If $35\%$ of a number is $175$, then what percent of $175$ is that number?

### Concept & Formula This is a two-step percentage problem. First, find the unknown base number using the given percentage. Then, calculate what percentage that new base number is relative to the original given value. The percentage formula is: $$\text{Percentage} = \left( \frac{\text{Part}}{\text{Base}} \right) \times 100$$ ### Step-by-Step Solution **Given:** * $35\%$ of a number $= 175$ **Calculation:** * Let the unknown number be $x$. $$\frac{35}{100} \times x = 175$$ * Solve for $x$: $$x = \frac{175 \times 100}{35}$$ * Simplify (since $35 \times 5 = 175$): $$x = 5 \times 100 = 500$$ * The question asks: What percent of $175$ is $500$? Here, $175$ is the new base. $$\text{Required Percentage} = \left( \frac{500}{175} \right) \times 100$$ * Simplify the fraction by dividing by $25$: $$\text{Required Percentage} = \left( \frac{20}{7} \right) \times 100 = \frac{2000}{7}\%$$ * Convert to a mixed number or decimal: $$\frac{2000}{7}\% \approx 285.71\%$$ ### Exam Strategy & Shortcut You can skip finding the absolute number entirely by using ratios! Let the number be $N$. We know $0.35N = 175$. The question asks for the ratio of $N$ to $175$, expressed as a percentage. Since $175 = 0.35N$, substituting this gives: $$\frac{N}{175} = \frac{N}{0.35N} = \frac{1}{0.35}$$ $$\frac{1}{0.35} = \frac{100}{35} = \frac{20}{7}$$ Convert the fraction $\frac{20}{7}$ to a percentage: $\frac{20}{7} \times 100 \approx 285.7\%$. ### Common Pitfall Students often misread "what percent of $175$ is that number" as "what percent of that number is $175$". If misread, you would just circle $35\%$ (Option a). Always pay close attention to what follows the word "of", as it identifies the base of the percentage. ### Final Answer **Therefore, the correct answer is None of these.**