Question

# A rectangular field 50 meters long and 42 meters broad contains a rectangular lawn inside is surrounded by a gravel path of uniform width. If the width of the path is 6m then the area of the path is(a) $240{{m}^{2}}$(b) $480{{m}^{2}}$(c) $720{{m}^{2}}$(d) $960{{m}^{2}}$

Hint: To solve this problem, we need to find all the dimensions of the rectangular lawn. We can find that as we know all the dimensions of the rectangular field and the width of the gravel path. The idea is to subtract the area of rectangular lawn from the area of the rectangular field.
Now, we have a rectangular field of 50m x 42m. And inside the field, we have a gravel path of width 6m from all sides. It can be shown pictorially as follows,

We aim to find the area of the gravel. To find this we need to get the dimensions of the rectangular lawn.
The rectangular lawn is surrounded by a 6m strip of gravel path.
Let's begin by finding the length of the rectangular lawn.
As we can see in the diagram,
Length (Rectangular field) = 6m + 6m + Length (Rectangular lawn) …………………...(i)
As it is given in the question the length of the rectangular field is 50m. Substituting the value in equation (i) we get,
50m = 6m + 6m + Length (Rectangular lawn)
Solving for the length of the rectangular lawn we get,
Length (Rectangular lawn) = 50m - 6m – 6m = 38m…………………(ii)
Let's go ahead by finding the width of the rectangular lawn.
As we can see in the diagram,
Width (Rectangular field) = 6m + 6m + Width (Rectangular lawn)…………………..(iii)
As it is given in the question the width of the rectangular field is 42m. Substituting the value in equation (iii) we get,
42m = 6m + 6m + Width (Rectangular lawn)
Solving for the Width of the rectangular lawn we get,
Width (Rectangular lawn) = 42m - 6m – 6m = 30m…………………(iv)
The area of the rectangular lawn is = length x width
Substituting the value of length and width from equation (ii) and (iv)
Area (Rectangular lawn) = Length(Rectangular lawn) x Width (Rectangular lawn)
Area = 38m x 30m = $1140{{m}^{2}}.........................(v)$
To find the area of the gravel path we also need to find the area of the rectangular field.
Area (Rectangular field) = Length (Rectangular field) x Width (Rectangular field)
As we know all the values of length and width of the rectangular field, we get,
Area (Rectangular field) = 50m x 42m = $2100{{m}^{2}}................(vi)$
The rectangular lawn and gravel path is inside the rectangular field so we can conclude that,
Area (Rectangular field) = Area (Rectangular lawn) + Area (Gravel path)
From equation (v) and (vi), substituting the values, we get,
$2100{{m}^{2}}=1140{{m}^{2}}+\text{Area (Gravel path)}$
Solving for Area (Gravel path) we get,
$\text{Area (Gravel path)}=2100{{m}^{2}}-1140{{m}^{2}}=960{{m}^{2}}$
Therefore, the area of the gravel path is $960{{m}^{2}}$ .
Hence, the correct option is (d).

Note: As the gravel path is made of rectangular strips we can also solve by calculating the dimensions of the strip and then finding the area individually. But that will be a very long and very much prone to error. The common mistake is while calculating the dimensions of the rectangular lawn. We need to subtract the width of the strip twice from the dimensions of the rectangular field as the strip is all around the lawn.