the monthly payment that amortizes a loan of A dollars in t yr when the interest rate is r per year, compounded monthly, is given by
P=f(A,r,t) = Ar/ 12[1-(1+ r/12)^-12t ]
Find the monthly payment for a home mortgage of 300,000 that will be amortized over 30 yr with an interest rate of 6%/year? And interest rate of 8%/year?
To find the monthly payment for a home mortgage of $300,000 that will be amortized over 30 years with an interest rate of 6% per year, we can substitute the given values into the formula for monthly payment:
P = f(A, r, t) = (Ar/12) [1 - (1 + r/12)^(-12t)]
A = $300,000
r = 6% = 0.06 (expressed as a decimal)
t = 30 years
Now we substitute these values into the formula and solve for P:
P = ($300,000 * 0.06/12) [1 - (1 + 0.06/12)^(-12 * 30)]
Simplifying further:
P = $1,500 * [1 - (1 + 0.06/12)^(-360)]
Calculating the exponential part first:
(1 + 0.06/12)^(-360) ≈ 0.55143
Now substitute back into the equation:
P = $1,500 * [1 - 0.55143]
Simplifying:
P = $1,500 * 0.44857
P ≈ $672.86
Therefore, the monthly payment for a home mortgage of $300,000 that will be amortized over 30 years with an interest rate of 6% per year is approximately $672.86.
To calculate the monthly payment with an interest rate of 8% per year, you can follow the same steps, but substitute r = 0.08:
P = f($300,000, 0.08, 30)
P = ($300,000 * 0.08/12) [1 - (1 + 0.08/12)^(-12 * 30)]
Calculate the exponential part:
(1 + 0.08/12)^(-360) ≈ 0.39784
Now substitute back into the equation:
P = $2,000 * [1 - 0.39784]
Simplifying:
P = $2,000 * 0.60216
P ≈ $1,204.32
Therefore, the monthly payment for a home mortgage of $300,000 that will be amortized over 30 years with an interest rate of 8% per year is approximately $1,204.32.
To find the monthly payment for a home mortgage of $300,000 that will be amortized over 30 years with an interest rate of 6% per year (compounded monthly), we can use the given formula:
P = f(A, r, t) = (A * r/12) / [1 - (1 + r/12)^(-12t)]
Replacing the variables with the given values:
A = $300,000
r = 6% = 0.06
t = 30
Let's calculate the monthly payment:
For an interest rate of 6% per year:
P = (300,000 * 0.06/12) / [1 - (1 + 0.06/12)^(-12*30)]
To calculate this, let's break down the formula further into steps:
Step 1: Calculate the monthly interest rate:
Divide the annual interest rate by 12.
r/12 = 0.06/12 = 0.005
Step 2: Calculate the number of months:
Multiply the number of years by 12.
t = 30 * 12 = 360
Step 3: Calculate the value inside the square brackets:
(1 + r/12)^(-12t) = (1 + 0.005)^(-12 * 360)
Step 4: Calculate the monthly payment:
P = (300,000 * 0.005) / [1 - (1 + 0.005)^(-12 * 360)]
Calculating each step and simplifying the expression, we get:
P = $1,798.65
Therefore, the monthly payment for a home mortgage of $300,000, amortized over 30 years with an interest rate of 6% per year, is $1,798.65.
Now, let's calculate the monthly payment for an interest rate of 8% per year using the same formula and steps:
Replacing the variables with the new values:
A = $300,000
r = 8% = 0.08
t = 30
P = (300,000 * 0.08/12) / [1 - (1 + 0.08/12)^(-12 * 30)]
Calculating each step and simplifying the expression, we get:
P = $2,202.29
Therefore, the monthly payment for a home mortgage of $300,000, amortized over 30 years with an interest rate of 8% per year, is $2,202.29.