Use the compound interest formulas \( A=P\left(1+\frac{r}{n}\right)^{n t} \) and \( A=P e^{r t} \) to answer the question below. Suppose that you have \( \$ 9000 \) to invest. Compare the following two investments. - \( 4.6 \% \) compounded monthly (which is 12 times per year) or \( 4.59 \% \) compounded continuously Which investment yields the greater return over 8 years? Select one: \( \$ 9000 \) invested at \( 4.6 \% \) compounded monthly over 8 years yields the greater return. \( \$ 9000 \) invested at \( 4.59 \% \) compounded continously over 8 years yields the greater return. Both investment plans yield the same return.

Let’s kick things off by calculating the returns for both investment options! For the first option, using the formula for compound interest compounded monthly: \[ A = P\left(1+\frac{r}{n}\right)^{nt} = 9000\left(1+\frac{0.046}{12}\right)^{12 \cdot 8} \] Calculating that gives us about \$13,731.77. Now for the second option, compounded continuously, we use: \[ A = Pe^{rt} = 9000 e^{0.0459 \cdot 8} \] This comes out to approximately \$13,765.68. Comparing the two, we see that \$9000 invested at \(4.59\%\) compounded continuously over 8 years yields a greater return! Now, you might be wondering why continuous compounding often gives a higher return. This is because with continuous compounding, interest is added to the principal at every possible instant, allowing your money to grow just a tad faster than with monthly compounding! Thinking about your own investments? Be mindful of these small differences, as they can lead to big gains over time. Always calculate your returns in various ways before settling on a plan!