SATII数学考试考题快递—Common Word Problems(III)

Exponential Growth and Decay

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:


	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


	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


	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 the information into the formula:

More Complicated Exponential Growth Problem


	A bank offers a 4.7% interest rate on all savings accounts, compounded monthly. If 1000 dollars is initially put into a savings account, how much money will the account hold two years later?

This problem is a bit tricky for the simple reason that the interest on the account is compounded monthly. This means that in the 2 years that question refers to, there will be 2

12 = 24 compoundings of interest. The time variable in the equation is affected by these monthly compoundings: it will be 24 instead of 2. Thus, our answer is:

Here’s another compounding problem:


	Sam puts 2000 dollars into a savings account that pays 5% interest compounded annually. Chris puts 2500 dollars into a different savings account that pays 4% annually. After 15 years, whose account will have more money in it, if no more money is added or subtracted from the principal?

Sam’s account will have $2000

1.05¹⁵ ≈ $4157.85 in it after 15 years. Chris’s account will have $2500

1.04¹⁵ ≈ $4502.36 in it. So, Chris’s account will still have more money in it after 15 years. Notice, however, that Sam’s account is gaining on Chris’s account.

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