Problem 68 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{5}{13}+\frac{2}{13}$$

Step-by-Step Solution

Answer

So the sum of the two fractions is \(\frac{7}{13}\).

Step 1: Identify the Problem

We are adding \(\frac{5}{13} + \frac{2}{13}\). Notice the denominators are the same. This means we can simply add the numerators.

Step 2: Add the Numerators

We add the numerators together, while leaving the denominator the same: \(\frac{5 + 2}{13} = \frac{7}{13}\).

Step 3: Simplify the Result

Since 7 and 13 are both prime numbers and thus, share no common factors other than 1, \(\frac{7}{13}\) is already at its simplest or lowest form.

Key Concepts

Common Denominator

When adding fractions, one of the first things to check is if the fractions share a common denominator. This is the number at the bottom of a fraction, indicating the total number of equal parts something is divided into. It's crucial to have the same denominator to accurately add fractions. Think of the common denominator as a shared language that helps us understand how to combine different parts into a whole.

• When fractions have the same denominator, you can directly add or subtract the numerators without changing the denominator.
• This is a simple process, but if the denominators are different, you'll first need to find a common denominator.

In this exercise, \(\frac{5}{13} + \frac{2}{13} \), the fractions already have a common denominator, 13, which makes the addition straightforward. If the denominators were different, finding the least common denominator would be necessary before proceeding.

Simplifying Fractions

After completing an arithmetic operation involving fractions, it's important to simplify the result. Simplifying is the process of making a fraction as simple as possible. This means reducing the fraction to its lowest terms. Why is this important? Because simple fractions are easier to understand and use in further calculations. To simplify, we divide both the numerator and the denominator by their greatest common divisor (GCD).

• If the numerator and the denominator have no common factors other than 1, the fraction is already simplified.
• This process makes it easier to compare, add, subtract, multiply, or divide fractions in future problems.

In the solution\(\frac{7}{13}\), both 7 and 13 are prime numbers, which means they don't have any common divisors with each other apart from 1. Therefore, \(\frac{7}{13}\) is already simplified.

Numerator Addition

Adding numerators is the exciting part of combining fractions when they already have a common denominator. The numerator, the top part of the fraction, tells us how many parts we are dealing with. For adding fractions like in our problem,\(\frac{5}{13} + \frac{2}{13}\), the process involves straightforward arithmetic:

• Since the denominators are the same, we simply add the numerators: 5 + 2.
• The sum here becomes the new numerator, and the common denominator remains unchanged.
• This gives us the expression\(\frac{7}{13}\), as we have 7 parts out of a possible 13.

The key is remembering that the denominator does not change during the addition of numerators; it stabilizes the fraction and keeps the proportions relevant. This rule ensures we correctly reflect the total number of parts in our solution.