Problem 43 - Prealgebra

Question

Find the following products. (Multiply.) $$\frac{3}{8} \cdot \frac{3}{5}$$

Step-by-Step Solution

Answer

The product is \( \frac{9}{40} \).

Step 1: Multiply the Numerators

To start, take the numerator of the first fraction, which is 3, and multiply it by the numerator of the second fraction, which is also 3. This gives you: 3 x 3 = 9.

Step 2: Multiply the Denominators

Next, take the denominator of the first fraction, 8, and multiply it by the denominator of the second fraction, 5. This results in: 8 x 5 = 40.

Step 3: Form the Fraction

Place the product of the numerators over the product of the denominators to form a single fraction. Thus, you have: \( \frac{9}{40} \).

Step 4: Simplify the Fraction (if Needed)

Check if the fraction can be simplified. In this case, 9 and 40 have no common factors other than 1, which means the fraction \( \frac{9}{40} \) is already in its simplest form.

Key Concepts

Multiplying Fractions

When you multiply fractions, the process is straightforward compared to other fraction operations. Instead of finding a common denominator as you do in addition or subtraction, you just multiply straight across. Here's how it works:

• Numerators: Multiply the numerators of the fractions together.
• Denominators: Multiply the denominators together as well.

Let's take an example: multiplying \( \frac{3}{8} \) by \( \frac{3}{5} \). First, you multiply the numerators: \( 3 \times 3 = 9 \). Then multiply the denominators: \( 8 \times 5 = 40 \). So, the product is \( \frac{9}{40} \).
A helpful tip is to multiply before you simplify, which keeps the process organized and prevent mistakes along the way.

Simplifying Fractions

Simplifying fractions makes them easier to read and work with. A fraction is in its simplest form when the numerator and the denominator have no common factor other than 1. Here's how to simplify:

• Find common factors: Look for the greatest common factor (GCF) of the numerator and denominator.
• Divide: Divide both by the GCF to reduce the fraction.

Consider our product \( \frac{9}{40} \). Check for common factors between 9 and 40. Both numbers only share 1 as a factor, meaning they cannot be reduced further.
Therefore, \( \frac{9}{40} \) is already in its simplest form. It’s essential to always check for possible simplification after multiplying two or more fractions.

Numerator and Denominator Multiplication

Understanding the multiplication of numerators and denominators individually is crucial in fraction multiplication. Here's a breakdown of why this method works well:

• Numerator: The product of the numerators determines how many parts of the whole we are dealing with.
• Denominator: The product of the denominators decides how many equal parts the whole is divided into.

When you multiply \( \frac{3}{8} \) by \( \frac{3}{5} \), you're determining how 9 parts out of 40 fit into the whole. This approach keeps the result in context and helps avoid errors in more complex calculations.
By practicing this multiplication process, you can tackle fraction multiplication problems with ease and confidence, understanding each step's significance to ensure accuracy every time.