Question

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

Answer 1

Hint: First, we will assume that the breadth of the original triangle is x and have the length of original rectangle is 7 more than its breadth we find the original length of the rectangle and then multiply the length by breadth to find the area of the rectangle. Now when the length is decreased by 4, we will subtract the original length by 4 to find the value of new length and when the breadth is increased by 3, so we will add the original breadth by 3 to find the value of new breadth.

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 the breadth is increased by 3 cm, the area of the new rectangle is the same as the area of the original rectangle.
Let us assume that the breadth of the original triangle is \[x\].
Since we know that the length of original rectangle is 7 more than its breadth, we find the original length of the rectangle, so we have
\[ \Rightarrow x + 7\]
Multiplying the length by breadth to find the area of the rectangle, we get
\[
   \Rightarrow \left( {x + 7} \right) \times x \\
   \Rightarrow {x^2} + 7x{\text{ ......eq.(1)}} \\
 \]
Now when the length is decreased by 4, so we will subtract the original length by 4 to find the value of new length, we get
\[
   \Rightarrow \left( {x + 7} \right) - 4 \\
   \Rightarrow x + 7 - 4 \\
   \Rightarrow x + 3 \\
 \]
So when the breadth is increased by 3, so we will add the original breadth by 3 to find the value of new breadth, we get
\[ \Rightarrow x + 3\]
Multiplying the new length by new breadth to find the area of the rectangle, we get
\[
   \Rightarrow \left( {x + 3} \right) \times \left( {x + 3} \right) \\
   \Rightarrow {\left( {x + 3} \right)^2} \\
 \]
Using the value, \[{\left( {a + b} \right)^2} = {a^2} + 2ab + {b^2}\] in the above equation, we get
\[
   \Rightarrow {x^2} + 2 \cdot 3x + {3^2} \\
   \Rightarrow {x^2} + 6x + 9{\text{ ......eq.(2)}} \\
 \]
Since we are given that the area remains the same, from equation (1) and equation (2), we get
\[ \Rightarrow {x^2} + 6x + 9 = {x^2} + 7x\]
Simplifying the above equation, we get
\[ \Rightarrow 9 = x\]

Therefore, the breadth is 9 cm and length will be \[9 + 7 = 16\] cm.

Note: There can be similar questions in which the area instead of being equal it will change. We know that the area of the rectangle is length times its breadth, do not get confused with that. We will start the solution by assuming the breadth of the rectangle by any arbitrary variable.