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 $m = \frac{{pa + qc + re + \cdots }}{{pb + qd + rf + \cdots }} = \frac{{a + c + e + \cdots }}{{b + d + f + \cdots }}$