Question

The Perimeter of a rectangle is 240 cm. If the length is decreased by 10 % and its breadth is increased by 20 %, we get the same perimeter. Find the length and breadth of the rectangle. ​

Answer 1

★ Concept

The above question (Problem on Mensuration) is based on the concept of Area and Perimeter as an application of linear equations to practical problems. Let's understand the question first, we're provided with Perimeter of rectangle and we are asked to calculate the original Length and Breadth of the rectangle. Also, a condition is provided to us that if the length is decreased by 10% and its breadth is increased by 20% then the perimeter will remain unaltered. Assume the length of the rectangle be 'x' cm in that case breadth would be (120 - x) cm. The length is decreased by 10% i.e new length is 9x/10 ; Breadth is increased by 20% i.e new breadth is 120(120 - x)/100. Then, Apply the formula for Perimeter of rectangle and just substitute the values for New length and Breadth in the subjected formula, in this way we'll get an equation in terms of x i.e original length and so does the value for original breadth.

Let's proceed with calculation !!

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Required Diagram

$\begin{gathered} \begin{gathered}\begin{gathered} \begin{gathered} \bf{ \pink{40\: cm }\:}\huge\boxed{ \begin{array}{cc} \: \: \: \\ \: \: \: \: \: \: \: \sf\footnotesize{ \: \: \: \: \: }\ \footnotesize{} \: \: \: \end{array}} \\ \: \: \: \: \: \: \: \: \bf{ \red{ \quad \:80 \: cm}} \end{gathered} \end{gathered} \end{gathered}\end{gathered}$

[The rectangle with its original length and breadth i.e 80cm and 40cm respectively].

Calculation

Given, Perimeter = 240, i.e. 2(L + B) = 240

⇒ L + B = 120 cm.

Let length of the rectangle be x cm. Then, Breadth of rectangle = (120 - x) cm.

The Length is decreased by 10%, so new length is

$\rm \: x - x \times \dfrac{10}{100} = x - \dfrac{x}{10} = \underline{\boxed{ \red{\bf\dfrac{9 x}{10} \: cm}}}$

Breadth is increased by 20%, so new breadth is

$\mapsto \rm (120 - x) + (120 - x) \times \dfrac{20}{100}$

$\mapsto \rm (120 - x) + (120 - x) \times \dfrac{1}{5}$

$\mapsto \rm \dfrac{5(120 - x) + (120 - x)}{5}$

$\rm \mapsto \dfrac{600 - 5x + 120 - x}{5} = \underline{\boxed{ \red{\bf\dfrac{720 - 6x}{5} \: cm}}}$

By the condition, Perimeter remains the same i.e, 240cm.

$2\left(\begin{array}{c} \sf \dfrac{9x}{10} + \dfrac{720x - 6x}{5} \end{array}\right) = 240$

$\rightarrow \sf \dfrac{9x}{10} + \dfrac{720 - 6x}{5} = 120$

$\sf \rightarrow \dfrac{9x + 1440 - 12x}{10} = 120$

$\sf \rightarrow \: 1440 - 3x = 120 \times 10 = 1200$

$\sf \rightarrow \: - 3x = 1200 - 1440 = - 240$

$\sf \: x = \cancel \dfrac{ - 240}{ - 3} \qquad \underline{\boxed{ \purple{\bf x = 80}}}$

∴ Length of the rectangle = x = 80 cm

Breadth = (120 - x) = 120 - 80 = 40 cm

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Additional Information

Perimeter of rectangle = 2(L + B), where 'L' refers to Length while 'B' refers to Breadth of rectangle.

✦ Note

Here, in this question we have taken new length as 90x/100 while New Breadth as 120(120 - x)/100.

Answer 2

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♦ Concept :-

This question is based on is based on the concept of Area and Perimeter. so in this we are given that The Perimeter of a rectangle is 240 cm. We are asked to find the length and breadth of the rectangle. If the length is decreased by 10 % and its breadth is increased by 20 %. Let I be the length and b be the breadth of the rectangle.

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♦ Solution :-

⟹ Given that the perimetre of a rectangle is 240 cm.

⟹ Let I be the length and b be the breadth of the rectangle.

$\sf \: ⟹2(l + b) = 240$

$\sf \: ⟹l + b = \frac{240}{2}$

$\sf \: ⟹l + b = 120 \: \: \: ...(1)$

⟹ Let L be the decreased length and B be the increased breadth. Given that Length is decreased by 10%

$\sf \: ⟹thus \: decreased \: in \: lenght \: = \frac{10}{100} \times L = \frac{1}{10}$

$\sf \: ⟹therefore \: L = L - \frac{L}{10} = \frac{10 \: L - L}{10} = \frac{9 \: L}{10}$

⟹ Given that Breadth is increased by 20%

$\sf \: ⟹thus \: increase \: in \: lenght \: = \frac{20}{100} \times b = \frac{b}{5}$

$\sf \: ⟹therefore \: b = b + \frac{b}{5} = \frac{5b + b}{5} = \frac{6b}{5}$

$\sf \: ⟹the \: perimeter \: of \: new \: rectangle \: is \: = 2( \frac{9L}{10} + \frac{6b}{5})$

$\sf \: ⟹given \: that \: the \: perimeter \: of \: new \: rectangle \: is \: = 120$

$\sf \: ⟹ 2( \frac{9L}{10} + \frac{6b}{5} ) = 240$

$\sf \: ⟹ \frac{ 9L}{10} + \frac{6b}{5} = 120$

$\sf \: ⟹ \frac{ 9L}{10} + \frac{12b}{10} = 120$

$\sf \: ⟹ \frac{ 9L + 12b}{10} = 120$

$\sf \: ⟹ 9L+ 12b = 1200...(2)$

Multiplying equation (1) by 9 and subtracting from (2), we have,

$\sf \:⟹ 3b = 120$

$\sf ⟹b = 40 cm$

$\sf \: ⟹L= 120-40=80$

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♦ Answer :-

Finally, we conclude that the value for length is 80cm and the breadth of the rectangle will be 40cm

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♦ Note :-

• scroll in (➜) direction to see the full answer of your question

• Your following diagram is given in above image

• L = lenght
• B = Breadth

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