Worksheet for practicing division of rational numbers with 20 problems.

To solve the problems involving the division of rational numbers, we use the rule:

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

Additionally, we need to consider the signs of the numbers:
- Dividing two numbers with the same sign results in a positive quotient.
- Dividing two numbers with different signs results in a negative quotient.

Let's solve each problem step by step.

---

Problem 1: \( \frac{1}{2} \div \frac{-2}{5} \)

\[
\frac{1}{2} \div \frac{-2}{5} = \frac{1}{2} \times \frac{5}{-2} = \frac{1 \cdot 5}{2 \cdot -2} = \frac{5}{-4} = -\frac{5}{4}
\]

Answer: \(\boxed{-\frac{5}{4}}\)

---

Problem 2: \( \frac{-1}{3} \div \frac{-1}{6} \)

\[
\frac{-1}{3} \div \frac{-1}{6} = \frac{-1}{3} \times \frac{6}{-1} = \frac{-1 \cdot 6}{3 \cdot -1} = \frac{-6}{-3} = 2
\]

Answer: \(\boxed{2}\)

---

Problem 3: \( \frac{1}{3} \div \frac{-10}{9} \)

\[
\frac{1}{3} \div \frac{-10}{9} = \frac{1}{3} \times \frac{9}{-10} = \frac{1 \cdot 9}{3 \cdot -10} = \frac{9}{-30} = -\frac{3}{10}
\]

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

---

Problem 4: \( \frac{-5}{7} \div \frac{1}{5} \)

\[
\frac{-5}{7} \div \frac{1}{5} = \frac{-5}{7} \times \frac{5}{1} = \frac{-5 \cdot 5}{7 \cdot 1} = \frac{-25}{7}
\]

Answer: \(\boxed{-\frac{25}{7}}\)

---

Problem 5: \( -1 \div \frac{-1}{3} \)

\[
-1 \div \frac{-1}{3} = -1 \times \frac{3}{-1} = \frac{-1 \cdot 3}{-1} = \frac{-3}{-1} = 3
\]

Answer: \(\boxed{3}\)

---

Problem 6: \( \frac{-20}{11} \div \frac{-1}{6} \)

\[
\frac{-20}{11} \div \frac{-1}{6} = \frac{-20}{11} \times \frac{6}{-1} = \frac{-20 \cdot 6}{11 \cdot -1} = \frac{-120}{-11} = \frac{120}{11}
\]

Answer: \(\boxed{\frac{120}{11}}\)

---

Problem 7: \( \frac{-8}{9} \div \frac{15}{13} \)

\[
\frac{-8}{9} \div \frac{15}{13} = \frac{-8}{9} \times \frac{13}{15} = \frac{-8 \cdot 13}{9 \cdot 15} = \frac{-104}{135}
\]

Answer: \(\boxed{-\frac{104}{135}}\)

---

Problem 8: \( \frac{-1}{2} \div \frac{4}{5} \)

\[
\frac{-1}{2} \div \frac{4}{5} = \frac{-1}{2} \times \frac{5}{4} = \frac{-1 \cdot 5}{2 \cdot 4} = \frac{-5}{8}
\]

Answer: \(\boxed{-\frac{5}{8}}\)

---

Problem 9: \( \frac{-1}{4} \div \frac{-11}{7} \)

\[
\frac{-1}{4} \div \frac{-11}{7} = \frac{-1}{4} \times \frac{7}{-11} = \frac{-1 \cdot 7}{4 \cdot -11} = \frac{-7}{-44} = \frac{7}{44}
\]

Answer: \(\boxed{\frac{7}{44}}\)

---

Problem 10: \( -2 \div \frac{-1}{7} \)

\[
-2 \div \frac{-1}{7} = -2 \times \frac{7}{-1} = \frac{-2 \cdot 7}{-1} = \frac{-14}{-1} = 14
\]

Answer: \(\boxed{14}\)

---

Problem 11: \( \frac{6}{11} \div \frac{-6}{5} \)

\[
\frac{6}{11} \div \frac{-6}{5} = \frac{6}{11} \times \frac{5}{-6} = \frac{6 \cdot 5}{11 \cdot -6} = \frac{30}{-66} = -\frac{5}{11}
\]

Answer: \(\boxed{-\frac{5}{11}}\)

---

Problem 12: \( \frac{-29}{19} \div \frac{-1}{2} \)

\[
\frac{-29}{19} \div \frac{-1}{2} = \frac{-29}{19} \times \frac{2}{-1} = \frac{-29 \cdot 2}{19 \cdot -1} = \frac{-58}{-19} = \frac{58}{19}
\]

Answer: \(\boxed{\frac{58}{19}}\)

---

Problem 13: \( -1 \div \frac{1}{5} \)

\[
-1 \div \frac{1}{5} = -1 \times \frac{5}{1} = \frac{-1 \cdot 5}{1} = -5
\]

Answer: \(\boxed{-5}\)

---

Problem 14: \( \frac{19}{12} \div \frac{-5}{4} \)

\[
\frac{19}{12} \div \frac{-5}{4} = \frac{19}{12} \times \frac{4}{-5} = \frac{19 \cdot 4}{12 \cdot -5} = \frac{76}{-60} = -\frac{19}{15}
\]

Answer: \(\boxed{-\frac{19}{15}}\)

---

Problem 15: \( \frac{-1}{2} \div \frac{9}{20} \)

\[
\frac{-1}{2} \div \frac{9}{20} = \frac{-1}{2} \times \frac{20}{9} = \frac{-1 \cdot 20}{2 \cdot 9} = \frac{-20}{18} = -\frac{10}{9}
\]

Answer: \(\boxed{-\frac{10}{9}}\)

---

Problem 16: \( \frac{-1}{3} \div \frac{5}{7} \)

\[
\frac{-1}{3} \div \frac{5}{7} = \frac{-1}{3} \times \frac{7}{5} = \frac{-1 \cdot 7}{3 \cdot 5} = \frac{-7}{15}
\]

Answer: \(\boxed{-\frac{7}{15}}\)

---

Problem 17: \( \frac{-27}{19} \div \frac{-2}{7} \)

\[
\frac{-27}{19} \div \frac{-2}{7} = \frac{-27}{19} \times \frac{7}{-2} = \frac{-27 \cdot 7}{19 \cdot -2} = \frac{-189}{-38} = \frac{189}{38}
\]

Answer: \(\boxed{\frac{189}{38}}\)

---

Problem 18: \( \frac{10}{7} \div \frac{-12}{7} \)

\[
\frac{10}{7} \div \frac{-12}{7} = \frac{10}{7} \times \frac{7}{-12} = \frac{10 \cdot 7}{7 \cdot -12} = \frac{70}{-84} = -\frac{5}{6}
\]

Answer: \(\boxed{-\frac{5}{6}}\)

---

Problem 19: \( \frac{-3}{2} \div \frac{15}{11} \)

\[
\frac{-3}{2} \div \frac{15}{11} = \frac{-3}{2} \times \frac{11}{15} = \frac{-3 \cdot 11}{2 \cdot 15} = \frac{-33}{30} = -\frac{11}{10}
\]

Answer: \(\boxed{-\frac{11}{10}}\)

---

Problem 20: \( \frac{-6}{5} \div \frac{-5}{4} \)

\[
\frac{-6}{5} \div \frac{-5}{4} = \frac{-6}{5} \times \frac{4}{-5} = \frac{-6 \cdot 4}{5 \cdot -5} = \frac{-24}{-25} = \frac{24}{25}
\]

Answer: \(\boxed{\frac{24}{25}}\)

---

Final Answers:

\[
\boxed{
\begin{array}{ll}
1. & -\frac{5}{4} \\
2. & 2 \\
3. & -\frac{3}{10} \\
4. & -\frac{25}{7} \\
5. & 3 \\
6. & \frac{120}{11} \\
7. & -\frac{104}{135} \\
8. & -\frac{5}{8} \\
9. & \frac{7}{44} \\
10. & 14 \\
11. & -\frac{5}{11} \\
12. & \frac{58}{19} \\
13. & -5 \\
14. & -\frac{19}{15} \\
15. & -\frac{10}{9} \\
16. & -\frac{7}{15} \\
17. & \frac{189}{38} \\
18. & -\frac{5}{6} \\
19. & -\frac{11}{10} \\
20. & \frac{24}{25} \\
\end{array}
}
\]