2019 AMC 8 第 22 题

Question

一家商店先把一件衬衫的原价提高某个百分比，再把新价格降低同样的百分比。已知最后价格是原价的

84\%

，价格提高和降低的百分比是多少？

A store increased the original price of a shirt by a certain percent and then decreased the new price by the same percent. Given that the resulting price was

84\%

of the original price, by what percent was the price increased and decreased?

16

20

28

36

40

答案：E

知识点：百分数平方差

难度评级：1240

小提示：

提高

x

再降低

x

相当于乘以

(1 + x)(1 - x)

An increase by

x

and decrease by

x

multiplies by

(1 + x)(1 - x)

大提示：

建立方程

1 - x^2 = 0.84

Set

1 - x^2 = 0.84

视频讲解：

Click to load, then click again to play

文字解答：

设这个百分比写成小数为

x

。涨价会把原价乘以

1 + x

。

之后降价会把新价格乘以

1 - x

。因此最终价格是原价乘以

(1 + x)(1 - x) = 1 - x^2 = 0.84\text{。}

于是

x^2 = 0.16 \Rightarrow x = 0.4\text{。}

所以这个百分比是

40\%

。

正确答案是 E。

Let

x

be the percent converted to a real number. Then the percent increase would multiply the original by

1 + x.

The percent decrease would multiply the new price by

1 - x.

The final price will then be the original price multiplied by

(1 + x)(1 - x) = 1 - x^2 = 0.84.

This tells us that

x^2 = 0.16 \Rightarrow x = 0.4.

This tells us that the percent would be

40\%.

Thus, the correct answer is E.