Problem 15 - Prealgebra

Question

Find each of the following products. (Multiply.) $$\frac{2}{5} \cdot \frac{3}{5} \cdot \frac{4}{5}$$

Step-by-Step Solution

Answer

The product is \( \frac{24}{125} \).

Step 1: Multiply the Numerators

The problem asks to multiply the fractions \( \frac{2}{5} \), \( \frac{3}{5} \), and \( \frac{4}{5} \). Start by multiplying the numerators. Multiply 2, 3, and 4 together: \( 2 \times 3 \times 4 = 24 \). So, the combined numerator is 24.

Step 2: Multiply the Denominators

Next, multiply the denominators. Each fraction has a denominator of 5, so multiply 5, 5, and 5 together: \( 5 \times 5 \times 5 = 125 \). Hence, the combined denominator is 125.

Step 3: Form the New Fraction

Combine the results of steps 1 and 2 to form the new fraction. The new fraction will be \( \frac{24}{125} \).

Step 4: Simplify the Fraction

Check if the fraction \( \frac{24}{125} \) can be simplified by finding the greatest common divisor (GCD). The GCD of 24 and 125 is 1, meaning \( \frac{24}{125} \) is already in its simplest form.

Key Concepts

Understanding Numerators

When dealing with fractions, the terms "numerator" and "denominator" are crucial. The numerator is the top part of a fraction and represents how many parts of the whole are being considered. In the fraction \( \frac{2}{5} \), the number 2 is the numerator. It indicates that we have 2 parts out of a total of 5 parts.

• For the multiplication exercise given, the numerators 2, 3, and 4 of the fractions \( \frac{2}{5} \), \( \frac{3}{5} \), and \( \frac{4}{5} \) are combined.
• This involves multiplying these numbers together, which for our example, gives us \( 2 \times 3 \times 4 = 24 \).

Always remember when multiplying fractions, you multiply the numerators of each fraction together to get the numerator of the result.

Exploring Denominators

The denominator of a fraction is the bottom number, and it tells you into how many equal parts the whole is divided. In \( \frac{2}{5} \), the number 5 is the denominator, indicating that the whole is divided into 5 equally sized parts.

• In our example, each fraction \( \frac{2}{5}, \frac{3}{5}, \frac{4}{5} \) has a denominator of 5.
• When multiplying these fractions, multiply the denominators: \( 5 \times 5 \times 5 = 125 \).

This results in 125 as the denominator for the new fraction. Always line up and multiply the denominators, just as you do with the numerators.

A Guide to Simplifying Fractions

Simplifying fractions means reducing them to their simplest form, where the numerator and denominator have no common factors other than 1. To do this, find the greatest common divisor (GCD) of both numbers and divide both parts of the fraction by this GCD.

• For instance, after multiplying, we got \( \frac{24}{125} \).
• To simplify, check if 24 and 125 have any common factors other than 1 by calculating the GCD, which is 1 in this case.

Since the GCD is 1, \( \frac{24}{125} \) is already in its simplest form. Simplifying is a key step that ensures fractions are easily comparable and understandable.