How long before we double or triple?

A common question whenever we are dealing with a quantity experiencing constant growth; i.e., the growth rate remains constant over time, is after how many growth periods that quantity will double? For instance:
• If my company grows at x percent per year, when will it have doubled?
• If I take on a debt at x percent annual interest, and I do not pay down my debt, when will it have doubled?
• If my farm animals eat this amount of food per year and my herd grows x percent per year, when will the herd consume double the amount of food?
• If I want to double my revenue in 10 years, how much will my revenue —on average— have to grow to make that target? (about 7%).

The formula for computing the so-called doubling period depends on the type of growth we are dealing with:
• For so-called discrete growth processes; i.e., processes where growth occurs once per period; e.g., interest being paid out on a bank account once per month or year, the formula is:
  T = ln(2) / ln(1 + r)^*, where
  • T: the number of periods needed to double the process starting value,
  • r: the process growth rate

• For so-called continuous growth processes; i.e., processes where growth occurs all the time; e.g., business or organic growth, the formula is:
  T = ln(2) / r^*

The '2' in the numerator of these formulas represents the 'doubling.' To compute the tripling, quadrupling, or other start value multiplier, we use the associated numbers instead (3 for tripling, 4 for quadrupling, etc. Hence, we can generalize the above formulas as follows:
• For discrete growth processes:
  T = ln(m) / ln(1 + r), where:
  • m: the desired multiplier of the starting value.

• For continuous growth processes:
  T = ln(m) / r

We can, of course, also compute the value of a process after it has grown a certain number of periods:

• For discrete growth processes with growth rate r:
  V_t = V₀ × (1 + r)^t, where
  • V_t: value of the process after t periods.
  • V_o: process starting value (t = 0).
• For continuous growth processes with growth rate r:
  V_t = V₀ × e^rt

Below you can play with these as follows:

• Select the type of process to model: Discrete or Continuous.
• Specify a Start value (V₀); i.e., the value the process has before it starts growing.
• Specify a Multiplier (m) value.
• Specify a Low and High growth rate.
• Specify for how many growth rates (n), equally spaced between the low and high growth rates, to compute the growth trajectory.
• Hit Compute!

Process type: Discrete Continuous Start value (V0): (0 < V0 ≤ 1000):    Multiplier (m): (0 < m ≤ 10): Low growth rate (percent): (0 < percent ≤ 100):    High growth rate (percent): (0 < percent ≤ 100): How many growth rates (n)? (2 ≤ n ≤ 10): Compute!

^* For percentage-based growth rates; e.g., 5% or 10%, use percent / 100; i.e., .05 or .10.