why do we multiply fractions to find a fraction of a fraction

Question

 why do we multiply fractions to find a fraction of a fraction

Answer 1

Example: ½ of ½ = ¼.  

 

Consider ½ ÷ ½; the answer is obviously 1, because anything divided by itself is 1 (ruling out division by 0, of course).

 

If you consider, now, ½(x) ÷ [½(x)], i.e. half of anything divided by itself, the answer is still 1.

But suppose x were equal to 2; then this problem would be simplified to 1 divided by 1. The effect of the x=2 on both numerator and denominator is to cancel the denominators of the two original fractions!

 

Now consider (a/b) ÷ (c/d).  Multiply each part (numerator & denominator) by d/c, the RECIPROCAL of the denominator.

This gives us (a/b)(d/c) ÷ 1. This constitutes a PROOF that the division by c/d is equivalent to MULTIPLICATION by its reciprocal.

 

We memorize this rule, as a shortcut, from here on!