Rectangle Sides: Perimeter 98cm, Ratio 3:4

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Finding Rectangle Side Lengths with Perimeter and Proportional Sides

Hey guys! Let's dive into a classic geometry problem: finding the side lengths of a rectangle when we know its perimeter and the proportion of its sides. This is a common problem in math, and we're going to break it down step by step so it's super easy to understand. So, if you've ever wondered how to tackle such a question, you're in the right place!

Understanding the Problem

Okay, so here's the deal. We've got a rectangle, and we know two important things about it:

  • Its perimeter is 98 cm. Remember, the perimeter is the total distance around the outside of the rectangle.
  • The lengths of its sides are proportional to 3 and 4. This means that if we call the shorter side 3x, the longer side will be 4x, where 'x' is a common multiplier. This proportionality is key to solving the problem.

Our mission, should we choose to accept it (and we do!), is to figure out the actual lengths of the sides. Seems doable, right? Let's jump in!

Setting Up the Equations

The first step in solving this problem is to translate the given information into mathematical equations. This might sound intimidating, but trust me, it's not as scary as it seems! We'll use the properties of rectangles and the concept of perimeter to build our equations.

  • Perimeter: We know the perimeter (P) of a rectangle is given by the formula: P = 2(length + width). In our case, P = 98 cm. We also know that the sides are proportional to 3 and 4, so we can represent the sides as 3x and 4x. Therefore, our equation for the perimeter becomes: 98 = 2(3x + 4x).
  • Proportionality: The fact that the sides are proportional to 3 and 4 is already built into our side representations (3x and 4x). This is the beauty of using the proportional relationship – it simplifies the problem. The sides of the rectangle are expressed in terms of a single variable, 'x', which is the scaling factor that maintains the correct proportion. By using this scaling factor, we ensure that whatever value we find for 'x', the ratio of the sides will always be 3:4. It's a clever way to capture the given information mathematically and set the stage for solving for the unknown side lengths. So, remember, when you see a problem involving proportional sides, think about introducing a common variable to represent those proportions. This approach can significantly streamline your calculations and lead you to the solution more efficiently.

Solving for 'x'

Now that we have our equation, 98 = 2(3x + 4x), it's time to roll up our sleeves and solve for 'x'. This is where our algebra skills come into play! Don't worry, it's just a matter of simplifying and isolating the variable. Solving for 'x' is a crucial step because once we know the value of 'x', we can easily determine the actual lengths of the rectangle's sides.

Let's break it down:

  1. Simplify the equation:
    • First, we can simplify the expression inside the parentheses: 3x + 4x = 7x
    • So, our equation becomes: 98 = 2(7x)
    • Next, multiply 2 by 7x: 98 = 14x
  2. Isolate 'x':
    • To get 'x' by itself, we need to divide both sides of the equation by 14:
      • 98 / 14 = 14x / 14
      • This simplifies to: 7 = x

So, we've found that x = 7! This means our common multiplier is 7. Give yourself a pat on the back – you've conquered a significant hurdle. But remember, we're not quite done yet. We still need to use this value to find the actual side lengths of the rectangle.

Calculating the Side Lengths

Alright, we've successfully found the value of 'x', which is 7. Now comes the exciting part: using this value to calculate the actual lengths of the rectangle's sides. Remember, we defined the shorter side as 3x and the longer side as 4x. So, all we need to do is substitute x = 7 into these expressions.

  1. Shorter Side:
    • The shorter side is 3x, so we substitute x = 7:
      • 3 * 7 = 21 cm
    • Therefore, the length of the shorter side is 21 centimeters. See how straightforward it is once you know the value of 'x'?
  2. Longer Side:
    • The longer side is 4x, so we substitute x = 7:
      • 4 * 7 = 28 cm
    • Therefore, the length of the longer side is 28 centimeters.

And there you have it! We've calculated the lengths of both sides of the rectangle. The shorter side is 21 cm, and the longer side is 28 cm. It's always a good idea to take a moment to appreciate what we've achieved. We started with a problem that might have seemed a bit abstract, but by breaking it down into smaller steps, setting up equations, and solving for the unknown, we arrived at a clear and concrete answer. This is the power of mathematical problem-solving – it allows us to unravel complex situations and find solutions using logical reasoning and established principles.

Verifying the Solution

Before we declare victory and move on, it's always a smart practice to verify our solution. This is like the quality control step in our problem-solving process, ensuring that our answer makes sense and satisfies the original conditions of the problem. Think of it as double-checking your work to avoid any sneaky errors.

So, how do we verify our solution in this case? We have two key pieces of information that we can use:

  1. Perimeter: We know the perimeter of the rectangle should be 98 cm. Let's calculate the perimeter using the side lengths we found (21 cm and 28 cm) and see if it matches.
    • Perimeter = 2(length + width)
    • Perimeter = 2(21 cm + 28 cm)
    • Perimeter = 2(49 cm)
    • Perimeter = 98 cm
    • Great! Our calculated perimeter matches the given perimeter. This gives us confidence that our side lengths are correct.
  2. Proportionality: We also know that the sides should be proportional to 3 and 4. Let's check if the ratio of our calculated side lengths is indeed 3:4.
    • Ratio of sides = Shorter side / Longer side
    • Ratio of sides = 21 cm / 28 cm
    • We can simplify this fraction by dividing both numerator and denominator by their greatest common divisor, which is 7:
    • Ratio of sides = (21 / 7) / (28 / 7)
    • Ratio of sides = 3 / 4
    • Excellent! The ratio of the sides is 3:4, as required. This further validates our solution.

By verifying our solution, we've not only confirmed that our answer is correct, but we've also reinforced our understanding of the problem and the concepts involved. This step is especially important in exams or assessments, where accuracy is crucial. So, always remember to take a few moments to verify your solutions – it's a valuable habit that can save you from making careless mistakes.

Key Takeaways

Let's recap the key steps we took to solve this problem. This will help solidify our understanding and make it easier to tackle similar problems in the future.

  1. Understand the Problem: Read the problem carefully and identify what information is given (perimeter, proportional sides) and what needs to be found (side lengths).
  2. Set Up Equations: Translate the given information into mathematical equations. Remember the formula for the perimeter of a rectangle and how to represent proportional sides using a common variable.
  3. Solve for the Unknown: Use algebraic techniques to solve for the unknown variable (in this case, 'x').
  4. Calculate the Solution: Substitute the value of the unknown variable back into the expressions for the side lengths to find their actual values.
  5. Verify the Solution: Check if your solution satisfies the original conditions of the problem (perimeter and proportionality). This crucial step helps ensure accuracy.

Tips and Tricks for Similar Problems

Now that we've mastered this problem, let's talk about some general tips and tricks that can help you solve similar problems more efficiently. These strategies are like secret weapons in your problem-solving arsenal, giving you an edge when you encounter challenging questions.

  • Visualize the Problem: Drawing a diagram can often help you visualize the problem and understand the relationships between different quantities. In this case, sketching a rectangle and labeling its sides can make the problem more concrete.
  • Break Down Complex Problems: If a problem seems overwhelming, try breaking it down into smaller, more manageable steps. This makes the problem less daunting and allows you to focus on each step individually.
  • Look for Patterns and Relationships: Math is full of patterns and relationships. Identifying these patterns can help you solve problems more quickly and efficiently. For example, recognizing that the sides of a rectangle are proportional allows you to represent them using a single variable.
  • Practice Regularly: The more you practice, the more comfortable you'll become with different problem-solving techniques. Regular practice also helps you develop your intuition and learn to recognize common problem types.
  • Don't Be Afraid to Ask for Help: If you're stuck on a problem, don't hesitate to ask your teacher, classmates, or online resources for help. Explaining your thought process to someone else can often help you identify where you're going wrong.

Wrapping Up

So, there you have it! We've successfully tackled a rectangle problem involving perimeter and proportional sides. We've learned how to set up equations, solve for unknowns, and verify our solutions. More importantly, we've developed a systematic approach to problem-solving that can be applied to a wide range of mathematical challenges. Remember, the key to mastering math is not just memorizing formulas, but understanding the underlying concepts and developing problem-solving skills.

Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!