A poll of 100 students was taken at a no-residential college to find out how they got to campus. The results were as follows: 28 said car pools;31 said buses;and 42 said they drove to school in their own car alone. In addition, 9 used both car pools and buses;10 said both car pools and using their own car; and 6 used buses and sometimes used their own car. Of the 100 respondents, only 4 used all three methods to get to school. Can you show me how to set up the problem and answer how many students used none of the three methods to get to school?
This is perfect for a Venn diagram. Draw three intersecting circles, where there is a non-empty space where all three circles intersect.
Start from the center and work your way out.
The center area contains 4.
9 used carpool and bus. Of those, 4 used all three methods, leaving 5 inside the carpool-bus intersection, but outside the center.
10 used carpool and own-car. Of those, 4 used all three methods, leaving 6 inside the carpool-self intersection, but outside the center.
That makes 4+5+6=15 who used carpool and some other method. So, that leaves 13 who used ONLY carpool.
Do similar figuring for only bus and only self-car.
Add up the numbers in all the sections of the diagram, and subtract that from the Universe of 100 students. Those will be the ones who used none of the methods.
To set up the problem, we can use a Venn diagram to visualize the relationships between the three methods of transportation: car pools, buses, and driving their own car alone.
First, let's start by representing the total number of students who took part in the poll, which is 100.
Next, we will assign variables to the number of students in each group. Let's use "C" for car pools, "B" for buses, and "D" for driving their own car alone.
According to the given information, we know that:
- 28 students said they used car pools (C = 28).
- 31 students said they used buses (B = 31).
- 42 students said they drove to school in their own car alone (D = 42).
- 9 students used both car pools and buses (C ∩ B = 9).
- 10 students used both car pools and their own car (C ∩ D = 10).
- 6 students used buses and sometimes used their own car (B ∩ D = 6).
- 4 students used all three methods of transportation (C ∩ B ∩ D = 4).
Now, we can use this information to determine the number of students who used none of the three methods.
To find the number of students who used none of the methods, we need to subtract the total number of students who used at least one method (car pools, buses, or driving their own car) from the total number of students polled (100).
To calculate this, we need to add up the number of students in each individual group (C, B, D), subtract the students in the overlapping groups (C ∩ B, C ∩ D, B ∩ D), and finally add back the students in the group that used all three methods (C ∩ B ∩ D) because they were double-counted in the overlapping groups.
So, the number of students who used none of the three methods can be calculated as follows:
None of the 3 methods = Total students - (C + B + D - C ∩ B - C ∩ D - B ∩ D + C ∩ B ∩ D)
Substituting the given values:
None of the 3 methods = 100 - (28 + 31 + 42 - 9 - 10 - 6 + 4)
None of the 3 methods = 100 - 80
None of the 3 methods = 20
Therefore, 20 students used none of the three methods to get to school.