P = (40% of A) + (65% of B) and Q = (50% of A) + (50% of B),
Where, A is greater than B.

Correct Option (D)

The given expressions are P = 0.40A + 0.65B and Q = 0.50A + 0.50B. It is also stated that A is greater than B (A > B).

To determine the relationship between P and Q, we can analyze their difference:

P - Q = (0.40A + 0.65B) - (0.50A + 0.50B)

P - Q = (0.40A - 0.50A) + (0.65B - 0.50B)

P - Q = -0.10A + 0.15B

P - Q = 0.15B - 0.10A

The relationship between P and Q depends on the sign of the expression (0.15B - 0.10A).

• Consider A = 100 and B = 50 (satisfies A > B):
  P - Q = 0.15(50) - 0.10(100) = 7.5 - 10 = -2.5. In this case, P < Q.
• Consider A = 10 and B = 9 (satisfies A > B):
  P - Q = 0.15(9) - 0.10(10) = 1.35 - 1 = 0.35. In this case, P > Q.

Since the relationship between P and Q (P > Q, P < Q, or P = Q) varies depending on the specific numerical values of A and B, even while maintaining the condition A > B, a definitive conclusion cannot be drawn with certainty. Thus, option (D) is the correct statement.

Incorrect Options:

Options (A), (B), and (C) propose fixed relationships between P and Q (P is greater than Q, Q is greater than P, or P is equal to Q, respectively).

As demonstrated in the analysis for the correct option, the difference (P - Q) can be positive or negative depending on the specific values of A and B, even when A > B. Therefore, none of these definitive statements can be universally concluded as correct.