3. Suppose that you decide to borrow $$ \$ 25,000 $$ for a new car. You can seled one of the following loans, each requiring regular monthly payments: Installment Loan A: three-year 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 tatal interest for the two loans.

Question

UpStudy AI · Accepted Answer

**Step 1.** Write the formula for the monthly payment of an installment loan. For a loan of principal $$ P $$, monthly interest rate $$ r $$ and $$ n $$ total monthly payments, the monthly payment $$ M $$ is given by

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

---

**Step 2.** Set up the parameters and calculations for each loan.

### Loan A (3-year loan at $$1.9\%$$ annual rate)

- Principal: $$ P = 25000 $$
- Annual interest rate: $$1.9\% = 0.019$$
- Monthly interest rate:

$$
  r_A = \frac{0.019}{12} \approx 0.00158333
  $$
  
- Number of months:

$$
  n_A = 3 \text{ years} \times 12 = 36.
  $$

Insert these into the monthly payment formula:

$$
M_A = 25000 \cdot \frac{0.00158333\,(1 + 0.00158333)^{36}}{(1 + 0.00158333)^{36} - 1}.
$$

#### Calculation for Loan A:

1. Compute $$(1 + 0.00158333)^{36}$$.  
   Approximating using the exponential method:
   
   $$
   (1.00158333)^{36} \approx e^{36 \cdot 0.00158333} \approx e^{0.057} \approx 1.0586.
   $$

2. Compute the numerator:

$$
   0.00158333 \cdot 1.0586 \approx 0.001676.
   $$

3. Compute the denominator:

$$
   1.0586 - 1 = 0.0586.
   $$

4. Finally, the monthly payment:

$$
   M_A \approx 25000 \cdot \frac{0.001676}{0.0586} \approx 25000 \cdot 0.02863 \approx 715.75.
   $$

Thus, the monthly payment for Loan A is approximately

$$
M_A \approx \$716.
$$

5. **Total interest for Loan A:**

Total amount paid over 36 months:

$$
   \text{Total Paid}_A = 36 \times 716 \approx 25776.
   $$

Total interest is

$$
   \text{Interest}_A = 25776 - 25000 \approx \$776.
   $$

---

### Loan B (5-year loan at $$2.4\%$$ annual rate)

- Principal: $$ P = 25000 $$
- Annual interest rate: $$ 2.4\% = 0.024 $$
- Monthly interest rate:

$$
  r_B = \frac{0.024}{12} = 0.002.
  $$
  
- Number of months:

$$
  n_B = 5 \text{ years} \times 12 = 60.
  $$

Insert these into the monthly payment formula:

$$
M_B = 25000 \cdot \frac{0.002\,(1 + 0.002)^{60}}{(1 + 0.002)^{60} - 1}.
$$

#### Calculation for Loan B:

1. Compute $$(1 + 0.002)^{60}$$.  
   Approximating:
   
   $$
   (1.002)^{60} \approx e^{60 \cdot \ln(1.002)} \approx e^{60 \cdot 0.001998} \approx e^{0.11988} \approx 1.1274.
   $$

2. Compute the numerator:

$$
   0.002 \cdot 1.1274 \approx 0.0022548.
   $$

3. Compute the denominator:

$$
   1.1274 - 1 = 0.1274.
   $$

4. The monthly payment is:

$$
   M_B \approx 25000 \cdot \frac{0.0022548}{0.1274} \approx 25000 \cdot 0.017687 \approx 442.18.
   $$

Thus, the monthly payment for Loan B is approximately

$$
M_B \approx \$442.
$$

5. **Total interest for Loan B:**

Total amount paid over 60 months:

$$
   \text{Total Paid}_B = 60 \times 442 \approx 26520.
   $$

Total interest is

$$
   \text{Interest}_B = 26520 - 25000 \approx \$1520.
   $$

---

**Step 3.** Comparison of the two loans.

- **Monthly Payments:**
  - Loan A: Approximately $$ \$716 $$ per month.
  - Loan B: Approximately $$ \$442 $$ per month.
  
  Loan A has a higher monthly payment.

- **Total Interest:**
  - Loan A: Approximately $$ \$776 $$ in total interest.
  - Loan B: Approximately $$ \$1520 $$ in total interest.
  
  Loan B results in more total interest paid over the life of the loan.

---

**Final Answers:**

a. For Loan A (3-year loan at $$1.9\%$$):
   - Monthly payment: $$ \approx \$716 $$
   - Total interest paid: $$ \approx \$776 $$

b. For Loan B (5-year loan at $$2.4\%$$):
   - Monthly payment: $$ \approx \$442 $$
   - Total interest paid: $$ \approx \$1520 $$

c. Comparison:
   - Loan A requires higher monthly payments but results in lower total interest.
   - Loan B has lower monthly payments but accumulates more total interest over the term.
**Loan A (3-year, 1.9%):** - Monthly Payment: \$716 - Total Interest: \$776 **Loan B (5-year, 2.4%):** - Monthly Payment: \$442 - Total Interest: \$1520 **Comparison:** - Loan A has higher monthly payments but less total interest. - Loan B has lower monthly payments but more total interest over the loan term.

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Solución

Loan A (3-year loan at annual rate)

Calculation for Loan A:

Loan B (5-year loan at annual rate)

Calculation for Loan B:

The Deep Dive

a. Loan A: 3-year Loan at 1.9%

b. Loan B: 5-year Loan at 2.4%

c. Comparison of Loans

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