Kylie Can Afford A $\$1310$-per-month House Loan Payment. If She Is Being Offered A 25-year House Loan With An APR Of $8.4\%$, Compounded Monthly, Which Of These Expressions Represents The Most Money She Can Borrow?A.

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When it comes to determining the maximum amount that can be borrowed for a house loan, several factors come into play. In this scenario, we are given that Kylie can afford a monthly house loan payment of $1310\$1310. Additionally, she is being offered a 25-year house loan with an APR of 8.4%8.4\%, compounded monthly. To find the most money she can borrow, we need to consider the relationship between the monthly payment, the interest rate, and the loan term.

The Formula for Monthly Payments

The formula for calculating the monthly payment on a loan is given by:

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

where:

  • MM is the monthly payment
  • PP is the principal loan amount (the amount borrowed)
  • rr is the monthly interest rate (APR divided by 12)
  • nn is the number of payments (the loan term in months)

Calculating the Monthly Interest Rate

First, we need to calculate the monthly interest rate. The APR is given as 8.4%8.4\%, so we can calculate the monthly interest rate as follows:

r=8.4%12=0.007%r = \frac{8.4\%}{12} = 0.007\%

Calculating the Number of Payments

Next, we need to calculate the number of payments. The loan term is given as 25 years, so we can calculate the number of payments as follows:

n=25 years×12 months/year=300 monthsn = 25 \text{ years} \times 12 \text{ months/year} = 300 \text{ months}

Representing the Most Money She Can Borrow

Now that we have the monthly interest rate and the number of payments, we can represent the most money Kylie can borrow using the formula for monthly payments. We can rearrange the formula to solve for PP:

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

Substituting the values we have calculated, we get:

P=1310[(1+0.007)300−10.007(1+0.007)300]P = 1310 \left[ \frac{(1+0.007)^{300} - 1}{0.007(1+0.007)^{300}} \right]

This expression represents the most money Kylie can borrow.

Conclusion

In conclusion, to find the most money Kylie can borrow, we need to consider the relationship between the monthly payment, the interest rate, and the loan term. By using the formula for monthly payments and rearranging it to solve for the principal loan amount, we can represent the most money she can borrow. The expression we derived represents the maximum amount Kylie can borrow, given her monthly payment and the loan terms.

Calculating the Maximum Borrowable Amount

To calculate the maximum borrowable amount, we can plug in the values we have calculated into the expression:

P=1310[(1+0.007)300−10.007(1+0.007)300]P = 1310 \left[ \frac{(1+0.007)^{300} - 1}{0.007(1+0.007)^{300}} \right]

Using a calculator or a computer program to evaluate this expression, we get:

P≈163,919.19P \approx 163,919.19

Therefore, the most money Kylie can borrow is approximately $163,919.19\$163,919.19.

Implications

This result has several implications. First, it means that Kylie can borrow up to $163,919.19\$163,919.19 at an APR of 8.4%8.4\%, compounded monthly, and still afford a monthly payment of $1310\$1310. Second, it highlights the importance of considering the relationship between the monthly payment, the interest rate, and the loan term when determining the maximum amount that can be borrowed. By using the formula for monthly payments and rearranging it to solve for the principal loan amount, we can represent the most money that can be borrowed.

Real-World Applications

This problem has several real-world applications. For example, when considering a mortgage, it is essential to determine the maximum amount that can be borrowed based on the monthly payment, the interest rate, and the loan term. This can help individuals make informed decisions about their financial situation and avoid over-borrowing. Additionally, this problem can be applied to other types of loans, such as car loans or personal loans, where the monthly payment, interest rate, and loan term are critical factors in determining the maximum amount that can be borrowed.

Conclusion

In our previous article, we explored the concept of calculating the maximum borrowable amount based on the monthly payment, interest rate, and loan term. We derived an expression that represents the most money Kylie can borrow, given her monthly payment and the loan terms. In this article, we will answer some frequently asked questions related to this topic.

Q: What is the formula for calculating the maximum borrowable amount?

A: The formula for calculating the maximum borrowable amount is given by:

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

where:

  • PP is the principal loan amount (the amount borrowed)
  • MM is the monthly payment
  • rr is the monthly interest rate (APR divided by 12)
  • nn is the number of payments (the loan term in months)

Q: How do I calculate the monthly interest rate?

A: To calculate the monthly interest rate, you need to divide the APR by 12. For example, if the APR is 8.4%, the monthly interest rate would be:

r=8.4%12=0.007%r = \frac{8.4\%}{12} = 0.007\%

Q: How do I calculate the number of payments?

A: To calculate the number of payments, you need to multiply the loan term in years by 12. For example, if the loan term is 25 years, the number of payments would be:

n=25 years×12 months/year=300 monthsn = 25 \text{ years} \times 12 \text{ months/year} = 300 \text{ months}

Q: What is the significance of the loan term in calculating the maximum borrowable amount?

A: The loan term is a critical factor in calculating the maximum borrowable amount. A longer loan term means that the borrower will have to make more payments, which can increase the total amount paid over the life of the loan. Conversely, a shorter loan term means that the borrower will have to make fewer payments, which can reduce the total amount paid over the life of the loan.

Q: Can I use this formula to calculate the maximum borrowable amount for other types of loans?

A: Yes, you can use this formula to calculate the maximum borrowable amount for other types of loans, such as car loans or personal loans. However, you need to ensure that the formula is adjusted to reflect the specific terms of the loan.

Q: What are some common mistakes to avoid when calculating the maximum borrowable amount?

A: Some common mistakes to avoid when calculating the maximum borrowable amount include:

  • Failing to consider the loan term and its impact on the total amount paid over the life of the loan
  • Failing to consider the interest rate and its impact on the total amount paid over the life of the loan
  • Failing to adjust the formula to reflect the specific terms of the loan
  • Failing to consider the borrower's credit score and its impact on the interest rate

Q: How can I use this formula to make informed decisions about my financial situation?

A: You can use this formula to make informed decisions about your financial situation by:

  • Calculating the maximum borrowable amount based on your monthly payment and the loan terms
  • Comparing the maximum borrowable amount to your income and expenses to determine whether you can afford the loan
  • Considering the impact of the loan on your credit score and your overall financial situation
  • Adjusting your budget and financial plans to reflect the loan terms and the maximum borrowable amount.

Conclusion

In conclusion, calculating the maximum borrowable amount is a critical step in making informed decisions about your financial situation. By using the formula and adjusting it to reflect the specific terms of the loan, you can determine the maximum amount you can borrow and make informed decisions about your financial situation.