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 (\(\(P_0\)\)): \($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 = $\)10,000
- \(\(r\)\) = monthly interest rate = 0.01
- \(\(n\)\) = number of payments = 24
4. Calculate total payment for Loan A:
\(\[
\text{Total Payment}_A = M_A \times n
\]\)
5. Calculate total interest paid for Loan A:
\(\[
\text{Total Interest}_A = \text{Total Payment}_A - P_0
\]\)
From calculations: \(\(\text{Total Interest}_A = 1297.63\)\)
### Loan B:
- Principal Amount (\(\(P_0\)\)): \($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 = $\)10,000
- \(\(r\)\) = monthly interest rate = 0.005
- \(\(n\)\) = number of payments = 60
4. Calculate total payment for Loan B:
\(\[
\text{Total Payment}_B = M_B \times n
\]\)
5. Calculate total interest paid for Loan B:
\(\[
\text{Total Interest}_B = \text{Total Payment}_B - P_0
\]\)
From calculations: \(\(\text{Total Interest}_B = 1599.68\)\)
### Absolute Difference in Total Interest:
1. Calculate the absolute difference:
\(\[
\text{Absolute Difference} = \left| \text{Total Interest}_A - \text{Total Interest}_B \right|
\]\)
\(\[
\text{Absolute Difference} = \left| 1297.63 - 1599.68 \right| = 302.05
\]\)
2. Round to the nearest ten dollars:
\(\[
\text{Rounded Difference} = 300.0
\]\)
Thus, the absolute difference between the total interest paid on both loans, to the nearest ten dollars, is \(\(300\)\).
