The area of a rectangle gets reduced by a 9 square units, if its length is reduced by 5 units and the breadth is increased by 3 units. The area is increased by 67 sqaure units if length is increased by 3 units and breadth is increased by 2 units. Find the perimeter of the rectangle.

Answer

To solve the problem step by step, we will define the variables, set up equations based on the conditions provided, and then solve those equations to find the dimensions of the rectangle and its perimeter.

### Step 1: Define Variables
Let:
- \( L \) = length of the rectangle
- \( B \) = breadth of the rectangle

### Step 2: Set Up the First Equation
According to the problem, the area of the rectangle decreases by 9 square units when the length is reduced by 5 units and the breadth is increased by 3 units.

The area of the rectangle is given by:
\[ A = L \times B \]

When the length is reduced by 5 and the breadth is increased by 3, the new area becomes:
\[ (L - 5)(B + 3) \]

Setting up the equation based on the information given:
\[ L \times B - 9 = (L - 5)(B + 3) \]

Expanding the right-hand side:
\[ L \times B - 9 = LB + 3L - 5B - 15 \]

Now, simplifying this equation:
\[ LB - 9 = LB + 3L - 5B - 15 \]

Cancelling \( LB \) from both sides:
\[ -9 = 3L - 5B - 15 \]

Rearranging gives us:
\[ 3L - 5B = 6 \] (Equation 1)

### Step 3: Set Up the Second Equation
The problem also states that the area increases by 67 square units when the length is increased by 3 units and the breadth is increased by 2 units.

The new area in this case is:
\[ (L + 3)(B + 2) \]

Setting up the equation based on this information:
\[ L \times B + 67 = (L + 3)(B + 2) \]

Expanding the right-hand side:
\[ LB + 67 = LB + 2L + 3B + 6 \]

Cancelling \( LB \) from both sides:
\[ 67 = 2L + 3B + 6 \]

Rearranging gives us:
\[ 2L + 3B = 61 \] (Equation 2)

### Step 4: Solve the System of Equations
Now we have a system of linear equations:
1. \( 3L - 5B = 6 \) (Equation 1)
2. \( 2L + 3B = 61 \) (Equation 2)

To eliminate one variable, we can multiply Equation 1 by 2 and Equation 2 by 3:
- \( 2(3L - 5B) = 2(6) \) → \( 6L - 10B = 12 \) (Equation 3)
- \( 3(2L + 3B) = 3(61) \) → \( 6L + 9B = 183 \) (Equation 4)

Now, we subtract Equation 3 from Equation 4:
\[ (6L + 9B) - (6L - 10B) = 183 - 12 \]
\[ 6L + 9B - 6L + 10B = 171 \]
\[ 19B = 171 \]
\[ B = 9 \]

### Step 5: Find the Length
Now that we have \( B = 9 \), we can substitute this value back into either Equation 1 or Equation 2 to find \( L \). Let's use Equation 2:
\[ 2L + 3(9) = 61 \]
\[ 2L + 27 = 61 \]
\[ 2L = 34 \]
\[ L = 17 \]

### Step 6: Calculate the Perimeter
The perimeter \( P \) of a rectangle is given by:
\[ P = 2(L + B) \]
Substituting the values of \( L \) and \( B \):
\[ P = 2(17 + 9) \]
\[ P = 2(26) \]
\[ P = 52 \]

### Final Answer
The perimeter of the rectangle is **52 units**.