You want to buy a camera, but you are $1000 short. Your favorite uncle, offers to lend you that money, if you pay him $1,200 two years from today. He compounds interest monthly. A greedy high school friend also offers to give you the money. However, she says there is going to be a loan processing fee of $20 that will be included in the loan amount (once you pay the processing fee, she lends you $1020). Your greedy friend is expecting to receive $1220 in two years, based on monthly compounding of interest.
(a) What is the monthly rate your favorite uncle is charging? What rate is your greedy friend charging?
(b) What effective annual rate are your favorite uncle and your greedy friend charging?
(c) Which one would you prefer to borrow from?

Explanation:

Future value after 24 months = 1200

present value = 1000

Let monthly rate of interest = r

1000 = 1200/( 1+r )²⁴

( 1+r )²⁴ = 1200/1000

( 1+r )²⁴ = 1.2

taking log on both sides

24 log( 1+r ) = log 1.2

24 log( 1+r ) = .07918

log( 1+r ) = .003299

( 1+r ) = 1.007625

r = .007625

monthly rate of interest in percent = .7625%

II Option

Future value after 24 months = 1220

present value = 1020 - 20 = 1000

Let monthly rate of interest = r

1000 = 1220/( 1+r )²⁴

( 1+r )²⁴ = 1220/1000

( 1+r )²⁴ = 1.22

taking log on both sides

24 log( 1+r ) = log 1.22

24 log( 1+r ) = .086359

log( 1+r ) = .003598

( 1+r ) = 1.008319

r = .008319

monthly rate of interest in percent = .8319%

b )

Effective annual rate of uncle = (1.007625)¹² -1

= 1.09543 - 1 = .09543

In percent = 9.543 %

Effective annual rate of greedy friend = ( 1.008319)¹² -1

= 1.1045 -1

= 10.45 %

c ) The first  one is cheaper so it is preferable.

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