Length and breadth of a rectangle are 24 cm and 8 cm. The side of a square is two-thirds of the length of the rectangle. The sum of their areas (in sq cm) is

Options

A. 545
B. 448
C. 455
D. 544

Answer

To solve the problem, we need to find the areas of both the rectangle and the square, and then sum those areas. Here’s a step-by-step breakdown:

### 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 = 24 cm
- Breadth of the rectangle = 8 cm

Substituting the values:
\[ \text{Area of rectangle} = 24 \, \text{cm} \times 8 \, \text{cm} = 192 \, \text{cm}^2 \]

### Step 2: Calculate the side of the square.
The side of the square is given as two-thirds of the length of the rectangle. Therefore:
\[ \text{Side of square} = \frac{2}{3} \times \text{length of rectangle} \]
Substituting the length of the rectangle:
\[ \text{Side of square} = \frac{2}{3} \times 24 \, \text{cm} = 16 \, \text{cm} \]

### Step 3: Calculate the area of the square.
The formula for the area of a square is:
\[ \text{Area of square} = \text{side} \times \text{side} \]
Substituting the side length:
\[ \text{Area of square} = 16 \, \text{cm} \times 16 \, \text{cm} = 256 \, \text{cm}^2 \]

### Step 4: Calculate the sum of the areas.
Now, we 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:
\[ \text{Sum of areas} = 192 \, \text{cm}^2 + 256 \, \text{cm}^2 = 448 \, \text{cm}^2 \]

### Final Answer:
The sum of the areas of the rectangle and the square is \(448 \, \text{cm}^2\).

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