Dividing Fractions II
Are fractions and division related?
Fractions or division?
Have you ever wondered just how division and fractions are related? Well, the may look quite different, but it turns out they both relate to sharing and parts of a whole. In fact, if you think about it, you can express any fraction as a division, and any division as a fraction! Here's one example, but it works for any digits.
$3$3 ÷ $4$4 = $\frac{3}{4}$34
Let's see how we can do this in this video.
There's more than one way to think of division. A problem such as $11$11÷$3$3 could describe $11$11 of something, shared into $3$3 equal groups, or into groups of $3$3. The number after the division sign is always the denominator of our fraction, and the number before, the numerator.
Written problems
What if our problem is not expressed as division, but is a written problem? Well, by thinking of division, we can then express the problem as a fraction. Let's see how we think of sharing available internet time, or dividing up hay to farmers, using fractions.
The total to be shared is the numerator, whereas the denominator represents how many parts we are sharing it into, or among.
Picturing division with fractions
A good way to understand what a question is asking us, is to imagine using pictures. We can use things like area models and number lines to picture things like division. When we want to divide a whole number by a unit fraction (a fraction with $1$1 as the numerator), it helps to imagine how many of those parts each whole ($1$1) contains.
Let's take a look at how we can visualise this, using number lines and fraction bars.
Need more?
If you're still a little unsure, watch this video using a clock. By thinking about how many $\frac{1}{4}$14 hour blocks there are in $1$1 hour, we see that multiplying the denominator of the unit fraction by the whole number works!
If we see a problem like $5$5 ÷ $\frac{1}{6}$16, it helps to think of how many sixths there are in $1$1 whole. Then, we can think about how many there are in 5 wholes.
Worked Example
Question 1
The number line below shows $4$4 wholes split into $\frac{1}{3}$13 size pieces.
If $4$4 is divided into pieces that are $\frac{1}{3}$13 of a whole each, how many pieces are there in total?
How many pieces would there be if we had $5$5 wholes?
How many pieces would there be if we split up $10$10 wholes?
Dividing unit fractions by whole numbers
We've seen how to divide whole numbers by unit fractions, but what if we work in the other direction? This time we'll start with a unit fraction, and then think about sharing it out. Our whole number tells us how many we need to share it among.
Picturing it
We can use number lines, fraction bars, area models, and even time to help us share unit fractions out. Let's take a look at how we might approach this. If you only want to watch one method, you might like to jump to these times, for your preferred approach:
- using time 0:00
- using number lines 1:48
- using fraction bars 2:25
The rule for dividing a unit fraction by a whole number is important, once you understand what the number problem means, so be sure to check that out, at 4:26 in the video.
Dividing means sharing into pieces, so the pieces get smaller. When our piece is a fraction to start with, it gets even smaller, so the denominator gets bigger. That's a good way to check your answer is reasonable, by making sure the denominator is larger.
Worked Example
Question 2
Let's use the image below to help us find the value of $\frac{1}{3}\div3$13÷3. This image shows the $1$1 whole split into $3$3 divisions of size $\frac{1}{3}$13.
Which image shows that each third has been divided into $3$3 parts?
ABCWhat is the size of the piece created when $\frac{1}{3}$13 is divided by $3$3?
Other fractions
Once you have mastered dividing whole numbers by unit fractions, it's just one more step to divide whole numbers by other fractions. If you can work out $8$8 ÷ $\frac{1}{5}$15, you only need to know one more step, and you'll be able to work out $8$8 ÷ $\frac{2}{5}$25!
Let's see how this works, and find out the simple rule that means we don't need to use number lines or fraction bars to help us. Oh, you'll also get to see why it's great to know how to find the reciprocal of a fraction!
Worked Example
Question 3
Evaluate $8\div\frac{3}{8}$8÷38. Write your answer as a mixed number.
Now you can use the rule for dividing by fractions:
$a$a ÷ $\frac{b}{c}$bc = $a$a × $\frac{c}{b}$cb
Dividing fractions by whole numbers
We've seen how to divide a whole number by any fraction, so now we can do the reverse, and divide any fraction by a whole number. We're still sharing our fractions, but we are starting with a larger part of something. It's handy to remember the reciprocal of a number, as we'll be using that to solve our problems.
Let's see how we can divide our fraction by a whole number, in this video.
When you need to divide a fraction by a whole number, you can multiply the fraction by the reciprocal of the whole number.
Worked Example
question 4
Evaluate $\frac{4}{7}\div2$47÷2 giving your answer as a simplified fraction.
Just fractions now
We've seen how to divide each of these things:
- whole numbers by unit fractions
- unit fractions by whole numbers
- whole numbers by any fractions, and
- any fractions by whole numbers
There's more?
Yes! It's now time to look at dividing fractions by fractions. No need to press the panic button though, we've done the hard yards already.
Dividing fractions by fractions
Dividing is like sharing, or seeing how many of something we have in our total. It can be tricky to really imagine what that means with fractions, so this video shows you how to divide fractions by fractions, and what it looks like.
Take-home message
Now that we have worked through why we do what we do, we're ready to now use the rule for dividing fractions. This short video summarises it, and means you can use this now when dividing fractions. If you have a mixed number, such as $1$1 $\frac{2}{5}$25 you can change it to an improper fraction, $\frac{7}{5}$75.
We can now use this rule when dividing fractions:
$\frac{a}{b}$ab ÷ $\frac{c}{d}$cd = $\frac{a}{b}$ab × $\frac{d}{c}$dc
Worked Example
QUESTION 5
Evaluate $\frac{2}{5}\div\frac{1}{7}$25÷17
Write your answer in the simplest form possible.