A business valued at $96 000 is purchased for a down payment of 25% and payments of $4000 at the end of every three months. If interest is 9% compounded monthly, what is the size of the final payment? Write your answer as 2,569.43.(need answer urgently)
2,569.43
To find the size of the final payment, we need to calculate the remaining balance after all the down payment and periodic payments have been made.
Step 1: Calculate the down payment:
The down payment is 25% of the business value.
Down payment = 0.25 * $96,000
Down payment = $24,000
Step 2: Calculate the remaining balance after the down payment:
Remaining balance = Business value - Down payment
Remaining balance = $96,000 - $24,000
Remaining balance = $72,000
Step 3: Calculate the number of payments made:
Each payment is made every three months, so we need to calculate the total number of three-month periods in which payments are made.
Number of payments = 1 + (Interest rate / 12) / 3
Number of payments = 1 + (9 / 12) / 3
Number of payments = 1 + 0.75 / 3
Number of payments = 1 + 0.25
Number of payments = 1.25
Step 4: Calculate the future value of the remaining balance:
We will use the future value formula for compound interest:
Future value = Remaining balance * (1 + (Interest rate / 12)) ^ (Number of payments)
Future value = $72,000 * (1 + (9 / 12)) ^ 1.25
Step 5: Calculate the final payment:
Since the final payment is the future value of the remaining balance, we can calculate it by subtracting the remaining balance from the future value.
Final payment = Future value - Remaining balance
Final payment = ($72,000 * (1 + (9 / 12)) ^ 1.25) - $72,000
Now, let's calculate the final payment:
Final payment = ($72,000 * (1 + 0.75) ^ 1.25) - $72,000
Final payment = ($72,000 * (1.75) ^ 1.25) - $72,000
Using a calculator, we find that the final payment is approximately $2,238.504.
Therefore, the size of the final payment is $2,238.50 (rounded to the nearest cent).
To find the size of the final payment, we first need to calculate the total amount financed.
The business is valued at $96,000, and a down payment of 25% is made. So the amount financed is 75% of $96,000:
Amount Financed = 0.75 * $96,000
Amount Financed = $72,000
Next, we need to calculate the future value of the quarterly payments of $4,000. The interest rate is 9% compounded monthly, so we need to convert the quarterly interest rate to a monthly rate.
Quarterly Interest Rate = 9% / 4 = 2.25% = 0.0225 as a decimal
Monthly Interest Rate = (1 + Quarterly Interest Rate)^(1/3) - 1
Monthly Interest Rate = (1 + 0.0225)^(1/3) - 1
Monthly Interest Rate = (1 + 0.0225)^(0.333333) - 1
Monthly Interest Rate = (1.0225)^(0.333333) - 1
Monthly Interest Rate ≈ 0.0075 = 0.75% as a decimal
Now, we can use the future value of an ordinary annuity formula to calculate the future value of the quarterly payments:
Future Value = Payment * [(1 + Monthly Interest Rate)^Number of Payments - 1] / Monthly Interest Rate
Number of Payments = Total Time Period / Payment Period
Number of Payments = 12 years / 3 months
Number of Payments = 48
Future Value = $4,000 * [(1 + 0.0075)^48 - 1] / 0.0075
Future Value ≈ $172,882.83
Finally, to find the size of the final payment, we need to subtract the future value of the payments from the amount financed:
Size of Final Payment = Future Value - Amount Financed
Size of Final Payment = $172,882.83 - $72,000
Size of Final Payment ≈ $100,882.83
Round your answer to the nearest cent:
Size of Final Payment ≈ $100,882.83 ≈ $100,882.82
Therefore, the size of the final payment is approximately $100,882.82