125% of 3060 - 85% of $x$ = 408

### Concept & Logic This is a linear equation problem involving percentages. The core strategy is to simplify the known numerical percentage term first, isolate the algebraic term containing the variable $x$, and then solve for $x$. ### Step-by-Step Solution * **Given:** Equation: $ 125\% \times 3060 - 85\% \times x = 408 $ * **Calculation:** Step 1: Simplify the first term. Convert 125% to a fraction for easier multiplication. $ 125\% = \frac{125}{100} = \frac{5}{4} $ Value = $ \frac{5}{4} \times 3060 = 5 \times 765 = 3825 $ Step 2: Substitute this value back into the equation and isolate the term with $x$. $ 3825 - 0.85x = 408 $ $ 3825 - 408 = 0.85x $ $ 3417 = 0.85x $ Step 3: Solve for $x$. Convert the decimal to a fraction to avoid decimal division errors. $ 0.85 = \frac{85}{100} = \frac{17}{20} $ $ 3417 = \left(\frac{17}{20}\right) \times x $ $ x = \frac{3417 \times 20}{17} $ $ x = 201 \times 20 $ $ x = 4020 $ ### Exam Strategy & Shortcut Use digital root (sum of digits) to quickly check options, or use unit digit analysis. The equation is $ 3417 = 0.85x $. This means $ x = \frac{3417}{0.85} $. Since 3417 ends in 7, and we are dividing by an expression that has a 5 in the denominator structure ($ \frac{17}{20} $ means we divide by 17 and multiply by 20). Multiplying 3417 by 20 gives a number ending in 40. Dividing by 17, the unit digit of the quotient must be 0, and the digit before it must be determined by what times 7 ends in 4. $ 2 \times 7 = 14 $. Thus, the answer must end in 20. Option (d) 4020 is the only plausible choice ending in 20. ### Common Pitfall Students often waste time trying to multiply $ 1.25 \times 3060 $ directly using decimals. Recognizing that 125% is exactly $ \frac{5}{4} $ turns a complex decimal multiplication into a very simple division by 4 followed by multiplication by 5. ### Final Answer **Therefore, the correct answer is 4020.**