Percentages

Percentages

Percent is really two words in one: per cent. A cent is a hundredth (compare: centi) so per cent means in every hundred.

You will no doubt have been given back work or exams with a mark written as a percentage. What does it actually mean?

100% = all correct
75% = three-quarters correct
50% = half correct
and so on.

This means that tests that might have a different number of marks can be correctly compared.

e.g. 42/60 and 45/80. The last mark is higher, but its percentage is lower!

Using Percentages

Here's a question: Fraser scored 75% on a physics test. If there were 80 marks in total, how many marks did he lose?

Well, 75% means 75/100. So 25/100 would be wrong, which is 1 in every 4 marks - in other words one quarter.

One quarter of 80 is 20.

In general, we can solve questions like this by making the percentage equivalent to a decimal and multiplying it by the total.

e.g. 30% of 50 is 0.30 × 50 = 15.

Percentages as Fractions

As well as writing percentages in decimal form, it's also possible to think of them as fractions. Starting with 1% (one hundredth) we can write 1% = 1/100.

Similarly we can write any percentage as a fraction out of a hundred, but we can often simplify the fraction:

Percentage Equivalent Fraction Simplest Fraction 80% 8/10 4/5 50% 5/10 1/2 25% 25/100 1/4 33·3% 333/1000 1/3

Note that in the last example, 1/3 would be a recurring decimal.

Testing Percentages

Let's look at a typical example involving percentages and see if you can do it. Grab a pencil, paper and maybe a calculator.

In a maths class of 30 pupils, 40% are girls. 1 out of 3 pupils also studies physics. Half the boys have a mobile phone, whilst 6 girls have them. Three of the girls have a boyfriend at the moment.

1. What percentage of the class are boys?
2. How many boys are there in the class?
3. What number of pupils also study physics?
4. What percentage of the class have a mobile phone?
5. What percentage of girls don't have a boyfriend?

Let's see how you got on! In a maths class of 30 pupils, 40% are girls. 1 out of 3 pupils also studies physics. Half the boys have a mobile phone, whilst 6 girls have them. Three of the girls have a boyfriend at the moment.

1. What percentage of the class are boys?
   Total percentage is always 100%
   Percentage of boys = 100% - 40% = 60%
2. How many boys are there in the class?
   60% of 30 is 0·6 × 30 = 18
3. What number of pupils also study physics?
   1/3 of 30 = 10
4. What percentage of the class have a mobile phone?
   Boys: 0·5 × 18 = 9
   Girls: 6
   Total = 6 + 9 = 15.
   Percentage = 15 ÷ 30 = 0·5 = 50%
5. What percentage of girls don't have a boyfriend?
   Number of girls without = 12 - 3 = 9
   Fraction of girls = 9/12 = 0·75
   Percentage of girls without a boyfriend = 75%