Answer :

Final answer:

To find the interest rate for an investment compounded quarterly, we use the compound interest formula and solve for the interest rate, yielding approximately 7.24%. The given values are $5000 initially and $20835 after 20 years. Through calculations, we determine the interest rate is around 7.24%.

Explanation:

Finding the Interest Rate for Quarterly Compounded Investments

To find the interest rate for an investment compounded quarterly, we use the compound interest formula:

A = P (1 + r/n)^(nt)

where:

  • A = the amount of money accumulated after n years, including interest.
  • P = the principal amount (the initial amount of money).
  • r = the annual interest rate (decimal).
  • n = the number of times compounding occurs per year.
  • t = the time in years.

Given:

  • P = $5000
  • A = $20835
  • n = 4 (quarterly compounding)
  • t = 20 years

The formula becomes:

20835 = 5000 (1 + r/4)^(4*20)

Solving for r:

  1. Divide both sides by 5000 to isolate the compound factor:
    A/P = (1 + r/4)^80
    20835/5000 = (1 + r/4)^80
    4.167 = (1 + r/4)^80
  2. Take the 80th root of both sides to get rid of the exponent:
    (4.167)^(1/80) = 1 + r/4
  3. Simplify:
    1.0181 ≈ 1 + r/4
  4. Solve for r:
    r/4 ≈ 0.0181
    r ≈ 4 * 0.0181
    r ≈ 0.0724, or 7.24%

Thus, the interest rate is approximately 7.24%.

Learn more about compound interest here:

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