A number reduced by 25% becomes 225. What percent should it be increased so that it becomes 390?

### Concept & Logic This is a two-step problem. First, find the original base number (100%) by reversing the initial percentage reduction. Second, calculate the percentage increase required to go from the original base number to the new target value. $$ \text{Percentage Increase} = \left( \frac{\text{Final Value} - \text{Original Base}}{\text{Original Base}} \right) \times 100 $$ ### Step-by-Step Solution * **Step 1: Find the original number.** Let the original number be $x$. A reduction of $25\%$ means the number becomes $100\% - 25\% = 75\%$ of itself. $$ 75\% \text{ of } x = 225 $$ $$ \frac{75}{100}x = 225 \implies \frac{3}{4}x = 225 $$ $$ x = 225 \times \frac{4}{3} = 75 \times 4 = 300 $$ The original number is $300$. * **Step 2: Find the required percentage increase.** The new target value is $390$. We need to find the increase from the original number ($300$) to the target ($390$). $$ \text{Absolute Increase} = 390 - 300 = 90 $$ $$ \text{Percentage Increase} = \left( \frac{90}{300} \right) \times 100 $$ $$ \text{Percentage Increase} = \frac{9}{30} \times 100 = 30\% $$ ### Exam Strategy & Shortcut Work purely in ratios or percentages. $75\% \equiv 225$. Divide both sides by $75$: $1\% \equiv 3$. So, $100\% \equiv 300$ (This is your base). The target is $390$, which is $90$ more than $300$. Since $1\% = 3$, then $90$ is exactly $30 \times 3$, meaning it represents $30\%$. ### Common Pitfall The most frequent error is applying the second percentage calculation on the intermediate value ($225$) instead of the original base number ($300$), or mistakenly trying to find the percentage directly from the difference between $390$ and $225$. ### Final Answer **Therefore, the correct answer is 30%.**