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Fundamental Idea About Rectangular Pathways:

Consider a rectangular park, as shown in the above figure. A uniform path is to be laid outside the park. How do we find the area of the path?

The uniform path, including the park, is also a rectangle. If we consider the path as the outer rectangle, then the park will be the inner rectangle.

Let $$l$$ and $$b$$ be the length and breadth of the park.

Area of the park (inner rectangle) $$=$$ $l×b$ $$sq. units$$.

Let $$w$$ be the width of the path. If $$L$$, $$B$$ are the length and breadth of the outer rectangle, then
$$L = l + 2w$$ and $$B = b + 2w$$.

Similarly for inner rectangle $$l = L - 2w$$ and $$b = B - 2w$$

Therefore:
The area of the rectangular pathway $$=$$ The area of the outer rectangle $$–$$ The area of the inner rectangle
The area of the rectangular pathway is $\left(\mathit{LB}-\mathit{lb}\right)\phantom{\rule{0.147em}{0ex}}$$$sq. units$$