Mean, Median, and Mode!
If the mean, median and mode for five positive integers were all equal to 5, what is the largest integer could be found?
The answer is 12.
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2 solutions
Moderator note:
Yes. This is just the application for Extremal Principle .
Bonus question : What would the answer be if we want to minimize the largest number?
There must be at least two fives found otherwise there would be no mode. Assume there are exactly two fives so that the largest number is as big as possible. Since the median is also five, we can say for sure that the middle number (from smallest to largest) is also five. But what about the other five? Should it be the second smallest or second largest number? To maximise the value of the largest number, we should let the other five be the second largest number.
The two smallest numbers need to be distinct if not there would be no mode so let them be 1 and 2. The total of all the known numbers is 1 + 2 + 5 + 5 = 1 3 . Since the mean is 5, the total is 5 × 5 = 2 5 . So the largest integer is 2 5 − 1 3 = 1 2 .