Problem 46 - Algebra 1: Concepts and Skills

Question

Add. Write the answer as a fraction or a mixed number in simplest form. $$ \frac{9}{14}+\frac{3}{14} $$

Step-by-Step Solution

Answer

The sum of the two fractions \( \frac{9}{14} \) and \( \frac{3}{14} \) is \( \frac{6}{7} \).

Step 1: Identify the Fractions to Add

The exercise provides two fractions to add, \( \frac{9}{14} \) and \( \frac{3}{14} \). Both fractions have the same denominator, 14.

Step 2: Add the Numerators

Adding fractions with the same denominator is straightforward: the denominator remains the same, and the numerators are added. \( \frac{9+3}{14} = \frac{12}{14} \).

Step 3: Simplify the Result

The fraction \( \frac{12}{14} \) can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which here is 2. So, \( \frac{12}{14} \) simplifies to \( \frac{6}{7} \).

Key Concepts

Common Denominator

When adding fractions, one of the key concepts to understand is the common denominator. A common denominator refers to a shared bottom number between fractions. In our exercise, both fractions, \( \frac{9}{14} \) and \( \frac{3}{14} \), already have the same denominator, 14.
This makes the process easier since you don't need to adjust either fraction to add them. However, if you ever encounter fractions with different denominators, you'll need to find a common one before proceeding with addition.
Here's how to find a common denominator for different fractions:

• Look at the denominators of each fraction.
• Determine the least common multiple (LCM) of these denominators.
• Adjust both fractions so they share this LCM as their new denominator.

Finding a common denominator allows for a valid comparison and operation between different fractions, ensuring the addition or subtraction is accurate.

Simplifying Fractions

Once you add fractions and get a result, the next step is often to simplify the fraction. Simplifying a fraction means reducing it to the lowest possible terms, making it as simple as possible.
In our solution, we finished with \( \frac{12}{14} \) after adding our fractions. This fraction was not in its simplest form, as both the numerator and denominator could be divided by the same number, which leads to the next part of our explanation.
Here’s how you simplify a fraction:

• Find the greatest common divisor (GCD) of the numerator and denominator.
• Divide both the numerator and the denominator by this GCD.
• The result is a simplified fraction.

A simplified fraction is much easier to understand and often preferred in answers, so always check your fractions to see if they can be reduced.

Greatest Common Divisor

The greatest common divisor (GCD) is a crucial tool for simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction.
In our example, we found the GCD for \( \frac{12}{14} \), which was 2. By dividing both the numerator (12) and the denominator (14) by 2, we found the simplified answer as \( \frac{6}{7} \).To find the GCD:

• List all the factors of both the numerator and the denominator.
• Identify the largest number that appears in both lists of factors.

Finding the GCD is essential not only for simplifying fractions but also in many other areas of mathematics. It's what makes your fraction "fully reduced," keeping equations and results clean and simple.