Time & Work and Pipes & Cistern

Time & Work

Basic Concepts:

  • Work: Work is often considered as a single task or job. For simplicity, work can be represented as a unit (e.g., 1 work).
  • Time: Time refers to the number of days, hours, or any unit required to complete the work.
  • Efficiency: Efficiency refers to the amount of work done per unit of time. If a person can complete a work in ‘n’ days, their efficiency is   ( i.e. they complete 1/n of the work in a day ).

 

  1. Work Done by Multiple People:
  • If A can complete a work in ‘a’ days and B can complete it in ‘b’ days, together they will complete the work in:

Combined Efficiency :

Time Taken Together :  days

  • The above formula can be extended to more than two people working together.
  1. Concept of Man-Days:
  • Man-Day is a concept used to calculate the total work done. For example, if 5 men complete a job in 10 days, the work done is 5 men × 10 days = 50 man-days.
  1. Working with Efficiency:
  • If A is twice as efficient as B, A will take half the time that B takes to complete the same work.
  • If A and B work together, their combined efficiency will be the sum of their individual efficiencies.
  1. Inverse Proportion:
  • Time and work are inversely proportional. If the number of workers increases, the time taken to complete the work decreases, provided the work remains constant.
  1. Pipes and Cisterns:
  • This is a variation of the time and work problem. Pipes filling a tank are considered as positive work, while pipes emptying a tank are considered as negative work.
  1. Work Done When More Than One Entity is Involved:
  • If A, B, and C together complete a work in ‘x’ days, and A alone can complete it in ‘a’ days, B alone in ‘b’ days, and C alone in ‘c’ days, then:

=

  1. Fraction of Work Done:
  • If a person works for a certain number of days and then stops, the work completed is calculated as:

Fraction of Work Done=

  1. Alternate Days or Fractional Work:
  • If A and B work on alternate days, the work done by each on their respective days should be calculated and then summed up to find the total time taken to complete the work.
  1. Questions on Time and Work in CAT:
  • Questions in the CAT exam might involve people or machines working together, people joining or leaving after certain periods, pipes filling and emptying a tank, or scenarios requiring the calculation of total work done or time taken under varying conditions.

NIMBUS MANTRA:

  • Understand the Efficiency: Always start by calculating the efficiency of each worker/machine.
  • Use the LCM Method: For problems involving different days to complete work, the LCM of the number of days can simplify calculations.
  • Practice with Fractions: Many time and work problems involve fractions, so being comfortable with them is crucial.
  • Focus on Conceptual Clarity: Ensure that the concept of work done, efficiency, and inverse proportionality is clear.

 

  1. Ram completes 60% of a task in 15 days and then takes the help of Rahim and Rachel. Rahim is 50% as efficient as Ram is and Rachel is 50% as efficient as Rahim is. In how many more days will they complete the work?
  1. 1and 1/3
  2. 8 and 1/3
  3. 5 and 5/7
  4. 7 and 5/7

  Correct Answer      Choice (3). They will complete the work in 5 and 5/7 more days.

Explanatory Answer

Ram completes 60% of the task in 15 days.
i.e., he completes 4% of the task in a day.

Rahim is 50% as efficient as Ram is.
Therefore, Rahim will complete 2% of the task in a day.

Rachel is 50% as efficient as Rahim is.
Therefore, Rachel will complete 1% of the task in a day.

Together, Ram, Rahim and Rachel will complete 4 + 2 + 1 = 7% of the work in a day.
Ram, working alone, had already completed 60% of the task.
They have another 40% of the task to be completed.
Therefore, they will take 40/7 or 5and5/7 more days to complete the task.

The correct answer is Choice (3).

  1. A tank is fitted with 8 pipes, some of which that fill the tank and others that empty the tank. Each of the pipes that fills the tank fills it in 8 hours, while each of those that empty the tank empties it in 6 hours. If all the pipes are kept open when the tank is full, it will take 6 hours to drain the tank. How many of these are fill pipes?
  1. 2 fill pipes
  2. 4 fill pipes
  3. 6 fill pipes
  4. 5 fill pipes

  Correct Answer      Choice (2). 4 of the 8 pipes are fill pipes .

Words to Mathematical Expressions and Equations

Let the number of fill pipes be ‘n’
Therefore, there will be (8 – n) waste pipes.

Each of the fill pipes can fill the tank in 8 hours.
Therefore, each of the fill pipes will fill 18th of the tank in an hour.

Hence, n fill pipes will fill n/8th of the tank in an hour.

Similarly, each of the waste pipes will drain the full tank in 6 hours.
∴ each of the waste pipes will drain 16th of the tank in an hour.

(8 – n) waste pipes will drain 8-n/6th of the tank in an hour.

Solve the equation to get the answer

Between the fill pipes and the waste pipes, they drain the tank in 6 hours.
That is, when all 8 of them are opened, 16th of the tank gets drained in an hour.

(Amount of water filled by fill pipes in 1 hour – Amount of water drained by waste pipes 1 hour) = 16th of the tank

n8 – 8 – n/6 = -16

Note: The right hand side has a negative sign because the tank gets drained.

Cross multiplying and solving the equations, 14n – 64 = -8
Or 14n = 56 or n = 4

The correct answer is Choice (2).

  1. Working together, A and B can do a job in 6 days. B and C can do the same job in 10 days, while C and A can do it in 7.5 days. How long will it take if all A, B, and C work together to complete the job?
  1. 8 days
  2. 5 days
  3. 3 days
  4. 7 days

  Correct Answer      Choice (2). If all 3 worked together, the will complete the job in 5 days.

Explanatory Answer

We know that if A and B work together, they can complete the job in 6 days. Therefore, if all three of them A, B and C work together the number of days it will take to complete the job will surely be less than 6 days. Hence, we can eliminate answer choices (1) and (4) right away.

Let A be the number of days that A will take to complete the job if A worked alone, B days for B to complete the job if B worked alone and C days for C to complete the job if C worked alone.

A and B can do a job in 6 days. They complete 1/6 of the job in a day.
i.e., 1/A + 1/B = 1/6 —— (1)
Similarly, B and C will complete 1/10th of the job in a day.
i.e., 1/B + 1/C = 1/10 —— (2)
And C and A will complete 17.5 or 215th of the job in a day
i.e., 1/C + 1/A = 2/15 —— (3).

Adding (1), (2) and (3) we get 1/A + 1/B + 1/B + 1/C + 1/C + 1/A = 1/6 + 1/10 + 2/15
=> 2/A + 2/B = 2/C = 5+3+4/30= 12/30
Or 1/A + 1/B+ 1/C = 6/30 = 1/5
i.e., working together, A, B and C complete 1/5th of the job in a day.
Therefore, they will complete the job in 5 days.

The correct answer is Choice (2).

  1. Four men and three women can do a job in 6 days. When five men and six women work on the same job, the work gets completed in 4 days. How long will a woman take to do the job, if she works alone on it?
  1. 18 days
  2. 36 days
  3. 54 days
  4. None of these

  Correct Answer      Choice (3). A man and woman will take 54 days to complete the job.

Explanatory Answer

Let the amount of work done by a man in a day be ‘m’ and the amount of work done by a woman in a day be ‘w’.

Therefore, 4 men and 3 women will do 4m + 3w amount of work in a day.
If 4 men and 3 women complete the entire work in 6 days, they will complete 16th16th of the work in a day.
Hence, 4m + 3w = 1/6 —– eqn (1)

From statement (2), we know 5 men and 6 women take 4 days to complete the job.
i.e., 5 men and 4 women working together will complete 1414th of the job in a day.
So, 5m + 6w = 1/4 —– eqn (2)

2 * eqn(1) – eqn (2):
8m + 6w – (5m + 6w) = 2 * 1616 – 1414
3m = = 1/12 or m = 1/36.

i.e., a man does 136th of the work in a day.

Substitute the value of m in eqn (1), we get 4 x 1/36 + 3w = 1/6
=> 3w = 1/6 – 1/9 = 3−2/18 = 1/18

Or w = 1/54.

i.e. a woman does 154th of the work in a day.
Hence, she will take 54 days to do the entire work.

The correct answer is Choice (3).

  1. Shyam can do a job in 20 days, Ram in 30 days and Singhal in 60 days. If Shyam is helped by Ram and Singhal every 3rdday, how long will it take for them to complete the job?
  1. 12 days
  2. 5 days
  3. 15 days
  4. 10 days

  Correct Answer      Choice (3). They will complete the work in 15 days

Explanatory Answer

Shyam completes the job in 20 days. So, Shyam does 1/20th of the job in a day.
Ram completes the job in 30 days. So, Ram does 1/30th of the job in a day.
Singhal completes the job in 60 days. So, Singhal does 1/60th of the job in a day.

As Shyam is helped by Ram and Singhal every third day, Shyam works for 3 days while Ram and Singhal work for 1 day in every 3 days.
If in 1 day Shyam completes 1/20th of the job, in 3 days he will complete 3/20th of the job.
Therefore, the amount of work done in 3 days by Shyam, Ram and Singhal = 3/20 + 1/30 + 1/60
= (9+2+1)/60 = 1/5th of the job in 3 days.

Hence, it will take them 5 times the amount of time = 3 * 5 = 15 days to complete the entire job.

The correct answer is Choice (3).

  1. Pipe A usually fills a tank in 2 hours. On account of a leak at the bottom of the tank, it takes pipe A 30 more minutes to fill the tank. How long will the leak take to empty a full tank if pipe A is shut?
  1. 2 hours 30 minutes
  2. 5 hours
  3. 4 hours
  4. 10 hours

  Correct Answer      Choice (4). The leak at the bottom of the tank will empty the tank in 10 hours.

Explanatory Answer

Pipe A fills the tank normally in 2 hours.
Therefore, it will fill 1/2 of the tank in an hour.
Let the leak take x hours to empty a full tank when pipe A is shut.
Therefore, the leak will empty 1/x of the tank in an hour.
The net amount of water that gets filled in the tank in an hour when pipe A is open and when there is a leak =
1/2 – 1/x of the tank. —– (1)

When there is a leak, the problem states that it takes two and a half hours to fill the tank. i.e. 5252hours.
Therefore, in an hour, 25th25th of the tank gets filled. —– (2)
Equating (1) and (2), we get 1/2 – 1/x = 2/5
=> 1/x = 1/2 – 2/5 = 1/10
=> x = 10 hours.

The problem can also be solved without putting pen to paper as follows.
Pipe A takes 2 hours to fill the tank. Therefore, it fills half the tank in an hour or 50% of the tank in an hour.
When there is a leak it takes 2 hours 30 minutes for the tank to fill. i.e 5/2 hours to fill the tank or 2/5th or 40% of the tank gets filled.
On account of the leak, (50 – 40)% = 10% of the water gets wasted every hour.
Therefore, the leak will take 10 hours to drain a full tank.

The correct answer is Choice (4).

  1. A, B and C can do a work in 5 days, 10 days and 15 days respectively. They started together to do the work but after 2 days A and B left. In how many days did C complete the remaining work?
  1. 1 day
  2. 3 days
  3. 5 days
  4. 4 days

  Correct Answer      Choice (4). C completed the remaining task in 4 days.

  1. X alone can do a piece of work in 15 days and Y alone can do it in 10 days. X and Y undertook to do it for Rs. 720. With the help of Z they finished it in 5 days. How much is paid to Z?
  1. Rs.360
  2. Rs.120
  3. Rs.240
  4. Rs.300

  Correct Answer      Choice (2). Z is paid Rs.120.

  1. A and B can do a piece of work in 21 and 24 days respectively. They started the work together and after some days A leaves the work and B completes the remaining work in 9 days. After how many days did A leave?
  1. 5
  2. 7
  3. 8
  4. 6

  Correct Answer      Choice (2). A left after 7 days.

  1. ‘A’ can complete a project in 20 days and ‘B’ can complete the same project in 30 days. If A and B start working on the project together and A quits 10 days before the project is completed, in how many days will the project be completed?
  1. 18 days
  2. 27 days
  3. 26.67 days
  4. 16 days

  Correct Answer      Choice (1). The project will be completed in 18 days.

  1. A father can do a certain job in x hours. His son takes twice as long to do the job. Working together, they can do the job in 6 hours. How many hours does the father take to do the job?
  1. 9
  2. 18
  3. 12
  4. 20
  5. 16

  Correct Answer      Choice (1). The father takes 9 hours to complete the task.

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