Question

一个房间里， $2/5$ 的人戴手套， $3/4$ 的人戴帽子。房间里至少有多少人同时戴帽子和手套？

In a room, $2/5$ of the people are wearing gloves, and $3/4$ of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and gloves?

$\ 3$

$\ 5$

$\ 8$

$\ 15$

$\ 20$

答案：A

知识点：容斥原理最小公倍数

难度评级：1610

解答：

因为房间中 $\dfrac 25$ 的人戴手套，所以总人数必须是 $5$ 的倍数。因为 $\dfrac 34$ 的人戴帽子，所以总人数必须是 $4$ 的倍数。因此总人数必须是 $20$ 的倍数。

由容斥原理，

所求比例等于 $\dfrac{2}{5} + \dfrac 34 -$ 至少戴一种物品的比例。要使同时戴两者的人数最少，就让至少戴一种的比例尽可能大，最多为 $1$ 。所以同时戴两者的比例至少为 $\dfrac{2}{5} + \dfrac 34- 1 = \dfrac 3{20}$ 。

取最小的正总人数 $20$ ，在这 $20$ 人中，同时戴两者的人数为 $3$ 。

所以正确答案是 A。

Since our room has $\dfrac 25$ of the people wearing gloves, the number of people must be a multiple of $5.$ Since our room has $\dfrac 34$ of the people wearing hats, the number of people must be a multiple of $4.$ Therefore, the people in the room must be a multiple of $20.$

Now, we can also use the following formula by the principle of inclusion exclusion: Fraction of people wearing both = Fraction of people wearing gloves + Fraction of people wearing hats - Fraction of people wearing either.

This makes our desired fraction equal to $\dfrac{2}{5} + \dfrac 34 -$ Fraction of people who wear either. If we wish to minimize the number who wear both, we maximize the fraction of people who wear either, up to $1.$ Therefore, the fraction of people that wear both is $\dfrac{2}{5} + \dfrac 34- 1 = \dfrac 3{20}.$

Since our number is a (positive) multiple of $20,$ we have the number of people wearing both as $3$ if we choose to have just $20$ people.

Therefore, A is the correct answer.

2010 AMC 8 第 20 题

答案：A

难度评级：1610

解答：

其他年份的第 20 题