This is one of the most straightforward chapters for students. Even a typical pupil could perform better in other chapters. This chapter focuses on a single concept, namely the concept of efficacy.
Consequently, the majority of problems share a fundamental similarity. Almost every aptitude test includes questions from this chapter. In average, two to three problems from this chapter have been posed on the CAT in previous years.
As it is very obvious to all of us that the relationship between work and time is direct.
As one can see, if a person or machine works for a longer period of time, more work will be produced, and if it works for a shorter period of time, less work will be produced, i.e., the output of a machine or person is directly proportional to time, provided that the worker maintains his or her efficacy.
Efficiency Concept
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Generally, time is taken into account when evaluating efficiency. The duration may be calculated in days, hours, minutes, months, etc.
If a person completes his work in 5 days, then his efficiency (per day) is 20%.
Efficiency will be (1/n)
Where n is the number of days taken to complete the work.
Efficiency is also the percentage of work completed in a day.
Basic efficiency conversion table:-
| No of days to complete | Work of 1 day/hour | Percentage Efficiency |
|---|---|---|
| 1 | 1/1 | 1.00 |
| 2 | 1/2 | 0.50 |
| 3 | 1/3 | 0.33 |
| 4 | 1/4 | 0.25 |
| 5 | 1/5 | 0.20 |
| 6 | 1/6 | 0.17 |
| 7 | 1/7 | 0.14 |
| 8 | 1/8 | 0.13 |
| 9 | 1/9 | 0.11 |
| 10 | 1/10 | 0.10 |
For faster and smarter calculation, it is always better to remember the table. It is similar to the percentage conversion table.
The problems of this chapter can be answered using two methods:-
- Unitary Method
- Percentage Efficiency
Unitary method and percentage method can be used alternatively for different set of questions.
At times relying on one method will not help you to get results faster.
Gauging the type of question and employing the method will help you solve question faster.
Both the methods should be at your finger tips so that avoid question traps and directly answer them without losing time and building pressure in the exam.
Efficiency and Time relationship
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Efficiency is inversely proportional to the time.
Example: if A is twice efficient as B, it means, A takes half the time to finish the same job as B requires working alone.
Negative Work Concept
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When one is working, other is destroying. This is all about Negative work.
Example: A man was constructing a wall in 4 days but a drunkard comes and breaks (1/4)th wall every night. So, the drunkard man is doing negative work by breaking the wall at night.
Inverse Proportion Concept
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This is the concept of product constancy i.e., the work allotted or undertaken is unchanged or fixed.
Because the efficiency or rate of work done in one unit of time (mentioned) is inversely related to time, if the rate of work done is larger, the time required is less, and if the rate of work done is lower, the time necessary for the same amount of work is more.
This product constancy approach is confined to constant work; if the quantity of work changes, it no longer works, and we must rely on the unitary method.
When more number of workers work on a project then we calculate the work based on man-hours or man-days invested in the project.
Relationship between work done by persons with different efficiencies
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In this example, the efficiencies of various individuals varies, but because they operate in a group, the efficiency of the group is necessary to determine the time required.
3 adults can complete a task in 4 days, but 12 boys can complete the same task in 3 days. It indicates that 12 man-days are required, or that 12 persons can complete the task in one day.
Similarly, we need 12 x 3 = 36 boys' days, or 36 boys can complete the same task in a single day.
Here we can see that 12 males and 36 youths are required to complete the work in one day. Therefore, we can conclude that the labour of 12 males is equivalent to that of 36 boys. Therefore, the efficacy of 12 men is equivalent to that of 36 boys, or one man is equivalent to three boys.
Thus, a man is three times as efficient as a boy, or two times more efficient than a boy.
Important Formulae
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Equating the work done by two groups of people working h1 and h2 hours per day.
n1.h1 = n2.h2
Equating the work done by the two different groups of people working h1 and h2 hours per day for d1 and d2 days respectively.
n1.h1.d1 = n2.h2. d2
Equating the work done by the two different groups of people working h1 and h2 hours per day for d1 and d2 days, with different efficiencies of respectively.
\(\frac{n_1.h_1.d_1}{\eta_1}= \frac{n_2.h_2.d_2}{\eta_2}\)
Previous year CAT questions
Ques 1: Working alone, the times taken by Anu, Tanu and Manu to complete any job are in the ratio 5 : 8 : 10. They accept a job which they can finish in 4 days if they all work together for 8 hours per day. However, Anu and Tanu work together for the first 6 days, working 6 hours 40 minutes per day. Then, the number of hours that Manu will take to complete the remaining job working alone is: (CAT 2022 Slot 2)
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Ans:
To find the ratio of efficiencies, lets find the LCM of (5,8,10)= 40
Ratio of efficiencies = \(\frac{1}{5}\) x40 : \(\frac{1}{8}\) x40: \(\frac{1}{10}\) x40 = 8:5:4
Assuming the efficiencies to be 8x, 5x and 4x for Anu, Tanu and Manu respectively per hour.
Total workdone if they work for 4 days by Anu, Tanu and Manu, 8 hours a day is
= ( 8x+5x+4x)x 4 x 8 = (17x).(32) = 544x
As Tanu and Tanu worked for 6 days, 6 hours 40 min a day
Work accomplished by them = (13x).(6).(6\(\frac{2}{3}\))= 520x
Remaining work is accomplished by Manu in 4 days will be = \(\frac{544x-520x}{4x}\) = 6 hours
Therefore, Manu can complete the task in 4 days by working 6 hour-day work.
Ques: A group of N people worked on a project. They finished 35% of the project by working 7 hours a day for 10 days. Thereafter. 10 people left the group and the remaining people finished the rest of the project in 14 days by working 10 hours a day. Then the value of N is (CAT 2022 Slot 3)
- 23
- 140
- 36
- 150
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Ans: (B)
35% work by N men working for 7 hours/day is equal to 65% work done by (N-10) men in 14 days for 10 hours/day
Equating the number of people, days and hours worked
13N = 14N -140
N =140
Ques 3: Bob can finish a job in 40 days, if he works alone. Alex is twice as fast as Bob and thrice as fast as Cole in the same job. Suppose Alex and Bob work together on the first day, Bob and Cole work together on the second day, Cole and Alex work together on the third day, and then, they continue the work by repeating this three-day roster, with Alex and bob working together on the fourth day, and so on. Inen, the total number of days Alex would have worked when the job gets finished is (CAT 2022 Slot 3)
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Ans:
As Alex is twice as fast, so time taken by Alex = 40/2 = 20 days
Thrice as fast as Cole so , Time taken by Cole = 20 x 3 = 60 days
Total work done by Bob, Alex and Cole = 40 + 20 + 60 = 120 units
Efficiency of Alex (A) = 120/20 = 6 units/day
Efficiency of Bob (B)= 120/40 = 3 units/day
Efficiency of Cole (C)= 120/60 = 2 units/day
Cycle:-
Day 1 – A+B = 9 units
Day 2 – B+C = 5 units
Day 3 – C+A = 8 units
Workdone per cycle = 22 units.
No of Cycles= 120/22 = 5.45 cycles.
So , there will be total of 5 cycles, Workdone during 5 cycles = 5 x 22 = 110 units
For 1st 15 days, 110 units of work was done.
For 16th day , workdone by (A+B) = 9 units
For 17th day, workdone by (B+C) = 1 unit
Aggregating upto 120 units. (110+19+1)
So, Alex worked for 10 days in 5 complete cycles and on 16th day = 10+1 = 11 days
Ques 4: Anu, Vinu and Manu can complete a work alone in 15 days, 12 days and 20 days, respectively. Vinu works everyday. Anu works only on alternate days starting from the first day while Manu works only on alternate days starting from the second day. Then, the number of days needed to complete the work is (CAT 2022 Slot 1)
- 8
- 6
- 5
- 7
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Ans: (D)
Taking the LCM of (15,12,20) = 60 units
Efficiency of Anu (A) = 60/15 = 4 units/day
Efficiency of Vinu (V) = 60/12 = 5 units/day
Efficiency of Manu (M) = 60/20 = 3 units/day
Calculating the workdone in every 2 days because Anu and Manu work on alternate days.
Work done on day 1 = A+V = 9 units
Work done on day 2 = M+V = 8 units
In every 2 day cycle, 17 units of work is done.
No of cycles = 60/17 = 3.52
So, the number of cycles will be 3 and the workdone during 3 cycles = 3 x 17 = 51 units.
Work done on 6th day = 9 units
Workdone on 7th day = 1 unit
Therefore, 7 days are required to finish the work.
Ques 5: Amar, Akbar and Anthony are working on a project. Working together Amar and Akbar can complete the project in 1 year, Akbar and Anthony can complete in 16months, Anthony and Amar can complete in 2 years. If the person who is neither the fastest nor the slowest works alone, the time in months he will take to complete the project is (CAT 2021 Slot 1)
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Ans:
By taking the LCM of (12,16,24) total work to be done can be found = 48 units
Based on time taken by Amar(a), Akbar (b) and Anthony (c) their respective efficiencies can be found.
a+b = 48/12 = 4 ………. (i)
b+c= 48/16 = 3 ……….. (ii)
c+a= 48/24 = 2 ………… (iii)
After adding all the above equations
a+b+c= 4.5 ………… (iv)
After solving all the 4 equations
a=1.5, b= 2.5, and c=0.5
Amar(a) is neither slowest or fastest with 1.5 units/month as his efficiency
Time required by Amar to complete the task alone = 48/1.5 = 32 months
Ques 6: Anil can paint a house in 60 days while Bimal can paint it in 84 days. Anil starts painting and after 10 days, Bimal and Charu join him. Together, they complete the painting in 14 more days. If they are paid a total of R 21000 for the job, then the share of Charu, in INR, proportionate to the work done by him, is (CAT 2021 Slot 2)
- 9100
- 9000
- 9150
- 9200
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Ans: (A)
Let the total work to be done = LCM (60, 84) = 420 units.
Efficiency of Anil = 420/60 = 7 units/day
Efficiency of Bimal = 420/84 = 5 units/day
Work done by Anil in 10 + 14 days = 24 x 7 = 168 units
Work done by Bimal in 14 days = 14 x 5 = 70 units
Work done by Charu = 420 - 70 - 168 = 182 units
Fraction of work done by Charu = 182/420 = 91/210
Payment received by Charu = 91/210 x 21,000 = Rs. 9,100
Ques 7: One day, Rahul started a work at 9 AM and Gautam joined him two hours later. They then worked together and completed the work at 5 PM the same day. If both had started at 9 AM and worked together, the work would have been completed 30 minutes earlier. Working alone, the time Rahul would have taken, in hours, to complete the work. (CAT 2021 Slot 1)
- 10
- 12
- 12.5
- 11.5
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Ans: (A)
Assuming the efficiency of Rahul be 'R' units per hour and for Gautam be
'G' units per day.
Initially, Rahul works for 8 hours and Gautam works for 6 hours.
Work done by them = 8R + 6G …… (1)
If both of them worked starting from 9 AM, they would have
completed the work in 7.5 hours.
Work done by them = 7.5R + 7.5G …… (2)
8R + 6G = 7.5R + 7.5G
5R = 1.5G
5 R = 3G
Work to be done = 8R + 6G = 8R + 2G = 10R
Time taken by Rahul to complete the work alone = 10R/R = 10hours.
Ques 8: Anil can paint a house in 12 days while Barn can paint it in 16 days. Anil, Barun, and Chandu undertake to paint the house for 2 24000 and the three of them together complete the painting in 6 days. If Chandu is paid in proportion to the work done by him, then the amount in INR received by him is (CAT 2021 Slot 3)
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Ans:
Let the area to be painted = LCM(12, 16) = 48 units
Efficiency of Anil = 48/12 = 4 units/day
Efficiency of Barun = 48/16 = 3 units/day
Work done by Anil in 6 days = 6 x 4 = 24 units
Work done by Anil in 6 days = 6 x 3 = 18 units
Remaining work done by Chandu = 48 - 24 - 18 = 6 units.
Payment received by Chandu = 24000/48 x 6 = Rs. 3,000
Ques 9: John takes twice as much time as Jack to finish a job. Jack and Jim together take one-thirds of the time to finish the job than John takes working alone. Moreover, in order to finish the job, John takes three days more than that taken by three of them working together. In how many days will Jim finish the job working alone? (CAT 2020 Slot 2)
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Ans:
Let John's efficiency be 1 unit/day.
Jack's efficiency is 2 units/day.
Jack and Jim's efficiency is thrice John's efficiency.
:: 2 + ejim = 3 x 1
eJim = 1
John takes three days more than that taken by three of them
Let the time taken by all three together is t
:. Total work to be done = 4 x t = 1 x (t + 3)
d= 1
Jack finished the work in d + 3 = 3+1= 4 days.
Since, Jack and Jim have same efficiency, Jim will also finish the work in 4 days.
Ques 10: A contractor agreed to construct a 6 km road in 200 days. He employed 140 persons for the work. After 60 days, he realized that only 1.5 km road has been completed. How many additional people would he need to employ in order to finish the work exactly on time? (CAT 2020 Slot 3)
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Ans:
In 60 days 1.5 kms out of 6 kms is built (1/4)th work done
Days required to complete the work = 4x 60 = 240 days, i.e.,
Extra days required = 240 -60 = 180 days.
Days remaining now is 200 - 60 = 140 days.
:: To complete the remaining work in 140 days, contractor should hire more people. Assuming x more men are employed.
The work which 140 men would take 180 more days, now need to be completed by 140 + x men in 140 days.
.: 180 x 140 = 140 x (140 + x)
2x = 40
:. Contractor hired 40 more people.
Ques 11: At their usual efficiency levels, A and B together finish a task in 12 days. If A had worked half as efficiently as she usually does, and B had worked thrice as efficiently as he usually does, the task would have been completed in 9 days. How many days would A take to finish the task if she works alone at her usual efficiency? (CAT 2019 Slot 1)
- 36
- 24
- 12
- 18
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Ans: (D)
LCM of 12 and 9 = 36 units.
Let the efficiency of A and B be a and b respectively.
Work done per day when A and B are working together = 36/12 = 3units.
a+ b=3 (i)
Work done per day when A is working at half efficieny and B is working at thrice efficiency = 36/9 = 4 units.
(a/2) + 3b = 4 ... (ii)
Solving (i) and (ii), we get; a = 2.
Time taken by A to complete the work = 36/2 = 18 days.
Ques 12: Three men and eight machines can finish a job in half the time taken by three machines and eight men to finish the same job. If two machines can finish the job in 13 days, then how many men can finish the job in 13 days? (CAT 2019 Slot 1)
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Ans:
Let the work done by one man and one machine per day be x and y respectively.
3 men and 8 machines can finish a job in half the time taken by 3 machines and 8 men to finish the same job.
efficiency of 3 men and 8 machines is twice that of 8 men and 3 machines.
(3x + 8y) = 2(8x + 3y)
13x = 2y.
So, work done by 13 men in a day = work done by 2 machines in a day
If two machines can finish the job in 13 days, same work will be done by 13 men in 13 days.
Ques 13: Anil alone can do a job in 20 days while Sunil alone can do it in 40 days. Anil starts the job, and after 3 days, Sunil joins him. Again, after a few more days, Bimal joins them and they together finish the job. If Bimal has done 10% of the job, then in how many days was the job done? (CAT 2019 Slot 2)
- 13
- 12
- 15
- 14
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Ans: (A)
Anil in one day can do 1/20th of the work
Sunil in one day can do 1/40th of the work
Anil would have done 3/20th of the work by the time Sunil joins
After Sunil joins, they both would be doing 3/40th of work everyday
Bimal joins them after some days and finishes 10% of the work (i.e. 1/10th of the work).
As, Anil alone had done 3/20th of the work in first 3 days and Bimal completes 1/10th of the work
Total they would have done 3/20 + 1/10 = 1/4th of the work.
Remaining work = 3/4th, which would be done by Anil and Sunil together.
Anil and Sunil together complete= 1/20th + 1/40th = 3/40th work inday.
Therefore, time taken to complete 3/4th of the work = 10 days.
Total number of days = 3 + 10 = 13 days
Ques 14: John gets Rs 57 per hour of regular work and Rs 114 per hour of overtime work. He works altogether 172 hours and his income from overtime hours is 15% of his income from regular hours. Then, for how man hours did he work overtime? (CAT 2019 Slot 2)
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Ans:
If John works the same number of reqular and over-time hours sav
The income would be 57p and 114p
If John works x' hours regular work and 'y' hours overtime
So, the income would be 57x and 114y
we are told that 114y is 15% of 57x
114y = 0.15 x 57x
y= 0.075x
As given in the question
x + y = 172
x + 0.075x = 1.075x = 172
x = 160
y= 172 – 160 = 12
Therefore, the number of hours he worked overtime is 12 hours.
Ques 15: Humans and robots can both perform a job but at different efficiencies. Fifteen humans and five robots working together take thirty days to finish the job, whereas five humans and fifteen robots working together take sixty days to finish it. How many days will fifteen humans working together (without any robot) take to finish it? (CAT 2018 Slot 1)
- 36
- 32
- 45
- 40
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Ans: (B)
Assuming work-done by a human and a robot in a day are 'h' and ‘r’ respectively.
Total work = 30(15h + 5r) = 60(5h + 15r)
15h + 5r = 10h + 30r
h= 5r
Let 15 humans take 'y' days to finish the job.
30(15h + 5r) = y x 15h
30(15h + h) = y × 15h
y = 32
How to Prepare Time & Work Questions for CAT?
- Go through the concepts and understand the relationship between the time and work done.
- Relationship between time and efficiency is of utmost importance keeping the amount of work constant.
- One formula provided in the article plays a very important role in solving maximum questions asked from this chapter.
- Interchangeability between fractions and percentage plays a crucial role in this chapter.
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