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?

Correct Answer: B

Let the incomes of A, B, C, and D be represented by $A$, $B$, $C$, and $D$, respectively.

From the given problem, we can set up the following mathematical statements:

1. The sum of income of A and B is more than that of C and D taken together:
$A+B>C+D$    --- (Equation 1)

2. The sum of income of A and C is the same as that of B and D taken together:
$A+C=B+D$    --- (Equation 2)

3. A earns half as much as the sum of the income of B and D:
$A=\frac{B+D}{2}$    --- (Equation 3)

Now, let's analyze these equations step-by-step to compare their incomes.

From Equation 3, we can rewrite it as:
$2A=B+D$

Substitute $B+D=2A$ into Equation 2:
$A+C=2A$
Subtracting $A$ from both sides gives:
$C=A$

So, the income of A is equal to the income of C ($A=C$).

Next, substitute $C=A$ into Equation 1:
$A+B>A+D$
Subtracting $A$ from both sides of the inequality gives:
$B>D$

Now, let's compare B's income with A's income.
We know from Equation 3 that:
$A=\frac{B+D}{2}$
Since $B>D$, the average of $B$ and $D$ must be strictly less than $B$:
$A=\frac{B+D}{2}<\frac{B+B}{2}=B$
Therefore, $B>A$.

Combining all our results:
$B>A=C$ and $B>D$.

Thus, B has the highest income among all four people.