Let $a$, $b$, $c$, $d$, $e$ be positive reals satisfying \begin{align*} a + b &= c \\ a + b + c &= d \\ a + b + c + d &= e.\end{align*} If $c=5$, compute $a+b+c+d+e$.

[i]Proposed by Evan Chen[/i]
1. We start with the given equations:
   \[
   a + b = c
   \]
   \[
   a + b + c = d
   \]
   \[
   a + b + c + d = e
   \]
   and the given value \( c = 5 \).

2. From the first equation, we substitute \( c \) with 5:
   \[
   a + b = 5
   \]

3. Using the second equation, we substitute \( a + b \) and \( c \):
   \[
   a + b + c = d \implies 5 + 5 = d \implies d = 10
   \]

4. Using the third equation, we substitute \( a + b \), \( c \), and \( d \):
   \[
   a + b + c + d = e \implies 5 + 5 + 10 = e \implies e = 20
   \]

5. Finally, we need to compute \( a + b + c + d + e \):
   \[
   a + b + c + d + e = 5 + 5 + 10 + 20 = 40
   \]

The final answer is \(\boxed{40}\).