To determine how much of the bucket is filled in 3 minutes, we'll first calculate the rate at which the garden hose fills the bucket and then use that rate to find the amount filled in 3 minutes.

Step 1: Calculate the rate of filling

The garden hose fills $\frac{3}{4}$ of a bucket in 12 minutes. The rate of filling is:

$\text{Filling Rate} = \frac{\text{Amount Filled}}{\text{Time}} = \frac{\frac{3}{4} \text{ of a bucket}}{12 \text{ minutes}} = \frac{3}{4} \times \frac{1}{12} = \frac{3}{48} = \frac{1}{16} \text{ of a bucket per minute}$

Step 2: Calculate the amount filled in 3 minutes

Using the filling rate, the amount filled in 3 minutes is:

$\text{Amount Filled} = \text{Filling Rate} \times \text{Time} = \frac{1}{16} \times 3 = \frac{3}{16} \text{ of a bucket}$

So, in 3 minutes, $\frac{3}{16}$ of the bucket is filled.

Would you like more details or have any questions?

Here are some related questions:

1. How long would it take to fill the entire bucket at the same rate?
2. How much of the bucket would be filled in 5 minutes?
3. If the flow rate doubles, how much of the bucket would be filled in 3 minutes?
4. What fraction of the bucket is filled after 8 minutes?
5. How long would it take to fill $\frac{1}{2}$ of the bucket?

Tip: When dealing with problems involving rates, always try to find the rate per unit of time first, which makes it easier to calculate how much is done in any other time interval.