Question

Given that \(X\) and \(Y\) are non-negative. What is the value of \(X\)?
I. \(2X + 2Y \le 40\)
II. \(X - 2Y \ge 20\)

• A. Statement I alone is sufficient
• B. Statement II alone is sufficient
• C. Both statements I and II together are sufficient, but neither alone is sufficient
• D. The question cannot be answered even with the help of both statements

Hint:

Statement I gives an upper limit for X and statement II gives a lower limit. See what happens when both hold together with X, Y at least 0.

Answer 1

The Correct Option is C

Step 1: Statement I alone:
It gives \(X + Y \le 20\). Many pairs work, for example \((X, Y) = (5, 5)\) and \((10, 0)\), so \(X\) is not fixed. Statement I alone is not sufficient.

Step 2: Statement II alone:
It gives \(X \ge 20 + 2Y\). Here \((X, Y) = (20, 0)\) and \((30, 0)\) both work, so \(X\) is not fixed. Statement II alone is not sufficient.

Step 3: Both statements together:
From I: \(X \le 20 - Y\). From II: \(X \ge 20 + 2Y\). So \(20 + 2Y \le X \le 20 - Y\), which gives \(3Y \le 0\). Since \(Y\) is non-negative, \(Y = 0\). Then \(20 \le X \le 20\), so \(X = 20\). The value of \(X\) is found only when both statements are used.

Final Answer:
The correct option is (3): both statements I and II together are sufficient, but neither alone is sufficient.