Problem 31 - Introductory Algebra for College Students

Question

The bar graph shows that average rent and mortgage payments in the United States have increased since \(1975,\) even after taking inflation into account. Exercises \(31-32\) are based on the information displayed by the graph. (GRAPH CANNOT COPY) In \(2008,\) mortgage payments averaged 1514 dollars per month. For the period shown, monthly mortgage payments increased by approximately 20 dollars per year. If this trend continues, how many years after 2008 will mortgage payments average 1714 dollars ? In which year will this occur?

Step-by-Step Solution

Answer

The mortgage payment will reach $1714 in 10 years after 2008, which is the year 2018.

Step 1: Identify given and required amounts

First, identify the starting point and the target. We are given that the average monthly mortgage payment in 2008 is $1514, and we need to find out when this will rise to $1714.

Step 2: Calculate the difference between the target amount and the given amount

The next step will be to calculate the difference between the given amount (1514$) and the target amount (1714$). The difference is $1714 - $1514 = $200.

Step 3: Calculate the number of years required for the increase

Since the average monthly mortgage payment increases by $20 per year, we can divide the difference ($200) by the rate of increase ($20) to find out how many years it will take. So, $200 / $20 = 10 years.

Step 4: Find the year in which this amount will be reached

The final step is to add the number of years found in step 3 to our starting year (2008). So, 2008 + 10 = 2018.

Key Concepts

Linear Functions

Linear functions are a fundamental part of algebra and are characterized by a constant rate of change, or slope. They have equations typically expressed in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. In this specific word problem, the concept of a linear function is demonstrated through the consistent increase of mortgage payments by \(20\) dollars each year.
This constant increase represents the slope of our linear function. The setting of this problem involves starting values from the year 2008, where the mortgage payment was \(1514\) dollars, acting as our initial y-value. Each year thereafter, the payment increases by \(20\) dollars, illustrating a steady and predictable linear growth.
In practical terms:

• For every year after 2008, the mortgage payment grows by \(20\) dollars.
• The relationship between year and payment amount can described as linear, with the year number as the \(x\)-coordinate.
• The annual payment can be predicted for any given year using the linear function framework.

Understanding this helps us see why linear functions are so powerful—they enable us to model real-world trends simply and predict future outcomes based on past trends.

Word Problem Solving

Word problem solving involves translating real-world problems into mathematical expressions and equations. This process requires identifying the information given, what needs to be found, and then forming a strategy to solve the problem.
In the given example, we're asked to predict when future mortgage payments will reach a specified amount. To solve word problems like this:

• Extract key figures (start year, initial mortgage amount, rate of increase).
• Define what is unknown—here, it's the number of years until the payments reach \(1714\) dollars.
• Use logical reasoning to set up equations or expressions based on the relationship between the information given.

Breaking down each component logically and tackling them step-by-step reduces complexity and aids in comprehensive problem solving. In essence, handling word problems effectively is about organization and methodical analysis.

Algebraic Equations

Algebraic equations form the backbone of problem-solving in algebra. They represent a relationship between different variables using mathematical expressions. In this exercise, once the parameters are identified—a \(20\) dollar increase each year and a \(1514\) dollar starting point—we translate these into a mathematical formula.
The steps are

• Determine the difference needed to reach our target (\(1714 - 1514 = 200\) dollars).
• Equation setup: The formula connecting the annual increment and total difference is expressed as \(20y = 200\), where \(y\) is the number of years needed.
• Solve the equation: Here, solving \(y = 200/20\) gives \(y = 10\), indicating a \(10\)-year span until reaching the payment target.

Setting up equations to model scenarios allows us to uncover unknowns systematically. They provide a structured pathway from input to insight, highlighting relationships and dependencies clearly.