146. We shall now give some examples. Before multiplying together the factors of the new numerator and the factors of the new denominator, it is advisable to examine if any factor occurs in both the numerator and denominator, as it may be struck out of both, and the result will thus be simplified; see Art. 137.

We might multiply together the first two factors, and

then multiply the product separately by b and by

add the results; but it is more convenient to

mixed quantity b+

ab

1 -a

and

reduce the Thus

147. As we have already done in former chapters, we must here give some results which the student must assume to be capable of explanation, and which he must use as rules in working examples which may be proposed. See Arts. 63 and 135.

XVIII. Division of Fractions.

148. Rule for dividing one fraction by another. Invert the divisor and proceed as in Multiplication.

149. The following is the usual demonstration of the Rule. Suppose we have to divide by; put

andy; therefore

a

= x,

151. Complex fractional expressions may be simplified by the aid of some or all of the rules respecting fractions which have now been given. The following are examples.

In this example the factors a-b and a+b are multiplied together, and the result a2-b2 is used instead of (a+b) (a−b); in general however the student will find it