Given two fraction, say, \(\frac{1}{3},\) and \(\frac{4}{5},\) we can multiply them together by multiplying the two numerators (the numbers upstairs) and the two denominators (the numbers downstairs), separately.

So for example:

\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}.\]

Or in general:

\[ \frac{P}{M} \times \frac{ Q}{N} = \frac{ P \times Q }{ M \times N}.\]

When we multiply fractions we are really taking fractions of fractions. Imagine that we have divided a pie into five equal parts. One slice falls on the floor and so we have four slices left; i.e., we have \(\frac{4}{5}\) of the original pie. Now even though we had cut out pie into five pieces, because we had expected five people to show up for our party, only three people actually show up.

We want to divide what is left of the pie up evenly among our three guests, and so each guest gets

\[ \frac{1}{3} \times \frac{4}{5} = \frac{ ( 1 \times 4 ) }{( 3 \times 5)} = \frac{ 4}{15}\]

of a pie.

In practice, this means that we would take each of the pieces we originally cut, and cut them all into three parts. Then we would give each guest four of these smaller pieces.

Because we had originally cut the pie into five pieces, and then cut each one of these pieces (except the one that fell on the floor) into three pieces, each little piece is equal to one fifteenth of a pie. Then we give each guest for each of these smaller pieces.

\[ \frac{ 4}{15}.\]