Multiplying Fractions Intuition

When multiplying two fractions we get the answer by multiplying the denominators to get the denominator in the answer, and multiplying the numerators to get the numerators in the answer.

Why does this work though? One way to think about it is to imagine that we instead had done this:

This equation makes intuitive sense. We take three fourths seven times and we end up with twenty-one fourths.

In our example though, we didn’t take three fourths seven times, we took three fourths half of seven times (because we multiplied seven halves by three fourths), and therefore it makes sense that the answer should be half of what we got when we took three fourths seven times.

and in the final step we get:

To see why the last step is true, think of twenty-one fourths as taking the number 21 and dividing it in to 4 parts. To get a number that is half as small as one of those parts (which is what we do when we divide by 2), we have to divide 21 in twice as many parts as we divided it in originally, in other words we have to divide it by 8 instead of 4.