Answer key for a "Dividing Fractions" math worksheet, displaying solutions to 12 problems with step-by-step calculations and final answers.

Problem: Solving Division of Fractions

The task involves solving division problems involving fractions. The general rule for dividing fractions is:

\[
\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}
\]

This means you take the reciprocal of the divisor (the second fraction) and then multiply the two fractions.

Let's solve each problem step by step and verify the solutions provided in the answer key.

---

#### 1. \( \frac{1}{5} \div \frac{2}{3} \)

- Take the reciprocal of \( \frac{2}{3} \), which is \( \frac{3}{2} \).
- Multiply: \( \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10} \).

Answer: \( \frac{3}{10} \)

---

#### 2. \( \frac{11}{3} \div 5 \frac{1}{4} \)

- Convert the mixed number \( 5 \frac{1}{4} \) to an improper fraction: \( 5 \frac{1}{4} = \frac{21}{4} \).
- Take the reciprocal of \( \frac{21}{4} \), which is \( \frac{4}{21} \).
- Multiply: \( \frac{11}{3} \times \frac{4}{21} = \frac{11 \times 4}{3 \times 21} = \frac{44}{63} \).

Answer: \( \frac{44}{63} \)

---

#### 3. \( \frac{23}{3} \div \frac{25}{4} \)

- Take the reciprocal of \( \frac{25}{4} \), which is \( \frac{4}{25} \).
- Multiply: \( \frac{23}{3} \times \frac{4}{25} = \frac{23 \times 4}{3 \times 25} = \frac{92}{75} \).

Answer: \( \frac{92}{75} \)

---

#### 4. \( 7 \frac{1}{2} \div \frac{21}{4} \)

- Convert the mixed number \( 7 \frac{1}{2} \) to an improper fraction: \( 7 \frac{1}{2} = \frac{15}{2} \).
- Take the reciprocal of \( \frac{21}{4} \), which is \( \frac{4}{21} \).
- Multiply: \( \frac{15}{2} \times \frac{4}{21} = \frac{15 \times 4}{2 \times 21} = \frac{60}{42} \).
- Simplify \( \frac{60}{42} \): Divide numerator and denominator by their greatest common divisor (GCD), which is 6: \( \frac{60 \div 6}{42 \div 6} = \frac{10}{7} \).

Answer: \( \frac{10}{7} \) or \( 1 \frac{3}{7} \)

---

#### 5. \( 8 \frac{2}{3} \div 2 \frac{7}{4} \)

- Convert the mixed numbers to improper fractions:
- \( 8 \frac{2}{3} = \frac{26}{3} \)
- \( 2 \frac{7}{4} = \frac{15}{4} \)
- Take the reciprocal of \( \frac{15}{4} \), which is \( \frac{4}{15} \).
- Multiply: \( \frac{26}{3} \times \frac{4}{15} = \frac{26 \times 4}{3 \times 15} = \frac{104}{45} \).
- Convert to a mixed number: \( \frac{104}{45} = 2 \frac{14}{45} \).

Answer: \( \frac{104}{45} \) or \( 2 \frac{14}{45} \)

---

#### 6. \( \frac{18}{4} \div \frac{17}{2} \)

- Take the reciprocal of \( \frac{17}{2} \), which is \( \frac{2}{17} \).
- Multiply: \( \frac{18}{4} \times \frac{2}{17} = \frac{18 \times 2}{4 \times 17} = \frac{36}{68} \).
- Simplify \( \frac{36}{68} \): Divide numerator and denominator by their GCD, which is 4: \( \frac{36 \div 4}{68 \div 4} = \frac{9}{17} \).

Answer: \( \frac{9}{17} \)

---

#### 7. \( 8 \frac{2}{4} \div 3 \frac{1}{2} \)

- Simplify the mixed numbers:
- \( 8 \frac{2}{4} = 8 \frac{1}{2} = \frac{17}{2} \)
- \( 3 \frac{1}{2} = \frac{7}{2} \)
- Take the reciprocal of \( \frac{7}{2} \), which is \( \frac{2}{7} \).
- Multiply: \( \frac{17}{2} \times \frac{2}{7} = \frac{17 \times 2}{2 \times 7} = \frac{34}{14} \).
- Simplify \( \frac{34}{14} \): Divide numerator and denominator by their GCD, which is 2: \( \frac{34 \div 2}{14 \div 2} = \frac{17}{7} \).
- Convert to a mixed number: \( \frac{17}{7} = 2 \frac{3}{7} \).

Answer: \( \frac{17}{7} \) or \( 2 \frac{3}{7} \)

---

#### 8. \( \frac{29}{4} \div \frac{16}{3} \)

- Take the reciprocal of \( \frac{16}{3} \), which is \( \frac{3}{16} \).
- Multiply: \( \frac{29}{4} \times \frac{3}{16} = \frac{29 \times 3}{4 \times 16} = \frac{87}{64} \).
- Convert to a mixed number: \( \frac{87}{64} = 1 \frac{23}{64} \).

Answer: \( \frac{87}{64} \) or \( 1 \frac{23}{64} \)

---

#### 9. \( \frac{3}{5} \div \frac{1}{2} \)

- Take the reciprocal of \( \frac{1}{2} \), which is \( \frac{2}{1} \).
- Multiply: \( \frac{3}{5} \times \frac{2}{1} = \frac{3 \times 2}{5 \times 1} = \frac{6}{5} \).
- Convert to a mixed number: \( \frac{6}{5} = 1 \frac{1}{5} \).

Answer: \( \frac{6}{5} \) or \( 1 \frac{1}{5} \)

---

#### 10. \( 4 \frac{3}{5} \div \frac{9}{4} \)

- Convert the mixed number \( 4 \frac{3}{5} \) to an improper fraction: \( 4 \frac{3}{5} = \frac{23}{5} \).
- Take the reciprocal of \( \frac{9}{4} \), which is \( \frac{4}{9} \).
- Multiply: \( \frac{23}{5} \times \frac{4}{9} = \frac{23 \times 4}{5 \times 9} = \frac{92}{45} \).
- Convert to a mixed number: \( \frac{92}{45} = 2 \frac{2}{45} \).

Answer: \( \frac{92}{45} \) or \( 2 \frac{2}{45} \)

---

#### 11. \( \frac{22}{3} \div 7 \frac{1}{2} \)

- Convert the mixed number \( 7 \frac{1}{2} \) to an improper fraction: \( 7 \frac{1}{2} = \frac{15}{2} \).
- Take the reciprocal of \( \frac{15}{2} \), which is \( \frac{2}{15} \).
- Multiply: \( \frac{22}{3} \times \frac{2}{15} = \frac{22 \times 2}{3 \times 15} = \frac{44}{45} \).

Answer: \( \frac{44}{45} \)

---

#### 12. \( 8 \frac{1}{2} \div 7 \frac{3}{4} \)

- Convert the mixed numbers to improper fractions:
- \( 8 \frac{1}{2} = \frac{17}{2} \)
- \( 7 \frac{3}{4} = \frac{31}{4} \)
- Take the reciprocal of \( \frac{31}{4} \), which is \( \frac{4}{31} \).
- Multiply: \( \frac{17}{2} \times \frac{4}{31} = \frac{17 \times 4}{2 \times 31} = \frac{68}{62} \).
- Simplify \( \frac{68}{62} \): Divide numerator and denominator by their GCD, which is 2: \( \frac{68 \div 2}{62 \div 2} = \frac{34}{31} \).
- Convert to a mixed number: \( \frac{34}{31} = 1 \frac{3}{31} \).

Answer: \( \frac{34}{31} \) or \( 1 \frac{3}{31} \)

---

Final Answer Key Verification

Comparing our solutions with the provided answer key:

1. \( \frac{3}{10} \)
2. \( \frac{44}{63} \)
3. \( \frac{92}{75} \)
4. \( \frac{10}{7} \) or \( 1 \frac{3}{7} \)
5. \( \frac{104}{45} \) or \( 2 \frac{14}{45} \)
6. \( \frac{9}{17} \)
7. \( \frac{17}{7} \) or \( 2 \frac{3}{7} \)
8. \( \frac{87}{64} \) or \( 1 \frac{23}{64} \)
9. \( \frac{6}{5} \) or \( 1 \frac{1}{5} \)
10. \( \frac{92}{45} \) or \( 2 \frac{2}{45} \)
11. \( \frac{44}{45} \)
12. \( \frac{34}{31} \) or \( 1 \frac{3}{31} \)

All solutions match the provided answer key.

Final Answer:

\[
\boxed{\text{All answers are correct as per the solution process.}}
\]