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Fractions & Decimals

Recurring Decimal to Fraction

Convert any recurring decimal to its exact fraction using the algebraic method — with every step shown. Enter the non-recurring part and the repeating cycle separately, and see why 0.9̄ really equals 1.

Recurring decimal converter

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Integer Non-recur Repeating ¯

The highlighted box is the repeating cycle. Leave non-recurring blank if the repeat starts immediately after the decimal point.

Step by step calculations

Enter the recurring digits in the converter to see the algebraic step by step calculations.

Recurring Decimal to Fraction Explained

Core concepts, classroom applications, and plain-English explanations for all audiences.

Where recurring decimals appear

Whenever you divide £1 among 3 people, each person gets 33.333…p. On a ruler, ⅓ inch or ⅙ inch cannot be measured precisely with decimal millimetres. In recipes, 0.333 cups is an approximation of ⅓ cup. Converting recurring decimals back to exact fractions guarantees 100% mathematical precision without rounding errors.

Worked example

"Convert 0.1666… (0.16̄) to a fraction."

Let x = 0.1666…

100x = 16.666… (move 2 places)

10x = 1.666… (move 1 place)

100x − 10x = 16.666… − 1.666…

90x = 15

x = 15/90 = 1/6

Verification: 1 ÷ 6 = 0.1666… ✓

Common mistake to avoid

  • ✗ Subtracting x instead of 10^p x: When there are non-recurring digits before the cycle (like 0.16̄), subtracting x leaves a decimal (16.666… − 0.166… = 16.5). Multiplying by both 100 and 10 aligns the tails so subtraction produces pure integers.
Good to know

The maths behind recurring decimals

x
The algebraic method
The key insight: if you call the recurring decimal x, then multiplying by a power of 10 shifts it so the recurring parts line up exactly. Subtracting removes the infinite tail, leaving a simple equation with an integer answer — which you can then solve for x and simplify.
Let x = 0.333...
10x = 3.333...
10x - x = 3.333... - 0.333...
9x = 3
x = 3/9 = 1/3
p q
Mixed recurring decimals
When the repeating part doesn't start right after the decimal point, you need two multiplications: one to move past the non-recurring part (multiply by 10^p) and one more to shift a full cycle (multiply by 10^(p+q)). Subtracting these eliminates the infinite tail cleanly.
Let x = 0.1666... (p=1, q=1)
100x = 16.666...
10x = 1.666...
90x = 15
x = 15/90 = 1/6
=1
Why 0.9̄ = 1 exactly
The most debated identity in elementary maths. Many people feel 0.999… must be slightly less than 1 — but this is wrong. The algebraic proof: let x = 0.999…, then 10x = 9.999…, so 10x − x = 9, giving 9x = 9, so x = 1. There is no gap between 0.9̄ and 1. They are the exact same number.
Let x = 0.999...
10x = 9.999...
9x = 9
x = 1 ✓

Also: 1/3 = 0.333...
3 × 1/3 = 1
3 × 0.333... = 0.999... = 1 ✓
∞
Every fraction has a recurring or terminating decimal
The reverse is also true: every recurring decimal is exactly equal to some fraction. This means recurring decimals are rational numbers — they can always be written as a/b. Only irrational numbers (like π and √2) have infinite decimals that never repeat.
0.(142857) = 1/7 (6-digit cycle)
0.(076923) = 1/13 (6-digit cycle)
0.(0588235294117647) = 1/17

Irrational: π = 3.14159265...
(never repeats — no fraction exists)