Finding The Equation For Rectangle Width Given Length

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When tackling geometry problems, particularly those involving rectangles, a solid grasp of the relationship between length and width is crucial. This article delves into a specific problem where the length of a rectangle is defined in terms of its width, and our mission is to identify the equation that accurately represents this relationship. The core concept revolves around translating word problems into mathematical equations, a fundamental skill in algebra and problem-solving. Translating word problems into mathematical equations is a fundamental skill. We will dissect the given information, break down the language, and construct the equation that bridges the gap between the verbal description and the symbolic representation. This process not only sharpens our algebraic abilities but also enhances our logical reasoning and critical thinking skills.

The problem states that the length of a rectangle is 4 units shorter than half the width, and we're given that the length is 18 units. The objective is to pinpoint the equation that can be used to determine w, the width of the rectangle. Let's dissect this statement piece by piece.

Keywords here are "4 units shorter than half the width." This phrase is the crux of the problem. It tells us that we need to perform two operations on the width: first, we need to halve it (divide by 2), and then we need to subtract 4 from the result. This order of operations is critical, as changing the order would lead to a different equation and, consequently, a different solution. The given information also explicitly states that the length of the rectangle is 18 units. This is our known value, and it will form the basis of our equation. We know that "the length of the rectangle is 4 units shorter than half the width", which means w/2 - 4 is equal to the length of the rectangle, which is 18. This understanding allows us to start building our equation.

To construct the equation, let's represent the width of the rectangle with the variable w. "Half the width" can be mathematically expressed as w/2. The phrase "4 units shorter than half the width" translates to subtracting 4 from half the width, giving us w/2 - 4. The problem states that this expression is equal to the length of the rectangle, which is given as 18 units. Therefore, we can write the equation as:

18 = w/2 - 4

This equation precisely captures the relationship described in the problem statement. It sets the length (18) equal to the expression representing “4 units shorter than half the width” (w/2 - 4). This is a linear equation in one variable, and solving it will give us the value of w, the width of the rectangle. This equation is the key to solving for the width, and it's crucial to understand how it was derived from the word problem.

Now, let's examine the options provided in the problem and see which one matches our derived equation:

  • 18 = w/2 - 4
  • 18 = 4 - w/2
  • 18 - 4 = w/2
  • 18 - w/2 = 4

Comparing these options to our equation (18 = w/2 - 4), we can clearly see that the first option, 18 = w/2 - 4, is the correct one. The other options misrepresent the relationship between the length and width as described in the problem. For instance, the second option, 18 = 4 - w/2, implies that half the width is being subtracted from 4, which is the opposite of what the problem states. The third and fourth options rearrange the terms but do not accurately reflect the initial condition that the length is 4 units shorter than half the width.

It's crucial to understand why the other options are incorrect to solidify our understanding of the problem. Let's break down the errors in each incorrect option:

  • 18 = 4 - w/2: This equation suggests that we are subtracting half the width from 4 to get the length. This contradicts the problem statement, which clearly says the length is 4 units shorter than half the width. This means we should be subtracting 4 from half the width, not the other way around.
  • 18 - 4 = w/2: This equation implies that the difference between the length and 4 units is equal to half the width. While this might seem related, it doesn't accurately capture the initial relationship. This equation could be interpreted as stating that half the width is 4 units more than the length, which is not what the problem describes.
  • 18 - w/2 = 4: This equation suggests that the difference between the length and half the width is 4 units. This is also a misinterpretation of the problem. It doesn't reflect the crucial detail that the length is less than half the width by 4 units. This equation could be rearranged to w/2 = 18 - 4, which is closer to the correct equation but still misses the initial framing of the relationship.

By understanding why these options are wrong, we gain a deeper understanding of the correct equation and the importance of accurately translating word problems into mathematical expressions. Accurately translating word problems is essential for mathematical proficiency.

While the problem only asks for the equation, let's take it a step further and solve for w to reinforce our understanding. We start with the equation:

18 = w/2 - 4

To isolate w, we first add 4 to both sides of the equation:

18 + 4 = w/2 - 4 + 4

22 = w/2

Next, we multiply both sides by 2 to get w by itself:

22 * 2 = (w/2) * 2

44 = w

Therefore, the width of the rectangle is 44 units. This step demonstrates the complete problem-solving process, from identifying the correct equation to finding the solution. Solving the equation provides a concrete answer and validates our equation.

This problem highlights several key concepts in mathematical problem-solving:

  1. Translating words into equations: The ability to convert verbal descriptions into mathematical expressions is crucial. Pay close attention to keywords like