Question

A survey is done among 200 people. 60% like tea and 72% like coffee. Let \(x\) be the number who like both. Let \(m\le x\le n\). Find the correct option.

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

Hint:

For two-set survey problems: \[ \text{Minimum intersection}=n(A)+n(B)-N \] and \[ \text{Maximum intersection}=\min(n(A),n(B)). \]

Answer 1

The Correct Option is A

Concept: For two sets, \[ n(A\cup B)=n(A)+n(B)-n(A\cap B) \] The intersection cannot exceed the smaller set and cannot be less than the excess over total population.

Step 1: Find numbers liking tea and coffee.
Tea: \[ 60\%\times200=120 \] Coffee: \[ 72\%\times200=144 \]

Step 2: Find minimum value of \(x\).
\[ 120+144-200=64 \] Thus \[ x\ge64 \] \[ m=64 \]

Step 3: Find maximum value of \(x\).
Maximum intersection equals smaller set. \[ x\le120 \] \[ n=120 \]

Step 4: Compute \(n-m\).
\[ 120-64=56 \] \[ \boxed{56} \] Hence option (A).