Problem 35 - Algebra A Combined Function

Question

Perform each indicated operation. Simplify if possible. \(\frac{2}{m}+1\)

Step-by-Step Solution

Answer

The simplified expression is \( \frac{2 + m}{m} \).

Step 1: Understanding the Expression

The expression given is \( \frac{2}{m} + 1 \). This is an addition operation between a fraction and a whole number.

Step 2: Finding a Common Denominator

In order to add the fraction \( \frac{2}{m} \) and the whole number 1, we need a common denominator. Since the denominator of \( \frac{2}{m} \) is \( m \), we convert 1 to a fraction with denominator \( m \), which is \( \frac{m}{m} \).

Step 3: Adding the Fractions

Now that both terms have the same denominator, we can add them: \( \frac{2}{m} + \frac{m}{m} = \frac{2 + m}{m} \).

Step 4: Simplifying the Expression

The expression \( \frac{2 + m}{m} \) is already simplified as there are no like terms to combine or common factors to cancel out.

Key Concepts

Fractions

Fractions are a way to represent parts of a whole. They consist of two numbers: a numerator and a denominator.
The numerator is the top number indicating how many parts we have, while the denominator is the bottom number showing into how many parts the whole is divided.
For example, in the fraction \( \frac{2}{m} \), 2 is the numerator, and \( m \) is the denominator. This tells us that we have 2 parts of a whole divided into \( m \) parts.
Using fractions allows us to perform operations like addition, subtraction, multiplication, and division with numbers that are not whole. When dealing with different denominators, it’s essential to find a way to make them the same so that fractions can be easily added or subtracted.

Common Denominator

When adding or subtracting fractions, they must have the same denominator. This is called having a 'common denominator'.
A common denominator is like a shared base or reference that lets us easily combine or compare fractions.
Here’s how you find a common denominator for fractions:

• Identify the denominators of the fractions involved. For \( \frac{2}{m} + 1 \), the denominators are \( m \) for the fraction, and for the number 1, we treat it as \( \frac{1}{1} \) with a denominator of 1.
• Choose a common denominator that both original denominators can divide into. Here, \( m \) is used.
• Convert each term so they share this common denominator. The whole number 1 becomes \( \frac{m}{m} \).

By having the fractions share the same denominator, the addition or subtraction becomes straightforward: simply add or subtract the numerators.

Simplifying Expressions

Simplifying expressions means making them as easy to understand as possible, with no redundant terms. For the expression \( \frac{2 + m}{m} \), we first need to make sure each term is in its simplest form.
An expression is considered simplified if:

• No further operations can be done.
• There are no like terms to combine.
• Every fraction is reduced to its simplest form.

In our example, \( \frac{2 + m}{m} \) is already simplified. The terms \( 2 \) and \( m \) inside the numerator are different and cannot be combined further. Also, there are no common factors to cancel with the denominator. Therefore, \( \frac{2 + m}{m} \) represents its simplest form, conveying all the information without unnecessary complexity.