10.3 Time & Work, Efficiency Ratios & Pipes and Cisterns

10.3 Time & Work, Efficiency Ratios & Pipes and Cisterns

Time and Work and its engineering counterpart, Pipes and Cisterns, form one of the highest-weightage arithmetic modules in the SBI Clerk Preliminary and Main examinations. Standard textbook approaches rely on fractions (assigning total work as 1 and tracking fractional daily outputs $\frac{1}{A} + \frac{1}{B}$), which inevitably creates tedious common-denominator arithmetic under exam pressure. The LCM Efficiency Method replaces fractional algebra with integer arithmetic, enabling candidates to solve multi-worker schedules, alternating work shifts, mid-way departures, and drainage leak problems rapidly.

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1. The LCM Efficiency Method Foundations

Instead of defining total work as $1$, let:

$\text{Total Work} = \text{LCM}(\text{Individual Completion Times})$

$\text{Daily Efficiency } (E) = \frac{\text{Total Work (in units)}}{\text{Individual Completion Time (in days)}}$

$\text{Time Taken to Complete Work} = \frac{\text{Remaining Work (in units)}}{\text{Combined Efficiency of Active Workers } (\sum E_i)}$

LCM Efficiency Reference Architecture

Suppose Associate A completes an administrative audit in 12 days, Associate B completes it in 15 days, and Associate C completes it in 20 days.

1. Compute $\text{LCM}(12, 15, 20) = 60\text{ units}$ (Total Work).
2. Establish individual daily efficiencies:
   • $E_A = \frac{60}{12} = +5\text{ units/day}$
   • $E_B = \frac{60}{15} = +4\text{ units/day}$
   • $E_C = \frac{60}{20} = +3\text{ units/day}$
3. Combined daily efficiency ($E_{\text{total}}$) = $5 + 4 + 3 = 12\text{ units/day}$.
4. Time taken working concurrently = $\frac{60}{12} = \mathbf{5\text{ days}}$.

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2. The Chain Rule & Workforce Equivalence

When workforce sizes, daily working hours, and physical output requirements vary simultaneously, the Chain Rule provides a single unifying equation:

$\frac{M_1 \times D_1 \times H_1 \times E_1}{W_1} = \frac{M_2 \times D_2 \times H_2 \times E_2}{W_2}$

Where:

• $M = \text{Number of workers}$
• $D = \text{Number of working days}$
• $H = \text{Operating hours per day}$
• $E = \text{Efficiency rating per worker}$
• $W = \text{Quantity of work accomplished (pages verified, meters dug, loans processed)}$

Workforce Equivalence Mechanics (Men, Women, Children)

Examinations frequently feature composite teams (e.g., "6 men or 10 women can complete a task"). Convert heterogeneous worker groups into a single equivalent metric:

$6M = 10W \implies 3M = 5W \implies \frac{E_M}{E_W} = \frac{5}{3}$

Assign integer efficiency ratings: each man possesses an efficiency of $E_M = 5$ units/day, while each woman possesses $E_W = 3$ units/day. Substitute these integer values directly into the Chain Rule.

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3. Worker Departures and Arrivals Mid-Way

When team members join or leave mid-way through a project, two systematic techniques replace fractional timelines:

1. Direct Deduction Method (Worker Leaves After $k$ Days)

• Calculate work completed by the initial team during the first $k$ days: $W_{\text{done}} = (\sum E_{\text{initial}}) \times k$.
• Determine remaining work: $W_{\text{rem}} = W_{\text{total}} - W_{\text{done}}$.
• Compute remaining days: $T_{\text{rem}} = \frac{W_{\text{rem}}}{\sum E_{\text{remaining}}}$.

2. Virtual Work Addition Method (Worker Leaves $k$ Days Before Completion)

• When a question states that "Worker A leaves 3 days before the scheduled completion", do not work backwards with variable $T$.
• Shortcut: Force Worker A to stay! Add Worker A's phantom output for those 3 days to the Total Work: $W_{\text{virtual}} = W_{\text{total}} + (E_A \times 3)$
• Divide $W_{\text{virtual}}$ by the combined efficiency of the entire team $(\sum E_{\text{all}})$. The result gives the total completion duration directly in a single step.

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4. Alternate-Day Work Schedules

In alternating-day scenarios, workers operate in sequential rotation (e.g., Worker A works on Day 1, Worker B on Day 2, Worker A on Day 3, and so forth).

Step-by-Step Cycle Tracking Algorithm

1. Define Cycle Length: For two alternating workers, 1 complete cycle $= 2\text{ days}$. For three workers, 1 complete cycle $= 3\text{ days}$.
2. Compute Cycle Work: Sum the individual efficiencies across the full cycle: $W_{\text{cycle}} = E_A + E_B$.
3. Compute Complete Cycles:
   $\text{Number of Full Cycles} = \left\lfloor \frac{W_{\text{total}}}{W_{\text{cycle}}} \right\rfloor$
4. Evaluate Cumulative Work & Time:
   $\text{Work Done in Full Cycles} = \text{Full Cycles} \times W_{\text{cycle}}$ $\text{Time Elapsed} = \text{Full Cycles} \times (\text{Days per Cycle})$
5. Allocate Remaining Units to Sequential Starters:
   $W_{\text{remainder}} = W_{\text{total}} - W_{\text{done in cycles}}$ Allocate $W_{\text{remainder}}$ to the worker whose turn opens the next cycle. If the remainder is less than that worker's daily output, the fractional time added is $\frac{W_{\text{remainder}}}{E_{\text{worker}}}$.

[!WARNING] The Overshoot Trap: Never divide total work directly by cycle work if a fractional remainder exists, because the work concludes the exact moment the cumulative threshold is reached. A worker does not complete a full extra day if only a small fraction of work remains.

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5. Pipes & Cisterns: Negative Efficiency Dynamics

Pipes and cisterns problems operate under identical mathematical principles as human labor, with one fundamental physical distinction:

$\text{Inlet Pipes contribute Positive Efficiency } (+E)$ $\text{Drainage Pipes / Structural Leaks contribute Negative Efficiency } (-E)$

Worked Pipe and Structural Leak Problem

Problem: A main supply tap fills an overhead water tank at an SBI residential quarters in 8 hours. Due to an accidental leak in the base of the tank, it requires 10 hours to fill the tank completely. If the tank is completely full and the inlet tap is turned off, how long will the leak take to empty the entire tank?

1. Establish Total Capacity: $\text{LCM}(8, 10) = 40\text{ units}$.
2. Determine Inlet Efficiency ($E_{\text{inlet}}$):
   $E_{\text{inlet}} = \frac{+40}{8} = +5\text{ units/hour}$
3. Determine Net Combined Efficiency ($E_{\text{net}}$):
   $E_{\text{net}} = E_{\text{inlet}} + E_{\text{leak}} = \frac{+40}{10} = +4\text{ units/hour}$
4. Isolate Leak Efficiency ($E_{\text{leak}}$):
   $+5 + E_{\text{leak}} = +4 \implies E_{\text{leak}} = 4 - 5 = -1\text{ unit/hour}$
5. Calculate Emptying Time:
   $\text{Time to Empty Full Tank} = \frac{\text{Total Capacity}}{|E_{\text{leak}}|} = \frac{40}{1} = \mathbf{40\text{ hours}}$