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Practical guides to the calculations behind everyday decisions: the formulas, worked examples, the assumptions each one makes on your behalf, and the mistakes that produce a confident-looking wrong answer.

Every article here starts from a question someone actually needs answered — what a pay rise is worth after inflation, whether the larger pack is really cheaper, how much a set of small subscriptions costs over a year, what a discount stacked on a discount actually comes to. Each one gives the formula, walks through a realistic example with numbers, and then does the part that matters most: explains where the calculation stops being reliable.

Where a calculator on this site covers the same ground, the article links to it, so you can read the reasoning first and then run your own figures. Results throughout are estimates for planning and educational purposes; rates, taxes, fees and personal circumstances change the outcome.

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How to get the most from these articles

Read the assumptions and limitations section before trusting any number for a real decision. Most of these calculations are short — a division, a multiplication, occasionally an exponent — and the arithmetic is rarely where things go wrong. What goes wrong is using a figure that does not match the situation: an advertised fuel-consumption number instead of your own, a total-spending figure where essential spending was needed, or a rate quoted on a different basis from the one you are comparing it against.

Several of the articles overlap on purpose. Percentage change, reverse percentages and stacked discounts are the same underlying idea applied to three different problems, and reading two of them together tends to make the third obvious.