How do you convert a repeating decimal to a fraction?

Set the decimal equal to a variable $x$. Multiply both sides by a power of 10 that shifts the repeating block one full cycle to the left. Subtract the original equation from the new one — the repeating digits cancel out, leaving a simple equation. Solve for $x$ and simplify the resulting fraction.

How do you convert 0.5 to a fraction?

Since 0.5 has one decimal place, write it as $\frac{5}{10}$. Then divide both the numerator and denominator by 5 to get $\frac{1}{2}$.

What fraction is 0.125?

The decimal 0.125 has three decimal places, so it equals $\frac{125}{1000}$. The GCF of 125 and 1000 is 125, so dividing both gives $\frac{1}{8}$.

Converting Decimals to Fractions — Definition, Formula & Examples

Converting decimals to fractions is the process of rewriting a decimal number as a fraction with a whole-number numerator and denominator. For example, 0.75 becomes

\frac{3}{4}

To convert a terminating decimal to a fraction, express the decimal as a ratio of the decimal's digits (without the decimal point) over a power of 10 determined by the number of decimal places, then simplify to lowest terms. For repeating decimals, an algebraic method is used: multiply by a power of 10 that shifts the repeating block, then subtract the original equation to eliminate the repeating portion and solve for the fractional equivalent.

Key Formula

\text{decimal with } n \text{ decimal places} = \frac{\text{digits after the decimal point}}{10^n}

Where:

$n$ = The number of digits after the decimal point
$10^n$ = The place-value denominator: 10 for one decimal place, 100 for two, 1000 for three, etc.

How It Works

For a terminating decimal, count the number of digits after the decimal point. Write the digits as the numerator and place a 1 followed by that many zeros as the denominator. Then simplify the fraction by dividing both numerator and denominator by their greatest common factor (GCF). For instance, 0.6 has one decimal place, so it becomes

\frac{6}{10}

, which simplifies to

\frac{3}{5}

. For a repeating decimal like

0.\overline{3}

, set

x = 0.333\ldots

, multiply both sides by 10 to get

10x = 3.333\ldots

, subtract the original equation to get

9x = 3

, and solve to find

x = \frac{1}{3}

. If the decimal has a whole-number part (like 2.75), convert only the decimal part and then add it back to the whole number, or convert the entire value at once.

Worked Example

Problem: Convert 0.375 to a fraction in simplest form.

Step 1: Count the decimal places. There are 3 digits after the decimal point.

0.375 \text{ has } n = 3 \text{ decimal places}

Step 2: Write the digits as the numerator and

10^3 = 1000

as the denominator.

0.375 = \frac{375}{1000}

Step 3: Find the greatest common factor of 375 and 1000. Both are divisible by 125.

\gcd(375, 1000) = 125

Step 4: Divide numerator and denominator by 125 to simplify.

\frac{375 \div 125}{1000 \div 125} = \frac{3}{8}

Answer:

0.375 = \dfrac{3}{8}

Another Example

This example shows the algebraic technique for repeating decimals, which cannot use the simple 'power of 10' method for terminating decimals.

Problem: Convert the repeating decimal

0.\overline{27}

to a fraction in simplest form.

Step 1: Let

x

equal the repeating decimal.

x = 0.272727\ldots

Step 2: The repeating block "27" has 2 digits, so multiply both sides by

10^2 = 100

100x = 27.272727\ldots

Step 3: Subtract the original equation from the new one to cancel the repeating part.

100x - x = 27.2727\ldots - 0.2727\ldots \implies 99x = 27

Step 4: Solve for

x

and simplify by dividing numerator and denominator by their GCF, which is 9.

x = \frac{27}{99} = \frac{27 \div 9}{99 \div 9} = \frac{3}{11}

Answer:

0.\overline{27} = \dfrac{3}{11}

Visualization

Why It Matters

This skill appears constantly in middle-school and pre-algebra courses when you need exact values instead of rounded decimals. Careers in engineering, pharmacy, and cooking rely on fraction equivalents for precise measurements. Standardized tests like the SAT and GRE frequently require switching between decimal and fraction forms to compare or simplify expressions.

Common Mistakes

Mistake: Using the wrong power of 10 as the denominator — for example, writing 0.25 as

\frac{25}{10}

instead of

\frac{25}{100}

Correction: Count the decimal places carefully. Two digits after the decimal point means the denominator is

10^2 = 100

Mistake: Forgetting to simplify the fraction after forming it.

Correction: Always find the GCF of the numerator and denominator and divide both by it.

\frac{75}{100}

should become

\frac{3}{4}

, not stay as

\frac{75}{100}

Mistake: Trying the terminating-decimal method on a repeating decimal, leading to an incorrect or never-ending numerator.

Correction: For repeating decimals, use the algebraic method: set

x

equal to the decimal, multiply to shift the repeating block, and subtract to eliminate it.

Check Your Understanding

Convert 0.45 to a fraction in simplest form.

Hint: Write as

\frac{45}{100}

and find the GCF of 45 and 100.

Answer:

\dfrac{9}{20}

Convert

0.\overline{6}

to a fraction.

Hint: Let

x = 0.666\ldots

, then multiply by 10 and subtract.

Answer:

\dfrac{2}{3}

Convert 2.08 to a mixed number in simplest form.

Hint: Convert just the 0.08 part:

\frac{8}{100}

simplifies to

\frac{2}{25}

Answer:

2\dfrac{2}{25}

Related Terms

Decimal — The starting value you convert from
Fraction — The resulting form after conversion
Numerator — Top number in the resulting fraction
Denominator — Bottom number, a power of 10 before simplifying
Fraction Rules — Rules used to simplify the resulting fraction
Improper Fraction — Result when decimal is greater than or equal to 1
Mixed Number — Alternative form for decimals greater than 1
Proper Fraction — Result when the decimal is between 0 and 1
Least Common Denominator — Used when comparing converted fractions
Ratio — Fractions and ratios share the same notation