The difference between the areas of a rectangel and square is 35 `cm^2`. If the rectangle's length and breadth are 50% more and 10% less respectively than the side of the square, what is the area of the rectangle?(in `cm^2`)

Options

A. 105
B. 100
C. 135
D. 145

Answer

To find the area of the rectangle given the conditions in the problem, we can follow these steps:

### Step 1: Define the side of the square
Let the side of the square be \( x \) cm.

### Step 2: Calculate the dimensions of the rectangle
- The length of the rectangle is 50% more than the side of the square:
\[ \text{Length} = x + 0.5x = \frac{3x}{2} \]
- The breadth of the rectangle is 10% less than the side of the square:
\[ \text{Breadth} = x - 0.1x = 0.9x = \frac{9x}{10} \]

### Step 3: Write the area of the rectangle and the area of the square
- The area of the rectangle is given by:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = \left(\frac{3x}{2}\right) \times \left(\frac{9x}{10}\right) \]
- The area of the square is:
\[ \text{Area of square} = x^2 \]

### Step 4: Set up the equation based on the difference in areas
According to the problem, the difference between the area of the rectangle and the area of the square is 35 cm²:
\[ \text{Area of rectangle} - \text{Area of square} = 35 \]
Substituting the areas:
\[ \left(\frac{3x}{2} \times \frac{9x}{10}\right) - x^2 = 35 \]

### Step 5: Simplify the equation
Calculating the area of the rectangle:
\[ \frac{3x}{2} \times \frac{9x}{10} = \frac{27x^2}{20} \]
So the equation becomes:
\[ \frac{27x^2}{20} - x^2 = 35 \]
To combine the terms, convert \( x^2 \) to have a common denominator:
\[ \frac{27x^2}{20} - \frac{20x^2}{20} = 35 \]
This simplifies to:
\[ \frac{7x^2}{20} = 35 \]

### Step 6: Solve for \( x^2 \)
Multiply both sides by 20:
\[ 7x^2 = 700 \]
Now divide by 7:
\[ x^2 = 100 \]

### Step 7: Find \( x \)
Taking the square root of both sides:
\[ x = 10 \text{ cm} \]

### Step 8: Calculate the area of the rectangle
Now substitute \( x \) back into the area of the rectangle:
\[ \text{Area of rectangle} = \frac{27x^2}{20} = \frac{27 \times 100}{20} = \frac{2700}{20} = 135 \text{ cm}^2 \]

### Final Answer
The area of the rectangle is \( 135 \text{ cm}^2 \).

---