Learn how to multiply fractions, whole numbers, and mixed numbers with clear examples and easy-to-follow steps.

The image you've shared is an educational guide from Mashup Math that explains how to multiply fractions in three different scenarios: multiplying fractions by fractions, by whole numbers, and by mixed numbers. Let's go through each case step-by-step and explain the solution process clearly.

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🔹 1. Multiplying Fractions by Fractions

Example:
$$
\frac{1}{2} \times \frac{3}{4}
$$

#### ✔ Steps:
1. Multiply the numerators: $1 \times 3 = 3$
2. Multiply the denominators: $2 \times 4 = 8$
3. Write the result: $\frac{3}{8}$

✔ Final Answer:
$$
\frac{1}{2} \times \frac{3}{4} = \frac{3}{8}
$$

> 📌 Rule: When multiplying two fractions, multiply numerator × numerator and denominator × denominator.

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🔹 2. Multiplying Fractions by Whole Numbers

Example:
$$
\frac{3}{8} \times 2
$$

#### ✔ Steps:
1. Rewrite the whole number as a fraction with denominator 1:
$$
2 = \frac{2}{1}
$$
2. Now multiply:
$$
\frac{3}{8} \times \frac{2}{1} = \frac{3 \times 2}{8 \times 1} = \frac{6}{8}
$$
3. Simplify the fraction:
$$
\frac{6}{8} = \frac{3}{4}
$$

✔ Final Answer:
$$
\frac{3}{8} \times 2 = \frac{3}{4}
$$

> 📌 Tip: Always rewrite whole numbers as fractions (e.g., $2 = \frac{2}{1}$) before multiplying.

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🔹 3. Multiplying Fractions by Mixed Numbers

Example:
$$
3\frac{2}{3} \times 4\frac{1}{5}
$$

#### ✔ Steps:
1. Convert both mixed numbers to improper fractions:

- $3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3}$
*(Wait — correction! The image shows $3\frac{2}{3} = \frac{21}{5}$? That’s incorrect. Let's fix this.)*

✘ Error in Image:
The image says $3\frac{2}{3} = \frac{21}{5}$ — but that’s wrong.
Let’s correct it.

Actually:
- $3\frac{2}{3} = \frac{(3 \times 3) + 2}{3} = \frac{11}{3}$
- $4\frac{1}{5} = \frac{(4 \times 5) + 1}{5} = \frac{21}{5}$

So the correct multiplication is:
$$
\frac{11}{3} \times \frac{21}{5}
$$

2. Multiply numerators and denominators:
$$
\frac{11 \times 21}{3 \times 5} = \frac{231}{15}
$$

3. Simplify:
- Divide numerator and denominator by GCF of 231 and 15 → GCF is 3.
$$
\frac{231 \div 3}{15 \div 3} = \frac{77}{5}
$$

4. Convert improper fraction to mixed number:
$$
\frac{77}{5} = 15\frac{2}{5}
$$

✔ Final Answer:
$$
3\frac{2}{3} \times 4\frac{1}{5} = 15\frac{2}{5}
$$

> 📌 Important: Always convert mixed numbers to improper fractions before multiplying.

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✔ Summary of Rules:

| Case | Method |
|------|--------|
| Fraction × Fraction | Multiply numerators and denominators directly |
| Fraction × Whole Number | Write whole number as $\frac{\text{number}}{1}$, then multiply |
| Fraction × Mixed Number | Convert mixed numbers to improper fractions first, then multiply |

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🔍 Final Notes:

- The image has a mistake in the third example: it incorrectly writes $3\frac{2}{3} = \frac{21}{5}$, which should be $\frac{11}{3}$. But the final answer $15\frac{2}{5}$ is correct because they used $\frac{21}{5} \times \frac{11}{3} = \frac{231}{15} = 15\frac{2}{5}$, so likely they meant $4\frac{1}{5} = \frac{21}{5}$ and $3\frac{2}{3} = \frac{11}{3}$ — just mislabeled.

✔ Corrected version:
$$
3\frac{2}{3} \times 4\frac{1}{5} = \frac{11}{3} \times \frac{21}{5} = \frac{231}{15} = 15\frac{2}{5}
$$

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✔ Final Answer Recap:

1. $\frac{1}{2} \times \frac{3}{4} = \boxed{\frac{3}{8}}$
2. $\frac{3}{8} \times 2 = \boxed{\frac{3}{4}}$
3. $3\frac{2}{3} \times 4\frac{1}{5} = \boxed{15\frac{2}{5}}$

Let me know if you'd like practice problems or a visual explanation! 😊