Convert Rational Numbers to a Fraction

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.

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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. \(\color{blue}{x = 0.363636\ldots}\); \(\color{blue}{100x = 36.363636\ldots}\); \(\color{blue}{99x = 36}\); \(\color{blue}{x = \frac{36}{99} = \frac{4}{11}}\)

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Frequently Asked Questions

Is every decimal a rational number?

Every terminating decimal (like 0.5) and every repeating decimal (like 0.333…) is rational. Non-repeating, non-terminating decimals — like π or √2 — are irrational and cannot be written as fractions.

How do I simplify a fraction after converting?

Find the Greatest Common Factor (GCF) of the numerator and denominator, then divide both by that number. For example, \(\color{blue}{\frac{24}{36}}\): \(\color{blue}{\text{ GCF } = 12}\), so \(\color{blue}{\frac{24}{36} = \frac{2}{3}}\).

What if the repeating decimal has digits before the repeat starts?

Use two equations: multiply once to shift just the repeating part and a second time to shift the whole decimal. Subtract to cancel the repeating portion. The algebra still works the same way.

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