   SEARCH HOME Math Central Quandaries & Queries  Question from lilly, a student: if i decide to loan a friend 12000 and he said that he will pay it back in a single payment of 16000 after 5 years. Which compounding period will make the interest rate be as low as possible? daily or yearly? how can i calculate each (yearly compound and daily compound)? thank you Hi Lilly,

The usual way this type of problem is given is to say

I have decided to loan a friend 12000 and he said that he will pay it back in a single payment after 5 years at a rate of r% per year. (r is usually given maybe 5% or 10% or 3%.)
a) If I compound yearly how much will he owe me in 5 years?
b) If I compound daily how much will he owe me in 5 years?

The answer to b) is larger than the answer to a) because in b) you have compounded 5 365 times and in a) you have only compounded 5 times. Suppose that the answer for a) was $16 000 then the answer for b) is larger and hence you can reduce the annual rate used for b) a little until you get an answer of$16 000 for that calculation also.

The expression used for this type of problem is

A = P(1 + r/n)nt

where P is the amount you lend your friend, A is the amount he owes you at the end of t years, r is the annual interest rate and n is the number of times per year you compound. In your example

P = $12 000, A =$16 000, t = 5, r is unknown (maybe 0.05, 0.10 or 0.03) and n is either 1 or 365.

Penny     Math Central is supported by the University of Regina and The Pacific Institute for the Mathematical Sciences.