Mathematics for FinanceMTP Dec 23 Series IQuestion 1397 of 512
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The population of a town increases by 2%\displaystyle 2\% of the population at the beginning of the year. The number of years by which the total increases in population would be 40%\displaystyle 40\% is

Options

A7 years
B10 years
C17 years
D19 years
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Correct Answer

✅ Option c — 17 years

All Options:

  • A7 years
  • B10 years
  • C17 years
  • D19 years

Detailed Solution & Explanation

**Derivation of Required Period** Given: - Annual growth rate (r\displaystyle r) = 2%\displaystyle 2\% per annum - Target population increase = 40%\displaystyle 40\% (so Amount A=1.40P0\displaystyle A = 1.40 P_0) **Step 1: Set up the growth equation** A=P0(1+r)tA = P_0(1 + r)^t 1.40P0=P0(1+0.02)t1.40 P_0 = P_0(1 + 0.02)^t 1.40=(1.02)t1.40 = (1.02)^t **Step 2: Solve for t\displaystyle t using logarithms** ln⁡(1.40)=tln⁡(1.02)\ln(1.40) = t \ln(1.02) t=ln⁡(1.40)ln⁡(1.02)t = \frac{\ln(1.40)}{\ln(1.02)} Using log values: ln⁡(1.40)≈0.336472\ln(1.40) \approx 0.336472 ln⁡(1.02)≈0.019803\ln(1.02) \approx 0.019803 t=0.3364720.019803≈16.99 years≈17 yearst = \frac{0.336472}{0.019803} \approx 16.99 \text{ years} \approx 17 \text{ years} Hence, **Option C** is the correct answer.

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