If $p$% of $p$ is 36, then $p$ is equal to

### Concept & Formula This is a fundamental percentage equation that loops a variable back onto itself, creating a quadratic relationship (a square). By applying the basic definition of a percentage, we can isolate the squared variable and solve by taking the square root. $$ p\% \text{ of } p = \frac{p}{100} \times p $$ ### Step-by-Step Solution * **Given:** * $p$% of $p = 36$ * **Calculation:** Translate the mathematical English into an algebraic equation. Replace "of" with multiplication and "$p$%" with $\frac{p}{100}$: $$ \frac{p}{100} \times p = 36 $$ Multiply the $p$ terms in the numerator: $$ \frac{p^2}{100} = 36 $$ Multiply both sides by 100 to isolate $p^2$: $$ p^2 = 36 \times 100 $$ $$ p^2 = 3600 $$ Take the square root of both sides to solve for $p$. (Since the options are positive and standard aptitude context implies positive values, we take the principal root): $$ p = \sqrt{3600} $$ $$ p = 60 $$ ### Exam Strategy & Shortcut You can solve this purely through mental math by understanding squares. The equation $\frac{p^2}{100} = 36$ means $p^2 = 3600$. You should instantly recognize that $\sqrt{36} = 6$. Because there are two zeros in 3600, its square root will have exactly half that amount of zeros (one zero). Therefore, the answer is just $6$ with a zero attached: $60$. ### Common Pitfall The most common trap is forgetting to multiply by the 100 from the percentage denominator, leading students to solve $p^2 = 36 \implies p = 6$. Always remember that the `%` symbol represents a hidden division by 100 that must be cleared algebraically. ### Final Answer Therefore, the correct answer is **60**.