50 students showed which of the following sports they liked:football,volley ball and hockey.24 liked football,21 liked volley ball and 18 liked hockey.9 liked football only,3 liked volley ball only and 2 liked hocky only.2 liked all three sports.(a)Draw a venn diagram to illustrate this information.(b) find the number of students who liked exactly two sports
see the related questions below
(a) The Venn diagram can be illustrated as follows:
_________
/ \
/ \
| F: 9 |
|___________ \
| | |
| | 21|
| V: 3 | |
|__________|__|
\ | /
H\ | /H
\ | /
\|
|
|
F: 15
F represents football, V represents volleyball, and H represents hockey. The numbers inside the circles represent the number of students who liked each sport. The numbers outside the circles represent the number of students who liked more than one sport.
(b) To find the number of students who liked exactly two sports, we need to add up the numbers outside the circles, excluding the students who liked all three sports.
The number of students who liked exactly two sports = Students who liked (Football and Volleyball) + Students who liked (Football and Hockey) + Students who liked (Volleyball and Hockey)
Students who liked (Football and Volleyball) = Number outside Football circle - Students who liked all three sports = 15 - 2 = 13
Students who liked (Football and Hockey) = Number outside Hockey circle - Students who liked all three sports = 6 - 2 = 4
Students who liked (Volleyball and Hockey) = Number outside Volleyball circle - Students who liked all three sports = 10 - 2 = 8
Total number of students who liked exactly two sports = 13 + 4 + 8 = 25
Therefore, 25 students liked exactly two sports.
To solve this problem, we can use a Venn diagram to visualize the information given.
(a) Drawing a Venn diagram:
To represent the three sports (football, volley ball, and hockey), draw three intersecting circles. Label one circle as "Football", another as "Volley ball", and the third as "Hockey".
Next, we can place the numbers inside the circles based on the given information:
- 24 students liked football, so write 24 in the Football circle.
- 21 students liked volley ball, so write 21 in the Volley ball circle.
- 18 students liked hockey, so write 18 in the Hockey circle.
Additionally, we know that:
- 9 students liked football only, so write 9 in the Football circle but outside the intersection areas.
- 3 students liked volley ball only, so write 3 in the Volley ball circle but outside the intersection areas.
- 2 students liked hockey only, so write 2 in the Hockey circle but outside the intersection areas.
- 2 students liked all three sports, so write 2 in the intersection area of the three circles.
Now, the Venn diagram should illustrate the given information.
(b) Finding the number of students who liked exactly two sports:
To find the number of students who liked exactly two sports, we need to find the sum of the numbers in the regions where two circles intersect.
Looking at the Venn diagram, we can add the numbers in the intersections:
- The intersection of Football and Volley ball has 2 students.
- The intersection of Football and Hockey has 2 students.
- The intersection of Volley ball and Hockey has 0 students.
Adding the numbers, we have 2 + 2 + 0 = 4 students who liked exactly two sports.
Therefore, the number of students who liked exactly two sports is 4.