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Recurring Decimal To Fraction Calculator

Work out recurring decimal to fraction instantly with clear inputs, formula shown and shareable results.

About Recurring Decimal To Fraction Calculator

For a decimal 0.a(b) with p non-repeating digits and q repeating digits, multiplying by 10^(p+q) and by 10^p and subtracting eliminates the tail, giving (ab − a) / (10^p(10^q − 1)).

Formula

Recurring decimal

0.a(b) = (ab − a) / (10^p · (10^q − 1)), p = length of a, q = length of b

Worked example

With the default values below, Recurring Decimal To Fraction Calculator returns Exact fraction of 13/6. Change any input above and every figure here is recalculated in your browser.

Inputs used

Whole part
2
Digits after the point that do not repeat
1
Repeating digit block
6

Results

Exact fraction
13/6
Fractional part alone
1/6
Decimal check
2.1666666667
Denominator before reducing
90

Step by step

  1. 1Values used
  2. 2Recurring decimal
  3. 3Exact fraction= 13/6
  4. 4Fractional part alone= 1/6
  5. 5Decimal check= 2.1666666667
  6. 6Denominator before reducing= 90

Assumptions and limits

  • The result is derived only from the 3 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to 10 decimal places, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

What is 0.1666… as a fraction?

With a = 1 and b = 6: (16 − 1)/(10 × 9) = 15/90 = 1/6.

What about 0.999…?

With no non-repeating digits and b = 9: (9 − 0)/9 = 1. Recurring nine really does equal one.

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