The expenses of carpeting a half of the floor were Rs. 759, but if the length had been 6 m less than it was , the expenses would have been Rs. 561. What is the length ?

Options

A. a. 21 m
B. b. 23 m
C. c. 45 m
D. d. 27 m

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the Variables
Let the length of the floor be \( x \) meters and the width be \( y \) meters.

### Step 2: Set Up the First Equation
According to the problem, the expenses for carpeting half of the floor is Rs. 759. The area of the floor is given by \( xy \), so the cost for half the area is:

\[ \frac{1}{2} \times xy = 759 \]

This simplifies to:

\[ xy = 1518 \quad \text{(Equation 1)} \]

### Step 3: Set Up the Second Equation
If the length is reduced by 6 meters, the new length becomes \( x - 6 \). The expenses for carpeting half of this new area is Rs. 561. Therefore, we have:

\[ \frac{1}{2} \times (x - 6) \times y = 561 \]

This simplifies to:

\[ (x - 6)y = 1122 \quad \text{(Equation 2)} \]

### Step 4: Solve the Equations
Now we have two equations:
1. \( xy = 1518 \)
2. \( (x - 6)y = 1122 \)

From Equation 1, we can express \( y \) in terms of \( x \):

\[ y = \frac{1518}{x} \]

Substituting this expression for \( y \) into Equation 2 gives:

\[ (x - 6) \left(\frac{1518}{x}\right) = 1122 \]

### Step 5: Simplify and Solve for \( x \)
Multiply both sides by \( x \) to eliminate the fraction:

\[ (x - 6) \times 1518 = 1122x \]

Expanding this gives:

\[ 1518x - 9108 = 1122x \]

Rearranging the equation:

\[ 1518x - 1122x = 9108 \]

This simplifies to:

\[ 396x = 9108 \]

Now, divide both sides by 396:

\[ x = \frac{9108}{396} = 23 \]

### Step 6: Conclusion
The length of the floor is \( 23 \) meters.

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