Problem 20 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{15}{20} $$

Step-by-Step Solution

Answer

The simplified fraction is \(\frac{3}{4}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{15}{20}\), we need to find the greatest common divisor (GCD) of the numerator (15) and the denominator (20). List the factors of each number: - Factors of 15: \(1, 3, 5, 15\)- Factors of 20: \(1, 2, 4, 5, 10, 20\).The greatest common factor is 5.

Step 2: Divide Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their greatest common divisor (5): - \(\frac{15}{5} = 3\)- \(\frac{20}{5} = 4\).This gives us the simplified fraction.

Step 3: Write the Simplified Fraction

After dividing both terms by the GCD, the simplified form of the fraction \(\frac{15}{20}\) is \(\frac{3}{4}\).

Key Concepts

Greatest Common Divisor

The greatest common divisor, often abbreviated as GCD, is a key concept in fraction simplification. It is the largest number that divides both the numerator and the denominator of a fraction without leaving a remainder. Finding the GCD helps us reduce fractions to their simplest forms.
To find the GCD of any two numbers, like the numerator 15 and the denominator 20 in our problem, you can list the factors of each number and identify the highest number they have in common.
For example:

• The factors of 15 are: 1, 3, 5, and 15.
• The factors of 20 are: 1, 2, 4, 5, 10, and 20.

The highest common factor here is 5. Hence, the GCD is 5. Once the GCD is determined, it sets the stage for simplifying the fraction efficiently by dividing both numerator and denominator by this number.

Numerator and Denominator

In a fraction, there are two main parts you need to understand: the numerator and the denominator.
The **numerator** is the number located above the fraction bar. It represents how many parts we have. For instance, in the fraction \(\frac{15}{20}\), the numerator is 15.
The **denominator** sits below the fraction bar. It represents the total number of equal parts that make up a whole. In our example, the denominator is 20.
The positions of these two numbers are crucial as they define the fraction's value. When simplifying fractions, we focus on finding a number that can divide both the numerator and the denominator, which typically is the GCD.

Simplified Fraction

A simplified fraction is the most reduced form of a fraction, where the numerator and the denominator have no common factor other than 1. Simplifying fractions makes them easier to understand and compare.
Once we have identified the GCD, the next step is to divide both the numerator and the denominator by this number. In our example with \(\frac{15}{20}\), the GCD is 5. So, we divide:

• Numerator: \(\frac{15}{5} = 3\)
• Denominator: \(\frac{20}{5} = 4\)

This process results in a simplified fraction: \(\frac{3}{4}\). This simplified form is much easier to interpret and use in further mathematical operations.