This doubling time calculator finds how long anything growing at a constant percentage rate takes to double — the exact answer ln2/ln(1+r), with the famous Rule of 70 and Rule of 72 estimates beside it so you can see what the shortcuts cost.

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The Clock Inside Every Exponential

Ask "how fast is it growing?" and the percentage is the answer; ask "how long until it's twice as big?" and you get the more visceral number — the doubling time. For growth at rate r per period, compounded, it is T = ln 2 / ln(1 + r): at 7%, ln2/ln(1.07) = 10.24 years. The folk rules approximate it: 70/r gives 10.0, 72/r gives 10.3 — bracketing the truth, which is why both survived: 70 (from ln2 ≈ 69.3, the continuous-compounding constant) is the better estimate at small rates, 72 at the 6–10% rates of finance, where its error is a rounding whisker and its divisibility (2, 3, 4, 6, 8, 9, 12) makes it mental arithmetic. The deeper fact the formula encodes is scale-independence: doubling time has no term for the amount, because exponential growth's character is entirely in the rate. A colony, a debt, a data set, a rumor — at a steady percentage, each doubles on the same clock regardless of current size, and each doubling adds as much as ALL previous growth combined. That last property is the one intuition chronically misses, and the one every doubling-time computation whispers.

The applications are wherever exponentials hide: populations (the world at ~0.9%/yr doubles in ~77 years; a city at 3%, in 23.4), epidemics in their early phase (cases doubling every 3 days is r ≈ 26%/day — the number that governed 2020's headlines), computing (Moore's law was a doubling-time observation, ~2 years), and debt, where doubling time is the creditor's clock against you (a 24% credit card balance, minimum payments aside, doubles in ~3.2 years). The mirror concept for decay is the half-life, T½ = ln(½)/ln(1+r) for negative r — same logarithm, opposite sign, and the calculator names it when you enter a negative rate. Two disciplines guard the tool: constant rate is an assumption, not a law — real growth saturates (logistic curves bend every exponential eventually), so a doubling time is a statement about the CURRENT regime; and periods must match the rate's period (an annual 7% uses years; monthly compounding at 0.55%/mo gives the same economy a doubling of 126 months = 10.5 years — units are the silent half of every answer).

Worked Examples (Verified)

  • Defaults: 7% → exact 10.24 periods (Rule of 70: 10.0; Rule of 72: 10.29).
  • Savings-account reality: 2% → 35.0 years (69.3/2 ≈ 34.7 continuous; exact discrete 35.00).
  • Brisk startup growth: 20% → 3.80 periods.
  • Hyper-growth: 100% → exactly 1 period — the definition, checking itself.
  • Slow population: 1% → 69.66 years — why 70 is the population literature's constant.

Doubling Time Discipline

  • Match the period: rate per year → answer in years. A monthly rate with an annual answer is the classic silent error.
  • Discrete vs continuous: ln2/ln(1+r) assumes per-period compounding; continuous growth uses ln2/r — the calculator's exact figure is the discrete one (the Rule-of-70 note shows the continuous constant).
  • A doubling time is a regime statement: extrapolated saturation-free growth eventually breaks physics, markets and petri dishes alike.
  • For decline, the same mathematics gives half-life — ask the decay question with the decay tool's signs.
  • Rule selection: 69.3/70 for continuous or low rates, 72 for 6–10% financial rates, 70 for populations by convention.

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Frequently Asked Questions

How do you calculate doubling time?

Exactly: T = ln 2 ÷ ln(1 + r), with r the growth rate per period as a decimal. At 7%: 0.6931 ÷ 0.06766 = 10.24 periods. The shortcuts: Rule of 70 (70 ÷ rate-in-percent) and Rule of 72 (72 ÷ rate) bracket the exact answer for quick mental work. This calculator returns all three — exact for the record, rules for the intuition — for any positive rate, and explains the decay (half-life) version when the rate is negative.

Should I use the Rule of 70 or the Rule of 72?

Depends on the rate and the flavor of growth. The mathematically pure constant is 69.3 (100 × ln 2), exact for CONTINUOUS compounding; 70 is its rounded form and the population/biology convention, most accurate at low rates (error at 2%: 35.0 exact vs 35.0 by 70 — nearly perfect). 72 compensates for discrete annual compounding and shines at 6–10% (at 8%: exact 9.01, rule of 72 gives 9.0 exactly). Pick by regime, or sidestep the choice: this page's exact figure is one glance away.

Why doesn't doubling time depend on the starting amount?

Because exponential growth multiplies: after time T the amount is A(1+r)^T, and setting the multiplier to 2 cancels A entirely. A gram of bacteria and a tonne of bacteria at the same rate double together. This scale-freedom is what makes doubling time the right unit for comparing growth across wildly different systems — and the source of its menace: a problem that has doubled quietly ten times is 1,024 times bigger, and the eleventh doubling adds more than the previous ten combined, whatever the units.

What is the doubling time of the world population?

At the recent growth rate of roughly 0.9% per year, about 77 years — down from ~35 years at the 1960s peak rate of ~2.1%. The trend itself is the story: the RATE has been falling for six decades, so the doubling time keeps lengthening, and demographic projections have the population peaking (~10.4 billion in the 2080s on UN median figures) rather than doubling again at all. Doubling time applied honestly includes watching the rate move — the calculator gives the current regime's clock, not a prophecy.

How does doubling time relate to half-life?

Same formula, opposite direction: half-life is the doubling time of a quantity growing at a negative rate — T½ = ln(0.5)/ln(1 + r) with r < 0, or equivalently ln2/|decay constant|. Carbon-14's 5,730-year half-life and a drug's 6-hour half-life are doubling-time arithmetic read in reverse. Enter a negative rate here and the calculator points you to the half-life form; the underlying logarithm doesn't care which way time's arrow of quantity points.

What doubling time does Moore's law imply?

About 2 years per doubling of transistor density, sustained from the 1970s through the 2010s — equivalent to a steady ~41% annual growth rate, an almost unheard-of regime for a physical industry, maintained for five decades. It's the canonical demonstration that a modest-looking doubling time, COMPOUNDED, is destiny: 25 doublings ≈ 33 million-fold. The same math in reverse explains why the law's slowing (to 3+ years recently) changes computing economics so profoundly — the exponent is the product.

How do I double money at a given interest rate?

Read the calculator's answer as your horizon: at 6% compounded annually, money doubles in 11.90 years (Rule of 72: 12 — close enough for planning); at 9%, 8.04 years. Then the caveats that separate arithmetic from outcomes: inflation taxes the nominal doubling (at 3% inflation, the REAL doubling at 6% nominal takes ~24 years, governed by the ~3% real rate), taxes and fees shave the effective rate, and no market return is the constant r the formula assumes — sequence matters in life, even when the endpoint math is clean.

Is this doubling time calculator free?

Yes — free, instant, no sign-up. Enter a growth rate per period for the exact doubling time plus the Rule of 70 and Rule of 72 estimates.