Converting Fractions, Decimals and Percentages
The Skill
Fractions, decimals and percentages are three different ways of writing the same value. Being able to convert fluently between them is essential for comparing values and solving problems.
Key Conversions
Fraction → Decimal: Divide the numerator by the denominator. $$\frac{3}{4} = 3 \div 4 = 0.75$$
Decimal → Percentage: Multiply by 100. $$0.75 \times 100 = 75\%$$
Percentage → Decimal: Divide by 100. $$75\% \div 100 = 0.75$$
Decimal → Fraction: Write as a fraction over a power of 10, then simplify. $$0.75 = \frac{75}{100} = \frac{3}{4}$$
Percentage → Fraction: Write over 100, then simplify. $$75\% = \frac{75}{100} = \frac{3}{4}$$
Fraction → Percentage: Convert to decimal first, then multiply by 100. $$\frac{3}{4} \times 100 = 75\%$$
Common Equivalents to Memorise
| Fraction | Decimal | Percentage |
|---|---|---|
| $\frac{1}{2}$ | 0.5 | 50% |
| $\frac{1}{4}$ | 0.25 | 25% |
| $\frac{3}{4}$ | 0.75 | 75% |
| $\frac{1}{5}$ | 0.2 | 20% |
| $\frac{1}{10}$ | 0.1 | 10% |
| $\frac{1}{3}$ | 0.333... | 33.3% |
| $\frac{1}{8}$ | 0.125 | 12.5% |
The Traps
Common misconceptions and how to avoid them.
Dividing numerator and denominator by different numbers "The Unequal Split"
The Mistake in Action
Simplify $\frac{12}{18}$
Wrong: $\frac{12}{18} = \frac{12 \div 2}{18 \div 3} = \frac{6}{6} = 1$
Why It Happens
Students know they need to divide both parts but don't realise it must be by the same number.
The Fix
When simplifying a fraction, you must divide the top and bottom by the same number.
Ask yourself: "What number goes into BOTH 12 AND 18?"
$\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}$
Spot the Mistake
Simplify $\frac{12}{18}$
$= \frac{12 \div 2}{18 \div 3}$
$= \frac{6}{6} = 1$
Click on the line that contains the error.
Moving the decimal point the wrong way "The Backwards Shift"
The Mistake in Action
Convert 0.35 to a percentage.
Wrong: $0.35 \div 100 = 0.0035\%$
Why It Happens
Students remember "something with 100" but forget whether to multiply or divide. They may think "percentages are smaller numbers" and divide instead of multiply.
The Fix
Remember: Percentages are the "big" version.
- To make a number look bigger (decimal → percentage): multiply by 100
- To make a number look smaller (percentage → decimal): divide by 100
Check with a value you know: $0.5 = 50\%$. Did 0.5 get bigger or smaller to become 50? It got bigger, so we multiplied.
Spot the Mistake
Convert 0.35 to a percentage
$0.35 \div 100$
$= 0.0035\%$
Click on the line that contains the error.
Thinking percentage means out of 100 literally "The Literal Hundred"
The Mistake in Action
Write $\frac{7}{20}$ as a percentage.
Wrong: "It's not out of 100, so I can't convert it"
Why It Happens
Students learn that "percent means out of 100" and interpret this too literally.
The Fix
"Out of 100" means we need to scale the fraction so the denominator becomes 100.
Method 1: Find an equivalent fraction with denominator 100 $$\frac{7}{20} = \frac{7 \times 5}{20 \times 5} = \frac{35}{100} = 35\%$$
Method 2: Divide then multiply by 100 $$\frac{7}{20} = 7 \div 20 = 0.35 \rightarrow 35\%$$
Spot the Mistake
Write $\frac{7}{20}$ as a percentage
The denominator is 20, not 100
So this cannot be written as a percentage
Click on the line that contains the error.
The Deep Dive
Apply your knowledge with these exam-style problems.
Level 1: Fully Worked
Complete solutions with commentary on each step.
Question
Write $\frac{3}{8}$ as a percentage.
Solution
Method 1: Via decimal
Step 1: Convert the fraction to a decimal by dividing: $$\frac{3}{8} = 3 \div 8 = 0.375$$
Step 2: Convert the decimal to a percentage by multiplying by 100: $$0.375 \times 100 = 37.5\%$$
Method 2: Direct calculation
Multiply the fraction by 100: $$\frac{3}{8} \times 100 = \frac{300}{8} = 37.5\%$$
Answer: 37.5%
Question
Write 65% as a fraction in its simplest form.
Solution
Step 1: Write the percentage as a fraction over 100: $$65\% = \frac{65}{100}$$
Step 2: Simplify by finding the highest common factor of 65 and 100.
- HCF = 5
Step 3: Divide both numerator and denominator by 5: $$\frac{65}{100} = \frac{65 \div 5}{100 \div 5} = \frac{13}{20}$$
Answer: $\frac{13}{20}$
Level 2: Scaffolded
Fill in the key steps.
Question
Put these values in order from smallest to largest: $$\frac{2}{5}, \quad 0.45, \quad 38\%$$
Level 3: Solo
Try it yourself!
Question
A shop reduces prices by $\frac{1}{5}$. Write this discount as: (a) a decimal (b) a percentage
Show Solution
(a) $\frac{1}{5} = 1 \div 5 = 0.2$
(b) $0.2 \times 100 = 20\%$
Or directly: $\frac{1}{5} \times 100 = 20\%$
Answers: (a) 0.2 (b) 20%
Question
Write $\frac{5}{6}$ as a decimal. Give your answer to 3 decimal places.
Show Solution
$\frac{5}{6} = 5 \div 6 = 0.8333...$
To 3 decimal places: 0.833
Note: This is a recurring decimal, which could be written as $0.8\dot{3}$
Examiner's View
Mark allocation: Conversion questions are typically worth 1-2 marks.
Common errors examiners see:
- Moving the decimal point the wrong way when converting to/from percentages
- Not simplifying fractions fully
- Writing 0.3 instead of 0.333... for 1/3
- Confusing which operation to use (×100 vs ÷100)
What gains marks:
- Showing the division for fraction → decimal
- Showing the ×100 or ÷100 step explicitly
- Fully simplified fractions
AQA Notes
AQA often combines conversions with ordering questions. Convert everything to decimals first for easy comparison.