Problem 4 - Beginning and Intermediate Algebra

Question

Add or subtract as indicated. Simplify the result if possible. See Examples 1 through 3. $$ \frac{3 p}{2 q}+\frac{11 p}{2 q} $$

Step-by-Step Solution

Answer

The simplified result is \( \frac{7p}{q} \).

Step 1: Identifying the Problem

We need to add two fractions: \( \frac{3p}{2q} \) and \( \frac{11p}{2q} \). These have the same denominator \( 2q \). Our task is to sum these fractions.

Step 2: Adding Fraction Numerators

Since the denominators are the same, we can directly add the numerators. This gives us: \( \frac{3p + 11p}{2q} \).

Step 3: Simplifying the Numerator

Combine the terms in the numerator: \( 3p + 11p = 14p \). This simplifies the fraction to \( \frac{14p}{2q} \).

Step 4: Simplifying the Entire Fraction

To simplify \( \frac{14p}{2q} \), divide both the numerator and the denominator by their greatest common divisor, which is 2. This gives us \( \frac{7p}{q} \).

Key Concepts

Adding Fractions

When adding fractions, the key is to ensure they have the same denominator. If they don't, you'll need to find a common denominator first. In our problem, we were lucky because the fractions already shared the same denominator: \( 2q \). This meant we could simply add the numerators together.

• Keep the denominator the same: \( \frac{a}{d} + \frac{b}{d} = \frac{a+b}{d} \).
• Add the numerators: In our case, \( 3p \) and \( 11p \) were added together to form \( 14p \).

After this step, the new fraction becomes \( \frac{14p}{2q} \). It's important to always simplify fractions after adding, if possible, to make them easier to understand and work with.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. After adding the numerators, our fraction was \( \frac{14p}{2q} \). Simplifying involves dividing both the numerator and denominator by their greatest common divisor (GCD).

• Check both numbers for any common factors they share.
• Divide the numerator and denominator by their largest shared factor. This gives the fraction in its simplest form.

In this exercise, since 14 and 2 both share a factor of 2, dividing both terms gives you \( \frac{7p}{q} \). Simplification not only makes the fraction easier to understand but also easier to use in further calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is the largest number that can divide two numbers without leaving a remainder. Finding the GCD is an essential skill for reducing and simplifying fractions.

• List out the factors of each number.
• Identify the largest factor that is common to both.

For our exercise, the numbers 14 and 2 share a common divisor of 2. This means that dividing both the numerator and the denominator by 2 will help reduce the fraction to its most straightforward expression, resulting in \( \frac{7p}{q} \). Finding the GCD ensures you're achieving the simplest and most efficient form of the fraction, aiding in clarity and ease of use in mathematics.