Multiplying and Dividing Rational numbers puzzle online exercise ... - Free Printable

Let's solve each problem step by step. The problems involve multiplication and division of fractions. Here are the solutions:

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1. \( \frac{2}{3} \cdot \frac{-3}{5} \)

Solution:
- Multiply the numerators: \( 2 \cdot (-3) = -6 \)
- Multiply the denominators: \( 3 \cdot 5 = 15 \)
- Result: \( \frac{-6}{15} \)
- Simplify the fraction: \( \frac{-6}{15} = \frac{-2}{5} \)

Answer: \( \boxed{-\frac{2}{5}} \)

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2. \( \frac{-9}{10} \cdot \frac{-1}{8} \)

Solution:
- Multiply the numerators: \( -9 \cdot (-1) = 9 \)
- Multiply the denominators: \( 10 \cdot 8 = 80 \)
- Result: \( \frac{9}{80} \)

Answer: \( \boxed{\frac{9}{80}} \)

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3. \( \frac{1}{7} \div \frac{5}{7} \)

Solution:
- Dividing by a fraction is equivalent to multiplying by its reciprocal.
- Reciprocal of \( \frac{5}{7} \) is \( \frac{7}{5} \).
- So, \( \frac{1}{7} \div \frac{5}{7} = \frac{1}{7} \cdot \frac{7}{5} \)
- Multiply the numerators: \( 1 \cdot 7 = 7 \)
- Multiply the denominators: \( 7 \cdot 5 = 35 \)
- Result: \( \frac{7}{35} \)
- Simplify the fraction: \( \frac{7}{35} = \frac{1}{5} \)

Answer: \( \boxed{\frac{1}{5}} \)

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

Solution:
- Dividing by a fraction is equivalent to multiplying by its reciprocal.
- Reciprocal of \( \frac{2}{5} \) is \( \frac{5}{2} \).
- So, \( \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \cdot \frac{5}{2} \)
- Multiply the numerators: \( 3 \cdot 5 = 15 \)
- Multiply the denominators: \( 4 \cdot 2 = 8 \)
- Result: \( \frac{15}{8} \)

Answer: \( \boxed{\frac{15}{8}} \)

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5. \( \frac{-1}{2} \div \frac{4}{5} \)

Solution:
- Dividing by a fraction is equivalent to multiplying by its reciprocal.
- Reciprocal of \( \frac{4}{5} \) is \( \frac{5}{4} \).
- So, \( \frac{-1}{2} \div \frac{4}{5} = \frac{-1}{2} \cdot \frac{5}{4} \)
- Multiply the numerators: \( -1 \cdot 5 = -5 \)
- Multiply the denominators: \( 2 \cdot 4 = 8 \)
- Result: \( \frac{-5}{8} \)

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

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6. \( \frac{4}{5} \cdot \frac{4}{5} \)

Solution:
- Multiply the numerators: \( 4 \cdot 4 = 16 \)
- Multiply the denominators: \( 5 \cdot 5 = 25 \)
- Result: \( \frac{16}{25} \)

Answer: \( \boxed{\frac{16}{25}} \)

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7. \( \frac{1}{9} \div \frac{1}{8} \)

Solution:
- Dividing by a fraction is equivalent to multiplying by its reciprocal.
- Reciprocal of \( \frac{1}{8} \) is \( \frac{8}{1} \).
- So, \( \frac{1}{9} \div \frac{1}{8} = \frac{1}{9} \cdot \frac{8}{1} \)
- Multiply the numerators: \( 1 \cdot 8 = 8 \)
- Multiply the denominators: \( 9 \cdot 1 = 9 \)
- Result: \( \frac{8}{9} \)

Answer: \( \boxed{\frac{8}{9}} \)

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8. \( \frac{3}{5} \div (-2) \)

Solution:
- Dividing by a number is equivalent to multiplying by its reciprocal.
- Reciprocal of \( -2 \) is \( -\frac{1}{2} \).
- So, \( \frac{3}{5} \div (-2) = \frac{3}{5} \cdot \left( -\frac{1}{2} \right) \)
- Multiply the numerators: \( 3 \cdot (-1) = -3 \)
- Multiply the denominators: \( 5 \cdot 2 = 10 \)
- Result: \( \frac{-3}{10} \)

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

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9. \( \frac{-3}{5} \cdot \frac{-3}{5} \)

Solution:
- Multiply the numerators: \( -3 \cdot (-3) = 9 \)
- Multiply the denominators: \( 5 \cdot 5 = 25 \)
- Result: \( \frac{9}{25} \)

Answer: \( \boxed{\frac{9}{25}} \)

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10. \( 10 \cdot \frac{2}{5} \)

Solution:
- Multiply the whole number by the fraction:
- \( 10 \cdot \frac{2}{5} = \frac{10 \cdot 2}{5} = \frac{20}{5} \)
- Simplify the fraction: \( \frac{20}{5} = 4 \)

Answer: \( \boxed{4} \)

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Final Answers:

1. \( \boxed{-\frac{2}{5}} \)
2. \( \boxed{\frac{9}{80}} \)
3. \( \boxed{\frac{1}{5}} \)
4. \( \boxed{\frac{15}{8}} \)
5. \( \boxed{-\frac{5}{8}} \)
6. \( \boxed{\frac{16}{25}} \)
7. \( \boxed{\frac{8}{9}} \)
8. \( \boxed{-\frac{3}{10}} \)
9. \( \boxed{\frac{9}{25}} \)
10. \( \boxed{4} \)