When does the rule of 72 start to lie?

Question

Use the rule of 72 to find the doubling time at $2\%$ a year, then judge how far off it is. Solve in your head.

What doubling time does the rule give at $2\%$ , and is it high or low versus the truth? Aim: under 30 seconds.

Accepted Answer

Rule of 72: \frac{72}{2} = 36 	ext{ years.} True doubling time: \ln 2 / \ln 1.02 = 0.693 / 0.0198 \approx 35.0 years. So the rule reads about one year high, it overstates by roughly 3\% at this low rate. The reason: the "correct" numerator is 100 \ln 2 \approx 69.3 near a 0\% rate, drifting up toward 72 only around 8\%. So at low rates the rule of 70 is sharper: \frac{70}{2} = 35 	ext{ years} \;(	ext{spot on}). Takeaway. The rule of 72 is tuned to be most accurate near 8\%. Below a few percent, switch to 70; well above 15\%, it starts overstating the speed of doubling. Knowing where the sh…

When does the rule of 72 start to lie?

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