Answer :

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

we know area of a rectangle is (length × breadth)

Therefore, original area of rectangle = xy

Condition 1:

The area of a rectangle gets reduced by 9 square units if its length is reduced by 5 units and the breadth is increased by 3 units.

(x - 5) (y + 3) = xy - 9

xy + 3x - 5y - 15 = xy - 9

3x - 5y = -9 + 15

3x - 5y = 6… (i)

Condition 2: If we increase the length by 3 units and breadth by 2 units, the area is increased by 67 square units.

(x+3) (y+2) = xy + 67

xy + 2x +3y + 6 = xy + 67

2x + 3y = 67-6

2x + 3y = 61 … (ii)

Multiplying (i) by 3 and (ii) by 5

3 (3x - 5y = 6)

9x - 15y = 18 … (iii)

5 (2x+3y = 61)

10x + 15y = 305 … (iv)

9x - 15y = 18

10x + 15y = 305

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

19x = 323

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

x = 323/19

x = 17

Substitute x = 17 in (i)

3x - 5y = 6

3(17) - 5y = 6

51 - 5y = 6

-5y = 6-51

-5y = -45

y = -45/-5

y = 9

.

^{ . }. The length is 17 units and the breadth is 9 units.

Verification:

Area of the rectangle is 17 x 9 = 153 square units

a. (x-5) (y+3) = (17-5) (9+3) = 12 × 12 = 144 = 153 - 9

b. (x+3) (y+2) = (17+3) (9+2) = 20 × 11= 220 = 153 + 67.

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