Multiplying and Dividing Rational Numbers - Elementary Studies - Free Printable

Let's solve each of the 16 problems on the worksheet "Dividing Rational Numbers" step by step.

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Key Rule for Dividing Rational Numbers:

To divide two rational numbers:
- Change the division into multiplication by the reciprocal.
- Apply the sign rules:
- Negative ÷ Negative = Positive
- Positive ÷ Negative = Negative
- Negative ÷ Positive = Negative

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Problem 1:

$$
-\frac{5}{8} \div \left(-\frac{15}{16}\right)
$$
- Convert to multiplication by reciprocal:
$$
= -\frac{5}{8} \times \left(-\frac{16}{15}\right)
$$
- Multiply numerators and denominators:
$$
= \frac{(-5)(-16)}{(8)(15)} = \frac{80}{120}
$$
- Simplify:
$$
\frac{80}{120} = \frac{2}{3}
$$
✔ Answer: $\boxed{\frac{2}{3}}$

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Problem 2:

$$
-\frac{1}{2} \div \left(-\frac{1}{4}\right)
$$
- Multiply by reciprocal:
$$
= -\frac{1}{2} \times \left(-\frac{4}{1}\right) = \frac{4}{2} = 2
$$
✔ Answer: $\boxed{2}$

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Problem 3:

$$
\frac{8}{13} \div (-6)
$$
- Write $-6$ as $-\frac{6}{1}$, then multiply by reciprocal:
$$
= \frac{8}{13} \times \left(-\frac{1}{6}\right) = -\frac{8}{78} = -\frac{4}{39}
$$
✔ Answer: $\boxed{-\frac{4}{39}}$

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Problem 4:

$$
6.5 \div (-0.1)
$$
- Convert decimals to fractions or use decimal division:
$$
6.5 \div (-0.1) = -65
$$
(Explanation: dividing by 0.1 is same as multiplying by 10; so $6.5 \times 10 = 65$, but negative → $-65$)

✔ Answer: $\boxed{-65}$

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Problem 5:

$$
0.88 \div (-4)
$$
- Divide:
$$
0.88 \div (-4) = -0.22
$$
✔ Answer: $\boxed{-0.22}$

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Problem 6:

$$
\left(-3\frac{2}{3}\right) \div 11
$$
- Convert mixed number:
$$
-3\frac{2}{3} = -\frac{11}{3}
$$
- Divide:
$$
-\frac{11}{3} \div 11 = -\frac{11}{3} \times \frac{1}{11} = -\frac{1}{3}
$$
✔ Answer: $\boxed{-\frac{1}{3}}$

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Problem 7:

$$
\left(-6\frac{4}{5}\right) \div \frac{17}{10}
$$
- Convert mixed number:
$$
-6\frac{4}{5} = -\frac{34}{5}
$$
- Multiply by reciprocal:
$$
-\frac{34}{5} \div \frac{17}{10} = -\frac{34}{5} \times \frac{10}{17}
$$
- Simplify:
$$
= -\frac{34 \cdot 10}{5 \cdot 17} = -\frac{340}{85} = -4
$$
✔ Answer: $\boxed{-4}$

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Problem 8:

$$
-19.8 \div (-0.6)
$$
- Divide:
$$
-19.8 \div (-0.6) = 33
$$
(Explanation: both negative → positive; $19.8 \div 0.6 = 33$)

✔ Answer: $\boxed{33}$

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Problem 9:

$$
-\frac{2}{5} \div \frac{3}{10}
$$
- Multiply by reciprocal:
$$
= -\frac{2}{5} \times \frac{10}{3} = -\frac{20}{15} = -\frac{4}{3}
$$
✔ Answer: $\boxed{-\frac{4}{3}}$

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Problem 10:

$$
2.52 \div (-3)
$$
- Divide:
$$
2.52 \div (-3) = -0.84
$$
✔ Answer: $\boxed{-0.84}$

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Problem 11:

$$
79.2 \div 9
$$
- Divide:
$$
79.2 \div 9 = 8.8
$$
✔ Answer: $\boxed{8.8}$

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Problem 12:

$$
-\frac{7}{9} \div \frac{5}{6}
$$
- Multiply by reciprocal:
$$
= -\frac{7}{9} \times \frac{6}{5} = -\frac{42}{45} = -\frac{14}{15}
$$
✔ Answer: $\boxed{-\frac{14}{15}}$

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Problem 13:

$$
\frac{5}{12} \div \left(-\frac{3}{8}\right)
$$
- Multiply by reciprocal:
$$
= \frac{5}{12} \times \left(-\frac{8}{3}\right) = -\frac{40}{36} = -\frac{10}{9}
$$
✔ Answer: $\boxed{-\frac{10}{9}}$

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Problem 14:

$$
-\frac{7}{24} \div \left(-\frac{14}{15}\right)
$$
- Multiply by reciprocal:
$$
= -\frac{7}{24} \times \left(-\frac{15}{14}\right) = \frac{105}{336}
$$
- Simplify:
$$
\frac{105 \div 21}{336 \div 21} = \frac{5}{16}
$$
✔ Answer: $\boxed{\frac{5}{16}}$

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Problem 15:

$$
-2.7 \div (-9)
$$
- Both negative → positive:
$$
2.7 \div 9 = 0.3
$$
✔ Answer: $\boxed{0.3}$

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Problem 16:

$$
-\frac{12}{15} \div \frac{7}{9}
$$
- First simplify $-\frac{12}{15} = -\frac{4}{5}$
- Then:
$$
-\frac{4}{5} \div \frac{7}{9} = -\frac{4}{5} \times \frac{9}{7} = -\frac{36}{35}
$$
✔ Answer: $\boxed{-\frac{36}{35}}$

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✔ Final Answers:

| Problem | Answer |
|--------|--------|
| 1 | $\frac{2}{3}$ |
| 2 | $2$ |
| 3 | $-\frac{4}{39}$ |
| 4 | $-65$ |
| 5 | $-0.22$ |
| 6 | $-\frac{1}{3}$ |
| 7 | $-4$ |
| 8 | $33$ |
| 9 | $-\frac{4}{3}$ |
| 10 | $-0.84$ |
| 11 | $8.8$ |
| 12 | $-\frac{14}{15}$ |
| 13 | $-\frac{10}{9}$ |
| 14 | $\frac{5}{16}$ |
| 15 | $0.3$ |
| 16 | $-\frac{36}{35}$ |

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