Why when you divide fraction you use multiplying?

Answer 1

We could simply answer, 'Because that's the rule that we are taught in school.' But I think that perhaps you would like a more complete explanation.

There are two reasons why you do this:

(1) it is easier, and

(2) it works.

First, let's make the method explicit. Let's say that you want to divide a/b by c/d. The result can be written as a fraction within a fraction, as follows:

(a/b) ÷ (c/d) = (a/b) / (c/d).

Now we want to simplify this larger resulting fraction by eliminating its denominator, (c/d). To do this, we need to observe three principles:

(1) We define the reciprocal of a given number as a second number which, when multiplied by the given number, gives 1. Thus, for example,

½ is the reciprocal of 2, because 2 × ½ = 1; and

1/7 is the reciprocal of 7, because 7 × 1/7 = 1.

Note, then, that the reciprocal of the fraction, a/b is b/a, because a/b × b/a = ab/ab = 1.

(2) Altering a fraction by multiplying (or dividing) its numerator and its denominator by the same number does not alter the value of the fraction. Example: a/b × n/n = an/bn = a/b. Note, when we divide them by the same number, we call it reducing the fraction.

(3) Any number divided by 1 is the number itself.

Now, returning to the original problem, dividing a/b by c/d, we apply the three principles above:

(a/b) ÷ (c/d) = (a/b) / (c/d)

= [(a/b) / (c/d)] × [(d/c) / (d/c)]

= [(a/b) × (d/c)] / [(c/d) × (d/c)]

= [(a/b) × (d/c)] / 1

= [(a/b) × (d/c)] = ad/bc.

But we don't have to go through these steps every time we encounter the problem. It suffices if we merely say, on each occasion,

(a/b) ÷ (c/d) = (a/b) × (d/c) = ad/bc.

And, so, we have the rule:

To divide by a fraction, one merely multiplies by its reciprocal.

There are certainly other methods for dividing fractions; but none is so simple and convenient as the one just shown.