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Decimal to Fraction Converter

Convert any decimal number to a simplified fraction with step-by-step breakdown.

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Decimal to Fraction Converter

Convert any decimal number to a simplified fraction with step-by-step breakdown.

Key values: 0.75 · = 3/4

Convert 0.333... (Repeating) to a Fraction

The classic repeating decimal — what fraction does 0.333... equal?

Key values: 0.333... · = 1/3

Documentation

Converting Decimals to Fractions

Every decimal that either terminates or eventually repeats can be written as a fraction. The method depends on which type of decimal you have.

Terminating Decimals

A terminating decimal has a finite number of digits. To convert it, write it over the appropriate power of 10 and simplify.

0.625 = \frac{625}{1000} = \frac{625 \div 125}{1000 \div 125} = \frac{5}{8}

In general, if the decimal has $n$ digits after the point:

\text{decimal} = \frac{\text{digits as integer}}{10^n}

Then reduce by dividing both by their GCD.

More Examples

Decimal	As fraction	Simplified
0.5	$\frac{5}{10}$	$\frac{1}{2}$
0.75	$\frac{75}{100}$	$\frac{3}{4}$
0.125	$\frac{125}{1000}$	$\frac{1}{8}$
0.4	$\frac{4}{10}$	$\frac{2}{5}$
2.35	$\frac{235}{100}$	$\frac{47}{20} = 2\frac{7}{20}$

Repeating Decimals

A repeating decimal has a block of digits that cycles forever. The conversion uses an elegant algebraic trick.

Purely Repeating

For $0.\overline{r}$ where $r$ has $k$ digits:

0.\overline{r} = \frac{r}{10^k - 1}

Why it works: Let $x = 0.\overline{r}$ . Multiply by $10^k$ :

10^k x = r.\overline{r} = r + x

Subtract: $10^k x - x = r$ , so $x = \frac{r}{10^k - 1}$ .

$0.\overline{3} = \frac{3}{9} = \frac{1}{3}$

$0.\overline{142857} = \frac{142857}{999999} = \frac{1}{7}$

Mixed Repeating Decimals

When some digits don't repeat, like $0.1\overline{6}$ (= $0.16666\ldots$ ), with $m$ non-repeating digits and $k$ repeating digits:

x = \frac{\text{(all digits)} - \text{(non-repeating digits)}}{\underbrace{9\ldots9}_{k}\underbrace{0\ldots0}_{m}}

For $0.1\overline{6}$ : $\frac{16 - 1}{90} = \frac{15}{90} = \frac{1}{6}$ .

Which Fractions Terminate?

A fraction $\frac{a}{b}$ in lowest terms produces a terminating decimal if and only if the prime factorization of $b$ contains only 2s and 5s:

b = 2^m \times 5^n \quad \text{for some } m, n \geq 0

Denominator	Factors	Terminates?
2, 4, 5, 8, 10, 16, 20, 25, 32, 40, 50, 100	Only 2 and/or 5	Yes
3, 6, 7, 9, 11, 12, 13, 14, 15	Contains 3, 7, 11, or 13	No — repeats

Why 2 and 5? Our number system is base 10, and $10 = 2 \times 5$ . A fraction terminates in base $b$ exactly when its denominator divides some power of $b$ . In base 10, that means denominators whose only prime factors are 2 and 5.

Frequently Asked Questions

How do I convert a terminating decimal to a fraction?

Write the decimal digits as the numerator over the appropriate power of 10, then simplify by dividing both by their GCD. For example, $0.625 = \frac{625}{1000}$ . $\gcd(625, 1000) = 125$ , so $\frac{625}{125} = 5$ and $\frac{1000}{125} = 8$ , giving $\frac{5}{8}$ .

How do I convert a repeating decimal to a fraction?

Use the algebraic method: let $x$ equal the repeating decimal, multiply by $10^k$ (where $k$ is the number of repeating digits), subtract the original equation, and solve. For $0.333\ldots$ : let $x = 0.333\ldots$ , then $10x = 3.333\ldots$ , subtract to get $9x = 3$ , so $x = \frac{1}{3}$ .

Which decimals can be written as fractions?

Every decimal that terminates or eventually repeats can be written as an exact fraction. These are precisely the rational numbers. Non-repeating, non-terminating decimals like $\pi$ (3.14159...) and $\sqrt{2}$ (1.41421...) are irrational and cannot be expressed as fractions.

How can I tell if a fraction will produce a terminating or repeating decimal?

A fraction $\frac{a}{b}$ in lowest terms terminates if and only if the denominator $b$ has no prime factors other than 2 and 5. For example, $\frac{1}{8}$ ( $8 = 2^3$ ) terminates as 0.125, but $\frac{1}{3}$ repeats because 3 is not a factor of any power of 10.

What is a mixed repeating decimal and how do I convert it?

A mixed repeating decimal has some non-repeating digits followed by a repeating block, like $0.1\overline{6}$ (= $0.1666\ldots$ ). The formula is: (all digits - non-repeating digits) / (9s for each repeating digit followed by 0s for each non-repeating digit). For $0.1\overline{6}$ : $\frac{16 - 1}{90} = \frac{15}{90} = \frac{1}{6}$ .