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54. Reduce 71.500 to tenths.
55. Change 19, 43.6, 0.64, and 53 to thousandths.
56. Express 15.600, 4.7, and 13 as hundredths.

57. Reduce 18.0156, 401.6, and 176.4700 to decimals having the least common denominator.

A Decimal to a Common Fraction. 58. How many halves in .5? In .50 ? 59. How many fourths in .25 ? In .75 ? In .750 ? 60. How many fifths in 0.4? In 0.6 ? In 0.8 ? 61. How many twentieths in 0.05 ? In 0.15 ? In 0.45 ?

WRITTEN EXERCISES. 62. Change .355 to a common fraction in its smallest terms. .355 = 35.5 = 774 Solution. — As .355 is 355 thou

sandths, it may be written 86, which, changed to its smallest terms, is zoo.

63. Change .225 to a common fraction in its smallest terms. 64. Change .875 to a common fraction in its smallest terms. 143. To change a decimal to a common fraction :

Rule. Omit the decimal point, write the denominator, and change the fraction to its smallest terms. Reduce to common fractions in smallest terms: 65. .025 70. .375

75. 9.374
66. .561
71. .368

76. 115.875
67. .054
72. 11.75

77. .01375
68. .66%
73. 4.437

78. 200.96
69. .625
74. .09375

79. .015625

A Common Fraction to a Decimal. 80. How many tenths in 1 ? In ? In } ? In 81. How many hundredths in 1 ? In 1? In å? 82. How many thousandths in 1 ? In 1? In 1 ?

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WRITTEN EXERCISES. 83. Change to a decimal. 8) 5.000

Solution. - is equal to 1 of 5. 5 equals 50 .625 tenths, or 5.0 ; } of 50 tenths is 6 tenths, with 2 tenths, equal to 20 hundredths, remaining. f of 20 hundredths is 2 hundredths, with 4 hundredths, equal to 40 thousandths, remaininy. of 40 thousandths is 5 thousandths. Ans. .625.

84. Change to a decimal.
85. Change ito a decimal of four places.
11) 8.0000

Solution. — li is equal to 1 of 8. As .7272,8 there are to be four places in the decimal,

we annex four decimal places of ciphers to the numerator, reducing it to 8.0000. 11 of 8.0000 is .727211.

86. Change to a complex decimal of three places. 87. Change it to a complex decimal of four places. 144. To reduce a common fraction to a decimal:

Rule. Annex decimal ciphers to the numerator, divide by the denominator, and point off as many decimal orders in the quotient as there are ciphers annexed.

Note 1. — When the division does not terminate, or has been carried as far as is desirable, the remainder may be expressed as a common fraction and made a part of the result.

NOTE 2. — When an approximate result is sufficient, a fraction of or more than } in a result may be rejected, and the last figure of the decimal be made to express 1 more. Thus,

.7272 & approximately expressed to the nearest ten-thousandth is .7273. NOTE 3. — When the fraction as a part of a decimal is unimportant, it may be omitted, and the incompleteness of the result simply marked by +.

Thus,

.7272+ may be written instead of .72721.
Reduce to decimals :
88. 7.
91. 18.

94. 484
89.
92. 35

95. 2;s:
90. 35.
93. ift.

96. 5138

97. Reduce 1' to a complex decimal of four places. 98. Reduce 1 to the nearest ten-thousandth. 99. Reduce 18 to an approximate decimal of four places. 100. Reduce 75 to a complex decimal of four places. 101. Reduce to the nearest millionth.

102. Change to the nearest thousandth, }, and to, and find the sum of the results.

103. Reduce to the nearest ten-thousandth, and add, 156, 203, 124, and 2.68.

104. Reduce 34 to the nearest millionth, and subtract the result from 15.057.

145. For rules for Addition and Subtraction of Decimals, see Arts. 38 and 47.

MULTIPLICATION.

ORAL EXERCISES. 105. How much is 3 times 2 tenths ? 3 times 0.3 ?

106. How much is 7 times 1 hundredth ? 7 times 187 ? 9 times 0.06 ?

107. How much is to of 1? Of 3?. It of y? of .3 ? 108. i X to? i X too ? 3 X.3? 4 X.3? .04 .3?

109. How many places of decimals in the product when tenths are multiplied by tenths ? When hundredths are multiplied by tenths ?

WRITTEN EXERCISES. 110. Multiply 31.5 by .07. 31.5 Solution. — As .07 is the same as to of 7, 31.5

multiplied by .07 is the same as ado of 7 times 31.5. 2.205

7 times 31.5 = 220.5, and to of 220.5, which is

found by removing the decimal point two places to the left (Art. 75), is 2.205.

111. What is the product of 671 by .305 ? 112. What is the product of 18.72 by 7.1 ?

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146. Rule for Multiplication of Decimals. Multiply as integers, and point off as many figures for decimals in the product as there are decimal places in both factors.

NOTE. – If there are not figures enough in the product, supply the deficiency by prefixing ciphers: 113.

114.
Multiply
126

3.18
By
.016

.00029
756

2862 126

636 Product 002016

.0009222 115. Multiply 5.64 by 45. 120. Multiply .563 by 47. 116. Multiply 96.5 by 100. 121. Multiply 19634 by .0073. 117. Multiply 6.34 by .0023. 122. Multiply .0703 by .0055. 118. Multiply 42.2 by 2.004. 123. Multiply.0505 by .001. 119. Multiply 1671 by.013. 124. Multiply.0076 by .017. 125. What is the product of one million by one millionth ? 126. What is the cost of 35.75 yards of cloth, $ 4.50 a yard ?

127. How much must be paid for 13.375 cords of wood, at $ 4.62 a cord ?

128. Wbat is the product of one hundred one thousandths by ten thousand one hundred one hundred-thousandths ?

DIVISION.

ORAL EXERCISES. 129. How many times 2. tenths in 8 tenths ? 3 tenths in 9 tenths ?

130. How much is t of 8 tenths ? of 9 tenths ?

131. How much is f of 15 hundredths ? 1 of 24? } of :45 ?

132. Divide .8 by .2; .63 by 7; .63 by .07.

133. The product of two factors is . 72, and one of the factors is 9. What is the other factor ?

134. The product of two factors is .72, and one of the factors is .9. What is the other factor ?

135. In .72 divided by 9, how many places of decimals are there in the quotient ? In .72 divided by .9 ?

WRITTEN EXERCISES. 136. Divide 46.48 by .4. 44.) 4644.8 Solution. — Both dividend and divisor may be 116.2 multiplied by 10 without changing the quotient

(Art. 72). This is done by moving the point one place to the right (Art. 75). Dividing as in integers, and placing the quotient point under the new dividend point, we have the quotient 116.2. 137. Divide .7935 by .23.

3.45 (23.) 479.35

Solution. — To make the divisor an integer,

we multiply both divisor and dividend by 100, 103 by moving the point in each two places to the 92 right. Dividing as in integers, and placing the 115

quotient point over the new dividend point, 115

we have 3.45 as the quotient required.

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138. Divide 1.264 by 4. 139. Divide .00115 by .05.

147. Rule for Division of Decimals. If the divisor is an integer, divide as in integers, and point off as many decimal places in the quotient as there are such places in the dividend.

If the divisor is a decimal, make it an integer by moving the decimal point a sufficient number of places to the right; move the decimal point in the dividend as many places to the right, annexing ciphers, if necessary, and then divide.

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