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 least number that must have liked both products?

Answer 1

Step 1: Given data:
Total number of consumers surveyed $=1000$
Number of consumers who like product $A=n\left(A\right)=720$
Number of consumers who like product $B=n\left(B\right)=450$

Step 2: Solve for $n(A\cap B)$
$\because n(A\cup B)=n\left(A\right)+\left(B\right)-n(A\cap B)$
For least $n(A\cap B),n(A\cap B)$ must be maximum.
$n(A\cup B)$ is maximum when it is equal to number of consumers surveyed.
So, $1000=720+450-n(A\cap B)$
$\Rightarrow n(A\cap B)=1170-1000$
$\Rightarrow n(A\cap B)=170$

Final answer:
Least number of consumers who liked both products is $170$.