Question

11. In a class of 150 students, the following results were obtained in a certain examination; 45 students
failed in Maths, 50 students failed in Physics, 48 students failed in Chemistry, 30 students failed
in both Maths and Physics, 32 failed in Physics and Chemistry. 35 failed in both Maths and
Chemistry, 25 failed in all the three subjects. Draw a Venn diagram corresponding to this data
and find the number of students who have failed in at least one subject.
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Answer 1

Answer:

Given that there are 100 students and only one student passed in all 3 subjects

So the number of students failed in at least one subject is 99

Let a be the number of students failed only in mathematics, b be number of students failed only in physics and c be number of students failed only in Biology.

Let x be number of students failed only in mathematics and physics, y be number of students failed only in physics and biology and z be the number of students failed only in biology and mathematics.

Let p be the number of students failed in all three subjects

By Venn diagram, we get 99=a+b+c+x+y+z+p

Number of students failed in exactly two subjects is 32=x+y+z

Students failed in mathematics is 50=a+x+z+p , students failed in physics is 45=b+x+y+p and students failed in Biology is 40=c+y+z+p

By adding above three equations, we get 135=a+b+c+2(x+y+z)+3p

⇒135=99+32+2p

⇒p=2