Problem:
In a room, of all the people are wearing gloves, and of the people are wearing hats. What is the minimum number of people in the room wearing both a hat and gloves?
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Answer Choices:
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Solution:
Because and of the people in the room are whole numbers, the number of people in the room is a multiple of both and . The least common multiple of and is , so the minimum number of people in the room is . If of people are wearing gloves, then people are wearing gloves. If of people are wearing hats, then are wearing hats. The minimum number wearing gloves and hats occurs if the not wearing hats are each wearing gloves. This leaves people wearing both gloves and hats.
If are wearing gloves and are wearing hats, then are wearing gloves and/or hats. There is a minimum of people in the room, so people are wearing both a hat and gloves.
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions