Problem 15 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{x}{18}+\frac{5}{18}\)

Step-by-Step Solution

Answer

The simplified sum is \( \frac{x + 5}{18} \).

Step 1: Identify the Denominators

Both fractions, \( \frac{x}{18} \) and \( \frac{5}{18} \), have the same denominator of 18. This allows us to add the fractions directly without needing to find a common denominator.

Step 2: Add the Numerators

Since the denominators are the same, we add the numerators together: \( (x + 5) \). This gives us a single fraction with the same denominator: \( \frac{x + 5}{18} \).

Step 3: Simplify the Fraction

Check if the resulting fraction \( \frac{x + 5}{18} \) can be simplified. Since there are no common factors between the numerator \( x + 5 \) and the denominator 18, the fraction is already in its simplest form.

Key Concepts

Addition of Fractions

Adding fractions may initially seem intimidating, but it's simpler than it looks. When you add fractions, you combine two separate parts into a single fraction. The vital point is that fractions must typically have the same denominator before they can be added. This consistent denominator acts like a common ground, ensuring the parts we are combining have the same basis. In our example, both \(\frac{x}{18}\) and \(\frac{5}{18}\) share the denominator 18. They are already ready for adding! Simply put, we save the denominator as it is, and focus on adding only the numerators. In this case, it means adding \(x\) and 5, leading to the combined fraction \(\frac{x + 5}{18}\). Following these straightforward steps turns the addition of fractions task easy-breezy!
Ul>li>Ensure the fractions have the same denominator.

Add up only the numerators.

Keep the denominator unchanged.

Common Denominators

The concept of common denominators is at the heart of working with fractions. Simply put, a common denominator between two fractions is the shared denominator that allows these fractions to be added or subtracted directly. Lack of a common denominator can make adding fractions challenging. Imagine trying to add slices of two cakes that are cut into different sizes. It would be much easier if the cakes were sliced into equal pieces!
In the given example, the fractions \(\frac{x}{18}\) and \(\frac{5}{18}\) already have a common denominator of 18. This is a convenient situation. Since they are already on the same terms, they streamline the addition process. However, when fractions don't share a denominator, we must convert them to have a common one by finding the least common multiple (LCM). This step is crucial in ensuring the fractions speak the same language, making addition or subtraction straightforward.
Keep in mind:

• Fractions with the same denominators are ready to be added directly.
• If denominators differ, find a common one by calculating the LCM.
• This simplifies the process of addition or subtraction.

Simplifying Fractions

The process of simplifying a fraction involves reducing it to its simplest form. A fraction is simplified when the numerator and denominator have no common factors other than 1. Think of it like making a recipe as deliciously simple as possible, removing all unnecessary elements!
In our exercise \(\frac{x + 5}{18}\), we check if the fraction can be simplified. This entails determining whether both the numerator \(x + 5\) and the denominator 18 have any shared factors. In this case, since there are no common factors, the fraction is already in its simplest form. Simplification is essential, as it makes fractions easier to read and work with, ensuring the result looks neater and saves computational space.
Steps to simplify:

Identify any common factors between the numerator and denominator.

Divide both by these factors to simplify the fraction.

If no common factors exist, the fraction is in its simplest form.