Problem 12 - Intermediate Algebra

Question

Multiply, and then simplify, if possible. See Objective 1. $$ \frac{13}{6} \cdot \frac{6}{21} $$

Step-by-Step Solution

Answer

The simplified product of the fractions is \( \frac{13}{21} \).

Step 1: Multiply the Numerators

Multiply the numerators of the two fractions. The numerators are 13 and 6. Calculation: \( 13 \times 6 = 78 \)

Step 2: Multiply the Denominators

Multiply the denominators of the two fractions. The denominators are 6 and 21. Calculation: \( 6 \times 21 = 126 \)

Step 3: Write the Resulting Fraction

Combine the results from Step 1 and Step 2 to get the product of the fractions. Thus, the fraction is: \( \frac{78}{126} \)

Step 4: Simplify the Fraction

Find the greatest common divisor (GCD) of 78 and 126 to simplify the fraction. The GCD of 78 and 126 is 6. Now, divide the numerator and the denominator by 6.Calculation:\( \frac{78 \div 6}{126 \div 6} = \frac{13}{21} \)

Key Concepts

Simplifying Fractions

When working with fractions, one key step is simplifying them to their most basic form. Simplification makes fractions easier to understand and work with. To simplify a fraction, you need to find the greatest common divisor (GCD) of the numerator and the denominator.
Dividing both by their GCD reduces the fraction to its simplest form, without changing its value. For example, to simplify the fraction \( \frac{78}{126} \), we first find the GCD, which is 6, and then divide both the numerator and the denominator by 6. So, \( \frac{78}{126} \) simplifies to \( \frac{13}{21} \).

Simplifying can greatly help in comparing fractions, performing arithmetic operations, and recognizing equivalent fractions.

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a crucial mathematical tool when dealing with fractions. The GCD is the largest positive integer that evenly divides both numbers, leaving no remainder.
Finding the GCD involves listing out the factors of each number and identifying the largest one they have in common.

For the example \( \frac{78}{126} \), the factors of 78 are 1, 2, 3, 6, 13, 26, 39, 78, and the factors of 126 are 1, 2, 3, 6, 7, 9, 14, 18, 21, 42, 63, 126.

• The common factors between the two are 1, 2, 3, and 6.
• The largest factor is 6, which is the GCD.

Using the GCD, we can simplify the fraction efficiently, as seen where we determined that dividing both 78 and 126 by 6 gives us \( \frac{13}{21} \).

Numerator and Denominator Multiplication

Multiplying fractions involves multiplying their numerators together and their denominators together. It’s a straightforward operation, but you must pay attention to the numbers involved to ensure accuracy.

• The numerator is the top number of the fraction, representing the parts being considered.
• The denominator is the bottom number, representing the total parts.

In the problem \( \frac{13}{6} \times \frac{6}{21} \), we perform two main calculations:

• Multiply the numerators: \( 13 \times 6 = 78 \).
• Multiply the denominators: \( 6 \times 21 = 126 \).

This gives us the fraction \( \frac{78}{126} \).
Often, after multiplying, the resulting fraction can be simplified. In this case, the ultimate goal is to find and divide by the GCD, resulting in \( \frac{13}{21} \). This step ensures the product of the fractions is as simplified as possible, making further calculations easier.