JEE MAIN - Mathematics (2024 - 4th April Morning Shift - No. 27)
In a survey of 220 students of a higher secondary school, it was found that at least 125 and at most 130 students studied Mathematics; at least 85 and at most 95 studied Physics; at least 75 and at most 90 studied Chemistry; 30 studied both Physics and Chemistry; 50 studied both Chemistry and Mathematics; 40 studied both Mathematics and Physics and 10 studied none of these subjects. Let $$m$$ and $$n$$ respectively be the least and the most number of students who studied all the three subjects. Then $$\mathrm{m}+\mathrm{n}$$ is equal to ___________.
Answer
45
Explanation
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$\begin{aligned} & 125 \leq \mathrm{m}+90-\mathrm{x} \leq 130 \\\\ & 85 \leq \mathrm{P}+70-\mathrm{x} \leq 95 \\\\ & 75 \leq \mathrm{C}+80-\mathrm{x} \leq 90 \\\\ & \mathrm{~m}+\mathrm{P}+\mathrm{C}+120-2 \mathrm{x}=210 \\\\ & \Rightarrow 15 \leq \mathrm{x} \leq 45 \& 30-\mathrm{x} \geq 0 \\\\ & \Rightarrow 15 \leq \mathrm{x} \leq 30 \\\\ & 30+15=45\end{aligned}$
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