Time & Work
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Work Rates & the LCM Method
One-day work
\frac{1}{T}
Combined time (two workers)
T = \frac{ab}{a + b}
Combined time (three workers)
T = \frac{abc}{ab + bc + ca}
Work done and left
\text{done} = \frac{t}{T}, \quad \text{left} = 1 - \frac{t}{T}
Efficiency, 'Twice as Good' & Ratio Cases
Efficiency and time
\frac{E_A}{E_B} = \frac{T_B}{T_A}
k-times worker
T_B = (k+1)T, \quad T_A = \frac{(k+1)T}{k}
Times from an efficiency ratio
E_A : E_B = a : b \Rightarrow T_A : T_B = b : a
Per cent more efficient
a\% \text{ more} \Rightarrow E_A : E_B = (100 + a) : 100
Pipes & Cisterns
Net speed
\text{net} = (\text{inlets}) - (\text{outlets})
Speeds in units per hour, with tank = LCM of the times.
Time to fill
T = \frac{\text{capacity}}{\text{net speed}}
Two inlets
T = \frac{a b}{a + b}
Both pipes fill.
One inlet, one outlet
T = \frac{a b}{b - a}
a = filling time, b = emptying time, b > a.
Leak time from two fill times
T_{\text{leak}} = \frac{t_1 t_2}{t_2 - t_1}
t₁ = normal time, t₂ = time with the leak.
Stage method
t_2 = \frac{\text{capacity} - \text{speed}_1 \times t_1}{\text{speed}_2}
Men–Days–Hours Chain & Provisions
MDH chain
M_1 D_1 H_1 E_1 = M_2 D_2 H_2 E_2 \quad (W_1 = W_2)
Men and days constant
M_1 D_1 = M_2 D_2
Provisions remaining
\text{days} = \frac{M_1 (D_{total} - t)}{M_2}
Work scaling
W_2 = W_1 \times \frac{M_2}{M_1} \times \frac{D_2}{D_1}
Joining / Leaving Mid-work, Alternate Days & Wages
Work done by A in t days
\frac{t}{T_A}
Leaves t days before the end
\frac{T - t}{T_A} + \frac{T}{T_B} = 1
Two-day cycle
\frac{1}{T_A} + \frac{1}{T_B} \text{ per 2 days}
Wage split
w_A = \text{total} \times \frac{1/T_A}{1/T_A + 1/T_B}