2IIM Quant Notes  Time and Work

rajesh_balasubramanian
Director, 2IIM Online CAT Preparation  IIT Madras  IIM Bangalore  CAT 100th percentile  CAT 2011, 2012 and 2014.
We will start this with an old joke. My friends tell me I procrastinate a lot. I do not know what that long word means. I will look it up tomorrow. CAT Preparation, like calling that old friend, or clearing up a cupboard is one of those things that gets pushed to the tomorrows. Anyone ambitious enough to think about the IIMs needs to break out of this. One of the better ways of motivating oneself is to pick up simple cues from the environment to kickstart projects. April 1 is universally acknowledged as a tribute to people’s folly. Ergo, an ideal day to start the process of deluding oneself that MBA is a big breakthrough in life. You need two things while preparing for CAT. Decent levels of motivation and an ability to laugh at oneself. We will be damned if we provided only one of these two :D
We will now solve some interesting questions from the topic Time and work, Pipes and Cisterns.
A drain pipe can drain a tank in 12 hours, and a fill pipe can fill the same tank in 6 hours. A total of n pipes – which include a few fill pipes and the remaining drain pipes – can fill the entire tank in 2 hours. How many of the following values could ‘n’ take ?
a) 24 b) 16 c) 33 d) 13 e) 9 f) 8 3
 4
 2
 1
Two drain pipes can drain the same volume that one fill pipe fills. This means that a DDF combination has to have a net volume effect of 0.
In spite of this, the tank still gets filled. Only the fill pipes can manage to fill the tank. In addition to all the net zero effect pipes, we need three more fill pipes in order to fill the tank in 2 hours.
So, we can have as many DDFs as we want, but we need one FFF at the end to ensure that the tank gets filled in 2 hours.
So the number of pipes will be → (D – D  F).......(D – D  F) + (F – F  F).
The number of pipes has to be a multiple of 3. Only options (a), (c) and (e) fit the description. Choice (A).A fill pipe can fill a tank in 20 hours, a drain pipe can drain a tank in 30 hours. If a system of n pipes (fill pipes and drain pipes put together) can fill the tank in exactly 5 hours, which of the following are possible values of n (More than one option could be correct)?
(1) 32 (2) 54 (3) 29 (4) 40 1 and 2 only
 1 and 3 only
 2 and 4 only
 2 and 3 only
3 fill pipes cancel out 2 drain pipes. Plus, you need an additional 4 fill pipes fill the tank in 5 hours. so the answer has to be 5k + 4.
Both 54 and 29 are possible.
Choice (D)Pipe A, B and C are kept open and together fill a tank in t minutes. Pipe A is kept open throughout, pipe B is kept open for the first 10 minutes and then closed. Two minutes after pipe B is closed, pipe C is opened and is kept open till the tank is full. Each pipe fills an equal share of the tank. Furthermore, it is known that if pipe A and B are kept open continuously, the tank would be filled completely in t minutes. Find t?
 18
 36
 27
 24
A is kept open for all t minutes and fills onethird the tank. Or, A should be able to fill the entire tank in '3t' minutes.
A and B together can fill the tank completely in t minutes. A alone can fill it in 3t minutes.
A and B together can fill 1/t of the tank in a minute. A alone can fill 1/3t of the tank in a minute. So, in a minute, B can fill 1/t  1/3t = 2/3t. Or, B takes 3t/2 minutes to fill an entire tank.
To fill onethird the tank, B will take t/2 minutes. B is kept open for t  10 minutes.
t/2 = t  10, t = 20 minutes.
A takes 60 minutes to fill the entire tank, B takes 30 minutes to fill the entire tank. A is kept open for all 20 minutes. B is kept open for 10 minutes.
C, which is kept open for 8 minutes also fills onethird the tank. Or, c alone can fill the tank in 24 minutes. Choice (D)Pipe A fills a tank at the rate of 100lit/min, Pipe B fills at the rate of 25 lit/min, pipe C drains at the rate of 50 lit/min. The three pipes are kept open for one minute each, one after the other. If the capacity of the tank is 7000 liters, how long will it take to fill the tank if
i. A is kept open first, followed by B and then C.
ii. B first, followed by A, and then C.
iii. B first, followed by C, and then A. 279.25 mins, 280 mins and 280 mins
 280 mins, 280 mins and 279.25 mins
 279 mins, 280 mins and 279.25 mins
 279.25 mins, 280 mins and 279 mins
i. A is kept open first, followed by B and then C
Each cycle of 3 minutes, 75 liters get filled. 100 + 25  50. So, after 3 minutes the tank would have 75 liters
6 mins  150 liters
9 mins  225 liters
30 mins  750 liters
270 mins  6750 liters
273 mins  6825 liters
279 mins  6975 liters
In the 280^{th} minute, pipe A would be open and it would fill the remaining 25 liters in 15 seconds. So, it would take 279 mins and 15 seconds to fill the tank.
The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.
ii. B first, followed by A, and then C
Each cycle of 3 minutes, 75 liters get filled. 25 + 100  50. So, after 3 minutes the tank would have 75 liters.
6 mins  150 liters
9 mins  225 liters
30 mins  750 liters
270 mins  6750 liters
273 mins  6825 liters
279 mins  6975 liters
In the 280^{th} minute, pipe B would be open and it would fill the remaining 25 liters in one minutes. So, it would take 280 mins.
iii. B first, followed by C, and then A
In this case also, Each cycle of 3 minutes, 75 liters get filled. 25 50 + 100. But there is a small catch here. In the first set of 3 minutes, we would fill up to about 100 liters. After 1 minute, we would be at 25 liters, after 2 minutes, we would be at 0 liters and in the third minute, the tank would be 100 liters full.
6 mins  175 liters
9 mins  250 liters
30 mins  775 liters
270 mins  6775 liters
273 mins  6850 liters
276 mins  6925 liters
279 minutes  7000 liters
So, it would take 279 mins to fill the tank.The most important thing in these type of questions is to think in terms of cycles till we reach close to the required target and then think in simple steps.
Answer Choice (D).Consider three friends A, B and C who work at differing speeds. When the slowest two work together they take n days to finish a task. When the quickest two work together they take m days to finish a task. One of them, if he worked alone would take thrice as much time as it would take when all three work together. How much time would it take if all three worked together?
 3mn/2(m+n)
 2mn/(m+n)
 4mn/3(m+n)
 5mn/3(m+n)
Let A < B < C in terms of efficiency.
B and C together take n days.
A and B together take m days.
One of them, if he worked alone would take thrice as much time as it would take when all three work together. This is a crucial statement. Now, if there are three people who are all equally efficient, for each of them it would take thrice as much time as for all three together.
Now, this tells us that the person who takes thrice as much time cannot be the quickest one. If the quickest one is only onethird as efficient as the entire team, the other two cannot add up to twothirds. By a similar logic, the slowest one cannot be the person who is onethird as efficient.
In other words, the person onethird as efficient = B
Let A, B and C together take x days. B alone would take 3x days
B and C together take n days. Or B + C in 1 day do 1/n of the task  > Eq(1)
A and B together take m days. Or, A + B in 1 day do 1/m of the task  > Eq(2)
B takes 3x days to do the task. Or, B, in one day, does 1/3x of the task  > Eq(3)Now, if we do (i) + (i) – (iii) we get
A + B + C do 1/n + 1/m  1/3x in a day. This should be equal to 1/x as all three of them complete the task in x days.
1/n + 1/m  1/3x = 1/x
1/n + 1/m = 4/3x
m+n/mn = 4/3x
Required time = 4mn/3(m+n)4 men and 6 women complete a task in 24 days. If the women are at least half as efficient as the men, but not more efficient than the men, what is the range of the number of days for 6 women and 2 men to complete the same task?
 30 to 33.6 days
 32 to 35 days
 33.6 to 35 days
 30 to 35 days
4m and 6w finish in 24 days.
In one day, 4m + 6w = 1/24 of task.
In these questions, just substitute extreme values to get the whole range
If a woman is half as efficient as man
4m + 3m = 1/24, 7m = 1/24 , m = 1/168
6w + 2m = 3m + 2m = 5m, 5m will take 168/5 days = 33.6 days
If a woman is as efficient as a man
4m + 6w finish in 24 days
10m finish 1/24 of task in a day
6w + 2m = 8m, 8m will take 240/8 = 30 days to finish the task.
So, the range = 30 to 33.6 days. The new team will take 30 to 33.6 days to finish the task. Choice (A)Pipes A, B and C can fill a tank in 30, 60 and 120 minutes respectively. Pipes B and C are kept open for 10 minutes, and then Pipe B is shut while Pipe A is opened. Pipe C is closed 10 minutes before the tank overflows. How long does it take to fill the tank?
 40 minutes
 28 minutes
 30 minutes
 36 minutes
Let us assume that the tank has a capacity of 120 litres. So, the pipes discharge the following amounts of water:
(A) 4 litres per minute
(B ) 2 litres per minute
(C) 1 litre per minute.
Part 1: B and C (3 litres/min) are kept open for 10 minutes, filling 3 × 10 = 30 litres. 90 litres remain to be filled in the tank.
Part 2: Now, B is shut and A is opened. Effectively, this means that A and C are filling the tank together (5 litres / minute). We don’t yet know how long A and C are open together.
Part 3: C is closed 10 minutes before the tank overflows. This means that only A works for the last 10 minutes, filling 40 litres ([email protected] litres/min)
Since 30 litres are filled in Part 1 and 40 litres in Part 3, the balance (50 litres) should have been filled in Part 2.
Working together, A and C fill 5 litres per minute in Part 2. This means that they would have taken 10 minutes to fill 50 litres.
So, the entire time it took to fill the tank is:
10 + 10 + 10 = 30 mins.
Answer choice (c)Alternate Solution
In one minute, A fills (1/30)th of the tank, B fills (1/60)th of the tank, and C fills (1/120)th of the tank.
(B + C) work for 10 minutes, followed by (A + C), which works for “t” minutes, followed by A, which work for 10 minutes. This ensures that the tank gets filled. This can be written in an equation form:
10 x (1/60 + 1/120) + t x (1/30 + 1/120) + 10 x (1/30) = 1
10 x (1/40) + t x (1/24) + 10 x (1/30) = 1.
Or, 1/4 + t/24 + 1/3 = 1. Or t = 10.
So, the entire tank was filled in 30 mins.
Answer choice (C).B takes 12 more days than A to finish a task. B and A start this task and A leaves the task 12 days before the task is finished. B completes 60% of the overall task. How long would B have taken to finish the task if he had worked independently?
 48 days
 36 days
 28 days
 32 days
Let us say A and B split their share of the task and started doing their respective shares simultaneously.
Let’s say A takes A days to finish the task. Therefore, B takes A + 12 days to finish the entire task.
A has to finish 40% of the task, since B is doing the rest. So A will only take 2A/5 number of days.
B only has to finish 60% of the task, so B will take 3(A+12)/5 number of days.
But as we know, B starts working along with A and finishes 12 days after A stops working.
So, 3(A+12)/5 = (2A)/5 + 12
3A + 36 = 2A + 60
A = 24; B = 36 days.
Answer choice (b)A and B together can finish a task in 12 days. If A worked half as efficiently as he usually does and B works thrice as efficiently as he usually does, the task gets completed in 9 days. How long would A take to finish the task if he worked independently?
 12 days
 24 days
 27 days
 18 days
Let A take ‘a’ days to complete the task and B take ‘b’ days to complete the task.
Thus in one day, A will complete (1/a)th of the task.
Similarly in one day, B will complete (1/b)th of the task.
So in one day, if A and B work together they will complete (1/a + 1/b)^{th} of the task.
Given that A and B together take 12 days to complete the task, then in one day A and B together complete (1/12)th of the task.
Thus, 1/a + 1/b = 1/12 ……Eqn. 1
If A worked half as efficiently as he usually does, then A will take twice the time as he usually takes, i.e., 2a days. Thus in one day, A completes (1/2a)^{th} of the task.
Similarly if B worked thrice as efficiently as he usually does, then B will take onethird the time he usually takes, i.e., b/3 days. Thus in one day, B completed (1/(b⁄3))^{th} or (3/b)^{th} of the task.
Thus when both of them work together, they will complete (1/2a + 3/b)^{th} of the task, given that A and B take 9 days to complete the task.
Thus, 1/a + 1/b = 1/12 ……Eqn. 1
1/2a + 3/b = 1/9 …… Eqn. 2
1/2a + 3/b = 1/9 …… Eqn. 2
Solving Equations 1 and 2 for ‘a’ we should get the answer,
From equation (1) we get 12(a + b) = ab
From equation (2), we get 9(b + 6a) = 2ab
Substituting ab as 12(a + b) in equation (2) we get 9b + 54a = 2 x 12 x ( a + b)
9b + 54a = 24a + 24b;
Or, 30a = 15b,
Or, b = 2a
Now, 12(a + b) = ab, or 12 x 3a = 2a^{2}
a = 18 days.
Answer choice (d).B takes 12 more hours than A to complete a task. If they work together, they take 16 fewer hours than B would take to complete the task. How long will it take A and B together to complete a task twice as difficult as the first one?
 16 hrs
 12 hrs
 14 hrs
 18 hrs
Let us assume that A takes ‘x’ hours to finish a task. Then, B takes ‘x+12’ hours to finish the same task. Given, if they work together, they take 16 fewer hours than B would take to complete the task = ‘x4’ hours.
A completes the task in ‘x’ hours = > A finishes 1/xth of the task in 1 hour. B finishes the 1/(x+12)th of the task in 1 hour. A and B finishes 1/(x4)th of the task in one hour .
Therefore, 1/x + 1/(x+12) = 1/(x4) . Solving for x,
(2x+12)(x4) = x^{2} + 12x.
x^{2} – 8x 48 =0.
x^{2} – 12x+4x 48 =0 = > x(x12)+4(x12)
(x12)(x+4) =0. X=12 or x=4. Only x =12 is possible, since x cannot be negative.
Therefore, when A and B work together they finish a task in x4 = 124 = 8 hours.
If the task is twice as difficult as the first one, they finish it in 2*8 = 16 hours.Pipe A can fill a tank in 12 hours. When it works along with Pipe B, it can fill the tank in 8 hours. In how many hours can pipe B fill the same tank independently?.
 8 hrs
 12 hrs
 24 hrs
 16 hrs
Let the capacity of the tank be C.
Rate of Pipe A = C/A
Rate of Pipe B = C/B
Rate of Pipe A and Pipe B together = C/A + C/B = C/8
C/12 + C/B = C/8
C/B = C/8  C/12 = C/24
Pipe B independently will take 24 hours to fill the tank.Number of units of a good that can be produced by a factory is directly proportional to the square of the number of workers, square root of the number of machines and to the number of hours put in. The factory produces 200 goods when 4 people work for 8 hours each with 4 machines. When 3 people work for 12 hours each with 9 machines, how many goods will be produced?
 K = 25/32
 K = 100/163
 K = 25/256
 K = 16/29
G α No of workers^{2}
G α Root of Number of machines
G α No of hours200 α 16 x 2 x 8
200 α 256
200 = k x 256
K = 200/256 = 25/32A can complete a task in 12 days. B can complete the task in 18 days. If A and B work on this same task in alternate days starting with A, in how many days do they finish the entire task?
 10.8 days
 14.33 days
 11 days
 8.4 days
A can complete a task in 12 days and B can complete the task in 18 days. So,
In one day, A can do 1/12 and B can do 1/18
In two days they can complete 1/12 + 1/18 = 5/36
In four days they can finish 10/36 and so on
We can have 7 such sets of 2days each.
So, at the end of 14 days, they would have done 35/36 of the task
On the 15th day A would begin work with 1/36 of the task to finish. He can finish 1/12 in a day. So, he would take onethird of a day. So, they can finish the whole task in 14.33 days.A can complete a task 4 hours lesser time than B takes to complete the same. If A and B together can complete the task in 288 minutes, how long does B alone take to complete the task?
 1 hr
 2 hrs
 3 hrs
 4 hrs
Let time taken by A be 'a' and time taken by B be 'b'
Then a = b  (4 * 60) minutes
a = b  240 minutes
Let time taken to complete the task by both together be 't' = 288 min
1/a + 1/b = 1/288
1/(b240) + 1/b = 288
(2b  240)/(b^{2 } 240b) = 288
2b  240 = 288b^{2}  288*240b
288b^{2}  288*240b + 240 = 0
b = 240 min
b = 4hrs