Suppose your gross monthly income is $5,100 and your current monthly payments are $625. If the bank will allow you to pay up to 36% of gross monthly income (less current monthly payments) for a monthly house payment, what is the maximum loan you can obtain if the rate for a 30-year mortgage is 4.65%? (Round your answer to the nearest cent.)
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To determine the maximum loan you can obtain based on the given information, we need to follow these steps:

1. Calculate the maximum monthly house payment allowed by the bank.
2. Use the maximum monthly house payment to find the maximum loan amount with the given mortgage rate and term.

### Step 1: Calculate the Maximum Monthly House Payment Allowed by the Bank

The bank allows you to pay up to 36% of your gross monthly income for housing, less your current monthly payments. 

Gross monthly income: $ \$5,100 $

Current monthly payments: $ \$625 $

Maximum percentage allowed for housing: $ 36\% $

First, calculate 36% of your gross monthly income:
\[ 0.36 \times 5100 = 1836 \]

Next, subtract your current monthly payments:
\[ 1836 - 625 = 1211 \]

So, the maximum monthly house payment allowed by the bank is $ \$1,211 $.

### Step 2: Calculate the Maximum Loan Amount

To find the maximum loan amount, we need to use the maximum monthly payment, the mortgage rate, and the term of the mortgage.

- Monthly house payment: $ \$1,211 $
- Annual interest rate: $ 4.65\% $ (monthly rate = $ \frac{4.65\%}{12} = 0.3875\% $)
- Term: 30 years (360 months)

The formula for the monthly mortgage payment is:
\[ M = \frac{P \cdot r \cdot (1+r)^n}{(1+r)^n - 1} \]

Where:
- $ M $ is the monthly payment (\$1,211).
- $ P $ is the loan principal (the amount we need to find).
- $ r $ is the monthly interest rate (0.00465 / 12 = 0.003875).
- $ n $ is the number of payments (360).

Rearranging the formula to solve for $ P $:
\[ P = \frac{M \cdot \left( (1 + r)^n - 1 \right)}{r \cdot (1 + r)^n} \]

Now, plug in the values:
\[ r = 0.003875 \]
\[ n = 360 \]
\[ M = 1211 \]

\[ P = \frac{1211 \cdot \left( (1 + 0.003875)^{360} - 1 \right)}{0.003875 \cdot (1 + 0.003875)^{360}} \]

### Calculation

Let's compute the values step-by-step:

1. Calculate $ (1 + r)^n $:
\[ (1 + 0.003875)^{360} \approx 3.89867 \]

2. Calculate $ (1 + r)^n - 1 $:
\[ 3.89867 - 1 = 2.89867 \]

3. Calculate $ r \cdot (1 + r)^n $:
\[ 0.003875 \cdot 3.89867 \approx 0.01511 \]

4. Finally, calculate $ P $:
\[ P = \frac{1211 \cdot 2.89867}{0.01511} \approx \frac{3511.30237}{0.01511} \approx 232332.59 \]

Thus, the maximum loan amount you can obtain is approximately $ \$232,332.59 $.

### Summary

The maximum loan you can obtain, given a monthly income of \$5,100, current monthly payments of \$625, a bank's allowance of up to 36% of gross monthly income for housing, and a 30-year mortgage rate of 4.65%, is approximately **\$232,332.59**.

Would you like further details or have any questions?

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Here are some relative questions to expand your understanding:

1. How does changing the mortgage rate affect the maximum loan amount?
2. What is the impact of increasing the term of the mortgage from 30 years to 40 years?
3. How would additional monthly payments, such as for insurance and taxes, affect the maximum loan amount?
4. What if your gross monthly income increased to \$6,000? How would this affect your maximum loan amount?
5. What is the difference in the maximum loan amount if the interest rate were 5% instead of 4.65%?
6. How do early repayments affect the total interest paid over the mortgage term?
7. What would be the maximum loan amount if the bank allowed up to 40% of gross monthly income for housing?
8. How does your credit score impact the interest rate offered by the bank?

**Tip:** Always ensure that you consider additional costs such as property taxes, insurance, and maintenance when calculating how much house you can afford.

Learn how to calculate the maximum mortgage loan you can obtain with a gross monthly income of $5,100, current monthly payments of $625, and a bank's allowance of up to 36% of gross monthly income for housing. Discover the detailed step-by-step solution with a 30-year mortgage rate of 4.65%.

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