Ratios, Variations and Proportions: formula sheet

Key formulas

Ratio

a : b = a ÷ b (b ≠ 0)

a is the antecedent, b the consequent. Both must be in the same unit.

Equivalent ratio

a : b = ka : kb = (a ÷ k) : (b ÷ k), k ≠ 0

Multiply or divide both terms by the same non-zero number.

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

Square root of both terms.

Sub-triplicate ratio

a : b → ∛a : ∛b

Cube root of both terms.

Inverse ratio

a : b → b : a

Interchange the terms.

Compounded ratio

(a : b) and (c : d) → ac : bd

For three ratios, multiply all antecedents and all consequents.

Comparing ratios

a : b > c : d if ad > bc

For positive terms. Cross-multiply the fractions a/b and c/d.

Proportion

a : b = c : d ⇔ a/b = c/d ⇔ ad = bc

Extremes are a and d. Means are b and c.

Fourth proportional

Fourth proportional to a, b, c = bc ÷ a

Taken in the order a : b = c : x.

Third proportional

Third proportional to a and b = b² ÷ a

From a : b = b : x.

Mean proportional

Mean proportional between a and b = √(ab)

For positive a and b, from a : x = x : b.

Continued proportion

a : b = b : c = c : d ⇒ b² = ac, c² = bd

Common ratio k gives b = ak, c = ak², d = ak³.

Invertendo

If a/b = c/d, then b/a = d/c

Flip both ratios.

Alternendo

If a/b = c/d, then a/c = b/d

Swap the means.

Componendo

If a/b = c/d, then (a + b)/b = (c + d)/d

Add 1 to both sides.

Dividendo

If a/b = c/d, then (a − b)/b = (c − d)/d

Subtract 1 from both sides.

Componendo and dividendo

If a/b = c/d, then (a + b)/(a − b) = (c + d)/(c − d)

Valid when a ≠ b and c ≠ d.

Direct variation

y ∝ x ⇒ y = kx ⇒ y₁ ÷ x₁ = y₂ ÷ x₂

The ratio y ÷ x is constant. Both quantities increase or decrease together.

Inverse variation

y ∝ 1/x ⇒ y = k ÷ x ⇒ x₁y₁ = x₂y₂

The product xy is constant. When one rises, the other falls.

Joint variation

y ∝ xz ⇒ y = kxz ⇒ y₁ ÷ (x₁z₁) = y₂ ÷ (x₂z₂)

y varies directly with each of x and z when the other is held fixed.

Mixed variation

y ∝ x/z ⇒ y = kx ÷ z

Put directly varying quantities on top and inversely varying ones at the bottom.

Constant of variation

k = y ÷ x (direct), k = xy (inverse), k = y ÷ (xz) (joint)

Find k from the first set of data. It stays the same for the second set.

Dividing a sum in a ratio

Share of first = (a ÷ (a + b)) × Total, for ratio a : b

For three parts a : b : c, divide by a + b + c.

Common multiplier method

Quantities = ax, bx, cx; find x from the given total or difference

Works for sharing, mixtures and ages.

Proportion rule

a : b = c : d ⇔ a × d = b × c

Product of extremes equals product of means.

Age problem with change

(A ± n) : (B ± n) = given ratio

Add or subtract the same number of years to both actual ages.

Mixture quantity

Quantity of an ingredient = (its part ÷ total parts) × total mixture

Use only when the ratio is by the same measure (all litres or all kg).

Change in ratio

(ax ± p) : (bx ± q) = m : n, so n(ax ± p) = m(bx ± q); solve for x

Here m : n is the given new ratio. Cross-multiply to get a linear equation in x, then solve for x.

Quick revision

• A ratio compares two quantities of the same kind and unit; it has no unit itself.
• Ratio a : b is written as the fraction a ÷ b, and b should not be zero.
• Duplicate ratio of a : b is a² : b²; triplicate is a³ : b³; sub-duplicate is √a : √b.
• Inverse ratio of a : b is b : a.
• Compound ratio of a : b and c : d is ac : bd.
• Proportion: a : b = c : d means ad = bc.
• Continued proportion: a : b = b : c means b² = ac, and b is the mean proportional between a and c.
• Third proportional to a and b is b² ÷ a.
• Direct variation: y = kx, so y ÷ x stays constant.
• Inverse variation: y = k ÷ x, so x × y stays constant.
• Joint variation: z = kxy; find k first, then substitute.
• If a quantity is shared in the ratio 3 : 2, take the parts as 3x and 2x and divide the total by 5 to find x.

Common mistakes

• Confusing duplicate ratio with double the ratio. Fix: Duplicate means squared. The duplicate ratio of 2 : 3 is 4 : 9, not 4 : 6.
• Mixing up duplicate and sub-duplicate ratios. Fix: Duplicate squares. Sub-duplicate is 'under' the square, so it takes the square root.
• Taking the third proportional to a and b as a²÷b. Fix: Remember a : b = b : x. The repeated term b is squared, so x = b² ÷ a.
• Using the average (a + b)/2 as the mean proportional. Fix: Mean proportional is √(ab), the geometric mean of the two numbers.
• Treating an inverse link as direct, for example more men means more days. Fix: Ask: if one quantity doubles, does the other double or halve? Halve means inverse.
• Using y = kx when the question says y varies as the square of x. Fix: Write exactly what is stated. 'y varies as x²' means y = kx².
• Adding years to the ratio terms instead of to the actual ages. Fix: Always write ages as 3x and 5x first, then form (3x + n) : (5x + n).
• Taking one part as the total. Fix: Divide by a + b (or a + b + c) to get the value of one part.

Exam tips

• Learn the five rules by heart: square, cube, square root, flip, multiply. Most MCQs are direct applications.
• Watch for questions that ask for the original ratio when the duplicate or sub-duplicate is given. Reverse the operation.
• When asked to arrange ratios in order, cross-multiply pairs. It is faster than finding a common denominator for two ratios.
• With no negative marking, always mark an answer. Eliminate options with terms in the wrong order first.
• Reduce your final answer to the lowest terms. Options are usually given that way.
• Questions are usually direct: find the fourth, third or mean proportional. Practise the three formulas until they are automatic.
• When a ratio such as (a + b)/(a − b) appears, componendo and dividendo usually gives the answer in one line.
• In continued proportion, put the common ratio as k and write terms as ak, ak², ak³ to solve proofs quickly.