Fractions as Operators

Fractions as Operators

GCSE(F)

A fraction can be seen as a division waiting to happen: for example, `frac(1)(4)` of a pizza means a whole pizza split, or divided, into four equal slices. Similarly, `frac(3)(4)` of a pizza means splitting a pizza into four slices, then taking three of these slices: in other words, 3 x `frac(1)(4)`, or `frac(3)(4)`.

When a fraction of an amount is required, the fraction acts as an operator or function on the amount. The original amount is multiplied by the numerator and divided by the denominator. Note that the multiplication and division can be calculated in either order.

For example, `frac(3)(5)` of 50 could be either:
• 50 x 3 ÷ 5, or
• 50 ÷ 5 x 3.
In this instance, the calculation is easier by dividing by 5 first and then multiplying by 3 to obtain an answer of 30. (Although BIDMAS says that Division happens before Multiplication, you can actually carry out a division and a multiplication in either order).

When using a mixed fraction as a multiplier, turn the mixed fraction into an improper fraction first. For example, 2 `frac(3)(8)` x 40:
= `frac(19)(8)` x 40
= `frac(19)(1)` x 5 (divide both 40 and 8 by 8)
= 95.

Examples

1. Three quarters of the passengers at a train station are waiting for a train to Birmingham. If there are 120 passengers at the station, how many are waiting for the train to Birmingham?

Answer: 90

120 x `frac(3)(4)`
= 120 x 3 ÷ 4
= 360 ÷ 4
= 90

2. After a flood in the stock room, `frac(3)(8)` of the boxes of chocolates were found to be damaged. If there were originally 400 boxes of chocolates, how many were damaged?

Answer: 150

`frac(3)(8)` x 400
= 3 x 50 (Divide 400 ÷ 8 = 50)
= 150