Loan Payment Formula (Installment Loans) Px (APR) 1-(1+APR) PMT= APR (ny) Suppose you have a credit card balance of $1,500 with an annual interest rate of 11%. You decide to pay off your balance over 1 year. How much will you need to pay each month? Assume you make no further credit card purchases. Round to the nearest cent
Loan Payment Formula (Installment Loans) Px (APR) 1-(1+APR) PMT= APR (ny) Suppose you have a credit card balance of $1,500 with an annual interest rate of 11%. You decide to pay off your balance over 1 year. How much will you need to pay each month? Assume you make no further credit card purchases. Round to the nearest cent
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Loan Payment Formula (Installment Loans)**
The formula provided to calculate the monthly payment for installment loans is:
\[ PMT = \frac{P \times \left(\frac{APR}{n}\right)}{1 - \left(1 + \frac{APR}{n}\right)^{-n \times t}} \]
Where:
- \( PMT \) is the monthly payment.
- \( P \) is the loan principal (the initial amount of the loan).
- \( APR \) is the annual percentage rate (annual interest rate).
- \( n \) is the number of payments per year.
- \( t \) is the loan term in years.
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### Example Calculation:
Suppose you have a credit card balance of $1,500 with an annual interest rate of 11%. You decide to pay off your balance over 1 year. How much will you need to pay each month? Assume you make no further credit card purchases.
1. **Loan Principal (P):** $1,500
2. **Annual Percentage Rate (APR):** 11% or 0.11
3. **Number of Payments per Year (n):** 12 (since payments are made monthly)
4. **Loan Term (t):** 1 year
First, plug these values into the formula:
\[ PMT = \frac{1500 \times \left(\frac{0.11}{12}\right)}{1 - \left(1 + \frac{0.11}{12}\right)^{-12}} \]
Calculate the monthly interest rate:
\[ \frac{0.11}{12} = 0.0091667 \]
Then, the calculation becomes:
\[ PMT = \frac{1500 \times 0.0091667}{1 - \left(1 + 0.0091667\right)^{-12}} \]
Simplify the denominator:
\[ 1 + 0.0091667 = 1.0091667 \]
Raise to the power of -12:
\[ 1.0091667^{-12} \approx 0.890997 \]
Subtract from 1:
\[ 1 - 0.890997 = 0.109003 \]
Finally, calculate the payment:
\[ PMT = \frac{1500 \times 0.0091667}{0.109003} \approx \frac{13.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9e14e833-b473-4144-adde-e217b311654e%2Fdad71783-b368-4103-8f1d-556f1b610a50%2Fwqtm8ii_processed.png&w=3840&q=75)
Transcribed Image Text:**Loan Payment Formula (Installment Loans)**
The formula provided to calculate the monthly payment for installment loans is:
\[ PMT = \frac{P \times \left(\frac{APR}{n}\right)}{1 - \left(1 + \frac{APR}{n}\right)^{-n \times t}} \]
Where:
- \( PMT \) is the monthly payment.
- \( P \) is the loan principal (the initial amount of the loan).
- \( APR \) is the annual percentage rate (annual interest rate).
- \( n \) is the number of payments per year.
- \( t \) is the loan term in years.
---
### Example Calculation:
Suppose you have a credit card balance of $1,500 with an annual interest rate of 11%. You decide to pay off your balance over 1 year. How much will you need to pay each month? Assume you make no further credit card purchases.
1. **Loan Principal (P):** $1,500
2. **Annual Percentage Rate (APR):** 11% or 0.11
3. **Number of Payments per Year (n):** 12 (since payments are made monthly)
4. **Loan Term (t):** 1 year
First, plug these values into the formula:
\[ PMT = \frac{1500 \times \left(\frac{0.11}{12}\right)}{1 - \left(1 + \frac{0.11}{12}\right)^{-12}} \]
Calculate the monthly interest rate:
\[ \frac{0.11}{12} = 0.0091667 \]
Then, the calculation becomes:
\[ PMT = \frac{1500 \times 0.0091667}{1 - \left(1 + 0.0091667\right)^{-12}} \]
Simplify the denominator:
\[ 1 + 0.0091667 = 1.0091667 \]
Raise to the power of -12:
\[ 1.0091667^{-12} \approx 0.890997 \]
Subtract from 1:
\[ 1 - 0.890997 = 0.109003 \]
Finally, calculate the payment:
\[ PMT = \frac{1500 \times 0.0091667}{0.109003} \approx \frac{13.
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