Question

Q. 1. The perimeter of a rectangle is 240 cm. if its 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

$\bf{\huge{\underline{\boxed{\rm{\red{Answer:}}}}}}$

Given :-

• Perimeter of the rectangle = 240 cm
• Length is decreased by 10%
• Breadth is increased by 20%

To find :-

• Length of the rectangle
• Breadth of the rectangle

Sóĺúťíóń :-

Let the length of the rectangle be x cm

Let the breadth of the rectangle be y cm

Perimeter = 240 cm, even after the length is decreased by 10% and breadth is increased by 20%

Perimeter of a rectangle = 2 ( l + b)

240 = 2 ( x+ y )

240 = 2x+ 2y

Dividing throughout by 2,

120 = x + y ------> 1

As per the question :-

Length is decreased by 10%

So, new length of the rectangle,

L = x - $\bf\large\frac{10}{100}$ × x

L = x - $\bf\large\frac{10x}{100}$

L = $\bf\large\frac{100x- 10x}{100}$

L = $\bf\large\frac{90x}{100}$

L = $\bf\large\frac{9x}{10}$

L = 0.9x

Breadth is increased by 20%

B = y + $\bf\large\frac{20}{100}$ × y

B = y + $\bf\large\frac{20y}{100}$

B = $\bf\large\frac{100y + 20y}{100}$

B = $\bf\large\frac{120y}{100}$

B = $\bf\large\frac{12y}{10}$

B = 1.2 y

Perimeter of the rectangle using the new length and breadth,

Perimeter = 2 ( l + b)

240 = 2 ( 0.9x + 1.2 y)

240 = 2 × 0.9x + 2 × 1.2y

240 = 1.8x + 2.4 y ----> 2

Since, perimeter remains unchanged, we will divide by whole equation again by 2,

120 = 0.9 x + 1.2y -----> 3

Multiplting equation 1 by -6,

-6 × 120 = -6 × x + (-6) × y

-720 = -6x - 6y -----> 4

Multiplying equation 3 by 5,

5 × 120 = 5 × 0.9x + 5 × 1.2y

600 = 4.5x + 6y -----> 5

Solve equations 4 and 5 simultaneously by elimination method.

Add equation 5 to 4

.....4.5x + 6y = 600

+ - 6x - 6y = - 720 )

------------------------------------

- 1.5x = - 120

x = $\bf\large\frac{-120}{1.5}$

x = $\bf\large\frac{120}{1.5}$

Multiplying RHS by 10,

x = $\bf\large\frac{1200}{15}$

x = 80 cm

•°• Length of the rectangle, x = 80 m

To find the breadth of the rectangle, use the formula for perimeter of the rectangle and plug in the values,

Perimeter = 240 cm

Length = 80 cm

Bredth = b

Perimeter of the rectangle = 2 ( l + b)

240 = 2 ( 80 + b)

240 = 2 × 80 + 2 × b

240 = 160 + 2b

240 - 160 = 2b

80 = 2b

$\bf\large\frac{80}{2}$ = b

40 = b

•°•Breadth of the rectangle = 40 cm

Length of the rectangle = 80 cm

$\bf{\huge{\underline{\boxed{\rm{\blue{Verification:}}}}}}$

Perimeter = 240 cm

Length = 80 cm

Breadth = 40 cm

Perimeter = 2 ( l + b)

240 = 2 ( 80 + 40)

240 = 2 × 80 + 2 × 40

240 = 160 + 80

240 = 240

LHS = RHS.

Hence verified.