To use the graph, simply select a growth rate per interval i and the curve gives you the approximation error for the Doubling-Time formula compared to an exponential calculation.  Thus agreement occurs when i ≈ 8 %.  At the 'self-referent' growth rate i = 17.65717 % per year the doubling-time = 72 / 17.65717 = 4.08 years, and the approximation error ≈ - 4.4%.

By the way, for a Tripling-Time formula, simply replace 72 with 114.  Using n = 114 / i, we find that, for example, an enterprise growing at 9 % per year will triple in size in twelve years and eight months, and the error in that estimate is the same as graphed above.

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our puzzle-master graduated from high-school in 1951.  The United States Census Bureau USCB reports that the human population of the world that year was 2,594,939,877, which was 37,311,223 higher than the year before.

Assuming a constant growth rate in population worldwide, in what year did the population double in size: 2 x 2,594,939,877 = 5,189,879,754?

First we calculate the population growth rate for 1951: i = 100 x 37,311,223 / 2,594,939,877 = 1.438% per year. Then we simply use the doubling-time formula: n = 72 / i = 72 / 1.438 = 50.077 years + 1951 ≈ 2001.  For solvers who like logarithms: n = log 2 / log (1 + i /100) = 48.548 + 1951 ≈ 2000.  That amounts to only a 3% 'approximation' error from the graph.