Problem 16 - Beginning and Intermediate Algebra

Question

Write the fraction in lowest terms. $$\frac{5}{9}$$

Step-by-Step Solution

Answer

\( \frac{5}{9} \) is already in its lowest terms.

Step 1: Determine the Greatest Common Divisor (GCD)

To reduce a fraction to its lowest terms, we need to find the greatest common divisor of the numerator and the denominator. Identify the greatest number that divides both 5 and 9 without leaving a remainder. Since 5 and 9 have no common divisors other than 1, their GCD is 1.

Step 2: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their greatest common divisor. Since the GCD of 5 and 9 is 1, we divide both by 1: \( \frac{5}{9} = \frac{5 \div 1}{9 \div 1} = \frac{5}{9} \).

Step 3: Confirm the Fraction is in Lowest Terms

Verify that the fraction is now in its simplest form. Since the only common factor of 5 and 9 is 1, \( \frac{5}{9} \) is already in its simplest form.

Key Concepts

Greatest Common Divisor

Understanding the concept of the Greatest Common Divisor (GCD) helps you in simplifying fractions. The GCD is the largest number that can divide both the numerator and the denominator of a fraction without leaving a remainder. To figure out the GCD, list all the factors of each number, then look for the greatest number that appears in both lists.
For example, the factors of 9 are 1, 3, and 9, while the factors of 5 are 1 and 5. The only common factor is 1, so the GCD of 5 and 9 is 1.
Knowing the GCD is crucial because it tells you how much you can simplify the fraction without changing its value. Finding the GCD is the first step to check if a fraction is already in its simplest form or if it needs further simplification.

Simplest Form

Once you have the GCD, you can use it to write the fraction in its simplest form, also known as lowest terms. This means reducing the fraction such that the numerator and denominator have no common divisors other than 1.
To do this, divide both the numerator and the denominator by the GCD. If they divide evenly, you have your fraction in simplest form. For example, when simplifying \( \frac{5}{9} \), since the GCD is 1, dividing both 5 and 9 by 1 does not change the fraction. Thus, \( \frac{5}{9} \) is already in its simplest form.
Keeping fractions in their simplest form makes them easier to work with and compare. It also presents a clearer picture of the relationship between the numerator and the denominator.

Numerator and Denominator

Fractions consist of two parts: a numerator and a denominator, separated by a slash. The numerator is the top number, indicating how many parts you have. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.
In the fraction \( \frac{5}{9} \), the numerator is 5 and the denominator is 9.
Understanding these terms is essential because they help you grasp the meaning of the fraction. In this instance, 5 out of 9 parts are being considered.

• The numerator tells you "how many."
• The denominator tells you "of what."

When simplifying fractions, you work with both the numerator and denominator to ensure the fraction is represented in its most concise form.