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:
- 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 - Take the 80th root of both sides to get rid of the exponent:
(4.167)^(1/80) = 1 + r/4 - Simplify:
1.0181 ≈ 1 + r/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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