Question

If the length of a rectangle is reduced by 5 units and its breadth is increased by 2 units then
the area is reduced by 80 sq. units. However if the length is increased by 10 units and breadth
is decreased by 5 units its area increases to 50 sq. units. Find the length and breadth of the
rectangle.​

Answer 1

Answer:

• Length = 40

1. Breadth = 30

Step-by-step explanation:

Given: As in first case

Area of rectangle gets reduced by 80 sq units

Its Length and Breadth reduced by 5 sq units and increased by 2 sq units respectively

★In second case

Its length is increased by 10 sq units and Breadth is decreased by 5 sq units.

Then, Area of rectangle will increase by 50 sq units.

To find:

Length and Breadth of rectangle

Solution: Let Length be 'x' and Breadth be 'y'.

Then, Area of rectangle will be Length x Breadth = xy

According to the question :

 In Case (1)

⟹ ( x–5)(y+2)=xy – 80

⟹ x(y+2) –5(y+2) = xy –80

⟹ xy + 2x – 5y –10 = xy –80

⟹ 2x – 5y –10 = –80

⟹ 2x – 5y = –80 + 10

⟹ 2x – 5y = –70 .................. ( equation 1)

→ In second case

⟹ (x+10)(y–5) = xy + 50

⟹ x(y–5) +10(y–5) = xy + 50

⟹ xy –5x + 10y –50 = xy + 50

⟹ –5x + 10y = 50 + 50

⟹ –5x + 10y = 100 ................. ( equation 2)

Now, Multiplying both sides of equation 1 by 2

=> 2(2x–5y) = 2(–70)

=> 4x – 10y = –140

Solving both equations ( eqn 1 and eqn 2)

=> –5x + 10y = 100

⠀⠀⠀4x – 10y = –140

__________________________

⠀⠀⠀–x = –40

or x = 40

Hence, We get Breadth of rectangle 'x' = 40 units

Now, Putting the value of 'x' in equation 2nd We got:

⟹ –5 x 40 + 10y = 100

⟹ –200 + 10y = 100

⟹ 10y = 100+200

⟹ 10y = 300

⟹ y = 30 units

Hence, We get Length of rectangle 'y' = 30 units.

Answer 2

$\large\red{\underline{\underline{\bf{\blue{Question}}}}}$

If the length of a rectangle is reduced by 5 units and its breadth is increased by 2 units then the area is reduced by 80 sq. units. However if the length is increased by 10 units and breadth is decreased by 5 units its area increases to 50 sq. units.

$\large\red{\underline{\underline{\bf{\blue{Solution}}}}}$

➨ Let length be 'x' units and breadth be 'y' units

We know that,

➨ Area of Rectangle = l × b

➨ Area of Original Rectangle = xy

So,

• ✞ According to 1st condition

➦ Length = ( x - 5) units

➦ Breadth = ( y + 2) units

➦ Area = (xy - 80) unit ²

We get,

➮ ( x - 5) ( y + 2) = xy - 80

➮ xy + 2x - 5y - 10 = xy - 80

➮ xy - xy + 2x - 5y = - 80 + 10

➮ 2x - 5y = - 70

➮ x = (-70 + 5y) / 2. ---(1)

• ✞ According to 2nd Condition

➦ Length = ( x + 10)

➦ Breadth = ( y - 5)

➦ Area = (xy + 50) unit²

We get,

➮ ( x + 10)(y - 5) = xy + 50

➮ xy - 5x + 10y - 50 = xy + 50

➮ xy - xy - 5x + 10y = 50 + 50

➮ - 5x + 10y = 100

➮ -5(x - 2y) = 100

➮ x - 2y = 100/-5

➮ x - 2y = -20

➮ (-70 + 5y) / 2 - 2y = 2 × -20. --(from -1)

➮ (-70 + 5y) / 2 - (2y ×2 )/2 = -40

➮ -70 + 5y - 4y = -40

➮ y = 70 - 40

➮ y = 30

Then,

➮ x = (-70 + 5y) /2

➮ x = ( -70 + 5×30)/2

➮ x = (-70 + 150)/2

➮ x = 80/2

➮ x = 40

Hence,

• Length = 40 unit

• Breadth = 30 unit