55 If a student loan in your freshman year is repaid plus 20% four years later, what was the effective interest rate?
55 If a student loan in your freshman year is repaid plus 20% four years later, what was the effective interest rate?
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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![**Question 55:**
If a student loan in your freshman year is repaid plus 20% four years later, what was the effective interest rate?
---
To solve this, we need to determine the effective annual interest rate that results in a total repayment of the principal amount plus 20% at the end of four years.
**Solution Approach:**
1. **Understanding the Problem:**
- Let the initial amount of the loan be \( P \).
- After four years, the total amount repaid is \( P + 0.2P = 1.2P \).
2. **Set Up the Equation:**
- Using the formula for compound interest:
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial loan).
- \( r \) is the annual interest rate.
- \( n \) is the number of years the money is invested or borrowed for.
- Here, \( A = 1.2P \), \( n = 4 \).
3. **Solve for \( r \):**
- \[
1.2P = P(1 + r)^4
\]
- Divide both sides by \( P \):
\[
1.2 = (1 + r)^4
\]
- Take the fourth root of both sides:
\[
1 + r = \sqrt[4]{1.2}
\]
- Subtract 1 from both sides to find \( r \):
\[
r = \sqrt[4]{1.2} - 1
\]
4. **Calculate the Effective Interest Rate:**
- Using a calculator, compute \( \sqrt[4]{1.2} \approx 1.0461 \).
- Thus, \( r \approx 1.0461 - 1 = 0.0461 \) or 4.61%.
Therefore, the effective annual interest rate is approximately **4.61%**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F892e817a-9b32-4eeb-b8fc-5dd7ffde6479%2F92f9d638-5bfb-43ac-8c8f-11d148ca0659%2Fr243j8n_processed.png&w=3840&q=75)
Transcribed Image Text:**Question 55:**
If a student loan in your freshman year is repaid plus 20% four years later, what was the effective interest rate?
---
To solve this, we need to determine the effective annual interest rate that results in a total repayment of the principal amount plus 20% at the end of four years.
**Solution Approach:**
1. **Understanding the Problem:**
- Let the initial amount of the loan be \( P \).
- After four years, the total amount repaid is \( P + 0.2P = 1.2P \).
2. **Set Up the Equation:**
- Using the formula for compound interest:
\[
A = P(1 + r)^n
\]
where:
- \( A \) is the amount of money accumulated after n years, including interest.
- \( P \) is the principal amount (initial loan).
- \( r \) is the annual interest rate.
- \( n \) is the number of years the money is invested or borrowed for.
- Here, \( A = 1.2P \), \( n = 4 \).
3. **Solve for \( r \):**
- \[
1.2P = P(1 + r)^4
\]
- Divide both sides by \( P \):
\[
1.2 = (1 + r)^4
\]
- Take the fourth root of both sides:
\[
1 + r = \sqrt[4]{1.2}
\]
- Subtract 1 from both sides to find \( r \):
\[
r = \sqrt[4]{1.2} - 1
\]
4. **Calculate the Effective Interest Rate:**
- Using a calculator, compute \( \sqrt[4]{1.2} \approx 1.0461 \).
- Thus, \( r \approx 1.0461 - 1 = 0.0461 \) or 4.61%.
Therefore, the effective annual interest rate is approximately **4.61%**.
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