2020 AMC 12B Problems/Problem 5
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 
more games and lost more games than team 
How many games has team played?
Solution
First, let us assign some variables. Let
![\[A_w=2x, A_l=x, A_g=3x,\]](http://latex.artofproblemsolving.com/5/4/d/54d46822f0eb0e91b0d42693d3ebc295fcb288b4.png)
![\[B_w=5y, B_l=3y, B_g=8y,\]](http://latex.artofproblemsolving.com/9/8/8/988577e90978966eca38b36a2750c16c1dd70f0a.png)
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:
![\[B_w=A_w+7\implies 5y=2x+7,\]](http://latex.artofproblemsolving.com/b/7/1/b71dfcc8ba4e52d0329f554f7409f9b81ff18c02.png)
![\[B_l=A_l+7\implies 3y=x+7.\]](http://latex.artofproblemsolving.com/6/e/e/6eed07f18f238db96911d75c247a9fb747592601.png)
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 
.
~Argonauts16
Solution 2
Using the information from the problem, we can note that team A has lost 
of their matches. Using the answer choices, we can find the following list of possible win-lose scenarios for , represented in the form 
for convenience:
![\[A \implies (14, 7)\]](http://latex.artofproblemsolving.com/d/d/7/dd7913bdd7b2fabe9c4f1f9484d3cf8ff94e9f1f.png)
![\[B \implies (18, 9)\]](http://latex.artofproblemsolving.com/2/4/9/2490f6f78298d1fa0fa19e7fc8ecc2610a93396b.png)
![\[C \implies (28, 14)\]](http://latex.artofproblemsolving.com/4/7/7/477cbbfa36f4586224389e2a5ec7879c24d9e47c.png)
![\[D \implies (32, 16)\]](http://latex.artofproblemsolving.com/2/1/7/2175e0a2947e4eb2b458cd89f4fcedac5843db49.png)
![\[E \implies (42, 21)\]](http://latex.artofproblemsolving.com/2/5/9/259fd170bf70c9f494952780279ab7aab5ba2000.png)
Thus, we have 5 matching scenarios, simply adding 7 to 
and subtracting 7 from 
. We can then test each of the five scenarios for 
and find that 
fits this description, making the answer 
.
Video Solution
~IceMatrix
See Also
| 2020 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 4 |
Followed by Problem 6 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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