Question

The area of a rectangle remains the same if the length is increased by 7 meters and the breadth is decreased by 3 meters. The area remains unaffected if the length is decreased by 7 meters and breadth is increased by 5 meters. Find the dimensions of the rectangle.

Answer 1

Hint - We will start by assuming two variables as x and y then make two different equations by using the information which is mentioned in the question then after solving those equations we will find the required values of x and y.

Complete step-by-step answer:
It is given that the area of a rectangle remains the same when the length is increased by 7 meters and the breadth is decreased by 3 meters and the area remains unaffected when the length is decreased by 7 meters and breadth is increased by 5 meters, so by using this we can make two different equations.
For which, let the length of the rectangle be $x$ units and the breadth be $y$ units.

We know that,
Area of the rectangle = Length $ \times $ Breadth
                                      $ = x \times y$
                                      $ = xy$ sq. units
Now, we will make equations from the given information.
According to first case the equation will be,
$\left( {x + 7} \right) \times \left( {y - 3} \right) = xy$……… (1)
And, according to the second case the equation will be,
$\left( {x - 7} \right) \times \left( {y + 5} \right) = xy$ ……... (2)
Taking equation (1),
$\left( {x + 7} \right) \times \left( {y - 3} \right) = xy$
$
   \Rightarrow xy - 3x + 7y - 21 = xy \\
   \Rightarrow - 3x + 7y - 21 = 0 \\
   \Rightarrow - 3x + 7y = 21 \\
 $
$ \Rightarrow 7y - 3x = 21$ …………. (i)
Now, taking equation (2),
$\left( {x - 7} \right) \times \left( {y + 5} \right) = xy$
$
   \Rightarrow xy + 5x - 7y - 35 = xy \\
   \Rightarrow 5x - 7y = 35 \\
 $
$ \Rightarrow - 7y + 5x = 35$ …………. (ii)
Adding (i) and (ii), we get
$7y - 3x = 21$
$ - 7y + 5x = 35$
$\overline {0 + 2x = 56} $
       $
  2x = 56 \\
  x = \dfrac{{56}}{2} \\
  x = 28 \\
 $
Putting $x = 28$ in equation (i), we get,
$
  7y - 3\left( {28} \right) = 21 \\
   \Rightarrow 7y - 84 = 21 \\
   \Rightarrow 7y = 21 + 84 \\
   \Rightarrow 7y = 105 \\
   \Rightarrow y = \dfrac{{105}}{7} \\
   \Rightarrow y = 15 \\
 $
Hence, length of the rectangle $ = x = 28$m
and, breadth of the rectangle $ = y = 15$m

Note - A rectangle is a quadrilateral with four straight sides and four right angles. It has unequal adjacent sides, in contrast to a square. These kinds of questions are very simple and easy to solve if one understands the question properly and knows how to solve the equations.