Math worksheet for practicing multiplication and division of fractions.

Problem: Solving the given expressions involving multiplication and division of fractions.

We will solve each expression step by step, ensuring that the final answer is in its lowest terms. Let's go through each problem systematically.

---

Problem 1:

$$
\frac{5}{4} \times \frac{7}{2} \times \frac{1}{7}
$$

#### Solution:
1. Multiply the numerators together:
$$
5 \times 7 \times 1 = 35
$$

2. Multiply the denominators together:
$$
4 \times 2 \times 7 = 56
$$

3. The resulting fraction is:
$$
\frac{35}{56}
$$

4. Simplify the fraction by finding the greatest common divisor (GCD) of 35 and 56, which is 7:
$$
\frac{35 \div 7}{56 \div 7} = \frac{5}{8}
$$

#### Final Answer:
$$
\boxed{\frac{5}{8}}
$$

---

Problem 2:

$$
\frac{4}{3} \div \frac{20}{3} \times \frac{11}{9}
$$

#### Solution:
1. Rewrite the division as multiplication by the reciprocal:
$$
\frac{4}{3} \times \frac{3}{20} \times \frac{11}{9}
$$

2. Multiply the numerators together:
$$
4 \times 3 \times 11 = 132
$$

3. Multiply the denominators together:
$$
3 \times 20 \times 9 = 540
$$

4. The resulting fraction is:
$$
\frac{132}{540}
$$

5. Simplify the fraction by finding the GCD of 132 and 540, which is 12:
$$
\frac{132 \div 12}{540 \div 12} = \frac{11}{45}
$$

#### Final Answer:
$$
\boxed{\frac{11}{45}}
$$

---

Problem 3:

$$
\frac{23}{12} \div \left( \frac{4}{3} \times \frac{13}{8} \right)
$$

#### Solution:
1. First, simplify the expression inside the parentheses:
$$
\frac{4}{3} \times \frac{13}{8} = \frac{4 \times 13}{3 \times 8} = \frac{52}{24}
$$

2. Simplify $\frac{52}{24}$ by finding the GCD of 52 and 24, which is 4:
$$
\frac{52 \div 4}{24 \div 4} = \frac{13}{6}
$$

3. Now, rewrite the original expression:
$$
\frac{23}{12} \div \frac{13}{6}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{23}{12} \times \frac{6}{13}
$$

5. Multiply the numerators and denominators:
$$
\frac{23 \times 6}{12 \times 13} = \frac{138}{156}
$$

6. Simplify the fraction by finding the GCD of 138 and 156, which is 6:
$$
\frac{138 \div 6}{156 \div 6} = \frac{23}{26}
$$

#### Final Answer:
$$
\boxed{\frac{23}{26}}
$$

---

Problem 4:

$$
\frac{11}{8} \div \left( \frac{21}{5} \times \frac{5}{6} \right)
$$

#### Solution:
1. First, simplify the expression inside the parentheses:
$$
\frac{21}{5} \times \frac{5}{6} = \frac{21 \times 5}{5 \times 6} = \frac{105}{30}
$$

2. Simplify $\frac{105}{30}$ by finding the GCD of 105 and 30, which is 15:
$$
\frac{105 \div 15}{30 \div 15} = \frac{7}{2}
$$

3. Now, rewrite the original expression:
$$
\frac{11}{8} \div \frac{7}{2}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{11}{8} \times \frac{2}{7}
$$

5. Multiply the numerators and denominators:
$$
\frac{11 \times 2}{8 \times 7} = \frac{22}{56}
$$

6. Simplify the fraction by finding the GCD of 22 and 56, which is 2:
$$
\frac{22 \div 2}{56 \div 2} = \frac{11}{28}
$$

#### Final Answer:
$$
\boxed{\frac{11}{28}}
$$

---

Problem 5:

$$
\frac{18}{11} \times \frac{11}{3} \div \frac{16}{5}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{18}{11} \times \frac{11}{3} = \frac{18 \times 11}{11 \times 3} = \frac{198}{33}
$$

2. Simplify $\frac{198}{33}$ by finding the GCD of 198 and 33, which is 33:
$$
\frac{198 \div 33}{33 \div 33} = \frac{6}{1} = 6
$$

3. Now, rewrite the expression:
$$
6 \div \frac{16}{5}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
6 \times \frac{5}{16} = \frac{6 \times 5}{16} = \frac{30}{16}
$$

5. Simplify the fraction by finding the GCD of 30 and 16, which is 2:
$$
\frac{30 \div 2}{16 \div 2} = \frac{15}{8}
$$

#### Final Answer:
$$
\boxed{\frac{15}{8}}
$$

---

Problem 6:

$$
\frac{3}{4} \times \frac{9}{4} \div \frac{5}{2}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{3}{4} \times \frac{9}{4} = \frac{3 \times 9}{4 \times 4} = \frac{27}{16}
$$

2. Now, rewrite the expression:
$$
\frac{27}{16} \div \frac{5}{2}
$$

3. Rewrite the division as multiplication by the reciprocal:
$$
\frac{27}{16} \times \frac{2}{5} = \frac{27 \times 2}{16 \times 5} = \frac{54}{80}
$$

4. Simplify the fraction by finding the GCD of 54 and 80, which is 2:
$$
\frac{54 \div 2}{80 \div 2} = \frac{27}{40}
$$

#### Final Answer:
$$
\boxed{\frac{27}{40}}
$$

---

Problem 7:

$$
\frac{13}{12} \times \frac{11}{4} \div \frac{13}{6}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{13}{12} \times \frac{11}{4} = \frac{13 \times 11}{12 \times 4} = \frac{143}{48}
$$

2. Now, rewrite the expression:
$$
\frac{143}{48} \div \frac{13}{6}
$$

3. Rewrite the division as multiplication by the reciprocal:
$$
\frac{143}{48} \times \frac{6}{13} = \frac{143 \times 6}{48 \times 13} = \frac{858}{624}
$$

4. Simplify the fraction by finding the GCD of 858 and 624, which is 78:
$$
\frac{858 \div 78}{624 \div 78} = \frac{11}{8}
$$

#### Final Answer:
$$
\boxed{\frac{11}{8}}
$$

---

Problem 8:

$$
\frac{5}{2} \div \left( \frac{13}{6} \div \frac{19}{9} \right)
$$

#### Solution:
1. First, simplify the expression inside the parentheses:
$$
\frac{13}{6} \div \frac{19}{9} = \frac{13}{6} \times \frac{9}{19} = \frac{13 \times 9}{6 \times 19} = \frac{117}{114}
$$

2. Simplify $\frac{117}{114}$ by finding the GCD of 117 and 114, which is 3:
$$
\frac{117 \div 3}{114 \div 3} = \frac{39}{38}
$$

3. Now, rewrite the original expression:
$$
\frac{5}{2} \div \frac{39}{38}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{5}{2} \times \frac{38}{39} = \frac{5 \times 38}{2 \times 39} = \frac{190}{78}
$$

5. Simplify the fraction by finding the GCD of 190 and 78, which is 2:
$$
\frac{190 \div 2}{78 \div 2} = \frac{95}{39}
$$

#### Final Answer:
$$
\boxed{\frac{95}{39}}
$$

---

Problem 9:

$$
\frac{7}{8} \times \frac{4}{3} \div \frac{2}{5}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{7}{8} \times \frac{4}{3} = \frac{7 \times 4}{8 \times 3} = \frac{28}{24}
$$

2. Simplify $\frac{28}{24}$ by finding the GCD of 28 and 24, which is 4:
$$
\frac{28 \div 4}{24 \div 4} = \frac{7}{6}
$$

3. Now, rewrite the expression:
$$
\frac{7}{6} \div \frac{2}{5}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{7}{6} \times \frac{5}{2} = \frac{7 \times 5}{6 \times 2} = \frac{35}{12}
$$

#### Final Answer:
$$
\boxed{\frac{35}{12}}
$$

---

Problem 10:

$$
\frac{10}{3} \times \frac{2}{9} \times \frac{6}{5}
$$

#### Solution:
1. Multiply the numerators together:
$$
10 \times 2 \times 6 = 120
$$

2. Multiply the denominators together:
$$
3 \times 9 \times 5 = 135
$$

3. The resulting fraction is:
$$
\frac{120}{135}
$$

4. Simplify the fraction by finding the GCD of 120 and 135, which is 15:
$$
\frac{120 \div 15}{135 \div 15} = \frac{8}{9}
$$

#### Final Answer:
$$
\boxed{\frac{8}{9}}
$$

---

Problem 11:

$$
\frac{24}{7} \div \left( \frac{16}{5} \div \frac{3}{5} \right)
$$

#### Solution:
1. First, simplify the expression inside the parentheses:
$$
\frac{16}{5} \div \frac{3}{5} = \frac{16}{5} \times \frac{5}{3} = \frac{16 \times 5}{5 \times 3} = \frac{80}{15}
$$

2. Simplify $\frac{80}{15}$ by finding the GCD of 80 and 15, which is 5:
$$
\frac{80 \div 5}{15 \div 5} = \frac{16}{3}
$$

3. Now, rewrite the original expression:
$$
\frac{24}{7} \div \frac{16}{3}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{24}{7} \times \frac{3}{16} = \frac{24 \times 3}{7 \times 16} = \frac{72}{112}
$$

5. Simplify the fraction by finding the GCD of 72 and 112, which is 8:
$$
\frac{72 \div 8}{112 \div 8} = \frac{9}{14}
$$

#### Final Answer:
$$
\boxed{\frac{9}{14}}
$$

---

Problem 12:

$$
\frac{4}{3} \times \frac{4}{3} \div \frac{20}{7}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{4}{3} \times \frac{4}{3} = \frac{4 \times 4}{3 \times 3} = \frac{16}{9}
$$

2. Now, rewrite the expression:
$$
\frac{16}{9} \div \frac{20}{7}
$$

3. Rewrite the division as multiplication by the reciprocal:
$$
\frac{16}{9} \times \frac{7}{20} = \frac{16 \times 7}{9 \times 20} = \frac{112}{180}
$$

4. Simplify the fraction by finding the GCD of 112 and 180, which is 4:
$$
\frac{112 \div 4}{180 \div 4} = \frac{28}{45}
$$

#### Final Answer:
$$
\boxed{\frac{28}{45}}
$$

---

Problem 13:

$$
\frac{11}{5} \div \left( \frac{5}{2} \div \frac{7}{2} \right)
$$

#### Solution:
1. First, simplify the expression inside the parentheses:
$$
\frac{5}{2} \div \frac{7}{2} = \frac{5}{2} \times \frac{2}{7} = \frac{5 \times 2}{2 \times 7} = \frac{10}{14}
$$

2. Simplify $\frac{10}{14}$ by finding the GCD of 10 and 14, which is 2:
$$
\frac{10 \div 2}{14 \div 2} = \frac{5}{7}
$$

3. Now, rewrite the original expression:
$$
\frac{11}{5} \div \frac{5}{7}
$$

4. Rewrite the division as multiplication by the reciprocal:
$$
\frac{11}{5} \times \frac{7}{5} = \frac{11 \times 7}{5 \times 5} = \frac{77}{25}
$$

#### Final Answer:
$$
\boxed{\frac{77}{25}}
$$

---

Problem 14:

$$
\frac{2}{3} \times \frac{4}{3} \div \frac{5}{8}
$$

#### Solution:
1. First, simplify the multiplication:
$$
\frac{2}{3} \times \frac{4}{3} = \frac{2 \times 4}{3 \times 3} = \frac{8}{9}
$$

2. Now, rewrite the expression:
$$
\frac{8}{9} \div \frac{5}{8}
$$

3. Rewrite the division as multiplication by the reciprocal:
$$
\frac{8}{9} \times \frac{8}{5} = \frac{8 \times 8}{9 \times 5} = \frac{64}{45}
$$

#### Final Answer:
$$
\boxed{\frac{64}{45}}
$$

---

Problem 15:

$$
\frac{1}{7} \times \frac{9}{4} \times \frac{17}{9}
$$

#### Solution:
1. Multiply the numerators together:
$$
1 \times 9 \times 17 = 153
$$

2. Multiply the denominators together:
$$
7 \times 4 \times 9 = 252
$$

3. The resulting fraction is:
$$
\frac{153}{252}
$$

4. Simplify the fraction by finding the GCD of 153 and 252, which is 9:
$$
\frac{153 \div 9}{252 \div 9} = \frac{17}{28}
$$

#### Final Answer:
$$
\boxed{\frac{17}{28}}
$$

---

Final Answers:

1. $\boxed{\frac{5}{8}}$
2. $\boxed{\frac{11}{45}}$
3. $\boxed{\frac{23}{26}}$
4. $\boxed{\frac{11}{28}}$
5. $\boxed{\frac{15}{8}}$
6. $\boxed{\frac{27}{40}}$
7. $\boxed{\frac{11}{8}}$
8. $\boxed{\frac{95}{39}}$
9. $\boxed{\frac{35}{12}}$
10. $\boxed{\frac{8}{9}}$
11. $\boxed{\frac{9}{14}}$
12. $\boxed{\frac{28}{45}}$
13. $\boxed{\frac{77}{25}}$
14. $\boxed{\frac{64}{45}}$
15. $\boxed{\frac{17}{28}}$