Question

The area of a rectangle gets reduced by $9$ square units if its length is reduced by $5$ units and the breadth is increased by $3$ units. If we increase the length by $3$ units and breadth by $2$ units, the area is increased by $67$ square units. Find the length and breadth of the rectangle.

• A. $17$ and $10$
• B. $17$ and $9$
• C. $17$ and $39$
• D. None of these

Answer 1

Answer: (B)

$17$ and $9$

Let the length and breadth of the rectangle be x and y units respectively. Then,
Area $=xy$ sq. units.

If length is reduced by $5$ units and the breadth is increases by $3$ units, then area is reduced by $9$ square units.
$\therefore xy-9=(x-5)(y+3)$
$\Rightarrow xy-9=xy+3x-5y-15$
$\Rightarrow 3x-5y-6=0$ ...(i)

When length is increased by $3$ units and breadth by $2$ units, the area is increased by $67$ sq. units.
$\therefore xy+67=(x+3)(y+2)$
$\Rightarrow xy+67=xy+2x+3y+6$
$\Rightarrow 2x+3y-61=0$ ...(ii)

Thus, we get the following system of linear equations:
$3x-5y-6=0$
$2x+3y-61=0$

By using cross-multiplication, we have
$\frac{x}{305+18}=\frac{-y}{-183+12}=\frac{1}{9+10}$

$\Rightarrow x=\frac{323}{19}=17$ and $y=\frac{171}{19}=9$

Hence, the length and breadth of the rectangle are $17$ units and $9$ units respectively.