Detailed Notes on Compound Interest Calculation

Financial Mathematics: Compound Interest

Problem Statement

  • You deposit a principal amount of P=4000P = 4000 dollars in an account.
  • The account earns an annual interest rate of r = 5 ext{%} = 0.05.
  • The interest is compounded quarterly, meaning the number of compounding periods per year is n=4n = 4.
  • You want to find the total amount in the account after t=10t = 10 years.

Formula for Compound Interest

  • The formula to calculate the future value of an investment compounded at regular intervals is given by: A=P(1+rn)ntA = P \left(1 + \frac{r}{n}\right)^{nt}
    • Where:
    • AA is the amount of money accumulated after n years, including interest.
    • PP is the principal amount (the initial amount of money).
    • rr is the annual interest rate (decimal).
    • nn is the number of times that interest is compounded per year.
    • tt is the number of years the money is invested or borrowed.

Applying the Formula

  1. Substitute the known values into the formula:

    • A=4000(1+0.054)4×10A = 4000 \left(1 + \frac{0.05}{4}\right)^{4 \times 10}

  2. Simplifying the expression:

    • Calculate the interest rate per period:
      • 0.054=0.0125\frac{0.05}{4} = 0.0125
    • Now substitute it back into the formula:
      • A=4000(1+0.0125)40A = 4000 \left(1 + 0.0125\right)^{40}

  3. Calculate the base:

    • 1+0.0125=1.01251 + 0.0125 = 1.0125
    • Thus, we have:
      • A=4000(1.0125)40A = 4000 \left(1.0125\right)^{40}

  4. Now calculate (1.0125)40\left(1.0125\right)^{40}:

    • Using a calculator or computational tool, we find:
      • (1.0125)40≈1.643619\left(1.0125\right)^{40} \approx 1.643619
    • Then substitute this value back into the expression:
      • A≈4000×1.643619A \approx 4000 \times 1.643619

  5. Calculate the final amount:

    • A≈6574.476A \approx 6574.476

  6. Rounding the answer:

    • Rounding to the nearest cent gives:
      • A≈6574.48A \approx 6574.48

Conclusion

  • After 10 years, the total amount in the account, including interest, is approximately 6574.486574.48.

Performance Score

  • Score achieved on the last attempt: 57.5/100
  • Total questions answered: 15/20
  • Incorrect answers: Question 13 received a score of 0 out of 5 points.