3 Set Venn Diagram

Let E, H, M be the sets of people speaking English, Hindi and Marathi. Hence n(M)=12 = n(E and M)+2+3+6. Hence, n(E and M)=1. Similarly, n(H)=n(H and M)+13+2+3=20. Hence, n(H and M)=2. As all employees speak at least one of these languages, the sum of all numbers=50. Hence n(E and H' and M')+1+2+3+6+2+13=50. Hence n(E and H' and M')+27=50. Hence the number of employees speaking only English is 23.

Let M, P and C be the sets of teachers of Maths, Physics and Chemistry.
Let x be the % who teach all three subjects. To maximize x, we assume that $$n(M \cap P \cap C') = n(M \cap C \cap P') = n(P \cap C \cap M')=0$$.
Hence, 100%=(n(M)-x)+(n(P)-x)+(n(C)-x)+x= 50%+32%+48%-2x=100%.
Hence x=15%.

Alternate solution:

Let the total number of teachers be 100.

Then, the total number of teachers from the Venn diagram = a + b + c + d + e + f + g = 100 ---(1)

Also given, a + d + e + g = 50 ----(2)

b + e + f + g = 48 ----(3)

c + d + f + g = 32 ---- (4)

Adding (2), (3), and (4) we get a + b + c + 2(d + e + f) + 3g = 130 ---- (5)

Subtracting (1) from (5) we get d + e + f + 2g = 30 ---- (6)

In equation (6) we need to maximize g. g becomes maximum when d + e + f = 0.

Thus, 2g = 3 => g = 15