Ex. 12. It is required to find the sum of 2a, 155. Reduce the fractions to a common denominator, if necessary, and then subtract the numerators from each other, and under the difference write the common denominator, and it will give the difference of the fractions required. Or, enclose the fractional quantity to be subtracted in a parenthesis; then, prefixing the negative sign, and performing the operation, observing the same remarks and rules as in addition, the result will be the difference required. The reason of this is evident; because, adding a negative quantity is equivalent to subtracting a positive one (Art. 63): thus, prefixing the negative it becomes sign to the fractional quantity a-b C (Art. 128); to the fractional quantity it " 5 3a+b y -5x+ ; and to the mixed quantity -3a+ y 5c X3b=15bc common denominator. Whence, + 12bx-6ab___5cx-5cy+12bx-6ab . required. is the difference Or, by prefixing the negative sign to the quantity dition, and the result will be the same as above. Ex. 3. From 2ab+ subtract 2ab α-x Here prefixing the negative sign to the quantity a + x a -30 tions is equivalent to the sum of 2ab+ and a+x a+x' - -2ab+ a 156. Multiply their numerators together for a new numerator, and their denominators together for a new denominator; reduce the resulting fraction to its lowest terms, and it will be the product of the fractions required. It has been already observed, (Art. 119), that when a fraction is to be multiplied by a whole quantity, the numerator is multiplied by that quantity, and the denominator is retained: ac α Thus, c=, and 10x b bor, which is the same, making an improper fraction of the integral quantity, and then proceeding according to the α C ac 2x 5 10x rule, we have, and X b b 1 b Hence, if a fraction be multiplied by its denomi α nator, the product is the numerator; thus, x6 = |