The GCSE Maths Foundation mistakes examiners flag every series

Calling a rise from £40 to £50 a 10% increase (using the new value as the base, or reading the gap as the percentage), or 'increase £80 by 15%' as £95, instead of a percentage change of the original amount.

Finding the original before a percentage change by taking the percentage off the new value (£384 after a 20% rise → £307.20), instead of dividing by the multiplier (384 ÷ 1.20 = £320).

Working out £8600 down 15% then 10% as 25% off (£6450), or 4% a year for 5 years as 20% off, instead of multiplying the multipliers (× 0.85 × 0.90; × 0.96 to the power 5).

Longer-is-bigger reasoning (0.45 > 0.6), ignoring the decimal point, or misplacing it in arithmetic results.

Treating a fraction as two integer streams, adding numerators and denominators independently.

Reading a quarter as '4%', or giving two fifths of 1020 as 40, treating the fraction's digits as the answer instead of dividing by the denominator and multiplying by the numerator.

Treating a rounded figure as exact (spending £12 at face value), or setting bounds a whole unit either side instead of half, so 8400 to the nearest 100 gives 8499 instead of 8449.

Reading a machine chain with BIDMAS (12 − 4 × 5 = −8 instead of (12 − 4) × 5 = 40), or reversing it by running the same operations again instead of inverting each one in reverse order.

Solving c/4 = 8 as c = 2 by dividing by 4 again, or moving a term across the equals sign without flipping its sign, instead of applying the inverse operation to both sides.

Listing the integers for −3 ≤ x < 2 with the wrong endpoints, or solving 5y + 14 ≥ 11 as the equation y = −0.6, instead of respecting the boundary and keeping the inequality.

Mashing unlike terms into one — 9x + y − 6x + y written as 5xy, or 12a + 15b factorised to 27ab — instead of collecting only like terms.

Multiplying out 5c(2d + 1) as 10cd, dropping the second term, instead of distributing the multiplier across every term inside the bracket.

Recording repeated multiplication as a coefficient (y × y × y = 3y) and repeated addition as a power (y + y + y = y³), or adding coefficients instead of multiplying (4 × 2c = 6c).

Taking the median of 7, 2, 9, 4, 5 as 9 (the middle of the unsorted list), or finding a missing value by averaging the numbers you can see instead of using total = mean × n.

Thinking a range of 12 is 'better' than a range of 4, or that adding 10 to every value changes the range, instead of reading the range as a measure of how spread out the data is.

Giving the mean when asked for the range of 3, 8, 5, 12, 7, or picking the repeated value 91 as the outlier instead of 159, treating the named measures as interchangeable.

Writing the tea sector as 15° because 15 people chose tea, or reading a 120° sector as 120 people, instead of converting between the angle (out of 360°) and the frequency (out of the total).

Saying Silver (15) is 10 times Gold (5) because 15 − 5 = 10, or counting a half pictogram symbol as a whole, instead of reading the value off the axis scale or the key.

Sharing £240 in 1 : 3 as 240 ÷ 3 = £80 because the total is divided by one ratio number, instead of by the sum of the parts (240 ÷ 4 = £60, so the larger share is £180).

Writing the fraction wearing glasses for 3 : 8 as 3/8 (part over part) instead of 3/11 (part over the whole, since 3 + 8 = 11 parts).

Writing 6 : 2 in the form n : 1 as 4 (doing 6 − 2) instead of 3 (6 ÷ 2), or scaling 2 : 3 to 4 : 5 by adding, instead of multiplying or dividing both parts by the same factor.

Treating perimeter and area as one fused 'size' — computing the wrong measure for what's asked, adding only the labelled sides of an L-shape (omitting the indented ones), or using the circumference formula (πd) where area (πr²) is needed.

Working out a triangle's area as base × height (e.g. 20 × 6.3 = 126) and stopping, instead of halving to get base × height ÷ 2 = 63 — you've found the bounding rectangle, not the triangle.

Applying a length ratio straight to area or volume — doubling the side and expecting the area to double (it goes ×4), or assuming two rectangles of equal area must have equal perimeters.

Reading a right-angled triangle wrongly — choosing sin/cos/tan without labelling opp/adj/hyp, or computing √(a² + b²) for a side that is shorter than the hypotenuse instead of √(c² − b²).

Treating a polygon's exterior angles like a triangle's — assuming they add to 180° instead of 360°, so a regular pentagon's exterior angle comes out wrong (it is 360 ÷ 5 = 72°, interior 108°) rather than the hexagon's 360 ÷ 6 = 60°.

Judging a quadrilateral's symmetry by appearance — giving a general parallelogram 2 lines of symmetry when it has 0 (rotational order 2), or a kite the wrong number when it has exactly 1 — and drawing plan and elevation views as a copy of the 3D picture rather than as flat 2D projections.

Reading a calculation strictly left to right, so 60 ÷ 2 + 4 becomes 60 ÷ 6 = 10 instead of 30 + 4 = 34, and treating 3 × (4 + 2) as 3 × 4 + 2 = 14 instead of finishing the bracket first for 3 × 6 = 18.

Judging −7 to be larger than −5 because 7 > 5, instead of reading the number line where further left is smaller, and losing the sign in directed-number arithmetic so −3 × 3 is taken as 9 rather than −9, or 5 − 7 as 2 rather than −2.

Treating a squared negative as still negative, so (−8)² is taken as −64 rather than 64, and reading −4² and (−4)² as identical instead of −16 versus 16 because only the bracketed form squares the sign.

Writing the nth-term rule as n plus the common difference (n + 3) instead of using the difference as the coefficient of n, so a sequence rising by 3 from 2 is 3n − 1 rather than n + 3.

Answering with a term-to-term step (add 4) when the nth-term rule was asked for, or the reverse, and treating a listed term as if it were the step between terms.

Continuing a geometric sequence (× r) or a Fibonacci sequence (add the two previous terms) by adding a constant difference (+ d), instead of spotting that the rule is multiplicative or additive-of-previous-terms.

Reading the coefficient of x straight off the equation as the gradient (calling 2y = 4x + 6 gradient 4) instead of rearranging to y = mx + c first, where m = 2 is the gradient.

Reading the y-intercept as an x-value (or the reverse), or finding an intercept without setting the other variable to zero, instead of using x = 0 for the y-intercept and y = 0 for the x-intercept.

Reflecting a point in an axis instead of the line stated in the question (or stopping at the point on the line), instead of measuring the same distance across the mirror line, so (4, −1) reflected in x = 6 becomes (8, −1).

Treating a fractional scale factor as if it could not shrink a shape (calling 6 enlarged by ⅓ equal to 8 or 18) instead of multiplying every length by the scale factor, so 6 × ⅓ = 2.

Swapping the components of a column vector when reversing it (turning (−3, 7) into (7, −3)) or writing the vector as a fraction, instead of negating each component in place to get (3, −7).

Finding the chance of both events by adding the probabilities (0.3 + 0.3 instead of 0.3 × 0.3 = 0.09), or leaving a tree branch without its 1 − p complement, instead of multiplying along the branches.

Reading '15 own a watch' as 15 watch-only when 7 of them also own a headset (so watch-only is 8), or taking a probability over one region instead of the whole set of 60.

Reading 'expected number' as a rounded guess instead of relative frequency × trials (0.35 × 200 = 70, not 'about 50'), or trusting the bigger relative frequency over the experiment with more trials.

Working out 30 miles in 20 minutes as 30 ÷ 20 = 1.5 mph because minutes are treated as hours, instead of converting the time first (20 minutes = ⅓ hour, so 90 mph), or reading a speed off a graph as a point not a gradient.

Comparing speeds or densities in different units as if directly comparable (24 s vs 28.8 km/h), comparing totals not rates, or working mass as 300 ÷ 2 = 150 instead of density × volume = 2 × 300 = 600 g.

Treating every proportion as direct, so 15 workers seem to take longer than 10, instead of recognising inverse proportion (the product is fixed: 10 × 9 = 90 worker-hours, so 15 workers take 6 hours).

Working 10³ as 10 × 3 = 30 because the index is read as a multiplier, instead of repeated multiplication (10 × 10 × 10 = 1000).

Working 19² as 19 × 2 = 38, or 1.5² as 3, because squaring is read as doubling instead of multiplying the number by itself (19 × 19 = 361).

Turning 2³ × 2⁴ into a power of 4 by multiplying the bases, or 'cancelling' the base to 1 when dividing powers, instead of keeping the base and adding or subtracting the indices (2³ × 2⁴ = 2⁷).