Question

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

• A. $\frac{1}{10}$
• B. $\frac{11}{20}$
• C. $\frac{1}{2}$
• D. none

Answer 1

Answer: (B)

$\frac{11}{20}$

We have,

$x+\frac{100}{x}>50⟹x^2+100>50x⟹x^2-50x+100>0⟹(x-25{)}^2-525>0$

$⟹x-25>\sqrt{525}$ or $x-25<\sqrt{525}$

$⟹x<25-\sqrt{525}$ or $25+\sqrt{525}$

As $x$ is positive and $\sqrt{525}=22.9$

We must have $x\le 2$ or $x\ge 48$

Thus 2 + 53 = 55 favorable cases.

Probability = $\frac{55}{100}=\frac{11}{20}$