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Samuel invested an amount with ABC bank at an interest 12% p.a. compounded monthly. His investment grew to R8450 at the end of 10 years. Determine the amount that Samuel initially invested.

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13. Samuel invested an amount with ABC bank at an interest 12% p.a. compounded monthly. His investment grew to R8450 at the end of 10 years. Determine the amount that Samuel initially invested.

  1. Identify the given values and the formula for compound interest.

    The future value is A=R8450A = \text{R}8450.
    The nominal annual interest rate is i=12%=0.12i = 12\% = 0.12.
    The interest is compounded monthly, so m=12m = 12.
    The time period is n=10n = 10 years.
    The formula for compound interest is A=P(1+im)n×mA = P\left(1 + \frac{i}{m}\right)^{n \times m}, where PP is the principal amount.

  2. Substitute the values into the formula and solve for PP.

    8450=P(1+0.1212)10×128450 = P\left(1 + \frac{0.12}{12}\right)^{10 \times 12} 8450=P(1+0.01)1208450 = P(1 + 0.01)^{120} 8450=P(1.01)1208450 = P(1.01)^{120} 8450=P(3.3003868949)8450 = P(3.3003868949)

  3. Calculate the principal amount PP.

    P=84503.3003868949P = \frac{8450}{3.3003868949} P≈2560.30P \approx 2560.30

    Samuel initially invested R2560.30.

    14. Calculate the effective interest rate per annum if an investment earns interest at a rate of 11.5% p.a., compounded monthly.

  4. Identify the given values and the formula for the effective interest rate.

    The nominal annual interest rate is i=11.5%=0.115i = 11.5\% = 0.115.
    The interest is compounded monthly, so m=12m = 12.
    The formula for the effective interest rate is ieff=(1+im)m−1i_{eff} = \left(1 + \frac{i}{m}\right)^m - 1.

  5. Substitute the values into the formula.

    ieff=(1+0.11512)12−1i_{eff} = \left(1 + \frac{0.115}{12}\right)^{12} - 1 ieff=(1+0.0095833333)12−1i_{eff} = (1 + 0.0095833333)^{12} - 1 ieff=(1.0095833333)12−1i_{eff} = (1.0095833333)^{12} - 1 ieff≈1.121339−1i_{eff} \approx 1.121339 - 1 ieff≈0.121339i_{eff} \approx 0.121339

  6. Convert the effective interest rate to a percentage.

    ieff≈0.121339×100%≈12.13%i_{eff} \approx 0.121339 \times 100\% \approx 12.13\%

    The effective interest rate is approximately 12.13%.

    15. Calculate the effective interest rate if interest is compounded at 5% p.a., compounded quarterly.

  7. Identify the given values and the formula for the effective interest rate.

    The nominal annual interest rate is i=5%=0.05i = 5\% = 0.05.
    The interest is compounded quarterly, so m=4m = 4.
    The formula for the effective interest rate is ieff=(1+im)m−1i_{eff} = \left(1 + \frac{i}{m}\right)^m - 1.

  8. Substitute the values into the formula.

    ieff=(1+0.054)4−1i_{eff} = \left(1 + \frac{0.05}{4}\right)^4 - 1 ieff=(1+0.0125)4−1i_{eff} = (1 + 0.0125)^4 - 1 ieff=(1.0125)4−1i_{eff} = (1.0125)^4 - 1 ieff≈1.0509453369−1i_{eff} \approx 1.0509453369 - 1 ieff≈0.0509453369i_{eff} \approx 0.0509453369

  9. Convert the effective interest rate to a percentage.

    ieff≈0.0509453369×100%≈5.09%i_{eff} \approx 0.0509453369 \times 100\% \approx 5.09\%

    The effective interest rate is approximately 5.09%.

    16. The nominal interest rate of an investment is 12,35 0/0 p.a., compounded monthly. Calculate the effective interest rate.

  10. Identify the given values and the formula for the effective interest rate.

    The nominal annual interest rate is i=12.35%=0.1235i = 12.35\% = 0.1235.
    The interest is compounded monthly, so m=12m = 12.
    The formula for the effective interest rate is ieff=(1+im)m−1i_{eff} = \left(1 + \frac{i}{m}\right)^m - 1.

  11. Substitute the values into the formula.

    ieff=(1+0.123512)12−1i_{eff} = \left(1 + \frac{0.1235}{12}\right)^{12} - 1 ieff=(1+0.0102916667)12−1i_{eff} = (1 + 0.0102916667)^{12} - 1 ieff=(1.0102916667)12−1i_{eff} = (1.0102916667)^{12} - 1 ieff≈1.131238−1i_{eff} \approx 1.131238 - 1 ieff≈0.131238i_{eff} \approx 0.131238

  12. Convert the effective interest rate to a percentage.

    ieff≈0.131238×100%≈13.12%i_{eff} \approx 0.131238 \times 100\% \approx 13.12\%

    The effective interest rate is approximately 13.12%.

    17. An investment earns interest at a rate of 7% per annum, compounded semi-annually. Calculate the effective annual interest rate on this investment.

  13. Identify the given values and the formula for the effective interest rate.

    The nominal annual interest rate is i=7%=0.07i = 7\% = 0.07.
    The interest is compounded semi-annually, so m=2m = 2.
    The formula for the effective interest rate is ieff=(1+im)m−1i_{eff} = \left(1 + \frac{i}{m}\right)^m - 1.

  14. Substitute the values into the formula.

    ieff=(1+0.072)2−1i_{eff} = \left(1 + \frac{0.07}{2}\right)^2 - 1 ieff=(1+0.035)2−1i_{eff} = (1 + 0.035)^2 - 1 ieff=(1.035)2−1i_{eff} = (1.035)^2 - 1 ieff=1.071225−1i_{eff} = 1.071225 - 1 ieff=0.071225i_{eff} = 0.071225

  15. Convert the effective interest rate to a percentage.

    ieff=0.071225×100%=7.1225%i_{eff} = 0.071225 \times 100\% = 7.1225\%

    The effective annual interest rate is 7.1225%.

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