In our daily life, we come across situations where we need to complete a particular job in a reasonable time. We have to complete the project earlier or later depending upon the needs. Accordingly, the men on duty have to be increased or decreased, i.e. the time allowed and the men engaged for a project are inversely proportional to each other, i.e. the more the number of men involved, the lesser is the time required to finish a job. We also come across situations where time and work or men and work are in direct proportion to each other.
For solving problems on time and work, we follow the following general rules: 1. If ‗A‘ can do a piece of working ‗A‘ will finish 1/nth work in one day.
2. If 1/n of a work is done by ‗A‘ in one day, then ‗A‘ will take n days to complete the full work. 3. If ‗ A‘ does 1/nth of a work in one hour then to complete the full work, ‗A‘ will take n/m hours. 4. If ‗ A‘ does three times faster work than ‗B‘ then ratio of work done by A and B is 3:1 and ratio of time
taken by A and B is1: 3.
5. A, B and C can do a piece of work in T1, T2 and T3, days, respectively. If they have worked for D1, D2 and
D3 days respectively, then
Amount of work dine by A= D1/T1
Amount of work dine by B= D2/T2
And, Mount of work done by C=D3/T3
Also, the amountof work done by A, B and C together
= D1/T1+D2/T2+D3/T3
Which will be equal to 1, if the work is complete?
SOME USEFUL SHORT-CUT METHODS
1. If A can do a piece do a piece of work in X days and B can do the same work in Y days, then both of them working together will do the same work in XY/ X+Y days.
Explanation
A‘s 1 day‘s work = 1/X B‘s 1 day‘s work = 1/Y
Then,(A+B)‘s 1 day‘s work = 1/X+1/Y= X+Y/XY
A and B together can complete the work on
= XY/ X+Y days.
Illustration 1 A can finish a piece of work by working alone in 6 days and B, while works alone, can finish the same work in 12days. If both of them work together, then in how many days, the work will be finished? Solution Here, X= 6 and y = 12.
Working together, A and B will complete the work
in = XY/ X+Y days = 612/6+12 days, i.e. 4days.
2. If A, B and C, while working alone, can complete a work in X, Y and Z days respectively, then they will together complete the work in XYZ/XY+YZ+ZX days.
Explanation
A‘s 1 day‘s work = 1/X B‘s 1 day‘s work = 1/Y C‘s 1 day‘s work = 1/Z
(A+B+C)‘s 1 days work
=1/X+1/Y+1/Z = (XY+YZ+ZX)/XYZ.
Illustration 2 A,B and C can complete a piece of work in 10, 15 and 18 days, respectively. In how many days would all of them complete the same work working together?
Solution Here, X=10, Y=15 and Z= 18. Therefore, the work will be completed in
= XYZ/ XY+YZ+ZX days.
= 101518/10 15 +15 18+18 10 days,
i.e. 2700/600 or, 4 ½ days.
3. Two persons A and B, working together , can complete a piece of work in X days. If A, working alone, can complete the work in XY/Y-X days.
Explanation
A and B together can complete the work in X days.
(A+B)‘s 1 day‘s work = 1/X
Similarly, A‘s 1day‘s work= 1/Y
There fore, B‘s 1 day‘s work =1/X-1/Y= Y-X/XY
B alone can complete the work in (XY/Y-X)days
B alone will complete the work in
Illustration 3 A and B working together take 15 days to complete a piece of work. If A alone can do this work in 20 days, how long would B take to complete the same work?
Solution Here, X = 15, and Y=20.
= XY/Y-X days =1520/20-15, i.e. 60 days
4. If A and B, working together, can finish a piece of work in X days, B and C in Y days, then a) A, B and C working together, will finish the job in (2XYZ/XY+YZ- ZX) days.
b) A alone will finish the job in (2XYZ/XY+YZ- ZX) days. c) B alone will finish the job in (2XYZ/ZX+XY- YZ) days. Explanation
(A+B)‘s 1 day‘s work = 1/X (B+C)‘s 1 day‘s work = 1/Y (C+A)‘s 1 day‘s work = 1/Z
So, [(A+B) + (B+C)+ (C+A)]‘s 1 day‘s work = 1/X+1/Y+1/Z or, 2(A+B+C)‘s 1day‘s work = (1/X+1/Y+1/Z)
or, (A+B+C)‘s 1day‘s work = ½ (1/X+1/Y+1/Z)
i.e (XY+YZ-XZ/2XYZ)
A, B and C working together, will complete the work in (2XYZ/XY+ZX-XY) days.
Also, A‘s 1 days work – (A+B+C)‘s 1day‘s work –(B+C)‘s 1 days work = ½ (1/X+1/Y+1/Z)- 1/Y
= ½ (1/X-1/Y+1/Z) = XY+YZ-XZ/2XYZ
So. A alone can do the work in (2XYZ/XY+YZ+XZ) days
Similarly, B alone can do the work in (2XYZ/YZ+XY+XY) days and C alone can do the work in(2XYZ/ZX+XY+YZ) days.
Illustration 4 A and B can do a piece of work in 12 days, B and C 15 days, C and A in 20 days. How long would each take separately to do the same work?
Solution Here, X = 12, Y=15 and Z=20.
A alone can do the work in = 2XYZ/XY+YZ-ZX
= 2121520/1215+1520-2012 days. or, 7200/240, i.e. 30 days.
B alone can do the work in = 2XYZ/ZY+ZX-XY days
= 2121520/1520+2012-1215 days or, 7200/360, i.e. 20 days.
C alone can do the work in = 2XYZ/ZX+XY-YZ
5. (a) If A can finish a work in X days and B is k times efficient than A, then the time taken by both A and B working together to complete the work is x/1+k.
(b) If A and B working together can finish a work in X days and B is k times efficient than A, then the time taken by
(i) A, working alone, to complete the work is (k+1)X. (ii) B, working alone, to complete the work is (k+1/k)X.
Illustration 5 Harbans Lal can do a piece of work in 24 days. If Bansi Lal works twice as fast as Harbans Lal, how long would they take to finsh the work working together?
Solution Here, X =24 and k=2.
Time taken by Harbans Lal and Bansi Lal, woking together, to complete the work
= (X/1+k)days.
= (24/1+2) days, i.e. 8 days
Illustration 6 A and B together can do a piece of work in 3 days. If A does thrice as much work as B in a given time, find how long A lone would take to do the work?
Solution Here, X = 3 and k = 3.
Time taken by A, working alone, to complete the work
= (k+1/k) X = (3+1/3)3 = 4days
6. If A working alone takes a days more than A and B working alone takes b days more than A and B
together, then the number of days taken by A and B, working together, to finish a job is given by ab.
Illustration 7 A alone would take 8 hours more to complete the job than if both A and B worked together. If B worked alone, he took 4 ½ hours more to complete the job than A and B worked together. What time would they take if both A and B worked together?
Solution Here, a = 8 and b = 9/2.
Time taken by A and B, working together, to complete the job
= ab days
= 8×9/2 or, 6days.
EXERCISE
1. Nishu and Archana can do a piece of work in 10 days and Nishu alone can do it in 12 days. In how many days can Archana do it alone?
1. 60 days 2. 30 days 3. 50 days 4. 45 days
2. Baba alone can do a piece of work in 10 days. Anshu alone can do it in 15 days. If the total wages for the work is Rs.50. How much should Baba be paid if they work together for the entire duration of his work?
1. 30/- 2. 20/- 3. 50/- 4. None
3. 4 men and 3 women finish a job in6 days, and 5 men and 7 women can do the same job in 4 days. How long will 1 man and 1 woman take to do the work?
1. 22(2/7) days 2. 25(1/2) days 3. 5(1/7) days 4. 12(7/22) days
4. A can do a piece of work in 10 days and B can do the same work in 20 days. With the help of C, they finish the work in 5 days. How long will it take for C alone to finish the work?
1. 20 days 2. 10 days 3. 35 days 4. 15 days
5. A can do a piece of work in 20 days. He works at it for 5 days and then B finishes it in 10 more days. In how many days will A and B together finish the work?
1. 8 days 2. 10 days 3. 12 days 4. 6 days
6. A and B undertake to do a piece of work for Rs.100. A can do it in 5 days and B can do it in 10 days. With the help of C, they finish it in 2 days. How much should C be paid for his contribution?
7. *Twenty workers can finish a piece of work in 30 days. After how many days should 5 workers leave the job so that the work is completed in 35 days?
1. 5 days 2. 10 days 3. 15 days 4. 20 days
8. Arun and Vinay together can do a piece of work in 7 days. If Arun does twice as much work as Vinay in a given time, how long will Aruna alone take to do the work?
1. 6.33 days 2. 10.5 days 3. 11 days 4. 72 days
9. X number of men can finish a piece of work in 30 days. If there were 6 men more, the work could be finished in 10 days less. What is the original number of men?
1. 10 2. 11 3. 12 4. 15
10. Sashi can do a piece of work in 25 days and Rishi can do it in 20 days. They work for 5 days and then Sashi
goes away. In how many more days will Rishi finish the work? (11 days)
1. 10 days 2. 12 days 3. 14 days 4. None
11. Raju can do a piece of work in 10 days, Vicky in 12 days and Tinku in 15 days. They all start the work together, but Raju leaves after 2 days and Vicky leaves 3 days before the work is completed. In how many days is the work completed?
1. 5 days 2. 6 days 3. 7 days 4. 8 days
12. Manoj takes twice as much time as Anjay and thrice as much as Vijjay to finish a piece overwork. Together they finish the work in 1 day. What is the time takes by Manoj to finish the work?
1. 6 days 2. 3 days 3. 2 days 4. None
13. In a company XYZ Ltd. A certain number of engineers can develop a design in 40 days. If there were 5 more engineers, it could be finished in 10 days less. How many engineers were there in the beginning?
1. 18 2. 20 3. 25 4. 15
14. If 12 men and 16 boys can do a piece of work in 5 days and 13 men and24 boys can do it in 4 days, compare the daily work done by a man with that done by a boy?
1. 1:2 2. 1:3 3. 2:1 4. 3:1
15. A can do a work in 10 days and B can do the same work in 20 days. They work together for 5 days and then a goes away. In how many more days will B finish the work?
1. 5 days 2. 6.5 days 3. 10 days 4. 8 1/3 days
16. 30 men working 5h a day can do a work in 16 days. In how many days will 20 men working 6h a day do the same work?
1. 22 1/2 days 2. 20 days 3. 21 days 4. None
17. ajay and Vijay undertake to do a piece of work for Rs. 200. ajay alone can do it in 24 days while Vijay alone can do it in 30 days. With the help of pradeep they finish the work in 12 days. How much should pradeep get for his work
1. 20/- 2. 100/- 3. 180/- 4. 50/-
18. *15 men could finish a piece of work in 210 days. But at the end of 100 days, 15 additional men are employed. In how many more days will the work be completed?
1. 80 days 2. 60 days 3. 55 days 4. 50 days
19. Ajay, vijay and Sanjay are employed to do a piece of work for Rs. 529. Ajay and Vijay together are supposed to do 19/23 of the work. Vijay & Sanjay together 8/23 of the work how much should Ajay be paid?
1. 245/- 2. 295/- 3. 300/- 4. 345/-
20. *In a fort there was sufficient food for 200 soldiers for 31 days. After 27 days 120 soldiers left the fort. For how many extra days will the rest of the food last for the remaining soldiers?
1. 12 days 2. 10 days 3. 8 days 4. 6 days
21. Ajay and Vijay cam do a piece of work in 28 days. With the help of Manoj, they can finish it in 21days. How long will Manoj take to finish the work alone.
1. 84 days 2. 80 days 3. 75 days 4. 70 days
22. Ajay and Vijay together can do a piece of work in 6 days. Ajay alone does it in 10 days. What time does Vijay require to do it alone?
1. 20 days 2. 15 days 3. 25 days 4. 30 days
23. A cistern is normally filled in 5 hours. However, it takes 6 hours when there is leak in its bottom. If the cistern is full, in what time shall the leak empty it?
1. 6h 2. 5h 3. 30h 4. 15h
24. *Pipe A and B running together can fill a cistern in 6 minutes. If B takes 5 minutes more than A to fill the cistern, then the time in which A and B will fill the cistern separately will be respectively?
1. 15 min, 20 min 2. 15 min, 10 min 3. 10 min, 15 min 4. 25 min, 20 min
25. A can do a work in 18 days, B in 9 days and C in 6 days. A and B start working together and after 2 days C joins them. In how many days will the job be completed?
1. 4.33 days 2. 4 days 3. 4.66 days 4. 5 days
26. 24 men working 8 h a day can finish a work in 10 days. Working at a rate of 10 h a day, the number of men required to finish the work in 6 days is
1. 30 2. 32 3. 34 4. 36
27. A certain job was assigned to a group of men to do it in 20 days. But 12 men did not turn up for the job and the remaining men did the job in 32 days. The original number of men in group was
1. 32 2. 34 3. 36 4. 40
28. *12 men complete a work in 18 days. 6 days after they had started working, 4 men join them. How many more days will all of them to complete the remaining work?
1. 10 days 2. 12 days 3. 15 days 4. 9 days
29. A cistern is normally filled in 6 h but takes 4H longer to fill because of a leak in its Bottom. If the cistern is full, the leak will empty it in how much time?
1. 15 h 2. 16 h 3. 20 h 4. None
30. There are two pipes in a tank. Pipe A is for filling the tank and Pipe B is for emptying the tank. If A can fill the tank in 10 hours and B can empty the tank in 15 hours hen find how many hours will nit take to completely fill a half empty tank?
1. 30 hours 2. 15 hours 3. 20 hours 4. 33.33 hours
31. Abbot can do some work in 10 days, Bill can do it in 20 days and Clinton can do it in 40 days. They start working in turn with Abbot starting to work on the first day followed by Bill on the second day and by Clinton on the third day and again by Abbot on the fourth day and so on till the work is completed fully. Find the time taken to complete the work fully?
1. 16 days 2. 15 days 3. 17 days 4. 16.5 days
32. A, B and C can do some work in 36 days. A and B together do twice as much work as C alone and A and C together can do thrice as much work as B alone. Find the time taken by C to do the whole work?
1. 72 days 2. 96 days 3. 108 days 4. 120 days
33. *There are three Taps A, B and C in a tank. They can fill the tank in 10 hrs, 20 hrs and 25 hrs respectively. At first, all of them are opened simultaneously. Then after 2 hours, tap C is closed and A and B are kept running
after the 4th hour, tap B is also closed. The remaining work is done by Tap A alone. Find the percentage of the
work done by Tap A by itself.
1. 32% 2. 52% 3. 75% 4. None
34. Two taps are running continuously to fill a tank. The 1st tap could have filled it in 5 hours by itself and the
second one by itself could have filled it in 20 hours. But the operator failed to realize that there was a leak in the tank from the beginning which caused a delay of one hour in the filling of the tank. Find the time in which the leak would empty a filled tank?
1. 15 hours 2. 20 hours 3. 25 hours 4. 40 hours
35. A can do some work in 24 days, B can do it in 32 days and C can do it in 60 days. They start working together. A left after 6 days and B left after working for 8 days. How many more days are required to complete the whole work?