a) Students excelling in all three subjects = 93 - 3x/2
b) Students excelling in two subjects = S ∩ M + S ∩ A + M ∩ A - 2 (subtracting the students excelling in all three subjects)
Let's break down the information given:
Let S = Students excelling in Science
Let M = Students excelling in Mathematics
Let A = Students excelling in Arts
Given:
Total students = 52
S ∩ M = 13 (Science and Mathematics)
S ∩ A = 16 (Science and Arts)
M ∩ A = 12 (Mathematics and Arts)
Students excelling in none = 2
Students excelling in Science only = 2x (Twice as many as Mathematics only)
Students excelling in Mathematics only = x
Students excelling in Arts only = x/6 (1/6th of Mathematics only)
Total students excelling in at least one subject = Total students - Students excelling in none
Using the principle of inclusion-exclusion, the formula for finding the total number of students excelling in at least one subject is:
Total = S + M + A - (S ∩ M) - (S ∩ A) - (M ∩ A) + (S ∩ M ∩ A)
From the given information, let's calculate the number of students excelling in at least one subject:
Total = S + M + A - 13 - 16 - 12 + (students excelling in all three subjects)
52 = S + M + A - 41 + (students excelling in all three subjects)
Students excelling in all three subjects = 52 - (S + M + A) + 41
Now, let's express S, M, and A in terms of x to solve:
S = 2x (Students excelling in Science only)
M = x (Students excelling in Mathematics only)
A = x/6 (Students excelling in Arts only)
Total = 2x + x + x/6 - 13 - 16 - 12 + (students excelling in all three subjects)
52 = 9x/6 - 41 + (students excelling in all three subjects)
52 = 3x/2 - 41 + (students excelling in all three subjects)
3x/2 + (students excelling in all three subjects) = 52 + 41
3x/2 + (students excelling in all three subjects) = 93
Now, solve for students excelling in all three subjects:
(students excelling in all three subjects) = 93 - 3x/2
Substitute this into the equation for the total number of students excelling in at least one subject:
52 = 3x/2 - 41 + 93 - 3x/2
52 = 52
This equation checks out and shows that the values for S, M, and A are consistent. So, the solution is:
a) Students excelling in all three subjects = 93 - 3x/2
b) Students excelling in two subjects = S ∩ M + S ∩ A + M ∩ A - 2 (subtracting the students excelling in all three subjects)