Time and work questions come as painters, tailors, machines, taps and leaking tanks, but almost all of them run on one idea: turn every time into a rate, add or subtract the rates, then turn the answer back into a time. This lesson covers rates, working together, finding one worker's time, joining and leaving, alternate days, efficiency and the units method, the M₁D₁H₁ rule, sharing wages, and pipes and cisterns, with worked examples and traps.
Times cannot be added; rates can. If a job takes 8 days, the worker does of it each day, and that fraction is the only thing you ever add or subtract. Turn every time into a rate, combine the rates, then turn the answer back into a time.
The quickest way to keep the fractions tidy is the units method: take the whole job as the LCM of the given times, so that every rate becomes a whole number of units a day.
It means measuring how much of the job someone finishes in one unit of time. A person who finishes a job in 8 days does of it a day. The whole job counts as 1, or as a convenient number of units.
Rate, work and time
Exam shortcut
Turn a part-job into a full-job rate before combining it with anyone else. Someone who does of a job in 6 days has a rate of , so the whole job would take them 21 days.
Add their rates and flip the result. If A takes days and B takes days, together they take days.
Example 1 · Working together
Harpreet can service a car in 9 hours and Dev in 18 hours. How long do they take together?
Rates: and .
Together: an hour.
Time: 6 hours. Shortcut: .
6 hours
Watch out
The together time must be less than the faster worker's time. An answer bigger than the quicker worker's time means days were added instead of rates. For three people, add three rates: 8, 12 and 24 days give , so 4 days.
Subtract rates. If A and B together take days and A alone takes , B's rate is .
Example 2 · One worker's time
Two cooks together prepare a wedding lunch in 10 hours. One of them alone would take 14 hours. How long would the other take?
Other cook's rate: .
So the other cook needs 35 hours.
35 hours
Subtracting the times, , is the classic wrong answer, and it is always far too small.
Exam shortcut
Given all three pair times, A with B, B with C and C with A, add the three pair rates and halve, because everyone appears in two pairs. That gives all three together; subtract a pair to find the third person.
Split the job into phases. Work out what each phase does at its own rate, and let the last phase finish whatever is left.
Example 3 · A worker joins
Mehul can build a bookshelf in 15 days and Zoya in 10. Mehul works alone for 5 days, then Zoya joins him. How many more days to finish?
Mehul's 5 days: done, so is left.
Together: a day.
Time for : days.
4 more days
The whole job takes 9 days.
Watch out
"How many more days", "after how many days did she leave" and "in how many days was the job finished" are three different numbers, and all three are usually among the options. When someone stops days before the end, let the total be and write .
Treat one turn each as a cycle. Find what a cycle does, count whole cycles until you are close to the end, and finish the last scrap carefully; it may take only part of a day.
Example 4 · Alternate days
Pallavi can paint a room in 4 days and Tushar in 6. They paint on alternate days, Pallavi first. How long does the room take?
One 2-day cycle: .
Two cycles, 4 days: done, left.
Day 5 is Pallavi's. At a day, takes of a day.
days
Watch out
Who starts matters: with Tushar first, the answer changes. And never just double the together time, because the job often ends part-way through a turn.
Efficiency is the rate, work per day, and it is inversely proportional to time: twice as efficient means half as long. The ratio of efficiencies is the ratio of times flipped. The fastest method is to give the slower worker a convenient number of units a day.
Example 5 · The units method
Farida is 60% more efficient than Bhushan. Together they paint a hall in 13 days. How long would Farida take alone?
Let Bhushan do 5 units a day. Farida does 60% more: 8 units a day.
Together: 13 units a day, so the job is units.
Farida alone: days.
days
| A is … more efficient than B | A's time as a fraction of B's |
|---|---|
| 25% | |
| 50% | |
| 100%, twice as efficient | |
| 200%, three times as efficient |
Watch out
Percentage efficiency is not percentage time. If A is 25% more efficient than B and B takes 20 days, A takes of 20, which is 16 days, only 20% less.
The total work, measured in person-hours, stays the same however it is split: workers times days times hours a day is equal on both sides. If the amount of work changes too, divide each side by its work.
Workers, days and hours
Example 6 · More workers, more cable
14 workers, working 5 hours a day, lay 350 m of cable in 10 days. How many days will 20 workers, working 7 hours a day, take to lay 600 m?
.
The left side is , so .
.
days
Exam shortcut
More workers or more hours mean fewer days; more work means more days. If your answer moves the wrong way, a factor is on the wrong side. When workers join partway, count worker-days: 25 workers for 36 days is 900 worker-days; after 12 days, 300 are done, and 40 workers finish the other 600 in 15 more days, 27 in all.
Wages are paid for work done, never for time spent or split equally. Each share is the fraction of the job that person did, times the total. With equal days, work done follows efficiency; with different days, it is efficiency times days.
Example 7 · Splitting the pay
Uma can stitch a set of costumes in 8 days and Vikram in 12. Uma works 2 days alone and Vikram finishes. The payment is ₹6,000. What does each get?
Uma's 2 days: of the job, so Vikram does .
Uma: . Vikram: .
Uma ₹1,500, Vikram ₹4,500
The efficiency ratio would be wrong, because they did not work for the same time.
They are time and work questions in which the workers are pipes. An inlet has a positive rate and an outlet or leak a negative one. Add the inlets, subtract the outlets, and flip.
Example 8 · Inlet and outlet
A tank has an inlet that fills it in 6 hours and an outlet that empties it in 10 hours. Both are opened on an empty tank. How long does it take to fill?
Net rate: an hour.
The tank fills in 15 hours.
15 hours
Exam shortcut
Rates
Changing teams
Efficiency and wages
Pipes
Rates in a moving setting are worked in time, speed and distance, and the ratio skills behind wage-sharing in ratio and proportion.
61 questions, grouped by type, easy to hard. Pick an answer and check it. The worked solution opens when you ask for it.
Meenakshi can paint a boundary wall on her own in 10 days. What fraction of the wall does she paint in 3 days?
AnswerC ·
In one day she paints of the wall.
In 3 days she paints of the wall.
Worth noting
is the part still left to paint, not the part done.
Aarav can tile a kitchen floor in 20 days and Gurpreet can tile the same floor in 30 days. If they work together, in how many days will the floor be tiled?
AnswerB · 12 days
Aarav's rate = of the floor per day; Gurpreet's = .
Together: per day.
So they need 12 days.
Worth noting
Add the RATES, never the days. 25 days is the average of the two times and 50 is their sum — both ignore that working together is faster than either alone.
Fatima can type a project report in 6 hours and Joseph can type it in 12 hours. How long will they take if they split the typing and work at the same time?
AnswerB · 4 hours
Fatima's rate = per hour; Joseph's = per hour.
Together: per hour.
So the report takes 4 hours.
Worth noting
The answer must be less than 6 hours, the faster person's time. That alone rules out 9 and 18 hours.
Two electricians, Harish and Nandini, together wire a house in 8 days. Harish alone would take 12 days. How many days would Nandini take alone?
AnswerD · 24 days
Together they do of the job per day; Harish does .
Nandini's rate = per day.
So Nandini alone needs 24 days.
Worth noting
Subtracting the days is the common slip. Subtract the rates, then flip.
Mohit has sanded of a wooden deck in 6 days. Working at the same rate, how many days does he need for the whole deck?
AnswerC · 16 days
of the deck takes 6 days.
Whole deck = days.
Worth noting
10 days is only the time for the remaining . Multiplying gives , which is upside down.
Sana has completed of a school survey in 8 days. At the same pace, how many more days will she need to finish it?
AnswerB · 12 days
in 8 days means the whole survey takes days.
The remaining takes days.
Worth noting
20 days is the whole survey, and 28 days adds the 8 already spent. The question asks only for the days still needed.
Rekha can pack a consignment of parcels in 4 hours and Prakash in 6 hours. What fraction of the consignment do they pack together in 1 hour?
AnswerD ·
Rekha packs per hour; Prakash packs .
Together: .
Worth noting
comes from adding the hours instead of the rates. is what is left, not what is done.
Lalremsiami can weave a shawl in 5 days and her apprentice Anjali in 20 days. Weaving together on the same loom, how many days will the shawl take?
AnswerA · 4 days
Rates: and per day.
Together: per day, so 4 days.
Worth noting
Together must be quicker than 5 days, the faster weaver's time.
Neha can check a pile of 60 answer sheets in 4 hours and Manoj can check the same pile in 6 hours. If they share the pile and work at the same time, how long will it take?
AnswerB · 2 hours 24 minutes
Together: of the pile per hour.
Time = hours = 2 hours + minutes = 2 hours 24 minutes.
Worth noting
0.4 hour is 24 minutes, not 40 minutes. 5 hours is the average of the two times, and 10 hours is their sum.
A can finish a piece of work in 12 days and B can finish the same work in 18 days. Working together, how long will they take?
AnswerA · 7.2 days
Work in one day rather than in whole jobs. A does of the work each day and B does .
Together they do of the work per day.
The whole job therefore takes days.
Worth noting
The answer must be less than 12, the faster worker's time alone. Averaging the two times gives 15 days, which is longer than either of them working alone.
Three volunteers are cleaning a beach. Priya alone would take 10 days, Tenzin 15 days and Omar 30 days. How many days will the job take if all three work together?
AnswerA · 5 days
Rates per day: .
With LCM 30: .
All three together finish in 5 days.
Worth noting
6 days is what Priya and Tenzin need without Omar. Every extra worker, however slow, shortens the time.
Ravi can repair a road stretch in 20 days and Sunita in 30 days. They work together for 6 days, after which Ravi is moved to another site and Sunita finishes the rest alone. How many days does the whole repair take?
AnswerD · 21 days
Together they do per day.
In 6 days: of the road.
Sunita does the other at per day: days.
Total = days.
Worth noting
15 days is only Sunita's part; the question asks for the whole repair, so add the first 6 days.
Kavya starts writing code for an app on her own; she could finish it alone in 24 days. After 6 days, Rohan joins her; he could finish the whole app alone in 12 days. How many days after Rohan joins is the app finished?
AnswerA · 6 days
In 6 days Kavya does of the app, leaving .
Together: per day.
Time for : days.
Worth noting
12 days is the total from the start . 9 days is Rohan alone doing the remaining three-quarters, forgetting Kavya keeps working.
Sneha can plaster a room in 6 days and Irfan in 9 days. They work on alternate days, with Sneha working on the first day. In how many days is the room plastered?
AnswerD · 7 days
Two-day cycle: .
3 cycles (6 days) finish , leaving .
Day 7 is Sneha's, and she does exactly in a day.
So the room is done in 7 days.
Worth noting
Alternate-day work is not the same as working together. days is the together time and doubling it gives , which is wrong because the last day finishes the job early.
Three friends are building a model city for an exhibition. Ananya and Bilal together would take 10 days, Bilal and Charu 15 days, and Charu and Ananya 12 days. How many days would all three take working together?
AnswerB · 8 days
Add the three pair rates: .
Each person is counted twice, so the three together do per day.
All three finish in 8 days.
Worth noting
Forgetting to halve gives 4 days. 12⅓ days is just the average of the three pair times.
Working in pairs on a mural, Deepa and Elango take 12 days, Elango and Farhan take 20 days, and Deepa and Farhan take 15 days. How many days would Farhan take working alone?
AnswerD · 60 days
Sum of pair rates: .
All three together: per day.
Farhan = all three − (Deepa + Elango) = .
Farhan alone takes 60 days.
Worth noting
Subtract the pair that does NOT contain Farhan. Removing the wrong pair gives Deepa's 20 days or Elango's 30 days instead.
Varun can dig a garden bed in 10 days and Lakshmi in 15 days. They start together, but Varun stops 5 days before the digging is finished and Lakshmi completes it. How many days does the digging take in all?
AnswerC · 9 days
Let the total be days. Lakshmi works all days; Varun works .
. Multiply by 30: .
, so days.
Worth noting
8 days comes from letting the wrong person stop early. 11 days wrongly adds the 5 days to the together time of 6 days.
Aditi can knit a blanket in 16 days. She knits for 4 days and then hands it to her grandmother, who finishes the rest in 18 days. How many days would the grandmother take to knit a whole blanket?
AnswerC · 24 days
Aditi's 4 days = of the blanket.
The grandmother does in 18 days.
Whole blanket: days.
Worth noting
18 days is the time for only of the blanket. 27 days comes from multiplying by instead of .
Three workers, Salim, Tanya and Uday, can build a shed alone in 30, 40 and 60 days. All three begin together, but after 4 days Uday leaves. How many more days do Salim and Tanya need to finish the shed?
AnswerA · 12 days
All three: per day.
In 4 days: done, so remains.
Salim + Tanya: per day.
Time: days.
Worth noting
16 days is the total from the start; the question asks for the days after Uday leaves.
Nikhil finishes of a data-entry job in 5 days, and Pooja finishes of the same job in 8 days. Working together, how long will they take for the whole job?
AnswerA · days
Nikhil's whole job: days, rate .
Pooja's whole job: days, rate .
Together: per day.
Time = days.
Worth noting
Convert each part-job into a full-job rate first. 13 days just adds the 5 and 8 days given.
Jaya can sew a set of curtains in 25 days and Karthik in 20 days. They work together for 5 days. What fraction of the job is still left?
AnswerB ·
Together: per day.
In 5 days: done.
Left: .
Worth noting
is the part finished; the question asks what is left.
A printing machine M can print a batch of posters in 5 hours and an older machine N in 8 hours. Only one machine runs at a time: they take turns for one hour each, and N runs the first hour. How long does the batch take?
AnswerB · hours
One two-hour cycle prints .
3 cycles (6 hours) print , leaving .
Hour 7 belongs to N, which prints per hour.
Time for : hour, so the total is hours.
Worth noting
Who starts matters. If M ran first, the last scrap would be done by M in hour, giving hours. Rounding up to 7 hours ignores that the job ends part-way through an hour.
Two tailors together stitch a batch of school uniforms in 6 days. Working alone, one of them takes 5 days more than the other. How many days would the slower tailor take alone?
AnswerC · 15 days
Let the faster tailor take days; the slower takes .
, so .
, i.e. , so .
The slower tailor takes days.
Worth noting
Check the answer: . 10 days is the faster tailor; 12 days wrongly assumes both are equal .
Two carpenters, Manoj and Rehana, can make a set of classroom benches together in 12 days. They work together for 5 days, then Rehana leaves and Manoj finishes the rest alone in 21 days. How many days would Rehana take to make the whole set alone?
AnswerA · 18 days
In 5 days together: done, leaving .
Manoj does in 21 days, so his rate is per day.
Rehana's rate = .
Rehana alone takes 18 days.
Worth noting
36 days is Manoj's time, not Rehana's. 28 days comes from treating Manoj's 21 days as the whole job instead of the remaining seven-twelfths.
Gopal can whitewash a school building in 16 days. Two helpers, Hema and Iqbal, could do it alone in 24 and 48 days. Gopal works every day, but the helpers join him only on every third day (day 3, day 6, day 9, …). In how many days is the building whitewashed?
AnswerB · 12 days
In each 3-day block Gopal does ; the helpers add .
One block = of the building.
4 blocks finish it, and the 4th block ends on a helper day, so the time is days.
Worth noting
8 days would be right only if all three worked every day. Always check that the last block really ends on the day you count.
A new harvesting machine works twice as fast as an old one. The old machine takes 30 days to harvest a farm. How many days does the new machine take?
AnswerA · 15 days
Twice as fast means twice the work per day.
So it needs half the time: days.
Worth noting
Time and efficiency are inversely related: double the speed, half the days. 60 days multiplies instead of dividing, and 45 days adds 50% instead of halving.
Asha can prepare a science exhibit in 18 days and Deepa can prepare it in 24 days. What is the ratio of Asha's efficiency to Deepa's efficiency?
AnswerB ·
Efficiency is work per day: Asha , Deepa .
Ratio = .
Worth noting
Efficiency is the inverse of time, so the ratio flips. is the ratio of their times.
A team of 12 workers can build a compound wall in 15 days. How many workers are needed to build the same wall in 9 days?
AnswerC · 20
Total work = worker-days.
Workers for 9 days = .
Worth noting
Fewer days need MORE workers. 7.2 comes from , treating it as direct proportion.
Rakesh can assemble a cupboard in 20 hours. Tanvi is 25% more efficient than Rakesh. How long will Tanvi take to assemble the same cupboard?
AnswerB · 16 hours
25% more efficient: Tanvi does of Rakesh's work per hour.
Time = hours.
Worth noting
25% more efficient does NOT mean 25% less time. hours is the trap; the time falls by only 20%.
8 tailors stitch 120 shirts in 5 days. How many shirts will 10 tailors stitch in 6 days at the same rate?
AnswerD · 180
One tailor stitches shirts a day.
10 tailors in 6 days: shirts.
Worth noting
150 shirts accounts only for the extra tailors, and 144 only for the extra day. Both changes increase the output.
Rajesh works three times as fast as his trainee Sameer. Together they can service all the fans in a hostel in 14 days. How many days would Sameer take alone?
AnswerD · 56 days
Let Sameer do 1 unit per day; Rajesh does 3 units. Together: 4 units per day.
Total work = units.
Sameer alone: days.
Worth noting
days is Rajesh's time. 42 days multiplies the together time by 3 instead of by 4 — the together rate is shares.
16 labourers working 9 hours a day can finish a building's flooring in 20 days. How many days will 24 labourers working 8 hours a day take for the same flooring?
AnswerB · 15 days
Total work = labour-hours.
New team per day: labour-hours.
Days = .
Worth noting
Ignoring the change in hours gives days. Use with every factor.
Anil can paint a house in 10 days and Bharti in 15 days. They paint it together and are paid ₹4,500 in all. If the money is shared by the work each does, how much does Anil get?
AnswerC · ₹2,700
Working the same number of days, work done is in the ratio of rates: .
Anil's share = .
Worth noting
Share by work, i.e. by efficiency, not by days. ₹1,800 is Bharti's share; splitting equally gives ₹2,250.
Ishita is 50% more efficient than Farooq. Together they can catalogue a library in 18 days. How many days would Ishita take alone?
AnswerB · 30 days
Let Farooq do 2 units per day; Ishita does 3 units. Together: 5 units per day.
Total work = units.
Ishita alone: days.
Worth noting
45 days is Farooq's time. 27 days wrongly treats efficiency as if it were time.
At a pottery unit, 3 men or 5 women can glaze a batch of pots in 20 hours. How long will 6 men and 5 women take together?
AnswerA · hours
6 men are twice 3 men, so 6 men alone take 10 hours: rate .
5 women alone take 20 hours: rate .
Together: , so time = hours.
Worth noting
Averaging 10 and 20 hours gives 15 hours and adding them gives 30 hours — both slower than the men alone, which is impossible when helpers are added.
Arjun can tile a terrace in 16 days and Bina in 24 days. Arjun works alone for 4 days, and then Bina finishes the job alone. The total payment is ₹7,200, shared by the work each did. How much does Bina get?
AnswerD · ₹5,400
Arjun's 4 days = of the job.
Bina does the remaining .
Bina's share = .
Worth noting
They did not work for the same time, so the efficiency ratio does not apply here; ₹2,880 comes from wrongly using it. ₹1,800 is Arjun's share.
An experienced mason is three times as efficient as a new mason and so takes 24 days less to build a wall. In how many days can they build the wall working together?
AnswerA · 9 days
If the experienced mason takes days, the new one takes .
, so and .
Together: , i.e. 9 days.
Worth noting
12 days is the experienced mason alone; together must be faster than that.
A contractor estimates that 45 workers will finish a metro-station floor in 32 days. If he hires 60 workers instead, how many days earlier will the floor be finished?
AnswerA · 8 days
Total work = worker-days.
With 60 workers: days.
Days saved = days.
Worth noting
24 days is the new time, not the saving. The question asks how many days EARLIER.
10 workers, working 6 hours a day, dig a 60 m long drainage trench in 12 days. How many days will 15 workers, working 8 hours a day, take to dig a 90 m trench of the same width and depth?
AnswerB · 9 days
Use .
, so .
days.
Worth noting
Forgetting that the second trench is longer gives 6 days. Putting the hours on the wrong side gives 16 days.
Chitra can plaster a hall in 12 days and Dinesh in 16 days. With the help of Ezhil, all three finish it in 4 days. The ₹3,600 payment is shared by the work each did. What is Ezhil's share?
AnswerC · ₹1,500
In 4 days Chitra does ; Dinesh does .
Ezhil does the rest: .
Ezhil's share = .
Worth noting
All three worked the same 4 days, but that does not mean equal pay: splitting three ways gives ₹1,200, which is Chitra's share, not Ezhil's.
A contractor plans to lay a road with 30 workers in 40 days. After 16 days, 10 more workers join. In how many days in all is the road finished?
AnswerC · 34 days
Total work = worker-days.
In 16 days: worker-days, leaving 720.
Now 40 workers: more days.
Total = days.
Worth noting
18 days counts only the time after the new workers join. 30 days assumes all 40 workers were there from day 1.
Kiran, Leela and Mustafa can build a stage alone in 10, 12 and 15 days. All three start together. Kiran leaves after 2 days, Leela leaves 3 days after that, and Mustafa finishes the stage. The ₹9,000 payment is shared by the work done. What is Mustafa's share?
AnswerC · ₹3,450
First 2 days, all three: .
Next 3 days, Leela + Mustafa: . Total so far .
Mustafa does the last in day, so he worked days.
Mustafa's work = ; share = .
Worth noting
Check: Kiran did and Leela , and . ₹3,750 is Leela's share and ₹1,800 is Kiran's.
Rupa works twice as efficiently as Sanjay. To finish a contract, Rupa works for 5 days and Sanjay for 8 days. The ₹5,400 payment is shared by the work each did. What is Rupa's share?
AnswerD · ₹3,000
Let Sanjay do 1 unit per day; Rupa does 2 units.
Rupa's work = units; Sanjay' units.
Rupa's share = .
Worth noting
Pay follows work = efficiency × days. Using only the efficiency ratio gives ₹3,600; using only the days ignores that Rupa is faster. ₹2,400 is Sanjay's share.
An overhead tank can be filled by pipe P in 12 hours and by pipe Q in 15 hours. If both pipes are opened together on an empty tank, how long will it take to fill?
AnswerA · 6 hours 40 minutes
Together: per hour.
Time = hours = 6 hours 40 minutes.
Worth noting
27 hours adds the two times and 13 hours 30 minutes averages them. Two pipes must fill faster than the quicker pipe alone.
A tap fills a water drum in 10 hours, but a crack at the bottom would empty the full drum in 30 hours. With the tap on and the crack leaking, how long does the empty drum take to fill?
AnswerA · 15 hours
Net rate = per hour.
So the drum fills in 15 hours.
Worth noting
A leak works against the tap, so subtract its rate. Subtracting the hours gives 20 hours, and 40 hours adds them.
A garden tap fills of a tub in 9 minutes. How long does it take to fill the whole tub?
AnswerC · 12 minutes
of the tub takes 9 minutes, so takes 3 minutes.
The whole tub takes minutes.
Worth noting
3 minutes is only the last quarter. minutes comes from multiplying 9 by instead of dividing.
A 1,200-litre tank is being filled at 40 litres per minute, but a leak lets out 10 litres per minute. How many minutes does the empty tank take to fill?
AnswerC · 40 minutes
Net inflow = litres per minute.
Time = minutes.
Worth noting
30 minutes ignores the leak, and 24 minutes adds the leak to the inflow.
An outlet can empty a full swimming-pool tank in 18 minutes. What fraction of the water is still in the tank after the outlet has run for 6 minutes?
AnswerD ·
In 6 minutes the outlet removes of the water.
Water left = .
Worth noting
is the fraction drained, not the fraction left.
One pipe fills a water tanker in 9 hours. How long will three identical pipes working together take to fill it?
AnswerA · 3 hours
Each pipe fills per hour; three fill per hour.
Time = 3 hours.
Worth noting
More identical pipes mean less time: divide, don't multiply. 27 hours multiplies by 3.
A 900-litre tank already holds 300 litres. An inlet adds 25 litres per minute while an outlet removes 15 litres per minute. With both open, how many minutes will the tank take to fill?
AnswerD · 60 minutes
Water still needed = litres.
Net inflow = litres per minute.
Time = minutes.
Worth noting
36 minutes fills the whole 900 litres with the inlet alone; 24 minutes ignores the outlet. Use the space left and the net flow.
A school's water tank has two inlet pipes that fill it in 20 and 30 minutes, and an outlet that empties it in 15 minutes. If all three are opened on an empty tank, how long will it take to fill?
AnswerD · 60 minutes
Inlets: per minute.
Outlet: per minute.
Net: , so 60 minutes.
Worth noting
12 minutes ignores the outlet; minutes treats the outlet as a third filler. An outlet's rate is always subtracted.
Pipe A can fill a cistern in 8 hours and pipe B in 12 hours. Both are opened, and after 2 hours pipe A is shut. How many more hours does pipe B take to fill the cistern?
AnswerA · 7 hours
Both together: per hour.
In 2 hours: filled, leaving .
B alone: hours.
Worth noting
9 hours is the total from the start. 10 hours forgets that A also filled part of the cistern in the first 2 hours.
A pump can normally fill a sump in 6 hours. Because of a leak, it now takes 8 hours to fill. If the sump is full and the pump is off, how long will the leak take to empty it?
AnswerD · 24 hours
Pump alone: per hour. Pump with leak: per hour.
Leak's rate = .
The leak empties the full sump in 24 hours.
Worth noting
2 hours subtracts the times . hours adds the rates, as if the leak helped the pump.
Pipes A and B can fill a tank in 30 and 45 minutes. Both are opened on an empty tank, and B is closed after some time. The tank is full exactly 20 minutes after the pipes were opened. For how many minutes was B open?
AnswerC · 15 minutes
A runs all 20 minutes: of the tank.
B filled the rest, : time = minutes.
Worth noting
18 minutes is how long both together would take for the whole tank. 10 minutes uses A's 30 minutes instead of B's 45 when converting B's third.
A full tank has an outlet that can empty it in 10 hours and an inlet that can fill it in 15 hours. If both are opened at the same time, how long will the tank take to become empty?
AnswerD · 30 hours
Emptying beats filling: net emptying rate = per hour.
The tank empties in 30 hours.
Worth noting
6 hours adds both rates as if both were emptying. 5 hours subtracts the times. Check: 30 hours is slower than the outlet alone, as it should be.
Two pipes can fill a storage tank in 20 minutes and 24 minutes. A drain at the bottom lets out 3 litres per minute. With both pipes and the drain open, the empty tank fills in 15 minutes. What is the capacity of the tank?
AnswerB · 120 litres
Pipes together: of the tank per minute.
With the drain the net is , so the drain removes of the tank per minute.
of the tank = 3 litres, so the tank holds litres.
Worth noting
Check: the pipes add 6 and 5 litres a minute, the drain removes 3, net 8 litres a minute, and minutes. 40 is the drain's emptying time in minutes, not the capacity.
A cistern has two inlets, which fill it in 15 minutes and 10 minutes, and one outlet. With all three open, the empty cistern fills in 18 minutes. How long would the outlet alone take to empty the full cistern?
AnswerC · 9 minutes
Inlets together: per minute.
Outlet = inlets − net = .
The outlet empties it in 9 minutes.
Worth noting
minutes comes from adding and . 6 minutes is just the two inlets together.
A tank is half full. Pipe X can fill the whole tank in 16 minutes and pipe Y can empty the whole tank in 24 minutes. If both are opened now, in how many minutes will the tank be full?
AnswerC · 24 minutes
Net fill rate = per minute.
Only half the tank needs filling: minutes.
Worth noting
48 minutes fills an empty tank; this one is already half full. 8 minutes ignores pipe Y.
A reservoir has two inlets that fill it in 10 hours and 15 hours, and an outlet that empties it in 12 hours. At 8 am the two inlets are opened on the empty reservoir. At 10 am the outlet is also opened by mistake and left open. At what time is the reservoir full?
AnswerB · 6 pm
Inlets together: per hour; in 2 hours is full.
With the outlet: per hour.
The remaining takes hours.
Full at 10 am + 8 hours = 6 pm.
Worth noting
2 pm ignores the outlet completely. 8 pm assumes the outlet was open from 8 am, and 10 pm counts 12 hours from 10 am as if nothing had been filled before.
A pipe can fill a tank in 30 minutes. After it has been running for 10 minutes, a leak starts at the bottom, and because of it the tank fills 10 minutes later than it should have. How long would the leak alone take to empty the full tank?
AnswerD · 90 minutes
In the first 10 minutes the pipe fills of the tank.
The remaining should take 20 minutes but takes 30, so the net rate is .
Leak = , so it empties the tank in 90 minutes.
Worth noting
45 minutes is the time the pipe-with-leak would take for a full tank, not the leak alone. Measure the net rate only over the part when the leak is actually running.
Pipe A fills a tank in 12 minutes and pipe B in 20 minutes. They are opened one at a time for a minute each, in turn, starting with A. How long does the empty tank take to fill?
AnswerA · minutes
Each 2-minute cycle fills .
7 cycles (14 minutes) fill , leaving .
Minute 15 is A's, and A fills per minute, so it needs of that minute.
Total = minutes.
Worth noting
Doubling the together time ( minutes) ignores that the tank fills before the 8th cycle ends. Starting with B instead would give a different answer, so read who goes first.
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