Find the length of the loan in months, if $400 is borrowed with an annual simple interest rate of 12% and with $420 repaid at the end of the loan.

Question

I 've tried this like 12 times and I still can't figure out the answer.

Answer 1

Hi Katherine,

 

Have you worked with logarithms?   If yes, I will show you how to do it that way at the end.   But this first way makes the whole process much more understandable.

 

The formula for the Amount Paid on a loan is this:

 

A = (InitAmt)(1+r)^t    

 

So for us, the InitAmt is $400

The rate is 12% per YEAR, but we are calculating by the MONTH, so take the interest rate and divide by 12 (b/c there are 12 months in a year) and this gives you the interest rate each MONTH.

 

So r = 1% for us.  And the formula becomes

 

A (the amount we have paid) = 400(1+1%)^t

 

And in these problems we must always convert 1% to 0.01, so the final formula is

 

A=400(1.01)^t

 

So just start at the moment the loan began, t=0

 

0 months:  A = $400(1.01)⁰ = 400 (1) = $400

 

after 1 month, 400(1.01)¹ = 400(1.01) = $404

 

after 2 months, 400(1.01)² = 400(1.0201) = $408.04

 

after 3 months, 400(1.01)³ = 400(1.030301) = $412.12

 

after 4 months, 400(1.01)⁴ = 400(1.04060401) = $416.24

 

after 5 months, 400(1.01)⁵ = 400(1.05101005) = $420.40

 

So it looks like it took 5 months

 

(or technically just under 5 months, since at exactly 5 months it was 40 cents over $420)

 

 

Ignore this if you have not learned logarithms:

 

400(1.01)^t = 420

 

(1.01)^t = 420/400

 

(1.01)^t = 1.05

 

t ln(1.01) = ln(1.05)

 

t = ln(1.05) / ln(1.01)

 

t = 4.903 months  or 4 months and 27 days  or approximately 5 months.