Problem 24 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{2}{5} \cdot \frac{5}{6}$$

Step-by-Step Solution

Answer

\( \frac{2}{5} \cdot \frac{5}{6} = \frac{1}{3} \)

Step 1: Understand the Multiplication of Fractions

When multiplying fractions, you multiply the numerators together to get a new numerator and the denominators together to get a new denominator. The product of two fractions is found as follows: \(\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}\).

Step 2: Set Up the Multiplication

Identify the numerators and denominators from each fraction: the numerator of \(\frac{2}{5}\) is 2, and its denominator is 5; the numerator of \(\frac{5}{6}\) is 5, and its denominator is 6. Set up the product as \( \frac{2 \cdot 5}{5 \cdot 6} \).

Step 3: Multiply the Numerators

Multiply the numerators (top numbers) of the fractions: \( 2 \cdot 5 = 10 \).

Step 4: Multiply the Denominators

Multiply the denominators (bottom numbers) of the fractions: \( 5 \cdot 6 = 30 \).

Step 5: Write Down the New Fraction

After the multiplication of numerators and denominators, you get a new fraction: \( \frac{10}{30} \).

Step 6: Simplify the Fraction

Simplify \( \frac{10}{30} \) by finding the greatest common divisor (GCD) of 10 and 30, which is 10. Divide both the numerator and the denominator by 10: \( \frac{10}{30} = \frac{10 \div 10}{30 \div 10} = \frac{1}{3} \).

Key Concepts

Area Model

An area model is a useful visual tool for understanding multiplication, particularly when dealing with fractions. It involves representing fractions as parts of a rectangle, where the total area is equal to 1 (the whole).

In our exercise, we have the fractions \( \frac{2}{5} \) and \( \frac{5}{6} \). To use an area model, imagine a rectangle divided into 5 equal columns to represent the denominator of \( \frac{2}{5} \), as the whole is divided into 5 parts. Out of these, 2 parts are shaded to represent the numerator. Next, divide this same rectangle into 6 equal rows to represent the denominator of \( \frac{5}{6} \) and shade 5 rows, showing its corresponding numerator.

Where the two shaded areas overlap is the fraction of the products, \( \frac{10}{30} \). This step helps visualize how multiplying fractions gives us a smaller part of the whole, as the overlap highlights the shared portion of the total area.

Numerator

The numerator is the top part of a fraction, indicating how many parts we have out of the total, which is represented by the denominator. It is essential to differentiate between the numerator and denominator to correctly perform operations like multiplication.

In the exercise \( \frac{2}{5} \times \frac{5}{6} \), the numerators are 2 and 5, respectively. To multiply these fractions, you begin by multiplying the numerators. This means finding the product of the top numbers:

• Multiply the numerators: \( 2 \times 5 = 10 \)

Clearly knowing what the numerator represents helps in understanding that the result, 10, shows there are 10 parts in the product before reduction or simplifying is performed.

Denominator

The denominator forms the bottom half of a fraction, illustrating into how many equal parts the whole is divided. It is key in understanding the portion size or the whole.

For the fractions \( \frac{2}{5} \) and \( \frac{5}{6} \) from our exercise, the denominators are 5 and 6, respectively. To find the product of these fractions, follow these steps:

• Multiply the denominators: \( 5 \times 6 = 30 \)

The resulting product, \( \frac{10}{30} \), shows the new denominator as the total divisions in the combined system of parts. Understanding this helps visualize the division of a whole in fractional multiplication that brings the two components together.

Simplifying Fractions

Simplifying, or reducing fractions, means making them easier to work with by finding their simplest form. This involves dividing both the numerator and the denominator by their greatest common divisor (GCD).

In our example \( \frac{10}{30} \), we simplify by identifying the GCD of 10 and 30, which is 10. Simplifying uses the formula:

• Divide both numbers: \( \frac{10 \div 10}{30 \div 10} = \frac{1}{3} \)

The simplest form \( \frac{1}{3} \) tells us that out of every three parts of the whole, we have one. Simplifying helps not only in obtaining a more useful form of the fraction but also in recognizing equivalent fractions that give the same value.