If $\$ 350$ is invested at $7 \%$ compounded (A) annually, (B) quarterly, (C) monthly, what is the amount after 5 years? How much interest is earned?

Step 1 ：Given that the principal amount (P) is \$350, the annual interest rate (r) is 7% or 0.07 in decimal, and the time (t) is 5 years.

Step 2 ：We need to calculate the amount after 5 years and the interest earned for different compounding periods using the formula for compound interest: \(A = P (1 + \frac{r}{n})^{nt}\), where n is the number of times that interest is compounded per year.

Step 3 ：For annual compounding (n=1), we substitute the values into the formula to get \(A_{annual} = 350 (1 + \frac{0.07}{1})^{1*5} = 490.89\). The interest earned is \(interest_{annual} = A_{annual} - P = 490.89 - 350 = 140.89\).

Step 4 ：For quarterly compounding (n=4), we substitute the values into the formula to get \(A_{quarterly} = 350 (1 + \frac{0.07}{4})^{4*5} = 495.17\). The interest earned is \(interest_{quarterly} = A_{quarterly} - P = 495.17 - 350 = 145.17\).

Step 5 ：For monthly compounding (n=12), we substitute the values into the formula to get \(A_{monthly} = 350 (1 + \frac{0.07}{12})^{12*5} = 496.17\). The interest earned is \(interest_{monthly} = A_{monthly} - P = 496.17 - 350 = 146.17\).

Step 6 ：\(\boxed{\text{Final Answer: The amount after 5 years when compounded annually is \$490.89 and the interest earned is \$140.89. When compounded quarterly, the amount is \$495.17 and the interest earned is \$145.17. When compounded monthly, the amount is \$496.17 and the interest earned is \$146.17. Therefore, the interest earned is highest when the compounding is done monthly.}}\)