Bernadette Took Out A Loan For $\$ 1250$ At A $10.8 \%$ APR, Compounded Monthly, To Buy A Refrigerator. If She Will Make Monthly Payments Of $\$ 85.50$[/tex\] To Pay Off The Loan, Which Of These Expressions Could Be

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Bernadette took out a loan for $1250 at a 10.8% APR, compounded monthly, to buy a refrigerator. If she will make monthly payments of $85.50 to pay off the loan, which of these expressions could be used to calculate the number of payments she will make?

Background Information

Before we dive into the problem, let's understand the key concepts involved. Bernadette has taken out a loan with the following characteristics:

  • Principal (P): The initial amount borrowed, which is $1250 in this case.
  • Annual Percentage Rate (APR): The interest rate charged on the loan, which is 10.8% in this case.
  • Compounding frequency: The interest is compounded monthly, meaning that the interest is applied once a month.
  • Monthly payment (M): The amount Bernadette will pay each month to pay off the loan, which is $85.50 in this case.

The Problem

We need to find an expression that can be used to calculate the number of payments Bernadette will make to pay off the loan. This is also known as the number of periods or number of payments.

The Formula

The formula to calculate the number of payments is based on the concept of present value and future value. The present value is the current value of the loan, which is the principal amount ($1250). The future value is the total amount paid, including the principal and the interest.

The formula to calculate the number of payments is:

n = ln(M / (M - rP)) / ln(1 + r/m)

Where:

  • n: The number of payments
  • M: The monthly payment ($85.50)
  • r: The monthly interest rate (APR / 12)
  • P: The principal amount ($1250)
  • m: The number of times interest is compounded per year (12)

Simplifying the Formula

To simplify the formula, we can substitute the values of M, r, P, and m into the equation.

n = ln(85.50 / (85.50 - 0.009)) / ln(1 + 0.009/12)

n = ln(85.50 / 85.41) / ln(1.00075)

n = ln(1.00009) / ln(1.00075)

n = 0.00009 / 0.00075

n = 0.12

Conclusion

The expression that could be used to calculate the number of payments Bernadette will make to pay off the loan is:

n = ln(M / (M - rP)) / ln(1 + r/m)

Where:

  • n: The number of payments
  • M: The monthly payment ($85.50)
  • r: The monthly interest rate (APR / 12)
  • P: The principal amount ($1250)
  • m: The number of times interest is compounded per year (12)

By using this expression, we can calculate the number of payments Bernadette will make to pay off the loan.

Example Use Case

Let's say we want to calculate the number of payments Bernadette will make to pay off the loan if she increases her monthly payment to $100. We can substitute the new value of M into the expression:

n = ln(100 / (100 - 0.009)) / ln(1 + 0.009/12)

n = ln(100 / 99.99) / ln(1.00075)

n = ln(1.00001) / ln(1.00075)

n = 0.00001 / 0.00075

n = 0.01333

Therefore, if Bernadette increases her monthly payment to $100, she will make approximately 13.33 payments to pay off the loan.

Conclusion

In conclusion, the expression that could be used to calculate the number of payments Bernadette will make to pay off the loan is:

n = ln(M / (M - rP)) / ln(1 + r/m)

Where:

  • n: The number of payments
  • M: The monthly payment ($85.50)
  • r: The monthly interest rate (APR / 12)
  • P: The principal amount ($1250)
  • m: The number of times interest is compounded per year (12)

Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about Bernadette's loan.

Q: What is the principal amount of the loan?

A: The principal amount of the loan is $1250.

Q: What is the annual percentage rate (APR) of the loan?

A: The annual percentage rate (APR) of the loan is 10.8%.

Q: How much will Bernadette pay each month?

A: Bernadette will pay $85.50 each month.

Q: How many payments will Bernadette make?

A: To calculate the number of payments Bernadette will make, we can use the expression:

n = ln(M / (M - rP)) / ln(1 + r/m)

Where:

  • n: The number of payments
  • M: The monthly payment ($85.50)
  • r: The monthly interest rate (APR / 12)
  • P: The principal amount ($1250)
  • m: The number of times interest is compounded per year (12)

Q: What is the monthly interest rate?

A: The monthly interest rate is 0.009 (APR / 12).

Q: How many times is the interest compounded per year?

A: The interest is compounded 12 times per year.

Q: What is the formula to calculate the number of payments?

A: The formula to calculate the number of payments is:

n = ln(M / (M - rP)) / ln(1 + r/m)

Q: Can I use this formula to calculate the number of payments for any loan?

A: Yes, you can use this formula to calculate the number of payments for any loan, as long as you know the principal amount, the annual percentage rate, the monthly payment, and the number of times interest is compounded per year.

Q: What if I want to increase the monthly payment?

A: If you want to increase the monthly payment, you can substitute the new value of M into the expression:

n = ln(M / (M - rP)) / ln(1 + r/m)

Q: Can I use this formula to calculate the number of payments for a loan with a different compounding frequency?

A: Yes, you can use this formula to calculate the number of payments for a loan with a different compounding frequency, as long as you know the principal amount, the annual percentage rate, the monthly payment, and the number of times interest is compounded per year.

Conclusion

In conclusion, we have answered some of the most frequently asked questions about Bernadette's loan. We have also provided the formula to calculate the number of payments, which can be used for any loan, as long as you know the principal amount, the annual percentage rate, the monthly payment, and the number of times interest is compounded per year.

Example Use Case

Let's say we want to calculate the number of payments for a loan with a principal amount of $2000, an annual percentage rate of 12%, a monthly payment of $150, and a compounding frequency of 12 times per year. We can substitute the values into the expression:

n = ln(150 / (150 - 0.012*2000)) / ln(1 + 0.012/12)

n = ln(150 / 149.84) / ln(1.001)

n = ln(1.00016) / ln(1.001)

n = 0.00016 / 0.001

n = 0.16

Therefore, for a loan with a principal amount of $2000, an annual percentage rate of 12%, a monthly payment of $150, and a compounding frequency of 12 times per year, the number of payments would be approximately 0.16 years, or 6 months.

Conclusion

In conclusion, we have provided the formula to calculate the number of payments for any loan, as long as you know the principal amount, the annual percentage rate, the monthly payment, and the number of times interest is compounded per year. We have also provided an example use case to demonstrate how to use the formula.