Question

unit 3 • finance
topic c: loans and financed purchases
use the given information to complete problems 8–10.
priit took out a loan for $50,000 in order to purchase some land. the following table
shows his repayment schedule.

quarter	previous balance	payment	interest payment	principal payment	new balance
2	$47,875.00	$3,000.00	$837.81	$2,162.19	$45,712.81
3	$45,712.81	$3,000.00	$799.97	$2,200.03	$43,512.79
4	$43,512.79	$3,000.00	$761.47	$2,238.53	$41,274.26
5	$41,274.26	$3,000.00	$722.30	$2,277.70	$38,996.56
6	$38,996.56	$3,000.00	$682.44	$2,317.56	$36,679.00
7	$36,679.00	$3,000.00	$641.88	$2,358.12	$34,320.88
8	$34,320.88	$3,000.00	$600.62	$2,399.38	$31,921.50
9	$31,921.50	$3,000.00	$558.63	$2,441.37	$29,480.12
10	$29,480.12	$3,000.00	$515.90	$2,484.10	$26,996.03
11	$26,996.03	$3,000.00	$472.43	$2,527.57	$24,468.46
12	$24,468.46	$3,000.00	$428.20	$2,571.80	$21,896.65
13	$21,896.65	$3,000.00	$383.19	$2,616.81	$19,279.85
14	$19,279.85	$3,000.00	$337.40	$2,662.60	$16,617.24
15	$16,617.24	$3,000.00	$290.80	$2,709.20	$13,908.05
16	$13,908.05	$3,000.00	$243.39	$2,756.61	$11,151.44
17	$11,151.44	$3,000.00	$195.15	$2,804.85	$8,346.59
18	$8,346.59	$3,000.00	$146.07	$2,853.93	$5,492.65
19	$5,492.65	$3,000.00	$96.12	$2,903.88	$2,588.77
20	$2,588.77	$2,634.08	$45.30	$2,588.78	$0.00

1. what is the quarterly interest rate?
2. find a recursive formula to model the new balance in terms of the previous balance.
3. suppose priit pays $3,600 each quarter instead. what formula would model this situation?

Explanation:

Problem 8

Step 1: Recall the interest formula

The interest payment for a period is calculated as \( \text{Interest} = \text{Previous Balance} \times r \), where \( r \) is the quarterly interest rate. We can use the first quarter's data: Previous balance = $50,000, Interest payment = $875.

Step 2: Solve for \( r \)

Using the formula \( 875 = 50000 \times r \), we solve for \( r \) by dividing both sides by 50000: \( r=\frac{875}{50000} \).

Step 3: Calculate the rate

\( r = 0.0175 \) or \( 1.75\% \).

Step 1: Analyze the new balance

The new balance is calculated as: \( \text{New Balance} = \text{Previous Balance} + \text{Interest Payment} - \text{Payment} \). But since \( \text{Interest Payment} = \text{Previous Balance} \times r \) (with \( r = 0.0175 \)) and \( \text{Principal Payment} = \text{Payment} - \text{Interest Payment} \), we can also write the new balance as: \( \text{New Balance} = \text{Previous Balance} \times (1 + r) - \text{Payment} \).

Step 2: Substitute values

We know \( r = 0.0175 \) and the payment is $3,000. So the recursive formula is: Let \( B_n \) be the new balance at quarter \( n \) and \( B_{n - 1} \) be the previous balance. Then \( B_n = B_{n - 1} \times (1 + 0.0175) - 3000 \), with the initial condition \( B_0 = 50000 \).

Step 1: Modify the payment in the formula

From the previous recursive formula, we only change the payment amount. The interest rate \( r = 0.0175 \) remains the same, but the payment is now $3,600 instead of $3,000.

Step 2: Write the new formula

Using the same structure as in Problem 9, the new recursive formula is: \( B_n = B_{n - 1} \times (1 + 0.0175) - 3600 \), with the initial condition \( B_0 = 50000 \).

Answer:

The quarterly interest rate is \( 1.75\% \) (or \( 0.0175 \) in decimal form).

Problem 9