Problem 8 - Beginning and Intermediate Algebra

Question

Add or subtract, as indicated. $$\frac{6}{5 c}+\frac{14}{5 c}$$

Step-by-Step Solution

Answer

The short answer is: \[\frac{20}{5c}\].

Step 1: Identify the common denominator

The common denominator in this case is already given: \(5c\).

Step 2: Add the numerators

Add the numerators of both fractions: \(6+14=20\).

Step 3: Combine the numerators and denominator

Combine the sum of the numerators with the common denominator: \(\frac{20}{5c}\).

Step 4: Simplify the fraction (if applicable)

In this case, the fraction is already in its simplest form: \[\frac{20}{5c}\].

Key Concepts

Common Denominator

When working with algebraic fractions, the concept of a common denominator is essential. It is the shared denominator between two or more fractions that allows us to perform addition or subtraction. In this exercise, you might notice that both fractions have the same denominator, \(5c\). This means they are already using a common denominator, simplifying our task of adding them.
It’s important to recognize when fractions share common denominators because it means we can focus entirely on the numerators when performing operations. For fractions that do not initially share a common denominator:

• Find the least common multiple (LCM) of all denominators involved.
• Adjust the fractions so that they each have this LCM as their denominator.

Once a common denominator is established, you can proceed with the addition or subtraction process.

Fraction Addition

Fraction addition might seem intimidating, but it becomes simple when you have a common denominator, as seen in this problem. In algebraic fractions, the principle remains the same as with numbers: add only the numerators, keeping the common denominator constant.
In our example, we had:

• First fraction: \(\frac{6}{5c}\)
• Second fraction: \(\frac{14}{5c}\)

Since both fractions share the denominator \(5c\), we simply add the numerators: \(6 + 14\). This sum, \(20\), forms the new numerator, giving us the combined fraction \(\frac{20}{5c}\).
Remember, when the denominators differ, adjust the fractions to have the same denominator first before adding.

Simplification of Fractions

After adding fractions, the next step is to simplify the result, if possible. Simplifying a fraction involves reducing it to its lowest terms. This happens when the numerator and the denominator share common factors that can be divided out. In our problem, the resulting fraction was \(\frac{20}{5c}\).
To simplify:

• Factor both the numerator and denominator. Here, the numerator \(20\) can be factored into \(5 \times 4\).
• Recognize the common factor between \(20\) and \(5c\), which is \(5\).
• Divide both the numerator and the denominator by this common factor to simplify the fraction.

Thus, dividing \(20\) by \(5\) results in \(4\), and dividing \(5c\) by \(5\) results in \(c\). Therefore, the simplified form is \(\frac{4}{c}\). Simplifying fractions helps in making calculations easier and results more understandable.