Fractions, Decimals & Percentages
🎯 In this topic you will
- Recognise and use the fact that fractions, decimals, and percentages can represent the same value
🧠 Key Words
- decimal
- denominator
- equivalent fraction
- numerator
- percentage
Show Definitions
- decimal: A number that uses a decimal point to show values smaller than one.
- denominator: The bottom number of a fraction that shows how many equal parts the whole is divided into.
- equivalent fraction: A fraction that represents the same value as another fraction.
- numerator: The top number of a fraction that shows how many parts are taken.
- percentage: A way of expressing a number as a fraction of 100.
You can write a percentage as a fraction or as a decimal.
$32\% = \tfrac{32}{100} = 0.32 \qquad 60\% = \tfrac{60}{100} = 0.6 \qquad 83\% = \tfrac{83}{100} = 0.83$
Sometimes you can write the fraction in a simpler form.
For $\tfrac{32}{100}$ you can divide the numerator and the denominator by $4$ to get $\tfrac{32}{100} = \tfrac{8}{25}$.
🔎 Reasoning Tip
$\dfrac{32}{100}$ is equivalent to $\dfrac{8}{25}$
For $\tfrac{60}{100}$ you can divide the numerator and the denominator by $20$ to get $\tfrac{60}{100} = \tfrac{3}{5}$.
You cannot simplify $\tfrac{83}{100}$ because the only common factor of 83 and 100 is 1.
In part b of Worked example 10.1, you could use algebra to write an expression linking the area of the pink region to the area of the circle; for example: $r = 0.3c$ or $r = \tfrac{3}{10}c$, where $r$ is the area of the pink region and $c$ is the area of the circle.
❓ EXERCISES
1. You could work with a partner on this question.
70% is equivalent to $ \tfrac{70}{100} $.
a. Show that $ \tfrac{70}{100} $ in its simplest form is $ \tfrac{7}{10} $.
The denominator of $ \tfrac{7}{10} $ is 10.
b. 70% as a fraction in its simplest form has 10 as its denominator.
25% = $ \tfrac{1}{4} $, so as a fraction in its simplest form it has 4 as a denominator.
Starting with a whole-number percentage, what other denominators can you find when you write the percentage as a fraction in its simplest form?
👀 Show answer
b. Examples: 50% = $ \tfrac{1}{2} $ (denominator 2), 20% = $ \tfrac{1}{5} $ (denominator 5), 12.5% = $ \tfrac{1}{8} $ (denominator 8), 33⅓% = $ \tfrac{1}{3} $ (denominator 3).
2. Write these percentages as fractions. Write your answers in the simplest form.
a. 60% b. 61% c. 62% d. 64% e. 65% f. 70%
👀 Show answer
b. $ \tfrac{61}{100} $
c. $ \tfrac{62}{100} = \tfrac{31}{50} $
d. $ \tfrac{64}{100} = \tfrac{16}{25} $
e. $ \tfrac{65}{100} = \tfrac{13}{20} $
f. $ \tfrac{70}{100} = \tfrac{7}{10} $
3. Joshua writes 0.3 = 3%.
a. Explain why this is incorrect.
b. Write 3% correctly as a decimal and as a fraction.
👀 Show answer
b. 3% = 0.03 = $ \tfrac{3}{100} $.
4. Write these percentages as decimals and as fractions.
a. 40% b. 4% c. 9% d. 90% e. 5%
👀 Show answer
b. 4% = 0.04 = $ \tfrac{4}{100} = \tfrac{1}{25} $
c. 9% = 0.09 = $ \tfrac{9}{100} $
d. 90% = 0.9 = $ \tfrac{90}{100} = \tfrac{9}{10} $
e. 5% = 0.05 = $ \tfrac{5}{100} = \tfrac{1}{20} $
5. Write these fractions as percentages.
a. $ \tfrac{1}{4} $ b. $ \tfrac{2}{5} $ c. $ \tfrac{4}{5} $ d. $ \tfrac{7}{50} $ e. $ \tfrac{7}{20} $ f. $ \tfrac{7}{25} $
👀 Show answer
b. $ \tfrac{2}{5}=0.4=40\% $
c. $ \tfrac{4}{5}=0.8=80\% $
d. $ \tfrac{7}{50}=0.14=14\% $
e. $ \tfrac{7}{20}=0.35=35\% $
f. $ \tfrac{7}{25}=0.28=28\% $
6. Here are two shapes:
a. Write down as a fraction, a percentage and a decimal the part of shape $1$ that is coloured pink.
b. Write down as a fraction, a percentage and a decimal the part of shape $1$ that is coloured blue.
c. Write down as a fraction, a percentage and a decimal the part of shape $2$ that is coloured pink.
d. Write down as a fraction, a percentage and a decimal the part of shape $2$ that is coloured blue.
e. The area of shape $1$ is $45\ \text{cm}^2$. Work out the area that is shaded pink.
f. The area of shape $2$ is $75\ \text{cm}^2$. Work out the area that is shaded blue.
👀 Show answer
a. Pink (shape $1$): $ \tfrac{3}{10}=0.3=30\% $.
b. Blue (shape $1$): $ \tfrac{4}{10}=\tfrac{2}{5}=0.4=40\% $.
For shape $2$ there are $25$ equal squares: the inner $3\times 3$ is white ($9$), the border has $16$ coloured of which blue corners $=4$ and pink $=12$.
c. Pink (shape $2$): $ \tfrac{12}{25}=0.48=48\% $.
d. Blue (shape $2$): $ \tfrac{4}{25}=0.16=16\% $.
e. Pink area (shape $1$): $ \tfrac{3}{10}\times 45=\tfrac{135}{10}=13.5\ \text{cm}^2 $.
f. Blue area (shape $2$): $ \tfrac{4}{25}\times 75=\tfrac{300}{25}=12\ \text{cm}^2 $.
7. Sort these percentages, fractions and decimals into five groups.
👀 Show answer
Group 1: $4\% = \tfrac{1}{25} = 0.04$
Group 2: $6\% = 0.06$
Group 3: $60\% = \tfrac{3}{5} = 0.6$
Group 4: $40\% = \tfrac{2}{5} = 0.4$
Group 5: $30\% = \tfrac{3}{10} = 0.3$
(Note: $ \tfrac{3}{50} = 0.06 = 6\% $, so it belongs in Group 2 as well.)
🧠 Think like a Mathematician
Task: Explore fractions with a numerator of 3 that convert neatly to whole-number percentages, and reflect on the method you used.
Questions:
👀 show answer
- a) Examples: - $\dfrac{3}{4} = 75\%$ - $\dfrac{3}{5} = 60\%$ - $\dfrac{3}{10} = 30\%$ - $\dfrac{3}{20} = 15\%$ - $\dfrac{3}{2} = 150\%$ These all give whole-number percentages.
- b) The list can be extended by checking denominators that make $\dfrac{3}{n} \times 100$ a whole number (i.e., denominators that divide 300 exactly).
- c) Yes, the same method works for other numerators: multiply by 100 and check which denominators divide evenly. For example, with numerator 2, look for denominators of 200; with numerator 5, look for denominators of 500.
❓ EXERCISES
9. Work out:
a. $ \tfrac{1}{2} $ of 60 grams
b. $ \tfrac{3}{4} $ of 60 grams
c. $ \tfrac{4}{5} $ of 60 grams
d. $ \tfrac{7}{10} $ of 60 grams
e. $ \tfrac{11}{20} $ of 60 grams
👀 Show answer
b. $ \tfrac{3}{4} \times 60 = 45 $ g
c. $ \tfrac{4}{5} \times 60 = 48 $ g
d. $ \tfrac{7}{10} \times 60 = 42 $ g
e. $ \tfrac{11}{20} \times 60 = 33 $ g
10. Work out:
a. 50% of $300
b. 20% of $300
c. 30% of $300
d. 15% of $300
👀 Show answer
b. $ \tfrac{20}{100} \times 300 = 60 $
c. $ \tfrac{30}{100} \times 300 = 90 $
d. $ \tfrac{15}{100} \times 300 = 45 $
11.
a. Copy and complete this diagram. 100% = 40 m.
b. Now add four more lines to your diagram. You choose the percentages.
👀 Show answer
50% = 20 m
25% = 10 m
20% = 8 m
10% = 4 m
(Other percentages can be chosen freely, e.g., 5% = 2 m, 75% = 30 m).
12. 30% of $70 = $21
a. How can you use this result to find 60% of $70?
b. What other percentages of $70 can you find using this result? Show your method each time.
c. Compare your results with a partner’s. Have you got different results?
👀 Show answer
b. From $30\% = 21$:
• $10\% = 7$ (divide by 3).
• $20\% = 14$ (double 10%).
• $40\% = 28$ (double 20%).
• $50\% = 35$.
• $70\% = 49$.
• $100\% = 70$.
c. Values should match across different methods.
13.
a. Write $ \tfrac{1}{4} $ as a percentage.
b. Use your result from part a to write $ \tfrac{1}{8} $ as a percentage.
c. Write equivalent percentages for other fractions with 8 as a denominator.
👀 Show answer
b. $ \tfrac{1}{8} = 12.5\% $.
c. With denominator 8:
$ \tfrac{2}{8} = 25\% $, $ \tfrac{3}{8} = 37.5\% $, $ \tfrac{4}{8} = 50\% $, $ \tfrac{5}{8} = 62.5\% $, $ \tfrac{6}{8} = 75\% $, $ \tfrac{7}{8} = 87.5\% $.
⚠️ Be careful! Fractions, Decimals & Percentages
- Don’t confuse decimals with percentages. $0.3=30\%$, not $3\%$. To get the percentage, multiply the decimal by 100.
- Always simplify fractions fully. $60\%=\tfrac{60}{100}=\tfrac{3}{5}$, not left as $\tfrac{60}{100}$.
- Check denominator factors. Some percentages reduce neatly (e.g. $25\%= \tfrac{1}{4}$), but others don’t (e.g. $83\%= \tfrac{83}{100}$).
- Fractions, decimals, and percentages are the same value in different forms. Don’t treat them as separate answers — they must match exactly.
- Use estimation. If $30\%$ of $60$ is about a third, your answer should be close to $20$ (not much larger or smaller).
- Be cautious with percentages over 100. $125\%=1.25=\tfrac{5}{4}$, which is greater than 1 — not less.