Let's solve this step-by-step.
### Step 1: Determine the individual rates of work
An experienced computer technician can set up a network in 12 hours, and an intern can set up the same network in 20 hours. To find their work rates, we need to determine how much of the network each can set up in one hour:
- Technician's rate:
[tex]\[
\text{Rate of Technician} = \frac{1 \text{ network}}{12 \text{ hours}} = \frac{1}{12} \text{ networks per hour}
\][/tex]
- Intern's rate:
[tex]\[
\text{Rate of Intern} = \frac{1 \text{ network}}{20 \text{ hours}} = \frac{1}{20} \text{ networks per hour}
\][/tex]
### Step 2: Calculate the combined rate of work
When they work together, their combined rate is the sum of their individual rates:
[tex]\[
\text{Combined rate} = \left( \frac{1}{12} + \frac{1}{20} \right) \text{ networks per hour}
\][/tex]
### Step 3: Find the sum of the rates
To add these fractions, we need a common denominator. The least common multiple of 12 and 20 is 60. Therefore:
[tex]\[
\frac{1}{12} = \frac{5}{60} \quad \text{and} \quad \frac{1}{20} = \frac{3}{60}
\][/tex]
Adding these together:
[tex]\[
\text{Combined rate} = \frac{5}{60} + \frac{3}{60} = \frac{8}{60} = \frac{2}{15} \text{ networks per hour}
\][/tex]
### Step 4: Calculate the time to complete the job together
The combined rate tells us how much of the network they can set up in one hour. To find out how long it will take them to set up one entire network, we take the reciprocal of the combined rate:
[tex]\[
\text{Time to complete the network together} = \frac{1}{\left( \frac{2}{15} \right)} = \frac{15}{2} = 7.5 \text{ hours}
\][/tex]
### Conclusion
If the experienced technician and the intern work together, it will take them:
[tex]\[
\boxed{7.5 \text{ hours}}
\][/tex]
to set up the network.
