# Exponential Decay in Real Life

Practical Uses of the Formula for Solving Everyday Math Problems

In mathematics, exponential decay occurs when an original amount is reduced by a consistent rate (or percentage of the total) over a period of time. One real-life purpose of this concept is to use the exponential decay function to make predictions about market trends and expectations for impending losses. The exponential decay function can be expressed by the following formula:

y = a(1-b)x
y: final amount remaining after the decay over a period of time
a: original amount
b: percent change in decimal form
x: time

But how often does one find a real world application for this formula? Well, people who work in the fields of finance, science, marketing, and even politics use exponential decay to observe downward trends in markets, sales, populations, and even poll results.

Restaurant owners, goods manufacturers and traders, market researchers, stock salesmen, data analysts, engineers, biology researchers, teachers, mathematicians, accountants, sales representatives, political campaign managers and advisers, and even small business owners rely on the exponential decay formula to inform their investment and loan-taking decisions.

## Percent Decrease in Real Life: Politicians Balk at Salt

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

However, too much of a good thing can be detrimental, especially when it comes to natural resources like salt. As a result, a lawmaker once introduced legislation that would force Americans to cut back on their consumption of salt. It never passed the House, but it still proposed that each year restaurants would be mandated to decrease sodium levels by two and a half percent annually.

In order to understand the implications of reducing salt in restaurants by that amount each year, the exponential decay formula can be used to predict the next five years of salt consumption if we plug in facts and figures into the formula and calculate the results for each iteration.

If all restaurants start out using a collective total of 5,000,000 grams of salt a year in our initial year, and they were asked to reduce their consumption by two and a half percent each year, the results would look something like this:

• 2010: 5,000,000 grams
• 2011: 4,875,000 grams
• 2012: 4,753,125 grams
• 2013: 4,634,297 grams (rounded to nearest gram)
• 2014: 4,518,439 grams (rounded to nearest gram)

By examining this data set, we can see that the amount of salt used goes down consistently by percentage but not by a linear number (such as 125,000, which is how much it is reduced by the first time), and continue to predict the amount restaurants reduce salt consumption by each year infinitely.

## Other Uses and Practical Applications

As mentioned above, there are a number of fields that use the exponential decay (and growth) formula to determine results of consistent business transactions, purchases, and exchanges as well as politicians and anthropologists who study population trends like voting and consumer fads.

People working in finance use the exponential decay formula to help with calculating compound interest on loans taken out and investments being made in order to evaluate whether or not to take those loans or make those investments.

Basically, the exponential decay formula can be used in any situation where an amount of something decreases by the same percentage every iteration of a measurable unit of time—which can include seconds, minutes, hours, months, years, and even decades. As long as you understand how to work with the formula, using the x as the variable for the number of years since Year 0 (the amount before decay occurs).

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