Time and work has a reputation for fractions, and it does not deserve it. There is one move that removes every fraction from the chapter: instead of saying “A does 1/12 of the job a day”, call the whole job 60 units and say “A does 5 units a day”. From that point on it is addition. All 23 questions the Punjab Police SI has asked on work and pipes since 2016 fall into four shapes. Combined work: add the unit rates. Pipes: a leak is a negative pipe, so subtract it. Men-days-hours: M₁D₁H₁ = M₂D₂H₂, one equation for every variation. Wages: pay follows the work done, which is the rate ratio.
23
SI questions
2016–2023
14
on time and work
combined and M-D-H
9
on pipes and cisterns
fill, empty, alternate
0
fractions needed
if you use LCM units
The LCM units method
Learn this once and the chapter is arithmetic.
KEY TERM
Job = LCM of the day counts
If A takes 12 days and B takes 15, let the job be LCM(12, 15) = 60 units. A does 5 units a day, B does 4, together 9, and the job takes 60/9 = 6⅔ days. The same table answers every variation: how much is left after three days, who joins when, and what each is paid. Rates add; times do not.
12 days, 15 days→LCM = 60 units→5 + 4 = 9 a day
QUICK CHECK
Arun and Monika can complete a work in 6 days when they work together. If Arun works alone, he can complete the work in 10 days.In how many days can Monika alone complete the work? (SI 29 Jun 2023)
Tap an option to answer
Answer: 15
Let the job be LCM(6, 10) = 30 units. Together they do 30 ÷ 6 = 5 units a day and Arun alone does 30 ÷ 10 = 3 units a day. So Monika does 5 − 3 = 2 units a day and needs 30 ÷ 2 = 15 days.
In fractions this is 1/6 − 1/10 = 1/15 — the same answer, but you had to find a common denominator to get there.
QUICK CHECK
S can complete a job in 12 days and U, in 18 days. S, U and K, working together, can complete the job in 6 days. In what time can K alone complete the job? (SI 29 Jul 2023)
Tap an option to answer
Answer: 36 days
Job = LCM(12, 18, 6) = 36 units. S does 3 a day, U does 2, and all three together do 36 ÷ 6 = 6 a day. So K does 6 − 3 − 2 = 1 unit a day and alone needs 36 days.
Whenever a question gives you the group rate and all but one member, subtract.
QUICK CHECK
A and B together can do a piece of work in 8 days. B and C together can do the same work in 12 days. A, B and C together can do the same work in 6 days. In how many days can B alone do that work? (SI 19 Aug 2021)
Tap an option to answer
Answer: 24 days
Job = LCM(8, 12, 6) = 24 units. A + B = 3 a day, B + C = 2 a day, and A + B + C = 4 a day.
C = (A + B + C) − (A + B) = 4 − 3 = 1 unit a day. Then B = (B + C) − C = 2 − 1 = 1 unit a day, so B alone needs 24 ÷ 1 = 24 days.
Subtract the pair rates from the group rate one at a time — never try to solve three simultaneous fractions.
QUICK CHECK
A and B can do a piece of work in 9 days and 12 days respectively. They work together for 3 days and then A leaves. In how many days after that will B complete the work alone? (SI 16 Oct 2022)
Tap an option to answer
Answer: 5 days
Job = LCM(9, 12) = 36 units. A does 4 a day and B does 3, so together 7 a day. In 3 days they finish 21 units, leaving 15. B alone clears 15 ÷ 3 = 5 days.
Read the last line carefully: it asks how long B takes after A leaves, not the total.
QUICK CHECK
A, B and C can complete a piece of work in 12, 15 and 10 days respectively. They work together for 2 days and then A leaves the work. B and C carry on working. In how many days will the work be completed? (SI 19 Aug 2021)
Tap an option to answer
Answer: 5 days
Job = LCM(12, 15, 10) = 60 units. A does 5 a day, B does 4 and C does 6, so all three do 15 a day. In 2 days they complete 30 units, leaving 30. B and C together do 10 a day, so they need 3 more days. The work is completed in 2 + 3 = 5 days.
This question asks for the total time, unlike the previous one — the two phrasings sit side by side in SI papers on purpose.
QUICK CHECK
X can cultivate 7/10 part of the land in 15 days. After that, he completes the remaining work in 4 days with the help of Y. In how many days will X and Y together do the same work? (SI 31 Jul 2023)
Tap an option to answer
Answer: 13 1/3 days
X does 7/10 of the land in 15 days, so X's daily rate is 7/150 of the land. The remaining 3/10 takes X and Y together 4 days, so their combined rate is (3/10) ÷ 4 = 3/40 of the land a day.
Working together on the whole job therefore takes 1 ÷ (3/40) = 40/3 = 13⅓ days. Option B (3/40) is the rate itself, offered as a trap for anyone who forgets the final reciprocal.
Pipes and cisterns
The same method, with one sign flipped.
QUICK CHECK
If pipe A can fill a cistern in 10 min, pipe B can fill the same cistern in 15 min, and pipe C can empty the full cistern in 18 min, then find the time taken to completely fill the cistern, if pipes A, B and C are opened together. (SI 9 Jul 2023)
Tap an option to answer
Answer: 9 min
Tank = LCM(10, 15, 18) = 90 units. A fills 9 a minute, B fills 6, and C empties 5. The net rate is 9 + 6 − 5 = 10 units a minute, so the tank fills in 90 ÷ 10 = 9 minutes.
QUICK CHECK
Pipe A can fill a tank in 5 hours, Pipe B in 10 hours and Pipe C in 30 hours. If all the pipes are open, in how many hours will the tank be filled ? (SI 2016)
Tap an option to answer
Answer: 3
Tank = LCM(5, 10, 30) = 30 units. The three pipes fill 6, 3 and 1 units an hour, a net 10 an hour, so the tank fills in 30 ÷ 10 = 3 hours.
All three are filling pipes here, so nothing is subtracted — check the wording before you reach for a minus sign.
QUICK CHECK
There is a hole at the bottom of a tank through which water leaks. A tap can fill the tank in 3 hours and the hole at the bottom can empty a completely filled tank in 5 hours. If both the tap and the hole remain open, how long will it take to fill the empty tank completely? (SI 29 Jun 2023)
Tap an option to answer
Answer: 7.5 hours
Tank = LCM(3, 5) = 15 units. The tap fills 5 an hour, the hole empties 3, so the net is 2 units an hour and the tank fills in 15 ÷ 2 = 7.5 hours.
A leak always makes the filling slower than the tap alone — if your answer is less than 3 hours you have added instead of subtracted.
QUICK CHECK
Two pipes A and B can separately fill a tank in 130 minutes and 195 minutes respectively. There is an outlet pipe C at the bottom of the tank. If all three pipes are opened together the tank fills in 234 minutes, then how long will pipe C take to empty the tank? (SI 20 Jun 2023)
Tap an option to answer
Answer: 117 minutes
Tank = LCM(130, 195, 234) = 1170 units. A fills 9 a minute and B fills 6, so together 15. With C open the tank still fills, at 1170 ÷ 234 = 5 units a minute. So C empties 15 − 5 = 10 units a minute and alone would drain the tank in 1170 ÷ 10 = 117 minutes.
QUICK CHECK
Two taps can fill a cistern in 18 hours and 24 hours, respectively. A third pipe can empty a full cistern in 21 hours. How long (in hours, correct to 1 decimal place) would it take to fill five-seventh of the empty tank if all three taps are operated simultaneously? (SI 29 Jul 2023)
Tap an option to answer
Answer: 14.4
Tank = LCM(18, 24, 21) = 504 units. The two taps fill 28 and 21 units an hour; the third empties 24. The net rate is 28 + 21 − 24 = 25 units an hour.
Five-sevenths of the tank is 504 × 5/7 = 360 units, which takes 360 ÷ 25 = 14.4 hours. Filling it completely would take 504 ÷ 25 = 20.16 hours — also on offer among the options.
QUICK CHECK
A pipe can empty a tank in 40 minutes. A second pipe, of twice the diameter of the first pipe, is added to empty that tank. In how many minutes can both together empty the tank? (SI 20 Jun 2023)
Tap an option to answer
Answer: 8 minutes
Twice the diameter means four times the cross-section, so the second pipe empties four times as fast. If the first pipe is 1 unit a minute, the tank is 40 units and the second pipe is 4 units a minute.
Together they remove 5 units a minute, so the tank empties in 40 ÷ 5 = 8 minutes. The trap answer 20 comes from assuming twice the diameter means twice the flow.
QUICK CHECK
Two pipes A and B can fill a tank in 15 minutes and 30 minutes respectively. Both pipes are opened at the same time. However, after 6 minutes, pipe A is closed. How much total time is needed to fill the tank? (SI 14 Jul 2023)
Tap an option to answer
Answer: 18 minutes
Tank = LCM(15, 30) = 30 units. A fills 2 a minute and B fills 1. For the first 6 minutes both run, filling 6 × 3 = 18 units. The remaining 12 units are left to B alone at 1 a minute, which takes 12 minutes.
The question asks for the total time, so the answer is 6 + 12 = 18 minutes, not 12.
QUICK CHECK
Two pipes A and B can fill a cistern in 4 hours and 6 hours, respectively. These pipes are opened alternatively for 1 hour each, beginning with A. In how much time will they fill the cistern? (SI 14 Jul 2023)
Tap an option to answer
Answer: 4 hours 40 min
Tank = LCM(4, 6) = 12 units. A fills 3 an hour and B fills 2, so one two-hour cycle adds 5 units. Two full cycles take 4 hours and fill 10 units, leaving 2.
The fifth hour is A's turn at 3 units an hour, so it needs 2/3 of an hour = 40 minutes. The total is 4 hours 40 minutes.
Never divide 12 by the cycle average — the remainder is always cleared by one pipe at its own rate.
CHECK YOURSELF
One pipe can fill a tank 11 times faster than another pipe. If both pipes together can fill this tank in 33 minutes, then the faster pipe alone will be able to fill the tank in ________. (SI 15 Jul 2023)
Show answer
If the slower pipe fills 1 unit a minute, the faster fills 11, so together they fill 12 units a minute. In 33 minutes they fill 12 × 33 = 396 units, which is the whole tank. The faster pipe alone would need 396 ÷ 11 = 36 minutes. Option D (396) is the tank size in units, and option C (121) is 11² — both are there to catch a half-finished calculation.
Men, days and hours
One equation, five questions.
QUICK CHECK
45 people working 6 hours a day can complete a project in 8 days. In how days will 54 people working 8 hours a day complete the same project? (SI 18 Aug 2021)
Tap an option to answer
Answer: 5
The job is 45 × 6 × 8 = 2,160 person-hours. With 54 people working 8 hours a day the daily output is 54 × 8 = 432 person-hours, so the job takes 2,160 ÷ 432 = 5 days.
Always reduce both sides to person-hours first; it removes any need to decide whether a quantity is directly or inversely proportional.
QUICK CHECK
A room is being constructed in a house which can be finished in 12 days when 39 workers work for 5 hours a day. If 30 workers work for 6 hours a day, then in how many days will they complete the construction work? (SI 29 Jul 2023)
Tap an option to answer
Answer: 13
Job = 39 × 12 × 5 = 2,340 worker-hours. With 30 workers at 6 hours a day the daily output is 180 worker-hours, so the room takes 2,340 ÷ 180 = 13 days.
QUICK CHECK
A solar project is being prepared in a science class which can be completed in 12 days if 5 students work 39 hours in a week. If 6 students work 30 hours in a week, then in how many days will they complete the project? (SI 20 Jun 2023)
Tap an option to answer
Answer: 13
Job = 5 × 12 × 39 = 2,340 student-hours, using the weekly hours consistently on both sides. With 6 students at 30 hours a week the output is 180 per day-unit, so the project takes 2,340 ÷ 180 = 13 days.
Note that the hours are given per week on both sides, so the week cancels and no conversion is needed.
QUICK CHECK
A work force of 420 men can complete a job in 9 months. How many more men should a contractor employ to complete the work in 7 months? (SI 29 Jun 2023)
Tap an option to answer
Answer: 120
The job is 420 × 9 = 3,780 man-months. To finish in 7 months you need 3,780 ÷ 7 = 540 men. The contractor already has 420, so he must hire 540 − 420 = 120 more.
The question asks for the extra men, not the total — 540 is offered as option D.
QUICK CHECK
A project can be completed by 120 workers in 30 days. However, the project manager brings in 20 more workers after 16 days. In how many days will the remaining work be completed? (SI 14 Jul 2023)
Tap an option to answer
Answer: 12
The project is 120 × 30 = 3,600 worker-days. In the first 16 days the original crew completes 120 × 16 = 1,920, leaving 1,680 worker-days.
From then on there are 140 workers, so the remainder takes 1,680 ÷ 140 = 12 days. Note the answer is the time for the remaining work, so the project runs 16 + 12 = 28 days in all.
Efficiency and wages
Pay follows work done, which follows the rate.
EFFICIENCY AND TIME ARE INVERSES
If Tanya is 25% more efficient than Sakshi, her rate is 1.25 times Sakshi's, so her time is Sakshi's divided by 1.25 — not reduced by 25%. Writing it as a ratio avoids the mistake: efficiencies 4 : 5 mean times 5 : 4. The same inversion explains why a pipe of twice the diameter empties four times as fast: flow follows area, which follows the square of the diameter.
QUICK CHECK
Sakshi can do a piece of work in 20 days. Tanya is 25 % more efficient than Sakshi. How many days will Tanya take to do the same piece of work ? (SI 2016)
Tap an option to answer
Answer: 16
Sakshi's rate is 1 job in 20 days. Tanya is 25% more efficient, so her rate is 1.25 × (1/20) = 1/16, and she takes 16 days.
In ratio form: efficiencies are 100 : 125 = 4 : 5, so times are 5 : 4. If Sakshi's 5 parts are 20 days, one part is 4 days and Tanya's 4 parts are 16. Reducing 20 by 25% would give 15 — option A, and wrong.
QUICK CHECK
A can complete a work in 12 days while B alone can complete it in 15 days. With C's help, they can finish it in 5 days. If they are paid ₹2,880 for the whole work, find C's share. (SI 2 Jul 2023)
Tap an option to answer
Answer: ₹720
Job = LCM(12, 15, 5) = 60 units. A does 5 a day and B does 4; all three together do 60 ÷ 5 = 12 a day, so C does 12 − 9 = 3 a day.
Wages split in the ratio of work done, which over the same 5 days is the ratio of rates: A : B : C = 5 : 4 : 3, twelve parts in all. C's share is 2,880 × 3/12 = ₹720.
CHECK YOURSELF
If 5 men can do a work in 2 days and 3 boys can do the same work in 5 days, then find the time taken to complete the same work by 10 men and 6 boys. (SI 2 Jul 2023)
Show answer
5 men take 2 days, so the job is 10 man-days and one man does 1/10 of it a day. 3 boys take 5 days, so the job is 15 boy-days and one boy does 1/15 a day. Ten men and six boys together do 10/10 + 6/15 = 1 + 2/5 = 7/5 of the job a day, so the time is 5/7 = 5/7 days. Men and boys have different rates, so they can only be combined after both are expressed as a fraction of the same job.
COMMON TRAPS
- Adding times instead of rates. A at 12 days and B at 15 days do not finish in 27 days, or in 13.5 — they finish in 6⅔.
- Giving the time after a worker leaves when the question wanted the total, or the other way round. Both phrasings appear in SI papers, often on the same page.
- Reading “25% more efficient” as “25% less time”. Divide by 1.25 to get 16 days from 20; subtracting 25% gives the wrong 15.
- Assuming twice the diameter means twice the flow. Area goes with the square, so the flow is four times.
- In alternating-pipe questions, dividing the tank by the average cycle rate. Finish the last part at the rate of whichever pipe's turn it actually is.
QUICK BYTES
- Job = LCM of the given day counts
- Rates add, times do not
- A leak is a pipe with a negative rate
- M₁D₁H₁ = M₂D₂H₂
- Efficiency ratio a : b → time ratio b : a
- Wages split in the ratio of work done
- Twice the diameter → four times the flow
- Alternating pipes: cycle total, then the remainder
REMEMBER
Make the job a number. The moment you read “A takes 12 days, B takes 15”, write LCM = 60 at the top of your rough sheet and convert everyone to units a day. Every later sentence — someone leaves, someone joins, a leak opens, wages are split — becomes an addition or a subtraction on that one number.