Question

Each of the numbers x, y, w, and z (not necessarily distinct) can have any of the values 2, 3, 9, or 14.
Column A: \(\frac{x}{y}\)
Column B: \(wz\)

• 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 "cannot be determined" questions, you only need to find two conflicting examples. A good strategy is to test the extreme values (maximum and minimum) available for the variables to see if you can change the outcome of the comparison.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This question asks us to compare two expressions whose values can change depending on which numbers from the given set are chosen for the variables. To determine the relationship, we should try to find cases where Column A is greater and cases where Column B is greater.
Step 2: Detailed Explanation:
The variables x, y, w, and z can each be 2, 3, 9, or 14. Let's test different scenarios to see how the comparison changes. The strategy is to try to maximize one quantity while minimizing the other, and vice versa.
Scenario 1: Try to make Column A large and Column B small.
To maximize \(\frac{x}{y}\), we should choose the largest possible value for \(x\) and the smallest possible value for \(y\).
Let \(x = 14\) and \(y = 2\). Then, Column A = \(\frac{14}{2} = 7\).
To minimize \(wz\), we should choose the smallest possible values for \(w\) and \(z\).
Let \(w = 2\) and \(z = 2\). Then, Column B = \(2 \times 2 = 4\).
In this scenario, Column A (7)>Column B (4).
Scenario 2: Try to make Column A small and Column B large.
To minimize \(\frac{x}{y}\), we should choose the smallest possible value for \(x\) and the largest possible value for \(y\).
Let \(x = 2\) and \(y = 14\). Then, Column A = \(\frac{2}{14} = \frac{1}{7}\).
To maximize \(wz\), we should choose the largest possible values for \(w\) and \(z\).
Let \(w = 14\) and \(z = 14\). Then, Column B = \(14 \times 14 = 196\).
In this scenario, Column A (\(\frac{1}{7}\))<Column B (196).
Step 3: Final Answer:
Since we found one case where Column A is greater than Column B, and another case where Column B is greater than Column A, the relationship between the two quantities is not fixed. Therefore, the relationship cannot be determined from the information given.