Question

In a beauty contest, half the number of experts voted for Miss A and two thirds voted for Miss B, 10 voted for both and 6 did not vote for either. Then how many experts were there in all?
(A). 20
(B). 24
(C). 22
(D). 26

Answer 1

Hint: To find the total number of experts in the contest, we can form an equation from the given information and then we can assign N as the total number of experts. This gives an equation in one variable which when solved gets the total number of experts in the beauty contest.

Complete step-by-step answer:
Given half the number of experts voted for Miss A
And two thirds voted for Miss B
And 10 voted for both Miss A and Miss B.
Let the total number of experts be N.
E is the set of experts who voted for Miss A.
F is the set of experts who voted for Miss B.
Since 6 did not vote for either, ${\text{n}}\left( {{\text{E}} \cup {\text{F}}} \right)$=N-6.
So from the above information given
Half the number of experts voted for Miss A,${\text{n}}\left( {\text{E}} \right) = \dfrac{{\text{N}}}{2}$
Two thirds number of experts voted for Miss B, ${\text{n}}\left( {\text{F}} \right) = \dfrac{2}{3}{\text{N}}$
And 10 voted for both Miss A and Miss B that is, ${\text{n}}\left( {{\text{E}} \cap {\text{F}}} \right)$=10
The intersection of two sets A and B, denoted by ${\rm A} \cap {\rm B}$, is the set containing all elements of A that also belong to B (or equivalently, all the elements of B that also belongs to A).
So from the formula of sets, If there are two sets P and Q, ${\text{n}}\left( {{\text{P}} \cap {\text{Q}}} \right)$represents the number of elements present in one of the sets P and Q. ${\text{n}}\left( {{\text{P}} \cup {\text{Q}}} \right)$represents the number of elements present in both the sets P & Q.
${\text{n}}\left( {{\text{P}} \cup {\text{Q}}} \right) = {\text{n}}\left( {\text{P}} \right){\text{ + n}}\left( {\text{Q}} \right){\text{ - n}}\left( {{\text{P}} \cap {\text{Q}}} \right)$
Therefore, from the above values
$ \Rightarrow {\text{n}}\left( {{\rm E} \cup {\text{F}}} \right) = {\text{n}}\left( {\text{E}} \right){\text{ + n}}\left( {\text{F}} \right){\text{ - n}}\left( {{\rm E} \cap {\text{F}}} \right)$
So, $${\text{N - 6 = }}\dfrac{{\text{N}}}{2} + \dfrac{2}{3}{\text{N - 10}}$$
$$ \Rightarrow {\text{N - }}\dfrac{{\text{N}}}{2} - \dfrac{2}{3}{\text{N = - 10 + 6}}$$
$$ \Rightarrow \dfrac{{{\text{6N - 3N - 4N}}}}{6}{\text{ = - 4}}$$
Solving the above equation gives$\dfrac{{\text{N}}}{6} = 4 \Rightarrow {\text{N = 24}}$
Therefore, total number of experts = 24.
Hence option B is the correct answer.

Note: If we have three sets P, Q, R then to solve we have${\text{n}}\left( {{\text{P}} \cup {\text{Q}} \cup {\text{R}}} \right) = {\text{n}}\left( {\text{P}} \right){\text{ + n}}\left( {\text{Q}} \right){\text{ + n(R)}} - {\text{ n}}\left( {{\text{P}} \cap {\text{Q}}} \right) - {\text{ n}}\left( {{\text{Q}} \cap {\text{R}}} \right) - {\text{ n}}\left( {{\text{R}} \cap {\rm P}} \right) + {\text{n}}\left( {{\text{P}} \cap {\text{Q}} \cap {\text{R}}} \right)$. Apart from these union and intersection of sets, we have Complement of sets and Cartesian product of sets.