To solve this problem, let's calculate the rate at which each hose fills the tank and then combine these rates to find the total time required to fill the tank when both hoses are used simultaneously.

Step 1: Determine the Filling Rates of Each Hose

• Hose 1: It takes 50 minutes to fill the entire tank.

  • Therefore, the rate of Hose 1 is $\frac{1}{50}$ of the tank per minute.

• Hose 2: It takes 40 minutes to fill the entire tank.

  • Therefore, the rate of Hose 2 is $\frac{1}{40}$ of the tank per minute.

Step 2: Combine the Rates

When both hoses are used together, their combined rate of filling the tank is the sum of their individual rates:

$\text{Combined Rate} = \frac{1}{50} + \frac{1}{40}$

To add these fractions, find the least common multiple (LCM) of the denominators (50 and 40), which is 200. Rewrite the fractions with a common denominator:

$\frac{1}{50} = \frac{4}{200}, \quad \frac{1}{40} = \frac{5}{200}$

Now, add the fractions:

$\text{Combined Rate} = \frac{4}{200} + \frac{5}{200} = \frac{9}{200}$

This means both hoses together fill $\frac{9}{200}$ of the tank per minute.

Step 3: Calculate the Time Needed to Fill the Tank

To find the total time needed to fill the tank when both hoses are working together, we take the reciprocal of the combined rate:

$\text{Time} = \frac{1}{\text{Combined Rate}} = \frac{1}{\frac{9}{200}} = \frac{200}{9} \, \text{minutes}$

Step 4: Simplify the Answer

$\frac{200}{9} \approx 22.22 \, \text{minutes}$

Conclusion

When both hoses are used together, the tank will be filled in approximately 22.22 minutes.

Would you like more details on any step, or do you have any questions?

Here are 5 related questions to explore:

1. How would the time change if the first hose took 30 minutes and the second hose took 20 minutes?
2. What is the formula for combining rates of work?
3. If there is a third hose that takes 60 minutes to fill the tank, how would you calculate the total time for all three hoses?
4. How can we convert this problem into a rate problem involving distance, speed, and time?
5. If one of the hoses has a blockage reducing its rate by half, how would that affect the time to fill the tank?

Tip: When combining rates, always add them together and take the reciprocal to find the total time required.