Convert Between Improper Fractions and Mixed Numbers

Converting between improper fractions and mixed numbers is a skill the GED Mathematical Reasoning test tests directly. An improper fraction like \(\color{blue}{\frac{7}{3}}\) and a mixed number like \(\color{blue}{2 \frac{1}{3}}\) represent exactly the same value — you need to move fluently between both forms to add, subtract, multiply, and divide fractions efficiently.

What Are Improper Fractions and Mixed Numbers?

An improper fraction has a numerator that is greater than or equal to its denominator (e.g., \(\color{blue}{\frac{7}{3}}\), \(\color{blue}{\frac{11}{4}}\)). It represents a quantity equal to or greater than 1. A mixed number combines a whole number with a proper fraction (e.g., \(\color{blue}{2 \frac{1}{3}}\), \(\color{blue}{2 \frac{3}{4}}\)). Both forms are valid ways to express the same value.

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How to Convert an Improper Fraction to a Mixed Number

Divide the numerator by the denominator

The quotient is the whole number; the remainder becomes the new numerator over the original denominator.

• \(\color{blue}{\frac{7}{3}}\): \(\color{blue}{7 \div 3 = 2}\) remainder \(\color{blue}{1}\). Mixed number: \(\color{blue}{2 \frac{1}{3}}\).
• \(\color{blue}{\frac{11}{4}}\): \(\color{blue}{11 \div 4 = 2}\) remainder \(\color{blue}{3}\). Mixed number: \(\color{blue}{2 \frac{3}{4}}\).
• \(\color{blue}{\frac{13}{5}}\): \(\color{blue}{13 \div 5 = 2}\) remainder \(\color{blue}{3}\). Mixed number: \(\color{blue}{2 \frac{3}{5}}\).
• \(\color{blue}{\frac{22}{7}}\): \(\color{blue}{22 \div 7 = 3}\) remainder \(\color{blue}{1}\). Mixed number: \(\color{blue}{3 \frac{1}{7}}\).

How to Convert a Mixed Number to an Improper Fraction

Multiply, add, keep the denominator

Multiply the whole number by the denominator, add the numerator of the fraction, and write the result over the original denominator.

• \(\color{blue}{2 \frac{1}{3}}\): \(\color{blue}{(2 \times 3) + 1 = 7}\). Improper fraction: \(\color{blue}{\frac{7}{3}}\).
• \(\color{blue}{2 \frac{3}{4}}\): \(\color{blue}{(2 \times 4) + 3 = 11}\). Improper fraction: \(\color{blue}{\frac{11}{4}}\).
• \(\color{blue}{2 \frac{5}{6}}\): \(\color{blue}{(2 \times 6) + 5 = 17}\). Improper fraction: \(\color{blue}{\frac{17}{6}}\).

Step-by-Step Summary

1. Improper fraction → mixed number: Divide numerator by denominator. \(\color{blue}{\text{ Quotient } = \text{ whole }}\) number; remainder over \(\color{blue}{\text{ denominator } = \text{ fractional }}\) part.
2. Mixed number → improper fraction: \(\color{blue}{(\text{ Whole } \times \text{ denominator }) + \text{ numerator } = \text{ new }}\) numerator. Keep the same denominator.
3. Simplify the fractional part if possible.
4. Check by converting back: if the round trip gives you the original, you are correct.

Watch: Mixed Numbers and Improper Fractions (Video Lesson)

Math Antics explains both conversions with a clear visual approach:

Worked Examples

Example 1: Convert \(\color{blue}{\frac{7}{3}}\) to a mixed number.

\(\color{blue}{7 \div 3 = 2}\) remainder \(\color{blue}{1}\). Answer: \(\color{blue}{2 \frac{1}{3}}\).
Check: \(\color{blue}{(2 \times 3) + 1 = 7}\) ✓

Example 2: Convert \(\color{blue}{\frac{11}{4}}\) to a mixed number.

\(\color{blue}{11 \div 4 = 2}\) remainder \(\color{blue}{3}\). Answer: \(\color{blue}{2 \frac{3}{4}}\).

Example 3: Convert \(\color{blue}{3 \frac{1}{7}}\) to an improper fraction.

\(\color{blue}{(3 \times 7) + 1 = 22}\). Answer: \(\color{blue}{\frac{22}{7}}\).

Example 4: Convert \(\color{blue}{\frac{17}{6}}\) to a mixed number.

\(\color{blue}{17 \div 6 = 2}\) remainder \(\color{blue}{5}\). Answer: \(\color{blue}{2 \frac{5}{6}}\).

More Practice: Mixed Numbers and Improper Fractions (Video)

Khan Academy reinforces both conversion directions with additional worked examples:

Exercises

1. Convert \(\color{blue}{\frac{9}{4}}\) to a mixed number.
2. Convert \(\color{blue}{\frac{15}{7}}\) to a mixed number.
3. Convert \(\color{blue}{\frac{20}{3}}\) to a mixed number.
4. Convert \(\color{blue}{4 \frac{1}{5}}\) to an improper fraction.
5. Convert \(\color{blue}{3 \frac{2}{9}}\) to an improper fraction.
6. Convert \(\color{blue}{5 \frac{3}{8}}\) to an improper fraction.

Answers

1. \(\color{blue}{9 \div 4 = 2}\) R \(\color{blue}{1}\) → \(\color{blue}{2 \frac{1}{4}}\)
2. \(\color{blue}{15 \div 7 = 2}\) R \(\color{blue}{1}\) → \(\color{blue}{2 \frac{1}{7}}\)
3. \(\color{blue}{20 \div 3 = 6}\) R \(\color{blue}{2}\) → \(\color{blue}{6 \frac{2}{3}}\)
4. \(\color{blue}{(4 \times 5) + 1 = 21}\) → \(\color{blue}{\frac{21}{5}}\)
5. \(\color{blue}{(3 \times 9) + 2 = 29}\) → \(\color{blue}{\frac{29}{9}}\)
6. \(\color{blue}{(5 \times 8) + 3 = 43}\) → \(\color{blue}{\frac{43}{8}}\)

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Frequently Asked Questions

When should I use an improper fraction versus a mixed number?

Use improper fractions when multiplying or dividing fractions (it simplifies the calculation). Use mixed numbers when the context asks for a “real-world” answer like a measurement or a GED final answer expressed in lowest terms.

Can an improper fraction have a denominator of 1?

Yes. Any whole number can be written as an improper fraction with denominator 1: \(\color{blue}{5 = \frac{5}{1}}\). Converting \(\color{blue}{\frac{5}{1}}\) gives whole number 5, remainder 0, so the mixed number is just \(\color{blue}{5}\).

How does this topic appear on the GED test?

GED questions may present an improper fraction and ask for a mixed number, or vice versa. Multiplication and division of mixed numbers always require converting to improper fractions first.

Related Topics

• Simplifying Fractions
• Convert Between Decimals and Mixed Numbers
• Multiplying Mixed Numbers
• Dividing Mixed Numbers
• Order of Decimals Mixed Numbers and Fractions