To find out how much flour the factory will use for 6 batches of cookies, we can multiply the amount of flour used for each batch by the number of batches.
The amount of flour used in each batch is \( \frac{1}{4} \) of a barrel.
So for 6 batches, we calculate:
\[ \text{Total flour} = 6 \times \frac{1}{4} \]
To perform the multiplication, we can rewrite it:
\[ \text{Total flour} = \frac{6}{1} \times \frac{1}{4} = \frac{6 \times 1}{1 \times 4} = \frac{6}{4} \]
Now, we can simplify \( \frac{6}{4} \):
\[ \frac{6}{4} = \frac{3}{2} \]
Thus, the total amount of flour used for 6 batches is \( \frac{3}{2} \) barrels, which can also be expressed as a mixed number:
\[ \frac{3}{2} = 1 \frac{1}{2} \]
Therefore, the factory will use \( \frac{3}{2} \) barrels of flour or \( 1 \frac{1}{2} \) barrels of flour for 6 batches.