Problem 28 - Elementary Algebra

Question

For exercises 27-34, evaluate. $$ \frac{1}{14}+\frac{5}{14} $$

Step-by-Step Solution

Answer

The answer is \( \frac{3}{7} \).

Step 1 - Identify Common Denominator

Verify that both fractions share the same denominator. Here, both the fractions \( \frac{1}{14} \) and \( \frac{5}{14} \) have the denominator 14.

Step 2 - Add the Numerators

Since the denominators are the same, add the numerators: \( 1 + 5 = 6 \).

Step 3 - Write the Sum Over the Common Denominator

Place the sum of the numerators over the common denominator: \( \frac{6}{14} \).

Step 4 - Simplify the Fraction

Simplify \( \frac{6}{14} \) by finding the greatest common divisor (GCD) of 6 and 14, which is 2. Divide both the numerator and denominator by 2: \( \frac{6 \div 2}{14 \div 2} = \frac{3}{7} \).

Key Concepts

Understanding Common Denominator

To add fractions successfully, they must share a common denominator. This is the number at the bottom of each fraction. In our exercise, both fractions have the same denominator, 14.
When fractions share a common denominator, it means they are segmented into the same number of parts. For example, \(\frac{1}{14}\) and \(\frac{5}{14}\) are both divided into 14 parts, making it simpler to add them together.
When fractions initially don't have a common denominator, you must find a number that both denominators can divide into evenly. This is typically achieved by identifying the least common multiple (LCM) of the denominators. After aligning them to a common denominator, the next step is to add the numerators.

The Role of Numerators in Fraction Addition

The numerator is the top number in a fraction, representing how many parts of the whole you have. In our example, the numerators are 1 and 5.
Once you have a common denominator, you can add these numerators directly. For \(\frac{1}{14} + \frac{5}{14}\), you add 1 and 5 to get 6. The resulting fraction is \(\frac{6}{14}\).
It’s important to remember that you never add the denominators together. Doing so would change the size of the parts you’re working with, leading to incorrect results. Always keep the denominators the same when adding fractions.

How to Simplify Fractions

Simplifying a fraction means reducing it to its simplest form, where the numerator and the denominator have no common factors other than 1. In the example \(\frac{6}{14}\), we need to simplify this fraction.
First, identify shared factors of the numerator and the denominator. Here, 6 and 14 share the factor 2.
To simplify, divide both the numerator and denominator by their greatest common divisor. In this case, \(\frac{6}{14}\) simplifies to \(\frac{3}{7}\). This is the simplest form of the fraction, meaning it can't be reduced any further.

Greatest Common Divisor (GCD) Basics

The greatest common divisor (GCD) is the largest number that divides both the numerator and the denominator without leaving a remainder. Finding the GCD is crucial for simplifying fractions.
In our scenario, the GCD of 6 and 14 is 2.
Here's a simple method to find the GCD:

• List out the factors of both numbers (e.g., factors of 6 are 1, 2, 3, 6 and factors of 14 are 1, 2, 7, 14).
• Identify the greatest factor that appears in both lists (in this case, 2).

Once you have the GCD, divide both the numerator and denominator by it to simplify the fraction. In our example, \(\frac{6}{14}\) becomes \(\frac{3}{7}\) after dividing by 2.