Problem 3 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{15 a}{b}+\frac{6 b}{5}\)

Step-by-Step Solution

Answer

Result is \(\frac{75a + 6b^2}{5b}\).

Step 1: Identify a Common Denominator

To add two fractions, it's necessary to have a common denominator. The denominators here are \(b\) and \(5\). The easiest way to find a common denominator is to multiply these together, giving us \(5b\) as the common denominator.

Step 2: Rewrite Each Fraction

Rewrite each fraction with the new common denominator \(5b\). For \(\frac{15a}{b}\), multiply both the numerator and the denominator by \(5\) to get \(\frac{75a}{5b}\). For \(\frac{6b}{5}\), multiply both the numerator and the denominator by \(b\) to get \(\frac{6b^2}{5b}\).

Step 3: Add the Fractions

Now that both fractions have the same denominator, add them: \(\frac{75a}{5b} + \frac{6b^2}{5b} = \frac{75a + 6b^2}{5b}\).

Step 4: Simplify the Expression

In this case, \(75a + 6b^2\) cannot be simplified further in a meaningful way because it contains different variables in separate terms. Therefore, the final expression cannot be reduced further.

Key Concepts

Common Denominator

When dealing with fractions, a common denominator is essential for performing operations like addition or subtraction. The denominator is the bottom part of a fraction, indicating into how many equal parts the whole is divided.
To add fractions, the denominators must be the same. In the original exercise, the fractions have denominators of \(b\) and \(5\).
To find a common denominator:

• Identify the two denominators: \(b\) and \(5\).
• Multiply them together to get the common denominator: \(5b\).

Using a common denominator is akin to finding a common language between fractions, making it essential for accurate addition and subtraction. Without it, the operation is impossible to perform correctly.

Fraction Addition

Adding fractions requires a common denominator, as explained earlier. Once both fractions have been rewritten to have this common denominator, the process of addition becomes straightforward. Here's how it works in practice for the exercise given:

• First, convert each fraction so they both share the common denominator of \(5b\). For \(\frac{15a}{b}\), multiply by \(\frac{5}{5}\) to get \(\frac{75a}{5b}\). For \(\frac{6b}{5}\), multiply by \(\frac{b}{b}\) to get \(\frac{6b^2}{5b}\).
• Now, simply add the numerators while keeping the common denominator fixed: \(\frac{75a + 6b^2}{5b}\).

Remember that once fractions share the same denominators, you only combine the tops (the numerators) while retaining the bottom number (the denominator). Fraction addition may seem daunting at first, but with practice, this method becomes second nature.

Simplifying Expressions

After successfully performing operations like addition, it’s important to consider simplifying the expression. Simplifying means reducing the expression to its most basic form while equivalent to the original.
For the expression \(\frac{75a + 6b^2}{5b}\), simplification involves checking if there's anything common that can be factored out across the numerator.

• Look at each term: \(75a\) and \(6b^2\). Are there common factors? In this case, they can’t be further simplified because they involve different variables without any common factors.
• Re-evaluate to ensure you can't factor anything out. Since the terms contain separate variables, no simplification is possible beyond this point.

This inability to simplify further is common when variable terms are involved. Focusing on simplification helps in neatly wrapping up the process, ensuring the solution is as concise as possible when presenting the final result.