3. Suppose that you decide lo borrow $$ \$ 25,000 $$ for a new car. You can select 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 total interest for the two loans.

Question

UpStudy AI · Accepted Answer

To solve this problem, we will use the formula for calculating the monthly payment $$ M $$ for an installment loan, which is given by:

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

where:
- $$ P $$ is the principal amount (the loan amount),
- $$ r $$ is the monthly interest rate (annual interest rate divided by 12),
- $$ n $$ is the total number of payments (loan term in months).

### Given Data:
- Principal $$ P = 25000 $$
- Loan A: 
  - Annual interest rate = $$ 1.9\% $$ 
  - Loan term = 3 years
- Loan B: 
  - Annual interest rate = $$ 2.4\% $$ 
  - Loan term = 5 years

### Step-by-step calculations:

#### a. Loan A
1. Convert the annual interest rate to a monthly interest rate:
   $$
   r_A = \frac{1.9\%}{12} = \frac{0.019}{12}
   $$
2. Calculate the total number of payments:
   $$
   n_A = 3 \times 12 = 36
   $$
3. Substitute $$ P $$, $$ r_A $$, and $$ n_A $$ into the monthly payment formula.

#### b. Loan B
1. Convert the annual interest rate to a monthly interest rate:
   $$
   r_B = \frac{2.4\%}{12} = \frac{0.024}{12}
   $$
2. Calculate the total number of payments:
   $$
   n_B = 5 \times 12 = 60
   $$
3. Substitute $$ P $$, $$ r_B $$, and $$ n_B $$ into the monthly payment formula.

#### c. Calculate total interest for both loans:
Total interest can be calculated as:
$$
\text{Total Interest} = (M \times n) - P
$$

Now, let's perform the calculations for both loans.

### Loan A Calculations:
1. Monthly interest rate for Loan A:
   $$
   r_A = \frac{0.019}{12}
   $$
2. Total number of payments for Loan A:
   $$
   n_A = 36
   $$
3. Monthly payment for Loan A:
   $$
   M_A = 25000 \frac{r_A(1 + r_A)^{36}}{(1 + r_A)^{36} - 1}
   $$

### Loan B Calculations:
1. Monthly interest rate for Loan B:
   $$
   r_B = \frac{0.024}{12}
   $$
2. Total number of payments for Loan B:
   $$
   n_B = 60
   $$
3. Monthly payment for Loan B:
   $$
   M_B = 25000 \frac{r_B(1 + r_B)^{60}}{(1 + r_B)^{60} - 1}
   $$

Now, I will calculate the monthly payments for both loans and their total interests.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(1+\left(\frac{0.024}{12}\right)\right)^{60}}{\left(\left(1+\left(\frac{0.024}{12}\right)\right)^{60}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(1+\left(\frac{0.024}{12}\right)\right)^{60}}{\left(1+\left(\frac{0.024}{12}\right)\right)^{60}-1}$$
- step2: Divide the terms:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(1+\frac{1}{500}\right)^{60}}{\left(1+\left(\frac{0.024}{12}\right)\right)^{60}-1}$$
- step3: Add the numbers:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(\frac{501}{500}\right)^{60}}{\left(1+\left(\frac{0.024}{12}\right)\right)^{60}-1}$$
- step4: Divide the terms:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(\frac{501}{500}\right)^{60}}{\left(1+\frac{1}{500}\right)^{60}-1}$$
- step5: Add the numbers:
  $$\frac{25000\left(\frac{0.024}{12}\right)\left(\frac{501}{500}\right)^{60}}{\left(\frac{501}{500}\right)^{60}-1}$$
- step6: Divide the terms:
  $$\frac{25000\times \frac{1}{500}\left(\frac{501}{500}\right)^{60}}{\left(\frac{501}{500}\right)^{60}-1}$$
- step7: Multiply:
  $$\frac{50\times \frac{501^{60}}{500^{60}}}{\left(\frac{501}{500}\right)^{60}-1}$$
- step8: Subtract the numbers:
  $$\frac{50\times \frac{501^{60}}{500^{60}}}{\frac{501^{60}-500^{60}}{500^{60}}}$$
- step9: Evaluate the power:
  $$\frac{\frac{501^{60}}{50^{59}\times 10^{60}}}{\frac{501^{60}-500^{60}}{500^{60}}}$$
- step10: Multiply by the reciprocal:
  $$\frac{501^{60}}{50^{59}\times 10^{60}}\times \frac{500^{60}}{501^{60}-500^{60}}$$
- step11: Rewrite the expression:
  $$\frac{501^{60}}{50^{59}\times 10^{60}}\times \frac{50^{60}\times 10^{60}}{501^{60}-500^{60}}$$
- step12: Reduce the numbers:
  $$\frac{501^{60}}{10^{60}}\times \frac{50\times 10^{60}}{501^{60}-500^{60}}$$
- step13: Rewrite the expression:
  $$\frac{501^{60}}{10^{60}}\times \frac{10\times 5\times 10^{60}}{501^{60}-500^{60}}$$
- step14: Reduce the numbers:
  $$\frac{501^{60}}{10^{59}}\times \frac{5\times 10^{60}}{501^{60}-500^{60}}$$
- step15: Rewrite the expression:
  $$\frac{501^{60}}{5^{59}\times 2^{59}}\times \frac{5\times 10^{60}}{501^{60}-500^{60}}$$
- step16: Reduce the numbers:
  $$\frac{501^{60}}{5^{58}\times 2^{59}}\times \frac{10^{60}}{501^{60}-500^{60}}$$
- step17: Rewrite the expression:
  $$\frac{501^{60}}{5^{58}\times 2^{59}}\times \frac{5^{60}\times 2^{60}}{501^{60}-500^{60}}$$
- step18: Reduce the numbers:
  $$501^{60}\times \frac{5^{2}\times 2}{501^{60}-500^{60}}$$
- step19: Multiply:
  $$\frac{501^{60}\times 50}{501^{60}-500^{60}}$$
Calculate or simplify the expression $$ 25000 * (0.019/12) * (1 + (0.019/12))^36 / ((1 + (0.019/12))^36 - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(1+\left(\frac{0.019}{12}\right)\right)^{36}}{\left(\left(1+\left(\frac{0.019}{12}\right)\right)^{36}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(1+\left(\frac{0.019}{12}\right)\right)^{36}}{\left(1+\left(\frac{0.019}{12}\right)\right)^{36}-1}$$
- step2: Divide the terms:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(1+\frac{19}{12000}\right)^{36}}{\left(1+\left(\frac{0.019}{12}\right)\right)^{36}-1}$$
- step3: Add the numbers:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(\frac{12019}{12000}\right)^{36}}{\left(1+\left(\frac{0.019}{12}\right)\right)^{36}-1}$$
- step4: Divide the terms:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(\frac{12019}{12000}\right)^{36}}{\left(1+\frac{19}{12000}\right)^{36}-1}$$
- step5: Add the numbers:
  $$\frac{25000\left(\frac{0.019}{12}\right)\left(\frac{12019}{12000}\right)^{36}}{\left(\frac{12019}{12000}\right)^{36}-1}$$
- step6: Divide the terms:
  $$\frac{25000\times \frac{19}{12000}\left(\frac{12019}{12000}\right)^{36}}{\left(\frac{12019}{12000}\right)^{36}-1}$$
- step7: Multiply:
  $$\frac{\frac{19\times 12019^{36}}{12\times 5^{106}\times 96^{36}}}{\left(\frac{12019}{12000}\right)^{36}-1}$$
- step8: Subtract the numbers:
  $$\frac{\frac{19\times 12019^{36}}{12\times 5^{106}\times 96^{36}}}{\frac{12019^{36}-12000^{36}}{12000^{36}}}$$
- step9: Multiply by the reciprocal:
  $$\frac{19\times 12019^{36}}{12\times 5^{106}\times 96^{36}}\times \frac{12000^{36}}{12019^{36}-12000^{36}}$$
- step10: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{12\times 5^{106}\times 96^{36}}\times \frac{12^{36}\times 1000^{36}}{12019^{36}-12000^{36}}$$
- step11: Reduce the numbers:
  $$\frac{19\times 12019^{36}}{5^{106}\times 96^{36}}\times \frac{12^{35}\times 1000^{36}}{12019^{36}-12000^{36}}$$
- step12: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{5^{106}\times 12^{36}\times 8^{36}}\times \frac{12^{35}\times 1000^{36}}{12019^{36}-12000^{36}}$$
- step13: Reduce the numbers:
  $$\frac{19\times 12019^{36}}{5^{106}\times 12\times 8^{36}}\times \frac{1000^{36}}{12019^{36}-12000^{36}}$$
- step14: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{5^{106}\times 12\times 8^{36}}\times \frac{125^{36}\times 8^{36}}{12019^{36}-12000^{36}}$$
- step15: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{5^{106}\times 12\times 8^{36}}\times \frac{5^{108}\times 8^{36}}{12019^{36}-12000^{36}}$$
- step16: Reduce the numbers:
  $$\frac{19\times 12019^{36}}{12\times 8^{36}}\times \frac{5^{2}\times 8^{36}}{12019^{36}-12000^{36}}$$
- step17: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{4\times 3\times 8^{36}}\times \frac{5^{2}\times 8^{36}}{12019^{36}-12000^{36}}$$
- step18: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{2^{2}\times 3\times 8^{36}}\times \frac{5^{2}\times 2^{108}}{12019^{36}-12000^{36}}$$
- step19: Reduce the numbers:
  $$\frac{19\times 12019^{36}}{3\times 8^{36}}\times \frac{5^{2}\times 2^{106}}{12019^{36}-12000^{36}}$$
- step20: Rewrite the expression:
  $$\frac{19\times 12019^{36}}{3\times 2^{108}}\times \frac{5^{2}\times 2^{106}}{12019^{36}-12000^{36}}$$
- step21: Reduce the numbers:
  $$\frac{19\times 12019^{36}}{3\times 2^{2}}\times \frac{5^{2}}{12019^{36}-12000^{36}}$$
- step22: Multiply the fractions:
  $$\frac{19\times 12019^{36}\times 5^{2}}{12\left(12019^{36}-12000^{36}\right)}$$
- step23: Multiply:
  $$\frac{475\times 12019^{36}}{12\times 12019^{36}-12\times 12000^{36}}$$
### Monthly Payments Calculation:

#### a. Loan A:
- Monthly payment $$ M_A \approx 714.97 $$

#### b. Loan B:
- Monthly payment $$ M_B \approx 442.58 $$

### Total Interest Calculation:
Now, let's calculate the total interest for both loans using the formula:

$$
\text{Total Interest} = (M \times n) - P
$$

#### For Loan A:
- Total payments for Loan A:
  $$
  \text{Total Payments}_A = M_A \times n_A = 714.97 \times 36
  $$
  
#### For Loan B:
- Total payments for Loan B:
  $$
  \text{Total Payments}_B = M_B \times n_B = 442.58 \times 60
  $$

Now, I will calculate the total payments and total interest for both loans.
Calculate the value by following steps:
- step0: Calculate:
  $$442.58\times 60$$
- step1: Multiply the numbers:
  $$26554.8$$
Calculate or simplify the expression $$ 714.97 * 36 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$714.97\times 36$$
- step1: Multiply the numbers:
  $$25738.92$$
### Total Payments Calculation:

#### a. Loan A:
- Total payments for Loan A:
  $$
  \text{Total Payments}_A \approx 25738.92
  $$

#### b. Loan B:
- Total payments for Loan B:
  $$
  \text{Total Payments}_B \approx 26554.80
  $$

### Total Interest Calculation:
Now, we can calculate the total interest for both loans:

#### For Loan A:
$$
\text{Total Interest}_A = \text{Total Payments}_A - P = 25738.92 - 25000 \approx 738.92
$$

#### For Loan B:
$$
\text{Total Interest}_B = \text{Total Payments}_B - P = 26554.80 - 25000 \approx 1554.80
$$

### Summary of Results:
- **Loan A:**
  - Monthly Payment: $$ \approx 714.97 $$
  - Total Interest: $$ \approx 738.92 $$

- **Loan B:**
  - Monthly Payment: $$ \approx 442.58 $$
  - Total Interest: $$ \approx 1554.80 $$

### Comparison:
- Loan A has a higher monthly payment but lower total interest compared to Loan B.
- Loan B has a lower monthly payment but results in higher total interest over the loan term.
**Loan A:** - Monthly Payment: \$714.97 - Total Interest: \$738.92 **Loan B:** - Monthly Payment: \$442.58 - Total Interest: \$1554.80 **Comparison:** - Loan A has higher monthly payments but lower total interest. - Loan B has lower monthly payments but higher total interest over the loan term.

Upstudy AI Solution

Answer

Solution

Given Data:

Step-by-step calculations:

a. Loan A

b. Loan B

c. Calculate total interest for both loans:

Loan A Calculations:

Loan B Calculations:

Monthly Payments Calculation:

a. Loan A:

b. Loan B:

Total Interest Calculation:

For Loan A:

For Loan B:

Total Payments Calculation:

a. Loan A:

b. Loan B:

Total Interest Calculation:

For Loan A:

For Loan B:

Summary of Results:

Comparison:

The Deep Dive

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