35568 ÷ $x$% of 650 = 456

### Concept & Formula This problem tests the order of operations combined with percentage calculations. The term "$x$% of 650" acts as the entire denominator for the initial division operation. $$ \frac{A}{\left(\frac{x}{100} \times B\right)} = C $$ ### Step-by-Step Solution * **Given:** Equation: $ 35568 \div (x\% \text{ of } 650) = 456 $ * **Calculation:** Step 1: Rearrange the equation to isolate the percentage term. If $ \frac{35568}{\text{Term}} = 456 $, then $ \text{Term} = \frac{35568}{456} $. $ x\% \text{ of } 650 = \frac{35568}{456} $ Step 2: Perform the division. $ \frac{35568}{456} = 78 $ So, $ x\% \text{ of } 650 = 78 $ Step 3: Solve for $x$. $ \left(\frac{x}{100}\right) \times 650 = 78 $ $ 6.5x = 78 $ $ x = \frac{78}{6.5} $ $ x = \frac{780}{65} $ $ x = 12 $ ### Exam Strategy & Shortcut Instead of heavy division upfront, look at unit digits and relative scale. You know $ 456 \times \text{Something} = 35568 $. The unit digit of 456 is 6, and the target ends in 8. $ 6 \times 3 = 18 $ or $ 6 \times 8 = 48 $. The multiplier likely ends in 8. Since $ 450 \times 80 = 36000 $, the multiplier must be exactly 78. Then solve $ 6.5 \times x = 78 $. $ 6 \times 12 = 72 $ and $ 0.5 \times 12 = 6 $. $ 72 + 6 = 78 $. Thus $ x = 12 $. ### Common Pitfall A major error is misinterpreting the order of operations as $ (35568 \div x\%) \times 650 $. The word "of" binds the percentage and 650 together tightly as a single term before the division occurs. ### Final Answer **Therefore, the correct answer is 12.**