Nicole B. answered • 03/23/21
Making High School Math Fun
This is an example of exponential growth. I will explain the algebra way a little later, but let's reason our way through this. First. you have $200 in year 0. Then, in year 1, you double it, so you have $400. You double it again, that's $800. What are you multiplying by each time to double it? It's 2. So, we multiply by 2 for every year that it doubles. That's $200*2*2*2*2*2.... for 16 years. How many times do we multiply by 2? 16 times. So, it's $200*2^16 (that little carrot means that the 16 is an exponent).
Now, let's look at this with algebra. So, you start with some amount (let's call it "a"). Then, you multiply it by your growth rate (let's call that "b"). You do this over time (we can call time "x"). So, we set it up as a*b^x (remember the carrot means that x is an exponent). a is our initial amount (or the amount we started with). b is the growth rate (or the number we multiply by over and over) and x is the time.
For this problem, a is $200 and b is 2, so it's 200*2^x. We plug in 16 for x because it's over 16 years, and we get where we were before.
Here's a side note on growth rate...a rate of 1 means that we have 100% of what we started with. So, it's not growing or shrinking. So, a rate of 2 means that we have 200% of what we started with (or double). Also, a rate of 0.5 would mean that we have 50% of what we started with, or half. So, if something was growing at a rate of 5%, we would have 105% of what we started with...which is a growth rate of 1.05.
Hope that helps!
