Calculate the total dollar amount paid for a house purchased for $200,000. The buyer paid $50,000 as down payment and the remaining $150,000 was obtained with a closed mortgage having a 25 year loan at 10% interest compounded semi-annually and a monthly payment period. Assume the buyer renews the loan after each term at the same interest rate.
To calculate the total dollar amount paid for the house, we need to consider the down payment, the mortgage amount, and the interest payments over the loan term.
First, let's calculate the mortgage amount. The buyer paid $200,000 for the house and made a down payment of $50,000. Therefore, the mortgage amount is $200,000 - $50,000 = $150,000.
Next, let's calculate the monthly payment for the mortgage. We'll use the formula for the monthly payment on a loan:
Monthly Payment = (P * r * (1 + r)^n) / ((1 + r)^n - 1),
where P is the principal (mortgage amount), r is the monthly interest rate, and n is the total number of payments.
The monthly interest rate is the annual interest rate divided by 12 and expressed as a decimal:
Monthly Interest Rate = 10% / 12 = 0.1 / 12 = 0.0083333.
The total number of payments is the number of years multiplied by 12:
Total Number of Payments = 25 years * 12 months/year = 300 months.
Now, using the formula, we can calculate the monthly payment:
Monthly Payment = ($150,000 * 0.0083333 * (1 + 0.0083333)^300) / ((1 + 0.0083333)^300 - 1).
Once we have the monthly payment, we can calculate the total dollar amount paid over the loan term. This amount depends on the number of times the loan is renewed. For each renewal, we need to calculate the new mortgage amount and recompute the monthly payment.
Since the buyer renews the loan after each term at the same interest rate, we can assume that the mortgage amount will remain the same for each renewal.
To find the total dollar amount paid, we'll multiply the monthly payment by the number of payments, taking into account the number of times the loan is renewed.
Total Dollar Amount Paid = (Monthly Payment * Total Number of Payments) + (Monthly Payment * Total Number of Payments * Number of Renewals).
Since it's not provided how many times the loan is renewed, this information is missing, and we cannot give a definite total dollar amount paid without knowing the number of renewals.
So, to calculate the total dollar amount paid for the house, we need to know the number of times the loan is renewed at the same interest rate.