You deposit 6,500 dollars into an account that pays 8% annual interest compounded monthly. How much money will be in the account (total balance) after 7 years?

Introduction:
This problem involves compound interest with monthly compounding. It tests your ability to correctly convert an annual nominal rate to a monthly rate, count the total number of months, and apply the compound interest formula over many compounding periods.

Given Data / Assumptions:

• Initial deposit (principal) P = 6,500 dollars
• Annual interest rate r = 8% per annum
• Interest is compounded monthly
• Investment period t = 7 years
• We need the final account balance after 7 years

Concept / Approach:
For monthly compounding, the periodic rate is the annual rate divided by 12, and the number of periods is 12 times the number of years. The compound amount is found by:
A = P * (1 + r_monthly)^nwhere:
r_monthly = r_annual / 12n = 12 * t

Step-by-Step Solution:
Step 1: Convert annual rate to monthly rate.r_annual = 8% = 0.08r_monthly = 0.08 / 12 ≈ 0.0066667Step 2: Compute number of compounding periods.t = 7 years, so n = 12 * 7 = 84 monthsStep 3: Apply the compound interest formula.A = 6,500 * (1 + 0.08 / 12)^84A ≈ 6,500 * (1.0066667)^84Using accurate calculation, A ≈ 11,358.24 dollars

Verification / Alternative check:
We can approximate the effective annual rate. For 8% compounded monthly, the effective annual rate is roughly:
(1 + 0.08 / 12)^12 - 1 ≈ 0.083This is about 8.3% per year. Over 7 years, the growth factor is around (1.083)^7 ≈ 1.746, and 6,500 * 1.746 ≈ 11,349, which is close to 11,358.24, confirming our detailed computation.

Why Other Options Are Wrong:

• 12,334 and 12,386 dollars: These exceed the correct amount and correspond to higher assumed effective rates.
• 15,789 dollars: Much too high, more than doubling the money at 8% over 7 years.
• 10,950 dollars: Too low, underestimates the impact of monthly compounding.

Common Pitfalls:
Some students mistakenly use 8% divided by 7 years, or they use annual compounding instead of monthly compounding. Others forget to multiply years by 12 to find the number of monthly periods. Always ensure that rate and time are expressed in consistent units when using the compound interest formula.

Final Answer:
The account balance after 7 years will be approximately 11,358.24 dollars.