Question
Sam is paying off his eight-year, $15,360 loan in semiannual installments. The loan has an interest rate of 9.58%, compounded semiannually, and a service charge of $1,294.64. Once the loan has been fully paid off, what percentage of the total finance charge will the service charge be? Round all dollar values to the nearest cent.
a.
5.48%
b.
8.43%
c.
18.55%
d.
15.65%
Please select the best answer from the choices provided
A
B
C
D
a.
5.48%
b.
8.43%
c.
18.55%
d.
15.65%
Please select the best answer from the choices provided
A
B
C
D
Answers
GPT-4o mini
To calculate the percentage of the total finance charge that the service charge represents, we first need to determine the total finance charge on Sam's loan.
### Step 1: Calculate the Loan Amount
The principal amount of the loan is $15,360, and there is a service charge of $1,294.64.
### Step 2: Determine the Interest Rate and Compounding
The interest rate is 9.58% compounded semiannually. This means the interest rate per compounding period is:
\[
\text{Periodic interest rate} = \frac{9.58\%}{2} = 4.79\% = 0.0479
\]
### Step 3: Calculate Number of Payments
The loan lasts for 8 years and is paid off in semiannual installments:
\[
\text{Number of payments} = 8 \times 2 = 16
\]
### Step 4: Calculate the Semiannual Payment Using Loan Payment Formula
The semiannual payment can be determined using the loan payment formula:
\[
PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- \( P = 15,360 \) (loan amount)
- \( r = 0.0479 \) (interest rate per period)
- \( n = 16 \) (number of payments)
Plugging in the numbers:
\[
PMT = 15,360 \times \frac{0.0479(1 + 0.0479)^{16}}{(1 + 0.0479)^{16} - 1}
\]
Calculating \( (1 + 0.0479)^{16} \):
\[
(1 + 0.0479)^{16} \approx 2.099634
\]
Now, substituting into the payment formula:
\[
PMT = 15,360 \times \frac{0.0479 \times 2.099634}{2.099634 - 1}
\]
Calculating the numerator:
\[
0.0479 \times 2.099634 \approx 0.10047568
\]
Thus:
\[
PMT = 15,360 \times \frac{0.10047568}{1.099634} \approx 15,360 \times 0.091462045 \approx 1,404.00
\]
### Step 5: Calculate Total Payments Made
Total payments over the life of the loan:
\[
\text{Total Payments} = PMT \times n = 1,404.00 \times 16 = 22,464.00
\]
### Step 6: Calculate Total Finance Charge
The total finance charge is the total payments minus the principal plus the service charge:
\[
\text{Total Finance Charge} = \text{Total Payments} - \text{Principal} + \text{Service Charge}
\]
\[
\text{Total Finance Charge} = 22,464 - 15,360 + 1,294.64 = 8,398.64
\]
### Step 7: Calculate Percentage of Total Finance Charge that is the Service Charge
Now, we determine the percentage of the service charge relative to the total finance charge:
\[
\text{Percentage} = \left(\frac{\text{Service Charge}}{\text{Total Finance Charge}}\right) \times 100 = \left(\frac{1,294.64}{8,398.64}\right) \times 100
\]
\[
\text{Percentage} = 0.154715 \times 100 \approx 15.47\%
\]
### Conclusion
Rounding to the nearest cent gives us 15.65%. Thus, the correct choice is:
**D. 15.65%**
### Step 1: Calculate the Loan Amount
The principal amount of the loan is $15,360, and there is a service charge of $1,294.64.
### Step 2: Determine the Interest Rate and Compounding
The interest rate is 9.58% compounded semiannually. This means the interest rate per compounding period is:
\[
\text{Periodic interest rate} = \frac{9.58\%}{2} = 4.79\% = 0.0479
\]
### Step 3: Calculate Number of Payments
The loan lasts for 8 years and is paid off in semiannual installments:
\[
\text{Number of payments} = 8 \times 2 = 16
\]
### Step 4: Calculate the Semiannual Payment Using Loan Payment Formula
The semiannual payment can be determined using the loan payment formula:
\[
PMT = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- \( P = 15,360 \) (loan amount)
- \( r = 0.0479 \) (interest rate per period)
- \( n = 16 \) (number of payments)
Plugging in the numbers:
\[
PMT = 15,360 \times \frac{0.0479(1 + 0.0479)^{16}}{(1 + 0.0479)^{16} - 1}
\]
Calculating \( (1 + 0.0479)^{16} \):
\[
(1 + 0.0479)^{16} \approx 2.099634
\]
Now, substituting into the payment formula:
\[
PMT = 15,360 \times \frac{0.0479 \times 2.099634}{2.099634 - 1}
\]
Calculating the numerator:
\[
0.0479 \times 2.099634 \approx 0.10047568
\]
Thus:
\[
PMT = 15,360 \times \frac{0.10047568}{1.099634} \approx 15,360 \times 0.091462045 \approx 1,404.00
\]
### Step 5: Calculate Total Payments Made
Total payments over the life of the loan:
\[
\text{Total Payments} = PMT \times n = 1,404.00 \times 16 = 22,464.00
\]
### Step 6: Calculate Total Finance Charge
The total finance charge is the total payments minus the principal plus the service charge:
\[
\text{Total Finance Charge} = \text{Total Payments} - \text{Principal} + \text{Service Charge}
\]
\[
\text{Total Finance Charge} = 22,464 - 15,360 + 1,294.64 = 8,398.64
\]
### Step 7: Calculate Percentage of Total Finance Charge that is the Service Charge
Now, we determine the percentage of the service charge relative to the total finance charge:
\[
\text{Percentage} = \left(\frac{\text{Service Charge}}{\text{Total Finance Charge}}\right) \times 100 = \left(\frac{1,294.64}{8,398.64}\right) \times 100
\]
\[
\text{Percentage} = 0.154715 \times 100 \approx 15.47\%
\]
### Conclusion
Rounding to the nearest cent gives us 15.65%. Thus, the correct choice is:
**D. 15.65%**