2.02 Dividing fractions
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$6รทโ3.
Since there are $3$3 equal groups of $2$2 in $6$6 we can say that $6\div3=2$6รทโ3=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}$6ร13โ. 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$6ร13โ=2.
Notice that we can take the same approach with $6\div3$6รทโ3 as with $6\times\frac{1}{3}$6ร13โ. This means that $6\div3=6\times\frac{1}{3}$6รทโ3=6ร13โ.
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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Dividing by fractions
If multiplying by a fraction is like dividing by a whole number, what is dividing by a fraction like?
Exploration
Evaluate $\frac{8}{9}\div\frac{2}{9}$89โรทโ29โ.
There are a few approaches we can take with this question. We could rephrase it as, "How many groups of $\frac{2}{9}$29โ make up $\frac{8}{9}$89โ?" 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}$29โ make up $\frac{8}{9}$89โ. First we can draw a diagram of $\frac{8}{9}$89โ.
Then we can split it into groups of $\frac{2}{9}$29โ.
We can see that there are $4$4 equal groups of $\frac{2}{9}$29โ in $\frac{8}{9}$89โ. So $\frac{8}{9}\div\frac{2}{9}=4$89โรทโ29โ=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}$89โร29โ=8ร29ร9โ. Following a similar process, we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8\div2}{9\div9}$89โรทโ29โ=8รทโ29รทโ9โ.
We evaluate each of these divisions, which gives us $\frac{8\div2}{9\div9}=\frac{4}{1}$8รทโ29รทโ9โ=41โ. 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$89โรทโ29โ=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}$89โ"?
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=31โ. 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}$29โ we swap the numerator, $2$2 and the denominator $9$9 to get $\frac{9}{2}$92โ.
Now we can say that $\frac{8}{9}\div\frac{2}{9}=\frac{8}{9}\times\frac{9}{2}$89โรทโ29โ=89โร92โ. To evaluate this we use the method for multiplying fractions. So $\frac{8}{9}\times\frac{9}{2}=\frac{8\times9}{9\times2}$89โร92โ=8ร99ร2โ.
If we cancel the $9$9s we get $\frac{8}{2}$82โ, and since $8=4\times2$8=4ร2 we can cancel the $2$2s to get $\frac{4}{1}$41โ 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.
Worked Example
Evaluate $\frac{7}{10}\div\frac{4}{11}$710โรทโ411โ.
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}$710โ and then draw $\frac{4}{11}$411โ on top of it.
We can see that $1\times\frac{4}{11}<\frac{7}{10}$1ร411โ<710โ and $2\times\frac{4}{11}>\frac{7}{10}$2ร411โ>710โ so we can't find the answer this way. It turns out that the number of groups of $\frac{4}{11}$411โ in $\frac{7}{10}$710โ 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}$710โรทโ411โ=7รทโ410รทโ11โ. 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}$411โ is $\frac{11}{4}$114โ. Multiplying by this give us $\frac{7}{10}\div\frac{4}{11}=\frac{7}{10}\times\frac{11}{4}$710โรทโ411โ=710โร114โ.
Then we can multiply the numerators and denominators, $\frac{7\times11}{10\times4}$7ร1110ร4โ and evaluate the multiplications, $\frac{77}{40}$7740โ. So $\frac{7}{10}\div\frac{4}{11}=\frac{77}{40}$710โรทโ411โ=7740โ.
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}$411โ in $\frac{7}{10}$710โ. It's also correct to say that $\frac{7}{10}\div\frac{4}{11}=\frac{7\div4}{10\div11}$710โรทโ411โ=7รทโ410รทโ11โ although this isn't fully simplified.
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What are fractions and what is division?
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$6รทโ3 is the same as $\frac{1}{3}$13โ of $6$6 or $\frac{6}{3}$63โ.
This applies to any kind of division. If we wanted to find $7\div4$7รทโ4 we can say that it is $\frac{1}{4}$14โ of $7$7 or $\frac{7}{4}$74โ.
This also works in the other direction. A fraction like $\frac{11}{12}$1112โ is the same as $11\div12$11รทโ12.
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}$7รทโ410รทโ11โ. That is, $\frac{7\div4}{10\div11}=\frac{7}{4}\div\frac{10}{11}$7รทโ410รทโ11โ=74โรทโ1011โ. Although if we wanted to fully simplify this, we would still need to multiply by the reciprocal of $\frac{10}{11}$1011โ.
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}$712โ we get $1\div\frac{7}{12}$1รทโ712โ. We can rearrange this to $\frac{1}{7}\div\frac{1}{12}$17โรทโ112โ, and since $\frac{1}{12}$112โ is the reciprocal of $12$12, we get $\frac{1}{7}\times12=\frac{12}{7}$17โร12=127โ. So we can find the reciprocal of any fraction by swapping the numerator and denominator.
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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.
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.
Practice questions
Question 1
Evaluate $\frac{1}{8}\div\frac{1}{5}$18โรทโ15โ
Give your answer as a fraction.
Question 2
Evaluate $\frac{2}{7}\div\frac{5}{3}$27โรทโ53โ
- Give your answer as a fraction.
Question 3
Evaluate $\frac{4}{5}\div\frac{36}{35}$45โรทโ3635โ
- Give your answer as a fraction.