Problem:
Teams and are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team has won of its games and team has won of its games. Also, team has won 7 more games and lost 7 more games than team . How many games has team played?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Suppose team has played games in total so that it has won games. It follows that team has played games in total so that it has won games.
We set up and solve an equation for team 's win ratio:
OR
If we consider the number of games team has played as and the number of games that team has played as , then we can set up the following system of equations:
The first system equated the number of wins of each team, while the second system equates the number of losses by each team. By multiplying the second equation by and solving the system, we get or answer choice .
OR
First, let us assign some variables. Let
where denotes number of games won, denotes number of games lost, and denotes total games played for . Using the given information, we can set up the following two equations:
We can solve through substitution, as the second equation can be written as , and plugging this into the first equation gives , which means . Finally, we want the total number of games team has played, which is .
The problems on this page are the property of the MAA's American Mathematics Competitions