Question

The area of a rectangle gets reduced by 9 square units, if its length is reduced by 5 units and breadth is increased by 3 units. If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units. Find the dimensions of the rectangle.​

Answer 1

$\sf \underline{\large{Solution} \: - }$

☯ Let's consider length and breadth of a rectangle be x and y units respectively.

Then,

• Area of Rectangle = xy units

Now,

If the length is reduced by 5 units and breadth is increased by 3 units, area of a rectangle gets reduced by 9 square units.

$\sf \implies \: xy - 9 = (x - 5)(y + 3)$

$\sf \implies \: xy - 9 = xy + 3x - 5y - 15$

$\sf \implies \: - 9 = 3x - 5y - 15$

$\sf \implies \: - 9 + 15 = 3x - 5y$

$\sf \implies \: 6 = 3x - 5y$

$\sf \implies \: 6 - 3x = - 5y$

$\sf \implies \: \boxed{ \sf y = \dfrac{6 - 3x }{ - 5} } \: \: \: \: \: \: \: \: \: \: \: \: (Equation : 1 )$

Now,

If we increase the length by 3 units and the breadth by 2 units, the area increases by 67 square units.

$\sf \implies \: xy + 67 = (x + 3)(y + 2)$

$\sf \implies \: xy + 67 = xy + 2x + 3y + 6$

$\sf \implies \: 67 = 2x + 3y + 6$

$\sf \implies \: 67 - 6 = 2x + 3y$

$\sf \implies \: \boxed{ \sf 61 = 2x + 3y } \: \: \: \: \: \: \: \: \: \: \: \: (Equation : 2)$

Now,

Putting the (Equation : 1) in (Equation : 2)

$\sf \implies \: 61 = 2x + 3y$

$\sf \implies \: 61 = 2x + 3 \bigg( \dfrac{6 - 3x }{ - 5} \bigg)$

$\sf \implies \: 61 = 2x + \bigg( \dfrac{18 - 9x}{ - 5} \bigg)$

$\sf \implies \: 61 = 2x - \bigg( \dfrac{18 - 9x}{ 5} \bigg)$

$\sf \implies \: 61 = \dfrac{2x}{1} - \dfrac{18 + 9x}{ 5}$

$\sf \implies \: 61 = \dfrac{10x - 18 + 9x}{ 5}$

$\sf \implies \: 61 = \dfrac{19x - 18}{ 5}$

$\sf \implies \: 61 \times 5 = 19x - 18$

$\sf \implies \: 305 = 19x - 18$

$\sf \implies \: 305 + 18 = 19x$

$\sf \implies \: 323 = 19x$

$\sf \implies \: x = \dfrac{323}{19}$

$\bf \implies \: x = 17$

Now,

Put the value of x in (Equation : 1)

$\sf \implies \: y = \dfrac{6 - 3x }{ - 5}$

$\sf \implies \: y = \dfrac{6 - 3(17) }{ - 5}$

$\sf \implies \: y = \dfrac{6 - 51}{ - 5}$

$\sf \implies \: y = \dfrac{ - 45}{ - 5}$

$\bf \implies \: y = 9$

Hence, the length of the rectangle is 17 units and breadth is 9 units.