Question

A market research group conducted a survey of 1000 consumers and reported that 720 consumers like product A and 450 consumers like product B. What is the leastnumber that must have liked both products?

Answer 1

TOTAL CONSUMERS=1000
CONSUMERS WHO LIKE A=720
CONSUMERS WHO LIKE B=450
TOTAL CONSUMERS WHO LIKE A & B=720+450=1170
i.e. LEAST NUMBER OF CONSUMERS WHO LIKE BOTH A & B=1170-1000=170

Answer 2

Hey !!

Let U be the set of consumers questioned, S be the set of consumers who liked the product A and T be the set of consumers who like the product B.

Given that,

          n (U) = 1000 , n (S) = 720 , n (T) = 450

So,

      n (S ∪ T) = n (S) + n (T) - n (S ∩ T)

        = 720 + 450 - n (S ∩ T) = 1170 - n (S ∩ T)

Therefore, n (S ∪ T) is maximum when n (S ∩ T) is least. But S ∪ T implies ≤ n (U) = 1000. So, maximum values of n (S ∩ T) is 1000. Thus, the least value of n (S ∩ T) is 170.

          Hence, the least number of consumers who liked both the products is ''170''.

GOOD LUCK !!