Ratio, Proportion & Partnership

Ratio — meaning, terms and properties; comparing ratios; duplicate, triplicate, sub-duplicate and compounded ratios; dividing a quantity in a given ratio; proportion — extremes and means, mean, third and fourth proportional, continued proportion; componendo and dividendo; direct and inverse variation; partnership — simple and compound, capital × time, working and sleeping partners, capital added or withdrawn during the year — with fully worked examples.

Ratio, Proportion & Partnership

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📑 Contents (6 sections)

Last reviewed 22 Sept 2026 · 6 min read

Meaning of a ratio

A ratio compares two quantities of the same kind by division: $a : b = \frac{a}{b}$ , where $a$ is the antecedent and $b$ the consequent. A ratio has no unit, so 4 kg : 800 g must first be written in one unit — 4000 g : 800 g = 5 : 1.

Multiplying or dividing both terms by the same non-zero number does not change a ratio: $6 : 9 = 2 : 3$ .
$a : b : c$ means the three quantities are in the proportion $a$ , $b$ , $c$ of one common part $k$ — write them as $ak$ , $bk$ , $c k$ and one equation finds $k$ .

Comparing two ratios

Cross-multiply: $a : b > c : d$ when $a d > b c$ . To compare $5 : 7$ and $7 : 10$ : $5 \times 10 = 50$ and $7 \times 7 = 49$ , so $5 : 7$ is greater.

Derived ratios

Name	For the ratio $a : b$
Duplicate	$a^{2} : b^{2}$
Sub-duplicate	$a : b$
Triplicate	$a^{3} : b^{3}$
Sub-triplicate	$3 a : 3 b$
Inverse	$b : a$
Compounded of $a : b$ and $c : d$	$a c : b d$

Proportion

FormulaProportion and its consequences

Four quantities are in proportion when $a : b = c : d$ , written $a : b :: c : d$ .

a \times d = b \times c (product of extremes = product of means)

Fourth proportional to $a$ , $b$ , $c$ is $\frac{b c}{a}$
Third proportional to $a$ and $b$ is $\frac{b ^{2}}{a}$
Mean proportional between $a$ and $b$ is $ab$
$a$ , $b$ , $c$ are in continued proportion when $b^{2} = a c$

FormulaComponendo and dividendo

If $\frac{a}{b} = \frac{c}{d}$ , then

\frac{a + b}{a - b} = \frac{c + d}{c - d}

This turns an equation such as $\frac{3 x + 2}{3 x - 2} = \frac{5}{3}$ into one step instead of cross-multiplication.

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