A survey shows that 63% of the Indians like Paneer whereas 76% like Tofu. If x% of the Indians like both Paneer and Tofu, then which of the following is/are correct?
A
Maximum value of x is 63.
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B
Maximum value of x is 76.
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C
Minimum value of x is 39.
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D
Minimum value of x is 0.
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Solution
The correct options are A Maximum value of x is 63. C Minimum value of x is 39. Let X denote the set of Indians who like Paneer and let Y denote the set of Indians who like Tofu. Let the population of India be 100. Then, n(A) = 63, n(B) = 76. We know that, n(A∪B)=n(A)+n(B)−n(A∩B)⇒n(A∪B)=63+76−n(A∩B)⇒n(A∪B)=139−xSincethetotalnumberofIndianswholikebothPaneerandTofuwillbelessthanorequalto100.∴n(A∪B)≤100⇒139−x≤100⇒x≥39.Theminimumvalueofxis39.Weknowthat,A∩B⊂AandA∩B⊂B∴n(A∩B)≤n(A)andn(A∩B)≤n(B)⇒n(A∩B)≤63andn(A∩B)≤76⇒n(A∩B)≤63⇒x≤63Maximumvalueofxis63