1. **Problem Statement:** In a class, 50% students passed in Maths, 55% passed in English, and 15% passed in both subjects. 60 students failed in both subjects. Find: (i) Total students appeared in the exam. (ii) Number of students passed in only 1 subject. (iii) Number of students passed in at least 1 subject. (iv) Number of students failed in only 1 subject. 2. **Formulas and Rules:** - Let total students be $N$. - Percentage passed in Maths = 50%, so students passed in Maths = $0.5N$. - Percentage passed in English = 55%, so students passed in English = $0.55N$. - Percentage passed in both = 15%, so students passed in both = $0.15N$. - Students failed in both = 60. 3. **Calculations:** (i) Students passed in at least one subject = students passed in Maths + students passed in English - students passed in both = $0.5N + 0.55N - 0.15N = 0.9N$. Since 60 students failed in both, total students = passed in at least one + failed in both = $0.9N + 60$. But total students = $N$, so: $$N = 0.9N + 60$$ $$N - 0.9N = 60$$ $$0.1N = 60$$ $$N = \frac{60}{0.1} = 600$$ (ii) Students passed in only 1 subject = (passed in Maths only) + (passed in English only) = $(0.5N - 0.15N) + (0.55N - 0.15N) = 0.35N + 0.4N = 0.75N$ $$0.75 \times 600 = 450$$ (iii) Students passed in at least 1 subject = $0.9N = 0.9 \times 600 = 540$ (iv) Students failed in only 1 subject = total students - (passed in both + failed in both + passed in only 1 subject) = $600 - (0.15N + 60 + 0.75N) = 600 - (0.15 \times 600 + 60 + 0.75 \times 600)$ = $600 - (90 + 60 + 450) = 600 - 600 = 0$ So, no student failed in only one subject. **Final answers:** (i) 600 students appeared. (ii) 450 students passed in only 1 subject. (iii) 540 students passed in at least 1 subject. (iv) 0 students failed in only 1 subject.