Work & Wages

Work and Wages problems are based on the simple concept of the rate of work.

Work Rate: This is the amount of work an individual can do in a unit of time (e.g., per day or per hour). The fundamental formula is: Total Work = Rate × Time.
Inverse Relationship: The time taken to complete a job is inversely proportional to the rate of work (efficiency). More efficient workers take less time.
Wages: Payment is directly proportional to the amount of work done by an individual, not necessarily the time they spent.

Unitary Method (Fraction Method): If a person A can do a work in 'x' days, then A's 1 day's work is \( \frac{1}{x} \). This method is fundamental but can lead to complex fraction calculations.
LCM Method (Efficiency Method): Assume the Total Work to be the LCM of the days taken by all individuals. From this, calculate the per-day work (efficiency) of each person. This method keeps calculations in whole numbers and is generally faster.
Combined Work Formula: The core formula for all problems is \( M_1 D_1 H_1 E_1 / W_1 = M_2 D_2 H_2 E_2 / W_2 \), where M=Men, D=Days, H=Hours, E=Efficiency, and W=Work.

Always prefer the LCM method over the fraction method. For example, if A takes 10 days and B takes 15 days, assume Total Work = LCM(10, 15) = 30 units. Then, A's efficiency = 3 units/day and B's efficiency = 2 units/day.
Negative Work: When dealing with pipes and cisterns, an outlet pipe or a leak does negative work. Its work rate should be subtracted from the others.
Alternate Days: If A and B work on alternate days, find the work done in a 2-day cycle and use it to get close to the total work.

Read the question carefully to determine if people are working together, in turns, or if someone leaves in between.
For wage distribution, always calculate the ratio of the total work done by each person. For example, if A works for 2 days at a rate of 3 units/day, their work is 6 units. Distribute wages based on this work unit ratio.