The debt is amortized by the periodic payment shown. Compute (a) the number of payments required to amortize the debt; (b) the outstanding principal at the time indicated Debt Principal Debt Payment $1480 $14,000 Payment Interval 1 month Interest Rate 9% 30039 Conversion Period monthly Outstanding Principal After: 7th payment
The debt is amortized by the periodic payment shown. Compute (a) the number of payments required to amortize the debt; (b) the outstanding principal at the time indicated Debt Principal Debt Payment $1480 $14,000 Payment Interval 1 month Interest Rate 9% 30039 Conversion Period monthly Outstanding Principal After: 7th payment
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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![### Amortization of Debt: Calculation Example
**Amortization Problem:**
The debt is amortized by the periodic payment shown. Compute (a) the number of payments required to amortize the debt; (b) the outstanding principal at the time indicated.
#### Debt Details:
| Debt Principal | Debt Payment | Payment Interval | Interest Rate | Conversion Period | Outstanding Principal After |
|----------------|--------------|------------------|---------------|-------------------|-----------------------------|
| $14,000 | $1480 | 1 month | 9% | monthly | 7th payment |
#### Given Information:
- **Debt Principal:** $14,000
- **Debt Payment:** $1480 per month
- **Payment Interval:** 1 month
- **Interest Rate:** 9% annually
- **Conversion Period:** monthly
- **Outstanding Principal After:** 7th payment
### Problem Questions:
(a) The number of payments required to amortize the debt is [ ]
*(Round the final answer up to the nearest whole number. Round all intermediate values to six decimal places as needed.)*
---
### Detailed Explanation for Users:
1. **Understanding the Debt Principal:** This is the initial amount borrowed, $14,000 in this case.
2. **Payment Description:** A fixed payment of $1480 is made every month.
3. **Interest Rate Details:** The interest rate is provided annually but needs to be converted to a monthly interest rate for the calculations. For a 9% annual interest rate, the monthly interest rate would be \( \frac{9\%}{12} \).
4. **Outstanding Principal After a Certain Payment:** We also need to find the remaining debt after specific payments made (after the 7th payment).
### Steps to Solve:
1. **Convert Annual Interest Rate to Monthly Interest Rate:**
\[ \text{Monthly Interest Rate} = \frac{9\%}{12} = 0.75\% \text{ per month} \]
2. **Calculate the Number of Payments (n):**
To determine the number of payments required to amortize the debt, the loan amortization formula can be used:
\[ P \times \frac{r(1+r)^n}{(1+r)^n - 1} = \text{Monthly Payment} \]
where:
- \( P \) is the principal amount ($14](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F91caee7e-a450-4823-b604-91c802978c64%2F7bd2faac-2a0e-40bf-8fa4-c8e15263d2f8%2Fp2om84a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Amortization of Debt: Calculation Example
**Amortization Problem:**
The debt is amortized by the periodic payment shown. Compute (a) the number of payments required to amortize the debt; (b) the outstanding principal at the time indicated.
#### Debt Details:
| Debt Principal | Debt Payment | Payment Interval | Interest Rate | Conversion Period | Outstanding Principal After |
|----------------|--------------|------------------|---------------|-------------------|-----------------------------|
| $14,000 | $1480 | 1 month | 9% | monthly | 7th payment |
#### Given Information:
- **Debt Principal:** $14,000
- **Debt Payment:** $1480 per month
- **Payment Interval:** 1 month
- **Interest Rate:** 9% annually
- **Conversion Period:** monthly
- **Outstanding Principal After:** 7th payment
### Problem Questions:
(a) The number of payments required to amortize the debt is [ ]
*(Round the final answer up to the nearest whole number. Round all intermediate values to six decimal places as needed.)*
---
### Detailed Explanation for Users:
1. **Understanding the Debt Principal:** This is the initial amount borrowed, $14,000 in this case.
2. **Payment Description:** A fixed payment of $1480 is made every month.
3. **Interest Rate Details:** The interest rate is provided annually but needs to be converted to a monthly interest rate for the calculations. For a 9% annual interest rate, the monthly interest rate would be \( \frac{9\%}{12} \).
4. **Outstanding Principal After a Certain Payment:** We also need to find the remaining debt after specific payments made (after the 7th payment).
### Steps to Solve:
1. **Convert Annual Interest Rate to Monthly Interest Rate:**
\[ \text{Monthly Interest Rate} = \frac{9\%}{12} = 0.75\% \text{ per month} \]
2. **Calculate the Number of Payments (n):**
To determine the number of payments required to amortize the debt, the loan amortization formula can be used:
\[ P \times \frac{r(1+r)^n}{(1+r)^n - 1} = \text{Monthly Payment} \]
where:
- \( P \) is the principal amount ($14
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