Arithmetic ReasoningLesson 7 of 12
Work and Production Rates
Combined effort, filling and emptying, and why rates add where times do not.
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Work problems are the topic where intuition is reliably wrong, which is why they appear. The fix is mechanical and it always works.
Why times cannot be added
If one pump fills a tank in 4 hours and another fills the same tank in 6, the two together do not take 10 hours, and they do not take 5. Adding times says that bringing in help makes the job slower, which is nonsense, and averaging them says the second pump contributed nothing.
What adds is work per hour.
- Pump A does 1/4 of the tank per hour.
- Pump B does 1/6 of the tank per hour.
- Together: 1/4 plus 1/6.
Common denominator 12: 3/12 plus 2/12 is 5/12 of the tank per hour.
If they do 5/12 of a tank each hour, a whole tank takes 12/5 hours, which is 2.4 hours, or 2 hours 24 minutes.
Flip the combined rate to get the combined time. That last step is the one people forget, and 5/12 as an answer looks like a fraction of an hour, so it passes a careless check.
The method, four steps
- Write each worker's rate as one over their time. Someone who takes 9 hours does 1/9 of the job per hour.
- Add the rates. Use a common denominator.
- Flip the sum. That is the combined time.
- Sanity check: the answer must be shorter than the fastest single time.
| A alone | B alone | Rates | Sum | Together |
|---|---|---|---|---|
| 2 h | 3 h | 1/2 + 1/3 | 5/6 | 6/5 = 1.2 h |
| 4 h | 4 h | 1/4 + 1/4 | 1/2 | 2 h |
| 6 h | 12 h | 1/6 + 1/12 | 3/12 | 4 h |
| 5 h | 20 h | 1/5 + 1/20 | 5/20 | 4 h |
The second row is worth noticing. Two identical workers halve the time, which is the one case intuition gets right, and it is a good check on the method: if you run the steps on two equal times and do not get half, you made an arithmetic error.
Filling and emptying at once
A drain is a worker with a negative rate.
A tank fills in 3 hours through the inlet and empties in 5 hours through the drain. If both are open on an empty tank, how long to fill it?
- Inlet: 1/3 per hour.
- Drain: minus 1/5 per hour.
- Net: 1/3 minus 1/5, which over 15 is 5/15 minus 3/15, or 2/15 per hour.
Flip: 15/2 hours, or 7.5 hours.
Check the sign before you flip. If the drain were faster than the inlet, the net rate would be negative and the tank would never fill - and a question that sets that up is asking whether you noticed, not asking for a number.
Partial work and staggered starts
One crew paints a hangar in 10 hours. They work 4 hours, then a second crew, which alone would take 15 hours, joins them. How much longer to finish?
Work in fractions of the job.
- First crew's rate: 1/10 per hour. In 4 hours they do 4/10, or 2/5 of the job.
- Remaining: 3/5.
- Combined rate: 1/10 plus 1/15. Over 30: 3/30 plus 2/30 is 5/30, or 1/6 per hour.
- Time for the remaining 3/5 at 1/6 per hour: divide 3/5 by 1/6, which is 3/5 times 6, or 18/5 hours, which is 3.6 hours.
Total elapsed is 7.6 hours, and the question asked how much longer, which is 3.6. Both are choices.
Production rates
Some questions state output directly rather than a time to finish, and those are easier - they are unit rate questions.
Machine A makes 40 parts an hour and machine B makes 25. How long to make 520 parts running both?
Combined output is 65 parts an hour. 520 divided by 65 is 8 hours.
No fractions needed. When the question gives you output per hour, just add the outputs. Only invert when the question gives you a time to complete a whole job. Telling those two cases apart is most of the skill.
| The question says | Do this |
|---|---|
| "finishes the job in 6 hours" | rate is 1/6 of the job per hour, add, then flip |
| "produces 30 units per hour" | add the outputs directly, then divide the target |
What you can skip
Across the 141 questions on this topic:
- More than two workers or pipes is rare. The add-the-rates method extends to three without change, so nothing new needs learning.
- Formula sheets for work problems. Everything is rate x time = work, with rates added when people or pipes work together and subtracted when a drain works against a fill.
Where people lose points
Adding the times. The error the topic exists to catch.
Averaging the times. Slightly less wrong and still wrong. The average of 4 and 6 is 5, and the real answer is 2.4.
Forgetting to flip. Stopping at 5/12 instead of 12/5.
Answering total elapsed time when the question asked how much longer, or the reverse, on staggered-start problems.
Sign errors on a drain. Adding the drain's rate instead of subtracting it gives a faster fill than the inlet alone, which is impossible.
Mixed units. One rate per hour, one per minute. Convert before adding.
Work one in under a minute
Three identical loaders can clear a pad in 12 hours. How long do four of them take?
Do not build a fraction. Think in loader-hours: the job is 3 times 12, which is 36 loader-hours. Four loaders take 36 divided by 4, or 9 hours.
The loader-hours shortcut works whenever the workers are identical, and it is much faster than the rate method. Reach for the rate method when they are not identical.
Where this leads
The combined-rate move is the same move as two vehicles closing a gap, and the fraction arithmetic is the same as everywhere else on the subtest.
Related lessonsReference
- Rate, Time and Distance - combining rates when the job is covering ground
- Unit Rate and Scaling - the production-rate half of this lesson in its general form
- Fractions, Decimals and Percents - adding fractions with unlike denominators, drilled on its own
- Reading a Word Problem - telling "how much longer" from "how long in total"
Practice this topic
Check that this lesson stuck. Answer questions on work and production rates only, and see the right answer and why after each one.
Practice Work and Production Rates questions