NBT cheat sheet

Percentage change = (new value − original value) ÷ original value × 100; a 20% increase followed by a 20% decrease leaves 0.96 of the original, a net 4% fall.

To split a quantity in the ratio a : b, divide it by (a + b) to find one part, then multiply by a and by b; R600 in 2 : 3 gives parts of R240 and R360.

Average speed = total distance ÷ total time; it is never the mean of two separate speeds unless the distances and times are identical.

Compound growth: A = P(1 + r)^n. Simple growth: A = P(1 + rn). Use the compound form whenever interest is added to the balance each period.

Conversion chain: 1 km = 1 000 m; 1 m³ = 1 000 L; 1 hectare = 10 000 m²; 1 km² = 100 hectares; 1 hour = 3 600 seconds.

To find a whole from a percentage, divide the part by the decimal rate: if 15% of x is 45, then x = 45 ÷ 0.15 = 300.

On a graph, the gradient of a line is the rate of change, and the area under a rate-versus-time graph gives the accumulated total (e.g. speed–time area = distance).

A valid inference must follow necessarily from the passage; a statement that merely could be true, or that adds outside knowledge, is not a supported inference.

Hedging words (may, suggests, appears, tends to) weaken a claim and are usually defensible; absolute words (always, never, all, proves) strengthen a claim and usually mark the incorrect option.

Quadratic formula: x = [−b ± √(b² − 4ac)] ÷ 2a for ax² + bx + c = 0. The discriminant b² − 4ac gives two real roots if positive, one if zero, none if negative.

Log laws: log(ab) = log a + log b; log(a/b) = log a − log b; log aⁿ = n log a; log_b a = log a ÷ log b.

Arithmetic sequence: Tₙ = a + (n − 1)d and Sₙ = (n/2)[2a + (n − 1)d]. Geometric sequence: Tₙ = arⁿ⁻¹ and Sₙ = a(1 − rⁿ)/(1 − r) for r ≠ 1.

Trigonometry: sin²θ + cos²θ = 1 and tan θ = sin θ ÷ cos θ; sine rule a/sin A = b/sin B = c/sin C; cosine rule a² = b² + c² − 2bc cos A.

Mensuration: circle area πr², circumference 2πr; cylinder volume πr²h; sphere volume (4/3)πr³ with surface area 4πr².

Differential calculus: d/dx (xⁿ) = nxⁿ⁻¹; product rule (uv)′ = u′v + uv′; chain rule dy/dx = dy/du × du/dx; solve f′(x) = 0 for stationary points.

Straight lines: gradient m = (y₂ − y₁)/(x₂ − x₁); y = mx + c; parallel lines have equal gradients; perpendicular lines satisfy m₁ × m₂ = −1.

Financial maths: A = P(1 ± r)ⁿ for growth or decay; future value of an annuity FV = x[(1 + i)ⁿ − 1] ÷ i, with i the rate per period and x the regular payment.

Probability: P(A or B) = P(A) + P(B) − P(A and B); P(not A) = 1 − P(A); independent events satisfy P(A and B) = P(A) × P(B).