Time and work questions measure how quickly people, machines, or pipes complete a task. The key is to convert every statement into a rate of work and combine those rates correctly.

Basic work concept

Treat the complete task as one unit of work. If a person finishes a task in n days, that person's one-day work is 1/n of the task. The time required is the reciprocal of the daily work rate.

• If A completes a job in 10 days, A's daily work is 1/10.
• If A completes 1/10 and B completes 1/15 each day, together they complete 1/10 + 1/15 = 1/6 per day.
• Therefore, together A and B complete the job in 6 days.

Work, rate, and time

• Work = Rate x Time
• Rate = Work / Time
• Time = Work / Rate
• For the same job, efficiency is inversely proportional to time.

Combined work

Add work rates when people or machines work together. If A takes a days and B takes b days, their combined one-day work is 1/a + 1/b. Their combined time is ab/(a + b) days.

When one worker leaves or joins partway through the task, calculate the work completed in each period separately and add the parts.

Efficiency and wages

• Efficiency is the amount of work completed in one unit of time.
• If A is twice as efficient as B, A's efficiency ratio to B is 2:1 and their time ratio is 1:2.
• For equal time, work done is proportional to efficiency.
• When workers contribute for the same time, wages are divided in the ratio of work done.

Pipes and cisterns

• An inlet pipe has a positive work rate because it fills the tank.
• An outlet pipe has a negative work rate because it empties the tank.
• Net rate = total inlet rates - total outlet rates.
• Use the same rate method as workers, treating the tank as one unit.

How to solve questions

1. Take the whole job or tank as one unit, unless an easier total work value is given.
2. Convert each person's or pipe's completion time into a one-day rate.
3. Add rates for work done together and subtract outlet rates.
4. For changing teams, find work done in each stage before finding the remaining work.
5. Check that more workers or higher efficiency gives less time for the same work.