Difference between revisions of "2020 AMC 12B Problems/Problem 5"

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<math>\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63</math>
 
<math>\textbf{(A) } 21 \qquad \textbf{(B) } 27 \qquad \textbf{(C) } 42 \qquad \textbf{(D) } 48 \qquad \textbf{(E) } 63</math>
  
==Solution==
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==Solution 1 (Two Variables)==
  
 
First, let us assign some variables. Let
 
First, let us assign some variables. Let
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~Argonauts16
 
~Argonauts16
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 +
==Solution 2 (One Variable)==
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Suppose team <math>A</math> has played <math>g</math> games in total so that it has won <math>\frac23g</math> games.
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It follows that team <math>B</math> has played <math>g+14</math> games in total so that it has won <math>\frac23g+7</math> games.
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We set up and solve an equation for team <math>B</math>'s win ratio:
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<cmath>\begin{align*}
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\frac{\frac23g+7}{g+14}&=\frac58 \\
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\frac{16}{3}g+56&=5g+70 \\
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\frac13g&=14 \\
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g&=\boxed{\textbf{(C) } 42}.
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\end{align*}</cmath>
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~MRENTHUSIASM
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==Solution 3 (Answer Choices: Substitutions)==
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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:
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<cmath>A \implies (14, 7)</cmath>
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<cmath>B \implies (18, 9)</cmath>
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<cmath>C \implies (28, 14)</cmath>
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<cmath>D \implies (32, 16)</cmath>
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<cmath>E \implies (42, 21)</cmath>
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Thus, we have 5 matching <math>B</math> scenarios, simply adding 7 to <math>w</math> and <math>l</math>. We can then test each of the five <math>B</math> scenarios for <math>\frac{w}{w+l} = \frac{5}{8}</math> and find that <math>(35, 21)</math> fits this description. Then working backwards and subtracting 7 from <math>w</math> and <math>l</math> gives us the point <math>(28, 14)</math>, making the answer <math>\boxed{\textbf{C}}</math>.
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==Solution 4 (Answer Choices: Observations)==
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Let's say that team <math>A</math> plays <math>n</math> games in total. Therefore, team <math>B</math> must play <math>n + 14</math> games in total (7 wins, 7 losses) Since the ratio of <math>A</math> is <cmath>\frac{2}{3} \implies n \equiv 0 \pmod{3}</cmath> Similarly, since the ratio of <math>B</math> is <cmath>\frac{5}{8} \implies n + 14 \equiv 0 \pmod{8}</cmath> Now, we can go through the answer choices and see which ones work:
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<cmath>\textbf{(A) } 21 \implies 21 + 14 = 35 \not \equiv 0\pmod{8}</cmath>
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<cmath>\textbf{(B) } 27 \implies 27 + 14 = 41 \not \equiv 0\pmod{8}</cmath>
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<cmath>\textbf{(C) } 42 \implies 42 + 14 = 56  \equiv 0\pmod{8}</cmath>
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<cmath>\textbf{(D) } 48 \implies 48 + 14 = 62 \not \equiv 0\pmod{8}</cmath>
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<cmath>\textbf{(E) } 63 \implies 63 + 14 = 77 \not \equiv 0\pmod{8}</cmath>
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So we can see <math>\boxed{\textbf{(C) } 42} \text{ \tiny nice}</math> is the only valid answer.
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 +
~herobrine-india
  
 
==Video Solution==
 
==Video Solution==
 +
 
https://youtu.be/WfTty8Fe5Fo
 
https://youtu.be/WfTty8Fe5Fo
  

Revision as of 09:32, 12 June 2021

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 1 (Two Variables)

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 (One Variable)

Suppose team $A$ has played $g$ games in total so that it has won $\frac23g$ games. It follows that team $B$ has played $g+14$ games in total so that it has won $\frac23g+7$ games.

We set up and solve an equation for team $B$'s win ratio: \begin{align*} \frac{\frac23g+7}{g+14}&=\frac58 \\ \frac{16}{3}g+56&=5g+70 \\ \frac13g&=14 \\ g&=\boxed{\textbf{(C) } 42}. \end{align*} ~MRENTHUSIASM

Solution 3 (Answer Choices: Substitutions)

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 $\boxed{\textbf{C}}$.

Solution 4 (Answer Choices: Observations)

Let's say that team $A$ plays $n$ games in total. Therefore, team $B$ must play $n + 14$ games in total (7 wins, 7 losses) Since the ratio of $A$ is \[\frac{2}{3} \implies n \equiv 0 \pmod{3}\] Similarly, since the ratio of $B$ is \[\frac{5}{8} \implies n + 14 \equiv 0 \pmod{8}\] Now, we can go through the answer choices and see which ones work:

\[\textbf{(A) } 21 \implies 21 + 14 = 35 \not \equiv 0\pmod{8}\] \[\textbf{(B) } 27 \implies 27 + 14 = 41 \not \equiv 0\pmod{8}\] \[\textbf{(C) } 42 \implies 42 + 14 = 56  \equiv 0\pmod{8}\] \[\textbf{(D) } 48 \implies 48 + 14 = 62 \not \equiv 0\pmod{8}\] \[\textbf{(E) } 63 \implies 63 + 14 = 77 \not \equiv 0\pmod{8}\]

So we can see $\boxed{\textbf{(C) } 42} \text{ \tiny nice}$ is the only valid answer.

~herobrine-india

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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