The Rule of 72 is a rule of thumb that states that the number of time periods (such as years) that it takes for a sum to double at compound interest is very nearly 72 divided by the percentage interest rate per period. Thus, for example, it takes (almost exactly) 9 years for a sum to double at 8% interest compounded yearly.

1. Using the rule, find the annual rate of increase for an investment that has quadrupled in 24 years. Compare to the exact value.

2. At what interest rate is the rule exact?

3. Justify the rule using mathematical analysis and a few numerical calculations.

If the interest is compounded annually, then it is only computed at the end of a full year.  (I think banks usually compound monthly, not annually)

The compound interest formula is P(1+r)^t
P=principal
r=rate
t=time in years

1.

Quadrupaling means doubling twice.  So the amount actually doubled in 12 years.  By the rule of 72 the interest is 6%.
According to the formula, (1.06)^12 = 2.012 and (10.6)^24 = 4.049 which is pretty close to 4

2.

The rule is not exact for any whole number of years.  Solving (1+r/100)^(72/r) = 2 gives r = 7.845% t = 9.176 years.
The closest to this is 9 years at 8%

3.

The rule is closest when r and t are both around 8.  See table below.

years rate amount
1       72       1.72
2       36       1.8496
3       24       1.9066
4       18       1.9388
5       14.4    1.9594
6       12       1.9738
7       10.286 1.9844
8        9        1.9926
9        8        1.9990
10      7.2     2.0042
11      6.546  2.0086
12      6        2.0122

-Jer 

Posted by Jer on 2005-01-25 17:46:57