After a 20 year period Josh's lump sum investment matures to an amount of R313550. How much did he invest if his money earned interest at a rate of 13,65% p.a compounded half yearly for the first 10years ,8,4% p.a compounded quarterly for the next five years and 7.2 p.a compounded monthly for the remaining period?

((x(1+.1365/2)^20)(1+.084/4)^20)(1+.072/12)^60 = 313500

x = 38582.10

To determine the initial investment amount, we will break down the calculation into three parts:

1. Calculation for the first 10 years:
- The interest rate is 13.65% per annum, compounded semi-annually.
- We are given that the investment amount matures to R313,550 after 20 years.
- To calculate the value after 10 years, we will use the compound interest formula: A = P(1 + r/n)^(nt), where:
- A = R313,550 (matured amount after 20 years)
- P = Initial investment amount (what we're trying to find)
- r = 13.65% (annual interest rate)
- n = 2 (semi-annual compounding)
- t = 10 (number of years for this calculation)
- Substituting the given values into the formula:
R313,550 = P(1 + 0.1365/2)^(2*10)
Simplifying the equation gives us:
R313,550 = P(1.06825)^(20)
- Solving for P (initial investment):
P = R313,550 / (1.06825^20)

2. Calculation for the next 5 years:
- The interest rate is 8.4% per annum, compounded quarterly.
- We will use the same formula as above, but with the new interest rate, the remaining maturity time of 5 years, and quarterly compounding.
- Substituting the given values into the formula:
R313,550 = P(1 + 0.084/4)^(4*5)
Simplifying the equation gives us:
R313,550 = P(1.021)^(20)
- Solving for P (initial investment):
P = R313,550 / (1.021^20)

3. Calculation for the remaining period:
- The interest rate is 7.2% per annum, compounded monthly.
- Using the same formula as above, but with the new interest rate, remaining maturity time, and monthly compounding:
- Substituting the given values into the formula:
R313,550 = P(1 + 0.072/12)^(12*5)
Simplifying the equation gives us:
R313,550 = P(1.006)^(60)
- Solving for P (initial investment):
P = R313,550 / (1.006^60)

By calculating the values for each of the three periods mentioned above, you can find the initial investment amount made by Josh.