The area of a rectangle is 4 times the area of a square. The length of the rectangle is 90 cm and the breadth of the rectangle is `2//3 rd` of the side of the square. What is the side of the square?

Options

A. 10 cm
B. 20 cm
C. 15 cm
D. Couldn't be determined

Answer

To find the side of the square given the conditions in the problem, we can follow these steps:

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

### Step 2: Find the area of the square
The area of the square can be calculated using the formula:
\[ \text{Area of square} = x^2 \]

### Step 3: Find the dimensions of the rectangle
The length of the rectangle is given as 90 cm. The breadth of the rectangle is \( \frac{2}{3} \) of the side of the square, which can be expressed as:
\[ \text{Breadth of rectangle} = \frac{2}{3}x \]

### Step 4: Find the area of the rectangle
The area of the rectangle can be calculated using the formula:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = 90 \times \frac{2}{3}x \]

### Step 5: Set up the equation based on the problem statement
According to the problem, the area of the rectangle is 4 times the area of the square. Therefore, we can write the equation:
\[ 90 \times \frac{2}{3}x = 4 \times x^2 \]

### Step 6: Simplify the equation
First, simplify the left side:
\[ 90 \times \frac{2}{3}x = 60x \]
So the equation becomes:
\[ 60x = 4x^2 \]

### Step 7: Rearrange the equation
Rearranging gives us:
\[ 4x^2 - 60x = 0 \]

### Step 8: Factor the equation
Factoring out \( 4x \):
\[ 4x(x - 15) = 0 \]

### Step 9: Solve for \( x \)
Setting each factor to zero gives us:
1. \( 4x = 0 \) → \( x = 0 \) (not a valid solution)
2. \( x - 15 = 0 \) → \( x = 15 \)

Thus, the side of the square is:
\[ \text{Side of the square} = 15 \text{ cm} \]

### Final Answer
The side of the square is \( 15 \) cm.

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