Decimal Fraction Conversion Calculator

Use this calculator to convert a decimal number to a fraction or mixed number. It can handle terminating decimals and repeating decimals when the repeating-digit count is specified. The result is reduced to lowest terms whenever possible.

A decimal that ends and a decimal that repeats forever must be interpreted differently. For example, the terminating value 0.33333333 is exactly 33333333/100000000, while the repeating decimal 0.333... is exactly 1/3. Enter the repeating information correctly when the digits continue indefinitely.

How to Use the Decimal to Fraction Converter

1. Enter the decimal value, including a minus sign if the value is negative.

2. For a terminating decimal, set the number of repeating decimal places to zero.

3. For a repeating decimal, enter the number of digits in the repeating block.

4. Read the simplified fraction and, when the absolute value is at least one, the mixed-number result.

Example: for 0.1666..., the non-repeating digit is 1 and the repeating block is 6, so the repeating-digit count is 1. The exact result is 1/6.

What Are Decimals and Fractions?

A decimal uses place value based on powers of ten. Digits to the right of the decimal point represent tenths, hundredths, thousandths, and smaller powers of ten.

A fraction is written as a numerator divided by a denominator. In a/b, a is the numerator and b is the denominator, with b not equal to zero. Fractions that represent the same value are equivalent; a fraction is in lowest terms when its numerator and denominator have no common factor greater than one.

A rational number can be written as a fraction of two integers. Its decimal representation either terminates or eventually repeats. A nonterminating, nonrepeating decimal cannot be represented exactly as a fraction of two integers.

How to Convert a Terminating Decimal to a Fraction

If a terminating decimal has n digits after the decimal point, remove the decimal point to form an integer numerator and use 10ⁿ as the denominator. Then divide the numerator and denominator by their greatest common divisor.

Decimal = Integer formed from the digits ÷ 10ⁿ

Example: Convert 0.375 to a Fraction

1. There are three decimal places, so the denominator is 10³ = 1000.

2. Write 0.375 as 375/1000.

3. The greatest common divisor of 375 and 1000 is 125.

4. Divide both values by 125: 375/1000 = 3/8.

0.375 = 3/8

Example: Convert −2.45 to a Fraction

Keep the negative sign, write the digits over 100, and reduce:

−2.45 = −245/100 = −49/20 = −2 9/20

How to Convert a Repeating Decimal to a Fraction

Parentheses are used below to identify a repeating block. For example, 0.(3) means 0.333..., and 0.1(6) means 0.1666....

Pure Repeating Decimal

If all decimal digits repeat and the repeating block has n digits, place the repeating block over a denominator containing n nines, then reduce the fraction.

0.(27) = 27/99 = 3/11

The same result can be derived algebraically. Let x = 0.272727.... Multiplying by 100 gives 100x = 27.272727.... Subtracting x removes the repeating part:

100x − x = 27, so 99x = 27 and x = 3/11

Decimal with Non-Repeating and Repeating Digits

For a decimal with m non-repeating decimal digits followed by a repeating block of n digits, use:

Fraction = (digits through one complete repeat − non-repeating leading digits) ÷ (10^m+n − 10^m)

Example: Convert 0.1(6) to a Fraction

There is one non-repeating digit and one repeating digit:

0.1(6) = (16 − 1) ÷ (100 − 10) = 15/90 = 1/6

Example: Convert 2.3(45) to a Fraction

Use the digits through one repeating block, including the whole-number part:

2.3(45) = (2345 − 23) ÷ (1000 − 10) = 2322/990 = 129/55 = 2 19/55

How to Convert a Fraction to a Decimal

Divide the numerator by the denominator:

Decimal value = Numerator ÷ Denominator

• 3/8 = 3 ÷ 8 = 0.375

• 7/4 = 7 ÷ 4 = 1.75

• 1/6 = 1 ÷ 6 = 0.1666...

• −11/5 = −11 ÷ 5 = −2.2

After a fraction is reduced to lowest terms, its decimal terminates only when the denominator has no prime factors other than 2 and 5. For example, 3/40 terminates because 40 = 2³ × 5, while 1/6 repeats because the denominator includes a factor of 3.

Fractions and Mixed Numbers

A proper fraction has an absolute numerator smaller than its denominator, such as 3/8. An improper fraction has an absolute numerator equal to or greater than its denominator, such as 17/5. A mixed number combines a whole-number part and a proper fraction.

Convert an Improper Fraction to a Mixed Number

1. Divide the numerator by the denominator.

2. Use the quotient as the whole-number part.

3. Use the remainder as the new numerator.

4. Keep the original denominator and reduce the fractional part if needed.

17/5 = 3 remainder 2 = 3 2/5

Convert a Mixed Number to an Improper Fraction

Multiply the whole number by the denominator, add the numerator, and place the result over the original denominator:

3 2/5 = (3 × 5 + 2)/5 = 17/5

For a negative mixed number, the minus sign applies to the entire value. Thus −3 2/5 = −17/5.

Decimal to Fraction Conversion Table

An ellipsis indicates that the displayed digit or digit block repeats forever. These repeating values are exact only when the repetition continues indefinitely.

Terminating Decimal Place Values

Exact Values, Rounded Values, and Approximation

A calculator can only return the exact intended fraction when the decimal input describes the intended value unambiguously. Consider these two inputs:

• 0.33333333 with no repeating digits is a terminating decimal and equals 33333333/100000000 exactly.

• 0.333... with 3 repeating forever equals 1/3 exactly.

If a decimal came from a rounded measurement, many fractions may lie close to it. For example, a displayed measurement of 0.33 does not prove that the underlying value is exactly 1/3. Treat a “simple fraction” result as an approximation unless the source value or repeating pattern is known exactly.

Common Conversion Mistakes

• Leaving the result unreduced, such as reporting 375/1000 instead of 3/8.

• Treating a finite string of repeated-looking digits as an infinite repeating decimal.

• Counting the non-repeating digits as part of the repeating block.

• Using zero as a fraction denominator.

• Dropping the negative sign when converting a negative decimal.

• Rounding a decimal before converting it when an exact value is required.

Decimal and Fraction FAQ

Can every decimal be converted to a fraction?

Every terminating decimal and every repeating decimal can be represented exactly as a fraction of integers. A nonterminating decimal with no repeating pattern is irrational and cannot be represented exactly as such a fraction.

Why does the calculator need the number of repeating places?

The repeat length identifies which digits continue forever. For example, 0.(3), 0.(27), and 0.1(6) have repeating blocks of one, two, and one digits respectively. Without that information, a typed digit string is normally interpreted as a terminating decimal.

What is the fastest way to convert a terminating decimal?

Remove the decimal point, place the resulting integer over 1 followed by one zero for each decimal place, and reduce. Thus 0.024 becomes 24/1000, which reduces to 3/125.

How do I know whether a fraction produces a terminating decimal?

First reduce the fraction. Its decimal terminates if the denominator contains only factors of 2 and 5. Otherwise, the decimal repeats.

What happens if the numerator is larger than the denominator?

The result is an improper fraction with an absolute value of at least one. It may also be displayed as a mixed number, such as 9/4 = 2 1/4.

Does 0.999... equal 1?

Yes. Let x = 0.999.... Then 10x = 9.999.... Subtracting gives 9x = 9, so x = 1. This applies to an infinite repeating decimal, not to a finite input such as 0.999.

References

• OpenStax Elementary Algebra: Decimals

• OpenStax Prealgebra 2e: Decimals and Fractions

• OpenStax Prealgebra 2e: Fractions and Mixed Numbers

• OpenStax College Algebra 2e: Repeating Decimals and Geometric Series