Mr. Green teaches band, choir, and math. This year, he has 57 students that take at least one of his classes. He teaches band to 25 students. There are 9 students who have Mr. Green for math and nothing else. Use the Venn diagram below:
A Venn diagram of three overlapping circles titled 'Band,' 'Choir,' and 'Math' is shown.
· The circle on the left titled 'Band' contains the number 11 in the part of the circle that is not overlapping with any other circle.
· The circle on the right titled 'Choir' contains the number 17 in the part of the circle that is not overlapping with any other circle.
· The circle above the other two circles titled 'Math' is empty in the part of the circle that is not overlapping with any other circle.
· The region where all three circles overlap in the middle of the diagram contains the number 3.
· The region where the 'Band' circle overlaps with the 'Math' circle and does not include the intersection of all three circles contains the number 7.
How many students take exactly 2 classes with Mr. Green?
To find the number of students who take exactly 2 classes, we need to add up the numbers in the overlapping regions of the Venn diagram.
- The region where the 'Band' and 'Math' circles overlap but do not include the intersection of all three circles contains 7 students.
- The region where the 'Choir' and 'Math' circles overlap but do not include the intersection of all three circles is empty (since the 'Math' circle is empty in that region).
- The region where the 'Band' and 'Choir' circles overlap but do not include the intersection of all three circles contains x students (which we want to find).
We can use the formula:
total = band + choir + math - (band and choir) - (band and math) - (choir and math) + (band and choir and math)
Plugging in the values we know:
57 = 25 + 17 + 0 - x - 7 - 0 + 3
Simplifying:
57 = 38 - x
x = 38 - 57
x = 19
Therefore, there are 19 students who take exactly 2 classes with Mr. Green.