AMC10 2017 A

AMC10 2017 A · Q14

AMC10 2017 A · Q14. It mainly tests Systems of equations, Percent.

Every week Roger pays for a movie ticket and a soda out of his allowance. Last week, Roger’s allowance was A dollars. The cost of his movie ticket was 20% of the difference between A and the cost of his soda, while the cost of his soda was 5% of the difference between A and the cost of his movie ticket. To the nearest whole percent, what fraction of A did Roger pay for his movie ticket and soda?

每周 Roger 从零用钱中支付电影票和苏打水的费用。上周，Roger 的零用钱是 A 元。电影票的价格是 A 与苏打水价格之差的 20%，而苏打水的价格是 A 与电影票价格之差的 5%。Roger 为电影票和苏打水支付的占 A 的分数，近似到最接近的整百分比是多少？

(A) 9% 9%

(B) 19% 19%

(D) 23% 23%

(E) 25% 25%

Answer

Correct choice: (D)

正确答案：（D）

Solution

Let \(M\) be the cost of Roger’s movie ticket, and let \(S\) be the cost of Roger’s soda. Then \(M = 0.20(A-S)\) and \(S = 0.05(A-M)\). Thus \(5M + S = A\) and \(M + 20S = A\). Solving the system for \(M\) and \(S\) in terms of \(A\) gives \(M = \frac{19}{99}A\) and \(S = \frac{4}{99}A\). The total cost of the movie ticket and soda as a fraction of \(A\) is \(\frac{23}{99} = 0.2323\ldots \approx 23\%\).

设 \(M\) 为 Roger 电影票的价格，\(S\) 为苏打水的价格。那么 \(M = 0.20(A-S)\) 且 \(S = 0.05(A-M)\)。于是 \(5M + S = A\) 且 \(M + 20S = A\)。解方程组得 \(M = \frac{19}{99}A\) 且 \(S = \frac{4}{99}A\)。电影票和苏打水总价占 A 的分数为 \(\frac{23}{99} = 0.2323\ldots \approx 23\%\)。

Topics