The population of a colony was 3600 three years back. It is 4800 right now. What will be the population three years down the line, if the rate of growth of population has been constant over the years and has been compounding annually?

### Concept & Logic This problem relies on the principle of **Constant Compounding Multipliers**. When a quantity grows at a constant compound rate over equal time intervals, it gets multiplied by a constant factor for each of those intervals. If the population grows from $P_1$ to $P_2$ in $t$ years, the multiplying factor is $\frac{P_2}{P_1}$. Over the next $t$ years, it will multiply by the exact same factor. ### Step-by-step Solution * **Given:** * Population 3 years ago = 3600 * Current population = 4800 * Time interval = 3 years * **Calculation:** First, find the growth multiplying factor over a 3-year period. $$\text{Factor} = \frac{\text{Current Population}}{\text{Past Population}}$$ $$\text{Factor} = \frac{4800}{3600} = \frac{4}{3}$$ * This means every 3 years, the population becomes $\frac{4}{3}$ times its previous value. * To find the population 3 years from now, multiply the current population by this same factor: $$\text{Future Population} = 4800 \times \frac{4}{3}$$ $$\text{Future Population} = 1600 \times 4 = 6400$$ ### Exam Strategy & Shortcut Recognize the symmetry in the time intervals: 3 years back to now, and now to 3 years future. When intervals are equal, the values form a Geometric Progression (G.P.). Let the populations be $A, B, C$. $$\frac{B}{A} = \frac{C}{B} \implies C = \frac{B^2}{A}$$ $$C = \frac{4800 \times 4800}{3600} = 4800 \times \frac{4}{3} = 6400$$ This G.P. shortcut bypasses solving for the actual percentage rate $R$, saving crucial minutes. ### Common Pitfall The most time-consuming mistake is trying to calculate the exact annual percentage rate $R$ by solving $3600\left(1 + \frac{R}{100}\right)^3 = 4800$. Since you need to find the cube root of $\frac{4}{3}$, this calculation becomes messy and completely unnecessary when you realize the time intervals are identical blocks of 3 years. ### Final Answer **Therefore, the correct answer is 6400.**