In a group of girls, 20 play volleyball, 21 play badminton and 18 play table tennis; 7 play volleyball only; 9 play badminton only; 6 play volleyball and badminton only and 2 play badminton and table-tennis only.
a)How many play all three games?
b)How many play volleyball and table-tennis only?
c)How many play table-tennis only?
d)How many girls are there altogether?
Have you tried making a Venn diagram?
Hint: let the intersection of all 3 circles be x
you can fill in the "volleyball only" and "badminton only" parts.
Fill in the rest
To find the answers to the questions, we can use the concept of overlapping sets and the principle of inclusion-exclusion.
a) To find how many play all three games, we need to find the intersection of all three sets (volleyball, badminton, and table tennis). We can start by adding up the number of girls who play each individual game: 20 + 21 + 18 = 59.
Now, let's subtract the number of girls who play only one game, the number who play two games only, and then we can find the number who play all three games.
According to the given information:
- 7 play volleyball only.
- 9 play badminton only.
- 6 play volleyball and badminton only.
- 2 play badminton and table-tennis only.
To find the number who play all three games, we can subtract these numbers from the total count of girls who play any of the games.
59 - (7 + 9 + 6 + 2) = 35.
Therefore, 35 girls play all three games.
b) To find how many play volleyball and table-tennis only, we need to subtract the number of girls who play both volleyball and badminton from the total count of girls who play volleyball.
According to the information, 6 girls play volleyball and badminton only.
So, the number of girls who play volleyball and table-tennis only is:
20 (girls who play volleyball) - 6 (girls who play both volleyball and badminton) = 14.
Therefore, 14 girls play volleyball and table-tennis only.
c) To find how many play table-tennis only, we need to subtract the number of girls who play both volleyball and table-tennis, and the number who play both badminton and table-tennis from the total count of girls who play table-tennis.
According to the information, 2 girls play badminton and table-tennis only.
So, the number of girls who play table-tennis only is:
18 (girls who play table-tennis) - 2 (girls who play both badminton and table-tennis) = 16.
Therefore, 16 girls play table-tennis only.
d) To find how many girls are there altogether, we can add up the number of girls who play each individual game, the number who play multiple games, and the number who play all three games.
Total count of girls = 20 (girls who play volleyball) + 21 (girls who play badminton) + 18 (girls who play table-tennis) - 7 (girls who play volleyball only) - 9 (girls who play badminton only) - 6 (girls who play volleyball and badminton only) - 2 (girls who play badminton and table-tennis only) + 35 (girls who play all three games) = 80.
Therefore, there are 80 girls altogether.