Percentages: Large & small
🎯 In this topic you will
- Recognise and use percentages, including percentages less than 1 or greater than 100
🧠 Key Words
- common factor
- mixed number
Show Definitions
- common factor: A number that divides exactly into two or more numbers.
- mixed number: A number made up of a whole number and a fraction together.
$50\% = 0.5 \qquad 50\% = \tfrac{50}{100} = \tfrac{1}{2}$
$5\% = 0.05 \qquad 5\% = \tfrac{5}{100} = \tfrac{1}{20}$
$0.5\% = 0.005 \qquad 0.5\% = \tfrac{0.5}{100} = \tfrac{5}{1000} = \tfrac{1}{200}$
0.5% is too small to show on a diagram.
To write $\tfrac{0.5}{100}$ as a fraction in its simplest form:
- First, multiply the numerator and denominator by 10 to get $\tfrac{5}{1000}$.
- Then divide the numerator and denominator by 5 to get $\tfrac{1}{200}$.
Percentages can be more than 100%.
Suppose that over 20 years the price of a house increases by 75%. You can draw a diagram to show this.
old price 100%
new price 175%
$100\% + 75\% = 175\%$
The new price is 175% of the old price.
In Worked example 10.3, you could use algebra to write an expression linking the child’s height now and the child’s height 2 years ago; for example: $n = 1.4t$ or $n = \tfrac{7}{5}t$, where $n$ is the child’s height now and $t$ is the child’s height 2 years ago.
❓ EXERCISES
1. Worked example 10.2 shows how to write 17.5% as a fraction. Choose some more percentages that end in .5. Write these percentages as fractions in their simplest form. What different denominators do you get? What is the largest denominator? What is the smallest denominator?
👀 Show answer
12.5% = $ \tfrac{125}{1000} = \tfrac{1}{8} $
22.5% = $ \tfrac{225}{1000} = \tfrac{9}{40} $
32.5% = $ \tfrac{325}{1000} = \tfrac{13}{40} $
47.5% = $ \tfrac{475}{1000} = \tfrac{19}{40} $
Largest denominator from these = 40, smallest = 8.
2. Write these percentages as decimals and as fractions. Write each fraction in its simplest form.
a. 7.5% b. 62.5% c. 1.5% d. 47.5% e. 32.5%
👀 Show answer
b. 62.5% = 0.625 = $ \tfrac{625}{1000} = \tfrac{5}{8} $
c. 1.5% = 0.015 = $ \tfrac{15}{1000} = \tfrac{3}{200} $
d. 47.5% = 0.475 = $ \tfrac{475}{1000} = \tfrac{19}{40} $
e. 32.5% = 0.325 = $ \tfrac{325}{1000} = \tfrac{13}{40} $
3. Copy and complete this table.
| $80 | $300 | $90 | |
|---|---|---|---|
| 100% | $80 | $300 | |
| 50% | $45 | ||
| 5% | |||
| 0.5% |
👀 Show answer
| $80 | $300 | $90 | |
|---|---|---|---|
| 100% | $80 | $300 | $90 |
| 50% | $40 | $150 | $45 |
| 5% | $4 | $15 | $4.50 |
| 0.5% | $0.40 | $1.50 | $0.45 |
4. Write these percentages as fractions in their simplest form.
a. 2% b. 0.2% c. 8% d. 0.8% e. 7% f. 0.7%
👀 Show answer
b. 0.2% = $ \tfrac{0.2}{100} = \tfrac{2}{1000} = \tfrac{1}{500} $
c. 8% = $ \tfrac{8}{100} = \tfrac{2}{25} $
d. 0.8% = $ \tfrac{0.8}{100} = \tfrac{8}{1000} = \tfrac{2}{250} = \tfrac{1}{125} $
e. 7% = $ \tfrac{7}{100} $
f. 0.7% = $ \tfrac{0.7}{100} = \tfrac{7}{1000} $
5.
a. Work out:
i. 1% of 600 kilograms
ii. 1.5% of 600 kilograms
iii. 3.5% of 600 kilograms
iv. 2.2% of 600 kilograms
b. In part a, why was it useful to work out 1% first?
👀 Show answer
i. $1\% = \tfrac{1}{100} \times 600 = 6$ kg
ii. $1.5\% = 1.5 \times 6 = 9$ kg
iii. $3.5\% = 3.5 \times 6 = 21$ kg
iv. $2.2\% = 2.2 \times 6 = 13.2$ kg
b. Once you know 1% = 6 kg, you can scale it to find other percentages quickly.
6.
a. Work out 1% of 7000 metres.
b. Use your answer to part a to work out:
i. 2% of 7000 metres
ii. 0.5% of 7000 metres
iii. 0.1% of 7000 metres
iv. 0.3% of 7000 metres
👀 Show answer
b.
i. $2\% = 2 \times 70 = 140$ m
ii. $0.5\% = \tfrac{1}{2} \times 70 = 35$ m
iii. $0.1\% = \tfrac{70}{10} = 7$ m
iv. $0.3\% = 3 \times 7 = 21$ m
7. How would you work out 1.3% of $7500? Can you think of any different methods?
👀 Show answer
$0.3\% = \tfrac{0.3}{100} \times 7500 = 22.5$
So $1.3\% = 75 + 22.5 = 97.5$
✅ Different methods: Multiply directly → $0.013 \times 7500 = 97.5$.
8.
a. Work out $ \tfrac{1}{3} $ of 100.
b. Write $ \tfrac{1}{3} $ as a percentage.
Zara says: $ \tfrac{1}{3} $ is 33%.
Arun says: $ \tfrac{1}{3} $ is 33.3%.
c. How could Zara be correct?
d. How could Arun be correct?
e. Write $ \tfrac{2}{3} $ as a percentage.
👀 Show answer
b. $ \tfrac{1}{3} = 0.333\ldots = 33.3\% $ (recurring).
c. Zara rounded $33.\overline{3}\%$ down to 33%.
d. Arun gave the value to 1 decimal place: 33.3%.
e. $ \tfrac{2}{3} = 0.666\ldots = 66.7\% $ (to 1 decimal place).
9. Write each of these mixed numbers as a percentage. The first one has been done for you.
a. $ 1 \tfrac{1}{2} = 150\% $
b. $ 1 \tfrac{1}{4} $
c. $ 1 \tfrac{3}{4} $
d. $ 1 \tfrac{3}{10} $
e. $ 1 \tfrac{7}{10} $
f. $ 2 \tfrac{1}{4} $
👀 Show answer
b. $ 1 \tfrac{1}{4} = \tfrac{5}{4} = 1.25 = 125\% $
c. $ 1 \tfrac{3}{4} = \tfrac{7}{4} = 1.75 = 175\% $
d. $ 1 \tfrac{3}{10} = \tfrac{13}{10} = 1.3 = 130\% $
e. $ 1 \tfrac{7}{10} = \tfrac{17}{10} = 1.7 = 170\% $
f. $ 2 \tfrac{1}{4} = \tfrac{9}{4} = 2.25 = 225\% $
10.
a. Write $ \tfrac{1}{5} $ as a percentage.
b. Write these mixed fractions as percentages:
i. $ 1 \tfrac{1}{5} $
ii. $ 1 \tfrac{3}{5} $
iii. $ 1 \tfrac{4}{5} $
iv. $ 2 \tfrac{3}{5} $
👀 Show answer
b.
i. $ 1 \tfrac{1}{5} = \tfrac{6}{5} = 1.2 = 120\% $
ii. $ 1 \tfrac{3}{5} = \tfrac{8}{5} = 1.6 = 160\% $
iii. $ 1 \tfrac{4}{5} = \tfrac{9}{5} = 1.8 = 180\% $
iv. $ 2 \tfrac{3}{5} = \tfrac{13}{5} = 2.6 = 260\% $
🧠 Think like a Mathematician
Task: Explore percentages of a given amount, including percentages greater than 100%.
Questions:
👀 show answer
- a) 30% of $60 = $0.3 \times 60 = 18$. So 30% of $60 is $18.
- b) Since 30% = $18, - 150% = $90 (because $18 \times 5$), - 200% = $120 (because $18 \times \tfrac{200}{30}$), - 300% = $180, and so on.
- c) The values are correct as long as they are exact multiples of $18, since each $18 represents 30%.
- d)$102 \div 60 = 1.7$, so $102 is 170% of $60.
❓ EXERCISES
12. Look at the diagram
a. What percentage of $50 is $40?
b. What percentage of $40 is $50?
👀 Show answer
b. $ \tfrac{50}{40} \times 100 = 125\% $
13. Marcus has 500 g of flour. He uses 200 g to bake bread and 120 g to make biscuits. Work out the percentage of the flour:
a. he uses for bread
b. he uses for biscuits
c. he does not use
👀 Show answer
b. $ \tfrac{120}{500} \times 100 = 24\% $
c. $ 100\% - (40\% + 24\%) = 36\% $
14. Sofia works for 40 hours in a week. She spends 25 hours in her office, 8 hours in the factory and the rest in a workshop. Work out the percentage of the time that she spends:
a. in her office
b. in the factory
c. in a workshop
👀 Show answer
b. $ \tfrac{8}{40} \times 100 = 20\% $
c. Remaining $ = 40 - (25+8) = 7$ hours → $ \tfrac{7}{40} \times 100 = 17.5\% $
15. Zara earns $250. She pays $95 in tax and pays a bill of $65. Work out the percentage of her money:
a. she pays in tax
b. she has left after both payments
👀 Show answer
b. Left = $250 - (95+65) = 90$
$ \tfrac{90}{250} \times 100 = 36\% $
16. An airline ticket costs $40. The price is increased by 200%.
a. Work out:
i. the increase
ii. the price after the increase
b. Find the number to complete this sentence: The new price is … times the original price
👀 Show answer
ii. Price after increase = $40 + 80 = 120$
b. New price = 3 times the original price
17. The mass of a puppy is 5 kg. As it grows, the mass increases by 300%.
a. Work out: i. the increase in mass ii. the new mass
b. Find the number to complete this sentence: The new mass is … times the original mass.
👀 Show answer
a.ii. New mass = $5 + 15 = 20$ kg
b. New mass is 4 times the original mass.
18.
a. The cost of a car increases by 5%. What percentage of the original price is the new price?
b. The cost of petrol increases by 80%. What percentage of the original price is the new price?
👀 Show answer
b. New price = $100\% + 80\% = 180\%$ of original
19. Ruqiyah is trying to answer this question: What percentage of $125 is $200?
She thinks the answer is a whole number. She cannot find the answer. Can you help her?
👀 Show answer
20. Vikram wants to work out 135% of $40 but he does not know how. Describe some different methods to do this calculation. Which method do you prefer? Why?
👀 Show answer
Method 2: Find 100% ($40$) and 35% ($14$), then add → $40+14=54$
Method 3: Break into 120% + 15% → $48+6=54$
✅ Preferred: Method 1 (direct multiplication), quickest and simplest.
⚠️ Be careful! Percentages Below 1% and Above 100%
- Percentages less than 1% are very small. For example, $0.5\%=0.005=\tfrac{1}{200}$, not $\tfrac{1}{20}$.
- Divide by 100, not 10. $5\%=\tfrac{5}{100}=0.05$, not $0.5$.
- More than 100% means “more than the whole.” $175\%$ of a number is the whole (100%) plus three quarters more (75%).
- Use decimals or fractions for clarity. $140\%=1.4=\tfrac{7}{5}$. Multiply either form by the original value to find the new total.
- Check results with a “times” statement. If a mass increases by $300\%$, the new mass is $4$ times the original (not $3$ times).
- Estimate to sense-check. $200\%$ of $40$ must be double: $80$. So $135\%$ of $40$ should be a bit more than $50$.