Chapter 8.5, Problem 65E

To determine

To find: the actual time to return the loan amount and the amount of the last payment.

Answer to Problem 65E

  $t=16\text{months}\$212.72$ .

Explanation of Solution

Given:

The borrow amount of diamond ring is $\$3500$ at $10\%$ annual interest compounded monthly.

The down payment of diamond ring is $\$500$ and the monthly installment is $\$213.59$ .

Calculation:

As borrow amount of diamond ring is $\$3500$ at $10\%$ annual interest compounded monthly. And the down payment of diamond ring is $\$500$ and the monthly installment is $\$213.59$ .Then,

  $\begin{array}{l}{a}_1=2000\\ d=500\\ r=\frac{10}{12}\%=0.0083\\ p=213.59\\ t=7\end{array}$

Therefore, the amount is to be paid $\$3500-500=\$3000$ . Then

  $a_n=1.0083a_{n-1}-213.59$

Now, create the table to show that the payment number and balance after payment:

Payment number	Balance after payment	Payment number	Balance after payment
1	$3000$	9	$1445.89$
2	$2811.31$	10	$1244.3$
3	$2621.05$	11	$1041.04$
4	$2429.21$	12	$836.09$
5	$2235.78$	13	$629.44$
6	$2040.75$	14	$421.07$
7	$1844.1$	15	$210.97$
8	$1645.82$	16	$-0.87$

It is notice that the values become negative in the 16th line, therefore there are 16 payments (16 months).

The negative value means that the 16th payment was actually done.

Therefore, the amount of the last payment is $213.59-0.87=212.72$

Hence, the actual time to return the loan is 16 months and amount of the last payment is $\$212.72$ .