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Key Stage 3 Mathematics: Introduction to Percentages (levels 5-7)
Percentages are simply fractions out of 100. You should learn these very common percentages:-
| Percentage | Fraction | Decimal |
| 50% | 1/2 | 0.5 |
| 25% | 1/4 | 0.25 |
| 75% | 3/4 | 0.75 |
| 10% | 1/10 | 0.1 |
| 20% | 1/5 | 0.2 |
| 30% | 3/10 | 0.3 |
| 40% | 2/5 | 0.4 |
| 60% | 3/5 | 0.6 |
| 70% | 7/10 | 0.7 |
| 80% | 4/5 | 0.8 |
| 90% | 9/10 | 0.9 |
| 33.333% | 1/3 | 0.3333... |
| 66.666% | 2/3 | 0.6666... |
Calculating percentages of quantities (non-calculator method)
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Example: |
Working: |
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Calculate 15% of £56
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10% of £56 = £5.60 (divide by 10) 5% of £56 = £2.80 Therefore 15% of £56 = £5.60 + £2.80 = £8.40 |
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Example: |
Working: |
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Calculate 43% of £120
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10% of £120 = £12 (divide by 10) So, 40% of £120 = £48
1% of £120 = £1.20 So, 3% of £120 = £3.60
Therefore, 43% of £120 = £48 + £3.60 = £51.60 |
Converting between fractions, decimals and percentages
(1) To convert a fraction to a percentage, change the denominator to 100:
Examples: 3/20 becomes 15/100 or 15% (multiply top and bottom by 5)
8/25 becomes 32/100 or 32% (multiply top and bottom by 4)
21/200 becomes 10.5/100 or 10.5% (divide top and bottom by 2)
(2) To convert a percentage to a fraction, write is as a fraction over 100 and simplify:
Examples: 45% becomes 45/100 = 9/20
48% becomes 48/100 = 24/50 = 12/25
8.5% becomes 8.5/100 = 17/200 (double top and bottom to clear decimal)
(3) To convert a decimal to a percentage, multiply it by 100:
Examples: 0.28 → 28%
0.05 → 5%
0.137 → 13.7%
(4) To convert a percentage to a decimal, divide it by 100:
Examples: 54% → 0.54
9% → 0.09
37.5% → 0.375
Finding a percentage of a quantity (calculator method)
We can find a percentage of a number by converting the percentage to a decimal and then multiplying:
Example: 55% of 760 = 0.55 × 760 = 418 (since 55% = 0.55)
Example 2: 6.5% of 250 = 0.065 × 250 = 16.25 (since 6.5% = 0.065)
Example 3 A type of cheese contains 21% fat. How much fat is there in 360g of cheese?
Solution: 21% of 360g = 0.21 ×360 = 75.6g of fat.
Changing a fraction to a decimal (calculator method)
In a fraction, such as 7/8, the line between the top and bottom numbers is short for divide. So 7/8 is equivalent to 7 ÷ 8.
We can therefore change a fraction to a decimal by simply dividing the top number by the bottom number on a calculator.
Example 1: 7/8 = 7 ÷ 8 = 0.875
Example 2: 19/32 = 19 ÷ 32 = 0.59375
Example 3: 5/18 = 5 ÷ 18 = 0.2777... = 0.278 (to 3 d.p.)
Finding one number as a percentage of another (using a calculator)
We can change a fraction to a percentage in two steps:
(1) change the fraction to a decimal by dividing the top number by the bottom number;
(2) multiply the decimal by 100 to convert to a percentage.
Example: Write 9/16 as a percentage.
Solution: 9/16 = 9 ÷ 16 = 0.5625 (changing to a decimal)
0.5625 × 100 = 56.25%
Therefore 9/16 = 56.25%
Example 2: Alexa scored 47 out of 60 in a test. What was her percentage mark?
Solution: As a fraction Alexa scored 47/60.
We can change this to a decimal: 47/60 = 47 ÷ 60 = 0.7833...
We then change to a percentage by multiplying by 100: 0.7833....× 100 = 78.3% (to 1 d.p.)
Example 3: Kieran examined the weather records of two towns last summer. In Clacton, it was sunny on 15 out of 24 days. In Bournemouth it was sunny for 21 out of 33 days. Which town had the greatest proportion of sunny days?
Solution: We can change the figures to percentages:-
Clacton: 15/24 = 0.625 = 62.5%
Bournemouth 21/33 = 0.63636... = 63.6% (to 1 d.p.)
So Bournemouth had a slightly greater proportion of sunny days.