Question 1035061: The rule of 72 is a rule of thumb for finding how long it takes money at interest to double:� If r is the annual interest rate, then the doubling time is approximately 72/(100r) years.
(a) Calculate the balance at the end of the predicted doubling time for each $1000, with annual compounding, for the small growth rates of 3%, 4%, and 6%
b) Repeat part (a) for the intermediate interest rates of 8% and 9%.
(c) Repeat part (a) for the larger interest rates of 12%, 24%, and 36%.
(d)What do you conclude about the rule of 72?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! f = p * (1+r)^n
f = future value
p = present value
r = interest rate per time period
n = number of time periods.
when you compound annually, r is the annual interest rate and n is the number of years.
if you divide both sides of the equation by p, you get:
f/p = (1+r)^n
if your present value doubles, then f/p = 2
you wind up with 2 = (1+r)^n
take the natural log of both sides of this equation to get:
ln(2) = ln(1+r)^n
since log(a^b) = b*log(a), your equation becomes:
ln(2) = n*ln(1+r).
solve for n to get n = ln(2) / ln(1+r).
when r = .03, this becomes n = ln(2) / ln(1.03)
when r = .04, this becomes n = ln(2) / ln(1.04)
when r = .06, this becomes n = ln(2) / ln(1.06)
solve for n and you get:
when r = .03, n = 23.45
when r = .04, n = 17.67
when r = .06, n = 11.90
using the rule of thumb, .....
when r = .03, n = 72/3 = 24
when r = .04, n = 72/4 = 18
when r = .06, n = 72/6 = 12
that's a pretty close estimate to the actual calculations.
rule of 72 looks like it works pretty good.
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