Hong Life CalculatorsUnderstanding the doubling time calculator — concept and principle
This calculator finds how long a quantity growing at a steady percentage rate takes to double. It works from a rate compounded each period, a continuous rate, or two measurements, and compares the exact answer with the rules of 70 and 72. It also gives the time to grow tenfold. Use it for savings, inflation, population and bacterial growth.
How it is calculated
Rate r compounded once per period: doubling time = ln 2 ÷ ln(1 + r)
Continuous rate k: doubling time = ln 2 ÷ k ≈ 0.693 ÷ k
From two measurements: k = ln(later ÷ start) ÷ elapsed time
Rule of 70 ≈ 70 ÷ rate in %; rule of 72 ≈ 72 ÷ rate in %
Time to grow 10-fold = ln 10 ÷ ln(1 + r)
- The time unit is whatever your period is: years, days or hours.
The principle
In exponential growth the size increases by the same percentage each period, so it takes the same time to double from 100 to 200 as from 1,000 to 2,000. Solving (1 + r)ᵗ = 2 for t gives the formula. Because ln 2 ≈ 0.693, dividing about 70 by the percentage rate gives a quick estimate; 72 is popular for yearly compounding at moderate rates because it fits them a little better and divides evenly by many numbers.
Worked example
Default inputs: growth rate 7 % per period, compounded each period.
- Exact doubling time = ln 2 ÷ ln 1.07 = 0.6931 ÷ 0.06766 = 10.24 periods.
- Rule of 70: 70 ÷ 7 = 10; rule of 72: 72 ÷ 7 = 10.29.
- Error of the rule of 72 = (10.29 − 10.24) ÷ 10.24 ≈ 0.4 %.
- Equivalent continuous rate k = ln 1.07 = 6.766 %.
- Time to grow 10-fold = ln 10 ÷ ln 1.07 = 34.03 periods.
| Growth rate (% per period) | Doubling time (periods) |
|---|---|
| 1.75 | 39.95 |
| 3.5 | 20.15 |
| 5.25 | 13.55 |
| 7 | 10.24 |
| 8.75 | 8.263 |
| 10.5 | 6.942 |
| 14 | 5.29 |
How to read the result
At 7 % a year, money roughly doubles in about 10 years and grows tenfold in about 34. The same logic applies to costs: 3 % inflation halves purchasing power in about 23 years. The rules of thumb are close at moderate rates but drift at very high ones, where the exact formula should be used.
Common mistakes
- Entering 7 % as 7 instead of 0.07 in your own formula (the calculator takes percent).
- Using the rule of 72 for continuous growth, where 70 (or 69.3) is better.
- Assuming growth stays exponential forever; real growth often slows.
- Mixing periods, such as a monthly rate with a yearly answer.
Frequently asked questions
How long does it take money to double at 7 %?
About 10.2 years with yearly compounding.
Why is it called the rule of 72?
Dividing 72 by the percentage rate approximates the doubling time for typical yearly rates.
What is the rule of 70?
The same idea using 70, closer to ln 2 × 100 ≈ 69.3, suited to continuous or small rates.
Sources: OpenStax, College Algebra 2e (2021), §6.7 Exponential and Logarithmic Models