Difference between revisions of "2020 AMC 12B Problems/Problem 5"
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Teams <math>A</math> and <math>B</math> are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team <math>A</math> has won <math>\tfrac{2}{3}</math> of its games and team <math>B</math> has won <math>\tfrac{5}{8}</math> of its games. Also, team <math>B</math> has won <math>7</math> more games and lost <math>7</math> more games than team <math>A.</math> How many games has team <math>A</math> played? | Teams <math>A</math> and <math>B</math> are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team <math>A</math> has won <math>\tfrac{2}{3}</math> of its games and team <math>B</math> has won <math>\tfrac{5}{8}</math> of its games. Also, team <math>B</math> has won <math>7</math> more games and lost <math>7</math> more games than team <math>A.</math> How many games has team <math>A</math> played? | ||
Revision as of 21:43, 7 February 2020
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 
.
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 | |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
