If Sandy can afford car payments of $$ \$ 380 $$ per month for 3 years, what is the price of a car that she can afford now? Assume an interest rate of 9.9 percent. Sandy can afford a car that costs $$ \$ \square $$ or tess. $$ 42.17 \%, 29.1 $$ of 69 points (Round to the nearest dollar as needed.)

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To find the price of the car that Sandy can afford, we need to calculate the present value of the car payments she can make. The formula for the present value of an annuity (which is what her car payments represent) is given by:

$$
PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r
$$

Where:
- $$ PV $$ = Present Value (price of the car)
- $$ PMT $$ = Monthly payment (\$380)
- $$ r $$ = Monthly interest rate (annual rate / 12)
- $$ n $$ = Total number of payments (months)

### Step 1: Calculate the monthly interest rate
The annual interest rate is 9.9%, so the monthly interest rate $$ r $$ is:

$$
r = \frac{9.9\%}{12} = \frac{0.099}{12}
$$

### Step 2: Calculate the total number of payments
Sandy is making payments for 3 years, which is:

$$
n = 3 \times 12 = 36 \text{ months}
$$

### Step 3: Substitute the values into the formula
Now we can substitute $$ PMT = 380 $$, $$ r $$, and $$ n $$ into the present value formula.

Let's calculate $$ r $$ and then the present value $$ PV $$.

First, calculate $$ r $$:

$$
r = \frac{0.099}{12}
$$

Now, we will calculate the present value using the formula.

Let's perform these calculations.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{0.099}{12}$$
- step1: Convert the expressions:
  $$\frac{\frac{99}{1000}}{12}$$
- step2: Multiply by the reciprocal:
  $$\frac{99}{1000}\times \frac{1}{12}$$
- step3: Reduce the numbers:
  $$\frac{33}{1000}\times \frac{1}{4}$$
- step4: Multiply the fractions:
  $$\frac{33}{1000\times 4}$$
- step5: Multiply:
  $$\frac{33}{4000}$$
Calculate or simplify the expression $$ 380 * (1 - (1 + (0.099/12))^(-36)) / (0.099/12) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{380\left(1-\left(1+\left(\frac{0.099}{12}\right)\right)^{-36}\right)}{\left(\frac{0.099}{12}\right)}$$
- step1: Remove the parentheses:
  $$\frac{380\left(1-\left(1+\left(\frac{0.099}{12}\right)\right)^{-36}\right)}{\frac{0.099}{12}}$$
- step2: Divide the terms:
  $$\frac{380\left(1-\left(1+\frac{33}{4000}\right)^{-36}\right)}{\frac{0.099}{12}}$$
- step3: Add the numbers:
  $$\frac{380\left(1-\left(\frac{4033}{4000}\right)^{-36}\right)}{\frac{0.099}{12}}$$
- step4: Subtract the numbers:
  $$\frac{380\times \frac{4033^{36}-4000^{36}}{4033^{36}}}{\frac{0.099}{12}}$$
- step5: Divide the terms:
  $$\frac{380\times \frac{4033^{36}-4000^{36}}{4033^{36}}}{\frac{33}{4000}}$$
- step6: Multiply the numbers:
  $$\frac{\frac{380\times 4033^{36}-380\times 4000^{36}}{4033^{36}}}{\frac{33}{4000}}$$
- step7: Multiply by the reciprocal:
  $$\frac{380\times 4033^{36}-380\times 4000^{36}}{4033^{36}}\times \frac{4000}{33}$$
- step8: Multiply the fractions:
  $$\frac{\left(380\times 4033^{36}-380\times 4000^{36}\right)\times 4000}{4033^{36}\times 33}$$
- step9: Multiply:
  $$\frac{1520000\times 4033^{36}-1520000\times 4000^{36}}{4033^{36}\times 33}$$
### Step 1: Calculate the monthly interest rate
The monthly interest rate $$ r $$ is approximately:

$$
r \approx 0.00825
$$

### Step 2: Calculate the present value
Now, substituting the values into the present value formula:

$$
PV = 380 \times \left(1 - (1 + 0.00825)^{-36}\right) / 0.00825
$$

Calculating this gives us:

$$
PV \approx 11793.82
$$

### Step 3: Round to the nearest dollar
Rounding $$ 11793.82 $$ to the nearest dollar, we find:

$$
PV \approx 11794
$$

Thus, Sandy can afford a car that costs approximately **\$11,794**.
Sandy can afford a car priced at approximately \$11,794.

If Sandy can afford car payments of \( \$ 380 \) per month for 3 years, what is the price of a car that she can afford now? Assume an interest rate of 9.9 percent. Sandy can afford a car that costs \( \$ \square \) or tess. \( 42.17 \%, 29.1 \) of 69 points (Round to the nearest dollar as needed.)

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