Solving Proportional Division Problems: A Case Study in Wood Segmentation

Ratios are a fundamental concept in mathematics and are frequently used to describe the relationship between different quantities. One practical application of ratios is in the segmentation of objects, such as wood, where the total length is divided according to a specific ratio. This article examines a problem involving the segmentation of wood according to a given ratio and provides a step-by-step solution.

Problem Statement

A piece of wood was cut into three smaller pieces such that the length of the pieces are in the ratio 2:3:5. Given that the total length of the wood is 70 dm (decimeters), what is the length of the longest piece?

Solution

To solve this problem, let's start by assigning variables to represent the length of each piece according to the given ratio:

Step 1: Define Variables

Let the length of the first piece be (2x), the second piece be (3x), and the third piece be (5x).

Step 2: Equation Setup

The total length of the three pieces is given as 70 dm. Therefore, we can set up the following equation:

[2x 3x 5x 70]

Step 3: Simplify the Equation

Combine like terms:

[1 70]

Step 4: Solve for (x)

[x frac{70}{10} 7]

Step 5: Calculate the Length of Each Piece

[2x 2 times 7 14 , text{dm}]

[3x 3 times 7 21 , text{dm}]

[5x 5 times 7 35 , text{dm}]

Conclusion

The length of the longest piece is 35 dm.

Verification

To verify the solution, we can add the lengths of the three pieces:

[14 21 35 70 , text{dm}]

The solution is correct.

Additional Examples and Solutions

Example 1

Consider a piece of wood with a total length of 50 inches, segmented into three parts in the ratio 2:3:5. We can solve this problem in a similar manner:

Step 1: Define Variables

Let the length of the first piece be (2k), the second piece be (3k), and the third piece be (5k).

Step 2: Equation Setup

[2k 3k 5k 50]

Step 3: Simplify the Equation

[10k 50]

Step 4: Solve for (k)

[k frac{50}{10} 5]

Step 5: Calculate the Length of Each Piece

[5k 5 times 5 25 , text{inches}]

The longest piece is 25 inches.

Example 2

Another method to solve the problem is to directly use the ratio. If you add the 3 parts of the ratio (2 3 5 10), and the total length is 50 inches, the longest part will be:

[50 div 10 times 5 25 , text{inches}]

This confirms the longest piece is 25 inches.

Conclusion

To summarize, when dealing with proportional division problems, it is essential to first define the variables and then set up the equation based on the given ratio. By solving the equation and calculating the lengths, we can determine the length of each segment. In the given problem, the longest piece of wood is 35 dm, and the longest piece in the second example is 25 inches.