Why it works
Doubling time is exactly ln(2) ÷ ln(1 + r), which is not something anyone computes at a dinner table. Because ln(2) is about 0.693 and ln(1 + r) is close to r for small rates, the relationship collapses to roughly 69.3 ÷ rate. The number 72 is used instead because it divides cleanly by 2, 3, 4, 6, 8, 9 and 12, and it happens to be more accurate in the 6 to 10% range where most real returns sit.
The approximation is excellent between 4% and 15%. Below 3% or above 20% it drifts, which is why this calculator shows the exact figure alongside the shortcut.
Using it in reverse
The rule works both ways. If an investment promises to double your money in three years, divide 72 by 3 to find the implied return: 24% per year, sustained. That is a useful scam detector, because sustained returns at that level are extraordinarily rare.
It also works on inflation. At 6% inflation, prices double in about twelve years, meaning today's 100 becomes 200 for the same basket of goods.