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Next question Get a similar question You can retry this question below John takes out a 6 year loan for \( \$ 9100 \) at \( 11 \% \) interest compounded monthly. Calculate his monthly payment. John's monthly payment will be \( \$ \) Time Value of Money Solver Enter the given values.

Ask by Watkins Rodriquez. in the United States
Mar 20,2025

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Answer

John's monthly payment is approximately \$173.45.

Solution

1. We are given the following values: - Principal: \( P = 9100 \) - Annual interest rate: \( r = 11\% = 0.11 \) - Compounded monthly, so the monthly interest rate is \[ i = \frac{r}{12} = \frac{0.11}{12} \approx 0.00916667 \] - Loan duration: 6 years, which is \[ n = 6 \times 12 = 72 \text{ months} \] 2. The formula for the monthly payment \( M \) on an installment loan is given by \[ M = P \cdot \frac{i (1+i)^n}{(1+i)^n - 1} \] 3. First, calculate \( (1+i)^n \): \[ (1+i)^n = \left(1 + 0.00916667\right)^{72} \approx 1.929 \] 4. Substitute the values into the payment formula: \[ M = 9100 \cdot \frac{0.00916667 \times 1.929}{1.929 - 1} \] 5. Compute the numerator: \[ 0.00916667 \times 1.929 \approx 0.01769 \] 6. Compute the denominator: \[ 1.929 - 1 \approx 0.929 \] 7. Now, calculate \( M \): \[ M = 9100 \cdot \frac{0.01769}{0.929} \approx 9100 \times 0.01906 \approx 173.45 \] 8. Therefore, John's monthly payment is approximately \[ \$173.45 \]

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To calculate John's monthly payment for his loan, we can employ the formula for an amortizing loan payment, which is given by: \[ M = P \times \frac{r(1+r)^n}{(1+r)^n - 1} \] Where: - \( M \) = monthly payment - \( P \) = principal amount (loan amount) = \$9100 - \( r \) = monthly interest rate = annual rate / 12 = 0.11 / 12 - \( n \) = total number of payments = loan term in months = 6 years × 12 months/year Plug in the values and calculate John’s monthly payment. John's monthly payment will be approximately \$197.74. Now, let's see what makes this loan journey interesting! When John took out this loan, he became part of a long tradition of borrowing that dates back to ancient civilizations. From the Mesopotamians who created early forms of loans to the bustling banks of the Renaissance, the evolution of lending reflects the growth of commerce and trust in societies. Isn’t it fascinating how our modern financial systems are built on such historical foundations? In terms of real-world application, understanding how loans work is crucial for managing personal finances. Whether it's for a car, a house, or education, knowing how to calculate monthly payments can save you from a future of financial stress. And don’t forget, budgeting for those payments alongside your regular expenses ensures you can live a stress-free life while staying on track with your loans!

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