Question

The area of a rectangle gets reduced by 8 sq. metre, if its length is reduced by 5 metre and breadth is increased by 3 metre. If we increase the length by 3 metre and breadth by 2 metre, the area is increased by 74 sq. metre. Find the length and breadth of the rectangle.?​

Answer 1

Solution :-

Let :-

Length of rectangle = x

Breadth of rectangle = y

Area = xy

According to the first condition :-

$\longrightarrow\sf (x-5)(y+3) = xy-8 \\\\\longrightarrow xy+3x-5y-15 = xy-8 \\\\\longrightarrow 3x-5y = -8+15 \\\\\longrightarrow 3x-5y = 7 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(1)$

According to the second condition :-

$\longrightarrow \sf (x+3)(y+2) = xy+74 \\\\\longrightarrow xy+2x+3y+6 = xy+74 \\\\\longrightarrow 2x+3y = 74-6 \\\\\longrightarrow 2x+3y = 68 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(2)$

Equation (1) × 2 :-

$\longrightarrow \sf 6x-10y=14 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(3)$

Equation (2) × 2 :-

$\longrightarrow \sf 6x+9y = 204 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ .....................(4)$

Equation (4) - Equation (2) :-

$\longrightarrow \sf 19y = 190 \\\\\longrightarrow \bf y = 10$

Substitute the value of y in eq (2) :-

$\longrightarrow \sf 2x + 3 \times 10 = 68 \\\\\longrightarrow 2x+30 = 68\\\\\longrightarrow 2x = 38 \\\\\longrightarrow \bf x = 19$

Length of the rectangle = 19 cm

Breadth of the rectangle = 10 cm