Q:

how to logically describe dividing by fraction

dividing by a whole number i can describe by simply saying split this "cookie" into two pieces, then you now have half a cookie. 
does anyone have an easy way to describe dividing by a fraction? 1/2 divided by 1/2 is 1

A:

$a \div b $ means "how many $b $s does it take to get $a $"
So "$2 \frac 12 \div \frac 12$" is "how many $\frac 12$s does it take to get $2\frac 12$?"  The answer is $5$.
So how many half cookies does it take to make half a cookie? The answer is one.

