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

Question

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

Bytelearn · Accepted Answer

Formulate Principle: We need to find the minimum overlap between the two groups of consumers who liked products A and B. This is a classic application of the principle of inclusion-exclusion. The formula for the principle of inclusion-exclusion is: Number liking both A and B = (Number liking A) + (Number liking B) - (Total number surveyed)Plug in Numbers: First, let's plug in the numbers we know into the formula: Number liking both A and B = 720 (liking A) + 450 (liking B) - 1000 (total surveyed)Perform Calculation: Now, let's do the calculation: Number liking both A and B = 720 + 450 - 1000 Number liking both A and B = 1170 - 1000 Number liking both A and B = 170Determine Minimum Number: The least number that must have liked both products is therefore 170. This is because if more than 170 liked both, the total number of consumers would exceed 1000 when adding those who liked only one product.

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

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