Out of all the patients in a hospital 89 % are found to be suffering from heart ailment and 98 % are…

Let $H$ and $L$ is set of people suffered from heart ailment and lungs infections respectively.

$n\left(H\right)=89\%$

$n\left(L\right)=98\%$

$n\left(H\cap L\right)=K\%$

$\max \left\{n\left(H\right), n\left(L\right)\right\}\le n\left(H\cup L\right)\le n\left(U\right)$

Where, $U$ is the universal set.

$\Rightarrow 98\le n\left(H\cup L\right)\le 100$

$\Rightarrow 98\le n\left(H\right)+n\left(L\right)-n\left(H\cap L\right)\le 100$

$\Rightarrow 98\le 89+98-k\le 100$

$\Rightarrow 0\le 89-k\le 2$

$\Rightarrow -89\le -k\le -87$

$\Rightarrow 87\le K\le 89$