ICSE Class 10 Maths: Ratio, Proportion & Variation — Notes 2026

Ratio Basics

• Ratio a:b = a/b; must be in simplest form; both quantities must have same units
• Compounded ratio: (a:b) × (c:d) = ac:bd; duplicate ratio: a²:b²; triplicate ratio: a³:b³
• Inverse ratio: b:a is inverse of a:b; sub-duplicate ratio: √a:√b

Proportion

• Four quantities a, b, c, d in proportion: a:b = c:d → ad = bc (product of extremes = product of means)
• Continued proportion: a:b = b:c → b² = ac; b is geometric mean of a and c
• Fourth proportional: if a:b = c:x, then x = bc/a

Properties of Proportion

• Componendo: if a/b = c/d then (a+b)/b = (c+d)/d
• Dividendo: if a/b = c/d then (a−b)/b = (c−d)/d
• Componendo-Dividendo: (a+b)/(a−b) = (c+d)/(c−d) — used to simplify ICSE problems

Direct Variation

• y ∝ x means y = kx; k is constant of variation
• Graph of direct variation: straight line through origin
• Example: speed is constant, distance ∝ time; if speed = 60 km/h, d = 60t

Inverse Variation

• y ∝ 1/x means xy = k; as x increases, y decreases proportionally
• Graph of inverse variation: rectangular hyperbola
• Example: workers × days = constant; 10 workers × 6 days = 5 workers × 12 days

Joint Variation

• z ∝ xy means z = kxy; z varies jointly with x and y
• Example: volume of cylinder V ∝ r²h; V = πr²h
• ICSE applications: cost problems, work problems, pressure-volume-temperature

Exam Strategy

• ICSE Maths paper typically has 2–3 questions from this chapter; often combined with algebra
• Componendo-Dividendo property: most frequently tested in ICSE long-answer questions
• Always verify proportion using cross-multiplication and show working clearly