A friend and I have a disagreement, and I am hoping someone here can help settle the matter.(adsbygoogle = window.adsbygoogle || []).push({});

The conditions:

8 homes along a dead-end street need to repair the sidewalk, and so they all agree that they will split the bill based on use. No homeowner walks past their own house towards the dead end, ever, but each homeowner goes for a walk towards the cross-street and back home once a day. The total bill is based on a straight per foot cost. There isnoflat part to the fee, like $5000 + $20.00 per foot or something like that.

I reason as follows:

All homeowners use the first 1/8 of the sidewalk, so they should all split the bill for that 1/8 of the sidewalk, which is 1/8 of the total bill. Thus every house pays 1/64 of the total bill for their use of that section of the sidewalk. The second 1/8 section is only used by 7 of the homeowners, and so the bill for that section of the sidewalk should be equally shared by all but the first homeowner on the block. Thus every homeowner but the first should pay 1/(8*7) of the total bill for their use of that section. This continues until all segments of the sidewalk are paid for equally by those who use it.

The fraction of the total cost paid by each home is:

H1: 1/(8*8) = 1/64

H2: 1/(8*8) + 1/(8*7) = 15/448

H3: 1/(8*8) + 1/(8*7) + 1/(8*6) = 73/1344

H4: 1/(8*8) + 1/(8*7) + 1/(8*6)+ 1/(8*5) = 533/6720

etc.

My friend says this:

Assign each 1/8 section of the road a unit U. Then the first homeowner uses 1*U units, the second homeowner uses 2*U units, the third homeowner uses 3*U units, etc. Thus we have U + 2*U + 3*U + ... +8*U = 36*U and so the fraction of the total cost paid by each home is:

H1: 1/36

H2: 2/36

H3: 3/36

H4: 4/36

etc.

The difference is shown graphically in the image. He says I would have the two homeowners at the end of the block overpaying, and I say he is having the first 6 homeowners paying for sidewalk they don't use.

So the question is, which is most fair andwhy?

Thanks!

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# B What is the fair way to distribute costs and why?

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