The length and the breadth of a rectangular garden are in the ratio 9 : 5.

Let the length and breadth of the garden be 9x m and 5x m, respectively,
Now,

Area of the garden $=(9 x \times 5 x)=45 x^{2}$

Length of the garden excluding the path $=(9 x-7)$

Breadth of the garden excluding the path $=(5 x-7)$

Area of the path $=45 x^{2}-[(9 x-7)(5 x-7)]$

$\Rightarrow 1911=45 x^{2}-\left[45 x^{2}-63 x-35 x+49\right]$

$\Rightarrow 1911=45 x^{2}-45 x^{2}+63 x+35 x-49$

$\Rightarrow 1911=98 x-49$

$\Rightarrow 1960=98 x$

$\Rightarrow x=\frac{1960}{98}$

$\Rightarrow x=20$

Thus, we have:

Length $=9 x=20 \times 9=180 \mathrm{~m}$

Breadth $=5 x=5 \times 20=100 \mathrm{~m}$