Solve the venn diagram question based on the data given. Two sets A & B intersect in such a way that, Ratio of Number of elements in only A to only B = 1:8 Ratio of Number of elements common to both A & B to only B = 1:4. Number of elements that belong to neither A or B is zero. Based on the data, the possible value for number of elements in AUB, can be,
98
140
112
132
Answers
Given : Two sets A & B intersect in such a way that, Ratio of Number of elements in only A to only B = 1:8
Ratio of Number of elements common to both A & B to only B = 1:4.
Number of elements that belong to neither A or B is zero.
To find : the possible value for number of elements in AUB
Solution:
Only A = n(A) - n(A ∩ B)
only B = n(B) - n(A ∩ B)
Ratio of Number of elements in only A to only B = 1:8
=> ( n(A) - n(A ∩ B)) / (n(B) - n(A ∩ B)) = 1/8
=> 8 n(A) - 8 n(A ∩ B) = n(B) - n(A ∩ B)
=> 8 n(A) = n(B) + 7n(A ∩ B)
n(A ∩ B) / ( n(B) - n(A ∩ B) ) = 1/4
=> 4 n(A ∩ B) = n(B) - n(A ∩ B)
=> n(B) = 5n(A ∩ B)
8 n(A) = n(B) + 7n(A ∩ B)
=> 8 n(A) = 5n(A ∩ B) + 7n(A ∩ B)
=> n(A) = 1.5n(A ∩ B)
n(AUB) = n(A) + n(B) - n(A ∩ B)
=> n(AUB)= 1.5n(A ∩ B) +5n(A ∩ B) - n(A ∩ B)
=> n(AUB) = 5.5n(A ∩ B)
132/5.5 = 24 given only integral value
Rest 98/5.5 , 140/5.5 and 112/5.5 Does not give integral value
Hence 132 is , the possible value for number of elements in AUB
n(A ∩ B) = 24
n(A) = 36 only A = 12
n(B) = 120 only B = 96
132 is the correct answer.
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Solution :-
given that,
→ only A : only B = 1 : 8
→ A∩B : only B = 1 : 4 = 2 : 8
so,
→ only A : A∩B : only B = 1 : 2 : 8
then,
→ |AUB| = |onlyA| + |A∩B| + |onlyB|
Now, let x = |onlyA|
so,
→ |AUB| = x + 2x + 8x = 11x
since , x is a natural number , |A∪B| must be a multiple of 11 .
checking options we get,
→ 132/11 = 12 quotient and 0 remainder .
hence, the possible value for number of elements in AUB is 132 .
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