Let's break down the problem step by step.

1. Total distance of the trip: $2100 \text{ kilometers}$

2. Distance Sathish drives alone:

   • He drives alone for the first $150$ kilometers and for the last $150$ kilometers. $150 + 150 = 300 \text{ kilometers}$

3. Distance Sathish drives with his friends:

   • Sathish picks up his friends after driving $150$ kilometers and drops them off when $150$ kilometers remain. Therefore, he drives with his friends for: $2100 - 300 = 1800 \text{ kilometers}$

4. Gas consumption of the car:

   • The car consumes $6$ liters of gas for every $100$ kilometers. Thus, for the entire trip, the car will consume: $\frac{6 \text{ liters}}{100 \text{ kilometers}} \times 2100 \text{ kilometers} = 126 \text{ liters of gas}$

5. Total gas cost:

   • The cost of gas is $\$1.20$ per liter. Therefore, the total cost of gas for the trip is: $126 \text{ liters} \times 1.20 = 151.20 \text{ dollars}$

6. Cost distribution:

   • When Sathish is alone: He drives $300$ kilometers alone. The amount of gas he uses during this time is: $\frac{6 \text{ liters}}{100 \text{ kilometers}} \times 300 \text{ kilometers} = 18 \text{ liters}$

     • The cost of gas for these $18$ liters is: $18 \text{ liters} \times 1.20 = 21.60 \text{ dollars}$
     • Sathish will pay this amount by himself.

   • When Sathish is with his friends: He drives $1800$ kilometers with his friends. The amount of gas used during this time is: $\frac{6 \text{ liters}}{100 \text{ kilometers}} \times 1800 \text{ kilometers} = 108 \text{ liters}$

     • The cost of gas for these $108$ liters is: $108 \text{ liters} \times 1.20 = 129.60 \text{ dollars}$
     • This cost is split evenly among Sathish and his two friends, i.e., 3 people. Each person will pay: $\frac{129.60}{3} = 43.20 \text{ dollars}$

7. Total amount Sathish pays:

   • For the portion he drives alone, he pays $21.60$ dollars.
   • For the portion with his friends, he pays $43.20$ dollars.
   • So, the total amount Sathish pays is: $21.60 + 43.20 = 64.80 \text{ dollars}$

Final Answer:

Sathish will pay $64.80 for the gas on the trip.

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Would you like more details or have any questions?

Here are 5 related questions to expand your understanding:

1. How much would Sathish's friends pay for the trip individually?
2. What if the gas price increased to $1.50 per liter, how much would Sathish pay then?
3. What is the total distance each person contributes towards the gas cost?
4. If Sathish drove an extra 100 kilometers alone, how would this affect his total cost?
5. How would the calculations change if there was a fourth person splitting the cost?

Tip: When splitting costs on trips, it's useful to divide the expenses based on time spent together to ensure fairness.