The Power of Doubling
Exponential Growth
Say, for your allowance, you were offered a choice between these two options:
Option 1: starting with a penny on the first day, and having it double every day (two on the
second, four on the third and so on for 30 days,)
Option 2: starting with $1 with $1 a day increase every day
Which would you choose? If calculated, the first scenario would give you somewhat more than a
million dollars in a month!
Week
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Option
1c 2c 4c 8c 16c 32c 64c $1.28 $2.56 $5.12 $10.24 $20.48 $40.96 $81.92 $163.84 $327.68 $655.36 $1310.72 $2621.44
1
Option
$1 $2 $3 $4 $5
2
$6
$7
$8
$9
$10
$11
$12
$13
$14
$15
$16
Etc.
Although the number of pennies increases slowly, eventually after you keep doubling your number
every day a major increase in growth is seen. That is the power of doubling, or exponential growth.
Exponential growth is defined generically as growth in which the increase of a quantity is proportional
to the size of the quantity that is already there (i.e. 24 and 48). Growth starts out slowly, but
increases greatly in time. All exponential growth shows a characteristic “J” curve when plotted.
True exponential growth occurs at a constant rate (i.e. 4% per year or 1% per month, etc.) per unit
time. We call it fixed rate exponential growth. HOWEVER, growth can still be considered
exponential even if the growth rate changes somewhat over a long period of time, staying fixed for
certain intervals within that period. Over a period of 100 years, the growth rate may vary…the first 10
years it may be fixed at 1%, increase and be fixed at 3.5% for the next 10 years, and then for the next
10 years be fixed at 2%, but it is still exponential growth. The characteristic J shape when the data
is plotted is the main indicator for determining whether growth is exponential or not.
A frequent measure of exponential growth is doubling time. This is the amount of time required for
the quantity to double assuming there is a constant rate of growth within that time period. The shorter
the doubling time is, the faster the rate of growth and the larger the growth rate.
The human population, like all populations of organisms, grows exponentially when unchecked.
Although it took 130 years (from 1800-1930) for the world population to double, it doubled again by
1975, a mere 45 years later!
We are going to demonstrate exponential population growth and determine the doubling time and
growth rate of a simulated population.
Procedure:
1. Label one small paper cup “parents”, and the other one “offspring.”
2. Label the large cup “bean pot.”
3. Place 10 beans in the parent cup and the rest in the bean pot. Each bean represents an
individual in a population.
4. Toss the two pennies:
*both heads up-toss again
*both tails up-one member of parent population has died. Remove a
bean from the parent cup
*one head and one tail-take one bean from the parent cup and one
from the bean pot. Place both in the offspring cup.
5. Continue tossing until there are no longer any beans in the parent cup. Count the number of
beans in the offspring cup. This is the population size of the 1st generation. Record this
number in the Population Data Table.
6. Now the offspring become the parents, so move all of the beans from the offspring cup to the
parents cup and begin generation 2. Repeat steps 5-7 until you have data for 6 generations.
Data and Analysis:
Population Data Table
Assume each generation lasts 30 years.
Generation Number
0 (start of simulation)
1
2
3
4
5
6
Population Size
10
Time (years)
0
30
1. Make a graph of your data, labeling “Time” (years) on the horizontal axis,
“Population Size” on the vertical axis, and a title at the top. Each box should
represent a 10 year increment.
2. Looking at your graph determine if this is considered exponential growth or linear. How
can you tell?
3. From your graph, determine the doubling times at the beginning and the end of your
simulation. In other words, how long does it take at the beginning of your graph for the
population to double? The end?
4. Are the doubling times the same? Explain what this means in relation to growth rates.
5. Use the doubling times from #3 and the Rule of 70 to determine the annual growth rates
at the beginning and end of the simulation (70/% = d.t. or 70/d.t. = %) Remember with
this equation that you don’t move decimal points for the percentages!!! Show your work.
6. The United States’ population is approximately 300 million people, growing at a 1%
growth rate. How many people would be added to the U.S. population in one year? In
other words, how many people will be added in a year if our population grows 1% a year?
In other words, what is 1% of 300 million? Show work.
7. Why might this be a problem? Be specific about its implications.
8. If a farmer stocked his pond with 100 fish in the year 2000 and their population grows at
a fixed rate of 10%, how many fish will he have in the year 2035? Hint: Use the Rule of
70!