Suppose you have $\$ 12,000$ to invest. Which of the two rates would yield the larger amount in 2 years: $10 \%$ compounded quarterly or $9.91 \%$ compounded continuously?
Which of the two rates would yield the larger amount in 2 years?
$10 \%$ compounded quarterly
$9.91 \%$ compounded continuously

Step 1 ：Given that the principal amount (P) is $12,000 and the time (t) is 2 years.

Step 2 ：For the first case, the annual interest rate (r1) is 10% or 0.1 and the number of times that interest is compounded per year (n1) is 4. We use the formula for compound interest: \(A = P (1 + \frac{r}{n})^{nt}\) to calculate the future value of the investment. Substituting the given values, we get \(A1 = 12000 (1 + \frac{0.1}{4})^{4*2}\) which simplifies to approximately \$14620.83.

Step 3 ：For the second case, the annual interest rate (r2) is 9.91% or 0.0991. We use the formula for continuously compounded interest: \(A = Pe^{rt}\) to calculate the future value of the investment. Substituting the given values, we get \(A2 = 12000 * e^{0.0991*2}\) which simplifies to approximately \$14630.47.

Step 4 ：Comparing the future values of the investment for both cases, we find that the investment would yield a larger amount when the interest rate is 9.91% compounded continuously.

Step 5 ：\(\boxed{\text{The rate of 9.91% compounded continuously would yield the larger amount in 2 years.}}\)