Split the bill on your vacation:The calculation of compensation of debts

Who owes whom and how much money? Cashinator knows (Download Cashinator-App). Cashinator calculates a mathematically optimal way to settle all debts in one go with as few transactions as possible.

However, it is often suspected that Cashinator is wrong with his calculations. Even if the travel costs are calculated manually, it is highly probable that a different result will be achieved. If person A has borrowed person B a lot of money, person B can but doesn't have to pay back the money to person A. I beg your pardon? Too abstract? Okay, then let's clarify this phenomenon with a concrete, simple case study:

Example

Maria, Susan and James are on vacation in threes. They make a total of three transactions:

Transaction 1:
Maria pays the accommodation, the costs are \$300. The situation is as follows:
1. Susan owes Maria \$100 and
2. James owes Maria \$100.

Transaction 2:
As Susan does not have enough money for the long awaited parachute jump, James lends her \$200. Now it looks like that:
1. Susan owes James \$200,
2. Susan owes Maria \$100 and
3. James owes Maria \$100.

Transaction 3:
Even the parachute jump of Susan does not change Maria's fear of flying, who spontaneously decides to go back with an overpriced train ticket worth \$200. The credit card does not work, Susan lends Maria the amount of \$200.

We do a simple calculation and we see that at the end of this journey the following money must be distributed, in order that all participants are balanced again:

1. Susan gives James \$200,
2. Maria gives Susan \$100,
3. James gives Maria \$100.

After these three compensation payments all participants would be balanced again. By the way, that's the bill that any of us would make.

Optimization

Three transactions are certainly correct and they were to be expected. But isnâ€™t there an even easier solution perhaps? Yes! Cashinator mathematically optimizes all transactions in such a way that a minimum number of compensation payments have to be made to balance the complete journey. Instead of three compensations Cashinator would suggest only one:

1. Susan gives James \$100.

Wow! That's all you have to do, everyone is balanced in one fell swoop! If you want to see, how that is possible, just look at the balance (especially at the end of the holiday):

Balance after transaction 1:
Maria +\$200, Susan -\$100, James -\$100.

Balance after transaction 2:
Maria: +\$200, Susan: -\$300, James +\$100.

Balance after transaction 3:
Maria: \$0, Susan: -\$100, James: +\$100.

Maria is balanced after transaction 3 without doing anything, nobody has to give her money and she doesn't have to pay money to anybody. Even though she was funding the accommodation. If Susan pays \$100 to James, the balance for all three participants solves perfectly to \$0.

Conclusion: After a single compensation payment all debts are settled and no further payment has to be made.

So what are you waiting for?

Try it out! Go on a journey, take the cashinator app with you! This app will make your life easier, I promise!