Converting a repeating decimal to a fraction

A repeating decimal represents an exact rational number. The algebra method works because a suitable power-of-ten shift lines up identical repeating tails; subtraction cancels them. The most common errors are shifting the decimal by the wrong number of places and forgetting to simplify the fraction.

One repeating digit

Convert 0.333… to a fraction. Let x = 0.333…. Multiply by 10: 10x = 3.333…. Subtract x = 0.333… from that equation: 9x = 3. Therefore x = 3/9 = 1/3. The repeating 3 appears in both decimal expressions in the same aligned position, so it cancels.

A repeating block with several digits

For 0.272727…, the repeating block has two digits. Let x = 0.272727…. Multiply by 100, not 10: 100x = 27.272727…. Subtract the original equation to get 99x = 27, so x = 27/99 = 3/11. The power of 10 must move one complete repeating block into alignment.

A mixed repeating decimal

For 0.1666…, the 1 is nonrepeating and the 6 repeats. Let x = 0.1666…. Multiply by 10 to move past the nonrepeating digit: 10x = 1.666…. Multiply again by 10 so the repeat lines up: 100x = 16.666…. Subtract 10x from 100x: 90x = 15, so x = 15/90 = 1/6. Equivalently, multiply once by 10^(m+n) and once by 10^m, where m is the number of nonrepeating decimal digits and n is the number of repeating digits.

A reliable procedure

1. Let the decimal equal x.
2. Count the nonrepeating digits and the digits in one complete repeating block.
3. Multiply x by powers of 10 until the repeating tails line up.
4. Subtract the equations to remove the repeating decimal.
5. Solve for x and reduce the fraction.
6. Check by dividing the numerator by the denominator to see whether the original repeating decimal returns.

Key takeaway

Use powers of 10 to align one full repeat block, subtract to cancel the infinite tail, and reduce. For mixed repeats, account for both the nonrepeating prefix and the repeating block.

Use place value to remove the repeating part

A repeating decimal represents a rational number, so it can be written as a fraction. Let x equal the decimal. Multiply by a power of 10 that shifts one complete repeating block to the left of the decimal point. Subtract the original equation from the shifted equation; the repeating tails then cancel. Solve the remaining linear equation and reduce the fraction.

For 0.272727…, let x = 0.272727…. The repeating block has two digits, so 100x = 27.272727…. Subtract x: 99x = 27. Therefore x = 27/99 = 3/11. Multiplying by 10 would not align the repeating block; the decimal tails would differ rather than cancel cleanly.

Handle a nonrepeating prefix

When digits before the repeating block do not repeat, use two shifts. For 0.1666…, let x = 0.1666…. One shift gives 10x = 1.666…. A second shift gives 100x = 16.666…. Subtract the first shifted equation from the second: 90x = 15, so x = 15/90 = 1/6. The shift difference, rather than the total number of digits after the decimal, determines the denominator.

For 2.1343434…, the first digit after the decimal is a nonrepeating prefix and 34 repeats. Set x = 2.1343434…. Shift past the prefix with 10x = 21.343434…, then past one full repeating block with 1000x = 2134.343434…. Subtract: 990x = 2113, giving x = 2113/990. Check by division or by reversing the algebra to confirm the repeating pattern.

Build the denominator from the shifts

For a pure repeating block of k digits, shifting by 10ᵏ and subtracting produces a denominator of 10ᵏ − 1: 9 for one digit, 99 for two, 999 for three. With a nonrepeating prefix of m digits and a repeating block of k digits, the subtraction denominator is 10ᵐ(10ᵏ − 1). The numerator comes from the two shifted whole-number parts, so write both equations before subtracting rather than trying to memorize a numerator rule.

A zero can be part of the repeating block. For 0.0454545…, the repeating block 45 begins after one prefix digit. The method gives 10x = 0.454545… and 1000x = 45.454545…. Subtract to get 990x = 45, so x = 45/990 = 1/22. Reducing the fraction is part of the final answer.

• Count the nonrepeating prefix and the full repeating block separately.
• Use powers of 10 that align identical repeating tails.
• Subtract the equations in the same order on both sides.
• Reduce the resulting fraction by a common factor.
• Check with decimal division, especially when the prefix includes zero.

Reduce after subtracting the aligned decimals

Once subtraction produces a fraction, reduce it by the greatest common factor. For 0.454545…, let x = 0.454545…. Since the two-digit block 45 repeats, 100x = 45.454545…. Subtract x to get 99x = 45, so x = 45/99 = 5/11. A calculator check should reproduce a repeating 45 pattern.

A repeating block with leading zeros still uses its full length when choosing the power of ten. For 0.007007007…, the block 007 has three digits: 1000x = 7.007007…, and subtracting x gives 999x = 7, or 7/999. Dropping the leading zeros from the block would use the wrong shift.

Watch for a decimal with both fixed and repeating digits

For 1.258888…, the prefix after the decimal is 25 and the repeating block is 8. Let x = 1.258888…. Shift two places to pass the prefix: 100x = 125.8888…. Shift one more place to align the repeating 8: 1000x = 1258.8888…. Subtract the first equation from the second: 900x = 1133, so x = 1133/900. Verify by long division or a calculator.

Keep the whole number in the variable before shifting. An alternate approach is to convert the fractional decimal part and then add the integer, but the equations must still align the repeating portion exactly. Reducing at the end keeps the algebra transparent and prevents a denominator error.