Dividing a fraction by a fraction

The cookie scenario below is an excellent example to visualize why
dividing a whole number by a fraction causes the answer to be
larger than the original whole number. But what about dividing a
fraction by a fraction? The scenario becomes incomprehensible when
the "5 cookies" become a half a cookie.

Do you have another example/scenario that can help students visualize
a problem such as:

1/3 / 1/2 = 2/3

The abstract concepts have been explained tremendously. Is there a
concrete way? If I have a third of a pie, and I want to divide that

third of a pie by 1/2, why does the answer become 2/3 of the pie??

Answer :

Lots of people find this confusing. If you divide 5 by 2, the
answer is 2.5. If you divide 5 by 1/2, do you expect the same thing
as dividing by 2?

If you divide by a number bigger than 1, it always reduces the number.
If you divide by 1, it doesn't change anything. Does that make you
think that dividing by a fraction less than 1 should INCREASE the
number?

How many kids can you serve with 5 cookies if each kid gets 2 cookies?
You can serve 2 kids (with enough left over for 1/2 a kid).

How many kids can you serve with 5 cookies if each kid gets 1/2 a
cookie? That's 5 divided by 1/2.

Maybe you could think about it this way. For the 5/2 = 2.5 you
could think of how many 2-cookie servings you can make out of
5 cookies. You get two full 2-cookie servings plus half of a
2-cookie serving. For the 5/half = 10 you could think of how
many half-cookie servings you could get out of 5 cookies.

For the (1/3)/(1/2) = 2/3 it's probably clearer to write it as
(2/6)/(3/6) = 2/3 and ask how times you could get a (3/6)-cookie
serving out of 2/6 of a cookie. You can't! You get **zero**
(3/6)-cookie servings. But you can get PART OF A (3/6)-cookie
serving. In fact you get exactly "two thirds of a (3/6)-cookie
serving. I'll leave it up to you to decide whether what I just
said is incomprehensible.

It may be clearer to keep it (1/3)/(1/2) = 2/3. Then say,
"how many cookie-halves can you get out of a third of
a cookie?" The answer would then be, "You can't get ANY cookie-
halves out of a third of a cookie, BUT you CAN get two thirds of
a cookie-half from a third of a cookie.

Source : mathforum