Given:
- Length of rectangular garden plot = 16m
- Breadth of rectangular garden plot = 24m

To find:
- New length and breadth of the rectangular garden after adding a uniform width border

Solution:

Step 1: Calculate the area of the rectangular garden plot:
Area = Length × Breadth = 16m × 24m = 384m²

Step 2: Let's assume the width of the border added is 'X' meters.

Step 3: Calculate the new length and breadth of the garden plot:
New Length = Length of garden plot + 2 × Width of the border
New Breadth = Breadth of garden plot + 2 × Width of the border

Substituting the given values:
New Length = 16m + 2X
New Breadth = 24m + 2X

Step 4: Calculate the area of the new rectangular garden plot:
New Area = New Length × New Breadth = (16m + 2X) × (24m + 2X)

Step 5: Equate the new area to the area of the original rectangular garden plot:
New Area = Area
(16m + 2X) × (24m + 2X) = 384m²

Step 6: Solve the equation to find the value of 'X':
(16m + 2X) × (24m + 2X) = 384m²
384m² + 48X + 32X + 4X² = 384m²
4X² + 80X = 0
4X(X + 20) = 0

Step 7: Find the possible values of 'X':
Either X = 0 (which means no border is added)
Or X = -20 (which is not possible in this context)

Step 8: Calculate the new length and breadth using the value of 'X':
If X = 0, then New Length = 16m + 2 × 0 = 16m and New Breadth = 24m + 2 × 0 = 24m
If X = -20, it is not possible in this context.

Step 9: Final Answer:
If no border is added (X = 0), the new length and breadth of the rectangular garden plot will remain the same as the original length and breadth, i.e., New Length = 16m and New Breadth = 24m.