To determine what time Jim left in the morning, we need to calculate the total time Jim spent traveling and visiting.

Step 1: Time spent walking to his friend's house

Jim walks at 3 miles per hour, and his friend lives 6 miles away. The time taken to walk to his friend's house is:

$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{6 \text{ miles}}{3 \text{ miles per hour}} = 2 \text{ hours}$

Step 2: Time spent visiting

Jim spends 90 minutes visiting, which is equal to:

$90 \text{ minutes} = 1.5 \text{ hours}$

Step 3: Time spent driving home

His friend drives him back home at 30 miles per hour. The distance is 6 miles, so the time taken to drive home is:

$\text{Time} = \frac{6 \text{ miles}}{30 \text{ miles per hour}} = 0.2 \text{ hours} = 12 \text{ minutes}$

Step 4: Total time spent traveling and visiting

Now, we add up all the time Jim spent walking, visiting, and driving home:

$\text{Total time} = 2 \text{ hours} + 1.5 \text{ hours} + 0.2 \text{ hours} = 3.7 \text{ hours}$

$3.7 \text{ hours} = 3 \text{ hours} + 42 \text{ minutes}$

Step 5: Time Jim arrived home

Jim arrives home at 2 p.m. If the total time spent traveling and visiting was 3 hours and 42 minutes, we subtract this from 2 p.m. to find the time he left in the morning:

$2:00 \text{ p.m.} - 3 \text{ hours} 42 \text{ minutes} = 10:18 \text{ a.m.}$

Thus, Jim left home at 10:18 a.m..

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Would you like more details or have any questions?
Here are 5 related questions you can explore:

1. How would the result change if Jim walked faster or slower to his friend's house?
2. What if Jim's friend lived 10 miles away—how would that affect his departure time?
3. How would the departure time be affected if Jim visited for 2 hours instead of 90 minutes?
4. What time would Jim have left if his friend drove him back at 20 miles per hour instead of 30?
5. If Jim returned home an hour earlier, how would that change the calculations?

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