Chapter 7.3, Problem 1MP

(a)

To determine

To find:The number of times per year the interest is compounded and the interest rate that is used to find the interest if amount is invested compounded quarterly.

Answer to Problem 1MP

The interest is compounded 4 times per year and the interest rate used is 0.95% if amount is invested compounded quarterly.

Explanation of Solution

Given information:Theinitial amount invested is $15,820 at annual interest rate 3.8%. The bank calculates the rate if compounded quarterly is $\frac{1}{4}$ of the annual interest rate.

Calculation:

There are 12 months in a year. Quarterly means every 3 months of a year.

If interest is compounded quarterly, then the interest earned is added to the principal amount every three months.

So, the number of times interest is compounded per year is:

  $\frac{12\text{months}}{3\text{months}}=4$

The quarterly interest rate is $\frac{1}{4}$ of the annual interest rate. So,

  $\begin{array}{c}\text{Interest}\text{rate}=\frac{1}{4}\times 3.8\%\\ =0.95\%\end{array}$

Therefore, the interest is compounded 4 times per year and the interest rate used is 0.95% if amount is invested compounded quarterly.

(b)

To determine

To find:The value of CD after one year and complete the given table.

Answer to Problem 1MP

The value of CD after one year is $16,429.781 and the complete table with values is Table (1).

Explanation of Solution

Given information:The initial amount invested is $15,820 at annual interest rate 3.8%.

Formula used: The balance of an account if it earns compound interest is calculated by the formula:

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

Here, A is the balance, r is the annual interest rate, P is the principal amount, n is the number of times interest compounded per year and t is the times in years.

Calculation:

From part (a), $r=0.95\%$ and $n=4$ . It is given that $P=\$15820$ .

For Quarter 1: $t=1$ and $n=1$ .

To confirm the values given in the table, substitute 15820 for P , 0.095 for r , 1 for n and 1 for t in the above formula.

  $\begin{array}{c}A=15820{\left(1+\frac{0.95}{100}\right)}^{1\cdot 1}\\ =15820\left(1.0095\right)\\ =15970.29\end{array}$

So, the ending principal is $15,970.29.

For Quarter 2: $t=1$ , $n=1$ and $P=\$15970.29$ .

To find the ending principal for quarter 2, substitute 15970.29 for P , 0.095 for r , 1 for n and 1 for t in the above formula.

  $\begin{array}{c}A=15970.29{\left(1+\frac{0.95}{100}\right)}^{1\cdot 1}\\ =15970.29\left(1.0095\right)\\ =16122.0078\end{array}$

So, the ending principal is $16122.0078.

The interest earned if principal amount is:

  $\$16122.0078-\$15970.29=\$151.7178$

For Quarter 3: $t=1$ , $n=1$ and $P=\$16122.0078$ .

To find the ending principal for quarter 3, substitute 16122.0078 for P , 0.095 for r , 1 for n and 1 for t in the above formula.

  $\begin{array}{c}A=16122.0078{\left(1+\frac{0.95}{100}\right)}^{1\cdot 1}\\ =16122.0078\left(1.0095\right)\\ =16275.1669\end{array}$

So, the ending principal is $16275.1669.

The interest earned if principal amount is:

  $\$16275.1669-\$16122.0078=\$153.1591$

For Quarter 4: $t=1$ , $n=1$ and $P=\$16275.1669$ .

To find the ending principal for quarter 4, substitute 16275.1669 for P , 0.095 for r , 1 for n and 1 for t in the above formula.

  $\begin{array}{c}A=16275.1669{\left(1+\frac{0.95}{100}\right)}^{1\cdot 1}\\ =16275.1669\left(1.0095\right)\\ =16429.781\end{array}$

So, the ending principal is $16429.781.

The interest earned if principal amount is:

  $\$16429.781-\$16275.1669=\$154.6141$

The complete table is:

Quarter	Starting Principal	Interest Earned	Ending Principal
1	$15,820.00	$150.29	$15,970.29
2	$15,970.29	$151.7178	$16,122.0078
3	$16,122.0078	$153.1591	$16,275.1669
4	$16,275.1669	$154.6141	$16,429.781

Table (1)

Therefore, the value of CD after one year is $16,429.781 and the complete table with values is Table (1).

(c)

To determine

To find:The simple interest earned onCD after one year.

Answer to Problem 1MP

The simple interest earned on CD after one year is $601.16.

Explanation of Solution

Given information:The initial amount invested is $15,820 at annual interest rate 3.8%.

Formula used: The simple interest can be calculated by the formula:

  $\text{Simple interest}=P\times r\times t$

Here, the principal is P , ttime in years and rate isr .

Calculation:

The annual interest rate is:

  $\frac{3.8}{100}=0.038$

To find the interest, substitute 15820 for P , 3.8 for r and 1 for t .

  $\begin{array}{c}\text{Simple interest}=15820\times 0.038\times 1\\ =601.16\end{array}$

Therefore, the simple interest earned on CD after one year is $601.16.