Length of a rectangle is how much percent more than its breadth if area of rectangular plot is 480 sq. m. and when each side has been 5 m longer, the area would have been increased by 245 sq. m.?

Options

A. A)`50% `
B. B)`20% `
C. C)`10% `
D. D)`80% `

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the variables
Let the length of the rectangle be \( L \) meters and the breadth be \( B \) meters.

### Step 2: Set up the equations
From the problem, we know:
1. The area of the rectangle is given by the equation:
\[ L \times B = 480 \quad \text{(1)} \]
2. When each side is increased by 5 meters, the new area becomes:
\[ (L + 5)(B + 5) = 480 + 245 = 725 \quad \text{(2)} \]

### Step 3: Expand equation (2)
Expanding equation (2):
\[ (L + 5)(B + 5) = LB + 5L + 5B + 25 = 725 \]
Substituting \( LB \) from equation (1):
\[ 480 + 5L + 5B + 25 = 725 \]
This simplifies to:
\[ 5L + 5B + 505 = 725 \]
Subtracting 505 from both sides gives:
\[ 5L + 5B = 220 \]
Dividing the entire equation by 5:
\[ L + B = 44 \quad \text{(3)} \]

### Step 4: Solve the system of equations
Now we have two equations:
1. \( L \times B = 480 \) (from equation (1))
2. \( L + B = 44 \) (from equation (3))

We can express \( B \) in terms of \( L \) from equation (3):
\[ B = 44 - L \]

Substituting this into equation (1):
\[ L(44 - L) = 480 \]
Expanding this gives:
\[ 44L - L^2 = 480 \]
Rearranging it into standard quadratic form:
\[ L^2 - 44L + 480 = 0 \]

### Step 5: Solve the quadratic equation
We can use the quadratic formula \( L = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = -44, c = 480 \):
\[ L = \frac{44 \pm \sqrt{(-44)^2 - 4 \cdot 1 \cdot 480}}{2 \cdot 1} \]
Calculating the discriminant:
\[ L = \frac{44 \pm \sqrt{1936 - 1920}}{2} \]
\[ L = \frac{44 \pm \sqrt{16}}{2} \]
\[ L = \frac{44 \pm 4}{2} \]
Calculating the two possible values for \( L \):
1. \( L = \frac{48}{2} = 24 \)
2. \( L = \frac{40}{2} = 20 \)

### Step 6: Find the breadth
Using \( L = 24 \):
\[ B = 44 - 24 = 20 \]
Using \( L = 20 \):
\[ B = 44 - 20 = 24 \]
Thus, the length \( L = 24 \) m and breadth \( B = 20 \) m.

### Step 7: Calculate the percentage increase
Now, we need to find how much percent more the length is than the breadth:
\[ \text{Difference} = L - B = 24 - 20 = 4 \]
The percentage increase is given by:
\[ \text{Percentage} = \left(\frac{\text{Difference}}{B}\right) \times 100 = \left(\frac{4}{20}\right) \times 100 = 20\% \]

### Final Answer
The length of the rectangle is **20%** more than its breadth.

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