Multiplying Fractions: Cross Cancelling - Variation Theory - Free Printable

You've uploaded an image containing 12 multiplication problems involving fractions. I'll solve each one step by step and explain the process.

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General Rule for Multiplying Fractions:

To multiply two fractions, multiply the numerators together and the denominators together:
\[
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
\]
Then simplify the result by canceling common factors (if any).

For mixed numbers (like in problems 10–12), convert them to improper fractions first.

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## ✔ Problem-by-Problem Solutions:

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1) $\frac{2}{9} \times \frac{3}{4}$

Multiply numerators: $2 \times 3 = 6$
Multiply denominators: $9 \times 4 = 36$
→ $\frac{6}{36}$

Simplify: Divide numerator and denominator by 6 → $\frac{1}{6}$

✔ Answer: $\boxed{\frac{1}{6}}$

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2) $\frac{2}{9} \times \frac{3}{8}$

Numerator: $2 \times 3 = 6$
Denominator: $9 \times 8 = 72$
→ $\frac{6}{72}$

Simplify: Divide by 6 → $\frac{1}{12}$

✔ Answer: $\boxed{\frac{1}{12}}$

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3) $\frac{4}{9} \times \frac{3}{16}$

Numerator: $4 \times 3 = 12$
Denominator: $9 \times 16 = 144$
→ $\frac{12}{144}$

Simplify: Divide by 12 → $\frac{1}{12}$

✔ Answer: $\boxed{\frac{1}{12}}$

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4) $\frac{4}{27} \times \frac{3}{16}$

Numerator: $4 \times 3 = 12$
Denominator: $27 \times 16 = 432$
→ $\frac{12}{432}$

Simplify: Divide numerator and denominator by 12 → $\frac{1}{36}$

✔ Answer: $\boxed{\frac{1}{36}}$

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5) $\frac{4}{27} \times \frac{9}{16}$

Numerator: $4 \times 9 = 36$
Denominator: $27 \times 16 = 432$
→ $\frac{36}{432}$

Simplify: Divide by 36 → $\frac{1}{12}$

*Alternatively, cross-cancel before multiplying:*
- 4 and 16 → 4 ÷ 4 = 1, 16 ÷ 4 = 4
- 9 and 27 → 9 ÷ 9 = 1, 27 ÷ 9 = 3
→ $\frac{1}{3} \times \frac{1}{4} = \frac{1}{12}$

✔ Answer: $\boxed{\frac{1}{12}}$

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6) $\frac{8}{27} \times \frac{9}{16}$

Cross-cancel:
- 8 and 16 → 8 ÷ 8 = 1, 16 ÷ 8 = 2
- 9 and 27 → 9 ÷ 9 = 1, 27 ÷ 9 = 3
→ $\frac{1}{3} \times \frac{1}{2} = \frac{1}{6}$

✔ Answer: $\boxed{\frac{1}{6}}$

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7) $\frac{8}{27} \times \frac{15}{16}$

Cross-cancel:
- 8 and 16 → 8 ÷ 8 = 1, 16 ÷ 8 = 2
- 15 and 27 → both divisible by 3 → 15 ÷ 3 = 5, 27 ÷ 3 = 9
→ $\frac{1}{9} \times \frac{5}{2} = \frac{5}{18}$

✔ Answer: $\boxed{\frac{5}{18}}$

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8) $\frac{16}{27} \times \frac{15}{16}$

Notice 16 in numerator and denominator → cancel out!

→ $\frac{1}{27} \times \frac{15}{1} = \frac{15}{27}$

Simplify: divide by 3 → $\frac{5}{9}$

✔ Answer: $\boxed{\frac{5}{9}}$

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9) $\frac{27}{8} \times \frac{16}{15}$

Cross-cancel:
- 27 and 15 → both divisible by 3 → 27 ÷ 3 = 9, 15 ÷ 3 = 5
- 16 and 8 → 16 ÷ 8 = 2, 8 ÷ 8 = 1
→ $\frac{9}{1} \times \frac{2}{5} = \frac{18}{5}$

Convert to mixed number: $3 \frac{3}{5}$

✔ Answer: $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$

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10) $3\frac{3}{8} \times \frac{16}{15}$

First, convert mixed number to improper fraction:
$3\frac{3}{8} = \frac{3 \times 8 + 3}{8} = \frac{27}{8}$

Now multiply: $\frac{27}{8} \times \frac{16}{15}$ — this is identical to problem 9!

So answer is same: $\frac{18}{5}$ or $3\frac{3}{5}$

✔ Answer: $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$

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11) $3\frac{3}{8} \times 1\frac{1}{15}$

Convert both to improper fractions:

- $3\frac{3}{8} = \frac{27}{8}$
- $1\frac{1}{15} = \frac{16}{15}$

Again, same as problem 9 and 10 → $\frac{27}{8} \times \frac{16}{15} = \frac{18}{5}$

✔ Answer: $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$

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12) $3\frac{3}{8} \times 1\frac{2}{30}$

First, simplify $1\frac{2}{30}$ → reduce $\frac{2}{30} = \frac{1}{15}$ → so it’s $1\frac{1}{15}$

Wait — that’s the same as problem 11!

So again, we have $3\frac{3}{8} \times 1\frac{1}{15} = \frac{27}{8} \times \frac{16}{15} = \frac{18}{5}$

✔ Answer: $\boxed{\frac{18}{5}}$ or $\boxed{3\frac{3}{5}}$

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## 📊 Final Answers Summary:

| Problem | Answer |
|---------|----------------|
| 1 | $\frac{1}{6}$ |
| 2 | $\frac{1}{12}$ |
| 3 | $\frac{1}{12}$ |
| 4 | $\frac{1}{36}$ |
| 5 | $\frac{1}{12}$ |
| 6 | $\frac{1}{6}$ |
| 7 | $\frac{5}{18}$ |
| 8 | $\frac{5}{9}$ |
| 9 | $\frac{18}{5}$ or $3\frac{3}{5}$ |
| 10 | $\frac{18}{5}$ or $3\frac{3}{5}$ |
| 11 | $\frac{18}{5}$ or $3\frac{3}{5}$ |
| 12 | $\frac{18}{5}$ or $3\frac{3}{5}$ |

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## 💡 Key Takeaways:

- Always look for cross-cancellation before multiplying — it saves time and reduces simplification steps.
- Mixed numbers must be converted to improper fractions before multiplying.
- Problems 9–12 are essentially the same calculation with different notations — great for recognizing patterns!

Let me know if you’d like a printable version or worksheet-style layout!