- In a graduating class of 236 students, 142 took algebra and 121 took chemistry. What

is the greatest possible number of students that could have taken both algebra and

chemistry?

The answer key says that the greatest possible intersection value should be the least of the two individual values i.e 121.

My doubt: if the total (236), chemistry (121) and algebra(142) are given, shouldnt the answer be 27 ( 121 +142 - 236)? .

Also, even if the intersection is 121, it doesnt add up to the total right ? (142 + 121 (chem) - 121 (intersection) != 236)

So when you suggest that 142+121-27 = 236, we are assuming that there are no students that do not take either of both classes.

Now, notice that the question asks for the greatest possible number of students that could have taken both classes. How can we achieve this? Visually (i.e. using a Venn Diagram), we can see that this is achieved if one circle is completely inside of the other, in this case, if the Algebra circle completely contains the Chemistry circle. This is what it means to take the least of the two individual values. Also, notice that 236\neq 142+121, so what if 142 took algebra, 121 chemistry, 121 chemistry and algebra, and 94 took none of those two classes?

hey Alex,

Thanks!. I understood how we get the greatest value but was confused as to why we do.

So the clue is the word "greatest possible " and the fact that they never mentioned there is no one who took both?