The length of a room floor exceeds its breadth by 20 m. The area of the floor remains unaltered when the lentgh is decresed by 10 m but the breadth is increased by 5m. The area of the floor (in square metres) is :

Options

A. 280
B. 325
C. 300
D. 420

Answer

To solve the problem step by step, let's denote the breadth of the room as \( x \) meters.

### Step 1: Define the Length
The length of the room is given to exceed the breadth by 20 meters. Therefore, we can express the length as:
\[ \text{Length} = x + 20 \]

### Step 2: Set Up the Area Equation
The area of the floor can be calculated using the formula:
\[ \text{Area} = \text{Length} \times \text{Breadth} \]
Substituting the expressions for length and breadth, we have:
\[ \text{Area} = x \times (x + 20) = x^2 + 20x \]

### Step 3: Adjust the Length and Breadth
According to the problem, the length is decreased by 10 meters and the breadth is increased by 5 meters. The new dimensions are:
- New Length = \( (x + 20) - 10 = x + 10 \)
- New Breadth = \( x + 5 \)

### Step 4: Set Up the New Area Equation
The area remains unchanged, so we can set up the equation:
\[ \text{Original Area} = \text{New Area} \]
This gives us:
\[ x^2 + 20x = (x + 10)(x + 5) \]

### Step 5: Expand the New Area
Now, let's expand the right side:
\[ (x + 10)(x + 5) = x^2 + 5x + 10x + 50 = x^2 + 15x + 50 \]

### Step 6: Set Up the Equation
Now we have:
\[ x^2 + 20x = x^2 + 15x + 50 \]

### Step 7: Simplify the Equation
Subtract \( x^2 \) from both sides:
\[ 20x = 15x + 50 \]
Now, subtract \( 15x \) from both sides:
\[ 5x = 50 \]

### Step 8: Solve for x
Dividing both sides by 5 gives:
\[ x = 10 \]

### Step 9: Calculate the Area
Now that we have the breadth, we can find the length:
\[ \text{Length} = x + 20 = 10 + 20 = 30 \]
Now, we can calculate the area:
\[ \text{Area} = \text{Length} \times \text{Breadth} = 30 \times 10 = 300 \text{ square meters} \]

### Final Answer
Thus, the area of the floor is:
\[ \boxed{300} \text{ square meters} \]

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