Doubling time

Examination

Examining the doubling time can give a more intuitive sense of the long-term impact of growth than simply viewing the percentage growth rate.

For a constant growth rate of r % within time t, the formula for the doubling time T_d is given by

${\displaystyle T_d=t\frac{\ln (2)}{\ln (1+\frac{r}{100})}}$

A common rule-of-thumb can be derived by Taylor series expanding the denominator ln(1+x) for x=0 using ${\displaystyle \ln (1+x)\approx x}$ and ignoring higher order terms (this corresponds to the assumption that ${\displaystyle r/100\ll 1}$, so it is a better approximation for smaller r):

${\displaystyle T_d=t\frac{\ln 2}{\frac{r}{100}}=t\frac{100\ln 2}{r}}$

${\displaystyle \approx t\frac{69.3}{r}\approx t\frac{70}{r}}$

This "Rule of 70" gives accurate doubling times to within 10% for growth rates less than 25% and within 20% for rates less than 60%. Larger growth rates result in the rule underestimating the doubling time by a larger margin.

Some doubling times calculated with this formula are shown in this table.

Simple doubling time formula:

${\displaystyle N(t)=N_0{2}^{t/T_d}}$

where

• N(t) = the number of objects at time t
• T_d = doubling period (time it takes for object to double in number)
• N₀ = initial number of objects
• t = time

For example, with an annual growth rate of 4.8% the doubling time is 14.78 years, and a doubling time of 10 years corresponds to a growth rate between 7% and 7.5% (actually about 7.18%).

When applied to the constant growth in consumption of a resource, the total amount consumed in one doubling period equals the total amount consumed in all previous periods. This enabled U.S. President Jimmy Carter to note in a speech in 1977 that in each of the previous two decades the world had used more oil than in all of previous history (The roughly exponential growth in world oil consumption between 1950 and 1970 had a doubling period of under a decade).

Given two measurements of a growing quantity, q₁ at time t₁ and q₂ at time t₂, and assuming a constant growth rate, the doubling time can be calculated as

${\displaystyle T_d=(t_2-t_1)\cdot \frac{\ln (2)}{\ln (\frac{q_2}{q_1})}.}$

Related concepts

The equivalent concept to doubling time for a material undergoing a constant negative relative growth rate or exponential decay is the half-life.

The equivalent concept in base-e is e-folding.

Cell culture doubling time

Cell doubling time can be calculated in the following way using growth rate (amount of doubling in one unit of time)

Growth rate:

${\displaystyle N(t)=N_0{e}^{rt}}$

or

${\displaystyle r=\frac{\ln \left(N(t)/N_0\right)}{t}}$

where

• ${\displaystyle N(t)}$ = the number of cells at time t
• ${\displaystyle N_0}$ = the number of cells at time 0
• ${\displaystyle r}$ = growth rate
• ${\displaystyle t}$ = time (usually in hours)

Doubling time:

${\displaystyle \text{doubling time}=\frac{\ln (2)}{\text{growth rate}}}$

The following is the known doubling time for the following cells: