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Time and Work

Theory

  1. The Fundamental Relationship:
    • The most basic formula is: Work = Rate × Time
    • This can be rearranged to find Rate or Time:
      • Rate = Work / Time (This is the amount of work done per unit of time)
      • Time = Work / Rate (This is the time taken to complete the work)
  2. Defining “Work”:
    • In most aptitude problems, the “Work” is a single, complete task (e.g., building a wall, filling a tank, finishing a report).
    • Often, it’s convenient to consider the total work as 1 unit or a quantity that is the Least Common Multiple (LCM) of the times given for individual workers (more on the LCM method later).
  3. Defining “Rate” (or Efficiency):
    • Rate is the amount of work a person (or pipe, or machine) can do in one unit of time (one day, one hour, one minute, etc.).
    • It’s a measure of their efficiency. A higher rate means they are more efficient and take less time to complete the same amount of work.
    • Key Concept: If a person takes ‘T’ days to complete the entire work, their rate (work done in one day) is 1/T of the total work.
  4. Inverse Proportion between Time and Rate/Workers:
    • For a fixed amount of work, Time and Rate are inversely proportional. If you double the rate, the time taken is halved.
    • Similarly, if all workers have the same efficiency, the Time taken is inversely proportional to the Number of Workers. More workers mean less time to complete the same task.
    • Example: If 10 men take 5 days to do a job, 20 men (with the same efficiency) will take 5/2 = 2.5 days. (Assuming work is constant: M1 * D1 = M2 * D2 if efficiency is same).
  5. Combining Rates when Working Together:
    • When multiple individuals or entities work together on the same task, their individual rates are added to find the combined rate.
    • If Person A’s rate is R_A (work per unit time) and Person B’s rate is R_B, their combined rate when working together is R_A + R_B.
    • The time taken to complete the work together would then be Total Work / (R_A + R_B).
    • Using the “work per unit time” concept: If A takes T_A time and B takes T_B time, their combined work in one unit of time is (1/T_A) + (1/T_B).
    • The total time taken together (T_together) is the reciprocal of their combined rate: 1 / [(1/T_A) + (1/T_B)]. This simplifies to (T_A * T_B) / (T_A + T_B) for two people.
  6. The Unitary Method (Work done in one unit of time):
    • This is the most common approach.
    • Steps:
      • Calculate the work done by each person (or entity) in one unit of time (day, hour, etc.). This is the reciprocal of the total time they take individually.
      • If they work together, add their individual rates (work per unit time).
      • The total time taken together is the reciprocal of the combined rate.
      • If only a fraction of work is done, multiply the rate by the time worked.
  7. The LCM Method (Alternative and Often Faster):
    • This method avoids fractions initially and is very useful when times for multiple workers are given.
    • Steps:
      • Find the LCM of the times taken by each individual worker.
      • Assume this LCM value represents the “Total Units of Work”.
      • Calculate the “Efficiency” (units of work per unit of time) for each worker by dividing the Total Units of Work by their individual time.
      • If they work together, add their efficiencies to get the combined efficiency.
      • The time taken to complete the work together is (Total Units of Work) / (Combined Efficiency).
  8. Problems with varying work or groups (M1D1W1 Formula):
    • These problems involve different groups of workers doing different amounts of work over different periods.
    • The principle is that the total amount of “worker-time units” required is proportional to the work done.
    • A common formula derived from Work = Rate × Time is: (M1 * D1 * T1 * E1) / W1 = (M2 * D2 * T2 * E2) / W2
      • M = Number of Men (or workers)
      • D = Number of Days
      • T = Time worked per day (e.g., hours/day)
      • E = Efficiency of the workers (often assumed constant, so E1=E2 and can be omitted)
      • W = Amount of Work
    • A simplified version, often used when efficiency and hours per day are constant, is M1 * D1 / W1 = M2 * D2 / W2.
    • If the work is the same (W1 = W2), this reduces to M1 * D1 = M2 * D2 (or M1 * D1 * T1 = M2 * D2 * T2 if hours per day vary).
  9. Pipes and Cisterns:
    • This is a direct application of Time and Work principles.
    • Filling a tank is the “work”.
    • Inlet pipes have positive rates (they do positive work by filling).
    • Outlet pipes (or leaks) have negative rates (they do negative work by emptying).
    • The net rate when multiple pipes are open is the sum of the individual rates (inlets add, outlets subtract).
    • Total Time = Capacity of Tank / Net Rate.

In summary, solving Time and Work problems hinges on:

  • Understanding the core relationship: Work = Rate × Time.
  • Calculating individual rates (work done per unit time).
  • Adding rates when people work together.
  • Recognizing the inverse relationship between time and rate/number of workers.
  • Using either the Unitary Method or the LCM Method consistently.
  • Applying the group work formula (MDT/W) when different groups and tasks are involved.
  • Treating emptying entities (like outlet pipes) as having negative rates.