Problem 91 - Algebra 1: Concepts and Skills

Question

Write the fraction in simplest form. (Skills Review p. 763) $$ \frac{20}{24} $$

Step-by-Step Solution

Answer

\(\frac{5}{6}\)

Step 1: Find the Greatest Common Divisor

Find the greatest common divisor (GCD) of both the numerator (20) and the denominator (24). By listing the factors of 20 and 24, it's clear that their GCD is 4.

Step 2: Divide by the GCD

Divide both the numerator and denominator by the GCD. In this case, divide both the numerator (20) and the denominator (24) by 4.

Step 3: Write the Simplified Fraction

After dividing, the fraction is simplified to \(\frac{5}{6}\). This cannot be simplified any further because 5 and 6 do not have any common factors other than 1.

Key Concepts

Greatest Common Divisor (GCD)

The Greatest Common Divisor (GCD), also known as the greatest common factor or highest common factor, is a fundamental concept when simplifying fractions. It's the largest number that can divide both the numerator and the denominator without leaving a remainder.

For example, when looking at the fraction \(\frac{20}{24}\), to determine the GCD, list all the factors of the numerator and denominator.

• Factors of 20: 1, 2, 4, 5, 10, 20
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

Both numbers share common factors, but the greatest one they share is 4. Knowing the GCD is crucial for simplifying fractions since it allows us to reduce the fraction to its simplest form.

Numerator and Denominator

In a fraction, the numerator is the number above the line, and it represents how many parts we are considering. The denominator is the number below the line, representing the total number of equal parts in the whole or in the set.

Using our example \(\frac{20}{24}\), 20 is the numerator, indicating the number of parts taken, while 24 is the denominator, representing the whole. When simplifying fractions, we adjust both the numerator and denominator to make the fraction easier to understand and use, often by identifying and dividing by the GCD.

Simplest Form

A fraction is in its simplest form when the numerator and the denominator are as small as possible and cannot be evenly divided by any number other than 1. This means the fraction has been reduced to its most basic representation, making it easier to grasp and work with.

After finding the GCD of 4 for the fraction \(\frac{20}{24}\), we divide the numerator and denominator by this number to simplify the fraction: \(\frac{20 \div 4}{24 \div 4} = \frac{5}{6}\). The resulting fraction, \(\frac{5}{6}\), is in its simplest form since 5 and 6 have no common divisors other than 1. Recognizing a fraction in simplest form is essential for understanding and solving problems with fractions effectively.