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To solve this problem, we need to calculate the future value of the initial deposit and the two additional deposits, considering that the interest is compounded monthly. Afterward, we will sum these values to find the total amount in the savings account at the end of three years.
### Step-by-Step Solution:
1. Identify the given values:
- Initial Deposit (P₀): R8,000
- Additional Deposit (P₁ and P₂): R2,500 each
- Annual Interest Rate (r): 7.6%
- Compound Frequency (n): Monthly, so n = 12
- Total Duration: 3 years
2. Convert the annual interest rate to a monthly rate:
[tex]\[ \text{Monthly Interest Rate (i)} = \frac{7.6\%}{12} = \frac{7.6}{100 \times 12} \approx 0.0063333 \][/tex]
3. Calculate the number of compounding periods for each deposit:
- For the initial deposit (P₀): 3 years
[tex]\[ \text{Periods} = 3 \times 12 = 36 \text{ months} \][/tex]
- For the first additional deposit (P₁), made after 1 year, hence it grows for 2 years:
[tex]\[ \text{Periods} = 2 \times 12 = 24 \text{ months} \][/tex]
- For the second additional deposit (P₂), made 1.5 years after the initial deposit, hence it grows for 1.5 years:
[tex]\[ \text{Periods} = 1.5 \times 12 = 18 \text{ months} \][/tex]
4. Calculate the future value of each deposit:
- Future Value of the Initial Deposit (P₀):
[tex]\[ FV_{\text{initial}} = P_0 \times \left(1 + i\right)^{36} \][/tex]
[tex]\[ FV_{\text{initial}} = 8000 \times \left(1 + 0.0063333\right)^{36} \approx 10041.46 \][/tex]
- Future Value of the First Additional Deposit (P₁):
[tex]\[ FV_{\text{first addition}} = P_1 \times \left(1 + i\right)^{24} \][/tex]
[tex]\[ FV_{\text{first addition}} = 2500 \times \left(1 + 0.0063333\right)^{24} \approx 2909.01 \][/tex]
- Future Value of the Second Additional Deposit (P₂):
[tex]\[ FV_{\text{second addition}} = P_2 \times \left(1 + i\right)^{18} \][/tex]
[tex]\[ FV_{\text{second addition}} = 2500 \times \left(1 + 0.0063333\right)^{18} \approx 2800.87 \][/tex]
5. Sum the future values to find the total amount in the account:
[tex]\[ \text{Total Amount} = FV_{\text{initial}} + FV_{\text{first addition}} + FV_{\text{second addition}} \][/tex]
[tex]\[ \text{Total Amount} = 10041.46 + 2909.01 + 2800.87 \approx 15751.34 \][/tex]
So, the total amount in the savings account at the end of three years is approximately R15,751.34.
### Step-by-Step Solution:
1. Identify the given values:
- Initial Deposit (P₀): R8,000
- Additional Deposit (P₁ and P₂): R2,500 each
- Annual Interest Rate (r): 7.6%
- Compound Frequency (n): Monthly, so n = 12
- Total Duration: 3 years
2. Convert the annual interest rate to a monthly rate:
[tex]\[ \text{Monthly Interest Rate (i)} = \frac{7.6\%}{12} = \frac{7.6}{100 \times 12} \approx 0.0063333 \][/tex]
3. Calculate the number of compounding periods for each deposit:
- For the initial deposit (P₀): 3 years
[tex]\[ \text{Periods} = 3 \times 12 = 36 \text{ months} \][/tex]
- For the first additional deposit (P₁), made after 1 year, hence it grows for 2 years:
[tex]\[ \text{Periods} = 2 \times 12 = 24 \text{ months} \][/tex]
- For the second additional deposit (P₂), made 1.5 years after the initial deposit, hence it grows for 1.5 years:
[tex]\[ \text{Periods} = 1.5 \times 12 = 18 \text{ months} \][/tex]
4. Calculate the future value of each deposit:
- Future Value of the Initial Deposit (P₀):
[tex]\[ FV_{\text{initial}} = P_0 \times \left(1 + i\right)^{36} \][/tex]
[tex]\[ FV_{\text{initial}} = 8000 \times \left(1 + 0.0063333\right)^{36} \approx 10041.46 \][/tex]
- Future Value of the First Additional Deposit (P₁):
[tex]\[ FV_{\text{first addition}} = P_1 \times \left(1 + i\right)^{24} \][/tex]
[tex]\[ FV_{\text{first addition}} = 2500 \times \left(1 + 0.0063333\right)^{24} \approx 2909.01 \][/tex]
- Future Value of the Second Additional Deposit (P₂):
[tex]\[ FV_{\text{second addition}} = P_2 \times \left(1 + i\right)^{18} \][/tex]
[tex]\[ FV_{\text{second addition}} = 2500 \times \left(1 + 0.0063333\right)^{18} \approx 2800.87 \][/tex]
5. Sum the future values to find the total amount in the account:
[tex]\[ \text{Total Amount} = FV_{\text{initial}} + FV_{\text{first addition}} + FV_{\text{second addition}} \][/tex]
[tex]\[ \text{Total Amount} = 10041.46 + 2909.01 + 2800.87 \approx 15751.34 \][/tex]
So, the total amount in the savings account at the end of three years is approximately R15,751.34.
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