Finding Rectangle Width: A Step-by-Step Guide

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Hey there, math enthusiasts! Ever stumbled upon a geometry problem and thought, "Whoa, where do I even begin?" Well, fear not! Let's tackle a classic: finding the width of a rectangle when we know its perimeter and length. It's like a puzzle, and we're the detectives! In this article, we'll break down the steps, explore the formulas, and make sure you understand this concept inside and out. Whether you're a student, a curious mind, or just someone who enjoys a good mental workout, this is for you. Let's get started and unravel the secrets of rectangle dimensions!

Understanding the Basics: What's a Rectangle, Anyway?

Before we dive into the calculations, let's quickly refresh our memory on what a rectangle actually is. A rectangle is a four-sided shape (a quadrilateral, to be precise) with a few special features. First, it has four right angles, meaning all its corners are 90 degrees. Second, its opposite sides are equal in length. We usually call the longer side the length (often denoted as 'l') and the shorter side the width (often denoted as 'w'). Think of a door, a table, or even your phone – they're all likely rectangles! Knowing these basics is crucial for understanding perimeter and how it relates to the length and width. Grasping these concepts lays the groundwork for solving our problem. Understanding the properties of a rectangle – its equal and parallel sides, and its right angles – allows us to apply the correct formulas and arrive at the correct solution. Remember, geometry is all about visualization and understanding the relationships between different parts of a shape. So, take a moment to picture a rectangle in your mind, and let's move on!

Decoding the Perimeter: The Magic Number

Alright, let's talk about the perimeter. The perimeter is simply the total distance around the outside of a shape. For a rectangle, you can find the perimeter by adding up the lengths of all four sides. But, since opposite sides are equal, we can use a handy formula: Perimeter = 2 * (length + width), or P = 2(l + w). In our problem, we're given the perimeter (P) and the length (l), and our mission is to find the width (w). So, we're essentially working backward, using the formula to solve for the unknown variable. It's like a treasure hunt where the perimeter is a map, and we need to find the width, which is the hidden treasure! This formula is the key to unlocking our solution, and understanding it is crucial. Remember, understanding perimeter helps you grasp the relationship between the sides of a shape and the space it encloses. A good understanding of perimeter and the formula allows us to tackle complex problems with greater confidence.

The Problem at Hand: Let's Get Specific!

Now, let's apply what we've learned to our specific problem. We're told that the perimeter of a rectangle is 160 mm, and the length is 50 mm. Our goal is to find the width. Here's how we can approach this problem, step by step:

  1. Write down what we know:
    • Perimeter (P) = 160 mm
    • Length (l) = 50 mm
    • Width (w) = ? (This is what we need to find!)
  2. Use the perimeter formula: P = 2(l + w)
  3. Substitute the known values into the formula: 160 mm = 2(50 mm + w)
  4. Simplify: Let's deal with this equation and isolate the width to solve it! The next step is dividing both sides by 2. This leaves us with 80 mm = 50 mm + w.
  5. Isolate the width: Subtract 50 mm from both sides: 80 mm - 50 mm = w
  6. Solve for the width: w = 30 mm.

And there you have it! The width of the rectangle is 30 mm. See? It's not rocket science, right? Just a few simple steps and a clear understanding of the formulas. You've successfully solved the problem by applying the basic formulas and understanding the relationship between the sides and the perimeter. It’s just like solving a puzzle by figuring out the missing piece. So, in our specific case, we substitute our known values into the formula and follow the steps to arrive at the final answer!

Step-by-Step Solution: The Detailed Breakdown

Let's break down the solution into manageable steps to ensure everyone follows along. Here's a more detailed explanation of how we arrived at the answer:

  1. Start with the Formula: The perimeter formula, P = 2(l + w), is our starting point. This formula describes the relationship between the perimeter, length, and width of a rectangle. It's the fundamental equation we'll use.
  2. Substitute Known Values: We know P = 160 mm and l = 50 mm. Substituting these values into the formula, we get 160 mm = 2(50 mm + w). This step replaces the general terms (P and l) with the specific values from our problem. Now our equation is tailored to our specific rectangle.
  3. Simplify the Equation: To isolate 'w', we need to simplify the equation. Divide both sides by 2, yielding 80 mm = 50 mm + w. This simplifies the equation, getting us closer to solving for the width. Remember, the goal is always to isolate the unknown variable on one side of the equation.
  4. Isolate the Width: Next, subtract 50 mm from both sides of the equation: 80 mm - 50 mm = w. This isolates 'w' on the right side of the equation, making it easier to find the solution.
  5. Calculate the Width: Finally, subtract 50 mm from 80 mm, to get the width of the rectangle. w = 30 mm. This provides the final answer to our problem. That's the width of the rectangle! Now, you know what the width is, all thanks to the power of formulas and step-by-step problem-solving.

Tips and Tricks: Making it Easier

Here are some helpful tips and tricks to make solving these kinds of problems a breeze:

  • Draw a Diagram: Always draw a simple diagram of the rectangle. Label the length, width, and perimeter. This visual aid helps you understand the problem better and avoids mistakes.
  • Double-Check Units: Make sure all measurements are in the same units (e.g., mm, cm, inches). If not, convert them before you start calculating. This prevents errors.
  • Practice Regularly: The more you practice, the more comfortable you'll become with the formulas and the problem-solving process. Try different examples with varying perimeters and lengths.
  • Break it Down: If you're having trouble, break the problem into smaller, more manageable steps. This makes the overall task less daunting.
  • Use a Calculator: While it's good to practice mental math, don't be afraid to use a calculator to double-check your answers and avoid silly mistakes.

Additional Examples: More Practice Makes Perfect

Let's solidify our understanding with a couple more examples. This is where you get to shine! Remember, the more you practice, the more confident you'll become.

Example 1: A rectangle has a perimeter of 100 cm and a length of 30 cm. Find the width.

  1. P = 100 cm, l = 30 cm, w = ?
  2. 100 cm = 2(30 cm + w)
  3. 50 cm = 30 cm + w
  4. w = 20 cm

Example 2: A rectangle has a perimeter of 200 mm and a length of 60 mm. What is the width?

  1. P = 200 mm, l = 60 mm, w = ?
  2. 200 mm = 2(60 mm + w)
  3. 100 mm = 60 mm + w
  4. w = 40 mm

Try these examples yourself first, and then check your answers. Doing practice problems can dramatically enhance your understanding of the concept, which will ensure you grasp all important topics. It helps you apply the formulas, and it lets you build confidence. Make sure you understand all the steps and feel comfortable with the process. It allows you to gain a solid grasp of this type of problem, enhancing your ability to solve similar mathematical challenges in the future! Practice and consistent effort are key to mastering any concept.

Real-World Applications: Where Does This Matter?

You might be wondering, "Why do I need to know this?" Well, knowing how to find the width of a rectangle has many real-world applications. From everyday tasks to more complex scenarios, this skill comes in handy. Here's why it's valuable:

  • Home Improvement: Planning to build a fence around your yard? You'll need to calculate the perimeter and understand the relationship between the length and width to determine the materials needed.
  • Interior Design: Designing a room? Knowing the dimensions of the walls and furniture is crucial for creating a functional and aesthetically pleasing space. This helps in arranging the furniture, and in working out the area of the room.
  • Construction: Architects and builders use these concepts daily to design and construct buildings. Precision in measuring and calculating dimensions is critical.
  • Gardening: Planning a garden? You'll need to know the perimeter to figure out how much fencing or edging you need.
  • General Problem-Solving: This type of problem hones your critical thinking and problem-solving skills, which are valuable in all aspects of life.

Conclusion: You've Got This!

Congratulations, guys! You've successfully navigated the process of finding the width of a rectangle. You've learned the key formulas, practiced with examples, and seen how this knowledge applies in the real world. Remember to always break down the problem into manageable steps, draw diagrams, and practice regularly. Now, you have the tools and understanding to tackle similar problems with confidence. Keep exploring, keep learning, and keep those math muscles strong! You've got this!