Two numbers are respectively $12 \frac{1}{2}\%$ and 25% more than a third number. The first number as a percentage of the second number is

### Concept & Strategy Like other successive comparison problems, defining the common reference point (the third number) as 100 simplifies the math. The goal is to find one number as a direct percentage of another, rather than finding a percentage increase/decrease. $$ \text{Percentage} = \frac{\text{Target Value}}{\text{Reference Value}} \times 100 $$ ### Step-by-Step Solution * **Given:** Number 1 is $12 \frac{1}{2}\%$ more than Number 3. Number 2 is 25% more than Number 3. * Let the third number be 100. * The first number = $100 + 12.5 = 112.5$. * The second number = $100 + 25 = 125$. * The question asks for the first number *as a percentage of* the second number. * Required Percentage = $\frac{112.5}{125} \times 100$. * To remove decimals, multiply numerator and denominator by 10: $\frac{1125}{1250} \times 100$. * Simplify: $\frac{1125}{1250} = \frac{9}{10}$. * $\frac{9}{10} \times 100 = 9 \times 10 = 90$. ### Exam Strategy & Shortcut Recognize the fractional equivalents: $12.5\%$ is $\frac{1}{8}$ and $25\%$ is $\frac{1}{4}$ (or $\frac{2}{8}$). Let the third number be 8 to avoid decimals entirely. First number = $8 + 1 = 9$. Second number = $8 + 2 = 10$. First as a % of second = $\frac{9}{10} = 90\%$. This ratio method is significantly faster. ### Common Pitfall Students frequently misread "first number as a percentage of the second" and mistakenly calculate the *difference* between the two numbers before dividing, solving for how much *less* the first is than the second. Read the final prompt carefully. ### Final Answer **Therefore, the correct answer is 90.**