AMC12 2001 A

AMC12 2001 A · Q3

AMC12 2001 A · Q3. It mainly tests Linear equations, Interest / growth (simple).

The state income tax where Kristin lives is charged at the rate of $p\%$ of the first $\$28000$ of annual income plus $(p + 2)\%$ of any amount above $\$28000$. Kristin noticed that the state income tax she paid amounted to $(p + 0.25)\%$ of her annual income. What was her annual income?

Kristin 所在州的所得税按如下方式征收：年收入的前 $\$28000$ 部分按 $p\%$ 征税，超过 $\$28000$ 的部分按 $(p + 2)\%$ 征税。Kristin 注意到她缴纳的州所得税相当于她年收入的 $(p + 0.25)\%$。她的年收入是多少？

(A) $28000$ $28000$

(B) $32000$ $32000$

(D) $42000$ $42000$

(E) $56000$ $56000$

Answer

Correct choice: (B)

正确答案：（B）

Solution

Let $A$, $T$ be Kristin's annual income and the income tax total, respectively. Notice that \begin{align*} T &= p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) \\ &= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000) \\ &= p\%\cdot A + 2\%\cdot(A - 28000) \end{align*} We are also given that \[T = (p + 0.25)\%\cdot A = p\%\cdot A + 0.25\%\cdot A\] Thus, \[p\%\cdot A + 2\%\cdot(A - 28000) = p\%\cdot A + 0.25\%\cdot A\] \[2\%\cdot(A - 28000) = 0.25\%\cdot A\] Solve for $A$ to obtain $A = \boxed{\textbf{(B) }\$32000}$. Note: You may also assume p = 0 for easy calculation.

设 $A$ 和 $T$ 分别为 Kristin 的年收入与所得税总额。注意到 \begin{align*} T &= p\%\cdot28000 + (p + 2)\%\cdot(A - 28000) \\ &= [p\%\cdot28000 + p\%\cdot(A - 28000)] + 2\%\cdot(A - 28000) \\ &= p\%\cdot A + 2\%\cdot(A - 28000) \end{align*} 又已知 \[T = (p + 0.25)\%\cdot A = p\%\cdot A + 0.25\%\cdot A\] 因此， \[p\%\cdot A + 2\%\cdot(A - 28000) = p\%\cdot A + 0.25\%\cdot A\] \[2\%\cdot(A - 28000) = 0.25\%\cdot A\] 解得 $A = \boxed{\textbf{(B) }\$32000}$。注：也可以令 $p = 0$ 以便计算。

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