Question
summer has an offer to buy an item with a sticker price of $13,200 by paying $460 a month for 36 months. what interest rate is summer being offered?
22.9%
5.7%
7.6%
11.4%
22.9%
5.7%
7.6%
11.4%
Answers
GPT-4o mini
To determine the interest rate being offered, we will compare the total cost of the payment plan to the sticker price and calculate the annual interest rate based on that information.
First, let's calculate the total payment Summer will make over the 36 months:
\[
\text{Total Payments} = \text{Monthly Payment} \times \text{Number of Months} = 460 \times 36 = 16,560
\]
Next, we find the total interest paid by subtracting the sticker price from the total payments:
\[
\text{Total Interest} = \text{Total Payments} - \text{Sticker Price} = 16,560 - 13,200 = 3,360
\]
From here, we can use the following formula for calculating the interest rate \( r \) with respect to a loan:
\[
P = \frac{r \times PV}{1 - (1 + r)^{-n}}
\]
Where:
- \( P \) = Monthly payment ($460)
- \( PV \) = Present value/loan amount ($13,200)
- \( n \) = Total number of payments (36)
However, this formula is not easy to solve directly for \( r \). Instead, we can use the trial and error method or utilize financial calculators or software to compute this more directly.
Since interest rates are limited to the options given, we can check the given interest rates directly through calculations.
Let’s calculate the effective monthly payment using simple estimates for each of the interest rates and see which one gives us a payment close to $460.
1. **For 5.7% APR:**
- Monthly interest = \( \frac{0.057}{12} ~= 0.00475 \)
- \[ M = \frac{PV \cdot i}{1 - (1 + i)^{-n}} \]
- \[ PV = 13,200; n = 36 \]
- Substituting \( i \):
- \[ M = \frac{13,200 \cdot 0.00475}{1 - (1 + 0.00475)^{-36}} \approx 397.51 \]
2. **For 7.6% APR:**
- Monthly interest = \( \frac{0.076}{12} \approx 0.00633 \)
- \[ M = \frac{13,200 \cdot 0.00633}{1 - (1 + 0.00633)^{-36}} \approx 412.38 \]
3. **For 11.4% APR:**
- Monthly interest = \( \frac{0.114}{12} \approx 0.0095 \)
- \[ M = \frac{13,200 \cdot 0.0095}{1 - (1 + 0.0095)^{-36}} \approx 461.47 \]
4. **For 22.9% APR:**
- Monthly interest = \( \frac{0.229}{12} \approx 0.01908 \)
- \[ M = \frac{13,200 \cdot 0.01908}{1 - (1 + 0.01908)^{-36}} \approx 644.32 \]
Among the calculated monthly payments, the one that closely matches the payment of $460 is with an interest rate of **11.4%**.
Thus, the answer is **11.4%**.
First, let's calculate the total payment Summer will make over the 36 months:
\[
\text{Total Payments} = \text{Monthly Payment} \times \text{Number of Months} = 460 \times 36 = 16,560
\]
Next, we find the total interest paid by subtracting the sticker price from the total payments:
\[
\text{Total Interest} = \text{Total Payments} - \text{Sticker Price} = 16,560 - 13,200 = 3,360
\]
From here, we can use the following formula for calculating the interest rate \( r \) with respect to a loan:
\[
P = \frac{r \times PV}{1 - (1 + r)^{-n}}
\]
Where:
- \( P \) = Monthly payment ($460)
- \( PV \) = Present value/loan amount ($13,200)
- \( n \) = Total number of payments (36)
However, this formula is not easy to solve directly for \( r \). Instead, we can use the trial and error method or utilize financial calculators or software to compute this more directly.
Since interest rates are limited to the options given, we can check the given interest rates directly through calculations.
Let’s calculate the effective monthly payment using simple estimates for each of the interest rates and see which one gives us a payment close to $460.
1. **For 5.7% APR:**
- Monthly interest = \( \frac{0.057}{12} ~= 0.00475 \)
- \[ M = \frac{PV \cdot i}{1 - (1 + i)^{-n}} \]
- \[ PV = 13,200; n = 36 \]
- Substituting \( i \):
- \[ M = \frac{13,200 \cdot 0.00475}{1 - (1 + 0.00475)^{-36}} \approx 397.51 \]
2. **For 7.6% APR:**
- Monthly interest = \( \frac{0.076}{12} \approx 0.00633 \)
- \[ M = \frac{13,200 \cdot 0.00633}{1 - (1 + 0.00633)^{-36}} \approx 412.38 \]
3. **For 11.4% APR:**
- Monthly interest = \( \frac{0.114}{12} \approx 0.0095 \)
- \[ M = \frac{13,200 \cdot 0.0095}{1 - (1 + 0.0095)^{-36}} \approx 461.47 \]
4. **For 22.9% APR:**
- Monthly interest = \( \frac{0.229}{12} \approx 0.01908 \)
- \[ M = \frac{13,200 \cdot 0.01908}{1 - (1 + 0.01908)^{-36}} \approx 644.32 \]
Among the calculated monthly payments, the one that closely matches the payment of $460 is with an interest rate of **11.4%**.
Thus, the answer is **11.4%**.