Perimeter of a rectangle is equal to the perimeter of a square whose side is twice of that of the breadth of the rectangle. What is the ratio of length of side of square to the length of rectangle?

Options

A. `2:3`
B. `6:5`
C. `7:9`
D. `14:9`

Answer

To solve the problem step by step, let's define the variables and then derive the required ratio.

### Step 1: Define the Variables
Let:
- \( b \) = breadth of the rectangle
- \( l \) = length of the rectangle
- \( s \) = side of the square

According to the problem, the side of the square is twice the breadth of the rectangle:
\[ s = 2b \]

### Step 2: Write the Perimeter Formulas
The perimeter of the rectangle is given by:
\[ P_{\text{rectangle}} = 2(l + b) \]

The perimeter of the square is given by:
\[ P_{\text{square}} = 4s \]

### Step 3: Set the Perimeters Equal
According to the problem, the perimeter of the rectangle is equal to the perimeter of the square:
\[ 2(l + b) = 4s \]

### Step 4: Substitute the Expression for \( s \)
Substituting \( s = 2b \) into the perimeter equation:
\[ 2(l + b) = 4(2b) \]
\[ 2(l + b) = 8b \]

### Step 5: Simplify the Equation
Now, simplify the equation:
\[ l + b = 4b \]
\[ l = 4b - b \]
\[ l = 3b \]

### Step 6: Find the Ratio
Now we need to find the ratio of the side of the square to the length of the rectangle:
\[ \text{Ratio} = \frac{s}{l} \]
Substituting \( s = 2b \) and \( l = 3b \):
\[ \text{Ratio} = \frac{2b}{3b} = \frac{2}{3} \]

### Final Answer
The ratio of the length of the side of the square to the length of the rectangle is:
\[ \frac{2}{3} \]

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