The value of $$\frac{1 + \sqrt{0.01}}{1 - \sqrt{0.1}}$$ is close to

### Concept & Formula Evaluate the individual square roots first. Identify which roots are rational (perfect squares with even decimal places) and which require approximation (odd decimal places). ### Step-by-Step Solution Evaluate the square root in the numerator: $\sqrt{0.01}$ has two decimal places and $1$ is a perfect square, so it is rational. $$\sqrt{0.01} = 0.1$$ Evaluate the square root in the denominator: $\sqrt{0.1}$ has one decimal place (odd), so it is irrational. $$\sqrt{0.1} = \sqrt{\frac{10}{100}} = \frac{\sqrt{10}}{10}$$ Knowing $\sqrt{10} \approx 3.162$: $$\frac{3.162}{10} = 0.3162$$ Substitute these values back into the main expression: $$\frac{1 + 0.1}{1 - 0.3162}$$ $$\frac{1.1}{0.6838}$$ Perform the final division: $$\frac{1.1}{0.6838} \approx 1.608$$ The value is closest to $1.6$. ### Exam Strategy & Shortcut Memorize $\sqrt{10} \approx 3.16$. This makes $\sqrt{0.1} = 0.316$. The expression becomes $\frac{1.1}{1 - 0.316} = \frac{1.1}{0.684}$. Notice that $0.684 \times 1.5 = 1.026$, and $0.684 \times 1.6 = 1.0944$. Since $1.0944$ is remarkably close to $1.1$, the quotient is roughly $1.6$. ### Common Pitfall Assuming $\sqrt{0.1} = 0.01$ or some other erroneous value due to misinterpreting decimal rules. Always remember that $\sqrt{0.1}$ is equivalent to $\sqrt{10}/10$. ### Final Answer **Therefore, the correct answer is 1.6.**