Problem 45 - A Graphical Approach to College Algebra

Question

Find each sum or difference. $$\frac{3}{p}+\frac{1}{2}$$

Step-by-Step Solution

Answer

The sum is \( \frac{6 + p}{2p} \).

Step 1: Determine a Common Denominator

To add fractions, they need to have the same denominator. The denominators here are \( p \) and \( 2 \). The common denominator will be the product of the two, which is \( 2p \).

Step 2: Convert Fractions to the Common Denominator

Convert \( \frac{3}{p} \) and \( \frac{1}{2} \) to have the same denominator \( 2p \). Multiply the numerator and denominator of \( \frac{3}{p} \) by \( 2 \) to get \( \frac{6}{2p} \), and multiply the numerator and denominator of \( \frac{1}{2} \) by \( p \) to get \( \frac{p}{2p} \).

Step 3: Add the Fractions

With a common denominator, you can add the fractions: \( \frac{6}{2p} + \frac{p}{2p} = \frac{6 + p}{2p} \).

Step 4: Simplify the Expression (if necessary)

Check if the expression \( \frac{6 + p}{2p} \) can be simplified further. Since 6 and \( p \) have no common factor, this is the simplest form.

Key Concepts

Understanding the Common Denominator

Adding fractions requires them to have the same denominator, called a common denominator. This ensures that all the fractions are counted in the same-sized portions, very much like adding pieces of a cake from the same tin.
To find a common denominator, you look at all the different denominators you have in the fractions you are adding. In most cases, it’s easiest to multiply these denominators to get a common one.

• In our example, the fractions are \( \frac{3}{p} \) and \( \frac{1}{2} \).
• The denominators are \( p \) and \( 2 \).
• Multiply them together to get \( 2p \) as the common denominator.

Now, each fraction needs to be converted to have this common denominator before they can be added together.

Working with Rational Expressions

A rational expression is simply a fraction that has polynomials in the numerator and/or the denominator. In the problem, fractions like \( \frac{3}{p} \) and \( \frac{1}{2} \) can be considered rational expressions.
These expressions follow the same rules as regular fractions when it comes to operations like addition, subtraction, multiplication, or division.

• To add the rational expressions \( \frac{3}{p} \) and \( \frac{1}{2} \), we need to make sure they share a common denominator.
• Each expression is adjusted so its denominator matches the common denominator, \( 2p \).
• This involves multiplying both the top (numerator) and bottom (denominator) by a factor so both fractions have \( 2p \) as the denominator.

This adjustment allows the fractions to align properly for addition.

Simplifying Algebraic Fractions

Once fractions share a common denominator and are added, the resulting expression sometimes needs to be simplified. Simplifying a fraction makes it easier to understand and work with, similar to reducing a fraction to its simplest form in basic arithmetic.
The goal is to make the numerator and the denominator have no common factors other than 1, which makes the fraction as simple as it can be.

• For the resulting fraction \( \frac{6+p}{2p} \), check if the numerator (6 + p) and the denominator (2p) have any common factors.
• In this example, they do not, as 6 and \( p \) do not share any factors, nor do 6+p and 2p together.

Thus, \( \frac{6+p}{2p} \) is already in its simplest form, completing the process!