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Tip: If you’re in a lab or real-world scenario, use Initial & final values. If you only know a percent growth rate, choose Growth rate (%).

Example: If something grows 5% per year, enter 5 and choose Years.

Continuous is common in science models (N(t)=N₀e^{rt}). Discrete is common in compounding (N(t)=N₀(1+r)^t).

Options:

Chips prefill common doubling-time scenarios and run the calculation.

Result:

No results yet. Enter values and click Calculate.

How to use this calculator

  1. Choose what you know.
    Pick Growth rate, Initial & final values, or Rule of 70/72.
  2. Enter your values.
    Add the rate or the measured values, and select the correct time unit.
  3. Pick a growth model (if needed).
    Use Continuous for exponential models and Discrete for compounding per period.
  4. Click Calculate.
    The calculator returns the doubling time, optional Rule-of-70/72 comparison, and steps.
  5. Use the mini visual to “see” doubling.
    The chart highlights the doubling point so students can connect the math to intuition.

Tip: Try Quick picks to see common examples like 5%/year growth or lab measurements.

How this calculator works

  • Rate mode: converts the percent rate to a decimal and uses td = ln(2)/r (continuous) or td = ln(2)/ln(1+r) (discrete).
  • Values mode: estimates the exponential rate from your data using r = ln(N/N0)/t, then computes doubling time.
  • Rule mode: uses the fast shortcuts 70/% and 72/% and compares them to the exact exponential result.

Reminder: Doubling time only makes sense when growth is roughly exponential (proportional to current size).

Formula & Equation Used

Continuous growth: td = ln(2) / r

Discrete growth: td = ln(2) / ln(1 + r)

Rate from data: r = ln(N / N0) / t

Rule of 70 / 72: td ≈ 70/% or 72/%

Example Problem & Step-by-Step Solution

Example 1 — 5% growth per year (continuous)

  1. Convert percent to decimal rate: r = 5% = 0.05
  2. Use continuous doubling-time formula: td = ln(2)/r
  3. Compute: td = 0.6931 / 0.05 = 13.86 years

Example 2 — Lab data: 100 → 220 in 3 days

  1. Compute growth rate from data: r = ln(N/N₀)/t = ln(220/100)/3
  2. Calculate: ln(2.2) ≈ 0.7885 → r ≈ 0.2628 per day
  3. Doubling time: td = ln(2)/r = 0.6931/0.2628 ≈ 2.64 days

Example 3 — Rule of 70/72 at 8% per year

  1. Rule of 70: td ≈ 70/8 = 8.75 years
  2. Rule of 72: td ≈ 72/8 = 9.00 years
  3. Exact (continuous): td = ln(2)/0.08 ≈ 8.66 years

Frequently Asked Questions

Q: What is doubling time?

Doubling time is the time required for a quantity to become twice as large under exponential growth. It’s commonly modeled using N(t)=N₀e^{rt} or N(t)=N₀(1+r)^t.

Q: What’s the difference between continuous and discrete growth?

Continuous growth assumes the quantity grows at every instant (e^{rt}). Discrete growth assumes growth happens once per period, like compounding interest ((1+r)^t).

Q: When does the Rule of 70/72 work well?

The Rule of 70/72 is a shortcut that’s usually most accurate for small to moderate growth rates. This calculator shows the exact doubling time so you can see the approximation error.

Q: Is doubling time the same as half-life?

They’re related but opposite ideas. Doubling time describes growth, while half-life describes decay. Both use logarithms and exponential models.

Q: Can doubling time change over time?

Yes. If the growth rate changes (due to limited resources, interventions, temperature changes, etc.), the doubling time changes too. This calculator assumes a constant growth rate during the time window.