The Mathematics Behind Fair Bill Splitting
Published: October 2024
Ever wondered how bill splitting calculators work? Let's explore the math:
**Equal Splitting**
The simplest case: Total amount ÷ Number of people = Amount per person
**Proportional Splitting**
When people consume different amounts, we calculate individual shares based on what each person ordered or used.
**Debt Minimization Algorithm**
This is where it gets interesting. Instead of having everyone pay the person who covered the bill, we use a graph theory approach to minimize transactions.
For example, if Alice owes Bob $10, Bob owes Charlie $10, and Charlie owes Alice $5, instead of making 3 transactions, we can settle everything with just one: Bob pays Charlie $10, and Alice pays Charlie $5.
Our algorithm uses a greedy matching approach:
1. Calculate net balances (who is owed money vs. who owes money)
2. Match the largest creditor with the largest debtor
3. Create a transaction for the minimum of their amounts
4. Repeat until all debts are settled
This mathematical approach ensures you make the fewest possible transactions while settling all debts fairly.