Math Core

Every repeating decimal is rational, so it must be equal to some fraction. But which one? You can't just put the digits over $10$ or $100$ , because the digits never stop. This lesson shows a neat algebra trick that makes the endless part disappear.

The big idea: subtract away the repeating part

Let's find the fraction equal to $0.\overline{7} = 0.777\ldots$

Call the number $x$ . Multiplying by $10$ shifts every digit one place to the left, but the tail of $7$ s still goes on forever:

\begin{aligned} 10x &= 7.777\ldots \\ x &= 0.777\ldots \end{aligned}

Both numbers have exactly the same endless tail after the decimal point. So when you subtract, the tails cancel:

\begin{aligned} 10x - x &= 7.777\ldots - 0.777\ldots \\ 9x &= 7 \\ x &= \frac{7}{9} \end{aligned}

So $0.\overline{7} = \dfrac{7}{9}$ . Check by dividing: $7 \div 9 = 0.777\ldots$ It works.

Converting a repeating decimal to a fraction

Let $x$ equal the repeating decimal.
Multiply by $10$ if one digit repeats, by $100$ if two digits repeat, by $1000$ if three repeat, and so on. This lines up the repeating parts.
Subtract $x$ from the new equation so the repeating tails cancel.
Solve for $x$ and simplify the fraction.

Longer repeating blocks

If two digits repeat, multiplying by $10$ doesn't line up the tails. You need to shift by the whole block, so multiply by $100$ .

Worked example: A two-digit block

Write $0.\overline{36}$ as a fraction in lowest terms.

Let $x = 0.363636\ldots$ Two digits repeat, so multiply by $100$ :

\begin{aligned} 100x &= 36.363636\ldots \\ x &= 0.363636\ldots \\ 99x &= 36 \\ x &= \frac{36}{99} = \frac{4}{11} \end{aligned}

Check: $4 \div 11 = 0.3636\ldots$ Correct.

Tip

Notice a pattern: $0.\overline{7} = \dfrac{7}{9}$ and $0.\overline{36} = \dfrac{36}{99}$ . When the repeating block starts right after the decimal point, the fraction is the block over the same number of $9$ s. For example, $0.\overline{123} = \dfrac{123}{999}$ . Always simplify afterward.

Decimals with a whole-number part

A number like $2.\overline{5}$ is just $2 + 0.\overline{5}$ . You can convert the decimal part and add, or use the same method on the whole number.

Worked example: A whole-number part

Write $2.\overline{5}$ as a fraction.

Let $x = 2.555\ldots$ Then $10x = 25.555\ldots$ Subtract:

\begin{aligned} 10x - x &= 25.555\ldots - 2.555\ldots \\ 9x &= 23 \\ x &= \frac{23}{9} \end{aligned}

Check with the other way: $2 + \dfrac{5}{9} = \dfrac{18}{9} + \dfrac{5}{9} = \dfrac{23}{9}$ . Both methods agree.

When the repeat starts later

In $0.1\overline{6} = 0.1666\ldots$ , the $1$ does not repeat. The trick still works; you just need two equations whose tails match.

Worked example: A delayed repeat

Write $0.1\overline{6}$ as a fraction in lowest terms.

Let $x = 0.1666\ldots$

$10x = 1.666\ldots$ (the repeating part now starts right after the decimal point)
$100x = 16.666\ldots$ (shifted one more place; same tail)

Subtract the first from the second:

\begin{aligned} 100x - 10x &= 16.666\ldots - 1.666\ldots \\ 90x &= 15 \\ x &= \frac{15}{90} = \frac{1}{6} \end{aligned}

Check: $1 \div 6 = 0.1666\ldots$ Correct.

Common mistake

The shortcut "block over $9$ s" only works when the repeat starts right after the decimal point. $0.1\overline{6}$ is not $\dfrac{16}{99}$ or $\dfrac{16}{90}$ . When some digits don't repeat, use the subtraction method and make sure the two numbers you subtract have exactly the same tail.

A surprising result

What fraction is $0.\overline{9} = 0.999\ldots$ ? Use the method: $10x = 9.999\ldots$ , so $9x = 9$ and $x = 1$ . The decimal $0.999\ldots$ is exactly equal to $1$ . It's not "just less than" $1$ : there is no number that fits between them.