A number increased by $37 \frac{1}{2}\%$ gives 33. The number is

### Concept & Formula When a number is increased by a certain percentage, you add that percentage to 100% of the original number. Converting complex percentages to their fractional equivalents simplifies the calculation immensely. $$ \text{Final Value} = \text{Original Value} \times \left(1 + \text{Fractional Equivalent}\right) $$ ### Step-by-Step Solution * **Given:** Percentage increase = $37 \frac{1}{2}\%$. Final value = 33. * First, convert $37 \frac{1}{2}\%$ to a standard fraction. * We know that $12.5\% = \frac{1}{8}$. * Since $37.5\% = 3 \times 12.5\%$, the fractional equivalent of $37 \frac{1}{2}\%$ is $\frac{3}{8}$. * Let the original number be $x$. * The number is increased by $\frac{3}{8}$ of itself. $$ \text{New Number} = x + \frac{3}{8}x = \frac{11}{8}x $$ * We are given that the new number is 33. $$ \frac{11}{8}x = 33 $$ * Solve for $x$: $$ x = 33 \times \frac{8}{11} $$ $$ x = 3 \times 8 = 24 $$ ### Exam Strategy & Shortcut Memorize standard fraction-percentage pairs. Recognizing $37.5\% = \frac{3}{8}$ is key. An increase of $\frac{3}{8}$ means the new value represents $11$ parts, while the original represents $8$ parts. $11 \text{ parts} = 33 \implies 1 \text{ part} = 3$. Original number ($8 \text{ parts}$) $= 8 \times 3 = 24$. This can be done mentally in under 5 seconds. ### Common Pitfall A common error is setting the equation as just $\frac{3}{8}x = 33$ (ignoring the words "increased by") and solving for $x = 88$, which severely misinterprets the question. ### Final Answer **Therefore, the correct answer is 24.**