How do you keep units consistent when estimating a school district's annual lunch spend?

## Units are the cheapest error detector you have In a Fermi chain, nearly every serious mistake is a unit mistake in disguise. You cannot check a guess for correctness, but you can always check a chain for *dimensional consistency* — whether the units multiply out to the units of the answer. This costs nothing and catches the errors that change an answer by a factor of two, twelve, or a thousand. The discipline: write each factor with its unit attached, treat the units as algebra, and cancel. ## The worked chain **Question:** what does a school district spend on lunches in a year? 1. **Enrolment:** 20,000 students. Unit: *students*. 2. **Participation rate:** about 60 percent of students buy lunch on a given day. Unit: *meals per day per student*. - 20,000 students x 0.6 meals/day/student = **12,000 meals per day**. 3. **School days in a year:** about 180. Unit: *days per year*. - 12,000 meals/day x 180 days/year = **2.16 million meals per year**. 4. **Cost per meal:** about 3.50 dollars, covering food, labour and overhead. Unit: *dollars per meal*. - 2.16 million meals/year x 3.50 dollars/meal = **about 7.6 million dollars per year**. Round to **8 million dollars a year**. Now read the cancellation back: students cancels against per-student, meals cancels against per-meal, days cancels against per-day, and what survives is dollars per year. That is the requested unit, so the chain is at least dimensionally sound. ## The traps this catches **Calendar days versus operating days.** Using 365 instead of 180 roughly doubles the answer. The unit *days per year* is not wrong, but the *kind* of day is. Whenever a factor is a count of periods, ask which periods actually count: school days, business days, opening hours, delivery days. **Stock versus flow.** Enrolled students is a **stock** — a quantity that exists at a point in time. Meals served is a **flow** — a quantity per unit of time. You cannot multiply a stock by a price and get an annual figure; you need a rate to convert the stock into a flow first. Here the participation rate does that job. Chains that produce absurd answers very often multiplied a stock by a price directly. **Per-what ambiguity.** A cost of 3.50 dollars can be per meal, per student per day, or per student per year, and the three differ by more than an order of magnitude. Writing the denominator explicitly forces you to notice which one you actually have. **Period mismatch.** A monthly figure multiplied by 365, or a weekly rate multiplied by 12, are silent factor-of-thirty and factor-of-four errors. Normalising everything to one period before multiplying removes the whole class. **Double counting the same conversion.** Applying a participation rate and then also using a *meals per student* figure that already embeds participation halves the answer. Each conversion should appear exactly once in the chain. ## Per-capita anchors carry units too Many estimates lean on an anchor rate: meals per student per day, cups per person per week, doctor visits per person per year. These are useful precisely because they are dimensionally explicit — *per person per year* tells you exactly what you must multiply by to reach a total. When an anchor rate feels wrong, check its denominator first: *per household* and *per person* differ by roughly the household size, and *per user* and *per active user* can differ by far more. ## Ordering the chain for readability A small habit that pays off in interviews: order the factors so that each intermediate product is itself a meaningful quantity you can sanity-check aloud. In the chain above, the intermediates are 12,000 meals a day and 2.16 million meals a year — both are numbers a listener can react to. If an intermediate is meaningless, such as *students times dollars*, the chain is probably ordered badly or missing a conversion. ## Sanity-checking the result against a known total Once the units check out, do a second, cruder check on magnitude. Eight million dollars across 20,000 students is 400 dollars per student per year, or a bit over 2 dollars per student per school day — consistent with the 3.50 dollars per meal and 60 percent participation you assumed, since 0.6 x 3.50 is 2.10. That circularity is fine as an arithmetic check; the real magnitude check is against a total you know independently, such as the district's overall budget. If your lunch estimate came out larger than a plausible total budget, something upstream is broken. ## What the interviewer is watching for They want to hear the units spoken. A candidate who says *twelve thousand meals per day, times a hundred and eighty school days per year* is auditable in real time. A candidate who says *twelve thousand times a hundred eighty times three fifty* has produced a number nobody, including themselves, can check.