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Sagot :
Let's go through a detailed, step-by-step solution to find the size of the monthly payment for Hunan's car loan. We'll break down the calculations as follows:
1. Identify the given information:
- Car price: [tex]$15,800 - Down payment percentage: 15% (or 0.15 in decimal form) - Annual interest rate: 7% (or 0.07 in decimal form) - Loan term: 5 years 2. Calculate the down payment: The down payment is a percentage of the car price. \[ \text{Down payment} = \text{Car price} \times \text{Down payment percentage} \] Substituting the values: \[ \text{Down payment} = 15800 \times 0.15 = 2370 \] 3. Calculate the loan amount: The loan amount is the car price minus the down payment. \[ \text{Loan amount} = \text{Car price} - \text{Down payment} \] Substituting the values: \[ \text{Loan amount} = 15800 - 2370 = 13430 \] 4. Converting the annual interest rate to a monthly interest rate: Since the interest is compounded monthly, we need to convert the annual interest rate to a monthly rate. \[ \text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12} \] Substituting the values: \[ \text{Monthly interest rate} = \frac{0.07}{12} = 0.005833 \quad (\text{rounded to six decimal places}) \] 5. Calculate the total number of monthly payments: Multiplying the number of years by the number of months in a year gives the total number of monthly payments. \[ \text{Total payments} = \text{Years} \times 12 \] Substituting the values: \[ \text{Total payments} = 5 \times 12 = 60 \] 6. Calculate the monthly payment using the annuity formula: The annuity formula to calculate the monthly payment is given by: \[ \text{Monthly payment} = \text{Loan amount} \times \frac{\text{Monthly interest rate} \times (1 + \text{Monthly interest rate})^{\text{Total payments}}}{(1 + \text{Monthly interest rate})^{\text{Total payments}} - 1} \] Substituting the values: \[ \text{Monthly payment} = 13430 \times \frac{0.005833 \times (1 + 0.005833)^{60}}{(1 + 0.005833)^{60} - 1} \] 7. Simplify the calculation: First, calculate \((1 + 0.005833)^{60}\): \[ (1 + 0.005833)^{60} \approx 1.416599 \] Now, calculate the numerator and the denominator: \[ \text{Numerator} = 0.005833 \times 1.416599 = 0.008265 \] \[ \text{Denominator} = 1.416599 - 1 = 0.416599 \] Finally, calculate the monthly payment: \[ \text{Monthly payment} = 13430 \times \frac{0.008265}{0.416599} \approx 13430 \times 0.01984 = 265.93 \] 8. Final answer: The size of the monthly payment is \(\$[/tex]265.93\).
So, Hunan's monthly payment for the car loan is $265.93.
1. Identify the given information:
- Car price: [tex]$15,800 - Down payment percentage: 15% (or 0.15 in decimal form) - Annual interest rate: 7% (or 0.07 in decimal form) - Loan term: 5 years 2. Calculate the down payment: The down payment is a percentage of the car price. \[ \text{Down payment} = \text{Car price} \times \text{Down payment percentage} \] Substituting the values: \[ \text{Down payment} = 15800 \times 0.15 = 2370 \] 3. Calculate the loan amount: The loan amount is the car price minus the down payment. \[ \text{Loan amount} = \text{Car price} - \text{Down payment} \] Substituting the values: \[ \text{Loan amount} = 15800 - 2370 = 13430 \] 4. Converting the annual interest rate to a monthly interest rate: Since the interest is compounded monthly, we need to convert the annual interest rate to a monthly rate. \[ \text{Monthly interest rate} = \frac{\text{Annual interest rate}}{12} \] Substituting the values: \[ \text{Monthly interest rate} = \frac{0.07}{12} = 0.005833 \quad (\text{rounded to six decimal places}) \] 5. Calculate the total number of monthly payments: Multiplying the number of years by the number of months in a year gives the total number of monthly payments. \[ \text{Total payments} = \text{Years} \times 12 \] Substituting the values: \[ \text{Total payments} = 5 \times 12 = 60 \] 6. Calculate the monthly payment using the annuity formula: The annuity formula to calculate the monthly payment is given by: \[ \text{Monthly payment} = \text{Loan amount} \times \frac{\text{Monthly interest rate} \times (1 + \text{Monthly interest rate})^{\text{Total payments}}}{(1 + \text{Monthly interest rate})^{\text{Total payments}} - 1} \] Substituting the values: \[ \text{Monthly payment} = 13430 \times \frac{0.005833 \times (1 + 0.005833)^{60}}{(1 + 0.005833)^{60} - 1} \] 7. Simplify the calculation: First, calculate \((1 + 0.005833)^{60}\): \[ (1 + 0.005833)^{60} \approx 1.416599 \] Now, calculate the numerator and the denominator: \[ \text{Numerator} = 0.005833 \times 1.416599 = 0.008265 \] \[ \text{Denominator} = 1.416599 - 1 = 0.416599 \] Finally, calculate the monthly payment: \[ \text{Monthly payment} = 13430 \times \frac{0.008265}{0.416599} \approx 13430 \times 0.01984 = 265.93 \] 8. Final answer: The size of the monthly payment is \(\$[/tex]265.93\).
So, Hunan's monthly payment for the car loan is $265.93.
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