Let the width of the gravel = x metres. Then,
Each side of the square flower bed
= (44 – 2x) metres.
Now, Area of the square field
= 44 x 44 = 1936 m²
and Area of the flower bed
= (44 – 2x)² m²
∴ Area of the gravel path
= Area of the field – Area of the flower bed
= 1936 – (44 – 2x)²
= 1936 – (1936 – 176x +4x²)
= (176x – 4x²) m²

Cost of laying the flower bed
= (Area of the flower bed) (Rate per sq. m)

Cost of gravelling the path
= (Area of the path) x (Rate per sq. m)
= 
It is given that the total cost of laying the flower bed and gravelling the path is Rs. 4904.
∴ 11 (22 – x)² + 6 (44x – x²) = 4904
⇒ 11(484 – 44x + x)² + (264x – 6x²) = 4904
⇒ 5x² – 220x + 5324 = 4908
5x² – 220x + 420 = 0
⇒ x² – 44x + 84 = 0
⇒ x² – 42x – 2x + 84 = 0
⇒ x (x – 42) – 2 (x – 42) = 0
⇒ (x – 2) (x – 42) = 0
⇒ x = 2 or x = 42
But, x ≠ 42, as the side of the square is 44 m.
Therefore, x = 2.
Hence, the width of the gravel path is 2 metres.