Ans. ***21 Ans. x + ile 6.552 and 44 Ane 19x – 14 · 6. 0+0.2 and 82+253 Ans. 4x + 10.17 Ans. 4x + 5.24 + ax + a'. 2 ax Ans. 24 +49 9. 47, ", and 2+ Ans. 2 + 4x +4 10. 3x + 24 and 2 – 84. Ans. 82 + 23: ac - and 1-ci Ans. 1+ ac – 6bd + Sac. 60 8ad 1 18. 4(1+) 162–a). and 2(1 –a) Ano. 17e SUBTRACTION OF FRACTIONS. 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. 1. What is the difference between and ? 2. Find the difference of the fractions 2,79 and 20 – 40. Here and Hence 3. Required the difference of 14€ and so Ans. 4. Required the difference of Ans. 5. Required the difference of 34 and Ans. 1832 6. From Ans. x + y subtracta X – Y 7. From ,, subtract vizat y — 2 Find the differences of the following: Ans. 242 +84.2.2062 – 35 3. MULTIPLICATION OF FRACTIONS. 93. Let and represent any two fractions. It has been shown ($ 81) that any quantity may be multiplied by a frac*tion by first multiplying by the numerator, and then dividing the result by the denominator. To multiply by we first multiply by c, giving ac; then we divide this result by d, which is done by multiplying the denominator by d. This gives for the product, ac; that is, G Hence the rule: 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. Exercises. 1. Multiply a + box by First, hence a? + bx x? - a'c + bcx. axãe ad Find the products of the following quantities : We have, by the rule, 2a va? – 62 - 2a (a? – 62) - 2a (a + b)(a - b) – 22 (a + b). a-6* 3^ 3(a - b) = 3(a - b) ^3 NOTE. — 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 factors. Ans. 22. DIVISION OF FRACTIONS. 94. Since =px, it follows that dividing by a quantity is equivalent to multiplying by its reciprocal. But the reciprocal of a fraction, , is a ($ 28): consequently, to divide 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. |