Problem 57 - Prealgebra

Question

A farmer is replacing several turbines on his windmills. He plans to replace \(x\) turbines, and he is going to get \(\$ 300\) off each turbine he buys. Also, he'll get a \(\$ 250\) rebate on his entire purchase. Write an expression that describes this situation and then simplify.

Step-by-Step Solution

Answer

The expression is \( 300x + 250 \).

Step 1: Define the Expression

Start by creating an expression for the total discount the farmer receives. The farmer receives a \(300 discount for each of the \( x \) turbines he purchases. Therefore, the expression for the discount from the turbines is \( 300x \). Additionally, he receives a one-time rebate of \)250 on his entire purchase.

Step 2: Combine Discounts into a Single Expression

To find the total discount the farmer receives, combine the individual discounts: the expression becomes \( 300x + 250 \). This expression accounts for both the consistent discount for each turbine and the one-time rebate.

Step 3: Simplify the Expression

This expression \( 300x + 250 \) is already in its simplest form as it combines all the given discounts without any like terms to combine further.

Key Concepts

Variables in Mathematics

In mathematics, variables are symbols used to represent unknown or changeable values in equations or expressions. They provide a way to generalize problems and formulas so that they can be applied to various situations, allowing for greater flexibility and understanding.
Variables are often denoted by letters such as \(x, y,\) and \(z\), but any symbol can technically serve as a variable.

• Purpose: They allow us to express relationships and formulate equations.
• Representation: Commonly letters of the alphabet are used to create expressions or equations.

In our example of the farmer buying turbines, \(x\) represents the number of turbines he's planning to purchase. This helps us calculate total cost, discounts, or other factors related to his purchase, all in one neat expression.

Mathematical Operations

Mathematical operations are the procedures or processes used to combine or change numbers and variables in expressions. The fundamental operations include addition, subtraction, multiplication, and division. Each has specific rules and properties that govern how they affect numbers and variables.

• Addition: Combining numbers or variables. For instance, adding the rebate \(250\) to the discount \(300x\).
• Subtraction: Taking away from a number or variable value.
• Multiplication: Repeated addition or combining. In our expression, \(300x\), it shows the $300 discount for each \(x\) turbine.
• Division: Distributing a value equally or finding how many times something fits into another.

The farmer's total discount situation employs these operations to combine the different parts of his discount into one cohesive expression.

Simplifying Expressions

Simplifying expressions is the process of combining like terms and organizing an expression into its simplest form. This makes them easier to handle and understand, especially when performing further calculations or solving equations.
In the case of the farmer's discount, the expression \(300x + 250\) is already simplified. Here’s why:

• Like Terms: These are terms in an expression that contain the same variables raised to the same power. In this case, there aren’t any like terms with \(300x\) and \(250\) to combine further.
• No Further Simplification: Since there are no like terms, each part of the expression is fully consolidated, reflecting both the per-turbine discount and the total rebate neatly together.

Simplifying helps in understanding the net effect of all discounts and rebates applied together, making it straightforward for later calculations or comparisons.