The most common maths mistakes, and what causes each one

Most wrong answers are not careless — they are a rule applied correctly in the wrong place. These are the specific mistakes people actually make, grouped by topic, with the misunderstanding behind each group.

Updated 30 August 2026

A mistake that gets repeated is almost never carelessness. It is a rule that works somewhere else, applied confidently in a place it does not hold — which is why telling someone to "be more careful" rarely fixes anything, and why naming the specific wrong rule usually does.

The lists below are the mistakes that actually recur, taken from the lessons on this site. They are written so you can recognise your own work in them. If one of them makes you wince, that is the useful one.

Fractions

The largest single source of later trouble. Nearly every one of these comes from treating a fraction as two separate numbers rather than as one number written as a division.

Equivalent & Simplifying Fractions

Adding the same number to the top and the bottom instead of multiplying: writing $3/4 = 4/5$ .
Changing only one part of the fraction, so $3/4$ becomes $3/12$ instead of $9/12$ .
Stopping half way when simplifying, leaving $9/12$ when $3/4$ was asked for.
Swapping the whole number and the remainder when converting, writing $17/5$ as $2 3/5$ instead of $3 2/5$ .
Using the leftovers as the denominator: calling 18 fletched out of 24 arrows "18/6".

Comparing Fractions

Comparing numerators alone and concluding that $3/10$ is bigger than $1/2$ .
Assuming a bigger denominator means a bigger fraction, so claiming $1/8 > 1/3$ .
Cross-multiplying but attaching each product to the wrong fraction.
Changing the denominator to the common one without scaling the numerator to match.
Giving the answer as the rewritten fractions $(9/24, 12/24$ , ...) instead of the originals the question asked about.

Adding & Subtracting Fractions

Adding the denominators as well as the numerators: writing $1/2 + 1/3 = 2/5$ .
Changing the denominator to the common one but carrying the old numerator across, so $1/2$ becomes $1/6$ .
Flipping a fraction subtraction round to avoid a negative instead of regrouping a whole.
Adding only the whole numbers of two mixed numbers and forgetting the fraction parts.
Leaving an answer unsimplified, or leaving an improper fraction where a mixed number was wanted.

Multiplying & Dividing Fractions

Multiplying straight across when the question is a division, instead of flipping the divisor first.
Flipping the first fraction rather than the divisor, which gives the reciprocal of the right answer.
Looking for a common denominator before multiplying, which is needed for addition but not here.
Multiplying a fraction by a whole number and multiplying the denominator too, so the fraction never changes.
Multiplying mixed numbers part by part instead of converting them to improper fractions first.

Decimals, percents and ratios

These three are one idea in three notations, and most errors here are conversion errors — moving between the notations without carrying the meaning across.

Decimals

Judging size by digit count and claiming $0.45 > 0.5$ .
Lining numbers up by their last digit instead of by the decimal point when adding.
Losing a decimal place in a product: writing $0.4 x 0.3 = 1.2$ instead of 0.12.
Dropping a zero placeholder, turning 3.072 into 3.72.
Writing $3/8$ as 0.3 by copying the numerator after the point.

Percents

Forgetting to divide by 100 and answering 600 for "40% of 15".
Giving the discount instead of the price actually paid.
Comparing the change with the new amount rather than with the original when finding a percent change.
Dividing the whole by the part when asked "what percent of a is b", getting the reciprocal by mistake.
Assuming $a 20%$ increase followed by $a 20%$ decrease returns to the starting value.

Ratios & Proportions

Adding the same number to both parts of a ratio instead of scaling them.
Cross-multiplying the wrong pair, so a numerator is matched with the denominator directly below it.
Choosing the cheapest total price rather than the lowest cost per item.
Dividing a total by one part of the ratio instead of by the sum of the parts.
Confusing a part-to-part ratio with a part-to-whole fraction.

Negative numbers and the order of operations

Sign errors and precedence errors are the two most common ways an otherwise correct method produces a wrong answer.

Negative Numbers

Ordering negatives by size alone and claiming $-9 > -2$ .
Getting the size of an answer right but the sign wrong — the classic sign error.
Treating "minus a minus" as another minus: writing 7 - $(-4) = 3$ .
Dropping the minus sign from a starting value in a word problem.

Order of Operations

Evaluating strictly left to right and getting $2 + 3 x 4 = 20$ .
Doing all multiplication before all division (or all addition before all subtraction) instead of going left to right.
Ignoring brackets, or resolving the outer bracket before the inner one.
Applying an exponent to a whole product, writing $3 x 4 2$ as 144.

Exponents

Multiplying the base by the exponent: reading $34$ as 12.
Multiplying the exponents when the powers are multiplied, or multiplying the bases as well.
Claiming $a 0 = 0$ , or that a negative exponent makes the answer negative.
Applying an exponent to a coefficient: reading $3 x 4 2$ as 144.

Algebra

Almost every mistake here is doing something to one side of an equation and not the other, or distributing over the wrong thing.

Variables & Expressions

Combining unlike terms, writing $3x + 4$ as 7x.
Reading 5x with $x = 9$ as the digits 59 instead of the product 45.
Distributing to only the first term inside a bracket: $4(x + 3) = 4x + 3$ .
Losing a sign when a negative factor multiplies a bracket, as in -2(3x - 5).
Squaring a negative incorrectly, treating $(-4) 2$ as -16.
Translating "5 less than n" as 5 - n.

One-Step Equations

Using the wrong inverse: answering 26 for $x + 19 = 7$ because the two numbers were added instead of subtracted.
Changing only one side of the equation, which breaks the balance.
Answering with the number on the right of the equals sign instead of the value of the unknown.
Subtracting the coefficient in $5x = 40$ and answering 35 instead of dividing to get 8.

Two-Step Equations

Dividing before subtracting: answering $19/2$ - 7 for $2x + 7 = 19$ instead of clearing the $+7$ first.
Stopping one step early and giving the value of 2x rather than the value of x.
Dividing the whole total by the rate in a fee-plus-rate problem, forgetting that part of the total was the one-off fee.
Losing the minus sign on a negative coefficient and getting the right size with the wrong sign.

Multi-Step Equations

Distributing to the first term only: writing $3(x + 4)$ as $3x + 4$ .
Forgetting that a minus outside a bracket multiplies both terms, so $-2(x + 5)$ becomes $-2x + 10$ instead of -2x - 10.
Moving a term across the equals sign without changing its sign.
Combining unlike terms, for example turning $5x + 3$ into 8x.

Factoring

Stopping too early — taking out the common factor but leaving a bracket like $x 2$ - 9 that still factors.
Taking out only the number and forgetting the shared x, writing $3(2x 2 + 3x)$ instead of $3x(2x + 3)$ .
Losing the common factor from the answer: writing $(x + 2)(x + 3)$ when the polynomial was $2x 2 + 10x + 12$ .
Getting the signs the wrong way round, so that (x - 2)(x - 5) is offered for $x 2 + 7x + 10$ .
Trying to factor a sum of squares such as $x 2 + 16$ as if it were a difference.
Giving the trinomial of a cube factorisation the wrong middle sign, so $x 3 + 27$ comes out as $(x + 3)(x 2 + 3x + 9)$ . SOAP says Opposite.
Writing the second factor of a cube as a perfect square — $(x + 3)(x 2 + 6x + 9)$ — when the middle term is 3x, not 6x.
Losing the sign of the second group, so $x 3 + 2x 2$ - 3x - 6 is offered as $(x + 2)(x 2 + 3)$ instead of $(x + 2)(x 2$ - 3).

Geometry

Formula substitution and unit errors, plus applying a theorem to a shape that does not satisfy its conditions.

Perimeter & Area

Reporting a perimeter when an area was asked for, or giving an area in plain (non-square) units.
Using the slanted side instead of the perpendicular height.
Forgetting the $1/2$ in the triangle formula, or adding the parallel sides of a trapezoid without averaging.
Subtracting instead of dividing when working backwards from an area to a missing side.

Volume & Surface Area

Reporting a surface area when a volume was asked for, or using the wrong units.
Counting only three faces of a box instead of six, or using the cube shortcut on a non-cube.
Doubling the radius of a cylinder instead of squaring it, or forgetting to multiply by the height.
Multiplying by 1000 when converting $cm 3$ to litres instead of dividing.
Leaving out the $1/3$ for a cone or a pyramid, which gives the whole cylinder or prism instead.
Treating a given diameter as the radius, so a cone comes out four times too big and a sphere eight times.
Squaring the radius of a sphere instead of cubing it, or borrowing the $4/3$ of a sphere for a cone.
Putting the slant height of a cone into the volume formula in place of the perpendicular height.

Pythagorean Theorem

Adding the legs without squaring them — $3 + 4$ is 7, not 5.
Stopping at $c 2$ and forgetting the square root.
Adding when the hypotenuse was already given, instead of subtracting.
Applying the theorem to a triangle that has no right angle.

Angles

Using 180° for complementary angles or 90° for supplementary ones.
Answering with x when the question asked for an angle (or the other way round).
Assuming a longer-looking arm means a bigger angle.
Treating two neighbouring angles at an intersection as equal — those are the opposite ones.

What to do with a mistake you recognise

Do not simply redo the problem. A mistake that is a wrong rule will come back, because the rule is still there.

Say the rule out loud in words. "I added the denominators" is a fixable sentence; "I got it wrong" is not.
Find the place the rule $*is*$ true. You add denominators nowhere, but you $*do*$ multiply them — knowing which operation the habit came from stops it firing next time.
Do two problems, not ten. One where the correct rule applies and one where the wrong rule would have been tempting. Volume does not fix a misconception; contrast does.