Final answer:
To find the effective annual rate of interest for the two investments, we need to calculate the compound amounts for each investment and set them equal to each other. Solving the equation, we find that the effective annual rate of interest for the investment is 5.67%.
Step-by-step explanation:
To find the effective annual rate of interest for the two investments, we need to calculate the compound amounts for each investment. Let's start with the first investment:
- In the first two years, the interest is compounded quarterly, so we use the formula A = P(1 + r/n)^(nt), where A is the compound amount, P is the principal, r is the nominal annual rate of interest, n is the number of compounding periods per year, and t is the number of years. Plugging in the values, we get A = P(1 + 0.05/4)^(4*2) = P(1.0125)^8.
- In the following three years, the interest is compounded quarterly again, but with a higher rate of 6%. We use the same formula again, but with the new rate, which gives us A = P(1 + 0.06/4)^(4*3) = P(1.015)^12.
For the second investment, the interest is compounded at a constant rate over the five years. Let's call this rate x. Using the formula again, we have A = P(1 + x/4)^(4*5) = P(1 + 0.25x)^20.
Since the compound amounts for both investments are the same at the end of five years, we can set up the equation P(1.0125)^8 * P(1.015)^12 = P(1 + 0.25x)^20. Simplifying the equation and solving for x, we find that x = 5.67%. Therefore, the effective annual rate of interest for the investment is 5.67% (option c).