The problem state that:

Quantity A: The integer ages of the three children range from 2 to 13, and no two children are the same age. The average of all three children.

Quantity B: 10

Compare A and B.

What does range from 2 to 13 means here?

My solution:

Lets assume ages are 2,3,4 then average 3, A < B

Lets assume ages are 11,12,13 then average 12, A > B

But the solution I have says

Since there are only three children and the range is from 2 to 13, one child must be 2, one must be 13, and the other child’s age must fall somewhere in between.

So basically I think I am confused with the range, help appreciated.


1 Answer 1


Suppose that $x_1$, $x_2$ and $x_3$ denote the three children, with $x_1$ being the youngest and $x_3$ the oldest such that $x_1<x_2<x_3$, where we have strict inequalities since no two children have the same age. A range from 2 to 13 then means that $x_1=2$ and $x_3=13$ -- we know that the minimum age amongst the three children is 2 and the maximum is 13. So the average is then: $$ A=\frac{x_1+x_2+x_3}{3}=\frac{2+x_2+13}{3} $$

The maximum value of $x_2$ is 12, since $x_2<x_3=13$. Hence the maximum possible average is: $$ \max A=\frac{2+12+13}{3}=9 $$ and thus $$ \max A < B. $$

  • $\begingroup$ So you want to say range denotes the lowest and highest values? Shouldn't range means possible lowest and highest? $\endgroup$ Oct 17, 2013 at 5:59
  • $\begingroup$ @amitchhajer Yes, that is what I am saying. See here: en.wikipedia.org/wiki/Range_%28statistics%29 "The range is the size of the smallest interval which contains all the data and provides an indication of statistical dispersion." I see why you'd think it's the set of possible ages, but it really depends on the context what type of range is meant. I think it's pretty clear from the answer that you provided that in this case it refers to the lowest and highest (actual) ages. $\endgroup$
    – hejseb
    Oct 17, 2013 at 6:27
  • $\begingroup$ Yes it states so from the answer, isn't the answer flawed? quite possible ! $\endgroup$ Oct 17, 2013 at 6:50
  • $\begingroup$ @amitchhajer Sure, but then your solution would be correct. The question is ambiguously stated I would say so then it is hard to know what exactly is meant. Where is this from? $\endgroup$
    – hejseb
    Oct 17, 2013 at 7:05
  • $\begingroup$ Manhattan book for gre quants. Anyways, i also think question is ambiguous. Thanks for replying. $\endgroup$ Oct 17, 2013 at 7:29

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