Consider a credit card with a balance of $8500$⁢8500 and an APR of 16.99%16.99%. In order to pay off the balance in 22 years, what monthly payment would you need to make? Round your answer to the nearest cent, if necessary.


To calculate the monthly payment needed to pay off a $8500 balance on a credit card with a 16.99% APR in 22 years, we use the same amortizing loan formula:

\[ M = \frac{P \cdot r \cdot (1 + r)^n}{(1 + r)^n - 1} \]

where:
- $ M $ is the monthly payment,
- $ P $ is the principal amount ($8500),
- $ r $ is the monthly interest rate (APR divided by 12),
- $ n $ is the number of monthly payments (22 years × 12 months/year = 264).

First, calculate the monthly interest rate:

\[ r = \frac{16.99\%}{12} = \frac{0.1699}{12} = 0.0141583 \]

Next, we plug these values into the formula:

\[ M = \frac{8500 \cdot 0.0141583 \cdot (1 + 0.0141583)^{264}}{(1 + 0.0141583)^{264} - 1} \]

Now we calculate each component step-by-step.

1. Calculate $(1 + r)^{264}$:

\[ (1 + 0.0141583)^{264} \approx 57.2966 \]

2. Multiply $P \cdot r \cdot (1 + r)^{264}$:

\[ 8500 \cdot 0.0141583 \cdot 57.2966 \approx 6903.3095 \]

3. Calculate the denominator $(1 + r)^{264} - 1$:

\[ 57.2966 - 1 = 56.2966 \]

4. Divide the result from step 2 by the result from step 3:

\[ M = \frac{6903.3095}{56.2966} \approx 122.65 \]

So, the monthly payment needed to pay off the balance in 22 years is approximately $122.65.

Would you like details on any part of this process or have any questions? Here are some questions you might ask next:

1. How does the length of the payment period affect the monthly payment?
2. What if the APR was different, like 10% or 20%?
3. Can you explain how the amortization formula works in detail?
4. How do changes in the principal amount impact the monthly payment?
5. What are the benefits and drawbacks of long-term loan repayment plans?

**Tip:** Always be mindful of the total interest paid over the life of a loan; longer repayment periods can lead to significantly higher total interest payments.

Discover how to calculate the monthly payment needed to pay off an $8500 credit card balance with a 16.99% APR over 22 years. This detailed solution uses the amortizing loan formula, breaking down each step to find the monthly payment. Suitable for advanced high school students and anyone interested in financial mathematics.

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