# How do you find the multiplier for the rate of exponential decay?

The very quick answer: Exponential decay problems will give you a decay rate percentage. **Multiplier = (100 – percent given)##-:##100.**

Understand the process explained below before going straight to the formula.

Finding the multiplier means first knowing what a multiplier is.

**Definition: A multiplier is the amount of stuff that remains after the stuff has decayed for a given unit of time.**

This definition might be confusing since it is wordy. To understand that definition and how to apply it mathematically, let’s do a problem.

**A town’s population is decreasing at a rate of 5% per year. What is the decay multiplier?**

ANSWER: In this problem, although I haven’t told you the starting size of the population, you can still say the starting population has 100% of its people. So before any time has passed, no decay has occurred, and the population is at its fullest, or 100%.

Next, the “town’s population is decreasing by 5% each year”. Thus, after 1 year, the city’s population will be 5% less of the original population. Well, 5% less than 100% is: ##100%-5%=95##%. Multipliers are written in decimal form, so the decimal form of 95% is: ##95-:100= 0.95##.

Here’s the definition of multiplier again: **A multiplier is the amount of stuff that remains after the stuff has decayed for the given amount of time.**

In this problem, the “given unit of time” is a year (as opposed to a second or an hour). After a year of decay, 0.95 of the original population remains. 0.95 is the multiplier.