Raju's factory kept increasing its output by the same percentage every year. Find the percentage if it is known that his output is doubled after two years.

### Concept & Formula This problem is based on the concept of **Compound Growth**, which follows the same mathematical structure as compound interest. If a quantity grows by a constant percentage every year, the final amount is calculated using the formula: $$A = P\left(1 + \frac{R}{100}\right)^n$$ Where $A$ is the final amount, $P$ is the initial amount, $R$ is the rate of increase, and $n$ is the number of years. ### Step-by-step Solution * **Given:** Let the initial output of the factory be $P$. After $n = 2$ years, the final output $A$ becomes $2P$ (since it doubled). * Let the constant percentage increase every year be $R\%$. * Applying the compound growth formula: $$2P = P\left(1 + \frac{R}{100}\right)^2$$ * Divide both sides by $P$: $$2 = \left(1 + \frac{R}{100}\right)^2$$ * Taking the square root on both sides: $$\sqrt{2} = 1 + \frac{R}{100}$$ * Rearranging to solve for $R$: $$\frac{R}{100} = \sqrt{2} - 1$$ * Multiply by 100 to get the percentage: $$R = 100(\sqrt{2} - 1)$$ ### Exam Strategy & Shortcut For problems where a value becomes $k$ times its original value in $n$ years at compound growth, you can directly use the relation: $$1 + \frac{R}{100} = (k)^{\frac{1}{n}}$$ Here, $k = 2$ and $n = 2$, so $1 + \frac{R}{100} = 2^{\frac{1}{2}} = \sqrt{2}$. This saves time writing out the full equation with $P$ and instantly gets you to the final steps. ### Common Pitfall A very common mistake is assuming simple interest instead of compound interest when a "constant percentage" is mentioned without specifying compounding. In real-world growth scenarios like population or factory output, the growth is calculated on the previous year's total, making it compound growth. Using simple interest ($P + P \times R \times \frac{2}{100} = 2P \rightarrow R = 50\%$) leads to a completely wrong answer. ### Final Answer **Therefore, the correct answer is $100(\sqrt{2} - 1)\%$.**