Jee mains pyq math

Question

A survey shows that 63% of the people in a city read newspaper A whereas 76% read newspaper B. If x% of the people read both the newspapers, then a possible value of x can be:

(1) 65

(2) 37

(3) 29

(4) 55

Answer 1

Suppose that there are 100 people are there in the city

Given n(A) = 63, n(B) = 76, n(A ∩ B) = x

The addition theorem on cardinal number of sets

 n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

Substitute the appropriate values 

we get

n(A ∪ B) = n(A) + n(B) − n(A ∩ B)

= 63 + 76 − x

= 139 − x

A ⊆ A ∪ B, B ⊆ A ∪ B

⇒ n(A) ≤ n(A ∪ B), n(B) ≤ n(A ∪ B)

Therefore,

76 ≤ 139 - x ≤ 100

-63 ≤ -x ≤ -39

39 ≤ x ≤ 63

Answer 2

P(A∪B)=P(A)+P(B)−P(A∩B)

Substituting the given values:

100≥63+76−x100 \geq 63 + 76 - x100≥63+76−x 100≥139−x100 \geq 139 - x100≥139−x x≥39x \geq 39x≥39

Thus, xxx must be at least 39. Among the given options, 37 is less than 39, so it is not possible. Now let's check if any of the remaining options fit.

• For x=29x = 29x=29, the equation becomes 100≥139−29100 \geq 139 - 29100≥139−29, which gives 100≥110100 \geq 110100≥110, which is not true.
• For x=37x = 37x=37, similarly, the result would not satisfy the inequality.
• For x=55x = 55x=55, 100≥139−55100 \geq 139 - 55100≥139−55, this gives 100≥84100 \geq 84100≥84, which is valid.

Therefore, the possible value of xxx is 55. The correct answer is (4).