2015 AMC 8 Problems/Problem 24

A baseball league consists of two four-team divisions. Each team plays every other team in its division $N$ games. Each team plays every team in the other division $M$ games with $N>2M$ and $M>4$ . Each team plays a 76 game schedule. How many games does a team play within its own division?

$\textbf{(A) } 36 \qquad \textbf{(B) } 48 \qquad \textbf{(C) } 54 \qquad \textbf{(D) } 60 \qquad \textbf{(E) } 72$

Solution 1

Note that the equation rewrites to $3N+4M=76$ .

Now remark that if $(m,n)$ is a solution to this equation, then so is $(m+3,n-4)$ . This is because $3(n-4)+4(m+3)=3n-12+4m+12=3n+4m=76.$ Thus, we can now take an "edge case" solution and work upward until both conditions ( $N>2M$ and $M>4$ ) are met.

We see by inspection that $(M,N)=(1,24)$ is a solution. By the above work, we can easily deduce that $(4,20)$ and $(7,16)$ are solutions. The last one is the intended answer (the next solution fails $N>2M$ ) so our answer is $3N=\boxed{48\textbf{ (B)}}$ .