Multiplying Fractions
To multiply two
fractions
you don't need a common denominator at all. Just multiply the numerators
together and multiply the denominators together:
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
For example, \frac{2}{3} \times \frac{4}{5} = \frac{2 \times 4}{3 \times 5} = \frac{8}{15}.
Top times top, bottom times bottom — that's the whole rule.
The little word "of" is the key to why this works: "of" means
multiply. So \frac{1}{2} of
\frac{1}{3} means \frac{1}{2} \times \frac{1}{3} = \frac{1}{6} —
half of a third is a sixth.
Simplify at the end
After multiplying, the answer may not be in lowest terms, so
simplify
it. For instance \frac{2}{3} \times \frac{3}{4} = \frac{6}{12} = \frac{1}{2}.
You can also cancel common factors first to keep the numbers small — the
3s cancel, leaving \frac{2}{4} = \frac{1}{2}
directly.
Notice the answer to \frac{1}{2} \times \frac{1}{3} is
smaller than either fraction you started with. Multiplying two
proper fractions (each less than one) always gives a smaller result —
you are taking a part of a part.
- Multiply numerator × numerator for the new top, and
denominator × denominator for the new bottom.
- The word "of" means ×:
\frac{1}{2} of \frac{1}{3} is
\frac{1}{6}.
- Multiplying two proper fractions gives a smaller answer.
- Simplify the result (or cancel common factors before multiplying).
See it as area
A unit square is a perfect picture of \frac{1}{2} \times \frac{1}{3}.
Slice it into thirds one way and into halves the other way, and the overlap of "one third"
and "one half" is a single small box — and the grid has six boxes in all, so that overlap
is \frac{1}{6}. Step through it.