2. Rational numbers

Lesson

When we divide whole numbers we split the whole number into equally sized parts and find the size of one part.

For example, consider $6\div3$6÷3.

Since there are $3$3 equal groups of $2$2 in $6$6 we can say that $6\div3=2$6÷3=2.

Now consider multiplication of fractions from the previous section. When we multiply by a fraction we are finding what that fraction of the quantity represents.

For example, consider $6\times\frac{1}{3}$6×13. Here we want to find how much is one third of six.

To do this, we split $6$6 into $3$3 equal groups and find the size of one group.

And we can see that $6\times\frac{1}{3}=2$6×13=2.

Notice that we can take the same approach with $6\div3$6÷3 as with $6\times\frac{1}{3}$6×13. This means that $6\div3=6\times\frac{1}{3}$6÷3=6×13.

This isn't a coincidence. Dividing by any number is the same as multiplying by the fraction one over that number.

We call the fraction made by taking $1$1 over a whole number a reciprocal.

If multiplying by a fraction is like dividing by a whole number, what is dividing by a fraction like?

Evaluate $\frac{8}{9}\div\frac{2}{9}$89÷29.

There are a few approaches we can take with this question. We could rephrase it as, "How many groups of $\frac{2}{9}$29 make up $\frac{8}{9}$89?" We could use the same approach as we used in multiplication, and divide the numerators and denominators separately. Or we could try to apply what we found with dividing whole numbers to fractions.

Let's start by asking, "How many groups of $\frac{2}{9}$29 make up $\frac{8}{9}$89. First we can draw a diagram of $\frac{8}{9}$89.

Then we can split it into groups of $\frac{2}{9}$29.

We can see that there are $4$4 equal groups of $\frac{2}{9}$29 in $\frac{8}{9}$89. So $\frac{8}{9}\div\frac{2}{9}=4$89÷29=4.

Let's try another way, by dividing the numerators and denominators separately.

In the previous section, we found that $\frac{8}{9}\times\frac{2}{9}=\frac{8\times2}{9\times9}$89×29=8×29×9. Following a similar process, we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8\div2}{9\div9}$89÷29=8÷29÷9.

We evaluate each of these divisions, which gives us $\frac{8\div2}{9\div9}=\frac{4}{1}$8÷29÷9=41. And since a denominator of $1$1 means that we have four wholes, we can say that $\frac{8}{9}\div\frac{2}{9}=4$89÷29=4.

Both of these methods give us the same result, so we can use either method to divide fractions. Let's try one more way.

The question we will ask is: "What fraction do we multiply by which gives the same result as dividing by $\frac{8}{9}$89"?

When we take the reciprocal of a whole number, we take the fraction of one over that whole number. However, we can also think of the whole number as a fraction over one. That is, $3=\frac{3}{1}$3=31. So to find the reciprocal we can swap the numerator and the denominator of a fraction.

We can do this with any fraction. So to find the reciprocal of $\frac{2}{9}$29 we swap the numerator, $2$2 and the denominator $9$9 to get $\frac{9}{2}$92.

Now we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8}{9}\times\frac{9}{2}$89÷29=89×92. To evaluate this we use the method for multiplying fractions. So $\frac{8}{9}\times\frac{9}{2}=\frac{8\times9}{9\times2}$89×92=8×99×2.

If we cancel the $9$9s we get $\frac{8}{2}$82, and since $8=4\times2$8=4×2 we can cancel the $2$2s to get $\frac{4}{1}$41 which is the same as $4$4.

All three methods give us the same result. It might seem like this third method is more complicated than the other two. However, in most situations it is actually the best one to use.

Evaluate $\frac{7}{10}\div\frac{4}{11}$710÷411.

**Think**: We have three methods we could use. Let's try all three and see which one is the easiest.

**Do**: We can draw $\frac{7}{10}$710 and then draw $\frac{4}{11}$411 on top of it.

We can see that $1\times\frac{4}{11}<\frac{7}{10}$1×411<710 and $2\times\frac{4}{11}>\frac{7}{10}$2×411>710 so we can't find the answer this way. It turns out that the number of groups of $\frac{4}{11}$411 in $\frac{7}{10}$710 is a fraction. This does mean something, but it will be easier to use a different method to answer this question.

We can also try dividing the numerators and denominators separately. This gives us $\frac{7}{10}\div\frac{4}{11}=\frac{7\div4}{10\div11}$710÷411=7÷410÷11. From here it isn't clear what to do, since $4$4 does not divide $7$7 and $11$11 does not divide $10$10.

So let's use the third method. The reciprocal of $\frac{4}{11}$411 is $\frac{11}{4}$114. Multiplying by this give us $\frac{7}{10}\div\frac{4}{11}=\frac{7}{10}\times\frac{11}{4}$710÷411=710×114.

Then we can multiply the numerators and denominators, $\frac{7\times11}{10\times4}$7×1110×4 and evaluate the multiplications, $\frac{77}{40}$7740. So $\frac{7}{10}\div\frac{4}{11}=\frac{77}{40}$710÷411=7740.

**Reflect**: In this question it was much easier to divide by multiplying by the reciprocal.

As we saw, the result of the division was a fraction. This does mean that there are $77$77 groups of one fortieth of $\frac{4}{11}$411 in $\frac{7}{10}$710. It's also correct to say that $\frac{7}{10}\div\frac{4}{11}=\frac{7\div4}{10\div11}$710÷411=7÷410÷11 although this isn't fully simplified.

To properly understand what happens when we divide by fractions we have to stop and ask what fractions and division really are.

The answer is that fractions are really another way to write division. As we saw earlier, $6\div3$6÷3 is the same as $\frac{1}{3}$13 of $6$6 or $\frac{6}{3}$63.

This applies to any kind of division. If we wanted to find $7\div4$7÷4 we can say that it is $\frac{1}{4}$14 of $7$7 or $\frac{7}{4}$74.

This also works in the other direction. A fraction like $\frac{11}{12}$1112 is the same as $11\div12$11÷12.

This explains how we can cancel common factors from the numerator and denominator, because canceling is really special kind of division. In fact, because any number divided by itself is $1$1, it also explains how we get equivalent fractions when we multiply the numerator and denominator by the same amount. That is, we are really multiplying the fraction by $1$1, which gives us the original fraction.

In the case of the last example, we can see the meaning in writing $\frac{7\div4}{10\div11}$7÷410÷11. That is, $\frac{7\div4}{10\div11}=\frac{7}{4}\div\frac{10}{11}$7÷410÷11=74÷1011. Although if we wanted to fully simplify this, we would still need to multiply by the reciprocal of $\frac{10}{11}$1011.

Finally, we can unite the two meanings of reciprocal. First we said that the reciprocal of a number is the fraction of $1$1 over that number. If we apply this to a fraction, for example, $\frac{7}{12}$712 we get $1\div\frac{7}{12}$1÷712. We can rearrange this to $\frac{1}{7}\div\frac{1}{12}$17÷112, and since $\frac{1}{12}$112 is the reciprocal of $12$12, we get $\frac{1}{7}\times12=\frac{12}{7}$17×12=127. So we can find the reciprocal of any fraction by swapping the numerator and denominator.

Did you know?

Notice that all fractions are the result of a division of one whole number by another whole number. There are some divisions which result in whole numbers, and others which do not, but in either case they can be written as a fraction.

We call the numbers which can be written as a division of whole number by another rational numbers. Rational numbers include both whole numbers and fractions. We'll explore rational numbers more in the coming chapters.

There are also numbers which cannot be written as a division of whole numbers. These are called irrational numbers. We'll explore these in year 8.

Summary

Fractions are a way of writing division of one whole number by another.

The reciprocal of a number is $1$1 divided by that number.

- The reciprocal of a whole number is a fraction of $1$1 over the whole number
- The reciprocal of a fraction can be found by swapping the numerator and denominator

To divide one fraction by another, multiply the first fraction by the reciprocal of the second.

Evaluate $\frac{1}{8}\div\frac{1}{5}$18÷15

Give your answer as a fraction.

Evaluate $\frac{2}{7}\div\frac{5}{3}$27÷53

- Give your answer as a fraction.

Evaluate $\frac{4}{5}\div\frac{36}{35}$45÷3635

- Give your answer as a fraction.