Rectangle Width Problem: Find The Length Fraction

Hey guys! Today, we're diving into a classic geometry problem that involves understanding the relationship between the width, length, and perimeter of a rectangle. We're going to break down the problem step-by-step, so you can easily grasp the concepts and apply them to similar problems in the future. Let's get started!

Understanding the Problem Statement

Let's start by understanding the problem. We know that the width of the rectangle is 4 cm. This is a crucial piece of information that we'll use later. The problem also states that this width represents 1/6 of the rectangle's perimeter. So, the key is understanding the relationship between the width and the perimeter. Our ultimate goal is to figure out what fraction of the length the width represents. To solve this problem, we'll need to use our knowledge of rectangles, perimeters, and fractions. Remember, the perimeter of a rectangle is the total distance around its outside, which can be calculated by adding up all the sides. Since rectangles have two lengths and two widths, the formula for the perimeter is often expressed as P = 2L + 2W, where P represents the perimeter, L represents the length, and W represents the width. We'll use this formula to help us find the missing piece of the puzzle: the fraction of the length that the width represents.

Breaking Down the Information

The initial information given is that the width of the rectangle is 4 cm. It's also stated that this width (4 cm) is equal to 1/6 of the total perimeter of the rectangle. This is a crucial piece of information because it gives us a direct link between a known value (the width) and the perimeter, which we can use to find the perimeter's actual value. Think of it like a clue in a puzzle – it helps us connect different parts of the problem. To clarify, 1/6 of the perimeter means that if you divide the perimeter into six equal parts, the width occupies one of those parts. This is a fractional relationship, which is a key concept in mathematics and essential for solving this problem. From this relationship, we can deduce that the full perimeter is six times the width. This is because if one-sixth of something is 4 cm, then the whole thing (the entire perimeter) is six times that amount. Understanding this connection is the foundation for the next steps in solving this problem. We're using the information we have to deduce something new – the total perimeter – which will be vital for finding the length.

The Goal: Finding the Fraction

The ultimate aim here is to determine what fraction of the rectangle's length is represented by its width. This isn't as straightforward as just knowing the width and the perimeter; it requires us to find the length first. Think of it like this: we know one part of the rectangle (the width), and we know something about the total distance around it (the perimeter). Now, we need to find the other part (the length) so we can compare it to the width. The phrase "what fraction of the length" is key here. It tells us that the final answer will be expressed as a fraction, with the width as the numerator (the top number) and the length as the denominator (the bottom number). For instance, if we find that the length is 12 cm, then the fraction we're looking for would be 4/12 (since the width is 4 cm). This fraction can then be simplified to its lowest terms, if necessary. However, before we can find this fraction, we need to calculate the length of the rectangle. This involves using the information we have about the width and the perimeter to work backward and isolate the length in our calculations. It's like solving a puzzle where we have to find the missing piece to complete the picture.

Calculating the Perimeter

Now that we understand the problem, let's calculate the perimeter. We know the width is 4 cm and that it is 1/6 of the perimeter. This means the full perimeter is 6 times the width. So, we can calculate the perimeter by multiplying the width (4 cm) by 6. This gives us a perimeter of 24 cm. Remember, the perimeter is the total distance around the rectangle, so 24 cm represents the sum of all four sides. Understanding this calculation is crucial because it allows us to move from a fractional relationship (1/6 of the perimeter) to a concrete value (24 cm). We've essentially used the information given to us to find a key piece of the puzzle. This value will be essential when we go to calculate the length because the perimeter is made up of both the lengths and the widths combined. By figuring out the total perimeter, we've taken a significant step towards isolating the length and eventually finding the fraction we need. It's like having the total cost of a meal and knowing the price of one item; you're then closer to figuring out the cost of the other items.

Using the Given Information

We are given that the width of the rectangle is 4 cm and that this width constitutes 1/6 of the entire perimeter. This is a direct relationship that we can use to find the full perimeter. Think of it like a proportion: if 4 cm is 1/6, then what is the whole (6/6)? To find the whole, we simply multiply the known part (4 cm) by the inverse of the fraction (which is 6/1 or simply 6). This is a common strategy in math problems involving fractions and proportions. When you know a fraction of a whole, you can always find the whole by multiplying the known part by the denominator of the fraction (in this case, 6). This principle applies not just to geometry but to various mathematical contexts. It's a fundamental tool for problem-solving. By recognizing this direct relationship and applying the appropriate mathematical operation (multiplication), we can quickly and accurately determine the perimeter of the rectangle. This highlights the importance of carefully reading the problem statement and identifying the key information and relationships that can help us solve the problem.

Multiplying to Find the Whole

To find the full perimeter, we multiply the width (4 cm) by 6. This is because the width represents one-sixth of the perimeter, and we want to find the value of the whole (6/6). The multiplication 4 cm * 6 gives us 24 cm. This result, 24 cm, is the total perimeter of the rectangle. We've now successfully calculated the perimeter using the given information and the principles of fractions. This calculation is a critical step in solving the problem because it provides us with the total distance around the rectangle. With this information, we can now move on to finding the length. Think of the perimeter as a pie that's been divided into slices. We knew the size of one slice (the width) and how many slices there were (6). By multiplying the size of one slice by the number of slices, we've figured out the size of the whole pie. This analogy helps visualize the concept of finding the whole when you know a fraction of it. Now that we know the "size of the whole pie" (the perimeter), we can use it to figure out the dimensions of the rectangle.

Determining the Length

Now that we know the perimeter is 24 cm, we can find the length. Remember, the perimeter of a rectangle is calculated as P = 2L + 2W. We know P is 24 cm and W is 4 cm. So, we can substitute these values into the formula: 24 cm = 2L + 2(4 cm). This equation allows us to isolate the length (L) and solve for it. Think of this as an algebraic equation where we're trying to find the value of an unknown variable. We're using the formula for the perimeter as a framework and plugging in the values we know. This is a common strategy in math: using formulas and known values to find unknowns. The equation now shows the relationship between the perimeter, the length, and the width in a clear mathematical form. By solving this equation, we'll be able to determine the length of the rectangle, which is another crucial piece of information we need to solve the original problem. It's like having a recipe where you know the total cooking time and the time for one step; you can then figure out the time for the remaining steps.

Using the Perimeter Formula

The formula for the perimeter of a rectangle is P = 2L + 2W, where P is the perimeter, L is the length, and W is the width. This formula is a fundamental concept in geometry and is essential for solving problems involving rectangles. It states that the perimeter is equal to twice the length plus twice the width. This makes sense because a rectangle has two sides of equal length (the lengths) and two sides of equal width (the widths). By understanding this formula, we can relate the perimeter to the dimensions of the rectangle. In this problem, we know the perimeter (P = 24 cm) and the width (W = 4 cm). We can substitute these values into the formula to create an equation that we can solve for the length (L). This substitution is a key step in applying the formula to a specific problem. It allows us to translate the geometric relationship into an algebraic equation that we can manipulate and solve. Think of the formula as a blueprint for the rectangle's perimeter. By plugging in the known values, we can use the blueprint to find the missing dimensions.

Substituting Values and Solving

We substitute the values we know into the perimeter formula: 24 cm = 2L + 2(4 cm). This simplifies to 24 cm = 2L + 8 cm. Now, we need to isolate L to find the length. To do this, we first subtract 8 cm from both sides of the equation: 24 cm - 8 cm = 2L. This gives us 16 cm = 2L. Finally, we divide both sides by 2 to solve for L: 16 cm / 2 = L. This results in L = 8 cm. Therefore, the length of the rectangle is 8 cm. We've now successfully calculated the length using the perimeter formula and algebraic manipulation. This process demonstrates how to use a formula and known values to solve for an unknown variable. It's a fundamental skill in mathematics and is applicable to various problem-solving scenarios. Think of it as unwrapping a gift: we're peeling away the layers to reveal the unknown value (the length) at the center. Each step we take brings us closer to the solution.

Finding the Fraction of Length

Now that we know the width (4 cm) and the length (8 cm), we can find the fraction of the length that the width represents. This is done by creating a fraction with the width as the numerator and the length as the denominator: Width / Length = 4 cm / 8 cm. This fraction represents the ratio of the width to the length. It tells us what portion of the length is occupied by the width. Think of it as comparing two quantities: the width is one quantity, and the length is another. The fraction shows how these quantities relate to each other. This fraction, 4/8, is the answer to the original question, but it can be simplified. Simplifying fractions makes them easier to understand and compare. It's like taking a complex idea and expressing it in its simplest form. In this case, we can simplify 4/8 to its lowest terms.

Creating the Fraction

To find the fraction of the length that the width represents, we divide the width by the length. We know the width is 4 cm and the length is 8 cm. So, we form the fraction 4 cm / 8 cm. This fraction represents the ratio of the width to the length. It tells us what proportion of the length is taken up by the width. Think of it as dividing a pie: the width is a slice of the pie, and the length represents the whole pie. The fraction shows what fraction of the whole pie the slice (width) represents. This step is crucial because it translates our geometric understanding into a mathematical expression. We're taking the dimensions of the rectangle and expressing their relationship as a fraction. This fraction is the direct answer to the question posed in the problem, but it's often best to simplify fractions to their lowest terms for clarity and ease of understanding. It's like translating a sentence from one language to another: we've found the direct translation (the fraction), but we can make it even clearer by simplifying it.

Simplifying the Fraction

The fraction 4/8 can be simplified by dividing both the numerator (4) and the denominator (8) by their greatest common divisor, which is 4. When we divide 4 by 4, we get 1. When we divide 8 by 4, we get 2. So, the simplified fraction is 1/2. This means that the width of the rectangle is 1/2 (one-half) of its length. We've now successfully found the fraction and simplified it to its lowest terms. This simplified fraction provides a clear and concise answer to the problem. It's easy to understand that the width is half the length, which gives us a good sense of the rectangle's proportions. Simplifying fractions is a fundamental skill in mathematics, and it's essential for expressing answers in their most understandable form. Think of it as polishing a gem: we're removing the rough edges to reveal the beautiful, simplified answer within. In this case, we've transformed the fraction 4/8 into the more elegant and easily understood form of 1/2.

Final Answer

So, guys, the width of the rectangle is 1/2 of its length. We successfully solved the problem by understanding the relationship between the width, perimeter, and length of a rectangle, using the perimeter formula, and simplifying fractions. Great job! This problem highlights the importance of breaking down complex problems into smaller, manageable steps. We started by understanding the given information, then calculated the perimeter, used that to find the length, and finally, expressed the relationship between the width and length as a simplified fraction. This step-by-step approach is a valuable problem-solving strategy that can be applied to various mathematical problems and real-life situations. Remember, practice makes perfect, so keep working on similar problems to sharpen your skills and build your confidence. You've got this!

Reviewing the Steps

Let's quickly review the steps we took to solve the problem. First, we understood the problem and identified the key information: the width is 4 cm, and it's 1/6 of the perimeter. Second, we calculated the perimeter by multiplying the width by 6, which gave us 24 cm. Third, we used the perimeter formula (P = 2L + 2W) and substituted the known values to find the length, which was 8 cm. Finally, we created a fraction (4/8) representing the ratio of the width to the length and simplified it to 1/2. By reviewing these steps, we can reinforce our understanding of the problem-solving process. It's like retracing your steps on a hike to make sure you understand the path. This review helps solidify the concepts in our minds and allows us to see the connections between the different steps. Each step built upon the previous one, leading us to the final answer. This structured approach is a powerful tool for tackling complex problems in mathematics and beyond.

Importance of Understanding Ratios

This problem beautifully illustrates the importance of understanding ratios and how they connect different parts of a geometric figure. We weren't just dealing with individual measurements; we were dealing with the relationship between those measurements. The fact that the width was 1/6 of the perimeter gave us a critical clue that allowed us to unlock the rest of the problem. Without understanding this ratio, it would have been much harder to find the perimeter and subsequently the length. Ratios are fundamental in mathematics and have wide-ranging applications in various fields, including geometry, algebra, and even real-world scenarios like cooking, construction, and finance. Being comfortable working with ratios allows you to make comparisons, solve proportions, and understand how different quantities relate to each other. This problem is a great example of how ratios can be used to solve practical problems, and it highlights the importance of developing a strong foundation in this key mathematical concept. It's like having a secret code that unlocks the solution; understanding ratios allows you to decipher the relationships between numbers and solve the puzzle.