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home/calculators/bill-splitter

Split the bill by what people actually had.

Add the items from the receipt and tick who shared each one. Service and tip are shared in proportion to what each person ordered, so the friend who only had a salad doesn't pay for the wine.

itemprice
M
T
P
S
person_add
£1.63£1.63£1.63£1.63close
£10.67£10.67£10.67close
£16.50close
£24.00close
£12.00close
£21.00close
£4.50close
addAdd item7 items · £116.50
service / tip£14.56
0%10%12.5%15%20%
share the tip
By what they hada share of each orderEquallysame tip for everyone
round each share up to
none10p50p£1
food & drink£116.50service 12.5%£14.56rounding extra+£1.94collected£133.00
MMaya£33.00sourdough, bottle, risotto
£28.79 food + £3.60 tip + £0.61 rounding
TTom£41.00sourdough, bottle, steak
£36.29 food + £4.54 tip + £0.17 rounding
PPriya£21.00sourdough, caesar, sparkling
£18.13 food + £2.27 tip + £0.60 rounding
SSam£38.00sourdough, bottle, fish
£33.29 food + £4.16 tip + £0.55 rounding
~/splitting-evenly

"Let's just split it evenly" costs someone money.

Using the bill above, here's what each person would pay with an even split compared with paying for what they had. The bars show who's subsidising whom. Sometimes it's a few pence and not worth the fuss; sometimes it isn't.

Maya
+£0.38
Tom
−£8.06
Priya
+£12.38
Sam
−£4.69
pays lesspays more
An even split of £32.77 each means Tom pays £8.06 less than their share and Priya pays £12.38 more.
~/one-card-paid

One person paid. Here's who owes them what.

Pick who put their card down. Everyone else gets one transfer to make, with a reference you can copy into a banking app.

MMaya paidTTom paidPPriya paidSSam paid
Tarrow_forwardM£41.00Tom sends £41.00 to Maya.Dinner Tom £41.00content_copyCopy
Parrow_forwardM£21.00Priya sends £21.00 to Maya.Dinner Priya £21.00content_copyCopy
Sarrow_forwardM£38.00Sam sends £38.00 to Maya.Dinner Sam £38.00content_copyCopy
~/where-the-penny-goes

£100 split three ways is £33.33, £33.33 and £33.34.

Exact shares rarely land on whole pennies, so someone has to round. Rounding each share on its own can collect a penny too much or too little. Five real methods are applied below to the same shares, in integer pennies, with the total checked against the bill.

Your bill£100 ÷ 3£50 ÷ 6£1 ÷ 7£87.40 by 7:4:4:1
personexact shareRound eachhalf upBanker’shalf to evenRound each upceilingPayer absorbsfloor, rest to firstLargest remainderHamilton
Person A£33.3333£33.33£33.33£33.34£33.34£33.34
Person B£33.3333£33.33£33.33£33.34£33.33£33.33
Person C£33.3333£33.33£33.33£33.34£33.33£33.33
collected£100.00£99.99−1p short£99.99−1p short£100.02+2p over£100.00exact£100.00exact
check_circleNo Alabama paradox near this totalSearched every total from £97.00 to £106.00 with these weights: no person’s share ever drops when the total rises by a penny. Whether it appears depends on the exact weights; the 7:4:4:1 preset shows one.
Which one to useLargest remainder (Hamilton’s method) always collects the exact total and never moves anyone more than one penny from their exact share. That is why this splitter’s transfers use it. Rounding each share up is simplest at the table: the extra pennies go to the waiter or to whoever paid.
~/fewest-transfers

Several people paid. What’s the fewest transfers that settles it?

The usual shortcut matches the biggest creditor with the biggest debtor until everyone is square, which never takes more than one transfer fewer than the number of people. The true minimum comes from splitting people into the most groups that already balance among themselves; finding that is NP-hard in general, so it’s solved here exactly by searching every grouping.

This billTrip example
Who paid how much at the till
Maya+£46.61owes £32.39 · paid
Tom+£11.23owes £40.83 · paid
Priya−£20.39owes £20.39 · paid
Sam−£37.45owes £37.45 · paid
Payments total £131.06 but the shares add to £131.06: £0.00 missing. Transfers need the two to match.Put the difference on the main payer
Biggest-first shortcut3Largest creditor paired with largest debtor, repeated.
Sam → Maya£37.45
Priya → Tom£11.23
Priya → Maya£9.16
at most 3 for 4 unsettled people
Proven minimum3Most zero-sum groups found by exhaustive search; each group of k needs k − 1 transfers.
Sam → Maya£37.45
Priya → Tom£11.23
Priya → Maya£9.16
1 balanced group · 4 − 1 = 3
check_circleThe shortcut is already optimal here: 3 transfers. It usually is when one person paid.
~/shapley-value

Sharing a taxi home: three fair-sounding splits, one that game theory backs.

Riders leave a taxi at different stops, so the fare depends on who travels furthest. The Shapley value (1953) charges each rider their average extra cost over every order in which the group could have formed. For a shared route it reduces to a simple rule: each stretch of road is shared equally by everyone still in the car. It’s computed below by that rule and by brute force over every ordering.

riders2345
£8.00 ÷ 3£14.70 ÷ 2£12.60 ÷ 1AnaBenCal
Fare to the last stop: £3.80 + £2.10 × 15 km = £35.30. Labels show each stretch’s cost and how many riders share it (the first stretch includes the flag fall).
riderkmaloneequalby distanceShapley
Ana2£8.00£11.77£2.72£2.67
Ben9£22.70£11.77£12.22£10.02
Cal15£35.30£11.77£20.37£22.62
fare£66.00£35.30£35.30£35.30
Brute force over all 6 orderings (3!) gives the same Shapley shares, largest difference 0. Every split collects the same £35.30.
Red means paying more than riding alone: Ana would be better off getting their own taxi. Shapley never does this.
~/reference

Tips, service charges and the rules behind them.

Customs vary more than the maths. The last column applies each country’s typical rate to the food on the bill above.

countryrestaurant customnoteson this bill
United Kingdom10-12.5%Often added as a discretionary service charge; it must be shown and can be removed. All tips must reach staff in full since October 2024.£14.56
United States18-22%Expected at table-service restaurants. Federal law lets employers count tips towards minimum wage (the tip credit).£23.30
Canada15-20%Usually worked out on the pre-tax amount; card machines suggest percentages.£20.97
Franceservice comprisMenu prices include service by law; leaving small change for good service is optional.£0.00
Germanyround up, 5-10%Say the total you want to pay when handing over money rather than leaving it on the table.£8.16
Italycoperto + optionalA per-person cover charge is common and itemised; tipping beyond it is not expected.£0.00
JapannoneTipping is not customary and can cause confusion.£0.00
Australiaoptional, ~10%Wages are higher and tipping is not expected; 10% for exceptional service.£11.65
All tips to workersEmployers must pass on all tips, gratuities and service charges to workers without deductions, other than tax.Employment (Allocation of Tips) Act 2023, in force 1 Oct 2024
Fair and transparentTips must be shared fairly under a written policy that staff can see, and paid by the end of the following month.Statutory Code of Practice
Records keptEmployers must keep records of tips received and paid out for three years; workers can request them.Employment (Allocation of Tips) Act 2023
VATA discretionary service charge or tip is outside the scope of VAT; a compulsory service charge is part of the price and taxed.HMRC guidance
Income tax and NITips are taxable income. Tips shared through an independent tronc can be free of National Insurance.HMRC E24
Rounding to the payerNothing in law sets how friends round. Largest remainder is the only common method that is always exact.Hamilton’s method

Timeline

1792Washington vetoes Hamilton’s largest-remainder apportionment, the first US presidential veto.
1880The Alabama paradox: a larger House would have given Alabama fewer seats.
1953Lloyd Shapley publishes the Shapley value for sharing gains in cooperative games.
1966US Fair Labor Standards Act amendments introduce the tip credit.
1973Littlechild and Owen apply the Shapley value to sharing runway costs: the airport game.
1987France requires restaurant prices to include service.
2004Verhoeff shows minimising transfers to settle debts is NP-hard.
2008UK Faster Payments makes instant bank transfers between friends routine.
2009Venmo launches in the US.
2011Splitwise launches its shared-expense app.
Jun 2023The UK Employment (Allocation of Tips) Act receives royal assent.
1 Oct 2024The UK tips law and its statutory code of practice come into force.