A sum of money invested under compound interest, compounded annually, doubles itself in 5 years. At the same annual compound interest rate, the number of years it will take to become 8 times its initial value is:

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### 1. Strategy & Theoretical Principles Under annual compounding, the accumulated amount $A$ after $t$ years with principal $P$ and annual interest rate $r$ is: $$A(t) = P (1 + r)^t$$ If the sum doubles in $t_1 = 5$ years: $$(1 + r)^5 = 2$$ ### 2. Analytical Formulation To find the time $t_2$ required to grow to 8 times the principal ($A(t_2) = 8 P$): $$(1 + r)^{t_2} = 8 = 2^3$$ ### 3. Step-by-Step Calculation Substitute $(1 + r)^5 = 2$ into $2^3$: $$(1 + r)^{t_2} = \left((1 + r)^5\right)^3 = (1 + r)^{15}$$ Equating exponents directly: $$t_2 = 15\text{ years}$$ ### 4. Distractor Forensics - Candidates frequently confuse compound interest with simple interest (where doubling in 5 years gives $r = 20\%$, requiring $(8-1)/0.20 = 35\text{ years}$). Under compounding, exponential powers add linearly across periods ($2^k \implies k \times 5 = 15$). ### 5. Exam Trap & Standard Reference - **Trap Alert:** Distinguish exponential growth in compound interest from linear growth in simple interest. - **Standard Reference:** GATE General Aptitude Official Syllabus - Quantitative Reasoning.

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