2020 AMC 12B Problem 5

Question

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

\tfrac23

of its games and team

B

has won

\tfrac58

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?

21

27

42

48

63

Answer: C

Concepts:system of equations fraction

Difficulty rating: 1290

Small hint:

Let

A

play

a

games and

B

play

b

games, then write both the win and loss conditions

Big hint:

Wins:

\tfrac58 b = \tfrac23 a + 7;

losses:

\tfrac38 b = \tfrac13 a + 7

Solution:

Let

a

be the number of games team

A

played and

b

the number team

B

played. Team

A

wins

\tfrac23 a

and loses

\tfrac13 a;

team

B

wins

\tfrac58 b

and loses

\tfrac38 b.

The conditions give

\tfrac58 b = \tfrac23 a + 7

and

\tfrac38 b = \tfrac13 a + 7.

Subtracting the equations gives

\tfrac14 b = \tfrac13 a,

so

b = \tfrac43 a.

Substituting into the loss equation:

\tfrac38 \cdot \tfrac43 a = \tfrac13 a + 7,

i.e.

\tfrac12 a = \tfrac13 a + 7,

so

\tfrac16 a = 7

and

a = 42.

Thus, the correct answer is C.

2020 AMC 12B Problem 5

Answer: C

Difficulty rating: 1290

Small hint:

Big hint:

Solution:

Problem 5 in other years