Problem 33 - Elementary and Intermediate Algebra

Question

Multiply. $$ -\frac{3}{8} \cdot\left(-\frac{2}{9}\right) $$

Step-by-Step Solution

Answer

The product is \[ \frac{1}{12} \].

Step 1: Identify the numbers

Identify the two fractions in the problem: \( -\frac{3}{8} \text{ and } -\frac{2}{9} \).

Step 2: Multiply the numerators

Multiply the numerators of the two fractions: \( -3 \cdot -2 \). \ This calculation results in \ 6.

Step 3: Multiply the denominators

Multiply the denominators of the two fractions: \( 8 \cdot 9 \). \ This calculation yields \ 72.

Step 4: Combine the results

Combine the results of the multiplication of the numerators and denominators to form a new fraction: \[ \frac{6}{72} \].

Step 5: Simplify the fraction

Simplify the fraction \[ \frac{6}{72} \]. \ Both the numerator and the denominator can be divided by their GCD, which is 6. Hence, \[ \frac{6 \div 6}{72 \div 6} = \frac{1}{12} \].

Key Concepts

fraction multiplication

Fraction multiplication is quite straightforward and involves straightforward steps. Each fraction has two parts: a numerator (the top part) and a denominator (the bottom part). To multiply two fractions, follow these steps:

• Multiply the numerators of the fractions.
• Multiply the denominators of the fractions.
• Combine the results to get your new fraction.

For example, in the provided problem, we multiplied the numerators as \( -3 \times -2 = 6 \) and the denominators as \( 8 \times 9 = 72 \). This resulted in a new fraction \( \frac{6}{72} \).

simplifying fractions

Simplifying fractions is the final step in fraction multiplication and many other fraction operations. A fraction is simplified when the numerator and the denominator have no common factors other than 1. To simplify fractions:

• Find the Greatest Common Divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by the GCD.

For example, in the exercise, the fraction \( \frac{6}{72} \) was simplified to \( \frac{1}{12} \) because the GCD of 6 and 72 is 6. Therefore, \( \frac{6 \text{ divided by } 6}{72 \text{ divided by } 6} = \frac{1}{12} \).

numerator and denominator

Understanding what numerators and denominators are is key to mastering fractions.

• The numerator is the top part of the fraction and represents how many parts you have.
• The denominator is the bottom part of the fraction and shows into how many parts the whole is divided.

For instance, in the provided example, \( -\frac{3}{8} \), 3 is the numerator and 8 is the denominator.

When multiplying, the numerators and denominators are treated separately. First, multiply the numerators together and then multiply the denominators together, finally simplifying if necessary. This helps in managing fractions effectively.