GCSE Maths (AQA) formulas

Number

Structure and calculation

• Order of operations: brackets first, then powers and roots, then × and ÷, then + and −.
• Negative numbers: the bigger the size, the smaller the number, e.g. −7 < −2.
• Fractions: change mixed numbers to improper fractions first. To multiply, multiply tops and multiply bottoms. To divide, turn the second fraction upside down and multiply.
• To add or subtract fractions, use a common denominator.
• HCF from prime factors: take the lowest power of each prime that is in both numbers.
• LCM from prime factors: take the highest power of every prime that appears in either number.
• Product rule: m choices then n choices gives m × n outcomes. If nothing can repeat, the number of choices drops by 1 each time.

Powers, roots, indices and standard form

• Multiply: add the indices (a^m × aⁿ = a^{m + n}). Divide: subtract them (a^m ÷ aⁿ = a^{m − n}).
• Power of a power: multiply the indices ((a^m)ⁿ = a^mn). Everything in the bracket is raised: (2x)³ = 8x³.
• a⁰ = 1 and a⁻ⁿ = 1/aⁿ.
• a^1/n is the nth root of a. a^m/n = (nth root of a)^m: take the root first, then the power.
• Standard form: A × 10ⁿ with 1 ≤ A < 10. n is positive for large numbers and negative for numbers less than 1.
• To multiply in standard form, multiply the A values and add the powers of 10. To divide, divide the A values and subtract the powers. Then adjust so that A is between 1 and 10.
• To estimate a root, find the square numbers on each side: √40 is between √36 = 6 and √49 = 7.

Surds and exact calculation

• √a × √b = √(ab) and √a ÷ √b = √(a/b).
• √a × √a = a.
• Simplify by taking out the largest square factor: √48 = √16 × √3 = 4√3.
• Only like surds can be added: 3√2 + 5√2 = 8√2. √a + √b is not √(a + b).
• (a + √b)(a − √b) = a² − b, with no surd left.
• To rationalise k/√a, multiply top and bottom by √a.
• To rationalise k/(a + √b), multiply top and bottom by (a − √b). Change the sign in the middle.

Fractions, decimals and percentages

• Terminating decimal to fraction: use place value, then simplify, e.g. 0.36 = 36/100 = 9/25.
• Fraction to decimal: divide the top by the bottom, e.g. 5/8 = 5 ÷ 8 = 0.625.
• Recurring decimal: let x = the decimal. Multiply by 10 if one digit repeats, by 100 if two digits repeat, then subtract to remove the repeating part.
• Quick check: one repeating digit is that digit over 9 (0.777… = 7/9). Two repeating digits are those digits over 99 (0.4545… = 45/99 = 5/11).
• Ratio a : b means the two parts are a/(a + b) and b/(a + b) of the whole.
• "Of" means multiply: 2/3 of 60 = 2/3 × 60 = 40, and 35% of 60 = 0.35 × 60 = 21.
• If you know a percentage of a number, divide to find 1% (or 10%) first, then multiply up.

Measures, estimation and bounds

• 1 m = 100 cm, so 1 m² = 100² = 10 000 cm² and 1 m³ = 100³ = 1 000 000 cm³.
• Time is not decimal: 1 hour = 60 minutes, so 0.1 hour = 6 minutes and 0.25 hour = 15 minutes.
• Significant figures are counted from the first digit that is not zero: 0.004 72 to 2 significant figures is 0.0047.
• Rounding: the bounds are half a unit each side. 7.4 to 1 decimal place gives 7.35 ≤ x < 7.45.
• Truncating: 6.8 truncated to 1 decimal place gives 6.8 ≤ x < 6.9.
• Largest value of a + b or a × b: use both upper bounds. Smallest: use both lower bounds.
• Largest value of a − b or a ÷ b: upper bound of a with lower bound of b. Smallest: lower bound of a with upper bound of b.

Algebra

Algebraic expressions

• Like terms have exactly the same letters and powers: 3x² and 5x² are like terms, 3x² and 3x are not.
• When substituting a negative number, put it in brackets: if a = −3 then a² = (−3)² = 9.
• Two brackets: (a + b)(c + d) = ac + ad + bc + bd. For three brackets, expand two first, then multiply by the third.
• x² + bx + c: find two numbers that multiply to c and add to b.
• ax² + bx + c: find two numbers that multiply to a × c and add to b, split the middle term, then factorise in pairs.
• Difference of two squares: a² − b² = (a + b)(a − b).
• A minus sign in front of a bracket changes every sign inside: −2(x − 4) = −2x + 8.

Formulae, identities, proof and functions

• Changing the subject: undo each operation with its inverse (+ and −, × and ÷, square and square root), the same on both sides.
• If the new subject appears twice: remove fractions, collect its terms on one side, factorise, then divide.
• Identity: expand one side and compare the x² terms, the x terms and the constants.
• For any integer n: 2n is even, 2n + 1 is odd, n and n + 1 are consecutive. A multiple of k is k × (an integer).
• Inverse function f⁻¹(x): write y = f(x), rearrange to make x the subject, then swap the letter y for x.
• Composite function: fg(x) = f(g(x)). Do g first, then f. fg(x) and gf(x) are usually different.

Coordinates and straight-line graphs

• Gradient = change in y ÷ change in x (m = (y₂ − y₁)/(x₂ − x₁)).
• Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).
• Length of a line segment = √((x₂ − x₁)² + (y₂ − y₁)²).
• Parallel lines: same m. Perpendicular lines: m₁ × m₂ = −1, so the new gradient is −1/m (turn it upside down and change the sign).
• Line with gradient m through (x₁, y₁): y − y₁ = m(x − x₁), or put the point into y = mx + c to find c.
• A line crosses the y-axis where x = 0 and the x-axis where y = 0.
• In a real context, m is the rate (for example cost per km) and c is the starting value (for example a fixed charge).

Quadratic and other graphs

• Roots: solve y = 0. y-intercept: put x = 0.
• Completing the square: x² + bx + c = (x + b/2)² − (b/2)² + c.
• y = (x + p)² + q has its turning point at (−p, q). Its x-coordinate is also halfway between the two roots.
• Reciprocal y = k/x: two separate branches that never touch the axes. Exponential y = a^x: passes through (0, 1) and never touches the x-axis.
• sin x and cos x stay between −1 and 1 and repeat every 360°. tan x repeats every 180°. sin x = sin (180° − x) and cos x = cos (360° − x).
• Translations: y = f(x) + a moves the graph up a. y = f(x + a) moves it left a (y = f(x − a) moves it right a).
• Reflections: y = −f(x) reflects in the x-axis (y-values change sign). y = f(−x) reflects in the y-axis (x-values change sign).

Graphs in context, gradients, areas and circles

• Distance-time graph: gradient = speed. A horizontal line means not moving. A steeper line means faster.
• Velocity-time graph: gradient = acceleration (m/s²). A horizontal line means constant velocity.
• Velocity-time graph: area under the graph = distance travelled. Split it into triangles, rectangles and trapezia.
• Area of a triangle = ½ × base × height. Area of a trapezium = ½ × (a + b) × h, where a and b are the parallel sides.
• Gradient at a point on a curve: draw a tangent there, then use change in y ÷ change in x for two points on the tangent.
• Circle, centre (0, 0), radius r: x² + y² = r². A point is on the circle if its coordinates fit the equation.
• Tangent at (a, b): the radius has gradient b/a, so the tangent has gradient −a/b.

Solving equations

• Linear: whatever you do to one side, do to the other side too.
• Fractions: multiply every term by the common denominator first.
• Factorising: if (x − p)(x − q) = 0 then x = p or x = q.
• Quadratic formula for ax² + bx + c = 0: x = (−b ± √(b² − 4ac)) ÷ (2a)
• Completing the square: x² + bx = (x + b/2)² − (b/2)²
• If b² − 4ac is negative there are no real solutions; if it is zero there is one repeated solution.
• Iteration: put x₀ into the formula to get x₁, put x₁ in to get x₂, and so on.

Simultaneous equations and inequalities

• Elimination: same signs subtract, different signs add.
• Linear with quadratic: substitute, solve the quadratic, then find the other letter for each solution.
• Multiplying or dividing an inequality by a negative number reverses the sign.
• Number line: open circle for < or >, filled circle for ≤ or ≥.
• Set notation: write the answer as {x : x > 3}.
• Quadratic with roots p < q and positive x² term: "< 0" gives p < x < q; "> 0" gives x < p or x > q.
• Graphs: dashed line for < or >, solid line for ≤ or ≥; test a point to find the correct side.

Sequences

• Linear sequence with common difference d and first term a: nth term = dn + (a − d).
• Quadratic sequence an² + bn + c: a = second difference ÷ 2.
• To find bn + c, subtract an² from each term and find the nth term of what is left.
• Geometric sequence: common ratio = any term ÷ the term before it.
• Square numbers: 1, 4, 9, 16, 25, … (n²). Cube numbers: 1, 8, 27, 64, 125, … (n³).
• Triangular numbers: 1, 3, 6, 10, 15, … with nth term n(n + 1) ÷ 2.
• Fibonacci-type: each term = sum of the two terms before it.

Ratio, proportion and rates of change

Units, scale and ratio

• 1 km = 1000 m, 1 m = 100 cm, 1 kg = 1000 g, 1 litre = 1000 ml = 1000 cm³.
• 1 m² = 100² = 10 000 cm²; 1 m³ = 100³ = 1 000 000 cm³.
• m/s to km/h: multiply by 3.6. km/h to m/s: divide by 3.6.
• 1 g/cm³ = 1000 kg/m³.
• Scale 1 : n: real length = map length × n; real area = model area × n².
• Sharing in the ratio a : b: one part = total ÷ (a + b); the first share is a/(a + b) of the total.
• x : y = a : b means x/y = a/b, so bx = ay.

Percentages

• A as a percentage of B = (A ÷ B) × 100
• Percentage change = (change ÷ original value) × 100
• Multiplier for an increase of r% = 1 + r/100; for a decrease of r% = 1 − r/100
• New value = original value × multiplier
• Original value = new value ÷ multiplier
• Simple interest = amount invested × rate × number of years ÷ 100
• Two changes one after the other: multiply the two multipliers.

Direct and inverse proportion

• y is directly proportional to x: y = kx, so y ÷ x is constant.
• y is inversely proportional to x: y = k/x, so x × y is constant.
• "Inversely proportional to x" means the same as "directly proportional to 1/x".
• Other forms: y = kx², y = k√x, y = k/x².
• If y is proportional to xⁿ and x is multiplied by c, then y is multiplied by cⁿ.
• Unitary method (direct): find the value for one item first, then multiply.

Compound measures and rates of change

• speed = distance ÷ time
• density = mass ÷ volume
• pressure = force ÷ area (N/m²)
• Gradient = change in y ÷ change in x = (y₂ − y₁)/(x₂ − x₁)
• Average rate of change between two points = gradient of the chord joining them.
• Instantaneous rate of change at a point = gradient of the tangent at that point.
• A negative gradient means the quantity is decreasing.

Growth and decay

• Growth: value after n periods = starting value × (1 + r/100)ⁿ
• Decay: value after n periods = starting value × (1 − r/100)ⁿ
• Doubling: value = starting value × 2ⁿ, where n = number of doubling periods.
• To find the rate: multiplier = nth root of (final value ÷ starting value).
• Iteration V_n+1 = aV_n + b: put V₀ in to get V₁, then V₁ in to get V₂.
• Compound interest gives more than simple interest after the first year at the same rate.

Geometry and measures

Angles, polygons and bearings

• Angles at a point add to 360°; angles on a straight line add to 180°; vertically opposite angles are equal.
• Parallel lines: alternate angles are equal, corresponding angles are equal, co-interior angles add to 180°.
• Angles in a triangle add to 180°; in a quadrilateral, 360°.
• Sum of interior angles of a polygon with n sides = (n − 2) × 180°
• Exterior angles of any polygon add to 360°; in a regular polygon each one = 360° ÷ n.
• Interior angle + exterior angle = 180°
• Bearing: from north, clockwise, three figures (070°). Bearing back = bearing ± 180°.

Transformations

• Reflection: x-axis (x, y) → (x, −y); y-axis (x, y) → (−x, y); line y = x (x, y) → (y, x); line y = −x (x, y) → (−y, −x).
• Rotation about the origin: 90° clockwise (x, y) → (y, −x); 90° anticlockwise (x, y) → (−y, x); 180° (x, y) → (−x, −y).
• Translation by the vector with top entry a and bottom entry b: (x, y) → (x + a, y + b).
• Enlargement, scale factor k: image = centre + k × (point − centre).
• To describe fully: reflection needs the mirror line; rotation needs angle, direction and centre; translation needs the vector; enlargement needs scale factor and centre.
• Points on the mirror line are invariant in a reflection; the centre is invariant in a rotation or an enlargement.

Circle theorems

• The angle at the centre is twice the angle at the circumference standing on the same arc.
• The angle in a semicircle (standing on a diameter) is 90°.
• Angles in the same segment (standing on the same arc, on the same side of the chord) are equal.
• Opposite angles of a cyclic quadrilateral add up to 180°.
• A tangent meets the radius at 90°, and the two tangents from one outside point are equal in length.
• Alternate segment theorem: the angle between a tangent and a chord equals the angle in the opposite (alternate) segment.
• The perpendicular from the centre to a chord cuts the chord in half.

Area, volume and surface area

• Area of a triangle = ½ × base × perpendicular height; area of a parallelogram = base × perpendicular height.
• Area of a trapezium = ½ × (a + b) × h, where a and b are the parallel sides and h is the distance between them.
• Circle: circumference = π × diameter (C = πd = 2πr); area = π × radius² (A = πr²).
• Volume of a prism = area of cross-section × length; volume of a cylinder = πr²h.
• Volume of a pyramid or cone = (1/3) × base area × vertical height; cone: V = (1/3)πr²h, curved surface area = πrl (l = slant height).
• Sphere: volume = (4/3)πr³; surface area = 4πr².
• Arc length = (θ/360) × 2πr; sector area = (θ/360) × πr², with θ in degrees.

Congruence and similarity

• Congruent triangles: SSS (three sides), SAS (two sides and the angle between them), ASA (two angles and a matching side), RHS (right angle, hypotenuse and one other side).
• AAA does not prove congruence: equal angles only show the triangles are similar.
• Scale factor = length on the new shape ÷ matching length on the old shape.
• Length scale factor k gives area scale factor k² and volume scale factor k³.
• Lengths in ratio a : b give areas in ratio a² : b² and volumes in ratio a³ : b³.
• To go from an area ratio to a length ratio take the square root; from a volume ratio take the cube root.
• A line parallel to one side of a triangle cuts off a smaller triangle that is similar to the whole triangle.

Pythagoras and trigonometry

• Pythagoras: a² + b² = c², where c is the hypotenuse. For a shorter side, subtract: a² = c² − b².
• sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent (SOH CAH TOA).
• To find an angle use the inverse function, e.g. θ = tan⁻¹(opposite ÷ adjacent).
• Diagonal of a cuboid: d² = length² + width² + height².
• sin 0° = 0, sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2, sin 90° = 1.
• cos 0° = 1, cos 30° = √3/2, cos 45° = √2/2, cos 60° = 1/2, cos 90° = 0.
• tan 0° = 0, tan 30° = 1/√3 (the same as √3/3), tan 45° = 1, tan 60° = √3.

Sine rule, cosine rule and area of a triangle

• Sine rule: a ÷ sin A = b ÷ sin B = c ÷ sin C.
• To find an angle, turn the sine rule over: sin A ÷ a = sin B ÷ b.
• Cosine rule for a side: a² = b² + c² − 2bc cos A.
• Cosine rule for an angle: cos A = (b² + c² − a²) ÷ (2bc).
• Area of a triangle = ½ × a × b × sin C, where C is the angle between sides a and b.
• A negative value of cos A means the angle is obtuse (between 90° and 180°).

Vectors

• Translation vector = (new x − old x, new y − old y), top entry first.
• Add or subtract column vectors entry by entry; multiplying by a number (a scalar) multiplies each entry.
• Vector AB = b − a, where a and b are the vectors from O to A and from O to B.
• Vector BA = −(vector AB).
• The midpoint M of AB has vector OM = ½(a + b).
• Two vectors are parallel when one is a multiple of the other, e.g. 6a − 3b = 3(2a − b).
• If vector PQ and vector QR are multiples of each other, then P, Q and R lie on one straight line.

Probability

Probability and expected outcomes

• Probability = number of successful outcomes ÷ total number of equally likely outcomes.
• Relative frequency = number of times the event happens ÷ number of trials.
• Expected number of times = probability × number of trials.
• The probabilities of all the possible (mutually exclusive) outcomes add up to 1.
• P(event does not happen) = 1 − P(event happens).
• More trials give a more reliable estimate of probability.
• In a frequency tree, the numbers on the branches from a point add up to the number at that point.

Combined events and conditional probability

• Independent events: P(A and B) = P(A) × P(B).
• Mutually exclusive events: P(A or B) = P(A) + P(B).
• Tree diagram: multiply along the branches, then add the results of the paths you want.
• Without replacement: the second probability has a denominator one smaller.
• P(at least one) = 1 − P(none).
• P(A given B) = P(A and B) ÷ P(B); from a table or Venn diagram, number in both A and B ÷ number in B.
• Two dice give 6 × 6 = 36 equally likely outcomes.

Statistics

Sampling and statistical diagrams

• Estimate for the population = (number in sample with the feature ÷ sample size) × population size.
• Pie chart angle = (frequency ÷ total frequency) × 360°.
• Frequency density = frequency ÷ class width.
• Frequency = frequency density × class width (the area of the bar).
• Plot cumulative frequency against the upper end of each class.
• For n values on a cumulative frequency graph: median at n/2, lower quartile at n/4, upper quartile at 3n/4.
• Number above a value = total − cumulative frequency at that value.

Averages, spread and scatter graphs

• Mean = sum of the values ÷ number of values, so sum of the values = mean × number of values.
• Median: put the values in order; it is the value in position (n + 1) ÷ 2. For two middle values, take the number halfway between them.
• Estimated mean from a grouped table = sum of (midpoint × frequency) ÷ total frequency. The modal class has the highest frequency.
• Range = largest − smallest; interquartile range (IQR) = upper quartile − lower quartile.
• Box plot: minimum, lower quartile, median, upper quartile, maximum, with 25% of the data in each of the four parts.
• An outlier is a value that lies far away from the rest of the data. It can distort the mean and the range.
• Positive correlation: both increase together; negative: one goes up as the other goes down. Correlation does not prove that one causes the other, and predictions outside the data range (extrapolation) are unreliable.