In a class of 100 students 70 passed in physics, 62 passed in mathemat

Question

In a class of 100 students 70 passed in physics, 62 passed in mathematics, 84 passed in English and 82 passed in chemistry. 37 students passed in all 4 subjects. What is the maximum number of students who could have failed all four subjects?

A. 10
B. 12
C. 13
D. 15
E. 17

Show Answer

Official Answer

C

A. 10
B. 12
C. 13
D. 15
E. 17

imeanup · Accepted Answer

Sum =70+62+84+82=298 
Total Pass =37*4=148 
Remaining = 298-148=150  students pass in 3 exams. i.e.  \frac{150}{3}=50 
Total max passed  =37+50=87 
Max failed students =100-87=13

IMO  Ans C

Krunaal · Answer

I did it this way - there are 100 students in total, since 37 passed in all 4 subjects,  63 have failed in at least one subject .

100 - 70 = 30 failed in Physics
100 - 62 = 38 failed in Mathematics
100 - 84 = 16 failed in English
100 - 82 = 18 failed in Chemistry

30 + 38 + 16 + 18 =  102 failures occurred, with some students failing in more than one subject.

Now, since 63 students led to 102 failures, there were overlaps of multiple students failing in more than one subject

63 = 102 - 3*(Students failing in all 4 subjects)

Students failing in all 4 subjects = 39/3 =  13

Answer C.

effatara · Answer

This was a new one for me (4 variables) so it took some time to figure out. Although the first answer is essentially correct, it needs elaboration to explain the assumptions made and the rationale behind them.
In order to determine the maximum number of students who failed we have to assume maximum overlap i.e. all those who passed, apart from the 37 who passed in all 4 subjects, passed in exactly 3 subjects. In other words, no student passed in exactly one or exactly two subjects. Let the number who passed in Math, Physics and Chemistry be denoted by 'mpc', Math Physics and English by 'mpe' and so on. Then:

Breakdown of students who passed in Physics: pmc + mpe + pec + 37 = 70
.................................................... Math: pmc + emc + mpe +37 = 62
.................................................... English: emc + pec + mpe + 37 = 84
.................................................... Chem: pmc + pec + emc + 37 = 82

3(pmc+mpe+pec+emc) + 148 = 298.....> (pmc+mpe+pec+emc)=50. Total number of students who passed is 50+37=87 so the number who failed in all subjects is 100-87=13.
ANS: C

P.S. From the above it can be calculated that: emc=17, pec=25, pmc=3 and mpe=5

gmattyfatty · Answer

First we add up the counts of all students that have passed a test, disregarding the subjects and any overlap.
Total Pass=70+62+84+82=298

Now, if 37 students passed all 4 tests, that means each category must include these same 37 students. Since we have 4 categories:
Total 4-Pass=37∗4=148

Now we subtract the numbers. The difference is the number of students that have passed at least a single test, but did not pass all N tests
Remaining=298−148=150

Now to solve the problem, we must maximize the overlap by assuming these kids all passed exactly 3 tests. It doesn't matter which tests exactly, and the numbers technically dont need to be evenly distributed. So we can just take the average
Avg 3-Pass = 150/3=50

At this point the original numbers and categories dont even matter anymore as we've slimmed the numbers down to include 100% of the kids that have passed, but we've now distributed them across only 3 tests instead of 4. So we can add these 2 numbers together
Total max passed =37+50=87

So out of 100 students, a maximum of 87 have all passed identical tests (100% overlap). And we can easily subtract these 2 to assume that all of the remaining kids have all failed all tests.
Max failed students=100−87=13

Bunuel  what do you think of this analysis? maybe you have a clearer approach or explanation

jcrafer · Answer

Help please... Why the answer is not 16?
There is a possible scenario where English includes all the the other sets: 
English includes Chemestry, Chemestry includes Physics, and math intersects Physic with 37 students, 
I think this is ok with the statement, and the number of people out of this set is 16, where am I wrong please.

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In a class of 100 students 70 passed in physics, 62 passed in mathemat

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