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Step-by-step Solution
Step 1: Identify the Sets and Given Percentages
Let the hospital’s entire patient population be represented by a universal set of 100%. Define:
• Set A: Patients suffering from heart ailment (89%)
• Set B: Patients suffering from lungs infection (98%)
We are told that K% of the patients have both ailments, i.e., they are in the intersection of sets A and B.
Step 2: Apply the Principle of Inclusion-Exclusion
The principle of inclusion-exclusion for sets A and B is given by:
$ n(A \cup B) = n(A) + n(B) - n(A \cap B) $
Here, $n(A \cup B)$ represents the percentage of patients in A or B or both. Since all patients are considered, $n(A \cup B) \leq 100\%$.
Rearranging,
$ n(A \cap B) = n(A) + n(B) - n(A \cup B) $
Because $n(A \cup B) \leq 100\%$, we have:
$ n(A \cap B) \geq n(A) + n(B) - 100\% $
Substituting the given percentages:
$ n(A \cap B) \geq 89\% + 98\% - 100\% = 87\% $
Step 3: Determine the Upper Bound
The proportion of patients suffering from both ailments cannot exceed the total percentage of those with either ailment. Thus:
$ n(A \cap B) \leq \min\{n(A), n(B)\} $
Since $n(A) = 89\%$ and $n(B) = 98\%$, the smaller group is $89\%$:
$ n(A \cap B) \leq 89\% $
Step 4: Combine Both Bounds
Combining the results yields:
$ 87\% \leq n(A \cap B) \leq 89\% $
This means that the intersection percentage must lie between 87% and 89% (inclusive).
Step 5: Conclusion
Any value of K% lying outside this range is not possible. In the given options, the set {79, 81, 83, 85} has values all below 87%. Therefore, these values are not possible for the percentage of patients suffering from both ailments.
