Doubles in eight years, then what?

Question

An investment doubles in 8 years with annual compounding. Solve in your head. What annual rate is that, and roughly how long would it take to grow tenfold? Aim: under 20 seconds.

Accepted Answer

The rate. Rule of 72 in reverse: rate \approx 72 / 8 = 9\%. (Exact: 2^{1/8} - 1 \approx 9.05\%.) Time to tenfold. Convert 10	imes into doublings. Since 10 \approx 2^{3.32}, growing tenfold is about 3.32 doublings, and each doubling takes 8 years: 3.32 	imes 8 \approx 26.5 	ext{ years}. Sanity check: the exact time is \dfrac{\ln 10}{\ln 1.09} = \dfrac{2.303}{0.0862} \approx 26.7 years. Takeaway. A doubling time pins down the rate (72 / 8 = 9\%) and, reused directly, the time for any multiple: 10	imes \approx 3.32 doublings because 2^{3.32} \approx 10, so about 3.32 doubling periods.

Doubles in eight years, then what?

Your answer

More Mental Math questions