Splitting the Bill Six Ways

This brain teaser is about splitting a shared expense after applying a percentage increase, framed as a group of people settling a restaurant bill with a tip. The candidate must reason about how to incorporate a fixed percentage adjustment into a total before dividing it equally, and recognize that there is a straightforward way to compute each person's share without multiple intermediate calculations. It presents a familiar real-world context and asks the candidate to translate that into a clean numerical procedure that avoids confusion around order of operations and rounding.

The question leans on basic percentage calculations, proportional reasoning, and comfort with decimal arithmetic. It indirectly tests understanding of linearity: that scaling a total and then dividing it is equivalent to combining those into a single factor. Interviewers watch for whether the candidate can set up the computation systematically, explain why the steps are valid, and carry out the arithmetic accurately. They may also observe how the candidate checks their result and whether they can generalize the approach beyond this specific restaurant scenario.

What it tests

When a percentage increase is applied to a total before splitting it among individuals, the process can always be represented as multiplying the original amount by a scaling factor (such as $1 + r$ for a tip rate $r$), then dividing by the number of people. This is a manifestation of the distributive property: scaling and splitting commute, so the order does not matter. The key is that both operations—percentage increase and division—are linear, so you can combine them into a single multiplication by $(1 + r)/n$. This principle generalizes to any scenario where a total is adjusted by a fixed rate and then shared equally, and it holds because both operations preserve proportionality. Recognizing this lets you avoid unnecessary intermediate steps and reduces calculation errors.