Doubling time calculator
Find how long growth takes to double (or triple), and compare the exact answer with the rule of 72.
A clear answer. Here’s how.
With growth compounding once per period, the time to multiply by M is ln(M) ÷ ln(1 + r). The rules of 72 and 70 estimate the doubling time as 72 ÷ r or 70 ÷ r with r in percent: they are mental-arithmetic shortcuts. 100 × ln(M) ÷ r is the exact answer for continuous growth and a good approximation for small rates. Solving for the rate uses r = M^(1 ÷ time) − 1. Periods can be years, months or any other interval, as long as the rate uses the same one.
How this is checked
4 reference checks passed These 4 cases were run when this page was built, on 20 October 2026, and all reproduce within the stated tolerance. The same cases are part of the automated tests that run on every change to Toolmoz. Engine version 1.
See the checks and where each expected answer comes from
Case 1
- Inputs
- Solve for timeGrowth rate per period (%) 7Growth multiple 2
- Expected, and what Toolmoz gave
- Exact time (periods) expected 10.244768, Toolmoz 10.244768351058712Rule of 72 estimate (periods) expected 10.2857, Toolmoz 10.285714285714286
- Where the expected answer comes from
- ln 2 ÷ ln 1.07 = 0.693147 ÷ 0.067659 = 10.2448 and 72 ÷ 7 = 10.2857, in 50-digit decimal arithmetic
- Allowed difference
- ±0.0001 absolute
Case 2
- Inputs
- Solve for timeGrowth rate per period (%) 100Growth multiple 2
- Expected, and what Toolmoz gave
- Exact time (periods) expected 1, Toolmoz 1
- Where the expected answer comes from
- Growth of 100% a period exactly doubles in one period
- Allowed difference
- ±1e-12 absolute
Case 3
- Inputs
- Solve for rateTime allowed (periods) 10Growth multiple 2
- Expected, and what Toolmoz gave
- Growth rate needed per period expected 7.177346, Toolmoz 7.177346253629313
- Where the expected answer comes from
- 2^(1 ÷ 10) − 1 = 1.071773463 − 1 = 7.1773%, from the definition of compound growth
- Allowed difference
- ±0.00001 absolute
Case 4
- Inputs
- Solve for timeGrowth rate per period (%) 8Growth multiple 3
- Expected, and what Toolmoz gave
- Exact time (periods) expected 14.274915, Toolmoz 14.274914586003176Rule of 72 estimate (periods) expected Only applies to doubling, Toolmoz Only applies to doubling
- Where the expected answer comes from
- ln 3 ÷ ln 1.08 = 1.098612 ÷ 0.076961 = 14.2749, and the rule of 72 does not apply to tripling
- Allowed difference
- ±0.001 absolute
What this shows: the engine reproduces independently worked values for these cases. It does not prove every possible input is right. To test an answer you were given, see how to check a number; to see what is and is not checked across Toolmoz, see the accuracy ledger.
See a worked example
Worked example
A fixed demonstration, separate from the values you enter above.
- Solve for
- Time to reach the multiple, given the growth rate
- Growth rate per period (%)
- 7
- Growth multiple
- 2
Example result
- Exact time (periods)
- 10.2448
- Rule of 72 estimate (periods)
- 10.2857
- Rule of 70 estimate (periods)
- 10
- Continuous-growth estimate, 100 × ln(multiple) ÷ rate (periods)
- 9.9021
- Rule of 72 error
- 0.4%
Exact time = ln(multiple) ÷ ln(1 + rate), with compounding once per period. The rules of 72 and 70 are mental-arithmetic shortcuts for doubling; 72 is most accurate near 8% and has many whole-number divisors.