Problem 23 - Elementary Algebra

Question

For exercises \(23-74\), evaluate. $$ \frac{1}{3} \cdot \frac{4}{9} $$

Step-by-Step Solution

Answer

\(\frac{4}{27}\)

Step 1: Identify the Numerators and Denominators

Identify the numerators and denominators of the fractions. For the fraction \(\frac{1}{3}\) the numerator is 1 and the denominator is 3. For the fraction \(\frac{4}{9}\) the numerator is 4 and the denominator is 9.

Step 2: Multiply the Numerators Together

Multiply the numerators of the fractions together: \(1 \times 4 = 4\)

Step 3: Multiply the Denominators Together

Multiply the denominators of the fractions together: \(3 \times 9 = 27\)

Step 4: Combine the Results

Combine the results from steps 2 and 3 to form the new fraction: \(\frac{4}{27}\)

Key Concepts

Numerator and Denominator

When working with fractions, it's essential to understand the terms 'numerator' and 'denominator'. These are the two main components of any fraction.

The numerator is the top part of the fraction. It indicates how many parts of the whole we have.
For example, in the fraction \(\frac{1}{3}\), the numerator is 1. This means we have one part out of the three equal parts that make up the whole.

The denominator is the bottom part of the fraction. It tells us into how many equal parts the whole is divided.
In \(\frac{1}{3}\), the denominator is 3, showing that the whole is divided into three equal parts.

Understanding these terms is crucial for fractions work, especially when performing operations such as addition, subtraction, multiplication, or division.

Multiplying Fractions

Multiplying fractions may seem tricky at first, but it's quite straightforward when you know the steps. Unlike addition or subtraction, you don't need a common denominator.

Here's a simple guide to follow:

• First, identify and separate the numerators and denominators of both fractions.
• Next, multiply the numerators together. This product becomes the numerator of the result.
• Afterward, multiply the denominators together. This product becomes the denominator of the result.

For instance, let's multiply \(\frac{1}{3}\) and \(\frac{4}{9}\).

- The numerators are 1 and 4. Multiplying these gives us \(1 \times 4 = 4\).

- The denominators are 3 and 9. Multiplying these gives us \(3 \times 9 = 27\).

So, \(\frac{1}{3} \times \frac{4}{9} = \frac{4}{27}\).

That's it! You now have a new fraction as the result of your multiplication.

Step-by-Step Solution

Let's break down the solution to our original exercise step by step for clarity.

**Step 1: Identify the Numerators and Denominators**
First, identify the numerators and denominators of the fractions. For \(\frac{1}{3}\), the numerator is 1, and the denominator is 3. For \(\frac{4}{9}\), the numerator is 4, and the denominator is 9.

**Step 2: Multiply the Numerators Together**
Multiply the numerators of both fractions: \(1 \times 4 = 4\). This product is the numerator of the new fraction.

**Step 3: Multiply the Denominators Together**
Multiply the denominators of both fractions: \(3 \times 9 = 27\). This product is the denominator of the new fraction.

**Step 4: Combine the Results**
Combine the numerator and denominator from the previous steps to form the new fraction: \(\frac{4}{27}\).

By following each step carefully, you can confidently solve fraction multiplication problems with ease.