Calculating investment time based on interest?

Question

Determine how much time is required for an investment to double in value if interest is earned at the annual rate of 6.75% compounded monthly. Confirm numerically.

Any help on this would be appreciated. Please explain completely so that I fully understand and can do similar problems on my own. Thanks so much and 10 points to a best answer.

Answer 1

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

P = principal

r = annual interest rate

n = number of times the interest is compounded per year

t = number of years

A = amount after time t

To double, A = 2P

Solve for t

2P = P * (1 + .0675/12)^12t

2 = (1 + 0.005625)^12t

log(2) = log(1.005625) * 12 * t

t = log(2)/(log(1.005625) * 12) = approx. 10.298 years

Confirm:

A = P * (1 + 0.0675/12)^(12*10.298)

A = P * (1.005625)^123.576

A = P * 2.0

As an aside, the "rule of 72" is a good approximation for annual compounding. Divide 72 by the interest rate to determine how long it takes money to double. 72/6.75 = 10.66 years