5. Suppose that you are thinking about buying a car and have narrowed down your choices to two options: The new-car option: The new car costs $$ \$ 28,000 $$ and can be financed with a four-year loan at $$ 2.45 \% $$. The used-car option: A three-year old model of the same car costs $$ \$ 16,000 $$ and can be financed with a four-year loan at $$ 3.15 \% $$. What is the difference in monthly payments between financing the new car and financing the used car?

Question

UpStudy AI · Accepted Answer

We begin by using the formula for the monthly payment on a fixed‐rate loan:

$$
M = \frac{P \, i}{1 - (1+i)^{-n}}
$$

where  
$$P$$ is the principal (the price of the car),  
$$i$$ is the monthly interest rate, and  
$$n$$ is the total number of monthly payments.

The total number of payments for a four‐year loan is

$$
n = 4 \times 12 = 48.
$$

---

### New-Car Option

1. **Principal**:  
   $$P_{\text{new}} = \$28,\!000$$

2. **Annual Interest Rate**:  
   $$r_{\text{new(annual)}} = 2.45\%$$  
   Convert to a monthly rate:
   $$
   i_{\text{new}} = \frac{0.0245}{12} \approx 0.00204167.
   $$

3. **Monthly Payment Calculation**:
   $$
   M_{\text{new}} = \frac{28000 \times 0.00204167}{1 - \left(1+0.00204167\right)^{-48}}.
   $$
   
   To approximate the denominator, first compute the factor $$\left(1+0.00204167\right)^{48}$$.  
   Using the exponential approximation:
   $$
   \left(1+0.00204167\right)^{48} \approx e^{48 \times 0.00204167} = e^{0.09792} \approx 1.1028.
   $$
   
   Then,
   $$
   1 - \left(1.00204167\right)^{-48} \approx 1 - \frac{1}{1.1028} \approx 1 - 0.9061 = 0.0939.
   $$
   
   The numerator is:
   $$
   28000 \times 0.00204167 \approx 57.17.
   $$
   
   Thus, the estimated monthly payment is:
   $$
   M_{\text{new}} \approx \frac{57.17}{0.0939} \approx 608.4.
   $$

---

### Used-Car Option

1. **Principal**:  
   $$P_{\text{used}} = \$16,\!000$$

2. **Annual Interest Rate**:  
   $$r_{\text{used(annual)}} = 3.15\%$$  
   Convert to a monthly rate:
   $$
   i_{\text{used}} = \frac{0.0315}{12} \approx 0.002625.
   $$

3. **Monthly Payment Calculation**:
   $$
   M_{\text{used}} = \frac{16000 \times 0.002625}{1 - \left(1+0.002625\right)^{-48}}.
   $$
   
   First, compute the factor:
   $$
   \left(1+0.002625\right)^{48} \approx e^{48 \times 0.002625} = e^{0.126} \approx 1.134.
   $$
   
   Then,
   $$
   1 - \left(1.002625\right)^{-48} \approx 1 - \frac{1}{1.134} \approx 1 - 0.8816 = 0.1184.
   $$
   
   The numerator is:
   $$
   16000 \times 0.002625 = 42.
   $$
   
   Thus, the estimated monthly payment is:
   $$
   M_{\text{used}} \approx \frac{42}{0.1184} \approx 354.7.
   $$

---

### Difference in Monthly Payments

Now, subtract the used-car payment from the new-car payment:

$$
\Delta M = M_{\text{new}} - M_{\text{used}} \approx 608.4 - 354.7 \approx 253.7.
$$

---

### Final Answer

The difference in monthly payments between financing the new car and financing the used car is approximately $$\$253.70$$.
The monthly payment for the new car is about \$608.40, and for the used car, it's about \$354.70. The difference is approximately \$253.70.

Upstudy AI Solution

Answer

Solution

New-Car Option

Used-Car Option

Difference in Monthly Payments

Final Answer

Bonus Knowledge

Related Questions

Latest Arithmetic Questions