equator
and the north pole ? How many degrees apart
are two places which are on exactly opposite
points of the equator ?
Recall your knowledge of meridians and parallels of
latitude by reference to your geography, if necessary.
364. How many degrees apart are two places,
one of which is on the equator, and the other half
Fia. 50. way between the equator and the south pole ?
365. How many degrees apart are two points, one of which
is at the south pole, and the other 30 from the north pole ?
366. Considering the circumference of the earth as 25,000
miles, how far apart are two places on the equator which are
60 apart ? 180 apart ? 120 apart ?
In the following problems the equator and meridian circles are assumed
to be circles 25,000 miles in circumference.
367. How long is an arc of the equator which is 50 ? 70 ?
368. How long is an arc of 50 of a circle of latitude which
is 1800 miles in circumference ?
ARCS AND ANGLES 407
369. An arc of 75 of a parallel of latitude 120 miles in cir-
cumference equals how many miles ?
370. How many miles from the equator is a place 20 north
of it ? About how many miles from the equator are you ?
371. If a ship sailed 2000 miles on the equator, through how
many degrees of longitude would it pass ?
372. If a ship sailed 200 miles on a circle of latitude, the
circumference of which was 4000 miles, through how many
degrees would it pass ?
373. How many degrees from the equator is a place that
is 500 miles north of it ?
374. A village is 2 10' 30" east of a certain city, and
another village is 17 49' 30" west of the city. How many
degrees apart are they ? If the circle of latitude upon which
they are situated is 1200 miles in circumference, how many
miles apart are the villages ?
375. What is the difference in longitude between a place
20 13' 48" east longitude, and a place 15 15' 55" west longi-
tude ? Draw a diagram to illustrate the problem.
Find the difference of longitude between :
376. A, 17 15' 30" east, and B, 16 16' 58" west.
377. C, 40 30' 20" east, and D, 50 10' 25" west.
378. E, 17 13' 21" west, and F, 19 18' 24" west.
379. G, 41 16' 29" east, and H, 31 17' 27" east.
380. J, 61 16' 38" east, and K, 15 15' 45" west.
381. L, 6' 17" east, and M, 20' 30" west.
382. A certain lighthouse is 6 15' 20" north of the equator,
and another lighthouse on the same meridian is 11 44' 40"
south of the equator. How many miles apart are they ?
408 MEASUREMENTS AND CONSTRUCTIONS
383. The earth, revolves on its axis once in 24 hr. Through
how many degrees does a point on the earth's surface turn in
one hour ?
384. A point on the equator revolves at the rate of how
many miles in one hour?
LONGITUDE AND TIME
As the earth's motion in revolving on its axis is from west to east,
when it is sunrise at a certain place, points west of that place are still in
darkness.
385. Suppose the sun to rise at 6 o'clock at a certain place,
how much less than 6 o'clock is the time of day at a place 15
west of that place ? What is the time 15 east of that place ?
386. When it is noon at Denver, is it A.M. or P.M. at Bos-
ton ? New York ? San Francisco ? Philadelphia ? Portland,
Oregon ? Portland, Me. ? Washington, D.C. ? St. Louis ?
387. Find the difference in longitude and the direction from
Chicago of a place whose real time is 2 hr. later than Chicago
time. 3 hr. earlier. 5J hr. earlier. 4 hr. 30 min. later ?
388. When it is noon at Buffalo, what time is it at a place
15 east of Buffalo ? 15 west ? 60 east ? 40 west ?
389. When it is midnight at St. Paul, what time is it at a
place 30 north of St. Paul ? 30 south ? 30 west ? 30 east ?
390. If a point moves 15 in 1 hr., or in 60 minutes of time,
how far will it move in one minute ?
391. If a point moves \ of a degree, or 15' in one minute of
time, how far will it move in one second of time ?
15 of longitude cause a difference of 1 hr. of time.
15' " " " 1 min. of time.
15" " " 1 sec. of time.
392. What is the difference in longitude of two places be-
tween which there is a difference of 2 hr. 10 min. 17 sec. of time ?
LONGITUDE AND TIME 409
Since a difference of 1 hr. of time is caused by a
hr. min. sec. difference of 15 in longitude, 1 min. of time by 15' of
2 10 17 longitude, and 1 sec. of time by 15" of longitude, we
15 multiply the number of hours, minutes, and seconds by
40 4 5 15 to find the corresponding number of degrees, minutes,
and seconds of longitude.
393. When it is 4 P.M. at Anda it is 6.30 P.M. at Roseville.
What is the difference in longitude, and which place is further
east?
Find difference in longitude and relative position of places
whose simultaneous time reckonings are as follows :
394. A, 3 hr. 30 min. P.M. B, 5 hr. 10 min. 30 sec. P.M.
395. C, 11 hr. 30 min. A.M. D, 12 hr. 30 min. P.M.
396. E, 10 hr. 30 min. 30 sec. A.M. F, 2 hr. 30 min. 45 sec. P.M.
397. G, Noon H, 10 hr. 30 min. 25 sec. A.M.
398. J, 9 hr. 30 min. P.M. K, Midnight.
399. L, 8 hr. 40 min. 30 sec. P.M. M, 5 hr. 20 min. 10 sec. P. M.
400. N, 7 hr. 30 min. 15 sec. A.M. 0, 1 hr. 10 min. 15 sec. P.M.
401. P, 9 hr. 20 min. 10 sec. A.M. Q, 3 hr. 5 min. 30 sec. P.M.
402. The longitude of the following places is reckoned from
the meridian of Greenwich.
Portland, Me.,
70 15' 18" W.
Chicago, 111.,
87 37' 0" W.
Boston, Mass.,
71 3' 50" W.
New Orleans, La. ,
90 5' 0" W.
New York City,
74 0' 36" W.
Omaha, Neb.,
95 56' 14" W.
Pittsburg, Pa.,
80 2' 0" W.
Paris, France,
2 20' 0" E.
Savannah, Ga.,
81 5' 26" W.
Rome, Italy,
12 28' 0" E.
Louisville, Ky.,
85 30' 0" W.
Vienna, Austria,
16 23' 0" E.
Nashville, Tenn., 86 49' 0" W. St. Petersburg, Russia, 30 18' 0" E.
The difference in time between Portland, Me., and a place
west of it is 2 hr. What is the longitude of that place ?
403. In what longitude is a place that has 2 hr. 10 min.
later time than Louisville ?
410 MEASUREMENTS AND CONSTRUCTIONS
404. In what longitude is a place where it is half past ten
in the morning at the same instant when it is noon at Pitts-
burg?
What is the longitude of a place which has :
405. 3 hr. 30 min. later time than Boston?
406. 2 hr. 20 min. earlier time than New York City ?
407. 1 hr. 10 min. earlier time than Nashville ?
408. 2 hr. 15 min. later time than New Orleans ? Louisville ?
409. 3 hr. 15 min. 30 sec. later time than New Orleans ?
410. 1 hr. 20 min. later time than Paris?
411. 1 hr. 40 min. later time than Kome ?
412. 3 hr. earlier time than Kome?
413. 4 hr. earlier time than St. Petersburg?
414. 3 hr. later time than Washington, D.C. ?
i
415. 4 hr. 30 min. later time than Boston ?
416. A difference of 75 45' 15" between two places causes
how much difference in time ?
15) 75 45' 15" Since 15 of longitude cause a difference of 1 hr. in
K o T~ time, 15' of longitude, 1 min. in time, and 15" of lon-
gitude 1 sec. in time, we divide the number of degrees,
minutes, and seconds of longitude by 15 to find the corresponding num-
bers of hours, minutes, and seconds of time.
417. A city is 30 15' 45" west of a certain meridian. What
is the difference in time between that city and all places on
that meridian ?
418. There are two meridians 24 48' 30" apart. When it is
2 P.M. at places on the western meridian, what time is it at
the places on the eastern meridian ?
What is the difference in time between the following places :
419. Portland, Me., and Omaha ?
420. Boston and Chicago ?
LONGITUDE AND TIME 411
421. Washington, B.C., and New Orleans ?
422. Pittsburg and Omaha ?
423. New York and Paris ?
424. Boston and St. Petersburg?
425. Washington and Rome ?
426. Nashville and Vienna ?
427. Two persons, one in Paris and the other in Rome,
agreed to read a certain poem at the same time. If the time
selected by the one in Paris is 9 P.M., at what time by the
clocks in Rome must the other person begin reading the poem ?
A system of standard time has been adopted in the United States by
which the difference in time between places differs by whole hours or
not at all. The meridians 60, 75, 90, 105 and 120 west from Green-
wich are called time meridians. Places within 7 east or 7 west of the
meridian of 75 have tlte time of that meridian, which is called Eastern
Time. The time within 7| either side of the meridian of 90 is called
Central Time. The time within 7| either side of the meridian of 105
is Mountain Time. The time within 7 either side of the meridian of
120 is Pacific Time.
428. How many hours by standard time are there between
places which have Eastern Time and those which have Moun-
tain Time ? Pacific Time ? Central Time ?
429. If school begins at 9 o'clock, how long have the chil-
dren in Denver been in school when the morning session begins
on the Pacific coast ? How long have the children in Boston
been in school ? The children in Chicago ?
430. If the afternoon session begins at 1 P.M. and closes at
3 P.M., in what part of the country is the afternoon session
just beginning when the children in Washington, D.C., are
being dismissed ?
431. A train entered a city at 10 P.M. Central Time. After a
stop of 10 min. the train left the city at 9 hr. 10 min. Mountain
Time. In what direction was the train running ? What was
the longitude of the city ?
432. Is the present time of Boston slower or faster than the
old local time ?
412 MEASUREMENTS AND CONSTRUCTIONS
MISCELLANEOUS EXERCISES
. simplify m m m m m.
100 50
2. Write a complex fraction in which each number in the
numerator is prime and each number in the denominator is
composite. Simplify it.
3. Make a fraction whose numerator is the only prime
between 90 and 100, and whose denominator is the product
of all the primes between 80 and 90. Express that fraction as
per cent.
Which is greater, and how much :
4. 3V441 or V3721 ? 5. 5V676 or 13V121?
6. What number squared equals 77,284 ?
7. Express 2 mi. 20 rd. 3 yd. as yards. As rods. As
miles.
8. At 7^ a foot, what is the cost of 2 mi. of telephone
wire?
9. At 7 / a foot, how much will it cost to fence a square lot
containing 1521 sq. rd. ?
10. If a = 4, how much is cc 2 ? V#? 5V#? V9x?
7V25z? aV64? V21 + a? VoT+45?
11 . If x = 9, and y = 16, how much is V# + \A/ ?
Solve.
12. 8:ar = aj:2. 13. 9 : a; = a?: 16.
Write mean proportionals between the following numbers
14. 2 and 32. 19. 2J and 10.
15. 2 and 18. 20. 12J and 50.
16. 2 and 50. 21. 8 and 75.
17. 8 and 4J. 22. 6J and 100.
18. 3 and 27. 23. f and 54.
MISCELLANEOUS EXERCISES 413
24. Name all the demominations in the table of metric
linear measure. Of metric square measure. Of metric cubic
measure. Of metric measure of capacity. Of metric weight.
Give the meaning of the prefixes in the metric tables.
25. What is the weight of a cubic centimeter of water ? Of
a liter of water ?
26. A tank 3 meters long and 2 meters wide is filled with
water to the depth of 1 meters. How many kiloliters of
water are in it ?
27. How many meters are there in the perimeter of a right
triangle whose base is 27 centimeters and altitude 120 centim-
eters ?
28. There are 2400 square decimeters in the surface of a
cube. How many cubic centimeters does it contain ?
29. Find the cost of digging a cellar 2 dekameters long, 1
dekameters wide, and 6 meters deep at 10 ^ a cubic meter.
30. The measure of a meter was found by taking as nearly
as possible 1 * of the distance from the equator to the
pole. When a man has traveled one kilometer north from the
equator, how far is he from each pole ?
31. What are the dimensions of a cube that holds a milli-
liter? A kiloliter ?
32. What is the unit of land measure in the metric system ?
What are its dimensions ?
33. One side of a piece of land in the form of a right tri-
angle is 2.7 kilometers, the side perpendicular to it is 3.6 kilo-
meters. What is the value of the land at $ 50 a hectare ?
What is the cost of fencing the land at 20 $ a meter ?
34. Give the dimensions of the unit of wood measure in the
metric system.
35. At $1.50 per stere, what is the value of a pile of wood
8 m. long, 4 m. wide, and 2 m. high ?
36. A circle is inscribed in a square whose side is 10 in.
Find the area of the circle.
414 MEASUREMENTS AND CONSTRUCTIONS
37. Write two fractions, such that the quotient of the greater
divided by the less is f .
38. Add 3 to each term of a proper fraction and determine
whether the resulting fraction is greater or less than the original
fraction.
39. Add 3 to each term of an improper fraction and deter-
mine whether the resulting fraction is greater or less than the
original fraction.
40. Name three perfect squares whose sum is 14. 21. 26.
29. 30. 35. 38. 42.
41. Name two perfect cubes whose sum is 9. 35. 65. 133.
42. The Kohinoor diamond weighs 103 carats. The Star of
Brazil weighs 125 carats. The smaller equals what per cent of
the larger ? The larger equals what per cent of the smaller ?
43. Virginia tobacco contains 1% nicotine. How much nic-
otine in a ton of it ?
44. Some India rubber was bought for 12^ a pound where
it was grown, and was sold at 50^ a pound in this country.
What was the per cent of increase in price?
45. How many degrees in an arc which is 40% of a circum-
ference? 60% of a circumference ? 50% of a quadrant?
46. If an arc of 45 is 17 in. long, how long is the whole
circumference ?
47. Find the number of which 40 is f %.
48. 120 rd. are 8% of how many rods? 12%? 6J%?
49. If the decoration of a building required 3000 cu. yd., 25
cu. ft of stone, and that was 7% of the whole amount used in
constructing the building, how much was used ?
50. Draw four isosceles right triangles whose equal sides
are each 6 in. Arrange them so as to form a square. An
oblong. A right triangle. A rhomboid. A trapezoid. An
irregular pentagon. An irregular hexagon. Find the per-
imeter and area of each figure.
MISCELLANEOUS EXERCISES 415
51. If the rainfall on a certain day were -J of an inch, how
many gallons of water would fall on an acre of land ?
52. A boat can sail 15 mi. an hour down the river, and 10 mi.
an hour up the river. If it sails down the river for 2 hr. and
then returns, how long a time will elapse between its departure
and its arrival ?
53. $ 1500 equal 25% more than A's money, and 25% less
than B's. How much more has B than A?
54. A box 28 in. by 18 in. by 14 in. is filled with packages
of coffee, each 7 in. by 3J in. by 3 in. Find the value of all
at 10^ per package.
55. If 5 boys do a piece of work in 8 hr., how long will it
take a man who works twice as fast as a boy ?
56. A dealer bought a number of stoves for $ 240, paying
the same amount for each. If he had bought another dozen of
stoves at the same price the cost of both invoices of stoves
would have been $ 360. How much did each stove cost ?
57. Mr. Dean earns $1.25 per day and his brother $1.75
per day. .How many days more are required for Mr. Dean to
earn $ 350 than for his brother to earn it ?
58. A man bought a cow for $35, a horse for 3J times as
much as the cow, and a wagon for $ 1 more than ^ of the cost
of the cow and the horse. How much was paid for both horse
and wagon?
59. A man and his wife received $ 270 each year from money
which they had at interest. The man received $ 30 more than
3 times as much as his wife. How much did each receive ?
60. The sum of $24,000 was divided between A, B, and C
so that A received f as much as B, and C $ 4000 less than A
and B together. How much did each receive ?
61. A man has two fields, containing 10 A. and 12 A.
respectively. Find the length, in rods, of the side of a square
field equal in area to both fields.
416 MEASUREMENTS AND CONSTRUCTIONS
62. Construct, if possible, triangles whose sides have the
following lengths. Measure their angles with a protractor,
and tell whether each triangle is right, acute angled, or obtuse
angled.
(a) 2 in., 3 in., 4 in. (6) 2 in., 4 in., 5 in. (c) 3 in., 4 in., 5 in.
(d) 3 in., 4 in., 8 in. (e) 3 in., 4 in., 7 in. (/) 3 in., 4 in., 6 in.
63. Three lines being given, when is it impossible to con-
struct a triangle with them ?
64. Can a pyramid be constructed whose base is 8 in. square
and whose slant height is 3 in. ? Explain.
65. A toy table 10 in. high is an exact model of a study
table 30 in. high. If a leg of the large table is 191 in. long,
how long is a leg of the small table ? If the area of the top
of the small table is 120 sq. in., what is the area of the top of
the large table ? If a drawer of the small table contains 40
cu. in., how many cubic inches does the corresponding drawer
of the large table contain ?
66. A train running at the rate of 40 mi. an hour starts
from Newburg, to go to Ironton, a distance of 245 mi. At the
same time another train going 30 mi. an hour starts from Iron-
ton to go to Newburg. In how many hours will they meet ?
67. Atmospheric pressure is computed to be 15 Ib. to the
square inch. At that rate how many pounds of pressure are
upon the top of a round table 15 in. in diameter ?
68. Suppose Fig. 38, page 401, to represent the upper sur-
face of a cheese 16 in. in diameter and 6 in. high, from which
a part has been cut whose upper surface is represented by the
sector AOB. How many cubic inches are in the part that is
left?
69. A cylindrical tank If ft. in diameter and 6 ft. high is
half full of water. Assuming that a cubic foot of water weighs
1000 oz., how many pounds of wajfrer are in the tank ?
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