Mia Salto wishes to determine how long it will take to repay a

$18,000 loan given that the lender requires her to make annual end-of-year installment payments of

$4,309

.

a. If the interest rate on the loan is

15 %,

how long will it take for her to repay the loan fully?

b. How long will it take if the interest rate is

12%?

c. How long will it take if she has to pay

19 %

annual interest?

d. Reviewing your answers in parts

a ,

b ,

and

c ,

describe the general relationship between the interest rate and the amount of time it will take Mia to repay the loan fully.

Answer: a. 10.2 years

b. 12.6 years

c. 8.2 years

d. n = ln 4.1773/ln (1+r)

Step-by-step explanation:

$18,000 loan

annual end-of-year installment payments of $4,309

F = P (1+r) ⁿ

a. r = 15% = 0.15

18000 = 4309 (1+0.15) ⁿ

18000/4309 = 1.15ⁿ

4.1773 = 1.15ⁿ

ln 4.1773 = ln 1.15ⁿ

ln 4.1773 = n*ln 1.15

n = ln 4.1773/ln 1.15

n = 10.2 years

b. r = 12% = 0.12

18000 = 4309 (1+0.12) ⁿ

18000/4309 = 1.12ⁿ

4.1773 = 1.12ⁿ

ln 4.1773 = ln 1.12ⁿ

ln 4.1773 = n*ln 1.12

n = ln 4.1773/ln 1.12

n = 12.6 years

c. r = 19% = 0.19

18000 = 4309 (1+0.19) ⁿ

18000/4309 = 1.19ⁿ

4.1773 = 1.19ⁿ

ln 4.1773 = ln 1.19ⁿ

ln 4.1773 = n*ln 1.19

n = ln 4.1773/ln 1.19

n = 8.2 years

d. The general relationship is n = ln 4.1773/ln (1+r) r as a decimal