bill and laura borrowed $15 000 at 6 per annum compounded monthly for 30 years to purchase a home. Their monthly payment is determined to be $899.63.
Find a recursive formula for their balance after each monthly payment has been made.
Determine Bill and Laura's balance after the first payment.
r = (6%/12)/100% = 0.005 = Monthly % rate expressed as a decimal.
I = 0.005 * 150000 = $750.00 = Int. for
1st month.
---Payments---Interest.-Prin.--Bal.
1. $899.33----750.00---149.33-149850.67
2. $899.33----749.25---150.08-149700.59
3. $899.33----748.50---150.83-149549.77
NOTE: The int. is always 0.005 times the
unpaid Bal.
To find the recursive formula for their balance after each monthly payment, we need to use the formula for compound interest:
Balance = Principal + Interest - Payment
The principal amount is the initial loan amount, which is $15,000. The interest rate per month can be calculated by dividing the annual interest rate by 12 (since it is compounded monthly), which gives us 6%/12 = 0.5% = 0.005. The payment amount is given as $899.63.
The interest for the first payment can be calculated as the balance at the start of the month multiplied by the monthly interest rate. So, the formula for the balance after each monthly payment is:
Balance[n] = Balance[n-1] + Balance[n-1] * 0.005 - $899.63
Now, let's calculate Bill and Laura's balance after the first payment:
Balance[1] = $15,000 + $15,000 * 0.005 - $899.63
Simplifying the expression:
Balance[1] = $15,000 + $75 - $899.63
Balance[1] = $14,175.37
Therefore, after the first payment, Bill and Laura's balance is $14,175.37.
To find a recursive formula for Bill and Laura's balance after each monthly payment, we need to understand how the balance changes over time.
Let's break down the problem step by step:
1. First, we need to determine the monthly interest rate. The annual interest rate is 6%, which means the monthly interest rate is 6% / 12 = 0.5%.
2. Next, calculate the total number of months for the loan. In this case, Bill and Laura have a 30-year loan, so the total number of months is 30 * 12 = 360.
3. Now, let's establish the recursive formula. We'll use the balance (B) after each payment.
Let B(n) represent Bill and Laura's balance after the nth payment. Since their monthly payment is $899.63, each month their balance decreases by this amount. However, the balance also increases due to the compounding effect of the interest.
4. To calculate their new balance after each payment, we'll use the following recursive formula:
B(n) = B(n-1) * (1 + r) - p
Where:
- B(n) is the balance after the nth payment
- B(n-1) is the balance after the (n-1)th payment
- r is the monthly interest rate
- p is the monthly payment
For the first payment, we need to find B(1). Since Bill and Laura haven't made any payments yet, their initial balance is $15,000. Therefore:
B(1) = $15,000 * (1 + 0.005) - $899.63
Calculating this equation will give you the answer.