Multiplying Rational Numbers Lesson Plans & Worksheets - Free Printable

The image you uploaded contains a worksheet titled "Dividing Rational Numbers." The problems involve dividing rational numbers, which are fractions or decimals. Below, I will solve the first few problems as examples and explain the general approach for solving these types of problems.

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General Approach to Dividing Rational Numbers

1. Understand the Problem: Identify the dividend (the number being divided) and the divisor (the number dividing the dividend).
2. Reciprocal Rule: To divide by a fraction, multiply by its reciprocal. The reciprocal of a fraction is obtained by swapping its numerator and denominator.
3. Simplify: Simplify the resulting expression if possible.

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Solutions to Selected Problems

#### Problem 1:
$$
\frac{3}{4} \div \frac{5}{8}
$$

- Step 1: Write the division as multiplication by the reciprocal.
$$
\frac{3}{4} \div \frac{5}{8} = \frac{3}{4} \times \frac{8}{5}
$$

- Step 2: Multiply the numerators and denominators.
$$
\frac{3 \times 8}{4 \times 5} = \frac{24}{20}
$$

- Step 3: Simplify the fraction.
$$
\frac{24}{20} = \frac{6}{5}
$$

- Final Answer:
$$
\boxed{\frac{6}{5}}
$$

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#### Problem 2:
$$
\frac{7}{9} \div \frac{2}{3}
$$

- Step 1: Write the division as multiplication by the reciprocal.
$$
\frac{7}{9} \div \frac{2}{3} = \frac{7}{9} \times \frac{3}{2}
$$

- Step 2: Multiply the numerators and denominators.
$$
\frac{7 \times 3}{9 \times 2} = \frac{21}{18}
$$

- Step 3: Simplify the fraction.
$$
\frac{21}{18} = \frac{7}{6}
$$

- Final Answer:
$$
\boxed{\frac{7}{6}}
$$

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#### Problem 3:
$$
-26 \div (-4)
$$

- Step 1: Divide the absolute values.
$$
\frac{26}{4} = 6.5
$$

- Step 2: Determine the sign. Since both numbers are negative, the result is positive.
$$
-26 \div (-4) = 6.5
$$

- Final Answer:
$$
\boxed{6.5}
$$

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#### Problem 4:
$$
-24 \div 6
$$

- Step 1: Divide the absolute values.
$$
\frac{24}{6} = 4
$$

- Step 2: Determine the sign. Since one number is negative and the other is positive, the result is negative.
$$
-24 \div 6 = -4
$$

- Final Answer:
$$
\boxed{-4}
$$

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#### Problem 5:
$$
75 \div (+15)
$$

- Step 1: Divide the absolute values.
$$
\frac{75}{15} = 5
$$

- Step 2: Determine the sign. Since both numbers are positive, the result is positive.
$$
75 \div (+15) = 5
$$

- Final Answer:
$$
\boxed{5}
$$

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#### Problem 6:
$$
\frac{3}{4} \div 6
$$

- Step 1: Write 6 as a fraction: $6 = \frac{6}{1}$.
$$
\frac{3}{4} \div \frac{6}{1} = \frac{3}{4} \times \frac{1}{6}
$$

- Step 2: Multiply the numerators and denominators.
$$
\frac{3 \times 1}{4 \times 6} = \frac{3}{24}
$$

- Step 3: Simplify the fraction.
$$
\frac{3}{24} = \frac{1}{8}
$$

- Final Answer:
$$
\boxed{\frac{1}{8}}
$$

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#### Problem 7:
$$
\frac{5}{6} \div \left(-\frac{2}{3}\right)
$$

- Step 1: Write the division as multiplication by the reciprocal.
$$
\frac{5}{6} \div \left(-\frac{2}{3}\right) = \frac{5}{6} \times \left(-\frac{3}{2}\right)
$$

- Step 2: Multiply the numerators and denominators.
$$
\frac{5 \times (-3)}{6 \times 2} = \frac{-15}{12}
$$

- Step 3: Simplify the fraction.
$$
\frac{-15}{12} = \frac{-5}{4}
$$

- Final Answer:
$$
\boxed{-\frac{5}{4}}
$$

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Summary

The key steps for dividing rational numbers are:
1. Convert division into multiplication by the reciprocal.
2. Multiply the numerators and denominators.
3. Simplify the resulting fraction.
4. Determine the sign based on the signs of the original numbers.

If you need solutions for more problems, please let me know!