Question

Roger 每周用零花钱买一张电影票和一杯汽水。上周 Roger 的零花钱是 $A$ 美元。电影票价格是 $A$ 与汽水价格之差的 $20\%$ ，汽水价格是 $A$ 与电影票价格之差的 $5\%$ 。四舍五入到最接近的整数百分比，Roger 买电影票和汽水共花了 $A$ 的百分之多少？

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?

$9\%$

$19\%$

$22\%$

$23\%$

$25\%$

答案：D

知识点：百分数方程组

难度评级：1480

解答：

设电影票价格为 $t$ ，汽水价格为 $s$ 。 $t = \dfrac{A - s}{5}$ $s = \dfrac{A - t}{20}$

第一式两边同乘五，得到 $5t = A - s$ 。代入 $s$ 的表达式：解得 $5t = A - \dfrac{A - t}{20}.$ $\begin{aligned} 5t &= A - \dfrac{A - t}{20} \\ 100t &= 20A - A + t \\ 99t &= 19A \\ t &= \dfrac{19A}{99}. \end{aligned}$

代入第二式可得 $s = \dfrac{A - \frac{19A}{99}}{20} = \dfrac{4A}{99}.$

两项费用相加得 $\dfrac{19A}{99} + \dfrac{4A}{99} = \dfrac{23A}{99} \approx 23 \%.$

所以正确答案是 D。

Let $t$ be the cost of the ticket and $s$ be the cost of the soda. Then we get the following equations. $t = \dfrac{A - s}{5}$ $s = \dfrac{A - t}{20}$

Cross-multiplying the first equation gives us $5t = A - s.$ Substituting in the expression for $s$ yields $5t = A - \dfrac{A - t}{20}.$ Solving yields $\begin{aligned} 5t &= A - \dfrac{A - t}{20} \\ 100t &= 20A - A + t \\ 99t &= 19A \\ t &= \dfrac{19A}{99}. \end{aligned}$

This also gives us $s = \dfrac{A - \frac{19A}{99}}{20} = \dfrac{4A}{99}.$

Adding together the costs gives us $\dfrac{19A}{99} + \dfrac{4A}{99} = \dfrac{23A}{99} \approx 23 \%.$

Thus, D is the correct answer.

2017 AMC 10A 第 14 题

答案：D

难度评级：1480

解答：

其他年份的第 14 题