How come '70' can always be used to approximate the doubling time for a population? Is there some mathematical explanation for this phenomenon?

Name: Rule of 70 to approximate population doubling time
Uploaded: 2021-04-29T19:49:28Z
Description: Practice using the rule of 70 to approximate population doubling time.

This formula is derived from a really well-known general formula to describe exponential change for a function. The formula is ``` Final Value = Initial Value * e^(Rate * Time) ``` Let's take rate = 1% which is 0.01 to test Rule of 70. Let initial population be x. Since final population is initial population doubled, then final population = 2x. 2x = x * e^(0.01t) 2x / x = x * e^(0.01t) / x 2 = e^(0.01t) ln2 = 0.01t t = ln2 / 0.01 t ≈ 69.3147 t ≈ 70 Notice when the rate is for example doubled, simply multiply 0.01 by 2, then it means divide t by half. This equation is also used in various aspect, such as describing rate of nucleus in a radioactive decay.

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Course: AP®︎/College Environmental science > Unit 2

Lesson 3: Human populations

Rule of 70 to approximate population doubling time

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Practice using the rule of 70 to approximate population doubling time. Created by Sal Khan.

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G. Tarun
Posted 2 years ago. Direct link to G. Tarun's post “Is the rate here the annu...”
Is the rate here the annual growth rate? And is it computed by dividing the increase in population (increased value minus initial value) by 100? And finally, are we assuming the population growth rate is stable in projections? Is it really the case that population growth rate is actually stable? Or does it vary? Or does it remain more or less stable to make rounding it off to one rate reliable?
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(3 votes)
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Cecilia
Posted 5 months ago. Direct link to Cecilia's post “How come '70' can always ...”
How come '70' can always be used to approximate the doubling time for a population? Is there some mathematical explanation for this phenomenon?
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- joshua
  Posted 5 months ago. Direct link to joshua's post “This formula is derived f...”
  This formula is derived from a really well-known general formula to describe exponential change for a function.
  
  The formula is
  Final Value = Initial Value * e^(Rate * Time)
  
  Let's take rate = 1% which is 0.01 to test Rule of 70.
  
  Let initial population be x.
  Since final population is initial population doubled, then final population = 2x.
  
  2x = x * e^(0.01t)
  2x / x = x * e^(0.01t) / x
  2 = e^(0.01t)
  ln2 = 0.01t
  t = ln2 / 0.01
  t ≈ 69.3147
  t ≈ 70
  
  Notice when the rate is for example doubled, simply multiply 0.01 by 2, then it means divide t by half.
  
  This equation is also used in various aspect, such as describing rate of nucleus in a radioactive decay.
  Comment on joshua's post “This formula is derived f...”
  (2 votes)

Video transcript

- [Instructor] When we're dealing with population growth rates an interesting question is, how long would it take for a given rate for the population to double? So we're gonna think about doubling time. Now if you were to actually calculate it precisely mathematically precisely, it gets a little bit mathy. You need to use a little bit of logarithms and you'll probably need a calculator, but I did that here in this spreadsheet by calculating the exact doubling time. So this is saying that if a population is growing at 1% a year, it's going to take almost 70 years for that population to double. But if that population is growing at 5% per year then it's going to take a little over 14 years for that population to double. If the population is growing at 10%, we know mathematically it's going to take a little bit over seven years for that population to double. Now, I was able to calculate this as I just mentioned using a little bit of fancy math, but what we see in this next column is there's actually a pretty easy way to approximate doubling time. And this is known as the rule of 70. And the rule of 70 is used in a lot of different areas, a lot of different subjects, people in finance would use it because once again, you're thinking about things growing at a certain percent every year, but you can also use it for things like population growth rates. So what we see with the rule of 70, and let me just write that down, rule of 70 is that you can approximate the doubling time by taking the number 70 and dividing it by the, not actually the percentage, but just the number of the percentage. So for example, this right over here is 70 divided by this one here, which is equal to 70. And notice this 70 is pretty close to 69.7. If you wanted to figure out or you wanted to approximate the doubling time, if the population is growing at 7% a year, well what you would say is, all right, what is 70 divided by seven? Well, that is equal to 10. So this would be your approximation. And if you were to do it in a mathematically precise way, it would be 10.2. So if you're taking, say an AP environmental science course and they're asking you for the, how long it takes for something to double let's say a population that's growing 7% a year, they're probably expecting you to use the rule of 70. So let's say that we have a population that is growing at 14% per year, and that would actually be a very huge growth rate. What I want you to do is pause this video and approximate how long would it take for that population to double? All right, now let's work through this together. So as I mentioned, we're approximating, we don't have to do calculate the exact doubling time. So if we're approximating, it's going to be 70 divided by the rate of growth. So in this situation, this is going to be 70 divided by 14, which is equal to five. So if a population is growing at 14%, it'll take it roughly five years to double.