Two numbers are respectively 20% and 25% lower than a third number. By how much percentage is the second number lower than the first?

### Concept & Strategy When multiple values are defined in relation to a single unknown "third number," the most efficient strategy is to assign the third number a convenient base value, typically 100, to bypass complex algebra. $$ \text{Required \%} = \frac{\text{Difference}}{\text{Base Value}} \times 100 $$ ### Step-by-Step Solution * **Given:** Two numbers are 20% and 25% lower than a third number. * Let the third number be exactly 100. * The first number is 20% lower: $100 - 20 = 80$. * The second number is 25% lower: $100 - 25 = 75$. * We must find the percentage by which the *second* number (75) is lower than the *first* number (80). * Absolute difference = $80 - 75 = 5$. * The comparison base is the *first number* (80). * Required Percentage = $\frac{5}{80} \times 100$. * Simplify the fraction: $\frac{1}{16} \times 100 = \frac{100}{16} = \frac{25}{4}$. * Convert to a mixed fraction: $6 \frac{1}{4}\%$. ### Exam Strategy & Shortcut Always use 100 as the reference point for the independent variable in percentage chain problems. It directly translates percentages into hard numbers, making the secondary comparison ($\frac{5}{80}$) immediately obvious. ### Common Pitfall Test-takers often spot the 5 unit difference between 80 and 75, and mistakenly select 5% as the answer because they subconsciously divide by 100 (the original base). The question explicitly shifts the base to the *first number* (80). ### Final Answer **Therefore, the correct answer is $6 \frac{1}{4}\%$.**