Problem 11 - Elementary Algebra

Question

For exercises \(5-48\), simplify. $$ \frac{4 n}{n+3}+\frac{n}{n+3} $$

Step-by-Step Solution

Answer

The simplified form is \( \frac{5n}{n+3} \).

Step 1: Identify the Common Denominator

Both fractions \( \frac{4n}{n+3} \) and \( \frac{n}{n+3} \) have the same denominator \( n+3 \).

Step 2: Combine the Numerators

Since the denominators are the same, combine the numerators: \( \frac{4n}{n+3} + \frac{n}{n+3} = \frac{4n + n}{n+3} \).

Step 3: Simplify the Numerator

Add the terms in the numerator: \( 4n + n = 5n \), so the expression simplifies to \( \frac{5n}{n+3} \).

Key Concepts

Common Denominator

When dealing with fractions in algebra, finding a common denominator is a crucial first step. The denominator is the bottom part of a fraction, and when adding or subtracting fractions, they must have the same denominator.
In our example, both fractions already share the common denominator of \( n + 3 \). This makes the process simpler because you don't have to manipulate either fraction to create a matching denominator. Identifying the common denominator early on saves time and helps avoid mistakes later in the problem.
In general, the steps to find a common denominator when they aren't the same include:

• Factor each of the denominators completely
• Determine the least common multiple (LCM) of these factors
• Adjust each fraction to have this LCM as the new denominator

Because our fractions already share the denominator \( n+3 \), we can move on to the next step.

Combining Numerators

After ensuring both fractions have a common denominator, we can combine their numerators. The numerator is the top part of a fraction. Combining numerators means adding or subtracting them while keeping the common denominator.
For our specific problem, we have:
\( \frac{4n}{n+3} \) and \( \frac{n}{n+3} \)
Since the denominators are the same, we add the numerators directly:
\( \frac{4n + n}{n+3} \)
This step consolidates the fractions into a single fraction, making it easier to work with.

If you're working with different fractions with like denominators in general, the process is the same:

• Keep the common denominator
• Add or subtract the numerators accordingly

Simplification Steps

Once the numerators are combined, the final step is to simplify. Simplifying a fraction means reducing it to its simplest form.
In our problem, the combined numerator was \(4n + n \):
Adding these terms together gives:
\(4n + n = 5n \)
So the fraction becomes:
\( \frac{5n}{n+3} \)
This is already in its simplest form, as there are no common factors between 5 and \(n+3\).
To simplify fractions in general, follow these steps:

• Combine like terms if necessary
• Factorize both the numerator and the denominator
• Cancel out any common factors

In conclusion, simplifying fractions in algebra requires understanding how to find common denominators, combine numerators, and reduce to the simplest form. The more you practice, the easier these steps will become!