Bob wishes to borrow some money. He needs to defer to the following formula, where M is the monthly payment, r is the monthly decimal interest rate, P is the amount borrowed, and t is the number of months it will take to repay the loan :
`M = (rP)/(1-((1)/(1+r))^(t))`
If Bob secures a loan of `$4,00` that he will pay back in 36 months with a monthly interest rate of 0.01, what is his monthly payment ?

Options

A. `$111.11`
B. `$119.32`
C. `$132.86`
D. `$147.16`

Explanation

To find Bob's monthly payment using the formula provided, we will follow these steps:

### Step 1: Identify the given values
- Amount borrowed (P) = $4000
- Monthly interest rate (r) = 0.01
- Number of months (t) = 36

### Step 2: Substitute the values into the formula
The formula for monthly payment (M) is:
\[ M = \frac{rP}{1 - \left( \frac{1}{1 + r} \right)^{t}} \]

Substituting the values we have:
\[ M = \frac{0.01 \times 4000}{1 - \left( \frac{1}{1 + 0.01} \right)^{36}} \]

### Step 3: Calculate the numerator
Calculate \( rP \):
\[ rP = 0.01 \times 4000 = 40 \]

### Step 4: Calculate the denominator
First, calculate \( 1 + r \):
\[ 1 + r = 1 + 0.01 = 1.01 \]

Now calculate \( \left( \frac{1}{1 + r} \right)^{t} \):
\[ \frac{1}{1.01} \approx 0.990099 \]
Now raise this to the power of 36:
\[ (0.990099)^{36} \approx 0.69892 \]

Now calculate the denominator:
\[ 1 - 0.69892 = 0.30108 \]

### Step 5: Calculate M
Now substitute back into the formula:
\[ M = \frac{40}{0.30108} \approx 132.855 \]

### Step 6: Round the answer
Rounding \( 132.855 \) to two decimal places gives us:
\[ M \approx 132.86 \]

### Final Answer
Bob's monthly payment is approximately **$132.86**.

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