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 solve the problem, we need to find the number of people who like both tea and coffee using the principle of inclusion-exclusion.

1. Understanding the Concepts:

- Total People: 50
- Number liking tea (T): 30
- Number liking coffee (C): 25
- Number liking neither: 10
- Use the formula for union: \[ |T \cup C| = |T| + |C| - |T \cap C| \] - People liking tea or coffee or both = total - neither

2. Given Values:

Total people = 50
Neither = 10 → People liking tea or coffee or both = 50 - 10 = 40
Tea (T) = 30
Coffee (C) = 25

3. Calculate Number Liking Both:

\[ |T \cup C| = 40 = 30 + 25 - |T \cap C| \] \[ |T \cap C| = 30 + 25 - 40 = 15 \]

Final Answer:

The number of people who like both tea and coffee is 15.