22. In the Maths Premier League, teams get 3 points for a win, 1 point for a draw and 0 points for a loss. Last year, my team played 38 games and got 80 points. We won more than twice the number of games we drew and more than five times the number of games we lost.
How many games did we draw?
A 8
B 9
C 10
D 11
E 14
SolUtion
D

Let the number of matches my team won, drew and lost be $x, y$ and $z$ respectively. Since we played 38 games in total and got 80 points, $x+y+z=38$ and $3 x+y=80$. The information in the question tells us that $x>2 y$ and $x>5 z$. Since $3 \times 27=81>80$, it follows that $x<27$. The possible values for $x, y$ and $z$ which satisfy the two equations are $(26,2,10),(25,5,8)$, $(24,8,6),(23,11,4),(22,14,2)$ and $(21,17,0)$. The only one of these combinations which also satisfies the two inequalities is $(23,11,4)$ and hence my team drew 11 matches.