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Mortgage Payment:

The equation below helps us determine the monthly payment for a mortgage.

P
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Payment=
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P - principal (the borrowed amount)

n - the number of times the loan is compounded in a year

t - the number of years

r - the interest rate but written as a decimal

11 answers

The formula you've provided is used to calculate the monthly mortgage payment based on the principal loan amount, interest rate, and loan term. The formula can be simplified for clarity. Here’s the correctly formatted equation for the monthly payment \(M\):

\[ M = P \cdot \frac{r/n}{1 - (1 + r/n)^{-nt}} \]

Where:

  • \(M\) = monthly payment
  • \(P\) = principal (the borrowed amount)
  • \(r\) = annual interest rate (as a decimal)
  • \(n\) = number of times that payments are compounded (typically 12 for monthly payments)
  • \(t\) = loan term in years

Here's a breakdown of the components:

  • \(P\) is the total amount borrowed.
  • \(r/n\) gives the interest rate per compounding period (monthly, if \(n = 12\)).
  • \(nt\) is the total number of payments made over the loan term.

Example Calculation

Suppose you have a mortgage of $200,000 with an annual interest rate of 5% (0.05 as a decimal) to be paid over 30 years. Then the calculation would be as follows:

  1. Convert the annual interest rate to a monthly rate: \[ r/n = 0.05/12 \approx 0.0041667 \]

  2. Determine the total number of payments: \[ nt = 12 \times 30 = 360 \]

  3. Substitute the values into the formula: \[ M = 200,000 \cdot \frac{0.0041667}{1 - (1 + 0.0041667)^{-360}} \]

  4. Calculate the denominator: \[ 1 - (1 + 0.0041667)^{-360} \approx 1 - (1.0041667)^{-360} \approx 1 - 0.2318 \approx 0.7682 \]

  5. Continue the calculation: \[ M \approx 200,000 \cdot \frac{0.0041667}{0.7682} \approx 200,000 \cdot 0.0054 \approx 1086.88 \]

So the monthly payment for this mortgage would be approximately $1,086.88.

If you have specific values for \(P\), \(r\), \(n\), and \(t\), I can help you compute the monthly payment!

John is going to make a house purchase and borrow
$
350
,
000
$350,000 and has two options.

John can have a mortgage for
30
30 years an interest rate of
5.5
%
5.5% OR a mortgage for
15
15 years at an interest rate of
5.25
%
5.25%

What is
r
r for the
30
30 loan?

To find \( r \) for the 30-year loan at an interest rate of 5.5%, you need to express the annual interest rate as a decimal. Here’s how to do that:

  1. Convert the percentage to a decimal: \[ r = \frac{5.5}{100} = 0.055 \]

Thus, for the 30-year loan, \( r \) is 0.055.

If you need further calculations such as monthly payments, feel free to ask!

What was your equation to find the monthly payment for the 30 year loan?

The equation to find the monthly mortgage payment \(M\) for a loan can be expressed using the formula:

\[ M = P \cdot \frac{r/n}{1 - (1 + r/n)^{-nt}} \]

Where:

  • \(M\) = monthly payment
  • \(P\) = principal (the borrowed amount)
  • \(r\) = annual interest rate (as a decimal)
  • \(n\) = number of times the loan is compounded per year (typically 12 for monthly payments)
  • \(t\) = loan term in years

Given your scenario for a 30-year loan with a principal of $350,000 and an annual interest rate of 5.5%, we will plug in the values as follows:

  1. Principal \(P\): $350,000
  2. Annual interest rate \(r\): 5.5% (as a decimal is \(0.055\))
  3. Number of compounding periods per year \(n\): 12 (for monthly payments)
  4. Loan term \(t\): 30 years

Substituting the values into the formula:

  1. Convert the annual interest rate to a monthly rate: \[ r/n = 0.055/12 \approx 0.00458333 \]

  2. Determine the total number of payments: \[ nt = 12 \times 30 = 360 \]

  3. Substitute these values into the formula: \[ M = 350,000 \cdot \frac{0.00458333}{1 - (1 + 0.00458333)^{-360}} \]

Calculate the denominator:

  1. Calculate \(1 + \frac{0.055}{12}\): \[ 1 + 0.00458333 \approx 1.00458333 \]

  2. Calculate \((1.00458333)^{-360}\): \[ (1.00458333)^{-360} \approx 0.167665 \]

  3. Calculate the denominator: \[ 1 - 0.167665 \approx 0.832335 \]

Substitute into the formula:

Now, substitute the calculations back into the formula: \[ M = 350,000 \cdot \frac{0.00458333}{0.832335} \]

Final Calculation:

Calculating the fraction: \[ \frac{0.00458333}{0.832335} \approx 0.00550 \]

Then, \[ M \approx 350,000 \cdot 0.00550 \approx 1925.63 \]

Thus, the monthly payment \(M\) for the 30-year loan at an interest rate of 5.5% is approximately $1,925.63.

If you would like to calculate the payment for the 15-year loan as well, just let me know!

Show how you find the total paid for the 30 year loan.

To find the total amount paid over the life of a 30-year loan, you can use the monthly mortgage payment we calculated and multiply it by the total number of payments over the loan's term.

Steps to Calculate the Total Amount Paid

  1. Monthly Payment: From our previous calculation, the monthly payment \(M\) is approximately $1,925.63.

  2. Total Number of Payments: For a 30-year loan with monthly payments: \[ \text{Total Number of Payments} = 12 \times 30 = 360 \]

  3. Total Amount Paid: Multiply the monthly payment by the total number of payments: \[ \text{Total Amount Paid} = M \times \text{Total Number of Payments} \] \[ \text{Total Amount Paid} = 1,925.63 \times 360 \]

Calculation:

\[ \text{Total Amount Paid} \approx 1,925.63 \times 360 \approx 692,825.80 \]

Summary:

Therefore, the total amount paid over the life of the 30-year loan is approximately $692,825.80.

Breakdown:

  • Total Payments: 360 payments made over 30 years.
  • Monthly Payment: $1,925.63
  • Total Paid: $692,825.80

If you have any further questions or need information on anything else, feel free to ask!

What was your equation to find the monthly payment for the 15 year loan?

To find the monthly payment for the 15-year loan, you can use the same mortgage payment formula:

\[ M = P \cdot \frac{r/n}{1 - (1 + r/n)^{-nt}} \]

Given Values for the 15-Year Loan:

  1. Principal \(P\): $350,000 (the amount borrowed).
  2. Annual interest rate \(r\): 5.25% (as a decimal, this is \(0.0525\)).
  3. Number of compounding periods per year \(n\): 12 (for monthly payments).
  4. Loan term \(t\): 15 years.

Steps to Calculate Monthly Payment:

  1. Convert the annual interest rate to a monthly rate: \[ r/n = \frac{0.0525}{12} \approx 0.004375 \]

  2. Determine the total number of payments: \[ nt = 12 \times 15 = 180 \]

  3. Substitute values into the formula: \[ M = 350,000 \cdot \frac{0.004375}{1 - (1 + 0.004375)^{-180}} \]

Calculate the Denominator:

  1. Calculate \(1 + \frac{0.0525}{12}\): \[ 1 + 0.004375 \approx 1.004375 \]

  2. Calculate \((1.004375)^{-180}\): \[ (1.004375)^{-180} \approx 0.385217 \]

  3. Calculate the denominator: \[ 1 - 0.385217 \approx 0.614783 \]

Substitute into the Formula:

Now substitute back into the formula: \[ M = 350,000 \cdot \frac{0.004375}{0.614783} \]

Final Calculation:

Calculating the fraction: \[ \frac{0.004375}{0.614783} \approx 0.0071187 \]

Then, calculating the monthly payment: \[ M \approx 350,000 \cdot 0.0071187 \approx 2,490.55 \]

Conclusion:

Thus, the monthly payment \(M\) for the 15-year loan at an interest rate of 5.25% is approximately $2,490.55.

If you have more questions or need assistance with further calculations, feel free to ask!

How how you find the total paid for the 15 year loan.

To find the total amount paid over the life of a 15-year loan, you can use the monthly mortgage payment we calculated and multiply it by the total number of payments over the loan's term.

Steps to Calculate the Total Amount Paid

  1. Monthly Payment: From our previous calculation, the monthly payment \(M\) for the 15-year loan at an interest rate of 5.25% was approximately $2,490.55.

  2. Total Number of Payments: For a 15-year loan with monthly payments: \[ \text{Total Number of Payments} = 12 \times 15 = 180 \]

  3. Total Amount Paid: Multiply the monthly payment by the total number of payments: \[ \text{Total Amount Paid} = M \times \text{Total Number of Payments} \] \[ \text{Total Amount Paid} = 2,490.55 \times 180 \]

Calculation:

\[ \text{Total Amount Paid} \approx 2,490.55 \times 180 \approx 448,249 \]

Summary:

Therefore, the total amount paid over the life of the 15-year loan is approximately $448,249.

Breakdown:

  • Total Payments: 180 payments made over 15 years.
  • Monthly Payment: $2,490.55
  • Total Paid: $448,249

If you have any further questions or need additional assistance, feel free to ask!