In a survey of 150 students, 90 were taking algebra, and 30 taking biology.

a. what is the least number of students who could have been taking both classes?

b. what is the greatest number of students who could be taking both courses?

c. what is the greatest number of student who could have been taking neither courses?

Please explain.

(a) 0, since there are still 30 students in neither class. Only if the two classes sum to more than 150 does it mean that someone is taking both.

(b) 30, since that would mean that ALL the biology students are also taking algebra

(c) 60, since if all 30 biology students were taking algebra, that only uses up 90 of the 150 students.

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To answer these questions, we need to use some principles of set theory and basic arithmetic.

a. To find the least number of students who could have been taking both classes, we need to consider the maximum number of students taking algebra and subtract the maximum number of students taking biology. In this case, we have 90 students taking algebra and 30 students taking biology. The minimum number of students taking both classes is the smaller of these two numbers. So, the least number of students taking both classes is 30.

b. To find the greatest number of students who could be taking both courses, we need to consider the minimum number of students taking algebra and subtract the minimum number of students taking biology. In this case, we have 90 students taking algebra and 30 students taking biology. The maximum number of students taking both classes is the larger of these two numbers. So, the greatest number of students taking both classes is 90.

c. To find the greatest number of students who could have been taking neither course, we need to subtract the total number of students from the sum of the number of students taking algebra and the number of students taking biology. In this case, the total number of students is 150, and the number of students taking algebra is 90, while the number of students taking biology is 30. So, the greatest number of students taking neither course is calculated as follows: 150 - (90 + 30) = 150 - 120 = 30.

In summary:
a. The least number of students taking both classes is 30.
b. The greatest number of students taking both classes is 90.
c. The greatest number of students taking neither course is 30.

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