A new homeowner is purchasing a living room set for $2,975 and must decide between two financing offers.
Offer 1: $250 down payment, 24.90% interest rate, compounded monthly, for 3 years, with no payments due for 6 months and then fixed payments of $139.05 for the remainder of the loan term
Offer 2: $400 down payment, 22.90% interest rate, compounded monthly, for 3 years, with no payments due for 12 months and then fixed payments of $165.76 for the remainder of the loan term
Part A: What is the total cost of offer 1? Explain which technology you used to solve and each step of your process.
Part B: What is the total cost of offer 2? Explain which technology you used to solve and each step of your process.
Part C: Which financing offer should the new homeowner choose? Explain your reasoning.
Part A: The total cost of offer 1 is $4,845.25. I used an online loan calculator to solve this problem. I entered the loan amount ($2,975), the interest rate (24.90%), the loan term (3 years), and the number of payments (30). I then clicked "Calculate" to get the total cost of the loan.
Part B: The total cost of offer 2 is $4,717.60. I used the same online loan calculator as in Part A. I entered the loan amount ($2,975), the interest rate (22.90%), the loan term (3 years), and the number of payments (36). I then clicked "Calculate" to get the total cost of the loan.
Part C: The new homeowner should choose offer 2. Offer 2 has a lower interest rate and a longer period of no payments due, which will result in a lower total cost.
Part A: Total cost for offer 1 is $3895.25 + $250 down payment = $4,145.25
Results:
Payment Every Month $108.20
Total of 36 Payments $3,895.25
Total Interest $1,170.25
Part B: Total Cost for offer 2 = $3954.79 + $400 down payment = $4,354.79
Results:
Payment Every Month $82.39
Total of 48 Payments $3,954.79
Total Interest $1,379.79
Part C: the homeowner should choose offer 1 because the monthly payment is higher but the total overall is less.
Part A: To calculate the total cost of offer 1, we need to consider the down payment, the interest on the loan, and the fixed payments for the loan term.
Step 1: Calculate the interest on the loan for the 3 years.
The interest rate is 24.90% compounded monthly, which means we need to calculate the monthly interest rate first. To do this, we divide the annual interest rate by 12.
Monthly interest rate = 24.90% / 12 = 0.0249
Next, we can use the formula for compound interest to calculate the total interest over 36 months (3 years).
Interest = Principal * (1 + Monthly interest rate)^n - Principal
Where:
Principal = Loan amount - Down payment
n = Number of compounding periods
Principal = $2,975 - $250 = $2,725
n = 36 (since there are 12 months in a year for 3 years)
Using the formula, we get:
Interest = $2,725 * (1 + 0.0249)^36 - $2,725
Interest ≈ $1,420.19
Step 2: Calculate the fixed payments for the loan term.
The monthly fixed payment is given as $139.05 for the remainder of the loan term.
Fixed payments = Monthly fixed payment * Number of months
Fixed payments = $139.05 * (36 - 6)
Fixed payments = $139.05 * 30
Fixed payments = $4,171.50
Step 3: Calculate the total cost.
Total cost = Down payment + Interest + Fixed payments
Total cost = $250 + $1,420.19 + $4,171.50
Total cost ≈ $5,841.69
Therefore, the total cost of offer 1 is approximately $5,841.69.
Part B: To calculate the total cost of offer 2, we follow a similar process as in Part A.
Step 1: Calculate the interest on the loan for the 3 years.
The interest rate is 22.90% compounded monthly.
Monthly interest rate = 22.90% / 12 = 0.0191
Principal = $2,975 - $400 = $2,575
n = 36 (since there are 12 months in a year for 3 years)
Using the compound interest formula, we get:
Interest = $2,575 * (1 + 0.0191)^36 - $2,575
Interest ≈ $1,119.62
Step 2: Calculate the fixed payments for the loan term.
The monthly fixed payment is given as $165.76 for the remainder of the loan term.
Fixed payments = $165.76 * (36 - 12)
Fixed payments = $165.76 * 24
Fixed payments = $3,977.76
Step 3: Calculate the total cost.
Total cost = Down payment + Interest + Fixed payments
Total cost = $400 + $1,119.62 + $3,977.76
Total cost ≈ $5,497.38
Therefore, the total cost of offer 2 is approximately $5,497.38.
Part C:
Comparing the total costs of offer 1 and offer 2:
Total cost of offer 1 = $5,841.69
Total cost of offer 2 = $5,497.38
The new homeowner should choose offer 2 because it has a lower total cost. By choosing offer 2, the homeowner will save approximately $344.31 compared to offer 1.
To find the total cost of each financing offer, we need to calculate the present value of the fixed payments along with the down payment.
For Part A: To find the total cost of offer 1, we need to calculate the present value of the fixed payments and the down payment.
Step 1: Calculate the present value of the fixed payments:
Using a financial calculator or spreadsheet software, we can calculate the present value of annuity formula, which is:
PV = PMT * [1 - (1 + r)^(-n)] / r
Where:
PMT = fixed payment per period ($139.05)
r = monthly interest rate (24.90% / 12 = 0.0249)
n = number of periods (3 years * 12 months per year = 36 months)
Plug in these values to find the present value of the fixed payments:
PV = $139.05 * [1 - (1 + 0.0249)^(-36)] / 0.0249 ≈ $3,737.28
Step 2: Add the down payment to the present value of the fixed payments:
Down payment = $250
Total cost of offer 1 = $3,737.28 + $250 = $3,987.28
Therefore, the total cost of offer 1 is approximately $3,987.28.
For Part B: To find the total cost of offer 2, we will follow a similar process.
Step 1: Calculate the present value of the fixed payments:
Using the same formula as before:
PV = $165.76 * [1 - (1 + 0.0229)^(-36)] / 0.0229 ≈ $4,319.64
Step 2: Add the down payment to the present value of the fixed payments:
Down payment = $400
Total cost of offer 2 = $4,319.64 + $400 = $4,719.64
Therefore, the total cost of offer 2 is approximately $4,719.64.
For Part C: Comparing the total costs of the two financing offers:
Offer 1: Total cost ≈ $3,987.28
Offer 2: Total cost ≈ $4,719.64
The new homeowner should choose offer 1 because it has a lower total cost compared to offer 2. Though offer 1 has a higher interest rate, the lower down payment and slightly lower fixed payments result in a lower total cost over the life of the loan.