Work and Pipe Problems: Quick Lesson, Shortcuts and UPCAT Practice
Work and Pipe Problems
A quick lesson, reliable shortcuts, worked examples, adaptive practice, and a five-form mastery check for work-rate problems.
Work-Rate Problems
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Turn completion time into a rate
A worker, machine, or pipe that finishes one whole job in t hours completes 1/t of the job per hour.
rate = 1 ÷ completion timeWorking together
Add rates that complete the same job.
combined rate = 1/a + 1/bFilling and draining
Add filling rates, but subtract a drain or leak because it removes work.
net rate = fill rate − drain rateFor any interval, use work completed = rate × time. If part of the job is already done, use remaining work = 1 − completed work.
Two shortcuts—and when not to use them
Why these shortcuts work
The first formula is the reciprocal of 1/a + 1/b. The second is the reciprocal of 1/a − 1/b. They are compact versions of the same rate model—not separate tricks to memorize.
Five forms you should recognize
A can finish in 6 hours and B in 3 hours. Their combined rate is 1/6 + 1/3 = 1/2 job per hour.
time = 1 ÷ 1/2 = 2 hoursA pipe fills a tank in 4 hours while a drain empties it in 12 hours. The net rate is 1/4 − 1/12 = 1/6.
net fill time = 6 hoursA can finish in 8 hours and works alone for 2 hours, completing 2/8 = 1/4. If B also takes 8 hours, their combined rate is 1/4. The remaining 3/4 takes 3 more hours.
Together, A and B finish in 4 hours. A alone needs 12 hours.
B's rate = 1/4 − 1/12 = 1/6, so B needs 6 hoursMachines taking 8 and 12 hours work together for 2 hours.
work done = 2(1/8 + 1/12) = 5/12Check before you commit
- Averaging times: combine rates, not completion times.
- Adding a drain: removal rates must be subtracted.
- Ignoring intervals: delayed starts require separate stages.
- Wrong target: distinguish total elapsed time from time after another worker joins.
- Mixed units: convert minutes and hours before combining rates.
- No reality check: two productive workers together must finish faster than either alone.
Do you need the lesson—or just practice?
Answer one original question in each form. This recommends your next step; it does not yet verify mastery.
Work at the level you need.
Foundations
Unit rates, combined rates, and immediate explanations.
Core Practice
Workers, pipes, delayed starts, and unknown rates.
UPCAT-Style Transfer
Apply the same model to machines, crews, and partial jobs.
Ready to verify this competency?
These are different from the starting questions. A score of 5/5 verifies the competency. A different set appears after an unsuccessful attempt.
Work-rate problems FAQ
Do I add the times when two people work together?
No. Convert each time into a unit rate, add the rates, and then take the reciprocal to find the combined time.
Why is a drain subtracted?
A drain reverses progress by removing water, so its rate reduces the positive filling rate.
Can I always use ab/(a+b)?
Only when both productive rates operate together for the entire interval. Use the partial-work method when starting or stopping times differ.
Should the combined time be smaller?
When both rates are productive, yes: the combined time must be less than either individual time.
Are pipe problems a separate competency?
No. Pipes are one representation of work-rate reasoning. Workers, machines, pumps, leaks, and production crews use the same unit-rate model.
Continue building rate reasoning.
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