The order of operations

Fractions

A fraction is a division that hasn’t been completed. 3/4 means ‘3 divided by 4’, which is 0.75.

the top number is the numerator: how many parts we take
the number at the bottom is the denominator: into how many parts it has been divided

Simplifying

If both the top and bottom are divided by the same number, we can simplify it — the value doesn’t change, only the way it’s written:

6/8  =  3/4     (We halved the top and bottom)
10/5 =  2/1  =  2

A simplified fraction is easier to read. This is the habit you should get into once you’ve finished your calculations.

You can only add parts of the same size. If the denominators are the same, it’s straightforward:

1/4 + 1/4  =  2/4  =  1/2

Otherwise, you must first convert them to the same denominator:

1/2 + 1/3  =  3/6 + 2/6  =  5/6

Common mistake: 1/2 + 1/3 does not equal 2/5. We never add the denominators together.

Multiplication is simpler than addition — it’s counter-intuitive, but true. We multiply the numerators together and the denominators together:

2/3 x 3/4  =  6/12  =  1/2

Dividing by a fraction means multiplying by its inverse:

1/2 : 1/4  =  1/2 x 4/1  =  4/2  =  2

This result is often surprising. Yet it can be read quite simply: ‘how many quarters are there in a half?’ There are indeed 2.