Question

The perimeter of a rectangle is 100m if the length is decreased by 2m and the breadth is increased by 3m , the area increases by 44m² . find the length and breadth of rectangle​

Answer 1

$\huge\underline\mathrm\green{Given:}$

The perimeter of a rectangle is 100m

Length is decreased by 2m

The breadth is increased by 3m , the area increases by 44m.²

$\huge\underline\mathrm\green{To\:Find:}$

Find the length and breadth of rectangle

$\huge\underline\mathrm\green{Solution:}$

Let the length of the given rectangle be x m

Therefore,

Perimeter of the given rectangle = 100m

=> perimeter = 2( length + breadth )

=> 100 = 2(x + breadth)

=> 100/2 = x + breadth

=> 50 = x + breadth

=> Breadth = (50 - x) m

Or

Area of the given rectangle= l×b

= x(50 - x) m²

New Length = (x-2)

New breadth = (50 - x + 3) = (53 - x) m

So,

The area of the new rectangle

= (x-2)(53-x)m²

According to the given condition

Area of the new rectangle - Area of given rectangle = 44

=> (x-2)(53-x) = x(50-x)+44

=> x(53-x)-2(53-x)-x(50-x) = 44

=> 53x-x²-106+2x-50x+x² = 44

=> 53x-50x+2x-106 = 44

=> 5x = 106+44

=> 5x = 150

=> x = 150/5 = 30

Hence,

the length of the given rectangle = 30m

breadth of the given rectangle

= (50-x) = 50 - 30 = 20m

$\large{\boxed{\bf{Required\:length=30m}}}$

$\large{\boxed{\bf{Required\:breadth=20m}}}$

━━━━━━━━━━━━━━━━━━━━━━━━━━

$\huge\underline\mathrm\green{Verification:}$

Area of the given rectangle

= 30×20 = 600m²

New length = (x-2)=(30-2) = 28m

New breadth = (53-x) = (53-30) = 23m

Area of the new rectangle - Area of the given rectangle

= (644-600)m² = 44m²

Answer 2

❥${\purple{\underline{\underline{\large{\mathtt{ANSWER:-}}}}}}$

Length is 30 m. and breadth is 20 m.

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❥${\purple{\underline{\underline{\large{\mathtt{EXPLANATION:-}}}}}}$

•Given:-

• Perimeter of rectangle= 100 m.
• the length is decreased by 2m and the breadth is increased by 3m , the area increases by 44m² .

•To find:-

• Length of the rectangle.
• Breadth of the rectangle.

•Solution :-

Let the length of the rectangle be X m and the breadth of the rectangle be y m.

Perimeter = 100 m.

We know,

$\large{\boxed{\sf{\red{Perimeter\:of\: rectangle=2(length+breadth)}}}}$

Perimeter of rectangle=2(l+b)

=2(x+y) m.

We know,

$\large{\boxed{\sf{\blue{Area\:of\: rectangle=length×breadth}}}}$

Area of rectangle =l×b

= xy m²

★According to the question,

${\sf{2(x+y)=100}}$

${\sf{→x+y=50}}$

${\sf{\green{→x=50-y.............(i)}}}$

_____________________________________________

ᴥ Length is decreased by 2 m.

${\sf{Length=(x-2)\:m}}$

ᴥ Breadth is increased by 3 m.

tex]{\sf{Breadth=(y+3)\:m}}[/tex]

Now, Area of rectangle,

(x-2)×(y+3) m.

= xy+3x-2y-6 m.

‡ Area increases by 44 m².

★ According to the question,

${\sf{xy+3x-2y-6 \:=\:xy+44}}$

${\sf{→3x-2y-6 \:=\:44}}$

${\sf{→3x-2y\:=\:44+6}}$

${\sf{→3x-2y \:=\:50}}$

† Putting the value of x= (50-y) †

${\sf{→3(50-y)-2y \:=\:50}}$

${\sf{→150-3y-2y \:=\:50}}$

${\sf{→-5y \:=\:50-150}}$

${\sf{→-5y \:=\:-100}}$

${\sf{→y \:=\:20}}$

Breadth = 20 m.

________________________________________________

✞ Putting the value of y=20 in the (i)no. equation✞

x = 50-y

→ x = 50-20

→ x = 30

Length = 30 m.

______________________________________________

❥${\underline{\underline{\large{\mathtt{VERIFICATION:-}}}}}$

Length = 30 m.

Breadth = 20 m.

Perimeter= 100 m.

So,

2(30+20)=100

→100 = 100 (Verified)

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