To solve this problem, let's determine how long it will take to fill the bathtub when both faucets are used together.

Solution:

1. Rates of the Faucets:

   • The cold water faucet fills the tub in 11 minutes, so its rate is:
     $\text{Rate of cold faucet} = \frac{1}{11} \text{ tubs per minute.}$
   • The hot water faucet fills the tub in 14 minutes, so its rate is:
     $\text{Rate of hot faucet} = \frac{1}{14} \text{ tubs per minute.}$

2. Combined Rate of Both Faucets: When both faucets are used together, their rates add up:
   $\text{Combined rate} = \frac{1}{11} + \frac{1}{14}.$

   To add these fractions, find the least common denominator (LCD):
   $\text{LCD of 11 and 14} = 154.$

   Convert the fractions: $\frac{1}{11} = \frac{14}{154}, \quad \frac{1}{14} = \frac{11}{154}.$

   Add the fractions: $\frac{1}{11} + \frac{1}{14} = \frac{14}{154} + \frac{11}{154} = \frac{25}{154}.$

   So, the combined rate is: $\frac{25}{154} \text{ tubs per minute.}$

3. Time to Fill the Tub: The time required to fill one tub is the reciprocal of the combined rate: $\text{Time} = \frac{1}{\frac{25}{154}} = \frac{154}{25}.$

   Simplify: $\text{Time} = 6.16 \, \text{minutes (exactly } \frac{154}{25}).$

Final Answer:

The bathtub will take $\frac{154}{25}$ minutes to fill if both faucets are used together.

Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How would the time change if the hot water faucet could fill the tub in 10 minutes instead of 14?
2. What is the combined rate if only one faucet is used for the first half and both for the second?
3. How would the problem change if the tub has a leak draining it at a constant rate?
4. How can you express this scenario graphically?
5. What would the calculation be if we rounded to 2 decimal places instead of keeping the exact value?

Tip: Whenever solving combined rate problems, always calculate rates per unit time and use their sum to determine the overall rate.