Exponential Decay and Percent Change

How to Calculate a Decay Factor

Exponential decay may be calculated using a decay factor.
Exponential decay may be calculated using a decay factor. Andrey Prokhorov, Getty Images

When an original amount is reduced by a consistent rate over a period of time, exponential decay is happening. Here is an explanation of how to work a consistent rate problem or calculate the decay factor. The key to understanding the decay factor is learning about percent change.

Here’s an exponential decay function:  

y = a(1-b)x

  • y: Final amount remaining after the decay over a period of time
  • a: The original amount
  • x: Time
  • The decay factor is (1-b).
  • The variable, b, is percent change in decimal form. Because this is an exponential decay factor, this article focuses on percent decrease.

Three Ways to Find Percent Decrease

  1. The percent decrease is mentioned in the story.
  2. The percent decrease is expressed in a function.
  3. The percent decrease is hidden in a set of data.

1. The percent decrease is mentioned in the story.

Example: The country of Greece is experiencing tremendous financial strain. They owe more money than they can repay. As a result, the Greek government is trying to reduce how much it spends. Imagine that an expert has told Greek leaders that they must cut spending by 20%.

  • What is the percent decrease, b, of Greece’s spending?  20%
  • What is the decay factor of Greece’s spending?
    Decay factor: (1 –b) =  (1 - .20) = (.80)

2. The percent decrease is expressed in a function.

Example:  As Greece reduces its government spending, experts predict that the country’s debt will decline.

Imagine if the country’s annual debt could be modeled by this function: 

y = 500(1-.30)x, where y is in billions of dollars, and x represents the number of years since 2009

  • What is the percent decrease, b, of Greece’s annual debt? 30%
  • What is the decay factor of Greece’s annual debt?
    Decay factor: (1 –b) = (1 - .30) = .70

    3. The percent decrease is hidden in a set of data.

    Example:  After Greece reduces government services and salaries, imagine that this data details the country’s projected annual debt.

    Greece’s Annual Debt

    • 2009: $500 Billion
    • 2010: $475 Billion
    • 2011:  $451.25 Billion
    • 2012: $428.69 Billion

    How to Calculate Percent Decrease

    A. Pick 2 consecutive years to compare: 2009:  $500 Billion; 2010:  $475 Billion

    B. Use this formula:

    Percent decrease  = (older– newer)/older:

    (500 Billion – 475 billion)/500 billion = .05 or 5%

    C. Check for consistency. Pick 2 other consecutive years: 2011: $451.25 Billion; 2012: $428.69 Billion

    (451.25 – 428.69)/451.25 is approximately .05 or 5%

    Percent Decrease in Real Life: Politicians Balk at Salt

    Salt is the glitter of American spice racks. Glitter transforms construction paper and crude drawings into cherished Mother’s Day cards; salt transforms otherwise bland foods into national favorites. The abundance of salt in potato chips, popcorn, and pot pie mesmerizes the taste buds.

    Unfortunately, too much flavor and bling can ruin a good thing. In the hands of heavy-handed adults, excess salt can lead to high blood pressure, heart attacks, and strokes.

    Recently, a lawmaker announced legislation that will force us in the land of the free and the brave to cut back on the salt that we crave.

    What if the salt reduction law passed, and we consumed less of the white stuff?

    Suppose that each year, restaurants will be mandated to decrease sodium levels by 2.5% annually, beginning in 2011. The predicted decline in heart attacks can be described by the following function: 

    y = 10,000,000(1-.10)x , where y represents the annual number of heart attacks after x years.

    Apparently, the legislation will be worth its salt. Americans will be afflicted with fewer strokes.

    Here are my fictional projections for annual strokes in America:

    • 2010: 7,000,000 strokes
    • 2011: 6,650,000 strokes
    • 2012: 6,317,500 strokes
    • 2013: 6,001,625 strokes

    (Note: The numbers were made up to illustrate the math calculation! Please contact your local salt expert or cardiologist for real data.)

    Questions

    1. What is the mandated percent decrease in salt consumption in restaurants?

    Answer: 2.5%
    Explanation:  Be careful, three different things -- sodium levels, heart attacks, and strokes -- are predicted to decrease. Each year, restaurants will be mandated to decrease sodium levels by 2.5% annually, beginning in 2011.

    2. What is the mandated decay factor for salt consumption in restaurants?

    Answer: .975
    Explanation: Decay factor: (1 - b) = (1-.025) = .975

    3. Based on predictions, what will be the percent decrease for annual heart attacks?

    Answer:  10%
    Explanation:  The predicted decline in heart attacks can be described by the following function: 

    y = 10,000,000(1-.10), where y represents the annual number of heart attacks after years.

    4. Based on predictions, what will be the decay factor for annual heart attacks?

    Answer: 0.90
    Explanation: Decay factor: (1 - b) = (1 - 0.10) = 0.90

    5. Based on these fictional projections, what will be the percent decrease for strokes in America?

    Answer:  5%
    Explanation:

    A. Choose data for 2 consecutive years:  2010: 7,000,000 strokes; 2011: 6,650,000 strokes

    B. Use this formula:  Percent decrease = (older – newer) / older

    (7,000,000 – 6,650,000)/7,000,000 = .05 or 5%

    C. Check for consistency and choose data for another set of consecutive years: 2012: 6,317,500 strokes; 2013: 6,001,625 strokes

    Percent decrease  = (older – newer) / older

    (6,317,500 – 6,001,625)/6,001,625 approximately .05 or 5%

    6. Based on these fictional projections, what will be the decay factor for strokes in the America?

    Answer: 0.95
    Explanation: Decay factor: (1 - b) = (1 - 0.05) = 0.95

    Edited by Anne Marie Helmenstine, Ph.D.