Recurring Decimals to Fractions
Line up the repeating tails, subtract, then simplify.
Make the infinite parts cancel
A recurring decimal has a block of digits that repeats forever. Multiplying by suitable powers of ten gives two decimals with identical tails; subtraction removes those tails exactly.
Read the repeating block
British dots mark the first and last digits of the repeating block, or just one digit when it repeats alone. The American bar covers the whole block. Digits before the marked block do not repeat.
$0.1[[overline{6}]]=0.16666[[ldots]]$
$0.[[overline{16}]]=0.161616[[ldots]]$
These are different numbers: the first repeats one digit after a prefix, while the second repeats a two-digit block. Ellipses here mean that the marked block continues forever.
One repeating digit
Convert $0.[[overline{6}]]$ to a fraction in simplest form.
Step 1
Let $x=0.[[overline{6}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$10x=6.[[overline{6}]]$
$x=0.[[overline{6}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$10x-x=6-0$
$9x=6$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{6}{9}]]$
$x=[[frac{2}{3}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
Two repeating digits
Convert $0.[[overline{27}]]$ to a fraction in simplest form.
Step 1
Let $x=0.[[overline{27}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=27.[[overline{27}]]$
$x=0.[[overline{27}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-x=27-0$
$99x=27$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{27}{99}]]$
$x=[[frac{3}{11}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
Three repeating digits
Convert $0.[[overline{125}]]$ to a fraction in simplest form.
Step 1
Let $x=0.[[overline{125}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$1000x=125.[[overline{125}]]$
$x=0.[[overline{125}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$1000x-x=125-0$
$999x=125$
Step 4
Divide by the coefficient of $x$.
$x=[[frac{125}{999}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
A leading zero in the block
Convert $0.[[overline{09}]]$ to a fraction in simplest form.
The block is zero-nine, not just nine. Its length is two, so shift two places. Keep the leading zero when counting digits.
Step 1
Let $x=0.[[overline{09}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=9.[[overline{09}]]$
$x=0.[[overline{09}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-x=9-0$
$99x=9$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{9}{99}]]$
$x=[[frac{1}{11}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
A prefix changes both powers of ten
A nonrepeating prefix
Convert $0.1[[overline{6}]]$ to a fraction in simplest form.
Shift once to move past the prefix, then once more to pass one full repeating block. Subtracting the original decimal from ten times itself would not align the tails.
Step 1
Let $x=0.1[[overline{6}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=16.[[overline{6}]]$
$10x=1.[[overline{6}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=16-1$
$90x=15$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{15}{90}]]$
$x=[[frac{1}{6}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
More than one whole
Convert $1.2[[overline{3}]]$ to a fraction in simplest form.
Step 1
Let $x=1.2[[overline{3}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=123.[[overline{3}]]$
$10x=12.[[overline{3}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=123-12$
$90x=111$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{111}{90}]]$
$x=[[frac{37}{30}]]$
Check the fraction by division: it must reproduce the prefix and the full repeating block, not just the first few rounded digits.
Choose the shifts, not a memorised denominator
Let $m$ be the number of nonrepeating decimal digits and $n$ the length of the repeating block. Compare $10^{m+n}x$ with $10^m x$ Both now have the same fractional tail.
$(10^{m+n}-10^m)x=N-K$
Here $N$ and $K$ are the integer parts of those two scaled decimals. Divide by the coefficient of $x$, then simplify. For a pure recurring decimal, $m=0$ so the smaller equation is the original $x$ Count zeros in a prefix or block; they still occupy places.
Subtract like quantities on both sides. Do not subtract just the displayed digits while leaving the multiplier of $x$ unchanged. Simplification comes after the exact fraction has been formed.
Optional insight: Why recurring nines equal one
This is an exact boundary case, not a rounding rule.
$x=0.[[overline{9}]]$
$10x=9.[[overline{9}]]$
$10x-x=9$
$9x=9$
$x=1$
Every finite string of nines is below one, but the infinite recurring decimal has no last digit and equals one exactly. This gives one number two decimal representations.
Practice Problems
Work without a calculator first. Write the two scaled equations, subtract, and give each fraction in simplest form. Open the answer only after trying.
Practice 1
Convert $0.[[overline{4}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=0.[[overline{4}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$10x=4.[[overline{4}]]$
$x=0.[[overline{4}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$10x-x=4-0$
$9x=4$
Step 4
Divide by the coefficient of $x$.
$x=[[frac{4}{9}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 2
Convert $0.[[overline{18}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=0.[[overline{18}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=18.[[overline{18}]]$
$x=0.[[overline{18}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-x=18-0$
$99x=18$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{18}{99}]]$
$x=[[frac{2}{11}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 3
Convert $0.[[overline{037}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=0.[[overline{037}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$1000x=37.[[overline{037}]]$
$x=0.[[overline{037}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$1000x-x=37-0$
$999x=37$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{37}{999}]]$
$x=[[frac{1}{27}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 4
Convert $0.2[[overline{7}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=0.2[[overline{7}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=27.[[overline{7}]]$
$10x=2.[[overline{7}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=27-2$
$90x=25$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{25}{90}]]$
$x=[[frac{5}{18}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 5
Convert $0.03[[overline{6}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=0.03[[overline{6}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$1000x=36.[[overline{6}]]$
$100x=3.[[overline{6}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$1000x-100x=36-3$
$900x=33$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{33}{900}]]$
$x=[[frac{11}{300}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 6
Convert $2.1[[overline{08}]]$ to a fraction in simplest form.
Show answer
Step 1
Let $x=2.1[[overline{08}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$1000x=2108.[[overline{08}]]$
$10x=21.[[overline{08}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$1000x-10x=2108-21$
$990x=2087$
Step 4
Divide by the coefficient of $x$.
$x=[[frac{2087}{990}]]$
Count every digit, including zeros; check the fraction after subtraction.
Practice 7
Let $x=0.12[[overline{3}]]$ Which two powers of ten align the tails? Complete the equations and find the fraction.
Show answer
Step 1
Let $x=0.12[[overline{3}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$1000x=123.[[overline{3}]]$
$100x=12.[[overline{3}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$1000x-100x=123-12$
$900x=111$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{111}{900}]]$
$x=[[frac{37}{300}]]$
A two-digit prefix requires the smaller scale to be a hundred; the one-digit block adds one more shift.
Practice 8
For $x=0.1[[overline{6}]]$, a learner writes $10x-x=1$ Explain why this subtraction does not remove the recurring tails, then repair the method.
Show answer
The tails are six-six-six recurring and one-six-six-six recurring, so they are not identical. The unmatched tails contribute to the difference, so it is not one:
$10x-x=1.5$
The actual difference is one and a half. Ignoring the unmatched fractional parts gives the wrong result. Use the visibly aligned pair instead:
Step 1
Let $x=0.1[[overline{6}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=16.[[overline{6}]]$
$10x=1.[[overline{6}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=16-1$
$90x=15$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{15}{90}]]$
$x=[[frac{1}{6}]]$
Only identical aligned tails may be cancelled digit for digit.
Practice 9
A learner claims $[[frac{1}{11}]]=0.[[overline{09}]]$ Verify the claim using division or the subtraction method.
Show answer
Step 1
Let $x=0.[[overline{09}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=9.[[overline{09}]]$
$x=0.[[overline{09}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-x=9-0$
$99x=9$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{9}{99}]]$
$x=[[frac{1}{11}]]$
The exact fraction simplifies to one eleventh, so the claim is correct. In division, the remainders cycle back and the block zero-nine repeats.
A leading zero belongs to the block; dropping it would describe a different value.
Practice 10
Complete the subtraction for $x=0.[[overline{24}]]$: $100x-x=24$ A learner gives $[[frac{24}{99}]]$ as the simplest form. Finish the solution.
Show answer
$99x=24$
$x=[[frac{24}{99}]]$
$x=[[frac{8}{33}]]$
The fraction is exact before simplification, but both parts still share a factor of three.
Interactive Quiz
Exit check: Choose an answer, then explain your method before opening the review.
1. For $x=0.2[[overline{5}]]$, which subtraction aligns the tails?
2. Which simplest-form fraction equals $0.[[overline{12}]]$?
3. Which decimal equals $[[frac{1}{6}]]$?
Review the exit quiz
Question 1: B. Move past the prefix, then one full block.
Step 1
Let $x=0.2[[overline{5}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=25.[[overline{5}]]$
$10x=2.[[overline{5}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=25-2$
$90x=23$
Step 4
Divide by the coefficient of $x$.
$x=[[frac{23}{90}]]$
Question 2: C. B counts only one repeating digit; A uses the denominator for a terminating decimal.
$[[frac{12}{99}]]=[[frac{4}{33}]]$
Question 3: A. Only the six repeats after the one. B repeats the entire block one-six, and C stops.
Final written exit: Convert $0.4[[overline{2}]]$ to a simplest-form fraction and explain why the two fractional tails cancel.
Check the written exit
Step 1
Let $x=0.4[[overline{2}]]$
Step 2
Scale to place the same repeating block immediately after each decimal point.
$100x=42.[[overline{2}]]$
$10x=4.[[overline{2}]]$
Step 3
The fractional tails are aligned and identical. Subtract the entire lower equation from the upper equation.
$100x-10x=42-4$
$90x=38$
Step 4
Divide by the coefficient of $x$ and simplify.
$x=[[frac{38}{90}]]$
$x=[[frac{19}{45}]]$