Problem 1 - Prealgebra

Question

Write each fraction in simplest form. If the fraction is already in simplest form, write simplified. $$\frac{2}{14}$$

Step-by-Step Solution

Answer

The simplified form of \( \frac{2}{14} \) is \( \frac{1}{7} \).

Step 1: Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we need to find the greatest common divisor of the numerator and the denominator. In this case, the numerator is 2 and the denominator is 14. The divisors of 2 are 1 and 2. The divisors of 14 are 1, 2, 7, and 14. The greatest common divisor of 2 and 14 is 2.

Step 2: Divide the Numerator by the GCD

Divide the numerator of the fraction by the GCD found in step 1. Here, divide 2 by 2, which gives us 1.

Step 3: Divide the Denominator by the GCD

Divide the denominator of the fraction by the GCD found in step 1. Here, divide 14 by 2, which gives us 7.

Step 4: Write the Simplified Fraction

Using the results from Steps 2 and 3, we write the fraction. The simplified form of \( \frac{2}{14} \) is \( \frac{1}{7} \).

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept when it comes to simplifying fractions. It refers to the largest number that can evenly divide both the numerator and the denominator of a fraction. By finding the GCD, you can determine how much you can "reduce" the fraction to its simplest form.

Here's how to find it:

• List all the divisors (those numbers that divide evenly) of the numerator.
• List all the divisors of the denominator.
• The greatest number that appears in both lists is your GCD.

For the fraction \( \frac{2}{14} \), we already know that the GCD is 2 because it's the largest number that divides both 2 and 14. Using the GCD, we can make the fraction as simple as possible.

Numerator and Denominator

Understanding the numerator and denominator is essential for working with fractions. A fraction is written with these two parts, and they represent portions of a whole.

The **numerator** is the top number:

• It tells you how many parts of the whole you are considering.

The **denominator** is the bottom number:

• It indicates the number of equal parts the whole is divided into.

For example, in the fraction \( \frac{2}{14} \):

• 2 is the numerator.
• 14 is the denominator.

This means we're considering 2 parts out of 14 equal parts. Knowing the role of the numerator and denominator helps in understanding how a fraction is reduced or simplified.

Fraction Reduction

Reducing or simplifying a fraction means getting it to its simplest form, where the numerator and the denominator have no common factors except for 1. The process involves using the greatest common divisor (GCD), which you've found, to divide both parts of the fraction.

Steps to reduce a fraction:

• Find the GCD of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.

For the fraction \( \frac{2}{14} \):

• GCD = 2
• Divide numerator (2) by GCD, giving you 1.
• Divide denominator (14) by GCD, giving you 7.

After applying these steps, \( \frac{2}{14} \) reduces to \( \frac{1}{7} \), which is in its simplest form. This helps in making calculations easier and understanding the value of the fraction straightforward.