Fraction to Decimal: Long Division Explained (Including Repeating Decimals)

The core idea

Every fraction is a division problem. The fraction a/b means "a divided by b". To convert a fraction to a decimal, you simply carry out that division.

For fractions with denominators like 2, 4, 5, 8, 10, 20, 25, or 50, the division terminates neatly. For others, the decimal repeats. Both cases are handled by the same long-division process.

Terminating decimal example

Convert 9/20 to a decimal.

Set up the division: 9 ÷ 20.

• 20 goes into 9 zero times. Write 0 and add a decimal point. Bring down a zero: 90.
• 20 goes into 90 four times (20 × 4 = 80). Remainder: 10.
• Bring down a zero: 100. 20 goes into 100 five times (20 × 5 = 100). Remainder: 0.

Result: 9/20 = 0.45.

The division ended with no remainder, so this is a terminating decimal. See the full worked example at 9/20 to Decimal.

Repeating decimal example

Convert 1/3 to a decimal.

1 ÷ 3:

• 3 goes into 1 zero times. Add decimal point and zero: 10.
• 3 goes into 10 three times (3 × 3 = 9). Remainder: 1.
• Bring down zero: 10. Same as before.

The remainder 1 will keep appearing, producing 0.333... forever. We write this as 0.3̄ (with a dot or bar over the 3) to show it repeats.

Result: 1/3 = 0.333... or 0.3̄.

How to recognise repeating decimals

A fraction in lowest terms produces a terminating decimal only when the denominator's prime factors are exclusively 2s and 5s.

• 1/4 = 1/(2²) → terminates: 0.25
• 1/8 = 1/(2³) → terminates: 0.125
• 1/5 = 1/5¹ → terminates: 0.2
• 1/20 = 1/(2² × 5) → terminates: 0.05
• 1/3 → has prime factor 3 → repeats
• 1/6 = 1/(2 × 3) → repeats (the factor of 3 causes it)
• 1/7 → has prime factor 7 → repeats

If there is any prime factor other than 2 or 5 in the denominator, the decimal will repeat.

Longer repeating patterns

Some fractions have repeating blocks longer than one digit:

1/7 = 0.142857142857... The block "142857" repeats. Written as 0.1̄4̄2̄8̄5̄7̄.

1/11 = 0.090909... The block "09" repeats. Written as 0.0̄9̄.

1/6 = 0.1666... Here, only the 6 repeats. The 1 is not part of the repeating block.

The long-division method step by step

Here is the general procedure:

1. Write the numerator inside the division bracket and the denominator outside.
2. If the numerator is smaller than the denominator, write "0." and attach a zero to the numerator.
3. Divide, note the quotient digit and the remainder.
4. Bring down another zero and repeat.
5. Stop when the remainder is 0 (terminating) or when you see a remainder you have seen before (repeating).

Tracking remainders for repeating decimals

The key insight: once a remainder repeats, the entire division pattern from that point onward will repeat. So keep a list of remainders as you work. When one reappears, you have found the repeating cycle.

Worked example: 5/12

12's prime factorisation is 2² × 3. Because of the 3, this will repeat.

5 ÷ 12:

• 12 into 50: 4 times (48), remainder 2.
• 12 into 20: 1 time (12), remainder 8.
• 12 into 80: 6 times (72), remainder 8.

Remainder 8 has appeared before, so the "6" will repeat.

5/12 = 0.416̄ = 0.4166...

Converting repeating decimals back to fractions

If you need to reverse the process:

• For 0.333...: let x = 0.333... Then 10x = 3.333... Subtract: 9x = 3, so x = 3/9 = 1/3.
• For 0.142857...: let x = 0.142857... Then 1,000,000x = 142857.142857... Subtract: 999,999x = 142857, so x = 142857/999999 = 1/7.

This algebraic trick works for any repeating decimal.

Common mistakes

Forgetting to write the zero before the decimal point. 9/20 is 0.45, not .45. Always include the leading zero.

Rounding too early. If a question asks for the exact decimal, write the repeating notation. If it asks for a rounded answer, specify the number of decimal places.

Mixing up terminating and repeating. Not all fractions with even denominators terminate. 1/6 has an even denominator but repeats because 6 = 2 × 3.

Dividing the wrong way round. 3/4 means 3 ÷ 4 = 0.75, not 4 ÷ 3 = 1.333... Make sure you divide numerator by denominator.

For more on converting between fractions, decimals, and percentages, see the full conversion guide.

Quick reference: common fractions as decimals

| Fraction | Decimal | Type | |----------|---------|------| | 1/2 | 0.5 | Terminating | | 1/3 | 0.333... | Repeating | | 1/4 | 0.25 | Terminating | | 1/5 | 0.2 | Terminating | | 1/6 | 0.1666... | Repeating | | 1/7 | 0.142857... | Repeating | | 1/8 | 0.125 | Terminating | | 3/4 | 0.75 | Terminating | | 2/3 | 0.666... | Repeating | | 5/6 | 0.8333... | Repeating |

FAQ

How many decimal places should I give? It depends on the context. For school maths, 2-4 decimal places or the exact repeating notation is usually expected. For practical use, round to the precision you need.

Is 0.999... equal to 1? Yes. This is a well-known result in mathematics. 0.999... = 1 exactly.

Can a fraction have a non-repeating, non-terminating decimal? No. Every fraction (ratio of two integers) produces either a terminating or repeating decimal. Non-repeating, non-terminating decimals represent irrational numbers like pi or the square root of 2, which cannot be written as fractions.

How do I convert a fraction to a decimal on a calculator? Simply divide: enter the numerator, press ÷, enter the denominator, and press =. The display will show the decimal, though it may round or truncate repeating parts.