Time and Work

“Time and Work is an important concept in competitive exams, involving the calculation of how long it takes for a person or a group to complete a task.”

1. Basic Concepts and Formulas

  • Work Done = (Rate of Work) × (Time Taken)
  • Rate of Work = 1 / Time Taken (for 1 complete unit of work)
  • If A can do a work in x days, then A’s rate of work = 1/x
  • If A can complete a work in x days and B in y days, then their combined work rate is: 1/x + 1/y
  • Time taken to finish a work by A and B together = (x × y) / (x + y)
  • If A and B work together, their combined rate of work = 1/x + 1/y
  • If A works alone for a certain time and then B joins, the total work can be calculated by adding individual work contributions.

2. Type-Wise Questions

  • Type 1: Work Done by One Person
    Example: A can complete a task in 12 days. How much work will A do in 4 days?
    Solution: Work Done = (4/12) = 1/3 of the task.
  • Type 2: Time Taken by One Person
    Example: A can complete a task in 15 days. How long will it take for A to complete the same task if he works for 5 hours a day?
    Solution: If A works 5 hours a day, Time Taken = 15 days / 5 hours = 3 hours per day. Therefore, A would take 3 days to finish the task.
  • Type 3: Work Done by Two People Together
    Example: A and B can finish a task in 6 days and 8 days, respectively. How much time will it take for them to finish the task together?
    Solution: Time Taken = (6 × 8) / (6 + 8) = 48 / 14 = 3.43 days.
  • Type 4: Time Taken by Two People Together
    Example: A and B can complete a task in 15 days and 20 days, respectively. How long will it take them to finish the task together?
    Solution: Time Taken = (15 × 20) / (15 + 20) = 300 / 35 = 8.57 days.
  • Type 5: Work Done by One Person Alone and then by Another
    Example: A can finish a task in 10 days. B can finish the same task in 15 days. A works for 5 days, then B joins. How many more days will they need to finish the task?
    Solution: Work done by A in 5 days = 5/10 = 1/2 of the task. Remaining work = 1 - 1/2 = 1/2. Combined work rate of A and B = 1/10 + 1/15 = 1/6. Time needed to finish remaining work = 1/2 ÷ 1/6 = 3 days.
  • Type 6: Work Done by Two People Working Alternately
    Example: A can finish a task in 12 days and B can finish it in 18 days. If A and B work alternately, starting with A, how long will it take to complete the task?
    Solution: In 2 days, A and B will complete 1/12 + 1/18 = 5/36 of the task. Therefore, in 36/5 = 7.2 days, the task will be completed.
  • Type 7: A Works for Part of the Time and B Completes the Rest
    Example: A can finish a task in 10 days. B can finish the same task in 15 days. A works for 6 days, then B works for the remaining part. How many days did B take to complete the task?
    Solution: Work done by A in 6 days = 6/10 = 3/5 of the task. Remaining work = 1 - 3/5 = 2/5. B completes the remaining work in 2/5 × 15 = 6 days.
  • Type 8: Work Done in Fractions of the Day
    Example: A can finish a task in 6 hours. B can finish it in 8 hours. If they work together for 4 hours, how much work is done?
    Solution: Combined rate = 1/6 + 1/8 = 7/24. Work done in 4 hours = 4 × 7/24 = 7/6 of the task.
  • Type 9: A and B Working at Different Rates
    Example: A can complete a task in 12 days. B works twice as fast as A. How long will it take them to complete the task together?
    Solution: B’s rate = 1/6. Combined rate = 1/12 + 1/6 = 3/12 = 1/4. Time taken = 1 ÷ 1/4 = 4 days.
  • Type 10: Work Done by Three People
    Example: A can complete a task in 10 days, B in 15 days, and C in 20 days. How long will it take for all three to finish the task together?
    Solution: Combined rate = 1/10 + 1/15 + 1/20 = 1/6. Time taken = 1 ÷ 1/6 = 6 days.

3. Tips to Remember

  • When two or more people work together, add their rates of work to find the combined rate.
  • If a person works for a part of the time and another works for the rest, calculate the work done separately by both.
  • If A works for some time, then another person joins, find their combined rate and calculate the remaining time needed.
  • Always convert the rate of work to a fraction of the total task, and ensure units of time match.
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