The length and the breadth of a rectangular garden are in the ratio 9 :5. A path 3.5 m wide, running all around inside it has an area of 1911 m2. Find the dimensions of the garden.
Given:
Length : Breadth 9 : 5
Width of the path = 3.5 m
Area of path = 1911 m2
Let,
Length of field = 9x
Breadth of field = 5x
Area of field = Length × Breadth
= 9x × 5x
= 45 x2
Therefore,
Length of field without path = 9x - (3.5 + 3.5)
= 9x - 7
Breadth of field without path = 5x - (3.5 + 3.5)
= 5x - 7
Therefore,
Area of field without path = Length without path × Breadth without path
= (9x - 7) × (5x - 7)
= 45x2 – 35x – 63x + 49
= 45x2 – 98x + 49
Now,
Area of path = Area of field - Area of field without path
⇒ 1911 = 45 x2 – (45x2 – 98x + 49)
⇒ 1911 = 45 x2 – 45x2 + 98x - 49
⇒ 1911 = 98x - 49
⇒ 98x = 1911 + 49
⇒ 98x = 1960
⇒ x = 20
Hence,
Length of field = 9x = 9 × 20 = 180 m
Breadth of field = 5x = 5 × 20 = 100 m