Question

the perimeter of a rectangle is 360m. if its length is decreased by 20% and breadth is increased by 25%. we get the same perimeter. find the dimensions of the rectangle.

Answer 1

$\huge\sf\pink{Answer}$

☞ Length = 100 m

☞ Breadth = 80 m

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$\huge\sf\blue{Given}$

✭ Perimeter of a rectangle is 360 m

✭ If the perimeter is decreased by 20% & the breadth is increased by 25% the perimeter is the same

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$\huge\sf\gray{To \:Find}$

◈ The dimensions of the rectangle?

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$\huge\sf\purple{Steps}$

We know that perimeter of a rectangle is given by,

$\underline{\boxed{\sf Perimeter_{Rectangle} = 2(l+b)}}$

• Perimeter = 360 m

➢ $\sf2(l+b) = 360$

➢ $\sf l+b = \dfrac{360}{2}$

➢ $\sf l+b = 180 \:\:\ -eq(1)$

☯ $\underline{\boldsymbol{According \ to \ the \ Question}}$

When the length is decreased by 20%

➝ $\sf l-\dfrac{20l}{100}$

➝ $\sf\dfrac{100l-20l}{100}$

➝ $\sf\dfrac{80l}{100}$

➝ $\sf\dfrac{8l}{10}$

When the breadth is increased by 25%

➝ $\sf b+\dfrac{25b}{100}$

➝ $\sf b+\dfrac{1b}{4}$

➝ $\sf\dfrac{4b+1b}{4}$

➝ $\sf\dfrac{5b}{4}$

So now given that their perimeter is 360

➳ $\sf2\bigg\lgroup \dfrac{8l}{10} + \dfrac{5b}{4}\bigg\rgroup = 360$

➳ $\sf2\bigg\lgroup\dfrac{16l+25b}{20}\bigg\rgroup = 360$

➳ $\sf16l+25b = 360\times 10$

➳ $\sf16l+25b = 3600 \:\:\ -eq(2)$

Multiplying eq(1) by 16 and subtracting it from eq(2)

➠ $\sf(16l+25b)-(16l+16b) = 3600-2880$

➠ $\sf16l+25b-16l-16b = 720$

➠ $\sf9b = 720$

➠ $\sf b = \dfrac{720}{9}$

➠ $\sf\red{Breadth = 80 \ m}$

Substituting the value of b in eq(1)

»» $\sf l+b =180$

»» $\sf l = 180-80$

»» $\sf \orange{Length = 100 \ m}$

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Answer 2

$\pink{\large{\underline{\underline{ \rm{Given: }}}}}$

✿ Perimeter of a rectangle is 360m.

✿ The length is decreased by 20% and breadth is increased by 25%.

✿ Perimeter is same in both the cases.

$\pink{\large{\underline{\underline{ \rm{To \: Find: }}}}}$

⚜ The dimensions i.e., length and breadth of a rectangle.

$\pink{\large{\underline{\underline{ \rm{Solution: }}}}}$

Let length and breadth of the rectangle be x and y.

We know the perimeter of rectangle = $\green{ \underline{ \boxed{ \rm{2(l + b)}}}}$

$\tt{\therefore{2(x + y) = 360}}$

$\tt{x + y = \frac{360}{2} }$

$\tt{x + y = 180}$

By Multiplying both sides by 16, We get:

$\tt{x \times 16 + y \times 16 = 180 \times 16}$

$\tt{16x + 16y = 2880}$ ..........①

As per the question :

When the length is decreased by 20%

$\tt{x - 20\%}$

$\tt{= x - \frac{20}{10}x }$

$\tt{ = \frac{100 - 20}{100} x}$

$\tt{ = \frac{80}{100} </u></strong><strong><u>x</u></strong><strong><u> = \frac{8}{10}</u></strong><strong><u>x</u></strong><strong><u> }$

When the length is increased by 25%

$\tt{y + \frac{25}{100} }$

$\tt{ = y + \frac{1}{4} }$

$\tt{ = \frac{4 + 1}{4} y}$

$\tt{ = \frac{5}{4} y}$

We know that the perimeter is 360°

We have,

$\tt{2( \frac{8}{10} x + \frac{5}{4} y )= 360 }$

$\tt{2( \frac{16x + 25y}{20}) = 360 }$

Now 2 and 20 gets cancel, we have:

$\tt{ \frac{16x + 25y}{10} = 360 }$

$\tt{16x + 25y = 360 \times 10}$

$\tt{16x + 25y = 3600 }$ .........②

Now Substract eq① from eq②:

$\tt{(16x + 25y) - (16x + 16y) = 3600 - 2800}$

$\tt{16x + 25y - 16x - 16y = 720}$

$\tt{9y = 720}$

$\tt{y = \frac{720}{9}}$

$\tt{y = 80 }$

∴ Breadth of a rectangle = $\green{ \underline{ \boxed{ \rm{80 \: m}}}}$

Substituting the value of y i.e., breath in eq①.

$\tt{l + b = 180}$

$\tt{l + b = 180 - 80}$

$\tt{l = 100 \: m}$

∴ Length of a rectangle = $\green{ \underline{ \boxed{ \rm{100 \: m}}}}$