To solve the equation
\[ \frac{30}{k} + 1 = 4, \]
we start by isolating the term involving \( k \). We can do this by subtracting 1 from both sides:
\[ \frac{30}{k} = 4 - 1, \]
which simplifies to
\[ \frac{30}{k} = 3. \]
Next, we eliminate the fraction by multiplying both sides by \( k \):
\[ 30 = 3k. \]
Now, we solve for \( k \) by dividing both sides by 3:
\[ k = \frac{30}{3} = 10. \]
Thus, the solution for \( k \) is
\[ \boxed{10}. \]