Confused about interest rate calculation: "Someone borrows $200 for 14 days...."

[Mathless]

I've seen 2 ways of solving a problem and I don't understand why one of them works. Here's the problem:

Someone borrows $200 for 14 days. They must repay $240 at the end of the loan. What is the annual interest rate? ​

When I used the formula I=PRT, I took the following steps:

• R=I/PT
• I = $40
• P = $200
• T = 14 days

Then I converted 14 days into 14/365 and got R = 40/(200 x .038). So R = 5.26, which is 526% per year interest.

Someone else did it with these steps:

• 40/200 = .2
• .2*365 = 73
• 73/14 = 5.21
• 5.21*100 = 521%

I understand that some of the difference between the answers is probably due to rounding. But my question is how does the second method work? It looks like step one gets the amount of interest charged as a ratio to the amount borrowed, but from there I'm lost. Could anyone walk me through what this does?

 

[Denis]

Lotsa ways to calculate similar problems.
I've seen a few as ridiculous as using 365.25 days (to accommodate leap years!)

Keep it simple:
2 weeks = 40, so 52 weeks = 1040
200(1 + r) = 1240
solve for r: r = 5.2
So rate = 5.2*100 = 520%

 

[Dr.Peterson]

It appears that they have, in effect, solve I = PRT for R to get R = I/(PT) = (I/P)*T; they calculated that by finding I/P, converting that to days and dividing by T in days. So it's fully equivalent to what you did.

The difference is therefore entirely in the rounding! Your method, unrounded until the end as it should be, is

• R=I/PT
• I = $40
• P = $200
• T = 14 days
• convert 14 days into 14/365 = 0.038356
• R = 40/(200 x .038356) = 5.2142857
• R = 5.21, which is 521% per year interest.

while theirs, likewise, is

• 40/200 = .2
• .2*365 = 73
• 73/14 = 5.2142857
• 5.21*100 = 521%

Same answer both ways! But your way involved more rounding.