Splitting a bill — rounding, weights and who paid, for everyone at once

In Board 1, the input card on the left is the total, 87,000 yen, and the four cards on the right are Split the bill cards. All use 24 people; only the rounding step differs. The line under each card shows each person's amount, the amount collected and the leftover, as in "3,700 each × 24 = 88,800 (left over 1,800)".

A bigger rounding step leaves more over

With 100-yen steps, 3,625 becomes 3,700, so each person pays 75 yen extra. Among 24 people that is 75 × 24 = 1,800 yen. With 1,000-yen steps it becomes 4,000, which is 375 yen extra each and 375 × 24 = 9,000 yen in all. The bigger the step, the easier the collecting, and the more is left over. What to do with it is for the organiser to decide.

Changing the number of people changes the share. In the 100-yen card of Board 1, with 25 people, 87,000 ÷ 25 = 3,480 yen rounds up to 3,500 each, which collects 87,500 yen and leaves 500 yen over.

With weights, each share is total × weight ÷ total weight

Not everyone pays the same. You can give each person a weight, more for the director and less for the newcomer, and divide by weight. Board 2 has six people with weights: Director 2, Senior 1.5, Tanaka, Suzuki and Sato 1 each, and Newcomer 0.5. The weights add up to 7. The total is 30,000 yen.

Each share is total × the person's weight ÷ total weight, rounded up to 100 yen. The director's is 30,000 × 2 ÷ 7 = about 8,571 yen, rounded up to 8,600 yen. Senior pays 6,500 yen, Tanaka, Suzuki and Sato 4,300 yen each, and Newcomer 2,200 yen. Together that is 30,200 yen, and the leftover is 200 yen, the difference from the total.

This calculation is a function called share, and the list column of weights is connected to it so that everyone is worked out at once. How functions work is in My functions in the card reference. The rounding is the 2 at the end of the formula; use 3 for 1,000-yen steps, or remove ceilto to skip rounding. Formulas are covered in Writing formulas and functions in the usage guide.

If Newcomer's weight is 1, the weights add up to 7.5 and one weight is 4,000 yen, so every share divides exactly and the leftover is 0 yen. Whether anything is left over depends on the weights.

When someone paid up front, everyone's balances add up to 0

When one person pays for the group, the bill needs settling. In Board 3, Tanaka paid 12,000 yen and Suzuki 3,000 yen, and Sato and Takahashi paid 0 yen. The total is 15,000 yen, and each person's share is 15,000 ÷ 4 = 3,750 yen.

Each person's balance is what they paid − their share. Tanaka's is +8,250 yen, Suzuki's is −750 yen, and Sato's and Takahashi's are −3,750 yen. People with a plus receive money; people with a minus pay.

Add up everyone's balances and you always get 0 yen. The 8,250 yen to be received equals what is to be paid: 750 + 3,750 + 3,750 = 8,250 yen. If the sum is not 0, the total or the number of people is wrong somewhere, so it works as a check.

If Suzuki had paid 9,000 yen, the total would be 21,000 yen and each share 5,250 yen; Tanaka would be +6,750 yen, Suzuki +3,750 yen, and Sato and Takahashi −5,250 yen. They still add up to 0 yen.

What this calculation cannot tell you

The boards give amounts when the total, the number of people and the weights are settled. Collecting the money in practice needs some decisions.

• The leftover — whether the organiser keeps it, carries it over or lowers the next bill is not fixed by any rule. Who absorbs the odd yen is for the group to decide.
• The weights — weights are numbers a group agrees on. The 2 and the 0.5 in Board 2 are placeholders.
• Who sends what to whom — Board 3 stops at each person's balance. Who sends how much to whom is not unique, and the combinations grow with the number of people.
• Itemised bills — amounts by what each person drank or ordered, service charges, coupon discounts and payment fees are not in the boards.

Read next — Rounding, rounding up and rounding down — finding approximate values The boards compare rounding up, down and to the nearest, and which place to round to.

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