If n(A)=4x+5, n(B)=3x+8, n(A∩B)=2x+1, and n(A∪B)=72, what is the value of x?
Answer and explanation
Correct answer: 12
The governing Venn-diagram relation is n(A ∪ B) = n(A) + n(B) − n(A ∩ B), since the common region is counted twice in the first two terms. Substitution gives 72 = (4x+5) + (3x+8) − (2x+1). Simplifying, 72 = 5x + 12, so 5x = 60 and x = 12. The values then give valid nonnegative region counts, confirming the result. Hence option A is uniquely correct.
Frequently asked questions
What is the correct answer to this question?
12
Why is this the correct answer?
The governing Venn-diagram relation is n(A ∪ B) = n(A) + n(B) − n(A ∩ B), since the common region is counted twice in the first two terms. Substitution gives 72 = (4x+5) + (3x+8) − (2x+1). Simplifying, 72 = 5x + 12, so 5x = 60 and x = 12. The values then give valid nonnegative region counts, confirming the result. Hence option A is uniquely correct.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Sets. Topic: Venn Diagrams.
