Question

A survey is done among a population of 200 people who like either tea or coffee. It is found that 60% of the population like tea and 72% of the population like coffee. Let x be the number of people who like both tea & coffee. Let \( m \leq x \leq n \), then choose the correct option.

• A. \( n - m = 56 \)
• B. \( n - m = 28 \)
• C. \( n - m = 32 \)
• D. \( n + m = 92 \)

Hint:

For two sets A and B within a universal set U: \[\begin{array}{rl} \bullet & \text{The maximum size of the intersection is \( \min(N(A), N(B)) \).} \\ \bullet & \text{The minimum size of the intersection is \( \max(0, N(A) + N(B) - N(U)) \).} \\ \end{array}\] Apply these formulas to quickly find the range of the intersection.

Answer 1

The Correct Option is A

Let T be the set of people who like tea and C be the set of people who like coffee.
Total population = 200.
Number of people who like tea, \( N(T) = 60% \) of 200 = \( 0.60 \times 200 = 120 \).
Number of people who like coffee, \( N(C) = 72% \) of 200 = \( 0.72 \times 200 = 144 \).
The number of people who like both tea and coffee is \( x = N(T \cap C) \). The problem states that everyone likes "either tea or coffee", which implies that \( N(T \cup C) = 200 \).
Using the principle of inclusion-exclusion: \[ N(T \cup C) = N(T) + N(C) - N(T \cap C) \] \[ 200 = 120 + 144 - x \] \[ 200 = 264 - x \] \[ x = 264 - 200 = 64 \] This interpretation implies that \( x \) is a single value, 64. However, the question asks for a range \( m \leq x \leq n \), which suggests the phrase "people who like either tea or coffee" might mean the survey was limited to this group, but not everyone in the population necessarily likes one of them.
Let's re-interpret the problem without assuming \( N(T \cup C) = 200 \).
Let's find the minimum and maximum possible values for the intersection \( x \).
Maximum intersection (\( n \)): The number of people who like both cannot be more than the number of people in the smaller group. \( n = x_{max} = \min(N(T), N(C)) = \min(120, 144) = 120 \).
Minimum intersection (\( m \)): The minimum overlap occurs when the union is maximized. The union cannot be larger than the total population of 200.
From the inclusion-exclusion principle: \[ N(T \cap C) = N(T) + N(C) - N(T \cup C) \] To find the minimum value of \( N(T \cap C) \), we must use the maximum possible value of \( N(T \cup C) \), which is 200.
\[ m = x_{min} = 120 + 144 - 200 = 264 - 200 = 64 \] So, the range for \( x \) is \( 64 \leq x \leq 120 \). This means \( m = 64 \) and \( n = 120 \).
Now we check the options: \[ n - m = 120 - 64 = 56 \] This matches option (A).