Dividing fractions

What if we wanted to divide $\frac{3}{5}$ by $\frac{1}{2}$ . How do we do that?

Let’s first, recall our definition of division.

To divide a number by another number is to calculate how many times the second number goes into the first number.

If we divide $\frac{3}{5}$ by 1, what do we get? We get $\frac{3}{5}$ . 1 goes into $\frac{3}{5}$ three fifth of a time.

If we divide it by $\frac{1}{2}$ , will we get a number smaller or bigger than $\frac{3}{5}$ ?

Have a think about it before looking at the answer below. The question is are there more halves or units in $\frac{3}{5}$ ?

There are more halves and so the number will be bigger because a number smaller than one will go into $\frac{3}{5}$ by more than $\frac{3}{5}$ .

It’s hard to see how many times $\frac{1}{2}$ goes into $\frac{3}{5}$ . We can see than $\frac{3}{5}$ is bigger than a half though so we know the answer is more than one.

It is easier if we use the common denominator 10. Now we can see that $\frac{1}{2}$ goes into $\frac{3}{5}$ once with $\frac{1}{10}$ left over

You can see here, that $\frac{1}{2}$ goes into $\frac{3}{5}$ once (the blue segments) with one fifth of a half left over (the red segment).

The red segment is $\frac{1}{10}$ and $\frac{1}{10}$ is a fifth of a half. You can see this by noticing that the non-blue segments are $\frac{1}{2}$ and there are five of them. The red segment is therefore a fifth of $\frac{1}{2}$ .

So, $\frac{3}{5}$ divided by $\frac{1}{2}$ is one and a fifth. This is the same as $\frac{6}{5}$ .

$\frac{3}{5} \div \frac{1}{2} = \frac{6}{5}$

This may seem complicated but, in practice, dividing fractions is easy.

Do you recall that we said that we can think of division as the opposite of multiplication? Well, when we divide by a fraction, we can swap the numerator with the denominator and multiply the fractions instead.

For example,

$\frac{3}{5} \div \frac{1}{2} = \frac{3}{5} \times \frac{2}{1} = \frac{3 \times 2}{5 \times 1} = \frac{6}{5}= 1\frac{1}{5}$

Dividing by a half is like multiplying by 2. Does this make sense to you?

Writing this in a general way

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c}$

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Numbers Part Two

Dividing fractions