Chapter 12, Problem 1ECP

To determine

The Periods, Rate, Compounded amount and Total interest, for two year; principal, $1,250; interest rate, 4%; and compounded semiannually.

Answer to Problem 1ECP

The Periods is $\underset{\_}{4}$.

Rate is $\underset{\_}{2\%}$.

Compounded Amount is $\underset{\_}{\$1,353}$.

Total interest is $\underset{\_}{\$103}$.

Explanation of Solution

Formula used:

$\text{No}\text{. of periods}=\text{No}\text{. of years}\times \text{No}\text{. of times the Interest is compounded per year}\mathrm{......}\left(1\right)$

$\text{Rate of each period}=\frac{\text{Annual Interest rate}}{\text{No}\text{. of times the Interest is compounded per year}}\mathrm{......}\left(2\right)$

$\text{Compound Amount}=\text{Compounded value of \$1}\times \text{Principal }\mathrm{......}\left(3\right)$

$\text{Compound Interest}=\text{Compound Amount}-\text{Principal }\mathrm{......}\left(4\right)$

Calculation:

It is given that the Interest is compounded semiannually.

Hence, the $\text{No}\text{. of times the interest is compounded per year}=2$.

Substitute, $\text{No}\text{. of times the interest is compounded per year}=2,\text{ No}\text{. of years}=2$ in equation (1)

$\begin{array}{c}\text{No}\text{. of periods}=\text{No}\text{. of years}\times \text{No}\text{. of times the Interest is compounded per year}\\ =2\times 2\\ =4\end{array}$

Hence, the number of periods is $\underset{\_}{4}$.

Substitute, $\text{Annual Interest rate}=4\%,\text{ No}\text{. of times the interest is compounded per year}=2$ in equation (2)

$\begin{array}{c}\text{Rate of each period}=\frac{\text{Annual Interest rate}}{\text{No}\text{. of times the Interest is compounded per year}}\\ =\frac{4}{2}\\ =2\end{array}$

Thus, the Rate of each period is 2%.

Note that, the no. of periods is 4 and the interest per period is 2%

Future value or Compounded value of $1 is $1.0824 with the help of Table 12.1.

Substitute, $\text{Compounded value of }\$\text{1}=\$1.0824$ in equation (3).

$\begin{array}{c}\text{Compound Amount}=\text{Compounded value of \$1}\times \text{Principal}\\ =1,250\times 1.0824\\ =1,353\end{array}$

Hence, the Compounded Amount is $\underset{\_}{\$1,353}$.

Substitute, $\text{Compounded Amount}=\$1,353,\text{ Principal}=\$1,250$ in equation (4).

$\begin{array}{c}\text{Compound Interest}=\text{Compound Amount}-\text{Principal}\\ =1,353-1,250\\ =103\end{array}$

Hence, the compound Interest is $\underset{\_}{\$103}$.