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# Using An Inverse Function To Derive The Rule of 70

The Rule of 70 states: If you invest a given amount of money at R% interest, it will take approximately 70/R years for your money to double. Likewise, if the inflation rate is R%, prices will double in approximately 70/R years.  So for example, if you invest \$10,000 at R=7% interest, it will take about 70/7 = 10 years for your money to double. In another example, if the inflation rate is R=3.5%, prices will double every 70/3.5 = 20 years.

Where Does This Rule Come From? How To Derive The Rule of 70
The equation governing the growth of money is A = P●ert where

r is the interest rate, given as a decimal,

P is the original amount of money,

t is equal to the number of years the money is invested,

e is the exponential base approximately equal to 2.718, and

A is the total amount of money after t years.

If the amount of money P is doubled, the we let A = 2P and the equation we get is

2P = P●ert, which simplifies to

2 = ert after dividing both sides by P.

Now, we solve this equation for t in terms of r.  Here is where we need the inverse function!

The inverse of the exponential function f(x) = ex is the natural log function g(x) = LN(x) where
(f o g)(x) = e LN(x) = x and (g o f)(x) = LN(ex) = x.

So if we take the natural log function of both sides of  2 = ert, we get

LN(2) = LN(ert) which simplifies to

LN(2) = rt

Since LN(2) = approximately 0.693, we can write 0.693 = rt and solving for t results in

0.693/r = t  where r is the rate as a decimal.  If we multiply the left side by 100/100, we get

69.3/(100r) = t.   If we let R = 100r where R is the rate given as a percent, we get

69.3/R = t.  To account for less than continuous compounding and to make a nicer number in our formula, we round the 69.3 up to 70 and our formula becomes: where R is the rate as a percent and t is the number of years for the money (or price) to double