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Fractions are used to show a division between two numbers, with the top being called a **numerator**, and the bottom the **denominator**. The numerator is the number which will be divided and the denominator the number we are dividing by.

When we have a fraction with a numerator less than the denominator, the fraction will be less than one and is known as a **proper fraction**. A few examples of these are: 12 12 , 411 411 , 1920 1920

If we wish to add together two fractions with the same denominator, we can simply add the numerator and put the answer over the common denominator. For example, 14+24=3414+24=34

The same as the above can be done when subtracting fractions that have a **common denominator**. This rule does not apply if the two fractions have different bottoms.

Any fraction which has the same number for its numerator and denominator is always equal to 1. This is because any number divided by itself equals 1. e.g. 66 66 , 1717 1717 , 100100 100100

When we have a fraction that is equal to a number above 1, the numerator must be more than the denominator and this is known as an **improper fraction** or top-heavy fraction. Examples of these are: 32 32 , 102 102 , 73 73

We can convert improper fractions into **mixed numbers **which are a mixture of both real numbers and fractions. To do this we must work out how many times the number in the denominator will be able to go into the number of the numerator. This is best explained with the help of an example:

Convert 114 114 into a mixed number.

We can see that 4 will go into 11 twice with a remainder of 11−(4×2)=311−(4×2)=3. So we can see that:

114=8+34=84+34=234 114=8+34=84+34=234

Here we have made use of the fact that two fractions with a common denominator can be added together. But we have used the rule in reverse to separate the numerator into one part which is equal to a whole number and a remainder.

### Equivalent fractions

Sometimes two fractions can be equal to each other even if they do not look at all similar on first glance. If we draw a visual representation of the fraction 48 48 it looks like this:

Now drawing a visual representation of 12 12 it looks likes this:

From the two images we can tell that 48 48 and 12 12 are equal to each other in that they both have the same amount shaded.

This is because 48=4×14×2=44×12=1×12=1248=4×14×2=44×12=1×12=12 since both 4 and 8 have a common factor in 4, they can be simplified and we then see that the two fractions are **equivalent**.

By coincidence, what we have also done is to reduce 48 48 into its simplest form of 12 12 . We have done this by finding common factors for the numerator and denominator and **reducing the fraction** by dividing both the top and bottom by their common factor.

**Example**

Reduce 1015 1015 into its simplest form.

We can see that 10 and 15 have a highest common factor of 5, so now we can divide both the numerator and denominator by this HCF to get the fraction in its simplest form of 23 23

**Multiplying fractions**

If we wish to multiply two fractions we can simply multiply the two numerators and then multiply the two denominators to give a new fraction.

For example, to multiply 23 23 by 25 25 we first multiply the two top numbers, then the two on the bottom. So we get 2×23×5 2×23×5 which equals 415 415 so we now know that 23 ×25=41523 ×25=415

### Dividing fractions

To divide two fractions, we must flip the second fraction and then multiply the two instead of dividing.

#### Example

Divide 59 59 by 23 23

As outlined above, to do 59 ÷2359 ÷23 we need to first flip the second fraction and then multiply. This becomes 59 ÷23=÷59×÷32=÷1518=÷5659 ÷23=÷59×÷32=÷1518=÷56 Here we have used the rules for multiplying as already described and have also simplified the fraction into its lowest form.

#### Fractions of a value

It is often required for people to work out a ‘fraction of’ a certain number. This is used a lot in real-life situations when purchasing products and other such things. To do this we simply multiply the fraction by the amount we want to split.

#### Example

Find of £35

To do this we simply multiply 35 by . This is

So the answer is **£14**. (Do not forget the currency that is used in your answer)

### Fractions within a fraction

To work out the value of a fraction inside another fraction, we must calculate the numerator and denominator separately first. By treating the top and bottom of the fraction as their own sums, we can simplify to give a more straightforward fraction for us to solve.

#### Example

Work out

To do this we must first look at the numerator and convert this from a mixed number, , to an improper fraction. Doing this gives .

So now we have

And by recognising that a fraction is really just a division, we can write this as . Which, from the rules explained above, equals .

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