Question

If the length of a rectangle is reduced by 5 units and its breadth is increased by 3 units, then the area of the rectangle is reduced by 9 square units. If length is reduced by 3 units and breadth is increased by 2 units, then the area of rectangle will increase by 67 square units. Then find the length and breadth of the rectangle.

Answer 1

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

Area of the rectangle = xy sq. units

length of the rectangle is reduced by 5 units

∴ length = x – 5

breadth of the rectangle is increased by 3 units

∴ breadth = y + 3

area of the rectangle is reduced by 9 square units

∴ area of the rectangle = xy – 9

According to the first condition,

(x – 5) (y + 3) = xy – 9

∴ xy + 3x – 5y – 15 = xy – 9

∴ 3x – 5y = -9 + 15

∴ 3x – 5y = 6   ...(i)

length of the rectangle is reduced by 3 units

∴ length = x – 3

breadth of the rectangle is increased by 2 units

∴ breadth = y + 2

area of the rectangle is increased by 67 square units

∴ area of the rectangle = xy + 61

According to the second condition,

(x – 3) (y + 2) = xy + 67

∴ xy + 2x – 3y – 6 = xy + 67

∴ 2x – 3y = 67 + 6

∴ 2x – 3y = 73   ...(ii)

Multiplying equation (i) by 3,

9x – 15y = 18   ...(iii)

Multiplying equation (ii) by 5,

10x – 15y = 365   ...(iv)

Subtracting equation (iii) from (iv),

10x – 15y = 365
9x – 15y = 18        
-     +           -           
  x    =     347

Substituting x = 347 in equation (ii),

2x – 3y = 73

∴ 2(347) – 3y = 73

∴ 694 – 73 = 3y

∴ 621 = 3y

∴ y = `621/3`

∴ y = 207

∴ The length and breadth of rectangle are 347 units and 207 units respectively.