CA Foundation · Quantitative Aptitude

Ratio and Proportion, Indices and Logarithms: formula sheet

Full chapter guide

Key formulas

Ratio as a fraction: a : b = a ÷ b = a/b (b ≠ 0); Both terms must be in the same unit. Multiplying or dividing both terms by the same non-zero number keeps the ratio unchanged.
Duplicate ratio: a : b → a² : b²; Square both terms.
Triplicate ratio: a : b → a³ : b³; Cube both terms.
Sub-duplicate ratio: a : b → √a : √b; Take the square root of both terms.
Sub-triplicate ratio: a : b → ∛a : ∛b; Take the cube root of both terms.
Inverse ratio: a : b → b : a; Interchange the terms.
Compounded ratio: (a : b) and (c : d) → ac : bd; Multiply antecedents together and consequents together.
Comparing ratios: a/b > c/d ⇔ ad > bc (for b, d > 0); Cross-multiply. Valid only when both denominators are positive.
Ratio of ratios, a : b and b : c: a : b : c; Make the common term equal by scaling, then join the ratios.
Proportion: a : b = c : d ⇔ a ÷ b = c ÷ d ⇔ ad = bc; Extremes are a and d. Means are b and c. Terms are non-zero.
Fourth proportional: Fourth proportional to a, b, c = bc ÷ a; Order matters: a : b = c : x gives x = bc ÷ a.
Continued proportion (three terms): a : b = b : c ⇔ b² = ac; The middle term b is the mean proportional.
Mean proportional: Mean proportional between a and c = √(ac); For positive a and c, take the positive root.
Third proportional: Third proportional to a and b = b² ÷ a; From a : b = b : x. Put the given terms in order.
Continued proportion (four terms): a : b = b : c = c : d ⇒ b² = ac, c² = bd, and with common ratio k: b = ak, c = ak², d = ak³; Use k to express every term in one variable.
Duplicate, triplicate, sub-duplicate ratios: Duplicate of a : b = a² : b²; triplicate = a³ : b³; sub-duplicate = √a : √b; If a, b, c are in continued proportion, a : c = a² : b² (the duplicate ratio of a : b).
Direct variation: x ∝ y ⇒ x = ky, so x₁/y₁ = x₂/y₂; Use when both quantities increase or decrease together.
Inverse variation: x ∝ 1/y ⇒ xy = k, so x₁y₁ = x₂y₂; Use when one goes up as the other goes down, such as men and days.
Joint variation: x ∝ yz ⇒ x = kyz, so x₁/(y₁z₁) = x₂/(y₂z₂); Quantity depends on the product of two others.
Sharing a total in a ratio: Share of A = a/(a + b) × Total, for ratio a : b; For three parts a : b : c, divide by a + b + c.
Mixture ratio after adding: New ratio = (a·k + added A) : (b·k + added B); Write the original quantities as ak and bk first.
Age problems: Present ages ak and bk; after n years: (ak + n) and (bk + n); The difference of ages stays the same over time.
Replacement in a mixture: Final quantity of original liquid = Initial × (1 − x/T)ⁿ; Here T is the total volume of the mixture. Each time, x of the mixture is drawn out and replaced by the other liquid, and this is done n times. Initial is the original liquid's initial quantity.
Product law: aᵐ × aⁿ = aᵐ⁺ⁿ; Same base only. Add the powers.
Quotient law: aᵐ ÷ aⁿ = aᵐ⁻ⁿ; Same base, a ≠ 0. Subtract the powers.
Power of a power: (aᵐ)ⁿ = aᵐⁿ; Multiply the powers.
Power of a product: (ab)ⁿ = aⁿ × bⁿ; Applies to each factor inside the bracket.
Power of a quotient: (a/b)ⁿ = aⁿ ÷ bⁿ; b ≠ 0.
Zero index: a⁰ = 1; For a ≠ 0. 0⁰ is not defined at this level.
Negative index: a⁻ⁿ = 1 ÷ aⁿ; a ≠ 0. Also (a/b)⁻ⁿ = (b/a)ⁿ.
Fractional index: a^(m/n) = (ⁿ√a)ᵐ = ⁿ√(aᵐ); For a > 0 in general. Take the root first to keep numbers small.
Equal bases rule: If aˣ = aʸ, then x = y; Valid when a > 0 and a ≠ 1.
Equal powers rule: If aˣ = bˣ with x ≠ 0, then a = b; For positive a and b.
Definition: aˣ = N ⇔ logₐ N = x; Valid for a > 0, a ≠ 1, N > 0.
Log of 1: logₐ 1 = 0; Because a⁰ = 1.
Log of the base: logₐ a = 1; Because a¹ = a.
Product law: logₐ (mn) = logₐ m + logₐ n; m and n must both be positive.
Quotient law: logₐ (m ÷ n) = logₐ m − logₐ n; m and n must both be positive.
Power law: logₐ (mⁿ) = n × logₐ m; m must be positive.
Inverse relation: a^(logₐ N) = N; Useful for quick simplification.
Reciprocal: logₐ b = 1 ÷ log_b a; Also written logₐ b × log_b a = 1.
Definition: log_b a = x ⇔ b^x = a; Needs a > 0, b > 0 and b ≠ 1.
Common logarithm: log x means log_10 x; Base 10. Then log 10 = 1 and log 1 = 0.
Natural logarithm: ln x means log_e x; Base e ≈ 2.718. ln e = 1 and ln 1 = 0.
Change of base: log_b a = log_c a ÷ log_c b; c is any valid new base. Common choices are 10 or e.
Reciprocal rule: log_b a = 1 ÷ log_a b; So log_b a × log_a b = 1. Valid when a, b ≠ 1.
Chain rule: log_a b × log_b c = log_a c; The middle term b cancels, like fractions. The base of the second log must match the number in the first.
Power of base: log_(b^n) a = (1 ÷ n) × log_b a; Also log_b (a^m) = m × log_b a.
Base-power identity: b^(log_b a) = a; Useful for quick simplification.
Link between ln and log: ln x = ln 10 × log x ≈ 2.3026 × log x; Use only when the value of ln 10 is given or standard.

Quick revision

Ratio a : b = a ÷ b, with b ≠ 0; it compares quantities of the same kind in the same unit.
Multiplying or dividing both terms of a ratio by the same non-zero number does not change it.
Proportion: a : b = c : d means ad = bc.
Continued proportion a : b = b : c gives b² = ac, so b = √(ac) is the mean proportional.
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: if a/b = c/d then (a + b)/(a − b) = (c + d)/(c − d), where the denominators are non-zero.
Indices: aᵐ × aⁿ = aᵐ⁺ⁿ; aᵐ ÷ aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 for a ≠ 0.
Negative index: a⁻ⁿ = 1/aⁿ; fractional index: a^(1/n) is the nth root of a.
Logarithm: logₐ x = y means aʸ = x, for a > 0, a ≠ 1 and x > 0.
Log laws: log(mn) = log m + log n; log(m/n) = log m − log n; log(mⁿ) = n log m; logₐ 1 = 0; logₐ a = 1.
Change of base: logₐ b = log b ÷ log a (any common base) = 1 ÷ logᵦ a.

Common mistakes

Squaring only one term for the duplicate ratio, or writing 2a : 2b. Fix: Duplicate means squared: a² : b². Apply the power to both terms.
Confusing sub-duplicate with duplicate. Fix: Remember that the prefix sub- means the root. Sub-duplicate is the square root, and duplicate is the square.
Mixing the order when finding the fourth or third proportional Fix: Write the proportion in the order the question gives. Fourth to a, b, c is a : b = c : x, so x = bc ÷ a.
Writing the mean proportional as (a + c) ÷ 2 Fix: Mean proportional is the geometric type: √(ac). Use b² = ac.
Using a direct relation when the relation is inverse, for example in men and days problems. Fix: Ask: if one quantity increases, should the other increase or decrease? If it decreases, use xy = constant.
Adding the same number to ratio parts when time passes, but forgetting to add to the actual ages (ak + n), not to the ratio numbers. Fix: Always write ages as 3k and 5k, then add years to 3k and 5k.
Adding the powers when the bases are different, such as 2³ × 3² = 6⁵. Fix: Check the bases first. Add powers only if the bases match. Otherwise, rewrite the bases or calculate separately.
Treating a⁻ⁿ as a negative number, so 2⁻³ = −8. Fix: A negative index means reciprocal. 2⁻³ = 1/2³ = 1/8.
Writing log (m + n) = log m + log n. Fix: The sum of logs equals the log of the product: log m + log n = log (mn). There is no simple rule for log (m + n).
Writing log m ÷ log n as log (m ÷ n). Fix: The quotient law applies to log m − log n, not to a ratio of two logs. A ratio of logs is a change of base.

Exam tips

Learn the four names by their roots: duplicate is square, triplicate is cube, sub-duplicate is square root, sub-triplicate is cube root. Questions test the names directly.
Check the units in word problems before doing any calculation. A unit trap is a common way to build a wrong option.
In comparison questions, expect options that look close. Use cross-multiplication, not guesswork.
When a ratio is given as a : b : c with a total, use the k method and verify that the parts add up to the total.
Compounded ratio questions are often quick. Cancel diagonally and finish in under a minute.
Most questions are direct: find the fourth, third or mean proportional. Practise these until they take under 30 seconds.
Watch for option traps like the arithmetic mean or a wrongly ordered answer.
Questions on 'a, b, c in continued proportion' often ask you to prove a relation like a : c = a² : b². Use b² = ac or substitute b = ak, c = ak².