How long will it take a sum of money invested at 5% p.a. S.I. to increase its value by 40%?

### Concept & Percentage Increase in Simple Interest When a sum increases its value by a certain percentage under simple interest, that increase is entirely due to the simple interest earned. $$SI = \text{Percentage Increase} \times P$$ $$T = \frac{SI \times 100}{P \times R}$$ ### Step-by-Step Solution * **Given:** The sum increases by $40\%$, meaning $SI = 40\%$ of $P = 0.4P$. Rate ($R$) = $5\%$ p.a. * **Calculation:** Let the principal be $P$. * Using the standard formula: $SI = \frac{P \times R \times T}{100}$ * Substitute the knowns: $0.4P = \frac{P \times 5 \times T}{100}$ * Divide both sides by $P$: $0.4 = \frac{5T}{100}$ * Multiply by 100: $40 = 5T$ * Solve for $T$: $T = \frac{40}{5} = 8$ years. ### Exam Strategy & Shortcut If the principal increases by $40\%$, and it grows by $5\%$ every single year, then the number of years is simply the total percentage increase divided by the annual percentage rate: $$T = \frac{40\%}{5\%} = 8 \text{ years}$$ ### Common Pitfall Students sometimes confuse "increase its value by 40%" with "becomes 40% of itself", setting the Amount to $0.4P$ instead of setting $SI$ to $0.4P$. ### Final Answer Therefore, the correct answer is **8 years**.