If $x$ is 25% less than $y$, then what percent is $y$ more than $x$?

### Concept & Formula To find out how much one value is more or less than another in percentage terms, we use the relative percentage formula. When comparing back to a reduced value, the new base becomes the smaller number. $$ \text{Required \%} = \left( \frac{\text{Difference}}{\text{Reference Value}} \right) \times 100 $$ ### Step-by-Step Solution * **Given:** $x$ is 25% less than $y$. * Let the value of $y$ be 100 to simplify the calculation. * Since $x$ is 25% less, $x = 100 - 25 = 75$. * The question asks: what percent is $y$ more than $x$? * First, find the absolute difference: $y - x = 100 - 75 = 25$. * Here, the reference base is $x$ (because of the phrase "than $x$"). * Substitute into the formula: $\frac{25}{75} \times 100$. * Simplify the fraction: $\frac{1}{3} \times 100 = 33 \frac{1}{3}\%$. ### Exam Strategy & Shortcut Use the fraction rule: If $A$ is $\frac{1}{x}$ less than $B$, then $B$ is $\frac{1}{x-1}$ more than $A$. Here, 25% is $\frac{1}{4}$. So $x$ is $\frac{1}{4}$ less than $y$. Therefore, $y$ is $\frac{1}{4-1} = \frac{1}{3}$ more than $x$. And $\frac{1}{3}$ is $33 \frac{1}{3}\%$. This solves the question in 2 seconds without writing anything. ### Common Pitfall The most common mistake is assuming that if $x$ is 25% less than $y$, then $y$ must simply be 25% more than $x$. Percentages are not symmetrical because the base of comparison changes (from $y$ to $x$). Always identify your new denominator. ### Final Answer **Therefore, the correct answer is $33 \frac{1}{3}\%$.**