Find the number of people (word problem)

Question

James is 22 years older than everyone's average year (including him). He's also 24 years older than everyone's average year (not including him). how many people are there in the room excluding James.

Answer 1

Let $J$ be James' age, and let there be $n$ people other than James, where the age of the $k$th person is $A_k$. Then we can let $S=\sum_{k=1}^{n} A_k$ to simplify our equations: $$J - 22 = \frac{J + \sum_{k=1}^{n} A_k }{n+1} = \frac{J + S }{n+1}$$ and $$J - 24 = \frac{\sum_{k=1}^{n} A_k }{n} = \frac{S}{n}$$

Solving for $J$ in the second equation and substituting into the first equation, we eliminate the dependence on $J$: $$ (24+\frac{S}{n})-22 = \frac{(24 + \frac{S}{n})+S}{n+1} $$ Now multiplying both sides by $n$, we have $$ S + 2n = \frac{24n +S+Sn}{n+1} = \frac{24n}{n+1} + S $$ Equating the far left and far right quantities and simplifying fully, we get $$ n (n - 11) = 0 $$ The result is that either $n=0$ (which is not true, since james cannot be 22 years older than his own age), or $n=11$. So the answer is $11$