Can that percentage exist over that many people?

A published rate carries its own denominator inside it. If someone reports 66.8% of 396 people, no whole number of people produces that figure — the fraction steps 66.7 → 66.9 and skips it. Paste any rate and the population it claims to be over. Your browser enumerates every possibility and tells you whether the claim is arithmetically reachable, and if not, which denominators would reach it.

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Why the decimal is the evidence

The number of digits after the point is not decoration — it constrains the denominator. A rate given to one decimal can only land on values a fraction with that denominator can actually hit. Small populations hit few values and skip most of the range; large populations hit nearly all of them. So when a reported rate falls in one of the gaps, the population it was computed over is not the population named beside it.

This is the ordinary way a denominator shrinks in public: a study enrols 534 people, 396 finish, 193 answer the question being reported — and the sentence in the abstract names the largest of the three. Nobody lies. The qualifier just falls off between the results section and the summary, and then off the pricing page entirely.

The question that comes before the arithmetic

On 21 August 2026 I ran this page over the ten published figures for how many US employers cover GLP-1 drugs. Nine came back green. At least five of those greens were worthless: those figures are weighted estimates of a national population, not tallies of the people who answered. A weighted numerator is a sum of fractional weights, so no whole number produces it — and the sample size printed beside it was never its denominator. The page was answering can this fraction exist? when the honest first question is is this a fraction at all?

So the order is now: (0) is the figure a count or an estimate? (1) if a count, is the number printed next to it really its denominator, or the size of some larger group? (2) only then, is it reachable? Skipping straight to (2) is what this tool used to do, and it is what most people do by hand. The tick-box above is step 0. The ten figures, one by one.

What the tool does and does not prove

• Proves: that a rate is unreachable over a stated denominator. That is a closed arithmetic fact, not an opinion — every integer is tested.
• Suggests: which denominators do reach it. Many will; the smallest ones are usually the interesting ones, and are often confirmable elsewhere in the same document (degrees of freedom in a correlation, a sample size in a table).
• Does not prove: intent, or that anyone did anything wrong. A rounding convention, a typo, or truncation instead of rounding will all show up here. The tool tests both rounding and truncation and says which one, if either, works.

The controls

An instrument that cannot return the other answer is not an instrument. Press Run controls: it must find a solution where one exists, fail where none does, detect a truncated figure that rounding alone would miss, and — the two added on 21 August — let the step-0 gate override a green the arithmetic would have handed out, while not becoming a blanket silencer that refuses every figure. If any control comes back DEAD, ignore every verdict on this page until it is fixed.