You deposit $6500 into an account paying 8% annual interest compounded monthly. How much money (to the nearest cent) will be in the account after 7 years?

Introduction / Context:
This question involves compound interest with monthly compounding on a dollar-denominated deposit. The nominal annual interest rate is 8%, the compounding frequency is 12 times per year, and the investment horizon is 7 years. We are asked for the future value of a single deposit at this rate and compounding schedule. This is a typical financial mathematics problem that reinforces the general compound interest formula with more frequent than annual compounding.

Given Data / Assumptions:

• Initial deposit P = $6500.
• Nominal annual interest rate r = 8% per year.
• Compounding frequency: monthly, i.e., n = 12 compounding periods per year.
• Time t = 7 years.
• Interest is compounded monthly at an effective rate of r/12 per month.
• We need to find the final account balance (amount) after 7 years.

Concept / Approach:
For compound interest with more frequent compounding, the amount after t years is given by A = P * (1 + r/(100n))^(n*t), where r is the nominal annual rate and n is the number of compounding periods per year. Here, r = 8, n = 12 and t = 7, so the monthly rate is 8/12 percent per month. We then compute the power over 12 * 7 = 84 periods. The resulting value gives the accumulated balance in the account, including both the principal and the interest earned.

Step-by-Step Solution:
Monthly interest rate = 8% / 12 = 0.6666...% per month. In decimal form, monthly rate = 0.08 / 12 ≈ 0.0066667. Number of months in 7 years = 12 * 7 = 84. Use A = 6500 * (1 + 0.08 / 12)^(84). Compute the growth factor: (1 + 0.08 / 12) ≈ 1.0066667. Raise this to the 84th power: (1.0066667)^(84) ≈ 1.747426 (approximate value). Therefore, A ≈ 6500 * 1.747426 ≈ 11,358.24. So the account balance after 7 years is approximately $11,358.24.

Verification / Alternative check:
We can check whether this value is reasonable. At simple interest of 8% for 7 years, the amount would be A_simple = 6500 * (1 + 0.08 * 7) = 6500 * 1.56 = 10,140. Under monthly compounding, we expect a slightly higher amount than under simple interest, but not dramatically larger. The result $11,358.24 represents about 174.7% of the original principal, which is consistent with a moderate increase due to compounding over 7 years at 8%. The answer also fits common financial tables or calculator outputs for this rate and horizon.

Why Other Options Are Wrong:
$12,334.00 and $12,386.00 are somewhat higher than the correct value and would correspond to either a higher effective interest rate or a longer duration. $15,789.00 is far too high relative to an 8% nominal rate and 7-year period and would require either a much higher rate or additional deposits. $10,000.00 is too low, since even simple interest would yield over $10,000. Only $11,358.24 matches the correct compound interest computation and is consistent with the modest boost from monthly compounding.

Common Pitfalls:
Some students mistakenly use 8% as the monthly rate instead of dividing by 12, which massively overestimates the final amount. Others forget to convert the percentage to a decimal before plugging into the formula or use t = 7 months instead of 7 years. It is also common to confuse the formula for a single lump-sum future value with formulas for annuities (regular monthly deposits). Always check that the given scenario involves a single deposit, and ensure that both the rate and the number of periods are expressed in consistent units.

Final Answer:
After 7 years, the account balance will be approximately $11,358.24.