Question

The length of a rectangle exceeds its breadth by 7 cm. If the length is decreased by 4 cm and breadth is increased by 3, the new rectangle is the same as the area of the original rectangle. Find the length and the breadth of the original rectangle.

Answer 1

Hint: The area of the rectangle is length times its breadth. Solution can be started by assuming the breadth of the rectangle any arbitrary variable.

Complete step-by-step answer:
We are given that the length of a rectangle exceeds its breadth by 7 cm. If the length is decreased by 4 cm and breadth is increased by 3, the new rectangle is the same as the area of the original rectangle.
Let us start by assuming the breadth of original triangle as $x$ i.e.
$b = x$ ….. (1)
The length of original rectangle is 7 more than its breadth, i.e.
$l = x + 7$ ……. (2)
From equation (1) and (2) we get the area of the rectangle as –
$A = l \times b$
$A = (x + 7) \times x$
$A = {x^2} + 7x$ ………… (3)
Now, the lengths and breadth of the rectangle are change as –
Length is decreased by 4 i.e.
New length (L) = Original length +7
                     $\begin{gathered}
   = (x + 7) - 4 \\
   = x + 3.............(4) \\
\end{gathered} $
New breadth (B) = Original breadth +3
                         $ = (x) + 3.......(5)$
The area of the rectangle now becomes –
\[A = L \times B\]
From equation (4) and (5) we have,
\[A = (x + 3) \times (x + 3)\]
Simplifying the above equation we have,
\[A = {x^2} + 3x + 3x + 9\]
\[A = {x^2} + 6x + 9\] ……….. (6)
It is given that the equation area remains the same. Therefore,
From equation (3) and (6) we get,
\[{x^2} + 6x + 9 = {x^2} + 7x\]
Subtracting both sides by \[{x^2}\]
\[6x + 9 = 7x\]
Subtracting both sides by \[6x\] we get,
\[9 = x\]
Therefore, putting the value of x in equation (1) and two we get the length and breadth as follow –
$b = x = 9$ And,
$l = x + 7 = 9 + 7 = 16$

Therefore, the length is 16cm and the breadth is 9cm.

Note: There can be similar questions in which the area instead of being equal it will change. In such cases we form a quadratic equation and solve it by middle term split. Middle term split method is used to split the variables into 2 parts.