The area of paper can be divided into 144 squares, but if the dimensions of each square, were reduced by 2 cm each, then the number of squares so formed are 400. The area of the paper initially was :

Options

A. `544 cm^2`
B. `1444 cm^2`
C. `3600 cm^2`
D. none of these

Answer

To solve the problem step by step, we need to find the area of the paper given the conditions about the squares.

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

### Step 2: Calculate the area of the paper
Since the area of the paper can be divided into 144 squares, the total area of the paper is:
\[ \text{Area} = 144 \times x^2 \]

### Step 3: Define the new dimensions of the square
If the dimensions of each square are reduced by 2 cm, the new side length becomes:
\[ x - 2 \text{ cm} \]

### Step 4: Calculate the area of the new squares
The area of each new square is:
\[ \text{Area of new square} = (x - 2)^2 \]

### Step 5: Calculate the total number of new squares
According to the problem, the total number of new squares formed is 400. Therefore, the total area of the paper can also be expressed as:
\[ \text{Area} = 400 \times (x - 2)^2 \]

### Step 6: Set up the equation
Since both expressions represent the area of the same paper, we can set them equal to each other:
\[ 144x^2 = 400(x - 2)^2 \]

### Step 7: Expand the equation
Expanding the right side:
\[ 144x^2 = 400(x^2 - 4x + 4) \]
\[ 144x^2 = 400x^2 - 1600x + 1600 \]

### Step 8: Rearranging the equation
Now, rearranging the equation gives:
\[ 0 = 400x^2 - 144x^2 - 1600x + 1600 \]
\[ 0 = 256x^2 - 1600x + 1600 \]

### Step 9: Simplifying the equation
Dividing the entire equation by 16 for simplicity:
\[ 0 = 16x^2 - 100x + 100 \]

### Step 10: Using the quadratic formula
We can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 16, b = -100, c = 100 \):
\[ x = \frac{100 \pm \sqrt{(-100)^2 - 4 \times 16 \times 100}}{2 \times 16} \]
\[ x = \frac{100 \pm \sqrt{10000 - 6400}}{32} \]
\[ x = \frac{100 \pm \sqrt{3600}}{32} \]
\[ x = \frac{100 \pm 60}{32} \]

### Step 11: Finding the possible values for x
Calculating the two possible values:
1. \( x = \frac{160}{32} = 5 \)
2. \( x = \frac{40}{32} = 1.25 \) (not valid since side length cannot be less than 2)

Thus, \( x = 5 \) cm.

### Step 12: Calculate the initial area of the paper
Now, substituting \( x \) back into the area formula:
\[ \text{Area} = 144 \times (5^2) = 144 \times 25 = 3600 \text{ cm}^2 \]

### Final Answer
The area of the paper initially was **3600 cm²**.

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