Problem 40 - Elementary Algebra Within Reach

Question

The height of a rectangular picture frame is \(1.5\) times the width \(w\). Write an algebraic expression that represents the perimeter of the picture frame.

Step-by-Step Solution

Answer

The algebraic expression that represents the perimeter of the picture frame in terms of \(w\) is \(P = 5w\).

Step 1: Understanding the Problem

A rectangular picture frame's sides consist of two widths \(w\) and two heights \(h\). By identifying the width as \(w\), the problem states that the height is \(1.5w\).

Step 2: Formulating the Perimeter Expression

The perimeter of the rectangle (sum of all sides) is calculated by adding two widths and two heights: \(P = 2w + 2h\). Given that \(h = 1.5w\), the perimeter equation can be rewritten in terms of \(w\) as \(P = 2w + 2(1.5w)\).

Step 3: Simplifying the Perimeter Expression

Solving the equation above gives \(P = 2w + 3w\). Hence, the expression for the perimeter in terms of \(w\) is simplified to \(P = 5w\).

Key Concepts

Rectangular Frame Algebra

When solving algebraic problems involving geometric shapes such as rectangular frames, understanding how their dimensions relate to algebraic terms is crucial. For our example, we're looking at a picture frame where its height is directly proportional to its width. This relationship is expressed by the height being 1.5 times the width.

In algebraic terms, we assign a variable, typically 'w', to represent the width of the frame. The height, being 1.5 times this width, is then written as '1.5w'. This is a fundamental step in translating English words into the language of algebra. By expressing one dimension in terms of another, we create a formula that can be manipulated and solved algebraically. A clear understanding of this relationship is vital for correctly setting up the equation required for finding the perimeter of the frame.

Writing Algebraic Expressions

One of the key skills in algebra is the ability to write expressions that accurately represent real-world situations. When it comes to geometry, this often involves creating an equation that models the dimensions of a shape.

For the perimeter of our rectangular frame, we start by writing the expression for the perimeter as the sum of all sides: initially, this looks like \(P = 2w + 2h\). However, since we know the relationship between height and width (\(h = 1.5w\)), we can substitute 'h' with '1.5w' in our original perimeter expression. This substitution is a crucial algebraic operation known as variable replacement and simplifies the equation to a single variable. Always ensure that the variables you use are clearly defined to prevent any confusion during calculation.

The real art of writing algebraic expressions lies in this simplification process. Each term should be as concise as possible, eliminating any redundant parts while preserving the expression's integrity with regard to what it represents: the frame's perimeter.

Perimeter Calculations

The perimeter is the total length around a two-dimensional shape. To calculate the perimeter of a rectangle, you simply add up the lengths of all four sides. Since a rectangle has two pairs of equal sides, its perimeter can be calculated using the formula \(P = 2l + 2w\), where 'l' stands for length and 'w' for width.

However, when an exercise twists the problem by providing a proportional relationship between the length and the width, understanding how to adapt the formula is important. In our case, after substituting the given height-to-width ratio into the formula, we combine like terms to arrive at \(P = 5w\). This is arithmetic simplification at work, combining numeric coefficients and maintaining the correct relationship between the variables.

It's beneficial for students to remember that algebra is merely a tool for solving problems with unknown variables. When you're asked to calculate the perimeter and provided with such relationships, focus on expressing the quantities in terms of a single variable for simplicity. Grasping this concept allows you to tackle a wide array of perimeter-related algebra problems with confidence.