Math Problem
I hated algebra until I became a young adult and found a use for it and now have a mathematical problem that I can't figure out and need some help with.
I'm renting a large house this summer for me and my wife along with relatives that I haven't seen for years. There are also relatives that I've never met that will be joining us; the children of my ostracized uncle who was the black sheep of the family that we reconnected with on social media.
There will be a total of seven of us. My wife and I will be staying there for 13 nights, but not everyone will stay there for that long and I'm looking for a formula that will give me a per person cost based on how long they stay.
The total cost is $3,572 for 13 nights which comes to $248 per night, $40 per person per night if everyone were to stay all 13 nights. But if you only stay 7 nights and pay $40 per night, the total would come up short, no? There would be 6 nights where they didn't pay $40/night.
a + b + c + d + e + f + g = $3,572 is as far as I can get.
Two btu per sq ft for degree difference for a slab
Comments


Ratio is correct.
To put it into algebra. For each person staying (ag) replace the letter with the number of nights they are staying so:
(a+b+c+d+e+f+g) * x = $3,572
where x is the cost per personnight.
x = $3,572 / (a+b+c+d+e+f+g)
Then the actual cost per person is
a*x = Cost for a
b*x = Cost for b
etc
1 
@Alan (California Radiant) Forbes
The answer above is so simple but I would have struggled with that myself. Not a "math mind" but I probably would have got their eventually……..maybe.
I hated school until I got to a community college. Had a pretty good math professor but have forgotten most of it.
1 
Allocating fixed costs vs variable costs.
If three stay for all 13 nights, and four stay for 7 nights, then there are 3x13 and 4x7 occupied man nights.
(3x13) + (4x7) = 67 occupied man nights.
$3572 / 67 = $53.31 per occupied man night.
But some of the $3572 was a fixed cost to set up the house for everyone. Hotels charge more for the first night to cover their cost of setting up the room. If you check out early, the hotel still covers their fixed costs. Longer stays get the lower incremental nightly rate, which reflects the hotel's variable costs.
Say the fixed cost of setting up the house and reservation is $300.
$3572  $300 = $3272
$3272 / 67 man nights = $48.84
$300 / 13 guests = $23.08
First nights for everyone are $48.84 + $23.08 = $71.92
Second through final nights are $48.84
Rounding to simplify.
First night is $72. Second through final night is $49. Rounding goes to Alan.
🤑
I always loved Math. Hated writing when I was young (before word processors). Teacher's always gave me grief because I didn't show my work. Did the Math in my head.
Some people need a calculator, I needed a word processor, spell checker, and printer.
I DIY.1 
Reading your proposal, those who stay a shorter time have to pay more per night because others would stay longer. Otherwise, those who stay longer would have to compensate for those leaving earlier.
WMno57 has a good point. Compare the total renting price for various length to find the setup cost.
Now if those leaving at the end have to clean everything including the mess of those who have already leaved…
Some people might, for various reasons, leave earlier then planed or not show up.
Good luck to find a cost sharing that would not make some people unhappy.
0 
This may or may not be accurate, YMMV.
Fill out the orange cells to taste, add to the table as necessary.
0 
The offspring of the ostracized uncle must be considered at a different rate, since black sheep consume more than white sheep. The percentage of the black sheep offspring depends on the generational displacement of the actual black sheep. A first cousin once removed pays a higher rate than a first cousin twice removed.
Then you need to figure the cosign of the factor for removed cousins. You can probably get the numbers to logically explain why you can stay for free while all the others foot the bill for you, by including any inheritance the black sheep uncle provided to the shades of gray removed by the offspring.
You are doing all the work, so you should get something out of it for yourself!
Edward Young Retired
After you make that expensive repair and you still have the same problem, What will you check next?
2 
LOL BowTieEd. In the 50's, my father and his brothers owned a gift store in Egypt that did quite well selling luggage, jewelry and leather goods to the welltodo. In 1956, the Suez Canal War broke out and all foreigners had to leave. When the black sheep brother left, he took all the money.
His grandson, who I will be meeting for the first time, says he knew nothing about what his grandfather did to the other brothers which I believe because his brothers and their families refused to have anything to do with him after what he did.
8.33 lbs./gal. x 60 min./hr. x 20°ΔT = 10,000 BTU's/hour
Two btu per sq ft for degree difference for a slab0 
@Alan (California Radiant) Forbes
I think every family has one.
0
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