The length and breadth of a rectangle are 48 cm and 21cm, respectively. The side of a square is two-third the length of the rectangle. The sum of their areas (in sq cm) is

Options

A. 2032
B. 2123
C. 2028
D. 2030

Answer

To solve the problem step by step, we will calculate the areas of the rectangle and the square, and then find their sum.

### Step 1: Calculate the area of the rectangle.
The formula for the area of a rectangle is:
\[ \text{Area of Rectangle} = \text{Length} \times \text{Breadth} \]
Given:
- Length of the rectangle = 48 cm
- Breadth of the rectangle = 21 cm

Now, substituting the values:
\[ \text{Area of Rectangle} = 48 \, \text{cm} \times 21 \, \text{cm} = 1008 \, \text{cm}^2 \]

### Step 2: Determine the side of the square.
The side of the square is given as two-thirds of the length of the rectangle.
\[ \text{Side of Square} = \frac{2}{3} \times \text{Length of Rectangle} \]
Substituting the length of the rectangle:
\[ \text{Side of Square} = \frac{2}{3} \times 48 \, \text{cm} = 32 \, \text{cm} \]

### Step 3: Calculate the area of the square.
The formula for the area of a square is:
\[ \text{Area of Square} = \text{Side}^2 \]
Now substituting the side of the square:
\[ \text{Area of Square} = 32 \, \text{cm} \times 32 \, \text{cm} = 1024 \, \text{cm}^2 \]

### Step 4: Calculate the sum of the areas.
Now, we will add the area of the rectangle and the area of the square:
\[ \text{Sum of Areas} = \text{Area of Rectangle} + \text{Area of Square} \]
Substituting the calculated areas:
\[ \text{Sum of Areas} = 1008 \, \text{cm}^2 + 1024 \, \text{cm}^2 = 2032 \, \text{cm}^2 \]

### Final Answer:
The sum of the areas of the rectangle and the square is \( 2032 \, \text{cm}^2 \).

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