8. Multiply and divide fractions

Lesson

Do you remember how to multiply a fraction by a whole number?

What is the value of $4\times\frac{3}{4}$4×34?

To multiply two fractions together, we'll start by thinking of the fractions as multiples of unit fractions, and work towards a more efficient strategy.

Let's take an example of $\frac{2}{3}\times\frac{4}{5}$23×45. We can rewrite these fractions as

$\frac{2}{3}=2\times\frac{1}{3}$23=2×13 and $\frac{4}{5}=4\times\frac{1}{5}$45=4×15

We can then multiply the whole parts together:

$\frac{2}{3}\times\frac{4}{5}$23×45 | $=$= | $2\times\frac{1}{3}\times4\times\frac{1}{5}$2×13×4×15 |

$=$= | $8\times\frac{1}{3}\times\frac{1}{5}$8×13×15 |

What can we do with the product of the unit fractions $\frac{1}{3}$13 and $\frac{1}{5}$15? Well, this is like taking one whole, dividing it into $3$3 pieces to get thirds, and then dividing *each of those* thirds into $5$5 pieces. The result is that the whole has been divided into $15$15 pieces.

We can now finish our multiplication:

$\frac{2}{3}\times\frac{4}{5}$23×45 | $=$= | $8\times\frac{1}{3}\times\frac{1}{5}$8×13×15 |

$=$= | $8\times\frac{1}{15}$8×115 | |

$=$= | $\frac{8}{15}$815 |

Do you notice the pattern that has happened here?

In a fraction, the denominator tells us the size of the pieces, and the numerator tells us how many pieces there are. When we multiply two fractions, the denominators *multiply together* to tell us the new size of the pieces, and the numerators *also multiply together* to tell us how many of the new pieces there are.

That is:

$\frac{2}{3}\times\frac{4}{5}$23×45 | $=$= | $\frac{2\times4}{3\times5}$2×43×5 |

$=$= | $\frac{8}{15}$815 |

Find the value of $\frac{1}{3}\cdot\frac{7}{10}$13·710.

Remember!

To multiply two fractions together, we:

- multiply the numerators to form the new numerator, and
- multiply the denominators to form the new denominator