This is a classic problem involving rates of work. Let's approach this step-by-step.

Step 1: Determine the rate of each faucet

• The cold water faucet can fill the tub in 7 minutes, so its rate is:

$\text{Cold water rate} = \frac{1}{7} \, \text{tub per minute}$

• The hot water faucet can fill the tub in 13 minutes, so its rate is:

$\text{Hot water rate} = \frac{1}{13} \, \text{tub per minute}$

Step 2: Combine the rates

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

$\text{Combined rate} = \frac{1}{7} + \frac{1}{13}$

To add these fractions, first find a common denominator. The least common denominator (LCD) of 7 and 13 is $7 \times 13 = 91$. So:

$\frac{1}{7} = \frac{13}{91}, \quad \frac{1}{13} = \frac{7}{91}$

Now, add the fractions:

$\text{Combined rate} = \frac{13}{91} + \frac{7}{91} = \frac{20}{91} \, \text{tub per minute}$

Step 3: Find the time to fill the tub

The combined rate is $\frac{20}{91}$ of a tub per minute. To find the time required to fill 1 whole tub, take the reciprocal of the combined rate:

$\text{Time} = \frac{91}{20} = 4.55 \, \text{minutes}$

Conclusion:

It will take approximately 4.55 minutes to fill the tub when both faucets are used together.

Would you like more details or have any questions?

Here are 5 related questions to explore:

1. How would the answer change if one faucet was faster than the other?
2. What is the general formula for combining two work rates?
3. How can this approach be extended to more than two faucets?
4. What if one faucet could only partially fill the tub before being shut off?
5. How would the rate change if the pressure in one faucet was reduced?

Tip: When combining rates, always remember to add the individual rates, then take the reciprocal to find the total time.