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].
### 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].
