Out of all the patients in a hospital 89% are found to be suffering from heart ailment and 98% are suffering from lungs infection. If K% of them are suffering from both ailments, then K can not belong to the set :

Solution

Maths sir · 3 steps

Pin down the smallest and largest possible overlap, then find the set with nothing in that window.

1. Largest overlap. The overlap cannot be bigger than the smaller group. At most all the heart patients also have the lung infection, so K ≤ 89.
2. Smallest overlap. Take the whole hospital as 100%. The union of the two groups can be at most 100%, so the overlap must make up the excess.

   K = n(H) + n(L) - n(H ∪ L) ≥ 89 + 98 - 100 = 87

   Option	Values inside 87 ≤ K ≤ 89	Can K belong?
   (a) {80, 83, 86, 89}	89	yes
   (b) {84, 86, 88, 90}	88	yes
   (c) {79, 81, 83, 85}	none	no
   (d) {84, 87, 90, 93}	87	yes

3. So. K must lie between 87 and 89. Every element of set (c) is below 87, so K can never be a member of it. The answer is (c).

The idea to keep. For two sets inside a universe, the overlap lies between n(A)+n(B)-n(U) and the smaller of n(A), n(B). Find that window first, then test the options.

Answer from the NTA final answer key. Solution by PrepWiser.