Problem 1 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Simplify the result if possible. See Examples 1 through 3. $$ \frac{a+1}{13}+\frac{8}{13} $$

Step-by-Step Solution

Answer

\(\frac{a+9}{13}\)

Step 1: Identify the Common Denominator

In a problem involving addition or subtraction of fractions, the first step is to check if they have a common denominator. Here, both fractions have the denominator of 13, which means they already have a common denominator.

Step 2: Add the Numerators

Since the denominators are the same, we can simply add the numerators while keeping the same denominator. The numerators are \(a+1\) and 8. Thus, the new numerator is \((a+1) + 8\).

Step 3: Simplify the Numerator

Combine like terms in the numerator. Add the number 1 to 8 to obtain 9, resulting in the new numerator \(a + 9\).

Step 4: Write the Final Expression

Combine the result from steps 2 and 3 into a single fraction: \(\frac{a+9}{13}\). Check if the fraction can be simplified further, but here it is in its simplest form.

Key Concepts

Understanding the Common Denominator in Fraction Addition

When adding fractions, the common denominator is crucial. It ensures that both fractions are measured under the same whole. Think of the denominator as the number of equal parts a whole is divided into. For two fractions with the same denominator, comparing or adding them becomes straightforward.

• In our example, the fractions \( \frac{a+1}{13} \) and \( \frac{8}{13} \) already share the same denominator: 13.
• This shared denominator allows us to focus solely on the numerators without further adjustment.

Simplifying this part of the process, thanks to the common denominator, saves a lot of time and reduces complex calculations. This simple step paves the way to efficiently combining the fractions.

Simplifying Fractions for Cleaner Answers

Once the fractions are combined, simplification can tidy up the result, making it more understandable. Simplifying usually involves one or both of these strategies:

• Combining Like Terms: Merge similar elements in an expression.
• Reducing the Fraction: Adjusting the fraction to its simplest form.

In our case, the expression \( (a+1) + 8 \) simplifies by adding 1 and 8 together, giving us a neat \( a+9 \). Little simplifications like these improve clarity.
In simple terms, simplifying makes sure that your final answer isn't just smoother to read but often reveals insights you might miss by leaving the terms unsimplified. Here, our final fraction \( \frac{a+9}{13} \) is already in simplest form, offering a straightforward conclusion.

Working with Algebraic Expressions in Fractions

Handling algebraic expressions within fractions can seem daunting, but it's just a matter of treating variables like regular numbers during arithmetic processes. Here's how to navigate through them:

• Identify and isolate expressions in the numerator.
• Add or subtract constants as you normally would.

The presence of an algebraic term like \( a \) in \( \frac{a+1}{13} \) doesn't change the process of fraction addition. When adding \( a+1 \) and 8, treat the algebraic term separately but follow typical addition rules for the constants. This means adding 1 and 8 to get 9 while keeping \( a \) unchanged.
Algebraic fractions help in solving large-scale problems efficiently by consolidating different parts of an equation into a single structure. Understanding these and becoming comfortable with them opens up a world of mathematical problems you can solve with ease.