Time and work

In short

Time and work problems convert "how long someone takes" into "how much is done per day", so that several workers' outputs can simply be added. One day's work is the reciprocal of the total time, so a person finishing a job in 10 days does 1/10 of it daily. The topic is standard in the quantitative-aptitude section of every Indian placement test (TCS NQT, Infosys, Wipro, Capgemini, Cognizant, Accenture), bank and SSC exams, and in B.Tech/BBA/MBA aptitude papers under CBCS. On the job the same arithmetic drives manpower planning, sprint velocity, shift staffing and project crashing — any question of the form "how many people for how many days".

Explained

Core idea: work as a rate

• Treat the whole job as 1 unit of work. If A finishes it in *n* days, A's rate is 1/n per day.
• Rates of people working together add: A (1/10) and B (1/15) together do 1/10 + 1/15 = 1/6 per day → 6 days.
• Time taken = total work ÷ combined rate. Rates add; times never add — the commonest error in the topic.
• Work is proportional to men × days × hours per day, so (M₁ × D₁ × H₁)/W₁ = (M₂ × D₂ × H₂)/W₂.
• Efficiency is rate expressed as a percentage or ratio: a worker doing 1/10 per day has efficiency 10% per day.
• Inverse relation: if time is in ratio a : b, efficiency is in ratio b : a. "A is twice as efficient as B" means A takes half the time.

The LCM (total-work) method — preferred in exams

• Instead of fractions, let total work = LCM of the given times, in arbitrary "units".
• A in 12 days, B in 18 days → total work = LCM(12, 18) = 36 units. A does 3 units/day, B does 2 units/day.
• Together 5 units/day → 36/5 = 7.2 days. All arithmetic stays in whole numbers; no fraction addition.
• Extends cleanly to three or more workers, alternate-day patterns, and "one leaves after x days".
• When the answer is asked as a fraction of work, divide units done by total units.

Standard question types

1. Together time, given individual times — add rates (or units/day).
2. One rate unknown — A and B together in 6 days, A alone in 10 → B's rate = 1/6 − 1/10 = 1/15 → 15 days.
3. Pairwise data (A+B, B+C, C+A) — add all three pair rates and halve to get A+B+C; subtract a pair to isolate the third person.
4. Someone leaves or joins midway — compute work done in the completed phase, then divide the remainder by the new rate.
5. Alternate days — treat two days as one cycle; find work per cycle, count full cycles, finish the remainder carefully (the last part-day is not rounded up to a full day).
6. Efficiency comparison — "A is 25% more efficient than B": rates 5 : 4, so times 4 : 5.
7. Men, women, boys equivalence — convert all into one unit (say, man-equivalents) using the given relation.
8. Wages and shares — wages are divided in the ratio of work actually done, which for equal working periods is the ratio of efficiencies.
9. Contract / penalty problems — extra men are added to recover lost days; compute the shortfall and the required rate.
10. Work-and-days with hours — bring hours per day into the chain rule.

Alternate days, and who starts first

• A alone 9 days, B alone 18 days; they work on alternate days with A starting. Units: LCM = 18, A = 2/day, B = 1/day.
• One cycle (A then B) = 3 units in 2 days. After 5 full cycles (10 days) = 15 units; 3 units remain.
• Day 11 is A's: 2 units → 17 done. Day 12 is B's: needs 1 unit, B does exactly 1 → finished on day 12.
• If the remainder is a part of someone's daily output, express it as a fraction of a day (for example 1/2 day) unless the question says work happens only in full days.
• The order matters: the faster worker starting usually finishes the job marginally sooner.

Efficiency, wages and group-change problems

• Negative work: if one agent undoes work (a destroyer, a leak, a careless worker), subtract that rate. Combined rate can be negative, in which case the job never finishes.
• Wages rule: share ∝ (rate × days worked). If A works 6 days at 3 units/day and B works 4 days at 2 units/day, wages split 18 : 8 = 9 : 4.
• Group replacement: if 10 men are replaced by 15 women and the job still takes the same time, 10 men ≡ 15 women, so 1 man ≡ 1.5 women.
• Fractional work: "what part of the work is left after x days" — multiply combined rate by x and subtract from 1.
• Man-days is the standard unit for costing: a job of 60 man-days can be 10 men × 6 days or 12 men × 5 days (ignoring coordination limits).

Assumptions, advantages and limitations

• The model assumes constant and independent rates — real teams have learning curves, fatigue, supervision overhead and tasks that cannot be split.
• Brooks's law (software practice) is the real-world caveat: adding people to a late project can delay it further, because training and communication costs rise.
• Some work is not parallelisable (one painter per ladder, a single server migration), so the man-days arithmetic gives a lower bound only.
• Advantage: the rate model is linear, so it composes — exactly why it underlies capacity planning, Gantt resource levelling and the Critical Path Method's crashing calculations.
• In Indian practice, "man-day" rates appear in CPWD / PWD schedule-of-rates estimation and in IT services effort estimation (person-months), where the same proportionality is used for billing.

Key points

• Unit work - take the whole job as 1; a worker finishing in n days does 1/n per day.
• Rates add, times do not - combined rate = sum of individual rates; combined time = 1 ÷ that sum.
• Together formula - two workers taking a and b days finish in ab/(a + b) days.
• LCM method - set total work = LCM of times to work in integer units; fastest and least error-prone.
• Efficiency - work per unit time; efficiency ∝ 1/time, so time ratio a : b ⇒ efficiency ratio b : a.
• Chain rule - M₁D₁H₁/W₁ = M₂D₂H₂/W₂ for men, days, hours and work.
• Man-days - total effort = men × days; constant for a given job under the ideal model.
• Pairwise trick - (A+B) + (B+C) + (C+A) = 2(A+B+C); halve to get the team rate.
• Remaining work - after t days, work left = 1 − t × (combined rate).
• Percent more efficient - "x% more efficient" means rate ratio (100 + x) : 100 and time ratio 100 : (100 + x).
• Alternate days - treat two days as one cycle; handle the final partial day explicitly.
• Negative work - a spoiler's rate is subtracted; if the net rate is ≤ 0 the work never completes.
• Wages - shared in the ratio of work actually done, not in the ratio of time spent.
• Equivalence - convert men, women and boys to a single unit before combining.
• Scaling - doubling the workforce halves the time only if the task is fully divisible.
• Sanity check - the combined time must be less than the fastest individual time.
• Brooks's law - the practical limit of the model: more people on a late task can slow it down.

Formulas and examples

Example 1 — Together, and one leaves midway. A can do a job in 20 days, B in 30 days. They start together; A leaves after 5 days. How long does B take to finish?

• Total work = LCM(20, 30) = 60 units. A = 3 units/day, B = 2 units/day.
• In 5 days together: (3 + 2) × 5 = 25 units. Remaining = 60 − 25 = 35 units.
• B alone: 35 / 2 = 17.5 days more; total project time = 5 + 17.5 = 22.5 days.

Example 2 — Pairwise rates. A and B together finish in 8 days, B and C in 12 days, C and A in 16 days. Find each one's time and the time for all three.

• LCM(8, 12, 16) = 48 units. A+B = 6, B+C = 4, C+A = 3 units/day.
• Sum = 13 = 2(A+B+C) → A+B+C = 6.5 units/day → all three take 48/6.5 = 7 5/13 days ≈ 7.38 days.
• A = 6.5 − 4 = 2.5 → 48/2.5 = 19.2 days; C = 6.5 − 6 = 0.5 → 48/0.5 = 96 days; B = 6.5 − 3 = 3.5 → 48/3.5 = 13 5/7 ≈ 13.71 days.
• Check: A+B = 2.5 + 3.5 = 6 ✔, B+C = 3.5 + 0.5 = 4 ✔, C+A = 0.5 + 2.5 = 3 ✔.

Example 3 — Efficiency and wages. A is 50% more efficient than B. Together they complete a job in 18 days and are paid ₹60,000 in total. Find B's time alone and each one's wage.

• Rates A : B = 150 : 100 = 3 : 2. Combined 5 parts ⇒ 18 days, so A alone (3 parts) takes 18 × 5/3 = 30 days, B alone 18 × 5/2 = 45 days.
• Both worked all 18 days, so wages split in the rate ratio 3 : 2 → A = ₹36,000, B = ₹24,000.

Example 4 — Chain rule with hours and a deadline. 15 workers working 8 hours a day build a wall in 12 days. The contractor must finish an identical wall in 9 days working 10 hours a day. How many workers are needed?

• M₁D₁H₁ = 15 × 12 × 8 = 1,440 worker-hours.
• M₂ = 1,440 / (9 × 10) = 16 workers (one more than before).

Example 5 — Alternate days with a spoiler. A builds a wall in 10 days; B (careless) demolishes it in 15 days. A works on day 1, B on day 2, alternately, A starting. When is the wall complete?

• LCM(10, 15) = 30 units. A = +3 units/day, B = −2 units/day. One 2-day cycle = +1 unit.
• After 26 days (13 cycles) = 13 units. Day 27 (A): 13 + 3 = 16. Day 28 (B): 16 − 2 = 14 … the net gain is only 1 unit per cycle.
• Work needed before A's final day = 30 − 3 = 27 units. 27 units take 27 cycles = 54 days; day 55 is A's turn and adds 3 → 55 days.

Where it is used

• IT services effort estimation (TCS, Infosys, Wipro, HCL): person-months, resource loading sheets and billing are the man-days model, scaled to project size.
• Construction and infrastructure (L&T, NCC, state PWD/CPWD): labour schedules and schedule-of-rates estimates quote man-days per unit of masonry, plaster or concrete.
• Manufacturing and operations (Maruti, Tata Steel, Bajaj): line balancing, shift planning and takt-time calculations depend on output per worker per hour.
• Workforce management in BPO and banking: Erlang/capacity models begin with "calls handled per agent per hour" — the same rate addition.
• Agile delivery roles: sprint velocity, burndown charts and "days remaining = work left ÷ team rate" are this topic in project-management vocabulary.
• Placement and competitive exams: 1-3 questions in most IT aptitude rounds, and a regular item in IBPS, SBI PO, SSC CGL, RRB and CAT quantitative sections.

Interview questions

• Q: Why can you not simply add the times taken by two workers? A: Time is inversely proportional to work rate; only rates (work per day) are additive, so you add 1/a + 1/b and invert.
• Q: A does a job in 12 days, B in 24 days — how long together? A: Rates 1/12 + 1/24 = 1/8, so 8 days.
• Q: If A is twice as efficient as B and together they take 10 days, how long does B take alone? A: Rates 2 : 1, combined 3 parts = 10 days, so B (1 part) takes 30 days and A takes 15 days.
• Q: What does "15 men can do a job in 20 days" tell you in man-days? A: The job is 300 man-days of effort, so any combination of men and days multiplying to 300 should finish it under the ideal model.
• Q: Why is the LCM method preferred over fractions? A: It replaces fraction addition with integer units, which is faster and far less error-prone under exam time pressure.
• Q: How are wages divided when two people work different numbers of days? A: In the ratio of work actually done — rate multiplied by days worked — not in the ratio of days alone.
• Q: How do you handle a worker who destroys work? A: Subtract that rate; if the net rate is zero or negative the job never finishes.
• Q: In an alternate-day problem, why can the answer be a fraction like 11.5 days? A: Because the last worker may need only part of a day to complete the remaining work, and that is counted as a fraction unless whole days are stipulated.
• Q: Doubling the team should halve the time — when is that false in practice? A: When the task is not divisible, or coordination and training overheads grow — Brooks's law, where adding people to a late project delays it further.
• Q: A and B together take 6 days, A alone 9 days. Find B. A: 1/6 − 1/9 = 1/18, so B alone takes 18 days.
• Q: 10 men ≡ 15 women on the same job — what is one man worth? A: 1.5 women, since their total outputs are equal; convert everything to one unit before combining mixed groups.
• Q: Why must the combined time always be less than the fastest individual time? A: The combined rate exceeds every individual rate, and time is its reciprocal — a useful instant sanity check on an answer.

Common mistakes

• Adding times instead of rates ("10 and 15 days ⇒ 25 days together") — the classic blunder.
• Forgetting to invert at the end: computing the combined rate 1/6 and reporting "1/6 days".
• Rounding a partial final day up to a whole day in alternate-day problems when the question asks for exact time.
• Mis-reading "A is 25% more efficient than B" as a time ratio of 125 : 100 instead of 100 : 125.
• Splitting wages in the ratio of days worked rather than work done.
• Dropping hours per day from the chain rule when the two situations have different working hours.
• Treating the man-days formula as exact in interviews about real projects, without mentioning indivisible tasks and coordination overhead.

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