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Author(s): Gregory L. Vogt, EdD, and Nancy P. Moreno, PhD.

Calculating Exponential Growth

Under favorable conditions and with sufficient time and resources, populations of all organisms, including infectious agents like viruses, have the potential to increase dramatically over time. Even slow-growing organisms can reach astounding population sizes if reproduction is unchecked. Charles Darwin used elephants, which breed very slowly, as a hypothetical example. Beginning with two elephants, which generally produce only six offspring during a reproductive span of 60 years, an elephant population would number only 54 individuals after 200 years. However, after 1,000 years, the population would have grown to 86,000,000 elephants!

Now consider another example, in which a parent cell divides into two daughter cells every 10 minutes. After 10 minutes, there would be two cells; after 20 minutes, four cells; after 30 minutes, eight cells, and so on. After three hours, there would be close to one million cells. When quantity increases by a fixed percentage at regular time intervals, we have what is referred to as exponential growth. On a graph, exponential growth is represented by an upward curve, not a straight line. In addition to the example of cell division, exponential growth can be observed in the accumulation of compound interest, and in the increasing levels of CO2 in the atmosphere. Untreated, HIV also is capable of exponential growth once it begins to replicate and spread within the human body.


Lead a class discussion about the meaning of exponential growth, as it relates to HIV. Due to exponential growth, the greater the number of HIV particles present, the faster they will increase in number. Use the following example.

If an HIV particle reproduces itself every minute, at the end of one minute, there will be two particles. After two minutes, there will be four particles; and after 10 minutes, the number will have grown to 1,024. In 20 minutes, there will be more than one million particles, and after 30 minutes, the population will have increased to more than one billion. This is “exponential” growth.

Tell students that there are many examples of exponential growth. Pose the following scenario to the class.

Imagine you have applied for a job. Your future employer offers a temporary position lasting just 30 days. Then, something amazing happens: you’re asked to decide if you’d rather be paid in dollars or pennies. 

If you choose to be paid in dollars, you will earn $1,000 on the first day of work, $2,000 on your second day, $3,000 on the third, and so on. For each of your 30 days of employment, your salary will be increased by $1,000. 

If you choose to be paid in pennies, you will earn one cent on the first day of work, two cents on your second day, four cents on the third day, and so on. Each day, your will salary will be exactly double the salary you earned the day before. Which payment plan will you select?

Give each student group the “Dollars or Cents” page, which includes the challenge just described. Allow time for students to discuss the options and select one of the job’s two possible “pay schedules.” Have students calculate their daily salaries, total income earned so far at the end of each day, and the amount of money they will earn for the full 30-day period.

Compare the final balances accrued by each salary schedule. If required for clarification, share the following information with students (also see the answer sheet at the end of this activity).

Being paid in dollars certainly seems like the smart choice. In just five days, you will earn $15,000. By the end of the next five days, your salary will reach $55,000. Adding $1,000 to your salary each day quickly builds up to a 30-day grand total of $465,000! Not bad for a temporary job.

On the other hand, it takes a lot of discipline (and quick calculations!) to choose to be paid in pennies. Initially, the pay will be dismal. By day 10, you will have only earned a total of only $10.23. It takes three weeks before your salary begins to pick up. On day 20, you will have earned $10.485.75. And from that point on, salary growth becomes spectacular. Just five days later, your salary will pass $335,000. By day 30, you will have earned $10,737,417.61!

Revisit your previous discussion of HIV replication. Ask students to explain how the salary analogy applies to virus multiplication within cells in the body. Or, ask each group of students to summarize what they learned about exponential growth by writing a paragraph in their science notebooks or as a homework assignment.

Funded by the following grant(s)

Science Education Partnership Award, NIH

Grant Number: 5R25RR018605