Mathematics for FinancePYQ May 19, ICAI SMQuestion 1286 of 512
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The annual birth rates per 1,000\displaystyle 1,000 are 39.4\displaystyle 39.4 and 19.4\displaystyle 19.4 respectively. The number of years which the population will be doubled assuming there is no immigration or emigration is

Options

A35\displaystyle 35 years
B30\displaystyle 30 years
C25\displaystyle 25 years
DNone of these
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Correct Answer

✅ Option a — 35\displaystyle 35 years

All Options:

  • A35\displaystyle 35 years
  • B30\displaystyle 30 years
  • C25\displaystyle 25 years
  • DNone of these

Detailed Solution & Explanation

**Derivation of Population Doubling Time** Given: - Birth rate = 39.4\displaystyle 39.4 per 1,000\displaystyle 1,000 - Death rate = 19.4\displaystyle 19.4 per 1,000\displaystyle 1,000 **Step 1: Calculate the net annual growth rate (r\displaystyle r)** Net Growth Rate per 1,000=39.4−19.4=20\text{Net Growth Rate per 1,000} = 39.4 - 19.4 = 20 r=201000=2%=0.02 per annumr = \frac{20}{1000} = 2\% = 0.02 \text{ per annum} **Step 2: Set up the population doubling equation** (1+r)t=2(1 + r)^t = 2 (1.02)t=2(1.02)^t = 2 **Step 3: Solve for t\displaystyle t using logarithms** t=ln⁡(2)ln⁡(1.02)≈0.6931470.019803≈35 yearst = \frac{\ln(2)}{\ln(1.02)} \approx \frac{0.693147}{0.019803} \approx 35 \text{ years} Hence, **Option A** is the correct answer.

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