1. Basic Ratios and Proportions
| Concept | Description / Rule | Formula / Example |
| Ratio | The ratio of two quantities $a$ and $b$ in the same units. | $a : b = \frac{a}{b}$ |
| Terms | Names for the first and second terms. | $a$ = antecedent; $b$ = consequent |
| Ratio Rule | Multiplication/division by a non-zero number doesn’t change the ratio. | $a : b = ka : kb$ |
| Proportion | The equality of two ratios ($a, b, c, d$ are in proportion). | $a : b :: c : d$ |
| Product Rule | Relationship between means and extremes. | $(b \times c) = (a \times d)$ |
| Type | Definition |
| Fourth Proportional | If $a : b = c : d$, then $d$ is the fourth proportional to $a, b, c$. |
| Third Proportional | If $a : b = b : c$, then $c$ is the third proportional to $a$ and $b$. |
| Mean Proportional | The mean proportional between $a$ and $b$ is $\sqrt{ab}$. |
| Operation | Description | Formula |
| Comparison | Comparing two different ratios. | $(a : b) > (c : d) \iff \frac{a}{b} > \frac{c}{d}$ |
| Compounded Ratio | The ratio of the products of the terms. | $(a : b), (c : d), (e : f) \to (ace : bdf)$ |
4. Duplicate and Triplicate Ratios
| Name | Formula |
| Duplicate Ratio | $a^2 : b^2$ |
| Sub-duplicate Ratio | $\sqrt{a} : \sqrt{b}$ |
| Triplicate Ratio | $a^3 : b^3$ |
| Sub-triplicate Ratio | $a^{1/3} : b^{1/3}$ |
| Componendo and Dividendo | If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a + b}{a – b} = \frac{c + d}{c – d}$ |
5. Variations
| Variation Type | Definition | Notation |
| Direct Proportion | $x$ is directly proportional to $y$. | $x = ky$ or $x \propto y$ |
| Inverse Proportion | $x$ is inversely proportional to $y$. | $xy = k$ or $x \propto \frac{1}{y}$ |
