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

Following is an exponential decay function: * *

y = a(1-b)^{x}

^{}where:

- "y" is the final amount remaining after the decay over a period of time
- "a" is the original amount
- "x" represents time
- The decay factor is (1-b).
- The variable, b, is the percent change in decimal form.

Because this is an exponential decay factor, this article focuses on percent decrease.

### Ways to Find Percent Decrease

Three examples help illustrate ways to find percent decrease:

#### Percent Decrease Is Mentioned in the Story

Greece is experiencing tremendous financial strain because it owes more money than it 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 percent.

- What is the percent decrease, b, of Greece's spending? 20 percent
- What is the decay factor of Greece's spending?

Decay factor:

(1 - b) = (1 - .20) = (.80)

#### Percent Decrease Is Expressed in a Function

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" means billions of dollars, and "x" represents the number of years since 2009.

- What is the percent decrease, b, of Greece's annual debt? 30 percent
- What is the decay factor of Greece's annual debt?

Decay factor:

(1 - b) = (1 - .30) = .70

#### Percent Decrease Is Hidden in a Set of Data

After Greece reduces government services and salaries, imagine that this data details Greece's projected annual debt.

- 2009: $500 billion
- 2010: $475 billion
- 2011: $451.25 billion
- 2012: $428.69 billion

### How to Calculate Percent Decrease

A. Pick two 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 percent

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

(451.25 - 428.69) / 451.25 is approximately .05 or 5 percent

### Percent Decrease in Real Life

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 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 would force U.S. citizens and residents to cut back on the salt they consume. What if the salt-reduction law passed, and Americans began to consume less of the mineral?

Suppose that each year, restaurants were mandated to decrease sodium levels by 2.5 percent annually, beginning in 2017. 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 would be worth its salt. Americans would be afflicted with fewer strokes. Here are fictional projections for annual strokes in America:

- 2016: 7,000,000 strokes
- 2017: 6,650,000 strokes
- 2018: 6,317,500 strokes
- 2019: 6,001,625 strokes

### Sample Questions

**What is the mandated percent decrease in salt consumption in restaurants?**

Answer: 2.5 percent

Explanation: Three different things-sodium levels, heart attacks, and strokes-are predicted to decrease. Each year, restaurants were mandated to decrease sodium levels by 2.5 percent annually, beginning in 2017.

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

Answer: .975

Explanation: Decay factor:

(1 - b) = (1 - .025) = .975

**Based on predictions, what would be the percent decrease for annual heart attacks?**

Answer: 10 percent

Explanation: 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.

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

Answer: .90

Explanation: Decay factor:

(1 - b) = (1 - .10) = .90

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

Answer: 5 percent

Explanation:

A. Choose data for two consecutive years: 2016: 7,000,000 strokes; 2017: 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 percent

C. Check for consistency and choose data for another set of consecutive years: 2018: 6,317,500 strokes; 2019: 6,001,625 strokes

Percent decrease = (older - newer) / older

(6,317,500 - 6,001,625) / 6,001,625 approximately .05 or 5 percent

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

Answer: .95

Explanation: Decay factor:

(1 - b) = (1 - .05) = .95

Edited by Anne Marie Helmenstine, Ph.D.