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Sagot :
To solve this problem, let's follow each step carefully:
1. Initial Cash Payment:
- The customer pays [tex]\(\frac{1}{3}\)[/tex] of the price in cash.
- For simplicity, let's assume the price of the television is [tex]$1. - The initial payment in cash is: \[ \text{Cash Paid} = \frac{1}{3} \times 1 = 0.3333 \] (in monetary terms, this is approximately $[/tex]0.33).
2. Remaining Amount after Initial Payment:
- After paying [tex]\(\frac{1}{3}\)[/tex] of the price, [tex]\(\frac{2}{3}\)[/tex] of the price remains to be paid.
- Thus, the remaining amount to be financed through the hire purchase scheme is:
[tex]\[ \text{Remaining Amount} = 1 - 0.3333 = 0.6667 \][/tex]
3. Interest Calculation:
- The interest rate given is [tex]\(4\%\)[/tex] per annum.
- The loan term is 24 months, which is [tex]\(2\)[/tex] years.
- The simple interest for the remaining amount over 2 years is calculated as:
[tex]\[ \text{Interest} = \text{Remaining Amount} \times \text{Interest Rate} \times \text{Loan Term} \][/tex]
[tex]\[ \text{Interest} = 0.6667 \times 0.04 \times 2 = 0.0533 \][/tex]
(this can't be simplified further in meaningful ways without losing precision).
4. Total Amount to be Repaid with Interest:
- Adding the interest to the remaining amount gives the total amount to be repaid:
[tex]\[ \text{Total Amount with Interest} = \text{Remaining Amount} + \text{Interest} \][/tex]
[tex]\[ \text{Total Amount with Interest} = 0.6667 + 0.0533 = 0.7200 \][/tex]
5. Monthly Instalment Calculation:
- The loan is spread over 24 months.
- The monthly instalment is calculated as:
[tex]\[ \text{Monthly Instalment} = \frac{\text{Total Amount with Interest}}{\text{Number of Months}} \][/tex]
[tex]\[ \text{Monthly Instalment} = \frac{0.7200}{24} = 0.0300 \][/tex]
(approximately [tex]$0.03). Part (a): The monthly instalment is approximately $[/tex]\[tex]$0.03$[/tex].
6. Total Amount Paid by the Customer:
- The total amount paid by the customer includes the initial cash payment and the total amount with interest over the loan term.
- Thus, the total amount paid is:
[tex]\[ \text{Total Amount Paid} = \text{Cash Paid} + \text{Total Amount with Interest} \][/tex]
[tex]\[ \text{Total Amount Paid} = 0.3333 + 0.7200 = 1.0533 \][/tex]
(approximately [tex]$1.05). Part (b): The total amount paid for the television is approximately $[/tex]\[tex]$1.05$[/tex].
1. Initial Cash Payment:
- The customer pays [tex]\(\frac{1}{3}\)[/tex] of the price in cash.
- For simplicity, let's assume the price of the television is [tex]$1. - The initial payment in cash is: \[ \text{Cash Paid} = \frac{1}{3} \times 1 = 0.3333 \] (in monetary terms, this is approximately $[/tex]0.33).
2. Remaining Amount after Initial Payment:
- After paying [tex]\(\frac{1}{3}\)[/tex] of the price, [tex]\(\frac{2}{3}\)[/tex] of the price remains to be paid.
- Thus, the remaining amount to be financed through the hire purchase scheme is:
[tex]\[ \text{Remaining Amount} = 1 - 0.3333 = 0.6667 \][/tex]
3. Interest Calculation:
- The interest rate given is [tex]\(4\%\)[/tex] per annum.
- The loan term is 24 months, which is [tex]\(2\)[/tex] years.
- The simple interest for the remaining amount over 2 years is calculated as:
[tex]\[ \text{Interest} = \text{Remaining Amount} \times \text{Interest Rate} \times \text{Loan Term} \][/tex]
[tex]\[ \text{Interest} = 0.6667 \times 0.04 \times 2 = 0.0533 \][/tex]
(this can't be simplified further in meaningful ways without losing precision).
4. Total Amount to be Repaid with Interest:
- Adding the interest to the remaining amount gives the total amount to be repaid:
[tex]\[ \text{Total Amount with Interest} = \text{Remaining Amount} + \text{Interest} \][/tex]
[tex]\[ \text{Total Amount with Interest} = 0.6667 + 0.0533 = 0.7200 \][/tex]
5. Monthly Instalment Calculation:
- The loan is spread over 24 months.
- The monthly instalment is calculated as:
[tex]\[ \text{Monthly Instalment} = \frac{\text{Total Amount with Interest}}{\text{Number of Months}} \][/tex]
[tex]\[ \text{Monthly Instalment} = \frac{0.7200}{24} = 0.0300 \][/tex]
(approximately [tex]$0.03). Part (a): The monthly instalment is approximately $[/tex]\[tex]$0.03$[/tex].
6. Total Amount Paid by the Customer:
- The total amount paid by the customer includes the initial cash payment and the total amount with interest over the loan term.
- Thus, the total amount paid is:
[tex]\[ \text{Total Amount Paid} = \text{Cash Paid} + \text{Total Amount with Interest} \][/tex]
[tex]\[ \text{Total Amount Paid} = 0.3333 + 0.7200 = 1.0533 \][/tex]
(approximately [tex]$1.05). Part (b): The total amount paid for the television is approximately $[/tex]\[tex]$1.05$[/tex].
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