Problem 34 - Prealgebra

Question

The largest ranch in the world is in the Australian Outback. It is about \(12,000\) square miles, which is five times the size of the largest United States ranch. Write and solve an equation to find the size of the largest United States ranch.

Step-by-Step Solution

Answer

The largest U.S. ranch is 2,400 square miles.

Step 1: Understanding the Given Information

The problem states that the largest ranch in the world is 12,000 square miles and this is five times the size of the largest United States ranch. We need to find the size of the largest U.S. ranch.

Step 2: Setting Up the Equation

Let the size of the largest U.S. ranch be denoted by the variable \( x \). According to the problem, the largest ranch in the world, which is 12,000 square miles, is equal to five times the size of the U.S. ranch. Thus, the equation is \( 5x = 12000 \).

Step 3: Solving the Equation

To find \( x \), divide both sides of the equation \( 5x = 12000 \) by 5. This gives \( x = \frac{12000}{5} \).

Step 4: Calculating the Solution

Perform the division \( 12000 \div 5 \). The result is \( x = 2400 \). Thus, the size of the largest United States ranch is 2,400 square miles.

Key Concepts

Problem Solving

Problem-solving is a critical skill in tackling algebraic equations. It involves identifying the information given, what is being asked, and how to logically piece together these elements. In our exercise, we started by examining the details: knowing the size of the largest ranch and figuring out how it relates to another unknown quantity.

The key steps in this problem-solving process include:

• Understanding what is known: the largest ranch spans 12,000 square miles.
• Identifying what needs to be found: the size of the U.S. ranch.
• Understanding how these relate: one is five times the other.

Breaking the problem into smaller, manageable parts helps us effectively tackle the main question.

Variable Representation

In mathematics, variables act as placeholders that represent unknown values. They allow us to translate real-world problems into algebraic expressions or equations. In our problem, the largest U.S. ranch size is unknown, so we assign it a variable, "\(x\)".

Why is this useful?

• It simplifies complex information into easier, manageable symbols.
• It aids in setting up equations that can be solved systematically.

This variable representation helped us relate the size of the U.S. ranch to the given information about the world's largest ranch.

Equation Solving

Equation solving involves determining the value of a variable that makes an equation true. Once we convert a situation into an algebraic equation using variable representation, the next step is to solve it. For our exercise, the equation turns out to be \(5x = 12000\).

Solving equations is often a step-by-step process:

• Setting up the right equation, using known quantities.
• Performing operations to isolate the variable on one side.
• Verifying the solution to ensure accuracy.

Each of these steps is crucial to correctly finding the unknown variable's value.

Division

Division is a fundamental mathematical operation needed to solve equations like the one in our exercise. When we have an equation like \(5x = 12000\), we need to use division to isolate the variable \(x\).

Here’s how it works:

• Divide both sides of the equation by the coefficient of \(x\), which is 5.
• Perform the division: \(x = \frac{12000}{5}\).
• Solve for \(x\): the result gives the size of the U.S. ranch as \(2,400\) square miles.

Performing this division accurately is key, as it directly leads to the solution of the problem.