Question:
"In an examination 70 % of the candidates passed in History and 50% in Geography and 20% students failed in both the subjects. If 500 students passed in both the subjects, the how many candidates appeared for the exam?"
Doubt:
I am unable to understand how to frame the parameters that I am supposed to take into consideration in a Venn Diagram. A slight guidance about the same about the thinking process will be most certainly welcome.
Let us define P as the set that contains all the candidates that appeared for the exam. Then, if we define sets H as the candidates that passed history, G the candidates that passed Geography then we know that $|H|=N*.70$, $|G|=N*.50$ and $|H^{c}\cap G^{c}|=.20*N$. Where we define || as the cardinality of a set and $|P|=N.$ That is N is the total number of candidates. We are given:
$$|H\cap G|=500.$$
Les us Consider how we can break down $P$
$$|P|=|H\cap G| + |H\cap G^c| + |H^c \cap G| + |H^{c}\cap G^{c}|.$$
That is the total number of applicants can be divided into the disjoint sets of:
Passed Both Tests, Failed Both Tests, Passed History Failed Geography, Failed History Passed Geography.
We know: $$|H|=|H\cap G|+|H\cap G^c|=.70* N$$ $$|G|=|H\cap G| + |H^{c}\cap G|=.50*N$$ Since $|H\cap G|=500$, this implies $$|H\cap G^{c}|=.70*N-500$$ $$|H^{c}\cap G|=.50*N-500$$.
If we combine everything we get
$$N=500 +.70*N-500 + .50*N-500 + .20*N$$
Solving for N
$$N=1250$$