3. Suppose that you decide to borrow $$ \$ 25,000 $$ for a new car, You can selec one of the following loans, each requiring regular monthly payments: Installment Loan A: threc-ycar loan at $$ 1.9 \% $$ Installment Loan B: five-year loan at $$ 2.4 \% $$. a. Find the monthly payments and the total interest for Loan A. b. Find the monthly payments and the total interest for Loan B. c. Compare the monthly payments and the total interest for the two loans.

Question

UpStudy AI · Accepted Answer

We use the standard loan amortization formula

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

where

- $$ M $$ is the monthly payment,
- $$ P $$ is the principal (here $$ P=25000 $$),
- $$ r $$ is the monthly interest rate (annual rate divided by 12),
- $$ n $$ is the total number of payments.

---

### a. Loan A (3‐year car loan at $$1.9\%$$)

1. **Determine parameters:**

- Annual interest rate: $$1.9\%$$.  
     Monthly rate: 
     $$
     r_A = \frac{0.019}{12} \approx 0.00158333.
     $$
   - Term: 3 years, so
     $$
     n_A = 3 \times 12 = 36 \text{ months}.
     $$

2. **Calculate the monthly payment:**

Substitute into the formula:
   
   $$
   M_A = 25000 \times \frac{0.00158333\,(1+0.00158333)^{36}}{(1+0.00158333)^{36} - 1}.
   $$
   
   First, compute the factor:
   
   $$
   (1+0.00158333)^{36} \approx 1.0586.
   $$
   
   Now, the numerator:
   
   $$
   0.00158333 \times 1.0586 \approx 0.001675.
   $$
   
   And the denominator:
   
   $$
   1.0586 - 1 = 0.0586.
   $$
   
   Thus, the fraction becomes approximately
   
   $$
   \frac{0.001675}{0.0586} \approx 0.02861.
   $$
   
   Finally,
   
   $$
   M_A \approx 25000 \times 0.02861 \approx 715.25.
   $$
   
   So, the **monthly payment for Loan A is approximately $$\$715.25$$**.

3. **Compute the total interest:**

Total amount paid over 36 months:
   
   $$
   \text{Total paid}_A = 36 \times 715.25 \approx 25749.
   $$
   
   Total interest is the difference from the principal:
   
   $$
   \text{Total Interest}_A \approx 25749 - 25000 \approx 749.
   $$
   
   Hence, the **total interest on Loan A is approximately $$\$749$$**.

---

### b. Loan B (5‐year loan at $$2.4\%$$)

1. **Determine parameters:**

- Annual interest rate: $$2.4\%$$.  
     Monthly rate:
     $$
     r_B = \frac{0.024}{12} = 0.002.
     $$
   - Term: 5 years, so
     $$
     n_B = 5 \times 12 = 60 \text{ months}.
     $$

2. **Calculate the monthly payment:**

Substitute into the amortization formula:
   
   $$
   M_B = 25000 \times \frac{0.002\,(1+0.002)^{60}}{(1+0.002)^{60} - 1}.
   $$
   
   First, compute the factor:
   
   $$
   (1+0.002)^{60} \approx 1.1276.
   $$
   
   Now the numerator:
   
   $$
   0.002 \times 1.1276 \approx 0.0022552.
   $$
   
   And the denominator:
   
   $$
   1.1276 - 1 = 0.1276.
   $$
   
   Thus, the fraction becomes approximately
   
   $$
   \frac{0.0022552}{0.1276} \approx 0.017673.
   $$
   
   Finally,
   
   $$
   M_B \approx 25000 \times 0.017673 \approx 441.82.
   $$
   
   So, the **monthly payment for Loan B is approximately $$\$441.82$$**.

3. **Compute the total interest:**

Total amount paid over 60 months:
   
   $$
   \text{Total paid}_B = 60 \times 441.82 \approx 26509.
   $$
   
   Total interest is the difference from the principal:
   
   $$
   \text{Total Interest}_B \approx 26509 - 25000 \approx 2509.
   $$
   
   Thus, the **total interest on Loan B is approximately $$\$2509$$**.

---

### c. Comparison

- **Monthly Payments:**  
  $$\$715.25$$ for the 3‑year loan (Loan A) versus $$\$441.82$$ for the 5‑year loan (Loan B).  
  The three‑year loan has a higher monthly payment.

- **Total Interest:**  
  Approximately $$\$749$$ for Loan A versus $$\$2509$$ for Loan B.  
  The 3‑year loan incurs significantly less total interest than the 5‑year loan.

In summary, while the 5‑year loan offers lower monthly payments, it results in paying over three times as much in interest as the 3‑year loan.
**Loan A (3-year at 1.9%):** - **Monthly Payment:** \$715.25 - **Total Interest:** \$749 **Loan B (5-year at 2.4%):** - **Monthly Payment:** \$441.82 - **Total Interest:** \$2509 **Comparison:** - Loan A has higher monthly payments but less total interest. - Loan B has lower monthly payments but more total interest.

Upstudy AI Solution

Answer

Solution

a. Loan A (3‐year car loan at )

b. Loan B (5‐year loan at )

c. Comparison

Mind Expander

For Loan A (3-year loan at 1.9%):

For Loan B (5-year loan at 2.4%):

Comparison:

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