Problem 4 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{18}{32}$$

Step-by-Step Solution

Answer

\(\frac{9}{16}\) is the fraction in its lowest terms.

Step 1: Determine the Greatest Common Divisor (GCD)

To reduce the fraction \( \frac{18}{32} \) to its lowest terms, we first need to find the greatest common divisor (GCD) of the numerator (18) and the denominator (32). List the factors of both numbers.- Factors of 18: 1, 2, 3, 6, 9, 18- Factors of 32: 1, 2, 4, 8, 16, 32The greatest common divisor is the largest factor that both numbers share, which is 2 in this case.

Step 2: Divide by the GCD

Now that we have determined the GCD, which is 2, we will divide both the numerator and the denominator of the fraction \( \frac{18}{32} \) by 2. \[\frac{18}{32} = \frac{18 \div 2}{32 \div 2} = \frac{9}{16}\]

Step 3: Verify the Simplified Fraction

To ensure that \( \frac{9}{16} \) is in its lowest terms, check if there are any common factors other than 1. The factors of 9 are 1, 3, 9, and the factors of 16 are 1, 2, 4, 8, 16. The only common factor is 1, confirming that \( \frac{9}{16} \) is simplified.

Key Concepts

Greatest Common Divisor (GCD)

When simplifying fractions, identifying the Greatest Common Divisor (GCD) between the numerator and denominator is key. The GCD is the largest whole number that can evenly divide both numbers involved in the fraction. Finding the GCD helps in breaking down a fraction into its simplest form, known as reducing to lowest terms.

Here's how you can find the GCD:

• List all the factors of the numerator.
• List all the factors of the denominator.
• Identify the largest factor that appears in both lists. This is your GCD.

For example, in the fraction \( \frac{18}{32} \), we found that the GCD is 2 because it is the highest number that both 18 and 32 can be divided by without leaving a remainder. Identifying the GCD is a fundamental step in fraction simplification.

Numerator and Denominator

Understanding the roles of the numerator and denominator is crucial in fraction simplification. A fraction consists of two parts:

• Numerator: This is the number on top of the fraction. It represents how many parts of the whole are being considered.
• Denominator: This is the number on the bottom of the fraction. It signifies the total number of equal parts the whole is divided into.

In our example, the fraction \( \frac{18}{32} \), 18 is the numerator, and 32 is the denominator. Simplification involves manipulating both the numerator and the denominator to achieve a fraction equivalent to the original but in its simplest form. You divide the numerator and the denominator by their GCD to reduce the fraction effectively.

Lowest Terms Reduction

Reducing a fraction to its lowest terms means rewriting it in its simplest form, where the numerator and the denominator have no common factors other than 1. This process is essential because it makes fractions easier to work with in calculations and provides a standardized form for comparison.

Here's how to reduce a fraction:

• Find the Greatest Common Divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by their GCD.
• Check to ensure no additional common factors exist beyond one.

In the example \( \frac{18}{32} \), we divided both 18 and 32 by their GCD, 2, to simplify the fraction to \( \frac{9}{16} \). After this step, verifying if \( \frac{9}{16} \) can be simplified further (no common factors besides 1) assures that the fraction is in its lowest terms.