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
W = \text{rate} \times \text{time}
Efficiency, 'Twice as Good' & Ratio Cases
Efficiency ⇄ 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}
Efficiency ratio from times
E_A : E_B = \frac{1}{T_A} : \frac{1}{T_B}
Pipes & Cisterns
Net filling rate
\frac{1}{T} = \sum \frac{1}{T_i^{\,+}} - \sum \frac{1}{T_j^{\,-}}
Fill + drain pair
T = \frac{ab}{b - a}
Work done in stage 1
W_1 = \text{rate}_1 \times t_1
Tank as LCM units
\text{tank} = \text{LCM of times; rates in units/hour}
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 × days constant
M_1 D_1 = M_2 D_2
Provisions remaining
\text{days} = \frac{M_1 (D_{\text{total}} - t)}{M_2}
Piece of work
\text{days for } \tfrac{k}{n} \text{ of work} = \frac{k}{n} T
Joining / Leaving Mid-work, Alternate Days & Wages
Work done by A in t days
\frac{t}{T_A}
Remaining work
1 - \frac{t}{T_A}
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}