Question 1196127: Please help me with the homework:
1. Compare the amounts on a principal of R5000.00 for the five and one- quarter period at 18% interest per annum if:
1.1.Simple interest is used for the odd period and compound interest for the rest of the term.
1.2. fractional compound is used for the full term.
Answer by ElectricPavlov(122)   (Show Source):
You can put this solution on YOUR website! To compare the amounts, we calculate the total amount (future value) for the principal of \( R5000.00 \) under the given conditions.
---
### **1.1. Simple Interest for Odd Period and Compound Interest for the Rest**
- **Given Data:**
- Principal (\( P \)): \( R5000.00 \)
- Interest rate (\( r \)): 18% per annum (\( 0.18 \))
- Total period: \( 5.25 \) years = \( 5 \) years + \( 0.25 \) year
- Simple interest for \( 0.25 \) year, compound interest for \( 5 \) years.
---
#### **Step 1: Calculate Simple Interest for 0.25 Year**
\[
\text{Simple Interest (SI)} = P \cdot r \cdot t
\]
\[
\text{SI} = 5000 \cdot 0.18 \cdot 0.25 = R225.00
\]
The new principal after 0.25 year becomes:
\[
P_{\text{new}} = P + \text{SI} = 5000 + 225 = R5225.00
\]
---
#### **Step 2: Calculate Compound Interest for 5 Years**
The compound interest formula is:
\[
A = P \cdot (1 + r)^t
\]
\[
A = 5225 \cdot (1 + 0.18)^5
\]
First, calculate \( (1 + 0.18)^5 \):
\[
(1 + 0.18)^5 \approx 2.1228
\]
\[
A = 5225 \cdot 2.1228 \approx R11093.67
\]
Thus, the total amount is:
\[
A = R11093.67
\]
---
### **1.2. Fractional Compound Interest for the Full Term**
Using the compound interest formula directly for \( 5.25 \) years:
\[
A = P \cdot (1 + r)^t
\]
\[
A = 5000 \cdot (1 + 0.18)^{5.25}
\]
First, calculate \( (1 + 0.18)^{5.25} \):
\[
(1 + 0.18)^{5.25} = 2.1228 \cdot (1 + 0.18)^{0.25}
\]
For \( (1 + 0.18)^{0.25} \):
\[
(1 + 0.18)^{0.25} \approx 1.0414
\]
\[
(1 + 0.18)^{5.25} \approx 2.1228 \cdot 1.0414 \approx 2.2092
\]
\[
A = 5000 \cdot 2.2092 \approx R11046.00
\]
---
### **Final Comparison**
1. **Simple Interest + Compound Interest (5.25 years):** \( R11093.67 \)
2. **Fractional Compound Interest (5.25 years):** \( R11046.00 \)
**Difference:**
\[
R11093.67 - R11046.00 = R47.67
\]
The amount using simple interest for the odd period is slightly higher due to the larger base principal at the start of compounding.
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