The present population of a country estimated to be 10 crores is expected to increase to 13.31 crores during the next three years. The uniform rate of growth is

### Concept & Formula This problem requires finding the unknown uniform annual growth rate when the initial value, final value, and time period are given. This is a direct application of the compound interest formula: $$ A = P \left(1 + \frac{R}{100}\right)^n $$ Where $A$ is the final population, $P$ is the initial population, $R$ is the rate of growth, and $n$ is the number of years. ### Step-by-Step Solution * **Given:** Initial Population ($P$) = $10\text{ crores}$ Final Population ($A$) = $13.31\text{ crores}$ Time ($n$) = $3\text{ years}$ * **Set up the equation:** $$ 13.31 = 10 \times \left(1 + \frac{R}{100}\right)^3 $$ * **Isolate the growth factor:** $$ \frac{13.31}{10} = \left(1 + \frac{R}{100}\right)^3 $$ $$ 1.331 = \left(1 + \frac{R}{100}\right)^3 $$ * **Solve for the rate ($R$):** Recognize that $1.331$ is the cube of $1.1$. (Since $11^3 = 1331$, shifting the decimal point gives $1.1^3 = 1.331$). $$ (1.1)^3 = \left(1 + \frac{R}{100}\right)^3 $$ Taking the cube root of both sides: $$ 1.1 = 1 + \frac{R}{100} $$ $$ 0.1 = \frac{R}{100} $$ $$ R = 0.1 \times 100 = 10\% $$ ### Exam Strategy & Shortcut **Cube Root Recognition:** In compound growth problems spanning 3 years, the ratio of the Final Value to the Initial Value will almost always form a perfect cube. Ratio = $\frac{13.31}{10} = \frac{1331}{1000}$. Take the cube root immediately: $\sqrt[3]{\frac{1331}{1000}} = \frac{11}{10}$. The fraction $\frac{11}{10}$ means an increase of $\frac{1}{10}$ per year. $\frac{1}{10}$ expressed as a percentage is exactly $10\%$. This avoids all formal algebra. ### Common Pitfall A common mistake is attempting to use the simple interest formula: calculating the total increase ($3.31\text{ crores}$) and dividing it by 3 years to get an average annual increase ($1.103\text{ crores}$), which is roughly $11\%$. Population growth is strictly a compounding phenomenon, not linear. ### Final Answer **Therefore, the correct answer is 10%.**