3.2 Time & Work; Pipes & Cisterns

Why Time & Work Matters

These problems test rate thinking — converting "X can do a job in N days" into "X does 1/N of the job per day". Once you express every worker as work-per-unit-time, addition solves most questions. RRB Group D typically asks 2-3 questions from this cluster, often mixed with pipes-and-cisterns.

Work-Rate Basics

If A can finish a job in n days, A's 1-day work = 1/n.

If A takes a days and B takes b days for the same job:

• Combined 1-day work = 1/a + 1/b = (a + b) / ab
• Time to finish together = ab / (a + b) days

Worked example 1 (working together)

A can do a job in 12 days, B in 18 days. Working together, how long will they take?

• 1-day work = 1/12 + 1/18 = 3/36 + 2/36 = 5/36
• Time = 1 / (5/36) = 36/5 = 7.2 days (7 1/5 days, or 7 days 4 hr 48 min)

The MDH Formula

When work is measured in man-days and the same kind of work is done by different teams, the relation is:

M1 · D1 · H1 / W1 = M2 · D2 · H2 / W2

where M = number of men, D = number of days, H = hours per day, W = work units.

Worked example 2 (man-days)

8 men can build a wall in 12 days of 8 hours each. How many days of 6 hours each will 16 men take for the same wall?

• (8 × 12 × 8) / 1 = (16 × D × 6) / 1
• 768 = 96D → D = 8 days

Efficiency and "k Times as Fast"

If A is twice as efficient as B, then in the same time A does 2 units of work for every 1 unit B does. If A finishes a job in n days, B finishes it in 2n days.

Worked example 3 (efficiency ratio)

A is thrice as efficient as B and together they finish a job in 12 days. How long will A alone take?

• Let B's 1-day work = 1/x; A's = 3/x
• Together: 1/x + 3/x = 4/x = 1/12 → x = 48
• A alone: 3/x = 3/48 = 1/16 → 16 days

Pipes & Cisterns

The logic is identical to time-and-work, but an emptying pipe does NEGATIVE work (it removes water).

• Fill pipe (fills in n hours): +1/n per hour
• Emptying pipe (empties in m hours): −1/m per hour
• Net per hour = 1/n − 1/m
• Time to fill = 1 / (net rate), provided net rate is positive

Worked example 4 (fill + leak)

A tap fills a tank in 6 hours; a leak empties it in 12 hours. With both open, how long to fill the tank?

• Net rate = 1/6 − 1/12 = 2/12 − 1/12 = 1/12 per hour
• Time = 1 / (1/12) = 12 hours

Diagram — Net Rate in a Tank with Fill + Leak

graph LR
  A["Tap A<br/>+1/6 per hour"] --> T[(Tank)]
  L["Leak L<br/>-1/12 per hour"] --> T
  T --> F["Net +1/12 per hour<br/>12 hours to fill"]

Worked Examples — Extended Scenarios

Worked example 5 (alternate days)

A can do a job in 10 days and B in 15 days. They work on alternate days, starting with A. In how many days is the job finished?

• A's 1-day work = 1/10; B's 1-day work = 1/15

• Work done in a 2-day cycle = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 Work in thirtieths so every step is a whole number: A does 3/30 per day, B does 2/30 per day, and one 2-day cycle completes 5/30.

• After 4 full cycles (8 days): 4 × 5/30 = 20/30 done; remaining 10/30.

• Day 9 (A): +3/30 → 23/30 done; remaining 7/30.

• Day 10 (B): +2/30 → 25/30 done; remaining 5/30.

• Day 11 (A): +3/30 → 28/30 done; remaining 2/30.

• Day 12 (B): +2/30 → 30/30. The job finishes exactly at the end of day 12, on B's turn.

The faster route confirms it: each 2-day cycle does 5/30 = 1/6 of the job, so 6 cycles = 12 days complete exactly 1 whole job. Two bookkeeping rules keep this from going wrong: track work done and work remaining as separate running totals, and stop the moment the cumulative work reaches 1 — do not finish the cycle out of habit.

Worked example 6 (worker leaves mid-job)

A and B together can finish a job in 12 days, B and C in 15 days, C and A in 20 days. How long will A alone take?

• (A + B) 1-day = 1/12; (B + C) 1-day = 1/15; (C + A) 1-day = 1/20
• Add all three: 2(A + B + C) = 1/12 + 1/15 + 1/20 = 5/60 + 4/60 + 3/60 = 12/60 = 1/5
• A + B + C 1-day work = 1/10
• A's 1-day work = (A + B + C) − (B + C) = 1/10 − 1/15 = 3/30 − 2/30 = 1/30
• A alone = 30 days

Pipes — Multiple Fills

When two or more pipes fill together and a third empties, add the fill rates and subtract the empty rate, all in "tanks per hour".

Worked example 7 (two fill + one empty)

Pipe A fills a tank in 6 hours, B in 3 hours, and a drain C empties it in 4 hours. With all three open, how long to fill?

• Net rate = 1/6 + 1/3 − 1/4 = 2/12 + 4/12 − 3/12 = 3/12 = 1/4 per hour
• Time = 1 / (1/4) = 4 hours

Quick Reference — Rate Conversion

Common Exam Traps

• "A and B work on alternate days" — add one day of A, then one day of B; each 2-day cycle completes (1/a + 1/b) of the job. Always check whether the final day finishes the job exactly or leaves a remainder to be completed by the next worker.
• An emptying pipe is always subtracted; if its rate exceeds the fill rate, the tank never fills.
• "A leaves after x days" — compute the work done in x days, then the remaining work is finished by the others; divide remaining work by their combined rate.
• Keep all rates in the same time unit (hours vs days) before adding or subtracting.
• "A is k times as efficient as B" means A's rate is k × B's rate, not that A's time is k × B's time — the times go the opposite way.
• When three pairs of workers are given, never solve for each worker individually first — add the three pair-equations and halve to get the combined 1-day work of all three, then subtract the pair that excludes the worker you need.
• A pipe that fills in "n minutes" has rate 1/n per minute — do not mix minutes and hours in the same equation.
• If the question says "the tank is already half full", only half the tank's capacity needs to be filled — multiply the fill time by 1/2.