Multiply and Divide Fractions Worksheets | Worsheets library - Free Printable

Problem: Multiplying Fractions and Mixed Numbers

The task involves solving 14 problems that require multiplying fractions and mixed numbers. Below is a step-by-step explanation of how to solve each problem.

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#### Step 1: Convert Mixed Numbers to Improper Fractions
If any of the numbers are mixed numbers (e.g., $3 \frac{1}{2}$), convert them to improper fractions first. The formula for converting a mixed number $a \frac{b}{c}$ to an improper fraction is:
$$
\text{Improper Fraction} = \frac{(a \times c) + b}{c}
$$

#### Step 2: Multiply the Fractions
To multiply two fractions, multiply the numerators together and the denominators together:
$$
\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}
$$

#### Step 3: Simplify the Result
After multiplying, simplify the resulting fraction if possible by dividing both the numerator and the denominator by their greatest common divisor (GCD).

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Solutions to Each Problem

#### Problem 1: $3 \frac{8}{9} \times \frac{9}{10}$
1. Convert $3 \frac{8}{9}$ to an improper fraction:
$$
3 \frac{8}{9} = \frac{(3 \times 9) + 8}{9} = \frac{27 + 8}{9} = \frac{35}{9}
$$
2. Multiply the fractions:
$$
\frac{35}{9} \times \frac{9}{10} = \frac{35 \times 9}{9 \times 10} = \frac{315}{90}
$$
3. Simplify $\frac{315}{90}$:
- GCD of 315 and 90 is 45.
- $\frac{315 \div 45}{90 \div 45} = \frac{7}{2}$
4. Convert $\frac{7}{2}$ back to a mixed number:
$$
\frac{7}{2} = 3 \frac{1}{2}
$$

Answer: $3 \frac{1}{2}$

---

#### Problem 2: $1 \frac{1}{3} \times \frac{2}{7}$
1. Convert $1 \frac{1}{3}$ to an improper fraction:
$$
1 \frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{3 + 1}{3} = \frac{4}{3}
$$
2. Multiply the fractions:
$$
\frac{4}{3} \times \frac{2}{7} = \frac{4 \times 2}{3 \times 7} = \frac{8}{21}
$$
3. The fraction $\frac{8}{21}$ is already in simplest form.

Answer: $\frac{8}{21}$

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#### Problem 3: $2 \frac{4}{10} \times \frac{4}{5}$
1. Simplify $2 \frac{4}{10}$ first:
- $\frac{4}{10}$ simplifies to $\frac{2}{5}$.
- So, $2 \frac{4}{10} = 2 \frac{2}{5}$.
2. Convert $2 \frac{2}{5}$ to an improper fraction:
$$
2 \frac{2}{5} = \frac{(2 \times 5) + 2}{5} = \frac{10 + 2}{5} = \frac{12}{5}
$$
3. Multiply the fractions:
$$
\frac{12}{5} \times \frac{4}{5} = \frac{12 \times 4}{5 \times 5} = \frac{48}{25}
$$
4. Convert $\frac{48}{25}$ back to a mixed number:
$$
\frac{48}{25} = 1 \frac{23}{25}
$$

Answer: $1 \frac{23}{25}$

---

#### Problem 4: $1 \frac{7}{4} \times \frac{6}{7}$
1. Convert $1 \frac{7}{4}$ to an improper fraction:
$$
1 \frac{7}{4} = \frac{(1 \times 4) + 7}{4} = \frac{4 + 7}{4} = \frac{11}{4}
$$
2. Multiply the fractions:
$$
\frac{11}{4} \times \frac{6}{7} = \frac{11 \times 6}{4 \times 7} = \frac{66}{28}
$$
3. Simplify $\frac{66}{28}$:
- GCD of 66 and 28 is 2.
- $\frac{66 \div 2}{28 \div 2} = \frac{33}{14}$
4. Convert $\frac{33}{14}$ back to a mixed number:
$$
\frac{33}{14} = 2 \frac{5}{14}
$$

Answer: $2 \frac{5}{14}$

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#### Problem 5: $1 \frac{1}{13} \times \frac{2}{12}$
1. Convert $1 \frac{1}{13}$ to an improper fraction:
$$
1 \frac{1}{13} = \frac{(1 \times 13) + 1}{13} = \frac{13 + 1}{13} = \frac{14}{13}
$$
2. Simplify $\frac{2}{12}$:
- $\frac{2}{12}$ simplifies to $\frac{1}{6}$.
3. Multiply the fractions:
$$
\frac{14}{13} \times \frac{1}{6} = \frac{14 \times 1}{13 \times 6} = \frac{14}{78}
$$
4. Simplify $\frac{14}{78}$:
- GCD of 14 and 78 is 2.
- $\frac{14 \div 2}{78 \div 2} = \frac{7}{39}$

Answer: $\frac{7}{39}$

---

#### Problem 6: $3 \frac{1}{9} \times \frac{1}{4}$
1. Convert $3 \frac{1}{9}$ to an improper fraction:
$$
3 \frac{1}{9} = \frac{(3 \times 9) + 1}{9} = \frac{27 + 1}{9} = \frac{28}{9}
$$
2. Multiply the fractions:
$$
\frac{28}{9} \times \frac{1}{4} = \frac{28 \times 1}{9 \times 4} = \frac{28}{36}
$$
3. Simplify $\frac{28}{36}$:
- GCD of 28 and 36 is 4.
- $\frac{28 \div 4}{36 \div 4} = \frac{7}{9}$

Answer: $\frac{7}{9}$

---

#### Problem 7: $3 \frac{3}{5} \times \frac{5}{8}$
1. Convert $3 \frac{3}{5}$ to an improper fraction:
$$
3 \frac{3}{5} = \frac{(3 \times 5) + 3}{5} = \frac{15 + 3}{5} = \frac{18}{5}
$$
2. Multiply the fractions:
$$
\frac{18}{5} \times \frac{5}{8} = \frac{18 \times 5}{5 \times 8} = \frac{90}{40}
$$
3. Simplify $\frac{90}{40}$:
- GCD of 90 and 40 is 10.
- $\frac{90 \div 10}{40 \div 10} = \frac{9}{4}$
4. Convert $\frac{9}{4}$ back to a mixed number:
$$
\frac{9}{4} = 2 \frac{1}{4}
$$

Answer: $2 \frac{1}{4}$

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#### Problem 8: $3 \frac{1}{4} \times \frac{7}{8}$
1. Convert $3 \frac{1}{4}$ to an improper fraction:
$$
3 \frac{1}{4} = \frac{(3 \times 4) + 1}{4} = \frac{12 + 1}{4} = \frac{13}{4}
$$
2. Multiply the fractions:
$$
\frac{13}{4} \times \frac{7}{8} = \frac{13 \times 7}{4 \times 8} = \frac{91}{32}
$$
3. Convert $\frac{91}{32}$ back to a mixed number:
$$
\frac{91}{32} = 2 \frac{27}{32}
$$

Answer: $2 \frac{27}{32}$

---

#### Problem 9: $2 \frac{1}{8} \times \frac{7}{9}$
1. Convert $2 \frac{1}{8}$ to an improper fraction:
$$
2 \frac{1}{8} = \frac{(2 \times 8) + 1}{8} = \frac{16 + 1}{8} = \frac{17}{8}
$$
2. Multiply the fractions:
$$
\frac{17}{8} \times \frac{7}{9} = \frac{17 \times 7}{8 \times 9} = \frac{119}{72}
$$
3. Convert $\frac{119}{72}$ back to a mixed number:
$$
\frac{119}{72} = 1 \frac{47}{72}
$$

Answer: $1 \frac{47}{72}$

---

#### Problem 10: $3 \frac{1}{6} \times \frac{1}{7}$
1. Convert $3 \frac{1}{6}$ to an improper fraction:
$$
3 \frac{1}{6} = \frac{(3 \times 6) + 1}{6} = \frac{18 + 1}{6} = \frac{19}{6}
$$
2. Multiply the fractions:
$$
\frac{19}{6} \times \frac{1}{7} = \frac{19 \times 1}{6 \times 7} = \frac{19}{42}
$$
3. The fraction $\frac{19}{42}$ is already in simplest form.

Answer: $\frac{19}{42}$

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#### Problem 11: $1 \frac{5}{8} \times \frac{4}{5}$
1. Convert $1 \frac{5}{8}$ to an improper fraction:
$$
1 \frac{5}{8} = \frac{(1 \times 8) + 5}{8} = \frac{8 + 5}{8} = \frac{13}{8}
$$
2. Multiply the fractions:
$$
\frac{13}{8} \times \frac{4}{5} = \frac{13 \times 4}{8 \times 5} = \frac{52}{40}
$$
3. Simplify $\frac{52}{40}$:
- GCD of 52 and 40 is 4.
- $\frac{52 \div 4}{40 \div 4} = \frac{13}{10}$
4. Convert $\frac{13}{10}$ back to a mixed number:
$$
\frac{13}{10} = 1 \frac{3}{10}
$$

Answer: $1 \frac{3}{10}$

---

#### Problem 12: $2 \frac{6}{12} \times \frac{7}{12}$
1. Simplify $2 \frac{6}{12}$ first:
- $\frac{6}{12}$ simplifies to $\frac{1}{2}$.
- So, $2 \frac{6}{12} = 2 \frac{1}{2}$.
2. Convert $2 \frac{1}{2}$ to an improper fraction:
$$
2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}
$$
3. Multiply the fractions:
$$
\frac{5}{2} \times \frac{7}{12} = \frac{5 \times 7}{2 \times 12} = \frac{35}{24}
$$
4. Convert $\frac{35}{24}$ back to a mixed number:
$$
\frac{35}{24} = 1 \frac{11}{24}
$$

Answer: $1 \frac{11}{24}$

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#### Problem 13: $3 \frac{1}{3} \times \frac{6}{5}$
1. Convert $3 \frac{1}{3}$ to an improper fraction:
$$
3 \frac{1}{3} = \frac{(3 \times 3) + 1}{3} = \frac{9 + 1}{3} = \frac{10}{3}
$$
2. Multiply the fractions:
$$
\frac{10}{3} \times \frac{6}{5} = \frac{10 \times 6}{3 \times 5} = \frac{60}{15}
$$
3. Simplify $\frac{60}{15}$:
- GCD of 60 and 15 is 15.
- $\frac{60 \div 15}{15 \div 15} = \frac{4}{1} = 4$

Answer: $4$

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#### Problem 14: $2 \frac{1}{2} \times \frac{1}{4}$
1. Convert $2 \frac{1}{2}$ to an improper fraction:
$$
2 \frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}
$$
2. Multiply the fractions:
$$
\frac{5}{2} \times \frac{1}{4} = \frac{5 \times 1}{2 \times 4} = \frac{5}{8}
$$
3. The fraction $\frac{5}{8}$ is already in simplest form.

Answer: $\frac{5}{8}$

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Final Answers

1. $3 \frac{1}{2}$
2. $\frac{8}{21}$
3. $1 \frac{23}{25}$
4. $2 \frac{5}{14}$
5. $\frac{7}{39}$
6. $\frac{7}{9}$
7. $2 \frac{1}{4}$
8. $2 \frac{27}{32}$
9. $1 \frac{47}{72}$
10. $\frac{19}{42}$
11. $1 \frac{3}{10}$
12. $1 \frac{11}{24}$
13. $4$
14. $\frac{5}{8}$

$$
\boxed{
\begin{aligned}
1. & \ 3 \frac{1}{2} \\
2. & \ \frac{8}{21} \\
3. & \ 1 \frac{23}{25} \\
4. & \ 2 \frac{5}{14} \\
5. & \ \frac{7}{39} \\
6. & \ \frac{7}{9} \\
7. & \ 2 \frac{1}{4} \\
8. & \ 2 \frac{27}{32} \\
9. & \ 1 \frac{47}{72} \\
10. & \ \frac{19}{42} \\
11. & \ 1 \frac{3}{10} \\
12. & \ 1 \frac{11}{24} \\
13. & \ 4 \\
14. & \ \frac{5}{8}
\end{aligned}
}
$$