Study Hall Games
.Explore
There are 5 people who do NOT play games. There are 5 people to play BACKGAMMON and CHECKERS and CHESS. There are 3 people to play BACKGAMMON and CHECKERS. There are 2 people to play BACKGAMMON and CHESS. 1 person would play CHESS and CHECKERS.
Altogether there are 21 students to play BACKGAMMON or CHESS.
There are 24 students who would play CHECKERS or CHESS.
There are 26 students who would play CHECKERS or BACKGAMMON.
Find how many only play BACKGAMMON? How many only play CHECKERS? How many only play CHESS? How many students were in the study hall?
Plan
Use a Venn Diagram to show the known students. Use guess and check to balance the numbers.
Solve
The Venn Diagram uses circles to show the numbers of students who participate in more than one game. The students who do not participate are outside the diagram. These are the numbers 1, 2, 3, 5 and 5 from the information above.
The numbers in the overlapping parts of the circle, 1+ 2 + 3 + 5 = 11, is the number of students who said they would play 2 or more games. Therefore, each 2 circles share 11 students.
From the data given, there are 21 students to play BACKGAMMON or CHESS. Since they share 11, that leaves 10 to be divided between BACKGAMMON and CHESS. There are 24 students who would play CHECKERS or CHESS, that leaves 13 to be divided between CHECKERS and CHESS. There are 26 students who would play CHECKERS or BACKGAMMON, that leaves 15 to be divided between CHECKERS and BACKGAMMON.
From the diagram, there are 6 students who only play backgammon, 9 who only play checkers and 4 who only play chess. Adding 11 (the ones who play more than one game) + 6 + 4 + 9 + 5(the ones who did not play any games) gives a total of 35 students in study hall.
Examine
The diagram and guess and check method gives a reasonable answer of 35 total students for a study hall. All of the conditions are met.