Paige decided to buy a new car worth $32,000, she traded in her old mini-van and received $10,000, which she used as a down payment. She finances the…
Paige decided to buy a new car worth $32,000, she traded in her old mini-van and received $10,000, which she used as a down payment. She finances the balance at 8% APR over 36 months. Before making her 24th payment, she decides to pay off the loan.
a.) Use the APR table for Monthly Payment plans. What is the total interest Paige would pay if all 36 payments were made?
b.)What were Paige’s monthly payments?
c.) How much interest will Paige save by paying off the loan early?
d.) What is the total amount due to pay off the loan?
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Let’s break down the problem step by step:
a.) To find the total interest Paige would pay if all 36 payments were made, we’ll use the APR table for Monthly Payment plans. Keep in mind that the interest will decrease with each payment as the loan amount reduces.
b.) To calculate Paige’s monthly payments, we’ll use the loan amount after the down payment and the APR. We can use the formula for calculating monthly loan payments:
Monthly Payment (PMT) = P * (r * (1 + r)^n) / ((1 + r)^n – 1)
Where:
P = Principal amount (loan amount after the down payment)
r = Monthly interest rate (APR / 12)
n = Total number of payments (36 in this case)
c.) To find out how much interest Paige will save by paying off the loan early, we’ll calculate the remaining interest after making 23 payments instead of 36.
d.) The total amount due to pay off the loan will be the remaining loan amount after 23 payments.
Now, let’s calculate the answers:
a.) Total interest if all 36 payments were made:
To find the total interest paid over 36 months, we can use the formula for the sum of an annuity:
Total Interest = (Monthly Payment * Total number of payments) – Principal Amount
b.) Paige’s monthly payments:
We need to find the monthly payment using the formula mentioned earlier.
c.) Interest saved by paying off the loan early:
To find this, we need to calculate the remaining interest after making 23 payments instead of 36.
Remaining Interest = (Monthly Payment * Remaining number of payments) – Remaining Principal Amount
d.) Total amount due to pay off the loan:
This will be the remaining loan amount after 23 payments.
Let’s start with b.) Paige’s monthly payments:
Given data:
Car price: $32,000
Down payment: $10,000
APR: 8%
Number of payments: 36
1. Calculate the principal amount (loan amount after down payment):
Principal Amount = Car price – Down payment
Principal Amount = $32,000 – $10,000 = $22,000
2. Calculate the monthly interest rate (r):
Monthly Interest Rate (r) = APR / 12
Monthly Interest Rate (r) = 8% / 12 = 0.08 / 12 = 0.0066667 (approx)
3. Calculate the monthly payment (PMT):
Monthly Payment (PMT) = $22,000 * (0.0066667 * (1 + 0.0066667)^36) / ((1 + 0.0066667)^36 – 1)
Monthly Payment (PMT) ≈ $666.06 (approx)
Now, let’s proceed to calculate the remaining values:
a.) Total interest if all 36 payments were made:
Total Interest = ($666.06 * 36) – $22,000
Total Interest ≈ $7,979.16 (approx)
c.) Interest saved by paying off the loan early:
Remaining Interest = ($666.06 * 12) – Remaining Principal Amount
Remaining Principal Amount = Loan amount after 23 payments
To calculate the remaining principal amount after 23 payments:
Remaining Principal Amount = $22,000 * ((1 + 0.0066667)^23 – (1 + 0.0066667)^36) / ((1 + 0.0066667)^36 – 1)
Remaining Principal Amount ≈ $10,430.51 (approx)
Now, calculate the remaining interest:
Remaining Interest ≈ ($666.06 * 12) – $10,430.51
Remaining Interest ≈ $2,991.67 (approx)
d.) Total amount due to pay off the loan:
Total amount due = Remaining Principal Amount + Remaining Interest
Total amount due ≈ $10,430.51 + $2,991.67 ≈ $13,422.18 (approx)
To summarize the answers:
a.) Total interest if all 36 payments were made: Approximately $7,979.16
b.) Paige’s monthly payments: Approximately $666.06
c.) Interest saved by paying off the loan early: Approximately $2,991.67
d.) Total amount due to pay off the loan: Approximately $13,422.18
