Difference between revisions of "2020 AMC 12B Problems/Problem 5"
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Using the information from the problem, we can note that team A has lost <math>\frac{1}{3}</math> of their matches. Using the answer choices, we can find the following list of possible win-lose scenarios for <math>A</math>, represented in the form <math>(w, l)</math> for convenience: | Using the information from the problem, we can note that team A has lost <math>\frac{1}{3}</math> of their matches. Using the answer choices, we can find the following list of possible win-lose scenarios for <math>A</math>, represented in the form <math>(w, l)</math> for convenience: |
Revision as of 13:56, 26 May 2021
Contents
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 1 (Two Variables)
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
.
~Argonauts16
Solution 2 (One Variable)
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:
~MRENTHUSIASM
Solution 3 (Answer Choices: Substitutions)
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:
Thus, we have 5 matching scenarios, simply adding 7 to
and
. We can then test each of the five
scenarios for
and find that
fits this description. Then working backwards and subtracting 7 from
and
gives us the point
, making the answer
.
Solution 4 (Answer Choices: Observations)
Let's say that team plays
games in total. Therefore, team
must play
games in total (7 wins, 7 losses) Since the ratio of
is
Similarly, since the ratio of
is
Now, we can go through the answer choices and see which ones work:
So we can see is the only valid answer.
~herobrine-india
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.