1.3 Lerato has R10 000 to invest. She invests the money In a bank for 3 years at an interest rate of $$ 7 \% $$ per annum compounded annually. She extends the period of investment for a further 2 years when the bank offers her an interest rate of $$ 8,5 \% $$ compounded monthly. At the end of this period, due to a decrease in the repo rate, the bank informs her that they will decrease the interest rate on her investment to $$ 8 \% $$ p.a simple interest Lerato therefore decides to leave her money in the bank for only one more year 1.3.1 Draw a timelline for the above investment 1.3.2 Calculate the value of the investment after 6 years.

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**Timeline**

- At $$ t = 0 $$: Lerato invests $$ R10\,000 $$.
- From $$ t = 0 $$ to $$ t = 3 $$ years: The investment grows at $$ 7\% $$ per annum compounded annually.
- From $$ t = 3 $$ to $$ t = 5 $$ years: The investment grows at $$ 8.5\% $$ per annum compounded monthly.
- From $$ t = 5 $$ to $$ t = 6 $$ years: The investment grows at $$ 8\% $$ per annum simple interest.

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**Step-by-Step Calculation**

1. **First Period (0 to 3 years – 7% per annum compounded annually):**

The compound interest formula is
   $$
   A = P\,(1 + r)^n
   $$
   where
   - $$ P = 10\,000 $$
   - $$ r = 0.07 $$
   - $$ n = 3 $$
   
   Thus,
   $$
   A_1 = 10\,000\,(1.07)^3
   $$
   Compute:
   $$
   (1.07)^3 \approx 1.225043
   $$
   so
   $$
   A_1 \approx 10\,000 \times 1.225043 \approx R12\,250.43
   $$

2. **Second Period (3 to 5 years – 8.5% per annum compounded monthly):**

For monthly compounding:
   - Annual rate $$ = 8.5\% $$ so the monthly rate is
     $$
     r_m = \frac{0.085}{12} \approx 0.0070833
     $$
   - The number of months in 2 years is $$ n_m = 24 $$.
   
   The amount after this period is given by:
   $$
   A_2 = A_1\,(1 + r_m)^{n_m} = 12\,250.43\,(1 + 0.0070833)^{24}
   $$
   Compute the monthly factor:
   $$
   (1.0070833)^{24} \approx 1.1845
   $$
   
   Thus,
   $$
   A_2 \approx 12\,250.43 \times 1.1845 \approx R14\,520.53
   $$

3. **Third Period (5 to 6 years – 8% per annum simple interest):**

For simple interest:
   $$
   A = P\,(1 + rt)
   $$
   where
   - $$ P = A_2 $$
   - $$ r = 0.08 $$
   - $$ t = 1 $$ year.

Therefore,
   $$
   A_3 = A_2\,(1 + 0.08) = 14\,520.53 \times 1.08
   $$
   Compute:
   $$
   A_3 \approx 14\,520.53 \times 1.08 \approx R15\,662.07
   $$

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**Final Answer**

After 6 years, the value of the investment is approximately $$ R15\,662.07 $$.
After 6 years, the investment is worth approximately R15,662.07.

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