Danielle is buying a house that costs $275000. She will finance the purchase with a 25 yr mortgage with an interest rate of 2.9%, compounded semi-annually. She must make a down payment of 15% of the purchase price.

a) How much will each payment be?
b) How much interest will Danielle end up paying by the time she has paid off the loan?
c) How much will she pay altogether for the house?

P = Po(1+r)^n

Po = 0.85 * 275,000 = $233,750

r = (2.9%/2)/100% = 0.0145 = Semi-APR
expressed as a decimal.

n = 2comp/yr * 25yr = 50 Compounding
periods.

a. P = 233,750(1.0145)^50 = 480,124.85.
480,124.50/50 = $9,602.50 Semi-annually.
0r 480,124.85/300mo = $1600.42 Monthly.

b. Int. = 480,124.85 - 233,750 = 246,374.85

c. $480,124.85

To find the answers, we need to follow these steps:

Step 1: Calculate the down payment.
Step 2: Calculate the loan amount.
Step 3: Calculate the number of payments.
Step 4: Calculate the interest rate per period.
Step 5: Calculate the monthly payment.
Step 6: Calculate the total interest paid.
Step 7: Calculate the total cost of the house.

a) How much will each payment be?

Step 1: Calculate the down payment.
Down payment = 15% of the purchase price
Down payment = 0.15 x $275,000
Down payment = $41,250

Step 2: Calculate the loan amount.
Loan amount = Purchase price - Down payment
Loan amount = $275,000 - $41,250
Loan amount = $233,750

Step 3: Calculate the number of payments.
Number of payments = 25 years x 2 (since interest is compounded semi-annually)
Number of payments = 50

Step 4: Calculate the interest rate per period.
Interest rate per period = Annual interest rate / Number of compounding periods per year
Interest rate per period = 2.9% / 2
Interest rate per period = 1.45%

Step 5: Calculate the monthly payment.
To calculate the monthly payment, we can use the formula for the present value of an annuity:

Monthly payment = (Loan amount x Interest rate per period) / (1 - (1 + Interest rate per period)^(-number of payments))

Using the formula, the monthly payment is:

Monthly payment = ($233,750 x (1.45/100)) / (1 - (1 + (1.45/100))^(-50))
Monthly payment = $893.28

Therefore, each payment will be approximately $893.28.

b) How much interest will Danielle end up paying by the time she has paid off the loan?

Total interest paid = (Monthly payment x Number of payments) - Loan amount
Total interest paid = ($893.28 x 50) - $233,750
Total interest paid = $44,664 - $233,750
Total interest paid = $56,914

Danielle will end up paying approximately $56,914 in interest.

c) How much will she pay altogether for the house?

Total cost of the house = Loan amount + Total interest paid + Down payment
Total cost of the house = $233,750 + $56,914 + $41,250
Total cost of the house = $331,914

Danielle will pay a total of approximately $331,914 for the house.

To answer these questions, we need to work through the calculations step by step. Let's start with the down payment.

a) Down Payment:
The down payment is 15% of the purchase price. So, to calculate the down payment amount, we need to multiply the purchase price by 0.15.
Down payment = $275,000 * 0.15 = $41,250

b) Loan Amount:
The loan amount is the total purchase price minus the down payment.
Loan amount = $275,000 - $41,250 = $233,750

Next, we can calculate the mortgage payment using the formula for a loan with compound interest:

M = P * (r(1+r)^n) / ((1+r)^n - 1)

Where:
M = Mortgage payment
P = Loan amount
r = Interest rate per period (in this case, semi-annually)
n = Total number of payment periods (in this case, 25 years * 2 payments per year = 50)

c) Mortgage Payment:
M = $233,750 * (0.0145(1+0.0145)^50) / ((1+0.0145)^50 - 1)

This calculation gives us the value of the mortgage payment.

To find the value of interest paid over time, we can use the formula:

Total Interest Paid = (M * Number of Payments) - P

Where:
M = Monthly Mortgage Payment
Number of Payments = Total Number of Payment Periods (in this case, 25 years * 12 payments per year = 300)
P = Loan Amount

d) Total Interest Paid:
Total Interest Paid = ($ Mortgage Payment * Number of Payments) - Loan Amount

Lastly, let's calculate the total amount paid for the house, including both the down payment and the total mortgage payments.

e) Total Cost of the House:
Total Cost of the House = Down Payment + (Number of Payments * Monthly Mortgage Payment)

Now, you can substitute the values into the formulas and calculate the answers to each question.