Calculating Rectangle Width: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic geometry problem: finding the width of a rectangle. We'll break down the steps, using the information given, to arrive at the solution. Let's get started! This is a common type of problem you might encounter, so understanding the process is super helpful. We're going to use the information about the perimeter and the relationship between the length and width to nail this problem. It's all about setting up the equations and solving them systematically. Keep in mind, the key here is to relate the given information (perimeter and the length-width ratio) to the formula for the perimeter of a rectangle. This will give us an equation that we can solve for the width. Understanding these concepts will help you solve a variety of related problems. Ready to boost your problem-solving skills? Let's jump in!

Understanding the Problem and Given Information

Alright, guys, let's first understand what we're dealing with. The problem tells us a rectangle has a perimeter of 108 units. Also, we're told the length of the rectangle is $\frac{5}{4}$ times its width. These two pieces of information are the keys to solving this puzzle. The perimeter is the total distance around the rectangle, and the relationship between the length and width gives us a way to relate these two dimensions to each other. The problem gives us the value of the perimeter and the relationship between length and width. The perimeter is the total length of the sides of the rectangle. The relationship between length and width is crucial. It gives us a direct link between the two dimensions. Let's represent the width with the variable 'w'. The length, being $\frac{5}{4}$ times the width, can be represented as $\frac{5}{4}w$. The perimeter is known and we need to find the width. Remember, the perimeter of a rectangle is calculated by adding up all the sides. This means adding the length and width, twice each. We have the perimeter, so we can use it to set up an equation. It's essential to clearly understand the given information. This includes the values and the relationships. The perimeter is the sum of all sides, and the length is described in terms of width.

Let's write down what we know:

• Perimeter: 108 units.
• Length: $\frac{5}{4}w$ (where 'w' is the width).

Now, let's move on to how we'll use this info.

Setting Up the Equation

Now that we've got a grip on what we're working with, let's construct our equation. The formula for the perimeter (P) of a rectangle is: P = 2 * (length + width). We know the perimeter is 108, the length is $\frac{5}{4}w$, and the width is 'w'. Substituting these into the perimeter formula, we get: 108 = 2 * ($\frac{5}{4}w$ + w). See how we've turned the words and relationships into a concrete mathematical equation? This is the heart of the problem. We're using the formula to translate the information into a form we can solve. Remember, the goal here is to isolate the variable 'w' (the width). This equation has one variable, so we can solve it. We'll simplify the equation by combining like terms and then solving for 'w'. It's like a puzzle where each step brings us closer to the solution. Make sure to correctly substitute the values into the formula. Double-check your work to avoid errors. Let's break down the equation step-by-step. It's very important to keep everything organized. We are substituting the given values into the perimeter formula.

So, we have: 108 = 2 * ($\frac{5}{4}w$ + w). Let's simplify it. First, combine the terms inside the parentheses. $\frac{5}{4}w$ + w can be written as $\frac{5}{4}w$ + $\frac{4}{4}w$, which equals $\frac{9}{4}w$. Now, our equation looks like this: 108 = 2 * ($\frac{9}{4}w$). Next, multiply 2 by $\frac{9}{4}w$, and you get: 108 = $\frac{18}{4}w$ or 108 = $\frac{9}{2}w$. We're getting closer to solving for 'w'. We're gradually simplifying the equation. It's all about following the rules of algebra. Make sure to carefully simplify the terms. Every calculation must be accurate.

Solving for the Width

Now, guys, we're ready to find the width. We have the simplified equation: 108 = $\frac{9}{2}w$. To isolate 'w', we need to get rid of the fraction $\frac{9}{2}$. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{9}{2}$, which is $\frac{2}{9}$. This is a crucial step. Remember that whatever you do to one side of the equation, you must do to the other side. This maintains the balance and ensures the equality holds true. Keep in mind that our goal is to get 'w' by itself on one side of the equation. The key here is to get rid of the fraction multiplying 'w'. Remember that when you multiply a number by its reciprocal, you get 1. This simplifies the equation. Make sure to multiply both sides by the reciprocal correctly. Double-check your math! Let's work through this step. We multiply both sides by $\frac{2}{9}$:

$\frac{2}{9}$ * 108 = $\frac{2}{9}$ * $\frac{9}{2}w$.

This simplifies to: 24 = w. So, the width 'w' of the rectangle is 24 units. We have successfully calculated the width. We've isolated the variable and found its value. Make sure to verify your answer. Always double-check your work. Let's move on to make sure everything is correct.

Verifying the Solution

Awesome! Now, let's double-check our answer to ensure everything is correct. We found that the width (w) is 24 units. The length is $\frac{5}{4}$ times the width. So, let's calculate the length: Length = $\frac{5}{4}$ * 24 = 30 units. Now, with the length and width, we can verify the perimeter using the perimeter formula: P = 2 * (length + width). Substituting our values, we get: P = 2 * (30 + 24) = 2 * 54 = 108 units. This matches the given perimeter, so our answer is correct! Verification is essential to make sure the solution is correct. Plugging the answer back into the original equation or formula is the best method. Be sure to properly substitute values and perform the calculations carefully. This step is a must. Always verify your solution, even when you're confident. Remember to always take the time to check your work. It's easy to make a mistake. It's an important step to make sure the answer is correct. Doing this shows us that our calculation is perfect. Keep up the great work!

Conclusion

So, there you have it! We've successfully calculated the width of the rectangle, step-by-step. We started with the given perimeter and the relationship between the length and width. Then, we used the perimeter formula, set up an equation, solved for the width, and finally, verified our answer. This kind of problem is a great example of how math concepts work in real-world scenarios. Hopefully, this guide has been helpful! Remember, practice makes perfect. Geometry problems are all about understanding the properties of shapes and applying formulas. Keep practicing, and you'll become a pro at solving these types of problems. This detailed explanation has helped us reach a conclusion. We started with a word problem. Then, we used our knowledge of geometry to set up and solve an equation. We also learned how to verify the answer. Remember to review the steps and concepts. Understanding is the key to mastering math. Congratulations, everyone! You've successfully solved the problem. This concludes our guide. Keep practicing and honing your math skills.