Rule of 72

At a coffee chat, a friend asks: “If my money grows at a steady $8\%$ per year, about how long until it doubles?” You don’t have a calculator. Give a quick estimate.

Answer

Let $$x$$ be the number of years. We want

\[ (1+r)^x = 2. \]

Taking logs,

x=\frac{\ln 2}{\ln(1+r)}.

For small $$r$$ ,

\ln(1+r)\approx r,

x\approx \frac{\ln 2}{r}.

Since $\ln 2\approx 0.69$ , we round to $$0.72$$ for easy mental math (72 has many divisors: $2,3,4,6,8,9,\dots$ ). Hence

x\approx \frac{72}{8}=9.

General formula is given by:

\text{Time to double}\approx \frac{72}{r},

and it works best for rates roughly between $5\%$ and $10\%$ (reasonable rates).