Basic Idea
Efficiency is inversely proportional to time. If a person completes a job in n days, in one day the person does one over n of the job. If A alone takes a days and B alone takes b days, together they take ab divided by a plus b days. For three workers, take the reciprocal of the sum of the individual rates.
If A and B together take t days and A alone takes a days, then B alone takes one over the difference between one over t and one over a.
The man-days rule says men times days times hours per day over work is the same on both sides. If the number of workers rises, the days fall.
Pipes and Tanks
An inlet pipe fills and counts as positive work. An outlet pipe or leak empties and counts as negative. When both are open, the net rate is the filling rate minus the emptying rate.
Unit Method
Take the LCM of all the times as the total work in units. Divide it by each person's time to get the daily units. Add the daily units and divide the total work by the sum.
Shortcuts
For two workers use ab over a plus b. For three workers use the unit method. If one person is k times as efficient as another, their times are in the ratio 1 to k. The remaining work is one minus the fraction already done.
Worked Examples
Example 1: A alone does a job in 10 days and B alone in 15 days. How long together?
Solution: 10 times 15 divided by 25 equals 6 days.
Example 2: A and B together finish in 12 days. A alone takes 20 days. How long does B take alone?
Solution: One twelfth minus one twentieth equals one thirtieth, so B takes 30 days.
Example 3: A pipe fills a tank in 8 hours and another empties it in 12 hours. How long to fill with both open?
Solution: One eighth minus one twelfth equals one twenty-fourth, so it takes 24 hours.
Quick Check
Question 1: What is the formula for two workers together?
Answer: ab divided by a plus b.
Question 2: Is an outlet pipe positive or negative work?
Answer: Negative.
Question 3: What is the unit method?
Answer: Take the LCM of times as total work and compute daily units.