Problem involving three sets
Q. In a city, three daily newspapers A B and C are published. 42% read newspaper A, 51% read nespaper B, 68% read newspaper C, 30% read newspaper A and B, 28% read neapaper B and C, 36% read A and C, 8% read none of them. Find the percent of people who read all the three papers. Represent the information in a venn diagram.
Let U represent the set of people under survey.
Here, n(A)=42% n(B)=51% n(C)=68% n(A⋂B)=30% n(B⋂C)=28% n(A⋂C)=36% n(A∪B∪C)'=8%
Now, n(A∪B∪C)= n(U) - n(A∪B∪C)' =100% - 8% =92%
n(A∪B∪C)= n(A) +n(B) + n(C)- n(B⋂C) - n (A⋂ B)- n (A⋂C) + n(A⋂B⋂C)
or, 92%= 42% +51% +68% -28% -30% -36% + n(A⋂B⋂C) or, 92%= 161% - 94 + n(A⋂B⋂C) or, 92% = 67% + n(A⋂B⋂C) or, 92% - 67%= n(A⋂B⋂C) or, 25% = n(A⋂B⋂C)
Therefore, 25% of the people read all of the three newspapers.