2010 AMC 8 Problems/Problem 20
Problem 20
In a room, 
of 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 a glove?
Solution
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, it follows that the number of people in the room must be divisible by 
and 
. Since we are trying to find the minimum , we should also use the smallest possible value for the number of people in the room. Similarly, as we are trying to minimize , we can assume that everyone in the room must be wearing either a hat or a glove or both. That is, there are no people present who wearing neither of the two items. Thus, we can say that there are 
people in the room, all of which are wearing at least a hat or a glove.
It follows that there are 
people wearing gloves and 
people wearing hats. Then by applying the Principle of Inclusion Exclusion (PIE), the total number of people in the room wearing either a hat or a glove or both is 
. Since we know that this equals , it follows that 
, which implies that 
. Thus, 
is the correct answer.
See Also
| 2011 AMC 8 (Problems • Answer Key • Resources) | ||
| Preceded by Problem 19 |
Followed by Problem 21 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
| All AJHSME/AMC 8 Problems and Solutions | ||
