The 160 students at a certain high school must choose at least one sci

Official Explanation

This is an Overlapping Sets question. Initially, it might seem that there are three possible sets (physics chemistry, and some other unnamed subject). Because the problem is presented as choosing physics, chemistry, both, or neither, however, you can use a Double-Set Matrix to solve. You could also try to think through this type of problem logically—if you feel very comfortable with this type of math.

The at least language indicates that you will also have to use Maximize and Minimize principles in order to solve.

Logic

There are 160 students total, but at least 15 choose some other subject entirely, so at most 160 – 15 =145 students choose some combination of chemistry and physics.

The problem indicates that 110 choose chemistry and 70 choose physics, for an apparent total of 180 students. But since 145 is the actual maximum, at least 180 – 145 = 35 students must take both classes. This is the answer for the first column.

The problem states that a total of 70 students choose physics, so it can’t be the case that more than 70 take both chemistry and physics; eliminate answers 85 and 90. Since more students (110) take chemistry, it’s possible that all 70 students who take physics also take chemistry, so the maximum possible number of students who take both is 70. This is the answer for the second column.


Double-Set Matrix
Start by setting up the matrix with the information you are given as well as any additional information you can infer.

	P	Not P	Total
C			110
Not C		≥ 15	50
Total	70	90	160

The center box (neither chemistry nor physics) must have at least 15, but is there a top limit? Since 50 students do not take chemistry, there cannot be more than 50 students who take neither subject. The “neither” box, therefore, has a range of 15 to 50.

	P	Not P	Total
C			110
Not C		15-50	50
Total	70	90	160

Using each end of that range, find the possible range of students who took both chemistry and physics. For instance, in the middle row (NOT C), take the difference between each end of the range in the middle column and the figure in the Total column. The difference between 15 and 50 is 35, and the difference between 50 and 50 is 0, so the range for the “P but NOT C” box is 0 to 35.

	P	Not P	Total
C	35-70		110
Not C	0-35	15-50	50
Total	70	90	160

If “P but NOT C is 0 to 35, then “C and P” must have a range of 35 to 70, in order for the Total P box to equal 70.
Therefore, the minimum possible number of students who take both C and P is 35, while the maximum possible number of students who take both is 70.

Note: You can also solve by calculating the range for the “C but NOT P” box (that range is 40 to 75) and then finding the range for the “C and P” box (which would still be 35 to 70).

Column 1: The correct answer is (B).
Column 2: The correct answer is (D).