# Percentages

The word ‘percentage’ is very familiar to us as it is used regularly in the media to describe anything from changes in the interest rate, to the number of people taking holidays abroad, to the success rate of the latest medical procedures or exam results. Percentages are a useful way of making comparisons, apart from being used to calculate the many taxes that we pay such as VAT, income tax, domestic fuel tax and insurance tax, to name but a few. So percentages are very much part of our lives. But what does percentage actually mean? Now ‘per cent’ means ‘out of 100’; and ‘out of’, in mathematical language, means ‘divide by’. So if you score 85% (using the symbol ‘%’ for percentage) on a test then, if there were a possible 100 marks altogether, you would have achieved 85 mark . So

85% = 85/100

Let us look at some other common percentage amounts, and their fraction and decimal equivalents.

75% = 75/100 = 3/4 = 0.75
50% = 50/100 = 1/2 = 0.5
25% = 25/100 = 1/4 = 0.25
10% = 10/100 = 1/10 = 0.1
5% = 5/100 = 1/20 = 0.05

It is worth noting that 50% can be found be dividing by 2, and that 10% is easily found by dividing by 10.

Now let us look at writing fractions as percentages. For example, say you get 18 marks out of 20 in a test. What percentage is this?

First, write the information as a fraction. You gained 18 out of 20 marks, so the fraction is 18/20. Since a percentage requires a denominator of 100, we can turn 18/20 into a fraction out of 100 by multiplying both numerator and denominator by 5:

18/20 = 18 × 5 / 20 × 5 = 90/100 = 90%

Since we are multiplying both the numerator and the denominator by 5, we are not changing the value of the fraction, merely finding an equivalent fraction.

In that example it was easy to see that, in order to make the denominator 100, we needed to multiply 20 by 5. But if it is not easy to see this, such as with a score of, say, 53 out of 68, then we simply write the amount as a fraction and then multiply by 100/100:

53/68 × 100/100 = 53 ÷ 68 × 100% = 77.94%

which is 78% to the nearest whole number. Although it is easier to use a calculator for this type of calculation, it is advisable not to use the % button at this stage. We shall look at using the percentage button on a calculator at the end of this unit.