When we travel we split stuff up. One friend pays for the gasoline, one pays for the food, one pays for the hotel, etc....it actually averages out to be pretty fair. It does depend on how far we're traveling though because the gasoline does get pretty expensive. If we go to a large city nearby it's usually about $120 in gasoline. The hotel is usually for one or two nights so it's about $150, and we don't eat at 5 star places so food costs about $150 for a couple of days.
I suggest you all sit down and figure out how much the trip is going to cost and split it 3 ways. If she continues to say she's not paying gasoline then consider paying her $X.xx per mile and be done with it. She can figure out the rest. Wear and tear is not very much. I'd just take someone else's vehicle just to fight the principle of it.
If the trip is free and the only cost is the gasoline then it depends on how far you're driving. If it's hundreds of miles and the gasoline would cost $300 round trip it's totally not fair for her to expect to get out of paying for $100 in gasoline. If the total gasoline bill is less than $100 because it's close to your home town then it's not worth it to fight over the issue. The difference between splitting $100 by 3 = $33.33 each compared to $100 by 2 at $50 is just not enough to make a friend difference.
So figure out the mileage, the food, the traveling expenses for everything that is jointly shared. Take off the gasoline and split it 3 ways then add the gasoline back in and split it 3 ways. If it's under $50 difference I'd say don't fight. If it's more than it's not fair she rides for free even if it is her car.
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I had to go do the math. I figured out that from Witchita KS to Dallas Tx it's just over 5 1/2 hours, and about 363 miles.
So if you are driving about 5 hours it's about 363 miles. I divided 360 miles by about 15 miles per gallon since it's an SUV. That gave me 360/15=24.
So it will take approximately 24 gallons of gasoline to get to your destination. Let's call it 25 so I can do the math in my head.
25 gallons of gas each way so about 50 gallons of gas for the trip.
If gasoline is $3.5 per gallon, averaged over the different towns and places you could purchase it, then the total cost of gasoline could be about 50 gallons of gasoline times $3.50 per gallon or an estimated cost of $175.00. Or $87.50 each for 2 of you or $58.33 each between the 3 of you. That's only a difference for the 2 of you of $29.17 or just about $30.
So either you and the 3rd lady pay the 1st lady $90 each for gasoline or the 3 of you ladies each pay $60 for gasoline.
OR you could work out a compromise of you 2 ladies paying her $70 each which would leave her to pay $35 for her own part. That gives her part responsibility for the gasoline but also gives her money for wear and tear on her vehicle.
But I do think that if this is just too much for the other woman perhaps taking a car which would give much much better gas mileage might be a better idea.