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# How to Calculate Doubling Time and Tripling Time

If a quantity is growing at a constant rate of increase, you can model the process with the exponential function

y(t) = AB^{t},

where A is the initial amount at time t = 0 and B is the growth factor. For example, if a population starts out at 1000 individuals and grows by 5% each year, the equation is

y(t) = 1000(1.05)^{t}.

Given a growth rate of R%, the growth factor is B = 1 + R/100. If you wish to find the doubling time or tripling time of the population, you only need to know the value of B or R.

### Doubling Time

To find the doubling time of a process, you must solve the equation 2A = AB^{t}for t. Dividing both sides by A yields 2 = B

^{t}. If you take the natural logarithm of both sides and isolate t, you arrive at

t = LN(2)/LN(B).

**Example:**A savings account grows at an annual rate of 3.25%. Since R = 3.25, we have B = 1.0325. Thus, the doubling time is

LN(2)/LN(1.0325) = 21.672 years, or about 21 years and 8 months.

### Tripling Time

To find the tripling time of a growth process, you must solve the equation 3A = AB^{t}for t. Again, dividing both sides by A yields 3 = B

^{t}. And if you take the natural logarithm of both sides and isolate t, you arrive at

t = LN(3)/LN(B).

**Example:**A bacterial culture grows at a rate of 4.1% per day. In this case, R = 4.1 and B = 1.041. Thus, the tripling time is

LN(3)/LN(1.041) = 27.341 days, or about 27 days 8 hours and 11 minutes.

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