The price of a commodity which was ₹ 250 three years ago is ₹ 2000 now. The annual rate of increase in the price is

### Concept & Formula To find the uniform annual percentage rate of increase over multiple years, we use the compound growth formula. $$ \text{Final Price} = \text{Initial Price} \times \left(1 + \frac{R}{100}\right)^n $$ By setting up the ratio of the final price to the initial price, we can extract the rate by taking the $n$-th root. ### Step-by-Step Solution * **Given:** Initial Price ($P_{initial}$) = $₹ 250$ Final Price ($P_{final}$) = $₹ 2000$ Time period ($n$) = $3\text{ years}$ * **Set up the compound growth equation:** $$ 2000 = 250 \times \left(1 + \frac{R}{100}\right)^3 $$ * **Isolate the growth multiplier:** $$ \frac{2000}{250} = \left(1 + \frac{R}{100}\right)^3 $$ $$ 8 = \left(1 + \frac{R}{100}\right)^3 $$ * **Solve for the annual rate ($R$):** We need to find the cube root of $8$. We know that $2^3 = 8$. $$ 2^3 = \left(1 + \frac{R}{100}\right)^3 $$ Taking the cube root of both sides: $$ 2 = 1 + \frac{R}{100} $$ $$ 2 - 1 = \frac{R}{100} $$ $$ 1 = \frac{R}{100} $$ $$ R = 100\% $$ ### Exam Strategy & Shortcut **The Multiplier Root Method:** Find the overall multiplier instantly: $2000 \div 250 = 8$. The price became $8$ times its original value over $3$ years. For $n = 3$, take the cube root: $\sqrt[3]{8} = 2$. A multiplier of $2$ means the price doubles every year. When a value doubles every year, the rate of increase is exactly $100\%$. This calculation can be done entirely in your head in 5 seconds. ### Common Pitfall Students often calculate the total percentage increase and erroneously divide it by the number of years. Total increase = $2000 - 250 = 1750$. Percentage increase = $(\frac{1750}{250}) \times 100 = 700\%$. Dividing $700\%$ by 3 yields $233.33\%$, which is mathematically incorrect because it ignores the compounding effect of the growing base price each year. ### Final Answer **Therefore, the correct answer is 100%.**