You want to buy a car. The loan amount will be $$ \$ 34,000 $$. The company is offering a $$ 5 \% $$ interest rate for 60 months ( 5 years). What will your monthly payments be? $$ \$ 547.2 $$

Question

UpStudy AI · Accepted Answer

To calculate the monthly payments for a car loan, we can use the formula for the monthly payment $$ M $$ on an amortizing loan:

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

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

### Given:
- Loan amount $$ P = 34000 $$
- Annual interest rate = $$ 5\% $$ or $$ 0.05 $$
- Loan term = $$ 60 $$ months

### Step 1: Calculate the monthly interest rate
$$
r = \frac{0.05}{12}
$$

### Step 2: Calculate the number of payments
$$
n = 60
$$

### Step 3: Substitute the values into the formula
Now we can substitute $$ P $$, $$ r $$, and $$ n $$ into the formula to find $$ M $$.

Let's calculate $$ r $$ and then $$ M $$.

1. Calculate $$ r $$:
$$
r = \frac{0.05}{12}
$$

2. Substitute into the payment formula:
$$
M = 34000 \frac{r(1 + r)^{60}}{(1 + r)^{60} - 1}
$$

Now, I will perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.05}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{1}{20}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{1}{20}\times \frac{1}{12}$$
- step3: Multiply the fractions:
  $$\frac{1}{20\times 12}$$
- step4: Multiply:
  $$\frac{1}{240}$$
Calculate or simplify the expression $$ 34000*(0.05/12)*(1+(0.05/12))^60/((1+(0.05/12))^60-1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{60}}{\left(\left(1+\left(\frac{0.05}{12}\right)\right)^{60}-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(1+\left(\frac{0.05}{12}\right)\right)^{60}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{60}-1}$$
- step2: Divide the terms:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(1+\frac{1}{240}\right)^{60}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{60}-1}$$
- step3: Add the numbers:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{60}}{\left(1+\left(\frac{0.05}{12}\right)\right)^{60}-1}$$
- step4: Divide the terms:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{60}}{\left(1+\frac{1}{240}\right)^{60}-1}$$
- step5: Add the numbers:
  $$\frac{34000\left(\frac{0.05}{12}\right)\left(\frac{241}{240}\right)^{60}}{\left(\frac{241}{240}\right)^{60}-1}$$
- step6: Divide the terms:
  $$\frac{34000\times \frac{1}{240}\left(\frac{241}{240}\right)^{60}}{\left(\frac{241}{240}\right)^{60}-1}$$
- step7: Multiply:
  $$\frac{\frac{17\times 241^{60}}{3\times 5^{58}\times 48^{60}}}{\left(\frac{241}{240}\right)^{60}-1}$$
- step8: Subtract the numbers:
  $$\frac{\frac{17\times 241^{60}}{3\times 5^{58}\times 48^{60}}}{\frac{241^{60}-240^{60}}{240^{60}}}$$
- step9: Multiply by the reciprocal:
  $$\frac{17\times 241^{60}}{3\times 5^{58}\times 48^{60}}\times \frac{240^{60}}{241^{60}-240^{60}}$$
- step10: Rewrite the expression:
  $$\frac{17\times 241^{60}}{3\times 5^{58}\times 48^{60}}\times \frac{3^{60}\times 80^{60}}{241^{60}-240^{60}}$$
- step11: Reduce the numbers:
  $$\frac{17\times 241^{60}}{5^{58}\times 48^{60}}\times \frac{3^{59}\times 80^{60}}{241^{60}-240^{60}}$$
- step12: Rewrite the expression:
  $$\frac{17\times 241^{60}}{5^{58}\times 3^{60}\times 16^{60}}\times \frac{3^{59}\times 80^{60}}{241^{60}-240^{60}}$$
- step13: Reduce the numbers:
  $$\frac{17\times 241^{60}}{5^{58}\times 3\times 16^{60}}\times \frac{80^{60}}{241^{60}-240^{60}}$$
- step14: Rewrite the expression:
  $$\frac{17\times 241^{60}}{5^{58}\times 3\times 16^{60}}\times \frac{5^{60}\times 16^{60}}{241^{60}-240^{60}}$$
- step15: Reduce the numbers:
  $$\frac{17\times 241^{60}}{3}\times \frac{5^{2}}{241^{60}-240^{60}}$$
- step16: Multiply the fractions:
  $$\frac{17\times 241^{60}\times 5^{2}}{3\left(241^{60}-240^{60}\right)}$$
- step17: Multiply:
  $$\frac{425\times 241^{60}}{3\times 241^{60}-3\times 240^{60}}$$
### Step 1: Calculate the monthly interest rate
The monthly interest rate $$ r $$ is calculated as follows:
$$
r = \frac{0.05}{12} \approx 0.00416667
$$

### Step 2: Calculate the monthly payment $$ M $$
Now substituting $$ P = 34000 $$, $$ r \approx 0.00416667 $$, and $$ n = 60 $$ into the payment formula:
$$
M = 34000 \frac{0.00416667(1 + 0.00416667)^{60}}{(1 + 0.00416667)^{60} - 1}
$$

Calculating this gives:
$$
M \approx 641.62
$$

### Conclusion
The monthly payment for the car loan will be approximately $$ \$ 641.62 $$.

The value you provided, $$ \$ 547.2 $$, seems to be incorrect based on the calculations.
The monthly payment is approximately \$641.62.

You want to buy a car. The loan amount will be \( \$ 34,000 \). The company is offering a \( 5 \% \) interest rate for 60 months ( 5 years). What will your monthly payments be? \( \$ 547.2 \)

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