Essay · Ratio
A ratio isn't a slice of a whole
why a percentage is allowed to pass 100%
A ratio is the size B would have if you called A's size 1 — a comparison between two quantities. A proportion is the special case where B is a part of A, which is the only case that has to land between 0 and 1. Because most ratios aren't parts of anything, a percentage can be more than 100%: 150% simply means 1.5 times as large.
Almost everyone meets ratios through one narrow example: a slice of pie. Three quarters of the class, forty percent of the budget, half the bottle. The example works, and it quietly installs a rule that isn't true — that the number has to sit between zero and one. Then a treadmill offers a 40% incline, an economy grows 250%, and the rule breaks in public.
What a ratio actually says
Take two quantities, A and B. Call A's size 1. Now ask what size B is, measured on that new scale. That answer is the ratio. It is a re-description of B in units of A, and it carries no promise that B came out of A.
Compare the sugar in a banana with the sugar in an apple and the banana comes out at
1.17. Above one, and not a contradiction — the banana was never a piece of the
apple. It's just 1.17 times as sweet by that measure. As a percentage, that's 117%.
Then what is a proportion?
A proportion is a ratio with an extra condition attached: the thing on top is contained in the thing on the bottom. Spending 40 of a 100-unit budget is a proportion, because the 40 came out of the 100. Under that condition the value genuinely can't exceed 1, and a percentage genuinely can't exceed 100%.
So the cap is not a property of ratios. It's a property of parts. Every proportion is a ratio; most ratios are not proportions.
| Quantity | What it compares | Can it pass 100%? |
|---|---|---|
| Share of a budget spent | Part against its own whole | No — it's a proportion |
| Probability of an event | Outcomes against all outcomes | No — it's a proportion |
| Market share | One firm against the whole market | No — it's a proportion |
| Road or treadmill grade | Rise against run | Yes — 100% grade is a 45° slope |
| Growth over a year | New size against old size | Yes — 250% growth is ordinary |
| Price of A against price of B | Two independent prices | Yes — nothing contains anything |
| Debt-to-GDP | A stock against a flow | Yes — several countries are past 200% |
Is a ratio the same as a fraction?
A fraction is notation; a ratio is a meaning that notation can carry. Writing 3/4
tells you nothing about whether the 3 came out of the 4. That's why 5/4 is perfectly
sensible as a comparison and impossible as a share — same symbols, different claim about the
world.
A rate is the same idea again, with the two quantities measured in different units: kilometres per hour, dollars per kilogram, rise per run. Grade is a rate written as a percent, which is why a 40% incline means climbing 40 metres for every 100 you travel forward, and not a 40° angle.
The part that pays off later: you don't need the absolute sizes
Here's the quiet consequence, and the reason this is chapter one of the series. Once you have a comparison index, the original sizes stop mattering. We never learn how many grams of sugar are in the banana or the apple. We never need to. The 1.17 survives whatever those numbers were.
That habit is what makes calculus possible. The derivative compares an input nudge with an output nudge, and both are too small to measure — but a comparison index doesn't need measurements, only a relationship. That's exactly what "marginal" means in an economics textbook.
Go stand on it
A 100% grade is the cleanest proof that a ratio isn't a share: it's a 45° hill, not a vertical wall. Walk a 100% grade in Open World Discover and then try 40% for comparison.