Question

The area of a rectangle gets reduced by $80$ sq units, if its length is reduced by $5$ units and the breadth is increased by $2$ units. If we increase the length by $10$ units and decrease the breadth by $5$ units then area increased by $50$ sq units. Find the sum of the length and the breadth of the rectangle.

Answer 1

Let the length and breadth of the rectangle be $a,b$ units respectively.

Then the area will be $ab$ square units.

Now if the length of the rectangle is reduced by $5$ units and breadth is increased by $2$ units then new length and breadth will be $(a-5)$ units and $(b+2)$ units.

Then new area will be $(a-5)(b+2)$.

Then according to the problem,

$(a-5)(b+2)-ab=-80$

or, $2a-5b=-70$.......(1).

Now if length of the rectangle is increased by $10$ units and breadth is decreased by $5$ units then new length and breadth will be $(a+10)$ units and $(b-5)$ units.

Then new area will be $(a+10)(b-5)$.

Then according to the problem,

$(a+10)(b-5)-ab=50$

or, $10b-5a=100$

or, $2b-a=20$

or, $4b-2a=40$......(2).

Now adding (1) and (2) we get

$-b=-30$

or, $b=30$.

Putting the value of $b$ in (1) we get, $a=40$.

Now $a+b=40+30=70$.