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?

• Let the incomes of A, B, C, and D be represented by $a, b, c,$ and $d$ respectively.

• From the condition 'A earns half as much as the sum of the income of B and D', we get $a = \frac{b+d}{2}$, which implies $2a = b + d$.

• From the condition 'The sum of income of A and C is the same as that of B and D', we get $a + c = b + d$.

• Substituting $b + d = 2a$ into the second equation gives $a + c = 2a$, which simplifies to $c = a$.

• From the condition 'The sum of income of A and B is more than that of C and D', we get $a + b > c + d$.

• Substituting $c = a$ into this inequality yields $a + b > a + d$, which simplifies to $b > d$.

• Since $2a = b + d$ and $b > d$, it follows that $b > a$ (because if $b$ were less than or equal to $a$, the sum $b+d$ would be less than or equal to $2a$ only if $d$ were sufficiently small, but specifically, $b = 2a - d$. Since $d < b$, $d < 2a - d \Rightarrow 2d < 2a \Rightarrow d < a$. Thus $b = 2a - d > 2a - a = a$). Therefore, $b$ is the highest income.