A building is priced at 125,000. If a down payment of 25,000 is made and a payment of 1,000 every month thereafter is required, how many months will it take to pay for the building? Interest is charged at a rate of 9% compounded monthly.
P = (Po*r*t)/(1-(1+r)^-t)
Po = 125000-25000 = $100000.
r = (9%/12)/100% = 0.0075 = Monthly %
rate expressed as a decimal.
Po*r*t = 100000*0.0075t = 750t
P = 1000t.
P = 750t/(1-1.0075^-t) = 1000t
Divide both sides by 750t:
1/(1-1.0075^-t) = 1.333
Cross multiply:
1.333(1-1.0075^-t^-t) = 1
1-1.0075^-t = 0.750.
1.0075^-t = 1-0.75 = 0.25
-t*Log 1.0075 = Log 0.25
-t = Log 0.25/Log 1.0075 = -185.53
t = 185.53 Months.
To calculate the number of months it will take to pay for the building, we need to consider the initial down payment, the monthly payments, and the interest rate.
Step 1: Calculate the remaining balance after the down payment:
Building price - Down payment = 125,000 - 25,000 = 100,000
Step 2: Calculate the monthly interest rate:
Annual interest rate / 12 = 9% / 12 = 0.09 / 12 = 0.0075
Step 3: Calculate the monthly payment needed to pay off the remaining balance:
Monthly payment = 1,000
Step 4: Calculate the number of months needed to pay off the remaining balance.
To solve this, we can use the formula for the future value of an annuity:
FV = P * ((1 + r)^n - 1) / r
Where:
FV = Future Value (remaining balance)
P = Monthly payment
r = Monthly interest rate
n = Number of months
Rearranging the formula to solve for n:
n = (log(FV * r + P) - log(P * r)) / log(1 + r)
Using the values from steps 2 and 3:
n = (log(100,000 * 0.0075 + 1,000) - log(1,000 * 0.0075)) / log(1 + 0.0075)
Using a calculator to solve the equation:
n ≈ 172.38
Rounding up to the nearest whole number, it will take approximately 173 months to pay off the building.
To determine the number of months it will take to pay for the building, we need to calculate the monthly payments and then solve for the number of months.
Step 1: Calculate the loan amount
The loan amount is the total cost of the building minus the down payment. In this case, the total cost of the building is $125,000 and the down payment is $25,000. So, the loan amount would be $125,000 - $25,000 = $100,000.
Step 2: Calculate the monthly interest rate
The yearly interest rate is 9%, which is compounded monthly. To calculate the monthly interest rate, we divide the yearly rate by 12 and convert it into a decimal. So, the monthly interest rate would be (9% / 12) / 100 = 0.0075.
Step 3: Calculate the monthly payment
To calculate the monthly payment, we can use the formula for the monthly payment on a loan:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
M = Monthly payment
P = Loan amount
r = Monthly interest rate
n = Number of monthly payments
In this case, P = $100,000 and r = 0.0075. We need to solve for n.
Let's rearrange the formula to solve for n:
n = log(M / (M - P * r)) / log(1 + r)
Step 4: Substitute the values and calculate the result
Substituting the values, we get:
n = log(1,000 / (1,000 - 100,000 * 0.0075)) / log(1 + 0.0075)
Using a calculator, we find that n is approximately 124.5 months.
However, since we cannot have fractions of months, we need to round up to the nearest whole number of months. Thus, it would take 125 months to pay for the building.