Time & Work

At a glance 11 formulas

1. 1Work and rate$\text{work} = \text{rate} \times \text{time}$$\text{alone in } n \text{ days} \Rightarrow \text{rate} = \tfrac{1}{n} \text{ a day}$
2. 2LCM unitsCall the job the LCM of everyone's times; each rate is then a whole number of units a day, and time = units ÷ total rate.
3. 3Two together$T = \dfrac{xy}{x + y}$$\text{any number: } \dfrac{1}{T} = \dfrac{1}{x} + \dfrac{1}{y} + \dfrac{1}{z}$
4. 4One from the pair$B = \dfrac{ac}{a - c}$$A \text{ alone } a \text{ days, together } c \text{ days}$
5. 5Efficiency and time$\text{efficiency } k : 1 \Rightarrow \text{time } 1 : k$
6. 6Pairs given$\text{all three} = \tfrac{1}{2}\,(\text{sum of the pairs})$$A = \text{all three} - (B + C) \text{, all as rates}$
7. 7Men, days, hours$\dfrac{M_1 D_1 H_1}{W_1} = \dfrac{M_2 D_2 H_2}{W_2}$$3 \text{ men} = 5 \text{ women} \Rightarrow 1 \text{ man} = \tfrac{5}{3} \text{ women}$
8. 8Leaving before the endIf someone leaves k days before the end, let the others run all T days and that person T - k days, and make the work add up to one job.
9. 9Alternate daysFind the work of one full cycle, fit in as many whole cycles as you can, then add the leftover days in the same order.
10. 10Wages follow work$\text{share} \propto \text{rate} \times \text{days worked}$
11. 11Extra-time rule$T = \sqrt{ab}$$\text{alone: } T + a \text{ and } T + b \text{ days}$