Vinay sold a watch at Rs. 75. He gets profit percentage numerical value same as the cost price of the watch, then cost price of the watch is:

Let the Cost Price ($\text{CP}$) be Rs. $x$.According to the problem, Profit $\%=x\%$.
$$\text{Selling Price (SP)}=\text{CP}\times \left(1+\frac{\text{Profit }\%}{100}\right)$$
$$75=x\times \left(1+\frac{x}{100}\right)$$
$$75=x+\frac{x^2}{100}$$
$$7500=100x+x^2⟹x^2+100x-7500=0$$
$$(x+150)(x-50)=0$$
Since Cost Price cannot be negative, $x=50$.Note: The official option printed in memory-based papers as 22.5 is a typo; the standard correct answer to this famous problem is Rs. 50.