Question

An integer x is chosen from the first 50 positive integers. The probability that, $x+\frac{100}{x}>50$, is:

• A. $\frac{1}{25}$
• B. $\frac{2}{25}$
• C. $\frac{1}{10}$
• D. None of these

Answer 1

The correct option is C

$\frac{1}{10}$

$xϵ\{1,2,\dots \dots ,50\}$

$x+\frac{100}{x}>50;\Rightarrow x^2-50x+100>0$

$\Rightarrow x=\frac{50\pm \sqrt{{50}^2-4\left(100\right)}}{2}$

$x=25\pm \frac{1}{2}\sqrt{2500-400};x=25\pm \frac{10\sqrt{21}}{2}$

$x=25\pm 5\sqrt{21}$

$x_1=2.0871$ and $x_2=47.912$

So only for $x=1,2,48,49,50$;

y is greater than 0 or $x+\frac{100}{x}>50$

$\Rightarrow p=\frac{n(1,2,48,49,50)}{50}=\frac{5}{50}=\frac{1}{10}$