Compound Interest Calculation

  • Problem Overview:

    • Determine the annual interest rate needed to grow an investment of $2,000 to $3,000 in one year with compounding occurring twice a year.

  • Key Components:

    • Initial Investment (Principal) P=2000P = 2000
    • Final Amount after 1 year A=3000A = 3000
    • Number of Compounding Periods per Year n=2n = 2
    • Time in years t=1t = 1

  • Formula for Compound Interest:
    A=P(1+rn)ntA = P(1 + \frac{r}{n})^{nt}
    where:

    • AA = total amount after interest
    • PP = principal amount
    • rr = annual interest rate (decimal)
    • nn = number of compounding periods per year
    • tt = number of years

  • Setting Up the Equation:

    • Substitute values into the compound interest formula:
      3000=2000(1+r2)2imes13000 = 2000(1 + \frac{r}{2})^{2 imes 1}

  • Simplifying the Equation:

    • Divide both sides by 2000:
      1.5=(1+r2)21.5 = (1 + \frac{r}{2})^{2}

  • Solving for rr:

    • Take the square root of both sides:
      extsqrt(1.5)=1+r2ext{sqrt}(1.5) = 1 + \frac{r}{2}
    • Solve for rr:
      r=2(extsqrt(1.5)−1)r = 2( ext{sqrt}(1.5) - 1)

  • Interpretation of Results:

    • Final result will provide the required annual interest rate to achieve the investment goal under the specified conditions.