11 If \( \frac{a}{2}=\frac{b}{5}=\frac{2 a+b}{k} \), then \( k=\ldots \)

To solve the equation \( \frac{a}{2}=\frac{b}{5}=\frac{2a+b}{k} \), we start by letting \( \frac{a}{2} = \frac{b}{5} = t \). This gives us: \[ a = 2t \quad \text{and} \quad b = 5t \] Now, substituting \( a \) and \( b \) into \( \frac{2a+b}{k} \), we have: \[ 2a + b = 2(2t) + 5t = 4t + 5t = 9t \] Thus, we can express the equality as: \[ \frac{9t}{k} = t \] Assuming \( t \neq 0 \), we can safely divide both sides by \( t \) to get: \[ \frac{9}{k} = 1 \] From which we conclude: \[ k = 9 \] Therefore, the value of \( k \) is \( \boxed{9} \).