Ex. 2. Reduce 5a 3x 7 to an improper fraction. Here 5a Xy=5ay; to this add the numerator with its proper sign, viz. -3x; and we shall have 5ay-3x. Here, x2 ×x=x3; adding the numerator a2-y2 with its proper sign: It is to be recollected that the affixed to the fraction x means that the sign whole of that fraction is to be subtracted, and consequently that the sign of each term of the numerator must be changed, when it is combined with x3, X3 x3-a2 -a2 + y2 hence the improper fraction required is 30 -a2 + y2 y3. ; (Art. 67), x the proposed mixed quantity x2. 3x2-a+7 Ex. 4. Reduce 5a2 + to an improper 2ax fraction. Here, 5a2 2ax=10a3x; adding the numerator 3x2-a+7 to this, and we have 10a3 x+3x2-a+7. 2 Here, 4x2 X2ac Sacr2, in adding the numerator with its proper sign; the sign - prefixed to the 3ab+c fraction signifies that it is to be taken nega 2ac tively, or that the whole of that fraction is to be subtracted; and consequently that the sign of each term of the numerator must be changed when it is 8acx2-3ab- -C combined with 8acx2; hence, the fraction required. Or, as 3ab -C is 2ac (Art. 67); hence the reason of chang ing the signs of the numerator is evident. a2x2 2 Ex. 6. Reduce r to an improper frac 2x2-a2 tion. Ans. To reduce an improper fraction to a whole or mixed quantity. RULE. 149. Observe which terms of the numerator are divisible by the denominator without a remainder, the quotient will give the integral part; and put the remaining terms of the numerator, if any, over the denominator for the fractional part; then the two joined together with the proper sign between them. will give the mixed quantity required. |