Let's solve this problem by finding how long it would take to fill the bucket when both hoses are used simultaneously.

Step 1: Calculate the rate of each hose

• Hose A: Fills the bucket in 2 minutes. So, its rate of filling is: $\text{Rate of Hose A} = \frac{1}{2} \text{ bucket per minute}.$
• Hose B: Fills the bucket in 3 minutes. So, its rate of filling is: $\text{Rate of Hose B} = \frac{1}{3} \text{ bucket per minute}.$

Step 2: Add the rates

When both hoses are working together, their rates add up: $\text{Combined Rate} = \frac{1}{2} + \frac{1}{3}.$

To add these fractions, we need a common denominator: $\frac{1}{2} = \frac{3}{6}, \quad \frac{1}{3} = \frac{2}{6}.$ So, the combined rate is: $\text{Combined Rate} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6} \text{ bucket per minute}.$

Step 3: Calculate the time to fill the bucket

Since the combined rate is $\frac{5}{6}$ of a bucket per minute, the time it takes to fill the bucket is the reciprocal of this rate: $\text{Time} = \frac{1}{\frac{5}{6}} = \frac{6}{5} \text{ minutes}.$

Step 4: Convert to minutes and seconds

$\frac{6}{5} \text{ minutes} = 1.2 \text{ minutes} = 1 \text{ minute and } 12 \text{ seconds}.$

Final Answer:

It would take 1 minute and 12 seconds to fill the bucket if both hoses were used at the same time.

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Do you want more details or have any questions?

Here are some related questions you might be interested in:

1. What is the rate of flow if Hose A was twice as fast as it currently is?
2. How long would it take if a third hose could fill the bucket in 5 minutes?
3. What if Hose B had to be turned off after 1 minute—how long would it take then?
4. How can we express this problem using algebraic equations?
5. What if the bucket's size was doubled—how long would it take to fill it?

Tip: Whenever two rates are combined, you can always sum them up to get the total rate.