Question

Among 200 people, 90 people like tea and 108 like coffee. 46 people like both tea and coffee. How many people like neither tea nor coffee?

• A. 44
• B. 46
• C. 48
• D. 42

Hint:

To solve set problems quickly, add the individual sets ($90 + 108 = 198$), subtract the overlap ($198 - 46 = 152$), and subtract this result from the total population ($200 - 152 = 48$).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
This is a set theory problem that can be solved using the principle of inclusion-exclusion for two sets.
We can define the sets of people who like tea and coffee and calculate the number of people who do not belong to either set.
Key Formula or Approach:
Let:
- \( U \) be the universal set of all surveyed people.
- \( T \) be the set of people who like tea.
- \( C \) be the set of people who like coffee.
We are given the following values:
- Total number of people, \( n(U) = 200 \)
- Number of people who like tea, \( n(T) = 90 \)
- Number of people who like coffee, \( n(C) = 108 \)
- Number of people who like both, \( n(T \cap C) = 46 \)
The formula to find the number of people who like at least one of the two drinks is:
\[ n(T \cup C) = n(T) + n(C) - n(T \cap C) \]
The number of people who like neither is:
\[ n(\text{neither}) = n(U) - n(T \cup C) \]

Step 2: Detailed Explanation:
Let us calculate the values step-by-step:
1. Find the total number of people who like at least one of the drinks:
\[ n(T \cup C) = 90 + 108 - 46 \]
\[ n(T \cup C) = 198 - 46 \]
\[ n(T \cup C) = 152 \]
So, $152$ out of the $200$ people like tea, coffee, or both.
2. Now, find the number of people who like neither tea nor coffee:
\[ n(\text{neither}) = 200 - 152 \]
\[ n(\text{neither}) = 48 \]
Therefore, there are $48$ people who like neither tea nor coffee.

Step 3: Final Answer:
The number of people who like neither tea nor coffee is 48.