Answer
See below
Work Step by Step
(a)
Calculation of value of the annuity can be done by using formula:
\[A=\frac{P\left[ {{\left( 1+\frac{r}{n} \right)}^{nt}}-1 \right]}{\left( \frac{r}{n} \right)}\]
Where A denotes the value of the annuity, P denotes the periodic deposit, R denotes the rate of interest, t denotes the number of years, and n denotes the number of times compounding is done in a year.
Compute the value of the annuity by substituting the values in the formula as mentioned below:
\[\begin{align}
& \text{P}=\frac{\text{A}\left( \frac{r}{n} \right)}{\left[ {{\left( 1+\frac{r}{n} \right)}^{\text{nt}}}-1 \right]} \\
& =\frac{\$25,000\left(\frac{0.0725}{4}\right)}{\left[{{\left(1+\frac{0.0725}{4}\right)}^{4\times5}}-1\right]}\end{align}\]
Simplify and solve the equation as follows:
\[\begin{align}
& P=\frac{\$25,000\left(0.018125\right)}{{{\left(1+0.018125\right)}^{20}}-1}\\&=\frac{\$453.125}{{{\left(1.018125\right)}^{20}}-1}\\&=\frac{\$453.125}{0.432261}\\&=\$1,049\end{align}\]
Hence, the value of the deposit which is paid at the end of every 3 months is\[\$1,049\]
(b)
Computation of the interest amount can be done by deducting the total of periodic deposit amount from the annuity value. Compute the interest amount as mentioned below:
\[\begin{align}
& \text{Amount of deposit in 5 years}=\text{Number of times in a year}\times \text{Number of years} \\
& \times \text{Amount of deposits} \\
& =4\times 5\times \$1,049\\&=\$20,980\end{align}\]
Compute the amount of interest using the equation as shown below:
\[\begin{align}
& \text{Amount of interest}=\text{Value of Annuity after 5 years}-\text{Amount deposited in 5 years} \\
& =\$25,000-\$20,980\\&=\$4,020\end{align}\]
Hence, the amount of deposit and interest at the end of 5 years are \[\$20,980\]and \[\$4,020\]respectively.
