Chapter 8 - Personal Finance - 8.4 Compound Interest - Exercise Set 8.4 - Page 522: 47

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(a) The amount can be computed with the help of the compound interest formula, by using the formula given below, where A, r, and t is given. The calculation for the interest when the interest is compounded once in a year, the amount earned in year 1 is as follows: \[\begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\$5,000{{\left(1+\frac{0.055}{1}\right)}^{1}}\\&=\$5,000\left(1.055\right)\\&=\$5,275\end{align}\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$5,275-\$5,000\\&=\$275\end{align}\] The calculation for the interest when the interest is compounded once in a year, the amount earned in year 5 is as follows: \[\begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\$5,000{{\left(1+\frac{0.055}{1}\right)}^{5}}\\&=\$5,000{{\left(1.055\right)}^{5}}\\&=\$5,000\times1.30696\end{align}\] \[=\$6,534.8\approx\$6,535\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$6,535-\$5,000\\&=\$1,535\end{align}\] The calculation for the interest when the interest is compounded once in a year, the amount earned in year 10 is as follows: \[\begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\$5,000{{\left(1+\frac{0.055}{1}\right)}^{10}}\\&=\$5,000{{\left(1.055\right)}^{10}}\\&=\$5,000\times1.708144\end{align}\] \[=\$8,540.72\approx\$8,541\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$8,541-\$5,000\\&=\$3,541\end{align}\] The calculation for the interest when the interest is compounded once in a year, the amount earned in year 15 is as follows: \[\begin{align} & A=P{{\left( 1+\frac{r}{n} \right)}^{n\times t}} \\ & =\$5,000{{\left(1+\frac{0.055}{1}\right)}^{20}}\\&=\$5,000{{\left(1.055\right)}^{20}}\\&=\$5,000\times2.917757\end{align}\] \[=\$14,588.79\approx\$14,589\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$14,589-\$5,000\\&=\$9,589\end{align}\] (b) The amount can be computed with the help of the compound interest formula, by using the formula given below, where A, r, and t is given. The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 1 is as follows: \[\begin{align} & A=P{{e}^{r\times t}} \\ & =\$5,000{{\left(2.71828\right)}^{0.055\times1}}\\&=\$5,000\times1.056541\\&=\$5,282.70\approx\$5,283\end{align}\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$5,283-\$5,000\\&=\$283\end{align}\] The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 5 is as follows: \[\begin{align} & A=P{{e}^{r\times t}} \\ & =\$5,000{{\left(2.71828\right)}^{0.055\times5}}\\&=\$5,000\times{{\left(2.71828\right)}^{0.275}}\\&=\$5,000\times1.31653\end{align}\] \[=\$6,582.65\approx\$6,583\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$6,583-\$5,000\\&=\$1,583\end{align}\] The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 10 is as follows: \[\begin{align} & A=P{{e}^{r\times t}} \\ & =\$5,000{{\left(2.71828\right)}^{0.055\times10}}\\&=\$5,000\times{{\left(2.71828\right)}^{0.55}}\\&=\$5,000\times1.733252\end{align}\] \[=\$8,666.26\approx\$8,666\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$8,666-\$5,000\\&=\$3,666\end{align}\] The calculation for the interest when the interest is compounded continuously throughout the year, the amount earned in year 20 is as follows: \[\begin{align} & A=P{{e}^{r\times t}} \\ & =\$5,000{{\left(2.71828\right)}^{0.055\times20}}\\&=\$5,000\times{{\left(2.71828\right)}^{1.1}}\\&=\$5,000\times3.004164\end{align}\] \[=\$15,020.82\approx\$15,021\] The interest is as follows: \[\begin{align} & \text{Interest}=\text{Amount}-\text{Principal} \\ & =\$15,021-\$5,000\\&=\$10,021\end{align}\]