What are Percentages?
Percentage means out of 100.
Percentages are written using the symbol %.
10% means 10 out of 100.
15% means 15 out of 100.
80% means 80 out of 100.
Percentages are useful because they allow us to easily compare things.
For example exam scores are often given as a percentage. If Charlie scored 84 out of 100 on one exam and 102 out of 120 on another, it is hard to compare his scores and say which exam he did better on. But if I change his scores to percentages I can see that he got 84% in his first exam and 85% in the second, so he did better in his second exam.
How did I change his scores to percentages?
Well the first one was easy, Charlie scored 84 out of 100. Remember percentages are out of 100, so 84 out of 100 is 84%.
On the second exam his score was 102 out of 120. To change this to a percentage we have to change 102 out of 120 in to the form:
? out of 100.
In a similar way to changing fractions we need to find a way to change 120 in to 100 by using division and multiplication:
120 ÷ 6 = 20
20 x 5 = 100
Now must apply the same steps to 102:
102 ÷ 6 = 17
17 x 5 = 85
So 102 out of 120 is 85%.
Finding Percentages of Numbers
To find the percentage of a number we can follow two simple steps:
- We divide the percentage by 100 to change it to a decimal. The quick way to do this is just by moving the decimal point – see the lesson that covers dividing by 100 if you don’t understand this.
- We times the decimal by the number we are trying to find a percentage of.
Let’s work through an example question:
Question: What is 40% of 20?
First we divide the percentage by 100 to form a decimal:
40 ÷ 100 = 0.4
Then we multiply the decimal by the number we want to find a percentage of:
0.4 x 20 = 8
(A good way to do this last step is to say that 4 x 20 = 80, so 0.4 x 20 = 8. See the lesson on multiplication if you don’t understand this).
So 40% of 20 = 8
Another way to tackle this type of question, especially when there are easy numbers involved is to say, well 40% is the same as 4 tenths. One tenth of 20 (20 ÷ 10) = 2, so 4 tenths is 4 x 2 = 8.
Question: ‘Spaceships R Us’ are having a sale and all their spaceships have 20% off their original price. What was the original price of a spaceship now on sale for £400?
This type of question can be a bit tricky to get your head round. You need to think that the original price is 100% and the sale price is a percentage of the original cost.
If 20% has been taken off the original price to get the sale price of £400 then this means £400 is 80% of the original price.
If £400 is 80% then we need to find 100% as this will be the original price.
To do this we must first work out a way to get from 80% to 100%.
I can see two easy ways to do this. Either we divide 80% by 8 to get 10% and then times by 10 to get 100%. Or we divide 80% by 4 to get 20% and then times by 5 to get 100%. Either way will work, i’ll use the first method:
400 ÷ 8 = £50
(So £50 is 10% of the original price. Now we times by 10 to get 100% of the price:)
50 x 10 = £500
You might be asked to show percentages visually by shading a certain percentage of a picture.
Question: Fill in 30% of the squares in this grid:
If we count the grid squares there are 10 so we just need to find 30% of 10 and fill in that many squares:
First convert the percentage to a decimal:
30 ÷ 100 = 0.3
Then times the decimal by the number we are finding the percentage of:
0.3 x 10 = 3
Another way to work it out would be to think, 10% of 10 is 1, so 30% must be 3
So we shade in 3 squares: