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1 solution
Moderator note:
Good explanation. Another way to solve this question is to assume that are the three numbers. Since is the number is the center, then is the median, and so, is also the mode and mean. So at least one of and must be equal to . What's left to show is to prove that as well.
Bonus question : What would the answer to this same problem if this time, we were given 4 numbers instead of 3?
Relevant wiki: Understanding Data - Problem Solving
We will show, that if the mean and the mode is equal, then all three numbers have to be equal.
(And the median will be equal as well as a consequence, but not as a requirement).
First of all, there is no mode (most common number), unless at least two of the numbers are equal.
If our numbers are a, b and c, then we can assume, that a = b (the other possible combinations can be assessed similarly (due to symmetry, by interchanging letters)) then, mode = a.
Since our mode = mean, therefore:
3 a + b + c = a
and since a = b :
2a + c = 3a
c = a
Which means, that a = b = c (all three numbers are equal).
Q. E. D.
If all three numbers are equal, then the median (middle number, one of the numbers if we have an odd number of values) has to be equal to these numbers (and to the mode and mean) as well.
However, it is not true, that if either (but not both) the median = mode (e.g. 3, 3, 5) or if the median = mean (e. g. 1, 3, 5), then all (three) values are equal.