Ratio, Proportion, Indices & Logarithms — Formula Sheet
Foundation chapter for Paper 3. Master these 10 identities and you'll answer 4–6 questions every attempt.
Last reviewed: 22 April 2026
Ratio
- •Ratio of a to b = a:b = a/b (b ≠ 0).
- •Duplicate ratio of a:b = a²:b². Sub-duplicate = √a : √b.
- •Triplicate ratio = a³:b³. Sub-triplicate = ∛a : ∛b.
- •Inverse ratio of a:b = b:a.
- •Compound ratio of (a:b) & (c:d) = ac : bd.
Proportion
- •a, b, c, d in proportion ⇔ a/b = c/d ⇔ ad = bc.
- •Mean proportional between a and c = √(ac).
- •Third proportional to a, b = b²/a.
- •Fourth proportional to a, b, c = bc/a.
- •Componendo & dividendo: if a/b = c/d then (a+b)/(a−b) = (c+d)/(c−d).
Indices
- •a^m × a^n = a^(m+n).
- •a^m ÷ a^n = a^(m−n).
- •(a^m)^n = a^(mn).
- •a^0 = 1, a^(−n) = 1/a^n, a^(1/n) = ⁿ√a.
- •(ab)^n = a^n × b^n.
Logarithms
- •log_a(mn) = log_a m + log_a n.
- •log_a(m/n) = log_a m − log_a n.
- •log_a(m^n) = n · log_a m.
- •log_a a = 1; log_a 1 = 0.
- •Change of base: log_b a = log_c a / log_c b.
- •log_b a × log_a b = 1.
- •Common log: base 10. Natural log: base e.
Formulas
- Continued proportion (a, b, c)
- a/b = b/c ⇒ b² = ac ⇒ b = √(ac)
- n-th root via log
- log ⁿ√x = (1/n) · log x
- Ratio to percentage
- a : b as % = a/(a+b) × 100
Must know before the exam
- ★Ratio has no units — always dimensionless.
- ★Zero and negative numbers can't appear inside a logarithm.
- ★log(a+b) ≠ log a + log b (common trap). That identity is only for multiplication.
- ★Any number to the power zero is 1 (except 0⁰, which is undefined).
Common mistakes & fixes
- ✗ Writing log(a−b) = log a − log b.
- ✓ Subtraction of logs applies to a/b, not a−b.
- ✗ Saying (a+b)² = a² + b².
- ✓ (a+b)² = a² + 2ab + b². Don't miss the middle term.
Lock it in with practice
Reading without practising is the #1 reason people forget in the exam. Solve a quick set while this is fresh.