The area of a rectangle remains the same if the length is increased by 7 metres and the breadth is decreased by 3 metres. The area remains unaffected if the length is decreased by 7 metres and breadth is increased by 5 metres. Find the dimensions of the rectangle.
The area of a rectangle remains the same if the length is increased by 7 metres and the breadth is decreased by 3 metres. The area remains unaffected if the length is decreased by 7 metres and breadth is increased by 5 metres.
To do:
We have to find the dimensions of the rectangle.
Solution:
Let the original length of the rectangle be $l$ and the breadth be $b$.
Area of the original rectangle $=lb$.
In the first case, the length is increased by 7 metres and the breadth is decreased by 3 metres.
New length $=l+7$ m
New breadth $=b-3$ m
The area formed by the new rectangle $=(l+7)(b-3)\ m^2$
According to the question,
$(l+7)(b-3)=lb$
$lb-3l+7b-21=lb$
$3l-7b=-21$.....(i)
In the second case, the length is decreased by 7 metres and breadth is increased by 5 metres.
New length $=l-7$ m
New breadth $=b+5$ m
The area formed by the new rectangle $=(l-7)(b+5)\ m^2$
According to the question,
$(l-7)(b+5)=lb$
$lb+5l-7b-35=lb$
$5l-7b=35$.....(ii)
Subtracting (i) from (ii), we get,
$5l-7b-(3l-7b)=35-(-21)$
$5l-3l-7b+7b=56$
$2l=56$
$l=28$ m
$3(28)-7b=-21$ (From (i))
$84+21=7b$
$b=\frac{105}{7}$
$b=15$ m
The dimensions of the rectangle are $28$ m (length) and $15$ m (breadth).