A sum of money at compound interest doubles itself in 15 years. It will become eight times of itself in

### Concept & Exponential Growth of Time Periods If a sum of money becomes $x$ times in $T$ years at compound interest, it will become $x^n$ times in $n \times T$ years. Time progresses linearly, while the amount multiplier progresses exponentially. $$Time = n \times T$$ ### Step-by-Step Solution 1. **Extract the given parameters:** The sum becomes 2 times ($x = 2$) in 15 years ($T = 15$). 2. **Identify the target multiplier:** We need the sum to become 8 times itself. Express 8 as a power of the base multiplier (2). $8 = 2^3$. Here, $n = 3$. 3. **Calculate the required time:** Required time = $n \times T$ Required time = $3 \times 15 = 45$ years. ### Exam Strategy & Shortcut Write down the sequence of powers and years: $2^1$ times $\rightarrow$ $15$ years $2^2$ times $\rightarrow$ $30$ years $2^3$ times $\rightarrow$ $45$ years. Target is 8 times ($2^3$), so the answer is 45 years. ### Common Pitfall A frequent mistake is multiplying the years by the target number directly (e.g., $15 \times 8 = 120$), or using the power incorrectly. Remember that it's the *exponent* that multiplies with the time. ### Final Answer Therefore, the correct answer is **45 years**.