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.

Here are the solutions to the problems: 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. Step 1: Identify the given values and the formula for compound interest. The future value is A = R8450. The nominal annual interest rate is i = 12\% = 0.12. The interest is compounded monthly, so m = 12. The time period is n = 10 years. The formula for compound interest is A = P(1 + (i)/(m))^n × m, where P is the principal amount. Step 2: Substitute the values into the formula and solve for P. 8450 = P(1 + (0.12)/(12))^10 × 12 8450 = P(1 + 0.01)^120 8450 = P(1.01)^120 8450 = P(3.3003868949) Step 3: Calculate the principal amount P. P = (8450)/(3.3003868949) P ≈ 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. Step 1: Identify the given values and the formula for the effective interest rate. The nominal annual interest rate is i = 11.5\% = 0.115. The interest is compounded monthly, so m = 12. The formula for the effective interest rate is i_eff = (1 + (i)/(m))^m - 1. Step 2: Substitute the values into the formula. i_eff = (1 + (0.115)/(12))^12 - 1 i_eff = (1 + 0.0095833333)^12 - 1 i_eff = (1.0095833333)^12 - 1 i_eff ≈ 1.121339 - 1 i_eff ≈ 0.121339 Step 3: Convert the effective interest rate to a percentage. i_eff ≈ 0.121339 × 100\% ≈ 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. Step 1: Identify the given values and the formula for the effective interest rate. The nominal annual interest rate is i = 5\% = 0.05. The interest is compounded quarterly, so m = 4. The formula for the effective interest rate is i_eff = (1 + (i)/(m))^m - 1. Step 2: Substitute the values into the formula. i_eff = (1 + (0.05)/(4))^4 - 1 i_eff = (1 + 0.0125)^4 - 1 i_eff = (1.0125)^4 - 1 i_eff ≈ 1.0509453369 - 1 i_eff ≈ 0.0509453369 Step 3: Convert the effective interest rate to a percentage. i_eff ≈ 0.0509453369 × 100\% ≈ 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. Step 1: Identify the given values and the formula for the effective interest rate. The nominal annual interest rate is i = 12.35\% = 0.1235. The interest is compounded monthly, so m = 12. The formula for the effective interest rate is i_eff = (1 + (i)/(m))^m - 1. Step 2: Substitute the values into the formula. i_eff = (1 + (0.1235)/(12))^12 - 1 i_eff = (1 + 0.0102916667)^12 - 1 i_eff = (1.0102916667)^12 - 1 i_eff ≈ 1.131238 - 1 i_eff ≈ 0.131238 Step 3: Convert the effective interest rate to a percentage. i_eff ≈ 0.131238 × 100\% ≈ 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. Step 1: Identify the given values and the formula for the effective interest rate. The nominal annual interest rate is i = 7\% = 0.07. The interest is compounded semi-annually, so m = 2. The formula for the effective interest rate is i_eff = (1 + (i)/(m))^m - 1. Step 2: Substitute the values into the formula. i_eff = (1 + (0.07)/(2))^2 - 1 i_eff = (1 + 0.035)^2 - 1 i_eff = (1.035)^2 - 1 i_eff = 1.071225 - 1 i_eff = 0.071225 Step 3: Convert the effective interest rate to a percentage. i_eff = 0.071225 × 100\% = 7.1225\% The effective annual interest rate is 7.1225%. Send me the next one 📸