Carpenter's Square Frame Puzzle: Maximize Size & Quantity

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Carpenter's Square Frame Puzzle: Maximize Size & Quantity

Let's dive into a fun woodworking problem! A carpenter has a bunch of wooden strips and wants to make square frames. The challenge is to figure out the largest possible size for the sides of these frames and how many frames he can make in total. This involves a bit of math, specifically finding the greatest common divisor (GCD) and then doing some division to see how many frames we can squeeze out of the available materials. Let's break it down step by step, guys!

Understanding the Problem

So, the carpenter has three types of wooden strips:

  • 20 strips of 150 cm each
  • 15 strips of 60 cm each
  • 12 strips of 240 cm each

The goal is to create square frames. This means each frame will have four sides of equal length. The main question here is: What's the longest possible length for each side of the square frame that allows the carpenter to use the strips efficiently? And, of course, how many frames can he build with that side length?

Finding the Greatest Common Divisor (GCD)

The greatest common divisor is the largest number that divides evenly into two or more numbers. In this case, we need to find the GCD of 150 cm, 60 cm, and 240 cm. This will tell us the largest possible side length for our square frames.

Let's list the factors of each number:

  • Factors of 150: 1, 2, 3, 5, 6, 10, 15, 25, 30, 50, 75, 150
  • Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
  • Factors of 240: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240

Looking at these factors, the largest number that appears in all three lists is 30. Therefore, the GCD of 150, 60, and 240 is 30.

Determining the Side Length of the Frames

The GCD we just found, 30 cm, is the maximum possible length for each side of the square frames. This is because we can cut each of the wooden strips into segments of 30 cm without any leftover pieces. So, each side of our square frame will be 30 cm long. A square frame has four sides, so we need four 30 cm pieces for each frame.

Calculating the Number of Frames

Now that we know the side length of each frame (30 cm), we can figure out how many frames the carpenter can build from each type of wooden strip.

Frames from 150 cm Strips

The carpenter has 20 strips that are each 150 cm long. Each of these strips can be cut into 150 / 30 = 5 pieces of 30 cm each. So, each 150 cm strip can provide enough material for 5 / 4 = 1.25 sides of a frame. Since we can't have a fraction of a frame side from a single strip, we need to figure out how many complete frames we can get. A single 150 cm strip yields one side of a frame with one 30cm piece remaining. However, to calculate total possible frames, it's easier to first calculate total 30cm pieces and then divide by four.

Total 30 cm pieces from 150 cm strips: 20 strips * 5 pieces/strip = 100 pieces.

Frames from 60 cm Strips

The carpenter has 15 strips that are each 60 cm long. Each of these strips can be cut into 60 / 30 = 2 pieces of 30 cm each. So, each 60 cm strip can provide enough material for 2 / 4 = 0.5 sides of a frame. Again, we need whole frames. A 60 cm strip yields one side with two strips. However, to calculate total possible frames, it's easier to first calculate total 30cm pieces and then divide by four.

Total 30 cm pieces from 60 cm strips: 15 strips * 2 pieces/strip = 30 pieces.

Frames from 240 cm Strips

The carpenter has 12 strips that are each 240 cm long. Each of these strips can be cut into 240 / 30 = 8 pieces of 30 cm each. So, each 240 cm strip can provide enough material for 8 / 4 = 2 sides of a frame. A 240 cm strip yields two complete sides, with no material remaining. However, to calculate total possible frames, it's easier to first calculate total 30cm pieces and then divide by four.

Total 30 cm pieces from 240 cm strips: 12 strips * 8 pieces/strip = 96 pieces.

Total Frames

Now, let's add up all the 30 cm pieces we can get from all the strips:

Total 30 cm pieces = 100 (from 150 cm strips) + 30 (from 60 cm strips) + 96 (from 240 cm strips) = 226 pieces.

Since each frame needs 4 pieces, we can make 226 / 4 = 56.5 frames. However, we can only make whole frames, so the carpenter can build a maximum of 56 frames.

Answer

Therefore, the largest possible size for the side of the square frames is 30 cm, and the carpenter can build 56 frames.

Let's Summarize what we've found

In this woodworking puzzle, we aimed to find the largest possible size for square frames that a carpenter could build from wooden strips of varying lengths. We also wanted to calculate how many frames he could construct. Here's a recap of our journey:

  1. Identifying the materials: The carpenter had 20 strips of 150 cm, 15 strips of 60 cm, and 12 strips of 240 cm.
  2. Defining the objective: The goal was to maximize the side length of the square frames while using the available strips efficiently.
  3. Finding the Greatest Common Divisor (GCD): We determined the GCD of 150, 60, and 240 to be 30. This GCD represents the largest possible side length of the square frames.
  4. Calculating the number of frames from each strip type:
    • 150 cm strips: Each strip could be cut into 5 pieces of 30 cm.
    • 60 cm strips: Each strip could be cut into 2 pieces of 30 cm.
    • 240 cm strips: Each strip could be cut into 8 pieces of 30 cm.
  5. Determining the total number of 30 cm pieces: We calculated that the carpenter had a total of 226 pieces of 30 cm.
  6. Calculating the maximum number of frames: Since each frame required 4 pieces, the carpenter could build a maximum of 56 complete frames.

Key Takeaways

  • GCD is crucial: The GCD helps determine the maximum possible size for the frame's sides, ensuring efficient use of materials.
  • Whole frames matter: We can only count complete frames, so we had to round down when calculating the total number of frames.

This problem highlights the importance of mathematical concepts like GCD in practical scenarios. By applying these concepts, we were able to optimize the use of the carpenter's materials and determine the maximum number of square frames he could build. Remember, guys, that understanding the problem is key. This includes identifying what you need to solve, as well as what the solution has to be.

Making it Real: Practical Applications

The problem we just solved isn't just a theoretical exercise. It has real-world applications in various fields, including:

  • Construction: Estimating material usage and minimizing waste.
  • Manufacturing: Optimizing production processes and cutting materials efficiently.
  • Resource Management: Allocating resources effectively to maximize output.

For instance, imagine a construction worker who needs to cut wooden planks to build rectangular structures. By finding the GCD of the plank lengths, they can determine the largest possible size for the rectangular sides, minimizing waste and maximizing the number of structures they can build.

The Value of Problem-Solving Skills

Solving problems like this not only enhances your mathematical skills but also improves your ability to think critically and solve real-world challenges. By breaking down complex problems into smaller, manageable steps, you can develop effective strategies for finding solutions.

So, the next time you encounter a similar problem, remember the steps we followed: understand the problem, identify the key concepts, apply the appropriate mathematical tools, and break down the problem into smaller parts.

Conclusion

Through this problem, we found that the carpenter could create 56 square frames with each side measuring 30 cm. This required using knowledge of factors and, ultimately, the Greatest Common Divisor. This is a great demonstration of how math skills can be applied to help in design and building.