The value of $$ \frac{3.157 \times 4126 \times 3.198}{63.972 \times 2835.121} $$ is closest to

### Strategy & Logic When a question asks what a complex decimal expression is "closest to," you are not expected to calculate the exact value. Instead, use the **Rounding and Approximation Strategy**. Round each complex number to a nearby integer or simple fraction that is easy to manipulate. ### Step-by-Step Solution **Given:** $$ \frac{3.157 \times 4126 \times 3.198}{63.972 \times 2835.121} $$ **Deduction (Approximation):** * Round the numerator terms: $3.157 \approx 3.15$ $4126 \approx 4000$ (or keep as 4100 for slightly better accuracy) $3.198 \approx 3.2$ * Round the denominator terms: $63.972 \approx 64$ $2835.121 \approx 2800$ * Rewrite the expression using these approximated values: $$ \frac{3.15 \times 4100 \times 3.2}{64 \times 2800} $$ * Group the compatible terms to simplify: 1. Notice that $3.2$ and $64$ are related. $3.2 / 64 = 1 / 20$. $$ \frac{3.15 \times 4100}{20 \times 2800} $$ * Simplify the large numbers: $4100 / 2800 \approx 41 / 28 \approx 1.46$. Alternatively, estimate the remaining multiplication: Numerator: $3.15 \times 4100 \approx 13000$ Denominator: $20 \times 2800 = 56000$ * Find the final approximate ratio: $$ \frac{13000}{56000} = \frac{13}{56} $$ * Since $13 \times 4 = 52$, the fraction $13/56$ is slightly less than $1/4$ ($0.25$). * Looking at the options ($0.002, 0.02, 0.2, 2$), the value $13/56$ (which is roughly $0.23$) is overwhelmingly closest to $0.2$. ### Exam Strategy & Shortcut **Extreme Rounding:** When the options differ by factors of $10$ (i.e., $0.002, 0.02, 0.2, 2$), you can round very aggressively because you only need to determine the correct order of magnitude. Numerator: $\sim 3 \times 4000 \times 3 = 36,000$ Denominator: $\sim 60 \times 3000 = 180,000$ Ratio: $36,000 / 180,000 = 36 / 180 = 1 / 5 = 0.2$. This mental check takes 10 seconds and guarantees the correct magnitude. ### Common Pitfall Attempting to precisely calculate the product of the decimals. The phrase "closest to" is the examiner's signal to drop precision and focus on magnitude. Students who try to calculate the exact numerator waste minutes and often make alignment errors. ### Final Answer **Therefore, the correct answer is 0.2.**