Question

from unit 2, lesson 1
in a video game, players form teams and work together to earn as many points as possible for their team. each team can have between 2 and 4 players. each player can score up to 20 points in each round of the game. han and three of his friends decided to form a team and play a round.
write an expression, an equation, or an inequality for each quantity described here. if you use a variable, specify what it represents.
a. the allowable number of players on a team
b. the number of points hans team earns in one round if every player earns a perfect score
c. the number of points hans team earns in one round if no players earn a perfect score
d. the number of players in a game with six teams of different sizes: two teams have 4 players each and the rest have 3 players each
e. the possible number of players in a game with eight teams

Explanation:

Step1: Define variable for team - size

Let $n$ be the number of players on a team. Since each team can have between 2 and 4 players, the inequality is $2\leq n\leq4$.

Step2: Calculate perfect - score points for Han's team

Han and three of his friends form a team of 4 players. Each player can score up to 20 points in a round. If every player earns a perfect score, the number of points Han's team earns is $4\times20 = 80$.

Step3: Define variable for non - perfect scores

Let $x_i$ be the score of the $i$ - th player on Han's team, where $0\leq x_i<20$ for $i = 1,2,3,4$. The number of points Han's team earns is $x_1 + x_2+x_3 + x_4$.

Step4: Calculate number of players in six - team game

Two teams have 4 players each and $6 - 2=4$ teams have 3 players each. The total number of players is $2\times4+4\times3=8 + 12=20$.

Step5: Define variable for team - size in eight - team game

Let $n_i$ be the number of players on the $i$ - th team, where $2\leq n_i\leq4$ for $i = 1,\cdots,8$. The total number of players $P$ satisfies $8\times2\leq P\leq8\times4$, so $16\leq P\leq32$.

Answer:

a. $2\leq n\leq4$, where $n$ is the number of players on a team.
b. 80
c. $x_1 + x_2+x_3 + x_4$, where $0\leq x_i<20$ for $i = 1,2,3,4$
d. 20
e. $16\leq P\leq32$, where $P$ is the total number of players in the game with eight teams.