Chapter 8.5, Problem 60PPS

(a)

To determine

To find: The monthly payment that must be made in order to pay off the debt in exactly three years and the total amount paid.

Answer to Problem 60PPS

The total amount to be paid is $\$11,239.56$ and the monthly payment that must be paid is $\$312.21$ .

Explanation of Solution

Given:

The formula for the payment that must be made is,

  $m=\frac{b\left(\frac{r}{m}\right)}{1-{\left(1+\frac{r}{n}\right)}^{-nt}}$

The credit card debt of the average American is $8600 and the annual rate is $18.3\%$ .

Calculation:

Consider the given formula for the payment to be made is,

  $m=\frac{b\left(\frac{r}{m}\right)}{1-{\left(1+\frac{r}{n}\right)}^{-nt}}$

Consider the average American is $8600 and the annual rate is $18.3\%$ .

Then,

  $\begin{array}{c}m=\frac{8600\left(\frac{0.813}{12}\right)}{1-{\left(1+\frac{0.813}{12}\right)}^{-3\left(2\right)}}\\ =312.21\end{array}$

Thus, the total amount that must be paid is,

  $312.21\times 36=\$11,239.56$

(b)

To determine

To find: The completed given table,

Answer to Problem 60PPS

The completed table is shown in Table 2

Explanation of Solution

Given:

The formula for the number of years necessary for the payment is,

  $t=\frac{\log \left(1-\frac{br}{mn}\right)}{-n\log \left(1+\frac{r}{n}\right)}$

The given table for the payment schedule is shown in Table 1

Payment (m)	Years (t)
$50
$100
$150
$200
$250
$300

Table 1

Calculation:

Consider the formula for the number of years necessary for the payment is,

  $t=\frac{\log \left(1-\frac{br}{mn}\right)}{-n\log \left(1+\frac{r}{n}\right)}$

Consider the payment is of $50. Then,

  $\begin{array}{c}t=\frac{\log \left(1-\frac{\left(8600-0.813\right)}{50.12}\right)}{-12\log \left(1+\frac{0.813}{12}\right)}\\ =\text{not}\text{real}\end{array}$

From the above equation the table for the payment schedule is,

Table 2

Payment (m)	Years (t)
$50	Non real
$100	Non real
$150	11.42
$200	5.87
$250	4.09
$300	3.2

(c)

To determine

To find: The graph for the table of part (b)

Answer to Problem 60PPS

The graph is shown in Figure 1

Explanation of Solution

Calculation:

From the table shown in Table 2

The graph for the payment schedule is shown in Figure 1

  

Figure 1

(d)

To determine

To find: Whether an individual is able to pay the debt and time for the payment.

Answer to Problem 60PPS

The value of $t$ is not equal to 0 so the individual cannot afford the value of $\$100$ .

Explanation of Solution

Given:

The money that the individual can afford to pay is $\$100$ .

Calculation:

Consider the given formula for the payment to be made is,

  $m=\frac{b\left(\frac{r}{m}\right)}{1-{\left(1+\frac{r}{n}\right)}^{-nt}}$

Consider money that the individual can afford to pay is $\$100$ .

  $\begin{array}{l}100=\frac{8600\left(\frac{0.183}{12}\right)}{1-{\left(1+\frac{0.183}{12}\right)}^{-12t}}\\ {\left(1.01525\right)}^{-12t}=-0.312\\ {\log }_{1.01525}{\left(1.01525\right)}^{-12t}={\log }_{1.01525}\left(-0.312\right)\end{array}$

From above, the value of $t$ is not equal to 0 so the individual cannot afford the value of $\$100$ .

(e)

To determine

To find: The minimum monthly payment that will work toward paying off the debt.

Answer to Problem 60PPS

The value of the minimum monthly payment is of $\$131.15$ .

Explanation of Solution

Consider to determine the monthly payment that will work toward paying off the debt, find the domain of $\log \left(1-\frac{br}{mn}\right)$ .

Then,

  $\begin{array}{l}1-\frac{br}{mn}>0\\ 1-\frac{\left(8600\right)\left(0.183\right)}{12m}>0\\ m>131.15\end{array}$