Answer:
To calculate the future value of Marisha's investment, we can break it down into three periods based on the different interest rates and compounding frequencies:
**Period 1 (First Year):**
- Principal amount (initial investment) = R2,500
- Annual interest rate = 7.5%
- Compounding frequency = Quarterly (4 times a year)
We can use the formula for compound interest in this period:
\[A1 = P \left(1 + \frac{r}{n}\right)^{n*t1}\]
Where:
- A1 is the future value after the first year.
- P is the principal amount (R2,500).
- r is the annual interest rate (7.5% or 0.075 as a decimal).
- n is the number of times the interest is compounded per year (quarterly, so 4).
- t1 is the number of years in this period (1 year).
Plug in the values:
\[A1 = 2,500 \left(1 + \frac{0.075}{4}\right)^{4*1} \]
Calculate A1.
**Period 2 (Second Year):**
- Principal amount (starting with A1 from the first year) = A1 from the previous calculation.
- Annual interest rate = 6%
- Compounding frequency = Monthly (12 times a year)
Now, we'll use the same compound interest formula, but with the new interest rate and compounding frequency:
\[A2 = A1 \left(1 + \frac{r}{n}\right)^{n*t2}\]
Where:
- A2 is the future value after the second year.
- A1 is the amount from the end of the first year.
- r is the new annual interest rate (6% or 0.06 as a decimal).
- n is the new compounding frequency (monthly, so 12).
- t2 is the number of years in this period (1 year).
Plug in the values:
\[A2 = A1 \left(1 + \frac{0.06}{12}\right)^{12*1}\]
Calculate A2.
**Period 3 (Remaining 3 Years):**
- Principal amount (starting with A2 from the second year) = A2 from the previous calculation.
- Annual interest rate = 8.3%
- Compounding frequency = Semi-annually (2 times a year)
Use the compound interest formula once more:
\[A3 = A2 \left(1 + \frac{r}{n}\right)^{n*t3}\]
Where:
- A3 is the future value after the remaining 3 years.
- A2 is the amount from the end of the second year.
- r is the new annual interest rate (8.3% or 0.083 as a decimal).
- n is the new compounding frequency (semi-annually, so 2).
- t3 is the number of years in this period (3 years).
Plug in the values:
\[A3 = A2 \left(1 + \frac{0.083}{2}\right)^{2*3}\]
Calculate A3.
Now, to find the total amount Marisha will have at the end of 5 years, simply add up the amounts from each period:
\[Total Amount = A1 + A2 + A3\]
Calculate the total amount, and you will have the answer.