UK Secondary (7-11)
Multiplying, Dividing Fractions
Lesson

## Multiplication

Suppose it takes $\frac{1}{5}$15 a cup of water to cook a bowl of rice. How much water is needed to cook $\frac{1}{2}$12 of a bowl of rice? To find out, you would have to multiply the two fractions: $\frac{1}{5}\times\frac{1}{2}=\frac{1}{10}$15×12=110.

Multiplying fractions is something that you may have to do when calculating areas, volumes, distances, etc. When you have to multiply numbers that are not whole numbers, you will likely have to multiply fractions.

Before we begin looking at the numbers, let's watch a quick video that shows what multiplying fractions is all about.

The video will explain the area model using the question $\frac{2}{5}\times\frac{3}{4}$25×34

To picture what it means to multiply fractions check out the following mathlet. Choose a red and a blue fraction, first try and work out what the answer will be and then slide the RED box to the left so that it overlaps the BLUE.

So the quick way to see the maths is to multiply both denominators together, this gives us the size of the new fraction that is formed.

Then multiply both numerators together, this gives us the size of the array that we are actually interested in.

Here are some fractions I just multiplied,

See how I just multiplied the numerators and the denominators together.

In the next 2 examples I multiplied numerators and denominators, but then also had to simplify the result.

Example 1

Example 2

This one had an improper fraction in the multiplication, and I turned it into a mixed fraction in the answer.

Example 3

So you can see from these examples that the process is pretty simple.

To MULTIPLY FRACTIONS

multiply both the numerators
multiply both the denominators
check how the answer needs to be displayed

##### Question 1

Evaluate $\frac{11}{9}\times\frac{2}{99}$119×299.

##### Question 2

Evaluate $\left(\frac{4}{9}\right)^2$(49)2, writing your answer in its simplest form.

## Division

Whole number division defines the process of splitting an amount or value into groups.  This means that questions like $12\div4$12÷​4 is similar to asking how many groups of $4$4 can fit in $12$12, or how many in each group if we divide $12$12 into $4$4 groups.  Mostly we shortcut this to saying things like "how many $4$4's in $12$12?".

### Visualising Fraction Division

Fraction division can be thought of the same way, except the answers may not be as obvious to us as whole number division.

A question like $\frac{3}{4}\div\frac{1}{2}$34÷​12 is still asking "how many $\frac{1}{2}$12's are in $\frac{3}{4}$34's?" but this is not as easy to imagine.

Let's look at a rectangle and divide it into quarters,

see what happens when we add the halves,

Now looking at the question, we wanted to know $\frac{3}{4}$34 and $\frac{1}{2}$12

What this is telling us is that there are $1\frac{1}{2}$112 halves in three quarters.

We can picture this one by seeing that $1\frac{1}{2}$112 of the RED size piece would cover the BLUE size piece.

This is quite the process in terms of visualisations, and imagine how complicated this could get if we were to do $13\frac{5}{11}\div\frac{7}{9}$13511÷​79

Fortunately there is a mathematical shortcut we can use to get an answer.  (That's not to say that the understanding of what is happening in fraction division isn't important.)

### Reciprocals

Before we get into how the mathematics works we need to know about this thing called a reciprocal.

Woah!, What is a reciprocal?

Well a reciprocal is defined as what you multiply something by to get the answer of $1$1.  In the case of a fraction, this means multiplying by the 'upside down' version of it.

The reciprocal of $\frac{1}{2}$12 is $\frac{2}{1}$21 or $2$2.

The reciprocal of $\frac{3}{4}$34 is $\frac{4}{3}$43

The reciprocal of $\frac{5}{6}$56 is $\frac{6}{5}$65.  Got the idea?

### Dividing using reciprocals

For the question we were looking at before $\frac{3}{4}\div\frac{1}{2}$34÷​12

We change the division, to be a multiplication of the reciprocal, so

 $\frac{3}{4}\div\frac{1}{2}$34​÷​12​ $=$= $\frac{3}{4}\times\frac{2}{1}$34​×21​ $=$= $\frac{3\times2}{4}$3×24​ $=$= $\frac{6}{4}$64​

##### Question 3

Evaluate $\frac{4}{9}\div\frac{4}{9}$49÷​49.