The sum of income of A and B is more than that of C and D taken together. The sum of income of A and C is the same as that of B and D taken together. Moreover, A earns half as much as the sum of the income of B and D. Whose income is the highest?

Here's a step-by-step analysis: 1. From the third statement: "A earns half as much as the sum of the income of B and D." This can be written as: A = (B + D) / 2 Multiplying by 2, we get: 2A = B + D (Equation 1) 2. From the second statement: "The sum of income of A and C is the same as that of B and D taken together." This can be written as: A + C = B + D (Equation 2) 3. Now, substitute (B + D) from Equation 1 into Equation 2: A + C = 2A Subtract A from both sides: C = A (Conclusion 1: C's income is equal to A's income) 4. From the first statement: "The sum of income of A and B is more than that of C and D taken together." This can be written as: A + B > C + D (Equation 3) 5. Substitute C = A (from Conclusion 1) into Equation 3: A + B > A + D Subtract A from both sides: B > D (Conclusion 2: B's income is greater than D's income) 6. Now we have two key relationships: C = A and B > D. We also know from Equation 1: 2A = B + D. Since B + D = 2A, and B > D, it implies that B must be greater than A, and D must be less than A. (If B were equal to A, then D would also have to be A to sum to 2A, which contradicts B > D. If B were less than A, then D would have to be greater than A to sum to 2A, which would contradict B > D). Therefore, B > A (Conclusion 3) and D A From Conclusion 1: A = C From Conclusion 4: A > D So, the order of incomes is: B > A = C > D. Therefore, B's income is the highest. The final answer is B