Time and Work

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What: Time and Work covers individual and combined work rates, pipes filling/emptying a cistern, men-days-work proportionality, and efficiency-based comparisons.
Why it matters: CDS papers from 2000 to 2023 average 3–5 questions per sitting — second-highest weightage in arithmetic word problems after TSD.
Key fact: If A finishes a job in $a$ days and B in $b$ days, together they finish in $\tfrac{ab}{a+b}$ days — the harmonic-mean style formula that solves dozens of CDS variants in one line.

Time and Work is the chapter where setting up the equation correctly wins the marks. The arithmetic is light; the trap is in the units (days vs hours, men vs man-days, work as a fraction). Master the "1/n per day" reflex and the chapter resolves itself.

This page is built from CDS Previous Year Questions across 2000–2023. Pair with Time, Speed and Distance (same rate-based reasoning) and Ratio and Proportion (efficiency ratios).

What This Topic Covers

CDS scope: (1) individual work rate — $1/n$ per day; (2) combined work — sum of rates; (3) the $ab/(a+b)$ formula; (4) pipes and cisterns — inflow positive, outflow negative; (5) men-days-work — direct proportion with men and days, inverse with required time; (6) efficiency comparison — "A is twice as efficient as B" type; and (7) work-with-rest — partial completion with one worker absent.

Why This Topic Matters

3–5 CDS questions per paper, all word problems.
The combined-rate identity is a one-line reflex used in nearly every question.
Pipes-and-cisterns is just signed work-rate — outflow is negative inflow.

Exam Pattern & Weightage

Year / Paper	No.	Subtopics Tested
2007-II	3	Combined work, pipes
2009-II	3	Men-days, efficiency
2010-I/II	4	Combined rate, pipes filling/emptying
2011-II	3	Men-days-work, partial completion
2014-I/II	3	A and B with one absent
2016-I/II	3	Three-worker combined, efficiency
2017-I/II	4	Pipes, men-days variants
2018-I/II	3	Combined work, efficiency
2019-II	3	A+B then A leaves, B completes
2020-I/II	3	Three-worker, pipes
2021-I/II	3	Mixed
2022-I / 2023-I	3	Mixed

⚡ CDS Alert

Pipe $A$ fills a tank in $a$ hours; pipe $B$ empties it in $b$ hours. Together they fill the tank in $\tfrac{ab}{b-a}$ hours (assuming $b > a$, so net filling). The minus sign is the trap — opposite to combined-work formula.

Core Concepts

Individual Work Rate

If A finishes a job in $n$ days, his rate is $1/n$ per day. Total work is conventionally set to 1 (or to LCM of given days for cleaner numbers).

Rate from Time $$\text{rate} = \frac{1}{\text{time}}, \qquad \text{time} = \frac{1}{\text{rate}}$$

Combined Work

Two Workers Together $$\frac{1}{T} = \frac{1}{a} + \frac{1}{b} \implies T = \frac{ab}{a + b}$$

Three Workers Together $$\frac{1}{T} = \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \implies T = \frac{abc}{bc + ac + ab}$$

Pipes and Cisterns

A filling pipe has positive rate (work added per unit time); an emptying pipe has negative rate. Sum the signed rates as usual.

One Fills, One Empties Pipe A fills in $a$ hours, pipe B empties in $b$ hours, both open.
Net fill time $= \tfrac{ab}{b - a}$ (only if $b > a$, else tank empties).

Men-Days-Work

Proportionality $$\frac{M_1 \cdot D_1 \cdot H_1}{W_1} = \frac{M_2 \cdot D_2 \cdot H_2}{W_2}$$

$M$ = men, $D$ = days, $H$ = hours per day, $W$ = work done. Used for proportional men-days problems: "If 10 men do a job in 12 days, how long for 15 men?"

Efficiency Comparisons

"A is twice as efficient as B" ⇒ A's rate = 2 × B's rate ⇒ A takes half the time. "A is $p\%$ more efficient than B" ⇒ A's rate = $(1 + p/100) \times$ B's rate.

⚠ Common Trap

Efficiency is inverse of time. So "A is 25% more efficient than B" means $t_A = t_B / 1.25 = 0.8 t_B$, not $0.75 t_B$. CDS plants this every other paper.

Worked Examples

Example 1 — Combined Work (2007-II)

Q: A can finish a job in 12 days, B in 18 days. How long together?

Apply $T = ab/(a+b) = 12 \cdot 18 / 30 = 216/30 = 7.2$ days.

Example 2 — Pipes Both Open (2010-II)

Q: Pipe A fills a tank in 6 hours, pipe B fills in 4 hours. Both open together — how long?

$T = 6 \cdot 4 / (6 + 4) = 24/10 = 2.4$ hours.

Example 3 — Pipe Fills, Leak Empties (2017-I)

Q: A tap fills a tank in 10 hours. A leak at the bottom can empty it in 15 hours. With both active, how long to fill?

Net fill time $= ab/(b - a) = 10 \cdot 15 / (15 - 10) = 150/5 = 30$ hours.

Example 4 — Men-Days Proportion (2009-II)

Q: 18 men can finish a job in 24 days. How many men are needed to finish in 12 days?

Apply $M_1 D_1 = M_2 D_2$: $18 \cdot 24 = M_2 \cdot 12 \implies M_2 = 36$ men.

Example 5 — A and B with A Leaving (2019-II)

Q: A and B together can complete a work in 10 days. A alone takes 15 days. They work together for 4 days, then A leaves. How long does B take to finish?

B's rate: $1/T_B = 1/10 - 1/15 = 1/30 \implies T_B = 30$ days.
In 4 days together, they complete $4/10 = 2/5$ of the work. Remaining = $3/5$.
B alone takes $(3/5) \cdot 30 = 18$ days.

Example 6 — Three Workers (2016-II)

Q: A, B, C can each complete a job in 6, 12, and 18 days. How long together?

Combined rate = $1/6 + 1/12 + 1/18$. LCM = 36. Rate = $6/36 + 3/36 + 2/36 = 11/36$.
$T = 36/11 \approx 3.27$ days.

Example 7 — Efficiency Comparison (2018-I)

Q: A is 25% more efficient than B. Together they finish a job in 18 days. How long does A alone take?

Let B's rate = 1. Then A's rate = 1.25. Combined rate = 2.25.
Their combined time = 18 days, so total work = $2.25 \cdot 18 = 40.5$ "units".
A alone: $40.5 / 1.25 = 32.4$ days.

How CDS Tests This Topic

Six archetypes: (1) combined work $ab/(a+b)$, (2) pipes (both filling, or fill+leak), (3) A and B together, then one leaves, (4) men-days-hours proportionality, (5) efficiency given in % terms, (6) three workers combined.

Exam Shortcuts (Pro-Tips)

Shortcut 1 — LCM Method

Set total work = LCM of given times. Each worker's rate becomes a clean integer per day. Skips fraction arithmetic. Example: A in 12, B in 18, C in 24 days. LCM = 72. Rates: 6, 4, 3 per day. Combined: 13 per day. $T = 72/13$ days.

Shortcut 2 — $ab/(a+b)$ Reflex

For two workers, the combined time is always $ab/(a+b)$. Apply mentally for any two given times. For three, use $abc/(bc + ca + ab)$.

Shortcut 3 — Pipe Sign Trick

Filling pipes add their rates (positive). Emptying pipes subtract (negative). Sum and take reciprocal for the net time.

Shortcut 4 — Efficiency to Time Conversion

If A is $p\%$ more efficient: $t_A = t_B / (1 + p/100)$. If A is $p\%$ less efficient: $t_A = t_B / (1 - p/100)$. Always inverse.

Shortcut 5 — Work-with-Rest

"A and B work for $t_0$ days, then A leaves. B finishes." Compute fraction done by combined effort, then divide remaining by B's rate alone.

Common Question Patterns

Pattern 1 — Combined Work

Two or three workers. Compute combined rate, take reciprocal.

Pattern 2 — Pipes Filling / Emptying

Sum signed rates. Filling positive, emptying negative.

Pattern 3 — Partial Work + Departure

Combined for some days, then one leaves. Compute fraction done; remaining fraction done by survivor.

Pattern 4 — Men-Days Proportionality

$M_1 D_1 H_1 = M_2 D_2 H_2$ (when work is same). Solve for unknown.

Pattern 5 — Efficiency-Based

"A is $p\%$ more efficient than B." Convert to rate ratio, then to time.

Preparation Strategy

Week 1. Master $ab/(a+b)$ reflex. Drill 20 combined-work problems. Practice the LCM method on multi-worker problems for cleaner arithmetic.

Week 2. Pipes and cisterns (signed rates), men-days-hours proportionality, and efficiency comparisons. Practice partial-work problems where one worker leaves mid-way.

Mock testing. Take timed CDS papers. Common slip-points: sign error in pipe problems, efficiency inverse confusion, and incorrect baseline in partial-work problems. Use CDS mock tests to drill speed.

Drill Time and Work at Speed

CDS mocks with combined work, pipes, efficiency, and partial-work problems. Six archetypes — six formulas — reflex.

Start Free Mock Test

Frequently Asked Questions

If A takes $a$ days and B takes $b$ days, how long together?

$T = ab/(a + b)$ days. Example: A in 6, B in 12 → together in $72/18 = 4$ days. This is the most-used formula in the chapter.

What is the LCM method?

Set total work = LCM of the given times. Each worker's rate becomes a clean integer "units per day". Skips fractions. Example: A in 6, B in 8, C in 12 days. LCM = 24. Rates: 4, 3, 2 units/day. Combined = 9 units/day. $T = 24/9 = 8/3$ days.

How do pipe-and-cistern problems work?

Each pipe has a rate. Filling pipes contribute positively; emptying pipes contribute negatively. Sum the signed rates; take the reciprocal for net time. If the sum is negative, the tank empties; if positive, it fills.

What if A and B work together then A leaves?

Compute the fraction of work done while both worked: $t_{\text{joint}} \cdot (\text{combined rate})$. Subtract from 1 to find the remaining fraction. Divide remaining by B's rate alone to find the additional time B takes.

If A is 25% more efficient than B, how do their times compare?

Efficiency is inverse of time. A's rate = 1.25 × B's rate, so A takes $t_B / 1.25 = 0.8 t_B$ — 20% less time, not 25%. The percentage decrease in time is always less than the percentage increase in efficiency.

How does men-days proportionality work?

$M_1 D_1 H_1 = M_2 D_2 H_2$ when the same total work is done. Holding hours constant: doubling men halves days, and vice versa. Solve algebraically for the unknown variable.

Which CDS Maths topics connect to Time and Work?

Time, Speed and Distance — same rate-time-quantity structure. Ratio and Proportion — efficiency comparisons. Average — average rate when multiple workers contribute.