Problem 50 - Algebra and Trigonometry with Analytic Geometry

Question

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Home mortgage Find the largest 25 -year home mortgage that can be obtained at an interest rate of \(7 \%\) if the monthly payment is to be \(\$ 1500\).

Step-by-Step Solution

Answer

The largest home mortgage is approximately $215,685.

Step 1: Convert Percent to Decimal

The interest rate is provided as a percentage. To use it in calculations, convert it to decimal form. Given that the interest rate is 7%, we calculate it as follows:\[ r = \frac{7}{100} = 0.07 \]

Step 2: Calculate Variable k

We will use the formula for \(k\), which depends on the interest rate \(r\), to obtain its value. Given that \(t = 25\) years, we calculate:\[ k = \left(1 + \frac{r}{12}\right)^{12t} = \left(1 + \frac{0.07}{12}\right)^{300} \]

Step 3: Substitute Known Values into the Payment Formula

We have the formula for monthly payments \(M\):\[ M = \frac{Lrk}{12(k-1)} \]We know \(M = 1500\), \(r = 0.07\), and \(k\) from the previous step. Substitute these values into the equation:\[ 1500 = \frac{L \times 0.07 \times k}{12(k-1)} \]

Step 4: Solve for L

Rearrange the formula to solve for \(L\):\[ L = \frac{1500 \times 12(k-1)}{0.07 \times k} \]Substitute the calculated value of \(k\) from Step 2 and compute the result to find \(L\).

Step 5: Calculate L with Evaluated Values

Plug the evaluated value of \(k\) into the rearranged formula from Step 4. Calculate the largest loan \(L\) that can be afforded with the given monthly payment.Compute the numeric result to find \(L\).

Key Concepts

Home Mortgage Calculation

Understanding home mortgage calculations is essential when determining how much you can borrow. These calculations help ensure that your monthly payments are affordable based on the loan's terms. A common method for calculating the mortgage involves using a specific mathematical formula. This formula takes into account the principal loan amount, the interest rate, and the loan term.

In our problem, the formula provided was used to find the largest home mortgage possible with a specified interest rate and monthly payment. The variables include:

• **L** - the loan amount.
• **r** - the annual interest rate (expressed as a decimal).
• **t** - the term of the loan in years.
• **M** - the monthly payment.

By understanding and manipulating this formula, one can efficiently assess how mortgage terms, interest rates, or payment changes can affect the overall loan structure.

Interest Rate Conversion

In many financial calculations, particularly those involving loans and mortgages, converting interest rates to decimal form is crucial. It simplifies mathematical operations and aligns them with standard formulas. This conversion ensures the rate is correctly interpreted within calculations.

To convert a percentage to a decimal format, simply divide by 100. For instance, an interest rate of 7% becomes:

• Divide 7 by 100: \( r = \frac{7}{100} = 0.07 \)

Using this form allows you to substitute the interest rate into formulas more easily and correctly evaluate expressions that are part of the mortgage calculations or other financial formulas.

Solving for Loan Amount

Determining the loan amount that fits within a budgeted monthly payment involves rearranging the standard loan payment formula. This rearrangement helps us solve for the variable **L**, representing the loan amount. Such problems typically arise when you know the maximum payment you can make and need to solve for how large of a loan this allows.

For our scenario, we start by substituting known values, including the interest rate and term, into the established formula:

• Using the formula \( M = \frac{Lrk}{12(k-1)} \), where **M** is the known monthly payment.

By rearranging this into **L = ...**, you effectively isolate the loan amount. This modification is key to making sure the calculated loan is sustainable within your financial constraints.

Monthly Payment Calculation

The computation of monthly payments involves using pre-defined formulas that account for the loan amount, interest rate, and term of the loan. This helps ensure that all variables are balanced, and payments remain consistent.

The original problem provides a formula that specifically connects these variables. By substituting numerical values for the known terms like the interest rate and loan duration, you can calculate what monthly payment will balance the loan correctly.

• In practice, you continually substitute and solve until the results align with your budget, such as ensuring the payment does not exceed your maximum affordable amount, e.g., $1500 in this problem.

Understanding these formulas enhances financial literacy, allowing you to predict future costs and effectively plan for your mortgage commitments.