Multiplying and Dividing Rational Numbers

Multiplying and dividing rational numbers (which include fractions, decimals, and integers) uses the exact same sign rules you already learned for integers. Once you know the sign of your answer, you just apply the standard rules for calculating with fractions and decimals.

The Sign Rules

Whether you are multiplying or dividing, always look at the signs of the two numbers first:

• Same Signs = Positive: If both numbers are positive or both are negative, the result is positive.
  • $(+) \times (+) = +$
  • $(-) \times (-) = +$
• Different Signs = Negative: If one number is positive and the other is negative, the result is negative.
  • $(+) \times (-) = -$
  • $(-) \times (+) = -$

Multiplying Rational Numbers

To multiply fractions, simply multiply the numerators (top numbers) together and the denominators (bottom numbers) together. Then, apply the sign rules and simplify if possible.

Example 1: $\left(-\frac{2}{3}\right) \times \frac{3}{4}$

1. Check the signs: One is negative, one is positive. The answer will be negative.
2. Multiply numerators: $2 \times 3 = 6$
3. Multiply denominators: $3 \times 4 = 12$
4. Combine and simplify: $-\frac{6}{12} = -\frac{1}{2}$

Example 2: $\left(-\frac{1}{2}\right) \times \left(-\frac{4}{5}\right)$

1. Check the signs: Both are negative. The answer will be positive.
2. Multiply: $\frac{1 \times 4}{2 \times 5} = \frac{4}{10}$
3. Simplify: $\frac{2}{5}$

Dividing Rational Numbers

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. To find the reciprocal, just flip the numerator and denominator. For decimals, divide as you normally would, then apply the sign rules.

Example 3: $(-5.2) \div 1.3$

1. Check the signs: One is negative, one is positive. The answer will be negative.
2. Divide the decimals: $5.2 \div 1.3 = 4$
3. Apply the sign: $-4$

Summary of Steps

1. Determine the sign of the final answer using the sign rules.
2. Perform the math (multiply straight across for fractions, or multiply by the reciprocal for division).
3. Simplify your final answer.