Express $\frac{1999}{2111}$ in decimal.

### Concept & Strategy Direct long division of $\frac{1999}{2111}$ under time constraints is inefficient. The ideal strategy is to use **Approximation and Percentage Error Estimation** to narrow down the options efficiently. ### Step-by-Step Solution * **Given:** The fraction $\frac{1999}{2111}$. * **Step 1: Analyze the proximity to 1** Notice how close the numerator is to the denominator: $$2111 - 1999 = 112$$ This can be rewritten as: $$\frac{1999}{2111} = \frac{2111 - 112}{2111} = 1 - \frac{112}{2111}$$ * **Step 2: Approximate the fractional difference** Let's estimate $\frac{112}{2111}$ by rounding the numbers to make mental math easier: $$\frac{112}{2111} \approx \frac{112}{2100} = \frac{16}{300} = \frac{5.33}{100} \approx 0.0533$$ * **Step 3: Subtract from 1** $$1 - 0.0533 = 0.9467$$ Comparing this result with the given choices, $0.946$ is the closest matching value. ### Exam Strategy & Shortcut Use benchmark percentages. $10\%$ of $2111$ is $\approx 211$. $5\%$ of $2111$ is $\approx 105.5$. The difference between our numerator and denominator is $112$, which is just slightly more than $5\%$ of the denominator (around $5.3\%$). Therefore, the value must be around $100\% - 5.3\% = 94.7\%$, or $0.947$. Looking at the choices, option (c) $0.946$ is the only one in the $0.94$ or $0.95$ range. ### Common Pitfall Panicking at the sight of odd four-digit divisions and diving straight into tedious long division calculations. This consumes valuable time and increases the risk of calculation slips. Always rely on structural approximations first. ### Final Answer **Therefore, the correct answer is 0.946.**