The ratio of powers of two motors is $$\frac{3\sqrt{x}}{\sqrt{x}+1}$$, that are capable of raising $$300$$ kg water in $$5$$ minutes and $$50$$ kg water in $$2$$ minutes respectively from a well of $$100$$ m deep. The value of $$x$$ will be

The power of each motor is given by $$P = \frac{mgh}{t}$$. For the first motor, $$P_1 = \frac{300 \times g \times 100}{5 \times 60} = \frac{30000g}{300} = 100g$$ watts, and for the second motor, $$P_2 = \frac{50 \times g \times 100}{2 \times 60} = \frac{5000g}{120} = \frac{125g}{3}$$ watts. Their ratio is $$\frac{P_1}{P_2} = \frac{100g}{\frac{125g}{3}} = \frac{300}{125} = \frac{12}{5}$$.

Setting this equal to the given expression, $$\frac{3\sqrt{x}}{\sqrt{x}+1} = \frac{12}{5}$$. Cross-multiplying gives $$15\sqrt{x} = 12(\sqrt{x}+1) = 12\sqrt{x} + 12$$, so $$3\sqrt{x} = 12$$, which means $$\sqrt{x} = 4$$ and therefore $$x = 16$$.

The correct answer is Option A: $$16$$.