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45. Add together, 9, 126, 1o, and 21 . 46. What is the sum of 10 +å tø++ ? 47. What is the sum of 1927 of a +215 + 48. What is the sum of 1 of 2: +37 +67+i ? 49. Find the sum of of a shilling and of a penny?

In this example, first reduce the of a shilling to pence, and the fraction of a penny.

50. Find the sum of fe of a gallon and / of a gill. 51. What is the sum of 5 days and 52 minutes ? 52. What is the sum of 4 of a cwt., 83 lb. and 3 jo oz.?

SUBTRACTION OF FRACTIONS.

As in addition of fractions we find the sum of their numerators, so in subtraction of fractions we find the difference of their numerators.

RULE. If either quantity be a compound fraction, reduce it to a simple fraction, and if the two fractions have different denominators, reduce them to a common denominator, Subtract the numerator of the subtrahend from the numerator of the minuend, and place the remainder over the common denominator.

When the minuend is a mixed number, and the fraction in the subtrahend is greater than that in the ininuend, subtract the numerator of the subtrahend from the denominator, and to the difference add the numerator of the minuend; and consider the integer of the minuend to be 1 less than it stands.

It is not always obvious, which of two fractions expresses the greater quantity. In such case, the fractions are denoted with a character between them, thus, žā un it; and the greater is discovered by reducing them to a common denominator. 53. What is the difference between 24 and 263? 72

Here the fraction in the sub

trahend is the greater, and we 27 247 56

are obliged to convert a unit into seventy-seconds to obtain a quar:

tity from which to subtract 54. What is the difference between it and 15?

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55. Perform subtraction on an H. 56. What will remain if 5113 be taken from 343 ? 57. Subtract of from 36%. 58. What is the difference between 4 15 and 101? 59. What will remain if sof be taken from a unit? 60. What is the difference between 41 and ? 61. 4ş-- 1 of of } is equal to what quantity?

MULTIPLICATION OF FRACTIONS.

CASE JI.

The following rules for multiplication of fractions, are based on the Propositions 1, and 11, stated in page 37.

CASE I. To multiply a fraction by a whole number.

RULE. Either multiply the numerator, or divide the denominator by the whole number.

To multiply a whole number by a fraction. RULE. Multiply the whole number. by the numerator, and divide the product by the denominator.

CASE III. To multiply a fraction by a fraction.

RULE. Multiply numerator by numerator, and denominator by denominator, for a new fraction.

When both factors are mixed numbers, it is generally more convenient to reduce them to improper fractions and then proceed according to the rule under Case 111.

The effect of multiplying any quantity by a proper fraction is, to give in the product, such a part of the quantity multiplied as the fraction indicates. Thus the product must be less than the multiplicand. This effect of the operation will appear consistent with the principle of multiplication, when it is considered, that multiplying any number by 1, gives only the same number in the prodict; and, therefore, multiplying by less than 1, must give a product less than the rumber inultiplied.

62. Multiply by 9. 3X9= 2589=2=828 63 Multiply 49 by š. (See rule under Case 11.) 64. Multiply by ž. (See rule under Case 111.) 65. Multiply 610 by 3). (Remark under Rule in.) 66. What is the product of 2 by 15?

67. What is the product of 9241 by ?
68. What is the product of by } ?
69. What is the product of 8l by 12.1?
70. Which is the most, 7 x 65, or, 65 X ?
71. What is the product of 294 1 by 25 ?

In this example, it will be most convenient to find the product of the whole numbers without regard to the fraction first; then find the product of the fraction in a separate operation, and, finally, add the two products together.

72. What is the product of 361 by 34?

73. How many square inches of paper in a sheet that is 14] inches long, and 113 inches wide ?

DIVISION OF FRACTIONS.

CASE I.

CASE II.

RULE.

The rules for division of fractions, like those for multiplication, are based on Propositions i, and 11.

To divide a fraction by a whole number. • RULE. Either divide the numerator, or multiply the denominator, by the whole number:

To divide a whole number by a fraction. RULE. Multiply the whole number by the denominator, and divide the product by the numerator. CASE III. To divide a fraction by a fraction.

Invert the divisor, and then proceed as in multiplying a fraction by a fraction.

Observe, that the operation of this last rule is, to mul. tiply the denominator of the dividend by the numerator of the divisor for a new denominator, and the numerator of the dividend by the denominator of the divisor for a new numerator.

Compound fractions are to be reduced to simple ones, and mixed numbers to improper fractions, before the adoption of either of the above rules.

74. Divide ja by 8. +8=Taxa=. Ans 75. Divide 14 by 15. (See rule under Case 11.) 76. Divide by. (See rule under Case 11.) 77 Divide the compound fraction of x by 6.

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78. Divide 325 by the mixed number 54.
79. What is the quotient of divided by 13?
80. What is the quotient of 57 divided by ??
81. What is the quotient of 1a divided by i ?
82. Divide of % by z of of
83. What is the quotient of 91 divided by 15 ?
84. What is the quotient of 206 divided by 917
85. How many times is 44 contained in 319 ?
86. How many times is 194 contained in 997?
87. How many times of an inch in % of a yard?
First, reduce the 1% of a yd. to the fraction of an inch
88. How many times a of a gill in 3 barrels ?

89. Suppose a wheel to be 11 feet in circumference how many times will it roll round in going 39 rods?

MISCELLANEOUS EXAMPLES.

In the following examples, all fractions which appear in the answers, must be reduced to their value in whole numbers of lower denominations, whenever there is opportunity for such reduction.

90. What distance will a car run in 9 hours, allowing its velocity to be 23 miles an hour ?

91. Suppose a car wheel to be 8 feet 7 inches in circumference, how many times will it turn round in running 46 miles ?

92. If 3 cwt. of sugar be taken from a hogshead containing 14 cwt. Iqr. 6 lb., how much will remain in the hogshead?

93. What is the sum of 16ş cwt., 7cwt. 3qr. 81lb., 2T. 194cwt., 2 cwt. 1}qr., and of a ton ?

94. A farmer owning 132 acres of land, sold 46 A. 3R. 12r. How much land had he remaining ?

95. What is the value of 36 acres of land, at $ 475

96. What is the value of 15į barrels of flour, at $4.62, per barrel !

97. What is the value of a load of wond, containing 6 feet, (ó of a cord,] at $5.25 per cord ? Or, what is of $ 5.25? Or, $5.25 X = ?

1

per acre?

3

98. How much land is there in a square lot, measuring 354rods on every side ? (See page 28.).

99. What quantity of land in a lot, which is 65 rods long and 47} rods wide ?

100. What quantity of wood is there in a pile, 1412 feet long, 312 feet wide, and 6 in feet high?

101. Suppose a lot of land to be 64 rods wide, how long must it be, to contain 1 acre ? (See PROB. V, page 21. Consider that 1 acre contains 160 rods.)

102. What quantity of loaf sugar must be sold at 19 cents per pound, that the price shall amount to $ 524?

103. What cubical quantity of earth must be removed, in digging a pit, 132 feet deep, 12 feet long, and I feet wide

? 104. What quantity of hewn timber is there in a stick that is 12 feet long, 2 feet deep, and foot wide ?

105. Suppose a stick of timber to be 112 foot deep, and 8 inches wide; what must be the length of the stick, in order that its quantity shall be 1 ton of hewn timber? (See PROB. VIII, page 22. Consider a ton as the pro duct of three factors.) 106. Suppose wood to be piled on a base 18 feet long

feet wide, what must be the height of the pile, to contain 91 cords?

107. What quantity of molasses in 4 casks, containing severally, 554 çal., 31gal., 27 }gal., and 58 17 gal.?

108. "What is the cost of 436 bushels of corn, at 62 cents per bushel ?

109. Suppose 6 gallons to have leaked from a logshead of wine, what is the value of the remainder of the wine, at 87į cents per gallon?

110. How many bottles, each holding 13 pint, are required for bottling 3 barrels of cider ?

111. Suppose 4 gallons of cider to have evaporated from a barrel; what number of bottles, each holding 1 pt. 3}gi., will be required to bottle the remainder ?

112. What is the value of 142 tons of coal, at 7 dollars

per 113. What is the value of of a bushel of wheat, at the rate of ã of a dollar per bushel? [ăx =?]

and 7

ton?

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