Consider the following two loans for Po = $10,000, each requiring regular monthly payments.
•Loan A: 2 year loan, annual interest rate of 12%.
•Loan B: 5 year loan, annual interest rate of 6%.
Calculate the absolute difference between the total interest paid on both loans. Give your answer to the nearest ten dollars.
Do not include the dollar sign in your answer.
Provide your answer below:



Answer :

Alright, let's break down the problem and find the absolute difference between the total interest paid on both loans, rounded to the nearest ten dollars.

### Loan A:

- Principal Amount ([tex]\(P_0\)[/tex]): [tex]$10,000 - Loan Term: 2 years - Annual Interest Rate: 12% Steps to calculate total interest for Loan A: 1. Convert the annual interest rate to a monthly rate: \[ \text{Monthly interest rate} = \frac{12\%}{12} = 1\% = 0.01 \] 2. Calculate the number of monthly payments: \[ \text{Number of payments} = 2 \times 12 = 24 \] 3. Calculate the monthly payment using the formula for an installment loan: \[ M_A = \frac{P_0 \times r \times (1 + r)^n}{(1 + r)^n - 1} \] Where: - \(M_A\) = monthly payment for Loan A - \(P_0\) = principal amount = $[/tex]10,000
- [tex]\(r\)[/tex] = monthly interest rate = 0.01
- [tex]\(n\)[/tex] = number of payments = 24

4. Calculate total payment for Loan A:
[tex]\[ \text{Total Payment}_A = M_A \times n \][/tex]

5. Calculate total interest paid for Loan A:
[tex]\[ \text{Total Interest}_A = \text{Total Payment}_A - P_0 \][/tex]
From calculations: [tex]\(\text{Total Interest}_A = 1297.63\)[/tex]

### Loan B:

- Principal Amount ([tex]\(P_0\)[/tex]): [tex]$10,000 - Loan Term: 5 years - Annual Interest Rate: 6% Steps to calculate total interest for Loan B: 1. Convert the annual interest rate to a monthly rate: \[ \text{Monthly interest rate} = \frac{6\%}{12} = 0.5\% = 0.005 \] 2. Calculate the number of monthly payments: \[ \text{Number of payments} = 5 \times 12 = 60 \] 3. Calculate the monthly payment using the formula for an installment loan: \[ M_B = \frac{P_0 \times r \times (1 + r)^n}{(1 + r)^n - 1} \] Where: - \(M_B\) = monthly payment for Loan B - \(P_0\) = principal amount = $[/tex]10,000
- [tex]\(r\)[/tex] = monthly interest rate = 0.005
- [tex]\(n\)[/tex] = number of payments = 60

4. Calculate total payment for Loan B:
[tex]\[ \text{Total Payment}_B = M_B \times n \][/tex]

5. Calculate total interest paid for Loan B:
[tex]\[ \text{Total Interest}_B = \text{Total Payment}_B - P_0 \][/tex]
From calculations: [tex]\(\text{Total Interest}_B = 1599.68\)[/tex]

### Absolute Difference in Total Interest:

1. Calculate the absolute difference:
[tex]\[ \text{Absolute Difference} = \left| \text{Total Interest}_A - \text{Total Interest}_B \right| \][/tex]
[tex]\[ \text{Absolute Difference} = \left| 1297.63 - 1599.68 \right| = 302.05 \][/tex]

2. Round to the nearest ten dollars:
[tex]\[ \text{Rounded Difference} = 300.0 \][/tex]

Thus, the absolute difference between the total interest paid on both loans, to the nearest ten dollars, is [tex]\(300\)[/tex].