Chapter 7.4, Problem 51E

(a)

To determine

To Prove: The formula are equivalent.

Explanation of Solution

Given:

The given formulas for the monthly payment are,

  $\begin{array}{l}M=\frac{Pi}{1-{\left(\frac{1}{1+i}\right)}^{12t}}\\ M=\frac{Pi{\left(i+1\right)}^{12t}}{{\left(1+i\right)}^{12t}-1}\end{array}$

The dollars borrowed to buy the car is $P$ and time to repay the loan is $t$ .

The monthly interest rate is $i$ .

Calculation:

Consider the given formulas are,

  $\begin{array}{l}M=\frac{Pi}{1-{\left(\frac{1}{1+i}\right)}^{12t}}\\ M=\frac{Pi{\left(i+1\right)}^{12t}}{{\left(1+i\right)}^{12t}-1}\end{array}$

Consider the first equation is evaluated as,

  $\begin{array}{c}M=\frac{Pi}{1-{\left(\frac{1}{1+i}\right)}^{12t}}\\ =\frac{Pi}{1-\frac{1^{12t}}{1+i^{12t}}}\\ =\frac{Pi\left(1+i^{12t}\right)}{\left(1+i^{12t}\right)\left(1-\frac{1^{12t}}{1+i^{12t}}\right)}\\ =\frac{Pi\left(1+i^{12t}\right)}{\left(1+i^{12t}\right)-1}\end{array}$

Thus, it is proved that the formula are equivalent.

(b)

To determine

To find: The monthly payment when you borrow $\$15500$ at the monthly interest rate of 0.5% and repay the loan over 4 years.

Answer to Problem 51E

Thus, the monthly loan repayment is of $\$364.02$ .

Explanation of Solution

Calculation:

Consider the formula to determine the monthly loan payment as,

  $M=\frac{Pi{\left(i+1\right)}^{12t}}{{\left(1+i\right)}^{12t}-1}$

Then, substitute the values to determine the monthly payment as,

  $\begin{array}{c}M=\frac{\left(15500\right)\left(0.005\right){\left(1+0.005\right)}^{12\cdot 4}}{{\left(1+0.005\right)}^{12\cdot 4}-1}\\ =\frac{77.5{\left(1.005\right)}^{48}}{{\left(1.005\right)}^{48}-1}\\ \approx 364.02\end{array}$