Solving a Ribbon Puzzle: Proportional Cutting and the 2:3:6 Ratio

Mathematics and logical reasoning can be quite handy when dealing with real-life problems, such as dividing a ribbon into segments according to a given ratio. In this article, we will walk through a classic example of how to solve such a problem, specifically using the 2:3:6 ratio. This will help you understand the methodology and apply it to other similar problems.

Understanding the Problem

The problem we are going to solve is as follows: 'A piece of ribbon is cut in the ratio 2:3:6. If the shortest piece is 78 cm long, what is the length of the longest piece?' Let's break down the problem step by step.

The Mathematical Approach

First, let's denote the lengths of the pieces using a variable

x

Such that:

Given that the shortest piece is 78 cm, we can set up the following equation:

2x 78

Step 1: Solving for x

To find the value of x, we simply divide both sides of the equation by 2:

x 78 / 2 39 cm

Step 2: Calculating the Length of the Longest Piece

Now that we know the value of x, we can find the length of the longest piece by multiplying x by 6:

6x 6 * 39 234 cm

Therefore, the length of the longest piece is 234 cm.

Alternative Approaches

There are a couple of alternative approaches to solving this problem. Let's walk through them briefly:

Method 2: Using the Ratio Directly

We can also start by letting the total length of the ribbon be 11 parts, where 2, 3, and 6 parts correspond to the shortest, middle, and longest pieces, respectively. Given that the shortest piece is 78 cm, we can set up the following relationship:

(11 parts)  (2 parts   3 parts   6 parts)

Now, the length of the shortest piece (2 parts) is 78 cm, so we can find the length of one part (x) as follows:

x 78 / 2 39 cm

Therefore, the length of the longest piece (6 parts) is:

6x 6 * 39 234 cm

Method 3: Using Ratios and Proportions

Another approach is to consider the total length of the ribbon completely. If 2 parts are equal to 78 cm, then the total length of the ribbon can be found as follows:

The total length of the ribbon is 11 parts.

(2 parts) 78 cm,(11 parts) 78 cm * (11/2) 429 cm

Therefore, the length of the longest piece (6 parts) is:

6 parts 429 cm * (6/11) 234 cm

Conclusion

By breaking down the problem into smaller steps and using the ratio provided, we can easily determine the lengths of the pieces of ribbon. This problem not only tests your understanding of ratios but also your ability to apply mathematical principles to real-world situations. Whether you prefer the direct method or one of the alternative approaches, the result remains the same: the longest piece of ribbon is 234 cm long.