$x$% of 450 + 46% of 285 = 257.1

### Concept & Formula This problem requires evaluating a linear equation involving percentages. First, isolate the unknown percentage term by calculating the known percentage value and subtracting it from the total sum on the right side. $$ \left( \frac{x}{100} \right) \times A + \left( \frac{B}{100} \right) \times C = D $$ ### Step-by-Step Solution * **Given:** Equation: $x$% of 450 + 46% of 285 = 257.1 * **Calculation:** Step 1: Simplify the known term (46% of 285). $ 46\% \text{ of } 285 = 0.46 \times 285 $ $ = 131.1 $ Step 2: Substitute this value back into the original equation. $ x\% \text{ of } 450 + 131.1 = 257.1 $ Step 3: Isolate the term with the unknown $x$. $ x\% \text{ of } 450 = 257.1 - 131.1 $ $ x\% \text{ of } 450 = 126.0 $ Step 4: Solve for $x$. $ \left( \frac{x}{100} \right) \times 450 = 126 $ $ 4.5x = 126 $ $ x = \frac{126}{4.5} = \frac{1260}{45} = 28 $ ### Exam Strategy & Shortcut To quickly find 46% of 285 mentally, split it into 50% minus 4%. 50% of 285 is half of 285, which is 142.5. 1% of 285 is 2.85, so 4% is $ 2.85 \times 4 = 11.4 $. Subtract the two: $ 142.5 - 11.4 = 131.1 $. Then subtract 131.1 from 257.1 to get 126. Finally, to solve $ 4.5x = 126 $, double both sides to remove the decimal: $ 9x = 252 $. Then divide: $ 252 / 9 = 28 $. ### Common Pitfall Calculation errors during the decimal multiplication of $ 0.46 \times 285 $ often lead students off track. Furthermore, forgetting to convert the percentage symbol into a denominator of 100 before isolating $x$ will result in answers off by a factor of 100. ### Final Answer **Therefore, the correct answer is 28.**