Decimal to Fraction Conversion: Formula, Rules and Examples

Decimal to Fraction Conversion changes a decimal number into an equivalent fraction in simplest form. Terminating decimals use a power of 10 as the denominator, while recurring decimals require algebraic conversion. This page covers the decimal to fraction formula, reduction rules, recurring decimal methods and solved examples.

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What Is Decimal to Fraction Conversion?

Decimal to fraction conversion expresses a decimal number as a fraction whose numerator and denominator are integers. The fraction is then reduced to its simplest form.

For a terminating decimal with n digits after the decimal point, remove the decimal point and divide the resulting integer by 10^n. For example, 0.375 = 375/1000 = 3/8. A recurring decimal is converted using an algebraic equation because its digits continue indefinitely.

Decimal to Fraction Conversion Formula & Tricks

Important Formulas

Terminating decimal formula
Decimal with n places = integer formed by removing the decimal point / 10^n

Count the digits after the decimal point, use 1 followed by n zeros as the denominator, and reduce the fraction.

Pure recurring decimal formula
0.overline{A} = A / (10^n - 1)

If an n-digit block A repeats immediately after the decimal point, place that block over a denominator containing n nines.

Mixed recurring decimal formula
0.B overline{A} = (BA - B) / [10^m(10^n - 1)]

Here B is the m-digit non-repeating part and A is the n-digit repeating block. BA means the digits of B followed by A.

Quick Tricks

Use the number of decimal places

For a terminating decimal, the denominator is 10, 100, 1000 and so on according to the number of decimal digits.

Example: 0.48 has two decimal places, so 0.48 = 48/100 = 12/25.
Cancel trailing zeros before reducing

Zeros at the end of a decimal can be removed without changing its value. This can make the fraction conversion shorter.

Example: 2.500 = 2.5 = 25/10 = 5/2.
Use nines for a repeating block

A repeating block of one, two or three digits is placed over 9, 99 or 999 respectively when it starts immediately after the decimal point.

Example: 0.overline{27} = 27/99 = 3/11.

Decimal to Fraction Conversion Concepts

Terminating Decimal to Fraction

A terminating decimal has a finite number of digits after the decimal point and is converted by using a power of 10 as the denominator.

Remove the decimal point, write the resulting number over 10 raised to the number of decimal places, and simplify. For a decimal greater than 1, the complete number is used as the numerator after removing the decimal point.

Example: 4.375 = 4375/1000 = 35/8.

Decimal Values Less Than 1

For a decimal between 0 and 1, the digits after the decimal point form the numerator, while the denominator is the corresponding power of 10.

The zero before the decimal point does not affect the numerator. After forming the fraction, divide the numerator and denominator by their greatest common divisor.

Example: 0.56 = 56/100 = 14/25.

Repeating Decimal Conversion

A pure recurring decimal is converted by placing its repeating digit block over an equal number of nines and then simplifying.

If x = 0.overline{A}, multiply x by 10^n, subtract x, and solve: 10^n x - x = A. Therefore, x = A/(10^n - 1).

Example: Let x = 0.overline{36}. Then 100x - x = 36, so 99x = 36 and x = 36/99 = 4/11.

Mixed Recurring Decimal Conversion

A mixed recurring decimal has a non-repeating part before the repeating block and is converted by subtracting two shifted equations.

If m digits do not repeat and n digits repeat, the denominator is 10^m(10^n - 1). The numerator is the digits formed by joining the non-repeating and repeating parts minus the non-repeating part.

Example: 0.1overline{6} = (16 - 1)/[10(9)] = 15/90 = 1/6.

Simplifying the Converted Fraction

The fraction obtained from a decimal must be reduced by dividing its numerator and denominator by their greatest common divisor.

A fraction is in simplest form when the numerator and denominator have no common factor other than 1. For terminating decimals, factor cancellation may be done directly with powers of 10.

Example: 0.625 = 625/1000. Dividing both terms by 125 gives 5/8.

Decimal to Fraction Conversion Video Lessons

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Practice Decimal to Fraction Conversion Questions

Practise published questions related to this topic.

1Find the value of m which satisfies (26/7)⁵ × (7/26)¹¹ × (26/7)¹⁹ = (7/26)^(9m+17)→ 2A clerk simplifies (√20)(√5) for budget entries. The result is:→ 3If (3⁴ × 2⁵ × 1080)/360 = 6^p, what is the value of P² - 5p + 1 ?→ 4Solve √(20+√(20+√(20+...)))→ 5The Simplified value of (∛4913/⁴√1296) × 3/51 × 168 is→ 6If 2^(x-1) + 2^(x+1) = 160, then the value of 2x + 1 is:→ 7Simplify: 72 + √32 - √18→ 8The value of √((5 + 2√6)) is:→ 9Simplify the following. 5¹⁵ ÷ 25^(13/2) ÷ 125^(2/3) + 5¹⁷ ÷ 125^(16/3) + 5³→ 10Find the value of m which satisfies (22/5)¹⁸ × (5/22)⁸ × (22/5)⁶ = (5/22)^(9m+4) .→

Decimal to Fraction Conversion Quick Quiz

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Quick Revision Notes

Decimal to Fraction Conversion: Quick Revision

Use these rules to convert and simplify decimal values.

  • A terminating decimal with n digits after the decimal point has denominator 10^n before simplification.
  • Remove the decimal point to form the numerator, then reduce the fraction.
  • A one-digit pure recurring block uses denominator 9; two digits use 99; three digits use 999.
  • For a mixed recurring decimal, use (joined digits - non-repeating digits) / [10^m(10^n - 1)].
  • Trailing zeros do not change a decimal value: 0.700 = 0.7.
  • Always express the final fraction in simplest form.

Decimal to Fraction Conversion FAQs

What is the decimal to fraction formula for 0.875?

There are three decimal places, so 0.875 = 875/1000. Dividing by 125 gives 7/8.

How is 0.overline{4} converted into a fraction?

A one-digit repeating block is placed over 9: 0.overline{4} = 4/9.

How do you convert 0.overline{27} into a fraction?

The repeating block has two digits, so 0.overline{27} = 27/99 = 3/11.

What is the fraction form of 0.2overline{5}?

Using the mixed recurring formula, 0.2overline{5} = (25 - 2)/[10(9)] = 23/90.

How do you convert 3.06 into a fraction?

3.06 = 306/100. Dividing numerator and denominator by 2 gives 153/50.

What is the fraction form of 0.04?

0.04 = 4/100 = 1/25. The two decimal places require a denominator of 100 before simplification.

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