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Convert a repeating decimal to a fraction

Shift one repeating block, then subtract

If a repeating block has $n$ digits, multiplication by $1 0^{n}$ aligns the repeating tails. Subtract the original number to remove the infinite tail, then solve and reduce the resulting fraction.

Problem

Write $1. \overline{15} = 1.151515 \dots$ as a fraction in simplest form.

Answer

\frac{38}{33}

The bar applies to both digits 15.

Step-by-step solution

1. Name the repeating decimal

Let $x = 1.151515 \dots$ .

2. Shift the two-digit block

Multiply by 100 to get $100 x = 115.151515 \dots$ . The repeating tails now match.

3. Subtract and solve

Subtracting gives $99 x = 114$ , so $x = 114/99$ .

4. Reduce the fraction

The numerator and denominator are divisible by 3. Hence $x = 38/33$ .

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