Repeating Decimals to Fractions

How do you convert a repeating decimal to a fraction?

Converting a repeating decimal into a fraction requires an algebraic method that eliminates the infinite decimal tail.

Because the digits go on forever, you cannot simply place the decimal over a power of ten as you would with terminating decimals. Instead, you create two equations that share the exact same infinite repeating part.

When you subtract the smaller equation from the larger one, the infinite tails perfectly cancel each other out, leaving a whole number. This method allows you to rewrite recurring decimals as exact fractions.

Use notation for the repeating block

Before converting, it is essential to identify exactly which digits repeat.

The repeating sequence is called the repetend. In mathematics, this infinite repetition is shown using either a horizontal bar or a dot placed above the repeating digits.

If only one digit repeats, a single dot or a short bar is placed over it. If a block of multiple digits repeats, a bar covers the entire repeating block, or dots are placed over the first and last digits of the block.

Set up two aligned equations

To begin the conversion, set a variable such as $x$ equal to the original repeating decimal.

Next, multiply both sides of the equation by a power of ten. The power of ten depends on how many digits are in the repeating block.

If one digit repeats, multiply by $10$. If two digits repeat, multiply by $100$. If three digits repeat, multiply by $1{,}000$. This shifts the decimal point exactly one full repeating block to the right, creating a new equation with the same infinite tail.

Subtract to eliminate repetition

Place the new multiplied equation directly above the original equation, ensuring the decimal points are perfectly aligned.

When you subtract the original equation from the multiplied one, the identical repeating digits after the decimal point cancel each other out completely.

This step removes the infinite decimal entirely, replacing it with a simple linear equation that contains only whole numbers.

Solve and simplify

After subtraction, you will be left with an equation of the form $9x = \text{integer}$, $99x = \text{integer}$, or $990x = \text{integer}$.

Divide both sides by the coefficient of $x$ to isolate the variable. The resulting expression is the fractional equivalent of the repeating decimal.

Finally, reduce the fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor. Always look for common factors like $2$, $3$, $5$, or $9$ when simplifying fractions.

Handle a non-repeating prefix

Sometimes a decimal contains a mixture of non-repeating and repeating digits, such as $0.8\overline{3}$.

To convert these, you must set up two new equations that both place the decimal point immediately after a repeating block. Multiply the original variable by a power of ten large enough to shift the non-repeating prefix to the left of the decimal. Then, multiply by another power of ten to shift one full repeating block past the decimal.

Subtracting these two new equations will perfectly align and eliminate the repeating tails.

Worked examples

Example 1: Converting a single repeating digit

Question: Write $0.\overline{5}$ as a fraction in simplest form.

Method:

1. Define the decimal as a variable: $x = 0.5555\dots$
2. Multiply by $10$ because one digit repeats: $10x = 5.5555\dots$
3. Subtract $x$ from $10x$ to get $9x = 5$.
4. Solve for $x$ to get $x = \dfrac{5}{9}$.

Answer: $\dfrac{5}{9}$

Check: Using division, $5 \div 9 = 0.555\dots$, which verifies the answer.

Example 2: Converting a two-digit repeating block

Question: Write $0.\overline{72}$ as a fraction in simplest form.

Method:

1. Define the decimal as a variable: $x = 0.727272\dots$
2. Multiply by $100$ because two digits repeat: $100x = 72.727272\dots$
3. Subtract $x$ from $100x$ to get $99x = 72$.
4. Solve for $x$ to get $x = \dfrac{72}{99}$.
5. Simplify the fraction by dividing the numerator and denominator by $9$ to get $x = \dfrac{8}{11}$.

Answer: $\dfrac{8}{11}$

Check: Multiplying $8$ by $9$ gives $72$, and $11$ by $9$ gives $99$, confirming the simplified form.

Example 3: Converting a decimal with a non-repeating prefix

Question: Write $0.1\overline{6}$ as a fraction in simplest form.

Method:

1. Define the decimal as a variable: $x = 0.16666\dots$
2. Multiply by $10$ to move the non-repeating prefix past the decimal point: $10x = 1.6666\dots$
3. Multiply the original variable by $100$ to move one full repeating block past the decimal point: $100x = 16.6666\dots$
4. Subtract $10x$ from $100x$ to get $90x = 15$.
5. Solve for $x$ to get $x = \dfrac{15}{90}$.
6. Simplify the fraction by dividing the numerator and denominator by $15$ to get $x = \dfrac{1}{6}$.

Answer: $\dfrac{1}{6}$

Check: Dividing $1$ by $6$ yields $0.1666\dots$, verifying the result.

Common mistakes

A frequent error is multiplying by the wrong power of ten. Always count the number of digits under the repetition bar. If there are two repeating digits, you must multiply by $100$, not $10$.

Another common mistake is misaligning the decimal points during subtraction, especially when a non-repeating prefix is present. Ensure the repeating tails perfectly match before subtracting. If the tails do not match, the subtraction will leave an unwanted decimal remainder rather than a whole number.

Finally, students often forget to simplify the resulting fraction. Every valid decimal expansion of rational numbers corresponds to a fraction in its lowest terms, a foundational property of rational numbers.

Frequently asked questions

What does the bar over a decimal mean?

A horizontal bar over one or more digits in a decimal indicates that those specific digits repeat infinitely. For example, $0.\overline{8}$ represents $0.8888\dots$ continuing without end.

Can every repeating decimal be converted to a fraction?

Yes, every repeating decimal represents a rational number, which means it can always be written as a fraction where the numerator and denominator are both integers.

Why do we get denominators with nines?

When you subtract an $x$ variable from a power of ten like $10x$ or $100x$, the resulting coefficient is a sequence of nines, such as $9x$ or $99x$. This coefficient becomes the denominator when you isolate the variable.