If the length and breadth of a rectangular plot are each increased by 1 m, then the area of the floor is increased by 21 sq m. If the length is increased by 1 m and breadth is decreased by 1 m, then the area is decreased by 5 sq m. What is is the perimeter of the floor ?

Options

A. 30 m
B. 32 m
C. 36 m
D. 40 m

Answer

To solve the problem step by step, we will denote the length of the rectangular plot as \( L \) meters and the breadth as \( B \) meters.

### Step 1: Set up the equations based on the problem statement.

1. **First Condition**: When both length and breadth are increased by 1 meter, the area increases by 21 square meters.
- The original area \( A = L \times B \).
- The new area after increasing both dimensions:
\[ (L + 1)(B + 1) = LB + 21 \]
- Expanding the left side:
\[ LB + L + B + 1 = LB + 21 \]
- Simplifying gives:
\[ L + B + 1 = 21 \]
- Thus, we can rearrange it to:
\[ L + B = 20 \quad \text{(Equation 1)} \]

2. **Second Condition**: When the length is increased by 1 meter and the breadth is decreased by 1 meter, the area decreases by 5 square meters.
- The new area in this case:
\[ (L + 1)(B - 1) = LB - 5 \]
- Expanding the left side:
\[ LB - L + B - 1 = LB - 5 \]
- Simplifying gives:
\[ -L + B - 1 = -5 \]
- Rearranging leads to:
\[ -L + B = -4 \quad \text{or} \quad B - L = 4 \quad \text{(Equation 2)} \]

### Step 2: Solve the system of equations.

We now have two equations:
1. \( L + B = 20 \) (Equation 1)
2. \( B - L = 4 \) (Equation 2)

From Equation 2, we can express \( B \) in terms of \( L \):
\[ B = L + 4 \]

Substituting this expression for \( B \) into Equation 1:
\[ L + (L + 4) = 20 \]
\[ 2L + 4 = 20 \]
Subtracting 4 from both sides:
\[ 2L = 16 \]
Dividing by 2:
\[ L = 8 \]

Now substituting \( L = 8 \) back into Equation 1 to find \( B \):
\[ 8 + B = 20 \]
\[ B = 20 - 8 = 12 \]

### Step 3: Calculate the perimeter.

The perimeter \( P \) of a rectangle is given by the formula:
\[ P = 2(L + B) \]
Substituting the values of \( L \) and \( B \):
\[ P = 2(8 + 12) = 2 \times 20 = 40 \text{ meters} \]

### Final Answer:
The perimeter of the floor is **40 meters**.

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