Difficulty: Medium
Correct Answer: 5387.42
Explanation:
Introduction / Context:
This is a standard compound interest problem involving quarterly compounding. The principal is 4000 dollars, the nominal annual interest rate is 6%, and the interest is compounded four times per year. We must find the total amount in the account after 5 years, taking into account compounding every quarter.
Given Data / Assumptions:
Concept / Approach:
For nominal rate R compounded m times per year, the periodic rate is R / m percent per period, and the number of periods is n = m * years. The amount is computed as:
A = P * (1 + (R / 100) / m)^n
Here R = 6, m = 4, and years = 5, so n = 20 quarters and periodic rate = 1.5% per quarter.
Step-by-Step Solution:
Periodic rate = 6 / (4 * 100) = 1.5 / 100 = 0.015.
Number of periods n = 4 * 5 = 20.
Amount factor = (1.015)^20.
Compute A = 4000 * (1.015)^20.
(1.015)^20 is approximately 1.346855.
A ≈ 4000 * 1.346855 ≈ 5387.42 dollars.
Verification / Alternative Check:
You can confirm using a calculator that (1.015)^20 ≈ 1.346855.
Multiplying by 4000 gives approximately 5387.42, so the result is consistent.
This value is clearly larger than the simple interest amount that would be obtained at 6% for 5 years, which shows the effect of quarterly compounding.
Why Other Options Are Wrong:
3387.42 is less than the principal and therefore clearly impossible after earning positive interest.
4387.42 is lower than the correct amount and seems to ignore full compounding.
6387.42 is too high and would correspond to a higher rate or longer time than given.
Common Pitfalls:
Some learners mistakenly use 6% per quarter instead of 1.5% per quarter.
Others compute 6% per year for 5 years as simple interest without compounding.
Forgetting to multiply the number of years by 4 to get the number of quarters is another typical mistake.
Final Answer:
The amount in the account after 5 years will be approximately 5387.42 dollars.
Discussion & Comments