÷n · Splitting the Bill
Settlement hinges on one baseline — fix each person's fair share first, and the transfers are mere differences.
A paid $90, B paid $30, C paid $0 — who should send whom how much to make it fair?
What are we trying to understand?
Every trip ends the same way — 'I paid the hotel, you bought dinner…' Settling from memory guarantees a fight. This is a moment for arithmetic, not feelings.
The principle is single: total ÷ people = each person's share (the baseline). Paid more than your share, you receive; less, you send. The settlement table is just 'paid − share'.
Layer on weights (the non-drinker) and minimum-transfer matching, and any messy settlement resolves mechanically. This is exactly what settlement apps compute.
Change one assumption at a time
The result follows the inputs and assumptions. Check the units and model before treating it as an actual outcome.
Predict first — on a 3-person trip, A paid $90, B paid $30, C paid $0. The fair settlement?
Change what each person paid and settle
Total ÷ 3 = the baseline (one share). Differences from it ARE the settlement.
$40
baseline (one share)
Read the calculation another way
More people means many back-and-forth transfers.
An n-person settlement needs at most n−1 transfers. Compute everyone's difference, then connect senders directly to receivers — any relay payment is pure waste.
Splitting 1/n is always fair.
With a non-drinker or a late joiner, 1/n is precisely unfair. True fairness weights each line item by who it applies to — fairness is a choice of computation.
Why the calculation works, and when
The settlement procedure, formalized.
The baseline (one share)
Settlement's starting point. $120 among 3 makes the share $40 — above or below this line decides receive or send.
Each person's difference
Positive receives, negative sends. All Δ sum to exactly 0 — which doubles as the audit check.
Weighted, per line item
Dinner split 3 ways, drinks split between the 2 who drank — only the denominator changes per item; the principle stays.
Try the idea in another setting
Trip settlement apps
Splitwise-style apps run exactly 'differences + minimum-transfer matching'. Know the principle and a sheet of paper does the same job.
Group dinners
The non-drinker's bar tab, the latecomer's share — weighted line items settle it without hurt feelings. Agreeing on the definition of fair comes first.
Group-buy shipping fees
Split the $10 shipping equally, or in proportion to each order? Equal vs proportional — both are 'fair'; it's a choice.
Roommate bills
Rent by room size, utilities per head — real settlements mix different rules per line item.
Now explain it in your own words
Follow the mathematics behind the calculation
Related lab
Ratios
Equal split is 1:1:1; weighted split is another ratio — settlement is ratio division in the field.
Go to the Ratios lab →Previous lab
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