Question 1193755: Please help me with the homework.
1. Compare the amounts accumulated on a principal of R10000 is invested from 10 march 20.3 to 1 July 20.5 at 16 and half% per annum compounded semi-annually,and credited on 1 January and 1 July if,
1.1. simple interest is used for the odd period and compound interest for the rest of the term;
1.2. fractional compounding is used for the term
Note ignore the slight differences between the numbers.
Answer by proyaop(69)   (Show Source):
You can put this solution on YOUR website! **1.1 Simple Interest for Odd Period, Compound Interest for Rest**
* **Calculate the odd period:**
* From March 10, 2020, to July 1, 2020, is approximately 3 months.
* **Calculate simple interest for the odd period:**
* Principal (P) = R10,000
* Rate (R) = 16.5% per annum = 0.165
* Time (T) = 3 months = 3/12 years = 0.25 years
* Simple Interest (SI) = P * R * T = R10,000 * 0.165 * 0.25 = R412.50
* **Calculate the amount after the odd period:**
* Amount = Principal + Simple Interest = R10,000 + R412.50 = R10,412.50
* **Calculate the remaining period:**
* From July 1, 2020, to July 1, 2025, is 5 years.
* **Calculate the amount after 5 years with compound interest:**
* Principal (P) = R10,412.50
* Rate (R) = 16.5% per annum compounded semi-annually = 0.165 / 2 = 0.0825 per half-year
* Number of periods (n) = 5 years * 2 periods/year = 10 periods
* Amount = P * (1 + R)^n = R10,412.50 * (1 + 0.0825)^10
* Amount ≈ R23,967.47
**1.2 Fractional Compounding**
* **Calculate the total number of days:**
* From March 10, 2020, to July 1, 2025, is approximately 1826 days.
* **Calculate the daily interest rate:**
* Annual interest rate = 16.5% = 0.165
* Daily interest rate = 0.165 / 365
* **Calculate the amount with daily compounding:**
* Amount = P * (1 + (R/365))^N
* Amount = R10,000 * (1 + (0.165/365))^1826
* Amount ≈ R24,178.12
**Comparison**
* **Method 1 (Simple + Compound):** R23,967.47
* **Method 2 (Fractional Compounding):** R24,178.12
**Conclusion**
Fractional compounding (daily compounding in this case) results in a slightly higher accumulated amount compared to using simple interest for the odd period and then switching to semi-annual compounding. This is because fractional compounding applies interest more frequently, leading to slightly higher returns.
**Note:**
* This calculation assumes a year has 365 days.
* Slight variations in the exact number of days and rounding may result in minor differences in the final amounts.
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