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Time & Work

Chapter Outline & Learning Goals

By the end of this chapter, you will be able to:

1. Convert "days to finish" into "work per day" (the single most important move in this chapter).
2. Solve problems of two or more people working together or alternately.
3. Handle efficiency ratios, wages distribution, and "leaving/joining midway" problems.
4. Crack pipes & cisterns — which is just Time & Work wearing a disguise.
5. Spot the traps examiners set in SSC CGL, IBPS PO and RRB NTPC questions.

The Hook: The Examiner's Favourite ATM

A fact that should wake you up: Time & Work is one of the most reliably asked topics in SSC CGL, CHSL, RRB NTPC and IBPS papers — and its questions are almost always solvable in under 30 seconds if you know one idea. Just one.

That idea is this: never think in days, think in "work per day".

Students who think in days get stuck in fractions like 1/6 + 1/8 and panic. Students who think in "units of work per day" finish the same question mentally. This chapter will turn you into the second kind of student.

The Struggle is Real (And It's Not Your Fault)

If Time & Work has ever made you feel slow, here's why: school taught you the formula *Work = Rate × Time*, but exams ask twisted versions — "A works for 3 days and leaves, then B finishes it..." Suddenly the neat formula feels useless.

The truth? Every single twisted question is built from the same three building blocks. Master the blocks, and the twists become free marks.

Building Block 1: Work Per Day (The Unit Method)

If A can finish a job in 10 days, then in 1 day A finishes 1/10 of the job.

That's it. That is the entire foundation.

• A finishes in 10 days → A's 1-day work = 1/10
• B finishes in 15 days → B's 1-day work = 1/15
• Together in 1 day = 1/10 + 1/15 = 5/30 = 1/6 → so together they need 6 days.

The LCM shortcut (use this in exams): Instead of fractions, assume the total work = LCM of the days.

Example: A in 10 days, B in 15 days. LCM(10, 15) = 30 units of work.

• A does 30/10 = 3 units/day
• B does 30/15 = 2 units/day
• Together = 5 units/day → Time = 30/5 = 6 days. No fractions, no fear.

🧠 Active Recall: A can do a job in 12 days, B in 18 days. Using LCM = 36, how many units does each do per day, and how long do they take together? (Answer: A = 3, B = 2, together = 36/5 = 7.2 days.)

Building Block 2: Efficiency and Ratios

Sometimes the exam doesn't give days — it gives a comparison: "A is twice as efficient as B."

Efficiency and time are inversely related. If A is twice as efficient as B, A takes half the time.

• Efficiency ratio A : B = 2 : 1 → Time ratio A : B = 1 : 2

Memory trick (weird but sticky): *"Fast person, small clock."* The faster worker's time number is always the smaller one. If you ever get confused about which ratio flips, picture a lazy tortoise with a giant clock and a speedy rabbit with a tiny watch. Efficiency flips time.

Example: A is 50% more efficient than B. B alone takes 30 days. Find A's time.

• Efficiency A : B = 3 : 2 → Time A : B = 2 : 3 → A takes (2/3) × 30 = 20 days.

🧠 Active Recall: A is thrice as efficient as B, and B takes 24 days. How long does A take? How long together? (Answer: A = 8 days; together: LCM 24 → A = 3, B = 1 unit/day → 24/4 = 6 days.)

Building Block 3: Leaving, Joining, and Alternating Days

This is where exams earn their money. The method never changes: track the work done, chunk by chunk.

Type 1 — Someone leaves midway:
A can finish in 10 days, B in 15 days. Both start, but A leaves after 2 days. When does the work end?

• Total work = 30 units. A = 3/day, B = 2/day.
• First 2 days together: 5 × 2 = 10 units done. Remaining = 20.
• B alone: 20/2 = 10 more days. Total = 2 + 10 = 12 days.

Type 2 — Alternate days:
A (10 days) and B (15 days) work on alternate days, A starting.

• Every 2 days: 3 + 2 = 5 units done. 30 units ÷ 5 = 6 cycles → 12 days.
• (If it doesn't divide evenly, finish the last partial day with a fraction — exams love this.)

🧠 Active Recall: In the alternate-days example above, what if the total work were 28 units instead of 30 (A still does 3 units/day, B still 2 units/day)? (Answer: 5 full cycles = 25 units in 10 days; day 11 is A's turn and only 3 units remain, which A completes in exactly 1 day → 11 days.)

Pipes & Cisterns: Time & Work in Disguise

A pipe filling a tank is a worker. A pipe emptying a tank is a worker with negative efficiency. That's the whole chapter.

Example: Pipe A fills a tank in 6 hours, pipe B empties it in 12 hours. Both open — when is the tank full?

• LCM(6, 12) = 12 units. A = +2/hour, B = −1/hour. Net = +1/hour → 12 hours.

Trap alert: If the net rate comes out negative, the tank will *never* fill. Examiners include this option to catch students who calculate blindly.

🧠 Active Recall: A fills in 4 hours, B fills in 6 hours, C empties in 12 hours. All open together — time to fill? (Answer: LCM 12 → 3 + 2 − 1 = 4/hour → 3 hours.)

Wages: Work Done = Money Earned

When people work together and get paid, the payment splits in the ratio of work done (efficiency × days worked), not just days present.

Example: A and B finish a job for ₹900. A alone could do it in 6 days, B in 9 days; they work together throughout.

• LCM 18 → A = 3/day, B = 2/day. Ratio of work = 3 : 2.
• A gets 3/5 × 900 = ₹540, B gets ₹360.

The Examiner's Three Favourite Traps

1. "A and B together take X days, A alone takes Y days" — find B by subtraction, not guessing: B's rate = combined rate − A's rate.
2. Man-days problems: "10 men finish in 12 days" → total work = 120 man-days. If men change, divide 120 by the new number. (Men × Days stays constant when work is fixed.)
3. Fractional last day in alternate-day questions — always check whether the work ends mid-cycle.

Summary Table

📌 Review this chapter after 2 days. Close the book and try to re-derive the summary table from memory — especially the LCM unit method and the efficiency-time flip. If you can rebuild the table, you own this chapter.