These types of word problems take the concept of percent change even further. In questions involving populations growing in size or the diminishing price of a car over time, you need to perform percent-change operations repeatedly. Solving these problems would be time-consuming without exponents. Here’s an example:
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If a population of 100 grows by 5% per year, how great will the population be in 50 years? | |
To answer this question, you might start by calculating the population after one year:
Or use the faster method we discussed in percent increase:
After the second year, the population will have grown to:
And so on and so on for 48 more years. You may already see the shortcut you can use to avoid having to do, in this case, 50 separate calculations. The final answer is simply:
In general, quantities like the one described in this problem are said to be growing exponentially. The formula for calculating how much an exponential quantity will grow in a specific number of years is:
Exponential decay is mathematically equivalent to negative exponential growth. But instead of a quantity growing at a constant percentage, the quantity shrinks at a constant percentage. Exponential decay is a repeated percent decrease. That is why the formulas that model these two situations are so similar. To calculate exponential decay:
The only difference between the two equations is that the base of the exponent is less than 1, because during each unit of time the original amount is reduced by a fixed percentage. Exponential decay is often used to model population decreases, as well as the decay of physical mass.
Let’s work through a few example problems to get a feel for both exponential growth and decay problems.
Simple Exponential Growth Problems
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A population of bacteria grows by 35% every hour. If the population begins with 100 specimens, how many are there after 6 hours? | |
The question, with its growing population of bacteria, makes it quite clear that this is an exponential growth problem. To solve the problem, you just need to plug the appropriate values into the formula for a repeated percent increase. The rate is .035, the original amount is 100, and the time is 6 hours:
Simple Exponential Decay Problem
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A fully inflated beach ball loses 6% of its air every day. If the beach ball originally contains 4000 cubic centimeters of air, how many cubic centimeters does it hold after 10 days? | |
Since the beach ball loses air, we know this is an exponential decay problem. The decay rate is .06, the original amount is 4000 cubic centimeters of air, and the time is 10. Plugging theinto the formula:
