AMC 8 2010 Problem 20
AMC8 2010 年第 20 题
基本计数原理★★★☆☆
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?
在一个房间里，2/5 的人戴着手套，3/4 的人戴着帽子。那么房间里同时戴着帽子和手套的人最少是多少个？
- A.3
- B.5
- C.8
- D.15
- E.20
正确答案 · Correct Answer
A
解析 · Solution
Solution 1
Let be the number of people wearing both a hat and a glove. Since the number of people wearing a hat or a glove must be whole numbers, the number of people in the room must be a multiple of 4 and 5. Since we are trying to find the minimum , we must use the least common multiple. lcm(4,5) = 20. Thus, we can say that there are 20 people in the room, all of which are wearing at least a hat or a glove. (Any people wearing neither item would unnecessarily increase the number of people in the room.) It follows that there are people wearing gloves and people wearing hats. Then by applying the Principle of Inclusion and Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is 8+15-x = 23-x, where is the number wearing both. Since everyone in the room is wearing at least one item (see above), 23-x = 20, and so x=3. Video by MathTalks https:
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