Question

m and n are positive integers.
Column A: m + n
Column B: mn

• 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:

When variables are introduced with limited constraints (like "positive integers"), always test a few different types of numbers. Include 1, small integers (like 2, 3), and see how the relationship changes. If you find more than one possible relationship (A>B, B>A, or A=B), the answer is always (D).

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
We need to compare the sum and the product of two positive integers, \(m\) and \(n\). Since no specific values are given, the relationship might not be constant.
Step 2: Key Formula or Approach:
The best approach is to test different cases for the positive integers \(m\) and \(n\) to see if the relationship between their sum and product is consistent. We should check cases involving the number 1, and cases with numbers greater than 1.
Step 3: Detailed Explanation:
Let's test several pairs of positive integers for \(m\) and \(n\).
Case 1: One of the integers is 1. Let \(m = 1\) and \(n = 3\).

Column A: \( m+n = 1+3 = 4 \)
Column B: \( mn = 1 \times 3 = 3 \)
In this case, Column A>Column B.
Case 2: Both integers are greater than 1. Let \(m = 2\) and \(n = 3\).

Column A: \( m+n = 2+3 = 5 \)
Column B: \( mn = 2 \times 3 = 6 \)
In this case, Column B>Column A.
Case 3: Both integers are equal to 2. Let \(m = 2\) and \(n = 2\).

Column A: \( m+n = 2+2 = 4 \)
Column B: \( mn = 2 \times 2 = 4 \)
In this case, Column A = Column B.
Since we have found cases where Column A is greater, Column B is greater, and the two columns are equal, no single relationship holds true for all positive integers \(m\) and \(n\).
Step 4: Final Answer:
The relationship between \(m+n\) and \(mn\) depends on the specific values of \(m\) and \(n\). Therefore, the relationship cannot be determined from the information given.