Average of age in a family

If you say $s$ is the sum of the ages, and $k$ is the number of children, then we have the following two equations:

$\frac s {k+2}=18$

$\frac {s-38}{k+1}=14$

Solving them gives $k=4$, so there are 4 children.

Let $X$ be the sum of the ages (of the family members) and let $n$ be the number of members. We have:

$18=\frac{X}{n}$ and $14=\frac{X-38}{n-1}$

We conclude $18n=X$ and $14(n-1)=X-38$. From this, we conclude $4n=24$ and so $n=6$.

Since $n$ is the number of family members, we can subtract 2 (parents) from $n$ and get the number of children, 4.

We know the average of everyone besides the father is $14$. So if the father were fourteen, the average of every one would still be $14$. Now let's increase the hypothetical $14$ year old father's age by $24$ years up to $38$. The family's average age increases by $4$ years to $18$. Therefore the father's age has a $4/24$ (that is, $1/6$) weight in determining the average, so there are $6$ people total in the family, and therefore $4$ children.