# Maths help - Compound interest

How do I rewrite the compound interest formula, to isolate r (annual interest rate) ?

A = P ( 1 + r) ^ n

P = Principal (initial) amount
r = Annual interest rate as decimal
n = Years
A = Amount after n years

I can divide through by P to get: A/P = (1+r) ^ n, but can’t remember what to do then with the power sign .

A = P(1+r)^n

A/P = (1+r)^n

ln (A/P) = n*ln(1+r)

(1/n)*ln(A/P) = ln(1+r)

e^((1/n)*ln(A/P)) = 1+r

e^((1/n)*ln(A/P))-1 = r

(A/P)^(1/n) - 1 = r

So if I have 1000, invested for 3 years at 10%, that’s would grow into 1000 * (1.1 ^ 4) = 1,464.10.
Plugging those figures back into yours:

(A/P)^(1/n) - 1 = r
(1,464.10 / 1000) ^ (1/4) - 1 = r
(1.46641 ^ 0.25) - 1 = r
1.1 - 1 = r
0.1 = r ( 10%!)

Ah, thanks a lot. I wasn’t going to get there on my own.

When I was 17, I knew all that log and e stuff. Long gone now .

Don’t even need logs or e^x to solve it. Just go straight to taking the nth root.

A = P(1+r)^n
A/P = (1+r)^n
Then, take the nth root of both sides of the equation:
(A/P)^(1/n) = r+1
(A/P)^(1/n) -1 = r

I’m lazy and always use this online Compound Interest Calculator [ 1728.org/compint.htm ]. You can use it to find any one of 1) rate, or 2) time needed, or 3) end total, or 4) principal needed. Easy for quick guidance.