To calculate the fortnightly installments for Martin's refrigerator purchase, we can use the formula for an installment loan, which is based on the amortization concept. The formula for calculating the installment \( M \) is:
\[ M = P \times \frac{r(1 + r)^n}{(1 + r)^n - 1} \]
Where:
- \( M \) = Fortnightly installment
- \( P \) = Principal amount (remaining balance)
- \( r \) = Interest rate per payment period
- \( n \) = Total number of payments (installments)
Given:
- Principal amount \( P = 2500 \)
- Annual interest rate \( = 10% = 0.10 \)
- Fortnightly interest rate \( r = \frac{0.10}{26} \) (since there are 26 fortnights in a year)
- Total number of payments \( n = 30 \text{ months} \times 2 \text{ fortnights per month} = 60 \text{ payments} \)
Calculating \( r \) and then \( M \)
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Calculate the fortnightly interest rate: \[ r = \frac{0.10}{26} \approx 0.0038461538461538 \]
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Calculate \( n \): \[ n = 30 \times 2 = 60 \]
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Putting these values into the installment formula: \[ M = 2500 \times \frac{0.0038461538461538(1 + 0.0038461538461538)^{60}}{(1 + 0.0038461538461538)^{60} - 1} \]
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Calculate \( (1 + r)^n \): \[ (1 + r)^{60} = (1 + 0.0038461538461538)^{60} \approx 1.261361 \]
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Calculate the installment \( M \): \[ M = 2500 \times \frac{0.0038461538461538 \times 1.261361}{1.261361 - 1} \] \[ = 2500 \times \frac{0.00484382}{0.261361} \] \[ \approx 2500 \times 0.01851 \approx 46.275 \]
Final Answer:
Thus, Martin's fortnightly installment for the refrigerator will be approximately $46.28.