Question

If in a rectangle, the length is increased and breadth reduced each by $2$ units, the area reduces by $28$ square units. If, however the length is reduced by $1$ unit and the breadth increased by $2$ units, the area increases by $33$ square units. Find the area of the rectangle.

Answer 1

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

Area of the rectangle=length$\times$breadth

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

From the given information, we have,

$(x+2)\times (y-2)=xy-28and(x-1)\times (y+2)=xy+33(x+2)\times (y-2)=xy-28=>xy-2x+2y-4=xy-28=>-2x+2y=-24=>-x+y=-12=>x=y+\mathrm{12....}\left(i\right)Also,(x-1)\times (y+2)=xy+33=>xy+2x-y-2=xy+33=>2x-y=\mathrm{35....}\left(ii\right)$

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

$2x-y=35=>2(y+12)-y=35=>2y+24-y=35=>y=11$

Substituting $y=11$ in equation (i), we get,

$x=y+12=>x=11+12=>x=23$

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

and breadth of rectangle $=y=11$ units

Area of rectangle $=xy=23\times 11=253$ square units