The area of a rectangle increases by 200 sq, m, if the length is increased by 8 m and the breadth by 3 m. The area increases by 255 sq. m, if the length is increased by 3 m and breadth by 8m. Find the length and the breadth of the rectangle.

Answer

To solve the problem, we need to set up equations based on the information provided about the rectangle's area changes. Let's denote the length of the rectangle as \( L \) meters and the breadth as \( B \) meters.

### Step 1: Set Up the Equations
1. The area of the rectangle is given by \( A = L \times B \).
2. When the length is increased by 8 meters and the breadth by 3 meters, the new area becomes:
\[ (L + 8)(B + 3) = LB + 200 \]
Expanding this gives:
\[ LB + 3L + 8B + 24 = LB + 200 \]
Simplifying, we get:
\[ 3L + 8B + 24 = 200 \]
Thus, the first equation is:
\[ 3L + 8B = 176 \quad \text{(Equation 1)} \]

3. When the length is increased by 3 meters and the breadth by 8 meters, the new area becomes:
\[ (L + 3)(B + 8) = LB + 255 \]
Expanding this gives:
\[ LB + 8L + 3B + 24 = LB + 255 \]
Simplifying, we get:
\[ 8L + 3B + 24 = 255 \]
Thus, the second equation is:
\[ 8L + 3B = 231 \quad \text{(Equation 2)} \]

### Step 2: Solve the System of Equations
Now we have a system of two equations:
1. \( 3L + 8B = 176 \) (Equation 1)
2. \( 8L + 3B = 231 \) (Equation 2)

We can solve these equations using the elimination or substitution method. Here, we will use the elimination method.

### Step 3: Multiply and Eliminate
To eliminate \( B \), we can multiply Equation 1 by 3 and Equation 2 by 8:
- From Equation 1:
\[ 9L + 24B = 528 \quad \text{(Equation 3)} \]
- From Equation 2:
\[ 64L + 24B = 1848 \quad \text{(Equation 4)} \]

Now, we can subtract Equation 3 from Equation 4:
\[ (64L + 24B) - (9L + 24B) = 1848 - 528 \]
This simplifies to:
\[ 55L = 1320 \]
Thus, we find:
\[ L = \frac{1320}{55} = 24 \quad \text{(Length)} \]

### Step 4: Substitute to Find Breadth
Now that we have \( L = 24 \), we can substitute this value back into either Equation 1 or Equation 2 to find \( B \). We'll use Equation 1:
\[ 3(24) + 8B = 176 \]
This simplifies to:
\[ 72 + 8B = 176 \]
Subtracting 72 from both sides gives:
\[ 8B = 104 \]
Thus, we find:
\[ B = \frac{104}{8} = 13 \quad \text{(Breadth)} \]

### Final Answer
The length of the rectangle is \( 24 \) meters and the breadth is \( 13 \) meters.