Problem 17 - Algebra A Combined Function

Question

Simplify each fraction. See Examples 8 through \(10 .\) $$ \frac{2}{4} $$

Step-by-Step Solution

Answer

The simplified fraction is \( \frac{1}{2} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{2}{4} \), we first need to find the greatest common divisor of the numerator (2) and the denominator (4). The common divisors of 2 and 4 are 1 and 2, so the greatest common divisor is 2.

Step 2: Divide Both Numerator and Denominator by the GCD

Now we divide both the numerator and the denominator of the fraction by the GCD we found in Step 1. Thus, we divide 2 by 2 and 4 by 2. This simplifies the fraction as follows:\[ \frac{2}{4} = \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \]

Step 3: Verify the Simplified Fraction

Ensure that the fraction \( \frac{1}{2} \) is in its simplest form by confirming that the numerator and the denominator have no common divisors other than 1. Since 1 and 2 have no common divisors besides 1, \( \frac{1}{2} \) is indeed the simplest form.

Key Concepts

Greatest Common Divisor

To simplify fractions, the Greatest Common Divisor (GCD) plays a crucial role. The GCD is the largest number that divides both the numerator and the denominator without leaving a remainder.
By finding this number, we can effectively reduce fractions. For instance, when simplifying the fraction \(\frac{2}{4}\), we first identify the GCD of 2 and 4.

• List the divisors of 2: 1, 2.
• List the divisors of 4: 1, 2, 4.

The greatest common divisor from these lists is 2. Understanding how to find the GCD is essential for simplifying fractions accurately.
If a fraction's numerator and denominator share a GCD greater than 1, division by this number will make the fraction simpler.

Numerator and Denominator

A fraction consists of two parts: the numerator, which is the top number, and the denominator, which is the bottom number. These parts represent how many parts out of a whole are being considered.
In \(\frac{2}{4}\), 2 is the numerator and 4 is the denominator.
Recognizing the role of each part is important. The numerator indicates the number of parts we have, while the denominator suggests the total parts that make a whole. For a fraction to be simplified effectively, both these parts need to be divisible by their GCD.
This approach ensures that the fraction maintains its value while reducing its terms to the simplest form.

Fraction Reduction

Fraction reduction involves simplifying a fraction to its lowest terms by dividing both the numerator and the denominator by their GCD.
After finding that the GCD of 2 and 4 is 2, as from the previous section, we can simplify \(\frac{2}{4}\) by dividing:

• Divide the numerator: \(2 \div 2 = 1\)
• Divide the denominator: \(4 \div 2 = 2\)

The fraction \(\frac{2}{4}\) thus reduces to \(\frac{1}{2}\). This process ensures the value of the fraction remains unchanged, but its numbers are minimized. After reduction, check that the new numerator and denominator have no common divisors other than 1 to confirm it is fully simplified. Always aim for the smallest and simplest components possible in fraction form.