Question

The width of a rectangle is two-third of its length. If the perimeter is 180 meters ,find the dimensions of the rectangle.

Answer 1

$\huge{\underline{\purple{\rm{Solution:}}}}$

$\small{\underline{\red{\rm{Given:}}}}$

$\rm{The\: perimeter\:of\: rectangle=180\:m}$

$\large{\underline{\red{\rm{Let:}}}}$

$\rm{The\: length\:of\: rectangle=x\:m}$

$\rm{The\: breadth\:of\: rectangle=\frac{2}{3}x\:m}$

$\small{\underline{\red{\rm{To\: Find:}}}}$

$\rm{The\: dimensions\:of\: rectangle}$

$\small{\underline{\red{\rm{As\:per\:the\: above\: information:}}}}$

$\rm\purple{\implies Perimeter=2(l+b)}$

$\rm{\implies 2(x+\dfrac{2}{3}x)=180}$

$\rm{\implies 2(\dfrac{3x+2x}{3})=180}$

$\rm{\implies 2(\dfrac{5x}{3})=180}$

$\rm{\implies \cancel{10}x=\cancel{180}*3}$

$\rm{\implies x=18*3}$

$\rm\red{\implies x=54\:m}$

Hence,

$\rm\orange{\implies The\: length\:of\: rectangle=x=54\:m}$

$\rm{\implies The\: breadth\:of\: rectangle=\dfrac{2}{3}x=\dfrac{2}{3}*54}$

$\rm{\implies The\: breadth\:of\: rectangle=2*18\:m}$

$\rm\purple{\implies The\: breadth\:of\: rectangle=36\:m}$

$\huge{\underline{\purple{\rm{Verification:}}}}$

$\rm{\implies Perimeter\:of\: rectangle=2(l+b)}$

$\rm{\implies 2(54+36)=180}$

$\rm{\implies 2*90=180}$

$\rm\green{\implies 180=180}$

$\rm{\implies Hence,\:Verified}$

Answer 2

Answer:

⋆ DIAGRAM :

$\setlength{\unitlength}{1.5cm}\begin{picture}(8,2)\thicklines\put(7.7,3){\large{A}}\put(7.5,2){\large{ \sf{}^{2n}\!/{}_{3} }}\put(7.7,1){\large{B}}\put(9.5,0.7){\sf{\large{n}}}\put(11.1,1){\large{C}}\put(8,1){\line(1,0){3}}\put(8,1){\line(0,2){2}}\put(11,1){\line(0,3){2}}\put(8,3){\line(3,0){3}}\put(11.1,3){\large{D}}\end{picture}$

$\rule{150}{1}$

Let the Length be n and Width be ⅔n of the Rectangle respectively.

$\underline{\bigstar\:\textbf{According to the Question :}}$

$:\implies\sf Perimeter_{(Rectangle)}=2(Length+Width)\\\\\\:\implies\sf 180\:m=2\bigg\lgroup n+\dfrac{2n}{3}\bigg\rgroup\\\\\\:\implies\sf 90\:m=\bigg\lgroup\dfrac{3n + 2n}{3}\bigg\rgroup\\\\\\:\implies\sf 90\:m= \dfrac{5n}{3} \\\\\\:\implies\sf 90\:m \times \dfrac{3}{5} = n\\\\\\:\implies\sf 18\:m \times 3 = n\\\\\\:\implies\underline{\boxed{\sf n = 54 \:m}}$

$\rule{180}{2}$

$\underline{\bigstar\:\textbf{Dimensions of the Rectangle :}}$

$\bullet\:\:\textsf{Length = n = \textbf{54 m}}\\\bullet\:\:\sf Width={}^{2n}\!/{}_{3}={}^{2(54)}\!/{}_{3}=\textsf{\textbf{36 m}}$