Time & Work: Complete Notes, Formulas, Examples, and Practice Questions

 

Time & Work: Complete Notes, Formulas, Examples, and Practice Questions

1. Work Efficiency: Core Concepts & Formulas

A. Basic Formulas

• Work Done = Time Taken × Rate of Work

• Rate of Work = 1 / Time Taken

• Time Taken = 1 / Rate of Work
  (If A can do a job in n days, A’s 1 day work = 1/n)2567

B. Combined Work

• If A can do a job in a days, B in b days, working together:

  • Time taken together = (ab)/(a + b) days

  • Combined 1-day work = (1/a) + (1/b) = (a + b)/ab234

C. Efficiency

• Efficiency is inversely proportional to time taken:

  • If A is twice as efficient as B, A will take half the time of B.

  • If efficiency ratio is X:Y, time ratio is Y:X24

D. Unit Approach

• Take total work as LCM of given days for easy calculation.

• Work done per day = Total work / total days34

E. Fractional Work

• If A does m/n of work in D days, total time = (n/m) × D days2

F. Alternate Days

• For workers alternating days, calculate work done in one cycle, then divide total work by work per cycle2.

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2. Pipes and Cisterns

• Inflow Rate: If a pipe fills a tank in x hours, rate = 1/x.

• Outflow Rate: If a pipe empties a tank in y hours, rate = -1/y.

• Net Rate:

  • If Pipe A fills in x hours and Pipe B empties in y hours,
    Net 1-hour work = 1/x – 1/y

  • Total time to fill/empty = 1 / (Net Rate)2

Example:
Pipe A fills in 4 hours, B empties in 6 hours:
Net rate = 1/4 – 1/6 = (3 – 2)/12 = 1/12
So, time to fill = 12 hours

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3. Wages Problems

• Wages are proportional to work done.

• If A and B work together and earn W, their share = (A’s work : B’s work)25

Example:
A does 2/3 of work, B does 1/3. For ₹600, A gets ₹400, B gets ₹200.

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4. Solved Examples

Example 1:
A can do a work in 10 days, B in 15 days. Together?
1-day work = 1/10 + 1/15 = (3 + 2)/30 = 5/30 = 1/6
Time = 6 days

Example 2:
A and B together do a work in 8 days. A alone in 12 days. B alone?
1/12 + 1/B = 1/8 ⇒ 1/B = 1/8 – 1/12 = (3 – 2)/24 = 1/24
B alone: 24 days3

Example 3 (Pipes):
Pipe A fills in 5 hours, B in 10 hours. Together?
1/5 + 1/10 = 3/10 per hour ⇒ 10/3 hours

Example 4 (Wages):
A and B complete a job in 12 days. A alone in 20 days. Share of wages (₹600)?
A’s 1-day work = 1/20; B’s = 1/30 (since 1/12 – 1/20 = 1/30)
A:B = 1/20 : 1/30 = 3:2
A gets ₹360, B gets ₹240

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5. Practice Questions (with answers for first 20)

Questions 1–20 (with answers)

1. A can do a job in 12 days, B in 18 days. Together?
   Ans: 7.2 days

2. A and B together do a work in 10 days. A alone in 15 days. B alone?
   Ans: 30 days

3. A can do a work in 8 days, B in 12 days. Together?
   Ans: 4.8 days

4. A, B, C can do a work in 6, 8, 12 days. Together?
   Ans: 2.18 days

5. If A is twice as efficient as B and B can do a work in 20 days, A alone?
   Ans: 10 days

6. A can do a work in 10 days. B is 50% more efficient. B alone?
   Ans: 6.67 days

7. Pipe A fills a tank in 4 hours, B empties in 6 hours. Together?
   Ans: 12 hours

8. Pipe A fills in 3 hours, B in 2 hours. Together?
   Ans: 1.2 hours

9. A and B complete a job in 15 days. A alone in 25 days. B alone?
   Ans: 37.5 days

10. A, B, C can do a work in 10, 20, 30 days. Together?
    Ans: 4.62 days

11. If A does 1/3 of work, B does 2/3. Total wage ₹600. A’s share?
    Ans: ₹200

12. A and B together can do a work in 6 days. A alone in 9 days. B alone?
    Ans: 18 days

13. A can do a work in 5 days, B in 10 days. Together?
    Ans: 3.33 days

14. Pipe A fills in 8 hours, B in 12 hours. Together?
    Ans: 4.8 hours

15. Pipe A fills in 6 hours, B empties in 8 hours. Together?
    Ans: 24 hours

16. A and B together do a work in 16 days. A alone in 24 days. B alone?
    Ans: 48 days

17. A, B, C can do a work in 20, 30, 60 days. Together?
    Ans: 8.57 days

18. If A is 3 times as efficient as B and B can do a work in 18 days, A alone?
    Ans: 6 days

19. A can do a work in 12 days, B in 16 days. Together?
    Ans: 6.86 days

20. If A and B together do a work in 7 days, A alone in 14 days, B alone?
    Ans: 14 days

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Questions 21–100 (Practice Only)

21. A can do a work in 15 days, B in 20 days. Together?

22. A, B, C can do a work in 9, 12, 18 days. Together?

23. If A is 4 times as efficient as B and B can do a work in 24 days, A alone?

24. Pipe A fills in 5 hours, B empties in 10 hours. Together?

25. Pipe A fills in 6 hours, B in 8 hours, C in 12 hours. Together?

26. A and B together do a work in 8 days. A alone in 12 days. B alone?

27. A can do a work in 7 days, B in 14 days. Together?

28. Pipe A fills in 9 hours, B empties in 18 hours. Together?

29. A, B, C can do a work in 15, 20, 30 days. Together?

30. If A is 1.5 times as efficient as B and B can do a work in 18 days, A alone?

31. A and B together do a work in 10 days. A alone in 25 days. B alone?

32. Pipe A fills in 8 hours, B in 10 hours. Together?

33. Pipe A fills in 4 hours, B empties in 12 hours. Together?

34. A, B, C can do a work in 12, 16, 24 days. Together?

35. If A is 2 times as efficient as B and B can do a work in 30 days, A alone?

36. A and B together do a work in 9 days. A alone in 27 days. B alone?

37. Pipe A fills in 7 hours, B in 14 hours. Together?

38. Pipe A fills in 5 hours, B empties in 15 hours. Together?

39. A, B, C can do a work in 18, 24, 36 days. Together?

40. If A is 5 times as efficient as B and B can do a work in 25 days, A alone?

41. A and B together do a work in 15 days. A alone in 30 days. B alone?

42. Pipe A fills in 10 hours, B in 20 hours. Together?

43. Pipe A fills in 6 hours, B empties in 18 hours. Together?

44. A, B, C can do a work in 24, 30, 40 days. Together?

45. If A is 2.5 times as efficient as B and B can do a work in 40 days, A alone?

46. A and B together do a work in 12 days. A alone in 36 days. B alone?

47. Pipe A fills in 12 hours, B in 15 hours. Together?

48. Pipe A fills in 8 hours, B empties in 16 hours. Together?

49. A, B, C can do a work in 30, 40, 60 days. Together?

50. If A is 3 times as efficient as B and B can do a work in 21 days, A alone?

51. A and B together do a work in 18 days. A alone in 36 days. B alone?

52. Pipe A fills in 14 hours, B in 28 hours. Together?

53. Pipe A fills in 10 hours, B empties in 20 hours. Together?

54. A, B, C can do a work in 36, 48, 72 days. Together?

55. If A is 4 times as efficient as B and B can do a work in 32 days, A alone?

56. A and B together do a work in 20 days. A alone in 40 days. B alone?

57. Pipe A fills in 16 hours, B in 32 hours. Together?

58. Pipe A fills in 12 hours, B empties in 24 hours. Together?

59. A, B, C can do a work in 40, 50, 100 days. Together?

60. If A is 2 times as efficient as B and B can do a work in 36 days, A alone?

61. A and B together do a work in 14 days. A alone in 21 days. B alone?

62. Pipe A fills in 18 hours, B in 36 hours. Together?

63. Pipe A fills in 15 hours, B empties in 30 hours. Together?

64. A, B, C can do a work in 60, 80, 120 days. Together?

65. If A is 3 times as efficient as B and B can do a work in 27 days, A alone?

66. A and B together do a work in 16 days. A alone in 32 days. B alone?

67. Pipe A fills in 20 hours, B in 40 hours. Together?

68. Pipe A fills in 24 hours, B empties in 48 hours. Together?

69. A, B, C can do a work in 80, 100, 200 days. Together?

70. If A is 2.5 times as efficient as B and B can do a work in 50 days, A alone?

71. A and B together do a work in 21 days. A alone in 42 days. B alone?

72. Pipe A fills in 28 hours, B in 56 hours. Together?

73. Pipe A fills in 20 hours, B empties in 40 hours. Together?

74. A, B, C can do a work in 100, 120, 200 days. Together?

75. If A is 4 times as efficient as B and B can do a work in 40 days, A alone?

76. A and B together do a work in 24 days. A alone in 48 days. B alone?

77. Pipe A fills in 32 hours, B in 64 hours. Together?

78. Pipe A fills in 36 hours, B empties in 72 hours. Together?

79. A, B, C can do a work in 120, 150, 300 days. Together?

80. If A is 3 times as efficient as B and B can do a work in 33 days, A alone?

81. A and B together do a work in 28 days. A alone in 56 days. B alone?

82. Pipe A fills in 36 hours, B in 72 hours. Together?

83. Pipe A fills in 40 hours, B empties in 80 hours. Together?

84. A, B, C can do a work in 150, 180, 360 days. Together?

85. If A is 2 times as efficient as B and B can do a work in 54 days, A alone?

86. A and B together do a work in 32 days. A alone in 64 days. B alone?

87. Pipe A fills in 48 hours, B in 96 hours. Together?

88. Pipe A fills in 60 hours, B empties in 120 hours. Together?

89. A, B, C can do a work in 180, 240, 480 days. Together?

90. If A is 5 times as efficient as B and B can do a work in 60 days, A alone?

91. A and B together do a work in 35 days. A alone in 70 days. B alone?

92. Pipe A fills in 70 hours, B in 140 hours. Together?

93. Pipe A fills in 80 hours, B empties in 160 hours. Together?

94. A, B, C can do a work in 200, 250, 500 days. Together?

95. If A is 4 times as efficient as B and B can do a work in 100 days, A alone?

96. A and B together do a work in 40 days. A alone in 80 days. B alone?

97. Pipe A fills in 90 hours, B in 180 hours. Together?

98. Pipe A fills in 100 hours, B empties in 200 hours. Together?

99. A, B, C can do a work in 250, 300, 600 days. Together?

100. If A is 3 times as efficient as B and B can do a work in 90 days, A alone?

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Practice these formulas and questions to master time and work, work efficiency, pipes and cisterns, and wage distribution for all competitive exams.