RULE. Multiply the integral part by the denominator of the frac tion; to the product add the numerator if the sign of the fraction is plus, and subtract it if the sign is minus, and under the result write the denominator. NOTE.-By By a change of the language, Examples 12-14 in Art. 87, and 11-13 in Art. 88, become examples under this case. Thus, Example 12, Art. 87, might be expressed as follows: Reduce to an improper fraction. mx + 7 x 18 a 2. Reduce x2+4—— to an improper fraction. 7 y 6. Reduce a2+b2 — (a+b) to an improper fraction. b NOTE.It must be remembered that the sign before the dividing line belongs to the fraction as a whole. NOTE. According to the same principle an integral quantity can be reduced to a fraction having any given denominator, by multiplying the quantity by the proposed denominator, and under the product writing the denominator. a2 12. Reduce 4 ax to a fraction whose denominator is 2. CASE VI. 90. To reduce an improper fraction to an integral or As the value of a fraction is the quotient arising from dividing the numerator by the denominator (Art. 82), we perform the indicated division. Hence, RULE. Divide the numerator by the denominator; if there is any remainder, place it over the divisor, and annex the fraction so formed with its proper sign to the quotient. 2. Reduce 3. Reduce quantity. ax - 4 bx to an integral or mixed quantity. Ans. a 4 b. 4 ab−3 a b x+y to an integral or mixed a b 6. Reduce tity. 7. Reduce quantity. 8. Reduce quantity. 9. Reduce 10. Reduce to an integral or mixed 7 Ans. 4y-2 a + 6 y x2+ax+a3 to an integral or mixed quan 91. To multiply a fraction by an integral quantity. Divide the denominator by the integral quantity when it can be done without remainder; otherwise, multiply the numerator by the integral quantity. NOTE. Any factor common to the denominator and multiplier may be cancelled from both before multiplying. NOTE. When a fraction is multiplied by a quantity equal to its denominator, the product is the numerator. x y =x+y, Ans. x2+2xy + y2 9. Multiply by (x-a)2. (x — a)2 a+b 10. Multiply by 2 - 2 x y + y2. х 92. To multiply an integral quantity by a fraction. (x2 + 2 x y + y2) × 4 = 4 (x2 + 2 x y + y2) 4(x2 + 2 x y + y2) ÷ (x + y) = 4 (x + y) We first multiply the multiplicand by the numerator 4; but the multiplier is 4(x + y); and therefore this product is x + y times too great, and this product divided by x + y must be the product sought. It is evident that the result would be the same if the division were performed first, and the multiplication afterward. Hence, RULE. Divide the integral quantity by the denominator when it can be done without remainder, and multiply the quotient by the numerator. Otherwise, multiply the integral quantity by the numerator, and divide the product by the denom inator. - 2. Multiply a3-3 a2 b + 3 a b2 — b3 by a 2 a b + b"" 2ab+b3° Ans. 7x (a - b). |