A consumer owes \( \$ 2,550 \) on a credit card with a \( 21.9 \% \) annual interest rate. Assuming the consumer makes fixed payments and does not charge any more purchases with the card, what is the total monthly payment needed to pay off the card in three years? Show all work and round your answer to the nearest penny.

To determine the total monthly payment needed to pay off the credit card debt in three years, we can use the formula for the monthly payment on an installment loan, which is given by: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where: - \( M \) is the monthly payment - \( P \) is the principal amount (the initial amount owed) - \( r \) is the monthly interest rate (annual interest rate divided by 12) - \( n \) is the total number of payments (number of months) ### Step 1: Identify the known values - \( P = 2550 \) - Annual interest rate = \( 21.9\% \) - Monthly interest rate \( r = \frac{21.9\%}{12} = \frac{0.219}{12} \) - Total number of payments \( n = 3 \text{ years} \times 12 \text{ months/year} = 36 \text{ months} \) ### Step 2: Calculate the monthly interest rate \[ r = \frac{0.219}{12} = 0.01825 \] ### Step 3: Substitute the values into the formula Now we can substitute \( P \), \( r \), and \( n \) into the formula to find \( M \): \[ M = 2550 \frac{0.01825(1 + 0.01825)^{36}}{(1 + 0.01825)^{36} - 1} \] ### Step 4: Calculate \( (1 + r)^n \) First, we need to calculate \( (1 + r)^{36} \): \[ (1 + 0.01825)^{36} \] ### Step 5: Calculate the monthly payment \( M \) Now we can calculate \( M \) using the values we have. Let's perform the calculations step by step. I'll calculate \( (1 + r)^{36} \) and then use it to find \( M \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.01825\right)^{36}\) - step1: Add the numbers: \(1.01825^{36}\) - step2: Convert the expressions: \(\left(\frac{4073}{4000}\right)^{36}\) - step3: Use the properties of exponents: \(\frac{4073^{36}}{4000^{36}}\) Calculate or simplify the expression \( 2550 * (0.01825 * 1.917605) / (1.917605 - 1) \). Calculate the value by following steps: - step0: Calculate: \(\frac{2550\left(0.01825\times 1.917605\right)}{\left(1.917605-1\right)}\) - step1: Remove the parentheses: \(\frac{2550\times 0.01825\times 1.917605}{1.917605-1}\) - step2: Multiply the terms: \(\frac{\frac{1427848683}{16000000}}{1.917605-1}\) - step3: Subtract the numbers: \(\frac{\frac{1427848683}{16000000}}{0.917605}\) - step4: Convert the expressions: \(\frac{\frac{1427848683}{16000000}}{\frac{183521}{200000}}\) - step5: Multiply by the reciprocal: \(\frac{1427848683}{16000000}\times \frac{200000}{183521}\) - step6: Reduce the numbers: \(\frac{1427848683}{80}\times \frac{1}{183521}\) - step7: Multiply the fractions: \(\frac{1427848683}{80\times 183521}\) - step8: Multiply: \(\frac{1427848683}{14681680}\) ### Step 6: Calculate the monthly payment \( M \) From the calculations, we found that: \[ (1 + r)^{36} \approx 1.917605 \] Now substituting this value into the monthly payment formula: \[ M = 2550 \frac{0.01825 \times 1.917605}{1.917605 - 1} \] Calculating the numerator: \[ 0.01825 \times 1.917605 \approx 0.035036 \] Calculating the denominator: \[ 1.917605 - 1 \approx 0.917605 \] Now substituting these values back into the formula for \( M \): \[ M = 2550 \frac{0.035036}{0.917605} \] Calculating the fraction: \[ \frac{0.035036}{0.917605} \approx 0.038157 \] Finally, calculating \( M \): \[ M \approx 2550 \times 0.038157 \approx 97.253767 \] ### Step 7: Round to the nearest penny Rounding \( 97.253767 \) to the nearest penny gives: \[ M \approx 97.25 \] ### Conclusion The total monthly payment needed to pay off the credit card in three years is approximately **$97.25**.