Problem 14 - Elementary and Intermediate Algebra

Question

Add and simplify the result, if possible. \(\frac{4}{s}+\frac{4}{s}\)

Step-by-Step Solution

Answer

The result of adding and simplifying \( \frac{4}{s} + \frac{4}{s} \) is \( \frac{8}{s} \).

Step 1: Identify the problem type

We have an addition problem with two fractions: \( \frac{4}{s} + \frac{4}{s} \). Both fractions have the same denominator, which simplifies the process.

Step 2: Add the numerators

Since the fractions have the same denominator, we can add them by simply adding their numerators: \( 4 + 4 = 8 \). Keep the common denominator \( s \).

Step 3: Write the new fraction

Combine the result from Step 2 into a single fraction: \( \frac{8}{s} \). This is the fraction resulting from the addition.

Step 4: Simplify the fraction (if possible)

Examine \( \frac{8}{s} \) to see if it can be simplified further. Since there are no common factors between 8 and \( s \) other than 1, the fraction is already in its simplest form.

Key Concepts

Simplifying Fractions

When we talk about simplifying fractions, we mean finding an equivalent fraction that has the smallest possible numerator and denominator. This process makes the fraction easier to read and understand. To simplify, you need to identify any common factors between the numerator and the denominator.

• For example, in the fraction \( \frac{8}{s} \), you would look for factors that are common to both 8 and \( s \).
• If such factors exist, divide both the numerator and the denominator by the greatest common factor (GCF).
• In this exercise, since there are no common factors other than 1 between 8 and \( s \), the fraction remains \( \frac{8}{s} \).

Simplifying doesn't change the value of the fraction; it only makes it look cleaner. Always make sure to check for opportunities to simplify when you add or subtract fractions.

Same Denominator

Adding fractions is easiest when the denominators are the same. This is because having the same denominator means you are essentially dealing with parts of the same whole.

• For instance, in the problem \( \frac{4}{s} + \frac{4}{s} \), both fractions share the same denominator \( s \).
• This simplifies the addition process significantly because you only need to add the numerators together.

If the denominators were different, you would first need to convert the fractions so that they have a common denominator. However, in this case, the shared denominator allows you to move directly to adding the numerators.

Numerator Addition

Once you have fractions with the same denominator, the process of adding them becomes simple. You keep the common denominator and add the numerators together.

How to Add Numerators

• Look at the numerators of both fractions, and add them: for \( \frac{4}{s} + \frac{4}{s} \), add 4 and 4 to get 8.
• The denominator remains unchanged, which in this example is \( s \).
• The resulting new fraction is \( \frac{8}{s} \).

Remember, only the numerators are added; the denominator is carried over as it is. This technique ensures that the measurement of the parts (or fractions) stays consistent.