TITLE: Counting number of students who have failed in all four subjects
QUESTION [14 upvotes]: In an examination of a certain class, at least $69\%$ of the students failed in P, at least $72\%$ failed in C, at least $80\%$ failed in M, and at least $85\%$ failed in B. How many at least must have failed in all the four subjects?

My attempt:
I'm familiar with the type of questions where the exact number of students is given for each set (like this). But in this question, the minimum number for each set is given instead!
Every attempt I make is leading me nowhere. I counted the number of students who have passed in each subject, but it still does not help.
I feel like I'm missing a specific train of thought that'll lead me to the correct answer. Thus, I need hints for realizing the idea of this problem. Thanks!

REPLY [30 votes]: I like to look at the complements, since that's more intuitive: How many students can possibly have passed in at least one subject? We have that $31\%$ passed P, $28\%$ passed C, $20\%$ passed M and $15\%$ passed B. If we assume no overlap between these four groups for maximum at-least-one-subject-passed percentage, we get that at most
$$
31\% + 28\% + 20\% + 15\% = 94\%
$$
passed on at least one exam (any overlap between the groups would've subtracted from this total). That means that at least $6\%$ failed on all exams.