Basics of Financial Mathematics Test 44

$${\textbf{Step  - 1: Framing the formula for finding the rate of decrease}}{\text{.}}$$

                   $${\text{Let the rate of increase in the population be }}r,$$

                   $${\text{We know, population increases geometrical progression so, we}}$$

                   $${\text{will use the compound interest formula}}{\text{.}}$$

                   $${\text{Given}},{\text{ }}P{\text{ }} = {\text{ }}16,000$$

                   $${\text{Time period }},{\text{ }}n{\text{ }} = {\text{ }}2$$

                   $${\text{Population after }}2{\text{ yrs }},{\text{ }}A{\text{ }} = {\text{ }}17640$$

                   $${\text{Using the formula of compound interest, we can write}}$$

                   $$A = P{\left( {1 + \dfrac{r}{{100}}} \right)^n}$$

                   $$17640 = 16000{\left( {1 + \dfrac{r}{{100}}} \right)^2}$$

$${\textbf{Step  - 2: Solving the above equation to find the rate of decrease}}{\text{.}}$$

                   $$\dfrac{{17640}}{{16000}} = {\left( {1 + \dfrac{r}{{100}}} \right)^2}$$

                   $$\dfrac{{441}}{{400}} = {\left( {1 + \dfrac{r}{{100}}} \right)^2}$$

                   $${\text{Taking square root on both the sides,}}$$

                   $$\dfrac{{21}}{{20}} = 1 + \dfrac{r}{{100}}$$

                   $$\dfrac{1}{{20}} = \dfrac{r}{{100}}$$

                   $$r = 5% $$

$${\textbf{Hence, the rate of decrease of population is 5% }}{\text{.}}$$