A position compounds at 9% a year. Use the rule of 72 for the doubling time, then say how far the rule is from the truth and where it is exact.

A position compounds at 9% a year. Use the rule of 72 for the doubling time, then say how far the rule is from the truth and where it is exact.

Approach: Apply the rule, then compare it against the exact expression for doubling time and describe how the error behaves as the rate moves away from the middle of the range.

8. The rule of 72 gives 72/9 = 8 years. The exact doubling time comes from the logarithm, t = ln(2)/ln(1 + r) = 0.6931/0.08618, which is 8.043 years, so the rule is 0.043 years short, about 0.5% too fast. The constant 72 works because ln(2) is 0.6931 and the ratio ln(1 + r)/r sits near 0.96 across the useful range, giving 0.6931/0.96 = 0.722. The rule is exact at 7.85% and the error grows on both sides: at 2% the truth is 35.0 years against 36 from the rule, and at 20% the truth is 3.80 years against 3.6. Use 70 for continuously compounded rates and 72 for annual rates between 6% and 10%, where the error stays under 1%.

Follow-up: What single constant would you use for a rule that estimates the time to triple, and how accurate is it at 9%?

Key concepts: rule of 72, doubling time, logarithm.