Question

\(p + q = 1\)
\(0<p<q\)
Column A: \(\frac{1}{pq}\)
Column B: 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:

Remember the property of reciprocals: if a positive number \(k\) is less than 1 (\(0<k<1\)), its reciprocal \(1/k\) will always be greater than 1.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This question asks us to compare a value derived from two variables, p and q, with 1. The variables are constrained by a given equation and inequalities. We need to determine the range of the product pq.
Step 2: Detailed Explanation:
We are given \(p+q=1\), with both p and q being positive numbers (\(0<p\)) and \(p<q\).
Since \(p<q\) and \(p+q=1\), it must be that \(p<0.5\) and \(q>0.5\). (If p=q=0.5, p+q=1, so to make p smaller, q must be larger).
Let's analyze the product \(pq\). Since both p and q are positive, their product \(pq\) is also positive.
Let's express the product in terms of one variable, say p. Since \(q = 1 - p\), the product is \(p(1-p) = p - p^2\).
We know that \(0<p<0.5\). Let's see what values the product can take.
This is a downward-facing parabola, with its maximum at \(p=0.5\). At \(p=0.5\), the product would be \(0.5 \times 0.5 = 0.25\).
As p approaches 0, the product \(p(1-p)\) also approaches 0.
Since \(0<p<0.5\), the product \(pq\) must be in the range \(0<pq<0.25\).
So, \(pq\) is a positive fraction that is always less than 1 (and even less than 0.25).
Comparison:
Column A is \(\frac{1}{pq}\). Since \(pq\) is a positive number between 0 and 0.25, its reciprocal, \(\frac{1}{pq}\), must be greater than \(\frac{1}{0.25}\), which is 4.
Therefore, the quantity in Column A is always greater than 4.
Column B is 1.
Since the quantity in Column A is always greater than 4, it is always greater than 1.
Step 3: Final Answer:
The quantity in Column A is greater.