Everyday math

Percentages, proportions, and order of operations are taught early and misapplied constantly, usually because the rules are remembered and the reasons are not. The errors are systematic rather than random, which means they are worth learning to spot.

Percentage changes do not cancel

This is the single most consequential property of percentages. A 50 percent gain followed by a 50 percent loss does not return you to your starting point. $100 rises to $150, then loses 50 percent of $150 — which is $75, not $50 — leaving $75. You are down 25 percent.

The reason is that each percentage applies to a different base. The gain applies to $100, the loss to $150. Reversing a 50 percent loss requires a 100 percent gain; reversing an 80 percent loss requires 400 percent. This asymmetry is why volatility erodes compounded returns and why the arithmetic average of a series of percentage changes overstates the actual outcome.

Combining successive changes requires multiplying the factors, not adding the percentages. A 20 percent rise followed by a 10 percent fall is 1.20 times 0.90, which is 1.08 — an 8 percent net rise, not 10. The same applies to stacked discounts: 30 percent off then 20 percent off is a 44 percent total reduction.

Reversing any percentage requires division. To recover a pre-discount price from a 30 percent sale, divide by 0.70. To get a pre-tax amount from a total including 8 percent tax, divide by 1.08. Subtracting the percentage instead is the most common percentage error in everyday use, and it always produces a number close enough to look correct.

Percentage points are not percentages

A rate moving from 2 percent to 3 percent has risen by one percentage point and by 50 percent. Both statements are accurate and describe the same event, which is exactly why the ambiguity gets exploited.

The relative figure sounds dramatic when the base is small. An interest rate going from 0.5 percent to 1 percent is a 100 percent increase, which is technically true and rhetorically misleading. In the other direction, a conversion rate improving from 2 percent to 2.4 percent is only 0.4 percentage points but a 20 percent lift in conversions, which is the commercially relevant framing.

The convention that removes the ambiguity is to use percentage points for absolute differences between rates and percent for relative change, and to state which you mean whenever the base is small enough for them to diverge sharply.

Proportion, and when it does not apply

A proportion states that two ratios are equal, and given three terms the fourth follows from cross multiplication: from a/b = c/x, x equals bc/a. It is the most broadly useful piece of arithmetic there is — recipe scaling, map distances, unit conversion, mixing ratios.

It assumes a straight line through the origin: doubling the input doubles the output, and zero input gives zero output. Plenty of real relationships fail one or both conditions, and applying a proportion anyway gives a confident wrong answer.

Anything with a fixed component fails the origin test — a taxi fare with a flag-fall, a utility bill with a standing charge, a job with a setup cost. These are linear but not proportional, and scaling them proportionally overcharges the small case and undercharges the large one.

Others are not linear at all. Cooking time scales with thickness rather than volume, so doubling a recipe into the same pan needs considerably longer rather than twice as long. Wind resistance rises with the square of speed. Area scales with the square of linear dimensions and volume with the cube, so doubling a room does not double the paint — it quadruples it. And doubling a team does not halve a project timeline.

The check is simple: does zero input plausibly give zero output, and is the ratio the same between two known pairs? If a 10-unit order costs $100 and a 20-unit order costs $180, the relationship is not proportional and a proportion will mislead.

Ratios, fractions, and parts

A 1:3 ratio means one part to three parts, so the first component is one quarter of the whole and not one third. Reading a ratio as a fraction is a frequent error, particularly in dilution and mixing.

Multi-term ratios work by totalling the parts. For 1:2:3 cement, sand, and aggregate across 12 cubic metres, the six parts make one part equal 2 cubic metres, giving 2, 4, and 6. The advantage of working in parts is that the specification holds at any total.

Dilution instructions are genuinely ambiguous in ordinary use. "Dilute 1:10" can mean one part concentrate to ten parts water, giving eleven parts total, or one part making up ten parts total. For chemicals where the difference matters, the product documentation settles it and guessing does not.

Order of operations

Multiplication and division bind more tightly than addition and subtraction, and are evaluated left to right among themselves. So 2 + 3 × 4 is 14, not 20.

A basic four-function calculator gives 20, because sequential entry applies each operation as the key is pressed with no lookahead. Scientific calculators, spreadsheets, and programming languages all give 14. Both are internally consistent, which is precisely why a figure checked on two devices disagrees.

Left-to-right among equal precedence matters more than it looks. 100 ÷ 5 × 2 is 40, since the division comes first. Reading it as 100 ÷ (5 × 2) gives 10. The expression as written has one correct answer; the intent behind it is frequently the other one, which argues for parenthesising anything ambiguous rather than relying on the reader knowing the rule.

Some notation is genuinely undecidable. 6 ÷ 2(1+2) evaluates to 9 or 1 depending on how implicit multiplication is ranked, and textbooks, calculator manufacturers, and computer algebra systems genuinely disagree. The expression is badly written rather than difficult, and the fix is parentheses rather than a rule.

One last note on precision: computed decimal results are approximate, because binary floating point cannot represent most decimal fractions exactly. 0.1 plus 0.2 gives 0.30000000000000004. The error is about one part in ten quadrillion and never matters for everyday arithmetic, but it matters for exact equality comparisons and for money, which is why financial systems store integer cents.

Frequently asked questions

Calculators referenced in this guide