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: Principal $P=12000$, time $t=3$ years, and the interest rates for each scenario.
2. For the first scenario, use the compound interest formula $A=P{(1+\frac{r}{n})}^{nt}$ with $r=0.0096$ and $n=12$ (compounded monthly).
3. Substitute the values into the formula: $A=12000{(1+\frac{0.0096}{12})}^{12\times 3}$.
4. For the second scenario, use the continuous compound interest formula $A=P{e}^{rt}$ with $r=0.0095$.
5. Substitute the values into the formula: $A=12000e^{0.0095\times 3}$.

Key Concepts

Compound Interest

Compound interest refers to the interest calculated on the initial principal and also on the accumulated interest from previous periods. This means that interest is earned on interest, leading to exponential growth of the investment over time. The formula A = P(1 + r/n)^(nt) is used for investments compounded at regular intervals, while A = Pe^(rt) is used for continuous compounding.

Compounding Frequency

Compounding frequency is the number of times interest is calculated and added to the principal balance in a given time period. In the formula A = P(1 + r/n)^(nt), 'n' represents the number of compounding periods per year. More frequent compounding (e.g., monthly vs. annually) generally results in a higher total amount due to the effect of earning interest on previously accrued interest.

Exponential Growth

Exponential growth occurs when the growth rate of a value is proportional to its current value, leading to rapid increases over time. In the context of investments, both compound interest formulas demonstrate how money can grow exponentially due to the reinvestment of interest. Understanding this concept is crucial for comparing different investment options and their potential returns over time.