Question

The monthly payment \(p\) on a mortgage varies directly with the amount borrowed \(B\). If the monthly payment on a 15-year mortgage is \(\$ 8.99\) for every \(\$ 1000\) borrowed, find a linear equation that relates the monthly payment \(p\) to the amount borrowed \(B\) for a mortgage with these terms. Then find the monthly payment \(p\) when the amount borrowed \(B\) is $$ 175,000$.

Short answer

The monthly payment is \p = 1573.25\.

Step-by-step solution

Identify the Relationship

The problem states that the monthly payment \(p\) varies directly with the amount borrowed \(B\). This direct variation can be written as \(p = kB\), where \(k\) is a constant of proportionality.

Determine the Constant of Proportionality

Given that the monthly payment is \$8.99\ for every \$$1000\ borrowed, we can substitute these values into the equation \(p = kB\) to find \(k\). Thus, \(8.99 = k \cdot 1000\). Solving for \(k\) gives \(k = \frac{8.99}{1000} = 0.00899\).

Formulate the Linear Equation

Now that the constant \(k\) is known, the linear equation that relates the monthly payment \(p\) to the amount borrowed \(B\) is \(p = 0.00899B\).

Calculate the Monthly Payment for Given Amount

When the amount borrowed \(B\) is \$175,000\, substitute \B = 175,000\ into the linear equation to find \(p\). \(p = 0.00899 \cdot 175,000\). Calculating this gives \(p = 1573.25\).

Key concepts

proportionality constant

In this exercise, when we talk about the monthly payment of a mortgage varying directly with the amount borrowed, we're referring to a concept called direct variation. Direct variation is a mathematical relationship between two variables, where one is a constant multiple of the other. This constant multiple is what we call the proportionality constant.

In our exercise, the proportionality constant is essential because it simplifies the relationship between the monthly payment and the amount borrowed. Given the problem statement, we know that for every \(1000 borrowed, the monthly payment is \)8.99. This allows us to find the constant by dividing the payment by the amount borrowed:

• 
  k = \frac{8.99}{1000} = 0.00899

This constant, 0.00899, shows how much the monthly payment increases for each dollar borrowed. With this constant, we can establish a direct correlation between any amount borrowed and the respective monthly payment in a straightforward manner.

linear equation

Now that we have our proportionality constant, we can create a linear equation to relate the monthly payment to the amount borrowed. A linear equation is basically a mathematical equation that makes a straight line when graphed. In this scenario, the linear equation simplifies to the formula:

• p = kB

Here,
p represents the monthly payment,
B is the amount borrowed, and
k is the constant we determined earlier (0.00899).

Incorporating our constant into this formula, we get:

• p = 0.00899B

This linear equation allows us to easily compute the monthly payment for any borrowed amount. For example, if the amount borrowed, B, is \(175,000, we substitute B into our equation:

• p = 0.00899 × 175000 = 1573.25

So, the monthly payment would be \)1573.25. This straightforward linear relationship simplifies our calculations and helps predict monthly payments accurately.

mortgage calculation

When it comes to mortgages, understanding how to calculate payments is crucial. Mortgages can be complex, but breaking them down into simpler concepts can help.

A mortgage is a loan used to purchase a home, and it typically involves regular monthly payments over a set period (like 15 or 30 years). These payments are used to pay off both the principal (the amount borrowed) and the interest (the cost of borrowing the money).

In our exercise scenario, we’re dealing with a simplified version where we assume interest rates are factored into the given payment per $1000 borrowed. This is why we could use a proportionality constant and a linear equation.

For real-world mortgage calculations, you would often use more detailed formulas taking into account variable interest rates, down payments, taxes, and insurance costs. However, the concept of direct variation and linear equations remains a useful foundation. Knowing the proportionality constant, representing mortgage terms, helps estimate payments quickly.

Remember:

• Use the linear equation p=0.00899B for any Quick mortgage payment calculation in this scenario.
• Always break down complex terms to understand and calculate easier.