A discount electronics store offers to let you pay for a ​$1000 stereo in 12 equal ​$95 installments. The store claims that since you repay ​$1140 in 1​ year, the ​$140 finance charge represents a 14​% annual rate. This would be true if you repaid the loan in a single payment at the end of the year. But since you start repayment after 1​ month, this is an amortized​ loan, and 14​% is not the correct rate. What is the annual nominal compounding rate for this​ loan? Use graphical approximation techniques or an equation solver to approximate the interest rate. Express the answer as a percentage.

We are tasked with finding the annual nominal compounding rate for an amortized loan. The stereo is priced at $1000, and you make 12 monthly payments of $95 each, resulting in a total repayment of $1140. However, the store claims a 14% interest rate, but since payments start after the first month, this needs to be recalculated as the loan is amortized.

To solve for the actual interest rate, we will use the formula for the monthly payment on an amortized loan:

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

Where:
- $ M $ is the monthly payment ($95 in this case),
- $ P $ is the principal (loan amount of $1000),
- $ r $ is the monthly interest rate (annual interest rate divided by 12, which is what we are solving for),
- $ n $ is the total number of payments (12 months).

We want to find the monthly interest rate $ r $, and then multiply it by 12 to get the annual nominal rate. To do this, we need to solve the equation for $ r $.

Rearranging the formula to solve for $ r $ is complex, so we will use an equation-solving technique to approximate the value of $ r $.

Let me calculate this.The approximate annual nominal compounding rate for this loan is 24.91%.

This rate is significantly higher than the claimed 14%, due to the fact that the loan is amortized and payments are spread throughout the year rather than being repaid all at once.

Would you like more details on the calculation or further clarification? 

Here are 5 related questions:
1. How is the amortized loan payment formula derived?
2. What is the difference between nominal and effective interest rates?
3. How does an amortized loan differ from a simple loan?
4. What is the impact of increasing the number of payments on the interest rate?
5. How does compounding frequency affect the total amount paid over the loan's duration?

**Tip**: When dealing with loans, always compare both the nominal and effective interest rates to fully understand the cost of borrowing.

Learn how to calculate the annual nominal compounding rate for a $1000 amortized loan with monthly payments of $95 over 12 months. Understand key concepts such as amortized loans, interest rates, and the loan repayment formula. This solution helps explain why the actual interest rate is higher than the store's claimed 14%.

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