This is a small update of my last piece. I wish that I had put this graph in that piece, because it completes it.
Over the interest rate range of 0% to 30%, the average absolute deviations from perfect doubling using the Rule of 72 was 2.794%. Given the simplicity of the Rule of 72, that is wonderful.
But the “Rule of K” is virtually exact. The average of absolute deviations from perfect doubling using the Rule of K was 0.036%.
Is this great? Well, with modern computers, exactitude is easy to come by. But if you are in a pinch to figure out the time to double, and all you have is a pencil and paper, the rule of K can do it with addition, subtraction, and division. No fancy powers or logarithms. A four-function calculator will handle it, which, if you are using a rate that does divide into 72 easily, you will still need for the calculation.
At 8%, the two are equal. Near 8%, the Rule of 72 is pretty good. The Rule of K gives an almost exact answer at the cost of a little complexity. Your choice depends on whether you need exactness or simplicity when all you have to work with is a four-function calculator.
The Rule of K: If R is the interest rate multiplied by 100, money doubles in K/R years, where K = 70 + (R – 2)/3