Problem 20 - Algebra for College Students

Question

Reduce each rational expression to its lowest terms. $$\frac{14}{91}$$

Step-by-Step Solution

Answer

The reduced form is \( \frac{2}{13} \).

Step 1: Identify the Greatest Common Divisor (GCD)

First, identify the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 14 and 91 is 7.

Step 2: Divide the Numerator by the GCD

Divide the numerator (14) by the GCD (7). \[ \frac{14}{7} = 2 \]

Step 3: Divide the Denominator by the GCD

Divide the denominator (91) by the GCD (7). \[ \frac{91}{7} = 13 \]

Step 4: Write the Reduced Form

Combine the results of the divisions to write the rational expression in its reduced form. \[ \frac{14}{91} = \frac{2}{13} \]

Key Concepts

Greatest Common Divisor

The Greatest Common Divisor (GCD) is a crucial concept in simplifying rational expressions. It refers to the largest number that can exactly divide both the numerator and the denominator without leaving a remainder. Finding the GCD can greatly simplify fractions. For example, in the fraction \( \frac{14}{91} \), the GCD of 14 and 91 is determined to be 7. Identifying the GCD correctly allows us to reduce the fraction more efficiently.
Here are steps to find the GCD:

• List all factors of both the numerator and the denominator.
• Identify the largest common factor.

This process helps in simplifying fractions quickly and accurately.

numerator

The numerator is the top part of a fraction. It tells how many parts we have out of a whole. In the fraction \( \frac{14}{91} \), 14 is the numerator.
When simplifying fractions, we first focus on the numerator to check for common factors with the denominator. For instance, to simplify \( \frac{14}{91} \), we divide the numerator (14) by the GCD, which is 7:
\[ \frac{14}{7} = 2 \] This gives us a simplified numerator. Always remember to include this step in fraction simplification.

denominator

The denominator is the bottom part of a fraction. It shows into how many parts the whole is divided. In the fraction \( \frac{14}{91} \), 91 is the denominator.
To reduce a fraction, we must simplify the denominator alongside the numerator. Using our example, we divide the denominator (91) by the GCD, which is 7:
\[ \frac{91}{7} = 13 \] This results in a simplified denominator. Simplifying both numerator and denominator is key to reducing fractions correctly.

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form, where the numerator and the denominator have no common divisors other than 1. To simplify a fraction, follow these steps:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by the GCD.

Using our example \( \frac{14}{91} \):
First, find the GCD of 14 and 91, which is 7.
Then, divide both the numerator and the denominator by 7:
\[ \frac{14 \div 7}{91 \div 7} = \frac{2}{13} \] The fraction \( \frac{14}{91} \) simplifies to \( \frac{2}{13} \). Breaking down each step ensures accurate and clear simplification.