# 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

• How many half cookies can you make out of a half cookie? – Clement C. Sep 4 '16 at 22:52
• I like to think of $1/2$ ÷ $1/4$ as asking "how many times does $1/4$ go into $1/2$? – littleO Sep 4 '16 at 23:34

$a \div b$ 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.

Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. I think this is the most intuitive approach when trying to teach this concept.

For multiplication we first split the cookie up into denominator-number of pieces, and then take numerator-number of pieces of those pieces.

• Except I would image most student would (and if they don't they should) respond with "why the hell should that be true?" Which i think is would the point of the question was. – fleablood Sep 4 '16 at 23:22
• Well that's intuition about what division and multiplication itself are, for multiplication we perform addition that many times, for division we want to know how many of that object fit into our number. – ZirconCode Sep 4 '16 at 23:28
• So why should multiplying by the reciprical answer that question? It can be explained, but right now it makes as much as adding the numerator and averaging the denominators or any other busy arithmetic for no conernable reason. – fleablood Sep 4 '16 at 23:32

1/2 is half of a cookie. 1/2 divided by 1/2 is simply seeing how many times half a cookie fits, or corresponds to half a cookie, which is one time. Or how many halves of a cookie you need to get half of a cookie, which is one (One half of a cookie) again.

Dividing a whole number (lets say one) into two pieces can be restated as "If one (lets say gallon) fills two containers, then how much does one container hold?". Of course the answer is $\frac{1}{2}$ gal.

Using that same language consider "If $\frac{5}{16}$ Gal. fills $\frac{3}{7}$ of a container, then how much does one container hold?". If we scale both numbers by a factor of seven, we see that $3$ containers will hold $\frac{5\times 7}{16}$ gal.. Finally, we scale by a factor of $\frac{1}{3}$, giving $3\times\frac{1}{3} = 1$ container contains $\frac{5\times 7}{16\times 3}$ gal..