Problem 4 - Prealgebra

Question

Find each product. Use an area model if necessary. $$\frac{3}{7} \cdot \frac{1}{6}$$

Step-by-Step Solution

Answer

\( \frac{3}{7} \times \frac{1}{6} = \frac{1}{14} \).

Step 1: Understand the Problem

We need to multiply two fractions: \( \frac{3}{7} \) and \( \frac{1}{6} \). Multiplying fractions involves multiplying the numerators together and the denominators together.

Step 2: Multiply the Numerators

Multiply the numerators of the fractions: \( 3 \times 1 = 3 \).

Step 3: Multiply the Denominators

Multiply the denominators of the fractions: \( 7 \times 6 = 42 \).

Step 4: Write the Fraction Result

Combine the results from Steps 2 and 3 to form the fraction: \( \frac{3}{42} \).

Step 5: Simplify the Fraction

Check if the fraction \( \frac{3}{42} \) can be simplified. Divide both the numerator and the denominator by their greatest common divisor, which is 3: \( \frac{3 \div 3}{42 \div 3} = \frac{1}{14} \).

Step 6: Final Result

The product of \( \frac{3}{7} \) and \( \frac{1}{6} \) is \( \frac{1}{14} \).

Key Concepts

Using the Area Model for Fraction Multiplication

The area model is a visual tool that helps with understanding the process of fraction multiplication by representing fractions as areas of rectangles. When we multiply two fractions, such as \( \frac{3}{7} \times \frac{1}{6} \), we can use rectangles to create a visual representation.

Imagine a rectangle divided into 7 equal parts vertically, where 3 parts are colored to represent \( \frac{3}{7} \). Next, divide the same rectangle horizontally into 6 equal parts, and color 1 of these horizontal sections to represent \( \frac{1}{6} \). The overlapping colored area, which represents the product of the two fractions, will consist of the unit fractions that overlap in both divisions.

• The total number of small rectangles created is 7 times 6, or 42, representing the denominator of the product.
• The overlapping colored area, representing the numerator, is 3, as 3 of the sections are colored as both vertical and horizontal overlap.

Using this approach, students can better visualize how multiplying fractions involves both numerator and denominator, reinforcing the concept of fraction multiplication.

Simplifying Fractions

Once we have multiplied the fractions and arrived at a result like \( \frac{3}{42} \), it's beneficial to simplify this fraction to its lowest terms. Simplifying fractions makes them easier to understand and work with.

Simplifying a fraction involves dividing both the numerator and the denominator by their greatest common divisor (GCD). In our example, to simplify \( \frac{3}{42} \), you identify the GCD of 3 and 42. The GCD is the largest number that can divide both 3 and 42 evenly.

• Find factors of 3: 1, 3.
• Find factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
• The greatest number common to both lists is 3.

Dividing both the numerator and the denominator by 3, we get \( \frac{1}{14} \). The result is the simplest form of the fraction, which is easier to read and compare with other numbers.

Understanding the Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions, as it helps reduce fractions to their simplest form. The GCD of two numbers is the largest number that divides each of them without leaving a remainder.

For example, to simplify \( \frac{3}{42} \), we needed the GCD of 3 and 42. Finding the GCD involves listing the factors of each number and identifying the largest factor they share.

• Factors of 3: 1, 3.
• Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42.
• The largest common factor is 3.

Once the GCD is identified, simplify the fraction by dividing both the numerator and the denominator by this common divisor. This process ensures the fraction is as simple as possible, enhancing clarity and understanding while performing calculations and comparisons.