Writing a Repeating Decimal as a Fraction

The four-step setup

Step 1: let $x$ equal the repeating decimal. For $0.\overline{7}$, write $x = 0.777\ldots$

Step 2: multiply both sides by $10$ for each repeating digit. One repeating digit means multiply by $10$; two repeating digits mean multiply by $100$. This slides the decimal point over exactly one full repeating block.

Step 3: subtract the original equation from the new one. Both decimals have the same infinite tail, so the tails cancel and only whole numbers survive.

Step 4: solve for $x$ and simplify the fraction.

Why the multiplier matters

The subtraction only works if the two decimals line up tail-to-tail. Multiplying $x = 0.454545\ldots$ by $100$ gives $100x = 45.4545\ldots$ — same tail, shifted by one whole block, so subtracting kills everything after the decimal point.

Multiply by the wrong power of $10$ and the tails don't match. $10x = 4.54545\ldots$ has its pattern out of phase with $x = 0.45454\ldots$, and the subtraction leaves a mess instead of a whole number. Count the repeating digits, then match the zeros: one digit, $10$; two digits, $100$; three digits, $1000$.

A shortcut worth knowing

When the repeating block starts right after the decimal point, the method always produces the block over a string of $9$s: $0.\overline{4} = \dfrac{4}{9}$, $0.\overline{45} = \dfrac{45}{99}$, $0.\overline{123} = \dfrac{123}{999}$.

That's a great answer check, but don't stop there — the fraction usually simplifies. $\dfrac{45}{99}$ reduces to $\dfrac{5}{11}$, and most multiple-choice answers are written in simplest form.

Worked examples

Try one yourself

Common questions