Multiplying and Dividing Fractions
Learn multiplication, reciprocal-based division, and cross-canceling to keep fraction arithmetic efficient.
Scaling a construction project to ¾ of its original size, splitting a ⅔-acre lot into 4 equal parcels, or calculating how many ⅛-cup servings are in a ¾-cup container — all require multiplying or dividing fractions. These operations appear in real estate, cooking, engineering, and finance every single day.
- Multiply fractions by multiplying numerators and denominators straight across
- Divide fractions using the Keep-Change-Flip (KCF) method with reciprocals
- Apply cross-canceling before multiplying to keep numbers small and work faster
Multiplication: Multiply numerators together, denominators together — no LCD needed.
\[ \frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d} \]Division: Flip the second fraction (take its reciprocal) and multiply.
\[ \frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c} = \frac{a \times d}{b \times c} \]The KCF rule makes division identical to multiplication — once you flip the second fraction. Always convert mixed numbers to improper fractions before multiplying or dividing.
Cross-Canceling: Work Smarter, Not Harder
Before multiplying, check if any numerator shares a factor with any denominator (across either fraction). Divide both by that shared factor first — this keeps numbers small and often eliminates the need to simplify afterward.
Cross-Canceling — \(\frac{4}{9} \times \frac{3}{8}\)
| Step | Action | Result |
|---|---|---|
| 1 | Check: 4 (numerator) and 8 (denominator) — GCF=4. Cancel: 4÷4=1, 8÷4=2 | \(\frac{\color{#1d4ed8}{1}}{9} \times \frac{3}{\color{#1d4ed8}{2}}\) |
| 2 | Check: 3 (numerator) and 9 (denominator) — GCF=3. Cancel: 3÷3=1, 9÷3=3 | \(\frac{1}{\color{#166534}{3}} \times \frac{\color{#166534}{1}}{2}\) |
| ✅ | Multiply what remains | \(\frac{1 \times 1}{3 \times 2} = \frac{1}{6}\) |
Without cross-canceling: \(\frac{4 \times 3}{9 \times 8} = \frac{12}{72}\) → simplify to \(\frac{1}{6}\). Same answer, more work.
Calculate \(\frac{2}{3} \times \frac{3}{5}\).
Cross-cancel: 3 in numerator and 3 in denominator → both become 1.
\[ \frac{2}{3} \times \frac{3}{5} = \frac{2}{1} \times \frac{1}{5} = \frac{2}{5} \]Calculate \(\frac{3}{4} \div \frac{9}{16}\).
- Keep \(\frac{3}{4}\), Change ÷ to ×, Flip \(\frac{9}{16}\) → \(\frac{16}{9}\).
- Cross-cancel: 3 and 9 share GCF=3 → 1 and 3. Then 4 and 16 share GCF=4 → 1 and 4.
- \(\frac{1}{1} \times \frac{4}{3} = \frac{4}{3}\).
- Convert: \(1\frac{1}{3}\).
Real estate developer Priya owns a \(\frac{2}{3}\)-acre lot and wants to split it into parcels each measuring \(\frac{1}{6}\) of an acre. How many parcels can she create?
- Number of parcels = \(\frac{2}{3} \div \frac{1}{6}\).
- KCF: \(\frac{2}{3} \times \frac{6}{1}\).
- Cross-cancel: 3 and 6 → 1 and 2.
- \(\frac{2}{1} \times \frac{2}{1} = 4\).
Priya can create 4 parcels from her lot.
Test yourself:
- Calculate \(\frac{5}{6} \times \frac{3}{10}\).
- Calculate \(\frac{7}{8} \div \frac{7}{4}\).
- What is the reciprocal of \(3\frac{1}{2}\)?
▶ Show Answers
- Cross-cancel 3 and 6 (÷3), 5 and 10 (÷5): \(\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}\).
- KCF: \(\frac{7}{8} \times \frac{4}{7}\). Cancel 7s and 4/8: \(\frac{1}{2}\).
- Convert \(3\frac{1}{2} = \frac{7}{2}\). Reciprocal = \(\frac{2}{7}\).
- Finding LCD for multiplication: ❌ You don't need an LCD here — that's only for addition/subtraction. ✅ For multiplication, just multiply straight across.
- Forgetting to flip when dividing: ❌ \(\frac{3}{4} \div \frac{2}{5} = \frac{3 \times 2}{4 \times 5}\) — wrong. ✅ Flip the second fraction first, then multiply.
- Cross-canceling same-fraction numerator and denominator: You can only cross-cancel across the two fractions diagonally, not within the same fraction.
- Multiplication: multiply numerators × numerators and denominators × denominators — no LCD needed.
- Division = KCF: Keep the first fraction, Change ÷ to ×, Flip the second fraction.
- Cross-cancel before multiplying to reduce numbers and skip a separate simplification step.
- Convert mixed numbers to improper fractions first before multiplying or dividing.
Engineers scaling blueprints, machinists calculating gear ratios, and contractors estimating material quantities all divide and multiply fractions routinely. A machinist who needs to cut \(\frac{3}{8}\)-inch pieces from a \(\frac{3}{4}\)-inch rod divides fractions to find how many pieces fit — and cross-canceling keeps the mental arithmetic clean under workshop conditions.
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Multiplying and Dividing Fractions — Quiz
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