Problem 103 - College Algebra

Question

Use the five-step strategy for solving word problems. Give a linear inequality that models the verbal conditions and then solve the problem. A city commission has proposed two tax bills. The first bill requires that a homeowner pay \(\$ 1800\) plus \(3 \%\) of the assessed home value in taxes. The second bill requires taxes of \(\$ 200\) plus \(8 \%\) of the assessed home value. What price range of home assessment would make the first bill a better deal?

Step-by-Step Solution

Answer

The first bill becomes a better deal for any house assessed at more than $32,000.

Step 1: Define the problem

Solving this problem requires formulating it into two linear equations representing the two tax bills, then comparing them.

Step 2: Formulate the equations

Let's denote \(P\) as the price of the house. The first bill requires a homeowner to pay \(1800 + 0.03P\) in taxes, and the second bill requires \(\$200 + 0.08P\). We need to find the range of \(P\) for which the tax required by the first bill is less than the tax required by the second bill, thus we get the inequality \(1800 + 0.03P < 200 + 0.08P\).

Step 3: Simplify the inequality

Let's first subtract \(0.03P\) from each side of the inequality, giving us \(1800 < 200 + 0.05P\). Then subtract 200 from each side, resulting in \(1600 < 0.05P\).

Step 4: Solve the inequality for P

To find out the price range, divide each side of the inequality by 0.05. This results in \(P > 32000\).

Step 5: Interpret the solution

The solution \(P > 32000\) for the price of the house indicates that the first bill would be a better deal for any house assessed at more than $32,000. Homeowners with houses valued at less than $32,000 would find the second bill to be a more attractive option.

Key Concepts

Word Problems

Word problems are math exercises presented in a written narrative. They usually involve real-life scenarios where you're asked to find a solution using mathematical concepts. In our exercise, the word problem revolves around choosing a better tax bill between two options based on a house's assessed value.

To tackle a word problem, a systematic approach helps. Here's a handy strategy when faced with such tasks:

• Understand the problem: Break down what's being asked.
• Identify what you know and what you need to find out.
• Translate the words into mathematical equations or inequalities.
• Solve the equations or inequalities.
• Interpret the results in context of the problem.

By consistently using these steps, word problems can become less intimidating and more manageable.

Tax Equations

Tax equations are mathematical expressions used to calculate the taxes someone needs to pay, often based on an initial amount plus a percentage of some value. In this example, we have two tax bills with different structures:

1. The first tax equation is set up as a fixed amount ( $1800 $) plus a percentage (3%) of the home's assessed value ( $P $). This is expressed as $1800 + 0.03P$ .

2. The second tax equation uses a smaller fixed amount ( $200 $) but a larger percentage (8%) of the same home value, expressed as $200 + 0.08P$ .

This scenario demonstrates how the balance between fixed charges and percentage rates can influence the total outcome. Understanding these can help individuals make better fiscal decisions, given different circumstances.

Inequality Solving

Inequality solving involves finding the range of values that satisfy a condition. In the context of our problem, solving an inequality helps determine when one option is better than another. Here, we aim to find when the first tax bill costs less than the second.

The inequality we start with is $1800 + 0.03P < 200 + 0.08P$ . Solving steps include:

• Simplifying both sides by removing common terms: $1800 - 200 < 0.08P - 0.03P$ .
• This simplifies to $1600 < 0.05P$ .
• Divide each side by 0.05 to isolate $P$ : $P > 32000$ .

The result tells us that homes worth over $32,000 $ should choose the first bill for lower taxes.

Home Assessment

Home assessment is the process of determining the value of a home. This estimated value is crucial for setting property taxes, insurance, and sale prices. In our scenario, home assessment directly influences the total tax cost based on the proposed tax bill options.

Higher home values will alter which tax bill is more favorable. For the first bill to be a better deal, the home's assessed value must be greater than $32,000 $, as derived from solving the inequality in the problem. Therefore, understanding how assessed values interact with cost calculations helps homeowners evaluate their financial obligations effectively.

This highlights the importance of accurate home assessments, ensuring that homeowners are neither overtaxed nor undertaxed relative to their property's value.