Problem 41 - Prealgebra

Question

A rectangle has width \(w .\) Its length is one less than twice its width. Write an expression in simplest form for its perimeter.

Step-by-Step Solution

Answer

The perimeter is \(6w - 2\).

Step 1: Understand the Problem

The problem involves finding the perimeter of a rectangle. It states that the width is \(w\) and that the length is one less than twice the width.

Step 2: Express the Length

The length of the rectangle is described as one less than twice the width. This translates to the expression for the length being \(2w - 1\).

Step 3: Recall the Perimeter Formula

The formula for the perimeter \(P\) of a rectangle is given by \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width.

Step 4: Substitute Length and Width

Substitute the expressions for length and width into the perimeter formula: \(P = 2((2w - 1) + w)\).

Step 5: Simplify the Expression

First calculate the expression inside the parentheses: \((2w - 1) + w = 3w - 1\). Then, find the perimeter: \(P = 2(3w - 1) = 6w - 2\).

Key Concepts

Expressions in Prealgebra

In prealgebra, an expression is a mathematical phrase made up of numbers, variables, and operation symbols that represents a specific value or calculation. Expressions do not contain an equality sign, unlike equations.

• A simple example is the expression \(2w\), which consists of the coefficient \(2\), the variable \(w\), and multiplication.
• Expressions can describe relationships, like in our exercise where the length of the rectangle is expressed as \(2w - 1\). This expression shows how the length depends on the width \(w\).

When dealing with expressions, you work through simplification and substitution. For example, simplifying \((2w - 1) + w\) involves combining like terms to get \(3w - 1\). Expressions form the basis for creating and solving equations by setting them equal to something and adding an equality.

Understanding Equations

Equations in prealgebra are statements that assert the equality of two expressions, using an equality sign \(=\). It forms the backbone of solving various mathematical problems, by balancing both sides to find the value of unknown variables.

• An example of an equation is the perimeter equation \(P = 2(l + w)\). This formula tells us that the product of twice the sum of length and width equals the perimeter.
• Equations require the substitution of known values to find the unknown. In the exercise, after substituting \(l = 2w - 1\) into the equation, we were able to simplify it to find \(P = 6w - 2\).

Solving equations often requires rearranging terms and simplifying until the variable of interest is isolated. Understanding the role of equations helps in applying mathematical operations systematically to solve problems.

Rectangle Perimeter

The perimeter of a rectangle is the total distance around the edge of the rectangle. For any rectangle, the perimeter \(P\) can be calculated using the formula: \(P = 2(l + w)\) where \(l\) stands for length and \(w\) stands for width.

• This formula is derived by recognizing that a rectangle has two pairs of equal sides: two lengths and two widths. Thus, adding them together to form \(l + w\) and then multiplying by two accounts for every side.
• In our problem, we substituted the expression for the length \((2w - 1)\) and the width \(w\) into the formula to find a perimeter expression in terms of \(w\), resulting in \(P = 6w - 2\).

Understanding how to calculate the perimeter is essential in geometric problems, spatial reasoning, and real-world scenarios like fencing an area or framing a picture. This example illustrates how expressions and equations work together to solve practical problems.