If the length of a rectangle decreases by 5 m and breadth increases by 3 m, then its area reduces 9 sq m. If length and breadth of this rectangle increased by 3 m and 2 m respectively, then its area increased by 67 sq m. What is the length of rectangle?

Options

A. 9 m
B. 15.6 m
C. 17 m
D. 18.5 m

Answer

To solve the problem, we need to set up equations based on the information provided in the question.

Let the length of the rectangle be \( x \) meters and the breadth be \( y \) meters.

### Step 1: Set up the first equation based on the first condition
According to the problem, if the length decreases by 5 meters and the breadth increases by 3 meters, the area reduces by 9 square meters.

The original area of the rectangle is:
\[ A_1 = x \times y \]

The new dimensions after the changes are:
- New length = \( x - 5 \)
- New breadth = \( y + 3 \)

The new area is:
\[ A_2 = (x - 5)(y + 3) \]

According to the problem:
\[ A_1 - A_2 = 9 \]
Substituting the areas:
\[ xy - (x - 5)(y + 3) = 9 \]

Expanding the equation:
\[ xy - (xy + 3x - 5y - 15) = 9 \]
\[ xy - xy - 3x + 5y + 15 = 9 \]
\[ -3x + 5y + 15 = 9 \]
\[ -3x + 5y = -6 \]
Dividing through by -1:
\[ 3x - 5y = 6 \quad \text{(Equation 1)} \]

### Step 2: Set up the second equation based on the second condition
Now, if the length increases by 3 meters and the breadth increases by 2 meters, the area increases by 67 square meters.

The new dimensions after these changes are:
- New length = \( x + 3 \)
- New breadth = \( y + 2 \)

The new area is:
\[ A_3 = (x + 3)(y + 2) \]

According to the problem:
\[ A_3 - A_1 = 67 \]
Substituting the areas:
\[ (x + 3)(y + 2) - xy = 67 \]

Expanding the equation:
\[ xy + 2x + 3y + 6 - xy = 67 \]
\[ 2x + 3y + 6 = 67 \]
\[ 2x + 3y = 61 \quad \text{(Equation 2)} \]

### Step 3: Solve the system of equations
Now we have two equations:
1. \( 3x - 5y = 6 \) (Equation 1)
2. \( 2x + 3y = 61 \) (Equation 2)

We can solve these equations simultaneously.

From Equation 1, express \( y \) in terms of \( x \):
\[ 5y = 3x - 6 \]
\[ y = \frac{3x - 6}{5} \]

Substituting \( y \) into Equation 2:
\[ 2x + 3\left(\frac{3x - 6}{5}\right) = 61 \]
Multiplying through by 5 to eliminate the fraction:
\[ 10x + 3(3x - 6) = 305 \]
\[ 10x + 9x - 18 = 305 \]
\[ 19x - 18 = 305 \]
\[ 19x = 323 \]
\[ x = \frac{323}{19} = 17 \]

### Conclusion
The length of the rectangle is \( 17 \) meters.