In Calculating The Monthly Payment For A Five-year Loan, What Value Should Be Used For $n$, The Number Of Periods Over Which The Loan Is Repaid, As It Appears In The Following Formula?$P = PV \cdot \frac{1}{1-(1+i)^{-x}}$A. 5 B.

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When it comes to calculating the monthly payment for a five-year loan, one of the key factors to consider is the number of periods over which the loan is repaid. This is denoted by the variable $n$ in the formula for calculating monthly payments. In this article, we will explore the correct value to use for $n$ in the given formula.

The Formula for Calculating Monthly Payments

The formula for calculating monthly payments is given by:

P=PV⋅11−(1+i)−xP = PV \cdot \frac{1}{1-(1+i)^{-x}}

where:

  • P$ is the monthly payment

  • PV$ is the present value of the loan (the initial amount borrowed)

  • i$ is the monthly interest rate (the interest rate divided by 12)

  • x$ is the number of periods over which the loan is repaid

Understanding the Number of Periods

The number of periods over which the loan is repaid, denoted by $x$, is a critical component of the formula. However, the question asks for the value of $n$, which is not explicitly defined in the formula. To determine the correct value for $n$, we need to consider the context in which the formula is being used.

The Correct Value for $n$

In the context of a five-year loan, the number of periods over which the loan is repaid is typically 60 months (5 years * 12 months/year). This is because the loan is repaid in equal monthly installments over a period of 5 years.

However, the formula for calculating monthly payments uses the variable $x$, which represents the number of periods over which the loan is repaid. To use the formula, we need to convert the number of years to the number of periods.

Converting Years to Periods

To convert the number of years to the number of periods, we can multiply the number of years by 12 (the number of months in a year). For a five-year loan, this would give us:

x=5 years×12 months/year=60 monthsx = 5 \text{ years} \times 12 \text{ months/year} = 60 \text{ months}

Using the Correct Value for $n$

Now that we have determined the correct value for $x$, we can use it in the formula to calculate the monthly payment.

P=PV⋅11−(1+i)−60P = PV \cdot \frac{1}{1-(1+i)^{-60}}

By using the correct value for $x$, we can ensure that our calculation of the monthly payment is accurate.

Conclusion

In conclusion, when calculating the monthly payment for a five-year loan, the correct value to use for $n$ is 60 months. This is because the loan is repaid in equal monthly installments over a period of 5 years, and we need to convert the number of years to the number of periods to use the formula.

Frequently Asked Questions

  • Q: What is the correct value for $n$ in the formula for calculating monthly payments? A: The correct value for $n$ is 60 months.
  • Q: Why do we need to convert the number of years to the number of periods? A: We need to convert the number of years to the number of periods to use the formula for calculating monthly payments.
  • Q: How do we convert the number of years to the number of periods? A: We multiply the number of years by 12 (the number of months in a year).

Additional Resources

In our previous article, we explored the formula for calculating monthly payments and determined the correct value to use for $n$ in the formula. However, we understand that there may be additional questions and concerns regarding this topic. In this article, we will address some of the most frequently asked questions related to calculating monthly payments for a five-year loan.

Q: What is the formula for calculating monthly payments?

A: The formula for calculating monthly payments is given by:

P=PV⋅11−(1+i)−xP = PV \cdot \frac{1}{1-(1+i)^{-x}}

where:

  • P$ is the monthly payment

  • PV$ is the present value of the loan (the initial amount borrowed)

  • i$ is the monthly interest rate (the interest rate divided by 12)

  • x$ is the number of periods over which the loan is repaid

Q: What is the correct value for $n$ in the formula?

A: The correct value for $n$ is 60 months, which is equivalent to 5 years * 12 months/year.

Q: Why do we need to convert the number of years to the number of periods?

A: We need to convert the number of years to the number of periods to use the formula for calculating monthly payments. This is because the formula uses the variable $x$, which represents the number of periods over which the loan is repaid.

Q: How do we convert the number of years to the number of periods?

A: We multiply the number of years by 12 (the number of months in a year). For example, to convert 5 years to the number of periods, we would multiply 5 by 12:

x=5 years×12 months/year=60 monthsx = 5 \text{ years} \times 12 \text{ months/year} = 60 \text{ months}

Q: What is the monthly interest rate ($i$)?

A: The monthly interest rate ($i$) is the interest rate divided by 12. For example, if the annual interest rate is 6%, the monthly interest rate would be:

i=6%12=0.005i = \frac{6\%}{12} = 0.005

Q: How do we calculate the monthly payment ($P$)?

A: To calculate the monthly payment ($P$), we can use the formula:

P=PV⋅11−(1+i)−xP = PV \cdot \frac{1}{1-(1+i)^{-x}}

where:

  • PV$ is the present value of the loan (the initial amount borrowed)

  • i$ is the monthly interest rate (the interest rate divided by 12)

  • x$ is the number of periods over which the loan is repaid

Q: What is the present value of the loan ($PV$)?

A: The present value of the loan ($PV$) is the initial amount borrowed. For example, if you borrow $10,000 to purchase a car, the present value of the loan would be $10,000.

Q: How do we use the formula to calculate the monthly payment?

A: To use the formula to calculate the monthly payment, we need to plug in the values for $PV$, $i$, and $x$. For example, if we want to calculate the monthly payment for a $10,000 loan with an annual interest rate of 6% and a repayment period of 5 years, we would use the following values:

  • PV=10,000PV = 10,000

  • i = 0.005$ (6% divided by 12)

  • x = 60$ (5 years \* 12 months/year)

Plugging these values into the formula, we get:

P=10,000⋅11−(1+0.005)−60P = 10,000 \cdot \frac{1}{1-(1+0.005)^{-60}}

Solving for $P$, we get:

P=193.41P = 193.41

Therefore, the monthly payment for a $10,000 loan with an annual interest rate of 6% and a repayment period of 5 years would be $193.41.

Conclusion

In conclusion, calculating monthly payments for a five-year loan requires a clear understanding of the formula and the values to use. By following the steps outlined in this article, you can calculate the monthly payment for your loan and make informed decisions about your finances.

Additional Resources