PROBLEM ON AREAS –
Example.1) If the length and breadth of a room are increased by 2 m each, its area is increased by 22 m². If the length is increased by 2 m and breadth decreased by 2 m, the area is decreased by 6 m². Find the area of the room.
Ans.) Let the length and breadth of the room is x meters and y meters respectively. Then the area of the room = xy m²
Case.1) As per the given situation, when length = (x + 2) m, and breadth = (y + 2) m
In this case, area of the room => [(x + 2) (y + 2)] m²
As per the given condition => (x + 2) (y + 2) – xy = 22
=> xy + 2x + 2y + 4 – xy = 22
=> 2 (x + y) = 22 – 4
=> x + y = 9 …………………(i)
Case.2) As per the given condition, when length = (x + 2) m, and breadth = (y – 2) m
In this case, area of the room => [(x + 2) (y – 2)] m²
As per the given condition => xy - [(x + 2) (y – 2)] = 6
=> xy – (xy + 2y – 2x – 4) = 6
=> xy – xy – 2y + 2x + 4 = 6
=> 2x – 2y = 6 – 4
=> 2 (x – y) = 2
=> x – y = 1 ………………………(ii)
Now, we will add (i) and (ii), and we get –
x + y = 9
x – y = 1
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2x = 10
=> x = 5
Now, we will substitute the value of x in (i), and we get –
x + y = 9
=> 5 + y = 9
=> y = 9 – 5
=> y = 4
Hence the area of the room is xy = (5 X 4) = 20 m² (Ans.)
Example.2) If the length and breadth of a room are decreased by 3 m each, its area is decreased by 24 m². If the length is increased by 4 m and breadth decreased by 1 m, the area is increased by 9 m². Find the area of the room.
Ans.) Let the length and breadth of the room is x meters and y meters respectively. Then the area of the room = xy m²
Case.1) As per the given situation, when length = (x – 3) m, and breadth = (y – 3) m
In this case, area of the room => [(x – 3) (y – 3)] m²
As per the given condition => xy – [(x – 3) (y – 3)] = 24
=> xy – (xy – 3y – 3x + 9) = 24
=> xy – xy + 3y + 3x – 9 = 24
=> 3 (x + y) = 24 + 9
=> x + y = 11 …………………(i)
Case.2) As per the given condition, when length = (x + 4) m, and breadth = (y – 1) m
In this case, area of the room => [(x + 4) (y – 1)] m²
As per the given condition => [(x + 4) (y – 1)] – xy = 10
=> (xy + 4y – x – 4) – xy = 10
=> xy + 4y – x – 4 – xy = 10
=> 4y – x = 10 + 4
=> 4y – x = 14 ………………………(ii)
Now, we will add (i) and (ii), and we get –
x + y = 11
– x + 4y = 14
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5y = 25
=> y = 5
Now, we will substitute the value of y in (i), and we get –
x + y = 11
=> x + 5 = 11
=> x = 11 – 5
=> x = 6
Hence the area of the room is xy = (5 X 6) = 30 m² (Ans.)