How many students appeared in at least two exams?
Total students = 1000 Let, students appear in exam Z only = a Total students appeared in exam Y = 360 Ratio of number of students appeared in exam X and Y only to students appeared in exam Y and Z only = 2:3 Students appeared in exam X and Z both = a/2 Number of students appeared in all three exams = (4/100) × 1000 = 40 Number of students appeared in Y exam only = No. of students appeared in Y and Z only = 3x Number of students appeared in exam X and Y only = (2/3) × 3x = 2x Now, 2x + 3x + 3x + 40 = 360 ⇒ x = 40 And, 12.5x + a + a/2 + a = 1000 => 5a/2 = 500 ⇒ a = 200 
Students appeared in at least two exams = 80 + 120 + 100 + 40 = 340
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