A pet shop sells cats and two types of birds: ducks and parrots. In the shop, $\tfrac{1}{12}$ of animals are ducks, and $\tfrac{1}{4}$ of birds are ducks. Given that there are $56$ cats in the pet shop, how many ducks are there in the pet shop?
1. Let \( C \) be the number of cats, \( D \) be the number of ducks, and \( P \) be the number of parrots. We are given that \( C = 56 \).

2. Let \( T \) be the total number of animals in the pet shop. We know that \( \frac{1}{12} \) of the animals are ducks:
   \[
   D = \frac{1}{12}T
   \]

3. We are also given that \( \frac{1}{4} \) of the birds are ducks. Let \( B \) be the total number of birds. Then:
   \[
   D = \frac{1}{4}B
   \]

4. From the information that \( \frac{1}{12} \) of the animals are ducks, we can express the total number of animals \( T \) in terms of the number of ducks \( D \):
   \[
   T = 12D
   \]

5. Since the total number of animals \( T \) is the sum of the number of cats and the number of birds:
   \[
   T = C + B
   \]
   Substituting \( T = 12D \) and \( C = 56 \):
   \[
   12D = 56 + B
   \]

6. We also know that \( D = \frac{1}{4}B \), so we can substitute \( B = 4D \) into the equation:
   \[
   12D = 56 + 4D
   \]

7. Solving for \( D \):
   \[
   12D - 4D = 56
   \]
   \[
   8D = 56
   \]
   \[
   D = \frac{56}{8}
   \]
   \[
   D = 7
   \]

8. Therefore, the number of ducks in the pet shop is \( \boxed{7} \).