A person borrows $3,000 on a bank credit card at a nominal

Chapter 5, Problem 27E

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Problem 27E

A person borrows $3,000 on a bank credit card at a nominal rate of 18% per year, which is actually charged at a rate of 1.5% per month.

a. What is the annual percentage rate (APR) for the card? (See Example 5.5.8 for a definition of APR.)

b. Assume that the person does not place any additional charges on the card and pays the bank $150 each month to pay off the loan. Let Bn be the balance owed on the card after n months. Find an explicit formula for Bn.

c. How long will be required to pay off the debt?

d. What is the total amount of money the person will have paid for the loan?

Example

Compound Interest with Compounding Several Times a Year

When an annual interest rate of i is compounded m times per year, the interest rate paid per period is i/m. For instance, if 3% = 0.03 annual interest is compounded quarterly, then the interest rate paid per quarter is 0.03/4 = 0.0075.

For each integer k ≥ 1, let Pk = the amount on deposit at the end of the kth period, assuming no additional deposits or withdrawals. Then the interest earned during the kth period equals the amount on deposit at the end of the (k –1)st period times the interest rate for the period:

The amount on deposit at the end of the kth period, Pk , equals the amount at the end of the (k –1)st period, Pk–1, plus the interest earned during the kth period

Suppose $10,000 is left on deposit at 3% compounded quarterly.

a. How much will the account be worth at the end of one year, assuming no additional deposits or withdrawals?

b. The annual percentage rate (APR) is the percentage increase in the value of the account over a one-year period. What is the APR for this account?

Solution

a. For each integer n ≥ 1, let Pn= the amount on deposit after n consecutive quarters, assuming no additional deposits or withdrawals, and let P0 be the initial $10,000. Then by equation (5.5.4) with i = 0.03 and m = 4, a recurrence relation for the sequence P0, P1, P2,…is

(1) Pk = Pk–1(1 + 0.0075) = (1.0075) • Pk–1 for all integers k≥ 1.

The amount on deposit at the end of one year (four quarters), P4, can be found by successive substitution:

(2) P0 = $10,000

(3) P1 = 1.0075• P0 = (1.0075) $10,000.00 = $10,075.00 by (1) and (2)

(4) P2 = 1.0075• P1 = (1.0075) $10,075.00 = $10,150.56 by (1) and (3)

(5) P3 = 1.0075• P2 ≅ (1.0075)•$10,150.56 = $10,226.69 by (1) and (4)

(6) P4 = 1.0075• P3≅ (1.0075) •$10,226.69 = $10,303.39 by (1) and (5)

Hence after one year there is $10,303.39 (to the nearest cent) in the account.

b. The percentage increase in the value of the account, or APR, is