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.

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}{0.333ldots}) to a fraction.

Let (color{blue}{x = 0.333ldots}). Then (color{blue}{10x = 3.333ldots}). Subtract: (color{blue}{10x – x = 3.333ldots – 0.333ldots}), 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}{0.666ldots}) to a fraction.

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

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

The repeat block is 2 digits (27), so multiply by 100: Let (color{blue}{x = 0.272727ldots}). (color{blue}{100x = 27.272727ldots}). 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}{0.111ldots}) to a fraction.
6. Convert (color{blue}{0.363636ldots}) 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 = 0.111ldots}); (color{blue}{10x = 1.111ldots}); (color{blue}{9x = 1}); (color{blue}{x = frac{1}{9}})
6. (color{blue}{x = 0.363636ldots}); (color{blue}{100x = 36.363636ldots}); (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