Question

Q:You have borrowed $8000 from the bank. Suppose you want to repay a fixed amount of money for each of the following n years (except possibly the last year), and the annual interest rate r does not change in these n years. For example, if r = 10% and you repay $4000 each year, then you will own the bank $(8000+800-4000) = $4800 next year, $(4800+480-4000) = $1280 two years after, and at the end of the third year you only need to repay $(1280 + 128) = $1408.
(a) If r = 20% and you want to repay $2500 each year (except possibly the last year). How many years do you need to repay all the money?
(d) If r = 24% and you want to repay $2000 each year (except possibly the last year). How many years do you need to repay all the money.

Answer 1

a)
amount at end of 1st year = 8000 + 1600 - 2500 = 7100
amount at end of 2nd year = 7100+1420-2500=6020
amount at end of 3rd year = 6020+1240-2500=4724
amount at end of 4th year = 4724+944.8-2500=3168.80
amount at end of 5th year = 3168.8 +633.76-2500= 1302.56 , which is less than the 2500 payment

amount owing at end of 6th year = 1302.56 + 260.51 = 1563.07

So you will need 5 full payments of $2500 plus a partial payment of 1563.07 at the end of the 6th year

checking with my formula:
outstanding after 5 years
= 8000(1.2)^5 - 2500(1.2^5 - 1)/.2
= 19906.56 - 18604.00
= 1302.56 <---- amount owing after 5 years, see above

adding one more interest = 1302.56 + 260.51
= 1563.07 <--- the final partial payment I found above

d) follow the same steps as I used in a)
of course since the rate is higher, and the payment is only 2000, it will take considerably longer.