The difference between the amount the amount of compound interest and simple interest accrued on an amount of ₹ 26000 at the end of 3 years is ₹ 2994.134. What is the rate of interest p.c.p.a.? (Bank P.O., 2005)

### Concept & Formula The difference between Compound Interest (CI) and Simple Interest (SI) for 3 years on a principal $P$ at a rate $R\%$ per annum is given by the formula: $$D_3 = P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right)$$ ### Step-by-Step Solution 1. **Extract Given Values:** Principal ($P$) = ₹ 26000 Time ($T$) = 3 years Difference ($D_3$) = ₹ 2994.134 Let $r = \frac{R}{100}$ 2. **Set up the Equation:** $$2994.134 = 26000 \times r^2 \times (3 + r)$$ $$r^2(3 + r) = \frac{2994.134}{26000}$$ $$r^2(3 + r) = 0.115159$$ 3. **Test the Given Options:** Instead of solving the cubic equation, substitute the options into $r = \frac{R}{100}$: * Option (a) 17%: $r = 0.17 \implies (0.17)^2 \times (3 + 0.17) = 0.0289 \times 3.17 = 0.091613$ (Incorrect) * Option (b) 19%: $r = 0.19 \implies (0.19)^2 \times (3 + 0.19) = 0.0361 \times 3.19 = 0.115159$ (Correct) * Option (c) 22%: $r = 0.22 \implies (0.22)^2 \times (3 + 0.22) = 0.0484 \times 3.22 = 0.155848$ (Incorrect) ### Exam Strategy & Shortcut Solving cubic equations in a competitive exam is too time-consuming. Always rely on option checking. Look at the unit digits or the approximate scale to quickly narrow down the choices. Since $10\%$ yields roughly $0.03$, and $20\%$ yields $0.128$, the answer must be very close to $20\%$. Thus, $19\%$ is the immediate logical candidate to test first. ### Common Pitfall Attempting to algebraically solve the cubic polynomial $r^3 + 3r^2 - 0.115159 = 0$ manually, which leads to massive time loss. ### Final Answer Therefore, the correct answer is **19%**.