Let the total number of muffins in the shop be \( x \).
The number of pumpkin chip muffins is given by \( \frac{2}{5}x \).
Since there are 9 fewer pumpkin chip muffins than lemon crumb muffins, we can express the number of lemon crumb muffins as:
\[ \text{Lemon crumb muffins} = \text{Pumpkin chip muffins} + 9 = \frac{2}{5}x + 9 \]
The total number of muffins is the sum of pumpkin chip muffins and lemon crumb muffins, which gives us the equation:
\[ x = \left( \frac{2}{5}x \right) + \left( \frac{2}{5}x + 9 \right) \]
Now, combine like terms on the right side:
\[ x = \frac{2}{5}x + \frac{2}{5}x + 9 \] \[ x = \frac{4}{5}x + 9 \]
Next, we want to isolate \( x \). Subtract \( \frac{4}{5}x \) from both sides:
\[ x - \frac{4}{5}x = 9 \] \[ \frac{1}{5}x = 9 \]
Now, multiply both sides by 5 to solve for \( x \):
\[ x = 45 \]
Thus, the total number of muffins in the shop is \( \boxed{45} \).
To verify, we can find the number of each type of muffin:
- Number of pumpkin chip muffins: \( \frac{2}{5} \times 45 = 18 \)
- Number of lemon crumb muffins: \( 18 + 9 = 27 \)
Finally, check the total: \[ 18 + 27 = 45 \] This confirms our solution is correct. The total number of muffins is indeed \( \boxed{45} \).