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About Utilia Math Guides
A calculator that returns a number and explains nothing leaves you unable to check it. Utilia math guides exist for the other half of the problem: they show the formula, walk through the arithmetic one step at a time, and say plainly where the result is and isn't meaningful.
Every guide is paired with the calculator that automates it, so you can read the method once and then stop doing it by hand. The formulas here are standard mathematics — nothing proprietary, nothing you couldn't verify in a textbook — and the guides are written to make that verification easy rather than to make the site look clever.
All guides are free and require no account. New articles are added as the Math calculator collection grows.
What These Guides Cover
Percentages are the arithmetic people most often get wrong under time pressure, and usually in the same two places: confusing a percentage of a number with a percentage change, and assuming a rise then a fall of the same percentage returns you to where you started. It doesn't — a 50% gain followed by a 50% loss leaves you 25% down. The percentage guide works through each formula separately and shows why the order matters.
Mean, median and mode are the other common trap. They are all called "the average" in ordinary speech, and on a symmetric data set they land close together — which is exactly why people trust the mean on data that isn't symmetric. One outlier moves the mean and leaves the median alone, which is why median household income and mean household income tell different stories about the same country. The central-tendency guide shows how to calculate each measure and, more importantly, how to tell which one is lying to you.
Each article states the formula in plain notation, gives at least one fully worked numerical example, and links directly to the calculator that performs it. If you spot an error in any formula or worked example, tell us — corrections are made quickly and the page records when it last changed.