Problem 73 - Algebra A Combined Function

Question

Perform each indicated operation. Write all results in lowest terms. $$ \frac{4}{5} \cdot \frac{7}{8} $$

Step-by-Step Solution

Answer

The product of \(\frac{4}{5} \cdot \frac{7}{8}\) is \(\frac{7}{10}\).

Step 1: Multiply the Numerator

Multiply the numerators of the fractions together. Here, multiply the numerators 4 and 7: \(4 \cdot 7 = 28\).

Step 2: Multiply the Denominator

Multiply the denominators of the fractions together. Here, multiply the denominators 5 and 8: \(5 \cdot 8 = 40\).

Step 3: Write the Product as a Single Fraction

Combine the products of the numerators and denominators to form a single fraction: \(\frac{28}{40}\).

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 28 and 40. The GCD is 4. Divide both the numerator and denominator by 4: \(\frac{28 \div 4}{40 \div 4} = \frac{7}{10}\).

Key Concepts

Numerator and Denominator

When multiplying fractions, it is essential to understand the roles of both the numerator and the denominator. The numerator of a fraction is the top number, representing how many parts of a whole are considered. The denominator, on the other hand, is the bottom number and it tells us into how many equal parts the whole is divided. In the example given, the fractions \(\frac{4}{5}\) and \(\frac{7}{8}\) involve numerators 4 and 7, and denominators 5 and 8, respectively. When multiplying these fractions, we first consider the numerators.

• Multiply the numerators together: \(4 \cdot 7 = 28\).
• Multiply the denominators together: \(5 \cdot 8 = 40\).

This operation leads to a new fraction \(\frac{28}{40}\), which then needs further simplification.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when simplifying fractions to their lowest terms. It is the largest number that divides both the numerator and the denominator without leaving a remainder. Let's break it down with our example fraction \(\frac{28}{40}\):

• List the factors of 28: 1, 2, 4, 7, 14, 28.
• List the factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.

By comparing these lists, we find that the greatest common factor is 4. This is the GCD, and it is vital for the next step, which is to simplify the fraction.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. This is done by dividing both by their greatest common divisor.For our fraction \(\frac{28}{40}\), we determined that the GCD is 4. Here’s how you simplify:

• Divide the numerator by the GCD: \(28 \div 4 = 7\).
• Divide the denominator by the GCD: \(40 \div 4 = 10\).

So, \(\frac{28}{40}\) simplifies to \(\frac{7}{10}\). By understanding these steps, you can effectively work through fraction multiplication and ensure your answers are always simplified for clarity.