LGCSE Mathematics (0178) cheat sheet

Order of operations: work through brackets, then indices, then multiplication and division left to right, then addition and subtraction left to right.

Standard form: a × 10^n where 1 ≤ a < 10; to multiply add the indices, to divide subtract them.

HCF is the product of shared prime factors; LCM is the product of the highest power of every prime appearing in either number.

Percentage change = (new − old)/old × 100; a 20% increase followed by a 20% decrease leaves 0.96 of the original value.

Simple interest = P × r × t/100; compound amount = P(1 + r/100)^t, so compound exceeds simple after the first period.

Recurring decimal to fraction: for 0.4̄ use 4/9; for 0.12̄ (one digit recurring) subtract the non-recurring part and divide by 90.

Rounding: if the next digit is 5 or more round up; for upper and lower bounds of a rounded value, half the unit above and below.

Ratio a : b splits a quantity Q into parts Qa/(a+b) and Qb/(a+b); direct proportion gives y = kx, inverse gives y = k/x.

Expand (x + a)(x + b) = x² + (a + b)x + ab; factorise by finding two numbers with that sum and product.

Quadratic formula: x = [−b ± √(b² − 4ac)]/2a; the discriminant b² − 4ac gives two, one or no real roots.

Straight line y = mx + c has gradient m and y-intercept c; parallel lines share m, perpendicular gradients multiply to −1.

Gradient between two points = (y₂ − y₁)/(x₂ − x₁); midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2).

Changing the subject: perform the same inverse operation to both sides, reversing the order of operations.

Simultaneous equations: eliminate a variable by adding or subtracting scaled equations, then substitute back.

Graph transformations: y = f(x) + a shifts up by a, y = f(x + a) shifts left by a, y = −f(x) reflects in the x-axis.

Composite function fg(x) means apply g first then f; the inverse f⁻¹ undoes f and its graph reflects in y = x.

Angles on a straight line sum to 180°, angles at a point to 360°, and vertically opposite angles are equal.

Parallel lines: corresponding and alternate angles are equal; co-interior angles sum to 180°.

Interior angle sum of an n-sided polygon = (n − 2) × 180°; exterior angle sum is always 360°.

Circle theorems: angle in a semicircle is 90°; angle at the centre is twice the angle at the circumference.

Tangent–radius: a tangent is perpendicular to the radius at the point of contact; tangents from an external point are equal.

Pythagoras: a² + b² = c² for the hypotenuse c; the converse tests whether a triangle is right-angled.

Right-triangle trigonometry: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent.

Sine rule a/sin A = b/sin B; cosine rule a² = b² + c² − 2bc cos A; area = ½ab sin C.

Rectangle area = length × width; parallelogram area = base × perpendicular height; triangle area = ½ × base × height.

Trapezium area = ½(a + b)h, where a and b are the parallel sides and h the perpendicular distance between them.

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

Cuboid volume = lwh, surface area = 2(lw + lh + wh); prism volume = cross-sectional area × length.

Cylinder volume = πr²h; curved surface area = 2πrh; total surface area = 2πr² + 2πrh.

Cone volume = ⅓πr²h and curved surface = πrl with slant height l = √(r² + h²); sphere volume = 4/3πr³, surface = 4πr².

Density = mass/volume; pressure = force/area; speed = distance/time, with units converted consistently.

Similar solids: lengths scale by k, areas by k², volumes by k³; enlargement volume factor is the cube of the length factor.

Distance between points = √[(x₂ − x₁)² + (y₂ − y₁)²], derived from Pythagoras on the coordinate grid.

Midpoint = ((x₁ + x₂)/2, (y₁ + y₂)/2); it lies on the line joining the two points.

Gradient m = (y₂ − y₁)/(x₂ − x₁); horizontal lines have m = 0 and vertical lines have undefined gradient.

Equation through a point with gradient m: y − y₁ = m(x − x₁); rearrange to y = mx + c if needed.

Parallel lines have equal gradients; perpendicular gradients satisfy m₁ × m₂ = −1.

To find where a line meets an axis, set the other coordinate to zero: y = 0 gives the x-intercept, x = 0 the y-intercept.

Intersection of two lines is found by solving their equations simultaneously; the solution is the shared point.

Line of symmetry of y = ax² + bx + c is x = −b/2a; its turning point lies on this vertical line.

Mean = sum of values ÷ number of values; median is the middle value in order; mode is the most frequent value.

For grouped data use midpoints of class intervals to estimate the mean; the median lies in the class containing the (n+1)/2th value.

Range = largest − smallest; interquartile range = upper quartile − lower quartile and measures the middle half of the data.

Probability of an event = favourable outcomes ÷ total equally likely outcomes; all probabilities lie between 0 and 1.

Complement rule: P(not A) = 1 − P(A); P(A or B) = P(A) + P(B) − P(A and B).

Independent events: P(A and B) = P(A) × P(B); without replacement, adjust the denominator for the second event.

Tree diagrams multiply along branches and add across branches for mutually exclusive outcomes.

Relative frequency = number of successes ÷ number of trials; it approaches theoretical probability as trials increase.