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Shared work
Two painters, three pumps, a faucet and a drain: “working together” problems look tricky because adding the times feels natural and is always wrong. Add the rates instead, and every one of them becomes the same short calculation.
Add rates, not times
Pat can paint a room alone in 3 hours, so each hour Pat paints 1/3 of it. Lou needs 4 hours, so Lou paints 1/4 an hour. Those amounts per hour are rates, and rates can be added: working together, they paint 1/3 + 1/4 = 7/12 of the room every hour.
Together is always faster than the fastest one alone
If your answer for “working together” is longer than the faster worker’s time, something is wrong. Adding the times (3 + 4 = 7 hours) or averaging them (3.5 hours) gives exactly that kind of impossible answer.
How to solve them
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1
Turn each time into a rate: 1 ÷ time
A job that takes 5 hours is done at a rate of 1/5 of the job per hour. Write every worker, pipe or machine as a rate before you do anything else.
Next step: TogetherAlone in 5 hoursRate per hour
1/5 of the job
Alone in 8 hoursRate per hour
1/8 of the job
Alone in 40 minutesRate per hour (40 min = 2/3 hour)
1 ÷ 2/3 = 1.5 jobs
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2
Working together: add the rates, then flip
Add the rates to get the combined rate. The time is that rate flipped upside down: if together they do 7/12 of the job per hour, the whole job takes 12/7 hours.
1/a + 1/b = 1/T
Next step: Two-worker shortcutExamplePat paints a room in 3 hours, Lou in 4 hours. How long together?
- Rates: 1/3 and 1/4 of the room per hour.
- 1/3 + 1/4 = 4/12 + 3/12 = 7/12 per hour.
- Time = 12/7 hours ≈ 1.7 hours, about 1 hour 43 minutes.
Answer 12/7 hours
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3
The shortcut for exactly two: product over sum
For two workers only, the time together is (a × b) ÷ (a + b). It is quick, but it works for two and only two.
T = (a × b) ÷ (a + b)
Next step: Three or moreExampleA 6-hour worker and a 12-hour worker, together
- (6 × 12) ÷ (6 + 12) = 72 ÷ 18 = 4.
- Check with rates: 1/6 + 1/12 = 2/12 + 1/12 = 3/12 = 1/4 per hour, so 4 hours.
Answer 4 hours
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4
Three or more workers: keep adding rates
With three workers the shortcut breaks. Go back to rates: add all three, then flip.
Next step: Fill vs drainExampleThree pumps fill a pool alone in 4, 6 and 12 hours. How long together?
- 1/4 + 1/6 + 1/12 = 3/12 + 2/12 + 1/12 = 6/12 = 1/2 pool per hour.
- Time = 2 hours.
- Product over sum would give (4 × 6 × 12) ÷ (4 + 6 + 12) = 288 ÷ 22 ≈ 13.1 hours: slower than one pump alone, which is impossible.
Answer 2 hours
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5
One fills, one drains: subtract the rate that works against you
A leak, a drain or a worker who undoes the job has a rate too. Subtract it instead of adding it.
Next step: Finishing a jobExampleA hose fills a tank in 3 hours. A leak empties a full tank in 9 hours. With both, how long does it take to fill?
- 1/3 − 1/9 = 3/9 − 1/9 = 2/9 of the tank per hour.
- Time = 9/2 = 4.5 hours.
Answer 4.5 hours
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6
Part of a job: work done = rate × time
Work done is rate × time, just like distance. When someone works for part of the time, add up the pieces until they make one whole job.
ExampleKai mows a field in 6 hours. After he mows for 2 hours, Ruth (3 hours alone) joins him. How much longer do they need?
- In 2 hours Kai mows 2 × 1/6 = 1/3 of the field, so 2/3 is left.
- Together: 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 field per hour.
- 2/3 ÷ 1/2 = 4/3 hours = 1 hour 20 minutes.
Answer 1 hour 20 minutes
4 traps in the answer choices
The test writes its wrong answers on purpose: each one is what you get if you make one of these mistakes.
Adding the times
A 2-hour painter and a 6-hour painter, together:
Wrong2 + 6 = 8 hours
Right1/2 + 1/6 = 2/3 per hour → 1.5 hours
Averaging the times
The same two painters:
Wrong(2 + 6) ÷ 2 = 4 hours
Right1.5 hours: faster than either alone
Product over sum with three workers
Workers who take 2, 3 and 6 hours alone:
Wrong(2 × 3 × 6) ÷ (2 + 3 + 6) ≈ 3.3 hours
Right1/2 + 1/3 + 1/6 = 1 → 1 hour
Adding a drain
Fills in 4 hours, drains in 8 hours, both open:
Wrong1/4 + 1/8 → 2.7 hours
Right1/4 − 1/8 = 1/8 → 8 hours
5 questions at test level
Pick an answer, then read why. No timer, no account.
A painter can paint a fence in 5 hours. What part of the fence does she paint in 2 hours?
Show the answer
C. Her rate is 1/5 of the fence per hour, and 2 hours × 1/5 = 2/5. 1/5 is one hour’s work, and 5/2 flips the rate. She paints 2/5.
One machine fills an order in 10 hours and a second machine fills it in 15 hours. Working together, how long do they take?
Show the answer
B. Rates: 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 of the order per hour, so together they need 6 hours. 25 adds the times and 12.5 averages them: both are slower than the faster machine alone. Together they take 6 hours.
Pipe A fills a tank in 3 hours, pipe B in 4 hours and pipe C in 12 hours. With all three open, how long does it take to fill the tank?
Show the answer
A. Add the three rates: 1/3 + 1/4 + 1/12 = 4/12 + 3/12 + 1/12 = 8/12 = 2/3 of the tank per hour. Time = 3/2 = 1.5 hours. 7.6 hours uses product over sum, which only works for two pipes. It takes 1.5 hours.
A faucet fills a sink in 10 minutes. The open drain empties a full sink in 15 minutes. With both open, how long does it take to fill the sink?
Show the answer
D. The drain works against the faucet: 1/10 − 1/15 = 3/30 − 2/30 = 1/30 of the sink per minute. That takes 30 minutes. 6 minutes adds the drain as if it helped. The sink fills in 30 minutes.
Rosa can sort a pile of mail in 4 hours; Dev can sort it in 6 hours. Rosa works alone for 1 hour, then Dev joins her. How many more hours until the mail is sorted?
Show the answer
B. In 1 hour Rosa sorts 1/4, so 3/4 is left. Together they sort 1/4 + 1/6 = 3/12 + 2/12 = 5/12 per hour. 3/4 ÷ 5/12 = 9/12 × 12/5 = 9/5 = 1.8. 2.4 hours would be the whole pile together. They need 1.8 hours more.
9 flashcards
Tap a card to flip it. Keep the ones you miss until they stick.
All 9 cards known.
You finished shared work.
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