2020 AMC 12B Problems/Problem 5
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 (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 2 (Two Variables)
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
~AnkitAMC
Solution 3 (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 3 (Answer Choices: Substitutions)
Using the information from the problem, we can note that team has lost
of their matches. Using the answer choices, we can construct the following list of possible win-lose scenarios for
represented in the form
for convenience:
Thus, we have
matching
scenarios, simply adding
to
and
We can then test each of the five
scenarios for
and find that
fits this description. Then working backwards and subtracting
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) | |
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