P = (40% of A) + (65% of B) and Q = (50% of A) + (50% of B) , where A is greater than B. In this context, which of the following statements is correct?

Let's analyze the expressions for P and Q: P = 0.40A + 0.65B Q = 0.50A + 0.50B We need to compare P and Q. Let's find the difference P - Q: 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 Now, let's analyze the sign of P - Q based on the given condition A > B. 1. If P > Q, then P - Q > 0: -0.10A + 0.15B > 0 0.15B > 0.10A 15B > 10A 3B > 2A A 100. If we choose A = 120 (which is Q. 2. If Q > P, then P - Q 1.5B Example: Let B = 100. Then A must be > 100. If we choose A = 160 (which is > 1.5 * 100 = 150), then: P = 0.40(160) + 0.65(100) = 64 + 65 = 129 Q = 0.50(160) + 0.50(100) = 80 + 50 = 130 In this case, Q > P. 3. If P = Q, then P - Q = 0: -0.10A + 0.15B = 0 0.15B = 0.10A 15B = 10A 3B = 2A A = 1.5B Example: Let B = 100. Then A must be > 100. If we choose A = 150 (which is = 1.5 * 100 = 150), then: P = 0.40(150) + 0.65(100) = 60 + 65 = 125 Q = 0.50(150) + 0.50(100) = 75 + 50 = 125 In this case, P = Q. Since the only given condition is A > B, and we have shown that P can be greater than Q, less than Q, or equal to Q depending on the specific values of A and B (as long as A > B), we cannot conclude with certainty which statement is correct. The final answer is D) None of the above can be concluded with certainty.