# Rule of 70/72

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When we’re working with growth rates (especially in the context of growth models), it’s good to know how many years it takes for something to double. That is, we want to know “n” such that with growth rate x, (1+x)n = 2. For example, (1+.07)10 ≈ 2 [at 7% things double in 10 years].

There’s a shortcut to do this estimate in your head: divide the growth rate into 70 or 72.

Why does this work? You need to know two facts and one consequence:

a.  log (2) = 0.693

b.  log (1+x) ≈ x for small x

c.  Hence: n log (1+x) = n x = log(2) = .693 or n = .693 / x.

If we use this in percentage growth terms, then we have n = 69.3 / x.

But this is a mere approximation. So let’s take advantage of that! – make life easier by rounding 69.3 to either 70 or 72. Why? – think about divisors! We can divide 70 by 1, 2, 5, 7 and 10. We can divide 72 by 1, 2, 3, 4, 6, 8, 9 and 12. Of the first dozen digits, only poor 11 is orphaned.

We’re rounding up, and hence dividing into a number larger than it should be, making our answer a bit too large. On the other hand, the approximation of log(1+x) is less accurate when x is larger (and is always too big), so works in the opposite direction). The table below compares the actual value and our rule-of-thumb approximation.

The bottom line: the approximation is really pretty good!

 Growth Rate Rule of Thumb Actual Error (in years) using 70 using 72 0.5 140 – 139.0 1.0 1 70 – 69.7 0.3 2 35 – 35.0 0.0 3 – 24 23.4 0.6 4 – 18 17.7 0.3 5 14 – 14.2 -0.2 6 – 12 11.9 0.1 7 10 – 10.2 -0.2 8 – 9 9.0 0.0 9 – 8 8.0 0.0 10 7 – 7.3 -0.3 12 – 6 6.1 -0.1 20 3.5 3.6 3.8 -0.2 to -.03

September 2015 (c) Michael Smitka