# Brush up your maths: Fractions

## What is a fraction?

So far we have dealt with whole numbers or integers, as found by counting. A fraction is one whole number divided by another. We write fractions as one number over another, with a horizontal line between them, like this:

3
–
4

or in a line of text as one number with a slash or solidus then the other number, like this: 3/4.

The fraction 3/4 or three quarters means 3 parts out of 4. The upper number, 3, is called the numerator and the lower number, 4, is the denominator.

To calculate a fraction of something, multiply by the numerator and divide by the denominator. For example, 3/4 of 12 is 9:

3
– × 12 = (3 × 12) ÷ 4 = 36 ÷ 4 = 9
4

### Exercise: fractions

What is a half as a fraction?

What is four fifths of 20?

## Simplifying fractions

Sometime the numerator and denominator have a common factor. We can make the fraction simpler by dividing top and bottom by the factor. For example:

15     3 × 5         3
–– = –––––– = ––
20     4 × 5         4

because the common factor of 15 and 20 is 5. We divide top and bottom by 5.

Simplify 27/30.

## Improper fractions

Fractions which are between 0 and 1 are called proper fractions. Improper fractions are those greater than 1. For example, 7/4 is an improper fraction. We can also express this as a whole number and a proper fraction: 1¾.

### Exercise: improper fractions

Express five and two thirds as an improper fraction.

Express

17
––
5

as an integer and proper fraction.

## Reciprocals

The reciprocal of a number is one divided by the number. The reciprocal of 2 is ½. Conversely, the reciprocal of ½ is 2.

To find the reciprocal of a fraction we turn it over, so that the numerator becomes the denominator and the denominator becomes the numerator.

4           5
For example, the reciprocal of   ––   is   ––
5           4

We can convert this to an integer and a proper fraction as 1¼.

### Exercise: reciprocals

What are the the reciprocals of 4 and of 5/6?

## Multiplying fractions

To multiply two fractions we simply multiply their numerators, multiply their denominators, and simplify if necessary:

2       1           2 × 1           2           1
–– × ––   =   –––––   =   ––   =   ––
3       2           3 × 2           6           3

### Exercise: multiplying fractions

Multiply four fifths by three quarters.

## Adding and subtracting fractions

If we want to add two fractions which have the same denominator we just add the numerators:

2       1           2 + 1           3
–– + ––   =   –––––   =   ––
5       5               5             5

If we want to add two fractions which have different denominators, we must first make them have the same denominator. We do this by finding the lowest common multiple of the denominators. We call this the lowest common denominator. For the fractions 2/9 and 5/6 the lowest common denominator is 18. To get this we find the factors of 6 and 9. Factors of 6 are 1, 2, 3, and 6. Factors of 9 are 1, 3, and 9. The highest common factor is 3. 6/3 = 2 and 9/3 = 3. The lowest common multiple is therefore 3 × 2 × 3 = 18. We then multiply the top and bottom of each fraction by the factor which will make the bottom the lowest common denominator:

2       5           2 × 2       5 × 3           4       15           4 + 15         19
–– + ––   =   ––––– + –––––   =   ––– + –––   =   –––––   =   –––
9       6           9 × 2       6 × 3           18     18             18             18

We can also write 19/18 as 1 1/18.

To subtract fractions, we do exactly the same except that we subtract the numerators after finding the lowest common denominator:

5       1           5 × 2       1 × 3           10       3           10 – 3         7
–– – ––   =   ––––– – –––––   =   ––– – –––   =   –––––   =   –––
9       6           9 × 2       6 × 3           18     18             18             18

### Exercise: adding and subtracting fractions

Add 3/4 and 2/3 then subtract 1/6.

Back to Brush up your maths main menu.