If $X$ is 90% of $Y$, then what percent of $X$ is $Y$?

### Concept & Formula To find what percentage one value is of another, you divide the target value by the base value (the value following the word "of") and multiply by 100. $$ \text{Required Percentage} = \left( \frac{\text{Target}}{\text{Base}} \right) \times 100 $$ Here, we need to express $Y$ as a percentage of $X$. So, $X$ is the base. ### Step-by-Step Solution * Given: $X = 90\%$ of $Y$. * This can be written algebraically as: $X = \frac{90}{100} \times Y$ or $X = \frac{9}{10}Y$. * To find what percent $Y$ is of $X$, we need to find the value of $\frac{Y}{X} \times 100$. * From the given equation, rearrange to solve for $\frac{Y}{X}$: $$ \frac{Y}{X} = \frac{10}{9} $$ * Now, convert this fraction to a percentage: $$ \text{Percentage} = \left( \frac{10}{9} \right) \times 100 = \frac{1000}{9}\% $$ * Convert the improper fraction to a mixed number: $$ \frac{1000}{9} = 111 \text{ with a remainder of } 1 $$ * This equals $111\frac{1}{9}\%$. ### Exam Strategy & Shortcut **Assume a Number:** The fastest way to solve variable-based percentage problems is to assume a convenient value for the base variable. Let $Y = 100$. Since $X$ is $90\%$ of $Y$, $X = 90$. The question asks: What percent of $X$ ($90$) is $Y$ ($100$)? $\frac{100}{90} \times 100 = \frac{1000}{9} = 111\frac{1}{9}\%$. This avoids algebraic manipulation entirely. ### Common Pitfall A very common mistake is assuming that percentage relationships are symmetric. Students often think that if $X$ is $90\%$ of $Y$, then $Y$ must be $100\% + 10\% = 110\%$ of $X$. Percentages are always relative to their base, so changing the base changes the scale completely. ### Final Answer **Therefore, the correct answer is $111\frac{1}{9}$%.**