Recurring decimal to fraction calculator
0.1363636… as a fraction
3/22
0.1363636… = 3/22 in its simplest form.
Working
- Let x = 0.1363636…, where 36 repeats after 1.
- Multiply by 10³ = 1,000 and by 10¹ = 10 so the repeating parts line up: 1,000x = 136.3636… and 10x = 1.3636….
- Subtract to remove the repeating part: 990x = 135.
- x = 135/990 = 3/22 in its simplest form (dividing by 45).
Your result
3/22. 0.1363636… = 3/22 in its simplest form.
Recurring decimals, step by step
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Turn a recurring decimal into a fraction by subtracting multiples so the repeat cancels: 0.1363636… gives 1000x − 10x = 135, so x = 135/990 = 3/22.
Questions people ask
How do you convert a recurring decimal to a fraction?
Call the decimal x. Multiply by 10, 100 or 1,000 so one copy of the repeat moves in front of the point, then subtract x (or a multiple of it) so the repeating part cancels. Solve and simplify. 0.444… gives 10x − x = 4, so x = 4/9.
What if only some digits repeat?
Use two multiples. For 0.1363636…, 1000x = 136.3636… and 10x = 1.3636…, so 990x = 135 and x = 3/22.
What do the dots mean?
A dot over one digit means that digit repeats. Dots over two digits mean everything from the first dot to the second repeats.
Is every recurring decimal a fraction?
Yes. Every recurring or terminating decimal is rational, which is why this method always gives a fraction.
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How we work this out
What it does
Turn any recurring decimal into a fraction in its simplest form, with the algebra written out the way exams ask for it.
A whole number, up to 8 digits before the repeat and up to 8 repeating digits.
Method
- Let x = the decimal.
- Multiply by 10 to the power of (digits before the repeat + repeat length), and by 10 to the power of the digits before the repeat.
- Subtract so the repeat cancels, divide, and simplify.