If one pair of opposite sides of a square are increased in length by 20% and the other pair of sides are increased in length by 50%, by what percent is the area of the rectangle that results greater than the area of the original square?

Options

A. `80%`
B. `77%`
C. `75%`
D. `70%`

Answer

To solve the problem, we will follow these steps:

### Step 1: Define the side length of the square
Let the side length of the square be \( a \).

### Step 2: Calculate the area of the original square
The area of the original square is given by:
\[ \text{Area of square} = a^2 \]

### Step 3: Determine the new dimensions of the rectangle
- One pair of opposite sides is increased by 20%. Therefore, the new length of these sides is:
\[ \text{New Length} = a + 0.20a = 1.2a \]
- The other pair of opposite sides is increased by 50%. Therefore, the new breadth of these sides is:
\[ \text{New Breadth} = a + 0.50a = 1.5a \]

### Step 4: Calculate the area of the rectangle
The area of the rectangle formed by these new dimensions is:
\[ \text{Area of rectangle} = \text{Length} \times \text{Breadth} = (1.2a) \times (1.5a) = 1.8a^2 \]

### Step 5: Find the difference in areas
Now, we need to find the difference between the area of the rectangle and the area of the original square:
\[ \text{Difference in areas} = \text{Area of rectangle} - \text{Area of square} = 1.8a^2 - a^2 = 0.8a^2 \]

### Step 6: Calculate the percentage increase in area
To find the percentage by which the area of the rectangle is greater than the area of the original square, we use the formula:
\[ \text{Percentage Increase} = \left( \frac{\text{Difference in areas}}{\text{Area of square}} \right) \times 100\% \]
Substituting the values we calculated:
\[ \text{Percentage Increase} = \left( \frac{0.8a^2}{a^2} \right) \times 100\% = 0.8 \times 100\% = 80\% \]

### Final Answer
The area of the rectangle is 80% greater than the area of the original square.

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