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2.01 Multiplying and dividing fractions and mixed numbers

Lesson

Multiplying fractions

Multiplying is like making groups of something, isn't it? Well, yes, it is. With whole numbers, it's quite simple to see what we are doing, and what our numbers represent. With fractions, it can be a little hard to visualize things. We've seen how we can multiply fractions by whole numbers, so now we can build on that to multiply fractions by fractions.

Let's see how we can use a rule to multiply fractions together.

 

Multiplying fractions by fractions

When we multiply two whole numbers together, we know that we are going to end up with more than we started with. When we multiple proper fractions together, we are going to end up with less than we started with. Let's see what it really means to multiply fractions together, in this video.

Remember!

When we multiply something, anything, by a proper fraction, we end up with less than we started with.

A fraction of a fraction

We've seen how we can multiply fractions by whole numbers, so now it's time to multiplication fractions together. In this video, we're going to use a nifty array system to work out our answer, and then see the shortcut we can use going forward. 

Your turn

In this applet, you can change the numerator and denominator for each fraction, and then see what happens when you multiply them. You can click the box to show the answer and check if you were correct.

Practice question

QUESTION 1

Evaluate $\frac{1}{2}\times\frac{11}{12}$12×1112.

 

The rule

Now you've seen how we multiply fractions, you can use the rule for multiplying fractions now. 

  Fraction 1 x Fraction 2  

 

 

  $\frac{a}{x}$ax x $\frac{b}{y}$by

=

$\frac{a\times b}{x\times y}$a×bx×y

Let's see how we can solve some of these, and how we can simplify our fractions.


Practice question

QUESTION 2

Evaluate $\frac{1}{5}\times\frac{4}{12}$15×412.

Did you know?

If you need to multiply more than two fractions together, you can! You follow the same process, but you multiply as many numerators and denominators as you have.

 

Different types of fractions

Now we can use the rule for multiplying fractions any denominator, and look at how to work with mixed numbers and improper fractions in more detail.

Remember!

A whole number can be expressed as a fraction, we just use $1$1 as its denominator. Multiplying fractions can be done in different ways, so you may like to change mixed numbers to improper fractions, before multiplying, but you don't have to. 

 

Practice question

QUESTION 3

Evaluate $2\times8\frac{1}{9}$2×819, writing your answer in its simplest form.

 

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.

Remember!

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.

Remember!

The total to be shared is the numerator, whereas the denominator represents how many parts we are sharing it into, or among.

 

Dividing with fractions

Now we've worked up to the rule for multiplying fractions by fractions, it's time to turn our attention to division with fractions. We'll start with whole numbers and fractions, and build our knowledge until we're confident dividing fractions by fractions.

Sharing parts

Well, that's what we're doing, really. Division is like sharing, or splitting into groups, and fractions are parts of a whole. So what does it really mean? Let's watch these videos to see what dividing whole numbers by fractions means. 

In the first video, we divide by unit fractions, to get a sense of what we are doing. If you are confident with this, maybe you can skip to the second video, where we are dividing by fractions that aren't unit fractions.

What does it mean?

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 visualize this, using number lines and fraction bars.

Practice question

Question 4

This number line shows that each whole is divided up into thirds.

0123

  1. Write the fraction that shows how big each division is:

  2. How many divisions are in $2$2 wholes?

  3. What is the result of $2\div\frac{1}{3}$2÷​13?

  4. How many third size pieces are there in $5$5 wholes?

Remember!

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.

Not a unit fraction

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!

Practice question

Question 5

The number line below shows $2$2 wholes split into $\frac{1}{5}$15 size pieces.

       

0123

  1. If $2$2 is divided into pieces that are $\frac{1}{5}$15 of a whole each, how many pieces are there in total?

  2. How many $\frac{2}{5}$25 size pieces are in $2$2 wholes?

The rule

Now you're ready to use the rule for dividing by fractions:

$a$a ÷ $\frac{b}{c}$bc = $a$a × $\frac{c}{b}$cb

 

 

 

Starting with fractions

With multiplication, it doesn't matter which order we multiply in, but that's not the case with division. We've seen how to divide whole numbers by fractions, now we can divide fractions by whole numbers. We have less than a whole, and we want to share it evenly. Let's use unit fractions again as our starting point, to see how we can do this.

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.

Remember!

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.

Practice question

Question 6

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.

     
  1. Which image shows that each third has been divided into $3$3 parts?

    A

    B

    C

    A

    B

    C
  2. What is the size of the piece created when $\frac{1}{3}$13 is divided by $3$3?

 

 

The next step

Now that we have seen how to divide a fraction by a whole number, it's just one more step to be able to 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.

The rule

When you need to divide a fraction by a whole number, you can multiply the fraction by the reciprocal of the whole number.

Practice question

Question 7

Evaluate $\frac{3}{5}\div4$35÷​4 giving your answer as a simplified fraction.

Are you ready

Are you thinking what I'm thinking? It's 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 summarizes 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.

The rule

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

Practice question

QUESTION 8

Evaluate $\frac{2}{5}\div\frac{1}{7}$25÷​17

Write your answer in the simplest form possible.

Mixed Numbers

When we multiply fractions by the rule, we are able to multiply our numerators together, and multiply our denominators together. When we have mixed numbers, we can express them as improper fractions first, and then we are able to multiply them by the rule. We don't actually have to, but it means we have less work to do. Who doesn't like that idea?

What about division?

Sure! We can also express mixed numbers as improper fractions, to help with division. Once we've done that, we can use the rule for dividing fractions. Remember that rule? We can multiply by the reciprocal of a fraction, when we need to divide by a fraction.

Seeing it in action

In this video, we multiply and divide mixed numbers, by expressing them as improper fractions.

Practice question

Question 9

Evaluate $2\frac{2}{5}\times1\frac{3}{5}$225×135, giving your answer as a mixed number in simplest form.

 

Both together!

This time, we have a multiplication and a division in the same problem, but you'll see it's possible to do it all by changing our mixed numbers (also called mixed numbers) to improper fractions.

Question 10

Evaluate $4\frac{5}{9}\times3\frac{4}{9}\div\frac{3}{5}$459×349÷​35, writing your answer in its simplest form.

 

Outcomes

6.NS.A.1

Interpret and compute quotients of fractions to solve mathematical problems and problems in real-world context involving division of fractions by fractions using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. In general, a/b ÷ c/d = ad/bc.

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