if 16times the area of a square (A) is equal to the perimeter (P) f the square, then A is ______.

Answer

To solve the problem, we need to find the area \( A \) of a square given that 16 times the area is equal to the perimeter of the square.

### Step-by-Step Solution:

1. **Define the side of the square**: Let the side length of the square be \( x \).

2. **Calculate the area of the square**: The area \( A \) of the square is given by the formula:
\[ A = x^2 \]

3. **Calculate the perimeter of the square**: The perimeter \( P \) of the square is given by the formula:
\[ P = 4x \]

4. **Set up the equation based on the problem statement**: According to the problem, 16 times the area is equal to the perimeter:
\[ 16A = P \]
Substituting the formulas for \( A \) and \( P \):
\[ 16(x^2) = 4x \]

5. **Simplify the equation**: Divide both sides by 4:
\[ 4x^2 = x \]

6. **Rearrange the equation**: Move all terms to one side:
\[ 4x^2 - x = 0 \]

7. **Factor the equation**: Factor out \( x \):
\[ x(4x - 1) = 0 \]

8. **Solve for \( x \)**: Set each factor to zero:
- \( x = 0 \) (not a valid solution since side length cannot be zero)
- \( 4x - 1 = 0 \) leads to:
\[ 4x = 1 \quad \Rightarrow \quad x = \frac{1}{4} \]

9. **Calculate the area \( A \)**: Substitute \( x \) back into the area formula:
\[ A = x^2 = \left(\frac{1}{4}\right)^2 = \frac{1}{16} \]

Thus, the area \( A \) is \( \frac{1}{16} \).

### Final Answer:
The area \( A \) is \( \frac{1}{16} \).