How to use the combined work rate calculator
- Choose to find the time working together or a missing worker time.
- Pick a time unit and list each worker or machine with the time it takes alone to do the whole job.
- Mark a row as undoing the job if it works against the others, such as a drain beside fill pipes.
- Read the combined time, the net rate, and the table with each worker's rate and share of the job.
Formula
Rate of one worker = 1 / time alone (the share of the job done per unit of time)
Net rate = sum of the rates that do the job - sum of the rates that undo it
Time together = 1 / net rate
Missing time = 1 / (1 / time together - sum of the other rates)
Rates add when people work together at their own steady pace. A worker who undoes the job has a negative rate. A time is the inverse of a rate, which is why two times combine by adding their reciprocals.
Two workers, two pipes and a missing time
A worker who takes 6 hours alone and one who takes 3 hours have rates of 1/6 and 1/3. Together they do 1/2 of the job per hour, so they finish in 2 hours, and the first does 1/3 of the work while the second does 2/3.
Two fill pipes that take 4 and 6 hours, with a drain that empties the tank in 12 hours, give 1/4 + 1/6 - 1/12 = 1/3 per hour, so the tank fills in 3 hours. If a team finishes in 2 hours and one member takes 6 hours alone, the other would take 1 / (1/2 - 1/6) = 3 hours alone.
Why rates, not times, add
If one person paints a room in 6 hours and another in 3, you cannot add the times. Think of how much each paints per hour instead. The first covers one sixth of the room an hour, the second a third. Together that is half the room per hour, so the job takes 2 hours. The page does exactly this with any number of workers.
For the share of a ratio between parts see the ratio calculator. A tank that drains is covered by the tank drain time calculator, and the flow rate calculator converts volume and time.
When a job never ends
If the rates that undo the job add up to at least as much as the rates that do it, the net rate is zero or negative and the job is never finished. The page says so. In the missing worker task, an error appears if the workers you listed already finish in the time you gave, because no extra worker is needed then.
Assumptions and limits
- Each worker keeps a constant pace the whole time, and the work is the same kind for everyone.
- Times alone are for the whole job, and they use the unit you picked.
- Workers do not slow or speed each other up. Real teams can do either.
- The share column is shown only when no row undoes the job.
Frequently asked questions
How do I calculate how long two people take to do a job together?
Add their rates, which are 1 divided by each time alone, then take 1 divided by the sum. Times of 6 and 3 hours give 1 / (1/6 + 1/3) = 2 hours.
How do I solve a tank filling and draining problem?
Give each fill pipe a positive rate and the drain a negative rate, and add them. Pipes of 4 and 6 hours and a drain of 12 hours fill the tank in 3 hours.
How do I find a missing worker time?
Subtract the known rates from the rate of the team and invert the result. If a team of two takes 2 hours and one member takes 6, the other takes 3 hours.
Can the combined time be longer than one worker alone?
Only if someone undoes the work. With helpers only, the combined time is always shorter than the quickest worker.