How much money must you deposit today in an account that pays 9% annual interest compounded monthly in order to have 12,000 dollars in the account after 6 years?

Introduction:
This is a present value problem under compound interest with monthly compounding. Instead of asking for the final amount from a known principal, it asks you to find how much needs to be deposited today to reach a target future value. Such questions are very important in financial planning and time value of money topics.

Given Data / Assumptions:

• Future value (target amount) A = 12,000 dollars
• Annual nominal interest rate r = 9% per annum
• Interest is compounded monthly
• Investment period t = 6 years
• We must find present value or principal P

Concept / Approach:
For monthly compounding, we use:
A = P * (1 + r_monthly)^nwhere:
r_monthly = r_annual / 12n = 12 * tTo find present value P, we rearrange the formula:
P = A / (1 + r_monthly)^n

Step-by-Step Solution:
Step 1: Convert annual rate to monthly.r_annual = 9% = 0.09r_monthly = 0.09 / 12 = 0.0075Step 2: Compute the number of months.t = 6 years, so n = 6 * 12 = 72 monthsStep 3: Use the present value formula.P = 12,000 / (1 + 0.0075)^72P = 12,000 / (1.0075)^72With accurate calculation, this gives:
P ≈ 7,007.08 dollars

Verification / Alternative check:
We can quickly check using an approximate effective annual rate. The effective annual rate is about:
(1 + 0.09 / 12)^12 - 1 ≈ 0.0938 or 9.38%Over 6 years, the rough growth factor is (1.0938)^6 ≈ 1.712. Dividing 12,000 by 1.712 gives approximately 7,010 dollars, which is close to the more precise value 7,007.08, confirming that our calculation is reasonable.

Why Other Options Are Wrong:

• 9,007 dollars and 8,007 dollars: These are too high as present values and would grow to more than 12,000 at 9% compounded monthly.
• 4,007 dollars: Far too low, would not grow to the target.
• 6,950 dollars: Slightly below the correct present value, reflecting a rounding or rate error.

Common Pitfalls:
Students sometimes mistakenly use annual compounding for a monthly compounding problem, or they forget to convert the rate to a monthly rate and time to months. Another error is to multiply by the future factor instead of dividing by it when finding present value. Always rearrange the compound interest formula carefully when solving for P.

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