What Are The Rules For Multiplying Fractions?

Multiplication is one of the simplest operations you can carry out on fractions, because you don't need to worry about whether the fractions have the same denominator or not; simply multiply the numerators together, multiply the denominators together and simplify the resulting fraction if need be. However, there are a few things to watch out for, including mixed numbers and negative signs.

The first, and most important, rule of multiplying fractions is that you only multiply numerator × numerator and denominator × denominator. If you have the two fractions 2/3 and 4/5, multiplying them together would create the new fraction:

\(\frac{2 × 4}{3 × 5}\)

Which simplifies to:

\(\frac{8}{15}\)

At this point you would simplify if you could but, since 8 and 15 don't share any common factors, this fraction cannot be simplified any further.

For more examples including the multiplication of fractions that need to be reduced, watch the video below:

If you multiply fractions with negative terms in them, make sure you carry those negative signs through your calculations. For example, if you're given the two fractions -3/4 and 9/6, you'd multiply them together to create the new fraction:

\(\frac{-3 × 9}{4 × 6}\)

Which works out to:

\(\frac{-27}{24}\)

Because −27 and 24 both share 3 as a common factor, you can factor 3 out of both numerator and denominator, leaving you with:

\(\frac{-9}{8}\)

Note that −9/8 represents a very different value from 9/8. If that negative sign had gotten lost along the way, your answer would have been wrong.

Take another look at the example just given. The second fraction, 9/6, is an improper fraction. Or in other words, its numerator was larger than its denominator. That doesn't change the way your multiplication works at all, although depending on your teacher or the strictures of the problem you're working, you might prefer to simplify the result of the last example, which is an improper fraction itself, into a mixed number:

\(\frac{-9}{8} = -1 \, \frac{1}{8}\)

This leads perfectly into a discussion of how to multiply mixed numbers: Convert the mixed number into an improper fraction and multiply as usual, just as described in the last example. For example, if you're given the fraction 4/11 and the mixed number 5 2/3 to multiply, you'd first multiply the whole number, 5, by 3/3 (that's the number 1 in the form of a fraction that has the same denominator as the fraction part of the mixed number) to convert it to a fraction:

\(5 × \frac{3}{3} = \frac{15}{3}\)

Then add in the fraction part of the mixed number, giving you:

\(5 \,\frac{2}{3} = \frac{15}{3} + \frac{2}{3} = \frac{17}{3}\)

Now you're ready to multiply the two fractions together:

\(\frac{17}{3} × \frac{4}{11}\)

Multiplying numerator and denominator gives you:

\(\frac{17 × 4}{3 × 11}\)

Which simplifies to:

\(\frac{68}{33}\)

You can't simplify the terms of this fraction any more, but if you wanted to, you could convert it back to a mixed number:

\(2 \, \frac{2}{33}\)

Here's a handy trick: If you know how to multiply by fractions, you already know how to divide by fractions, too. Just flip the second fraction upside down and multiply that instead of doing any dividing. So if you have:

\(\frac{3}{4} ÷ \frac{2}{3}\)

It's the same thing as writing:

\(\frac{3}{4} × \frac{3}{2}\)

which you can then multiply as usual.