UPCAT Formula Bank

Unit 1Exam technique

Unit conversions

Area units: 1 m² = 10 000 cm²; Square the length factor: 100 × 100. Convert lengths first, then multiply.; Lesson »
Volume units: 1 m³ = 1 000 000 cm³; 1 L = 1000 cm³; Lesson »
Minutes to hours: hours = minutes ÷ 60, so 45 min = 0.75 h; 7 h 45 min is 7.75 h, never 7.45 h.; Lesson »

Unit 2Number sense and shortcuts

Signs and order of operations

Powers of negatives: (−a)ⁿ > 0 for even n, < 0 for odd n; −a² = −(a²); (−2)⁴ = 16 but −2⁴ = −16: the bracket decides whether the minus is raised.; Lesson »
Order of operations (GEMDAS): grouping → exponents → × and ÷ left to right → + and − left to right; × does not outrank ÷, and + does not outrank −; ties go left to right.; Lesson »

Ratio and proportion

Proportion (cross-multiplying): a/b = c/d ⟹ ad = bc; Keep the labels lined up: sugar over flour on both sides.; Lesson »
Sharing in a ratio: one part = total ÷ (sum of the ratio terms); ₱4,500 in 2 : 3 : 4 → 9 parts of ₱500.; Lesson »
Part to whole: in a ratio a : b, the first share is a/(a + b) of the whole; Sugar to flour 2 : 5 means sugar is 2/7 of the mix, not 2/5.; Lesson »
Direct and inverse proportion: direct: y₁/x₁ = y₂/x₂; inverse: x₁y₁ = x₂y₂; More workers, fewer days: the product (worker-days) stays constant.; Lesson »

Decimals and percents

Repeating decimal to fraction: a block of n repeating digits goes over n nines: 0.454545… = 45/99 = 5/11; 0.3̄ = 3/9 = 1/3; 0.123123… = 123/999.; Lesson »
Base, rate, percentage: P = R × B; The base follows of; the percentage follows is.; Lesson »
Percent change: percent change = (new − old) ÷ old × 100%; Always divide by the original value.; Lesson »
Percent increase and decrease: increase by r: × (1 + r); decrease by r: × (1 − r); 20% more than 50 is 50 × 1.20 = 60.; Lesson »

Profit, markup and commission

Profit and loss: profit = SP − C; loss = C − SP; Lesson »
Markup rate: markup rate = profit ÷ cost; Markup, gain and profit rate are on cost unless the selling price is named.; Lesson »
Margin: margin = profit ÷ selling price; For a profit, the margin is always smaller than the markup.; Lesson »
Selling price from cost: SP = C(1 + markup rate); at a loss, SP = C(1 − loss rate); Going backwards, divide: C = SP ÷ (1 + markup rate).; Lesson »

Primes and divisibility

Divisibility by 3 and 9: divisible by 3 (or 9) ⟺ digit sum divisible by 3 (or 9); Lesson »
Divisibility by 4 and 8: by 4: last two digits divisible by 4; by 8: last three digits divisible by 8; Lesson »
Divisibility by 6 and 11: by 6: divisible by 2 and by 3; by 11: alternating digit sum divisible by 11; There is no simple test for 7; just divide.; Lesson »
Primality test: n is prime if no prime ≤ √n divides it; 1 is not prime; 2 is the only even prime; 91 = 7 × 13.; Lesson »

GCF, LCM and divisors

GCF and LCM from primes: GCF: lowest power of each shared prime; LCM: highest power of every prime; Lesson »
GCF × LCM: GCF(a, b) × LCM(a, b) = a × b; Holds for two numbers only.; Lesson »
Number of divisors: n = p^a q^b r^c has (a + 1)(b + 1)(c + 1) divisors; A lone prime still counts: its exponent 1 gives a factor of 2.; Lesson »
Sum of divisors: (1 + p + … + p^a)(1 + q + … + q^b)…; 360 = 2³ · 3² · 5 → 15 × 13 × 6 = 1170.; Lesson »
Odd number of divisors: odd divisor count ⟺ n is a perfect square; Lesson »
LCM or GCF in word problems: events repeating together → LCM; largest equal pieces or groups → GCF; LCM ≥ both numbers; GCF ≤ both numbers.; Lesson »

Kinds of numbers and notation

Combining rationals and irrationals: rational + irrational = irrational; (non-zero rational) × irrational = irrational; Two irrationals can give a rational: √2 · √2 = 2.; Lesson »
Scientific notation: a × 10ⁿ, with 1 ≤ a < 10; Positive n for numbers above 1, negative n for numbers below 1.; Lesson »

Integer and digit problems

Consecutive integers: n, n + 1, n + 2; consecutive even or odd: n, n + 2, n + 4; Odd integers still step by 2.; Lesson »
Middle-term shortcut: middle term = sum ÷ number of terms (odd count of terms); Three integers summing to 54: the middle is 18.; Lesson »
Two-digit number and its reverse: 10t + u; reversed 10u + t; (10u + t) − (10t + u) = 9(u − t); A reversal gap that is not a multiple of 9 is impossible.; Lesson »

Clocks and cycles

Angle between clock hands: angle = |30H − 5.5M|; If it exceeds 180°, take 360° minus it.; Lesson »
Units-digit cycles: 2: 2,4,8,6; 3: 3,9,7,1; 7: 7,9,3,1; 8: 8,4,2,6; 4: 4,6; 9: 9,1; 0, 1, 5, 6 fixed; Divide the exponent by the cycle length; remainder 0 means the last entry.; Lesson »
Remainder shortcut: if b = m − 1, then bⁿ ÷ m leaves remainder 1 (n even) or m − 1 (n odd); b behaves like −1 when dividing by m: 2¹⁰ ÷ 3 leaves 1.; Lesson »

Complex numbers

Powers of i: i² = −1, i³ = −i, i⁴ = 1; iⁿ depends on the remainder of n ÷ 4; Remainder 0 → 1. Only the last two digits of n matter.; Lesson »
Conjugate product: (a + bi)(a − bi) = a² + b²; Lesson »

Unit 3Algebra

Exponents

Product rule: a^m · aⁿ = a^{m + n}; Lesson »
Quotient rule: a^m ÷ aⁿ = a^{m − n}; Subtracting a negative exponent adds: x^{−4 − (−3)} = x⁻¹.; Lesson »
Power of a power: (a^m)ⁿ = a^mn; Lesson »
Power of a product: (ab)ⁿ = aⁿ bⁿ; The coefficient is raised too: (3x²)² = 9x⁴.; Lesson »
Zero and negative exponents: a⁰ = 1 (a ≠ 0); a⁻ⁿ = 1/aⁿ; A negative exponent moves the factor across the fraction bar; it is not a minus.; Lesson »

Radicals

Product of roots: √(ab) = √a · √b (a, b ≥ 0); But √(a + b) ≠ √a + √b: √25 ≠ 3 + 4.; Lesson »
Square root of a square: √(x²) = |x|; √((−3)²) = 3, not −3.; Lesson »
Rationalising the denominator: a/√b = a√b/b; a/(√b − c) = a(√b + c)/(b − c²); With two terms, multiply by the conjugate: flip only the middle sign.; Lesson »

Special products and expansion

Square of a binomial: (a ± b)² = a² ± 2ab + b²; (a + b)² is never a² + b².; Lesson »
Difference of squares: (a + b)(a − b) = a² − b²; Lesson »
Binomial theorem, general term: (r + 1)th term of (a + b)ⁿ = C(n, r) a^{n − r} b^r; The 4th term has r = 3. The coefficient includes everything but the variable.; Lesson »
Terms in a binomial expansion: (a + b)ⁿ has n + 1 terms; the exponents in each term add to n; Coefficients come from row n of Pascal's triangle.; Lesson »
Sum of coefficients: sum of the coefficients of (ax + by)ⁿ = (a + b)ⁿ; Set every letter to 1.; Lesson »

Factoring

AC method: ax² + bx + c: find p, q with pq = ac and p + q = b; split bx into px + qx and group; Lesson »
Perfect square trinomial test: a² ± 2ab + b² = (a ± b)²: root both ends, check the middle is 2ab; The last term is always positive; the middle sign goes in the bracket.; Lesson »
Completing a perfect square: x² + bx + (b/2)² = (x + b/2)²; Halve b and square it; works only with a bare x².; Lesson »
Sum and difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²); SOAP: Same, Opposite, Always Positive. The trinomial never factors further.; Lesson »

Polynomial division

Remainder theorem: P(x) ÷ (x − a) leaves remainder P(a); For (x + 3) substitute −3.; Lesson »
Factor theorem: (x − a) is a factor of P(x) ⟺ P(a) = 0; Factor, root and x-intercept say the same thing.; Lesson »
Candidate integer roots: an integer root of a polynomial with integer coefficients divides the constant term; Lesson »
Division algorithm: dividend = divisor × quotient + remainder; Stop when the remainder's degree is below the divisor's.; Lesson »

Rational and radical expressions

Cancelling: cancel common factors only, never terms: (x + 3)(x − 3)/((x + 3)(x + 2)) = (x − 3)/(x + 2); Excluded values come from the original denominators, cancelled ones included.; Lesson »
Extraneous roots (radical): √(…) ≥ 0, so check every root of a squared equation in the original; Squaring can only add roots, never lose them.; Lesson »
Quadratic in disguise: if the middle power is half the top one, let u = that power: x⁴ − 13x² + 36 = 0 → u² − 13u + 36 = 0; Return to x at the end; reject u < 0 when u = x² or u = √x.; Lesson »

Inequalities and absolute value

Flipping the inequality: multiplying or dividing by a negative reverses the inequality sign; Adding or subtracting never flips it.; Lesson »
Interval notation: < or > → open dot, ( ); ≤ or ≥ → closed dot, [ ]; ∞ always takes ( ); −1 < x ≤ 3 is (−1, 3].; Lesson »
Absolute value equation: |u| = a (a ≥ 0) ⟹ u = a or u = −a; |u| = negative has no solution.; Lesson »
Absolute value inequalities: |u| < a ⟺ −a < u < a; |u| > a ⟺ u < −a or u > a; Less than: one stretch. Greater than: two pieces joined by or.; Lesson »

Systems and variation

Special results in a system: a false statement (0 = 5) → no solution; an identity (0 = 0) → infinitely many; Lesson »
Direct variation: y = kx, so k = y/x stays constant; Lesson »
Inverse variation: y = k/x, so k = xy stays constant; Double x and y halves.; Lesson »

Unit 4Quadratics

Solving

Zero product property: AB = 0 ⟹ A = 0 or B = 0; Get zero on one side first. Never divide both sides by x.; Lesson »
Extracting square roots: x² = k (k > 0) ⟹ x = ±√k; (x − h)² = k ⟹ x = h ± √k; k = 0 gives one root; k < 0 gives no real root.; Lesson »
Quadratic formula: x = (−b ± √(b² − 4ac))/(2a); The whole numerator sits over 2a.; Lesson »
Completing the square: x² + bx = c ⟹ (x + b/2)² = c + (b/2)²; Add (b/2)² to both sides. Divide by a first if a ≠ 1.; Lesson »

Roots

Discriminant: D = b² − 4ac: D > 0 two real roots; D = 0 one repeated root; D < 0 no real roots; Integer a, b, c with a perfect-square D: rational roots.; Lesson »
Sum and product of the roots: r₁ + r₂ = −b/a; r₁r₂ = c/a; Only the sum carries a minus. A negative product means opposite signs.; Lesson »
Equation from its roots: x² − (sum)x + (product) = 0; Lesson »
Conjugate roots: irrational roots p ± √q and complex roots p ± qi come in conjugate pairs; Roots 2 ± √3: sum 4, product 1, so x² − 4x + 1 = 0.; Lesson »

The parabola

Vertex: x = −b/(2a), then y = f(−b/(2a)); −b/(2a) gives when; substitute to get how high.; Lesson »
Vertex form: y = a(x − h)² + k, vertex (h, k); (x − 2)² means h = +2.; Lesson »
Opening and extremum: a > 0 opens up, vertex is a minimum; a < 0 opens down, vertex is a maximum; |a| > 1 narrower, |a| < 1 wider.; Lesson »
Axis and y-intercept: axis x = (r₁ + r₂)/2, midway between the roots; y-intercept = c; Lesson »
Largest rectangle for a fixed perimeter: perimeter P gives maximum area (P/4)², a square; Lesson »
Quadratic inequality: for r₁ < r₂: (x − r₁)(x − r₂) < 0 ⟹ r₁ < x < r₂; > 0 ⟹ x < r₁ or x > r₂; Never square-root an inequality: x² < 4 is −2 < x < 2.; Lesson »

Unit 5Word problems

Motion

Catching up: distances equal: r₁(t + head start) = r₂t; The vehicle that left first has the bigger time.; Lesson »
Opposite directions: gap = (r₁ + r₂)t; Lesson »
Same direction: gap = (r₁ − r₂)t; Lesson »
Average speed: average speed = total distance ÷ total time; Lesson »
Round-trip average speed: equal distances at a and b: 2ab/(a + b); equal times: (a + b)/2; The round-trip answer always leans toward the slower speed.; Lesson »
Boat and current: downstream = b + c; upstream = b − c; Same for plane and wind.; Lesson »
Still-water and current speeds: b = (down + up)/2; c = (down − up)/2; Lesson »

Work, mixtures, money

Working together: 1/A + 1/B = 1/t, so t = AB/(A + B); Add the rates, never the hours. A drain's rate is subtracted.; Lesson »
Coin value: total value = Σ (number of coins × face value); Letters count coins, not pesos.; Lesson »
Mixture: C₁V₁ + C₂V₂ = C(V₁ + V₂); Volumes add, percentages never do. Pure water is 0%.; Lesson »
Two investments: r₁x + r₂(total − x) = total interest; Write the equation on the interest, with rates as decimals.; Lesson »

Unit 6Coordinate geometry

Points and segments

Distance formula: d = √((x₂ − x₁)² + (y₂ − y₁)²); Lesson »
Midpoint formula: M = ((x₁ + x₂)/2, (y₁ + y₂)/2); Lesson »
Missing endpoint: other end = 2 × midpoint − known end; Do x and y separately.; Lesson »
Shoelace formula (triangle): area = ½|(x₁y₂ + x₂y₃ + x₃y₁) − (y₁x₂ + y₂x₃ + y₃x₁)|; List the vertices in order around the shape. Do not forget the ½.; Lesson »

Lines

Slope: m = (y₂ − y₁)/(x₂ − x₁); Use the same point first on top and bottom.; Lesson »
Point-slope form: y − y₁ = m(x − x₁); Lesson »
Slope-intercept and standard form: y = mx + b; Ax + By = C; Lesson »
Intercept form: x/a + y/b = 1 (x-intercept a, y-intercept b); Lesson »
Parallel and perpendicular: parallel: m₁ = m₂; perpendicular: m₁m₂ = −1; Flip and switch the sign. Horizontal and vertical lines are perpendicular too.; Lesson »
Slope from standard form: Ax + By = C has slope −A/B; 3x + y = 7 has slope −3.; Lesson »

Circles

Circle, centre-radius form: (x − h)² + (y − k)² = r²; (y + 2)² means k = −2; the right side is r², not r.; Lesson »
Circle, general form: x² + y² + Dx + Ey + F = 0: centre (−D/2, −E/2), r = √(D²/4 + E²/4 − F); Or complete the square twice, adding to both sides.; Lesson »

Transformations of points

Translation: (x, y) → (x + a, y + b); Lesson »
Reflections: x-axis: (x, −y); y-axis: (−x, y); line y = x: (y, x); origin: (−x, −y); The axis you reflect over keeps its coordinate.; Lesson »
Rotations about the origin: 90° counterclockwise: (−y, x); 180°: (−x, −y); 270° counterclockwise: (y, −x); 90° clockwise = 270° counterclockwise.; Lesson »
Dilation: (x, y) → (kx, ky); The only one of the four that changes size.; Lesson »

Unit 7Graph analysis

Reading graphs

Sign of the slope: rising → m > 0; falling → m < 0; horizontal → m = 0; vertical → undefined; Steeper than 45° means |m| > 1.; Lesson »
Solutions of f(x) = k: number of solutions = number of points where y = k meets the graph; Zeros of f are the x-intercepts. A touch counts.; Lesson »
Domain and range from a graph: domain = shadow on the x-axis; range = shadow on the y-axis; Filled dot → [ ], open dot → ( ). Scan the whole curve for the extremes.; Lesson »
Vertical line test: a graph is a function ⟺ no vertical line meets it more than once; Lesson »
Horizontal line test: a function is one-to-one (has an inverse function) ⟺ no horizontal line meets it more than once; Failing it does not stop a graph being a function.; Lesson »
Number of solutions of a linear system: different slopes → 1; same slope, different intercept → 0; same line → infinitely many; Lesson »

Transforming graphs

Shifts: y = f(x − h) + k moves the graph right h and up k; Inside the bracket the sign is reversed; outside it is as written.; Lesson »
Reflection in the x-axis: y = −f(x): (x, y) → (x, −y); Lesson »
Reflection in the y-axis: y = f(−x): (x, y) → (−x, y); No change for an even function such as x² or |x|.; Lesson »
Vertical stretch: y = a·f(x): every y multiplied by a; Lesson »
Horizontal stretch: y = f(bx): every x divided by b; f(4x) squeezes toward the y-axis: (4, 2) → (1, 2).; Lesson »
Combined transformation: y = a·f(b(x − h)) + k: (x, y) → (x/b + h, ay + k); Stretch before shift.; Lesson »

Unit 8Geometry and trigonometry

Angles and triangles

Angle pairs: complementary: sum 90°; supplementary: sum 180°; vertical angles equal; round a point: 360°; A linear pair is supplementary. C before S, 90 before 180.; Lesson »
Angle sum of a triangle: ∠A + ∠B + ∠C = 180°; Lesson »
Exterior angle: exterior angle = sum of the two remote interior angles; Lesson »
Triangle inequality: |a − b| < c < a + b; Equal to the sum is a flat line, not a triangle.; Lesson »
Acute, right or obtuse by sides: c longest: c² = a² + b² right; c² < a² + b² acute; c² > a² + b² obtuse; Always square the longest side.; Lesson »
Congruence tests: SSS, SAS, ASA, AAS, HL prove congruence; SSA and AAA do not; In SAS the angle must be the included one.; Lesson »
Parallel lines and a transversal: corresponding, alternate interior and alternate exterior angles are equal; co-interior angles sum to 180°; Only when the lines are parallel.; Lesson »

Polygons and quadrilaterals

Interior angle sum: (n − 2) × 180°; Lesson »
Regular polygon angles: each interior = (n − 2) × 180°/n; each exterior = 360°/n; Exterior angles of any polygon sum to 360°.; Lesson »
Number of diagonals: n(n − 3)/2; Lesson »
Parallelogram properties: opposite sides and angles equal; consecutive angles sum to 180°; diagonals bisect each other; Lesson »
Diagonals of special parallelograms: rectangle: equal; rhombus: perpendicular; square: equal and perpendicular; Every square is a rhombus; not every rhombus is a square.; Lesson »
Median of a trapezoid: m = (b₁ + b₂)/2; Lesson »
Area of a trapezoid: A = ½(b₁ + b₂)h; h is the perpendicular gap, never a slanted leg.; Lesson »
Area of a kite or rhombus: A = ½d₁d₂; Works whenever the diagonals are perpendicular.; Lesson »

Similarity and right triangles

Similar triangles: AA, SSS (sides in ratio) or SAS (two sides in ratio, included angle equal); A line parallel to one side cuts off a similar triangle.; Lesson »
Lengths and areas of similar figures: scale factor k: lengths and perimeters × k, areas × k²; Lesson »
Pythagorean theorem: a² + b² = c² (c the hypotenuse); Missing leg: a = √(c² − b²).; Lesson »
Pythagorean triples: (3, 4, 5), (5, 12, 13), (8, 15, 17), (7, 24, 25) and their multiples; Lesson »
45°–45°–90° triangle: sides 1 : 1 : √2; hypotenuse = leg × √2; A square's diagonal is s√2.; Lesson »
30°–60°–90° triangle: sides 1 : √3 : 2 (opposite 30°, 60°, 90°); Find the short side first: half the hypotenuse.; Lesson »
Equilateral triangle height: h = (s√3)/2; Lesson »
Regular hexagon area: A = (3√3/2)s²; Six equilateral triangles.; Lesson »

Areas

Rectangle and parallelogram: A = bh; Perpendicular height only.; Lesson »
Triangle: A = ½bh; Lesson »
Heron's formula: A = √(s(s − a)(s − b)(s − c)), s = (a + b + c)/2; 13, 14, 15 gives 84.; Lesson »
Triangle from two sides and the included angle: A = ½ab sin C; Lesson »

Circles

Circumference and area: C = 2πr = πd; A = πr²; Check whether you were given the radius or the diameter.; Lesson »
Arc length and sector area (degrees): arc = (θ/360°) × 2πr; sector = (θ/360°) × πr²; Perimeter of a sector = arc + 2r.; Lesson »
Central and inscribed angles: central angle = its arc; inscribed angle = ½ its arc; an angle in a semicircle = 90°; Inscribed angles on the same arc are equal.; Lesson »
Cyclic quadrilateral: opposite angles sum to 180°; Lesson »
Tangent facts: tangent ⊥ radius at the point of contact; two tangents from one point are equal; Lesson »
Intersecting chords: a · b = c · d (pieces of one chord × pieces of the other); Lesson »
Secants and tangents from an outside point: whole₁ × outside₁ = whole₂ × outside₂; tangent: t² = whole × outside; Whole means the entire secant, not just the inside part.; Lesson »

Solids

Prism and cylinder volume: V = Bh; cylinder V = πr²h; box V = lwh; cube V = s³; Lesson »
Pyramid and cone volume: V = ⅓Bh; cone V = ⅓πr²h; Do not drop the ⅓.; Lesson »
Sphere: V = (4/3)πr³; SA = 4πr²; Lesson »
Box and cube surface area: box: 2(lw + lh + wh); cube: 6s²; Lesson »
Cylinder surface area: SA = 2πr² + 2πrh; Lesson »
Cone surface area and slant height: SA = πr² + πrl, with l² = r² + h²; Volume uses h; the curved surface uses l.; Lesson »
Volumes of similar solids: scale factor k: volumes × k³; Radius tripled → volume × 27.; Lesson »
Ratios between similar figures: k = √(area ratio) = ∛(volume ratio); Go through the side ratio: areas 9 : 16 → k = 3 : 4 → volumes 27 : 64.; Lesson »

Angle measure

Degrees and radians: π rad = 180°; degrees → radians × π/180; radians → degrees × 180/π; π/6 = 30°, π/4 = 45°, π/3 = 60°, π/2 = 90°.; Lesson »
Arc length and sector area (radians): s = rθ; A = ½r²θ; θ must be in radians.; Lesson »
Reference angle: QI: θ; QII: 180° − θ; QIII: θ − 180°; QIV: 360° − θ; Always measured to the x-axis, always acute.; Lesson »
Signs by quadrant (ASTC): QI all +; QII sin +; QIII tan +; QIV cos +; All Students Take Chemistry. csc, sec, cot share the signs of sin, cos, tan.; Lesson »

Trigonometry

SOH-CAH-TOA: sin θ = opp/hyp; cos θ = adj/hyp; tan θ = opp/adj; Pick the ratio holding the side you know and the side you want.; Lesson »
Exact trig values: 30°: sin ½, cos √3/2, tan √3/3; 45°: sin √2/2, cos √2/2, tan 1; 60°: sin √3/2, cos ½, tan √3; Sines of 0°, 30°, 45°, 60°, 90° are √0/2, √1/2, √2/2, √3/2, √4/2.; Lesson »
Reciprocal identities: csc θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ; csc goes with sin, sec with cos.; Lesson »
Quotient identities: tan θ = sin θ/cos θ; cot θ = cos θ/sin θ; Lesson »
Pythagorean identity: sin²θ + cos²θ = 1; 1 − cos²θ = sin²θ, not (1 − cos θ)².; Lesson »
Pythagorean identities with tan and cot: 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ; Divide sin²θ + cos²θ = 1 by cos²θ or by sin²θ.; Lesson »
Law of Sines: a/sin A = b/sin B = c/sin C; Needs a side with its opposite angle. Solving for an angle may give two (θ and 180° − θ).; Lesson »
Law of Cosines: c² = a² + b² − 2ab cos C; Use for SAS or SSS. Take the square root at the end.; Lesson »
Angle from three sides: cos C = (a² + b² − c²)/(2ab); A negative cosine means C is obtuse.; Lesson »

Triangle centres

Centroid on a median: vertex to centroid : centroid to midpoint = 2 : 1; AG = ⅔ of the median.; Lesson »
Centroid from coordinates: G = ((x₁ + x₂ + x₃)/3, (y₁ + y₂ + y₃)/3); Lesson »

Unit 9Sequences and series

Arithmetic

nth term (arithmetic): aₙ = a₁ + (n − 1)d; n terms means n − 1 jumps.; Lesson »
Arithmetic series (ends known): Sₙ = n(a₁ + aₙ)/2; Lesson »
Arithmetic series (d known): Sₙ = (n/2)[2a₁ + (n − 1)d]; Lesson »
Sum of the first n whole numbers: 1 + 2 + … + n = n(n + 1)/2; 1 + 2 + … + 100 = 5050.; Lesson »

Geometric

nth term (geometric): aₙ = a₁ · r^{n − 1}; Find r by dividing consecutive terms.; Lesson »
Finite geometric series: Sₙ = a₁(1 − rⁿ)/(1 − r), r ≠ 1; Lesson »
Infinite geometric series: S = a₁/(1 − r), only when |r| < 1; With r negative, 1 − r is bigger than 1.; Lesson »

Means

Arithmetic and geometric mean: AM = (a + b)/2; GM = √(ab); Lesson »
Inserting k means: k means → k + 2 terms, k + 1 gaps: d = (b − a)/(k + 1); r^{k + 1} = b/a; Divide by the gaps, not the blanks. An even root gives ±r.; Lesson »
AM–GM inequality: AM ≥ GM for positive a, b; equal only when a = b; Lesson »
Harmonic sequences and mean: harmonic ⟺ the reciprocals are arithmetic; HM = 2ab/(a + b); Flip, use the arithmetic rules, flip back. HM is never the plain average.; Lesson »

Patterns, Σ and recursion

Constant second differences: constant second differences → the nth term is quadratic in n; 2, 6, 12, 20, 30 = n(n + 1).; Lesson »
Number of terms in a Σ: terms = stop − start + 1; Both endpoints count.; Lesson »
Σ rules: Σ c = cn; Σ (ak + b) = aΣk + Σb; Lesson »
Fibonacci-type recursion: aₙ = aₙ₋₁ + aₙ₋₂ (needs two seeds); Reaching term n from a₁ takes n − 1 applications of the rule.; Lesson »

Unit 10Counting, probability and statistics

Counting

Fundamental counting principle: and → multiply the options; or → add them; Lesson »
Arrangements of n different objects: n! = n × (n − 1) × … × 1; 0! = 1; 5! = 120.; Lesson »
Arrangements with repeats: n!/(a! b! c! …), one factorial per repeated item; LEVEL: 5!/(2! · 2!) = 30.; Lesson »
Circular arrangements: (n − 1)!; a ring or bracelet that can be flipped: (n − 1)!/2; Numbered seats: n!.; Lesson »
Permutations (order matters): P(n, r) = n!/(n − r)! = n(n − 1)…(n − r + 1); Lesson »
Combinations (order does not matter): C(n, r) = n!/(r!(n − r)!) = P(n, r)/r!; Committee, team, select → C; rank, arrange, officers → P.; Lesson »
Symmetry of combinations: C(n, r) = C(n, n − r); Handshakes among n people: C(n, 2).; Lesson »
Union of two sets: n(A ∪ B) = n(A) + n(B) − n(A ∩ B); Fill a Venn diagram from the overlap outwards.; Lesson »

Probability

Classical probability: P(E) = favourable outcomes ÷ total outcomes; Outcomes must be equally likely. Two dice: 36 outcomes; n coins: 2ⁿ.; Lesson »
Addition rule: P(A or B) = P(A) + P(B) − P(A and B); Mutually exclusive events: P(A and B) = 0.; Lesson »
Multiplication rule: P(A and B) = P(A) · P(B | A); independent: P(A) · P(B); Without replacement, adjust both the top and the bottom.; Lesson »
Conditional probability: P(A | B) = P(A and B)/P(B); What follows given goes underneath.; Lesson »
Complement and at least one: P(not A) = 1 − P(A); P(at least one) = 1 − P(none); 4 coin tosses: P(at least one head) = 1 − (½)⁴ = 15/16.; Lesson »
Odds and probability: odds in favour a : b ⟹ P = a/(a + b); odds against = b : a; Odds 3 : 7 means P = 3/10, not 3/7.; Lesson »
Expected value: E = Σ (value × probability); A fair game has E = 0. Subtract a certain cost without weighting it.; Lesson »
Geometric probability: P = favourable length or area ÷ total length or area; Half the radius → a quarter of the area.; Lesson »
Binomial probability: P(exactly k of n) = C(n, k) p^k (1 − p)^{n − k}; Lesson »

Statistics

Mean from a frequency table: mean = Σ fx ÷ Σ f; Divide by the number of values, not the number of rows.; Lesson »
Median and mode: median: middle value of the sorted list (average the two middle values if n is even); mode: most frequent; The median resists outliers; the mean does not.; Lesson »
Weighted mean: weighted mean = Σ (value × weight) ÷ Σ weight; Lesson »
Combined mean of two groups: (n₁x̄₁ + n₂x̄₂)/(n₁ + n₂); Never average two averages unless the groups are the same size.; Lesson »
Quartiles and IQR: Q1, Q3 = medians of the lower and upper halves; IQR = Q3 − Q1; With an odd count, leave the median out of both halves.; Lesson »
Outlier fences: outlier if below Q1 − 1.5 × IQR or above Q3 + 1.5 × IQR; Lesson »
Variance and standard deviation: σ² = Σ (x − x̄)²/n; σ = √(σ²); Deviations from the mean always sum to 0. σ is never negative.; Lesson »
z-score: z = (x − μ)/σ; Lesson »
Empirical rule: within 1σ: 68%; within 2σ: 95%; within 3σ: 99.7%; Halve by symmetry: above +2σ is 2.5%; mean to +1σ is 34%.; Lesson »

Unit 11Functions, exponentials and logarithms

Functions

Domain restrictions: denominator ≠ 0; expression under an even root ≥ 0; Cube roots accept negatives.; Lesson »
Composition: (f ∘ g)(x) = f(g(x)); g runs first. In general f ∘ g ≠ g ∘ f.; Lesson »
Inverse functions: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x; To find f⁻¹: write y = f(x), swap x and y, solve for y.; Lesson »
Graph of an inverse: y = f⁻¹(x) is the reflection of y = f(x) in the line y = x; The −1 is a label, not a power.; Lesson »

Exponentials

Equal bases: b^m = bⁿ ⟹ m = n (b > 0, b ≠ 1); Rewrite both sides as powers of one base: 9 = 3², 27 = 3³.; Lesson »
Fractional exponents: a^1/n = ⁿ√a; √2 = 2^1/2, 1/8 = 2⁻³.; Lesson »
Exponential growth and decay: A = A₀(1 + r)^t for growth; A = A₀(1 − r)^t for decay; The multiplier is what you keep: 12% loss → 0.88.; Lesson »
Half-life: A = A₀(½)^t/h, h = half-life; The exponent counts half-lives, not hours.; Lesson »
Doubling time: t = log 2 ÷ log(1 + r); Independent of the starting amount.; Lesson »

Logarithms

Definition of a logarithm: logₐ x = y ⟺ a^y = x; The argument x must be positive; a > 0, a ≠ 1.; Lesson »
Product law: logₐ(MN) = logₐ M + logₐ N; Lesson »
Quotient law: logₐ(M/N) = logₐ M − logₐ N; Lesson »
Power law: logₐ(M^k) = k logₐ M; (log M)² is not 2 log M; there is no law for log(M + N).; Lesson »
Special values: logₐ 1 = 0; logₐ a = 1; Lesson »
Change of base: logₐ x = log x ÷ log a; Solves a^y = x when no common base exists.; Lesson »

Rational functions

Vertical asymptote or hole: denominator zero that survives cancelling → vertical asymptote; one that cancels → hole; Factor top and bottom first.; Lesson »
Horizontal asymptote: deg top < deg bottom: y = 0; equal: y = ratio of leading coefficients; top > bottom: none; This is also the limit as x → ±∞.; Lesson »

Unit 12Basic calculus

Limits and continuity

Evaluating a limit: substitute first; if you get 0/0, factor, cancel and substitute again; 0/0 is an instruction, not an answer.; Lesson »
Limit of a reciprocal power: c/xⁿ → 0 as x → ∞ (n > 0); Divide top and bottom by the highest power of x.; Lesson »
Continuity at a point: f(a) exists, lim f(x) as x → a exists, and they are equal; Removable (hole), jump, or infinite discontinuity.; Lesson »

Derivatives

Meaning of f′(a): f′(a) = slope of the tangent at x = a; f′ > 0 rising, f′ < 0 falling, f′ = 0 flat.; Lesson »
Power rule: d/dx (xⁿ) = nx^{n − 1}; Lesson »
Constant, multiple and sum rules: d/dx (c) = 0; d/dx (cf) = cf′; d/dx (f + g) = f′ + g′; Differentiate a polynomial term by term.; Lesson »
Critical points: turning points occur where f′(x) = 0; f′ from + to − → maximum; − to + → minimum.; Lesson »
Product rule: (uv)′ = u′v + uv′; Lesson »
Quotient rule: (u/v)′ = (u′v − uv′)/v²; Low d-high minus high d-low, over low squared.; Lesson »
Chain rule: d/dx [uⁿ] = nu^{n − 1} · u′; Outside derivative, inside kept, times the inside derivative.; Lesson »
Derivative of a root: d/dx √u = u′/(2√u); Lesson »
Tangent line: y − f(a) = f′(a)(x − a); The slope is the number f′(a), not the formula f′(x).; Lesson »

Integrals

Reverse power rule: ∫ xⁿ dx = x^{n + 1}/(n + 1) + C (n ≠ −1); Always add C. Check by differentiating.; Lesson »
Definite integral: ∫ from a to b of f(x) dx = F(b) − F(a); Top limit minus bottom limit; no C needed.; Lesson »
Area below the axis: a region below the x-axis contributes a negative integral; For total area, split at the crossing and flip that part's sign.; Lesson »

Unit 13Financial math

Interest

Simple interest: I = Prt; Rate as a decimal, time in years.; Lesson »
Simple interest amount: A = P + I = P(1 + rt); Interest or amount? Both appear in the choices.; Lesson »
Time in years: t = months/12 = days/360 (banker's year); Use 365 only if the question says so.; Lesson »
Compound interest: A = P(1 + r/n)^nt, n = compoundings per year; Divide the rate and multiply the time together.; Lesson »
Compound interest earned: interest = A − P; Lesson »

Discounts, tax and installments

Successive discounts: final price = P(1 − r₁)(1 − r₂); 20% then 10% off is 28% off, not 30%.; Lesson »
Adding and removing 12% VAT: with VAT = base × 1.12; base = price with VAT ÷ 1.12; Never take 12% off the VAT-inclusive price.; Lesson »
Installment total: total = down payment + monthly payment × months; Lesson »
Finance charge: finance charge = installment total − cash price; rate = charge ÷ cash price; Lesson »

Formula bank

Unit 1Exam technique

Unit conversions

Unit 2Number sense and shortcuts

Signs and order of operations

Ratio and proportion

Decimals and percents

Profit, markup and commission

Primes and divisibility

GCF, LCM and divisors

Kinds of numbers and notation

Integer and digit problems

Clocks and cycles

Complex numbers

Unit 3Algebra

Exponents

Radicals

Special products and expansion

Factoring

Polynomial division

Rational and radical expressions

Inequalities and absolute value

Systems and variation

Unit 4Quadratics

Solving

Roots

The parabola

Unit 5Word problems

Motion

Work, mixtures, money

Unit 6Coordinate geometry

Points and segments

Lines

Circles

Transformations of points

Unit 7Graph analysis

Reading graphs

Transforming graphs

Unit 8Geometry and trigonometry

Angles and triangles

Polygons and quadrilaterals

Similarity and right triangles

Areas

Circles

Solids

Angle measure

Trigonometry

Triangle centres

Unit 9Sequences and series

Arithmetic

Geometric

Means

Patterns, Σ and recursion

Unit 10Counting, probability and statistics

Counting

Probability

Statistics

Unit 11Functions, exponentials and logarithms

Functions

Exponentials

Logarithms

Rational functions

Unit 12Basic calculus

Limits and continuity

Derivatives

Integrals

Unit 13Financial math

Interest

Discounts, tax and installments