Tough Percentage Puzzle

clartharlear

At college, 70% of the students studied Maths, 75% studied English, 85% studied French and 80% studied German.
What percentage at least must have studied all 4?

Initially, you’d think that it would be 70% AND 75% AND 85% AND 80%, which would give you 37.5%, but that doesn’t take into account that doing one subject doesn’t have any effect on whether any of the other subjects are done.

I’ve seen another solution here, but I don’t find it convincing, though I believe the numerical answer.
70 +75 + 85+ 80 = 310
310 ÷ 3 = 100 remainder 10
3 subjects each student, 10 at least 4

I’ve been trying to draw Venn diagrams and Karnaugh maps with 4 variables where the outcome is F = ABCD, but I don’t know how to work that with the percentages.

Help would be much appreciated!

Thomas_S_Hunterson

I think the Inclusion-Exclusion principle should help here.

/edit: I just stumbled upon a very easy but indirect way to do it.

It involves looking at the people not studying particular subjects, the exclusion

Mellor

look at it this way, the highest is 70%,
so at least 30% couldn't of done all 4, (30% didn't do Maths)
consider all subjects this way, what if they wre all different people

10%

clartharlear

You're both marvellous, thanks.