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Sets and Relations question

2020 · 4 Sep · Shift 1 · Q22
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Sets and Relations question

2020 · 4 Sep · Shift 1 · Q22

JEE MainMathematicsSets and RelationsMCQ+4 / −1
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. A
    37
  2. B
    65
  3. C
    29
  4. D
    55
View written solutionFree

Correct answer: D

  1. Let AAA be the set of people who read newspaper A, and BBB be the set of people who read newspaper B.

    Given: n(A)=63%,n(A)=63\%,n(A)=63%, n(B)=76%,n(B)=76\%,n(B)=76%, and n(A∩B)=x%.n(A\cap B)=x\%.n(A∩B)=x%.

  2. We use the basic set relation: n(A∪B)=n(A)+n(B)−n(A∩B).n(A\cup B)=n(A)+n(B)-n(A\cap B).n(A∪B)=n(A)+n(B)−n(A∩B).

    Since n(A∪B)n(A\cup B)n(A∪B) cannot exceed 100%100\%100%, 63+76−x≤100.63+76-x \le 100.63+76−x≤100.

    So, 139−x≤100139-x \le 100139−x≤100 x≥39.x \ge 39.x≥39.

  3. Also, the number of people reading both newspapers cannot exceed the number reading either one individually. Hence, x≤min⁡(63,76)=63.x \le \min(63,76)=63.x≤min(63,76)=63.

  4. Therefore, the possible range of xxx is: 39≤x≤63.39 \le x \le 63.39≤x≤63.

  5. Now check the options:

    • A: 373737 37<3937<3937<39 Not possible.

    • B: 656565 65>6365>6365>63 Not possible.

    • C: 292929 29<3929<3929<39 Not possible.

    • D: 555555 39≤55≤6339 \le 55 \le 6339≤55≤63 Possible.

  6. Hence, the possible value is: 55\boxed{55}55​

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