Question

the length of a rectangle exceeds its breadth by 9 CM if the length and breadth is increased by 3 cm the area of the new rectangle will be 84 CM square more than that of the given rectangle find the length and breadth of the given rectangle check your solution.

Answer 1

$\huge{\text{Solution - }}$

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Let the breadth of the given rectangle = x cm

Then, its length = ( x + 9 ) cm

$\textsf{area \: of \: the \: given \: rectangle}$

= length × breadth
$((x + 9) \times x) {cm}^{2}$

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New breadth = ( x + 3 ) cm

New length = [( x + 9 ) + 3 ] cm
=> ( x + 12 ) cm

$\textsf{area \: of \: new \: rectangle}$

= length × breadth
$((x + 12)(x + 3)) \: {cm}^{2}$

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$\textbf{according \: to \: question}$

(area of the new rectangle) - ( area of given rectangle ) = 84 sq. cm

$= > (x + 12)(x - 3) - x(x + 9) = 84 \\ = > ({x}^{2} + 15x + 36) - ( {x}^{2} + 9x) = 84 \\ = > 6x + 36 = 84 \\ = > 6x = 48 \\ = > x = \frac{48}{6} \\ = > x = 8$

Thus, the breadth = 8 cm
And, length = ( 8 + 9 ) cm = 17 cm.

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$\textbf{checking}$

In the given rectangle, we have

length = 17 cm , breadth = 8 cm.

Area of given rectangle = ( 17 × 8) sq.cm
$136 {cm}^{2}$

The new length = ( 17 + 3 ) cm = 20 cm,
new breadth = (8 + 3 ) cm = 11 cm.

The area of new rectangle = ( 20 × 11 ) sq.cm
$= > 220 {cm}^{2}$

Now,

(area of the new rectangle) - ( area of given rectangle ) 84 sq.cm

$= > (220 - 136) {cm}^{2} = 84 {cm}^{2} \\ = > 84 {cm}^{2} = 84 {cm}^{2}$

LHS = RHS

So, Length = 17 cm and breadth = 8 cm

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$\huge{\text{THANKS}}$

Answer 2

Let the breadth of the given rectangle = x cm

Then, its length = ( x + 9 ) cm

\textsf{area \: of \: the \: given \: rectangle}area  of  the  given  rectangle 

= length × breadth 
\begin{lgathered}((x + 9) \times x) {cm}^{2} \\\end{lgathered}((x+9)×x)cm2​ 

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New breadth = ( x + 3 ) cm 

New length = [( x + 9 ) + 3 ] cm 
=> ( x + 12 ) cm

\textsf{area \: of \: new \: rectangle}area  of  new  rectangle 

= length × breadth 
((x + 12)(x + 3)) \: {cm}^{2}((x+12)(x+3))cm2 

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\textbf{according \: to \: question}according  to  question 

(area of the new rectangle) - ( area of given rectangle ) = 84 sq. cm

\begin{lgathered}= > (x + 12)(x - 3) - x(x + 9) = 84 \\ = > ({x}^{2} + 15x + 36) - ( {x}^{2} + 9x) = 84 \\ = > 6x + 36 = 84 \\ = > 6x = 48 \\ = > x = \frac{48}{6} \\ = > x = 8\end{lgathered}=>(x+12)(x−3)−x(x+9)=84=>(x2+15x+36)−(x2+9x)=84=>6x+36=84=>6x=48=>x=648​=>x=8​ 

Thus, the breadth = 8 cm 
And, length = ( 8 + 9 ) cm = 17 cm.

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\textbf{checking}checking 

In the given rectangle, we have 

length = 17 cm , breadth = 8 cm.

Area of given rectangle = ( 17 × 8) sq.cm
136 {cm}^{2}136cm2 

The new length = ( 17 + 3 ) cm = 20 cm,
new breadth = (8 + 3 ) cm = 11 cm.

The area of new rectangle = ( 20 × 11 ) sq.cm
= > 220 {cm}^{2}=>220cm2 

Now,

(area of the new rectangle) - ( area of given rectangle ) 84 sq.cm

\begin{lgathered}= > (220 - 136) {cm}^{2} = 84 {cm}^{2} \\ = > 84 {cm}^{2} = 84 {cm}^{2}\end{lgathered}=>(220−136)cm2=84cm2=>84cm2=84cm2​ 

LHS = RHS 

So, Length = 17 cm and breadth = 8 cm