92. Fractions can be subtracted only when they have the same unit; that is, a common denominator. In that case, the numerator of the minuend, minus that of the subtrahend, will indicate the number of times that the common unit is to be taken in the difference. Hence the rule: Reduce the two fractions to a common denominator. Then subtract the numerator of the subtrahend from that of the minuend, for a new numerator, and write the remainder over the common denominator. Exercises. 3 1. What is the difference between 2 and 2? 7 24 14 10 5 56 56 56 28 2. Find the difference of the fractions (x-a)x3c=3cx-3 ac (2 a − 4 x) × 2b=4ab8b}, the numerators; 2bx3c6bc, the common denominator. 3 cx-3 ac 4 ab - 8 bx 3 cx-3ac-4 ab + 8 bx Here and Hence 6 bc 6 bc MULTIPLICATION OF FRACTIONS. 93. Let and represent any two fractions. It has been % d shown (§ 81) that any quantity may be multiplied by a fraction by first multiplying by the numerator, and then dividing the result by the denominator. с To multiply by we first multiply by c, giving; then we divide this result by d, which is done by multiplying the ac denominator by d. This gives for the product, that is, bd' If there are mixed quantities, reduce them to a fractional form; then Multiply the numerators together, for a new numerator; and the denominators, for a new denominator. NOTE. a2 — b2 _ 2 a (a2 — b2) _ 2 a (a+b)(a−b) _ 2a (a + b). 3 3 (a - b) _ = 3 (a - b) 3 After indicating the operation, we factored both numerator and denominator, and then canceled the common factors, before performing the multiplication. This should be done whenever there are common DIVISION OF FRACTIONS. px 1 =p×7, it follows that dividing by a quantity 10 is equivalent to multiplying by its reciprocal. But the reciprocal of a fraction, is (§ 28): consequently, to divide ď d с any quantity by a fraction, we invert the terms of the divisor, and multiply by the resulting fraction. Hence Whence the following rule for dividing one fraction by another: Reduce mixed quantities to fractional forms. Invert the terms of the divisor, and multiply the dividend by the resulting fraction. NOTE. The same remarks as were made on factoring and reducing, under the head of "Multiplication," are applicable in division. |