Time and Work

Figure 1: Time and work problems involve efficiency, pipes, and cisterns.

1. Basic Concepts

KEY FORMULAS

If A can do a work in X days, A's 1 day work = 1/X
If A's 1 day work = 1/X, A can finish work in X days
If A and B can do work in X and Y days: Together = XY/(X+Y) days
If A, B, C can do in X, Y, Z days: Together = XYZ/(XY+YZ+ZX) days

ADVANCED FORMULAS

If A is a times efficient than B and A finishes in X days: Together = aX/(a+1) days
If A alone takes X days more than A+B together, and B alone takes Y days more: Together = √(XY) days

2. Pipes and Cisterns

KEY CONCEPTS

Inlet: Pipe that fills a tank
Outlet: Pipe that empties a tank
If a pipe fills in x hours: Part filled in 1 hour = 1/x
If a pipe empties in y hours: Part emptied in 1 hour = 1/y

If a pipe fills in x hours and another empties in y hours:

Net part filled in 1 hour = (1/x - 1/y)

Time to fill = xy/(y-x) hours (when y > x)

REMEMBER

For pipes filling a tank with different diameters: The amount of water flowing is proportional to the square of the diameter.

3. Exercise (Questions 1-10)

Q1. A can complete a work in 12 days. A and B together can complete it in 8 days. In how many days can B alone complete the work?

(a) 15 days
(b) 18 days
(c) 24 days
(d) 28 days

Answer: (c) 1/B = 1/8 - 1/12 = 1/24 ⇒ B = 24 days

Q2. I can complete one-third of the work in 8 days. In how many days can I complete that work?

(a) 18
(b) 12
(c) 24
(d) Cannot be determined

Answer: (a) 8 × 3 = 24 days

Q3. A can finish in 18 days and B in half the time. Working together, what part of work can they finish in a day?

(a) 1/6
(b) 1/9
(c) 2/5
(d) 2/7

Answer: (a) 1/18 + 1/9 = 3/18 = 1/6

Q4. A and B can finish in 10 days, B and C in 18 days. A worked 5 days, B 10 days, C 15 days finished remaining. In how many days could C alone finish?

(a) 30 days
(b) 15 days
(c) 45 days
(d) 24 days

Answer: (c) 5/10 + 5/18 + 10/C = 1 ⇒ C = 45 days

Q5. Two pipes A and B can fill a cistern in 10 and 15 minutes. Both opened, but at end of 3 minutes, B turned off. How much time will the cistern take to fill?

(a) 6 min
(b) 8 min
(c) 10 min
(d) 12 min

Answer: (b) In 3 min, filled = 3(1/10+1/15) = 1/2. Remaining 1/2 by A in 5 min. Total = 8 min

Q6. A sum of ₹25 was paid for a work which A can do in 32 days, B in 20 days, B+C in 12 days and D in 24 days. How much did C receive if all work together?

(a) ₹3
(b) 16/3
(c) ₹3
(d) 17/3

Answer: (b) C's work = 1/12 - 1/20 = 1/30. Total = 1/32+1/20+1/30+1/24 = 5/32. C receives = (1/30)/(5/32)×25 = 16/3

Q7. Sunil and Pradeep can complete a work in 5 and 15 days. They both work for one day and then Sunil leaves. In how many days will remaining work be completed by Pradeep?

(a) 11 days
(b) 12 days
(c) 15 days
(d) 8 days

Answer: (a) 1 day work = 1/5+1/15 = 4/15. Remaining = 11/15. Pradeep = 11/15 × 15 = 11 days

Q8. If 6 men and 8 boys can do work in 10 days while 26 men and 48 boys in 2 days, time taken by 15 men and 20 boys is:

(a) 4 days
(b) 5 days
(c) 6 days
(d) 7 days

Answer: (a) Let man = x, boy = y. 6x+8y=1/10, 26x+48y=1/2. Solving x=1/100, y=1/200. 15 men+20 boys = 15/100+20/200=1/4. Time = 4 days

Q9. I can complete a work in 9 days. After they have worked for 6 days, 6 more men join them. How many days will they take to complete remaining work?

(a) 2 days
(b) 3 days
(c) 4 days
(d) 5 days

Answer: (a) 12 men's 1 day work = 1/9. In 6 days = 2/3. Remaining = 1/3. 18 men's 1 day work = 1/6. Time = 2 days

Q10. Three taps A, B and C can fill a tank in 12, 15 and 20 hours. If A is open all the time and B and C open for one hour each alternately, the tank will be full in:

(a) 6 hrs
(b) 6⅔ hrs
(c) 7 hrs
(d) 7½ hrs

Answer: (c) (A+B)=3/20, (A+C)=2/15 per hour. In 2 hrs = 17/60. In 6 hrs = 51/60. Remaining 9/60 = 3/20. Total = 7 hrs

4. Exercise (Questions 11-20)

Q11. 26 men can complete a work in 17 days. How many more men must be hired to complete in 13 days?

(a) 9
(b) 8
(c) 6
(d) 18

Answer: (b) 17:13 :: 26:x ⇒ x=34. Extra = 34-26=8

Q12. Work done by A in one day is half of B, and B is half of C. If C alone can complete in 7 days, in how many days can A, B and C together complete?

(a) 28
(b) 14
(c) 4
(d) 21

Answer: (c) Ratio A:B:C = 1:2:4. C=7 days, B=14, A=28. Together = 1/28+1/14+1/7 = 1/4 ⇒ 4 days

Q13. 16 men can complete a work in 8 days. In how many days can 12 men complete?

(a) 10
(b) 9⅓
(c) 10⅔
(d) Cannot be determined

Answer: (c) 16×8/12 = 32/3 = 10⅔ days

Q14. A can complete a work in 12 days. A and B together in 8 days. In how many days can B alone complete?

(a) 15 days
(b) 18 days
(c) 24 days
(d) 28 days

Answer: (c) 1/B = 1/8 - 1/12 = 1/24 ⇒ B=24 days

Q15. 12 Men can complete one-third of work in 8 days. In how many days can 16 men complete that work?

(a) 18
(b) 12
(c) 24
(d) Cannot be determined

Answer: (a) 12 men complete whole in 24 days. 16 men = 12×24/16 = 18 days

Q16. Computer A takes 3 minutes, B takes 5 minutes. A, B and C process average 14 inputs in one hour. How many minutes does C take for one input?

(a) 10
(b) 4
(c) 6
(d) 8

Answer: (c) 1/3 + 1/5 + 1/C = 42/60 = 7/10 ⇒ 1/C = 1/6 ⇒ C=6 min

Q17. 21 binders can bind 1400 books in 15 days. How many binders required to bind 800 books in 20 days?

(a) 7
(b) 9
(c) 12
(d) 14

Answer: (b) 21×15×800/(1400×20) = 9

Q18. A and B can finish in 10 days, B and C in 18 days. A worked 5 days, B 10 days, C 15 days finished remaining. In how many days could C alone finish?

(a) 30 days
(b) 15 days
(c) 45 days
(d) 24 days

Answer: (c) 5/10 + 5/18 + 10/C = 1 ⇒ C=45 days

Q19. A is 30% more efficient than B. How much time will they take together to complete a job which A alone could have done in 23 days?

(a) 11 days
(b) 13 days
(c) 20/17 days
(d) None of these

Answer: (b) A:B = 130:100 = 13:10. B time = 23×13/10 = 299/10. Together = 1/23 + 10/299 = 1/13 ⇒ 13 days

Q20. Sunil and Pradeep can complete in 5 and 15 days. They both work for one day and then Sunil leaves. In how many days will remaining work be completed by Pradeep?

(a) 11 days
(b) 12 days
(c) 15 days
(d) 8 days

Answer: (a) 1 day work = 1/5+1/15=4/15. Remaining=11/15. Pradeep=11/15×15=11 days

5. Exercise (Questions 21-30)

Q21. Two pipes A and B can fill in 15 and 12 hours. Pipe B alone kept open for 3/4 of time and both pipes for remaining. In how many hours will the tank be full?

(a) 18 h
(b) 20 h
(c) 10 h
(d) 13.5 h

Answer: (c) 1/12×3x/4 + 1/15×x/4 + 1/12×x/4 = 1 ⇒ x=10 h

Q22. If 6 men and 8 boys can do work in 10 days while 26 men and 48 boys in 2 days, time taken by 15 men and 20 boys is:

(a) 4 days
(b) 5 days
(c) 6 days
(d) 7 days

Answer: (a) x=1/100, y=1/200. 15 men+20 boys = 15/100+20/200=1/4. Time = 4 days

Q23. 12 men complete a work in 9 days. After 6 days, 6 more men join. How many days will they take to complete remaining?

(a) 2 days
(b) 3 days
(c) 4 days
(d) 5 days

Answer: (a) 12 men in 6 days = 2/3. Remaining=1/3. 18 men's 1 day = 1/6. Time=2 days

Q24. Three taps A, B and C can fill in 12, 15 and 20 hours. If A open all time and B and C open one hour alternately, tank will be full in:

(a) 6 hrs
(b) 6⅔ hrs
(c) 7 hrs
(d) 7½ hrs

Answer: (c) (A+B)=3/20, (A+C)=2/15. In 2 hrs=17/60. In 6 hrs=51/60. Remaining=9/60=3/20. Total=7 hrs

Q25. Two pipes A and B alone can fill in 36 and 45 min. Waste pipe C can empty in 30 min. A and B opened, after 7 min C opened. In how much time will tank be full?

(a) 39 min
(b) 30 min
(c) 20 min
(d) 36 min

Answer: (a) In 7 min = 7(1/36+1/45)=7/20. Remaining=13/20. (1/36+1/45-1/30)=1/60. Time=60×13/20=39 min

Q26. A is half as good a workman as B and together they finish in 14 days. In how many days working alone will B finish?

(a) 20 days
(b) 21 days
(c) 22 days
(d) None of these

Answer: (b) 1/B + 1/2B = 1/14 ⇒ 3/2B = 1/14 ⇒ B=21 days

Q27. A can do 3/4 of a work in 12 days. In how many days can he finish 1/8 of the work?

(a) 6 days
(b) 5 days
(c) 3 days
(d) 2 days

Answer: (d) 12×4/3×1/8 = 2 days

Q28. A man is twice as fast as a woman. Together they do work in 8 days. In how many days each will do if engaged alone?

(a) man-14, woman-28
(b) man-12, woman-24
(c) man-10, woman-20
(d) None of these

Answer: (b) 1/x + 1/2x = 1/8 ⇒ x=12 (man), woman=24

Q29. 24 men working 8 hours a day can finish in 10 days. Working 10 hours a day, number of men required to finish in 6 days is:

(a) 30
(b) 32
(c) 34
(d) 36

Answer: (b) 24×10×8 = m×6×10 ⇒ m=32

Q30. A tyre has two punctures. First alone would make flat in 9 minutes, second in 6 minutes. How long does it take both punctures together to make it flat?

(a) ½ min
(b) 3½ min
(c) 3⅗ min
(d) 4¼ min

Answer: (c) 1/9 + 1/6 = 5/18 ⇒ 18/5 = 3⅗ min

6. Exercise (Questions 31-59)

Q31. A man, woman or boy can do a job in 20, 30 or 60 days. How many boys must assist 2 men and 8 women to do work in 2 days?

(a) 15 boys
(b) 8 boys
(c) 10 boys
(d) None of these

Answer: (b) 2×(1/20)+8×(1/30)+x×(1/60)=1/2 ⇒ x=8

Q32. If 15 women or 10 men can complete a project in 55 days, in how many days will 5 women and 4 men working together complete?

(a) 75
(b) 8
(c) 9
(d) 85

Answer: (a) 15W = 10M. 5W+4M = 5W+6W=11W. 15×55/11=75 days

Q33. A can do in 10 days, B in 15 days. They work together for 5 days and rest by C in 2 days. If they get ₹450, how should they divide?

(a) 225,150,75
(b) 200,150,100
(c) 250,100,100
(d) 175,175,100

Answer: (a) Ratio = 5/10 : 5/15 : 2/12 = 3:2:1. Shares = 225,150,75

Q34. After working for 8 days, Anil finds only 1/3 work done. He employs Rakesh who is 60% efficient. How many more days will Anil take to complete?

(a) 15 days
(b) 12 days
(c) 10 days
(d) 8 days

Answer: (c) Anil's 1 day = 1/24. Rakesh = 1/40. Together = 1/15. Remaining = 2/3. Time = 15×2/3=10 days

Q35. X can do in 15 days. If joined by Y who is 50% more efficient, in what time will X and Y together finish?

(a) 10 days
(b) 6 days
(c) 18 days
(d) Data insufficient

Answer: (b) Y = 1/15×1.5 = 1/10. Together = 1/15+1/10=1/6 ⇒ 6 days

Q36. If 12 men or 15 women or 18 boys can do work in 15 days of 8 hours, find how many men assisted by 5 women and 6 boys will finish same work in 16 days of 9 hours.

(a) 6 men
(b) 2 men
(c) 8 men
(d) 4 men

Answer: (b) 1 man = 1.5 boys, 1 woman = 1.2 boys. 5W+6B=12B. Total boys needed=18×15×8/(16×9)=15. Already 12, need 3 boys = 2 men

Q37. 2 men and 3 boys can do in 10 days while 3 men and 2 boys in 8 days. In how many days can 2 men and 1 boy do the work?

(a) 12 days
(b) 11 days
(c) 15 days
(d) 13 days

Answer: (a) 2x+3y=1/10, 3x+2y=1/8. Solving x=7/200, y=1/100. 2x+y=2/25 ⇒ 12.5 days

Q38. A man can do in 5 days but with son's help in 3 days. In what time can son do alone?

(a) 6 days
(b) 7 days
(c) 7.5 days
(d) 8 days

Answer: (c) 1/3 - 1/5 = 2/15 ⇒ 7.5 days

Q39. Pipe A can fill in 5 hours, B in 10 hours, C in 30 hours. If all open, in how many hours will tank be filled?

(a) 2
(b) 2.5
(c) 3
(d) 3.5

Answer: (c) 1/5+1/10+1/30 = 1/3 ⇒ 3 hours

Q40. Two taps can fill in 12 and 18 minutes. Both open for 2 minutes and first turned off. In how many minutes more will tank be filled?

(a) 15 min
(b) 20 min
(c) 11 min
(d) 13 min

Answer: (d) 2(1/12+1/18)=5/18. Remaining=13/18. Second tap = 18×13/18=13 min

Q41. Fill pipe A is 3 times faster than B and takes 10 minutes less time. Find when cistern will be full if B only opened.

(a) 20 min
(b) 18 min
(c) 15 min
(d) 10 min

Answer: (c) B = x, A = x/3. x - x/3 = 10 ⇒ x=15 min

Q42. Three fill pipes A, B, C can fill separately in 3, 4, 6 minutes. A opened first, after 1 min B, after 2 min C. Find time when cistern full?

(a) 2 min
(b) 4 min
(c) 3 min
(d) None of these

Answer: (a) x/3 + (x-1)/4 + (x-2)/6 = 1 ⇒ x = 19/9 ≈ 2.11 min

Q43. Water flows at 3 m/s through pipe of radius 4 cm. How many hours to fill tank 40m×30m×8m?

(a) 176.6 hours
(b) 120 hours
(c) 135.5 hours
(d) None of these

Answer: (a) Volume per sec = π×0.04²×3 = 0.0151 m³. Tank volume = 9600 m³. Time = 9600/0.0151/3600 ≈ 176.6 hrs

Q44. 40 men can cut 60 trees in 8 hrs. If 8 men leave, how many trees cut in 12 hours?

(a) 72
(b) 60
(c) 48
(d) 36

Answer: (a) 40×8×W2 = 32×12×60 ⇒ W2=72

Q45. 10 horses and 15 cows eat grass of 5 acres. How many acres will feed 15 horses and 10 cows, supposing a horse eats as much as 2 cows?

(a) 40/7 acres
(b) 39/8 acres
(c) 40/11 acres
(d) 25/9 acres

Answer: (a) 10 horses+15 cows = 35 cows. 15 horses+10 cows = 40 cows. Acres = 5×40/35 = 40/7

Q46. 4 pipes can fill in 15, 20, 30 and 60 hours. First opened at 6 am, second at 7 am, third at 8 am, fourth at 9 am. When will reservoir be full?

(a) 11 am
(b) 12 pm
(c) 12.30 pm
(d) 1.00 pm

Answer: (d) t/15 + (t-1)/20 + (t-2)/30 + (t-3)/60 = 1 ⇒ t=7 hours ⇒ 1 pm

Q47. Two pipes can fill in 14 and 16 hours. Due to leakage, 32 minutes extra taken. If cistern is full, in what time would leak empty it?

(a) 110 hr
(b) 112 hr
(c) 115 hr
(d) 100 hr

Answer: (b) Together = 112/15 hrs. With leak = 8 hrs. Leak = 15/112 - 1/8 = 1/112 ⇒ 112 hrs

Q48. A tank is filled in 5 hours by pipes A, B and C. C is twice as fast as B and B twice as fast as A. How much time will A alone take?

(a) 20 hrs
(b) 25 hrs
(c) 35 hrs
(d) Cannot be determined

Answer: (c) 1/x + 2/x + 4/x = 1/5 ⇒ 7/x = 1/5 ⇒ x=35 hrs

Q49. 12 buckets fill a tank when capacity is 13.5 litres. How many buckets needed if capacity is 9 litres?

(a) 8
(b) 15
(c) 16
(d) 18

Answer: (d) Tank = 12×13.5 = 162 litres. Buckets = 162/9 = 18

Q50. A cistern has a leak that would empty in 8 hours. A tap admits 6 litres/min and it is emptied in 12 hours. The cistern can hold:

(a) 7860 litres
(b) 6840 litres
(c) 8640 litres
(d) 8840 litres

Answer: (c) Leak = 1/8 - 1/12 = 1/24. In 1 min = 6 litres. Capacity = 6×60×24 = 8640 litres

Q51. Rekha can do in 40 days. Surekha is 25% more efficient. Number of days taken by Surekha is:

(a) 24 days
(b) 28 days
(c) 32 days
(d) 35 days

Answer: (c) Efficiency ratio = 100:125 = 4:5. Time ratio = 5:4. Surekha = 40×4/5 = 32 days

Q52. A can do in 12 days and B in 15 days. If they work together for 6 days, fraction of work left is:

(a) 1/4
(b) 1/5
(c) 1/10
(d) 1/12

Answer: (c) 6(1/12+1/15)=9/10. Left = 1/10

Q53. A alone can complete in 15 days and B alone in 20 days. Starting with A, work on alternate days. Total work completed in:

(a) 17 days
(b) 16 days
(c) 14 days
(d) 13 days

Answer: (a) 2 days work = 7/60. 8 pairs = 14/15. Remaining = 1/15 by A. Total = 17 days

Q54. A alone can complete in 12 days and B alone in 48 days. Starting with B, work on alternate days. Total work completed in:

(a) 18 days
(b) 19 days
(c) 19½ days
(d) Data insufficient

Answer: (c) 2 days work = 5/48. 9 pairs = 15/16. Remaining = 1/16. B's 1 day = 1/48, remaining 1/24 by A in 1/2 day. Total = 19½ days

Q55. 4 men can do in 10 days, 2 women in 15 days, 5 children in 12 days. In how many days can 8 men, 5 women and 15 children together complete?

(a) 2 days
(b) 3 days
(c) 4 days
(d) None of these

Answer: (d) 8/40 + 5/30 + 15/60 = 1/5+1/6+1/4 = 37/60 ⇒ 60/37 days

Q56. 48 workers can reap a field in 9 days. If work to be completed in 6 days, extra workers required are:

(a) 36
(b) 32
(c) 24
(d) 22

Answer: (c) 48×9/6 = 72. Extra = 72-48=24

Q57. 4 men and 6 boys can do in 8 days and 6 men and 4 boys in 7 days. If daily wages of a boy be ₹20, what will be weekly wages of a man?

(a) 280
(b) 275
(c) 270
(d) 265

Answer: (a) 4M+6B=1/8, 6M+4B=1/7. Solving M=2B. Man's wage = 2×20=40. Weekly = 40×7=280

Q58. A is thrice as good as B and finishes 30 days less than B. Find time together.

(a) 45 days
(b) 11¼ days
(c) 15 days
(d) 22½ days

Answer: (b) Ratio times = 1:3. B-A=30 ⇒ 3x-x=30 ⇒ x=15. A=15, B=45. Together = 1/15+1/45=4/45 ⇒ 11¼ days

Q59. A takes twice as much time as B to do a work. Working together work completed in 8 days. A can do alone in:

(a) 24 days
(b) 12 days
(c) 16 days
(d) 28 days

Answer: (a) Ratio times A:B = 2:1. 1/2B + 1/B = 1/8 ⇒ B=12, A=24

Complete Answer Key

1. (c)	2. (a)	3. (a)	4. (c)	5. (b)
6. (b)	7. (a)	8. (a)	9. (a)	10. (c)
11. (b)	12. (c)	13. (c)	14. (c)	15. (a)
16. (c)	17. (b)	18. (c)	19. (b)	20. (a)
21. (c)	22. (a)	23. (a)	24. (c)	25. (a)
26. (b)	27. (d)	28. (b)	29. (b)	30. (c)
31. (b)	32. (a)	33. (a)	34. (c)	35. (b)
36. (b)	37. (a)	38. (c)	39. (c)	40. (d)
41. (c)	42. (a)	43. (a)	44. (a)	45. (a)
46. (d)	47. (b)	48. (c)	49. (d)	50. (c)
51. (c)	52. (c)	53. (a)	54. (c)	55. (d)
56. (c)	57. (a)	58. (b)	59. (a)

Arithmetic - Time and Work

Time and Work

1. Basic Concepts

2. Pipes and Cisterns

3. Exercise (Questions 1-10)

4. Exercise (Questions 11-20)

5. Exercise (Questions 21-30)

6. Exercise (Questions 31-59)

Complete Answer Key

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Arithmetic - Time and Work

1. Basic Concepts

2. Pipes and Cisterns

3. Exercise (Questions 1-10)

4. Exercise (Questions 11-20)

5. Exercise (Questions 21-30)

6. Exercise (Questions 31-59)

Complete Answer Key

JKAS.IN