Problem 8 - Introductory Algebra for College Students

Question

add or subtract as indicated. Simplify the result, if possible. $$\frac{5}{x}+\frac{13}{x}$$

Step-by-Step Solution

Answer

\(\frac{18}{x}\)

Step 1: Identify the Numerator and Denominator

The two fractions are \(\frac{5}{x}\) and \(\frac{13}{x}\). The denominators are the same in both fractions, which is 'x'. The numerators are '5' and '13' respectively.

Step 2: Perform the Addition

The addition of fractions with the same denominator is performed by adding the numerators and keeping the denominator the same. Therefore, \(\frac{5}{x}+\frac{13}{x}=\frac{5+13}{x}\).

Step 3: Simplify the Result

The addition in the numerator results in \(\frac{18}{x}\). Since the numerator and denominator do not have any common factor other than 1, the fraction cannot be simplified further.

Key Concepts

Simplifying Fractions

When working with fractions, simplifying them is an important step that can make calculations easier in the future. Simplifying a fraction means reducing it to its simplest form. This involves making the numerator and the denominator as small as possible while still keeping the value of the fraction the same. To simplify a fraction, follow these steps:

• Identify the greatest common factor (GCF) of both the numerator and the denominator.
• Divide both the numerator and the denominator by their GCF.
• Write the result as the simplified fraction.

For example, if you have the fraction \(\frac{10}{15}\), the GCF of 10 and 15 is 5. Dividing both by 5, we get \(\frac{2}{3}\). The fraction \(\frac{18}{x}\) in the given problem could not be simplified further because 18 and \(x\) don't share a common factor (assuming \(x\) is not 18 or a multiple of any factor of 18).
Keep in mind, simplifying is mainly useful when dealing with large numbers or where further calculations are required.

Common Denominator

In fraction arithmetic, a common denominator is needed to add or subtract fractions. Fractions like \(\frac{5}{x}\) and \(\frac{13}{x}\) already have the same denominator, which simplifies the process. This means they share the same base over which their values are compared or combined.

When fractions don't initially have common denominators, you'll need to find one. Here's how to find a common denominator:

• List the multiples of each denominator.
• Identify the smallest common multiple they both share.
• Convert the fractions so they each have this common denominator.

For example, to add \(\frac{2}{3}\) and \(\frac{4}{5}\), the denominators 3 and 5 have no common factors, so their least common multiple is 15. You would convert both fractions to have a denominator of 15 before proceeding with addition.
Finding the common denominator is crucial to ensuring fractions are properly managed and summed or subtracted accurately.

Numerator and Denominator

The concepts of numerator and denominator are central to understanding fractions. The numerator is the top number in a fraction and represents the number of parts being considered. The denominator is the bottom number and signifies the total number of equal parts in the whole.

For instance, in \(\frac{5}{x}\), 5 is the numerator, indicating that there are 5 parts taken out of \(x\). The denominator \(x\) suggests \(x\) total parts form a whole. These roles stay consistent across all arithmetic with fractions.

When adding or subtracting fractions like in the original exercise, the numerators are combined if the denominators are the same. The denominator remains unchanged, acting as a sort of common ground for the operation:

• Add the numerators together: \(5 + 13 = 18\).
• Keep the same denominator: \(x\).

This results in \(\frac{18}{x}\), where 18 is the new numerator after addition. Understanding these components ensures you can handle basic fraction operations with confidence and clarity.