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. \(\color{blue}{(5 \times 8) + 3 = 43}\) → \(\color{blue}{\frac{43}{8}}\)

Frequently Asked Questions

How do I convert an improper fraction into a mixed number?

Divide the numerator by the denominator. The quotient becomes the whole number, and the remainder goes over the original denominator. For 17/5, 17 ÷ 5 = 3 remainder 2, so 17/5 = 3 2/5. Check by working backward: 3 × 5 + 2 = 17. Simplify the fraction part if you can.

What are improper fractions and mixed numbers?

An improper fraction has a numerator greater than or equal to its denominator, like 9/4 or 6/6, so its value is at least 1. A mixed number shows the same amount as a whole number plus a proper fraction, like 2 1/4. Both forms are correct; mixed numbers are easier to picture, while improper fractions are easier to multiply and divide.

How do you convert mixed numerals to improper fractions?

Multiply the whole number by the denominator, add the numerator, and keep the same denominator. For 4 2/3, multiply 4 × 3 = 12, add 2 to get 14, and write 14/3. Converting to an improper fraction first is the usual step before multiplying or dividing mixed numbers on the GED Math test.

How do you convert improper fractions to mixed numbers on a calculator?

Many scientific and graphing calculators have a fraction key that switches between the two forms, so check your model’s manual. With any basic calculator, divide the numerator by the denominator: 23 ÷ 6 = 3.833…, so the whole number is 3. Then find the remainder: 23 − 3 × 6 = 5, giving 3 5/6.

How do you explain converting improper fractions to mixed numbers to kids?

Use pizza slices. If each pizza has 4 slices and you have 11 slices, that is 11/4. Group the slices into whole pizzas: two full pizzas use 8 slices, leaving 3 slices. So 11/4 is the same as 2 whole pizzas and 3/4 of another, or 2 3/4. Drawing circles makes the idea stick for 4th and 5th graders.

Related Topics

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