Section 11, arithmetic

Time & Work (with Pipes & Cisterns)

Turn every time into a rate with the LCM trick and most work questions become simple addition. CAT’s versions add people joining or leaving midway, efficiency comparisons, leaks and wages.

How CAT asks it

Workers with different speeds finish a job in phases; pipes fill and leaks empty a tank opened at different times; “A is 50% more efficient than B”; men–women–children equivalence; wages split by work done; workers who alternate days or drop out.

Concepts you must know

If A finishes a job in a days, A’s rate is 1/a of the job per day. Rates add when people work together.
LCM trick: total work = LCM of the individual times, so rates become whole units per day. A in 12 days and B in 18 days: work = 36 units, A = 3 and B = 2 per day, together 5, so 36/5 = 7.2 days.
Efficiency is the inverse of time: if A is twice as efficient as B, A takes half the time.
Man-days: men × days × hours per unit of work stays constant.
Pipes: inlets add, outlets and leaks subtract.
Wages are proportional to work done, not days present.
Alternate days: find one cycle’s work, count the full cycles, then finish the leftover.
The man-days relation:

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

Thought process

Set total work = LCM of the given times and write each worker’s rate in units per day.
Split the timeline into phases, each with a fixed set of workers.
The work in all phases adds up to the total; solve for the unknown time.
If someone leaves “x days before completion”, work backwards from the end.

Patterns CAT repeats

Numbered so your error log can point at them: “Pattern 2”.

1Combined work, then someone leaves or joins.
2Pair sums: A + B in x days, B + C in y, C + A in z. Adding all three gives twice the total rate.
3Efficiency statements: “50% more efficient” means rates in the ratio 3 : 2.
4Man-days and equivalences between men, women and children.
5Pipes and leaks, including pipes opened or closed at different times.
6Alternate-day cycles.
7Wages split by work done.
8A changing workforce, such as one worker leaving each day (an AP sum).

Worked examples

Attempt each one before opening its solution, then compare methods.

Example 1, LCM + leaving
A can finish a job in 12 days and B in 15. They work together for 4 days, then A leaves. How long does B take to finish?
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Work = 60 units; A = 5 and B = 4 per day. Four days together = 36 units, leaving 24 for B at 4 a day. Answer: 6 more days (10 in total).
Example 2, pair sums
A + B finish in 10 days, B + C in 12 and C + A in 15. How long would A take alone?
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Work = 60. A + B = 6, B + C = 5, C + A = 4, so 2(A + B + C) = 15 and A + B + C = 7.5. A = 7.5 − 5 = 2.5 a day. Answer: 24 days.
Example 3, efficiency
A is 50% more efficient than B. Together they finish in 18 days. How long would A take alone?
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A : B = 3 : 2, so 5 units a day for 18 days = 90 units. A alone: 90 ÷ 3. Answer: 30 days.
Example 4, man-days
12 men working 8 hours a day build a wall in 10 days. How many days would 16 men working 6 hours a day take to build a wall twice as long?
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12 × 8 × 10 ÷ 1 = 16 × 6 × D ÷ 2, so 960 = 48D. Answer: 20 days.
Example 5, leak
A pipe fills a tank in 6 hours; with a leak it takes 8 hours. How long would the leak alone take to empty a full tank?
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Work = 24. The pipe adds 4 an hour and the net is 3, so the leak removes 1 an hour. Answer: 24 hours.
Example 6, pipe closed midway
Pipe P fills a tank in 20 minutes and Q in 30. Both are opened, Q is closed after a while, and the tank fills in 15 minutes in total. When was Q closed?
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Work = 60; P = 3 and Q = 2 per minute. P alone does 45 in 15 minutes, so Q contributed 15 at 2 a minute. Answer: after 7.5 minutes.
Example 7, alternate days
A finishes a job in 10 days and B in 15. They work on alternate days, A first. How many days does the job take?
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Work = 30; A = 3, B = 2, so each 2-day cycle does 5. After 10 days, 25 is done. Day 11 (A) reaches 28 and day 12 (B) reaches 30. Answer: 12 days.
Example 8, wages
A, B and C can finish a job in 6, 8 and 12 days. Working together, they are paid ₹3,600. Find C’s share.
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Work = 24; rates 4, 3 and 2 (total 9). C’s share = 2/9 × 3,600. Answer: ₹800.
Example 9, shrinking team
10 workers start a 45-unit job, each doing 1 unit a day, and one worker leaves at the end of each day. How many days does the job take?
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10 + 9 + 8 + 7 + 6 + 5 = 45. Answer: 6 days.

Traps

Treating a leak as positive work.
Splitting wages by days present instead of work done.
“Leaves 3 days before completion”: the leaver worked (total − 3) days. Work backwards from the end.

Practice set

Timed, about 2.5 minutes per question. Log every miss in the error log.

1
A can do a job in 20 days and B in 30. How long together?
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12 days: 60 units at 3 + 2 a day.
2
A and B together take 12 days, and A alone takes 20. How long does B alone take?
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30 days: 60 units, A + B = 5 and A = 3, so B = 2.
3
Two pipes fill a tank in 10 and 15 hours; a third empties it in 20 hours. With all three open, how long does it take to fill?
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8 4/7 hours: 60 units at 6 + 4 − 3 = 7 an hour.
4
6 men or 10 women can finish a job in 15 days. How long will 3 men and 5 women take?
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15 days: 3 men = 5 women, so the team equals 10 women.
5
A works twice as fast as B, and together they take 12 days. How long does B alone take?
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36 days: rates 2 : 1 and 36 units in total.
6
A + B take 8 days, B + C take 12 and A + B + C take 6. How long do A and C take together?
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8 days: 24 units; A + B + C = 4 and B + C = 2 give A = 2; A + B = 3 gives B = 1, so C = 1 and A + C = 3.

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