, 2021

Team $P$ played 27 games which included 10 wins and 14 losses. Thus, Team $P$ had $27-10-14=3$ ties at the end of the season. Team $Q$ had 2 more wins than Team $P$, or $10+2=12$ wins. Team $Q$ had 4 fewer losses than Team $P$, or $14-4=10$ losses. Since Team $Q$ played 27 games, they had $27-12-10=5$ ties. At the end of the season, Team $Q$ had a total of $(2\times12)+(0\times10)+(1\times5)$ or 29 points. Solution 1 Assume that Team $R$ finished the season with exactly 6 ties. Since 6 ties contribute 6 points to their points total, then Team $R$ earned the remaining $25-6=19$ points as a result of their wins. However, each win contributes 2 points to the total, and thus it is not possible to earn an odd number of points from wins. Therefore, Team $R$ could not have finished the season with exactly 6 ties. Solution 2 Assume that Team $R$ finished the season with exactly $w$ wins. If Team $R$ finished with exactly 6 ties, then they finished the season with a total of $(2\times w)+(1\times6)$ or $2w+6=2(w+3)$ points (they earn 0 points for losses). Since $w$ is an integer, then $w+3$ is an integer and so $2(w+3)$ is an even integer. However, this is not possible since Team $R$ finished the season with 25 points, an odd number of points. Therefore, Team $R$ could not have finished the season with exactly 6 ties. Solution 1 Let the number of losses that Team $S$ had at the end of the season be $\ell$. Team $S$ had 4 more wins than losses and thus finished the season with $\ell+4$ wins. Since Team $S$ played 27 games, then each of their remaining $27-\ell-(\ell+4)=23-2\ell$ games resulted in a tie. Therefore, Team $S$ finished the season with a total of $(2\times(\ell+4)) + (0\times \ell) + (1\times (23-2\ell))$ or $2\ell+8+23-2\ell=31$ points. Solution 2 Each of the 4 teams played 27 games, 2 teams played in each game, and so the season finished with a total of $\frac{4\times27}{2}=54$ games played. Each of the 54 games resulted in a total of 2 points being awarded (either 2 points to a winning team and 0 to the losing team or 1 point to each of the two teams that tied). Thus, the total points earned by all 4 teams at the end of the season was $2\times54=108$. The table shows that Team $P$ finished with 23 points, Team $R$ had 25 points, and in part (b) we determined that Team $Q$ had 29 points at the end of the season. Therefore, Team $S$ finished the season with $108-23-25-29=31$ points.