There are two rectangular fields of same area. The length of first rectangular field is x% less than the length of the second field and breadth of the first field is ( 5x)% greater than the breadth of the second field. What is the value of x ?

Options

A. A)15
B. B)25
C. C)50
D. D)80

Answer

To solve the problem, let's break it down step by step.

### Step 1: Define the dimensions of the second rectangular field
Let the length of the second rectangular field be \( L_2 = 100 \) meters and the breadth be \( B_2 = 100 \) meters.

### Step 2: Express the dimensions of the first rectangular field
According to the problem:
- The length of the first rectangular field is \( x\% \) less than the length of the second field:
\[ L_1 = L_2 - \left(\frac{x}{100} \times L_2\right) = 100 - \frac{x}{100} \times 100 = 100 - x \]

- The breadth of the first rectangular field is \( (5x)\% \) greater than the breadth of the second field:
\[ B_1 = B_2 + \left(\frac{5x}{100} \times B_2\right) = 100 + \frac{5x}{100} \times 100 = 100 + 5x \]

### Step 3: Set up the equation for the areas of both fields
Since the areas of both rectangular fields are equal, we can set up the equation:
\[ L_1 \times B_1 = L_2 \times B_2 \]
Substituting the values we found:
\[ (100 - x)(100 + 5x) = 100 \times 100 \]
This simplifies to:
\[ (100 - x)(100 + 5x) = 10000 \]

### Step 4: Expand the left-hand side
Expanding the left-hand side:
\[ 10000 + 500x - 100x - 5x^2 = 10000 \]
This simplifies to:
\[ 10000 + 400x - 5x^2 = 10000 \]

### Step 5: Simplify the equation
Subtract \( 10000 \) from both sides:
\[ 400x - 5x^2 = 0 \]

### Step 6: Factor the equation
Factoring out \( x \):
\[ x(400 - 5x) = 0 \]
This gives us two solutions: \( x = 0 \) or \( 400 - 5x = 0 \).

### Step 7: Solve for \( x \)
From \( 400 - 5x = 0 \):
\[ 5x = 400 \implies x = \frac{400}{5} = 80 \]

### Conclusion
Thus, the value of \( x \) is \( 80 \).

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