$$ \frac{0.0203 \times 2.92}{0.0073 \times 14.5 \times 0.7} = x $$

### Concept & Formula Eliminate decimals by equating the total number of decimal places in the numerator and the denominator. Once converted to integers, rely on prime factorization and spotting multiples to simplify the expression rapidly. ### Step-by-Step Solution **Given:** $$ \frac{0.0203 \times 2.92}{0.0073 \times 14.5 \times 0.7} $$ **Calculation:** * Count the decimal places in the numerator: $0.0203$ (4 places) + $2.92$ (2 places) = $6$ total decimal places. * Count the decimal places in the denominator: $0.0073$ (4 places) + $14.5$ (1 place) + $0.7$ (1 place) = $6$ total decimal places. * The decimal places are perfectly balanced (6 on top, 6 on bottom). Remove the decimals completely. * Rewrite as whole numbers: $$ \frac{203 \times 292}{73 \times 145 \times 7} $$ * Simplify by dividing numbers that are obvious multiples: 1. Divide $203$ by $7$: $$ 203 / 7 = 29 $$ 2. Look at $292$ and $73$. Notice that $70 \times 4 = 280$ and $3 \times 4 = 12$, so: $$ 292 / 73 = 4 $$ * Substitute these simplified values back into the expression: $$ \frac{29 \times 4}{145} $$ * Notice the relationship between $29$ and $145$. Since $30 \times 5 = 150$, then $29 \times 5$ is $145$. $$ \frac{145}{29} = 5 $$ * This leaves $4$ in the numerator and $5$ in the denominator: $$ \frac{4}{5} $$ * Convert the fraction to a decimal: $$ \frac{4}{5} = 0.8 $$ ### Exam Strategy & Shortcut **Estimation and Unit Digits:** After stripping the decimals, you have $(203 \times 292) / (73 \times 145 \times 7)$. If you quickly estimate: $(200 \times 300) / (70 \times 150 \times 7) \approx 60000 / 70000 \approx 6/7$, which is slightly less than 1. Looking at the options, $0.8$ is the only logical choice that is slightly less than 1 (1.45, 2.40, and 3.25 are far too large). ### Common Pitfall A major trap is trying to multiply out the decimals as they are, like $0.0203 \times 2.92 = 0.059276$. This takes immense time and almost guarantees an arithmetic mistake. Always balance and drop decimals first. ### Final Answer **Therefore, the correct answer is 0.8.**