In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie?
(A) 3
(B) 4
(C) 5
(D) 7
(E) 10
In a softball league, after each team has played every other team 4 times, the total accumulated points are: Lions 22, Tigers 19, Mounties 14, and Royals 12. If each team received 3 points for a win, 1 point for a tie and no points for a loss, how many games ended in a tie?
(A) 3
(B) 4
(C) 5
(D) 7
(E) 10

## Solution

When every team plays every other team there are $3+2+1=6$ games. Since each team plays each of the other teams 4 times, there are $4(6)=24$ games.

When there is a winner 3 points are awarded. If each of the 24 games had winners there would be $24 \times 3=72$ points awarded. The actual point total is $22+19+14+12=67$.

When there are ties, only $1+1=2$ points are awarded and so every point below 72 represents a tie. Thus, the number of ties is $72-67=5$.