Solving For Length In Terms Of Width A Step-by-Step Guide

In the realm of geometry and spatial reasoning, understanding the relationships between different dimensions of objects is crucial. In this comprehensive guide, we will delve into a specific problem that explores the connection between the length and width of a rectangular box. This type of problem often appears in standardized tests and serves as a fundamental concept in various mathematical and real-world applications. By carefully analyzing the problem statement and applying basic algebraic principles, we can effectively determine the relationship between these dimensions and express it in a clear and concise mathematical form.

Keywords: length, width, box, relationship, equation, algebra, problem-solving, mathematical expression, standardized tests, geometry, spatial reasoning, dimensions, mathematical applications, real-world applications, problem statement, algebraic principles, mathematical form, algebraic manipulation, variables, constants, mathematical modeling, problem analysis, quantitative reasoning, critical thinking.

The core of our discussion lies in deciphering the following problem: "The length, in inches, of a box is 3 less than twice its width. Which of the following gives the length, $L$ in inches, in terms of the width w?" To effectively tackle this problem, we need to carefully dissect the given information and translate the verbal description into a mathematical equation. The key phrase here is "3 less than twice its width." This phrase reveals the relationship between the length and the width, indicating that the length is obtained by first doubling the width and then subtracting 3. This is a classic example of how mathematical problems can be presented in a descriptive format, requiring us to convert the words into mathematical symbols and operations.

Keywords: problem statement, analysis, deciphering, inches, box, 3 less than twice its width, mathematical equation, verbal description, mathematical symbols, operations, mathematical problems, descriptive format, word problems, mathematical translation, algebraic expressions, quantitative relationships, mathematical reasoning, problem interpretation, analytical skills, mathematical problem-solving, mathematical concepts, algebraic representation, mathematical precision, mathematical formulation, problem comprehension.

To derive the equation that represents the relationship between the length ($L$) and the width ($w$), we can break down the problem statement into smaller, manageable steps. Firstly, we identify that "twice its width" translates to $2w$. Secondly, "3 less than" implies a subtraction of 3 from the previous expression. Therefore, combining these two parts, we get the expression $2w - 3$. This expression directly represents the length ($L$) in terms of the width ($w$). Hence, the equation that captures the relationship is $L = 2w - 3$. This step-by-step approach helps in converting the verbal description into a symbolic representation, making the problem more accessible and solvable.

Keywords: step-by-step solution, derive, equation, represents, length (L), width (w), twice its width, 2w, 3 less than, subtraction, expression, 2w - 3, symbolic representation, mathematical expression, equation formulation, problem accessibility, problem solvability, mathematical steps, algebraic steps, mathematical transformation, verbal to symbolic, quantitative steps, problem-solving strategies, mathematical reasoning, algebraic manipulation, mathematical precision, mathematical clarity, problem simplification.

Now that we have derived the equation $L = 2w - 3$, we can compare it with the given options to identify the correct answer. The options provided are:

• A. $L = 2w + 3$
• B. $L = 2 + 3w$
• C. $L = 2w - 3$
• D. $L = 3 - 2w$

By direct comparison, it is evident that option C, $L = 2w - 3$, matches the equation we derived. Therefore, option C is the correct answer. This process of comparing the derived equation with the given options emphasizes the importance of accuracy and precision in the problem-solving process. It also highlights the need to double-check the derived equation to ensure it aligns perfectly with the problem statement and the available choices.

Keywords: identifying the correct answer, derived equation, comparing, options, direct comparison, matches, option C, $L = 2w - 3$, accuracy, precision, problem-solving process, double-check, equation alignment, problem statement, available choices, mathematical verification, algebraic verification, solution validation, answer confirmation, mathematical correctness, problem-solving confidence, quantitative comparison, equation matching, problem conclusion.

To further solidify our understanding, let's analyze why the other options are incorrect. Option A, $L = 2w + 3$, represents a scenario where the length is 3 more than twice the width, which contradicts the problem statement. Option B, $L = 2 + 3w$, implies a different relationship altogether, where the length is 2 plus three times the width, which is also inconsistent with the given information. Option D, $L = 3 - 2w$, suggests that the length is obtained by subtracting twice the width from 3, which is the reverse of the relationship described in the problem. Understanding why incorrect options are wrong is as important as knowing why the correct option is right. This critical analysis helps in developing a deeper understanding of the underlying mathematical concepts and strengthens problem-solving skills.

Keywords: incorrect options, analyzing, understanding, option A, $L = 2w + 3$, 3 more than twice the width, contradicts, problem statement, option B, $L = 2 + 3w$, different relationship, inconsistent, given information, option D, $L = 3 - 2w$, reverse relationship, problem description, critical analysis, deeper understanding, mathematical concepts, problem-solving skills, error analysis, misconception identification, algebraic reasoning, logical deduction, quantitative comparison, problem interpretation, solution justification.

The concept of relating length and width extends far beyond textbook problems. In real-world applications, this relationship is crucial in various fields. For instance, in construction, architects and engineers need to accurately calculate the dimensions of rooms, buildings, and other structures. Similarly, in manufacturing, the dimensions of boxes, containers, and packaging materials are critical for efficient storage and transportation. In graphic design, understanding the relationship between the dimensions of a canvas or a layout is essential for creating visually appealing designs. By grasping the fundamental principles of relating length and width, we can apply this knowledge to solve practical problems in diverse areas. Moreover, this problem can be extended to more complex scenarios involving three-dimensional objects, different geometric shapes, and varying constraints, providing a rich platform for mathematical exploration and application.

Keywords: real-world applications, extensions, textbook problems, construction, architects, engineers, dimensions, rooms, buildings, structures, manufacturing, boxes, containers, packaging materials, storage, transportation, graphic design, canvas, layout, visually appealing designs, fundamental principles, practical problems, diverse areas, three-dimensional objects, geometric shapes, constraints, mathematical exploration, mathematical application, dimensional analysis, geometric calculations, problem generalization, mathematical modeling, real-world relevance, mathematical thinking, quantitative reasoning.

In conclusion, this problem exemplifies the importance of translating verbal descriptions into mathematical equations and applying algebraic principles to solve real-world problems. By carefully analyzing the problem statement, breaking it down into smaller steps, and identifying the correct equation, we can effectively determine the relationship between the length and width of a box. The ability to solve such problems is crucial for success in mathematics and various other fields that rely on quantitative reasoning and problem-solving skills. Furthermore, understanding why incorrect options are wrong and exploring the real-world applications of these concepts enhances our comprehension and analytical abilities. This comprehensive guide aims to provide a solid foundation for tackling similar problems and fostering a deeper appreciation for the power and versatility of mathematical thinking.

Keywords: conclusion, exemplifies, translating, verbal descriptions, mathematical equations, algebraic principles, real-world problems, problem statement analysis, smaller steps, correct equation, determining, relationship, length, width, quantitative reasoning, problem-solving skills, incorrect options analysis, real-world applications exploration, comprehension, analytical abilities, comprehensive guide, solid foundation, similar problems, mathematical thinking, mathematical versatility, mathematical power, mathematical problem-solving, algebraic understanding, quantitative analysis, mathematical reasoning, problem-solving strategies.