Multiplying Fractions

Of the four things you can do to a pair of fractions, multiplying is by some distance the easiest: multiply the tops, multiply the bottoms, done. No common denominator, no flipping, no rearranging.

Which raises the question most pages skip. Adding fractions demands that the bottoms match. Why does multiplying not?

The answer is the overlap

The square is one whole. The first fraction cuts it into columns and shades some of them; the second cuts it into rows and shades some of those. The strongly shaded rectangle is where the two shadings overlap, and that rectangle is the answer.

With 3/4 and 2/3 the square is cut into 4 columns and 3 rows, which makes 12 small cells. The overlap covers 3 columns by 2 rows, so 6 cells. Six of twelve:

3/4 * 2/3 = (3*2)/(4*3) = 6/12 = 1/2

The two numbers on top multiplied because the overlap is that many cells across by that many down. The two bottoms multiplied because the whole square got cut into that many cells altogether. Nothing else could happen.

And now the common-denominator question answers itself. The two cuts run in different directions, so they never have to agree. Adding needs matching pieces because you are counting pieces together; here you are not counting anything together, you are taking a part of a part.

It means the word "of"

Read the times sign as of and multiplication of fractions stops needing to be remembered:

• half of a half is a quarter — 1/2 * 1/2 = 1/4
• two thirds of three quarters is a half — 2/3 * 3/4 = 1/2
• a third of 60 is 20 — 1/3 * 60 = 20

This is also the reason a word problem like "three fifths of the class of 30" is a multiplication and not something more complicated.

One consequence catches people out. Multiplying by a fraction below 1 makes things smaller, because you are taking a part of them. 3/4 * 8 = 6, and 6 is less than 8. That is not a mistake; it is what "three quarters of" means.

Whole numbers and mixed numbers

A whole number is a fraction with 1 underneath, and once it is written that way the same rule covers it:

4/5 * 10 = 4/5 * 10/1 = 40/5 = 8

A mixed number has to become an improper fraction first, because there is no way to multiply the whole part and the fraction part separately and get the right answer. 2 + 1/2 is 5/2, so

5/2 * 2/3 = 10/6 = 5/3

If you are tempted to multiply the 2 and the 2 and then the halves and the thirds, try it with 2 1/2 times 2 1/2: the honest answer is 6 1/4, and the shortcut gives 4 1/4.

Cancel first, and the numbers stay small

Because the whole product is one fraction, any top can be cancelled against any bottom before you multiply. Take 3/4 * 8/9:

• the 3 and the 9 share a factor of 3, leaving 1 and 3
• the 4 and the 8 share a factor of 4, leaving 1 and 2

which leaves 1/1 * 2/3 = 2/3 — the right answer, with 72 never written down. Multiplying first gives 24/36, which reduces to the same 2/3 after a bit more work. Cancelling early is not a different method, just less arithmetic.

Try one

2/3*5/4

If you got 10/12, that is the right amount written in the wrong form — reduce it. If you reached for a common denominator, that step belongs to adding, and this page is the reason it can be skipped here.