If you work through this section you should be able to:
A percentage, like a fraction, is used to represent some number of parts of a whole; but in this case, hundredths of the whole.
'Per cent' means per hundred. A percentage symbol % represents a fraction with 100 as the denominator, meaning 'over 100'.
10%, then means 10 out of 100 (where 100 represents 1 whole), or simply 10 hundredths.
If you are unfamiliar with fractions, you might want to work through the Fractions section first.
To change a percentage into a fraction:
Examples showing how to convert a percentage into a fraction
To find out how to convert fractions to percentages, see the Fractions section.
To change a percentage into a decimal:
Simply divide by 100, or move the decimal point two places to the left, inserting zeros as required. Remember, we usually put a zero before the decimal point if there is not a whole number part.
Examples showing how to convert percentages to decimals
For whole number percentages, the decimal point comes directly after the last number, i.e. 75% can be considered 75.0%, so 75% = 0.75; 135% can be considered 135.0%, so 135% = 1.35.
To find out how to convert decimals to percentages, see the Decimals section.
To calculate the percentage of a quantity, e.g. 60% of £20:
60% of £20
60% = 0.6
0.6 × 20 = £12
24% of 250 = 24/100 × 250
= 24/2 × 5 (cancelled by 50)
= 12/1 × 5 (cancelled by 2)
= 60
There are times when you may need to work out a percentage change. For example, when shops have sales it is usual to decrease the original price by a given percent.
Mrs Banks was looking for a new dress and came across a sign in a dress shop window - everything reduced by 25% off marked price.
She went into the shop and found a dress she liked. The marked price was £20.
How much did she have to pay for the dress?
Work out the percentage of the quantity.
0.25 × 20 = 5
Subtract it from the original price.
20 − 5 = £15
This is often used as a way of comparing changes by relating the change to original values or comparing proportions between different groups.
It is difficult to compare fractions that have different denominators, and expressing them as percentages makes them into their respective equivalent fractions with the common denominator of 100 (see the Fractions section for more information on working with fractions).
Rule: Express the proportions as fractions, then multiply by 100.
There are two ways of doing this (see 'Converting fractions to percentages' in the Fractions section) and we will use both methods in the example below.
In one group of students, 3 out of 4 passed their exams, whilst in another group of 7 students, 5 passed. Which group had the better pass rate?
5 is greater than 3 but we need to consider the total number taking the exam. So, we will express the total number passing as a percentage of the number in the group.
Method 1 (group 1): Multiply by and cancel if possible.
= 75%
Method 2 (group 2): Convert the fraction to a decimal and multiply by 100 by moving the decimal point 2 places to the right.
= 0.714 (to 3 decimal places)
0.714 × 100 = 71.4%
Answer: It is now easy to see that the pass rate was better in the first group.
When you are dealing with quantities or measurements with units involved, both must be expressed in the same units, so be prepared to change if necessary.
An item increases in price from £2.80 to £3.50. What is the percentage rise in price? In other words, what is the rise as a percentage of the original price?
Rise = £3.50 - £2.80 = 70p
Original price = £2.80 = 280p (changing price to pence)
% increase
=
=
=
= 25%
A 25km boat race is re-measured and found to be 100m too short. What percentage reduction is this?
Note: The 25 kilometres must be expressed as metres.
% reduction
=
=
=
= 0.4%
Interest is a percentage of an amount added to the original amount over a given time period. For example, in a savings account the bank may give you a certain amount of interest per month or per year on the amount you have invested with them.
Simple interest is interest that remains the same every year. The actual interest is only added on to the final amount at the end of the given period.
3 years simple interest on £300 at 4% p.a. (per annum, or year)
Work out the percentage of the given quantity
4% of £300
= 0.04 × 300
= 12
Interest for one year = £12
Multiply by the amount of time
3 × 12 = £36
The formula for simple interest is:
Using the same example as above:
= 36
Compound interest is calculated differently from simple interest. Interest may be added every month, every year, or even every day.
Each calculation of interest is a percentage of the original amount and any previous interest combined. So, each time it is calculated, the amount on which the interest is being paid increases.
Compound interest on £2000 invested for 2 years at 5% p.a.
1st year - original capital: 2000
Interest: 0.05 × 2000 = 100
2nd year - new capital: 2100 (original capital plus £100 interest)
Interest: 0.05 × 2100 = 105
Answer: New capital: 2205 (2nd year's new capital plus £105 interest)
If you want to work out exactly how much of the new amount is interest, simply subtract the original amount from the latest capital.
So in the example above, £2205 − £2000 = £205.
A ratio is a comparison between two similar quantities.
For example, if in a small company there are 11 unskilled workers and 3 skilled workers, the ratio of unskilled to skilled is 11 to 3.
This ratio is written 11:3
Ratios should be expressed in their simplest form. For example, a larger company that has 400 unskilled workers and 100 skilled workers would have a ratio of unskilled to skilled of 400:100, simplifying to 4:1 (for an explanation of simplifying see 'Simplifying fractions' in the Equivalent fractions section).
A specific feature of a ratio is that it gives no indication of the absolute size. Ratios are most frequently used in situations where the proportions of a specified mixture remain the same, regardless of the overall size of the product.
Concrete is made up of cement, sand and aggregate in the ratio 2:3:5.
Calculate how much of each component is needed to make 1500 kg of concrete.
Total number of parts is 2 + 3 + 5 = 10
= is cement.
is sand.
= is aggregate.
Amount of cement = × 1500 = 300 kg
Amount of sand = × 1500 = 450 kg
Amount of aggregate = × 1500 = 750 kg
Add the weights together to check that they add up to the original 1500 kg total.
A chemical analysis on impurities in river water passing a given point showed that in 100 parts of impurities, 60 were dangerous, 30 were harmless, and 10 were unidentified.
If 140 kg of impurities flowed past the given point one day, how many kilograms of each type would be present in the water?
Amount of dangerous impurities = × 140 = 84 kg
Amount of harmless impurities = × 140 = 42 kg
Amount of unidentified impurities = × 140 = 14 kg
Again, check your answers by adding them together and seeing that they total 140 kg.
You can download a version of this Percentages and ratios activity in Word format:
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