Question

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, i

Accepted Answer

D. Let $m$ = only Math, $b$ = only Biology, $p$ = only Physics. Given overlapping regions: Math & Physics = 6 Physics & Biology = 15 Math & Biology = 0 All three = 0. Total students majoring in Math $N_M = m + 6$. Total students majoring in Bio $N_B = b + 15$. Combined average of Math and Bio majors: $\frac{45(m + 6) + 60(b + 15)}{(m + 6) + (b + 15)} = 50$ $45m + 270 + 60b + 900 = 50m + 50b + 1050$ $10b + 120 = 5m \implies m = 2b + 24$. Total students in class $\le 60$: $m + 6 + 15 + b + p \le 60 \implies (2b + 24) + b + 21 + p \le 60 \implies 3b + 45 + p \le 60$. $p = 15 - 3b$. To maximize $p$, minimize $b \ge 0 \implies b = 0$. Max possible $p = 15$.

Class Majors Venn Diagram and Averages

Explanation

Practise this under exam conditions