Convert Rational Numbers to a Fraction

How to convert rational numbers to a fraction, step by step: the rule in plain language, solved examples worked one line at a time, and practice questions so you can try it yourself before moving on.

Free tool: try the problems from this lesson in our Fraction Calculator.

Every rational number can be expressed as a fraction. Converting between forms, decimal, mixed number, or repeating decimal, is a key GED Math skill that shows up in calculation, comparison, and word-problem questions. This lesson walks you through each conversion type with clear, repeatable steps.

What Is a Rational Number?

A rational number is any number that can be written as \(\color{blue}{\frac{a}{b}}\) where \(\color{blue}{a}\) and \(\color{blue}{b}\) are integers and \(\color{blue}{b \ne 0}\). This includes all integers (since, for example, \(\color{blue}{5 = \frac{5}{1}}\)), all terminating decimals, all repeating decimals, and all mixed numbers.

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Converting Rational Numbers to Fractions

1. Converting a Terminating Decimal to a Fraction

Count the number of decimal places. Use that as a power of 10 for the denominator, then simplify.

• \(\color{blue}{0.6 = \frac{6}{10} = \frac{3}{5}}\)
• \(\color{blue}{0.75 = \frac{75}{100} = \frac{3}{4}}\)
• \(\color{blue}{0.125 = \frac{125}{1000} = \frac{1}{8}}\)

2. Converting a Mixed Number to a Fraction

Multiply the whole number by the denominator, add the numerator, and keep the same denominator.

Formula: \(\color{blue}{a \frac{b}{c} = \frac{(a \times c + b)}{c}}\)

• \(\color{blue}{2 \frac{3}{4} = \frac{(2 \times 4 + 3)}{4} = \frac{11}{4}}\)
• \(\color{blue}{5 \frac{1}{3} = \frac{(5 \times 3 + 1)}{3} = \frac{16}{3}}\)

3. Converting a Repeating Decimal to a Fraction

Use algebra. Let \(\color{blue}{x}\) equal the repeating decimal. Multiply both sides by a power of 10 that shifts one full repeat to the left of the decimal. Subtract to eliminate the repeating part. Solve for \(\color{blue}{x}\).

Example: Convert \(\color{blue}{0.333\ldots}\) to a fraction.

Let \(\color{blue}{x = 0.333\ldots}\). Then \(\color{blue}{10x = 3.333\ldots}\). Subtract: \(\color{blue}{10x – x = 3.333\ldots – 0.333\ldots}\), so \(\color{blue}{9x = 3}\), giving \(\color{blue}{x = \frac{3}{9} = \frac{1}{3}}\).

Step-by-Step Summary

1. Terminating decimal: Write digits over the correct power of 10, then simplify by dividing numerator and denominator by their GCF.
2. Mixed number: Multiply whole \(\color{blue}{\text{ number } \times \text{ denominator } + \text{ numerator }}\); put result over same denominator.
3. Repeating decimal: Set equal to \(\color{blue}{x}\), multiply to shift the repeat, subtract, solve for \(\color{blue}{x}\), simplify.

Watch: Converting Any Fraction (Video Lesson)

Math Antics explains how to convert between fraction forms with clear visual steps:

Worked Examples

Example 1: Convert \(\color{blue}{0.4}\) to a fraction in simplest form.

\(\color{blue}{0.4 = \frac{4}{10}}\). GCF of 4 and 10 is 2. Simplify: \(\color{blue}{\frac{4}{10} = \frac{2}{5}}\).

Example 2: Convert \(\color{blue}{3 \frac{2}{5}}\) to an improper fraction.

\(\color{blue}{3 \frac{2}{5} = \frac{(3 \times 5 + 2)}{5} = \frac{(15 + 2)}{5} = \frac{17}{5}}\).

Example 3: Convert \(\color{blue}{0.666\ldots}\) to a fraction.

Let \(\color{blue}{x = 0.666\ldots}\). \(\color{blue}{10x = 6.666\ldots}\). Subtract: \(\color{blue}{9x = 6}\), so \(\color{blue}{x = \frac{6}{9} = \frac{2}{3}}\).

Example 4: Convert \(\color{blue}{0.272727\ldots}\) to a fraction.

The repeat block is 2 digits (27), so multiply by 100: Let \(\color{blue}{x = 0.272727\ldots}\). \(\color{blue}{100x = 27.272727\ldots}\). Subtract: \(\color{blue}{99x = 27}\), so \(\color{blue}{x = \frac{27}{99} = \frac{3}{11}}\).

More Practice: Converting Repeating Decimals Video

This Khan Academy lesson shows the algebraic method for turning repeating decimals into exact fractions:

Exercises

1. Convert \(\color{blue}{0.8}\) to a fraction in simplest form.
2. Convert \(\color{blue}{0.35}\) to a fraction in simplest form.
3. Convert \(\color{blue}{4 \frac{3}{8}}\) to an improper fraction.
4. Convert \(\color{blue}{2 \frac{5}{6}}\) to an improper fraction.
5. Convert \(\color{blue}{0.111\ldots}\) to a fraction.
6. Convert \(\color{blue}{0.363636\ldots}\) to a fraction.

Answers

1. \(\color{blue}{0.8 = \frac{8}{10} = \frac{4}{5}}\)
2. \(\color{blue}{0.35 = \frac{35}{100} = \frac{7}{20}}\)
3. \(\color{blue}{4 \frac{3}{8} = \frac{(4 \times 8 + 3)}{8} = \frac{35}{8}}\)
4. \(\color{blue}{2 \frac{5}{6} = \frac{(2 \times 6 + 5)}{6} = \frac{17}{6}}\)
5. \(\color{blue}{x = 0.111\ldots}\); \(\color{blue}{10x = 1.111\ldots}\); \(\color{blue}{9x = 1}\); \(\color{blue}{x = \frac{1}{9}}\)
6. \(\color{blue}{x = 0.363636\ldots}\); \(\color{blue}{100x = 36.363636\ldots}\); \(\color{blue}{99x = 36}\); \(\color{blue}{x = \frac{36}{99} = \frac{4}{11}}\)

Frequently Asked Questions

How do you convert rational numbers to fractions?

It depends on the form. For a terminating decimal, put the digits over a power of 10 and simplify: 0.75 = 75/100 = 3/4. For a mixed number, multiply the whole number by the denominator and add the numerator: 2 3/4 = 11/4. For an integer, write it over 1, so 5 = 5/1.

Is a rational number a fraction?

Every rational number can be written as a fraction a/b, where a and b are integers and b is not 0. That is actually the definition of a rational number. So integers like −7, terminating decimals like 0.125 and repeating decimals like 0.333… are all rational, because they equal −7/1, 1/8 and 1/3.

Can a fraction be a rational number?

A fraction made of two integers with a nonzero denominator, like 5/8 or −11/4, is always a rational number. A fraction that uses an irrational number, such as π/2 or √2/3, is not rational, even though it looks like a fraction. The top and bottom both have to be integers.

How do you convert a rational number into a mixed fraction?

Start with an improper fraction, then divide the numerator by the denominator. The quotient is the whole number and the remainder becomes the new numerator. For 16/3, 16 ÷ 3 = 5 remainder 1, so 16/3 = 5 1/3. For a decimal like 2.6, first write 26/10 = 13/5, then 13 ÷ 5 = 2 remainder 3, giving 2 3/5.

How do you convert a repeating decimal to a rational number?

Set the decimal equal to x, multiply by a power of 10 to shift one repeating block, then subtract. For x = 0.444…, 10x = 4.444…, so 10x − x = 4, which gives 9x = 4 and x = 4/9. For a two-digit repeat like 0.272727…, multiply by 100 instead: 99x = 27, so x = 27/99 = 3/11.

How do you write a rational number in p/q form?

Write it as one integer p over a nonzero integer q, then simplify so p and q share no common factor. For example, 1.25 = 125/100 = 5/4, and −3 = −3/1. In p/q form the sign usually goes with the numerator, so q stays positive.

Related Topics

• Repeating Decimals
• Convert Between Fractions and Decimals
• Convert Between Improper Fractions and Mixed Numbers
• Simplifying Fractions
• Using Number Lines to Represent Rational Numbers