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Class Majors Venn Diagram and Averages

XAT 2017 · Quantitative Ability & Data Interpretation · Hard · Venn Diagrams

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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?

  1. A.

    30

  2. B.

    25

  3. C.

    20

  4. D.

    15

  5. E.

    None of the above

Answer

D

Explanation

Let mm = only Math, bb = only Biology, pp = only Physics. Given overlapping regions: Math & Physics = 6 Physics & Biology = 15 Math & Biology = 0 All three = 0.

Total students majoring in Math NM=m+6N_M = m + 6. Total students majoring in Bio NB=b+15N_B = b + 15.

Combined average of Math and Bio majors: 45(m+6)+60(b+15)(m+6)+(b+15)=50\frac{45(m + 6) + 60(b + 15)}{(m + 6) + (b + 15)} = 50 45m+270+60b+900=50m+50b+105045m + 270 + 60b + 900 = 50m + 50b + 1050 10b+120=5m  ⟹  m=2b+2410b + 120 = 5m \implies m = 2b + 24.

Total students in class ≤60\le 60: m+6+15+b+p≤60  ⟹  (2b+24)+b+21+p≤60  ⟹  3b+45+p≤60m + 6 + 15 + b + p \le 60 \implies (2b + 24) + b + 21 + p \le 60 \implies 3b + 45 + p \le 60. p=15−3bp = 15 - 3b. To maximize pp, minimize b≥0  ⟹  b=0b \ge 0 \implies b = 0. Max possible p=15p = 15.

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