$\text{ A natural number is chosen at random from the first }100\text{ natural numbers. The probability that }x+\frac{100}{x}>50\text{ is }$

We have,

$\begin{array}{l}\mathrm{x}+\frac{100}{\mathrm{x}}>50\\ \mathrm{or} {\mathrm{x}}^2+100>50\mathrm{x}\\ \mathrm{or} (\mathrm{x}-25{)}^2>525\\ \Rightarrow \mathrm{x}-25<\sqrt{525}\text{ or }\mathrm{x}-25>\sqrt{525}\\ \Rightarrow \mathrm{x}<25-\sqrt{525}\text{ or }25+\sqrt{525}\end{array}$

$\text{ As }x\text{ is a positive integer and }\sqrt{525}=22.91,\text{ we must have }$

$x\le 2\text{ or }x\ge 48\text{. Thus, the favorable number of cases is }2+53==55$

$\text{ Hence, the required probability is }55/100=11/20\text{. }$