The sum of income of A and B is more than that of C and D when 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?

Correct Option (B)

The problem provides three conditions regarding the incomes of A, B, C, and D:

1. A + B > C + D
2. A + C = B + D
3. A = $\frac{1}{2}$(B + D)

From condition (3), we can deduce that B + D = 2A.

Substitute this expression for (B + D) into condition (2):
A + C = 2A
This simplifies to C = A.

Now, substitute C = A into condition (1):
A + B > A + D
Subtracting A from both sides yields B > D.

To compare A and B, recall the relation B + D = 2A. Since B > D, we can substitute D = 2A - B into the inequality B > D:
B > 2A - B
2B > 2A
B > A.

Therefore, the established relationships are B > A, A = C, and B > D. Combining these, it is evident that B's income is greater than A's, C's, and D's. Hence, B has the highest income.

Incorrect Options:

Options A, C, and D are incorrect. The derived relationships demonstrate that B's income is strictly greater than A's (B > A), and A's income is equal to C's (A = C). Additionally, B's income is greater than D's (B > D). These findings collectively indicate that A, C, and D do not possess the highest income among the individuals.