Question

The area of a rectangle gets reduced by 10 square units if its length is reduced by 4 units and breadth is increase by 2 units. If we increased the length by 3 units and breadth by 4 units, the area is increased by 96 square units. Find the length and breadth of the rectangle. ​

Answer 1

AnswEr :

Let the Length be x and Breadth be y of Rectangle and therefore, Area = xy

⋆ The area of a rectangle gets reduced by 10 square units if its length is reduced by 4 units and breadth is increase by 2 units.

⇝ Area = Length × Breadth

⇝ ( xy - 10 ) = ( x - 4 ) × ( y + 2 )

⇝ ( xy - 10 ) = x( y + 2 ) - 4( y + 2 )

⇝ xy - 10 = xy + 2x - 4y - 8

⇝ xy - 2x + 4y - xy = 10 - 8

⇝ 4y - 2x = 2 — eq.( I )

⋆ If we increased the length by 3 units and breadth by 4 units, the area is increased by 96 square units.

⇝ Area = Length × Breadth

⇝ ( xy + 96 ) = ( x + 3 ) × ( y + 4 )

⇝ ( xy + 96 ) = x( y + 4 ) + 3( y + 4 )

⇝ xy + 96 = xy + 4x + 3y + 12

⇝ xy + 96 - xy - 12 = 4x + 3y

⇝ 3y + 4x = 84 — eq.( II )

• Multiplying ( I ) by 2 and Adding Both :

⇒ 8y – 4x = 4

⇒ 3y + 4x = 84

_________________

⇒ 11y = 88

⇒ y = 88 /11

⇒ y = 8 unit = Breadth

• Putting the Value of y = 8 in ( I )

⇒ 4y - 2x = 2

⇒ 4 × 8 - 2x = 2

⇒ 32 - 2 = 2x

⇒ 30 /2 = x

⇒ x = 15 unit = Length

∴ Length and Breadth is 15 unit and 8 unit.

Answer 2

Let length of rectangle = x units

And width of rectangle = y units

Area of rectangle = length * width = x*y

The area of a rectangle gets reduced by 10 square units, if its length is reduced by 4 units and breadth is increased by 2 units.

So

→ Decrease the length by 4 unit so new length = x - 4

→ Increase the width by 2 unit so new width = y + 2

→ New area is reduced by 10 units.

So, new area = xy – 10

Plug the value in formula length * width = area we get

(x - 4)(y + 2) = xy - 10

Xy + 2x – 4y – 8 = xy – 10

2x - 4y = -2

$\large\red{\boxed{\sf x-2y=(-1)}}$ ---------Equation(1)

If we increase the length by 3units and the breadth by 4 units, the area increases by 96 square units.

→ Increase the length by 3 unit so new length = x +3

→ Increase the width by 2 unit so new width = y + 4

→ New area is increased by 96 units

So, new area = xy + 96

Plug the value in formula length * width = area we get

(x +3)(y + 4) = xy + 96

Xy + 4x + 3y + 12 = xy + 96

$\large\red{\boxed{\sf 4x+3y = 84}}$ ---------Equation(2)

Now ,

Multiply Equation (1) by 4 and than subtract from Equation (2) we get,

(4x+3y) - 4(x-2y) = 84 - 4(-1)

→ 4x+3y - 4x +8y = 84 +8

→ 11y = 88

→ y = 8 units .....

Putting in Equation (1) or (2) now we get,

x - 2y = -1

→ x = -1 + 2×8

→ x = 15 units ..........

So, Length of Rectangle is 15 units , and it's breadth will be 8 units ......

This will satisfy the above conditon ......

(Hope it helps you)