During the school year, the 10 members of the chess club play a total of 900 games of chess during their practice matches. Each member plays against every other member $N$ times. What is the value of $N$?
Since there are 10 club members, there are $\binom{10}{2} = \frac{10\cdot 9}{2} = 45$ pairings of members.  Thus, each pair must have played $\frac{900}{45} = \boxed{20}$ games.