Question

Column A: r
Column B: \(\frac{1}{r}\)

• 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:

Remember this key property of reciprocals: For a positive number \(x\), if \(x<1\), then \(1/x>1\). If \(x>1\), then \(1/x<1\). If \(x=1\), then \(1/x = 1\).

Answer 1

The Correct Option is B

Step 1: Understanding the Concept:
We are asked to compare a number, \(r\), with its reciprocal, \( \frac{1}{r} \), based on its position on a number line.
Step 2: Key Formula or Approach:
From the number line diagram, we can determine the range of \(r\). The point labeled \(r\) is between 0 and 1. So, we have the inequality \( 0<r<1 \). We can then analyze the behavior of the reciprocal function in this interval.
Step 3: Detailed Explanation:
Method 1: Analysis
The number line shows that \(r\) is a positive number that is less than 1.
Let's analyze the reciprocal, \( \frac{1}{r} \). When you take the reciprocal of a positive number between 0 and 1, the result is always a number greater than 1.
For example, if \( r = \frac{1}{100} \), then \( \frac{1}{r} = 100 \).
Since \(r<1\) and \( \frac{1}{r}>1 \), it must be true that \( \frac{1}{r}>r \).
Method 2: Test a value
Let's pick an approximate value for \(r\) from the diagram, say \( r = 0.5 \) or \( \frac{1}{2} \).
Column A: \( r = \frac{1}{2} \).
Column B: \( \frac{1}{r} = \frac{1}{1/2} = 2 \).
Comparison: We compare \( \frac{1}{2} \) (Column A) with 2 (Column B). Since \( \frac{1}{2}<2 \), the quantity in Column B is greater.
Step 4: Final Answer:
For any number \(r\) between 0 and 1, its reciprocal \(1/r\) will be greater than 1, and thus greater than \(r\). The quantity in Column B is greater.