Time-Work

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Core Formula: $$\text{Work} = \text{Rate} \times \text{Time}$$ or equivalently: $$\text{Time} = \frac{\text{Work}}{\text{Rate}}$$

If A can complete a job in $n$ days, A’s work rate = $\frac{1}{n}$ job per day.

Addition of Rates:

• If A and B work together: Combined rate = $\frac{1}{n_A} + \frac{1}{n_B}$
• Time taken together: $T = \frac{1}{\frac{1}{n_A} + \frac{1}{n_B}} = \frac{n_A \times n_B}{n_A + n_B}$

Key Principles:

• Work is directly proportional to time (for constant rate)
• Work is additive: If A does $\frac{1}{3}$ of job and B does $\frac{1}{4}$, together they’ve done $\frac{1}{3} + \frac{1}{4} = \frac{7}{12}$
• If rates change, calculate work done in each phase separately

⚡ CAT Tip: Convert everything to rates (jobs per day) before solving. This avoids confusion with “part of work” calculations. Also, use the “man-days” concept: if $M$ men can do $W$ work in $D$ days, then $M \times D \propto W$.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding.

Standard Work Problem Types

Type 1: Two Workers If A can do a job in $a$ days and B can do it in $b$ days:

• Working together: $\frac{1}{a} + \frac{1}{b} = \frac{a+b}{ab}$ jobs per day
• Time taken: $\frac{ab}{a+b}$ days

Type 2: Three or More Workers For A, B, C with times $a, b, c$:

• Time together: $\frac{abc}{ab + bc + ca}$ days

Type 3: Work with Rest Periods If A works for 4 days, rests for 1 day, and completes work in 16 days total, calculate actual productive days.

Type 4: Successive Jobs If A and B do a job, then A alone does another job: calculate each phase separately and add times.

Efficiency Concept If men work at the same rate, “efficiency” is proportional to work rate. If A is $x%$ more efficient than B:

• If B takes $n$ days, A takes $\frac{n}{1 + x/100}$ days
• Ratio of times = $100:(100+x)$

Example Problem: If 12 men can complete a work in 18 days, and after 6 days, 4 men leave, how many more days needed?

• Work done in 6 days = $12 \times 6 = 72$ man-days
• Total work = $12 \times 18 = 216$ man-days
• Remaining work = $216 - 72 = 144$ man-days
• 8 men at work: days needed = $144/8 = 18$ more days

⚡ Common Mistake: Students confuse “number of workers” with “work rate.” Always convert to a common unit (man-days or job per day) before calculating.

Type 5: Pipes and Cisterns This is work with inflow/outflow:

• Pipe filling = positive work
• Pipe emptying = negative work
• Net rate = sum of individual rates
• Formula: Same as workers, just with pipes instead

If pipe A fills in $a$ minutes, B fills in $b$ minutes, and drain C empties in $c$ minutes: Net part filled in 1 minute = $\frac{1}{a} + \frac{1}{b} - \frac{1}{c}$

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for serious exam preparation.

Advanced Work-Time Techniques

LCM Method for Multiple Workers Instead of fractions, use LCM of all time periods as total work units:

1. Find LCM of all individual times → this becomes total work
2. Calculate each person’s work per unit time
3. Add/subtract to find combined rate
4. Calculate total time

Example: A in 6 days, B in 8 days, C in 12 days

• LCM(6, 8, 12) = 24 units of work
• A’s rate = 24/6 = 4 units/day
• B’s rate = 24/8 = 3 units/day
• C’s rate = 24/12 = 2 units/day
• Combined = 9 units/day
• Time = 24/9 = 8/3 days

Negative Work (Outflow Problems) When a pipe drains while others fill:

• Net rate = Sum of filling rates - Sum of draining rates
• Critical point: If net rate = 0, tank never fills (or drains completely)
• If drain rate > fill rate initially, tank empties

Circumstantial Variations

1. Variable Working Hours: If work time changes during project, convert to effective working days
2. Partial Workers Leaving: Calculate work done, then recalculate remaining rate
3. New Workers Joining: Add their contribution to remaining work
4. Efficiency Changes: If productivity changes (say, due to fatigue), use weighted average rates

Work Distribution Problems If work is distributed in ratio $x:y:z$ and times are $a, b, c$:

• Work done per day by each = $1/a, 1/b, 1/c$
• Since $W_x:W_y:W_z = x:y:z$, we have $\frac{W_x}{1/a} : \frac{W_y}{1/b} : \frac{W_z}{1/c} = x:y:z$
• Simplify to find actual work amounts

Chain Rule Application For related work problems (where work of one group depends on another’s output):

• If A’s work in $x$ days = B’s work in $y$ days, then A:B = $y:x$
• Use this ratio to solve complex interdependencies

CAT 2023 Question Pattern Time-Work typically appears as:

• 1-2 questions in QA section
• Often combined with efficiency or wages
• Sometimes in DILR as scheduling problems

Problem-Solving Framework

Step 1: Convert all given information to rates (jobs per day) Step 2: Identify if the problem involves positive work, negative work, or both Step 3: Calculate combined rate if multiple workers involved Step 4: Use $T = W/R$ to find time, or rearrange for other quantities Step 5: Check for special conditions (leaving, joining, rest days)

⚡ Advanced Strategy: For “who will finish first” type questions, compare their daily output directly. For “work alternation” problems (A works 2 days, B works 3 days, etc.), calculate how much work gets done in each complete cycle, then find remaining work.

Wages Problems Work in ratio $x:y$ means wages in ratio $x:y$. Calculate total work, find each person’s share, then divide wages proportionally.

Example: A and B together complete work in $d$ days. A leaves after $x$ days. B finishes remaining in $y$ days. Find A and B’s rates:

• Total work = A’s rate $\times d$ + B’s rate $\times d$
• A’s contribution = A’s rate $\times x$
• B completes: B’s rate $\times y$
• Solve two equations for two unknowns

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📊 CAT Exam Essentials

Detail	Value
Sections	VARC (24 Qs), DILR (20 Qs), QA (22 Qs)
Time	2 hours (40 min per section)
Total	66 questions, 198 marks
Marking	+3 correct, −1 wrong (MCQ); no penalty for TITA
Mode	Computer-based, multiple sessions
Percentile	Normalized — 99+ needed for top IIMs

🎯 High-Yield Topics for CAT

• Reading Comprehension — 16-20 marks in VARC
• Para Summary + Odd Sentence — 8-12 marks
• DI Sets (Tables + Caselets) — 10-15 marks in DILR
• Arithmetic (Percentages + Profit/Loss) — 8-12 marks in QA
• Geometry + Mensuration — 6-10 marks
• Logarithm + Sequences — 6-10 marks

📝 Previous Year Question Patterns

• Q: “The passage is primarily concerned with…” [2024 VARC — RC passage]
• Q: “If f(x) = x² - 5x + 6, the value of f(3) is…” [2024 QA — Arithmetic]
• Q: “How many ways can 5 people be arranged around a round table…” [2024 DILR — Circular]

💡 Pro Tips

• VARC is the top priority — strong RC skills can push you to 99+ percentile quickly
• DILR: attempt 2 full sets out of 4-5 sets — accuracy matters more than coverage
• QA: arithmetic (time-speed-work) + geometry carry ~40% of QA marks
• Take 3-4 full mocks before the exam to find your section-wise pacing

🔗 Official Resources

• IIM CAT Official
• [CAT Syllabus](https://iimcat.ac.in/exam pattern)

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