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$6Γ·β3.

Since there are $3$3 equal groups of $2$2 in $6$6 we can say that $6\div3=2$6Γ·β3=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}$6Γ13β. 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$6Γ13β=2.

Notice that we can take the same approach with $6\div3$6Γ·β3 as with $6\times\frac{1}{3}$6Γ13β. This means that $6\div3=6\times\frac{1}{3}$6Γ·β3=6Γ13β.

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.

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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}$89βΓ·β29β.

There are a few approaches we can take with this question. We could rephrase it as, "How many groups of $\frac{2}{9}$29β make up $\frac{8}{9}$89β?" 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}$29β make up $\frac{8}{9}$89β. First we can draw a diagram of $\frac{8}{9}$89β.

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

We can see that there are $4$4 equal groups of $\frac{2}{9}$29β in $\frac{8}{9}$89β. So $\frac{8}{9}\div\frac{2}{9}=4$89βΓ·β29β=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}$89βΓ29β=8Γ29Γ9β. Following a similar process, we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8\div2}{9\div9}$89βΓ·β29β=8Γ·β29Γ·β9β.

We evaluate each of these divisions, which gives us $\frac{8\div2}{9\div9}=\frac{4}{1}$8Γ·β29Γ·β9β=41β. 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$89βΓ·β29β=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}$89β"?

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=31β. 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}$29β we swap the numerator, $2$2 and the denominator $9$9 to get $\frac{9}{2}$92β.

Now we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8}{9}\times\frac{9}{2}$89βΓ·β29β=89βΓ92β. To evaluate this we use the method for multiplying fractions. So $\frac{8}{9}\times\frac{9}{2}=\frac{8\times9}{9\times2}$89βΓ92β=8Γ99Γ2β.

If we cancel the $9$9s we get $\frac{8}{2}$82β, and since $8=4\times2$8=4Γ2 we can cancel the $2$2s to get $\frac{4}{1}$41β 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}$710βΓ·β411β.

**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}$710β and then draw $\frac{4}{11}$411β on top of it.

We can see that $1\times\frac{4}{11}<\frac{7}{10}$1Γ411β<710β and $2\times\frac{4}{11}>\frac{7}{10}$2Γ411β>710β so we can't find the answer this way. It turns out that the number of groups of $\frac{4}{11}$411β in $\frac{7}{10}$710β 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}$710βΓ·β411β=7Γ·β410Γ·β11β. 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}$411β is $\frac{11}{4}$114β. Multiplying by this give us $\frac{7}{10}\div\frac{4}{11}=\frac{7}{10}\times\frac{11}{4}$710βΓ·β411β=710βΓ114β.

Then we can multiply the numerators and denominators, $\frac{7\times11}{10\times4}$7Γ1110Γ4β and evaluate the multiplications, $\frac{77}{40}$7740β. So $\frac{7}{10}\div\frac{4}{11}=\frac{77}{40}$710βΓ·β411β=7740β.

**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}$411β in $\frac{7}{10}$710β. It's also correct to say that $\frac{7}{10}\div\frac{4}{11}=\frac{7\div4}{10\div11}$710βΓ·β411β=7Γ·β410Γ·β11β although this isn't fully simplified.

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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$6Γ·β3 is the same as $\frac{1}{3}$13β of $6$6 or $\frac{6}{3}$63β.

This applies to any kind of division. If we wanted to find $7\div4$7Γ·β4 we can say that it is $\frac{1}{4}$14β of $7$7 or $\frac{7}{4}$74β.

This also works in the other direction. A fraction like $\frac{11}{12}$1112β is the same as $11\div12$11Γ·β12.

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}$7Γ·β410Γ·β11β. That is, $\frac{7\div4}{10\div11}=\frac{7}{4}\div\frac{10}{11}$7Γ·β410Γ·β11β=74βΓ·β1011β. Although if we wanted to fully simplify this, we would still need to multiply by the reciprocal of $\frac{10}{11}$1011β.

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}$712β we get $1\div\frac{7}{12}$1Γ·β712β. We can rearrange this to $\frac{1}{7}\div\frac{1}{12}$17βΓ·β112β, and since $\frac{1}{12}$112β is the reciprocal of $12$12, we get $\frac{1}{7}\times12=\frac{12}{7}$17βΓ12=127β. So we can find the reciprocal of any fraction by swapping the numerator and denominator.

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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}$18βΓ·β15β

Give your answer as a fraction.

Evaluate $\frac{2}{7}\div\frac{5}{3}$27βΓ·β53β

- Give your answer as a fraction.

Evaluate $\frac{4}{5}\div\frac{36}{35}$45βΓ·β3635β

- Give your answer as a fraction.