Mathematics for FinanceMTP Dec 23 Series IQuestion 3949 of 507
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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 c17 years

All Options:

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

Detailed Solution & Explanation

Let the initial population be P\displaystyle P. The annual growth rate of the population is 2%\displaystyle 2\% p.a. (so i=0.02\displaystyle i = 0.02). We want to find the number of years (t\displaystyle t) by which the total increase in population would be 40%\displaystyle 40\%. An increase of 40%\displaystyle 40\% means the final population will be: A=P+0.40P=1.40PA = P + 0.40P = 1.40P The formula for population growth is: A=P(1+i)tA = P(1+i)^t Substituting the values: 1.40P=P(1.02)t1.40P = P(1.02)^t 1.02t=1.401.02^t = 1.40 Taking natural logarithms on both sides: tln(1.02)=ln(1.40)t \ln(1.02) = \ln(1.40) t0.336470.0198017 yearst \approx \frac{0.33647}{0.01980} \approx 17 \text{ years} Thus, the number of years is approximately 17\displaystyle 17. Hence, **Option C** is the correct answer.

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