How much money would you need to deposit today at 9% annual interest compounded monthly in order to have $12,000 in the account after 6 years?

Introduction / Context:
This is a present value problem involving compound interest with monthly compounding. Instead of asking for the future value of a current deposit, the question provides a desired future amount and asks how much must be deposited today to reach that amount. The nominal annual interest rate is 9%, and interest is compounded monthly for 6 years. Such questions are common in financial planning and time value of money calculations.

Given Data / Assumptions:

• Desired future amount A = $12,000.
• Nominal annual interest rate r = 9% per annum.
• Compounding frequency: monthly (n = 12 times per year).
• Time t = 6 years.
• We assume a single lump-sum deposit today with no additional contributions.
• We must find the present value P that grows to $12,000 under these conditions.

Concept / Approach:
With monthly compounding, the future value of a lump sum is A = P * (1 + r/(100n))^(n*t). To find the required present deposit P for a given future amount A, we rearrange the formula to P = A / (1 + r/(100n))^(n*t). Here, r = 9, n = 12 and t = 6. The monthly interest rate is thus 9/12 percent per month. By computing the growth factor over 72 months and dividing the target amount by this factor, we obtain the necessary deposit today.

Step-by-Step Solution:
Monthly interest rate = 9% / 12 = 0.75% per month. In decimal form, monthly rate = 0.09 / 12 = 0.0075. Number of months in 6 years = 12 * 6 = 72. Growth factor over 6 years = (1 + 0.0075)^(72). This factor is approximately (1.0075)^(72) ≈ 1.7123 (approximate value). Present value P = 12,000 / 1.7123 ≈ 7,007.08. Hence, you must deposit approximately $7,007.08 today.

Verification / Alternative check:
We can verify that this amount is plausible. At simple interest of 9% for 6 years, the amount would be A_simple = P * (1 + 0.09 * 6) = P * 1.54. For P around $7,000, that gives about $10,780, which is less than 12,000, so compounding monthly should require a deposit somewhat below $12,000 / 1.54 ≈ $7,792 but above $12,000 / 2 ≈ $6,000. The computed value of about $7,007.08 fits comfortably in this expected range and matches more precise calculations using financial calculators or spreadsheets.

Why Other Options Are Wrong:
$9,007 and $8,007 are too high; depositing that much would grow to significantly more than $12,000 under 6 years of monthly compounding at 9%. $4,007 is much too low and would not have enough time to grow to $12,000 at this rate. $6,500 would also fall short. Only $7,007.08 corresponds to the correct present value when we solve the compound interest equation exactly.

Common Pitfalls:
Mistakes include using 9% as the monthly rate instead of dividing by 12, which dramatically underestimates the required deposit, or confusing present value and future value formulas. Some may also incorrectly treat 6 years as 6 months when counting periods. Always ensure consistency in time units and interest rate units, and correctly rearrange the formula to solve for P rather than A.

Final Answer:
You must deposit approximately $7,007.08 today to have $12,000 after 6 years at 9% compounded monthly.