If you deposit $4,000 into an account paying 6% annual interest compounded quarterly, how much money will be in the account at the end of 5 years (that is, what will be the accumulated amount including interest)?

Introduction:
This is a standard compound interest problem involving quarterly compounding in a dollars based investment context. It tests the correct use of the compound interest formula with a nominal annual rate converted into a rate per quarter and the number of compounding periods over several years.

Given Data / Assumptions:
Principal P = $4,000. Nominal annual interest rate r = 6% per annum. Interest is compounded quarterly. Time t = 5 years.

Concept / Approach:
For quarterly compounding, we use: Rate per quarter = r / 4 = 6 / 4 = 1.5% = 0.015. Number of quarters n = 4 * 5 = 20. The amount formula becomes: A = P * (1 + 0.06 / 4)^(4 * 5) = 4000 * (1.015)^20. We then compute or use known approximations for (1.015)^20 to find A.

Step-by-Step Solution:
Step 1: Convert annual rate to quarterly. Rate per quarter = 0.06 / 4 = 0.015. Step 2: Determine the number of compounding periods. n = 4 * 5 = 20 quarters. Step 3: Apply the compound interest formula. A = 4000 * (1.015)^20. Using standard compound tables or a precise calculation, (1.015)^20 ≈ 1.346855. A ≈ 4000 * 1.346855 ≈ 5387.42. Thus, the accumulated amount is about $5,387.42.

Verification / Alternative check:
We can sanity check by approximating. A 6% annual rate for 5 years with annual compounding would roughly give a multiplier near (1.06)^5 ≈ 1.338. Quarterly compounding should be slightly more than this, and 1.346855 is reasonably close and slightly bigger, which matches expectations. Multiplying by 4000 gives a value just above $5,350, so $5,387.42 is plausible.

Why Other Options Are Wrong:
$3,387.42 is less than the principal and clearly impossible after 5 years of positive interest. $4,387.42 corresponds to a very small overall gain inconsistent with 6% over 5 years. $6,387.42 or $7,487.42 would require much higher effective rates or longer times than those given.

Common Pitfalls:
Learners sometimes mistakenly use the annual rate directly as 6% for 20 periods, or they use 6% for 5 periods instead of 1.5% for 20 periods. Others forget to convert 6% into 0.06 before dividing. Careful handling of the rate and number of periods is essential for quarterly compounding problems.

Final Answer:
The amount in the account after 5 years will be approximately $5,387.42.