Question

The length of a rectangle is 5 cm less than twice its breadth . if the perimeter of rectangle is 80 cm , find its area . solve equation by the cross multiplication method.

Answer 1

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

$\:\:$

$\large{\green{\underline \bold{\tt{Given :-}}}}$

$\:\:$

• The length of a rectangle is 5 cm less than twice its breadth

$\:\:$

• Perimeter of rectangle is 80 cm

$\:\:$

$\large{\red{\underline \bold{\tt{To \: Find :-}}}}$

$\:\:$

• Its Area

$\:\:$

$\large{\orange{\underline{\tt{Solution :-}}}}$

$\:\:$

• Let the length of rectangle be "x"

• Let the breadth of rectangle be "y"

$\:\:$

$\purple{\underline \bold{According \: to \: the \ question :}}$

$\:\:$

The length of a rectangle is 5 cm less than twice its breadth

$\:\:$

➜ x = 2y - 5

$\:\:$

➜ x - 2y + 5 = 0 ------- (1)

$\:\:$

Also, perimeter of rectangle is 80 cm

$\:\:$

$\underline{\bold{\texttt{Perimeter of rectangle :}}}$

$\:\:$

➠ 2( length + breadth ) --------- (2)

$\:\:$

$\underline{\bold{\texttt{Perimeter of given rectangle :}}}$

$\:\:$

• length = x

• breadth = y

$\:\:$

⟮ Putting these values in (2) ⟯

$\:\:$

➠ 2( length + breadth )

$\:\:$

➜ 2(x + y) = 80

$\:\:$

➜ x + y = 40

$\:\:$

➜ x + y - 40 = 0 ------- (3)

$\:\:$

$\underline{\bold{\texttt{Cross multiplication :}}}$

$\:\:$

For,

$\:\:$

$\bf a_1x + b_1y + c_1 = 0$

$\bf a_2x + b_2x + c_2 = 0$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { b_1c_2 - b_2c_1 } = \dfrac { y } { c_1a_2 - c_2a_1 } = \dfrac { 1 } { b_2a_1 - b_1a_2 }---(4)}$

$\:\:$

For ,

$\:\:$

x - 2y + 5 = 0

x + y - 40 = 0

$\:\:$

• $\sf a_1 = 1$

• $\sf b_1 = -2$

• $\sf c_1 = 5$

• $\sf a_2 = 1$

• $\sf b_2 = 1$

• $\sf c_2 = -40$

$\:\:$

⟮ Putting these values in (4) ⟯

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { b_1c_2 - b_2c_1 } = \dfrac { y } { c_1a_2 - c_2a_1 } = \dfrac { 1 } { b_2a_1 - b_1a_2 }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { (-2 × -40) - (1 × 5)} = \dfrac { y } { (5 × 1) - (-40 × 1) } = \dfrac { 1 } { (1 × 1) - (-2 × 1) }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { (80) - (5)} = \dfrac { y } { (5) - (-40) } = \dfrac { 1 } { (1) - (-2) }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { 75} = \dfrac { y } { 45 } = \dfrac { 1 } { 3 }}$

$\:\:$

We got ,

$\:\:$

• $\sf \dfrac { x } { 75} = \dfrac { 1 } { 3 }$ ------ (5)

$\:\:$

• $\sf \dfrac { y } { 45 } = \dfrac { 1 } { 3 }$ ------- (6)

$\:\:$

⟮ Solving equation (5) ⟯

$\:\:$

➜ $\sf \dfrac { x } { 75} = \dfrac { 1 } { 3 }$

$\:\:$

➜ $\sf x = \dfrac { 75 } { 3 }$

$\:\:$

➨ x = 25

$\:\:$

• Hence length of rectangle is 25 cm

$\:\:$

⟮ Solving equation (6) ⟯

$\:\:$

➜ $\sf \dfrac { y } { 45 } = \dfrac { 1 } { 3 }$

$\:\:$

➜ $\sf y = \dfrac { 45 } { 3 }$

$\:\:$

➨ y = 15

$\:\:$

• Hence breadth of rectangle is 15 cm

$\:\:$

$\underline{\bold{\texttt{Area of rectangle :}}}$

$\:\:$

➠ Length × Breadth ----- (7)

$\:\:$

$\underline{\bold{\texttt{Area of given rectangle :}}}$

$\:\:$

• Length = 25 cm

• Breadth = 15 cm

$\:\:$

⟮ Putting these values in (7) ⟯

$\:\:$

➠ Length × Breadth

$\:\:$

➜ 25 × 15

$\:\:$

➨ 375 sq cm

$\:\:$

• Hence area of rectangle is 375 sq. cm

Answer 2

$\huge{\blue{\bold{\underline{\underline{Answer :}}}}}$

$\:\:$

$\large{\green{\underline \bold{\tt{Given :-}}}}$

$\:\:$

• The length of a rectangle is 5 cm less than twice its breadth

$\:\:$

• Perimeter of rectangle is 80 cm

$\:\:$

$\large{\red{\underline \bold{\tt{To \: Find :-}}}}$

$\:\:$

• Its Area

$\:\:$

$\large{\orange{\underline{\tt{Solution :-}}}}$

$\:\:$

• Let the length of rectangle be "x"

• Let the breadth of rectangle be "y"

$\:\:$

$\purple{\underline \bold{According \: to \: the \ question :}}$

$\:\:$

The length of a rectangle is 5 cm less than twice its breadth

$\:\:$

➜ x = 2y - 5

$\:\:$

➜ x - 2y + 5 = 0 ------- (1)

$\:\:$

Also, perimeter of rectangle is 80 cm

$\:\:$

$\underline{\bold{\texttt{Perimeter of rectangle :}}}$

$\:\:$

➠ 2( length + breadth ) --------- (2)

$\:\:$

$\underline{\bold{\texttt{Perimeter of given rectangle :}}}$

$\:\:$

length = x

breadth = y

$\:\:$

⟮ Putting these values in (2) ⟯

$\:\:$

➠ 2( length + breadth )

$\:\:$

➜ 2(x + y) = 80

$\:\:$

➜ x + y = 40

$\:\:$

➜ x + y - 40 = 0 ------- (3)

$\:\:$

$\underline{\bold{\texttt{Cross multiplication :}}}$

$\:\:$

For,

$\:\:$

$\bf a_1x + b_1y + c_1 = 0$

$\bf a_2x + b_2x + c_2 = 0$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { b_1c_2 - b_2c_1 } = \dfrac { y } { c_1a_2 - c_2a_1 } = \dfrac { 1 } { b_2a_1 - b_1a_2 }---(4)}$

$\:\:$

For ,

$\:\:$

x - 2y + 5 = 0

x + y - 40 = 0

$\:\:$

$\sf a_1 = 1$

$\sf b_1 = -2$

$\sf c_1 = 5$

$\sf a_2 = 1$

$\sf b_2 = 1$

$\sf c_2 = -40$

$\:\:$

⟮ Putting these values in (4) ⟯

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { b_1c_2 - b_2c_1 } = \dfrac { y } { c_1a_2 - c_2a_1 } = \dfrac { 1 } { b_2a_1 - b_1a_2 }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { (-2 × -40) - (1 × 5)} = \dfrac { y } { (5 × 1) - (-40 × 1) } = \dfrac { 1 } { (1 × 1) - (-2 × 1) }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { (80) - (5)} = \dfrac { y } { (5) - (-40) } = \dfrac { 1 } { (1) - (-2) }}$

$\:\:$

$\footnotesize{ \sf \longmapsto \dfrac { x } { 75} = \dfrac { y } { 45 } = \dfrac { 1 } { 3 }}$

$\:\:$

We got ,

$\:\:$

$\sf \dfrac { x } { 75} = \dfrac { 1 } { 3 }$ ------ (5)

$\:\:$

$\sf \dfrac { y } { 45 } = \dfrac { 1 } { 3 }$ ------- (6)

$\:\:$

⟮ Solving equation (5) ⟯

$\:\:$

➜ $\sf \dfrac { x } { 75} = \dfrac { 1 } { 3 }$

$\:\:$

➜ $\sf x = \dfrac { 75 } { 3 }$

$\:\:$

➨ x = 25

$\:\:$

Hence length of rectangle is 25 cm

$\:\:$

⟮ Solving equation (6) ⟯

$\:\:$

➜ $\sf \dfrac { y } { 45 } = \dfrac { 1 } { 3 }$

$\:\:$

➜ $\sf y = \dfrac { 45 } { 3 }$

$\:\:$

➨ y = 15

$\:\:$

Hence breadth of rectangle is 15 cm

$\:\:$

$\underline{\bold{\texttt{Area of rectangle :}}}$

$\:\:$

➠ Length × Breadth ----- (7)

$\:\:$

$\underline{\bold{\texttt{Area of given rectangle :}}}$

$\:\:$

Length = 25 cm

Breadth = 15 cm

$\:\:$

⟮ Putting these values in (7) ⟯

$\:\:$

➠ Length × Breadth

$\:\:$

➜ 25 × 15

$\:\:$

➨ 375 sq cm

$\:\:$

• Hence area of rectangle is 375 sq. cm