splitting accommodation costs between people when some of them stay for fewer days We're 4 people and we are staying 7 nights, for a total cost of 546.
One of us however is leaving 1 day earlier.
Initially I thought the problem was very simple.
I reasoned that the 3 of us staying for the full period should pay $C_1$, and the person leaving earlier should pay $C_2$, under the following conditions:
$3 \cdot C_1 + C_2 = 546$
$C_2 = \frac 6 7 \cdot C_1$
The result was:
$C_1 = \frac {1274} 9 \approx 141.56$
$C_2 = \frac {364} 3 \approx 121.33$
I was happy with this, but then I thought: how am I going to explain it to the group?
So I tried a more 'intuitive' approach, which I believed was equivalent.
I reasoned that the first 6 nights should be split equally between all 4 of us, whereas the last night should only be split between the 3 of us who were still staying at that point.
Cost of each night:
$\frac {546} 7 = 78$
Cost of the first 6 nights:
$78 \cdot 6 = 468$
Split equally between 4 of us:
$\frac {468} 4 = 117$
Cost of the last night split only between 3 of us:
$\frac {78} 3 = 26$
Which would give:
$C_1 = 117 + 26 = 143$
$C_2 = 117$
The first condition above is still met:
$3 \cdot 143 + 117 = 546$
but the second is not:
$\frac 6 7 \cdot 143 \approx 122.57 \ne 117$
I can see that the two approaches are algebraically different, but I don't understand why they are conceptually different.
How can looking at 1 day at a time be different from looking at the total? Shouldn't the person staying 1 day less always be paying 6/7 of what the others pay? Or should the 6/7 condition perhaps be applied not to the cost each of the 3 people staying 7 days pays, but (somehow) to the total?
I'm sure there is a simple explanation, but I can't immediately see it. Any idea?
And do you think either of the two approaches is 'fairer'?
Thanks!

The early bird gets a refund of 546/(4*7) = 19.5 to be paid by the other three at 6.50 each.
