The price of an article was increased by $r$%. Later the new price was decreased by $r$%. If the latest price was ₹ 1, then the original price was

Concept & Formula When a value undergoes a successive percentage increase of $r\%$ and then a decrease of $r\%$, the final value is always less than the original value. We can determine the initial value by setting up a multiplier equation. The multiplier for an $r\%$ increase is $(1 + \frac{r}{100})$ and for an $r\%$ decrease is $(1 - \frac{r}{100})$. $$Final Value = Original Value \times \left( 1 + \frac{r}{100} \right) \times \left( 1 - \frac{r}{100} \right)$$ Step-by-Step Solution * **Given:** The percentage increase is $r\%$ and the percentage decrease is $r\%$. The final price is ₹ 1. * Let the original price of the article be $x$. * First, the price is increased by $r\%$: New Price = $x \times \left( 1 + \frac{r}{100} \right)$ * Then, this new price is decreased by $r\%$: Latest Price = $x \times \left( 1 + \frac{r}{100} \right) \times \left( 1 - \frac{r}{100} \right)$ * We know from algebra that $(a + b)(a - b) = a^2 - b^2$. Applying this to our multipliers: Latest Price = $x \times \left( 1 - \frac{r^2}{10000} \right)$ * We are given that the Latest Price is ₹ 1: $$x \times \left( \frac{10000 - r^2}{10000} \right) = 1$$ * Solving for $x$ (the original price): $$x = \frac{10000}{10000 - r^2}$$ Exam Strategy & Shortcut Memorize the net effect of an equal percentage increase and decrease: it is always a net decrease of $\frac{r^2}{100}\%$. This means the final value is $\left(100 - \frac{r^2}{100}\right)\%$ of the original. Converting this percentage to a fraction gives $\frac{10000 - r^2}{10000}$. Since the final value is 1, the original value is simply the reciprocal: $\frac{10000}{10000 - r^2}$. Common Pitfall A very common mistake is assuming that an $r\%$ increase followed by an $r\%$ decrease brings the price back to the original value (leading to Option A). Always remember that the decrease is calculated on a larger base value, resulting in a net loss. Final Answer **Therefore, the correct answer is ₹ $\left( \frac{10000}{10000 - r^2} \right)$.**