A rectangle has area $A$ and perimeter $P$.  The largest possible value of $\tfrac A{P^2}$ can be expressed as $\tfrac mn$, where $m$ and $n$ are relatively prime positive integers.  Compute $m+n$.
1. Let the length and width of the rectangle be \( l \) and \( w \) respectively. The area \( A \) of the rectangle is given by:
   \[
   A = l \cdot w
   \]
   The perimeter \( P \) of the rectangle is given by:
   \[
   P = 2(l + w)
   \]

2. We need to maximize the expression \( \frac{A}{P^2} \). Substituting the expressions for \( A \) and \( P \), we get:
   \[
   \frac{A}{P^2} = \frac{l \cdot w}{(2(l + w))^2} = \frac{l \cdot w}{4(l + w)^2}
   \]

3. To simplify the problem, let \( l + w = k \). Then the perimeter \( P \) becomes \( 2k \), and the expression for \( \frac{A}{P^2} \) becomes:
   \[
   \frac{A}{P^2} = \frac{l \cdot w}{4k^2}
   \]

4. Since \( l + w = k \), we can express \( w \) in terms of \( l \) and \( k \):
   \[
   w = k - l
   \]
   Substituting this into the expression for \( A \), we get:
   \[
   A = l \cdot (k - l) = lk - l^2
   \]

5. The expression for \( \frac{A}{P^2} \) now becomes:
   \[
   \frac{A}{P^2} = \frac{lk - l^2}{4k^2}
   \]

6. To find the maximum value of \( \frac{lk - l^2}{4k^2} \), we need to find the critical points by taking the derivative with respect to \( l \) and setting it to zero:
   \[
   \frac{d}{dl} \left( \frac{lk - l^2}{4k^2} \right) = \frac{k - 2l}{4k^2} = 0
   \]
   Solving for \( l \), we get:
   \[
   k - 2l = 0 \implies l = \frac{k}{2}
   \]

7. Substituting \( l = \frac{k}{2} \) back into the expression for \( A \), we get:
   \[
   A = \left( \frac{k}{2} \right) \left( k - \frac{k}{2} \right) = \frac{k}{2} \cdot \frac{k}{2} = \frac{k^2}{4}
   \]

8. Substituting \( A = \frac{k^2}{4} \) and \( P = 2k \) into the expression for \( \frac{A}{P^2} \), we get:
   \[
   \frac{A}{P^2} = \frac{\frac{k^2}{4}}{(2k)^2} = \frac{\frac{k^2}{4}}{4k^2} = \frac{1}{16}
   \]

9. The largest possible value of \( \frac{A}{P^2} \) is \( \frac{1}{16} \). Therefore, \( m = 1 \) and \( n = 16 \), and \( m + n = 1 + 16 = 17 \).

The final answer is \( \boxed{17} \).