Mathematics for FinanceMTP Dec 23 Series IQuestion 3948 of 507
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The annual birth and death rates per 1,000\displaystyle 1,000 are 39.4\displaystyle 39.4 and 19.4\displaystyle 19.4 respectively. The number of years in 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 a35\displaystyle 35 years

All Options:

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

Detailed Solution & Explanation

Given parameters: * Birth rate per 1,000\displaystyle 1,000 = 39.4\displaystyle 39.4 * Death rate per 1,000\displaystyle 1,000 = 19.4\displaystyle 19.4 The net annual growth rate of the population per 1,000\displaystyle 1,000 is: Net Growth Rate per 1000=39.419.4=20\text{Net Growth Rate per 1000} = 39.4 - 19.4 = 20 Thus, the annual percentage growth rate (r\displaystyle r) is: r=201,000×100%=2% p.a. (or i=0.02)r = \frac{20}{1,000} \times 100\% = 2\% \text{ p.a. (or } i = 0.02\text{)} Let the initial population be P\displaystyle P. We want to find the number of years (t\displaystyle t) in which the population will be doubled (2P\displaystyle 2P): P(1+i)t=2PP(1+i)^t = 2P (1.02)t=2(1.02)^t = 2 Taking natural logarithms on both sides: tln(1.02)=ln(2)t \ln(1.02) = \ln(2) t0.693150.0198035 yearst \approx \frac{0.69315}{0.01980} \approx 35 \text{ years} We can check: (1.02)351.999892(1.02)^{35} \approx 1.99989 \approx 2 Thus, the population will double in approximately 35\displaystyle 35 years. Hence, **Option A** is the correct answer.

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