Problem 15 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{8}{36}$$

Step-by-Step Solution

Answer

\( \frac{2}{9} \) is the simplest form.

Step 1: Identify the GCD

Find the Greatest Common Divisor (GCD) of the numerator and the denominator. The numerator is 8 and the denominator is 36. The factors of 8 are 1, 2, 4, 8 and the factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36. The greatest common factor is 4.

Step 2: Divide by the GCD

Divide both the numerator and the denominator by the GCD found in step 1. Divide 8 by 4 to get 2 and 36 by 4 to get 9. The fraction becomes \( \frac{2}{9} \).

Step 3: Verify the Result

Check if \( \frac{2}{9} \) is in its simplest form by ensuring that the only common factor between 2 and 9 is 1. They have no other common factors, confirming that the fraction is simplified.

Key Concepts

Greatest Common Divisor (GCD)

Understanding the greatest common divisor is key when simplifying fractions. The GCD of two numbers is the biggest number that divides both without leaving a remainder. For example, if we want to simplify the fraction \(\frac{8}{36}\), we first need to find the GCD of 8 and 36.

• The factors of 8 are 1, 2, 4, and 8.
• The factors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, and 36.

Comparing these lists, we see that the highest common factor is 4. This number, 4, is the greatest common divisor of 8 and 36. Once you identify the GCD, you can use it to simplify the fraction by dividing both the numerator and the denominator by this number.

Fraction Simplification

Simplifying fractions is about making them as simple as possible without changing their value. To do this, you divide both the top and bottom of the fraction by their greatest common divisor.
In our example, the fraction is \(\frac{8}{36}\). First, using the GCD, which is 4, we divide the numerator (8) and the denominator (36) by 4.

• \(8 \div 4 = 2\)
• \(36 \div 4 = 9\)

So, \(\frac{8}{36}\) simplifies to \(\frac{2}{9}\). This process makes the fraction easier to work with while keeping the same value. It's like reducing clutter to find what really matters.

Numerator and Denominator

When working with fractions, it's important to understand the roles of the numerator and the denominator.

• The numerator is the top number of a fraction, representing the number of parts being considered.
• The denominator is the bottom number, representing the total number of equal parts in a whole.

In \(\frac{8}{36}\), the 8 is the numerator, telling us you're considering 8 parts of something that's divided into 36 equal parts, which is the denominator.
Simplifying the fraction means altering these numbers without changing the actual quantity of the fraction. By dividing them by the greatest common divisor, we keep the ratio the same, providing a cleaner and more understandable representation.