Problem 54 - Algebra for College Students

Question

Set up an algebraic equation and solve each problem. The total value of a house and a lot is \(\$ 168,000\). If the ratio of the value of the house to the value of the lot is 7 to 1 , find the value of the house.

Step-by-Step Solution

Answer

The value of the house is $147,000.

Step 1: Understand the Problem

We know the total value of the house and the lot is $168,000, and the ratio of the value of the house to the lot is 7 to 1. We need to find the value of the house.

Step 2: Define Variables

Let the value of the lot be \(x\). Then the value of the house is \(7x\) because the ratio of the house to the lot is 7:1.

Step 3: Set Up the Equation

The total value is the sum of the house and the lot, which can be expressed as: \(7x + x = 168,000\).

Step 4: Simplify the Equation

Combine like terms in the equation: \(8x = 168,000\).

Step 5: Solve for x

Divide both sides of the equation by 8 to solve for \(x\): \(x = \frac{168,000}{8}\). After calculating, \(x = 21,000\).

Step 6: Find the Value of the House

Since the value of the house is \(7x\), substitute \(x = 21,000\) into this expression: \(7 \times 21,000 = 147,000\).

Key Concepts

Ratios in Algebra

Ratios are a way of comparing two quantities, often denoted as "a to b" or "a:b". In algebra, ratios help us set up relationships between variables. Let's explore how they're used, specifically in terms of finding unknown values. When given a ratio like 7:1, it means for every 7 units of one part, we have 1 unit of the other. In the given problem, this ratio helps us express the relationship between the house value and the lot value.

Here's how it works:

• Identify the individual parts of the ratio.
• Express these parts as variables or multiples of a variable.

In the problem, if we represent the lot value as the variable \( x \), then the house value, according to the 7:1 ratio, is \( 7x \). This expression is crucial for setting up equations, allowing us to solve for unknowns.

Solving Algebraic Equations

Solving algebraic equations involves finding the unknown value that makes the equation true. It's a structured process that requires setting up an equation based on known data and simplifying it to find the solution. Let's break down the approach used in the problem:

• Start by expressing the total relationship or statement you're trying to solve (total value of the house and lot is \( 168,000 \)).
• Create an equation that represents this relationship. In the case study: \( 7x + x = 168,000 \)
• This equation combines the values of the house and lot as derived from the ratio.
• Simplify the equation by combining like terms where \( 7x + x \) becomes \( 8x \).

Finally, solve for \( x \) using standard operations like division. In this step, we determined \( x = 21,000 \). With this value of \( x \), further calculations provided us with the solution for the house's value using \( 7x \).

The structured method ensures clarity and confidence in solving complex algebraic problems.

Variable Definition in Algebra

In algebra, a variable is a symbol—often a letter like \( x \)—that represents an unknown number. Defining variables clearly is a vital first step in problem-solving. It involves understanding what each variable represents in the context of the problem. Here's how to apply it effectively:

• Identify what you need to find—the unknowns in the equation.
• Assign a variable to represent those unknowns. In our scenario, we assign \( x \) to the value of the lot.
• Express other quantities in terms of \( x \) using known relationships or ratios (like the house's value as \( 7x \) for a 7:1 ratio).

This strategic assignment of variables is powerful in breaking down and solving problems. With clear definitions, the algebraic structure of the equation becomes more manageable, ensuring a consistent approach across various problems.

Therefore, defining variables not only helps unlock solutions but also enhances comprehension of the overall mathematical scenario in a problem.