Newton uses a credit card with a 18.6% APR, compounded monthly, to pay for a cruise totaling $1,920.96. He can pay $720 per month on the card. What will the total cost of this purchase be?
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You didn't tell us how long he is paying for the trip.
To calculate the total cost of Newton's purchase, we need to consider the monthly payments he makes on the credit card with an 18.6% Annual Percentage Rate (APR), compounded monthly.
First, let's find out the number of months it will take for Newton to pay off his total balance of $1,920.96.
We can start by using the formula for compound interest to find the monthly interest rate.
Monthly Interest Rate = (1 + APR)^(1/12) - 1
In this case, Newton's APR is 18.6%, so we have:
Monthly Interest Rate = (1 + 0.186)^(1/12) - 1
Now, we can calculate the monthly interest rate:
Monthly Interest Rate = (1.186)^(1/12) - 1
≈ 0.0148 or 1.48%
Next, Newton is making monthly payments of $720 on the credit card. We need to determine how many months it will take to pay off the balance.
Let's solve for the number of months using the monthly payment, remaining balance, and the monthly interest rate.
Balance After n Months = Principal * (1 + Monthly Interest Rate)^n - Payment * (((1 + Monthly Interest Rate)^n) - 1) / Monthly Interest Rate
In this case, Newton's principal is $1,920.96 and his payment is $720.
The equation becomes:
$0 = $1,920.96 * (1 + 0.0148)^n - $720 * (((1 + 0.0148)^n) - 1) / 0.0148
We can use numerical methods or trial and error to find the number of months it will take to pay off the debt. Solving this equation, we find n ≈ 3.943 months.
This means it will take approximately 3.943 months to pay off the entire balance.
Now, let's calculate the total cost of the purchase.
Total Cost = (Monthly Payment * Number of Months) + Principal
Plugging in the values, we have:
Total Cost = ($720 * 3.943) + $1,920.96
≈ $2,842.33
Therefore, the total cost of Newton's purchase, including interest, will be approximately $2,842.33.