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Sagot :
Let's solve the problem step-by-step using the given information:
1. Understand the given data:
- Current balance (principal, [tex]\( P \)[/tex]): \[tex]$18,875 - Annual interest rate: 11.6% (0.116 as a decimal) - Number of months to pay off: 24 2. Convert the annual interest rate to a monthly interest rate: - Monthly interest rate \( (i) \) = Annual interest rate / 12 - \( i = 0.116 / 12 \) - \( i \approx 0.0096667 \) 3. Use the given formula for the monthly payment: - The formula is: \[ \text{Monthly Payment} = \frac{P \times i \times (1 + i)^{n}}{(1 + i)^{n} - 1} \] where: - \( P \) = Principal amount (balance owed) - \( i \) = Monthly interest rate - \( n \) = Number of months 4. Substitute the known values into the formula: - Principal, \( P = 18,875 \) - Monthly interest rate, \( i \approx 0.0096667 \) - Number of months, \( n = 24 \) 5. Calculate the numerator and denominator separately: \[ \text{Numerator} = P \times i \times (1 + i)^{n} \] \[ \text{Denominator} = (1 + i)^{n} - 1 \] 6. Plug the values into the equations: - Numerator: \[ \text{Numerator} = 18,875 \times 0.0096667 \times (1 + 0.0096667)^{24} \approx 229.85 \] - Denominator: \[ \text{Denominator} = (1 + 0.0096667)^{24} - 1 \approx 0.2597 \] 7. Divide the numerator by the denominator to find the monthly payment: \[ \text{Monthly Payment} = \frac{229.85}{0.2597} \approx 884.99 \] Seth's approximate monthly payment will be \$[/tex]884.99.
Given the options:
- A. \[tex]$834.75 - B. \$[/tex]884.99
- C. \[tex]$868.35 - D. \$[/tex]753.97
The correct answer is B. Seth's approximate monthly payment will be \$884.99.
1. Understand the given data:
- Current balance (principal, [tex]\( P \)[/tex]): \[tex]$18,875 - Annual interest rate: 11.6% (0.116 as a decimal) - Number of months to pay off: 24 2. Convert the annual interest rate to a monthly interest rate: - Monthly interest rate \( (i) \) = Annual interest rate / 12 - \( i = 0.116 / 12 \) - \( i \approx 0.0096667 \) 3. Use the given formula for the monthly payment: - The formula is: \[ \text{Monthly Payment} = \frac{P \times i \times (1 + i)^{n}}{(1 + i)^{n} - 1} \] where: - \( P \) = Principal amount (balance owed) - \( i \) = Monthly interest rate - \( n \) = Number of months 4. Substitute the known values into the formula: - Principal, \( P = 18,875 \) - Monthly interest rate, \( i \approx 0.0096667 \) - Number of months, \( n = 24 \) 5. Calculate the numerator and denominator separately: \[ \text{Numerator} = P \times i \times (1 + i)^{n} \] \[ \text{Denominator} = (1 + i)^{n} - 1 \] 6. Plug the values into the equations: - Numerator: \[ \text{Numerator} = 18,875 \times 0.0096667 \times (1 + 0.0096667)^{24} \approx 229.85 \] - Denominator: \[ \text{Denominator} = (1 + 0.0096667)^{24} - 1 \approx 0.2597 \] 7. Divide the numerator by the denominator to find the monthly payment: \[ \text{Monthly Payment} = \frac{229.85}{0.2597} \approx 884.99 \] Seth's approximate monthly payment will be \$[/tex]884.99.
Given the options:
- A. \[tex]$834.75 - B. \$[/tex]884.99
- C. \[tex]$868.35 - D. \$[/tex]753.97
The correct answer is B. Seth's approximate monthly payment will be \$884.99.
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