G3 Mathematics revision reference

This selected lookup is for 2027 G3 Mathematics (K310). It combines formulas with their conditions and notation. It is a study aid, rather than a list of what an examination supplies.

Number, ratio, percentages and rates

• Standard form: A × 10ⁿ, with 1 ≤ A < 10 and integer n for a positive number. For a negative number, put the minus sign in front.
• Decimal places count places after the decimal point. Significant figures begin with the first nonzero digit. Keep guard digits while working and round the final result as requested.
• In a ratio a:b, the shares of the total are a/(a + b) and b/(a + b) for positive a,b. Convert quantities to matching units first.
• Direct proportion: y = kx; inverse proportion: y = k/x, with x nonzero. Find the constant k from a known pair.
• For a scale factor k in length, the area scale factor is k².
• Percentage change: divide the change by the original value, then multiply by 100%.
• A percentage increase of p% uses multiplier 1 + p/100; a decrease uses 1-p/100. For a reverse percentage, divide the final amount by the appropriate nonzero multiplier.
• Average speed is total distance divided by total time. It is not usually the average of the separate speeds.
• To convert km/h to m/s, divide by 3.6; reverse the conversion by multiplying by 3.6.

Example: after a 20% discount, a price is $48. The original price was 48÷0.8 = 60 dollars. Adding 20% to $48 would use the wrong whole.

For HCF and LCM, write the numbers as products of primes. HCF uses shared prime factors at their smallest powers; LCM uses every needed prime factor at its largest power.

Read more: Number and proportion.

Algebra and equations

For expressions where the operations are defined:

Expand to remove brackets; factorise to write a product. Cancel common factors, not terms in a sum. Keep every exclusion from an original denominator, even after cancellation.

For ax² + bx + c = 0, with a ≠ 0:

Over the real numbers, the discriminant D = b²-4ac gives two distinct roots if D > 0, a repeated root if D = 0, and no roots if D < 0.

For simultaneous equations, substitution replaces one unknown expression; elimination combines equations to remove an unknown. Check all resulting values in both original equations. In an inequality, multiplying or dividing both sides by a negative number reverses its sign.

For a nonvertical straight line, y = mx + c, where m is gradient and c is the y-intercept. A quadratic written y = a(x-h)² + k has vertex (h,k) and axis x = h; its vertex is a minimum for a > 0 and a maximum for a < 0.

Indices

For positive a,b and real exponents where defined:

• a^m aⁿ = a^(m + n); a^m÷ aⁿ = a^(m-n).
• (a^m)ⁿ = a^mn; (ab)ⁿ = aⁿbⁿ.
• a⁰ = 1; a⁻ⁿ = 1/aⁿ.
• a^(1/n) = nth root of a for a positive integer n.

These rules describe powers, not sums: a^m + aⁿ does not become a^(m + n).

When solving a fractional equation, list excluded inputs before clearing denominators. Substitute possible roots into the original equation. Squaring or multiplying by an expression can introduce a value the original equation does not allow.

Read more: Algebra and Graphs.

Sets, matrices, coordinates and vectors

• x∈ A: x belongs to A; A⊆ B: every element of A belongs to B.
• A∪ B: in either set or both; A∩ B: in both; A': outside A within the stated universal set; varnothing: empty set.
• n(A): number of elements in a finite set. Do not confuse an element with a subset.
• Matrices add entry by entry when their orders match. An m × n matrix times an n × p matrix gives an m × p matrix. Each product entry is a row–column sum of products; matrix multiplication is generally not commutative.

For distinct points A(x₁,y₁) and B(x₂,y₂):

Parallel nonzero vectors are scalar multiples. A negative multiplier reverses direction. For similar figures with positive length scale factor k, areas scale by k² and volumes by k³.

Read more: Sets and matrices, Coordinates and vectors and Geometry.

Geometry, right triangles and mensuration

Congruent figures have the same shape and size. Similar figures have equal corresponding angles and proportional corresponding lengths.

Angles on a straight line sum to 180°; angles at a point sum to 360°. Vertically opposite angles are equal. For parallel lines cut by a transversal, corresponding angles and alternate angles are equal; interior angles on the same side sum to 180°.

For a right triangle with hypotenuse c:

Relative to an acute angle θ:

Choose the sides relative to the stated angle. Use degree mode when the angle is given in degrees.

Plane figures, with perpendicular height h:

• Triangle: A = (1/2)bh.
• Parallelogram: A = bh.
• Trapezium, parallel sides a,b: A = (1/2)(a + b)h.
• Circle, radius r: C = 2π r, A = π r².
• Sector, angle θ in degrees: arc length = θ/360 2π r; area = θ/360 π r².

Solids, using perpendicular height h:

• Prism: V = cross-sectional area × length.
• Pyramid: V = 1/3 × base area × h.
• Cylinder: V = π r²h; curved surface area = 2π rh.
• Cone: V = (1/3)π r²h; curved surface area = π rl, where l is slant height.
• Sphere: V = (4/3)π r³; surface area = 4π r².
• Cuboid, length l, width w, height h: V = lwh; surface area = 2(lw + lh + wh).
• Cube, side s: V = s³; surface area = 6s².

Add exposed bases to a curved surface area when the question asks for total surface area. A cone’s slant height and perpendicular height do different jobs.

Units: 1 m² = 10 000 cm² and 1 m³ = 1 000 000 cm³. Square or cube the length conversion factor. Use any value of π or final precision stated in the question; otherwise follow your paper’s instructions.

A convex n-gon has interior-angle sum (n-2)180°. Exterior angles, taken once at each vertex in the same direction, sum to 360°.

Circle and triangle relationships

• An angle at the centre is twice an angle at the circumference standing on the same arc.
• Angles in the same segment are equal; the angle in a semicircle is 90°.
• Opposite angles of a cyclic quadrilateral add to 180°.
• A tangent is perpendicular to the radius at the point of contact. Tangents from the same external point have equal lengths.

For a triangle, side a is opposite angle A, and similarly for b,B and c,C:

The cosine-rule angle lies between the two sides in its product. An inverse sine may give two possible triangle angles; check the remaining angles and the data.

Radians and segments

In the last two formulas, θ must be in radians. For a minor segment with 0 < θ ≤ π:

This subtracts the triangle from the sector. Check whether the required region is the minor or major segment before applying it.

Read more: Geometry, Mensuration and Trigonometry.

Statistics and probability

Statistical measures

• Mean: x bar = (∑ fx)/(∑ f) for values x with frequencies f.
• For grouped intervals, replace x with each class midpoint to estimate the mean.
• Median: the middle value after ordering, or the mean of the two middle values for an even number of data.
• Mode: a value with the highest frequency; there may be more than one.
• Range: maximum minus minimum. Interquartile range: Q₃-Q₁.
• Standard deviation: σ = square root of ((∑ fx²)/(∑ f)-x bar²) for the data distribution.

Use the calculator’s population standard deviation for this data-distribution formula. The sample estimate used in A-Level inference has a different denominator.

Read scales and class boundaries carefully before extracting quartiles from a cumulative frequency graph. A mean or spread alone does not describe the whole shape of a distribution.

Probability

For equally likely elementary outcomes, count those in event A as favourable:

For independent events, P(A∩ B) = P(A)P(B). For mutually exclusive events, P(A∩ B) = 0. These conditions mean different things. On a probability tree, multiply along a path and add the probabilities of disjoint successful paths. Without replacement, later branch probabilities usually change.

Read more: Statistics and probability.