To solve this problem, we use the concept of rates. Let's denote the following:

1. The slower computer completes the task in 55 minutes, so its work rate is $\frac{1}{55}$ of the job per minute.
2. The faster computer completes the task in 45 minutes, so its work rate is $\frac{1}{45}$ of the job per minute.
3. When both computers work together, their combined work rate is $\frac{1}{55} + \frac{1}{45}$.

The time it takes for both computers to complete the job together is the reciprocal of their combined rate.

Step-by-Step Solution:

1. Calculate the combined rate: $\frac{1}{55} + \frac{1}{45}$ To add these, find the least common denominator (LCD) of 55 and 45, which is 495.

   $\frac{1}{55} = \frac{9}{495}, \quad \frac{1}{45} = \frac{11}{495}$

   Adding these fractions: $\frac{9}{495} + \frac{11}{495} = \frac{20}{495}$

2. Combined work rate: $\frac{20}{495}$

3. Time taken for both computers working together is the reciprocal of the combined rate: $\text{Time} = \frac{495}{20} = 24.75 \text{ minutes}.$

Final Answer:

It will take 24.75 minutes (or 24 minutes and 45 seconds) for both computers to complete the job together.

Would you like more details on the calculation steps or an explanation of the concept? Here are some related questions to explore:

1. How does the combined work rate affect the total time when one machine is much slower?
2. How do you find the least common denominator for adding fractions?
3. What happens to the total time if one computer's rate improves?
4. Can you explain how rates are related to proportions in solving similar problems?
5. How would the result change if the times for each computer were expressed in seconds instead of minutes?

Tip: Always check units to ensure consistency when solving work-rate problems!