Martin wants to buy a new bedroom set that costs $$ \$ 1429 $$ including tax. Unfortunately, he doesn't have $$ \$ 1429 $$, so he secures a 2 -year loan from the furniture store at $$ 9 \% $$ interest to be repaid in 24 equal monthly instaliments. Find the monthly payment. Round your intermediate steps to the nearest cent. The monthly payment is $$ \$ \square $$

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UpStudy AI · Accepted Answer

To find the monthly payment for Martin's loan, we can use the formula for the monthly payment on an installment loan, which is given by:

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

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

### Step 1: Identify the known values
- $$ P = 1429 $$
- Annual interest rate = $$ 9\% $$ or $$ 0.09 $$
- Monthly interest rate $$ r = \frac{0.09}{12} $$
- Number of payments $$ n = 24 $$

### Step 2: Calculate the monthly interest rate
$$
r = \frac{0.09}{12} = 0.0075
$$

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

$$
M = 1429 \frac{0.0075(1 + 0.0075)^{24}}{(1 + 0.0075)^{24} - 1}
$$

### Step 4: Calculate $$ (1 + r)^n $$
First, we need to calculate $$ (1 + r)^{24} $$:

$$
(1 + 0.0075)^{24}
$$

### Step 5: Calculate the monthly payment $$ M $$
Now we can calculate $$ M $$ using the values we have. Let's perform the calculations.

I'll calculate $$ (1 + 0.0075)^{24} $$ and then use it to find $$ M $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\left(1+0.0075\right)^{24}$$
- step1: Add the numbers:
  $$1.0075^{24}$$
- step2: Convert the expressions:
  $$\left(\frac{403}{400}\right)^{24}$$
- step3: Use the properties of exponents:
  $$\frac{403^{24}}{400^{24}}$$
Calculate or simplify the expression $$ 1429 * (0.0075 * 1.196414) / (1.196414 - 1) $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{1429\left(0.0075\times 1.196414\right)}{\left(1.196414-1\right)}$$
- step1: Remove the parentheses:
  $$\frac{1429\times 0.0075\times 1.196414}{1.196414-1}$$
- step2: Multiply the terms:
  $$\frac{12.822567045}{1.196414-1}$$
- step3: Subtract the numbers:
  $$\frac{12.822567045}{0.196414}$$
- step4: Convert the expressions:
  $$\frac{\frac{2564513409}{200000000}}{\frac{98207}{500000}}$$
- step5: Multiply by the reciprocal:
  $$\frac{2564513409}{200000000}\times \frac{500000}{98207}$$
- step6: Reduce the numbers:
  $$\frac{2564513409}{400}\times \frac{1}{98207}$$
- step7: Multiply the fractions:
  $$\frac{2564513409}{400\times 98207}$$
- step8: Multiply:
  $$\frac{2564513409}{39282800}$$
### Step 6: Calculate the monthly payment

From the calculations, we found that:

$$
(1 + 0.0075)^{24} \approx 1.196414
$$

Now substituting this value into the monthly payment formula:

$$
M = 1429 \frac{0.0075 \times 1.196414}{1.196414 - 1}
$$

Calculating the numerator:

$$
0.0075 \times 1.196414 \approx 0.008973105
$$

Calculating the denominator:

$$
1.196414 - 1 \approx 0.196414
$$

Now substituting these values into the formula:

$$
M \approx 1429 \frac{0.008973105}{0.196414} \approx 1429 \times 0.045707
$$

Calculating $$ M $$:

$$
M \approx 65.28
$$

### Final Answer
The monthly payment is approximately $$ \$ 65.28 $$.

Thus, the final answer is:

The monthly payment is $$ \$ 65.28 $$.
The monthly payment is \$65.28.

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