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

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

Componendo & Dividendo: If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$

Average (Mean): $\text{Average} = \frac{\text{Sum of all observations}}{\text{Number of observations}}$
Weighted Average: $\frac{w_1x_1 + w_2x_2 + \dots + w_nx_n}{w_1 + w_2 + \dots + w_n}$
Average Speed: $\frac{\text{Total Distance}}{\text{Total Time}}$. Not $\frac{v_1+v_2}{2}$.
If a person travels distance D at speed $v_1$ and returns at speed $v_2$: Average speed = $\frac{2v_1v_2}{v_1+v_2}$
If a person travels 3 equal distances at speeds $v_1, v_2, v_3$: Average speed = $\frac{3}{\frac{1}{v_1} + \frac{1}{v_2} + \frac{1}{v_3}}$
Average of first 'n' natural numbers: $\frac{n+1}{2}$
Average of first 'n' even numbers: $n+1$
Average of first 'n' odd numbers: $n$

Work Rate: If a person can do a piece of work in $n$ days, then in one day, the person does $\frac{1}{n}$ of the work.
Combined Work: If A does work in $x$ days and B in $y$ days, together they do it in $\frac{xy}{x+y}$ days.
Men, Days, Work: $\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$ (where H is hours per day)
Efficiency: Work done per unit time. Efficiency $\propto \frac{1}{\text{Time taken}}$.
Pipes and Cisterns: (Similar to Time and Work)
- Inlet pipe fills a tank in $x$ hours, fills $\frac{1}{x}$ of tank in 1 hour.
- Outlet pipe empties a tank in $y$ hours, empties $\frac{1}{y}$ of tank in 1 hour.
- Net part filled in 1 hour (if both open): $\frac{1}{x} - \frac{1}{y}$ (if $y>x$, tank fills; if $x>y$, tank empties)
- If multiple inlets and outlets: $\frac{1}{T} = \sum \frac{1}{T_{inlets}} - \sum \frac{1}{T_{outlets}}$

Speed: $\text{Speed} = \frac{\text{Distance}}{\text{Time}}$
Distance: $\text{Distance} = \text{Speed} \times \text{Time}$
Time: $\text{Time} = \frac{\text{Distance}}{\text{Speed}}$
Conversion: $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$; $1 \text{ m/s} = \frac{18}{5} \text{ km/hr}$
Relative Speed:
- Same direction: $|v_1 - v_2|$
- Opposite direction: $v_1 + v_2$
Trains:
- Time taken by a train of length $L$ to pass a pole/man: $\frac{L}{\text{Speed}}$
- Time taken by a train of length $L_T$ to pass a platform/bridge of length $L_P$: $\frac{L_T + L_P}{\text{Speed}}$
- Time taken by two trains of length $L_1, L_2$ and speeds $v_1, v_2$ to pass each other:
  - Same direction: $\frac{L_1 + L_2}{|v_1 - v_2|}$
  - Opposite direction: $\frac{L_1 + L_2}{v_1 + v_2}$
Boats and Streams:
- Speed of boat in still water = $u$
- Speed of stream = $v$
- Downstream Speed: $S_D = u+v$
- Upstream Speed: $S_U = u-v$
- $u = \frac{S_D + S_U}{2}$
- $v = \frac{S_D - S_U}{2}$