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In a class of 60, along with English as a common subject, students can opt to major in Mathematics, Physics, Biology or a combination of any two. 6 students major in both Mathematics and Physics, 15 major in both Physics and Biology, but no one majors in both Mathematics and Biology. In an English test, the average mark scored by students majoring in Mathematics is 45 and that of students majoring in Biology is 60. However, the combined average mark in English, of students of these two majors, is 50. What is the maximum possible number of students who major ONLY in Physics?
Let us note down the information given:
No person can major in all 3 subjects. 6 students major in both Mathematics and Physics, 15 major in both Physics and Biology, but no one majors in both Mathematics and Biology. There are 60 students in total.
It has been given that average marks scored by students majoring in Maths in English is 45.
Average marks scored by students majoring in Biology in English is 60.
But the combined average marks scored by students majoring in Maths and Biology in English is 50.
=> $$\dfrac{45*(a+6) + 60*(c+15)}{a+6+c+15} = 50$$
$$45a+270+60c+900 = 50a+50c+1050$$
$$5a=10c+120$$
$$a=2c+24$$
To maximize b, we have to minimize 'a' and 'c'. The least value that 'c' can take is 0.
The corresponding value of a is 24.
24+6+b+15 = 60
=> b = 15.
Therefore, the maximum number of students who could have majored only in physics is 15. Therefore,option D is the right answer.
