2020 AMC 12B Problems/Problem 5

Problem

Teams $A$ and $B$ are playing in a basketball league where each game results in a win for one team and a loss for the other team. Team $A$ has won $\tfrac{2}{3}$ of its games and team $B$ has won $\tfrac{5}{8}$ of its games. Also, team $B$ has won $7$ more games and lost $7$ more games than team $A.$ How many games has team $A$ played?

$\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63$

Solution

First, let us assign some variables. Let

\[A_w=2x, A_l=x, A_g=3x,\] \[B_w=5y, B_l=3y, B_g=8y,\]

where $X_w$ denotes number of games won, $X_l$ denotes number of games lost, and $X_g$ denotes total games played for $X\in \{A, B\}$. Using the given information, we can set up the following two equations:

\[B_w=A_w+7\implies 5y=2x+7,\] \[B_l=A_l+7\implies 3y=x+7.\]

We can solve through substitution, as the second equation can be written as $x=3y-7$, and plugging this into the first equation gives $5y=6y-7\implies y=7$, which means $x=3(7)-7=14$. Finally, we want the total number of games team $A$ has played, which is $A_g=3(14)=\boxed{\textbf{(C) } 42}$.

~Argonauts16

Solution 2

Using the information from the problem, we can note that team A has lost $\frac{1}{3}$ of their matches. Using the answer choices, we can find the following list of possible win-lose scenarios for $A$, represented in the form $(w, l)$ for convenience:

\[A \implies (14, 7)\] \[B \implies (18, 9)\] \[C \implies (28, 14)\] \[D \implies (32, 16)\] \[E \implies (42, 21)\]

Thus, we have 5 matching $B$ scenarios, simply adding 7 to $w$ and $l$. We can then test each of the five $B$ scenarios for $\frac{w}{w+l} = \frac{5}{8}$ and find that $(35, 21)$ fits this description. Then working backwards and subtracting 7 from $w$ and $l$ gives us the point $(28, 14)$, making the answer $\boxed{\textbf{C}}$.

Video Solution

https://youtu.be/WfTty8Fe5Fo

~IceMatrix

See Also

2020 AMC 12B (ProblemsAnswer KeyResources)
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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