Question

if the length of a rectangle is decreased by 2 metres and the breadth increased by 2 metres the area would increase 8 metre square. if the length is decreased by 3 metres and the breadth decreased by 1 metre the area would decrease by 27 square metre. what are the length and breadth?​

Answer 1

length=12m, breadth=6m

let length be 'l' & breadth be 'b'

Area of rectangle= length×breadth

1st Condition:

(l-2) (b+2)=lb+8

lb+2l-2b-4=lb+8

2l-2b=12

l-b=6 ---------(1)

2nd condition:

(l-3)(b-1)= lb-27

lb-l-3b+3=lb-27

l+3b=30 --------(2)

subtract (1) from (2)

l+3b-(l-b)=30-6

4b=24

b=6

put b=6 in (1)

l-6=6

l=12

So, length is 12 m & breadth is 6m

Answer 2

Given Question :-

• if the length of a rectangle is decreased by 2 metres and the breadth increased by 2 metres, the area would increase by 8 metre square. if the length is decreased by 3 metres and the breadth decreased by 1 metre, the area would decrease by 27 square metre. What are the length and breadth?

CALCULATION :-

• Let Length (L) of the rectangle be 'x' metre

and

• Let Breadth (B) of the rectangle be 'y' metre.

So,

• Area of rectangle, A = xy

Case - 1.

$\large \underline{\tt \:{ According \: to \: statement }}$

• if the length of a rectangle is decreased by 2 metres and the breadth increased by 2 metres, the area would increase by 8 metre square.

So,

• Length of rectangle = (x - 2) metres

• Breadth of rectangle = (y + 2) metres

• Area of rectangle = xy + 8

Now,

$\rm :\implies\:(x - 2)(y + 2) = xy + 8$

$\rm :\longmapsto\:xy + 2x - 2y - 4 = xy + 8$

$\rm :\longmapsto\:2x - 2y = 12$

$\rm :\implies\:x - y = 6$

$\rm :\implies\: \boxed{ \tt \:x = y + 6 } - - (1)$

Case - 2

$\large \underline{\tt \:{ According \: to \: second \: condition }}$

• If the length is decreased by 3 metres and the breadth decreased by 1 metre, the area would decrease by 27 square metre.

So,

• Length of rectangle = (x - 3) metre

• Breadth of rectangle = (y - 1) metre

• Area of rectangle = xy - 27

Now,

$\rm :\implies\:(x - 3)(y - 1) = xy - 27$

$\rm :\longmapsto\:xy - x - 3y + 3 = xy - 27$

$\rm :\implies\:x + 3y = 30$

$\rm :\implies\:y + 6 + 3y = 30 \: \: \: \: \: \: \{using \: {eq}^{n} \: (1) \}$

$\rm :\implies\:4y = 24$

$\rm :\implies\: \boxed{ \bf\:y \: = \: 6 } - - (2)$

• On substituting the value of y in equation (1), we get

$\rm :\implies\:x = 6 + 6$

$\rm :\implies\: \boxed{ \bf \:x \: = \: 12 }$

$\begin{gathered}\begin{gathered}\bf \: Hence, - \begin{cases} &\sf{Length_{(rectangle)} \: = x \: = \: 12 \: m} \\ &\sf{Breadth_{(rectangle)} \: = \: y \: = \: 6 \: m} \end{cases}\end{gathered}\end{gathered}$