Ratio and Proportion Formulas
1. If $$\frac{a}{b} = \frac{c}{d}$$ then $$ad = bc$$
2. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{a}{c} = \frac{b}{d}$$
3. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{b}{a} = \frac{d}{c}$$
4. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{{a + b}}{b} = \frac{{c + d}}{d}$$
5. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{{a – b}}{b} = \frac{{c – d}}{d}$$
6. If $$\frac{a}{b} = \frac{c}{d}$$ then $$\frac{{a + b}}{{a – b}} = \frac{{c + d}}{{c – d}}$$ is called the Componendo-Dividendo Rule.
7. If $$\frac{a}{b} = \frac{b}{c}$$ then $$\frac{a}{c} = \frac{{{a^2}}}{{{b^2}}}$$
8. If $$\frac{a}{b} = \frac{c}{d}$$ and $$a = c$$ then $$b = d$$
9. If $$\frac{a}{{b + c}} = \frac{b}{{c + a}} = \frac{c}{{a + b}}$$ and $$a + b + c \ne 0$$ then $$a = b = c$$
10. If $$\frac{a}{b} = \frac{c}{d}$$ then \[ \frac{{{a^2} + {b^2}}}{{{c^2} + {d^2}}} = \frac{{{b^2}}}{{{a^2}}}\]
11. If $$ax = by = cz$$ then \[ \frac{{{a^2} + {b^2} + {c^2}}}{{{x^2}{y^2} + {y^2}{z^2} + {z^2}{x^2}}} = \frac{{ab + bc + ca}}{{xyz(x + y + z)}} \]
12. If $$\frac{a}{b} = \frac{c}{d}$$ then \[ \frac{{{a^2} + {b^2}}}{{{a^2} – {b^2}}} = \frac{{ac + bd}}{{ac – bd}} \]
13. If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \frac{{la + mc + ne}}{{lb + md + nf}}$$
14. If $$\frac{a}{b} = \frac{c}{d}$$ then \[ \frac{{a – b}}{a} = \frac{{c – d}}{c}\]
15. If $$\frac{a}{b} = \frac{c}{d}$$ then \[ \frac{{pa + qb}}{{ra + sb}} = \frac{{pc + qd}}{{rc + sd}} \]
16. If $$\frac{a}{b} = \frac{c}{d} = \frac{e}{f} = \cdots = m$$ then \[ m = \frac{{pa + qc + re + \cdots }}{{pb + qd + rf + \cdots }} = \frac{{a + c + e + \cdots }}{{b + d + f + \cdots }} \]