Math calculators

Two different kinds of arithmetic

These five tools look like one category but split cleanly in two, and the mistakes they attract are quite different.

Proportional maths — percentages, ratios and averages — is about relationships between quantities. The arithmetic is simple; the errors come from applying it to the wrong base, or combining results that cannot be combined.

Calendar maths — age and date difference — looks simpler still and is not. Months have unequal lengths, leap years interrupt the pattern, and "between" two dates can mean two different counts depending on whether you include the endpoints.

Where the mistakes cluster

• Percentages do not add across periods. A 10% rise followed by a 10% fall leaves you down 1%, not level, because the second percentage applies to a larger base.
• The mean is not always the average you want. One outlier drags it; the median does not move. A week of sales with a closed day is the classic case.
• Ratios do not add. Doubling 3:2 gives 6:4, not 5 parts doubled.
• Counting days inclusively. A stay from the 1st to the 4th is three nights, not four — the source of endless booking and deadline confusion.

Each tool page covers its own version of this in more depth, and the guide on percentages in everyday life works through the ones that show up in headlines, pay rises and sale prices.