Rule of 72 derived

I first learned about the rule 72 was during a financial literacy seminar in one of the insurance companies here in cebu. I can't help but wonder how was this discovered thus i endeavored to get the bottom of this. Hmmmm besides this is in a way a blog with the Rule of 72 as the primary theme then it would just be fitting to check out how this was derived

Let’s start with Php 1 since it’s naturaly easy to work with.

Assuming we have Php 1 and a yearly interest rate (R). After one year we have:
1 * (1+R)
where R is the interest per year

Thus if we are going to derive for the 2nd year, the equation would now look like :
1 * (1+R) * (1+R) = 1 * (1+R)^2

and for the 3rd year we get:
1 * (1+R) * (1+R) * (1+R) = 1 * (1+R)^3

while for the 4th year we get:
1 * (1+R) * (1+R) * (1+R)* (1+R) = 1 * (1+R)^4

Extending to N years we get the equation of:
1 * (1+R)^N

Whew...so far we are still half of the journey.Lets proceed:

Since we are looking for how long it would take to double which is 2 php we then get the equation of:
1 * (1+R)^N = 2

Lets proceed in simplyfying above:
a.) 1 * (1+R)^N = 2
b.) (1+R)^N = 2

Hmmmm lets work on using log here:
c.) ln( (1+R)^N ) = ln(2)
d.) N * ln(1+R) = .693
e.) N * R = .693 [For small R, ln(1+R) ~ R]
basically we just assumed that ln(1+R)~ R because actual results comes really close.
however as we use bigger rates, accuracy becomes worse.

f:) N = .693 / R

multiplying by 100 to clean up the formula:

N = 69.3 / R

There you go...our magic number is 69.3...hmmm maybe because 72 is divisible by many wide array of such as 2,3,4,6,7,8 and 9numbers thus is its a preferred number to use.

whew... this blogged is giving me a headache with all this numbers....

hope you enjoyed guyz..