Question

a random sample of students are asked about their preferences for games.
complete the two - way table so that it has the characteristics listed.

• 122 people responded to the survey.
• 50 of the people who said they prefer board games said they also prefer two - player games.
• 68 people prefer two - player games.
• 56 people prefer card games rather than board games.

Explanation:

Step1: Find the number of people who prefer board games

Total people = 122. People who prefer card games = 56. So, people who prefer board games = \(122 - 56=66\)

Step2: Find the number of people who prefer two - player board games

People who prefer two - player games = 68. Let \(x\) be the number of people who prefer two - player board games. People who prefer two - player card games = \(68 - x\). Also, people who prefer board games = 66. Let \(y\) be the number of people who prefer 3 or more player board games. So \(x + y=66\). People who prefer card games = 56. Let \(z\) be the number of people who prefer 3 or more player card games. So \((68 - x)+z = 56\Rightarrow z=x - 12\). And total people \(x + y+(68 - x)+z=122\). Substitute \(y = 66 - x\) and \(z=x - 12\) into the equation: \(x+(66 - x)+(68 - x)+(x - 12)=122\). Simplify: \(66 + 68-12=122\) (which is a check). Now, since 50 people who prefer board games also prefer two - player games.

Step3: Fill the table

Number of people who prefer two - player board games: 50. Number of people who prefer 3 or more player board games: \(66 - 50 = 16\). Number of people who prefer two - player card games: \(68 - 50=18\). Number of people who prefer 3 or more player card games: \(56-(68 - 50)=38\)

2 Players	3 or More Players
Card Games	18	38

Answer:

2 Players	3 or More Players
Card Games	18	38