Question

对消费者来说，单次 $n\%$ 折扣比下面任何一种折扣都更划算：

（一）连续两次 $15\%$ 折扣；

（二）连续三次 $10\%$ 折扣；

（三）先 $25\%$ 折扣，再 $5\%$ 折扣。

$n$ 的最小正整数值是多少？

For the consumer, a single discount of $n\%$ is more advantageous than any of the following discounts:

(1) Two successive $15\%$ discounts.

(2) Three successive $10\%$ discounts.

(3) A $25\%$ discount followed by a $5\%$ discount.

What is the smallest possible positive integer value of $n?$

$\ \ 27$

$\ 28$

$\ 29$

$\ 31$

$\ 33$

答案：C

知识点：百分数不等式

难度评级：1540

解答：

三种连续折扣方案分别使顾客支付标价的 $\begin{aligned} 0.85^2&=0.7225,\\ 0.9^3&=0.729,\\ 0.75(0.95)&=0.7125 \end{aligned}$ 倍。因此它们的实际折扣分别为 $27.75\%$ 、 $27.1\%$ 和 $28.75\%$ 。

单次 $n\%$ 折扣必须大于这三个折扣，所以 $n>28.75$ 。符合条件的最小正整数是 $29$ 。

所以正确答案是 C。

The three successive-discount plans leave the customer paying $\begin{aligned} 0.85^2&=0.7225,\\ 0.9^3&=0.729,\\ 0.75(0.95)&=0.7125 \end{aligned}$ times the listed price. Their effective discounts are therefore $27.75\%$ , $27.1\%$ , and $28.75\%$ .

A single $n\%$ discount must exceed all three, so $n>28.75$ . The smallest positive integer that works is $29$ .

Thus, the correct answer is C .

2014 AMC 10B 第 11 题

答案：C

难度评级：1540

解答：

其他年份的第 11 题