Question

In a group of 50 people, 30 like tea, 25 like coffee, and 10 like neither. How many like both tea and coffee?

• A. 15
• B. 5
• C. 10
• D. 20

Hint:

For problems involving overlapping groups, use the inclusion-exclusion formula: $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. Draw a Venn diagram to visualize the sets, and account for those outside the sets (who like neither) to ensure the total matches. Always verify by checking if the numbers are consistent with the total population.

Answer 1

The Correct Option is A

To determine the number of individuals who prefer both tea and coffee, the principle of inclusion-exclusion will be applied.

1. Key Information:

- Total individuals surveyed: 50
- Individuals who prefer tea (T): 30
- Individuals who prefer coffee (C): 25
- Individuals who prefer neither: 10
- The formula for the union of two sets is: \[ |T \cup C| = |T| + |C| - |T \cap C| \] - The count of people who like tea or coffee or both is calculated by subtracting those who like neither from the total number of people.

2. Provided Data:

Total individuals = 50
Individuals liking neither = 10. Therefore, individuals liking tea or coffee or both = 50 - 10 = 40.
Number liking tea (T) = 30
Number liking coffee (C) = 25

3. Calculation of Overlap:

The number of people who like tea or coffee or both is 40. Using the union formula: \[ |T \cup C| = 40 = 30 + 25 - |T \cap C| \] Solving for the number of people who like both: \[ |T \cap C| = 30 + 25 - 40 = 15 \]

Conclusion:

There are 15 individuals who like both tea and coffee.