Two ways to write the same amount
A fraction and a decimal can describe exactly the same quantity. The fraction 3/4 and the decimal 0.75 are the same number, written in different notations. Being able to move between them is useful when comparing amounts, measuring, working with money and reading data.
The core idea is simple: a fraction is a division. The line in 3/4 means “3 divided by 4”. So converting a fraction to a decimal means carrying out that division.
From fraction to decimal
Step 1. Write the fraction as numerator ÷ denominator. Step 2. Divide, adding zeros after the decimal point as needed. Step 3. Stop when the division ends, or when a pattern repeats.
Example: 3/4
3 ÷ 4 does not divide evenly, so write 3 as 3.00. Then 30 ÷ 4 = 7 remainder 2, and 20 ÷ 4 = 5 remainder 0. The result is 0.75.
Example: 3/8
Using long division: 30 ÷ 8 = 3 remainder 6, then 60 ÷ 8 = 7 remainder 4, then 40 ÷ 8 = 5 remainder 0. The result is 0.375.
Example: a mixed number, 2 3/5
Convert only the fractional part and keep the whole number. 3 ÷ 5 = 0.6, so 2 3/5 = 2.6.
Example: 7/20
An alternative route is to rewrite the fraction with a denominator of 10, 100 or 1000. Since 20 × 5 = 100, multiply the top and bottom by 5: 7/20 = 35/100 = 0.35. Fractions with denominators such as 2, 4, 5, 8, 10, 20 and 25 can usually be done this way.
Terminating and repeating decimals
Some divisions end, such as 1/8 = 0.125. Others never end but repeat a pattern.
- 1/3 = 0.333…, with the 3 repeating forever.
- 2/3 = 0.666…
- 1/6 = 0.1666…, where only the 6 repeats.
- 1/7 = 0.142857142857…, where the block 142857 repeats.
A helpful rule: when a fraction is written in lowest terms, its decimal ends if the denominator has no prime factors other than 2 and 5. So 3/8 (8 = 2 × 2 × 2) and 7/20 (20 = 2 × 2 × 5) terminate, while 1/3, 1/6 and 1/7 do not. This is a well-known fact about how our base-ten system works, and it explains why some fractions look tidy as decimals and others do not.
A calculator will round a repeating decimal, so 1/3 may show as 0.3333333. That is an approximation. If you need an exact value, keep the fraction.
From decimal to fraction
For decimals that end, use place value.
Step 1. Count the digits after the decimal point. Step 2. Write the digits (without the point) over 10, 100, 1000 and so on, matching the number of digits. Step 3. Simplify by dividing top and bottom by their greatest common factor.
Example: 0.625
Three digits after the point, so the denominator is 1000: 625/1000. Both numbers divide by 125, since 625 ÷ 125 = 5 and 1000 ÷ 125 = 8. The answer is 5/8.
Example: 0.12
Two digits: 12/100. Dividing both by 4 gives 3/25.
Example: 3.4
Keep the 3 as a whole number and convert 0.4 = 4/10 = 2/5. So 3.4 = 3 2/5. As an improper fraction this is 17/5, since 3 × 5 + 2 = 17.
Converting repeating decimals to fractions
This takes one extra idea, using a little algebra. Say x = 0.333… Multiply both sides by 10: 10x = 3.333… Subtract the first from the second: 10x − x = 3.333… − 0.333…, so 9x = 3, and x = 3/9 = 1/3.
For a two-digit repeat, such as 0.7272…, let x = 0.7272… and multiply by 100: 100x = 72.7272… Subtracting gives 99x = 72, so x = 72/99. Dividing top and bottom by 9 gives 8/11. You can check by dividing 8 ÷ 11 and seeing 0.7272… emerge. The method relies on setting up and solving a small equation, which is explained in the article on linear equations.
Common conversions worth remembering
| 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… |
Knowing these helps you estimate. If a price is 3/4 of 20, you can reason that 0.75 × 20 = 15 without much effort. These equivalents also connect directly to percentage calculations: 0.75 is 75%, and 0.2 is 20%.
Common mistakes
- Dividing the wrong way round. For 3/8 the calculation is 3 ÷ 8, not 8 ÷ 3.
- Misreading place value, such as writing 0.05 as 5/10 instead of 5/100.
- Forgetting to simplify, leaving 625/1000 when 5/8 is the same amount.
- Treating a rounded calculator result as exact. Rounding is fine for measurements but should be remembered as rounding.
A good habit is to check your answer with number sense: 3/8 should be a little less than a half, so a decimal of 0.375 is plausible, while 3.75 or 0.0375 would not be.
Using tools
Spreadsheets can display a decimal as a fraction through number formatting, and they store decimals internally, which is why spreadsheet basics are a natural companion to this topic. Be aware that software handles many fractions as approximations, so 1/3 stored as a decimal is not exactly one third. This is a consequence of how computers store numbers, which the topic of binary numbers touches on. For learning purposes, though, doing the division by hand a few times builds understanding that a calculator does not.