Statistics Range from x to y find average

Question

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.

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 $$ \max A < B. $$