Multiply & Divide Fractions

Introduction

Procedural fluency with fractions means carrying out steps accurately, efficiently, and consistently. In this chapter you’ll multiply and divide fractions and mixed numbers, simplify results, and present answers in lowest terms or as mixed numbers.

Multiplying Fractions

To multiply:
[latex]\dfrac{a}{b}\times\dfrac{c}{d}=\dfrac{ac}{bd}[/latex]

Example 1
[latex]\dfrac{3}{4}\times\dfrac{2}{5}=\dfrac{6}{20}=\dfrac{3}{10}[/latex]

Steps

1. Multiply numerators.
2. Multiply denominators.
3. Reduce to lowest terms.

Example 2
[latex]\dfrac{5}{6}\times\dfrac{3}{4}=\dfrac{15}{24}=\dfrac{5}{8}[/latex]

Tip. You don’t need common denominators for multiplication; simplify before or after multiplying.

Multiplying Mixed Numbers

Rule: Convert to improper fractions, multiply, then convert back.

Example
[latex]1\dfrac{2}{3}\times 2\dfrac{1}{2}=\dfrac{5}{3}\times\dfrac{5}{2}=\dfrac{25}{6}=4\dfrac{1}{6}[/latex]

Steps

1. Convert each mixed number:
   [latex]1\dfrac{2}{3}=\dfrac{5}{3}[/latex], [latex]2\dfrac{1}{2}=\dfrac{5}{2}[/latex]
2. Multiply: [latex]\dfrac{5}{3}\times\dfrac{5}{2}=\dfrac{25}{6}[/latex]
3. Convert to mixed: [latex]\dfrac{25}{6}=4\dfrac{1}{6}[/latex]

Dividing Fractions

Keep–Change–Flip: keep the first fraction, change to multiplication, flip the second (reciprocal).

[latex]\dfrac{a}{b}\div\dfrac{c}{d}=\dfrac{a}{b}\times\dfrac{d}{c}[/latex]

Example 1
[latex]\dfrac{3}{4}\div\dfrac{2}{5}=\dfrac{3}{4}\times\dfrac{5}{2}=\dfrac{15}{8}=1\dfrac{7}{8}[/latex]

Steps

1. Write the first fraction.
2. Flip the second fraction.
3. Multiply.
4. Reduce.

Dividing Mixed Numbers

Example
[latex]2\dfrac{1}{4}\div\dfrac{3}{5}=\dfrac{9}{4}\times\dfrac{5}{3}=\dfrac{45}{12}=3\dfrac{9}{12}=3\dfrac{3}{4}[/latex]

Simplifying & Converting

Reduce to lowest terms using GCD.
Example [latex]\dfrac{12}{20}=\dfrac{3}{5}[/latex]

Improper → Mixed
[latex]\dfrac{17}{5}=3\dfrac{2}{5}[/latex]

H5P Practice

Note: Enter answers as plain text like 3/10, 15/8, or 3 3/4 (a space between whole and fraction). This renders cleanly even if MathJax isn’t active inside H5P.

H5P 1: Fill in the Blank — Multiply Fractions

Prompt: Solve and reduce to lowest terms.
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Items & Solutions

1. [latex]\dfrac{3}{4}\times\dfrac{2}{5}[/latex] → 3/10
2. [latex]\dfrac{7}{8}\times\dfrac{4}{9}[/latex] → 7/18
3. [latex]\dfrac{5}{6}\times\dfrac{3}{4}[/latex] → 5/8

H5P 2: Multiple Choice — Multiply Mixed Numbers

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Item
[latex]1\dfrac{1}{2}\times 2\dfrac{1}{3}[/latex]
Options:
a) 2 1/2 b) 3 1/2 ✅ c) 4 1/3 d) 3 2/3
Work: [latex]\dfrac{3}{2}\times\dfrac{7}{3}=\dfrac{21}{6}=3\dfrac{1}{2}[/latex]

H5P 3: Multiple Choice — Divide Fractions

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Item
[latex]\dfrac{3}{4}\div\dfrac{2}{5}[/latex]
Options:
a) 5/6 b) 15/8 ✅ c) 8/15 d) 3/10
Work: keep–change–flip → [latex]\dfrac{3}{4}\times\dfrac{5}{2}=\dfrac{15}{8}[/latex]

H5P 4: Fill in the Blank — Divide Mixed Numbers

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Items & Solutions

1. [latex]2\dfrac{1}{4}\div\dfrac{3}{5}[/latex] → 3 3/4
2. [latex]1\dfrac{2}{3}\div\dfrac{4}{9}[/latex] → 3 3/8
   (Work shown in body above.)

H5P 5: Word Problems — Fraction Operations

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1. A recipe uses [latex]\dfrac{2}{3}[/latex] cup per batch. How much for 4 batches? → 2 2/3
2. A rope [latex]\dfrac{3}{4}[/latex] yd long is cut into [latex]\dfrac{1}{8}[/latex]-yd pieces. How many pieces? → 6

Common Errors & Fixes

Reflection Prompts (for pre-service teachers)

• What classroom language will you use to cue “keep–change–flip”?
• How will you require and check simplification?
• What student work samples would you use to target each common error?

Summary

Fluency with fraction multiplication/division comes from clear steps + lots of practice: convert mixed numbers correctly, multiply across or keep–change–flip, and always simplify.