Questions: Suppose that on January 1 you have a balance of 5500 on a credit card whose APR is 15%, which you want to pay off in 1 year. Assume that you make no additional charges to the card after January 1. a. Calculate your monthly payments. b. When the card is paid off, how much will you have paid since January 1? c. What percentage of your total payment from part (b) is interest? a. The monthly payment is 461.65. (Do not round until the final answer. Then round to the nearest cent as needed.) b. The total paid since January 1 is . (Use the answer from part (a) to find this answer. Round to the nearest cent as needed.) c. The percentage of the total paid that is interest is %. (Use the answer from part (b) to find this answer. Round to one decimal place as needed.)

Suppose that on January 1 you have a balance of 5500 on a credit card whose APR is 15%, which you want to pay off in 1 year. Assume that you make no additional charges to the card after January 1. a. Calculate your monthly payments. b. When the card is paid off, how much will you have paid since January 1? c. What percentage of your total payment from part (b) is interest? a. The monthly payment is 461.65. (Do not round until the final answer. Then round to the nearest cent as needed.) b. The total paid since January 1 is . (Use the answer from part (a) to find this answer. Round to the nearest cent as needed.) c. The percentage of the total paid that is interest is %. (Use the answer from part (b) to find this answer. Round to one decimal place as needed.)
Transcript text: Suppose that on January 1 you have a balance of $\$ 5500$ on a credit card whose APR is $15 \%$, which you want to pay off in 1 year Assume that you make no additional charges to the card after January 1 . a Calculate your monthly payments. b. When the card is paid off, how much will you have paid since January 1? c. What percentage of your total payment from part (b) is interest? a. The monthly payment is $\$ 461.65$. (Do not round until the final answer. Then round to the nearest cent as needed.) b. The total paid since January 1 is $\$ \square$. (Use the answer from part (a) to find this answer. Round to the nearest cent as needed.) c. The percentage of the total paid that is interest is $\square \%$. (Use the answer from part (b) to find this answer. Round to one decimal place as needed.)
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Solution

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Solution Steps

To solve this problem, we need to calculate the monthly payment required to pay off a credit card balance over a year with a given annual percentage rate (APR). We will use the formula for the monthly payment on an amortizing loan. Once we have the monthly payment, we can calculate the total amount paid over the year and determine the interest paid as a percentage of the total payment.

  1. Monthly Payment Calculation: Use the formula for monthly payments on an amortizing loan: \[ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} \] where \( M \) is the monthly payment, \( P \) is the principal amount ($5500), \( r \) is the monthly interest rate (APR/12), and \( n \) is the number of payments (12 for one year).

  2. Total Payment Calculation: Multiply the monthly payment by the number of months to get the total amount paid over the year.

  3. Interest Percentage Calculation: Subtract the principal from the total payment to find the total interest paid. Then, calculate the percentage of the total payment that is interest.

Step 1: Monthly Payment Calculation

To calculate the monthly payment \( M \) for a credit card balance of \( P = 5500 \) with an annual percentage rate (APR) of \( 15\% \), we first convert the APR to a monthly interest rate: \[ r = \frac{0.15}{12} = 0.0125 \] Using the formula for monthly payments: \[ M = \frac{P \times r \times (1 + r)^n}{(1 + r)^n - 1} \] where \( n = 12 \) (the number of months), we find: \[ M = \frac{5500 \times 0.0125 \times (1 + 0.0125)^{12}}{(1 + 0.0125)^{12} - 1} \approx 496.4207 \]

Step 2: Total Payment Calculation

The total amount paid over the year is calculated by multiplying the monthly payment by the number of months: \[ \text{Total Payment} = M \times n = 496.4207 \times 12 \approx 5957.0486 \]

Step 3: Interest Calculation

The total interest paid is the difference between the total payment and the principal: \[ \text{Total Interest} = \text{Total Payment} - P = 5957.0486 - 5500 \approx 457.0486 \]

Step 4: Interest Percentage Calculation

The percentage of the total payment that is interest is calculated as: \[ \text{Interest Percentage} = \left( \frac{\text{Total Interest}}{\text{Total Payment}} \right) \times 100 \approx \left( \frac{457.0486}{5957.0486} \right) \times 100 \approx 7.6724\% \]

Final Answer

  • a. The monthly payment is approximately \( M \approx 496.42 \).
  • b. The total paid since January 1 is approximately \( 5957.05 \).
  • c. The percentage of the total paid that is interest is approximately \( 7.7\% \).

Thus, the final answers are: \[ \boxed{M \approx 496.42} \] \[ \boxed{\text{Total Payment} \approx 5957.05} \] \[ \boxed{\text{Interest Percentage} \approx 7.7\%} \]

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