# Finding $5$ numbers with mode $3$, median $7$ and mean $6$ [closed]

I got this problem to solve in a non calculator paper test and didn't know how to solve it.

The mode of five numbers is $3$. The median is $7$. The mean is $6$. Work out the $5$ numbers.

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## closed as off-topic by TMM, Michael Hoppe, Davide Giraudo, Sami Ben Romdhane, user127.0.0.1Mar 2 at 13:34

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You might want to review this mathforum.org/library/drmath/view/57586.html –  Amzoti Apr 30 '13 at 15:03

The median is 7, so when ordered the list of five numbers looks like

 _ _ 7 _ _


Since there are more threes than any other number, and now it is clear that there can be at most two threes, we see that in fact we have

 3 3 7 _ _


If we call $a$ and $b$, in order, the two numbers we still do not know, we have $$(3+3+7+a+b)/5=6$$ because of the hypothesis on the mean, so that $a+b=17$. The only options if we restrict to integer numbers are $7$, $10$ and $8$, $9$. The first option makes us uneasy about the mode being $3$... so we pick the other.

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