Introduction to Percentages %
The term ‘per cent’ means ‘out of a hundred’. In mathematics, percentages are used like fractions and decimals, as ways to describe parts of a whole. When you are using percentages, the whole is considered to be made up of a hundred equal parts. The symbol % is used to show that a number is a percentage, and less commonly the abbreviation ‘pct’ may be used.
You will see percentages almost everywhere: in shops, on the internet, in advertisements and in the media. Being able to understand what percentages mean is a key skill that will potentially save you time and money and will also make you more employable.
The Meaning of Percentages
Percentage is a term from Latin, meaning ‘out of one hundred’.
You can therefore consider each ‘whole’ as broken up into 100 equal parts, each one of which is a single percent.
The box below shows this for a simple grid, but it works the same way for anything: children in a class, prices, pebbles on the beach, and so on.
The grid below has 100 cells.
- Each cell is equal to 1% of the whole (the red cell is 1%).
- Two cells are equal to 2% (the green cells).
- Five cells are equal to 5% (the blue cells).
- Twenty five cells (purple cells) are equal to 25% of the whole or one quarter (¼).
- Fifty cells (yellow cells) are equal to 50% of the whole or half (½).
How many unshaded (white) cells are there? What is the percentage of unshaded cells?
Answer: There are two ways to work this out.
- Count the white cells. There are 17 of them. Out of 100 cells, 17% are therefore white.
- Add up the number of other cells, and take them from 100. There is one red cell, two green, five blue, 25 purple, and 50 yellow. That adds up to 83. 100−83 = 17. Again, out of 100 cells, 17 are white, or 17%.
It is easy to work out the percentage when there are 100 individual ‘things’ making up the whole, as in the grid above. But what if there are more or less?
The answer is that you convert the individual elements that make up the whole into a percentage. For example, if there had been 200 cells in the grid, each percentage (1%) would be two cells, and every cell would be half a percent.
We use percentages to make calculations easier. It is much simpler to work with parts of 100 than thirds, twelfths and so on, especially because quite a lot of fractions do not have an exact (non-recurring) decimal equivalent. Importantly, this also makes it much easier to make comparisons between percentages (which all effectively have the common denominator of 100) than it is between fractions with different denominators. This is partly why so many countries use a metric system of measurement and decimal currency.
Finding the Percentage
The general rule for finding a given percentage of a given whole is:
Work out the value of 1%, then multiply it by the percentage you need to find.
This is easiest to understand with an example. Let’s suppose that you want to buy a new laptop computer. You have checked local suppliers and one company has offered to give you 20% off the list price of £500. How much will the laptop cost from that supplier?
In this example, the whole is £500, or the cost of the laptop before the discount is applied. The percentage that you need to find is 20%, or the discount offered by the supplier. You are then going to take that off the full price to find out what the laptop will cost you.
Start by working out the value of 1%
One percent of £500 is £500 ÷ 100 = £5.
Multiply it by the percentage you are looking for
Once you have worked out the value of 1%, you simply multiply it by the percentage you are looking for, in this case 20%.
£5 × 20 = £100.
You now know that the discount is worth £100.
Complete the calculation by adding or subtracting as necessary.
The price of the laptop, including the discount, is £500−20%, or £500−£100 = £400.
The easy way to work out 1% of any number
1% is the whole (whatever that may be) divided by 100.
When we divide something by 100, we simply move the place values two columns to the right (or move the decimal point two places to the left).
You can find out more about numbers and place values on our Numbers page, but here’s a quick recap:
£500 is made up of 5 hundreds, zero tens and zero units. £500 also has zero pence (cents if you are working in dollars) so could be written as £500.00, with zero tenths or hundredths.
When we divide by 100, we move our number two columns to the right. 500 divided by 100 = 005, or 5. Leading zeros (zeros on the ‘outside left’ of a number, such as those in 005, 02, 00014) have no value, so we do not need to write them.
You can also think of this as moving the decimal point two places to the left.
This rule applies to all numbers, so £327 divided by 100 is £3.27. This is the same as saying that £3.27 is 1% of £327. £1 divided by 100 = £0.01, or one pence. There are one hundred pence in a pound (and one hundred cents in a dollar). 1p is therefore 1% of £1.
Once you have calculated 1% of the whole, you can then multiply your answer to the percentage you are looking for (see our page on multiplication for help).
Mental Maths Hacks
As your maths skills develop, you can begin to see other ways of arriving at the same answer. The laptop example above is quite straightforward and with practise, you can use your mental maths skills to think about this problem in a different way to make it easier. In this case, you are trying to find 20%, so instead of finding 1% and then multiplying it by 20, you can find 10% and then simply double it. We know that 10% is the same as 1/10th and we can divide a number by 10 by moving the decimal place one place to left (removing a zero from 500). Therefore 10% of £500 is £50 and 20% is £100.
A useful mental maths hack is that percentages are reversible, so 16% of 25 is the same as 25% of 16. Invariably, one of those will be much easier to work out in our head…try it!
Use our Percentage Calculators to quickly solve your percentage problems.
Working with Percentages
We calculated a 20% discount in the example above and then subtracted this from the whole to work out how much a new laptop would cost.
As well as taking a percentage away, we can also add a percentage to a number. It works exactly the same way, but in the final step, you simply add instead of subtracting.
For example: George is promoted and gets a 5% pay rise. George currently earns £24,000 a year, so how much will he earn after his pay rise?
Work out 1% of the whole
The whole in this example is George's current salary, £24,000. 1% of £24,000 is 24,000 ÷ 100 = £240.
Multiply that by the percentage you are looking for
George is getting a 5% pay rise, so we need to know the value of 5%, or 5 times 1%.
£240 × 5 = £1,200.
Complete the calculation by adding to the original amount
George’s pay rise is £1,200 per year. His new salary will therefore be £24,000 + £1,200 = £25,200.
Percentages over 100%
It is possible to have percentages over 100%. This example is one: George’s new salary is actually 105% of his old one.
However, his old salary is not 100% of his new one. Instead, it is just over 95%.
When you are calculating percentages, the key is to check that you are working with the correct whole. In this case, the ‘whole’ is George’s old salary.
Percentages as Decimals and Fractions
One percent is one hundredth of a whole. It can therefore be written as both a decimal and a fraction.
To write a percentage as a decimal, simply divide it by 100.
For example, 50% becomes 0.5, 20% becomes 0.2, 1% becomes 0.01 and so on.
We can calculate percentages using this knowledge. 50% is the same as a half, so 50% of 10 is 5, because five is half of 10 (10 ÷ 2). The decimal of 50% is 0.5. So another way of finding 50% of 10 is to say 10 × 0.5, or 10 halves.
20% of 50 is the same as saying 50 × 0.2, which equals 10.
17.5% of 380 = 380 × 0.175, which equals 66.5.
George’s salary increase above was 5% of £24,000. £24,000 × 0.05 = £1,200.
The conversion from decimal to percentage is simply the reverse calculation: multiply your decimal by 100.
0.5 = 50%
0.875 = 87.5%
To write a percentage as a fraction, put the percentage value over a denominator of 100, and divide it down into its lowest possible form.
50% = 50/100 = 5/10 = ½
20% = 20/100 = 2/10 = 1/5
30% = 30/100 = 3/10
It is possible to convert fractions to percentages by converting the denominator (the bottom number of the fraction) into 100.
However, it is harder to convert fractions to percentages than percentages to fractions because not every fraction has an exact (non-recurring) decimal or percentage.
If the denominator of your fraction does not divide a whole number of times into 100, then there will not be a simple conversion. For example, 1/3, 1/6 and 1/9 do not make ‘neat’ percentages (they are 33.33333%, 16.66666% and 11.11111%).
Working out Percentages of a Whole
So far we have looked at the basics of percentages, and how to add or subtract a percentage from a whole.
Sometimes it is useful to be able to work out the percentages of a whole when you are given the numbers concerned.
For example, let’s suppose that an organisation employs 9 managers, 12 administrators, 5 accountants, 3 human resource professionals, 7 cleaners and 4 catering staff. What percentage of each type of staff does it employ?
Start by working out the whole.
In this case, you do not know the ‘whole’, or the total number of staff in the organisation. The first step is therefore to add together the different types of staff.
9 managers + 12 administrators + 5 accountants + 3 HR professionals + 7 cleaners + 4 catering staff = 40 members of staff.
Work out the proportion (or fraction) of staff in each category.
We know the number of staff in each category, but we need to convert that to a fraction of the whole, expressed as a decimal. The calculation we need to do is:
Staff in Category ÷ Whole (See our division page for help with division sums or use a calculator)
We can use managers as an example:
9 managers ÷ 40 = 0.225
In this case it can be helpful if, instead of thinking of the division symbol ‘÷’ as meaning ‘divided by’, we can substitute the words ‘out of’. We use this often in the context of test results, for example 8/10 or ‘8 out of 10’ correct answers. So we calculate the ‘number of managers out of the whole staff’. When we use words to describe the calculation, it can help it to make more sense.
Convert the fraction of the whole into a percentage
0.225 is the fraction of staff that are managers, expressed as a decimal. To convert this number to a percentage, we need to multiply it by 100. Multiplying by 100 is the same as dividing by a hundred except you move the numbers the other way on the place values scale. So 0.225 becomes 22.5.
In other words, 22.5% of the organisation’s employees are managers.
We then do the same two calculations for each other category.
- 12 administrators ÷ 40 = 0.3. 0.3 × 100 = 30%.
- 5 accountants ÷ 40 = 0.125. 0.125 × 100 = 12.5%.
- 3 HR professionals ÷ 40 = 0.075. 0.075 × 100 = 7.5%.
- 7 cleaners ÷ 40 = 0.175. 0.175 × 100 = 17.5%.
- 4 catering staff ÷ 40 = 0.1. 0.1 × 100 = 10%.
TOP TIP! Check you have a total of 100%
When you have finished calculating your percentages, it is a good idea to add them together to make sure that they equal 100%. If they don't, then check your calculations.
In summary, we can say that the organisation is made up of:
|Roles||Number of Staff||% of Staff|
It can be useful to show percentage data representing a whole on a pie chart. You can quickly see the proportions of categories of staff in the example.
For more on pie charts and other types of graphs and charts see our page: Graphs and Charts.
Points to remember
- Percentages are a way to describe parts of a whole.
- They are a bit like decimals, except that the whole is always split into 100, instead of tenths, hundredths, thousandths and so on of a unit.
- Percentages are designed to make calculations easier.
Further Reading from Skills You Need
Part of The Skills You Need Guide to Numeracy
This eBook covers proportion looking at numbers as parts of other numbers, as parts of a larger whole, or in relation to other numbers. The book covers fractions and decimals, ratio and percentages with worked examples for you to try and develop your skills.
Whether you want to brush up on your basics, or help your children with their learning, this is the book for you.