GCSE Mathematics – GCSEGuide

Number

Fractions, Decimals & Percentages

You must be able to convert freely between fractions, decimals, and percentages.

To convert a fraction to a decimal: divide numerator by denominator. e.g. 3/4 = 0.75
To convert a decimal to a percentage: multiply by 100. e.g. 0.75 → 75%
Percentage increase/decrease: Multiply original by (1 ± rate). e.g. 20% increase → × 1.2
Reverse percentage: divide by the multiplier. e.g. after 20% increase, divide by 1.2

Percentage Change % change = (change ÷ original) × 100

Exam Tip:

In "find the original value" questions, always divide by the multiplier — never subtract the percentage directly.

Indices & Standard Form

x⁰ = 1 (any non-zero number to the power 0 equals 1)
x⁻ⁿ = 1/xⁿ (negative index means reciprocal)
x^(1/n) = n-th root of x
Multiplying powers: xᵃ × xᵇ = x^(a+b)
Dividing powers: xᵃ ÷ xᵇ = x^(a−b)
Power of a power: (xᵃ)ᵇ = x^(ab)

Standard Form A × 10ⁿ where 1 ≤ A < 10 and n is an integer

Surds

√a × √b = √(ab)
√a ÷ √b = √(a/b)
Rationalise the denominator: multiply top and bottom by the surd. e.g. 1/√3 = √3/3
Conjugate pairs: multiply by (a − b√c) to rationalise e.g. 1/(1+√2) × (1−√2)/(1−√2)

Algebra

Solving Linear Equations & Inequalities

Rearrange to isolate the unknown. When multiplying or dividing an inequality by a negative number, flip the inequality sign.

Changing the subject Perform the same operation to both sides. Work in reverse BIDMAS.

Sequences

Arithmetic sequence: nth term = a + (n−1)d, where a = first term, d = common difference
Geometric sequence: nth term = ar^(n−1), where r = common ratio
To find if a value is in a sequence, set the nth term equal to that value and check if n is a positive integer

Graphs

y = mx + c: m is gradient, c is y-intercept
Gradient = rise ÷ run = (y₂ − y₁)/(x₂ − x₁)
Parallel lines have equal gradients
Perpendicular lines: gradients multiply to −1 (negative reciprocal)

Quadratics

Factorising: find two numbers that multiply to c and add to b in ax² + bx + c
Completing the square: x² + bx = (x + b/2)² − (b/2)²

Quadratic Formula x = (−b ± √(b² − 4ac)) / 2a

Discriminant: b² − 4ac > 0 → two real roots | = 0 → one repeated root | < 0 → no real roots

Geometry & Measures

Angle Rules

Angles on a straight line: 180°
Angles around a point: 360°
Vertically opposite angles are equal
Alternate angles are equal (Z-angles)
Co-interior angles sum to 180° (C-angles)
Interior angle of regular polygon = (n−2) × 180° ÷ n
Exterior angle of regular polygon = 360° ÷ n

Area & Volume

Key Formulae Circle area = πr² | Circumference = 2πr Trapezium area = ½(a+b)h Cylinder volume = πr²h Sphere volume = (4/3)πr³ | Surface area = 4πr² Cone volume = (1/3)πr²h | Curved SA = πrl

Pythagoras & Trigonometry

Pythagoras' Theorem a² + b² = c² (c is the hypotenuse)

SOH CAH TOA sin θ = O/H | cos θ = A/H | tan θ = O/A

Sine rule: a/sin A = b/sin B = c/sin C
Cosine rule: a² = b² + c² − 2bc cos A
Area of triangle: ½ab sin C

Transformations

Translation: described by a column vector
Reflection: state the mirror line
Rotation: state centre, angle, and direction
Enlargement: state centre and scale factor. SF < 0 → inverted image

Statistics & Probability

Averages & Spread

Mean: sum of values ÷ number of values
Median: middle value when ordered (n+1)/2 th term
Mode: most frequent value
Range: largest − smallest
Interquartile range (IQR): Q3 − Q1

Probability

Probability P(event) = number of favourable outcomes / total outcomes P(A or B) = P(A) + P(B) − P(A and B) P(A and B) = P(A) × P(B) [if independent]

Tree diagrams: multiply along branches, add between paths
Venn diagrams: use for combined event probabilities
Relative frequency (experimental probability) = frequency ÷ total trials

Representing Data

Histograms: frequency density = frequency ÷ class width (not frequency on y-axis)
Cumulative frequency: running total — read off median (50%) and quartiles (25%, 75%)
Box plots: show min, Q1, median, Q3, max
Scatter graphs: describe correlation (positive/negative/none) and strength

Ratio & Proportion

To share in ratio a:b, find total parts (a+b), divide total by parts, multiply
Direct proportion: y = kx → doubling x doubles y
Inverse proportion: y = k/x → doubling x halves y
Speed = distance ÷ time
Density = mass ÷ volume
Pressure = force ÷ area