In a question on Venn diagrams, we try to evaluate the word problem in small chunks without trying to consume the whole question in one go.
When the Venn diagram is being drawn, we try to populate the common regions first followed by the exclusive regions i.e. we proceed outwards from the inner regions.
We also ensure that we don’t ignore the numbers that can be outside the circles but inside the rectangle.
Considering all the above, we can draw a Venn diagram for the given question which looks like the one below:
Attachment:
25th April 2020 - Reply 4 - 1.jpg
The question says that equal number of students enrolled for the English, Math and Science clubs. This essentially means that the total number of people in each of these circles should be the same.
Therefore,
a + 44 = b + 42 which gives us a = b – 2
c + 40 = b + 42 which gives us c = b + 2.
Total number of people inside the circles = 345 – People who opted for none of the clubs = 345 – 43 = 302.
Therefore, a + b + c + 14 + 16 + 14 + 12 = 302 OR a + b + c = 246.
Substituting the values of a and c in the above equation and simplifying, we get b = 82, a = 80 and c = 84.
We do not want the students enrolled in the Science Club, therefore, we consider the regions outside the Science circle (marked in red) i.e. a+16 + b = 80 + 16 + 82 = 178.
Attachment:
25th April 2020 - Reply 4 - 2.jpg
The correct answer option is C.
Hope that helps!
Thanks for the solution, Sir. I understood what you have done. However, I have a doubt about how the question is phrased - "BOTH English and Maths but not Science. Should the answer then not be 16? That is the only area where students are enrolled in both but not in the third subject.