Question

The area of a rectangle remains the same if the length is increased by $7$ metres and the breadth is decreased by $3$ meters. The area remains unaffected if the length is decreases by $7$ metres and breadth is increased by $5$ metres. Find the dimensions of the rectangle.

Answer 1

Let the length of the rectangle be $x$ metres and the breadth be $y$ metres.

Area of the rectangle=length$\times$breadth

$=x\times y=xy$ sq. metres

From the given information, we have,

$(x+7)\times (y-3)=xyand(x-7)\times (y+5)=xy(x+7)\times (y-3)=xy=>xy-3x+7y-21=xy=>-3x+7y-21=0=>7y=3x+\mathrm{21....}\left(i\right)Also,(x-7)\times (y+5)=xy=>xy+5x-7y-35=xy=>5x-7y-35=\mathrm{0....}\left(ii\right)$

Substituting equation (i) in equation (ii), we get,

$5x-7y-35=0=>5x-3x-21-35=0=>2x=56=>x=28$

Substituting $x=28$ in equation (i), we get,

$7y=3x+21=>7y=3\left(28\right)+21=>7y=105=>y=\frac{105}{7}$

Therefore, length of rectangle $=x=23$ metres

and breadth of rectangle $=y=15$ metres