Question

x >0
Column A: \(\frac{1}{x}+\)1
Column B: \(\frac{1}{x+1}\)

• A. The quantity in Column A is greater.
• B. The quantity in Column B is greater.
• C. The two quantities are equal.
• D. The relationship cannot be determined from the information given.

Hint:

For quantitative comparison questions with variables and constraints (like \(x>0\)), first try to reason about the possible range of values for each expression. If A is always greater than 1 and B is always less than 1, you have your answer without complex algebra. Testing simple numbers is a great way to confirm your reasoning.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to compare two algebraic expressions involving a variable \(x\), which is given to be positive. 
Step 2: Key Formula or Approach: 
We can compare the two expressions by simplifying Column A and then using logical reasoning or by testing with a simple value for \(x\). 
Step 3: Detailed Explanation: 
Method 1: Algebraic Comparison 
Since \(x>0\), \(x+1>1\), which means its reciprocal \( \frac{1}{x+1} \) must be less than 1. So, Column B is a positive number less than 1. 
For Column A, since \(x>0\), the term \( \frac{1}{x} \) is positive. Therefore, \( \frac{1}{x} + 1 \) must be greater than 1. 
So, Column A is greater than 1, and Column B is less than 1. 
Method 2: Testing a Value 
Let's choose a simple positive value for \(x\), for example, \(x=1\). 
Column A: \( \frac{1}{1} + 1 = 1 + 1 = 2 \). 
Column B: \( \frac{1}{1+1} = \frac{1}{2} \). 
In this case, \(2>\frac{1}{2}\), so Column A is greater. 
Comparison: The quantity in Column A is always greater than 1, while the quantity in Column B is always between 0 and 1. Therefore, Column A is always greater. 
Step 4: Final Answer: 
For any positive \(x\), Column A is greater than 1 and Column B is less than 1. Thus, the quantity in Column A is greater.