# Doubling Time: Definition & Calculation

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• 0:04 Doubling Time
• 0:58 The Rule of 70
• 1:35 Use and Implications
• 3:36 Lesson Summary
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Lesson Transcript
Instructor
Christopher Muscato

Chris has a master's degree in history and teaches at the University of Northern Colorado.

Expert Contributor
Kathryn Boddie

Kathryn earned her Ph.D. in Mathematics from UW-Milwaukee in 2019. She has over 10 years of teaching experience at high school and university level.

How long would it take a population to double? That's an important question, and in this lesson, we are going to explore one of the easy ways that statisticians can calculate that number.

## Doubling Time

There are a lot of people in this world, and that number just keeps increasing. Wouldn't it be nice if we had a simple formula to help us predict just how large our population could actually get? Well, we do. Doubling time is the amount of time it takes for a value to double itself at a consistent rate of growth. It can be applied to any value that increases at a consistent rate, but we very often use it to study human population growth.

Around the world, people in similar situations tend to reproduce at similar rates, and so the rates of growth throughout human history have been surprisingly consistent. So, ideas like doubling time are very useful to help us prepare for the future. What sort of infrastructure will we need? How much space will a population need? What amount of resources will they use? These questions all require an understanding of future population sizes. So, this is an important field of study. Let's get to it, on the double.

## The Rule of 70

To figure out how long it would take a population to double at a single rate of growth, we can use a simple formula known as the Rule of 70. Basically, you can find the doubling time (in years) by dividing 70 by the annual growth rate.

Imagine that we have a population growing at a rate of 4% per year, which is a pretty high rate of growth. By the Rule of 70, we know that the doubling time (dt) is equal to 70 divided by the growth rate (r). That means our formula would look like this:

• dt = 70 / r
• dt = 70 / 4
• dt = 17.5

## Use and Implications

So, now we know that at an annual growth rate of 4%, a population will double in 17.5 years. It's very important to note that when plugging in the formula, you enter the growth rate as a whole number, not as a decimal. Even though 4% is usually seen in equations as .04, for example, in this case you just enter it as 4.

This same formula can be used in reverse. Say you want to know what rate of population growth would have to be sustained in order to double your population in 10 years. In this case, we know the doubling time (10 years), but not the annual growth rate. So, let's plug that back into the formula.

• 10 = 70 / r
• 10r = 70
• r = 70 / 10
• r = 7

So, if you want to double your population in 10 years, you have to maintain an annual growth rate of 7%.

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## Practice Problems - Doubling Time

In the following problems, students will calculate the doubling time for different populations using the rule of 70. The annual rate of growth will also be solved for given the doubling time of the population.

## Practice Problems

1. A population of rabbits grows at an annual growth rate of 14%. What is the doubling time of the rabbit population? How long will it take for the rabbit population to quadruple?

2. If a population of people has a doubling time of 19 years, what is the annual growth rate of the population? Round to 3 decimal places.

3. Your friend calculated the doubling time for a population with annual growth rate of 5% as follows: dt = 70 / 0.05 = 1,400 years. What is wrong with this calculation? What is the correct doubling time?

## Solutions

1. To calculate the doubling time using the rule of 70, we have dt = 70 / 14 = 5. So the doubling time of the rabbit population is 5 years. This means that the population doubles every 5 years and so the population will quadruple (double twice) in 5*2 = 10 years.

2. Using the formula dt = 70 / r we can solve for the annual growth rate. We have 19 = 70 / r and so 19r = 70. Dividing by 19 gives r = 70 / 19 which is about 3.684%.

3. Your friend mistakenly converted the percentage to its decimal form, 0.05, rather than using 5 in the calculation. The correct doubling time is dt = 70 / 5 = 14 years.

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