How to Convert a Decimal to a Fraction

What fraction is 0.2727272727…, with "27" repeating forever? It looks hard. The answer is 3/11, and there's an algebra trick that finds it in two lines.

Converting decimals to fractions comes up in recipes, measurements, woodworking, finance and math class. There are three situations: decimals that end, decimals that repeat, and long decimals you want to approximate. Each has its own method.

Every Repeating Decimal Is Exactly a Fraction

It's tempting to think 0.333… is only approximately 1/3, since it never ends. In fact, every decimal that terminates or eventually repeats is exactly equal to some fraction, and every fraction produces a terminating or repeating decimal. The endless repetition isn't an approximation; it's just how base 10 writes certain fractions.

Only irrational numbers, like π and √2, have decimals that never end and never repeat, and those can't be written as fractions at all.

Method 1: Terminating Decimals

Step 1: Write the decimal digits over the matching power of 10.

Step 2: Simplify by dividing by the greatest common factor.

Example: 0.375

Three decimal places means a denominator of 1,000:

0.375 = 375/1000

The GCF of 375 and 1,000 is 125:

375 ÷ 125 = 3
1000 ÷ 125 = 8
0.375 = 3/8

Find GCFs quickly with the GCF calculator.

Example: 2.45

2.45 = 245/100 = 49/20 = 2 9/20

Method 2: Repeating Decimals

Step 1: Let x equal the decimal.

Step 2: Multiply by 10ⁿ, where n is the length of the repeating block.

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

Step 4: Solve for x and simplify.

Example: 0.272727…

The repeating block "27" has 2 digits, so multiply by 100:

100x = 27.272727…
    x =  0.272727…
  99x = 27
    x = 27/99 = 3/11

Shortcut: a pure repeating decimal equals the repeating block over the same number of 9s. 0.272727… = 27/99. 0.142857142857… = 142857/999999 = 1/7.

Method 3: Decimals That Start Repeating Later

Example: 0.1666… (only the 6 repeats)

Multiply to move the decimal point past the non-repeating part, and again past one repeating block:

100x = 16.666…
 10x =  1.666…
 90x = 15
   x = 15/90 = 1/6

The general idea: line up two copies whose repeating tails match exactly, and subtract. See the division tables for more repeating decimals.

Method 4: Approximating Long Decimals

Some decimals, like measurements or π, need a simple fraction that's close enough.

Example: 3.14159

Useful approximations come from continued fractions:

3.14159 = 3 + 0.14159
1/0.14159 ≈ 7.0626 → 3 + 1/7 = 22/7

Continuing the process gives better approximations: 333/106, then 355/113, which matches π to six decimal places. For woodworking, you might instead round to the nearest 1/16 or 1/32 of an inch.

Common Decimal–Fraction Pairs

An Insider Reference: Simon Stevin's Decimals

Fractions are thousands of years old, but decimal notation for fractions is relatively recent in Europe. In 1585, Flemish mathematician and engineer Simon Stevin published De Thiende ("The Tenth"), a short pamphlet arguing that everyday calculations, from measuring cloth to computing interest, would be much easier using tenths, hundredths and thousandths.

He even proposed that governments adopt decimal systems for coins, weights and measures, more than two centuries before the metric system. His notation was clumsy, with circled numbers marking each place, but the idea caught on. Scottish mathematician John Napier helped popularize the decimal point in the early 1600s.

Earlier mathematicians in the Islamic world, such as al-Kashi in the 15th century, had already worked extensively with decimal fractions.

Two Concepts Worth Knowing

Simplest Form

A fraction is in simplest form when the numerator and denominator share no common factor other than 1. Divide both by their GCF to get there.

Which Fractions Terminate?

A fraction in simplest form has a terminating decimal only if its denominator's prime factors are just 2s and 5s (the prime factors of 10). 3/8 terminates; 1/3, 1/6 and 3/11 repeat. See the prime factorization tool.

Quick Answer: How Do You Convert a Decimal to a Fraction?

For a terminating decimal, write the digits over a power of 10 and simplify: 0.375 = 375/1000 = 3/8. For a repeating decimal, set x equal to it, multiply by 10ⁿ (n = length of the repeating block), subtract to cancel the repeat, and solve: 0.2727… gives 99x = 27, so x = 3/11.

Try Them Yourself

• GCF Calculator: simplify fractions
• Division Tables: see fractions as decimals
• Prime Factorization Tool: predict terminating decimals
• Basic Calculator: check your conversions
• Why 0.999... Is Exactly Equal to 1: a famous repeating decimal
• Why Some Numbers Cannot Be Written as Fractions: decimals that never repeat

Convert 0.4545…, 0.0833… and 0.6875 to fractions. Then predict, from each fraction's denominator, which ones should repeat.