ACT (Enhanced)MathematicsMedium
A financial analyst is modeling the growth of an investment. If an initial investment of $5,000 grows at an annual rate of 4% compounded quarterly, which of the following expressions represents the value of the investment after 't' years?
- A5000 * (1 + 0.04/4)^(4t)
- B5000 * (1 + 0.04)^t
- C5000 * (1 + 0.04t)^4
- D5000 * (1 + 4*0.04)^t
Show answer & explanationAnswer & explanation
Correct answer: A. 5000 * (1 + 0.04/4)^(4t)
The formula for compound interest is A = P(1 + r/n)^(nt), where P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years. Here, P=5000, r=0.04, and n=4 (quarterly).
Why the other options are wrong
- B. This represents annual compounding, not quarterly.
- C. This incorrectly places 't' inside the parenthesis and raises it to the power of 4, rather than using 'nt' as the exponent.
- D. This incorrectly multiplies the rate by 'n' before adding to 1, and 'n' is not in the exponent.
Compound Interest Formula
A = P(1 + r/n)^(nt), where A is the future value of the investment/loan, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the time in years.
- Interest is earned on both the principal and accumulated interest.
- Higher 'n' leads to faster growth for the same 'r'.
- Used for investments, loans, and population growth models.
Memory trick: Money grows like a tree, compounding branches over time.