CCEA GCSE Maths cheat sheet

Order of operations: evaluate brackets first, then indices, then multiplication and division from left to right, then addition and subtraction from left to right.

The HCF is the product of the lowest common prime powers in the prime factorisations; the LCM is the product of the highest prime powers that appear.

Standard form writes a number as a × 10^n with 1 ≤ a < 10 and n an integer.

Percentage change = (new value − original value) ÷ original value × 100%; a multiplier of 1 + r/100 increases by r%.

Compound interest or depreciation: A = P(1 ± r/100)^n, using + for growth and − for decay.

To convert a recurring decimal to a fraction, subtract equations for x and 10^n x to eliminate the repeating block.

√(ab) = √a√b for a, b ≥ 0; rationalise a/√b by multiplying numerator and denominator by √b.

For a value rounded with half-interval d, lower bound = value − d and upper bound = value + d; for positive division use smallest numerator and largest denominator for the lower bound.

In a ratio a:b, total parts = a + b; share = amount × part ÷ total. Direct proportion has y = kx; inverse proportion has y = k/x.

Quadratic formula: the solutions of ax² + bx + c = 0 are x = (−b ± √(b² − 4ac)) ÷ 2a.

Difference of two squares: a² − b² = (a − b)(a + b).

Index laws: a^m × a^n = a^(m+n); a^m ÷ a^n = a^(m−n); (a^m)^n = a^(mn); a^0 = 1; a^(−n) = 1/a^n.

A straight line y = mx + c has gradient m and y-intercept c; gradient = (y₂ − y₁) ÷ (x₂ − x₁).

Perpendicular lines have gradients m₁ and m₂ with m₁m₂ = −1.

Completing the square: x² + bx + c = (x + b/2)² + c − b²/4.

When solving an inequality, multiplying or dividing both sides by a negative number reverses the inequality sign.

For a linear sequence with first term a and common difference d, nth term = a + (n − 1)d = dn + (a − d).

fg(x) means apply g first, then f; the inverse function reverses the operation and swaps x and y before rearranging.

Circle: circumference = 2πr; area = πr².

Sector arc length = θ/360 × 2πr; sector area = θ/360 × πr².

Pythagoras: a² + b² = c² for a right-angled triangle with hypotenuse c.

Trig ratios: sin θ = opposite ÷ hypotenuse; cos θ = adjacent ÷ hypotenuse; tan θ = opposite ÷ adjacent.

Sine rule: a/sin A = b/sin B = c/sin C. Cosine rule: a² = b² + c² − 2bc cos A.

Volume of a prism = area of cross-section × length; volume of a cone = 1/3πr²h; volume of a sphere = 4/3πr³.

Sum of interior angles of an n-sided polygon = (n − 2) × 180°; sum of exterior angles = 360°.

Similar shapes: length scale factor k gives area scale factor k² and volume scale factor k³.

Bearings are measured clockwise from north and written with three digits; density = mass ÷ volume; speed = distance ÷ time.

Mean = sum of values ÷ number of values.

Estimated mean from grouped data uses class midpoints: Σ(frequency × midpoint) ÷ Σfrequency.

P(not A) = 1 − P(A); for mutually exclusive events P(A or B) = P(A) + P(B).

For independent events P(A and B) = P(A) × P(B); tree diagram branch probabilities multiply along a path and add across paths.

Cumulative frequency graph: median at total frequency ÷ 2; lower quartile at total ÷ 4; upper quartile at 3 × total ÷ 4; IQR = upper − lower quartile.

Frequency density = frequency ÷ class width; in a histogram, frequency = area of the bar.

Stratified sample: number from a group = (group size ÷ population) × sample size.

Correlation describes the direction and strength of a linear relationship; a line of best fit passes through the mean point (x̄, ȳ).

A single counterexample disproves a general statement; a general proof must cover all cases.

Consecutive integers can be written n and n + 1; consecutive even integers 2n and 2n + 2.

The identity a² − b² = (a − b)(a + b) proves many number statements.

To check a calculation, round each value to 1 significant figure and compare the estimate with the exact answer.

In a Venn diagram, P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

For direct proportion use y = kx; for inverse proportion use y = k/x; find k from a known pair.

When using a formula, substitute consistent units before calculating; if the length conversion factor is L, area and volume factors are L² and L³ (cm↔m uses L = 100).