Question

DER THINKING SKILLS
1. The perimeter of a rectangle is 240 cm. If its length in decreased by 10% and
breadth in increased by 20%, we get the same perimeter. Find the length
and breadth of the rectangle.
QUESTIONS find by one variable​

Answer 1

❍ Let the Length be l cm and Breadth be b cm of the Rectangle.

¤ We're given with the perimeter of the rectangle, which is 240 cm. To find out the perimeter of the rectangle the formula is given by :

❏ $\underline{\boxed{\sf{Perimeter = 2\Big(Length + Breadth\Big)}}}$

$\dashrightarrow\sf 2\Big(Length + Breadth\Big) = 240 \\\\\\\dashrightarrow\sf l + b = \cancel\dfrac{240}{2}\\\\\\\dashrightarrow\sf l + b = 120\\\\\\\dashrightarrow\sf l = 120 - b\qquad\qquad\bigg\lgroup\sf eq^n\:(1) \bigg\rgroup$

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$\underline{\bigstar\:\textsf{According to the Question Now :}}$

• If the length of the rectangle is decreased by 10 % and the breadth of the rectangle is increased by 20 %, we get the same perimeter.

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$\dashrightarrow\sf l - l \times \cancel\dfrac{10}{100}\\\\\\\dashrightarrow\sf l - l \dfrac{1}{10}\\\\\\\dashrightarrow\sf \dfrac{10l - l}{10}\\\\\\\dashrightarrow{\pmb{\bf{\dfrac{9l}{10}}}}$

&

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$\dashrightarrow\sf b + b \times \cancel\dfrac{20}{100}\\\\\\\dashrightarrow\sf b + b \times \dfrac{1}{5}\\\\\\\dashrightarrow\sf \dfrac{5b + b}{5}\\\\\\\dashrightarrow{\pmb{\bf{ \dfrac{6b}{5}}}}$

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✇ Now, by putting the value of length and breadth in the above formula —

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$\dashrightarrow\sf 2\Bigg\{\dfrac{9l}{10} + \dfrac{6b}{5}\Bigg\} = 240\\\\\\\dashrightarrow\sf \Bigg\{\dfrac{9l + 12b}{10}\Bigg\} = 120\\\\\\\dashrightarrow\sf 9l + 12b = 1200\\\\\\\dashrightarrow\sf 3\Big\{3l + 4b\Big\} = 1200\\\\\\\dashrightarrow\sf 3l + 4b = 400 \\\\{\scriptsize\qquad\bf{\dag}\:\:\textsf{Putting value of l from eq.(1)}}\\\\\dashrightarrow\sf 3\Big\{120 - b\Big\} + 4b = 400\\\\\\\dashrightarrow\sf 360 - 3b + 4b = 400 \\\\\\\dashrightarrow\sf b = 400 - 360 \\\\\\\dashrightarrow\underline{\boxed{\pmb{\frak{b = 40}}}}\;\bigstar$

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✇ We know the, value of Breadth, now putting value of 'b' in eqⁿ ( 1 ) to find the value of 'l' —

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$\dashrightarrow\sf l = 120 - b\\\\\\\dashrightarrow\sf l = 120 - 40\\\\\\\dashrightarrow\sf \underline{\boxed{\pmb{\frak{l = 80}}}}\;\bigstar$

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∴ Hence, Length and Breadth of the rectangle are 80 cm and 40 cm respectively.

Answer 2

                                                                                   

Given :-

The perimeter of a rectangle is 240 cm. If its length in decreased by 10% and

breadth in increased by 20%, we get the same perimeter.

To Find :-

Length and breadth

Solution :-

Let the length be x and breadth be y

For length

Length decreased by 10%

x - 10% of x

x - 10/100 of x

x - 10/100 × x

x - 1/10 × x

x - x/10

10x - x/10

9x/10

For Breadth

Breadth increased by 20%

y + 20% of y

y + 20/100 of y

y + 20/100 × y

y + 2/10 × y

y + 2y/10

10y + 2y/10

12y/10

Given,

Perimeter is 240 cm

240 = 2(x + y)

240/2 = x + y

120 = x + y

120 - y = x (1)

240 = 2(9x/10 + 12y/10)

240/2 = (9x + 12y/10)

120 = 9x + 12y/10

120 × 10 = 9x + 12y

1200 = 9x + 12y

Divide both sides by 3

1200/3 = 9x + 12y/3

400 = 3x + 4y

400 = 3(120 - y) + 4y

400 = 360 - 3y + 4y

400 - 360 = -3y + 4y

40 cm = y

Now

For length

x + y = 120

x + 40 = 120

x = 120 - 40

x = 80 cm