Rule of 70 You have to know this for the test – and to compound daily interest on a savings account!

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.) Formula 70/r Where r = rate of increase

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.

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...

Complete the chart Year#hectares at start of year #hectares cleared Total hectares at end of year

Complete the chart Year#hectares at start of year #hectares cleared Total hectares at end of year

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.

Linear vs Exponential "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!

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!)

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 new goats ) = 350 goats?

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

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 = 968 goats on Jan 1, 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!

Year# Goats on Jan 1Goats added (10%) during the year 1 (2002) (2009 # of goats doubled)

Finish the calculations until year 26 (December 31, 2025)

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!

Exponential growth: Farmer Jones and his goats. This is exponential growth. Each year the total increases by a certain percent. This is very different from linear growth!

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.

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.