yeah i understood this but group example seems illusion here and out of context it doesn't not represent real existence of that number,next if 100/25 is 4 than it means size is reduced so, somewhere it should be cancelling each other to get such tiny reduced value.but where it cancelled and how this cancellation make real value

It seems like you're referring to the concept of division and reduction in size, especially in terms of how one quantity (like 100) gets reduced when divided by another quantity (like 25), resulting in a smaller number (in this case, 4). I’ll break this down step-by-step:

### Division as Size Reduction:
When you divide one number by another, you're essentially determining how many times the second number can fit into the first. 

**Example:**  
In the case of $ \frac{100}{25} $, you're asking: "How many times does 25 fit into 100?"

- $ \frac{100}{25} = 4 $, meaning that 25 fits exactly 4 times into 100.
- The size is reduced from 100 to 4 because you're scaling down the original value of 100 by a factor of 25.

### Cancellation Explained:
When you talk about "cancellation" in mathematics, it often refers to simplifying expressions. Division can be seen as a process of canceling factors. For $ \frac{100}{25} $, the numbers can be broken down into their factors:

- $ 100 = 25 \times 4 $
- $ \frac{100}{25} = \frac{25 \times 4}{25} $

Here, the common factor (25) cancels out, leaving just the number 4. So the "cancellation" occurs in the numerator and denominator where the shared factor is removed.

**Visualizing this Cancellation:**

Imagine you have 100 objects divided into groups of 25. You can form 4 groups, each containing 25 objects. The "cancellation" is the process of recognizing that these groups can be reduced or counted as one unit (25), thus leaving you with only 4 groups.

### Real Existence and Interpretation:
When you ask about "real existence" of this number, you're asking how this mathematical operation relates to the real world. In practice, division by 25 means you're scaling something down by a factor of 25, which can happen in various contexts:

- **Money:** If you have $100 and you divide it by $25, you can make 4 equal purchases.
- **Geometry:** If a length of 100 cm is divided into pieces of 25 cm, you get 4 pieces.

In each case, the result (4) represents the scaled-down reality.

### Key Point:
The cancellation happens because the two numbers (100 and 25) share a common relationship that allows one to "fit into" the other. The real-world interpretation of the result is that you're reducing or partitioning the original quantity into smaller units.

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**Would you like more details on any part of this explanation?**

### 5 Related Questions:
1. What is the mathematical meaning of a factor?
2. How does cancellation work in algebraic fractions?
3. Can you give a more complex example of cancellation in division?
4. How does division apply in geometric scaling problems?
5. How is the concept of reduction used in different fields like physics or economics?

**Tip:** When dividing large numbers, look for common factors to simplify the calculation before performing the operation!

Explore how division and cancellation work through the example of 100 divided by 25. Understand the concept of size reduction, cancellation of common factors, and how division represents real-world scaling. Suitable for students in Grades 6-8.

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