Fractions to decimals
A fraction is a division: 3/4 means 3 ÷ 4.
Method 1: Divide
- 3/4 = 3 ÷ 4 = 0.75
- 5/8 = 5 ÷ 8 = 0.625
- 2/3 = 2 ÷ 3 = 0.666… = 0.6̄ (repeating)
Method 2: Make the denominator a power of 10
- 3/5 = 6/10 = 0.6
- 7/20 = 35/100 = 0.35
- 3/25 = 12/100 = 0.12
This works when the denominator can be multiplied up to 10, 100, 1,000 …
Terminating or repeating?
| Type | Example | When it happens |
|---|---|---|
| Terminating decimal | 1/8 = 0.125 | Denominator (in simplest form) has only 2s and 5s as prime factors |
| Repeating decimal | 1/3 = 0.333… | Denominator has any other prime factor (3, 7, 11 …) |
Repeating decimals use bar notation: a bar over the repeating digits. 0.272727… is written 0.27 with a bar over 27.
Common conversions to know
| Fraction | Decimal |
|---|---|
| 1/2 | 0.5 |
| 1/4 | 0.25 |
| 3/4 | 0.75 |
| 1/5 | 0.2 |
| 1/8 | 0.125 |
| 1/10 | 0.1 |
| 1/3 | 0.333… |
| 2/3 | 0.666… |
Decimals to fractions
Use place value. The last digit’s place gives the denominator.
- Write the digits over 10, 100, 1,000 …
- Simplify to simplest form.
| Decimal | Place value | Fraction | Simplified |
|---|---|---|---|
| 0.6 | tenths | 6/10 | 3/5 |
| 0.45 | hundredths | 45/100 | 9/20 |
| 0.125 | thousandths | 125/1,000 | 1/8 |
| 2.75 | hundredths | 2 75/100 | 2 3/4 |
Mixed numbers
Convert the fraction part only: 3 2/5 = 3 + 0.4 = 3.4.
Worked examples
Example 1: Convert 7/8 to a decimal.
7 ÷ 8 = 0.875.
Example 2: Convert 0.36 to a fraction.
36/100 → ÷ 4 → 9/25.
Example 3: Will 5/12 terminate?
12 = 2² × 3. It has a factor of 3: repeating (0.41666…).
Common mistakes
- Dividing the wrong way: 3/4 is 3 ÷ 4, not 4 ÷ 3.
- Using the wrong denominator: 0.45 is 45/100, not 45/10.
- Not simplifying.
- Rounding a repeating decimal and calling it exact.
Try it yourself
- 3/8 as a decimal
- 9/20 as a decimal
- 0.85 as a fraction
- 0.004 as a fraction
- Does 7/40 terminate?
Answers: 1) 0.375 2) 0.45 3) 17/20 4) 1/250 5) yes (0.175)
Quick recap
- Fraction → decimal: divide, or scale the denominator to a power of 10.
- Decimal → fraction: use place value, then simplify.
- Denominators with only 2s and 5s give terminating decimals.