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08 Jan 2021, 02:06
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C
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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
A. 10
B. 12
C. 13
D. 15
E. 17
08 Jan 2021, 22:26
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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
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
General Discussion
06 Feb 2021, 23:11
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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
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
