Question

The area of a rectangle gets reduced by 9 square units. If it's length is reduced by 5 units and breath is increased by 3units .the area is increasing by 67 square units. If length is increasing by 3 unit and breath is increasing by 2 units then find the perimeter of the rectangle

Answer 1

Answer:

● perimeter of rectangle is 52 units.

Step-by-step explanation:

Let ,

length and breadth of Rectangle be l and b respectively.

then, area of rectangle will be l b

First case :

when length is reduced by 5 units and breadth is increased by 3 units then, area of rectangle is reduced by 9 sq. units.

so,

→ ( l - 5 ) ( b + 3 ) = l b - 9

→ l b + 3 l - 5 b - 15 = l b - 9

→ 3 l - 5 b - 15 = - 9

→ 3 l - 5 b = 6

→ 3 l = 6 + 5 b

→ l = ( 6 + 5 b ) / 3  ____eqn①

Second case :

when length is increased by 3 units and breadth is increased by 2 units then area of rectangle is increased by 67 sq. units.

so,

→ ( l + 3 ) ( b + 2 ) = l b + 67

→ l b + 2 l + 3 b + 6 = l b + 67

→ 2 l + 3 b = 67 - 6

→ 2 l + 3 b = 61

[ using equation ① ]

→ 2 ((6+5b)/3) + 3 b = 61

→ (12 + 10 b)/3 + 3 b = 61

[ multiplying by 3 both sides ]

→ 12 + 10 b + 9 b = 183

→ 19 b = 183 - 12

→ 19 b = 171

→ b = 171 / 19

→ b = 9 units

[ putting value of b in eqn ① ]

→ l = ( 6 + 5 b ) / 3

→ l = ( 6 + 5 (9) ) / 3

→ l = 51 / 3

→ l = 17 units

Therefore,

Length and breadth of rectangle are 8 units and 17 units respectively.

so,

→ perimeter of rectangle = 2 ( l + b )

→ perimeter of rectangle = 2 ( 9 + 17 )

→ perimeter of rectangle = 52 units.

Therefore,

perimeter of rectangle is 52 units.

Answer 2

Let , the area of rectangle be " l × b "

First condition :

The area of a rectangle gets reduced by 9 sq. units , If it's length is reduced by 5 units and breath is increased by 3 units

Thus ,

(l - 5)(b + 3) = lb - 9

lb + 3l - 5b - 15 = lb - 9

3l - 5b = 6 ---- (i)

Second condition :

The area of a rectangle gets increased by 67 sq. units , If it's length is increased by 3 units and breath is increased by 2 units

Thus ,

(l + 3)(b +2) = lb + 67

lb + 2l + 3b + 6 = lb + 67

2l + 3b = 61 ---- (ii)

Multiply eq (i) by 2 and eq (ii) by 3 , we get

6l - 10b = 12

and

6l + 9b = 183

Subtract eq (iii) from eq (iv) , we get

6l + 9b - (6l - 10b) = 183 - 12

9b + 10b = 171

19b = 171

b = 171/19

b = 9 units

Put the value of b = 9 in eq (i) , we get

2l + 3(9) = 61

2l = 61 - 27

2l = 34

l = 34/2

l = 17 units

$\sf \therefore \underline{The \: length \: and \: breadth \: of \: rectangle \: are \: 17 \: units \: and \: 9 \: units }$

Now , the perimeter of rectangle is given by

Perimeter = 2(l + b)

Thus ,

Perimeter = 2(17 + 9)

Perimeter = 2(26)

Perimeter = 52 units

$\sf \therefore \underline{The \: perimeter \: of \: rectangle \: is \: 52 \: units }$