A ratio compares amounts, written with a colon: 2 : 3.

• Equivalent: multiply/divide both sides equally — 2:3 = 4:6
• Simplify: divide by a common factor — 10:15 = 2:3
• As 1 : n: divide both by the first — 4:10 = 1:2.5

Direct proportion

Use the unitary method — find the value of one first. 3 pens cost £1.50 → 1 pen = 50p → 5 pens = £2.50.

Scale factor

Similar shapes are the same shape resized. Scale factor = new ÷ old; multiply every length by it.

Scale diagrams

A scale like 1 cm : 1 m links a drawing to real life.

Coordinates are (x, y) — along first, then up.

In y = mx + c: m = gradient (steepness), c = where it crosses the y-axis.

• on a straight line → 180°
• around a point → 360°
• vertically opposite → equal
• in a triangle → 180°; quadrilateral → 360°

On parallel lines: corresponding angles are equal (F), alternate angles are equal (Z), and co-interior angles add to 180° (C).

Form: "5 more than n" = n + 5; "3 lots of n" = 3n.

Expand: 3(x + 2) = 3x + 6. Factorise: 6x + 9 = 3(2x + 3).

Solve like equations. On a number line: open circle for < >, filled circle for ≤ ≥.

• Mixed ↔ improper: 7/3 = 2⅓
• Add/subtract: common denominator — ½ + ⅓ = 5/6
• Multiply: tops × tops, bottoms × bottoms — 2/3 × 4/5 = 8/15
• Divide: multiply by the reciprocal (keep, flip, multiply) — 2/3 ÷ 4/5 = 2/3 × 5/4 = 5/6
• Reciprocal: flip it; a number × its reciprocal = 1

• Power: 2³ = 2 × 2 × 2 = 8
• Root: √25 = 5
• Multiplying: add indices — x³ × x² = x⁵
• Dividing: subtract indices — x⁵ ÷ x² = x³
• anything to the power 0 = 1

FDP: ½ = 0.5 = 50% · ¼ = 0.25 = 25% · 1/10 = 10%

% of an amount: 30% of 50 = 0.3 × 50 = 15.

Multipliers: increase 20% → ×1.2; decrease 15% → ×0.85.

• Cuboid: length × width × height
• Prism: area of cross-section × length
• Density: mass ÷ volume (g/cm³)