Question

1.The area of a rectangular field gets increased by 25 square units if its length is reduced by 5 units and the breadth is increased by 10 units.
If we increase the length by 8 units and decrease the breadth by 3 units, the area is reduced by 14 sq units.
i) Find the length and breadth of the rectangular field.
ii) If one tree needs 2 sq units to plant, in which rectangular field how many trees can be planted. ​

Answer 1

        $\text{\huge \bf \underline{Answer:}}$

  $i) \; \; \; \text{Let the original length and breadth of rectangle be }x \text{ and }y$

$\implies \: \text{CASE 1:}$

        $\text{Length}(l_1) = (x - 5)\: \text{units}$

        $\text{Breadth}(b_1) = (y + 10) \: \text{units}$

        $\text{We know that area of rectangle} = \text{Length} \times \text{Breadth}$

        $\text{Area}(a_1) = (xy + 25)\: \text{sq. units}$

$\implies \: (x-5)(y+10) = xy + 25$

$\implies \: xy + 10x - 5y - 50 = xy + 25$

$\implies \: xy + 10x - 5y - 50 - xy - 25 = 0$

$\implies \: 10x - 5y - 75 = 0$

$\implies \: 5(2x - y - 15) = 0$

$\implies \: 2x - y - 15 = 0 \text{ ------ (i)}$

$\implies \: \text{CASE 2:}$

        $\text{Length}(l_2) = (x + 8)\: \text{units}$

        $\text{Breadth}(b_2) = (y - 3) \: \text{units}$

        $\text{Area}(a_2) = (xy - 14)\: \text{sq. units}$

$\implies \: (x+8)(y-3) = xy - 14$

$\implies \: xy - 3x + 8y - 24 = xy - 14$

$\implies \: xy - 3x + 8y - 24 - xy + 14 = 0$

$\implies \: - 3x + 8y - 10 = 0$

$\implies \: 3x - 8y + 10 = 0 \text{ ------ (ii)}$

        $\text{Applying elimination method}$

$\implies \: \text{Multiplying (i) by 8}$

$\implies \: 8(2x - y - 15) = 0$

$\implies \: 16x - 8y - 120 = 0 \text{ ------ (iii)}$

        $\text{Substracting (ii) from (iii)}$

$\implies \: 16x - 8y - 120 - \{3x - 8y + 10\} = 0$

$\implies \: 16x - 8y - 120 - 3x + 8y - 10 = 0$

$\implies \: 13x - 130 = 0$

$\implies \: 13x = 130$

$\implies \: x = \dfrac{130}{13}$

$\implies \: \boxed{\boxed{ x= \text{Length} = 10 \: \text{units}}}$

        $\text{Substituting in (i)}$

$\implies \: 2(10) - y - 15 = 0$

$\implies \: 20- y - 15 = 0$

$\implies \: -y + 5= 0$

$\implies \: -y = -5$

$\implies \: \boxed{\boxed{y = \text{Breadth} = 5 \: \text{units}}}$

 $ii) \; \; \; \text{Area of rectangular field} = xy = 10 \times 5 = 50 \: \text{sq. units}$

        $\text{Area occupied by one tree} = 2 \: \text{sq. units}$

$\implies \: \boxed{\boxed{\text{Number of trees that can be planted} = \dfrac{50}{2} = 25}}$