# How many failed both classes

I think this might be too easy compared to other questions on this site but I really can't find the solution.

60% of students passed class A
70% of students passed class B
80% of students passed at least one of both classes


a. What % of students passed both classes?

b. What % of students passed class A but failed class B?

I actually have the answers for both questions but no idea how to get there (a: 50% and b: 10%). Also, if anyone has a link to some theory about this kind of problems or more similar exercises I'd appreciate that a lot! Not sure how to look this up.

And another question (but I am not sure if I have all the numbers necessary for this one): If 51.8% of population has obesity and 48% of the population are male,.... what % of the male population has obesity if there's 10% more females (than males) without obesity? Can this be solved with these numbers? I might have forgotten some part of the question so I'm not sure about this one.

• Did you try drawing a Venn diagram? It sometimes help to visualize... and at the same time you should immediately see an algebraic relation you probably already know. – Glen_b Dec 14 '13 at 23:43
• Andrew drew one and that helped a lot. Any idea about the second question though? Not sure if one can be drawn for that one. – blop Dec 15 '13 at 1:06
• Yes, a diagram can be drawn that should help you (though it can be solved without one); you just need to see that Male/Female and Obsese/Not-Obese are pairs of exhaustive, mutually exclusive events (i.e. complementary events) so both sets completely divide the space. See here for a diagram with a similar example with a pair of such complementary events. – Glen_b Dec 15 '13 at 19:05
• That cleared up a lot, I'll remember that. Thank you – blop Dec 15 '13 at 21:13

x + w + y = 80% (Equation 1).
x + w = 60% (Equation 2).
w + y = 70% (Equation 3).

If you understand and solve the above equations, you will find the answers.

The second part of the question (copy-pasted from comments):

Q: ... Any idea about the second question, if it can be solved? I tried working with a venn diagram for that as well but I don't really see a common area where two circles could cross. But then again, it's possible I'm missing data for that question... – blop Dec 15 '13 at 0:40

A: ... A hint would be to start by the constraint that Female which are not obese (F_not) needs to be 10% greater than not obese Male (M_not). So, F_not = 1.1*M_not. Then, we have F_not + M_not = 48.2%. Can you figure out what it's next? Hope you can solve this. – Andre Silva Dec 15 '13 at 2:53

Q: I was actually trying something like M_ob = F_ob + 0.1 Also, I didn't think of doing 1-0.5180 to get the total population without obesity. Anyway, combining your two equations I got M_not = 22.95% so males without obesity is 1-0.2295=77.05%. Is that correct? It's strange that I didn't need to use the 48% of the population is male number. ... – blop Dec 15 '13 at 12:46

A: ... Actually, I believe you gave 1 step in the right direction and after that, 1 step in the wrong direction. M_not + M_obese = 48 (so you need to find M_obese, but can't assure you 100% this is the correct answer). If we think that if was not for the constraint of F_not = 1.1*M_not, the exercise would be simpler to solve. All we'd need to do is to multiply 48% of men by 51.8% of obeses. That was my understanding of this exercise. Regards. – Andre Silva Dec 15 '13 at 14:04

• Thanks, that helped a lot! Any idea about the second question, if it can be solved? I tried working with a venn diagram for that as well but I don't really see a common area where two circles could cross. But then again, it's possible I'm missing data for that question... – blop Dec 15 '13 at 0:40
• I was actually trying something like M_ob = F_ob + 0.1 Also, I didn't think of doing 1-0.5180 to get the total population without obesity. Anyway, combining your two equations I got M_not = 22.95% so males without obesity is 1-0.2295=77.05%. Is that correct? It's strange that I didn't need to use the 48% of the population is male number. Thanks again for your help! – blop Dec 15 '13 at 12:46

I generally draw a table in these types of situations:

In the first, I just show all the possibilities and that totals must be 100%

In the second, I add the facts about total passing rates for A and B

In the third, I add the joint failure rate (100 - 80%)

In the fourth, I calculate the joint pass/fail combinations from the passing rates numbers and overall totals

In the fifth, I calculate the complementary pass/fail combinations from the joint rates and overall totals

In the sixth, I calculate the pass/pass rate based on the totals and the previously calculated complementary pass/fail combination rates - there are two ways to do this, and thankfully they agree, suggesting I can still subtract despite having worked as an engineering manager for a few years back in the 2000s