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
• 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

Solution :

$\bf{\red{\underline{\bf{Given\::}}}}$

If the length of a rectangle is reduced by 5 units and it's 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 it's area increased to 50 sq.units.

$\bf{\red{\underline{\bf{To\:find\::}}}}$

The length and breadth of the rectangle.

$\bf{\red{\underline{\bf{Explanation\::}}}}$

Let the length of rectangle be r units

Let the breadth of rectangle be m units

Area of rectangle = rm

We know that formula of the area of rectangle :

$\boxed{\bf{Area=Length\times breadth}}}}$

A/q

$\longrightarrow\sf{(r-5)(m+2)=(rm-80)}\\\\\longrightarrow\sf{\cancel{rm}+2r-5m-10=\cancel{rm}-80}\\\\\longrightarrow\sf{2r-5m-10=-80}\\\\\longrightarrow\sf{2r-5m-10+80=0}\\\\\longrightarrow\sf{2r-5m+70=0...........................(1)}$

&

$\longrightarrow\sf{(r+10)(m-5)=(rm+50)}\\\\\longrightarrow\sf{\cancel{rm}-5r+10m-50=\cancel{rm}+50}\\\\\longrightarrow\sf{-5r+10m-50=50}\\\\\longrightarrow\sf{-5r+10m-50-50=0}\\\\\longrightarrow\sf{-5r+10m-100=0}\\\\\longrightarrow\sf{-5(r-2m+20)=0}\\\\\longrightarrow\sf{r-2m+20=0.........................(2)}$

$\underline{\underline{\bf{\pink{Using\:Substitution\:method\::}}}}$

From equation (2),we get;

$\implies\sf{r-2m+20=0}\\\\\implies\sf{r-2m=-20}\\\\\implies\sf{r=-20+2m.........................(3)}$

Putting the value of r in equation (1),we get;

$\implies\sf{2(-20+2m)-5m+70=0}\\\\\implies\sf{-40+4m-5m+70=0}\\\\\implies\sf{-40-m+70=0}\\\\\implies\sf{-40-m=-70}\\\\\implies\sf{-m=-70+40}\\\\\implies\sf{\cancel{-}m=\cancel{-}30}\\\\\implies\sf{\orange{m=30\;units}}$

Putting the value of m in equation (3),we get;

$\implies\sf{r=-20+2(30)}\\\\\implies\sf{r=-20+60}\\\\\implies\sf{\orange{r=40\:units}}$

Thus;

$\underbrace{\bf{The\:length\:of\:rectangle\:=r=40\:units}}}}}\\\underbrace{\bf{The\:breadth\:of\:rectangle\:=m=30\:units.}}}}}$