TITLE: Average of age in a family
QUESTION [15 upvotes]: Some days ago my friend sent me this problem, and I couldn't solve it. It's a pretty simple problem, but I'm struggling with it.
It reads:

The average age in a family (mother, father and their children) is $18$. If we don't take the father, who is $38$, into the average, it drops to $14$. How many kids are in that family?

So how many kids are in that family? Any help is very much appreciated.

REPLY [3 votes]: We have one person of age $38$ and an unspecified number of people with average age $14.$
The average age of the entire group is $18.$
When averaging any set of numbers, the sum of all deviations from the average (taking deviations above average as positive, deviations below as negative)
will be zero.
The father has a deviation of $20$ years above the average,
so the total net deviation of all other members of the family from the average age is $-20.$
But the average deviation of the other $n$ members of the family from the whole-family average is $14 - 18 = -4.$
In order for $n$ people with an average deviation of $-4$ to add up to a total net deviation of $-20,$ we must have $n = (-20)/(-4) = 5.$
Therefore there are $5$ other family members, consisting of the mother and $4$ children.