IBKA OF THE UNIVERSITY CALIFORNIA MULTIPLICATION. 25 92. To multiply an integer by a fraction, or to find a fractional part of an integer. 93. PRINCIPLE.-Multiplying by a fraction is taking such part of the multiplicand as the fraction is of a unit. Ex. If 1 ton of hay cost $18, what will of a ton cost? ANALYSIS.-If 1 ton cost $18, of a ton will cost of $18. of $18 is 3 times of $18. of $18 is $41 (taking † is the same as dividing by 4), and 3 times $4 is $131. Or, of $18 is 4 of 3 times $18. 3 times $18 is $54. of $54 is $131. Ex. Find the product of 175 and 84. 94. RULE.-Multiply by the numerator of the fraction, and divide the product by the denominator. Or, Divide by the denominator of the fraction and multiply the quotient by the numerator. When the multiplier is a mixed number, multiply by the fraction and integer separately, and add the results. EXAMPLES. 95. 1. Find the cost of 8 yds. of ribbon at 25 cts. a yard. 2. What is the cost of 423 pounds of butter at 26 cts. a pound. 3. Required the value of 483 yards of flannel at 75 cts. a yard. 96. To multiply a fraction by a fraction.* Ex. At $3 a pound, what will of a pound of tea cost? Ex. What is the value of 8 × 8 × ×? 97. RULE.-Reduce integers and mixed numbers to improper fractions. Cancel all factors common to the numerators and denominators. *The practical methods of multiplying one mixed number by another are given under Art. 108. Multiply the remaining numerators together for the numerator, and the remaining denominators for the denominator. Reduce the following compound fractions (55) to simple ones. 99. To divide a fraction by an integer. 100. PRINCIPLE.-Dividing the numerator or multiplying the denominator by a number divides the value of the fraction by that number (57, 2). Ex. What cost 1 pound of tea, if 5 pounds cost $31? the first operation, decreases the number of parts, their size remaining the same; multiplying the denominator divides the fraction by decreasing the size of the parts, their number remaining the same. denominator of the fraction by the integer. When the dividend is a mixed number, divide the integer and the fraction separately, and add the results. 103. To divide by a fraction. 104. The Reciprocal of a number is 1 divided by that number. Thus, the reciprocal of 4 is 1 divided by 4, or 1. The Reciprocal of a Fraction is 1 divided by that fraction. 105. PRINCIPLE. 1 divided by a fraction is the fraction inverted. Thus, 1 divided by is. This principle may be demonstrated as follows: In 1 there are 4 fourths. 1 fourth is contained in 4 fourths 4 times. Since is 3 times, is contained in 1 as many times as . Hence, is contained in 1 of 4 times, or times. The reciprocal of a fraction is the fraction inverted. Ex. At $a yard, how many yards of cloth can be bought for $5? Or, $ is contained in $1 times (Prin.), and in $5, 5 times or 20, equal to 6 times. Ex. At $a yard, how many yards of cloth can be bought for $5 ? times (Prin.), and in $§, § times or {g, equal to 13 times. Ex. If 6 yards of cloth cost $5, what will 1 yard cost? OPERATIONS. 5 ÷ (5÷20) × 3 = † Or, 5= { × ab = £f = } Or, 5 1 × = 1 = v 4 ANALYSIS. 6 yards are equal to 20 yards. Since 20 yards cost $5, of a yard will cost of $5 or $1, and or 1 yard will cost 3 times $1 or $4. Or, the price per yard equals the cost, divided by the quantity as an abstract number. 5 divided by 2o equals 5 times 1 divided by 20, or 5 times (Prin.), equal to . |