If 20% of $A$ = 50% of $B$, what percentage of $A$ is $B$?

### Concept & Formula This problem requires converting a percentage equality into a direct ratio between two variables, and then expressing one variable as a percentage of the other. To find what percentage one number is of another, we create a fraction of the two numbers and multiply by 100. $$ \text{Required Percentage} = \left( \frac{\text{Part}}{\text{Base}} \right) \times 100 $$ ### Step-by-Step Solution * **Given:** * 20% of $A = 50$% of $B$ * **Calculation:** First, convert the percentages into a mathematical equation: $$ 0.20 \times A = 0.50 \times B $$ Alternatively, using fractions (which is often cleaner): $$ \frac{20}{100} \times A = \frac{50}{100} \times B $$ Multiply both sides by 100 to eliminate the denominators: $$ 20A = 50B $$ $$ 2A = 5B $$ The question asks "what percentage of $A$ is $B$?". This means $A$ is our base. We need to solve for $B$ in terms of $A$: $$ B = \frac{2}{5} A $$ To express this fraction as a percentage, multiply by 100: $$ \text{Percentage} = \left( \frac{2}{5} \right) \times 100\% = 2 \times 20\% = 40\% $$ ### Exam Strategy & Shortcut Use the Ratio Method for instant solving. If $20\% \times A = 50\% \times B$, you can instantly invert the coefficients to find the ratio. $A:B = 50:20$, which simplifies to $A:B = 5:2$. The question asks for $B$ as a percentage of $A$. That is just $\frac{B}{A} \times 100$. Looking at our ratio, $\frac{B}{A} = \frac{2}{5}$. Since $\frac{1}{5} = 20\%$, then $\frac{2}{5} = 40\%$. No equations needed! ### Common Pitfall A very frequent mistake is calculating $A$ as a percentage of $B$ instead of $B$ as a percentage of $A$. Reading "what percentage of $A$ is $B$" carefully is crucial. The word following "of" is always your base (denominator). If you mistakenly calculated $\frac{5}{2}$, you would get 250%, which isn't an option here, but in harder exams, it will be a trap option. ### Final Answer Therefore, the correct answer is **40**.