Question

In the length is increased by 3 cm and the breadth is increased by 2cm , the area of a rectangle increase by 67cm² . But, if the length is reduced by 5cm and the breadth is increased by 3cm , the area reduces by 9cm². The area of the rectangle is

Answer 1

Answer:

$\large{\underline{\boxed{\sf Area \: of \: rectangle = 153 \: cm^2 }}}$

Step-by-step explanation:

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

Area of rectangle = xy cm²

Given that;

The length is increased by 3 cm.

• Increased length = (x + 3) cm

The breadth is increased by 2 cm.

• Increased breadth = (y +2) cm

Area of rectangle is increased by 67 cm².

• Increased area = (xy + 67) cm²

New area = Increased(Length * breadth)

⇒ (x + 3)(y + 2) = xy + 67

⇒ xy + 2x + 3y + 6 = xy + 67

⇒ xy + 2x + 3y  - xy = 67 - 6

⇒ 2x + 3y = 61 .... (i)

_______________________

Second situation;

Length is reduced by 5 cm

• Decreased length = (x - 5) cm

Breadth is increased by 3 cm

• Increased breadth = (y + 3) cm

Area reduces by 9 cm²

• Decreased area = (xy - 9) cm²

New area = Decreased length * increased length

⇒ (x - 5)(y + 3) = xy - 9

⇒ xy + 3x - 5y - 15 = xy - 9

⇒ xy + 3x - 5y - xy = 15 - 9

⇒ 3x - 5y = 6 .... (ii)

_______________________

We got two equations here,

• 2x + 3y = 61
• 3x - 5y = 6

On multiplying equation (i) by 3 and equation (ii) by 2.

(i) 3(2x + 3y) = 3 * 61

⇒ 6x + 9y = 183  .... (iii)

(ii) 2(3x - 5y) = 2 * 6

⇒ 6x - 10y = 12..... (iv)

On Subtracting equation (iii) from (iv) -

⇒ 6x - 10y - 6x - 9y = 12 - 183

⇒ - 19y = - 171

⇒ y = $\sf \dfrac{171}{19}$

⇒ y = 9

On Substituting the value of y in (i),

⇒ 2x + 3y = 61

⇒ 2x + 3 * 9 = 61

⇒ 2x = 61 - 27

⇒ 2x = 34

⇒ x = $\sf \dfrac{34}{2}$

⇒ x = 17

_______________________

Therefore,

Length of rectangle = 17 cm

Breadth of rectangle = 9 cm

Area of rectangle = xy

                             = 17 * 9

                             = 153 cm²

_______________________

Hence, the area of rectangle is 153 cm².

Answer 2

Answer:-

l = 17cm

b = 9cm

A = 153cm²

Solution:-

Let length of rectangle be l and breadth be b.

Then,

Area of rectangle will be l × b

Case 1:-

If the length is increased by 3 cm and breadth is increased by 2 cm area of rectangle increases by 67cm².

$(l +3) (b+2) = lb + 67$

$lb + 2l +3b +6 = lb + 67$

$\implies$ $2l +3b = 67 -6$

$\implies$ $2l +3b = 61$------eq1.

Case 2:-

If the length is reduced by 5 cm and breadth is increased by 3 cm, the area reduces by 9cm².

$(l -5) (b+3) = lb -9$

$lb+3l -5b -15 = lb -9$

$\implies$ $3l -5b = -9 + 15$

$\implies$ $3l -5b = 6$ ------eq2.

By multiplying eq1. by 3 and eq.2 by 2.

$(2l + 3b =61) \times3 \\(3l -5b =6 )\times 2$

$6l + 9b = 183 \\ 6l -10b = 12$

Subtract eq. 1 and eq. 2

$6l -6l +9b +10b = 183-12$

$\implies$ $19b = 171$

$\implies$ $b =\dfrac {171}{19}$

$\implies$ $b = 9 cm$

put the value of b in eq.1

$\implies$ $6l +9\times9 = 183$

$\implies$ $6l +81 = 183$

$\implies$ $6l = 183-81$

$\implies$ $6l = 102$

$\implies$ $l = \dfrac{102}{6}$

$\implies$ $l = 17 cm$

hence area of rectangle is l. b

= 17 × 9

= 153cm²