$(23.6\%$ of $1254) - (16.6\%$ of $834) = $x$

### Concept & Formula When dealing with heavy multi-digit decimal multiplication in multiple-choice exams, utilizing the Digital Sum (also known as digit root or casting out nines) is the most effective way to identify the correct answer. $$ \text{Digital Sum}(A \times B) = \text{Digital Sum}(A) \times \text{Digital Sum}(B) $$ ### Step-by-Step Solution * **Given:** Evaluate $(23.6\% \text{ of } 1254) - (16.6\% \text{ of } 834)$ * **Calculation:** Let's use the Digital Sum (DS) method. Find the DS of the first part: $23.6 \times 1254$ $$ DS(23.6) = 2 + 3 + 6 = 11 \rightarrow 1 + 1 = 2 $$ $$ DS(1254) = 1 + 2 + 5 + 4 = 12 \rightarrow 1 + 2 = 3 $$ $$ DS(\text{Part } 1) = 2 \times 3 = 6 $$ * Find the DS of the second part: $16.6 \times 834$ $$ DS(16.6) = 1 + 6 + 6 = 13 \rightarrow 1 + 3 = 4 $$ $$ DS(834) = 8 + 3 + 4 = 15 \rightarrow 1 + 5 = 6 $$ $$ DS(\text{Part } 2) = 4 \times 6 = 24 \rightarrow 2 + 4 = 6 $$ * Subtract the digital sums: $$ \text{Total DS} = 6 - 6 = 0 \text{ (which is equivalent to 9)} $$ * Check the Digital Sum of the options: (a) $153.5 \rightarrow 1+5+3+5 = 14 \rightarrow 5$ (b) $155.5 \rightarrow 1+5+5+5 = 16 \rightarrow 7$ (c) $157.5 \rightarrow 1+5+7+5 = 18 \rightarrow 9$ (Matches!) (d) $159.5 \rightarrow 1+5+9+5 = 20 \rightarrow 2$ ### Exam Strategy & Shortcut As demonstrated, calculating the digital sum takes less than 20 seconds and cleanly singles out $157.5$ as the only mathematically viable option, entirely bypassing the horrifying manual calculation of $0.236 \times 1254$ and $0.166 \times 834$. ### Common Pitfall Attempting to solve this using standard multiplication. Even with approximation, $0.24 \times 1250 - 0.17 \times 830 = 300 - 141 = 159$. Approximation leads you to guess $159.5$, which is incorrect. Only digital sum provides exact verification here. ### Final Answer **Therefore, the correct answer is 157.5.**