A line divides an equilateral triangle `ABC` of side 1 unit into two parts with same perimeter and different are `S_(1)` and `S_(2)` sq. units. If maximum value of `(S_(1))/(S_(2))=m/n` (where `m` and `n` coprime), then `|m-n|` is _____

Explanation

To solve the problem, we need to find the maximum value of the ratio \( \frac{S_1}{S_2} \) where \( S_1 \) and \( S_2 \) are the areas of two parts of an equilateral triangle divided by a line, and both parts have the same perimeter.

### Step-by-step Solution:

1. **Understanding the Triangle**:
- Let \( ABC \) be an equilateral triangle with side length 1 unit. The total area \( S_3 \) of triangle \( ABC \) is given by the formula:
\[ S_3 = \frac{\sqrt{3}}{4} \times (1^2) = \frac{\sqrt{3}}{4} \text{ square units.} \]

2. **Dividing the Triangle**:
- A line divides the triangle into two parts \( S_1 \) and \( S_2 \) such that both parts have the same perimeter. Let the lengths of the segments created by the division be \( x \), \( y \), and \( z \) for the respective sides.

3. **Setting Up the Perimeter Condition**:
- The perimeter of the triangle is 3 units. The perimeters of the two parts can be expressed as:
\[ P_1 = x + y + z \quad \text{and} \quad P_2 = (1 - x) + (1 - y) + z. \]
- Setting these equal gives:
\[ x + y + z = 2 - (x + y) \implies 2(x + y) + z = 2. \]
- Rearranging gives:
\[ x + y = \frac{3}{2}. \]

4. **Finding Areas**:
- The area \( S_1 \) can be expressed using the formula for the area of a triangle:
\[ S_1 = \frac{1}{2} \times x \times y \times \sin(60^\circ) = \frac{\sqrt{3}}{4} xy. \]
- Since \( S_2 = S_3 - S_1 \), we have:
\[ S_2 = \frac{\sqrt{3}}{4} - S_1. \]

5. **Maximizing the Ratio**:
- We want to maximize the ratio:
\[ \frac{S_1}{S_2} = \frac{\frac{\sqrt{3}}{4} xy}{\frac{\sqrt{3}}{4} - \frac{\sqrt{3}}{4} xy} = \frac{xy}{1 - xy}. \]
- Let \( k = xy \). From the earlier step, we know \( x + y = \frac{3}{2} \). By the AM-GM inequality:
\[ \frac{x + y}{2} \geq \sqrt{xy} \implies \left(\frac{3}{4}\right)^2 \geq k \implies k \leq \frac{9}{16}. \]

6. **Finding Maximum Value**:
- The maximum value of \( \frac{S_1}{S_2} \) occurs when \( k = \frac{9}{16} \):
\[ \frac{S_1}{S_2} = \frac{\frac{9}{16}}{1 - \frac{9}{16}} = \frac{\frac{9}{16}}{\frac{7}{16}} = \frac{9}{7}. \]

7. **Final Calculation**:
- Here, \( m = 9 \) and \( n = 7 \) are coprime. We need to find \( |m - n| \):
\[ |m - n| = |9 - 7| = 2. \]

### Final Answer:
The value of \( |m - n| \) is **2**.