Praxis Core Academic Skills for Educators: Mathematics (5733)Algebra and FunctionsMedium
A financial analyst is modeling the value of an investment. The value V(t) in dollars after t years is given by V(t) = 1000(1.05)^t. What is the annual percentage increase of this investment?
- A105%
- B5%
- C1.05%
- D10.5%
Show answer & explanationAnswer & explanation
Correct answer: B. 5%
In an exponential growth model V(t) = P(1 + r)^t, 'r' represents the annual growth rate as a decimal. Here, 1 + r = 1.05, so r = 0.05. Converting this decimal to a percentage gives 5%.
Why the other options are wrong
- A. This implies a 105% increase, which would mean the investment more than doubles each year.
- C. This is incorrect; 1.05 is the growth factor, not the percentage.
- D. This incorrectly adds the 1 to the decimal rate.
Exponential Growth Rate
The percentage by which a quantity increases per unit of time in an exponential growth model.
- Derived from the growth factor 'b' in y = a * b^x, where b = 1 + r.
- To find the rate (r), subtract 1 from the growth factor (b - 1).
- Expressed as a decimal for calculations, converted to percentage for interpretation.
Memory trick: Growth factor is 1 + rate; subtract 1 to find the rate!