Problem Solving
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7 min readWork-rate questions ask how long two or more workers, machines, or pipes take to complete a job. The core relationship is the same as distance-rate: work equals rate times time. The difference is that the total work is usually one job, and rates add when workers work together.
If a worker completes a job in hours, the work rate is of the job per hour. In general,
When two workers work together, their rates add:
For example, if A completes a job in 6 hours and B completes it in 3 hours, their combined rate is job per hour, so together they finish in 2 hours.
Convert every time to a rate before combining. Times do not add when workers work together. Rates do. This single habit prevents most work-rate errors.
Example. Machine A can fill a tank in 8 hours. Machine B can fill it in 12 hours. How long do they take working together?
Combined rate: tank per hour.
Time: hours, or 4 hours 48 minutes.
If more than two workers work together, add all the rates. For three workers with times , , and , the combined time satisfies .
When the combined time and one individual time are known, solve for the other rate. If A and B together finish in 4 hours and A alone takes 6 hours, then
so , and B alone takes 12 hours.
Example. Working alone, P can complete a report in 10 hours. Working with Q, P completes it in 6 hours. How long would Q take alone?
Q alone takes 15 hours.
If one worker leaves or joins partway through, split the job into phases. Calculate the work completed in each phase, add the amounts, and set the total equal to 1.
Example. A can paint a room in 6 hours and B can paint it in 4 hours. They work together for 2 hours, then A leaves. How much longer does B need to finish?
Combined rate: room per hour. In 2 hours, they complete of the room. Remaining: . B's rate is , so the time is hour, or 40 minutes.
Pipe problems use the same logic. A pipe that fills a tank has a positive rate; a drain has a negative rate. If an inlet fills in 5 hours and a drain empties in 10 hours, the net rate is tank per hour, so the tank fills in 10 hours.
If both the inlet and drain are open and the drain is faster, the net rate is negative and the tank never fills. The GMAT may ask for the time to empty the tank, or it may state that the tank is being drained.
Sometimes the job is not "1" but a number of units, such as 120 widgets. In that case, use with actual units. If a machine produces 30 widgets per hour, its rate is 30, not 1/30. The combined rate is the sum of the production rates.
Example. Machine A produces 40 parts per hour, and machine B produces 60 parts per hour. How long do they take to produce 500 parts together?
Combined rate: 100 parts per hour. Time: hours.
The unit method avoids fractions and is often faster when the problem gives explicit production rates. Use the fraction-of-job method when the problem gives completion times instead of rates.
Some questions combine work rates with payment. If workers are paid in proportion to the work they complete, find each worker's share of the total work and apply that fraction to the total pay. If A and B work together for the same time, their shares are in the ratio of their rates.
Fractions are not the only way to solve work problems. Choose a total amount of work that is divisible by the given times. If A takes 6 hours and B takes 4 hours, let the job be 12 units. A's rate is units per hour, and B's rate is units per hour. Together they complete 5 units per hour, so the job takes hours. This method avoids fractions and makes the arithmetic easier to check.
Example. A can complete a task in 10 hours, B in 15 hours, and C in 30 hours. How long do they take working together?
Use a job of 30 units. A's rate is 3, B's rate is 2, and C's rate is 1. Combined rate: 6 units per hour. Time: hours.
The common-unit method works because the total job can be any convenient number. Choose the least common multiple of the times to keep the rates as integers. If the times are 8 and 12, use 24 units. If the times are 5 and 7, use 35 units.
Some problems describe workers who do not start or stop at the same time. Split the timeline into periods when the set of workers is constant. For each period, multiply the combined rate by the time, then add the work completed. The total must equal one job, or the required number of units.
For example, if A works alone for 2 hours, then B joins for 3 hours, calculate A's work for 2 hours, then the combined work for 3 hours. Do not apply the combined rate to the entire 5 hours. The composition of the workforce changed.
If a worker leaves before the job is finished, the remaining workers continue at their own combined rate. Do not include the departed worker's rate in the final period. This is the most common error in multi-phase work problems.
If a problem gives a combined time and asks for the ratio of two individual rates, set the individual times as and . The ratio of the rates is , and the combined rate is . Solve for using the combined time.
| Situation | Equation |
|---|---|
| Work | |
| Rate from time | |
| Two workers together | |
| Three workers together | |
| Inlet and drain | |
| Partial work | add the work from each phase |
Work-rate questions become simple when every time is converted to a rate. Add rates, multiply by time, and keep the phases separate. If the answer is longer than the slower worker's individual time, something is wrong.
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