How to Divide Mixed Numbers? (+FREE Worksheet!)

Dividing mixed numbers is a fraction skill that appears frequently on the GED Mathematical Reasoning test. The method builds on what you already know about dividing fractions: convert, flip, and multiply. Once you can divide simple fractions, dividing mixed numbers requires only one extra step — converting to improper fractions first.

What Does It Mean to Divide Mixed Numbers?

Dividing by a mixed number asks “how many times does this value fit into that value?” For example, \(\color{blue}{3 \frac{1}{2} \div 1 \frac{3}{4}}\) asks how many \(\color{blue}{1 \frac{3}{4}}\)-unit lengths fit into \(\color{blue}{3 \frac{1}{2}}\) units. The answer is found by converting both to improper fractions, then multiplying the first fraction by the reciprocal (the flipped version) of the second.

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How to Divide Mixed Numbers

Step 1: Convert both to improper fractions

• \(\color{blue}{3 \frac{1}{2} = \frac{7}{2}}\)
• \(\color{blue}{1 \frac{3}{4} = \frac{7}{4}}\)

Step 2: Multiply by the reciprocal

Division becomes multiplication when you flip the second fraction (the divisor):

\(\color{blue}{\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}}\)

• \(\color{blue}{\frac{7}{2} \div \frac{7}{4} = \frac{7}{2} \times \frac{4}{7} = \frac{28}{14} = 2}\)

Step 3: Simplify

Reduce the result and, if greater than 1, convert to a mixed number.

Step-by-Step Summary

1. Convert each mixed number to an improper fraction.
2. Keep the first fraction, \(\color{blue}{\text{ change } \div \text{ to }}\) ×, flip the second fraction (use its reciprocal).
3. Cross-cancel common factors if possible.
4. Multiply numerators; multiply denominators.
5. Simplify and convert back to a mixed number if needed.

Watch: Dividing Mixed Numbers (Video Lesson)

Khan Academy walks through dividing mixed numbers with clear worked examples:

Dividing Mixed Numbers – Worked Examples

Example 1: Divide \(\color{blue}{3 \frac{1}{2} \div 1 \frac{3}{4}}\).

Convert: \(\color{blue}{3 \frac{1}{2} = \frac{7}{2}}\), \(\color{blue}{1 \frac{3}{4} = \frac{7}{4}}\).
Flip and multiply: \(\color{blue}{\frac{7}{2} \times \frac{4}{7} = \frac{28}{14} = 2}\).

Example 2: Divide \(\color{blue}{2 \frac{2}{3} \div 1 \frac{1}{3}}\).

Convert: \(\color{blue}{2 \frac{2}{3} = \frac{8}{3}}\), \(\color{blue}{1 \frac{1}{3} = \frac{4}{3}}\).
Flip and multiply: \(\color{blue}{\frac{8}{3} \times \frac{3}{4} = \frac{24}{12} = 2}\).

Example 3: Divide \(\color{blue}{4 \frac{1}{2} \div 1 \frac{1}{2}}\).

Convert: \(\color{blue}{4 \frac{1}{2} = \frac{9}{2}}\), \(\color{blue}{1 \frac{1}{2} = \frac{3}{2}}\).
Flip and multiply: \(\color{blue}{\frac{9}{2} \times \frac{2}{3} = \frac{18}{6} = 3}\).

Example 4: Divide \(\color{blue}{5 \frac{1}{4} \div 2 \frac{1}{8}}\).

Convert: \(\color{blue}{5 \frac{1}{4} = \frac{21}{4}}\), \(\color{blue}{2 \frac{1}{8} = \frac{17}{8}}\).
Flip and multiply: \(\color{blue}{\frac{21}{4} \times \frac{8}{17} = \frac{168}{68} = \frac{42}{17}}\).
Convert: \(\color{blue}{42 \div 17 = 2}\) R \(\color{blue}{8}\). Answer: \(\color{blue}{2 \frac{8}{17}}\).

More Practice: Dividing Mixed Numbers and Fractions (Video)

This Khan Academy video covers dividing mixed numbers and fractions together, with additional real-world examples:

Exercises for Dividing Mixed Numbers

1. \(\color{blue}{3 \frac{1}{2} \div 1 \frac{3}{4}}\)
2. \(\color{blue}{2 \frac{2}{3} \div 1 \frac{1}{3}}\)
3. \(\color{blue}{4 \frac{1}{2} \div 1 \frac{1}{2}}\)
4. \(\color{blue}{5 \frac{1}{4} \div 2 \frac{1}{8}}\)
5. \(\color{blue}{6 \div 1 \frac{1}{2}}\)

Answers

1. \(\color{blue}{\frac{7}{2} \times \frac{4}{7} = \frac{28}{14} = 2}\)
2. \(\color{blue}{\frac{8}{3} \times \frac{3}{4} = \frac{24}{12} = 2}\)
3. \(\color{blue}{\frac{9}{2} \times \frac{2}{3} = \frac{18}{6} = 3}\)
4. \(\color{blue}{\frac{21}{4} \times \frac{8}{17} = \frac{168}{68} = \frac{42}{17} = 2 \frac{8}{17}}\)
5. \(\color{blue}{\frac{6}{1} \times \frac{2}{3} = \frac{12}{3} = 4}\)

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Frequently Asked Questions

Why do I flip the second fraction when dividing?

Dividing by a fraction is the same as multiplying by its reciprocal. This is because \(\color{blue}{a \div (\frac{b}{c}) = a \times (\frac{c}{b})}\). The mathematical justification is that multiplying by the reciprocal reverses the division — the two operations cancel each other out.

Can I divide a mixed number by a whole number?

Yes. Convert the whole number to a fraction (e.g., \(\color{blue}{4 = \frac{4}{1}}\)), then flip and multiply. For example, \(\color{blue}{3 \frac{1}{2} \div 4 = \frac{7}{2} \div \frac{4}{1} = \frac{7}{2} \times \frac{1}{4} = \frac{7}{8}}\).

How does dividing mixed numbers appear on the GED test?

GED problems may ask how many equal portions fit into a total (e.g., how many 1 \(\color{blue}{\frac{1}{2}}\)-foot pieces can be cut from a 6-foot board), or they may present division directly in an expression. Recognizing the “how many fit?” structure helps you identify the operation.

Related Topics

• Multiplying Mixed Numbers
• Convert Between Improper Fractions and Mixed Numbers
• Simplifying Fractions
• Word Problems Fractions
• Order of Decimals Mixed Numbers and Fractions