Problem 2 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{1}{3} \cdot \frac{2}{5}$$

Step-by-Step Solution

Answer

The product is \( \frac{2}{15} \).

Step 1: Set up the problem

To solve the problem \( \frac{1}{3} \cdot \frac{2}{5} \), we'll multiply the numerators and then the denominators separately.

Step 2: Multiply the numerators

The product of the numerators is \( 1 \cdot 2 = 2 \).

Step 3: Multiply the denominators

The product of the denominators is \( 3 \cdot 5 = 15 \).

Step 4: Form the fraction

Combine the results to form a new fraction: \( \frac{2}{15} \).

Key Concepts

Understanding the Area Model

The area model is a visual way to learn and understand fraction multiplication. It helps students to see how parts of two fractions combine to make a new fraction. Imagine a rectangle that is divided into a grid based on the fractions we are working with.
For our problem, we use fractions \( \frac{1}{3} \) and \( \frac{2}{5} \). Here's how you can visualize it:

• Step 1: Draw a rectangle and divide it into three equal parts vertically. This division represents the denominator of the first fraction, \( 3 \).
• Step 2: Shade one part to show the numerator \( 1 \) of \( \frac{1}{3} \).
• Step 3: Now, horizontally divide the same rectangle into five equal parts. This division shows the denominator of the second fraction, \( 5 \).
• Step 4: Shade two horizontal parts to represent the numerator \( 2 \) of \( \frac{2}{5} \).

The overlap of these shaded areas is crucial. It represents \( \frac{2}{15} \), the product of the original fractions. This method makes the multiplication of fractions much easier to grasp.

The Role of Numerators

Numerators tell us how many parts of a whole we are considering. In our example where we multiply \( \frac{1}{3} \) by \( \frac{2}{5} \), the numerators are \( 1 \) and \( 2 \).
When multiplying fractions, we multiply the numerators straight across, combining the parts from each fraction.

Here are some steps to think about:

• Take the numerator of the first fraction \( (1) \), and multiply it by the numerator of the second fraction \( (2) \).
• The result is the numerator of the new fraction \( (2) \).

This process lets us determine how many parts we are mixing together from each fraction. The product of the numerators thus gives us the count for the new fraction's "part." It's simple once you break it down to these steps.

The Role of Denominators

Denominators indicate the total number of parts that make up the whole. They are essential for understanding the size of the parts we are dealing with.
In the fraction multiplication \( \frac{1}{3} \times \frac{2}{5} \), the denominators are \( 3 \) and \( 5 \). To find the new denominator:

• Multiply the denominator of the first fraction \( (3) \), by the denominator of the second fraction \( (5) \).
• The result is \( 15 \), which becomes the denominator of the new fraction \( (\frac{2}{15}) \).

This product tells us how many small parts fit into the whole and is a key factor in determining the size of each part in our resulting fraction. The denominator keeps the fraction balanced in its context, showing just how each fraction fits into the greater unit whole.

Steps in Fraction Simplification

Fraction simplification, often called reducing the fraction, is the process of breaking down a fraction into its simplest form. This typically involves finding the greatest common divisor (GCD). However, in our case, \( \frac{2}{15} \) is already in its simplest form since 2 and 15 have no common factors other than 1.

Here's how you generally approach fraction simplification:

• Step 1: Check both the numerator and the denominator for common factors.
• Step 2: Use the greatest common divisor (GCD) to divide both the numerator and the denominator.
• Step 3: Rewrite the fraction with the simplified numerator and denominator.

If the result is the same after the division, the fraction is already as simple as it gets. Simplification makes it easier to work with fractions and compare them to others.