Use the compound interest formulas A = P (1+ r/n)^nt and A =Pe^rt to solve exercises 53-56. Round answers to the nearest cent. Suppose that you have \$12,000 to invest. Which investment yields the greater return over 3 years: 0.96% compounded monthly or 0.95% compounded continuously?

Verified step by step guidance

1. Identify the given values: the principal amount \(P = 12000\), the time period \(t = 3\) years, the interest rates \(r_1 = 0.0096\) (0.96%) for monthly compounding, and \(r_2 = 0.0095\) (0.95%) for continuous compounding.
2. For the investment compounded monthly, use the formula \(A = P \left(1 + \frac{r}{n}\right)^{nt}\), where \(n = 12\) (months per year). Substitute the values to get \(A_1 = 12000 \left(1 + \frac{0.0096}{12}\right)^{12 \times 3}\).
3. For the investment compounded continuously, use the formula \(A = Pe^{rt}\). Substitute the values to get \(A_2 = 12000 \times e^{0.0095 \times 3}\).
4. Calculate the expressions inside the parentheses and exponents separately for both formulas, but do not compute the final numerical values yet.
5. Compare the two amounts \(A_1\) and \(A_2\) after evaluating the expressions to determine which investment yields the greater return over 3 years.

Key Concepts

Compound Interest Formula (Periodic Compounding)

This formula, A = P(1 + r/n)^(nt), calculates the amount of money accumulated after interest is compounded periodically. Here, P is the principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the time in years. It helps determine the future value of an investment with discrete compounding intervals.

Continuous Compounding Formula

The formula A = Pe^(rt) models interest compounded continuously, where e is Euler’s number (~2.718). This represents the limit of compounding frequency increasing indefinitely. It is used to find the future value when interest is added constantly, providing a slightly higher return than periodic compounding at the same rate.

Comparing Investment Returns

To determine which investment yields a greater return, calculate the final amounts using both compounding methods with the given rates and time. Comparing these results shows which option grows the principal more. Rounding to the nearest cent ensures practical financial interpretation.