GCE O/L Mathematics cheat sheet

Order of operations (BIDMAS): work through Brackets, Indices, Division and Multiplication (left to right), then Addition and Subtraction (left to right).

Highest common factor (HCF) is the product of shared prime factors; lowest common multiple (LCM) is the product of the highest power of every prime appearing in either number.

A number is divisible by 3 if its digit sum is divisible by 3, by 9 if its digit sum is divisible by 9, and by 4 if its last two digits form a multiple of 4.

Standard form: a × 10^n where 1 ≤ a < 10 and n is an integer; multiply the front numbers and add the indices, divide and subtract the indices.

For fractions: a/b + c/d = (ad + bc)/(bd); to divide, multiply by the reciprocal, a/b ÷ c/d = a/b × d/c.

Percentages: percentage change = (change ÷ original) × 100; a multiplier of 1.15 increases by 15%, and 0.85 decreases by 15%.

Surds: √(ab) = √a × √b and √(a/b) = √a ÷ √b; rationalise 1/√a by multiplying numerator and denominator by √a.

Rounding and bounds: for a value rounded to the nearest unit d, the upper bound is value + d/2 and the lower bound is value − d/2.

Expand brackets by multiplying every term inside by the factor outside: a(b + c) = ab + ac; for double brackets (x + a)(x + b) = x² + (a + b)x + ab.

Factorising quadratics x² + bx + c: find two numbers multiplying to c and adding to b, giving (x + p)(x + q).

Quadratic formula: for ax² + bx + c = 0, x = [−b ± √(b² − 4ac)] / (2a); the discriminant b² − 4ac gives two, one, or no real roots when positive, zero, or negative.

Straight line y = mx + c has gradient m and y-intercept c; gradient = (y₂ − y₁)/(x₂ − x₁); parallel lines share the same gradient, perpendicular gradients multiply to −1.

Rearrange a formula by performing the same inverse operation on both sides, working to isolate the required variable.

Rules of indices: 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.

Solve simultaneous linear equations by elimination (add or subtract the equations to remove a variable) or by substitution.

An inequality solved like an equation reverses its sign when both sides are multiplied or divided by a negative number.

Angle sum of a polygon with n sides is (n − 2) × 180°; each interior angle of a regular polygon is (n − 2) × 180° ÷ n.

Circle: circumference = 2πr, area = πr²; arc length = (θ/360) × 2πr and sector area = (θ/360) × πr² for angle θ in degrees.

Volume and surface area: cuboid V = lwh; cylinder V = πr²h; cone V = (1/3)πr²h; sphere V = (4/3)πr³.

Pythagoras: in a right-angled triangle a² + b² = c², where c is the hypotenuse opposite the right angle.

Similar shapes: corresponding lengths scale by k, areas by k², and volumes by k³.

Bearings are measured clockwise from north, written with three digits (e.g. 045°), and a back bearing differs by 180°.

Congruent triangles are proved by SSS, SAS, ASA (with AAS) or RHS; AAA proves similarity, not congruence.

Density = mass ÷ volume, pressure = force ÷ area, and speed = distance ÷ time are the standard compound-measure formulas.

Mean = total of all values ÷ number of values; median is the middle value of ordered data; mode is the most frequent value.

For grouped data, estimate the mean using midpoints of class intervals: mean = Σ(f × midpoint) ÷ Σf.

Cumulative frequency curves are plotted at upper class boundaries; the median is read at half the total frequency, quartiles at one-quarter and three-quarters.

Interquartile range = upper quartile − lower quartile and measures the spread of the middle 50% of data.

A histogram with unequal class widths plots frequency density = frequency ÷ class width on the vertical axis.

Correlation describes the strength and direction of a linear relationship on a scatter graph; the line of best fit passes through the mean point.

Pie chart sector angle for a category = (category frequency ÷ total frequency) × 360°.

Probability scale runs from 0 (impossible) to 1 (certain); P(not A) = 1 − P(A).

For mutually exclusive events, P(A or B) = P(A) + P(B); probabilities of all outcomes in a sample space sum to 1.

For independent events, P(A and B) = P(A) × P(B).

Theoretical probability = number of favourable outcomes ÷ total number of equally likely outcomes.

Relative frequency = number of times an event occurred ÷ total number of trials; it estimates probability and tends towards the theoretical value as trials increase.

Tree diagrams multiply along branches and add probabilities of separate branches that give the same combined outcome.

When items are taken without replacement, the denominator and numerator of later probabilities change to reflect the reduced total.

A ratio a : b is simplified by dividing both parts by their highest common factor; to share a quantity in a ratio, divide by the total number of parts.

Direct proportion: y = kx, so y₁/x₁ = y₂/x₂; the graph is a straight line through the origin.

Inverse proportion: y = k/x, so x₁y₁ = x₂y₂; doubling one quantity halves the other.

Scale drawings: map distance ÷ real distance gives the scale factor, e.g. 1 : 50 000 means 1 cm represents 0.5 km.

Compound interest: amount = P × (1 + r/100)^n, where r is the annual rate as a percentage and n the number of years.

Average speed = total distance ÷ total time; for a journey in stages, divide total distance by total time, not by averaging the speeds.

Best-buy comparisons use unit cost = price ÷ quantity, expressed per gram, per litre or per item.

In a right-angled triangle: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent (SOHCAHTOA).

Use the inverse functions sin⁻¹, cos⁻¹ and tan⁻¹ to find an angle from two known side lengths.

Exact values: sin 30° = 0.5, cos 60° = 0.5, tan 45° = 1, sin 45° = cos 45° = √2/2, sin 60° = cos 30° = √3/2.

Sine rule: a/sin A = b/sin B = c/sin C, used when given a matching side-and-angle pair.

Cosine rule: a² = b² + c² − 2bc cos A, used with two sides and the included angle or with all three sides.

Area of any triangle = ½ab sin C, where C is the angle between sides a and b.

Angles are measured in degrees unless radians are stated; ensure the calculator is in degree mode before evaluating.

The angle of elevation is measured upwards from the horizontal to an object; the angle of depression is measured downwards from the horizontal.