Solving work-rate problems with combined effort

Work-rate questions describe how quickly people or machines complete a task. The central idea is rate = work ÷ time. When two workers cooperate, they contribute work at the same time, so their rates add. Their times do not add because both are working during the same interval.

Convert completion times into rates

If a person completes one job in 6 hours, the rate is 1/6 job per hour. If a second person completes the same job in 3 hours, that rate is 1/3 job per hour. Together, their rate is 1/6 + 1/3 = 1/2 job per hour. The time for one job is work ÷ rate: 1 ÷ 1/2 = 2 hours.

Use a consistent unit for the whole job

Before combining rates, confirm that the workers are doing the same job and that the time unit matches. One worker’s output may be measured per hour and another per minute; convert first. If a problem gives amounts completed rather than time per job, divide the amount by elapsed time to obtain each rate. If each person contributes at a different stage, determine whether their work overlaps or occurs sequentially.

Handle one worker joining or leaving

Suppose a worker begins alone and another joins later. Calculate the amount completed during the first interval, subtract it from the total job, and use the combined rate for the remaining work. Keep elapsed time and remaining work in separate steps. This avoids treating the full job as if both workers were present from the start.

A reliable setup

1. Define one complete job as 1 unit of work.
2. Convert each person’s completion time into a rate per time unit.
3. Add rates only for people working at the same time.
4. Multiply rate by time to find completed work, or divide work by rate to find time.
5. Check that the answer is reasonable: a combined effort should finish sooner than either person alone if both rates are positive.

Common errors

• Adding 6 hours and 3 hours instead of adding 1/6 and 1/3 job per hour.
• Combining a rate per minute with a rate per hour without conversion.
• Using the combined rate during a period when only one worker was present.
• Forgetting that a rate is a fraction of the job, not a number of hours.

Key takeaway

Convert each completion time into a rate, add the rates during overlapping work, and divide remaining work by the applicable rate. This makes even multi-stage work problems manageable.

Convert each worker’s time to a rate

A work rate measures the fraction of a job completed per unit of time. If one person finishes a job in 6 hours, that person’s rate is 1/6 job per hour. If another finishes it in 4 hours, the second rate is 1/4 job per hour. Working together, their rates add when they work independently on the same job without interfering: 1/6 + 1/4 = 5/12 job per hour.

Time is the reciprocal of the combined rate: 1 ÷ (5/12) = 12/5 hours, or 2.4 hours. Do not add the times 6 and 4; that would describe two separate jobs in sequence, not one job completed together. A quick check helps: the combined time must be less than 4 hours, because the faster worker alone takes 4 hours.

Use a common job and consistent units

The “whole job” can be a tank, a report, a set of items, or a distance, but define it consistently. A pump fills 3/5 of a tank per hour while a drain empties 1/4 tank per hour. The net fill rate is 3/5 − 1/4 = 12/20 − 5/20 = 7/20 tank per hour. The time to fill one tank from empty at that constant net rate is 20/7 hours. Subtract rates when one process undoes the other.

If one worker joins later or leaves early, split the job into time intervals. Determine the fraction completed during each interval using rate × time, then add or subtract those fractions. For example, if a machine works alone for 2 hours at 1/8 job per hour, it completes 1/4 of the job. The remaining fraction is 3/4; a second machine’s later contribution is calculated against that same whole job.

Set up an equation when the rate is unknown

If a person completes a job in t hours, the rate is 1/t job per hour. If one worker takes 5 hours and the other takes x hours, their combined rate is 1/5 + 1/x. If together they finish in 2 hours, set 1/5 + 1/x = 1/2, solve for x, and then check whether the result is physically plausible. Multiplying through by the common denominator clears fractions, but keep track of units.

Rates must describe the same job and the same time unit. Convert minutes to hours or hours to minutes before adding rates. Also confirm the problem assumes constant rates; real tasks may not progress uniformly, but word problems generally provide or imply a constant-rate model.

• Change completion times to job-per-time rates before combining.
• Add rates for independent work on one job; subtract rates for opposing processes.
• Multiply rate by time to find the fraction completed.
• Use one consistent definition of the whole job and one time unit.
• Check that the combined time is sensible relative to each worker alone.

Handle a partial job or a leak explicitly

If a worker has already completed a fraction of the job, subtract that fraction from 1 to find the remaining work. If two workers complete 1/4 and 1/3 of a task, respectively, the remaining fraction is 1 − 1/4 − 1/3 = 5/12. Their work rates can then be used to find the time to finish, assuming constant rates and no overlap in what remains.

A leaking tank is an opposing rate. If a pipe fills 1/5 of a tank per hour and a leak drains 1/20 per hour, the net rate is 4/20 − 1/20 = 3/20 tank per hour. From empty, time to fill is 20/3 hours. If the tank already contains water, calculate the remaining fraction first.