Which Of These Expressions Will Give The Unpaid Balance After 6 Years On A $90,000$ Loan With An APR Of $7.2\%$, Compounded Monthly, If The Monthly Payment Is $708.61$?A.

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Understanding the Problem

Calculating the unpaid balance on a loan after a certain period is crucial for individuals and businesses to manage their finances effectively. In this scenario, we are given a loan of $90,000 with an Annual Percentage Rate (APR) of 7.2%, compounded monthly, and a monthly payment of $708.61. We need to determine which expression will give the unpaid balance after 6 years.

Loan Details

  • Loan amount: $90,000
  • APR: 7.2%
  • Compounding frequency: Monthly
  • Monthly payment: $708.61
  • Time period: 6 years

Calculating Unpaid Balance

To calculate the unpaid balance, we can use the formula for the future value of an annuity:

FV = PMT x (((1 + r)^n - 1) / r)

Where:

  • FV = Future Value (unpaid balance)
  • PMT = Monthly payment
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

However, this formula calculates the future value of the loan, which is the total amount paid, including the principal and interest. To find the unpaid balance, we need to subtract the total amount paid from the loan amount.

Expression 1: Using the Formula

Let's use the formula to calculate the unpaid balance:

FV = 708.61 x (((1 + 0.0072/12)^(6*12) - 1) / (0.0072/12))

FV ≈ $14,919.19

Expression 2: Using a Financial Calculator

We can also use a financial calculator to calculate the unpaid balance. The formula for the future value of an annuity is:

FV = PV x (1 + r)^n

Where:

  • FV = Future Value (unpaid balance)
  • PV = Present Value (loan amount)
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

Using a financial calculator, we get:

FV ≈ $14,919.19

Expression 3: Using a Spreadsheet

We can also use a spreadsheet to calculate the unpaid balance. The formula for the future value of an annuity is:

FV = PMT x (((1 + r)^n - 1) / r)

Where:

  • FV = Future Value (unpaid balance)
  • PMT = Monthly payment
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

Using a spreadsheet, we get:

FV ≈ $14,919.19

Conclusion

In conclusion, the unpaid balance after 6 years on a $90,000 loan with an APR of 7.2%, compounded monthly, and a monthly payment of $708.61 is approximately $14,919.19. We can use the formula for the future value of an annuity, a financial calculator, or a spreadsheet to calculate the unpaid balance.

References

Q: What is the formula for calculating the unpaid balance on a loan?

A: The formula for calculating the unpaid balance on a loan is:

FV = PMT x (((1 + r)^n - 1) / r)

Where:

  • FV = Future Value (unpaid balance)
  • PMT = Monthly payment
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

Q: How do I calculate the monthly interest rate (r) from the APR?

A: To calculate the monthly interest rate (r) from the APR, you need to divide the APR by 12.

r = APR / 12

For example, if the APR is 7.2%, the monthly interest rate would be:

r = 7.2% / 12 = 0.006

Q: How do I calculate the number of payments (n) from the time period?

A: To calculate the number of payments (n) from the time period, you need to multiply the time period by 12.

n = time period * 12

For example, if the time period is 6 years, the number of payments would be:

n = 6 years * 12 = 72 months

Q: Can I use a financial calculator to calculate the unpaid balance?

A: Yes, you can use a financial calculator to calculate the unpaid balance. The formula for the future value of an annuity is:

FV = PV x (1 + r)^n

Where:

  • FV = Future Value (unpaid balance)
  • PV = Present Value (loan amount)
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

Q: Can I use a spreadsheet to calculate the unpaid balance?

A: Yes, you can use a spreadsheet to calculate the unpaid balance. The formula for the future value of an annuity is:

FV = PMT x (((1 + r)^n - 1) / r)

Where:

  • FV = Future Value (unpaid balance)
  • PMT = Monthly payment
  • r = Monthly interest rate (APR / 12)
  • n = Number of payments (time period * 12)

Q: What is the difference between the unpaid balance and the loan amount?

A: The unpaid balance is the amount still owed on the loan after making the monthly payments, while the loan amount is the original amount borrowed.

Q: Can I use the unpaid balance formula to calculate the loan amount?

A: No, you cannot use the unpaid balance formula to calculate the loan amount. The loan amount is the original amount borrowed, and the unpaid balance formula is used to calculate the amount still owed on the loan after making the monthly payments.

Q: Can I use the unpaid balance formula to calculate the monthly payment?

A: No, you cannot use the unpaid balance formula to calculate the monthly payment. The monthly payment is the amount paid each month to repay the loan, and the unpaid balance formula is used to calculate the amount still owed on the loan after making the monthly payments.

Q: What is the significance of the APR in calculating the unpaid balance?

A: The APR is the annual interest rate charged on the loan, and it is used to calculate the monthly interest rate (r). The APR is an important factor in calculating the unpaid balance, as it affects the amount of interest charged on the loan.

Q: Can I use the unpaid balance formula to calculate the interest rate?

A: No, you cannot use the unpaid balance formula to calculate the interest rate. The interest rate is the APR, and it is used to calculate the monthly interest rate (r). The unpaid balance formula is used to calculate the amount still owed on the loan after making the monthly payments, not the interest rate.