Questions: Question 6, 5.3.15 HW Score: Part 1 of 3 Points: Suppose a credit card was used to make a 7500 purchase at 17.4% interest with a monthly payment of 186. Complete the following. (a) Calculate the time it will take to pay off the debt making only the given payment each month (b) Calculate the total interest paid. (c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month. (a) If only the given payment is made each month, it will take payments to pay off the debt (Round up to the nearest integer as needed.)

Question 6, 5.3.15
HW Score:
Part 1 of 3
Points:

Suppose a credit card was used to make a 7500 purchase at 17.4% interest with a monthly payment of 186. Complete the following.
(a) Calculate the time it will take to pay off the debt making only the given payment each month
(b) Calculate the total interest paid.
(c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month.
(a) If only the given payment is made each month, it will take payments to pay off the debt
(Round up to the nearest integer as needed.)

Transcript text: Question 6, 5.3.15 HW Score: Part 1 of 3 Points: Suppose a credit card was used to make a $\$ 7500$ purchase at $17.4 \%$ interest with a monthly payment of $\$ 186$. Complete the following. (a) Calculate the time it will take to pay off the debt making only the given payment each month (b) Calculate the total interest paid. (c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month. (a) If only the given payment is made each month, it will take $\square$ payments to pay off the debt (Round up to the nearest integer as needed.)

Solution

Solution Steps

To solve this problem, we need to break it down into three parts:

(a) Calculate the time it will take to pay off the debt making only the given payment each month. (b) Calculate the total interest paid. (c) Find the amount of money saved over the lifetime of the debt if twice the amount of the given monthly payment is paid each month.

Solution Approach

For part (a), we need to use the formula for the number of payments to pay off a loan: \[ N = \frac{\log\left(\frac{P}{P - r \cdot B}\right)}{\log(1 + r)} \] where $ P $ is the monthly payment, $ r $ is the monthly interest rate, and $ B $ is the loan balance.
For part (b), we calculate the total interest paid by summing up all the payments and subtracting the principal.
For part (c), we repeat the calculation for the number of payments and total interest with the doubled monthly payment.

Step 1: Calculate the Number of Payments

To determine the number of payments $ N $ required to pay off the debt, we use the formula: \[ N = \frac{\log\left(\frac{P}{P - r \cdot B}\right)}{\log(1 + r)} \] where:

$ P = 186 $
$ r = \frac{0.174}{12} \approx 0.0145 $
$ B = 7500 $

Substituting the values, we find: \[ N \approx 62 \]

Step 2: Calculate the Total Interest Paid

The total amount paid over the life of the loan is given by: \[ \text{Total Paid} = N \cdot P = 62 \cdot 186 = 11532 \] The total interest paid is then calculated as: \[ \text{Total Interest Paid} = \text{Total Paid} - B = 11532 - 7500 = 4032 \]

Step 3: Calculate Savings with Double the Monthly Payment

If the monthly payment is doubled to $ 372 $, we recalculate the number of payments: \[ N_{\text{double}} = \frac{\log\left(\frac{372}{372 - r \cdot B}\right)}{\log(1 + r)} \approx 25 \] The total amount paid with the doubled payment is: \[ \text{Total Paid}_{\text{double}} = N_{\text{double}} \cdot 372 = 25 \cdot 372 = 9300 \] The total interest paid with the doubled payment is: \[ \text{Total Interest Paid}_{\text{double}} = \text{Total Paid}_{\text{double}} - B = 9300 - 7500 = 1800 \] The savings from making double payments is: \[ \text{Savings} = \text{Total Interest Paid} - \text{Total Interest Paid}_{\text{double}} = 4032 - 1800 = 2232 \]