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Dr McConeghy Environmental Science NotesThe "Rule of 70" for Exponential Growth |
The "Rule of 70" is a math trick borrowed from business where it is used for predicting "future value of an investment."
It says, to calculate the future value of an investment that is increasing exponentially, divide the number 70 by the annual growth rate. The answer is the number of years it will take for the principal to double.
The Rule of 70 is a "rule of thumb" -- it is not exactly accurate -- it is an approximation.
p.s can you decide to get rich ? You can if you understand the rule of 70 and exponential growth.
Some things increase in a "linear" way and some increase "exponentially." The Rule of 70 does not work for all calculations. It works for problems where you have a value increasing by a per cent. The rule of 70 does not work for values that are increasing linearly, where the value is increasing by a set amount.
look below for a more detailed explanation of Linear vs Exponential
Use the "Rule of 70" whenever you have some amount that is increasing by a percent... the Rule of 70 tells you the answer to the question, when will you have twice as much? ... divide the number 70 by the rate of increase. The result is the number of years before the principal "doubles" (you have twice as much.)
OK, this is the kind of problem that any ambitious business person would want to calculate!
Suppose that I am raising valuable goats. As a goat farmer, my main source of income is selling my goats, and my main business asset is my goats... I take good care of my business, so the goats are increasing at the rate of about 10% per year...
My assets are increasing at 10% per year, how many assets (goats) should I expect to have about 20 to 25 years from now?
How are you going to solve this problem.?
Here is the solution... (and an example of the most common error that students make!)
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A Correct answer, or A Common error... You can't multiply the growth rate by the number of years! Instead, Use the Rule of 70 or, Calculate
the whole
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A Common Error Students tend to make an error which leads them into trouble! Can you figure out the future value by multiplying the growth rate times the number of years? Careful! That works for Linear Growth, but not for Exponential Growth... Suppose we try to use that kind of calculation to solve this problem. Farmer Jones Raises Goats Start with 100 goats increasing by 10% (that's exponential) annually for 25 years. OK, 10% of 100 goats is 10 goats. Take 10 goats times 25 years, that gives 250 goats. So, Total Goats = (100 original goats + 250 new goats ) = 350 goats? NO! That's WRONG! It just doesn't work. The answer should be more like 900 to 1000 goats, not 350. You can't do it that way. It doesn't give the right answer! Instead, you have to do it the way described below... Let's Figure the answer by the Rule of 70 The Rule of 70 says, to calculate a future value, divide the number 70 by the rate of increase. The result of that division represents the number of years before you "double" or have twice as much. OK, so we have 100 goats increasing at a rate of 10% per year. First, we divide 70 by rate of increase, namely, 10% 70/10 = 7 Seven what? Seven years before we have twice as much. The result is the number of years to double, that is, 7 years to double. How does this solve our question? We know that he has 100 goats right now. But they will double in 7 years. 100 goats at the start of 2002, double in 7 years, so 200 goats at the start of 2009 and 200 goats in 2009 will double in 7 years, so he will have 400 goats at the start of 2016 and 400 goats in 2016 will double in 7 years, so he will have 800 goats at the start of 2023. If you want to be more accurate, you can say, they are increasing by 10% each year, so if we have 800 goats on January 1, 2023, we can expect to have 80 more, a total of 880 goats on January 1, 2024, and the next year he would add 10% more, so 10% of 880 is 88 more goats, right? That would be a total of 880 + 88 = 968 goats on Jan 1, 2025. We understand that this is an approximation. It is not exactly accurate. Can we get an exact answer? This is tedious to calculate (which is why we want to do Rule of 70!) To get a more exact answer, we have to take the original amount and multiply by 10% each year for 25 years! Like this:
Obviously, if you had calculated the answer by the wrong method, you would be seriously far off! By using the Rule of 70 we can get a quick answer that is fairly accurate, an approximation. How far off is it? About 1 year off. The number for 2024 from the rule of 70 calculation is about equal to the number for 2025 in the long calculation. That is pretty good for a 25 year prediction! For those who are interested, the Rule of 70 is based on logarithms. You may have studied those in algebra or pre-calculus in High School. In this case the Rule of 70 is somewhat inaccurate because of the period of the compounding. It really is intended for financial calculations where the interest of an investment is compounded daily. If the interest is only compounded annually, then the rule gives a growth rate that is a little too fast. We could correct this by using a different number, say 72, 74 or 76 instead of 70. But let's remember, this is just an approximation. We do it because it gives a quick answer to a problem that otherwise would be difficult. We can live with a 4% or 5% error in this kind of problem. |
"Linear growth" means that the original value increases periodically by a set amount.
"Exponential growth" means that the original value increases periodically by a set percentage.
The difference between "amount" and "percentage" is a BIG difference!
Consider the following examples:
Linear Growth: A farmer wants to increase the size of his farm. Farmer Jones clears some land.
He starts with a farm that measures 10 hectares in size. (A "hectare" is a space 100meters by 100meters, about the size of a large soccer field, or about equal to 2 acres. It is the standard measure of land size used around the world.)
Each year he clears off 1 more hectare of new land and makes a new field.
He can only work hard enough to clear off 1 hectare of new field each year. So, each year his farm increases by 1 hectare. The first year he clears 1 hectare and each year after that he clears 1 more...
| Year | #hectares at start of year | #hectares cleared off | total hectares at end of year |
| 1 | 10 to start | 1 | 11 |
| 2 | 11 | 1 | 12 |
| 3 | 12 | 1 | 13 |
| 4 | 13 | 1 | 14 |
| 5 | 14 | 1 | 15 |
So, how much land does he have? In years 6 to 20 he will have 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, and at the end of 20 years he has 30 hectares of land cleared off.
Each year it will increase by 1 hectare. That is, a certain amount. Each year the total increases by a certain amount. Linear growth.
Exponential growth: Farmer Jones and his goats.
Goats!
A farmer wants to raise goats. He starts with a herd of 100 goats. Each year some will die and more new goats will be born. If he takes good care of them, the net total number of goats will increase about 10% each year.
This is exponential growth. Each year the total increases by a certain percent. This is very different from linear growth!
The long way to figure out the size of his goat herd:
Count how many goats he has at the start of the year and calculate 10% of that number. That is how much you can expect the goats to increase by the end of the year. Of course, you must do the calculation again for each new year...
| Year | Goats at start of year | Goats added (10%, rounded off) |
| 1 | 100 |
10 |
| 2 | 110 | 11 |
| 3 | 121 | 12 |
| 4 | 133 | 13 |
| 5 | 146 | 15 |
| 6 | 161 | 16 |
| 7 | 177 | 17 |
so, this is the calculation that we just did above in the discussion of Rule of 70 -- you have about 1000 goats at the end of the 25 years.
Notice! Each year the number of new goats is different -- the goats increase by a set percent not a set amount..
notice that while the farm went from 10 to 30 hectares of cleared land, the goats went from 100 to 1000.
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Lookin' Good!
You see that, given a few years, compound or "exponential" growth will usually outrun linear growth.
Fortunately, when we want to calculate exponential growth, you don't have to calculate it by doing the percentages over and over. There is a simpler way. The Rule of 70.
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