Problem 12 - Beginning and Intermediate Algebra

Question

Write the fraction in lowest terms. $$\frac{3}{6}$$

Step-by-Step Solution

Answer

The fraction \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator. The GCD is the largest number that can divide both the numerator and the denominator without leaving a remainder. In the fraction \( \frac{3}{6} \), the numerator is 3 and the denominator is 6.

Step 2: Find the GCD of 3 and 6

We list out the divisors for each number. The divisors of 3 are 1 and 3, and the divisors of 6 are 1, 2, 3, and 6. The common divisors are 1 and 3. The greatest of these is 3, so the GCD of 3 and 6 is 3.

Step 3: Divide by the GCD

Now that we have found the GCD, we divide both the numerator and the denominator by this number to simplify the fraction. Divide 3 by 3 to get 1, and divide 6 by 3 to get 2. Thus, \( \frac{3}{6} \) simplifies to \( \frac{1}{2} \).

Key Concepts

Greatest Common Divisor

Understanding the greatest common divisor, or GCD, is a key concept in simplifying fractions. The GCD is simply the largest number that divides two numbers exactly, without leaving any remainder.
Let's break this down:

• Divide: To find the GCD, identify the numbers that can divide both the numerator and the denominator.
  
• Largest Number: The GCD is the largest such number that fits this rule.
  

In our example with \( \frac{3}{6} \):

• Determine divisors of 3: 1, 3
  
• Determine divisors of 6: 1, 2, 3, 6
  
• Common divisors are 1 and 3.
  
• The greatest is 3, so the GCD of 3 and 6 is 3.
  

By using the GCD, you ensure the fraction gets fully simplified.

Simplifying Fractions

Simplifying fractions makes them easier to work with and understand. It involves reducing the numerator and denominator to their smallest possible values using the greatest common divisor (GCD).

Here's how:

• Identify the GCD of the fraction’s numerator and denominator.
  
• Divide both the numerator and the denominator by this GCD.
  

This process shortens the fraction while retaining its equal value.

In the case of \( \frac{3}{6} \), we divide both by their GCD, which is 3:

• \( \frac{3}{3} = 1 \)
  
• \( \frac{6}{3} = 2 \)
  

The simplified fraction is \( \frac{1}{2} \).
Simplifying helps you better compare fractions and perform calculations like addition or subtraction.

Numerator and Denominator

Every fraction is divided into two parts: the numerator and the denominator. Understanding these terms helps in many mathematical operations, including simplifying fractions.

• Numerator: The top number in a fraction. It signifies how many parts of a whole you have.
  
• Denominator: The bottom number. It represents the total number of equal parts the whole is divided into.
  

When simplifying a fraction, you aim to make the numerator and denominator as small as possible, keeping them integers and still representing the same value as before.
In our fraction \( \frac{3}{6} \):

• 3 is the numerator, showing 3 parts of something.
  
• 6 is the denominator, indicating the whole is divided into 6 equal parts.
  

By dividing both by their GCD (3), we reduce our understanding of the fraction to \( \frac{1}{2} \). This means for every 2 parts of the whole, we have 1 part, streamlining calculations and comparisons.